Mechanical Vibrations and Shocks: v.1-5

Figure 5.23. Resonance phase

Since Hrd ~1 for h small, for an excitation by force on the mass, The response is controlled in a dominating way by the stiffness of the system. In this domain where h is small with respect to the unit, the dimensioning

Response of a linear single-degree-of-freedom mechanical system

183

calculations of structure in statics can be carried out by taking the values of HRD at the frequency of the vibration, in order to take account at the same time of the static load and of the small dynamic amplification. These calculations can possibly be supplemented by a fatigue analysis if this phenomenon is considered to be important [HAL 75]. - for h = 1, the maximum value of q(0) is

and the phase is

The amplitude of the response is a function of the damping £. It is larger if £ is 71

smaller. The movement is out of phase by — with respect to the excitation. 2 If the excitation is a force, at the resonance,

Here, analysis must be of the dynamic type, the response being able to be equal to several times the equivalent static excitation.

184

Sinusoidal vibration

where

If the excitation is a force, we have

where Q = pulsation of the excitation. The response is primarily a function of the mass m. It is smaller than the equivalent static excitation. According to whether h satisfies one or the other of these three conditions, one of the three elements stiffness, damping or mass thus has a dominating action on the resulting movement of the system [BLA 61] [RUB 64].

Figure 5.24. Fields of dynamic amplification factor

Response of a linear single-degree-of-freedom mechanical system

185

Particular case where

(The positive root for h < 1 is chosen in order to preserve at q(6) the same sign for £, = 0 rather than for £ which is very small in the expression [5.66]).

The variations of qmax versus h are represented in Figure 5.25. It is noted that, when h tends towards 1, qmax tends towards infinity. It is necessary here to return to the assumptions made and to remember that the system is considered linear, which supposes that the amplitude of the variations of the response q remains small. This curve q max (h) thus does not make sense in the vicinity of the asymptote.

Figure 5.25. Variations of qmax versus h

186

Sinusoidal vibration

Figure 5.26. Dynamic amplification factor for, = 0

Figure 5.27. Phase for £ = 0

The case where £, = 0 is an ideal case: in practice, frictions are never negligible in the vicinity of resonance (apart from resonance, they are sometimes neglected at first approximation to simplify the calculations). As h varies, qmax changes sign while passing through infinity. To preserve at the reduced amplitude the character of an always positive amplitude (the temporal response being symmetrical with respect to the time axis), an abrupt phase shift of value 7t is introduced into the passage of h = 1. The phase q is zero in the interval 0 < h < 1; it is then equal to ± n for h > 1 (the choice of the sign is uninportatnt). If the value - is taken in (1, 00), then for example, for 0 < h < 1:

and, for

Response of a linear single-degree-of-freedom mechanical system

187

Particular case where Here

or

with

and

NOTE: The resonance frequency, defined as the frequency for which the response is at a maximum, has the following values: Table 5.2. Resonance frequency and maximum of the transfer function Response Displacement

Velocity

Acceleration

Resonance frequency

Amplitude of the relative response

188

Sinusoidal vibration

(the natural frequency of the system being equal to h = -\/l - £ ). For the majority of real physical systems, % is small and the difference between these frequencies is negligible.

Figure 5.28. Resonance frequency versus £

Figure 5.29. Error made by always considering h = 1

Figure 5.30. Peak amplitude of the transfer function

Figure 5.31. Error made by always taking l/2£

Response of a linear single-degree-of-freedom mechanical system

189

5.6. Responses

5.6.1. Movement transmissibility Here

The maximum amplitude of q(0) obtained for sin(h 6 - q>) = 1, occurring for is equal to

If the excitation is an absolute displacement of the support, the response is the absolute displacement of the mass m. The movement transmissibility is defined as the ratio of the amplitude of these two displacements:

For certain applications, in particular in the case of calculations of vibration isolators or package cushioning, it is more useful to know the fraction of the force amplitude applied to m which is transmitted to the support through the system [BLA 61] [HAB 68]. Then a force transmission coefficient or force transmissibility Tf is defined by

Tf = Tm = HAD is then obtained according to Table 5.1.

190

Sinusoidal vibration

5.6.2. Variations in amplitude The amplitude H AD (h) is at a maximum when that

Figure 5.32. Transmissibility

This derivative is zero if h = 0 or if

i.e. for

or, since

. for h such

Response of a linear single-degree-of-freedom mechanical system

Figure 5.33. Frequency of the maximum of transmissibility versus E,

191

Figure 5.34. Maximum oftransmissibility versus £,

yielding HAD max

When h tends towards zero, the amplitude H^ tends towards 1 (whatever £). When h -» oo, HAJQ -> 0. From the relation [5.147] is drawn

yielding h < 1. The locus of the maxima thus has as an equation H

AD

This gives the same law as that obtained for relative displacement.

192

Sinusoidal vibration

Figure 5.35. Locus of maximum oftransmissibility versus h

Case oft, = 0 With this assumption, H^ pass

through

for

Hnr). For all values of^, all the curves HAE

and

for

all the curves are above

Indeed,

H^j)

HAD

Indeed, the condition

is carried out only if In the same way, for h

5.6.3. Variations in phase

If

all the curves are below the straight line

H^

Response of a linear single-degree-of-freedom mechanical system

193

Figure 5.36. Phase variations

for

The denominator is zero if or, since

In this case, tan

and

All the curves have, for £, < 1, a point of inflection at h = 1. The slope at this point gets larger as £ gets smaller. when

194

Sinusoidal vibration

<j> then gets smaller as £, gets larger.

Figure 5.37. Phase versus E, for h = 1

For

5.7. Graphical representation of transfer functions The transfer functions can be plotted in a traditional way on linear or logarithmic axes, but also on a four coordinate nomographic grid, which makes it possible to directly reach the transfer functions of the displacements, the velocities and the accelerations. In this plane diagram at four inputs, the frequency is always carried on the abscissa. Knowing that HRV = Q HRD and that Hj^ = H RV , from the ordinate, along the vertical axis the following can be read:

Response of a linear single-degree-of-fireedom mechanical system

195

- either the velocity (Figure 5.38). Accelerations are then located on an axis of negative slope (-45°) with respect to the axis of the velocities while the amplitude of the displacements are on an axis at 45° with respect to the same vertical axis. Indeed (Figure 5.39):

However, a line at 45° with respect to the vertical axis,

O'K is thus proportional to log

Figure 5.38. Four coordinate diagram

- the amplitude of the displacements. A similar calculation shows that the axis of the velocities forms an angle of + 45° with respect to the horizontal line and that of the accelerations an angle of 90° with respect to the axis of the velocities.

196

Sinusoidal vibration

Figure 5.39. Construction of the four input diagram

Chapter 6

Non-viscous damping

6.1. Damping observed in real structures In real structures, damping, which is not perfectly viscous, is actually a combination of several forms of damping. The equation of movement is in c consequence more complex, but the definition of damping ratio £ remains —, c c where cc is the critical damping of the mode of vibration considered. The exact calculation of £, is impossible for several reasons [LEV 60]: probably insufficient knowledge of the exact mode of vibration, and of the effective mass of the system, the stiffnesses, the friction of the connections, the constant c and so on. It is therefore important to measure these parameters when possible. In practice, non-linear damping can often be compared to one of the following categories, which will be considered in the following paragraphs: - damping force proportional to the power b of the relative velocity z; - constant damping force (Coulomb or dry damping), which corresponds to the case where b = 0; - damping force proportional to the square of the velocity (b = 2); - damping force proportional to the square of the relative displacement; -hysteretic damping, with force proportional to the relative velocity and inversely proportional to the excitation frequency.

198

Sinusoidal vibration

Such damping produces a force which is opposed to the direction or the velocity of the movement. 6.2. Linearization of non-linear hysteresis loops - equivalent viscous damping Generally, the differential equation of the movement can be written [DEN 56]:

with, for viscous damping, f(z, z) = c z. Because of the presence of this term, the movement is no longer harmonic in the general case and the equation of the movement is no longer linear, such damping leads to nonlinear equations which make calculations complex in a way seldom justified by the result obtained. Except in some particular cases, such as the Coulomb damping, there is no exact solution. The solution of the differential equation must be carried out numerically. The problem can sometimes be solved by using a development of the Fourier series of the damping force [LEV 60]. Very often in practice damping is fortunately rather weak so that the response can be approached using a sinusoid. This makes it possible to go back to a linear problem, which is easier to treat analytically, by replacing the term f(z, z) by a force of viscous damping equivalent c^ z; by supposing that the movement response is sinusoidal, the equivalent damping constant ceq of a system with viscous damping is calculated which would dissipate the same energy per cycle as nonlinear damping. The practice therefore consists of determining the nature and the amplitude of the dissipation of energy of the real damping device, rather than substituting in the mathematical models with a viscous damping device having a dissipation of equivalent energy [CRE 65]. It is equivalent of saying that the hysteresis loop is modified. In distinction from structures with viscous damping, nonlinear structures have non-elliptic hysteresis loops Fd(z) whose form approaches, for example, those shown in Figures 6.1 and 6.2 (dotted curve).

Non-viscous damping

199

Figure 6.1. Hysteresis loops of non-linear systems

Linearization results in the transformation of the real hysteresis loop into an equivalent ellipse (Figure 6.2) [CAU 59], [CRE 65], [KAY 77] and [LAZ 68].

Figure 6.2. Linearization of hysteresis loop

Equivalence thus consists in seeking the characteristics of a viscous damping which include: - the surface delimited by me cycle Fd(z) (same energy dissipation); - the amplitude of the displacement zm. The curve obtained is equivalent only for the selected criteria. For example, the remanent deformation and the coercive force are not exactly the same. Equivalence leads to results which are much better when the non-linearity of the system is lower.

200

Sinusoidal vibration

This method, developed in 1930 by L. S. Jacobsen [JAC 30], is general in application and its author was able to show good correlation with the computed results carried out in an exact way when such calculations are possible (Coulomb damping [DEN 30aJ) and with experimental results. This can, in addition, be extended to the case of systems with several degrees of freedom.

Figure 6.3. Linearization of hysteresis loop

If the response can be written in the form z(t) = zm sin(Qt - cp), the energy dissipated by cycle can be calculated using

Energy AEd is equal to that dissipated by an equivalent viscous damping c^, if

[HAB 68]:

i.e. if [BYE 67], [DEN 56], [LAZ 68] and [THO 65a]:

Non-viscous damping

The transfer function of a one-degree-of-freedom system general way

In addition,

yielding

(or in a more

can be written while replacing ceq by this value in the

relation established, for viscous damping:

(since

201

\ and for the phase

202

Sinusoidal vibration is the energy dissipated by the cycle, the amplitude of the equivalent force

applied is

6.3. Main types of damping 6.3.1. Damping force proportional to the power b of the relative velocity Table 6.1. Expressions for damping proportional to power b of relative velocity

Damping force

Equation of the hysteresis loop

Energy dissipated by damping during a cycle

Equivalent viscous damping Equivalent damping ratio

Amplitude of the response

Non-viscous damping

203

Phase of the response

References in Table 6.1: [DEN 30b], [GAM 92], [HAB 68], [JAC 30], [JAC 58], [MOR 63a], [PLU 59], [VAN 57] and [VAN 58]. Relation between b and the parameter J the B. J. Lazan expression It has been shown [JAC 30] [LAL 96] that if the stress is proportional to the relative displacement zm (a = K zm), the coefficient J of the relation of B.J. Lazan (D = J a ) is related to the parameter b by

J depends on parameters related to the dynamic behaviour of the structure being considered (K and co0).

6.3.2. Constant damping force If the damping force opposed to the movement is independent of displacement and velocity, the damping is known as Coulomb or dry damping. This damping is observed during friction between two surfaces (dry friction) applied one against the other with a normal force N (mechanical assemblies). It is [BAN 77], [BYE 67], [NEL 80] and [VOL 65]: - a function of the materials in contact and of their surface quality; - proportional to the force normal to the interface; - mainly independent of the relative velocity of slipping between two surfaces; -larger before the beginning of the relative movement than during the movement in steady state mode.

204

Sinusoidal vibration

Figure 6.4. One-degree-of-freedom system with dry friction

The difference between the coefficients of static and dynamic friction is in general neglected and force N is supposed to be constant and independent of the frequency and of the displacement. A one-degree-of-freedom system damped by dry friction is represented in Figure 6.4. Table 6.2. Express ions for a constant damping force

Damping force Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping Equivalent damping ratio

Amplitude of the response

Non-viscous damping

205

Phase of the response

[BEA 80], [CRE 61], [CRE 65], [DEN 29], [DEN 56], [EAR 72], [HAB 68], [JAC 30], [JAC 58], [LEV 60], [MOR 63b], [PAI 59], [PLU 59], [ROO 82], [RUZ 57], [RUZ 71], [UNO 73] and [VAN 58].

6.3.3. Damping force proportional to the square of velocity A damping of this type is observed in the case of a body moving in a fluid .2 Z

(applications in fluid dynamics, the force of damping being of the form Cx p A —) 2

or during the turbulent flow of a fluid through an orifice (with high velocities of the fluid, from 2 to 200 m/s, resistance to the movement ceases to be linear with the velocity). When the movement becomes fast [BAN 77], the flow becomes turbulent and the resistance non-linear. Resistance varies with the square of the velocity [BAN 77], [BYE 67] and [VOL 65]. Table 6.3. Expressions for quadratic damping

Damping force Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping Equivalent damping ratio

206

Sinusoidal vibration

Amplitude of the response

Phase of response

[CRE 65], [HAB 68], [JAC 30], [RUZ 71], [SNO 68] and [UNG 73]. The constant (3 is termed the quadratic damping coefficient. It is characteristic of the geometry of the damping device and of the properties of the fluid [VOL 65]. 6.3.4. Damping force proportional to the square of displacement Table 6.4. Expressions for damping force proportional to the square of the displacement

Damping force

Equation of hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping

Equivalent damping ratio

Non-viscous damping

207

Amplitude of response

Phase of response

Such damping is representative of the internal damping of materials, of the structural connections, and cases where the specific energy of damping can be expressed as a function of the level of stress, independent of the form and distribution of the stresses and volume of the material [BAN 77], [BYE 67], [KIM 26] and [KIM 27]. 6.3.5. Structural or hysteretic damping Table 6.5. Expressions for structural damping Damping coefficient function of Q

Damping force

Equation of the hysteresis loop Energy dissipated by damping during a cycle Equivalent viscous damping

Damping force proportional to the displacement

Complex stiffness

208

Sinusoidal vibration

Equivalent damping ratio

Amplitude of response

Phase of response

This kind of damping is observed when the elastic material is imperfect when in a system the dissipation of energy is mainly obtained by deformation of material and slip, or friction in the connections. Under a cyclic load, the curve Q, s of the material forms a closed hysteresis loop rather than only one line [BAN 77]. The dissipation of energy per cycle is proportional to the surface enclosed by the hysteresis loop. This type of mechanism is observable when repeated stresses are applied to an elastic body causing a rise in temperature of the material. This is called internal friction, hysteretic damping, structural damping or displacement damping. Various formulations are used [BER 76], [BER 73], [BIR77], [BIS 55], [CLO 80], [GAN 85], [GUR 59], [HAY 72], [HOB 76], [JEN 59], [KIM 27], [LAL 75], [LAL 80], [LAZ 50], [LAZ 53], [LAZ 68], [MEI 67], [MOR 63a], [MYK 52], [PLU 59], [REI 56], [RUZ 71], [SCA 63], [SOR 49] and [WEG 35]. 6.3.6 Combination of several types of damping . If several types of damping, as is often the case, are simultaneously present combined with a linear stiffness [BEN 62] [DEN 30a], equivalent viscous damping can be obtained by calculating the energy AEdi dissipated by each damping device and by computing ceq [JAC 30] [JAC 58]:

Non-viscous damping

209

Example Viscous damping and Coulomb damping [JAC 30], [JAC 58], [LEV 60] and [RUZ71]

Fm F c Q

= maximum F(t) (excitation) = frictional force = viscous damping ratio = pulsation of the excitation

6.3.7. Validity of simplification by equivalent viscous damping The cases considered above do not cover all the possibilities, but are representative of many situations. The viscous approach supposes that although non-linear mechanisms of damping are present their effect is relatively small. It is thus applicable if the term for viscous damping is selected to dissipate the same energy by cycle as the system with nonlinear damping [BAN 77]. Equivalent viscous damping tends to underestimate the energy dissipated in the cycle and the amplitude of a steady state forced vibration: the real response can larger than envisaged with this simplification. The decrease of the transient vibration calculated for equivalent viscous damping takes a form different from that observed with Coulomb damping, with a damping force proportional to the square of the displacement or with structural damping. This difference should not be neglected if the duration of the decrease of the response is an important parameter in the problem being considered.

210

Sinusoidal vibration

The damped natural frequency is itself different in the case of equivalent viscous damping and in the non-linear case. But this difference is in general so small that it can be neglected. When damping is sufficiently small (10%), the method of equivalent viscous damping is a precise technique for the approximate solution of non-linear damping problems.

6.4. Measurement of damping of a system All moving mechanical systems dissipate energy. This dissipation is often undesirable (in an engine, for example), but can be required in certain cases (vehicle suspension, isolation of a material to the shocks and vibrations and so on). Generally, mass and stiffness parameters can be calculated quite easily. It is much more difficult to evaluate damping by calculation because of ignorance of the concerned phenomena and difficulties in their modelling. It is thus desirable to define this parameter experimentally. The methods of measurement of damping in general require the object under test to be subjected to vibration and to measure dissipated vibratory energy or a parameter directly related to this energy. Damping is generally studied through the properties of the response of a one-degree-of-freedom mass-spring-damping system [BIR 77] [CLO 80] [PLU 59]. There are several possible methods for evaluating the damping of a system: - amplitude of the response or amplification factor; - quality factor; - logarithmic decrement; - equivalent viscous damping; - complex module; Af -bandwidth —. f

6.4.1. Measurement of amplification factor at resonance The damping of the one-degree-of-freedom system tends to reduce the amplitude of the response to a sine wave excitation. If the system were subjected to no external forces, the oscillations response created by a short excitation would attenuate and disappear in some cycles. So that the response preserves a constant amplitude, the

Non-viscous damping

211

excitation must bring a quantity of energy equal to the energy dissipated by damping in the system. The amplitude of the velocity response z is at a maximum when the frequency of the sinewave excitation is equal to the resonance frequency f0 of the system. Since the response depends on the damping of the system, this damping can be deduced from measurement of the amplitude of the response, the one degree of linear freedom system supposedly being linear:

or

For sufficiently small £, it has been seen that with a weak error, the amplification 2

factor, defined by HRD =

co0 zm

, was equal to Q. The experimental determination

*m of E, can thus consist of plotting the curve HRD or HRV and of calculating £, from the peak value of this function. If the amplitude of the excitation is constant, the sum of potential and kinetic energies is constant. The stored energy is thus equal to the

maximum of one or the other; it will be, for example Us = — k z^ . The energy 2 dissipated during a cycle is equal to [5.87] AE<j = TccQz^, yielding, since it is supposed that £} = co0:

i.e.

212

Sinusoidal vibration

NOTE: The measurement of the response/excitation ratio depends on the configuration of the structure as much as the material. The system is therefore characterized by this rather than the basic properties of the material. This method is not applicable to non-linear systems, since the result is a function of the level of excitation.

6.4.2. Bandwidth or >/2 method Another evaluation method (known as Kennedy-Pancu [KEN 47]) consists of measuring the bandwidth Af between the half-power points relating to one peak of the transfer function [AER 62], with the height equal to the maximum of the curve HRD (or H RV ) divided by V2 (Figure 6.5). From the curve H RV (h), we will have if hj and h2 are the abscissae of the halfpower points:

where (f0 = peak frequency, h}

and

If T0 is the natural period and T1] and T2 are the periods corresponding to an attenuation of

•ft , damping c is given by 2

Non-viscous damping

213

Figure 6.5. Bandwidth associated with resonance

since

and

and

i.e., with the approximation

From the curve HRD, these relations are valid only if ^ is small. The curve could also be used for small £.

H^

6.4.3. Decreased rate method (logarithmic decrement) The precision of the method of the bandwidth is often limited by the non-linear behaviour of the material or the reading of the curves. Sometimes it is better to use the traditional relation of logarithmic decrement, defined from the free response of the system after cessation of the exciting force (Figure 6.6).

214

Sinusoidal vibration

Figure 6.6. Measurement of logarithmic decrement [BUR 59]

The amplitude ratio of two successive peaks allows the calculation of the logarithmic decrement 8 from

In addition, the existence of the following relation between this decrement and damping ratio is also shown

The measurement of the response of a one-degree-of-freedom system to an impulse load thus makes it possible to calculate 5 or £, from the peaks of the curve [FOR 37] and [MAC 58]:

Non-viscous damping

215

Figure 6.7. Calculation of damping from §

The curve of Figure 6.7 can be use to determine £, from 5. In order to improve the precision of the estimate of 8, it is preferable to consider two non-consecutive peaks. The definition then used is:

where zmi and zmn are, respectively, the first and the n* peak of the response (Figure 6.8). In the particular case where £, is much lower than 1, from [6.29] is obtained:

yielding

and

216

S inusoidal vibration

with

with 4 being small

yielding the approximate value £a

The error caused by using this approximate relation can be evaluated by plotting £a~£ the curve according to £ (Figure 6.8) or that giving the exact value of £ £ according to the approximate value £a (Figure 6.9). This gives

Figure 6.8. Error related to the approximate relation £(5)

Figure 6.9. Exact value of£> versus approximate value ^

Non-viscous damping

217

yielding

and

The specific damping capacity p, the ratio of the specific energy dissipated by damping to the elastic deformation energy per unit of volume, is thus equal to

In a more precise way, p can be also written

while assuming that U^ is proportional to the square of the amplitude of the response. For a cylindrical test-bar,

(potential energy + kinetic energy)

i.e., since constant

218

Sinusoidal vibration 2

Ute is thus proportional to zm yielding, from [6.31] and [6.41] for two successive peaks:

Figure 6.10. Specific damping capacity versus £,

Figure 6.11. Specific damping capacity versus 6

The use of the decrement to calculate p from the experimental results supposes that 8 is constant during n cycles. This is not always the case. It was seen that damping increases as a power of the stress, i.e. of the deformation and it is thus desirable to use this method only for very low levels of stress. For 8 small, we can write [6.45] in the form of a series:

If

we find

The method of logarithmic decrement takes no account of the non-linear effects. The logarithmic decrement 8 can be also expressed according to the resonance peak amplitude Hmax and its width Af at an arbitrary height H [BIR 77], [PLU 59]. F. Forester [FOR 37] showed that

Non-viscous damping

219

Figure 6.12. Bandwidth at amplitude H

Setting as ne the number of cycles such that the amplitude decreases by a factor e (number of Neper), it becomes

where te= time to reach the amplitude

If the envelope Z(t) of the response

(which is roughly a damped sinusoid) is considered this gives,

and if the amplitude in decibels is expressed as

220

Sinusoidal vibration

For a value of H such that

yielding

a relation already obtained. The calculation

of the Q factor from this result and from the curve H(f) can lead to errors if the damping is not viscous. In addition, it was supposed that the damping was viscous. If this assumption is not checked different values of £are obtained depending on the peaks chosen, particularly for peaks chosen at the beginning or end of the response [MAC 58]. Another difficulty can arise in the case of a several degrees-of-freedom system for which it can be difficult to excite only one mode. If several modes are excited, the response of a combination of several sinusoids to various frequencies will be presented.

6.4.4. Evaluation of energy dissipation under permanent sinusoidal vibration An alternative method can consist of subjecting the mechanical system to harmonic excitation and to evaluate, during a cycle, the energy dissipated in the damping device [CAP 82], where the quantity is largely accepted as a measure of the damping ratio. This method can be applied to an oscillator whose spring is not perfectly elastic, this then leads to the constant k and c of an equivalent simple oscillator.

Non-viscous damping

221

Figure 6.13. Force in a spring It has been seen that, if a one-degree-of-freedom mechanical system is subjected to a sinusoidal force F(t) = Fm sinQ t such that the pulsation is equal to the natural pulsation of the system (co0), the displacement response is given by

where

The force Fs in the spring is equal to Fs = k z(t) and the force F^ in the damping device to Fd = cz = 2 m £ Q z = 2 k £ z m s i n Q t , yielding Fd according toz:

Figure 6.14. Damping force versus displacement

222

Sinusoidal vibration

This function is represented by an ellipse. During a complete cycle the potential energy stored in the spring is entirely restored. On the other hand, there is energy AE
Figure 6.15. Total force versus elongation

The superposition of Figures 6.13 and 6.14 makes it possible to plot F = Fs + F<j against z (Figure 6.15). From these results, the damping constant c is measured as follows: - by plotting the curve F(Z) after moving the system out of equilibrium (force F applied to the mass); - by taking the maximum deformation zm . It is supposed here that the stiffness k is linear and k is thus calculated from the slope of the straight line plotted midway at the centre of the ellipse. The surface of the ellipse gives AE^ , yielding

NOTES: 1. When zm increases, the spring has an increasingly non-linear behaviour in general and the value o/£, obtained grows.

Non-viscous damping

223

• 2. The energy dissipated by the cycle (&E$) depends on the form, dimensions and the distribution of the dynamic stresses. It is preferable to consider the specific damping energy D, which is a basic characteristic of the material (damping energy per cycle and unit of volume by assuming a uniform distribution of the dynamic stresses in the volume V considered) [PLU 59].

where AEd is in Joules/cycle andD is in Joules/cycle/m3, Some examples of different values of £ are given in Table 6.6 [BLA 61] and [CAP 82].

Rubber-type materials, with weak damping The dynamic properties of Neoprene show a very weak dependence on the frequency. The damping ratio of Neoprene increases more slowly at high frequencies than the damping ratio of natural rubber [SNO 68].

Table 6.6. Examples of damping values

Material Welded metal frame

£ 0.04

Bolted metal frame

0.07

Concrete

0.010

Prestressed concrete

0.05

Reinforced concrete

0.07

High-strength steel (springs) Mild steel

0.637 10"3 to 1.27 10'3

3.18 10-3

Wood

7.96 10-3 to 3 1.8 10-3

Natural rubber for damping devices

1.59 10-3 to 12.7 10-3

Bolted steel

0.008

Welded steel

0.005

224

Sinusoidal vibration

Rubber-type materials, with strong damping The dynamic module of these materials increases very rapidly with the frequency. The damping ratio is large and can vary slightly with the frequency.

6.4.5. Other methods Other methods were developed to evaluate the damping of the structures such as, for example, that using the derivative to the resonance of the phase with respect to the frequency (Kennedy-Pancu improved method) [BEN 71].

6.5. Non-linear stiffness We considered in paragraph 6.3 the influence of non-linear damping on the response of a one-degree-of-freedom system. The non-linearity was thus brought about by damping. Another possibility relates to the non linearities due to the stiffness. It can happen that the stiffness varies according to the relative displacement response. The restoring force, which has the form F = -k z, is not linear any more and can follow a law such as, for example, F = k z + r z where k is the preceding constant and where r determines the rate of non-linearity. The stiffness can increase with relative displacement (hardening spring) (Figure 6.16) or decrease (softening spring) (Figure 6.17) [MIN 45].

Figure 6.16. Hardening spring

Figure 6.17. Softening spring

A well-known phenomenon of jump [BEN 62] can then be observed on the transfer function.

Non-viscous damping

Figure 6.18. Transmissibility against increasing frequency

225

Figure 6.19. Transmissibility against decreasing frequency

When the frequency increases slowly from zero, the transmissibility increases from the intercept 1 up to point A while passing through D and then decreases to B (Figure 6.18). If, on the contrary, resonance is approached from high frequencies by a slow sinusoidal sweeping at decreasing frequency, the transfer function increases, passes through C and moves to D near the resonance, and then decreases up to 1 asf tends towards zero (Figure 6.19).

Figure 6.20. Influence damping

It should be noted that the area CA is unstable and therefore cannot represent the transfer function of a physical system. The shape of the curve depends on the amplitude of the force of excitation, like the frequency of resonance. The mass can vibrate at its natural frequency with an excitation frequency much larger (phenomenon known as resonance of the nth order) [DUB 59].

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Chapter 7

Swept sine

7.1. Definitions 1.1.1. Swept sine A swept sine can be defined as a function characterized by a relation of the form:

where - the phase <(> is in general zero; - E(t) is a time function characteristic of the sweep mode; - ^(t) is generally an acceleration, sometimes a displacement, a velocity or a force. The pulsation of the sinusoid can be defined like the derivative of the function under the symbol sine [BRO 75], [HAW 64], [HOK 48], [LEW 32], [PIM 62], [TUR 54] and [WHI 72], i.e. by:

We will see that the most interesting sweep modes are: - the linear sweep, for which f of form f = a t + (3;

228

Sinusoidal vibration

-the logarithmic sweep (which should rather be termed exponential) if t/T if --f ij ee '-, - the hyperbolic sweep (or parabolic, or log log) if:

1

1 — = at

f, f

These sweeps can be carried out at an increasing frequency or a decreasing frequency. The first two laws are the most frequently used in laboratory tests. Other laws can however be met some of which have been the subject of other published work [SUZ 78a], [SUZ 79], [WHI 72].

7.1.2. Octave - number of octaves in frequency interval (Tj, f^) An octave is the interval ranging between two frequencies whose ratio is two. The number of octaves ranging between two frequencies fjand f2 is such mat:

f

yielding

(logarithms in both cases being base e or base 10). 7.1.3. Decade A decade is the interval ranging between two frequencies whose ratio is ten. The number of decades nd ranging between two frequencies fj and f2 is such that:

yielding

Swept sine

229

The relation between the number of decades and the number of octaves ranging between two frequencies

7.2. 'Swept sine' vibration in the real environment Such vibrations are relatively rare. They are primarily measured on structures and equipment installed in the vicinity of rotating machines, at tunes of launching, stop or speed changes. They were more particularly studied to evaluate their effects during transition through the resonance frequency of a material [HAW 64], [HOK 48], [KEY 71], [LEW 32], [SUZ 78a], [SUZ 78b] and [SUZ 79].

7.3. 'Swept sine' vibration in tests Material tests were and are still very often carried out by applying an excitation of the sine type to the specimen, the objectives being: - Identification of the material: the test is carried out by subjecting the material to a swept sine having in general a rather low and constant amplitude (not to damage the specimen), about 5 ms-2, the variation of the frequency with time being rather small (close to one octave per minute) in order to study the response at various points of the specimen, to emphasize the resonance frequencies and to measure the amplification factors; - The application of a test defined in a standard document (MIL STD 810 C, AIR 7304, GAM T 13...), the test being intended to show that the material has a certain standard robustness, independent or difficult to relate to the vibrations which the material will undergo in its service life; -The application of a specification which as well as being feasible covers vibrations in its future real environment. The swept sine can have a constant level over all the frequency band studied (Figure 7.1-a) or can be composed of several constant levels at various frequency intervals (Figure 7.1-b).

230

Sinusoidal vibration

Figure 7.1. Examples of swept sines

Sweep is most frequently logarithmic. The specification sometimes specifies the direction of sweep: increasing or decreasing frequency. Either the sweep rate (number of octaves per minute), or the sweep duration (from lowest frequency to highest or in each frequency band) is specified. The level is defined by the peak value of the sinusoid or the peak-to-peak amplitude. As for the sine test at constant frequency (a redundant expression because the definition of a sinusoid includes this assumption; this terminology is however commonly used for better distinguishing these two vibration types. This test is also termed the dwell test), the parameter characterizing the amplitude can be a displacement (a displacement is sometimes specified at very low frequencies, this parameter being easier to measure in this frequency domain), more rarely a velocity, and, in general, an acceleration. The the sweep rate is, in general, selected to be sufficiently low to enable the response of the equipment to the test to reach a very strong percentage of the level obtained in steady operation under pure sinusoidal excitation. If the sweep is fast, it can be estimated that each resonance is excited one after the other, in a transient way, when the frequency sweeps the interval ranging between the half-power points of each peak of the transfer function of the material. We will see (Volume 3) how this method can be used to measure the transfer functions. hi this direction, the swept sine is a vibration of which the effects can be compared with those of a shock (except the fact that under a shock, all the modes are excited simultaneously) [CUR 55]

Swept sine

231

The swept sinusoidal vibration tests are adapted badly to simulate random vibrations, whose amplitude and phase vary in a random way and with which all the frequencies are excited simultaneously.

7.4. Origin and properties of main types of sweepings 7.4.1. The problem We know that for a linear one-degree-of-freedom mechanical system the damping ratio is given by:

the Q factor by:

the resonance frequency by:

and the width Af of the peak of the transfer function between the half-power points by:

Figure 7.2. One-degree-of-freedom system

232

Sinusoidal vibration

NOTE: We saw that the maximum of the transfer function actually equal to

The mechanical systems have, in general, rather -weak damping so that the approximation Hm = Q (which is the exact result for the maximum of the transfer function acceleration - relative velocity instead here of the function acceleration relative displacement) can be used. One additionally recalls that the half-power points are defined because of mechanical-electric analogy, from the transfer function acceleration-relative velocity. Writing:

for the sweep rate around the resonance frequency, f0, the time spent in the band Af is given roughly by:

and the number of cycles performed by:

When such a system is displaced from its equilibrium position and then released (or when the excitation to which it is subjected is suddenly stopped), the displacement response of the mass can be written in the form:

where T is a time constant equal to:

i.e. according to [7.10]:

It will be supposed in the following that the Q factor is independent of the natural frequency f0, in particular in the case of viscoelastic materials. Different reasoning can take into account a variation of Q with f0 according to various laws [BRO 75]; this leads to the same laws of sweeping. In a swept sine test, with the frequency varying according to time the response of a mechanical system is never perfectly permanent. It is closer to the response which the system would have under permanent stress at a given frequency when the sweep rate is slower. To approach as closely as possible this response in the vicinity of the resonance frequency, it is required that the time At spent in Af be long compared to the constant T, a condition which can be written [MOR 76]:

1*1-

rcfp

HQ 2

[7.22]

The natural frequency f0 can be arbitrary in the band considered (f l s f2) and, whatever its value, the response must be close to Q times the input to the resonance. To calculate the sweep law f(t) let us generalize f0 by writing fas:

It can be seen that the sweep rate varies as NOTE: The derivative f is positive for increasing frequency sweep, negative for decreasing frequency sweep.

1. It is supposed here that Af is sufficiently small (i.e. £ is small) to be able to equalize with little error the slope of the tangent to the curve f(t) and the slope of the chord relating to the interval Af. We will see that this approximation is indeed acceptable in practice.

234

Sinusoidal vibration

7.4.2. Case n°l: sweep where time At spent in each interval Af is constant for all values of f0 Here, since

it is necessary that

the constant y has the dimension of time, and

if we set

Sweeping at frequency increasing between f1 and f2 We deduce from [7.26] t

The constant T1 is such that, for

where TI is the time needed to sweep the interval between two frequencies whose ratio is e. Relations [7.24] and [7.25] lead to

Swept sine

235

Sweep at decreasing frequency

the constant T1 having the same definition as previously.

Expression for E(t) Increasing frequency:

i.e. [HAW 64] and [SUN 75]:

Decreasing frequency:

Later in this paragraph, and except in a specified particular case, we will consider only sweepings at increasing frequency, the relations being for the other case either identical or very easy to rewrite. We supposed above that f1 is always, whatever the sweep direction, the lowest frequency, and f2 always the highest frequency. Under this assumption, certain relations depend on the sweep direction. If, on the contrary, it is supposed simply that f1. is the initial frequency of sweep and f2 the final frequency, whatever the direction, we obtain the same relations independently of the direction; relations in addition identical to those established above and in what follows in the case of an increasing frequency. Time t can be expressed versus the frequency f according to:

236

Sinusoidal vibration

In spite of the form of the relations [7.27] and [7.30], the sweep is known as logarithmic, by referring to he expression [7.35]. The time necessary to go from frequency f1 to frequency f2 is given by:

which can still be written:

The number of cycles carried out during time t is given by:

i.e., according to

The number of cycles between f1 and f2 is:

which can be also written, taking into account [7.36],

The mean frequency (or average frequency or expected frequency):

The number of cycles AN performed in the band Af between the half-power points (during time At) is written [7.41]:

Swept sine

237

thus varies like f0 yielding

Starting also from

The time At spent in an interval Af is:

yielding another expression of AN:

The number of cycles N1 necessary to go from frequency f1 to a resonance frequency f0 is:

238

Sinusoidal vibration

or

This number of cycles is carried out in time:

or

If the initial frequency f1 is zero, we have N1 = N0 given by:

or

It is not possible, in this case, to calculate the time t0 necessary to go from 0 to fo-

Sweep rate According to the sweep direction, we have:

Swept sine

239

The sweep rate is expressed, generally, in number of octaves per minute. The number of octaves between two frequencies f1 and f2 is equal to [7.4]:

yielding the number of octaves per second:

(ts being expressed in seconds) and the number of octaves per minute:

If we set f = f2 in [7.35] for t = ts, we obtain:

From [7.59] and from [7.61]:

yielding another expression of the sweep law:

240

Sinusoidal vibration

or

according to the definition of R.

Figure 7.3. Sweep duration

Figure 7.3 shows the variations of ts versus the ratio f2/f1, for Rom equal to 0.1 - 0.2 - 0.3 - 0.5 - 1 and 2 Oct.-min (relation [7.60]). In addition we deduce from [7.58] and [7.62] the relation:

Case where the sweep rate Rdm is expressed in decades per minute By definition and according to [7.6]

Swept sine

241

or

The time spent between two arbitrary frequencies in the swept interval f1, f2 Setting fA and fB (> fA) the limits of a frequency interval located in (fl, f 2 ). The time tB - tA spent between fA and fB is calculated directly starting from [7.35] and [7.36] for example:

Time spent between the half-power points of a linear one-degree-of-freedom system Let us calculate the time At between the half-power points using the relations established for small £. The half-power points have as abscissae and

between these points:

This relation can be written:

i.e., since

respectively, i.e.

yielding, starting from [7.69], the time At spent

242

Sinusoidal vibration

Figure 7.4 shows the variations of

versus

It is noted that, for

is very close to 1.

Figure 7.4. Validity of approximate expression of the time spent between the half-power points

(whereAt is expressed in seconds) [SPE 61] [SPE 62] [STE 73]. The number of cycles in this interval is equal to

Swept sine

243

Figure 7.5. Time spent between the half-power points

It is noted that At* given by [7.70] tends towards the value given by [7.47] as Q increases. Figure 7.5 shows the variations of At* versus Q for Rom (oct/min) equal to 4, 2,1,1/2, 1/3, 1/4, 1/6 and 1/8 respectively.

t s (s) Figure 7.6. Sweep duration between two frequencies

Figure 7.6 gives the sweep time necessary to go from f1 to f2 for sweep rates Rom (oct/min) equal to 4, 2, 1,1/2,1/3,1/4, 1/6 and 1/8.

244

Sinusoidal vibration

Number of cycles per octave If fA and fg are two frequencies separated by one octave:

The number of cycles in this octave is equal to [7.41]

i.e., according to

Time to sweep one octave Let us set t2 for this duration

Time to sweep l/nth octave

Swept sine

245

7.4.3. Case n°2: sweep with constant rate If we wish to carry out a sweep with a constant rate, it is necessary that constant, i.e., since f

where is a constant with dimension of a time squared.

. The time spent in the band Af delimited by the half-power points varies like the natural frequency f0.

where

is a constant.

Increasing frequency sweep

The constant a is such thatf = f2 when t = ts, yielding [BRO 75], [HOK 48], [LEW 32], [PIM 62], [TUR 54], [WHI 72] and [WHI 82]:

This sweep is known as linear. Decreasing frequency sweep

246

Sinusoidal vibration

Calculation of the Junction Eft) [SUN 75] Increasing frequency:

Decreasing frequency:

at

E(t) = 2 7 c t

{

2

+f 2

[7.93] }

Sweep rate This is equal, depending on the direction of sweep, with

7.4.4. Case n°3: sweep ensuring a number of identical cycles AN in all intervals Af (delimited by the half-power points) for all values of f0 With this assumption, since the quantity

must be constant, the parameter (3 must be itself constant, yielding:

Swept sine

where

247

The sweep rate varies like the square of the instantaneous

frequency. This expression is written [BIC 70] [PAR 61]:

Increasing frequency sweep between f1 and f2 By integration,

(at t = 0, we suppose that f = f1.,the starting sweep frequency) i.e. [PAR 61]:

or, since, for

In this case, little used in practice, the sweep is known as hyperbolic [BRO 75] (also the termed parabolic sweep, undoubtedly because of the form of the relation [7.96], and log-log sweep [ELD 61] [PAR 61]). NOTE: In spite of the form of the denominator of the expression [7.99], the frequency f cannot be negative. For that, it would be necessary that 1 - a f1 t < 0, i.e.

i.e. that

248

Sinusoidal vibration

Decreasing frequency sweeping between f2 and f1

For

yielding:

Expression for E(t) The function E(t) in the sine term can be calculated from expression [7.2] of f(t):

- Increasingfrequencysweep [CRU 70] [PAR 61]

i.e. taking into account [7.99]

Swept sine

249

- Decreasingfrequencysweep We have in the same way:

Sweep rate Increasing frequency:

Decreasing frequency:

Tables 8.2 to 8.8 at the end of the Chapter 8 summarize the relations calculated for the three sweep laws (logarithmic, linear and hyperbolic).

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Chapter 8

Response of a one-degree-of-freedom linear system to a swept sine vibration

8.1 Influence of sweep rate An extremely slow sweep rate makes it possible to measure and plot the transfer function of the one-degree-of-freedom system without distortion and to obtain correct values for the resonance frequency and Q factor. When the sweep rate increases, it is noted that the transfer function obtained differs more and more from the real transfer function. The deformations of the transfer function result in (Figure 8.1): - a reduction of the maximum 8H ; - a displacement of the abscissa of the maximum 8fr; - a displacement 8fm of the median axis of the curve (which loses its symmetry); - an increase in the bandwidth Af (interval between the half-power points). When the sweep rate increases: - one first of all observes on the time history response the appearance of beats which are due to interference between the free response of the mechanical system relatively important after resonance and the excitation 'swept sine' imposed on the system [BAR 48] [PIM 62]. The number and the importance of these beats are weaker since the damping is greater;

252

Sinusoidal vibration

- then, as if the system were subjected to a shock, the sweep duration decreases. The largest peak of the response occurs for t > tb (residual 'response' observed when the duration of the shock is small compared to the natural period of the system). We will see an example of this in paragraph 8.2.3.

Figure 8.1. Deformation of the transfer Junction related to a large sweep rate (according to [REE 60])

Figure 8.2. Sweep rate influence on the response of a one-degree-of-freedom system

8.2. Response of a linear one-degree-of-freedom excitation

system to a swept sine

8.2.1. Methods used for obtaining response The calculation of the response of a linear one-degree-of-freedom system cannot be carried out entirely in an analytical way because of the complexity of the equations (except in certain particular cases). Various methods were proposed to solve the differential equation of the movement (analogue [BAR 48], [MOR 65] and [REE 60], numerical [HAW 64]), using the Fourier transformation [WHI 72], the

Response of a linear system to a swept sine vibration

253

Laplace transformation [HOK 48], the convolution integral [BAR 48], [LEW 32], [MOR 65], [PAR 61], [SUN 75] and [SUN 80], a series [BAR 48], [MOR 65], [PAR 61], [SUN 75], the Fresnel integrals [DIM 61] [HOK 48], [LEW 32] and [WHI 72], asymptotic developments [KEV 71], techniques of variation of the parameters [SUZ 78a], [SUZ 78b] and [SUZ 79], numerical integration,... In general, the transient period of the beginning of the sweep, which relates to only a low number of cycles compared to the total number of cycles of sweep, is neglected. However, it will be preferable to choose the initial frequency of sweep at least an octave below the first resonance frequency of the material to be sure to be completely free from this effect [SUN 80].

8.2.2. Convolution integral (or Duhamel's integral) We will see in the following paragraphs that the choice of the initial and final frequencies of sweep influence the amplitude of the response, which is all the more sensitive since the sweep rates are larger. If the excitation is an acceleration the differential equation of the movement of a linear one-degree-of-freedom system is written:

The solution can be expressed in the form of Duhamel's integral:

if

variable of integration). The excitation x(t) is given by:

where E(t) is given, according to the case involved by [7.91], [7.32] for a logarithmic sweep or by [7.107] for a hyperbolic sweep (increasing frequency). If we set

254

Sinusoidal vibration

and

these expressions can be written respectively in reduced form:

where

and

This yields, being a variable of integration:

It is noted that the reduced response q(0) is a function of the parameters £ , 0S, h1 and h2 only and is independent of the natural frequency f0. Numerical calculation of Duhamel 's integral: Direct calculation of q(0) from numerical integration of [8.7] is possible, but it: - requires a number of points of integration all larger since the sweep rate is smaller; - sometimes introduces, for the weak rates, singular points in the plot of q(0), which do not necessarily disappear on increasing the number of points of integration (or changing the X-coordinate).

Response of a linear system to a swept sine vibration

255

It is in this manner that the results given in the following paragraphs were obtained. Integration was carried out by Simpson's method.

8.2.3. Response of a linear one-degree-of freedom system to a linear swept sine excitation The numerical integration of expression [8.7] was carried out for various values of h1 and h2, for £ = 0.l, with between 400 and 600 points of calculation (according to the sweep rate) with, according to the sweep direction, E(0) being given by [8.4] if the sweep is at an increasing frequency or by

if the frequency is decreasing. On each curve response q(0), we have to note: - the highest maximum; - the lowest minimum (it was noted that these two peaks always follow each other); - the frequency of the excitation at the moment when these two peaks occur; - the frequency of the response around these peaks starting from the relation fR =

, with A6 being the interval of time separating these two consecutive 2 A6

peaks. The results are presented in the form of curves in reduced coordinates with: - on the abscissae, the parameter r\ defined by:

We will see that this is used by the majority of authors [BAR 48], [BRO 75], [CRO 56], [CRO 68], [GER 61], [KHA 57], [PIM 62], [SPE 61], [TRU 70], [TRU 95] and [TUR 54], in this form or a very close form Since, for a linear sweep and according to the direction of sweep,

256

Sinusoidal vibration

or

we have:

If frequency and time are themselves expressed in reduced form, r\ can be written:

with for linear sweep with increasing frequency

and with decreasing frequency:

yielding:

and

- on the ordinates the ratio G of the largest positive or negative peak (in absolute value) of the response q(0) to the largest peak which would be obtained in steady state mode

Response of a linear system to a swept sine vibration

257

Calculations were carried out for sweeps at increasing and decreasing frequency. These showed that: - for given r\ results differ according to the values of the limits h1 and h2 of the sweep; there is a couple hl, h2 for which the peaks of the response are largest. This phenomenon is all the more sensitive since r\ is larger (n > 5 ); - this peak is sometimes positive, sometimes negative; -for given n, sweep at decreasing frequency leads to responses larger than sweep at increasing frequency. Figure 8.3 shows the curves G(n) thus obtained. These curves are envelopes of all the possible results.

Figure 8.3. Attenuation of the peak versus the reduced sweep rate

In addition, Figure 8.4 shows the variations with TJ of the quantity

where

difference between the peak frequency of the transfer function

measured with a fast sweep and resonance frequency with a very slow sweep.

measured

258

Sinusoidal vibration

Figure 8.4. Shift in the resonance frequency

The frequency fp is that of the excitation at the moment when the response passes through the highest peak (absolute value). Af is the width of the resonance peak measured between the half-power points (with a very slow sweep). The values of the frequencies selected to plot this curve are those of the peaks (positive or negative) selected to plot the curve G(T|) of Figure 8.3 (for sweeps at increasing frequency). NOTE: These curves have been plotted for n varying between 0.1 and 100. Thisis a very important domain. To be convincing, it is enough to calculate for various values of n the number of cycles Nb carried out between h1 and h2 for given Q. This number of cycles is given by:

In addition, we showed [8.16] that

Response of a linear system to a swept sine vibration

259

yielding, since

Example

If n = 0.1 there are Ns = 250 cycles and if n = 10, there are Ns = 2.5 cycles.

For the higher values of r\ and for certain couples h1, h2, it can happen that the largest peak occurs after the end of sweep (t > t s ). There is, in this case, a 'residual' response, the system responding to its natural frequency after an excitation of short duration compared to its natural period ('impulse response'). The swept sine can be considered as a shock.

260

Sinusoidal vibration

Example

With these data, the duration tb is equal to 20.83 ms. Figure 8.5 shows the swept sine and the response obtained (velocity: f = 960 Hz/s).

Figure 8.5. Example of response to a fast swept sine

It is noted that, for this rate, the excitation resembles a half-sine shock of duration ts and amplitude 1. On the shock response spectrum of this half-sine (Figure 8.6), we would read in the ordinate (for f0 = 20 Hz on abscissa) an amplitude of the response of the onedegree-of-freedom system (f0 = 20 Hz, Q = 5) equal to 1.22 m/s2, a value which is that raised above on the curve in Figure 8.5.

Response of a linear system to a swept sine vibration

261

Figure 8.6. Shock response spectrum of a half-sine shock

For the same value of r|, and with the same mechanical system, we can obtain, by taking f1 = 1 Hz and f2 =43.8 Hz, an extreme response equal to 1.65 m/s2 (Figure 8.7). In this case the duration has as a value 44.58 ms.

Figure 8.7. Response for the same r\ value and for other limit frequencies of the swept domain

262

Sinusoidal vibration

Note on parameter r\ As defined by [8.9], this parameter r\ is none other than the quantity TC/U. of the relation [7.22]. If we calculate the number of cycles AN according to r\ carried out in the band Af delimited by the half-power points, we obtain, according to the sweep mode: - linear sweep

yielding

- logarithmic sweep

- hyperbolic sweep

Response of a linear system to a swept sine vibration

263

For given Q and t|, the number of cycles carried out in the band Af is thus identical. As a consequence, the time At spent in Af is, whatever the sweep mode, for T| and Q constant

The expressions of the parameters considered in Chapter 7 expressed in terms of n) are given in Tables 8.2 to 8.7 at the end of this chapter.

Figure 8.8. Validity of approximate expressions for attenuation G(TI)

A good approximation of the curve at increasing frequency can be obtained by considering the empirical relation (Figure 8.8):

To represent the curve G(T|) relating to sweeps at decreasing frequency, one can use in the same interval the relation:

264

Sinusoidal vibration

When damping tends towards zero, the time necessary for the establishment of the response tends towards infinity. When the sweep rate is weak, F.M. Lewis [LEW 32] and D.L. Cronin [CRO 68] stated the response of an undamped system as:

i.e., if sweep is linear, by:

NOTE: For the response of a simple system having its resonance frequency f0 outside the swept frequency interval (fl, f2 in steady state mode, or for an extremely slow sweep, the maximum generalized response is given:

-for

When the sweep rate is faster it is possible to obtain an approximate value of the response by successively combining [8.23] and [8.27], [8.23] and [8.28]:

Response of a linear system to a swept sine vibration

265

where r\ is given by [8.11].

8.2.4. Response of a linear one-degree-of-freedom sine

system to a logarithmic swept

The calculation of Duhamel's integral [8.7] was carried out under the same conditions as in the case of linear sweep, with:

or

according to the direction of sweep, with £ = 0.1, for various values of the sweep rate, the limits h1 and h2 being those which, for each value of r\, lead to the largest response (in absolute value). The curves G(T|) thus obtained were plotted on Figure 8.9, T| being equal to:

266

Sinusoidal vibration

Figure 8.9. Attenuation versus reduced sweep rate

where

These curves can be represented by the following empirical relations (for 0 < T\ < 100): -for increasing frequencies:

-for decreasing frequencies:

Figure 8.10 shows the calculated curves and those corresponding to these relations. The remarks relating to the curves G(T|) for the linear sweep case apply completely to the case of logarithmic sweep.

Response of a linear system to a swept sine vibration

Figure 8.10. Validity of the approximate expressions for attenuation G(r|)

The number of cycles between h1 and h2 is given here by:

yielding, starting from [8.33]:

267

268

Sinusoidal vibration

Example Table 8.1. Examples of sweep durations for given values of n

I f f 1 =10 Hz Q=5 f2 = 30 Hz

f0 = 20 Hz

T\

Ns

0.1

250 2.5 0.417 0.25

10 60 100

ts () 137.33 0.1373 0.02289 0.01373

Figure 8.11 shows the swept sine (log) for increasing frequency and the response calculated with these data for n = 60. It is possible to find other limits of the swept range (f l, f2) leading to a larger response.

Figure 8.11. Example of response to a fast swept sine

Response of a linear system to a swept sine vibration

269

The curves G(n) obtained in the case of linear and logarithmic sweeps at increasing and decreasing frequencies are superimposed on Figure 8.12. We obtain very similar curves (for a given sweep direction) with these two types of sweeps.

Figure 8.12. Comparison of the attenuation of linear and logarithmic sweeps

8.3. Choice of duration of swept sine test In this paragraph, the duration of the tests intended to simulate a certain particular swept sine real environment will not be considered. During an identification test intended to measure the transfer function of a mechanical system, it is important to sweep slowly so that the system responds at its resonance with an amplitude very close to the permanent response, whilst adjusting the duration of sweep to avoid prohibitive test times . It was seen that a good approximation to the measure of the resonance peak can be obtained if n is sufficiently small; J.T. Broch [BRO 75] advised, for example, that n < 0.1, which ensures an error lower than 1 - G = 1% . For a given 1 - G0 error, the curve G(T|) makes it possible to obtain the limit value n0 of n not to exceed:

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Sinusoidal vibration

Table 8.2. Minimal sweeping duration versus sweeping mode

Linear sweep

Logarithmic sweep

Hyperbolic sweep

It is noted that, in this last case, ts is independent of f0. In both other cases, f0 being in general unknown, one will take for f0 the value of the swept frequency range which leads to the largest duration ts.

8.4. Choice of amplitude To search for resonance frequency the amplitude of the excitation must be: - sufficiently high to correctly 'reveal' the peaks of resonance in the response. If the structure is linear, the values of Q measured are independent of the sweep level; - sufficiently weak not to damage the specimen (by exceeding an instantaneous stress level or by fatigue). The choice must thus be made by taking account of the levels of the real vibratory environments. If the structure is not linear, the value of Q measured depends on the level of the excitation. Generally, Q decreases when the level of the excitation increases. If we wish to use in calculations an experimental transfer function, we will have to measure this function using an excitation in which the level is close to those of the real environments that the structure will exist in. One often has to choose two levels (or more).

Response of a linear system to a swept sine vibration

271

8.5. Choice of sweep mode The more common use of swept sine is for the determination of the dynamic properties of a structure or of a material (natural frequencies, Q factors). For this type of test, the sweep rate must be sufficiently slow so that the response reaches a strong percentage of the response in steady state excitation (it will be seen however (Volume 3) that there are methods using very fast sweeps). The relations allowing the determination of the test duration are based on the calculations carried out in the case of a linear single-degree-of-freedom system. It is admitted then that if this condition is conformed to, the swept sine thus defined will also create in a several degrees-of-freedom system responses very close to those which one would obtain in steady state mode; this assumption can be criticized for structures having modes with close frequencies. For this use, it may be worthwhile to choose a sweep mode like that which leads to the weakest duration of test, for the same percentage G of steady state response. According to the relations given in paragraph 8.3, it appears unfortunately that the mode determined according to this criterion is a function of the natural frequency f0 to be measured in the swept frequency interval. Generally a logarithmic sweep is the preferred choice in practice. The hyperbolic sweep, little used to our knowledge, presents the property appropriate to carry out a constant number of cycles in each interval Af delimited by the half-power points of a linear one-degree-of-freedom system, whatever the frequency of resonance (or a constant number of cycles of amplitude higher than P % of the Q factor [CRE 54]). This property can be used to simulate the effects of a shock (the free response of a system is made with the same number of cycles whatever the resonance frequency f0, provided that the Q factor is constant) or to carry out fatigue tests. In this case, it must be noted that, if the number of cycles carried out at each resonance is the same, the damage created by fatigue will be the same as if the excitation produces a maximum response displacement zm (i.e. a stress) identical to all the frequencies. Gertel [GER 61] advises this test procedure for materials for which the life duration is extremely long (equipment installed on various means of transport, such as road vehicles, aircraft). The sweep duration ts can be defined a priori or by imposing a given number AN of cycles around each resonance, the duration ts being then calculated from:

272

Sinusoidal vibration

by introducing the value of the highest Q factor measured during the identification tests (search for resonance) or a value considered representative in the absence of such tests. The limits f1 and f2 delimit a frequency range which must include the principal resonance frequencies of the material. This type of sweep is sometimes used in certain spectral analyzers for the study of experimental signals whose frequency varies with time [BIC 70]. To simulate an environment of short duration At, such as the propulsion of a missile, on a material of which the resonance frequencies are little known, it is preferable to carry out a test where each resonance is excited during this time At (logarithmic sweep) [PIM 62]. The total duration of sweep is then determined by the relation [7.37]:

NOTE: The test duration ts thus calculated can sometimes be relatively long and this, more especially as it is in general necessary to subject to the vibrations the specimen on each of its three axes. Thus, for example, ifQ = 10, At = 20 s, f1 =10 Hz and f2 = 2000 Hz

yielding a total test duration of 3 x 1060 s = 3180 s (53 min). C.E. Crede and E.J. Lunney [CRE 56] recommend that a sweep is carried out in several frequency bands simultaneously to save time. The method consists of cutting out the signal (swept sine) which would be held between times t1 and t2 in several intervals taken between t1 and ta: ta and tb, ... ,tn and t2, and to apply to the specimen the sum of these signals. Knowing that the material is especially sensitive to the vibrations whose frequency is located between the half-power points, it can be considered that only the swept component which has a frequency near to the frequency of resonance will act notably on the behaviour of the material, the others having little effect. In addition, if the specimen has several resonant frequencies, all will be excited simultaneously in conformity with reality.

Response of a linear system to a swept sine vibration

273

Another possibility consists of sweeping the frequency range quickly and in reducing the sweep rate in the frequency bands where the dynamic response is important to measure the peaks correctly. ' C.F. Lorenzo [LOR 70] proposes a technique of control being based on this principle, usable for linear and logarithmic sweeps, making it possible to reduce with equal precision the test duration by a factor of about 7.5 (for linear sweep). The justification for a test with linear sweep clearly does not appear, except if we accept the assumption that the Q factor is not constant whatever the natural frequency f0. IfQ can vary according to a law Q = constant x f0 (the Q factor being often an increasing function of the natural frequency), it can be shown that the best mode of sweep is the linear sweep.

274

Sinusoidal vibration Table 8.3. Summary of sweep expressions

Type of sweep Sweep rate

Constant

Law f(t)

Cst

Hyperbolic

Logarithmic

Libear

Response of a linear system to a swept sine vibration Table 8.4. Summary of sweep expressions

Type of sweep

Hyperbolic

Logarithmic

Number of cycles carried out during ts Between the frequencies f1 and f2

Sweep duration Between the frequencies f1 and f2

constant Interval of time spent in the band Af

Linear

275

276

Sinusoidal vibration Table 8.5. Summary of sweep expressions

Type of sweep Number of cycles carriec out in the interval Af (between the . half-power points) of a one-degreeof-freedom system

Number of cycles to be carried out between fj and f0 (resonance frequency)

Time tj between fj and f0

Hyperbolic constant

Logarithmic curve

Linear

Response of a linear system to a swept sine vibration Table 8.6. Summary of sweep expressions Type of sweep Time t1 between f1 and

fo

Number of cycles . to carry out between fj = 0 and f0

Time t0 between fj = 0 and f0

Mean frequency Time spent between faa and

Hyperbolic

Logarithmic curve

Linear

277

278

Sinusoidal vibration Table 8.7. Summary of sweep expressions

Type of sweep

Numbers of cycles per octave (fA = lower frequency of the octave)

Time • necessary to sweep an octave

Hyperbolic

Logarithmic curve

Linear

Response of a linear system to a swept sine vibration Table 8.8. Summary of sweep expressions Type of sweep

Time necessary to sweep l/n* octave

Sweep rate

Hyperbolic

Logarithmic curve

Linear

279

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Appendix

Laplace transformations

A.I. Definition Consider a real continuous function f(t) of the real definite variable t for all t > 0 and set

(provided that the integral converges). The function f(t) is known as 'original' or 'object', the function F(p) as 'image' or 'transform'. Example Consider a step function applied to t = 0 and of amplitude fm. The integral [A.I] gives simply

282

Sinusoidal vibration

A.2. Properties In this paragraph some useful properties of this transformation are given, without examples. A.2.1. Linearity

A.2.2. Shifting theorem (or time displacement theorem)

Figure A.1. Shifting of a curve with respect to the variable t

Let us set f(t)a transformable function and let us operate a translation parallel to the axis 0t, of amplitude T (T > 0). If F(p) is the transform of f(t), the transform of f(t - T) is equal to

(formula of the translation on the right or shifting theorem). Application A rectangular shock can be considered as being created by the superposition of f two levels, one of amplitude fm applied at time t = 0 (transform: —, cf. preceding P example), the other of amplitude —fm applied at time t = T, of transform —— e P

yielding the expression of the transform

Appendix

283

A.2.3. Complex translation

This result makes it possible to write directly the transform of f(t) e that of f(t) is known.

a

when

A.2.4. Laplace transform of the derivative off(t) with respect to time The transform of the derivative f'(t) of f(t) with respect to t is equal to

where F(p) is the Laplace transform of f(t) and f(0 + ) is the value of the first derivative of f(t) for t = 0 (as t approaches zero from the positive side). In a more general way, the transform of the nth derivative of f(t) is given by

where

are the successive derivatives of f(t) for

t = 0 (as t approaches zero from the positive side).

A.2.5. Derivative in the p domain The nth derivative of the transform F(p) of a function f(t) with respect to the variable p is given by

284

Sinusoidal vibration

A.2.6. Laplace transform of the integral of a function f(t> with respect to time If lim

when

and to the order n

A.2.7. Integral of the transform F(p) The inverse transform of the integral of F(p) between p and infinite is equal to:

When integrating n times between the same limits, it becomes

A.2.8. Scaling theorem If a is a constant,

Appendix

285

A.2.9. Theorem of damping or rule of attenuation

The inverse transform of F(p + a) is thus e at

e-

f(t). It is said that the function

damps the function f(t) when a is a positive real constant.

A.3. Application of Laplace transformation to the resolution of linear differential equations In the principal use made of Laplace transformation, the interest resides in the property relative to derivatives and integrals which becomes, after transformation, products or quotients of the transform F(p) of f(t) by p or its powers. Let us consider, for example, the second order differential equation

where a and b are constants. Let us set Q(P) and F(p) the Laplace transforms of q(t) and f(t) respectively. From the relations of the paragraph A.2.4, we have

where q(0) and q(0) are the values of q(t) and its derivative for t = 0. Because of the linearity of the Laplace transformation, it is possible to transform each member of the differential equation [A. 19] term by term:

by replacing each transform by its expression this becomes

286

Sinusoidal vibration

Let us expand the rational fraction

into partial

fractions; while noting that p1 and p2 are the roots of the denominator 2 p + ap -t- b, we have

with

yielding

i.e.

where is a variable of integration. In the case of a system initially at rest, q(0) = q(0) = 0 and

Appendix

287

A.4. Calculation of inverse transform: Mellin-Fourier integral or Bromwich transform Once the calculations are carried out in the domain of p, where they are easier, it is necessary to return to the time domain and to express the output variables as a function of t. We saw that the Laplace transform F(p) of a function f(t) is given by [A.1]

The inverse transformation is defined by the integral known as the MellinFourier integral

and calculated, for example, on a Bromwich contour composed of a straight line parallel to the imaginary axis, of positive abscissea C [ANG 61], C being such that all the singularities of the function F(p) ep t are on the left of the line [BRO 53]; this contour thus goes from C - i oo to C + i oo. If the function F(p) ep t has only poles, then the integral is equal to the sum of the corresponding residues, multiplied by 2 n i. If this function has singularities other than poles, it is necessary to find, in each case, a equivalent contour to the Bromwich contour allowing calculation of the integral [BRO 53] [QUE 65]. The two integrals [A.1] and [A.31] establish a one-to-one relationship between the functions of t and those of p. These calculations can be in practice rather heavy and it is prefered to use when possible tables of inverse transform providing the transforms of the most current functions directly [ANG 61] [DIT 67], [HLA 69] and [SAL 71]. The inverse transformation is also done using these tables after having expressed results as a function of p in a form revealing transforms whose inverse transform appears in Table A.1.

288

Sinusoidal vibration

Example Let us consider the expression of the response of a one-degree-of-freedom damped system subjected to a rectangular shock of amplitude unit of the form f(t) = 1 and of duration T. For this length of time T, i.e. for t < T, the Laplace transform is given by (Table A. 1):

NOTE: After the end of the shock, it would be necessary to use the relation

Relation [A.24] applies with a = 2 £ and b = 1, yielding

yielding, using Table A. 1

Appendix

A.5. Laplace transforms Table A.1. Laplace transforms

Function f(t) 1 t

eat

sin a t cos a t sh a t ch a t

Transform L[f(t)]=F(p)

289

290

Sinusoidal vibration

t2

tn

sin t cos t

a t - sin a t

sin a t - a t cos a t

t sin a t

sin a t + a t cos a t

t cos a t

a sin b t - b sin at

1 • a t - a t cos a t)\ —r- /(sin 2a

cos a t - cos b t

b 2 -a 2 e- a t -e- b t b-a

Appendix

291

292

Sinusoidal vibration

These transforms can be used to calculate others, for example starting from decompositions in partial fractions such as

where, in these relations, there arises, a =

A.6. Generalized impedance - the transfer function If the initial conditions are zero, the relation [A.24] can be written

i.e., while setting

Appendix

293

By analogy with the equation which links together the current l(Q) (output variable) and the tension E(Q) (input variable) in an electrical supply network in sinusoidal mode

Z(p) is called the generalized impedance of the system, andZ(Q) is the transfer impedance of the circuit. The inverse of A(p) of z(p), l/Z(p), is the operational admittance. The function A(p) is also termed the transfer function. It is by its intermediary that the output is expressed versus the input:

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Vibration tests: a brief historical background

Vibration tests on aircraft were developed from 1940 to verity the resistance of parts and equipment prior to their first use [BRO 67]. Such tests became necessary as a result of: -the increasing complexity of on-board flight equipment which was more sensitive to vibrations; - improved performance of aircraft (and, more generally, of vehicles), to the extent that the sources of vibration initially localized in engines became extended substantially outwards to the ambient medium (aerodynamic flows). The chronology of such developments can be summarized as follows [PUS 77]: 1940

Measurement of resonance frequencies. Self-damping tests. Sine tests (at fixed frequency) corresponding to the frequencies created by engines running at a permanent speed. Combined tests (temperature, humidity, altitude).

1946

First electrodynamic exciter [DEV 47] [IMP 47].

1950

Swept sine tests to simulate the variations in engine speed or to excite all the resonance frequencies of the test item of whatever value.

1953

Specifications and tests with random vibrations (introduction of jet engines, simulation of jet flows and aerodynamic turbulences with continuous spectra). These tests were highly controversial until the 1960's [MOR 53]. To overcome the insufficient power of such installations, attempts were

296

Sinusoidal vibration

made to promote swept narrow-band random variations in the frequency domain of interest [OLS 57]. 1955

First publications on acoustic vibrations (development of jet rockets and engines, effect of acoustic vibrations on their structures and equipment).

1957

First acoustic chambers [BAR 57] [COL 59] [FRI 59].

1960

The specification of random vibration became essential and the determination of equivalence between random and sinusoidal vibrations.

1965

J.W. Cooley and J.W. Tukey's algorithm for calculating FFTs [COO 65].

1967

Increasing number of publications on acoustic vibrations.

1970

Tri-axial test facility [DEC 70]. Development of digital control systems.

1976

Extreme response spectra and fatigue damage spectra developed; useful in writing specifications (a method in four stages starting from the life cycle profile).

1984

Account taken of the tailoring of tests in certain standard documents (MIL[STD 810 D], [GAM 92] (development of specifications on the basis of measuring the real environment).

1995

Customization of the product tailored to its environment. Account taken of the environment throughout all the phases of the project (according to the R.G. Aero 00040 Recommendation).

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302

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[HOP 12] [IMP 47] [JAC30] [JAC 58] [JEN 59] [JON 69]

[JON 70]

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Index

absorbed energy 17 accomodation ultimate stress 15 amplification factor 120,175 amplitude, reduction with damping 82, 83 et seq analastic stress 15 angular frequency 45 aperiodic critical response 69, 70 damped response 68 bandwidth 170,173 method 212 Bromwich transform 287 coercive force 32 conservative system 87 convolution integral 55 Coulomb damping 200 friction 27 coupled modes 36 critical impact loading 5 critical modes 177 cyclical stress, limits to sensitivity 22 damped aperiodic mode 64 exponential motion 65 oscillatory mode 71,72 pseudo-sinusoidal 72 sinusoidal 72 damper combination 26 parallel 26 series 27 damping 15, 16 capacity 17,23 Coulomb 203

critical 56,59 dry 203 energy 23 force 22, 39 internal 18 linear 21 measurement of 210 non-linear 27 non-viscous 197etseq period, influence on 85 quadratic 21,205 ratio 45 reduction with damping 82 sinusoidal 131 structural 18 subcritical 58 supercritical 56,60 zero 86,91 decade 228 decreased rate method 213 decrement and damping values 81 degree of freedom (dof) 34 single system 41,47 vibrating 44 dissonance 170 dry friction 28 interfaces 22,27 Duhamel integral 53, 55,116 dwell test 230 dynamic amplification factor 175,178, 184 dynamic effects 1 elastic solid 25 load, behaviour under 2,3 envelope, point of contact 75 equivalent damping constant 198 equivalent sine test 141

310

Sinusoidal vibration equivalent spring constant 9 stiffness 9 excitation 47 acceleration 47,48 on support 40 arbritary, response to linear dof system 4 et seq absolute response 57 acceleration, defined by 53 differential equations 53 force versus time, defined by 43 on mass, force 49 natural oscillations 63 periodic, spectrum of 135 sinusoidal 143 et seq swept sine 251 et seq velocity, defined by 50 force transmissibility 189 form factor 134 four coordinate diagram 195 friction, viscous 22 gain, gain factor 120 generalised impedance, transfer function 292,293 half-power point 169 hysteresis 17,18 loop 16 impact 3 impulse decline 118 responses 114 impulse and step responses 89 et seq response of mass spring system 102 by relative displacement 90 to unit step function 89 use of 112 zero damping indicia! admittance 92 inertia! force 38 Kelvin-Voigt model 26 Kennedy-Pancu method 212 Laplace transforms, transformations 281 et seq complex translation 283 damping, rule of attenuation 285 definition 281 derivative, p domain, wit time 283 generalised impedance 292 integral wrt time, of transform 284

inverse 287 linear differential eqns, resolution of 285 linearity 282 operational admittance 293 properties 282 scaling theorem 284 shifting theorem 282 transfer function 293 transfer impedance 293 linear one dof mechanical system 37 critical aperiodic mode 68 critical damping 56 damped aperiodic mode 64 damped oscillatory mode 71 definitions 41 differential equations 53 excitation by acceleration 47 displacement or velocity on support 50 force v time 43 hysteresis 164 natural oscillations 63 reduced form 49 response to sinusoidal vibration 143 subcritical damping 54 supercritical damping 60 transfer function 119,124 measurement of 130 linear single dof system Jfesponse to arbritary excitation 40 et seq to sinusoidal excitation 143 et seq absolute response eqns of motion 144 periodic excitation 152 reduced response 151 relative response 144 steady state 159 transient response 155 absolute 159 relative 155 to swept sine vibration 251 et seq steady state response 159 transfer function 194 transient response 155 linearization hysteresis loop 199 non-linear system 198 logarithmic decrement 77 method 213 lubricated interfaces 22 lumped parameter systems 33 example 39 n dof system 38

Index mass 6 damping and spring 41 spring equivalent 8 symbol 7 mass-spring system, response to unit impulse excitation 102etseq mathematical model 34 Maxwell model 26 mean square value 133 mean value 132 mechanical system, elements of 6 distributed 32 lumped parameter 32,33 Mellin-Fourier integral 287 microelastic ultimate stress 15 mode 35 coupled 37 modulus of elasticity dynamic 30 secant, static, and tangent 29 moment of inertia 6 movement, differential equation 53 natural frequency 45 natural period 45 natural pulsation 45 non-viscous damping 197 et seq bandwidth method 212 combination of 198 et seq, 208 constant damping force 203,204 damping, main types 202 energy dissipated 220 equivalent viscous 198,209 force v time 43 hysteretic 207 measurement 210 non-linear, linearization of 198 non-linear stiffness 224 in real structures 197 and relative velocity 202 resonance, amplification factor 210 square of displacement 206 square of velocity 205 structural 207 Nyquist diagram 124 octave 228 operational admittance 293 peak factor 134 phase factor 120 plastic deformation 1 ,28,29 pseudo-period 77 pseudo-pulsation, and damping 73 pseudo sinusoidal movement 155

311

quadratic damping 205 quality factor 88 quasi periodic signals 138 reduced variables 52 relative response 53 relative transmissibility 175 remanant deformation 32 response absolute 57, 62, 74, 126 calculation 122 absolute parameter 108 defined by absolute displacement 97 defined by relative velocity, 126 defined by velocity or acceleration 97 expression for 90 impulse 104,105 impulsive 105 mass-spring system to unit impulse 102 relative 62, 74 calculation 121 rheology 25 rms value 133 Rocard's integral 116 scaling theorem 284 secant modulus 31 of elasticity 32 selectivity, system 171 shifting theorem 282 shock absorber damping 25 sine test at constant frequency 230 single dof system, natural oscillations 63 sinusoid spectrum 134 sinusoidal vibration 131 et seq absolute response 147,149 amplification factor at resonance, 210 amplitude variation 189 critical damping 145 energy dissipated 165,220 by damping 167,168 equations of motion 144 relative response 144,149 influence damping 225 mean value 132 movement transmissibility 189,189 overdamped system 146 periodic excitation 135 response to 152 vibrations 139 phase variations 180,192 velocity 174 quasiperiodic signal 138 reduced response 151

312

Sinusoidal vibration resonance frequency 187 response amplitude, variation in 175 rms value 133 strong damping 224 tests 139 periodic and sinusoidal 139 specific damping capacity 217 spectrum discrete 134 lines 134 spring constant 7 equivalent 8 force in 221 response 8 symbol 7 static effects 1 static modulus of elasticity 29 steady state response 159 absolute 160 relative 159 step responses 89 et seq, 92,115 and impulse responses, use of 112 stiffness 7 bilinear 14 equivalent 9 examples 11,12 linear 14 non-linear 14 parallel 8 in rotation 12 series 10 strain 18 anelastic 20 energy 23 reversible 20 stress cycle 17 . stress-strain diagram 4 successive maxima, peaks 76, 78 superposition integral 55 sweep duration 230 logarithmic 236 rate 230,238 swept sine 227 et seq constant intervals case 234 constant rate case 245 definition 227

half-power points 242 hyperbolic or parabolic sweep 228,247 identical intervals case 246 increasing frequency 234 response of single dof system 251 etseq vibration 229,251 etseq tension 5 total damping energy 17 traction, dynamic and static 5 transfer function, 187,194 definitions 124 excitation and response 127 impedance 293 measurement 130 transient response absolute 159 linear 227 logarithmic sweep 228 origin and properties 231 vibrations 229 transmissibility 120, 160,190 factor 160 transmittance 160 relative 155,157,158 underdamped system 145 vehicle suspension, response to sinusoidal excitation 153 velocity amplitude variations 162 hysteresis loop 164 quality factor 162 velocity phase, variations in 174 viscous damping 164 velocity response, amplitude 163 viscous damping constant 24 Voigt model 38 weak damping 223 weight function 105 yield stress 15 zero damping 91

Mechanical Shock

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Mechanical Vibration & Shock

Mechanical Shock Volume II

Christian Lalanne

HPS

HERMES PENTON SCIENCE

First published in 1999 by Hermes Science Publications, Paris First published in English in 2002 by Hermes Penton Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: Hermes Penton Science 120 Pentonville Road London N1 9JN © Hermes Science Publications, 1999 © English language edition Hermes Penton Ltd, 2002 The right of Christian Lalanne to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A CIP record for this book is available from the British Library. ISBN 1 9039 9604 X

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn www.biddies,co.uk

Contents

Introduction List of symbols 1 Shock analysis 1.1. Definitions 1.1.1. Shock 1.1.2. Transient signal 1.1.3. Jerk 1.1.4. Bump 1.1.5. Simple (or perfect) shock 1.1.6. Half-sine shock 1.1.7. Terminal peak saw tooth shock (TPS) or final peak saw tooth shock (FPS) 1.1.8. Initial peak saw tooth shock (IPS) 1.1.9. Rectangular shock 1.1.10. Trapezoidal shock 1.1.11. Versed-sine (or haversine) shock 1.1.12. Decaying sinusoidal pulse 1.2. Analysis in the tune domain 1.3. Fourier transform 1.3.1. Definition 1.3.2. Reduced Fourier transform 1.3.3. Fourier transforms of simple shocks 1.3.3.1. Half-sine pulse 1.3.3.2. Versed-sine pulse 1.3.3.3. Terminal peak saw tooth pulse (TPS) 1.3.3.4. Initial peak saw tooth pulse (IPS) 1.3.3.5. Arbitrary triangular pulse 1.3.3.6. Rectangular pulse

xi xiii 1 1 1 2 2 2 3 3 3 3 3 3 3 4 4 4 4 6 7 7 8 9 10 12 14

vi

Mechanical shock

1.4

2

1.3.3.7. Trapezoidal pulse 1.3.4. Importance of the Fourier transform Practical calculations of the Fourier transform 1.4.1. General 1.4.2. Case: signal not yet digitized 1.4.3. Case: signal already digitized

Shock response spectra domains 2.1. Main principles 2.2. Response of a linear one-degree-of-freedom system 2.2.1. Shock defined by a force 2.2.2. Shock defined by an acceleration 2.2.3. Generalization 2.2.4. Response of a one-degree-of-freedom system to simple shocks 2.3. Definitions 2.4. Standardized response spectra 2.5. Difference between shock response spectrum (SRS) and extreme response spectrum (ERS) 2.6. Algorithms for calculation of the shock response spectrum 2.7. Subroutine for the calculation of the shock response spectrum 2.8. Choice of the digitization frequency of the signal 2.9. Example of use of shock response spectra 2.10. Use of shock response spectra for the study of systems wi th se ver al degress of fr ee dom

3 Characteristics of shock response spectra 3.1. Shock response spectra domains 3.2. Characteristics of shock response spectra at low frequencies 3.2.1. General characteristics 3.2.2. Shocks with velocity changed from zero 3.2.3. Shocks with AV = 0 and AD = 0 at end of pulse 3.2.4. Shocks with AV = 0 and AD = 0 at end of pulse 3.2.5. Notes on residual spectrum 3.3. Characteristics of shock response spectra at high frequencies 3.4. Damping influence 3.5. Choice of damping 3.6. Choice of frequency range 3.7. Charts 3.8. Relation of shock response spectrum to Fourier spectrum 3.8.1. Primary shock response spectrum and Fourier transform 3.8.2. Residual shock response spectrum and Fourier transform 3.8.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra 3.9. Characteristics of shocks of pyrotechnic origin

15 17 18 18 18 20 23 23 26 26 27 28 33 37 40 47 47 48 52 53

59 59 60 60 TT60 69 72 74 75 77 78 80 81 81 81 83 86 88

Contents

3.10. Care to be taken in the calculation of spectra 3.10.1. Influence of background noise of the measuring equipment.. 3.10.2. Influence of zero shift 4 Development of shock test specifications 4.1. General 4.2. Simplification of the measured signal 4.3. Use of shock response spectra 4.3.1. Synthesis of spectra 4.3.2. Nature of the specification 4.3.3. Choice of shape 4.3.4. Amplitude 4.3.5. Duration 4.3.6. Difficulties 4.4. Other methods 4.4.1. Use of a swept sine 4.4.2. Simulation of shock response spectra using a fast swept sine 4.4.3. Simulation by modulated random noise 4.4.4. Simulation of a shock using random vibration 4.4.5. Least favourable response technique 4.4.6. Restitution of a shock response spectrum by a series of modulated sine pulses 4.5. Interest behind simulation of shocks on a shaker using a shock spectrum 5. Kinematics of simple shocks 5.1. General 5.2. Half-sine pulse 5.2.1. Definition 5.2.2. Shock motion study 5.2.2.1. General expressions 5.2.2.2. Impulse mode 5.2.2.3. Impact mode 5.3. Versed-sine pulse 5.3.1. Definition 5.3.2. Shock motion study 5.4. Rectangular pulse 5.4.1. Definition 5.4.2. Shock motion study 5.5. Terminal peak saw tooth pulse 5.5.1. Definition 5.5.2. Shock motion study 5.6. Initial peak saw tooth pulse 5.6.1. Definition

vii

90 90 92 95 95 96 98 98 99 100 101 101 105 107 107 108 112 113 114 115 117 121 121 121 121 122 122 124 126 136 136 137 139 139 139 142 142 143 145 145

viii

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5.6.2.

Shock motion study

6 Standard shock machines 6.1. Main types 6.2. Impact shock machines 6.3. High impact shock machines 6.3.1. Lightweight high impact shock machine 6.3.2. Medium weight high impact shock machine 6.4. Pneumatic machines 6.5. Specific test facilities 6.6. Programmers 6.6.1. Half-sine pulse 6.6.2. Terminal peak saw tooth shock pulse 6.6.3. Rectangular pulse - trapezoidal pulse 6.6.4. Universal shock programmer 6.6.4.1. Generating a half-sine shock pulse 6.6.4.2. Generating a terminal peak saw tooth shock pulse 6.6.4.3. Trapezoidal shock pulse 6.6.4.4. Limitations 7 Generation of shocks using shakers 7.1. Principle behind the generation of a simple shape signal versus time 7.2. Main advantages of the generation of shock using shakers 7.3. Limitations of electrodynamic shakers 7.3.1. Mechanical limitations 7.3.2. Electronic limitations 7.4. The use of the electrohydraulic shakers 7.5. Pre-and post-shocks 7.5.1. Requirements 7.5.2. Pre- or post-shock 7.5.3. Kinematics of the movement for symmetrical preand post-shock 7.5.3.1. Half-sine pulse 7.5.3.2. TPS pulse 7.5.3.3. Rectangular pulse 7.5.3.4. IPS pulse 7.5.4. Kinematics of the movement for a pre-shock or a post-shock alone 7.5.5. Abacuses 7.5.6. Influence of the shape of pre- and post-pulses 7.5.7. Optimized pre- and post-shocks 7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General

145 149 149 151 160 160 162 163 164 165 165 173 180 181 182 182 183 183 189 189 190 191 191 193 193 193 193 195 198 198 206 207 208 208 212 213 216 220 220

Contents

7.6.2. 7.6.3.

Influence of the pre- and post-shocks on the time history response of a one-degree-of-freedom system Incidence on the shock response spectra

ix

220 223

8 Simulation of pyroshocks 8.1. Simulations using pyrotechnic facilities 8.2. Simulation using metal to metal impact 8.3. Simulation using electrodynamic shakers 8.4. Simulation using conventional shock machines

227 227 230 231 232

9 Control of a shaker using a shock response spectrum 9.1. Principle of control by a shock response spectrum 9.1.1. Problems 9.1.2. Method of parallel filterss 9.1.3. Current numerical methods 9.2. Decaying sinusoid 9.2.1. Definition 9.2.2. Response spectrum 9.2.3. Velocity and displacement 9.2.4. Constitution of the total signal 9.2.5. Methods of signal compensation 9.2.6. Iterations 9.3. D.L. Kern and C.D. Hayes' function 9.3.1. Definition 9.3.2. Velocity and displacement 9.4. ZERD function 9.4.1. Definition 9.4.1.1. D.K. Fisher and M.R. Posehn expression 9.4.1.2. D.O. Smallwood expression 9.4.2. Velocity and displacement 9.4.3. Comparison of the ZERD waveform with a standard decaying sinusoid 9.4.4. Reduced response spectra 9.4.4.1. Influence of the damping n of the signal 9.4.4.2. Influence of the Q factor 9.5. WAVSIN waveform 9.5.1. Definition 9.5.2. Velocity and displacement 9.5.3. Response of a one-degree-of-freedom system 9.5.3.1. Relative response displacement 9.5.3.2. Absolute response acceleration 9.5.4. Response spectrum 9.5.5. Time history synthesis from shock spectrum 9.6. SHOC waveform 9.6.1. Definition

235 235 235 236

237 239 239 239 242 243 244 250 251 251 252 253 253 253 254 255 257 257 257 258 259 259 260 262 263 265 265 266 267 267

x

Mechanical shock

9.6.2. 9.6.3.

Velocity and displacement Response spectrum 9.6.3.1. Influence of damping n of the signal 9.6.3.2. Influence of the Q factor on the spectrum 9.6.4. Time history synthesis from shock spectrum 9.7. Comparison of the WAVSIN, SHOC waveforms and decaying sinusoid 9.8. Use of a fast swept sine 9.9. Problems encountered during the synthesis of the waveforms 9.10. Criticism of control by a shock response spectrum 9.11. Possible improvements 9.11.1. IBS proposal 9.11.2. Specification of a complementary parameter 9.11.2.1. Rms duration of the shock 9.11.2.2. Rms value of the signal 9.11.2.3. Rms value in the frequency domain 9.11.2.4. Histogram of the peaks of the signal 9.11.2.5. Use of the fatigue damage spectrum 9.11.3. Remarks on the characteristics of response spectrum 9.12. Estimate of the feasibility of a shock specified by its SRS 9.12.1 C.D. Robbins and E.P. Vaughan's method 9.12.2. Evaluation of the necessary force, power and stroke

270 271 271 272 273 274 274 278 280 282 283 284 284 286 287 288 288 288 289 289 291

Appendix. Similitude in mechanics A1. Conservation of materials A2. Conservation of acceleration and stress

297 297 299

Mechanical shock tests: a brief historical background

301

Bibliography

303

Index

315

Synopsis of five volume series

319

Introduction

Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (material handling, transportation, etc.). This kind of environment, although of extremely short duration (from a fraction of a millisecond to a few dozen milliseconds) is often severe and cannot be neglected. The initial work into shocks was carried out in the 1930s on earthquakes and their effect on buildings. This resulted in the notion of the shock response spectrum. Testing on equipment started during World War II. Methods continued to evolve with the increase in power of exciters, making it possible to create synthetic shocks, and again in the 1970s, with the development of computerization, enabling tests to be directly conducted on the exciter employing a shock response spectrum. After a brief recapitulation of the shock shapes most widely used in tests and of the possibilities of Fourier analysis for studies taking account of the environment (Chapter 1), Chapter 2 presents the shock response spectrum with its numerous definitions and calculation methods. Chapter 3 describes all the properties of the spectrum, showing that important characteristics of the original signal can be drawn from it, such as its amplitude or the velocity change associated with the movement during the shock. The shock response spectrum is the ideal tool for drafting specifications. Chapter 4 details the process which makes it possible to transform a set of shocks recorded in the real environment into a specification of the same severity, and presents a few other methods that have been proposed in the literature. Knowledge of the kinematics of movement during a shock is essential to the understanding of the mechanism of shock machines and programmers. Chapter 5 gives the expressions for velocity and displacement according to time for classic shocks, depending on whether they occur in impact or impulse mode.

xii

Mechanical shock

Chapter 6 describes the principle of shock machines currently most widely used in laboratories and their associated programmers. To reduce costs by restricting the number of changes in test facilities, specifications expressed in the form of a simple shock (half-sine, rectangle, saw tooth with a final peak) can occasionally be tested using an electrodynamic exciter. Chapter 7 sets out the problems encountered, stressing the limitations of such means, together with the consequences of modification, that have to be made to the shock profile, on the quality of the simulation. Pyrotechnic devices or equipment (cords, valves, etc.) are very frequently used in satellite launchers due to the very high degree of accuracy that they provide in operating sequences. Shocks induced in structures by explosive charges are extremely severe, with very specific characteristics. Their simulation in the laboratory requires specific means, as described in Chapter 8. Determining a simple shape shock of the same severity as a set of shocks, on the basis of their response spectrum, is often a delicate operation. Thanks to progress in computerization and control facilities, this difficulty can occasionally be overcome by expressing the specification in the form of a response spectrum and by controlling the exciter directly from that spectrum. In practical terms, as the exciter can only be driven with a signal that is a function of time, the software of the control rack determines a time signal with the same spectrum as the specification displayed. Chapter 9 describes the principles of the composition of the equivalent shock, gives the shapes of the basic signals most often used, with their properties, and emphasizes the problems that can be encountered, both in the constitution of the signal and with respect to the quality of the simulation obtained. Containers must protect the equipment carried in them from various forms of disturbance related to handling and possible accidents. Tests designed to qualify or certify containers include shocks mat are sometimes difficult, not to say impossible, to produce, given the combined weight of the container and its content. One relatively widely used possibility consists of performing shocks on scale models, with scale factors of the order of 4 or 5, for example. This same technique can be applied, although less frequently, to certain vibration tests. At the end of this volume, the Appendix summarizes the laws of similarity adopted to define the models and to interpret the test results.

List of symbols

The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have another meaning locally which will be defined in the text to avoid any confusion. a

max a( t) Ac

Maximum value of a(t) Component of shock x( t) Amplitude of compensation signal A(0) Indicial admittance b parameter b of Basquin's relation N ob = C c Viscous damping constant C Basquin's law constant (N ob = C) d(t) Displacement associated with a(t) D Diameter of programmer D(f 0 ) Fatigue damage e Neper's number E Young's modulus or energy of a shock ERS Extreme response spectrum E(t) Function characteristic of swept sine f Frequency of excitation f0 Natural frequency

F(t) Prms Fm g h

External force applied to system Rms value of force Maximum value of F(t)

h(t) H HR

Acceleration due to gravity Interval (f/f 0 ) or thickness of the target Impulse response Drop height Height of rebound

H( )

Transfer function

i

v=f

IPS 3(Q) k K ^nns im t(t) t(t\

Initial peak saw tooth Imaginary part of X(O) Stiffness or coefficient of uncertainty Constant of proportionality of stress and deformation Rms value of #(t) Maximum of l(t) Generalized excitation (displacement) First derivative of ^(t)

xiv

Mechanical shock

"i(t) L L(Q) m n N p q0 q0 q(0) q( 0) q( 0) Q Q( p) Re Rm R(Q) 9?(Q) s S SRS s( ) t td tj tr tR T T0 TPS u(t) u(t) u(t) vf Vj VR

Second derivative of t(i) Length Fourier transform of ^(t) Mass Number of cycles undergone by test-bar or material Number of cycles to failure Laplace variable or percentage of amplitude of shock Value of q(0) for 0=0 Value of q(0) for 0=0 Reduced response First derivative of q( 0) Second derivative of q( 0) Q factor (quality factor) Laplace transform of q (0) Yield stress Ultimate tensile strength Fourier transform of the system response Real part of X(Q) Standard deviation Area Shock response spectrum Power spectral density Time Decay time to zero of shock Fall duration Rise time of shock Duration of rebound Vibration duration Natural period Terminal peak saw tooth Generalized response First derivative of u(t) Second derivative of u(t) Velocity at end of shock Impact velocity Velocity of rebound

v(t) v( ) xm x(t) x(t) x(t) \ms xm Xm X(Q) y(t)

y(t) y(t)

zm zs zsup z(t)

z(t)

Velocity x(t) or velocity associated with a(t) Fourier transform of v(t) Maximum value of x(t) Absolute displacement of the base of a one-degree-offreedom system Absolute velocity of the base of a one-degree-of-£reedom system Absolute acceleration of the base of a one-degree-offreedom system Rms value of x(t) Maximum value of x(t) Amplitude of Fourier transform X(Q) Fourier transform of x(t) Absolute response of displacement of mass of a one-degree-of-freedom system Absolute response velocity of the mass of a one-degreeof-freedom system Absolute response acceleration of mass of a one-degree-of-freedom system Maximum value of z(t) Maximum static relative displacement Largest value of z( t) Relative response displacement of mass of a one-degree-of-freedom system with respect to its base Relative response velocity

List of symbols

z(t)

Relative response acceleration

a 5(t) AV (ft (j)(Q) TI

Coefficient of restitution Dirac delta function Velocity change Dimensionless product f0 T Phase Damping factor of damped sinusoid Relative damping of compensation signal Reduced excitation Laplace transform of A,( ) 3.14159265... Reduced time (co0 t) Reduced decay time

TJC X.( ) A(p) K 6 0d

6m 60 p a crcr am T T! t2 i nns coc co0 Q

Reduced rise time Value of 0 for t = t Density Stress Crushing stress Maximum stress Shock duration Pre-shock duration Post-shock duration Rms duration of a shock Pulsation of compensation signal Natural pulsation (2 n f0) Pulsation of excitation (27Cf)

£

Damping factor

xv

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Chapter 1

Shock analysis

1.1. Definitions 1.1.1. Shock Shock is defined as a vibratory excitation with duration between once and twice the natural period of the excited mechanical system.

Figure 1.1. Example of shock

Shock occurs when a force, a position, a velocity or an acceleration is abruptly modified and creates a transient state in the system considered.

2

Mechanical shock

The modification is normally regarded as abrupt if it occurs in a time period which is short compared to the natural period concerned (AFNOR definition) [NOR]. 1.1.2. Transient signal This is a vibratory signal of short duration (of a fraction of second to a few tens of seconds), the mechanical shock, for example, in the air-braking phase on aircraft etc.

Figure 1.2. Example of transient signal 1.1.3. Jerk A jerk is defined as the derivative of acceleration with respect to time. This parameter thus characterizes the rate of variation of acceleration with time.

1.1.4. Bump A bump is a simple shock which is generally repeated many times when testing (AFNOR) [NOR]. Example The GAM EG 13 (first part - Booklet 43 - Shocks) standard proposes a test characterized by a half-sine: 10 g, 16 ms, 3000 bumps (shocks) per axis, 3 bumps a second [GAM86].

Shock analysis

3

1.1.5. Simple (orperfect) shock Shock whose signal can be represented exactly in simple mathematical terms, for example half-sine, triangular or rectangular shock. 1.1.6. Half-sine shock Simple shock for which the acceleration-time curve has the form of a half-period (part positive or negative) of a sinusoid. 1.1.7. Terminal peak saw tooth shock (TPS) or final peak saw tooth shock (FPS) Simple shock for which the acceleration-time curve has the shape of a triangle where acceleration increases linearly up to a maximum value and then instantly decreases to zero. 1.1.8. Initial peak saw tooth shock (IPS) Simple shock for which the acceleration-time curve has the shape of a triangle where acceleration instantaneously increases up to a maximum, and then decreases to zero. 1.1.9. Rectangular shock Simple shock for which the acceleration-time curve increases instantaneously up to a given value, remains constant throughout the signal and decreases instantaneously to zero. 1.1.10. Trapezoidal shock Simple shock for which the acceleration-time curve grows linearly up to a given value, remains constant during a certain time after which it decreases linearly to zero.

4

Mechanical shock

1.1.11. Versed-sine (or haversine) shock Simple shock for which the acceleration-time curve has the form of one period of the curve representative of the function [1 - cos( )], with this period starting from zero value of this function. It is thus a signal ranging between two minima. 1.1.12. Decaying sinusoidal pulse A pulse comprised of a few periods of a damped sinusoid, characterized by the amplitude of the first peak, the frequency and damping:

This form is interesting, for it represents the impulse response of a one-degreeof-freedom system to a shock. It is also used to constitute a signal of a specified shock response spectrum (shaker control from a shock response spectrum).

1.2. Analysis in the time domain A shock can be described in the time domain by the following parameters: - the amplitude x(t); - duration t; - the form. The physical parameter expressed in terms of time is, in a general way, an acceleration x(t), but can be also a velocity v(t), a displacement x(t) or a force F(t). In the first case, which we will particularly consider in this volume, the velocity change corresponding to the shock movement is equal to

1.3. Fourier transform 1.3.1. Definition The Fourier integral (or Fourier transform) of a function x(t) of the real variable t absolutely integrable is defined by

Shock analysis

5

The function X(Q) is in general complex and can be written, by separating the real and imaginary parts ${(£1) and 3(Q):

or

with

and

Thus is the Fourier spectrum of is the phase.

I the energy spectrum and

The calculation of the Fourier transform is a one-to-one operation. By means of the inversion formula or Fourier reciprocity formula., it is shown that it is possible to express in a univocal way x(t) according to its Fourier transform X(Q) by the relation

(if the transform of Fourier X(Q) is itself an absolutely integrable function over all the domain).

NOTES. I. For

dt

6

Mechanical shock

The ordinate at f = 0 of the Fourier transform (amplitude) of a shock defined by an acceleration is equal to the velocity change AV associated with the shock (area under the curve x(t)). 2. The following definitions are also sometimes found [LAL 75]:

In this last case, the two expressions are formally symmetrical. The sign of the exponent of exponential is sometimes also selected to be positive in the expression for X(Q) and negative in that for x(t).

1.3.2. Reduced Fourier transform The amplitude and the phase of the Fourier transform of a shock of given shape can be plotted on axes where the product f T (T = shock duration) is plotted on the abscissa and on the ordinate, for the amplitude, the quantity A(f )/xm r . In the following paragraph, we draw the Fourier spectrum by considering simple shocks of unit duration (equivalent to the product ft) and of the amplitude unit. It is easy, with this representation, to recalibrate the scales to determine the Fourier spectrum of a shock of the same form, but of arbitrary duration and amplitude.

Shock analysis 1.3.3. Fourier transforms of simple shocks 1.3.3.1. Half-sine pulse

Figure 1.3. Real and imaginary parts of the Fourier transform of a half-sine pulse Amplitude [LAL 75]:

Phase:

(k positive integer) Real part:

Imaginary part:

Figure 1.4. Amplitude and phase of the Fourier transform of a half-sine shock pulse

7

8

Mechanical shock

1.3.3.2. Versed-sine pulse

Figure 1.5. Real and imaginary parts of the Fourier transform of a versed-sine shock pulse

Amplitude:

Phase:

Real part:

Imaginary part:

Shock analysis

Figure 1.6. Amplitude and phase of the Fourier transform of a versed-sine shockpulse

1.3.3.3. Terminal peak saw tooth pulse (TPS) Amplitude:

Figure 1.7. Real and imaginary parts of the Fourier transform of a TPS shockpulse

9

10

Mechanical shock Phase:

Real part:

Imaginary part:

Figure 1.8. Amplitude and phase of the Fourier transform of a TPS shock pulse

1.3.3.4. Initial peak saw tooth pulse (IPS) Amplitude:

Shock analysis

Phase:

Figure 1.9. Real and imaginary parts of the Fourier transform of an IPS shock pulse

Real part:

Figure 1.10. Amplitude and phase of the Fourier transform of an IPS shock pulse

11

12

Mechanical shock

Imaginary part:

1.3.3.5. Arbitrary triangular pulse

If tr = the rise time and t^ = decay time. Amplitude:

Phase:

Real part:

Shock analysis

Figure 1.11. Real and imaginary parts of the Fourier transform of a triangular shock pulse

13

Figure 1.12. Real and imaginary parts of the Fourier transform of a triangular shock pulse

Imaginary part:

Figure 1.13. Amplitude and phase of the Fourier transform of a triangular shock pulse

14

Mechanical shock

Figure 1.14. Amplitude and phase of the Fourier transform of a triangular shock pulse

1.3.3.6. Rectangular pulse

Figure 1.15. Real and imaginary parts of the Fourier transform of a rectangular shock pulse

Amplitude:

Phase:

Shock analysis Real part:

Imaginary part:

Figure 1.16. Amplitude and phase of the Fourier transform of a rectangular shock pulse

1.3.3.7. Trapezoidal pulse Amplitude:

15

16

Mechanical shock Phase:

Real part:

Imaginary part:

Figure 1.17. Real and imaginary parts of the Fourier transform of a trapezoidal shock pulse

Shock analysis

17

Figure 1.18. Amplitude and phase of the Fourier transform of a trapezoidal shock pulse

1.3.4. Importance of the Fourier transform The Fourier spectrum contains all the information present in the original signal, in contrast, we will see, to the shock response spectrum (SRS). It is shown that the Fourier spectrum R(Q) of the response at a point in a structure is the product of the Fourier spectrum X(Q) of the input shock and the transfer function H(Q) of the structure: R(Q) = H(Q) X(Q)

[1-40]

The Fourier spectrum can thus be used to study the transmission of a shock through a structure, the movement resulting at a certain point being then described by its Fourier spectrum. The response in the time domain can be also expressed from a convolution utilizing the 'input' shock signal according to the time and the impulse response of the mechanical system considered. An important property is used here: the Fourier transform of a convolution is equal to the scalar product of the Fourier transforms of the two functions in the frequency domain. It could be thought that this (relative) facility of change in domain (time <=> frequency) and this convenient description of the input or of the response would make the Fourier spectrum method one frequently used in the study of shock, in particular for the writing of test specifications from experimental data. These mathematical advantages, however, are seldom used within this framework, because when one wants to compare two excitations, one runs up against the following problems:

18

Mechanical shock

- The need to compare two functions. The Fourier variable is a complex quantity which thus requires two parameters for its complete description: the real part and the imaginary part (according to the frequency) or the amplitude and the phase. These curves in general are very little smoothed and, except in obvious cases, it is difficult to decide on the relative severity of two shocks according to frequency when the spectrum overlap. In addition, the phase and the real and imaginary parts can take positive and negative values and are thus not very easy to use to establish a specification; - The signal obtained by inverse transformation has in general a complex form impossible to reproduce with the usual test facilities, except, with certain limitations, on electrodynamic shakers. The Fourier transform is used neither for the development of specifications nor for the comparison of shocks. On the other hand, the one-to-one relation property and the input-response relation [1.40] make it a very interesting tool to control shaker shock whilst calculating the electric signal by applying these means to reproduce with the specimen a given acceleration profile, after taking into account the transfer function of the installation. 1.4. Practical calculations of the Fourier transform 1.4.1. General Among the various possibilities of calculation of the Fourier transform, the Fast Fourier Transform (FFT) algorithm of Cooley-Tukey [COO 65] is generally used because of its speed (Volume 3). It must be noted that the result issuing from this algorithm must be multiplied by the duration of the analysed signal to obtain the Fourier transform. 1.4.2. Case: signal not yet digitized Let us consider an acceleration time history x(t) of duration T which one wishes to calculate the Fourier transform with nFT points (power of 2) until the frequency f max . According to the Shannon's theorem (Volume 3), it is enough that the signal is sampled with a frequency fsamp = 2 f max , i.e. that the temporal step is equal to

The frequency interval is equal to

Shock analysis

19

To be able to analyse the signal with a resolution equal to Af , it is necessary that its duration is equal to

yielding the temporal step

If n is the total number of points describing the signal

and one must have

yielding

The duration T needed to be able to calculate the Fourier transform with the selected conditions can be different to the duration T from the signal to analyse (for example in the case of a shock). It cannot be smaller than T (if not shock shape would be modified). Thus, if we set

the condition

T leads to

i.e. to

If the calculation data (n FT and f max ) lead to a too large value of Af , it will be necessary to modify one of these two parameters to satisfy to the above condition. If it is necessary that the duration T is larger than T, zeros must be added to the signal to analyse between T and T, with the temporal step At. The computing process is summarized in Table 1.1.

20

Mechanical shock

Table 1.1. Computing process of a Fourier transform starting from a non-digitized signal Data: - Characteristics of the signal to be analysed (shape, amplitude, duration) or one measured signal not yet digitized. - The number of points of the Fourier transform (npj-) and its maximum frequency (f max ). T

samp. -

fmax

Condition to avoid the aliasing phenomenon (Shannon's theorem) . If the measured signal can contain components at frequencies higher than f max , it must be filtered using a low-pass filter before digitalization. To take account of the slope of the filter beyond f max , it is preferable to choose fsamp = 2.6 fmax (Volume 3).

At = 2 fmax fmax At =

n

Temporal step of the signal to be digitized (time interval between two points of the signal). Frequency interval between two successive points of the Fourier transform.

FT

n = 2 npr

Number of points of the signal to be digitized.

T = n At

Total duration of the signal to be treated.

If T > t, zeros must added between T and T. If there are not enough points to represent correctly the signal between 0 and T, fmax must t>e increased. f 1 The condition Af = -^L < - must be satisfied (i.e. T > T ): T "FT - if fmax is imposed, take npj (power of 2) > T f max . npr

- if n FT is imposed, choose fmax < —— . T

1.4.3. Case: signal already digitized If the signal of duration T were already digitized with N points and a step 5T, the calculation conditions of the transform are fixed:

Shock analysis

(nearest power of 2)

and

(which can thus result in not using the totality of the signal). If however we want to choose a priori fmax and npj, the signal must be resampled and if required zeros must be added using the principles in Table 1.1.

21

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Chapter 2

Shock response spectra domains

2.1. Main principles A shock is an excitation of short duration which induces transitory dynamic stress structures. These stresses are a function of: - the characteristics of the shock (amplitude, duration and form); - the dynamic properties of the structure (resonance frequencies, Q factors). The severity of a shock can thus be estimated only according to the characteristics of the system which undergoes it. The evaluation of this severity requires in addition the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are: - The exceeding of a value threshold of the stress in a mechanical part, leading to either a permanent deformation (acceptable or not) or a fracture, or at any rate, a functional failure. - If the shock is repeated many times (e.g. shock recorded on the landing gear of an aircraft, operation of an electromechanical contactor, etc), the fatigue damage accumulated in the structural elements can lead in the long term to fracture. We will deal with this aspect later on. The severity of a shock can be evaluated by calculating the stresses on a mathematical or finite element model of the structure and, for example, comparison with the ultimate stress of the material. This is the method used to dimension the structure. Generally, however, the problem is rather to evaluate the relative severity of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a specification etc). This comparison would be

24

Mechanical shock

difficult to carry out if one used a fine model of the structure, and besides this is not always available, in particular at the stage of the development of the specification of dimensioning. One searches for a method of general nature, which leads to results which can be extrapolated to any structure. A solution was proposed by M.A. Biot [BIO 32] in 1932 in a thesis on the study of the earthquakes effects on the buildings; this study was then generalized to analysis of all kinds of shocks. The study consists of applying the shock under consideration to a 'standard' mechanical system, which thus does not claim to be a model of the real structure, composed of a support and of N linear one-degree-of-freedom resonators, comprising each one a mass mi, a spring of stiffness kj and a damping device Cj, chosen such that the fraction of critical damping

is the same for all N

resonators (Figure 2.1).

Figure 2.1. Model of the shock response spectrum (SRS)

When the support is subjected to the shock, each mass nij has a specific movement response according to its natural frequency

and to the

chosen damping £, while a stress GJ is induced in the elastic element. The analysis consists of seeking the largest stress cr mj observed at each frequency in each spring. A shock A is regarded as more severe than a shock B if it induces in each resonator a larger extreme stress. One then carries out an extrapolation, which is certainly criticizable, by supposing that, if shock A is more severe than shock B when it is applied to all the standard resonators, it is also more

Shock response spectra domains

25

severe with respect to an arbitrary real structure (which cannot be linear nor having a single degree of freedom). NOTE: A study was carried out in 1984 on a mechanical assembly composed of a circular plate on which one could place some masses and thus vary the number of degrees of freedom. The stresses generated by several shocks of the same spectra (in the frequency range including the principal resonance frequencies), but of different shapes [DEW 84], were measured and compared. One noted that for this assembly whatever the number of degrees of freedom, — two pulses of simple form (with no velocity change) having the same spectrum induce similar stresses, the variation not exceeding approximately 20 %. It is the same for two oscillatory shocks; — the relationship between the stresses measured for a simple shock and an oscillatory shock can reach 2. These results were supplemented by numerical simulation intended to evaluate the influence ofnon linearity. Even for very strong non-linearity, one did not note for the cases considered, an important difference between the stresses induced by two shocks of the same spectrum, but of different form. A complementary study was carried out by B.B. Petersen [PET 81] in order to compare the stresses directly deduced from a shock response spectrum with those generated on an electronics component by a half-sine shock envelop of a shock measured in the environment and by a shock of the same spectrum made up from WA VSIN signals (Chapter 9) added with various delays. The variation between the maximum responses measured at five points in the equipment and the stresses calculated starting from the shock response spectra does not exceed a factor of 3 in spite of the important theoretical differences between the model of the response spectrum and the real structure studied. For applications deviating from the assumptions of definition of the shock response spectrum (linearity, only one degree of freedom), it is desirable to observe a certain prudence if one wishes to estimate quantitatively the response of a system starting from the spectrum [BOR 89]. The response spectra are more often used to compare the severity of several shocks. It is known that the tension static diagram of many materials comprises a more or less linear arc on which the stress is proportional to the deformation. In dynamics, this proportionality can be allowed within certain limits for the peaks of the deformation (Figure 2.2).

26

Mechanical shock

If mass-spring-damper system is supposed to be linear, it is then appropriate to compare two shocks by the maximum response stress am they induce or by the maximum relative displacement zm that they generate, since:

Figure 2.2. Stress-strain curve

zm is a function only of the dynamic properties of the system, whereas am is also a function, via K, of the properties of the materials which constitute it. The curve giving the largest relative displacement zsup multiplied by oo0 according to the natural frequency f0, for a given £ damping, is the shock response spectrum (SRS). The first work defining these spectra was published in 1933 and 1934 [BIO 33] [BIO 34], then in 1941 and 1943 [BIO 41] [BIO 43]. The shock response spectrum, then named the shock spectrum, was presented there in the current form. This spectrum was used in the field of environmental tests from 1940 to 1950: J.M. Frankland [FRA 42] in 1942, J.P. Walsh and R.E. Blake in 1948 [WAL 48], R.E. Mindlin [MIN 45]. Since then, there have beenmany works which used it as tool of analysis and for simulation of shocks [HIE 74], [KEL 69], [MAR 87] and [MAT 77].

2.2. Response of a linear one-degree-of-freedom system 2.2.1. Shock defined by a force Being given a mass-spring-damping system subjected to a force F(t) applied to the mass, the differential equation of the movement is written as:

Shock response spectra domains

27

Figure 2.3. Linear one-degree-of-freedom system subjected to a force

where z(t) is the relative displacement of the mass m relative to the support in response to the shock F(t). This equation can be put in the form:

where

(damping factor) and

(natural pulsation).

2.2.2. Shock defined by an acceleration Let us set as x(t) an acceleration applied to the base of a linear one-degree-offreedom mechanical system, with y(t) the absolute acceleration response of the

28

Mechanical shock

mass m and z(t) the relative displacement of the mass m with respect to the base. The equation of the movement is written as above:

Figure 2.4. Linear one-degree-of-freedom system subjected to acceleration

i.e.

or, while setting z(t) = y(t) - x(t):

2.2.3. Generalization Comparison of the differential equations [2.3] and [2.8] shows that they are both of the form

where /(t) and u(t) are generalized functions of the excitation and response.

Shock response spectra domains

29

NOTE: The generalized equation [2.9] can be -written in the reduced form:

where

£m = maximum of l(t)

Resolution The differential equation [2.10] can be integrated by parts or by using the Laplace transformation. We obtain, for zero initial conditions, an integral called Duhamel 's integral:

where

variable of integration. In the generalized form, we deduce that

where a is an integration variable homogeneous with time. If the excitation is an acceleration of the support, the response relative displacement is given by:

and the absolute acceleration of the mass by:

30

Mechanical shock

Application Let us consider a package intended to protect a material from mass m and comprising a suspension made up of two elastic elements of stiffness k and two dampers of damping constant c.

Figure 2.5. Model of the package

Figure 2.6. Equivalent model

We want to determine the movement of the mass m after free fall from a height of h = 5 m, by supposing that there is no rebound of the package after the impact on the ground and that the external frame is not deformable (Figure 2.5). This system is equivalent to the model in Figure 2.6. We have (Volume 1, Chapter 3):

Shock response spectra domains

and

31

32

Mechanical shock

where

and

With the chosen numerical values, it becomes:

Shock response spectra domains

From this it is easy to deduce the velocity z(t) and the acceleration z(t) from successive derivations of this expression. The first term corresponds to the static deformation of the suspension under load of 100 kg.

2.2.4. Response of a one-degree-of-freedom system to simple shocks Half-sine pulse

Versed-sine pulse

33

34

Mechanical shock

Rectangular pulse

Initial peak saw tooth pulse

Shock response spectra domains

Terminal peak saw tooth pulse

Arbitrary triangular pulse

35

36

Mechanical shock

Trapezoidal pulse

Figure 2.7. Trapezoidal shock pulse

where

Shock response spectra domains

37

For an isosceles trapezoid, we set 6r = 00 - 6d . If the rise and decay each have a duration equal to 10% of the total duration of the trapezoid, we have

2.3. Definitions Response spectrum A curve representative of the variations of the largest response of a linear onedegree-of-freedom system subjected to a mechanical excitation, plotted against its natural frequency f0 =

for a given value of its damping ratio.

Absolute acceleration shock response spectrum In the most usual cases where the excitation is defined by an absolute acceleration of the support or by a force applied directly to the mass, the response of the system can be characterized by the absolute acceleration of the mass (which could be measured using an accelerometer fixed to this mass): the response spectrum is then called the absolute acceleration shock response spectrum. This spectrum can be useful when absolute acceleration is the parameter easiest to compare with a characteristic value (study of the effects of a shock on a man, comparison with the specification of an electronics component etc). Relative displacement shock spectrum In similar cases, we often calculate the relative displacement of the mass with respect to the base of the system, displacement which is proportional to the stress created in the spring (since the system is regarded as linear). In practice, one in general expresses in ordinates the quantity co0 zsup called the equivalent static acceleration. This product has the dimension of an acceleration, but does not represent the acceleration of the mass, except when damping is zero; this term is then strictly equal to the absolute acceleration of the mass. However, when damping is close to the current values observed in mechanics, and in particular when 2 4 = 0.05, one can assimilate as a first approximation co0 zsup to the absolute acceleration ysup of the mass m [LAL 75].

38

Mechanical shock

Very often in practice, it is the stress (and thus the relative displacement) which seems the most interesting parameter, the spectrum being primarily used to study the behaviour of a structure, to compare the severity of several shocks (the stress created is a good indicator), to write test specifications (it is also a good comparison between the real environment and the test environment) or to dimension a suspension (relative displacement and stress are then useful). The quantity co0 zsu is termed pseudo-acceleration. In the same way, one terms pseudo-velocity the product o)0 zsup. 2 The spectrum giving co0 zsup versus the natural frequency is named the relative

displacement shock spectrum. In each of these two important categories, the response spectrum can be defined in various ways according to how the largest response at a given frequency is characterized.

Primary positive shock response spectrum or initial positive shock response spectrum The highest positive response observed during the shock.

Primary (or initial) negative shock response spectrum The highest negative response observed during the shock.

Secondary (or residual shock) response spectrum The largest response observed after the end of the shock. Here also, the spectrum can be positive or negative.

Positive (or maximum positive) shock response spectrum The largest positive response due to the shock, without reference to the duration of the shock. It is thus about the envelope of the positive primary and residual spectra.

Shock response spectra domains

39

Negative (or maximum negative) shock response spectrum The largest negative response due to the shock, without reference to the duration of the shock. It is in a similar way the envelope of the negative primary and residual spectra.

Example

Figure 2.8. Shock response spectra of a rectangular shock pulse

Maximax shock response spectrum Envelope of the absolute values of the positive and negative spectra. Which spectrum is the best? The damage is supposed proportional to the largest value of the response, i.e. to the amplitude of the spectrum at the frequency considered, and it is of little importance for the system whether this maximum zm takes place during or after the shock. The most interesting spectra are thus the positive and negative spectra, which are most frequently used in practice, with the maximax spectrum. The distinction between positive and negative spectra must be made each time the system, if disymmetrical, behaves differently, for example under different tension and compression. It is, however, useful to know these various definitions so as to be able to correctly interpret the curves published.

40

Mechanical shock

2.4. Standardized response spectra For a given shock, the spectra plotted for various values of the duration and the amplitude are homothetical. It is thus interesting, for simple shocks to have a standardized or reduced spectrum plotted in dimensionless co-ordinates, while plotting on the abscissa the product f0 t (instead of f0) or co0 t and on the ordinate the spectrum/shock pulse amplitude ratio co0 zm /xm , which, in practice, amounts to tracing the spectrum of a shock of duration equal to 1 s and amplitude 1 m/s2.

Figure 2.9. Standardized SRS of a half-sine pulse

These standardized spectra can be used for two purposes: - plotting of the spectrum of a shock of the same form, but of arbitrary amplitude and duration; - investigating the characteristics of a simple shock of which the spectrum envelope is a given spectrum (resulting from measurements from the real environment). The following figures give the spectra of reduced shocks for various pulse forms, unit amplitude and unit duration, for several values of damping. To obtain the spectrum of a particular shock of arbitrary amplitude xm and duration T (different from 1) from these spectra, it is enough to regraduate the scales as follows: - for the amplitude; by multiply the reduced values by xm; - for the abscissae, replace each value <}> (= f0 T ) by f0 =

Shock response spectra domains

41

We will see later on how these spectra can be used for the calculation of test specifications.

Half-sine pulse

Figure 2.10. Standardized positive and negative relative displacement SRS of a half-sine pulse

Figure 2.11. Standardized primary and residual relative displacement SRS of a half-sine pulse

42

Mechanical shock

Figure 2.12. Standardized positive and negative absolute acceleration SRS of a half-sine pulse

Versed-sine pulse

Figure 2.13. Standardized positive and negative relative displacement SRS of a versed-sine pulse

Shock response spectra domains

Figure 2.14. Standardized primary and residual relative displacement SRS of a versed-sine pulse

Terminal peak saw tooth pulse

Figure 2.15. Standardized positive and negative relative displacement SRS of a TPS pulse

43

44

Mechanical shock

Figure 2.16. Standardized primary and residual relative displacement SRS of a TPS pulse

Figure 2.17. Standardized positive and negative relative displacement SRS of a TPS pulse with zero decay time

Shock response spectra domains Initial peak saw tooth pulse

Figure 2.18. Standardized positive and negative relative displacements SRS of an IPS pulse

Figure 2.19. Standardized primary and residual relative displacement SRS of an IPS pulse

45

46

Mechanical shock

Figure 2.20. Standardized positive and negative relative displacement SRS of an IPS with zero rise time

Rectangular pulse

Figure 2.21. Standardized positive and negative relative displacement SRS of a rectangular pulse

Shock response spectra domains

47

Trapezoidal pulse

Figure 2.22. Standardized positive and negative relative displacement SRS of a trapezoidal pulse 2.5. Difference between shock response spectrum (SRS) and extreme response spectrum (ERS) A spectrum known as of extreme response spectrum (ERS) and comparable with the shock response spectrum (SRS) is often used for the study of vibrations (Volume 5). This spectrum gives the largest response of a linear single-degree-of-freedom system according to its natural frequency, for a given Q factor, when it is subjected to the vibration under investigation. In the case of the vibrations, of long duration, this response takes place during the vibration: the ERS is thus a primary spectrum. In the case of shocks, we in general calculate the highest response, which takes place during or after the shock.

2.6. Algorithms for calculation of the shock response spectrum Various algorithms have been developed to solve the second order differential equation [2.9] ([COL 90], [COX 83], [DOK 89], [GAB 80], [GRI 96], [HAL 91], [HUG 83a], [IRV 86], [MER 91], [MER 93], [OHA 62], [SEI 91] and [SMA 81]). One which leads to the most reliable results is that of F. W. Cox [COX 83] (Section 2.7.). Although these calculations are a priori relatively simple, the round robins that were carried out ([BOZ 97] [CHA 94]) showed differences in the results, ascribable

48

Mechanical shock

sometimes to the algorithms themselves, but also to the use or programming errors of the software.

2.7. Subroutine for the calculation of the shock response spectrum The following procedure is used to calculate the response of a linear singledegree-of-freedom system as well as the largest and smallest values after the shock (points of the positive and negative SRS, primary and residual, displacements relative and absolute accelerations). The parameters transmitted to the procedure are the number of points defining the shock, the natural pulsation of the system and its Q factor, the temporal step (presumably constant) of the signal and the array of the amplitudes of the signal. This procedure can be also used to calculate the response of a one-degree-of-freedom system to an arbitrary excitation, and in particular to a random vibration (where one is only interested in the primary response).

Shock response spectra domains

49

Procedure for the calculation of a point of the SRS at frequency f0 (GFABASIC) From F. W. Cox [COX 83] PROCEDURE S_R_S(npts_signal%,wO,Q_factor,dt,VAR xppO) LOCAL i%,a,a1 ,a2,b,b1,b2,c,c1,c2,d,d2,e,s,u,v,wdt,w02,w02dt LOCAL p1d,p2d,p1a,p2a,pd,pa,wtd,wta,sd,cd,ud,vd,ed,sa,ca,ua,va,ea ' npts_signal% = Number of points of definition of the shock versus time ' xpp(npts_signal%) = Array of the amplitudes of the shock pulse ' dt= Temporal step ' wO= Undamped natural pulsation (2*PI*fO) ' Initialization and preparation of calculations psi=l/2/Q_factor // Damping ratio w=wO*SQR(l-psiA2) // Damped natural pulsation d=2*psi*wO d2=d/2 wdt=w*dt e=EXP(-d2*dt) s=e*SIN(wdt) c=e*COS(wdt) u=w*c-d2*s v=-w*s-d2*c w02=wOA2 w02dt=w02*dt 1 Calculation of the primary SRS ' Initialization of the parameters srcajprim_min=lE100 // Negative primary SRS (absolute acceleration) srca_prim_max=-srcajprim_mm // Positive primary SRS (absolute acceleration) srcd_prim_min=srca_prim_min // Negative primary SRS (relative displacement) srcdjprim_max=-srcd_prim_min // Positive primary SRS (relative displacement) displacement_z=0 // Relative displacement of the mass under the shock velocity_zp=0 // Relative velocity of the mass ' Calculation of the sup. and inf. responses during the shock at the frequency fO FOR i%=2 TO npts_signal% a=(xpp(i%-1 )-xpp(i%))/w02dt b=(-xpp(i%-1 )-d*a)/w02 c2=displacement_z-b c 1 =(d2*c2-t-velocity_zp-a)/w displacement_z=s*cl+c*c2+a*dt+b velocity_zp=u * c1 + v* c2+a responsedjprim=-displacement_z*w02 // Relative displac. during shock x square of the pulsation

50

Mechanical shock i

responsea_prim=-d*velocity_zp-displacement_z*w02 // Absolute response accel. during the shock ' Positive primary SRS of absolute accelerations srcaj3rim_max=ABS(MAX(srca_prim_max,responsea_prim)) ' Negative primary SRS of absolute accelerations srca_prim_min=MIN(srca_prim_min,responsea_prim) ' Positive primary SRS of the relative displacements srcd_prim_max=ABS(MAX(srcd_prim_max,responsed_prim)) ' Negative primary SRS of the relative displacements srcdjprim_min=MrN(srcd_prim_min,responsed_prim) NEXT i% ' Calculation of the residual SRS 1 Initial conditions for the residual response = Conditions at the end of the shock srca_res_max=responsea_prim // Positive residual SRS of absolute accelerations srcajres_min=responseajprim //Negative residual SRS of absolute accelerations srcd_res_max=responsedjprim // Positive residual SRS of the relative displacements srcd_res_min=responsed_prim // Negative residual SRS of the relative displacements ' Calculation of the phase angle of the first peak of the residual relative displacement c 1 =(d2 *displacement_z+velocity_zp)/w c2=displacement_z al=-w*c2-d2*cl a2=w*cl-d2*c2 pld=-al p2d=a2 IFpld=0 pd=PI/2*SGN(p2d) ELSE pd=ATN(p2d/pld) ENDIF IF pd>=0 wtd=pd ELSE wtd=PI+pd ENDIF ' Calculation of the phase angle of the first peak of residual absolute acceleration bla=-w*a2-d2*al b2a=w*al-d2*a2 pla=-d*bla-al*w02 p2a=d*b2a+a2*w02 IFpla=0 pa=PI/2*SGN(p2a)

Shock response spectra domains ELSE pa=ATN(p2a/pla) ENDIF IFpa>=0 wta^a ELSE wta=PI+pa ENDIF FOR i%=l TO 2 // Calculation of the sup. and inf. values after the shock at the frequency fO ' Residual relative displacement sd=SIN(wtd) cd=COS(wtd) ud=w*cd-d2*sd vd=-w*sd-d2*cd ed=EXP(-d2*wtd/w) displacementd_z=ed*(sd*c 1 +cd*c2) velocityd_zp=ed*(ud*c 1 +vd*c2) ' Residual absolute acceleration sa=SIN(wta) ca=COS(wta) ua=w*ca-d2*sa va=-w*sa-d2*ca ea=EXP(-d2*wta/w) displacementa_z=ea*(sa*c 1 +ca*c2) velocitya_zp=ea*(ua*c 1 +va*c2) 1 Residual SRS srcd_res=-displacementd_z*w02 // SRS of the relative displacements srca_res=-d*velocitya_zp-displacementa_z*w02 // SRS of absolute accelerations srcd_res_max=MAX(srcd_res_max,srcd_res) // Positive residual SRS of the relative displacements srcd_res_min=MIN(srcd_res_min,srcd_res)//Negative residual SRS of the relative displacements srca_res_max=MAX(srca_res_max,srca_res) // Positive residual SRS of the absolute accelerations srca_res_min=MIN(srca_res_min,srca_res) // Negative residual SRS of the absolute accelerations wtd=wtd+PI wta=\vta+PI NEXT i% srcdj)os=MAX(srcd_prim_max,srcd_res_max) // Positive SRS of the relative displacements srcd_neg=MrN(srcdjprim_min,srcd_res_min) // Negative SRS of the relative displacements

51

52

Mechanical shock

srcd_maximax=MAX(srcd_pos,ABS(srcd_neg)) // Maximax SRS of the relative displacements srcajpos=MAX(srca_prim_max,srca_res_max) // Positive SRS of absolute accelerations srca_neg=MIN(srca_prim_min,srca_res_min) //Negative SRS of absolute accelerations srca_maximax=MAX(srcajpos,ABS(srca_neg)) // Maximax SRS of absolute accelerations RETURN

2.8. Choice of the digitization frequency of the signal The frequency of digitalization of the signal has an influence on the calculated response spectrum. If this frequency is too small: -The spectrum of a shock with zero velocity change can be false at low frequency, digitalization leading artificially to a difference between the positive and negative areas under the shock pulse, i.e. to an apparent velocity change that is not zero and thus leading to an incorrect slope in this range. Correct restitution of the velocity change (error of about 1% for example) can require, according to the shape of the shock, up to 70 points per cycle. - The spectrum can be erroneous at high frequencies. The error is here related to the detection of the largest peak of the response, which occurs throughout shock (primary spectrum). Figure 2.23 shows the error made in the stringent case more when the points surrounding the peak are symmetrical with respect to the peak. If we set

it can be shown that, in this case, the error made according to the sampling factor SF is equal to [SIN 81] [WIS 83]

Shock response spectra domains

Figure 2.23. Error made in measuring the amplitude of the peak

53

Figure 2.24. Error made in measuring the amplitude of the peak plotted against sampling factor

The sampling frequency must be higher than 16 times the maximum frequency of the spectrum so that the error made at high frequency is lower than 2% (23 times the maximum frequency for an error lower than 1%). The rule of thumb often used to specify a sampling factor equal 10 can lead to an error of about 5%. The method proposing a parabolic interpolation between the points to evaluate the value of the maximum does not lead to better results.

2.9. Example of use of shock response spectra Let us consider as an example the case of a package intended to limit to 100 m/s2 acceleration on the transported equipment of mass m when the package itself is subjected to a half-sine shock of amplitude 300 m/s2 and of duration 6 ms. One in addition imposes a maximum displacement of the equipment in the package (under the effect of the shock) equal to e = 4 cm (to prevent that the equipment coming into contact with the wall of the package). It is supposed that the system made up by the mass m of the equipment and the suspension is comparable to a one-degree-of-freedom system with a Q factor equal to Q = 5. We want to determine the stiffness k of the suspension to satisfy these requirements when the mass m is equal to 50 kg.

54

Mechanical shock

Figure 2.25. Model of the package

Figures 2.26 and 2.27 show the response spectrum of the half-sine shock pulse being considered, plotted between 1 and 50 Hz for a damping of £, = 0.10 (= 1/2 Q). The curve of Figure 2.26 gives zsu on the ordinate (maximum relative displacement of the mass, calculated by dividing the ordinate of the spectrum ODO zsup by co0). The spectrum of Figure 2.27 represents the usual curve G)0 z sup (f 0 ). We could also have used a logarithmic four coordinate spectrum to handle just one curve.

Figure 2.26. Limitation in displacement

Figure 2.27. Limitation in acceleration

Figure 2.26 shows that to limit the displacement of the equipment to 4 cm, the natural frequency of the system must be higher or equal to 4 Hz. The limitation of acceleration on the equipment with 100 m/s2 also imposes f0 < 16 Hz (Figure 2.27). The range acceptable for the natural frequency is thus 4 Hz < f0 < 16 Hz.

Shock response spectra domains

55

Knowing that

we deduce that

2.10. Use of shock response spectra for the study of systems with several degrees of freedom By definition, the response spectrum gives the largest value of the response of a linear single-degree-of-freedom system subjected to a shock. If the real structure is comparable to such a system, the SRS can be used to evaluate this response directly. This approximation is often possible, with the displacement response being mainly due to the first mode. In general, however, the structure comprises several modes which are simultaneously excited by the shock. The response of the structure consists of the algebraic sum of the responses of each excited mode. One can read on the SRS the maximum response of each one of these modes, but one does not have any information concerning the moment of occurrence of these maxima. The phase relationships between the various modes are not preserved and the exact way in which the modes are combined cannot be known simply. In addition, the SRS is plotted for a given constant damping over all the frequency range, whereas this damping varies from one mode to another in the structure. With rigour, it thus appears difficult to use a SRS to evaluate the response of a system presenting more than one mode. But it happens that this is the only possible means. The problem is then to know how to combine these 'elementary' responses so as to obtain the total response and to determine, if need be, any suitable participation factors dependent on the distribution of the masses of the structure, of the shapes of the modes etc. Let us consider a non-linear system with n degrees of freedom; its response to a shock can be written as:

56

Mechanical shock

where n = total number of modes an = modal participation factor for the mode n h n (t)= impulse response of mode n x(t) = excitation (shock) (j)

- modal vector of the system

a = variable of integration If one mode (m) is dominant, this relation is simplified according to

The value of the SRS to the mode m is equal to

The maximum of the response z(t) in this particular case is thus

When there are several modes, several proposals have been made to limit the value of the total response of the mass j of the one of the degrees of freedom starting from the values read on the SRS as follows. A first method was proposed in 1934 per H. Benioff [BEN 34], consisting simply of adding the values with the maxima of the responses of each mode, without regard to the phase. A very conservative value was suggested by M.A. Biot [BIO 41] in 1941 for the prediction of the responses of buildings to earthquakes, equal to the sum of the absolute values of the maximum modal responses:

Shock response spectra domains

57

The result was considered sufficiently precise for this application [RID 69]. As it is not very probable that the values of the maximum responses take place all at the same moment with the same sign, the real maximum response is lower than the sum of the absolute values. This method gives an upper limit of the response and thus has a practical advantage: the errors are always on the side of safety. However, it sometimes leads to excessive safety factors [SHE 66]. In 1958, S. Rubin [RUB 58] made a study of undamped two-degrees-of-freedom systems in order to compare the maximum responses to a half-sine shock calculated by the method of modal superposition and the real maximum responses. This tsudy showed that one could obtain an upper limit of the maximum response of the structure by a summation of the maximum responses of each mode and that, in the majority of the practical problems, the distribution of the modal frequencies and the shape of the excitation are such that the possible error remains probably lower than 10%. The errors are largest when the modal frequencies are in different areas of the SRS, for example, if a mode is in the impulse domain and the other in the static domain. If the fundamental frequency of the structure is sufficiently high, Y.C. Fung and M.V. Barton [FUN 58] considered that a better approximation of the response is obtained by making the algebraic sum of the maximum responses of the individual modes:

Clough proposed in 1955, in the study of earthquakes, either to add to the response of the first mode a fixed percentage of the responses of the other modes, or to increase the response of the first mode by a constant percentage. The problem can be approached differently starting from an idea drawn from probability theory. Although the values of the response peaks of each individual mode taking place at different instants of time cannot, in a strict sense, being treated in purely statistical terms, Rosenblueth suggested combining the responses of the modes by taking the square root of the sum of the squares to obtain an estimate of the most probable value [MER 62]. This criterion, used again in 1965 by F.E. Ostrem and M.L. Rumerman [OST 65] in 1955 [RID 69], gives values of the total response lower than the sum of the absolute values and provides a more realistic evaluation of the average conditions. This idea can be improved by considering the average of the sum of the absolute values and the square root of the sum of the squares (JEN 1958). One can also choose to define positive and negative limiting values starting from a system of

58

Mechanical shock

weighted averages. For example, the relative displacement response of the mass j is estimated by

where the terms

are the absolute values of the maximum responses of each

mode and p is a weighting factor [MER 62].

Chapter 3

Characteristics of shock response spectra

3.1. Shock response spectra domains Three domains can be schematically distinguished in shock spectra: - An impulse domain at low frequencies, in which the amplitude of the spectrum (and thus of the response) is lower than the amplitude of the shock. The shock here is of very short duration with respect to the natural period of the system. The system reduces the effects of the shock. The characteristics of the spectra in this domain will be detailed in Section 3.2. - A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping. All occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), the natural period of the system being small compared with the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. The real shocks having necessarily a rise time different from zero, this restriction remains theoretical. - An intermediate domain, in which there is dynamic amplification of the effects of the shock, the natural period of the system being close to the duration of the shock. This amplification, more or less significant depending on the shape of the shock and the damping of the system, does not exceed 1.77 for shocks of traditional simple shape (half-sine, versed-sine, terminal peak saw tooth (TPS)). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid.

TTTTT echanical shock

3.2. Characteristics of shock response spectra at low frequencies 3.2.1. General characteristics In this impulse region - The form of the shock has little influence on the amplitude of the spectrum. We will see below that only (for a given damping) the velocity change AV associated with the shock, equal to the algebraic surface under the curve x(t), is important. -The positive and negative spectra are in general the residual spectra (it is necessary sometimes that the frequency of spectrum is very small, and there can be exceptions for certain long shocks in particular). They are nearly symmetrical so long as damping is small. 2 -The response (pseudo-acceleration co0 zsup or absolute acceleration y sup ) is lower than the amplitude of the excitation. There is an 'attenuation'. It is thus in this impulse region that it would be advisable to choose the natural frequency of an isolation system to the shock, from which we can deduce the stiffness envisaged of the insulating material:

(with m being the mass of the material to be protected). - The curvature of the spectrum always cancels at the origin (f0 = 0 Hz) [FUN 57]. The characteristics of the SRS are often better demonstrated by a logarithmic chart or a four coordinate representation.

3.2.2. Shocks with velocity changed from zero For the shocks simple in shaoe primary spectrum at low frequencies.

, the residual spectrum is larger than the

For an arbitrary damping £ it can be shown that the impulse response is given by

where z(t) is maximum for t such as

0, i.e. fort such that

Characteristics of shock response spectra

61

yielding

The SRS is thus equal at low frequencies to sin(arctan

i.e.

1 and the slope tends towards AV. The slope p of the spectrum at the origin is then equal to:

The tangent at the origin of the spectrum plotted for zero damping in linear scales has a slope proportional to the velocity change AVcorresponding to the shock pulse. If damping is small, this relation is approximate.

62

Mechanical shock

Example Half-sine shock pulse 100 m/s2, 10 ms, positive SRS (relative displacements). The slope of the spectrum at the origin is equal to (Figure 3.1):

yielding

a value to be compared with the surface under the half-sine shock pulse:

Figure 3.1. Slope of the SRS at the origin

With the pseudovelocity plotted against to0, the spectrum is defined by

Characteristics of shock response spectra

63

When O>Q tends towards zero, co0 zsup tends towards the constant value AV cp(^). Figure 3.2 shows the variations of cp(^) versus £.

Figure 3.2. Variations of the function
Example IPS shock pulse 100 m/s2, 10 ms), Pseudovelocity calculated starting from the positive SRS (Figure 3.3).

Figure 3.3. Pseudovelocity SRS of a TPS shock pulse

It is seen that the pseudovelocity spectrum plotted for £ = 0 tends towards 0.5 at low frequencies (area under TPS shock pulse).

64

Mechanical shock The pseudovelocity G)0 zsup tends towards AV when the damping tends towards

zero. If damping is different from zero, the pseudovelocity tends towards a constant value lower than AV. 2 The residual positive SRS of the relative displacements (o>0 zsup) decreases at low frequencies with a slope equal to 1, i.e., on a logarithmic scale, with a slope of 6dB/octave(£ = 0).

The impulse absolute response of a linear one-degree-of-freedom system is given by (Volume 1, relation 3.85):

where

If damping is zero,

The 'input' impulse can be represented in the form

as long as

The response which results

The maximum of the displacement takes place during the residual response, for

yielding the shock response spectrum

Characteristics of shock response spectra

65

and

A curve defined by a relation of the form y = a f slope n on a logarithmic grid:

is represented by a line of

The slope can be expressed by a number N of dB/octave according to

The undamped shock response spectrum plotted on a log-log grid thus has a slope at the origin equal to 1, i. e. 6 dB/octave.

Terminal peak saw tooth pulse 10 ms, 100 m/s2

Figure 3.4. TPS shock pulse

66

Mechanical shock

Figure 3.5. Residual positive SRS (relative displacements) of a TPS shock pulse

The primary positive SRS o>0 zsu_ always has a slope equal to 2 (12 dB/octave) (example Figure 3.6) [SMA 85].

Figure 3.6. Primary positive SRS of a half-sine shock pulse

The relative displacement zsup tends towards a constant value z0 = xm equal to the absolute displacement of the support during the application of the shock pulse (Figure 3.7). At low resonance frequencies, the equipment is not directly sensitive to accelerations, but to displacement:

Characteristics of shock response spectra

67

Figure 3.7. Behaviour of a resonator at very low resonance frequency

The system works as soft suspension which attenuates accelerations with large displacements [SNO 68]. This property can be demonstrated by considering the relative displacement response of a linear one-degree-of-freedom system given by Duhamel's equation (Volume 1, Chapter 2):

After integration by parts we obtain

68

Mechanical shock

The mass m of an infinitely flexible oscillator and therefore of infinite natural period (f0 = 0), does not move in the absolute reference axes. The spectrum of the relative displacement thus has as an asymptotic value the maximum value of the absolute displacement of the base. Example Figure 3.8 shows the primary positive SRS z sup (f 0 ) of a shock of half-sine shape 100 m/s2,10 ms plotted for £ = 0 between 0.01 and 100 Hz.

Figure 3.8. Primary positive SRS of a half-sine (relative displacements)

The maximum displacement xm under shock calculated from the expression x(t) for the acceleration pulse is equal to:

The SRS tends towards this value when

Characteristics of shock response spectra

69

For shocks of simple shape, the instant of time t at which the first peak of the response takes place tends towards

, tends towards zero [FUN 57].

The primary positive spectrum of pseudovelocities has, a slope of 6 dB/octave at the low frequencies. Example

Figure 3.9. Primary positive SRS of a TPS pulse (four coordinate grid)

3.2.3. Shocks for AV = 0 and AD * 0 at end of pulse In this case, for £, = 0 : - The Fourier transform of the velocity for f = 0, V(0), is equal to

Since acceleration is the first derivative of velocity, the residual spectrum is equal to co0 AD for low values of co0. The undamped residual shock response spectrum thus has a slope equal to 2 (i.e. 12 dB/octave) in this range.

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Mechanical shock

Example Shock consisted by one sinusoid period of amplitude 100 m/s2 and duration 10

Figure 3.10. Residual positive SRS of a 'sine 1 period' shockpulse

- The primary relative displacement (positive or negative, according to the form of the shock) zsup tends towards a constant value equal to xm, absolute displacement corresponding to the acceleration pulse x(t) defining the shock:

Characteristics of shock response spectra

71

Example Let us consider a terminal peak saw tooth pulse of amplitude 100 m/s2 and duration 10 ms with a symmetrical rectangular pre- and post-shock of amplitude 10 m/s2. The shock has a maximum displacement given by (Chapter 7):

At the end of the shock, there is no change in velocity, but the residual displacement is equal to

Using the numerical data of this example, we obtain xm = -4.428 mm We find this value of xm on the primary negative spectrum of this shock (Figure 3.11). In addition, ^residua] = -0-9576 10"4 mm

Figure 3.11. Primary negative SRS (displacements) of a TPS pulse with rectangular pre- and post-shocks

72

Mechanical shock

3.2.4. Shocks with AV = 0 and AD = 0 at end of pulse For oscillatory type shocks, we note the existence of the following regions [SMA 85] (Figure 3.12): -just below the principal frequency of the shock, the spectrum has, on a logarithmic scale, a slope characterized by the primary response (about 3); - when the frequency of spectrum decreases, its slope tends towards a smaller value of 2; - when the natural frequency decreases further, one observes a slope equal to 1 (6 dB/octave) (residual spectrum). In a general way, all the shocks, whatever their form, have a spectrum of slope of 1 on a logarithmic scale if the frequency is rather small.

Figure 3.12. Shock response spectrum (relative displacements) of a ZERD pulse

The primary negative SRS o0 zsup has a slope of 12 dB/octave; the relative displacement zsup tends towards the absolute displacement xm associated with the shock movement x(t).

Characteristics of shock response spectra

Examples

Figure 3.13. Primary negative SRS of a half-sine pulse with half-sine pre- and post-shocks

Figure 3.14. Primary negative SRS (displacements) of a half-sine pulse •with half-sine pre- and post-shocks

73

74

Mechanical shock

If the velocity change and the variation in displacement are zero the end of the shock, but if the integral of the displacement has a non zero value AD, the undamped residual spectrum is given by [SMA 85]

for small values of co0 (slope of 18 dB/octave). Example

Figure 3.15. Residual positive SRS of a half-sine pulse with half-sine pre- and post-shocks

3.2.5. Notes on residual spectrum Spectrum of absolute displacements When co0 is sufficiently small, the residual spectrum of an excitation x(t) is identical to the corresponding displacement spectrum in one of the following ways [FUN 61]:

a) b) c) However, contrary to the case (c) above, if

Characteristics of shock response spectra

75

but if there exists more than one value t_ of time in the interval 0 x(t) dt = 0, then the residual spectrum is equal to

which

while

the spectrum of the displacements is equal to the largest values of [FUN 61].

If AV and AD are zero at the end of the shock, the response spectrum of the absolute displacement is equal to 2 x(t) where x(t) is the residual displacement of the base. If x(t) = 0, the spectrum is equal to the largest of the two quantities where t = tp is the time when the integral is cancelled. The absolute displacement of response is not limited if the input shock is such that AV * 0. Relative displacement When (00 is sufficiently small, the residual spectrum and the spectrum of the displacements are identical in the following cases: a) if X(T) * 0 at the end of the shock, b) if X(T) = 0, but x(t) is maximum with t = T . If not, the residual spectrum is equal to X(T), while the spectrum of the displacements is equal to the largest absolute value of x(t).

3.3. Characteristics of shock response spectra at high frequencies The response can be written, according to the relation [2.16]

while setting

76

Mechanical shock

We want to show that

Let us set

Integrating by parts:

w(t) tends towards

such that

when co0 tends towards infinity. Let us show that

. constant.

If the function x(t) is continuous, the quantity

Characteristics of shock response spectra

77

tends towards zero as u tends towards zero. There thus exists r\ e [o, t] such that and we have

The function x(t) is continuous and therefore limited at

when, for

, and we have

Thus for

At high frequencies, o>0 z(t) thus tends towards x(t) and, consequently, the shock response spectrum tends towards xm, a maximum x(t).

3.4. Damping influence Damping has little influence in the static region. Whatever its value, the spectrum tends towards the amplitude of the signal depending on time. This property is checked for all the shapes of shocks, except for the rectangular theoretical shock

78

Mechanical shock

which, according to damping, tends towards a value ranging between one and twice the amplitude of the shock. In the impulse domain and especially in the intermediate domain, the spectrum has a lower amplitude when the clamping is greater. This phenomenon is not great for shocks with velocity change and for normal damping (0.01 to 0.1 approximately). It is marked more for oscillatory type shocks (decaying sine for example) at frequencies close to the frequency of the signal. The peak of the spectrum here has an amplitude which is a function of the number of alternations of the signal and of the selected damping. 3.5. Choice of damping The choice of damping should be carried out according to the structure subjected to the shock under consideration. When this is not known, or studies are being carried out with a view to comparison with other already calculated spectra, the outcome is that one plots the shock response spectra with a relative damping equal to 0.05 (i.e. Q = 10). It is about an average value for the majority of structures. Unless otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping (with the reservations of the preceding paragraph), this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram.

NOTE. In practice, the most frequent range of variation of the Q factor of the structures lies between approximately 5 and 50. There is no exact relation which makes it possible to obtain a shock response spectrum of given Q factor starting from a spectrum of the same signal calculated with another Q factor. M.B. Grath and W.F. Bangs [GRA 72] proposed an empirical method deduced from an analysis of spectra of pyrotechnic shocks to carry out this transformation. It is based on curves giving, depending on Q, a correction factor, amplitude ratio of the spectrum for Q factor with the value of this spectrum for Q = 10 (Figure 3.16). The first curve relates to the peak of the spectrum, the second the standard point (non-peak data). The comparison of these two curves confirms the greatest sensitivity of the peak to the choice ofQ factor. These results are compatible with those of a similar study carried out by W.P. Rader and W.F. Bangs [RAD 70],which did not however distinguish between the peaks and the other values.

Characteristics of shock response spectra

79

Figure 3.16. SRS correction factor of the SRS versus Q factor

To take account of the dispersion of the results observed during the establishment of these curves and to ensure reliability, the authors calculated the standard deviation associated -with the correction factor (in a particular case, a point on the spectrum plotted for Q = 20; the distribution of the correction factor is not normal, but near to a Beta or type I Pearson law). Table 3.1. Standard deviation of the correction factor

Q 5 10 20 30 40 50

Standard points 0.085 0 0.10 0.15 0.19 0.21

Peaks 0.10 0 0.15 0.24 0.30 0.34

The results show that the average is conservative 65% of time,and the average plus one standard deviation 93%. They also indicate that modifying the amplitude of the spectrum to take account of the value of Q factor is not sufficient for fatigue analysis. The correction factor being determined, they proposed to calculate the number of equivalent cycles in this transformation using the relation developed by J.D. Crum and R.L. Grant [CRU 70] (cf. Section 4.4.2.) giving the expression for the

80

Mechanical shock

response (OQ z(t) depending on the time during its establishment under a sine wave excitation as:

(where N = number of cycles carried out at time t).

Figure 3.17. SRS correction factor versus Q factor This relation, standardized by dividing it by the amount obtained for the particular case where Q = 10, is used to plot the curves of Figure 3.17 which make it possible to readN, for a correction factor and given Q. They are not reliable for Q < 10, the relation [3.23] being correct only for low damping.

3.6. Choice of frequency range It is customary to choose as the frequency range: - either the interval in which the resonance frequencies of the structure studied are likely to be found; -or the range including the important frequencies contained in the shock (in particular in the case of pyrotechnic shocks).

Characteristics of shock response spectra

81

3.7. Charts There are two spectral charts: - representation (x, y), the showing value of the spectrum versus the frequency (linear or logarithmic scales); -the four coordinate nomographic representation (four coordinate spectrum). One notes here on the abscissae the frequency

on the ordinates the

pseudovelocity co0 zm and, at two axes at 45° to the two first, the maximum relative displacement zm and the pseudo-acceleration co0 zm. This representation is interesting for it makes it possible to directly read the amplitude of the shock at the high frequencies and, at low frequencies, the velocity change associated with the shock (or if AV = 0 the displacement).

Figure 3.18. Four coordinate diagram

3.8. Relation of shock response spectrum to Fourier spectrum 3.8.1. Primary shock response spectrum and Fourier transform The response u(t) of a linear undamped one-degree-of-freedom system to a generalized excitation ^(t) is written [LAL 75] (Volume 1, Chapter 2):

82

Mechanical shock

We suppose here that t is lower than T.

which expression is of the form

with

where C and S are functions of time t. u(t) can be still written:

with

The function

is at a maximum when its derivative is zero

This yields the maximum absolute value of u(t)

where the index P indicates that it is about the primary spectrum. However, where the Fourier transform of £(t), calculated as if the shock were non-zero only between times 0 and t with co0 the pulsation is written as

and has as an amplitude under the following conditions:

Characteristics of shock response spectra

Comparison of the expressions of

83

shows that

In a system of dimensionless coordinates, with

The primary spectrum of shock is thus identical to the amplitude of the reduced Fourier spectrum, calculated for t < T [CA V 64]. The phase <|>L

of the Fourier spectrum is such that

However, the phase <j>p is given by [3.28]

where k is a positive integer or zero. For an undamped system, the primary positive shock spectrum and the Fourier spectrum between 0 and t are thus related in phase and amplitude. 3.8.2. Residual shock response spectrum and Fourier transform The response can be written, whatever value of t

84

Mechanical shock

which is of the form B1 sin co0t + B2 cos
where the constant C is equal to

and the phase R is such that

The residual spectrum, expressed in terms of displacement, is thus given by the maximum value of the response:

The Fourier transform of the excitation l(i) is by definition equal to:

or since outside (0, T), the function t(t) is zero

Characteristics of shock response spectra

85

This expression can be written, by expressing the exponential function according to a sine and a cosine term as,

where R(n) is the real part of the Fourier integral and I(Q) the imaginary part. L(Q) is a complex quantity whose module is given by

Let us compare the expressions of DR(Q) and of |L(Q)|. Apart from the factor w0 and provided that one changes O0 into Q, these two quantities are identical. The natural frequency of the system o>0 can take an arbitrary value since the simple mechanical system is not yet chosen, equal in particular to Q. We thus obtain the relation

The phase is given by

71

71

Only the values of <j>L € (-—,+ —) will be considered. Comparison of <j)R and (|>L 2 2 show that

For an undamped system, the Fourier spectrum and the residual positive shock spectrum are related in amplitude and phase [CA V 64]. NOTE: If the excitation is an acceleration,

and if, in addition,

the Fourier transform of x (t), we have [GER 66], [NAS 65]:

86

Mechanical shock

yielding

with VR (oo) being the pseudovelocity spectrum. The dimension of |L(p)| is that of the variable of excitation ^t) multiplied by time. The quantity Q |L(Q)| is thus that of l ( i ) . If the expression of l{t) is standardized by dividing it by its maximum value lm , it becomes, in dimensionless form

With this representation, the Fourier spectrum of the signal identical to its residual shock spectrum

for zero damping [SUT 68].

3.8.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra Let us consider the Fourier spectra (amplitude) of two shocks, one being an isosceles triangle shape and the other TPS (Figure 3.19), like their positive shock response spectra, for zero damping (Figure 3.20).

Characteristics of shock response spectra

87

Figure 3.19. Comparison of the Fourier transform amplitudes of a TPS pulse and an isosceles triangle pulse

Figure 3.20. Comparison of the positive SRS of a TPS pulse and an isosceles triangle pulse

It is noted that the Fourier spectra and shock response spectra of the two impulses have the same relative position as long as the frequency remains lower than f =1.25 Hz, the range for which the shock response spectrum is none other than the residual spectrum, directly related to the Fourier spectrum. On the contrary, for f > 1.25 Hz, the TPS pulse has a larger Fourier spectrum, whereas the SRS (primary spectrum) of the isosceles triangle pulse is always in the form of the envelope.

88

Mechanical shock

The Fourier spectrum thus gives only one partial image of the severity of a shock by considering only its effects after the end of the shock (and without taking damping into account).

3.9. Characteristics of shocks of pyrotechnic origin The aerospace industry uses many pyrotechnic devices such as explosive bolts, squib valves, jet cord, pin pushers etc. During their operation these devices generate shocks which are characterized by very strong acceleration levels at very high frequencies which can be sometimes dangerous for the structures, but especially for the electric and electronic components involved. These shocks were neglected until about 1960 approximately but it was estimated that, in spite of their high amplitude, they were of much too short duration to damage the materials. Some incidents concerning missiles called into question this postulate. An investigation by C. Moening [MOE 86] showed that the failures observed on the American launchers between 1960 and 1986 can be categorized as follows: - due to vibrations: 3; - due to pyroshocks: 63. One could be tempted to explain this distribution by the greater severity of the latter environment. The Moening study shows that it was not the reason, the causes being: - the partial difficulty in evaluating these shocks a priori; - more especially the lack of consideration of these excitations during design, and the absence of rigorous test specifications.

Figure 3.21. Example ofapyroshock

Characteristics of shock response spectra

89

Such shocks have the following general characteristics: - the levels of acceleration are very important; the shock amplitude is not simply related to the quantity of explosive used [HUG 83b]. Reducing the load does not reduce the consequent shock. The quantity of metal cut by a jet cord is, for example, a more significant factor; - the signals assume an oscillatory shape; - in the near-field, close to the source (material within about 15 cm of point of detonation of the device, or about 7 cm for less intense pyrotechnic devices), the effects of the shocks are primarily related to the propagation of a stress wave in the material; - the shock is then propagated whilst attenuating in the structure. The mid-field (material within about 15 cm and 60 cm for intense pyrotechnic devices, between 3 cm and 15 cm for less intense devices) from, which the effects of this wave are not yet negligible and combine with a damped oscillatory response of the structure at its frequencies of resonance, is to be distinguished from the far-field, where only this last effect persists; - the shocks have very close components according to three axes; their positive and negative response spectra are curves that are coarsely symmetrical with respect to the axis of the frequencies. They begin at zero frequency with a very small slope at the origin, grow with the frequency until a maximum located at some kHz, even a few tens of kHz, is reached and then tend according to the rule towards the amplitude of the temporal signal. Due to their contents at high frequencies, such shocks can damage electric or electronic components; - the a priori estimate of the shock levels is neither easy nor precise. These characteristics make them difficult to measure, requiring sensors that are able to accept amplitudes of 100,000 g, frequencies being able to exceed 100 kHz, with important transverse components. They are also difficult to simulate. The dispersions observed in the response spectra of shocks measured under comparable conditions are often important (3 dB with more than 8 dB compared to the average value, according to the authors [SMI 84] [SMI 86]), The reasons for this dispersion are in general related to inadequate instrumentation and the conditions of measurement [SMI 86]: - fixing the sensors on the structure using insulated studs or wedge which act like mechanical filters; - zero shift, due to the fact that high accelerations make the crystal of the accelerometer work in a temporarily non-linear field. This shift can affect the calculation of the shock response spectrum (cf. Section 3.10.2.); - saturation of the amplifiers;

90

Mechanical shock

- resonance of the sensors. With correct instrumentation, the results of measurements carried out under the same conditions are actually very close. The spectrum does not vary with the tolerances of manufacture and the assembly tolerances.

3.10. Care to be taken in the calculation of spectra 3.10.1. Influence of background noise of the measuring equipment The measuring equipment is gauged according to the foreseeable amplitude of the shock to be measured. When the shock characteristics are unknown, the rule is to use a large effective range in order not to saturate the conditioning module. Even if the signal to noise ratio is acceptable, the incidence of the background noise is not always negligible and can lead to errors of the calculated spectra and the specifications which are extracted from it. Its principal effect is to increase the spectra artificially (positive and negative), increasing with the frequency and Q factor. Example

Figure 3.22. TPSpulse with noise (rms value equal to one-tenth amplitude of the shock)

Figure 3.23 shows the positive and negative spectra of a TPS shock (100 m/s2, 25 ms) plotted in the absence of noise for an Q factor successively equal to 10 and 50, as well as the spectra (calculated in the same conditions) of a shock (Figure 3.22) composed of mis TPS pulse to which is added a random noise of rms value 10 m/s2 (one tenth of the shock amplitude).

Characteristics of shock response spectra

91

Figure 3.23. Positive and negative SRS of the TPS pulse and with noise

Due to its random nature, it is practically impossible to remove the noise of the measured signal to extract the shock alone from it. Techniques, however have been developed to try to correct the signal by cutting off the Fourier transform of the noise from that of the total signal (subtraction of the modules, conservation of the phase of the total signal) [CAI 94].

92

Mechanical shock

3.10.2. Influence of zero shift One very often observes a continuous component superimposed on the shock signal on the recordings, the most frequent origin being the presence of a transverse high level component which disturbs the operation of the sensor. If this component is not removed from the signal before calculation of the spectra, it can it also lead to considerable errors [BAG 89] [BEL 88]. When this continuous component has constant amplitude, the signal treated is in fact a rectangle modulated by the true signal. It is not thus surprising to find on the spectrum of this composite signal the characteristics, more or less marked, of the spectra of a rectangular shock. The effect is particularly important for oscillatory type shocks (with zero or very small velocity change) such as, for example, shocks of pyrotechnic origin. In this last case, the direct component has as a consequence a modification of the spectrum at low frequencies which results in [LAL 92a]: - the disappearance of the quasi-symmetry of the positive and negative spectra characteristic of this type of shocks; - appearance of more or less clear lobes in the negative spectrum, similar to those of a pure rectangular shock. Example

Figure 3.24. 24. pyrotechnic shock with zero shift

Characteristics of shock response spectra

93

The example treated is that of a pyrotechnic shock on which one artificially added a continuous component (Figure 3.24). Figure 3.25 shows the variation generated at low frequencies for a zero shift of about 5%. The influence of the amplitude of the shift on the shape of the spectrum (presence of lobes) is shown in Figure 3.26.

Figure 3.25. Positive and negative SRS of the centered and non-centered shocks

94

Mechanical shock

Figure 3.26. Zero shift influence on positive and negative SRS

Under certain conditions, one can try to center a signal presenting a zero shift constant or variable according to time, by addition of a signal of the same shape as this shift and of opposite sign [SMI 85]. This correction is always a delicate operation which supposes that only the average value was affected during the disturbance of measurement. In particular one should ensure that the signal is not saturated.

Chapter 4

Development of shock test specifications

4.1. General The first tests of the behaviour of materials in response to shocks were carried out in 1917 by the American Navy [PUS 77] and [WEL 46]. The most significant development started at the time of the World War II with the development of specific free fall or pendular hammer machines. The specifications are related to the type of machine and its adjustments (drop height, material constituting the programmer, mass of the hammer). Given certain precautions, this process ensures a great uniformity of the tests. The demonstration is based on the fact that the materials, having undergone this test successfully resist well the real environment which the test claims to simulate. It is necessary to be certain that the severity of the real shocks does not change from one project to another. It is to be feared that the material thus designed is more fashioned to resist the specified shock on the machine than the shock to which it will be really subjected in service. Very quickly specifications appeared imposing contractually the shape of acceleration signals, their amplitude and duration. In the mid-1950s, taking into account the development of electrodynamic exciters for vibration tests, and the interest in producing mechanical shocks, the same methods were developed (it was that time that simulation vibrations by random vibrations under test real conditions were started). This testing on a shaker, when possible, indeed presents a certain number of advantages [COT 66]; vibration and impact tests on the same device, the possibility of carrying out shocks of very diverse shapes, etc.

96

Mechanical shock

In addition the shock response spectrum became the tool selected for the comparison of the severity of several shocks and for the development of specifications, the stages being in this last case the following: - calculation of shock spectra of transient signals of the real environment; - plotting of the envelope of these spectra; - searching for a signal of simple shape (half-sine, saw tooth etc) of which the spectrum is close to the spectrum envelope. This operation is generally delicate and cannot be carried out without requiring an over-test or an under-test in certain frequency bands. In the years 1963/1975 the development of computers gave way to a method, consisting of giving directly the shock spectrum to be realized on the control system of the shaker. Taking into account the transfer function of the test machine (with the test item), the software then generates on the input of the test item a signal versus time which has the desired shock spectrum. This makes it possible to avoid the last stage of the process. The shocks measured in the real environment are in general complex in shape; they are difficult to describe simply and impossible to reproduce accurately on the usual shock machines. These machines can generate only simple shape shocks such as rectangle, half-sine, terminal peak saw tooth pulses. Several methods have been proposed to transform the real signal into a specification of this nature. 4.2. Simplification of the measured signal This method consists of extracting the first peak, the duration being defined by time when the signal x(t) is cancelled for the first time, or extraction of the highest peak.

Figure 4.1. Taking into account the largest peak

Development of shock test specifications

97

The shock test specification is then described in the form of an impulse of amplitude equal to that of the chosen peak in the measured signal, of duration equal to the half-period thus defined and whose shape can vary, while approaching as early as possible that of the first peak (Figure 4.1). The choice can be guided by the use of an abacus making it possible to check that the profile of the shock pulse remains within the tolerances of one of the standardized forms [KIR 69]. Another method consists of measuring the velocity change associated with the shock pulse by integration of the function x(t) during the half-cycle with greater amplitude. The shape of the shock is selected arbitrarily. The amplitude and the duration are fixed in order to preserve the velocity change [KIR 69] (Figure 4.2).

Figure 4.2. Specification with same velocity change

The transformation of a complex shock environment into a simple shape shock, realizable in the laboratory, is under these conditions an operation which utilizes in an important way the judgement of the operator. It is rare, in practice, that the shocks observed are simple, with a form easy to approach, and it is necessary to avoid falling into the trap of over-simplification.

Figure 4.3. Difficulty of transformation of real shockpulses

98

Mechanical shock

In the example in Figure 4.3, the half-sine signal can be a correct approximation of the relatively "clean" shock 1; but the real shock 2, which contains several positive and negative peaks, cannot be simulated by just one unidirectional wave. It is difficult to give a general empirical rule to ensure the quality of simulation in laboratory carried out according to this process and the experimental quality is important. It is not shown that the criterion of equivalence chosen to transform the complex signal to a simple shape shock is valid. It is undoubtedly the most serious defect. This method lends itself little to statistical analysis which would be possible if one had several measurements of a particular event and which would make it possible to establish a specification covering the real environment with a given probability. In the same way, it is difficult to determine a shock enveloping various shocks measured in the life profile of the material.

4.3. Use of shock response spectra 4.3.1. Synthesis of spectra The most complex case is where the real environment, described by curves of acceleration against time, is supposed to be composed of p different events (handling shock, inter-stage cutting shock on a satellite launcher), with each one of these events itself being characterized by ri successive measurements. These ri measurements allow a statistical description of each event. The folowing procedure consists for each one: - To calculate the shock response spectrum of each signal recorded with the damping factor of the principal mode of the structure if this value is known, if not with the conventional value 0.05. In the same way, the frequency band of analysis will have to envelop the principal resonance frequencies of the structure (known or foreseeable frequencies). - If the number of measurements is sufficient, to calculate the mean spectrum m (mean of the points at each frequency) as well as the standard deviation spectrum, then the standard deviation/mean ratio, according to the frequency; if it is insufficient, to make the envelope of the spectra. - To apply to the mean spectrum or the mean spectrum + 3 standard deviations a statistical uncertainty coefficient k, calculated for a probability of tolerated maximum failure (cf Volume 5), or contractual (if one uses the envelope).

Development of shock test specifications

99

Each event thus being synthesized in only one spectrum, one proceeds to an envelope of all the spectra obtained to deduce a spectrum from it covering the totality of the shocks of the life profile. After multiplication by a test factor (Volume 5), this spectrum will be used as reference 'real environment' for the determination of the specification.

Table 4.1. Process of developing a specification from real shocks measurements Event #1 Handling shock r1 measured data

Mean and standard deviation spectra or envelope

Calculation of

r1 S.RS.

Event #2 Landing shock T2 measured data

Calculation of

Event #p Ignition shock r_ measured data

Calculation of

T2 S.RS.

rp S.R.S.

-

k (m + 3 s) or k env.

Mean and standard k (m + 3 s) deviation spectra or k env. or envelope Mean and standard deviation spectra or envelope

k (m + 3 s) or k env.

—

Envelope X

Envelope

Test factor

The reference spectrum can consist of the positive and negative spectra or the envelope of their absolute value (maximax spectrum). In this last case, the specification will have to be applied according to the two corresponding half-axes of the test item.

4.3.2. Nature of the specification According to the characteristics of the spectrum and available means, the specification can be expressed in the form of: -A simple shape signal according to time realizable on the usual shock machines (half-sine, T.P.S., rectangular pulse). There is an infinity of shocks having a given response spectrum. The fact that this transformation is universal is related to its very great loss of information, since one retains only the largest value of the response according to time to constitute the SRS at each natural frequency. One can thus try to find a shock of simple form, to which the spectrum is closed to the reference spectrum, characterized by its form, its amplitude and its duration. It is in

100

Mechanical shock

general desirable that the positive and negative spectra of the specification respectively cover the positive and negative spectra of the field environment. If this condition cannot be obtained by application of only one shock (particular shape of the spectra, limitations of the facilities), the specification will be made up of two shocks, one on each half-axis. The envelope must be approaching the real environment as well as possible, if possible on all the spectrum in the frequency band retained for the analysis, if not in a frequency band surrounding the resonance frequencies of the test item (if they are known). - A shock response spectrum. In this last case, the specification is directly the reference SRS.

4.3.3. Choice of shape The choice of the shape of the shock is carried out by comparison of the shapes of the positive and negative spectra of the real environment with those of the spectra of the usual shocks of simple shape (half-sine, TPS, rectangle) (Figure 4.4).

Figure 4.4. Shapes of the SRS of the realizable shocks on the usual machines

Development of shock test specifications

101

If these positive and negative spectra are nearly symmetrical, one will retain a terminal peak saw tooth, whilst remembering, however, that the shock which will be really applied to the tested equipment will have a non-zero decay time so that its negative spectrum will tend towards zero at very high frequencies. This disadvantage is not necessarily onerous, if for example a preliminary study could show that the resonance frequencies of the test item are in the frequency band where the spectrum of the specified shock envelops the real environment. If only the positive spectrum is important, one will choose any form, the selection criterion being the facility for realization, or the ratio between the amplitude of the first peak of the spectrum and the value of the spectrum at high frequencies: approximately 1.65 for the half-sine pulse (Q = 10), 1.18 for the terminal peak saw tooth pulse, and no peak for the rectangular pulse.

4.3.4. Amplitude The amplitude of the shock is obtained by plotting the horizontal straight line which closely envelops the positive reference SRS at high frequency.

Figure 4.5. Determination of the amplitude of the specification

This line cuts the y-axis at a point which gives the amplitude sought (one uses here the property of the spectra at high frequencies, which tends in this zone towards the amplitude of the signal in the time domain).

4.3.5. Duration The shock duration is given by the coincidence of a particular point of the reference spectrum and the reduced spectrum of the simple shock selected above (Figure 4.6).

102

Mechanical shock

Figure 4.6. Determination of the shock duration

One in general considers the abscissa f01 of the first point which reaches the value of the asymptote at the high frequencies (amplitude of shock). Table 4.2 joins together some values of this abscissa for the most usual simple shocks according to the Q factor [LAL 78].

Table 4.2. Values of the dimensionless frequency corresponding to the first passage of the SRS by the amplitude unit f 01

£

Half-sine

Versed-sine

2 3

0.2500

0.413

0.542

IPS /

0.1667

0.358

0.465

0.564

0.219

4

0.1250

0.333

0.431

0.499

0.205

Q

Rectangle 0.248

5

0.1000

0.319

0.412

0.468

0.197

6 7 8 9 10 15 20 25 30 35 40 45 50

0.0833

0.310

0.400

0.449

0.192

0.0714 0.0625

0.304 0.293

0.392

0.437

0.188

0.385

0.427

0.185

0.0556 0.0500

0.295

0.381 0.377

0.421

0.183

0.415

0.181

00

0.293

0.0333

0.284

0.365

0.400

0.176

0.0250

0.280

0.360

0.392

0.174

0.0200

0.277

0.357

0.388

0.173

0.0167

0.276

0.354

0.385

0.172

0.0143

0.275

0.353

0.383

0.171

0.0125

0.274

0.0111 0.0100

0.273 0.272

0.352 0.351

0.382 0.380

0.170 0.170

0.0000

0.267

0.350

0.379

0.170

0.344

0.371

0.167

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NOTES: 1. If the calculated duration must be rounded (in milliseconds), the higher value should always be considered, so that the spectrum of the specified shock remains always higher or equal to the reference spectrum. 2. It is in general difficult to carry out shocks of duration lower than 2 ms on standard shock machines. This difficulty can be circumvented for very light equipment with a specific assembly associated with the shock machine (dual mass shock amplifier, Section 6.2). One will validate the specification by checking that the positive and negative spectra of the shock thus determined are well enveloped by the respective reference spectra and one will verify, if the resonance frequencies of the test item are known, that one does not over-test exaggeratedly at these frequencies.

Example Let us consider the positive and negative spectra characterizing the real environment plotted (Figure 4.7) (result of a synthesis).

Figure 4.7. SRS of the field environment

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Mechanical shock

It is noted that the negative spectrum preserves a significant level in all the frequency domain (the beginning of the spectrum being excluded). The most suitable simple shock shape is the terminal peak saw tooth. The amplitude of the shock is obtained by reading the ordinate of a straight line enveloping the positive spectrum at high frequencies (340 m/s2). The duration is deduced from the point of intersection of this horizontal line with the curve (point of lower frequency), which has as an abscissa equal to 49.5 Hz (Figure 4.8). One could also consider the point of intersection of this horizontal line with the tangent at the origin.

Figure 4.8. Abscissa of the first passage by the unit amplitude

One reads on the dimensionless spectrum of a TPS pulse (same damping ratio) the abscissa of this point: f0 T = 0.415, yielding, so that f0 = 49.5 Hz

The duration of the shock will thus be (rounding up)

which slightly moves the spectrum towards the left and makes it possible to bettei cover the low frequencies. Figure 4.9 shows the spectra of the environment and those of the TPS pulse thus determined.

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Figure 4.9. SRS of the specification and of the real environment

NOTE: In practice, it is only at this stage that the test factor can be applied to the shock amplitude.

4.3.6. Difficulties This method leads easily to a specification when the positive spectrum of reference increases regularly from the low frequencies to a peak value not exceeding approximately 1.7 times the value of the spectrum at the highest frequencies, and then decreases until it is approximately constant at high frequencies. This shape is easy to envelop since it corresponds to the shape of the spectra of normal simple shocks.

Figure 4.10. Case of a SRS presenting an important peak

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Mechanical shock

In practice it can happen that the first peak of the reference spectrum is much larger, that this spectrum has several peaks, and that it is almost tangential to the frequency axis at the low frequencies etc. In the first case (Figure 4.10), a conservative method consists of enveloping the whole of the reference spectrum. After choosing the shape as previously, one notes the coordinates of a particular point, for example: the amplitude Sp of the peak and its abscissa fp.

Figure 4.11. Coordinates of the peak of the dimensionless SRS of the selected shock

On the dimensionless positive spectrum of the selected signal, plotted with the same damping ratio, one reads the coordinates of the first peak:
- amplitude

Figure 4.12. Under-testing around the peak in the absence of resonance in this range

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Such a shock can over-test mostly at the frequencies before and after the peak. To avoid this, if one knows that the material does not have any resonance in the frequency band around the peak, a solution consists of adjusting the spectrum of the simple shock on the high frequency part of the reference spectrum, while cutting the principal peak (Figure 4.12). NOTE: In general it is not advisable to choose a simple shock shape as a specification when the real shock is oscillatory in nature. In addition to over-testing at low frequencies (the oscillatory shock is with quasi velocity change), the amplitude of the simple shock thus calculated is sensitive to the value of the Q factor in the intermediate frequency range. A specification using an oscillatory shock does not present this disadvantage (but presupposes that the shock is realizable on the exciter).

4.4. Other methods Other methods were used for simulation of the shocks using their response spectrum. We will quote some of them in the following paragraphs. 4.4.1. Use of a swept sine In the past and sometimes still today, shocks (often shocks of pyrotechnic origin, such as the separation between two stages of a satellite launcher using a flexible linear shaped charge) were simulated by a swept sine defined from the response spectrum of the shock [CUR 55] [DEC 76] [HOW 68]. The objective of this test was not the rigorous reproduction of the responses caused by the shock. This approach was used because it had proved its effectiveness as a stress screening test, the materials thus qualified as behaving well in the presence of real pyrotechnic shocks [KEE 74], but also because this type of test is well understood, easy to carry out, to control and is reproducible. The test was defined either in a specified way (5 g between 200 and 2000 Hz), or by research of the characteristics of a swept sine whose extreme response spectrum envelops the spectrum of the shock considered [CUR 55], [DEC 76], [HOW 68], [KEE 74] and [KER 84]. The sweeping profile is obtained in practice by dividing the response spectrum of the shock by the quality factor Q chosen for the calculation of the spectrum. The disadvantages of this process are multiple: - The result is in general very sensitive to the choice of the damping factor chosen for the calculation of the spectrum. It is therefore very important to know the

108

Mechanical shock

factor for transformation, which also implicitly also that if there are several resonances, the Q factor varies little with the frequencies. - A very short phenomenon, which will induce the response of few cycles, is replaced by a vibration of much large duration, which will produce a relatively significant number of cycles of stress in the system and will be able to thus damage the structures sensitive to this phenomenon in a non-representative manner [KER 84]. - The maximum responses are the same, but the acceleration signals x(t) are very different. In a sinusoidal test, the system reaches the maximum of its response at its resonance frequency. The input is small and it is the resonance which makes it possible to reach the necessary response. Under shock, the maximum response is obtained at a frequency more characteristic of the shock itself [CZE 67]. - The swept sine individually excites resonances, one after another, whereas a shock has a relatively broad spectrum and simultaneously excites several modal responses which will combine. The potential mechanisms of failure related to the simultaneous excitation of these modes are not reproduced. 4.4.2. Simulation of shock response spectra using a fast swept sine J.R. Pagan and A.S. Baran [FAG 67] noted in 1967 that certain shapes of shock, such as the terminal peak saw tooth pulse excite the high frequencies of resonance of the shaker and suggested the use of a fast swept sine wave to avoid this problem. They saw moreover two advantages there: there is neither residual velocity nor residual displacement and the specimen is tested according to two directions in the same test. The first work carried out by J.D. Crum and R.L Grant [CRU 70] [SMA 74a] [SMA 75], then by R.C. Rountree and C.R. Freberg [ROU 74] and D.H. Trepess and R.G. White [TRE 90] uses a drive signal of the form:

where A(t) and E(t) are two time functions, the derivative of <j)(t) being the instantaneous pulsation of x(t). The response of a linear one-degree-of-freedom mechanical system to a sinewave excitation of frequency equal to the natural frequency of the system can be written in dimensionless form (Volume 1, Chapter 5) as:

Development of shock test specifications

109

If damping is weak, this expression becomes

Since the excitation frequency is equal to the resonance frequency, the number of cycles carried out at time t is given by:

For an excitation defined by an acceleration

The relative displacement response z(t) is at a maximum when cos yielding:

The response o>oz m depends only on the values of Q and N (for xm fixed). Being given a shock measured in the real environment, J.D. Crum and R.L. Grant [CRU 70] plotted the ratio of the response spectra calculated for Q = 25 and Q = 5 versus frequency f0. Their study, carried out on a great number of shocks, shows that this ratio varies little in general around a value a. The specification is obtained by plotting a horizontal linear envelope of each spectrum (in the ratio a). In sinusoidal mode, the ratio

is, for Q given, only a function of N. With a

swept sine excitation, one obtains a spectrum of constant amplitude if the number of cycles AN carried out between the half-power points is independent of the natural frequency f0, i.e. if the sweeping is hyperbolic. J.D. Crum and R.L. Grant expressed their results according to the parameter N'= Q AN.

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Mechanical shock

If the sweep rate were weak, the ratio would be equal to 5 or 25 according to choice of Q (whatever, the sweep mode). To obtain spectra in the ratio a (in general lower than 5), a fast sweep should be used therefore. The hyperbolic swept sine is defined as follows, starting from a curve giving the ratio to responses for Q = 25 and Q = 5 versus N' and of

versus N'.

-The desired ratio a allows one to define N'= N'0 and N'0 gives using the two preceding curves. - Knowing the envelope spectrum w02 zm specified for Q = 5, one deduces from it the necessary amplitude xm. - The authors have given for an empirical rule the sweep starting from a frequency f1 lower by 25% than the lowest frequency of the spectrum of the specified shock and finishing at a frequency f2 higher by 25% than the highest frequency of the specified spectrum. The excitation is thus defined by:

with:

if the sweep is at increasing frequencies, or by:

for a sweep at decreasing frequencies. - The sweep duration is given by:

The durations obtained are between a few hundreds of milliseconds and several seconds.

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It is possible to modulate the amplitude xm according to the frequency to satisfy a specification which would not be a horizontal line and to vary N'0 to better follow the variations of the ratio a of the spectra calculated for Q = 25 and Q = 5 [CRU 70] [ROU 74]. The formulation of Routree and Freberg is more general. It is based on the relations:

The modifiable parameters are a, (3, f0, R and y where: - a is the initial value of A(t) (with t = 0); -(3 characterizes the variations of the amplitude A(t) according to time (or according to f); - f(t) is the instantaneous frequency, equal to f0 for t = 0; - R and y characterize the variations oft versus time. If y = 0, the law f(t) is linear, with a sweep rate equal to R. If y = 1, sweep is exponential, such that f = e

Rt

If y = 2, sweep is hyperbolic (as in the assumptions of Crum and Grant)

Advantages These methods: - produce shocks pulses well adapted for the reproduction on a shaker;

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Mechanical shock

- allow the simulation of a spectrum simultaneously for two values of the Q factor.

Drawbacks These methods lead to shock pulses which do not resemble the real environment at all These techniques were developed to simulate spectra which can be represented by a straight line on log log scales and they adapt badly to spectra with nother shapes.

4.4.3. Simulation by modulated random noise It was recognized that the shocks measured in the seism domain have a random nature. This is why many proposals [BAR 73], [LEV 71] were made to seek a random process which, after multiplication by an adequate window, provides a shock comparable with this type of shock. The aim is to determine a wave form showing the same statistical characteristics as the signal measured [SMA 74a], [SMA 75]. This wave form is made up of a nonstationary modulated random noise having the same response spectrum as the seismic shock to be simulated. It is, however, important to note that this type of method allows reproduction of a specified shock spectrum only in one probabilistic sense. L.L.Bucciarelli and J.Askinazi [BUG 73] proposed using an excitation of this nature to simulate pyrotechnic shocks with an exponential window of the form:

where g(t) is a deterministic function of the time, which characterizes the transitory nature of the phenomenon

and n(t) is a stationary broad band noise process with average zero and power spectral density Sn(Q).

Development of shock test specifications

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Being given a whole set of measurements of the shock, one seeks to determineSn(Q) and the time constant (3 to obtain the best possible simulation. The function Sn(o) is calculated from:

where E[X(Q) X*(Q)] is the mean value of the squares of the amplitudes of the Fourier spectra of the shocks measured. The constant B must be selected to be lower than the smallest interesting frequency of the shock response spectrum. N.C. Tsai [TSA 72] was based on the following process: - choice of a sample of signal x(t); - calculation of the shock response spectrum of this sample; - being given a white noise n(t), addition of energy to the signal by addition of sinusoids to n(t) in the ranges where the shock spectrum is small; in the ranges where the shock spectrum is large, filtering of n(t) with a filter attenuating a narrow band (—I I—I L); - calculation of the shock spectrum of the modified signal n(t); and repetition of the process until reaching the desired shock spectrum. Although interesting, this technique is not the subject of marketed software and is thus not used in the laboratory. NOTE: J.F. Unruth [UNR 82] suggested simulating the seisms while controlling the shock spectrum, the signal reconstituted being obtained by synthesis from the sum of pseudo-random noises into 1/6 octave. Each component of narrow band noise is the weighted sum of 20 cosine functions out of phase whose frequencies are uniformly distributed in the band considered. The relative phases have a random distribution in the interval [0, n].

4.4.4. Simulation of a shock using random vibration The probability that a maximum of w02 z(t) is lower than w02 zm over the duration T is equal to 1 - Pp(w02 zm) with PP being the distribution function of the peaks of the response. The number of cycles to be applied during the test is equal approximately to f0 T. If these peaks are supposed independent, the probability PT that all the

114

Mechanical shock

0

"7

maxima of w02 z(t) are lower than w02 zm is then

I

/

"7

1-Pp(w02

\ rO

zm)

. The

probability that a maximum of O>Q z(t) is higher than w02 zm is thus equal to

i4-4fo-FUse of a narrow band random vibration A narrow band random vibration can be applied to the material at a single frequency or several frequencies simultaneously. This process has some advantages [KER 84]: - the number of cycles exceeding a given level can be limited; - several resonances can be excited simultaneously; - amplification at resonance is reduced compared to the slow swept sine (the response varies as /Q instead of Q). But the nature of the vibration does not make it possible to ensure the reproducibility of the test.

4.4.5. Least favourable response technique Basic assumption It is supposed that the Fourier spectrum (amplitude) is specified, which is equivalent to specifying the undamped residual shock spectrum (Section 3.8.2.). It is shown that if the transfer function between the input and the response of the test item (and not that of the shaker) can be characterized by:

then the peak response of the structure will be maximized by the input [SMA 74a] [SMA 75]:

where Xe(Q) is the module of the specified Fourier transform and

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The calculation of the above expressions is relatively easy today. The phase angle 6 of the transfer function is measured using a test. With this function and the specified module X e (Q), one calculates the input x(t). The method supposes simply that the studied system is linear with a critical response well defined. There is no assumption on the number of degrees of freedom or on damping. It guarantees that the largest possible response peak will be reached, in practice, at about 1 to 2.5 times the response with the real shock (guarantee of a conservative test) [SMA 72] [WIT 74]. The techniques of the shock spectrum cannot give this insurance for systems to several degrees of freedom. The method requires important calculations and thus numerical means. An alternative can be found in supposing that H(Q) = 1 and to calculate the input to be applied to the specimen so that:

With Xe(Q) being a real positive function and x(t)a real even function. An input thus defined will resemble a SHOC waveform (Chapter 9). This input is independent of the characteristics of the test item and thus eliminates the need for defining the transfer function H(Q). The only necessary parameter is the module of the Fourier transform (or the undamped residual shock spectrum). A series of tests showed that this approach is reasonable [SMA 72].

4.4.6. Restitution of a shock response spectrum by a series of modulated sine pulses This method, suggested by D.L. Kern and C.D. Beam [KER 84], consists of applying a series of modulated sine wave shocks sequentially. The retained waveform resembles the response versus time of the mass of a one-degree-offreedom system base-excited when it is subjected to an exponentially decayed sine wave excitation; it has as an approximate equation

elsewhere where Q = 2 n f T] = damping of the signal A = Q e T| xm xm = amplitude of x(t) e = Neper number

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Mechanical shock

Figure 4.13. Shock waveform (D.L Kern and CD. Hayes)

The choice of r\ must meet two criteria: - to be close to 0.05, a value characteristic of many complex structures; - to allow that the maximum of x(t) (the largest peak) takes place at the same time as the peak of the envelope of x(t).

Figure 4.14. Coincidence of the peaks of the signal and its envelope

The interesting point of this approach, which takes again a proposal of J.T.Howlett and D.J.Martin [HOW 68] containing purely sinusoidal impulses, is in the facility of determination of the characteristics of each sinusoid, since each one of them is considered separately, contrary to the case of a control per spectrum (Chapter 9). The shocks are easy to create and to realize.

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The adjustable parameters are the amplitude and possibly the number of cycles. The number of frequencies is selected so that the point of intersection of the spectra of two adjacent signals is not lower by more than 3 dB than the amplitude of the peak of the spectrum (plotted for a damping equal to 0.05). Like the slowly swept sine, this method does not make it possible to excite all resonances simultaneously. We will see in Chapter 9 how this waveform can be used to constitute a complex drive signal restoring the whole of the spectrum.

4.5. Interest behind simulation of shocks on a shaker using a shock spectrum The data of a shock specification for a response spectrum has several advantages: - the response spectrum should be more easily exploitable for dimensioning of the structure than the signal x(t) itself; - this spectrum can result directly from measurements of the real environment and does not require, at the design stage to proceed to an often delicate equivalence with a signal of simple shape; - the spectrum can be treated in a statistical way if one has several measurements of the same phenomenon, it can be the envelope of several different transitory events and can be increased by a uncertainty coefficient; -the reference most commonly allowed to judge quality of the shock simulation is comparison of the response spectra of the specification with the shock carried out. In a complementary way, when the shock tests can be carried out using a shaker, one can have direct control from a response spectrum: - The search for a simple form shock of a given spectrum compatible with the usual test facilities is not always a simple operation, according to the shape of the reference spectrum resulting from measurements of the real environment. - The shapes of the specified spectra can be very varied, contrary to those of the spectra of the usual shocks (half-sine, triangles, rectangles etc) carried out on the shock machines. One can therefore improve the quality of simulation and reproduce shocks difficult to simulate with the usual means (case of the pyroshocks for example) [GAL 73] and [ROT 72]. - Taking into account the oscillatory nature of the elementary signals used, the positive and negative spectra are very close, which makes a reversal of the test item [PAI 64] useless.

118

Mechanical shock

- In theory, simple shape shocks created on a shock machine are reproducible, which makes it possible to expect uniform tests from one laboratory to another. In practice, one was obliged to define tolerances on the shapes of the signals to take account of the distortions really measured and difficult to avoid. The limits are rather broad (+15%) and can result, however, in accepting two shocks included within these limits likely to have very different effects (which one can evaluate with the shock spectra) [FAG 67].

Figure 4.15. Nominal half-sine and its tolerances

Figure 4.16. Shock located between the tolerances

Figure 4.17. SRS of the nominal half-sine and the tolerance limits

Figures 4.15 and 4.17 show as an example a nominal half-sine (100 m/s2, 10 ms) and its tolerance limits, as well as the shock spectra of the nominal shock and each lower and upper limit. Figure 4.16 represents a shock made up of the sum of the nominal half-sine and of a sinusoid of amplitude 15 m/s2 and frequency 2500 Hz.

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The spectrum of this signal is superimposed on the spectra of the tolerance limits in Figure 4.18. Although this composite signal remains within the tolerances, it is noted that it has a spectrum very different from the spectra of the tolerance limits for small £, in a frequency band around 250 Hz and mat the negative spectra of the tolerance limits intersect and thus do not delimit a well defined domain [LAL 72].

Figure 4.18. SRS of the shock of Figure 4.16 and of the tolerance limits

With mis some practical advantages are added: - sequence of the shock and vibration tests without disassembly and with the same test fixture (saving of time and money); - maintenance of the test item with its normal orientation during the test.

120

Mechanical shock

These the last two points are not, however, specific with spectrum control, but more generally relate to the use of a shaker. Control by the spectrum, however, increases the capacities of simulation because of the possibility of the choice of the shape of the elementary waveforms and of their variety.

Chapter 5

Kinematics of simple shocks

5.1. General The shock test is in general specified by an acceleration varying with time. This profile of acceleration can be obtained with various velocity and displacement profiles depending on the initial velocity of the table supporting the specimen, leading theoretically to various types of programme. All shock test facilities are in other respects limited in respect of force (i.e. in acceleration, taking into account the mass of the whole of the moving element, made up of the table, the test fixure, the armature assembly in the case of a shaker and the specimen), velocity and displacement. It is thus useful to study the kinematics of the principal shock pulses carried out classically on the machines, namely the half-sine (or versed-sine), the terminal peak saw tooth and the rectangle (or the trapezoid).

5.2. Half-sine pulse 5.2.1. Definition The excitation, zero for t < 0 and t > T can be written in the interval (0, T), in the form

122

Mechanical shock

where xm is the amplitude of the shock and T its duration. The pulsation is equal to This expression becomes, in generalized form, l(t) = lm sin Q t.

According to the type of excitation, l ( t ) is then a force

an acceleration

In reduced (dimensionless) form, and with the notations used in preceding chapters, the definition of shock can be

Note that h

Figure 5.1. Half-sine shock

5.2.2. Shock motion study 5.2.2.1. General expressions The motion study during the application of the shock is useful for the choice of the programmer and the test facility which will make it possible to carry out the

Kinematics of simple shocks

123

specification. We will limit ourselves, in what follows, to the most general case where the shock is defined by an acceleration pulse x(t) [LAL 75]. With the signal of acceleration

corresponds

by

integration

to

the

instantaneous

velocity

constant. Let us suppose that at the initial moment t = 0, the velocity is equal to Vj:

The constant is thus equal to

and the velocity to

At the moment t = T of the end of the shock, the velocity vf has as an expression

i.e., since Q t = n,

The body subjected to this shock thus undergoes a velocity change

It is the area delimited by the curve x(t) and the time axis between 0 and T.

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Mechanical shock

Figure 5.2. Velocity change of a half-sine

The displacement is calculated by a second integration; we will take for initial conditions t = 0, x = 0 , a s i t i s practically always the case in these problems. This yields

To further the study of this movement x(t), it is preferable to particularize the test conditions. Two cases arise; the velocity vi being able to be: - either zero before the beginning of the shock: the object subjected to the shock, initially at rest, comes under the effect of the impulse a velocity vf = AV; - or arbitrary nonzero: the specimen has a velocity which varies during the shock duration T from a value Vi to a value vf for t = T; it is then said that there is impact NOTE: This refers mostly to shocks obtained on shock machines. This classification can be open to confusion insofar as the shocks can be carried out on exciters with a pre-shock and/or post-shock which communicates to the carriage (table, fixture and test item) a velocity before the application of the shock itself (we will see in Chapter 7 the need for a pre-shock and/or a post-shock to cancel the table velocity at the end of movement). 5.2.2.2. Impulse mode Since Vi = 0

Kinematics of simple shocks

125

The velocity increases without changing sign from 0 to vf. In the interval (0, T), the displacement is thus at a maximum for t = T:

It is the area under the curve v(t) in (0, T). Equations [5.4], [5.10] and [5.11] describe the three curves x(t), v(t) and x(t) in this interval.

126

Mechanical shock Table 5.1. Kinematics of a half-sine shock generated by an impulse

Velocity

Acceleration

Maximum at

Displacement

Maximum at

Maximum at

Zero slope at

Zero slope at equal to

Inflection point at

and for

Inflection point at

5.2.2.3. Impact mode General case The initial velocity Vi is arbitrary, zero here. The body subjected to the shock arrives on the target with the velocity v i ,touches the target (which has a programmer intended to shape the acceleration x(t) according to a half-sine) between time t = 0 and t = T. Several cases can arise. At time t = t at the end of the shock, the velocity vf can be: - either zero (no rebound);

Kinematics of simple shocks

127

-or arbitrary, different from zero. It is said there is rebound with velocity V R (= v f ). We suppose that the movement is carried out along only one axis, the velocity having a different direction from the velocity of impact. The velocity change is equal, in absolute terms, to AV = VR - Vi . The most general case is where VR is arbitrary:

with

coefficient of restitution). The velocity change AV, equal to makes it possible to calculate Vi:

i.e., in algebraic value, and by definition

The velocity v(t), given by [5.6], is thus written:

i.e., since QT = n:

and the displacement:

To facilitate the study, we will consider some particular cases where the rebound velocity is zero, where it is equal (and opposite) to the impact velocity and finally Vi

where it is equal to -—. 2

128

Mechanical shock

Impact without rebound The rebound velocity is zero (a = 0). The mobile arrives on the target with velocity Vi at time t = 0, undergoes the shock x(t) for time i and stops at t = T.

and

The maximum displacement xm throughout the shock takes place here also for t = t since v(t) passes from -v4 to 0 continuously, without a change in sign. Moreover, since VR = 0, x remains equal to xm for t > t.

This value of x(t) is equal to the area under the curve v(t) delimited by the curve (between 0 and T) and the two axes of coordinates.

Kinematics of simple shocks

129

Table 5.2. Kinematics of a half-sine shock carried out by impact without rebound

Acceleration

Maximum at

Velocity

Displacement^

Maximum at

Zero at

Zero slope when

Inflection point at

and

Zero slope at

and equal

Inflection point at

Velocity of rebound equal and opposite to the velocity of impact (perfect rebound) After impact, the specimen sets out again in the opposite direction with a velocity equal to the initial velocity (a = 1 and VR = -Vi). It then becomes

and

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Mechanical shock

The velocity varies from Vj to VR = -Vj when t varies from 0 to T. Let us take again the general expressions [5.18] and [5.19] for v(t) and x(t) and set a = 1:

and, since x = 0 with t = 0 by assumption

Table 5.3. Kinematics of a half-sine shock carried out by impact with perfect rebound

Acceleration

Velocity

Displacement

Kinematics of simple shocks

131

dx

The displacement is maximum for t = tm

corresponding to — = 0, so that dt

T

If K = 0, tm = —. The maximum displacement xm thus has as a value 2

In the case of a perfect rebound (VR = -Vi), the amplitude xm of the displacement is smaller by a factor n than if VR = 0. It is pointed out that the amplitude xm is none other than the area ranging between the curve v(t) and the T

two axes of coordinates, in the time interval (0, —): 2

Velocity of rebound equal and opposed to half of the impact velocity 1 In this case, a = —. The mobile arrives at the programmer with a velocity vi5 2 meets it at time t = 0, undergoes the impact for the length of time T, rebounds and v i sets out again in the opposite direction with a velocity VR = -—: 2

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Mechanical shock

Let us set a = — in the general expressions [5.18] and [5.19] of v(t) and x(t); it 2 then becomes

and

The maximum displacement takes place when v(t) = 0, i.e. when t = tm such that

We will take, in (0, T),

This value of

yielding

lies between the two values

The hatched area under the curve v(t) is equal to xm.

and

Kinematics of simple shocks

133

Table 5.4. Kinematics of a half-sine shock caused by impact with 50% rebound velocity

Velocity

Acceleration

Maximum at

Displacement

Maximum at

Zero at

Zero slope when

and

Zero slope at equal to

in

and to

in

Inflection point at

Summary chart — remarks on the general case of an arbitrary rebound velocity

All these results are brought together in Table 5.5, One can note that: - the maximum displacements required in the case of impulse and the case of impact without rebound are equal; -the maximum displacement in the case of a 100% rebound velocity is smaller by a factor TC; the energy spent by the corresponding shock machine will thus be smaller [WHI].

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Mechanical shock

Locus of the maxima The velocity of rebound is, in the general case, a fraction of the velocity of impact:

However

or

and

or, since

x(t) is at a maximum when v(t) = 0, i.e. when is positive when 0 < t < t and since

for

Thus

The locus of maxima, given by the parametric representation t m (a), xm (a), can be expressed according to a relation x m (t m ) while eliminating a between the two relations:

Kinematics of simple shocks

135

The locus of the maxima is an arc of the curve representative of this function in T

the interval — < tm < T . 2

Table 5.5. Summary of the conditions for the realization of a half-sine shock

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

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Mechanical shock

5.3. Versed-sine pulse 5.3.1. Definition The versed-sine* (or haversine** ) shape consists of an arc of sinusoid ranging between two successive minima.

Figure 5.3. Haversine shock pulse

It can be represented by

for elsewhere Generalized form

We set here

One minus Cosine One half of one minus Cosine

Reduced form

for

for

elsewhere

elsewhere

Kinematics of simple shocks a General expressions

(it is supposed that x(o) = 0). Impulse mode

v

Table 5.6. Velocity and displacement for carrying out a versed-sine shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

137

138

Mechanical shock

(by preserving the notation VR = -a vi). Table 5.6 gives the expressions for the velocity and the displacement using the same assumptions as for the half-sine pulse.

Table 5.7. Summary of the conditions for the realization of a haversine shock pulse

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

Kinematics of simple shocks

5.4. Rectangular pulse 5.4.1. Definition

Figure 5.4. Rectangular shock pulse

for elsewhere

Generalized form

for elsewhere Reduced form

for elsewhere

5.4.2. Shock motion study General expressions

139

140

Mechanical shock

Impact

Impulse

Table 5.8. Velocity and displacement: rectangular shock pulse

Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

Kinematics of simple shocks

141

Table 5.9. Summary of the conditions for the realization of a rectangular shock pulse

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

142

Mechanical shock

5.5. Terminal peak saw tooth pulse 5.5.1. Definition

Figure 5.5. Terminal peak saw tooth pulse

for elsewhere Generalized form

for elsewhere Reduced form

for elsewhere

Kinematics of simple shocks 5.5.2. Shock motion study General expressions

Impulse

Impact

Table 5.110. Velocity and displacement to carry out a TPS shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

143

144

Mechanical shock Table 5.11. Summary of the conditions for the realization of a TPS shock

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

5.6. Initial peak saw tooth pulse 5.6.1. Definition

Figure 5.6. IPS shock pulse

for elsewhere Generalized form

for elsewhere Reduced form

for elsewhere

5.6.2. Shock motion study General expressions

146

Mechanical shock

Impact

Impulse

Table 5.12. )Velocity and displacement needed to carry out an IPS shock pulse Velocity Impact without rebound Impact with perfect rebound Impact with 50% rebound

Displacement

Kinematics of simple shocks

147

Table 5.13. Summary of the conditions for the realization of an IPS shock

Impulse

Impact without rebound

Impact with perfect rebound

Impact with rebound to 50% of the initial velocity

Whatever the shape of the shock, perfect rebound leads to the smallest displacement (and to the lowest drop height). With traditional shock machines, this

148

Mechanical shock

cannot be really exploited, since one is not able to choose the kinematics of the shock.

Chapter 6

Standard shock machines

6.1. Main types The first specific machines developed at the time of World War II belong to two categories: -Pendular type machines, equipped with a hammer which, after falling in a circular motion, strike a steel plate which is fixed to the specimen (high-impact machine) [CON 51] [CON 52] [VIG 61a]. The first of these machines was manufactured in England in 1939 to test the light equipment which was subjected, on naval ships, to shocks produced by underwater explosions (mines, torpedoes). Several models were developed in the United States and in Europe to produce shocks on equipment of more substantial mass. These machines are still used (cf. Figure 6.1). - Sand drop machines are made up of a table sliding on two vertical guide columns and free falling into a sand box, characteristics of the shock obtained being a function of the shape and the number of wooden wedges fixed under the table, as well as the granularity of sand (cf. Figure 6.2) [BRO 61] [LAZ 67] [VIG 61b]. NOTES: An alternative to this machine which was used simply comprised a wooden table supporting the specimen, under which a series of wooden wedges was fixed. The table was released from a given height, without guidance, and impacted the sand in the box.

150

Mechanical shock

Figure 6.1. Sand-drop shock testing machine

Figure 6.2. Sand-drop impact simulator

Standard shock machines

151

The test facilities now used are classified as follows: -Free fall machines, derived from the sand-drop machines, the impact being made on a programmer adapted to the shape of the specified shock (elastomer discs, conical or cylindrical lead pellets, pneumatic programmers etc). To increase the impact velocity, which is limited by the drop height, i.e. by the height of the guide columns, the fall can be accelerated by the use of bungee cords. - Pneumatic machines, the velocity being derived from a pneumatic actuator. - Electrodynamic exciters, the shock being specified either by the shape of a temporal signal, its amplitude and its duration, or by a shock response spectrum. - Exotic machines, designed to carry out non-realizable shocks by the preceding methods, generally because their amplitude and duration characteristics are not compatible with the performances from these means, because the desired shapes, not being normal, are not possible with the programmers delivered by the manufacturers. A shock machine, whatever its standard, is primarily a device allowing modification over a short time period of the velocity of the material to be tested. Two principal categories are usually distinguished: - "impulse" machines, which increase the velocity of the test item during the shock. The initial velocity is in general zero. The air gun, which creates the shock during the setting of velocity in the tube, is an example; - "impact" machines, which decrease the velocity of the test item throughout the shock and/or which change its direction. 6.2. Impact shock machines Most machines with free or accelerated drops belong to this last category. The machine itself allows the setting of velocity of the test item. The shock is carried out by impact on a programmer which formats the acceleration of braking according to the desired shape. The impact can be without rebound when the velocity is zero at the end of the shock, or with rebound when the velocity changes sign during the movement. The laboratory machines of this type consists of two vertical guide rods on which the table carrying test item (Figure 6.3) slides. The impact velocity is obtained by gravity, after the dropping of the table from a certain height or using bungee cords allowing one to obtain a larger impact velocity.

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Mechanical shock

Figure 6.3. Elements of a shock test machine

In all cases, whatever the method for realization of the shock, it is useful to consider the complete movement of the test item between the moment when its velocity starts to take a nonzero value and that where it again becomes equal to zero. One thus always observes the presence of a pre-shock and/or a post-shock. Let us consider a free fall shock machine for which the friction of the shock table on the guidance system can be neglected. The necessary drop height to obtain the desired impact velocity vi' is given by:

if M = the mass of the moving assembly of the machine (table, fixture, programmer) m = mass of the test item g = acceleration of gravity (9.81 m/s2). yielding

Standard shock machines

153

These machines are limited by the possible drop height, i.e. by the height of the columns and the height of the test item when the machine is provided with a gantry. It is difficult to increase the height of the machine due to overcrowding and problems with guiding the table. One can increase, however, the impact velocity using a force complementary to gravity by means of bungee cords tended before the test and exerting a force generally directed downwards. The acceleration produced by the cords is in general much higher than gravity which then becomes negligible. This idea was used to design horizontal [LON 63] or vertical machines [LAV 69] [MAR 65], this last configuration being less cumbersome.

Figure 6.4. Use of elastic cords

The Collins machine is an example. Its principle of operation is illustrated in Figure 6.4. The table is guided by two vertical columns in order to ensure a good position of the test item at impact. When the carriage is accelerated by elastic cords, the force applied to the table is due to gravity and to the action of these cords. One has then, if Th is the tension of the elastic cord at the instant of dropping and Tj the tension of the cord at the time of the impact:

154 Mechanical shock

(neglecting the kinetic energy of the elastic cords).

Figure 6.5. Principle of operation using elastic cords

Figure 6.6. Principle behindpendular shock test machine

If machine is of the pendular type, the impact velocity is obtained from

i.e.

where L is the length of the arm of the pendulum and a is the angle of drop.

Standard shock machines

155

During impact, the velocity of the table changes quickly and forces of great amplitude appear between the table and machine bases. To generate a shock of a given shape, it is necessary to control the amplitude of the force throughout the stroke during its velocity change. This is carried out using a shock programmer. Universal shock test machine

Figure 6.7. MRL universal shock test machine (impact mode)

Figure 6.8. MRL universal shock test machine (impulse mode)

The MRL Company (Monterey Research Laboratory) markets a machine allowing the carrying out of shocks according to two modes: impulse and impact [BRE 66]. In the two test configurations, the test item is installed on the upper face of the table. The table is guided by two rods which are fixed at a vertical frame.

156

Mechanical shock

To carry out a test according to the impact mode (general case), one raises the table by the height required by means of a hoist attached to the top of the frame, by the intermediary assembly for raising and dropping (Figure 6.7). By opening the blocking system in a high position, the table falls under the effect of gravity or owing to the relaxation of elastic cords if the fall is accelerated. After rebound on the programmer, the table is again blocked to avoid a second impact. NOTE: A specific device has been developed in order to make it possible to test relatively small specimens with very short duration high acceleration pulses (up to 100,000 g, 0.05 ms) on shock machines which would not otherwise be capable of generating these pulses. This shock amplifier ("Dual mass shock amplifier", marketed by MRL) consists of a secondary shock table (receiving the specimen) and a massive base which is bolted to the top of the carriage of the shock machine. When the main table impacts and rebounds from the programmer on the base of the machine (shock duration of about 6 ms), the secondary table, initially maintained above its base by elastic shock cords, continues downward, stretching the shock cords. The secondary table impacts on an high density felt programmer placed at the base of the shock amplifier, the generating the high acceleration shock. The impulse mode shocks (Figure 6.8) are obtained while placing the table on the piston of the programmer (used for the realization of initial peak saw tooth shock pulses). The piston of this hydropneumatic programmer propels the table upward according to an appropriate force profile to produce the specified acceleration signal. The table is stopped in its stroke to prevent its falling down a second time on the programmer. Pre- and post-shocks The realization of shocks on free or accelerated fall machines imposes de facto pre- shocks and/or post-shocks, the existence of which the user is not always aware, but which can modify the shock severity at low frequencies (Section 7.6). The movement of shock starts with dropping the table from the necessary height to produce the specified shock and finishes with stopping the table after rebound on the programmer. The pre-shock takes place during the fall of the table, the post-shock during its rebound. Freefall Let us set a as the rate of rebound (coefficient of restitution) of the programmer. If AV is the velocity change necessary to carry out the specified shock (AV = J x(t) dt), the carriage rebound velocity and the carriage impact velocity are

Standard shock machines es

related by [5.15] AV = v R - v i and v, =

AV

157

. One deduces from this the

1 + ct necessary drop height

where g = acceleration due to gravity.

Figure 6.9. Movement of the table

The movement of the table of the machine since the moment of its realease until impact is given by

yielding, at impact, the instant of time

where tjis the duration of the pre-shock, which has as an amplitude -g. Since the rebound velocity is equal, in absolute terms, to VR = a vi? the rebound of the carriage assembly occurs until a height HR is reached so that

158

Mechanical shock

and it lasts

The whole of the movement thus has the characteristics summarized in Figure 6.10.

Figure 6.10. Shock performed

A cceleratedfall Let us set m as the total impacting mass (table + fixture + test item), and k as the stiffness of the elastic cords.

Figure 6.11. Movement of the table during accelerated fall

Standard shock machines The differential equation of the movement

has as a solution

where

yielding

At impact, z = 0 and t = t; such that:

In addition

The impact velocity is equal to

yielding

and the duration of the pre-shock is

159

160

Mechanical shock

After the shock, the rebound is carried out with velocity VR = a v, . We have in the same way

and

6.3. High impact shock machines 6.3.1. Lightweight high impact shock machine This machine was developed in 1939 to simulate the effects of underwater explosions (mines) on the equipment onboard military ships. Such explosions, which occur basically large distances from the ships, create shocks which are propagated in all the structures. The high impact shock machine was reproduced in the United States in 1940 for use with light equipment; a third machine was built in 1942 for heavier equipment of masses ranging between 100 and 2500 kg [VIG 6la] (Section 6.3.2).

Standard shock machines

161

Figure 6.12. High impact shock machine for lightweight equipment

The procedure consisted not of specifying a shock response spectrum or a simple shape shock, but rather of the machine being used, the method of assembly, the adjustment of the machine etc. The machine consists of a welded frame of standard steel sections, of two hammers, one sliding vertically, the other describing an arc of a circle in a vertical plane, according to a pendular motion (Figure 6.12). A target plate carrying the test item can be placed to receive one or the other of the hammers. The combination of the two movements and the two positions of the target makes it possible to deliver shocks according to three perpendicular directions without disassembling the test item. Each hammer weighs approximately 200 kg and can fall a maximum height of 1.50 m [CON 52]. The target is a plate of steel of 86 cm x 122 cm x 1.6 cm, reinforced and stiffened on its back face by I-beams. In each of the three impact positions of the hammer, the target plate is assembled on springs in order to absorb the energy of the hammer with a limited displacement (38 mm to the maximum). Rebound of the hammer is prevented. Several intermediate standardized plates simulate various conditions of assembly of the equipment on board. These plates are inserted between the target and the equipment tested to provide certain insulation at the time of impact and to restore a shock considered comparable with the real shock.

162

Mechanical shock

The mass of the equipment tested on this machine should not exceed 100 kg. For fixed test conditions (direction of impact, equipment mass, intermediate plate), the shape of the shock obtained is not very sensitive to the drop height. The duration of the produced shocks is about 1ms and the amplitudes range between 5000 and 10000 m/s5. 6.3.2. Medium weight high impact shock machine This machine was designed to test equipment whose mass, including the fixure, is less than 2500 kg (Figure 6.13). It consists of a hammer weighing 1360 kg which swings through an arc of a circle at an angle greater than 180° and coming to strike an anvil at its lower face. Under the impact, this anvil, fixed under the table carrying the test item, moves vertically upwards. The movement of this unit is limited to approximately 8 cm at the top and 4 cm at the bottom ([CON 51], [LAZ67], [VIG 47] and [VIG 61b] by stops which stop it and reverse its movement. The equipment being tested is fixed on the table via a group of steel channel beams (and not directly to the rigid anvil structure), so that the natural frequency of the test item on this support metal structure is about 60 Hz. The shocks obtained are similar to those produced with the machine for light equipment. It is difficult to accept a specification which would impose a maximum acceleration. It is easier 'to control' starting from a velocity change, the function drop height of the hammer and total mass of the moving assembly (anvil, fixture and test item) [LAZ 67].

Figure 6.13. High impact machine for medium weight equipment

Standard shock machineses

163

The shocks carried out on all these facilities are not very reproducible, sensitive to the ageing of the machine and the assembly (the results can differ after dismantling and reassembling the equipment on the machine under identical conditions, in particular at high frequencies) [VIG 6la]. These machines can also be used to generate simple shape shocks such as halfsine or T.P.S. pulses [VIG 63], while inserting between the hammer and the anvil carrying the test item either an elastic or plastic material. One thus obtains durations of about 10 ms at 20 ms for the half-sine pulse and 10 ms for the TPS pulse.

6.4. Pneumatic machines Pneumatic machines in general consist of a cylinder separated in two parts by a plate bored to let pass the rod of a piston located lower down (Figure 6.14). The rod crosses the higher cylinder, comes out of the cylinder and supports a table receiving the test item.

Figure 6.14. The principle of pneumatic machines

164

Mechanical shock

The surface of the piston subjected to the pressure is different according to whether it is on the higher face or the lower face, as long as it is supported in the higher position on the Teflon seat [THO 64]. Initially, the moving piston, rod and table rose by filling the lower cylinder (reference pressure). The higher chamber is then inflated to a pressure of approximately five times the reference pressure. When the force exerted on the higher face of the piston exceeds the force induced by the pressure of reference, the piston releases. The useful surface area of the higher face increases quickly and the piston is subjected in a very short time to a significant force exerted towards the bottom. It involves the table which compresses the programmers (elastomers, lead cones etc) placed on the top of the body of the jack. This machine is assembled on four rubber bladders filled with air to uncouple it from the floor of the building. The body of the machine is used as solid mass of reaction. The interest behind this lies in its performance and its compactness.

6.5. Specific test facilities When the impact velocity of standard machines is insufficient, one can use other means to obtain the desired velocity: - Drop testers, equipped for example with two vertical (or inclined) guide cables [LAL 75], [WHI], [WHI 63]. The drop height can reach a few tens of metres. It is wise to make sure that the guidance is correct and in particular, that friction is negligible. It is also desirable to measure the impact velocity (photo-electric cells or any other device). - Gas guns, which initially use the expansion of a gas (often air) under pressure in a tank to propel a projectile carrying the test item towards a target equipped with a programmer fixed at the extremity of a gun on a solid reaction mass [LAZ 67], [LAL 75], [WHI], [WHI 63] and [YAR 65]. One finds the impact mode to be as above. It is necessary that the shock created at the time of the velocity setting in the gun is of low amplitude with regard to the specified shock carried out at the time of the impact. Another operating mode consists of using the phase of the velocity setting to program the specified shock, the projectile then being braked at the end of the gun by a pneumatic device, with a small acceleration with respect to the principal shock. A major disadvantage of guns is related to the difficulty of handling cables instrumentation, which must be wound or unreeled in the gun, in order to follow the movement of the projectile. -Inclined-plane impact testers [LAZ 67], [VIG 61b]. These were especially conceived to simulate shocks undergone during too severe handling operations or in

Standard shock machines

165

trains. They are made up primarily ofa carriage on which the test item is fixed, travelling on an inclined rail and coming to run up against a wooden barrier.

Figure 6.15. Inclined plane impact tester (CONBUR tester)

The shape of the shock can be modified y using elastomeric "bumpers' or springs. Tests of this type are often named 'CONBUR tests'.

6.6. Programmers We will describe only the most-frequently used programmers used to carry out half-sine, terminal peak saw tooth and trapezoid shock pulses. 6.6.1. Half-sine pulse These shocks are obtained using an elastic material interposed between the table and the solid mass reaction.

Shock duration The shock duration is calculated by supposing that the table and the programmer, for this length of time, constitute a linear mass-spring system with only one-degreeof-freedom. The differential equation of the movement can be written

where m = mass of the moving assembly (table + fixture + test item) k = stiffness constant of the programmer i.e.

166

Mechanical shock

The solution of this equation is a sinusoid of period T =

2n

. It is valid only

co0

during the elastomeric material compression and its relaxation, so long as there is contact between the table and the programmer, i.e. during a half-period. If t is the shock duration, we thus have

This expression shows that, theoretically, the duration can be regarded as a function alone of the mass m and of the stiffness of the target. It is in particular independent of the impact velocity. The mass m and the duration i being known, we deduce from it the stiffness constant k of the target:

Maximum deformation of the programmer If Vj is the impact velocity of the table and xm the maximum deformation of the programmer during the shock, it becomes, by equalizing the kinetic loss of energy and the deformation energy during the compression of the programmer

yielding

Shock amplitude k

From [6.25], one has, in absolute terms, m xm = k xm, yielding xm = xm — m

and, according to [6.30]

Standard shock machines

167

where the impact velocity YJ is equal to

with g = acceleration of gravity (1 g = 9.81 m/s2) H = drop height This relation, established theoretically for perfect rebound, remains usable in practice as long as the rebound velocity remains higher than approximately 50% of the impact velocity. Having determined k from m and T, it is enough to act on the impact velocity, i.e. on the drop height, to obtain the required shock amplitude. Characteristics of the target For a cylindrical programmer, we have

where S and L are respectively the cross-section and the height of the programmer and where E is Young's modulus of material in compression. Depending on the materials available, i.e. possible values of E, one chooses the values of L and S which lead to a realizable programmer (by avoiding too large a height to diameter ratio to eliminate the risks from buckling). When the table has a large surface, it is possible to place four programmers to distribute the effort. The cross-section of each programmer is then calculated starting from the value of S determined above and divided by 4. The elasticity modulus which intervenes here is the dynamic modulus, which is in general larger than the static modulus. This divergence is mainly a function of the type of elastomeric material used, although other factors such as the configuration, the deformation and the load can have an effect The ratio dynamic modulus Ed to static modulus Es ranges in general between 1 and 2. It can exceed 2 in certain cases [LAZ 67]. The greatest values of this ratio are observed with most damped materials. For materials such as rubber and Neoprene, it is close to unity.

168

Mechanical shock

Figure 6.16. High frequencies at impact

Figure 6.17. Impact module with conical impact face (open module)

If the surface of impact is plane, a wave created at the time of the impact is propagated in the cylinder and makes several up and down excursions. From it at the beginning of the signal the appearance of a high frequency oscillation which distorts the desired half-sine pulse results. To avoid this phenomenon, the front face of the programmer is designed to be slightly conical, in order to insert the load material gradually (open module). The shock thus created is between a half-sine and a versed-sine pulse.

Propagation time of the shock wave So that the target can be regarded a simple spring and not as a system with distributed constants, it is necessary that the propagation time of the shock wave through the target is weak with respect to the duration T of the shock. If a is the velocity of the sound in material constituting the target and h its height, this condition is written:

i.e., since a

(E = Young's modulus, p= density) and

Standard shock machines

If the mass of the target is equal to Mc = hSp necessary then that

169

(S = cross-section), it is

i.e.

Rebound The coefficient of restitution is a function of the material. The smallest rebounds are obtained with the elastic materials that are most strongly damped. The metal springs have small damping and thus produce significant rates of rebound, often about 75%. The elastomers vary greatly, with the rate of rebound which can be located as being between 0 and 75% of the drop height. The coefficient of restitution is also a function of the configuration and of the deformation of elastic material. The targets which are made up of very soft material, presenting great deformations, lead in general to significant rebounds, whereas the elastomeric materials, which are stiff and thin, are calculated to become deformed only by a few hundredths of millimetre, and produce only very little rebound [LAZ 67]. A not very substantial rebound can mean that the material of the programmer reacts during the impact like a viscoelastic material, the table taking a rebound velocity higher than the relaxation velocity of the material [BRO 63]. To create a perfectly half-sine shock pulse with this type of programmer, one needs a perfect rebound, with a rebound velocity equal to the impact velocity. It is necessary thus that damping is zero. The shock pulse obtained under these conditions is symmetrical. When the rate of rebound decreases, the return of acceleration to zero (relaxation) is faster than the rise of acceleration.

170

Mechanical shock

Figure 6.18. Distortsion of the half-sine pulse related to the damping of the material

A good empirical rule is to limit the maximum dynamic deformation of the programmer from 10 to 15% of its initial thickness. If this limit is exceeded, the shape obtained risks non-linear tendencies.

Standard shock machines

171

Example

Realization of a half-sine shock 300 m/s2, 10 ms. It is supposed that the mass of the moving assembly (table + fixture + test item) is equal to 600 kg. The elastomeric programmers often have a coefficient of restitution k (VR = -k Vj) of about 50 %. From [6.28]

The impact velocity is calculated from [6.31]:

2

which leads to the drop height H =

V:

_3

« 47 10

m. During the impact, the 2g elastomeric target will be deformed to a height equal to [6.30]:

The velocity change during the shock is equal to 2 2 ~> AV = - xm t = — 300 10 ^ » 1.91 m/s. It is checked that AV = 2 v{. 7i 7i with L being the height of the target, its diameter D is calculated from ES . u L

f\

")

If the target is an elastomer of hardness 60 Shore, then E » 4 10 N/m yielding, if L = 0.1 m, D « 0.137 m. It remains to check that the stress in material does not exceed the acceptable value.

172

Mechanical shock

NOTES: 1. The relations [6.27] and [6.31] were established by supposing that the material of the target is perfectly elastic and that the rebound is perfect. If it is not the case, these relations give only one approximation of xm and T (or k and vi). In difficult cases, it is undoubtedly quicker to carry out a first test, to measure the values of xm and i obtained, then to correct k and Vj using

i.e., according to the drop height

Index 1 corresponds to the first shock carried out, index 2 with the required shock. These relationship remain usable as long as there is a certain rebound and as long as the shock remains symmetrical. It is unfortunately difficult to maintain the same shape of the shock when one tries to modify its amplitude and its duration. Thus, when a rubber target is deformed by more than 30% approximately its length at rest, its characteristic force-displacement becomes non-linear, which leads to a distortion of the profile of the shock [BRO 63] [WHI] [WHI63]. 2. For a confined material (liquid for example), we have k =

^dv Sp

V (Edv = bulk dynamic modulus, V = volume of the liquid contained and Sp = effective area of the piston compressing the liquid).

The manufacturers provide cylindrical modules made up of an elastomer sandwiched between two metal plates. The programmer is composed of a stacked modules of various stiffnesses (Figure 6.19). It is enough for a relatively low number of different modules to cover a broad range of shock durations by combinations of these elements [BRE 67], [BRO 66a], [BRO 66b] and [GRA 66].

Standard shock machines es

173

Figure 6.19. Distribution of the modules (half-sine shock pulse)

The modules are in general distributed between the bottom of the table and the top of the solid mass of reaction to regularly distribute the load at the time of the shock in the lower part of the table. One thus avoids exciting its bending mode at lower frequency and amplifying the vibrations due to resonance of the table. The programmers for very short duration shock are made up of a high-strength thermoplastic material and with a large numbers of modules. The selected plastic is highly resilient and very hard. It is used within its yield stress and can thus be useful almost indefinitely. Reproducibility is very good. The programmer is composed of a cylinder of this material stuck on a plane circular plate screwed to the lower part of the table of the shock machine.

6.6.2. Terminal peak saw tooth shock pulse Programmers using crushable materials We showed that, at the time of a shock by impact without rebound, the deflection varies according to time according to the law

which can be written, since x(t) = xm — and F(t) = -m x(t),

174

Mechanical shock

To generate a terminal peak saw tooth shock pulse, any target made up of an inelastic material (crushable material) with a curve dynamic deflection-load which follows a cubic law is thus appropriate [WHI]. To obtain a perfect TPS shock pulse, it is necessary that

and, by integration

Knowing that |F = m x(t) = acr S(t), it becomes

where S(t) = surface of the programmer in contact with the table at time t. acr = crush stress of material constituting the target. The law S(t) is thus relatively complicated. If we set SQ =

—, we can write: °cr

and

Standard shock machines

Example Mass of the unit table + fixture + test item: 400 kg Maximum acceleration: 500 m/s2 Shock duration: 10 ms acr = 760 kg/cm2 = 760 104 kg/cm2

yielding

and

Figure 6.20 shows the variations of S(x) with x.

Figure 6.20. Evolution of the impact area according to the crushed length

175

176

Mechanical shock

It is supposed that S = A, x (A, = constant) and we have at time t

yielding

Let us set

This gives

which has as a solution x(t)= x m sinh£2t. Differentiating twice, we have successively x(t)= QxmcoshQt and x(t) = Q x m sinhQt. With this assumption, the rise curve is not perfectly linear. In practice, the approximation is sufficient. So that F(t)or x(t) presents a constant slope, it is thus enough that the cross-section of the programmer increases linearly according to the distance to its top (point of impact), i.e. to define a cone. For t = 0, x = 0 and x = Vj = AV . For t = T, X(T) = xm, yielding

i.e.

These relations make it possible in theory to determine the characteristics of the target. The calculations are however complex, with Q being related to X. Although it is possible to determine by calculation the required load deformation characteristic, according to a particular law of acceleration, it is very difficult to use this information in practice. The difficulty rests in the determination of the form of the programmer and the characteristic of the dynamic crushing to produce a given

Standard shock machines

177

shock. For each machine and each shock, it is necessary to carry out preliminary tests to check that the programmer is well calculated. The programmers are destroyed with each test. It is thus a relatively expensive method. One prefers to use, if possible, a universal programmer (Section 6.6.4). The material generally used is lead or honeycomb. The cones can be calculated as follows: - crushed length:

yielding the height of the cone h > 1.2 xm (to allow material to become deformed to the necessary height); - force maximum:

yielding the cross-section Sm of the cone at height xm:

When all the kinetic energy of the table is dissipated by crushing of lead, acceleration decreases to zero. The shock machine must have a very rigid solid mass of reaction, so that the time of decay to zero is not too long and satisfies the specification. The speed of this decay to zero is a function of the mass of reaction and of the mass of the table: if the solid mass of reaction has a not negligible elasticity, this time, already non-zero because of the imperfections inherent in the programmer, can become too long and unacceptable. For lead, the order of magnitude of acr is 760 kg/cm2 (7.6 107 N/m2 = 76 MPa). The range of possible durations lies between 2 and 20 ms approximately.

Penetration of a steel punch in a lead block Another method of generating a terminal using the penetration of a punch of required lead. The punch is fixed under the table of solid reaction mass. The velocity setting of

peak saw tooth shock pulse consists of form in a deformable material such as the machine, the block of lead on the the table is obtained, for example, by

178

Mechanical shock

free fall [BOC 70], [BRO 66a] and [ROS 70]. The duration and the amplitude of the shock are functions of the impact velocity and the point angle of the cone.

Figure 6.21. Realization of a TPS shock by punching of a lead block

Figure 6.22. Penetration of the steel punch in a lead block

The force which tends to slow down the table during the penetration of the conical punch in the lead is proportional to the greatest section S(x) which is penetrated, at distance x from the point. If cp is the point angle of the cone

yielding, in a simplified way, if m is the total mass of the moving assembly, by equalizing the inertia and braking forces in lead

with a being a constant function of the crush stress of lead (by supposing that only this parameter intervenes and that the other phenomena such as steel-lead friction are negligible). Let us set a

If v is the carriage velocity at the time t and Vj the impact velocity, this relation can be written

Standard shock machines

179

yielding

The constant of integration b is calculated starting from the initial conditions: for x = 0, v = Vi yielding

Let us write [6.45] in the form

it becomes by integration:

If we set

and

we obtain

Acceleration then results from [6.44]:

We have in addition v = vi yi - y . The velocity of the table is cancelled when all its kinetic energy is dissipated by the plastic deformation of lead. Then, y = 1 and

180

Mechanical shock

Knowing that vi = <J2 g H (H = drop height),

the shock duration is not very sensitive to the drop height. From these expressions, we can establish the relations:

and

In addition,

As an indication, this method allows us to carry out shocks of a few hundreds to a few thousands of grams, with durations from 4 to 10 ms approximately (for a mass m equal to 25 kg).

6.6.3. Rectangular pulse - trapezoidal pulse This test is carried out by impact. A cylindrical programmer consists of a material which is crushed with constant force (lead, honeycomb) or using the universal programmer. In the first case, the characteristics of the programmer can be calculated as follows:

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181

- the cross-section is given according to the shock amplitude to be realized using the relation

yielding

-starting from the dynamics of the impact without rebound, the length of crushing is equal to

and that of the programmer must be at least equal to 1.4 xm, in order to allow a correct crushing of the matter with constant force. One can say that the shock amplitude is controlled by the cross-section of the programmer, the crush stress of material and the mass of the total carriage mass. The duration is affected only by the impact velocity. For this pulse shape also, it is possible to use the penetration of a rigid punch in a crushable material such as lead. The two methods produce relatively disturbed signals, because of impact between two plane surfaces. They are adapted only for shocks of short duration, because of the limits of deformation. A long duration indeed requires a plastic deformation over a big length; but it is difficult to maintain constant the force of resistance on such a stroke. The honeycombs lend themselves better to the realization of a long duration [GRA 66]. One could also use the shearing of a lead plate. 6.6.4. Universal shock programmer The MTS Monterey programmer known as universal can be used to produce half-sine, TPS and trapezoidal shock pulses after various adjustments. This programmer consists of a cylinder fixed under the table of the machine, filled with a gas under pressure and in the lower part of a piston, a rod and a head (Figure 6.23).

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Mechanical shock

6.6.4.1. Generating a half-sine shock pulse The chamber is put under sufficient pressure so that, during the shock, the piston cannot move (Figure 6.23). The shock pulse is thus formatted only by the compression of the stacking of elastomeric cylinders (modular programmers) placed under the piston head. One is thus brought back to the case of Section 6.6.1.

Figure 6.23. Universal programmer MTS (half-sine and rectangle pulse configuration)

Figure 6.24. Universal programmer MTS (TPS pulse configuration)

6.6.4.2. Generating a terminal peak saw tooth shock pulse The gas pressure (nitrogen) in the cylinder is selected so that, after compression of elastomer during duration T, the piston, assembled in the cylinder as indicated in Figure 6.24, suddenly is released for a force corresponding to the required maximum acceleration xm. The pressure which was exerted before separation over the whole area of the piston applies only after separation to one area equal to that of the rod, producing a negligible resistant force.

Standard shock machines

183

Acceleration thus passes very quickly from xm to zero. The rise phase is not perfectly linear, but corresponds rather to an arc of versed-sine (since if the pressure were sufficiently strong, one would obtain a versed-sine by compression of the elastomer alone).

Figure 6.25. Realization of a TPS shock pulse

6.6.4.3. Trapezoidal shock pulse The assembly here is the same as that of the half-sine pulse (Figure 6.23). At the time of the impact, there is: - compression of the elastomer until the force exerted on the piston balances the compressive force produced by nitrogen. This phase gives the first part (rise) of the trapezoid; - up and down displacement of the piston in the part of the cylinder of smaller diameter, approximately with constant force (since volume varies little). This phase corresponds to the horizontal part of the trapezoid; - relaxation of elastomer: decay to zero acceleration. The rise and decay parts are not perfectly linear for the same reason as in the case of the TPS pulse. 6.6.4.4. Limitations Limitations of the shock machines The limitations are often represented graphically by straight lines plotted in logarithmic scales delimiting the domain of realizable shocks (amplitude, duration). The shock machine is limited by [IMP]: - the allowable maximum force on the table. To carry out a shock of amplitude xm, the force generated on the table, given by

184

Mechanical shock

must be lower or equal to the acceptable maximum force F,^. Knowing the total carriage mass, the relation [6.55] allows calculation of the possible maximum acceleration under the test conditions:

This limitation is represented on the abacus by a horizontal line xm = constant;

Figure 6.26. Abacus of the limitations of a shock machine

-the maximum free fall height H or the maximum impact velocity, i.e. the velocity change AV of the shock pulse. If VR is the rebound velocity, equal to a percentage a of the impact velocity, we have

yielding

where a is a function of the shape of the shock and of the type of programmer used. In practice, there are losses of energy by friction during the fall and especially in the programmer during the realization of the shock. To take account of these losses is difficult to calculate analytically and so one can set;

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185

[6.58] where P takes into account at the same time losses of energy and rebound. As an example, the manufacturer of machine IMPAC 60 x 60 (MRL) gives, according to the type of programmers [IMP]: Table 6.1. Loss coefficient (3 Programmer

Value of P

Elastomer (half-sine pulse)

0.556

Lead (rectangle pulse)

0.2338 1.544

Lead (TPS pulse)

Figure 6.27. Drop height necessary to obtain a given velocity change

The limitation related to the drop height can be represented by parallel straight lines on a diagram giving the velocity change AV as a function of the drop height in logarithmic scales. The velocity change being, for all simple shocks, proportional to the product xm T , we have

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Mechanical shock

yielding, while setting a =

[6.59] Table 6.2. Amplitude x duration limitation Programmer

Waveform Half-sine TPS Rectangle

(x (m/s) v m m T) 'max

Elastomer

17.7

Lead cone

10.8

Universal programmer

7.0

Universal programmer

9.2

On logarithmic scales (xm,T), the limitation relating to the velocity change is represented by parallel inclined straight lines (Figure 6.26). Limitations of programmers Elastomeric materials are used to generate shocks of -half-sine shape (or versed-sine with a conical frontal module to avoid the presence of high frequencies); - TPS and rectangular shapes, in association with a universal programmer. Elastomer programmers are limited by the allowable maximum force, a function of Young's modulus and their dimensions (Figure 6.26) [JOU 79]. This limitation is in fact related to the need to maintain the stress lower than the yield stress of material, so that the target can be regarded as a pure stiffness. The maximum stress a max developed in the target at the time of the shock can be expressed according to Young's modulus E, to the maximum deformation xm and to the thickness h of the target according to

with, for an impact with perfect rebound, the elastic ultimate stress

. It is necessary that, if Re is

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187

i.e.

Exa mple MR1. IMP AC 60 x 60 shock machine Table 6.3. Examples of the characteristics of half-sine programmers Maximum force (kN) Type

Colour

Hard

Diameter 150.5 mm

Diameter 295 mm

Red

667

2 224

Mean

Blue

445

1201

Soft

Green

111

333

Taking into account the mass of the carriage assembly, this limitation can be transformed into maximum acceleration (Fm = m x m ). Thus, without a load, with a programmer made out of a hard elastomer with diameter 295 mm and a table mass of 3000 g, we have xm « 740 m/s2. With four programmers used simultaneously, maximum acceleration is naturally multiplied by four. This limitation is represented on the abacus of Figure 6.26 by the straight lines of greater slope. The universal programmer is limited [MRL]: - by the acceptable maximum force; -by the stroke of the piston: the relations established in the preceding paragraphs, for each waveform, show that displacement during the shock is always proportional to the product xm T (Figure 6.28).

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Mechanical shock

(

Figure 6.28. Stroke limitation of universal programmers

This information is provided by the manufacturer. In short, the domain of the realizable shock pulses is limited on this diagram by straight lines representative of the following conditions:

Table 6.4. Summary of limitations on the domain of realizable shock pulses xm = constant

Acceptable force on the table or on the universal programmer

xm T = constant

Drop height (AV)

xm I2 = constant

Piston stroke of the universal programmer

xm t4 = constant

Acceptable force for elastomers

Chapter 7

Generation of shocks using shakers

In about the mid 1950s with the development of electrodynamic exciters for the realization of vibration tests, the need for a realization of shocks on this facility was quickly felt. This simulation on a shaker, when possible, indeed presents a certain number of advantages [COT 66].

7.1. Principle behind the generation of a simple shape signal versus time The objective is to carry out on the shaker a shock of simple shape (half-sine, triangle, rectangle etc) of given amplitude and duration similar to that made on the normal shock machines. This technique was mainly developed during the years 1955-1965 [WEL 61]. The transfer function between the electric signal of the control applied to the coil and acceleration to the input of the test item is not constant. It is thus necessary to calculate the signal of control according to this transfer function and the signal to be realized. One of the first methods used consisted in compensating for the system using analogue filters gauged in order to obtain a transfer function equal to H-1 (Q) (if H(Q) is the transfer function of the shaker-test item unit). The compensation must relate at the same time to the amplitude and the phase [SMA 74a]. One of the difficulties of this approach resides in the time and work needed to compensate for the system, with, in addition, a not always satisfactory result obtained.

190

Mechanical shock

The digital methods seemed to be much better. The process is as follows [FAV 69], [MAG 71]: - measurement of the transfer function of the installation (including the fixture and the test item) using a calibration signal; - calculation of the Fourier transform of the signal specified at the input of the test item; - by division of this transform by the transfer function, calculation of the Fourier transform of the signal of control; - calculation of the control signal vs time, by inverse transformation. Transfer function The measurement of the transfer function of the installation can be made using a calibration signal of the shock type, random vibration or sometimes fast swept sine [FAV 74]. In all cases, the procedure consists of measurement and calculation of the signal of control to -n dB (-12, -9, -6 and/or -3). The specified level is applied only after several adjustments on a lower level. These adjustments are necessary because of the sensitivity of the transfer function to the amplitude of the signal (nonlinearities). The development can be carried out using a dummy item representative of the mass of the specimen. However, and in particular if the mass of the test specimen is significant (with respect to that of the moving element), it is definitely preferable to use the real test item or a model with dynamic behaviour very near it. If random vibration is used as the calibration signal, its rms value is calculated in order to be lower than the amplitude of the shock (but not too distant in order to avoid the effects of any non-linearities). This type of signal can result in application to the test item of many substantial peaks of acceleration compared with the shock itself.

7.2. Main advantages of the generation of shock using shakers The realization of the shocks on shakers has very interesting advantages: - possibility of obtaining very diverse shocks shapes; - use of the same means for the tests with vibrations and shocks, without disassembly (saving of time) and with the same fixtures [HAY 63], [WEL 61]; - possibility of a better simulation of the real environment, in particular by direct reproduction of a signal of measured acceleration (or of a given shock spectrum);

Generation of shocks using shakers

191

- better reproducibility than on the traditional shock machines; - very easy realization of the test on two directions of an axis; - saves using a shock machine. In practice, however, one is rather quickly limited by the possibilities of the exciters which therefore do not make it possible to generalize their use for shock simulation.

7.3. Limitations of electrodynamic shakers 7.3.1. Mechanical limitations Electrodynamic shakers have limited performances in the following fields [MIL 64], [MAG 72]: - The maximum stroke of the coil-table unit (according to the machines being used, 25.4 or 50.8 mm peak to peak). Motion study of the coil-table assembly during the usual simple form shocks (half-sine, terminal peak saw tooth, rectangle) show that the displacement is always carried out the same side compared with the equilibrium position (rest) of the coil. It is thus possible to improve the performances for shock generation by shifting this rest position from the central value towards one of the extreme values [CLA 66] [MIL 64] [SMA 73].

Figure 7.1. Displacement of the coil of the shaker

- The maximum velocity [YOU 64]: 1.5 to 2 m/s in sine mode (in shock, one can admit a larger velocity with non-transistorized amplifiers (electronic tubes), because these amplifiers can generally accept a very short overvoltage). During the movement of the moving element in the air-gap of the magnetic coils, there is an electromotive force produced which is opposed to the voltage supply. The velocity must thus have a value such as this emf is lower than the acceptable maximum

192

Mechanical shock

output voltage of the amplifier. The velocity must in addition be zero at the end of the shock movement [GAL 73], [SMA 73]. - Maximum acceleration, related to the maximum force. The limits of velocity, displacement and force are not affected by the mass of the specimen. J.M. McClanahan and J.R. Fagan [CLA 65] consider that the realizable maxima shock levels are approximately 20% below the vibratory limit levels in velocity and in displacement. The majority of the authors agree that the limits in force are, for the shocks, larger than those indicated by the manufacturer (in sine mode). The determination of the maximum force and the maximum velocity is based, in vibration, on considerations of fatigue of the shaker mechanical assembly. Since the number of shocks which the shaker will carry out is very much lower than the number of cycles of vibrations than it will undergo during its life, the parameter maximum force can be, for the shock applications, increased considerably. Another reasoning consist of considering the acceptable maximum force, given by the manufacturer in random vibration mode, expressed by its rms value. Knowing that, one can observe random peaks being able to reach 4.5 times this value (limitation of control system), one can admit the same limitation in shock mode. One finds other values in the literature, such as: - < 4 times the maximum force in sine mode, with the proviso of not exceeding 300 g on the armature assembly [HUG]; - more than 8 times the maximum force in sine mode in certain cases (very short shocks, 0.4 ms for example) [GAL 66]. W.B. Keegan [KEE 73] and D.J. Dinicola [DIN 64] give a factor of about 10 for the shocks of duration lower than 5 ms. The limitation can also be due to: - The resonance of the moving element (a few thousands Hertz). Although it is kept to the maximum by design, the resonance of this element can be excited in the presence of signals with very short rise time. -The resistance of the material. Very great accelerations can involve a separation of the coil of the moving component.

Generation of shocks using shakers

193

7.3.2. Electronic limitations 1. Limitation of the output voltage of the amplifier [SMA 74a] which limits coil velocity. 2. Limitation of the acceptable maximum current in the amplifier, related to the acceptable maximum force (i.e. with acceleration). 3. Limitation of the bandwidth of the amplifier. 4. Limitation in power, which relates to the shock duration (and the maximum displacement) for a given mass. Current transistor amplifiers make it possible to increase the low frequency bandwidth, but do not handle even short overtensions well and thus are limited in mode shock [MIL 64].

7.4. The use of the electrohydraulic shakers Shocks are realizable on the electrohydraulic exciters, but with additional stresses: - contrary to the case of the electrodynamic shakers, one cannot obtain via these means shocks of amplitude larger than realizable accelerations in the steady mode; - the hydraulic vibration machines are in addition strongly non-linear [FAV 74].

7.5. Pre- and post-shocks 7.5.1. Requirements The velocity change

shock duration) associated with

shocks of simple shape (half-sine, rectangle, terminal peak saw tooth etc) is different from zero. At the end of the shock, the velocity of the table of the shaker must however be zero. It is thus necessary to devise a method to satisfy this need. One way of bringing back the variation of velocity associated with the shock to zero can be the addition of a negative acceleration to the principal signal so that the area under the pulse has the same value on the side of positive accelerations and the side of negative accelerations. Various solutions are possible a priori: - a pre-shock alone; - a post-shock alone;

194

Mechanical shock

- pre- and post-shocks, possibly of equal durations.

Figure 7.2. Possibilities for pre- and post-shocks positioning Another parameter is the shape of these pre- and post-shocks, the most used shapes being the triangle, the half-sine and the rectangle.

Figure 7.3. Shapes of pre- and post-shock pulses

Due to discontinuities at the ends of the pulse, the rectangular compensation is seldom satisfactory [SMA 85]. One often prefers a versed-sine applied to all the signal (Hann window) which has the advantages of being zero and smoothed at the ends (first zero derivative) and to present symmetrical pre- and post-shocks. In all the cases, the amplitude of pre- and post-shocks must remain small with respect to that of the principal shock (preferably lower than approximately 10%), in order not to deform too much the temporal signal and consequently, the shock spectrum. For a given shape of pre- and post-shocks, this choice thus imposes the duration.

Generation of shocks using shakers

195

7.5.2. Pre- or post-shock As an example, treated below is the case of a terminal peak saw tooth shock pulse (amplitude unit, duration equal to 1) with rectangular pre- and/or post-shocks (ratio p of the absolute values of the pre- and post-shocks amplitude and of the principal shock amplitude equal to 0.1).

Figure 7.4. Terminal peak saw tooth with rectangular pre- and post-shocks

Figure 7.4 shows the signal as a function of time. The selected parameter is the duration il of the pre-shock.

Figure 7.5. Influence of pre-shock duration on velocity during the shock

It is important to check that the velocity is always zero at the beginning and the end of the shock (Figure 7.5). Between these two limits, the velocity remains positive

196

Mechanical shock

when there is only one post-shock (TJ = 0) and negative for a pre-shock alone (T! = 5.05).

Figure 7.6. Influence of pre-shock duration on displacement during the shock

Figure 7.6 shows the displacement corresponding to this movement for the same values of the duration tj of the pre-shock between TJ = 0 and TJ = 5.05 s. Figure 7.7 shows that the residual displacement at the end of the shock is zero for T! » 2.4 s. The largest displacement during shock, envelope of the residual displacement and of the maximum displacement, is given according to TJ in Figure 7.8 (absolute values). This displacement has a minimum at t1 « 2s.

Figure 7.7. Influence of the pre-shock duration on the residual displacement

Figure 7.8. Influence of the pre-shock duration on the maximum displacement

Generation of shocks using shakers

197

Figure 7.9. TPS pulse with pre-shock alone (1) and post-shock alone (2)

If we compare the kinematics of the movements now corresponding to the realization of a TPS shock with only one pre-shock (1) and only one post-shock (2), we note, from Figures 7.9 to 7.11, that [YOU 64]: - the peak amplitude of the velocity is (in absolute value) identical; - in (2), the acceleration peak takes place when the velocity is very large. It is thus necessary to be able to provide the maximum force when the velocity is significant [MIL 64]; - in (1) to the contrary, the velocity is at a maximum when acceleration is zero.

Figure 7.10. Velocity curve with pre-shock alone (1) and post-shock alone (2)

Figure 7.11. Displacement curve with preshock alone (I) and post-shock alone (2)

Solution (1), which requires a less powerful power amplifier, thus seems preferable to (2). The use of symmetrical pre- and post-shocks is however better, because of a certain number of additional advantages [MAG 72]:

198

Mechanical shock

- the final displacement is minimal. If the specified shock is symmetrical (with respect to the vertical line

this residual displacement is zero [YOU 64];

- for the same duration x of the specified shock and for the same value of maximum velocity, the possible maximum level of acceleration is twice as big; - the maximum force is provided at the moment when acceleration is maximum, i.e. when the velocity is zero (one will be able to thus have the maximum current). The solution with symmetrical pre-post-shocks requires minimal electric power.

Figure 7.12. Kinematics of the movement with pre-shock alone (I), symmetrical preand post-shocks (2) and post-shock alone (3)

7.5.3. Kinematics of the movement for symmetrical pre- and post-shock 7.5.3.1. Half-sine pulse Half-sine pulse with half-sine pre- and post-shocks Duration of pre- and post-shocks [LAL 83]:

Generation of shocks using shakers

199

Figure 7.13. Half-sine -with half-sine symmetrical pre- and post-shocks The following relations give the expressions of the acceleration, the velocity and the displacement as a function of time in each interval of definition of the signal.

Velocity

Displacement

200

Mechanical shock

Half-sine pulse with triangular pre- and post-shocks

Figure 7.14. Half-sine with triangular symmetrical pre- and post-shocks

Duration of pre- and post-shocks

Generation of shocks using shakers

201

Rise time of pre-shock

(by supposing that the slope of the segment joining the top of the triangle to the foot of the half-sine is equal to the slope of the half-sine at its origin).

202

Mechanical shock

Generation of shocks using shakers Half-sine pulse with rectangular pre- and post-shocks Duration of pre- and post-shocks

Figure 7.15. Half-sine with rectangular symmetrical pre- and post-shocks

203

204

Mechanical shock

Generation of shocks using shakers

205

[7.43]

[7.44]

The expressions of the largest velocity during the movement and those of the maximum and residual displacements are brought together in Table 7.1.

Table 7.1. Half-sine ~ maximum velocity and displacement - residual displacement Symmetrical pre- and post-shocks Half-sine

Maximum velocity

Maximum displacement

Residual displacement

Half-sine

Triangles

Rectangles

Similar expressions can be established for the other shock shapes (TPS, rectangle, IPS pulses). The results appear in Tables 7.2 to 7.4.

206

Mechanical shock

7.5.3.2. TPS pulse Table 7.2. TPS pulse - maximum velocity and displacement — residual displacement Symmetrical pre- and post-shocks Half-sine

Triangles

Rectangles

TPS

Durations of pre- and postshocks

Maximum velocity Maximum displacement Residual displacement

1. TJ is the total duration of the pre-shock (or post-shock if they are equal). 12 is tne duration of the first part of the pre-shock when it is composed of two straight-line segments (or of the last part of the post-shock). 13 is the total duration of the post-shock when it is different from Tj.

Generation of shocks using shakers

207

7.5.3.3. Rectangular pulse Table 7.3. Rectangular pulse — maximum velocity and displacement - residual displacement Symmetrical pre- and post-shocks Half-sine Rectangle

Durations of pre-and postshocks Maximum velocity Maximum displacement Residual displacement

Triangles

Rectangles

208

Mechanical shock

7.5.3.4. IPS pulse Table 7.4. IPS pulse - maximum velocity and displacement - residual displacement Symmetrical pre and post-shocks Half-sine

Triangles

Rectangles

IPS

Durations of pre- and postshocks

Maximum velocity Maximum displacement Residual displacement

7.5.4. Kinematics of the movement for a pre-shock or a post-shock alone In the case of a pre-shock or a post-shock alone, the maximum velocity, equal to the velocity change AV related to the shock, takes place at the time of transition between the compensation signal and the shock itself. The displacement starts from zero and reaches its largest value at the end of the movement (without changing sign). Like the velocity, it is negative with a pre-shock and positive with a postshock. Tables 7.5-7.8 bring together the expressions for this displacement according to the shape of the principal shock and of that of the compensation signal, with the same notations and conventions as those in the preceding paragraphs.

Generation of shocks using shakers

209

Table 7.5. Half-sine withpre- or post-shock only - maximum velocity and displacement — residual displacement Pre-shock or post-shock only Half-sine

Half-sine

Triangle

Duration of pre-shock or post-shock

Maximum velocity Residual displacement

Pre-shock

Post-shock:

Rectangle

210

Mechanical shock Table 7.6. TPS with pre- or post-shock only - maximum velocity and displacement - residual displacement Pre-shock or post-shock only TPS

Half-sine

Rectangle

Triangle

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:

Duration of pre-shock or post-shock

Maximum velocity

Post-shock:

Pre-shock: Pre-shock:

Residual displacement

Pre-shock:

Post-shock: Post-shock:

Pre-shock:

Post-shock:

Generation of shocks using shakers Table 7.7. Rectangular pulse withpre- or post-shock only — maximum velocity and displacement — residual displacement Pre-shock or post-shock only Rectangle

Half-sine

Triangle

Duration of e pre-shock or post-shock Maximum velocity Residual displacement

Pre-shock:

Post-shock:

Rectangle

211

212

Mechanical shock Table 7.8. IPS pulse with pre- or post-shock only - maximum velocity and displacement — residual displacement Pre-shock or post-shock only IPS

Half-sine

Duration of pre-shock or post-shock

Maximum velocity

Residual displacement

Rectangle

Triangle

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:

Pre-shock:

Post-shock:

Pre-shock:

Pre-shock:

Post-shock:

Post-shock:

Pre-shock:

Post-shock:

7.5.5. Abacuses For a given shock and for given pre- and post-shocks shapes, we can calculate, by integration of the expressions of the acceleration, the velocity and the displacement as a function of time, as well as the maximum values of these parameters, in order to compare them with the characteristics of the facilities. This work was carried out for pre- and post-shocks - respectively half-sine, triangular and rectangular [LAL 83] in order to establish abacuses allowing quick

evaluation of the possibility of realization of a specified shock on a given test facility (characterized by its limits of velocity and of displacement). These abacuses are made up of straight line segments on logarithmic scales (Figure 7.16): -AA', corresponding to the limitation of velocity: the condition vm < VL (V L = acceptable maximum velocity on the facility considered) results in a relationship of the form xmt < constant (independent of p); - CC, DD, etc, greater slope corresponding to the limitation in displacement for various values of p (p = 0.05, 0.10, 0.25, 0.50 and 1.00). A particular shock will be thus realizable on shaker only if the point of coordinates T, xm (duration and amplitude of the shock considered) is located under these lines, this useful domain increasing when p increases.

Figure 7.16. Abacus of the realization domain of a shock

7.5.6. Influence of the shape of pre- and post-pulses The analysis of the velocity and the displacement varying with time associated with some simple shape shocks shows that [LAL 83]: - For all the shocks having a vertical axis of symmetry, the residual displacement is zero. - For a shock of given amplitude, duration and shape, the maximum displacement during movement is most important with half-sine pre- and postshocks. It is weaker in the case of the triangles, then rectangles.

214

Mechanical shock

- Triangular pre-post-shocks lead to the greatest duration of signal, the rectangle giving the smallest duration. Under these criteria of displacement and duration, it is thus preferable to use rectangular or triangular pre- and post-shocks. The rectangle has however the disadvantage of having slope discontinuities which make its reproduction difficult and which in addition can excite resonances at high frequencies. It seems, however, interesting to try to approach this form [MIL 64]. - The maximum displacement decreases, as one might expect, when p increases. It seems, however, hazardous to retain values higher than 0.10 (although possible with certain control systems), the total shock communicated to the specimen being then too deformed compared with the specification, which results in response spectra appreciably different from those of the pure shocks [FRA 77].

Example Half-sine shock with half-sine symmetrical pre- and post-shocks. Electrodynamic shaker

Figure 7.17. Half-sine pulse with half-sine symmetrical pre- and post-shocks

Generation of shocks using shakers

Figure 7.18. Acceleration, velocity and displacement during the shock of Figure 7.17

Figure 7.19. Abacus for a half-sine shock with half-sine pre- and post-shocks

215

216

Mechanical shock

Figure 7.20. Comparison between maximum displacements obtained with the typical shape of pre- and post-shocks

7.5.7. Optimized pre- and post-shocks At the time of the realization of a shock on shaker, the displacement starts from the equilibrium position, passes through a maximum, then returns to the initial position. One uses in fact only half of the available stroke. For better use of the capacities of the machine, we saw (Figure 7.1) that it is possible to shift the zero position of the table. Another method was developed [FAN 81] in order to fulfill the following objectives: - to take into account the tolerances on the shape of the signal allowed by the standards (R.T. Fandrich refers to standard MIL-STD 8IOC); - to best use the possibilities of the shaker. The solution suggested consists of defining: 1. A pre-shock made up of the first two terms of the development in a Fourier series of a rectangular pulse (with coefficients modified after a parametric analysis), of the form:

Generation of shocks using shakers

217

Figure 7.21. Optimizedpre-shock

The rectangular shape is preferred for the reasons already mentioned, the choice of only the first two terms of the development in series being intended to avoid the disadvantages related to slope discontinuities. The pre-shock consists of one period of this signal, each half-period having a different amplitude: - positive arch:

- negative arch:

where xm is the amplitude of the shock to be realized (in m/s2) and f is the fundamental frequency of the signal, estimated from the relationship [7.45] where g = 9.81 m/s2. This expression is calculated by setting the maximum displacement during the pre-shock lower than the possible maximum displacement on the shaker (for example 1.27 cm). This maximum displacement takes place at the end of the first arch, comparable at first approximation with a rectangle. duration, the maximum displacement is equal to yielding, if f

is its

218 if

Mechanical shock m

The total duration of the pre-shock is thus equal to

The factor of 0.05 corresponds to the tolerance limit of the quoted standard before the principal shock (5%). The constant 0.24 is the reduced amplitude of the first arch, the real amplitude for a shock of maximum value xm being equal to 0.24 (0.046 x m ). The second arch has unit amplitude. The table being, before the test, in equilibrium in a median position, the objective of this pre-shock is two fold: - to give to the velocity, just before the principal shock, a value close to one of the two limits of the shaker, so that during the shock, the velocity can use all the range of variation permitted by the machine (Figure 7.22);

Figure 7.22. Velocity during the optimizedpre-shock and the shock

Figure 7.23. Acceleration, velocity and displacement during the pre-shock

Generation of shocks using shakers

219

- to place, in the same way, the table as close as possible to one of the thrusts so that the moving element can move during the shock in all the space between the two thrusts (limitation in displacement equal, according to the machines, to 2.54 or 5.08 cm).

Figure 7.24. Acceleration, velocity and displacement during thepre-shock and the shock (half-sine) 2. A post-shock composed of one period of a signal of the shape K ty sin(2 7i fj t) where the constants K, y and fj are evaluated in order to cancel the acceleration, the velocity and the displacement at the end of the movement of the table.

Figure 7.25. Overall movement for a half-sine shock

220

Mechanical shock

The frequency and the exponent are selected in order to respect the ratio of the velocity to the displacement at the end of the principal shock. The amplitude of the post-shock is adjusted to obtain the desired velocity change. Figure 7.25 shows the total signal obtained in the case of a principal shock halfsine 30 g, 11 ms. This methodology has been improved to provide a more general solution [LAX 01]. 7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General The specification of shock is in general expressed in the form of a signal varying with time (half-sine, triangle etc). We saw the need for an addition of pre- and/or post-shock to cancel the velocity at the end of the shock when it is carried out on an electrodynamic shaker. There is no difference in principle between the realization of a shock by impact after free or accelerated fall and the realization of a shock on a shaker. On a shock machine of the impact type, the test item and the table have zero velocity at the beginning of the test. The free or accelerated fall corresponds to the pre-shock phase. The rebound, if it exists, corresponds to the post-shock. The practical difference between the two methods lies in the characteristics of shape, duration and amplitude of pre- and post-shocks. In the case of impact, the duration of these signals is in general longer than in the case of the shocks on the shaker, so that the influence on the response appears for systems of lower natural frequency.

7.6.2. Influence of the pre- and post-shocks on the time history response of a onedegree-of-freedom system To highlight the problems, we treated the case of a specification which can be realized on a shaker or on a drop table, applied to a material protected by a suspension with a 5 Hz natural frequency and with a Q factor equal to 10.

Nominal shock - half-sine pulse

ms.

Generation of shocks using shakers 221

Shock on shaker - identical pre-shock and post-shock; - half-sine shape; - amplitu - duration such that:

Shock by impact -freefall

- shock with rebound to 5 - velocity of impact: - velocity of rebou

- drop height

222

Mechanical shock The duration of the fall is tj where

Duration of the rebound

Figure 7.26. Influence of the realization mode of a half-sine shock on the response of a one-degree-of-freedom system

Figure 7.26 shows the response a>0 z(t) of a one-degree-of-freedom system ( f 0 = 5 H z , £ = 0.05): - for z0 = z0 = 0 (conditions of the response spectrum); - in the case of a shock with impact; - in the case of a shock on shaker. We observed in this example the differences between the theoretical response at 5 Hz and the responses actually obtained on the shaker and shock machine. According to the test facility used, the shock applied can under-test or over-test the

Generation of shocks using shakers

223

material. For the estimate of shock severity one must take account of the whole of the signal of acceleration.

7.6.3. Incidence on the shock response spectra In Figure 7.27, for £ = 0.05 , is the response spectrum of: -the nominal shock, calculated under the usual conditions of the spectra (z0 = z0 = 0); - the realizable shock on shaker, with its pre- and post-shocks, - the realizable shock by impact, taking of account of the fall and rebound phases.

Figure 7.27. Influence of the realization mode of a half-sine shock on the SRS One notes in this example that for: - f<} < 10 Hz, the spectrum of the shock by impact is lower than the nominal spectrum, but higher than the spectrum of the shock on the shaker, - 10 Hz < f0 < 30 Hz, the spectrum of the shock on the shaker is much overestimated, - f0 > 30 Hz, all the spectra are superimposed. This result appears logical when we remember that the slope of the shock spectrum at the origin is, for zero damping, proportional to the velocity change associated with the shock. The compensation signal added to bring back to zero the velocity change thus makes the slope of the spectrum at the origin zero. In addition, the response spectrum of the compensated signal can be larger than the spectrum of the theoretical signal close to the frequency corresponding to the inverse of the duration of the compensation signal. It is thus advisable to make sure that the

224

Mechanical shock

variations observed are not in a range which includes the resonance frequencies of the test item. This example was treated for a shock on shaker carried out with symmetrical pre- and post-shocks. Let us consider the case where only one pre-shock or one postshock is used. Figure 7.28 shows the response spectra of: - the nominal signal (half-sine, 500 m/s2, 10 ms); - a shock on a shaker with only one post-shock (half-sine, p = 0.1) to cancel the velocity change; - a shock on a shaker with a pre-shock alone; - a shock on a shaker with identical pre- and post-shocks.

Figure 7.28. Influence of the distribution of pre- and post-shocks on the SRS of a half-sine shock

It is noted that: -the variation between the spectra decreases when pre-shock or post-shock alone is used. The duration of the signal of compensation being then larger, the spectrum is deformed at a lower frequency than in the case of symmetrical pre- and post-shocks; - the pre-shock alone can be preferred with the post-shock, but the difference is weak. On the other hand, the use of symmetrical pre- and post-shocks has the already quoted well-known advantages.

Generation of shocks using shakers

225

NOTE: In the case of heavy resonant test items, or those assembled in suspension, there can be a coupling between the suspended mass m and the mass M of the coiltable -fixture unit, with resulting modification of the natural frequency according to the rule:

Figure 7.29 shows the variations of 0 /f 0 according to the ratio m / M. For m close to M, the frequency f0 can increase by a factor of about 1.4. The stress undergone by the system is therefore not as required.

Figure 7.29. Evolution of the natural frequency in the event of coupling

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Chapter 8

Simulation of pyroshocks

Many works have been published on the characterization, measurement and simulation of shocks of pyrotechnic origin (generated by bolt cutters, explosive valves, separation nuts, etc) [ZIM 93]. The test facilities suggested are many, ranging from traditional machines to very exotic means. The tendency today is to consider that the best simulation of shocks measured in near-field (cf. Section 3.9) can be obtained only by subjecting the material to the shock produced by the real device (which poses the problem of the application of an uncertainty factor to cover the variability of this shock). For shocks in the mid-field , simulation can be carried out either using real the pyrotechnic source and a particular mechanical assembly or using specific equipment using explosives, or by impacting metal to metal if the structural response is more important. In the far-field, when the real shock is practically made up only of the response of the structures, a simulation on a shaker is possible (when use of this method allow).

8.1. Simulations using pyrotechnic facilities If one seeks to carry out shocks close to those experienced in the real environment, the best simulation should be the generation of shocks of a comparable nature on the material concerned. The simplest solution consists of making functional real pyrotechnic devices on real structures. Simulation is perfect but [CON 76], [LUH 76]:

228

Mechanical shock - It can be expensive and destructive.

-One cannot apply an uncertainty factor without being likely to create unrealistic local damage (a larger load, which requires an often expensive modification of the devices can be much more destructive). To avoid this problem, an expensive solution consists of carrying out several tests in a statistical matter. One often prefers to carry out a simulation on a reusable assembly, the excitation still being pyrotechnic in nature. Several devices have been designed. Some examples of which are described below: 1. A test facility made up of a cylindrical structure [IKO 64] which comprises a 'consumable' sleeve cut out for the test by an explosive cord (Figure 8.1). Preliminary tests are carried out to calibrate 'the facility' while acting on the linear charge of the explosive cord and/or the distance between the equipment to be tested (fixed on the structure as in the real case if possible) and the explosive cord.

Figure 8.1. Barrel tester for pyroshock simulation

2. For a large-sized structure subjected to this type of shock, one in general prefers to make the real pyrotechnical systems placed on the structure as they could under operating conditions. The problem of the absence of the uncertainty factor for the qualification tests remains. 3. D.E. White, R.L. Shipman and W.L. Harlvey proposed placing a greater number of small explosive charges near the equipment to be tested on the structure, in 'flowers pots'. The number of pots to be used axis depends on the amplitude of the shock, of the size of the equipment and of the local geometry of the structure. They are manufactured in a stainless steel pipe which is 10cm in height, 5 cm in interior diameter, 15 cm in diameter external and welded to approximately 13 mm steel base plates [CAR 77], [WHI65].

Simulation of pyroshocks

229

Figure 8.2. 'Flower pot 'provided with an explosive charge

A number of preliminary shots, reduced as a result of experience one acquires from experiment, are necessary to obtain the desired shock. The shape of the shock can be modified within certain limits by use of damping devices, placing the pot more or less close to the equipment, or by putting suitable padding in the pot. If, for example, one puts sand on the charge in the pot, one transmits to the structure more low frequency energy and the shape of the spectrum is more regular and smoother. One can also place a crushable material between the flower pot and the structure in order to absorb the high frequencies. When the explosive charge necessary is substantial, this process can lead to notable permanent deformations of the structure. The transmitted shock then has an amplitude lower than that sought and, to compensate, one can be tempted to use a larger charge with the following shooting. To avoid entering this vicious circle, it is preferable, with the next shooting, either to change the position of the pots, or to increase the number by using weaker charges. The advantages of this method are the following: - the equipment can be tested in its actual assembly configuration; - high intensity shocks can be obtained simultaneously along the three principal axes of the equipment. There are also some drawbacks: -no analytical method of determination a priori of the charge necessary to obtain a given shock exists; - the use of explosive requires testing under specific conditions to ensure safety; -the shocks obtained are not very reproducible, with many influential parameters; - the tests can be expensive if, each time, the structure is deformed [AER 66].

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Mechanical shock

4. A test facility made up of a basic rectangular steel plate (Figure 8.3) suspended horizontally. This plate receives on its lower part, directly or by the intermediary of an 'expendable' item, an explosive load (chalk line, explosive in plate or bread).

Figure 8.3. Plate with resonant system subjected to detonation

A second plate supporting the test item rests on the base plate via four elastic supports. Tests carried out by this means showed that the shock spectrum generated at the input of the test item depends on: - the explosive charge; - the nature and thickness of the plate carrying the test item; - the nature of the elastic supports and their prestressing; - the nature of material of the base plate and its dimensions; constituting -the mass of the test item [THO 73]. The reproducibility of the shocks is better if the load is not in direct contact with the base plate.

8.2. Simulation using metal to metal impact The shock obtained by a metal to rnetal impact has similar characteristics to those of a pyrotechnical shock in an intermediate field: great amplitude; short duration; high frequency content; shock response spectrum comparable with a low frequency slope of 12 dB per octave etc. Simulation is in general satisfactory up to approximately 10 kHz.

Simulation of pyroshocks

231

Figure 8.4. Simulation by metal to metal impact (Hopkinson bar)

The shock can be created by the impact of a hammer on the structure itself, a Hopkinson bar or a resonant plate [BAI 79], [DAY 85], [DAV 92] and [LUH 81].

Figure 8.5. Simulation by the impact of a ball on a steel beam

With all these devices, the amplitude of the shock is controlled while acting on the velocity of impact. The frequency components are adjusted by modifying the resonant geometry of system (length of the bar between two points of fixing, the addition or removal of runners, etc) or by the addition of a deformable material between the hammer and the anvil. To generate shocks of great amplitude, the hammer can be replaced by a ball or a projectile with a plane front face made out of steel or aluminium, and launched by a pneumatic gun (air or nitrogen) [DAV 92]. The impact can be carried out directly on the resonant beam or to a surmounted plate of a resonant mechanical system composed of a plate supporting the test item connecting it to the impact plate. 8.3. Simulation using electrodynamic shakers The possibilities of creating shocks using an electrodynamic shaker are limited by the maximum stroke of the table and more especially by the acceptable maximum force. The limitation relating to the stroke is not very constraining for the pyrotechnical shocks, since they are at high frequencies. There remains a limitation on the maximum acceleration of the shock [CAR 77], [CON 76], [LUH 76] and

232

Mechanical shock

[POW 76]. If, with the reservations of Section 4.3.6, one agrees to cover only part of the spectrum, then when one makes a possible simulation on the shaker; this gives a better approach to matching the real spectrum. Exciters have the advantage of allowing the realization of any signal shape such as shocks of simple shapes [DIN 64], [GAL 66], but also random noise or a combination of simple elementary signals with the characteristics to reproduce a specified response spectrum (direct control from a shock spectrum, of. Chapter 9). The problem of the over-testing at low frequencies as previously discussed is eliminated and it is possible, in certain cases, to reproduce the real spectrum up to 1000 Hz. If one is sufficiently far away from the source of the shock, the transient has a lower level of acceleration and the only limitation is the bandwidth of the shaker, which is about 2000 Hz. Certain facilities of this type were modified in the USA to make it possible to simulate the effects of pyrotechnical shocks up to 4000 Hz. One can thus manage to simulate shocks whose spectrum can reach 7000 g [MOE 86]. We will see, however, in Chapter 9 the limits and disadvantages of this method.

8.4. Simulation using conventional shock machines We saw that, generally, the method of development of a specification of a shock consists of replacing the transient of the real environment, whose shape is in general complex, by a simple shape shock, such as half-sine, triangle, trapezoid etc, starting from the 'shock response spectrum' equivalence criterion (with the application of a given or calculated uncertainty factor^ to the shock amplitude) [LUH 76]. With the examination of the shapes of the response spectra of standard simple shocks, it seems that the signal best adapted is the terminal peak saw tooth pulse, whose spectra are also appreciably symmetrical. The research of the characteristics of such a triangular impulse (amplitude, duration) having a spectrum envelope of that of a pyrotechnical shock led often to a duration of about 1ms and to an amplitude being able to reach several tens of thousands of ms"1. Except in the case of very small test items, it is in general not possible to carry out such shocks on the usual drop tables: - limitation in amplitude (acceptable maximum force on the table); - duration limit: the pneumatic programmers do not allow it to go below 3 to 4 ms. Even with the lead programmers, it is difficult to obtain a duration of less than 2 ms. However spectra of the pyrotechnical shocks with, in general, averages close

1 cf. Volume 5.

Simulation of pyroshocks

233

to zero have a very weak slope at low frequencies, which leads to a very small duration of simple shock, of about one millisecond (or less); - the spectra of the pyrotechnical shocks are much more sensitive to the choice of damping than simple shocks carried out on shock machines. To escape the first limitation, one accepts, in certain cases, simulation of the effects of the shock only at low frequencies, as indicated in Figure 8.6. The 'equivalent' shock has in this case a larger amplitude since fa, the last covered frequency is higher.

Figure 8.6. Need for a TPS shock pulse of very short duration

Figure 8.7. Realizable durations lead to an over-test

With this approximation, the shape of the shock has little importance, all the shocks of simple shape having in the zone which interests us (impulse zone) symmetrical spectra. One however often chooses the terminal peak saw tooth to be able to reach, with lead programmers, levels of acceleration difficult to obtain with other types of programmers. This procedure, one of the first used, is open to criticism for several reasons: -if the tested item has only one frequency fa, simulation can be regarded as correct (insofar as the test facilities are able to carry out the specified shock perfectly). But very often, in addition to a fundamental frequency ff of rather low resonance such as one can realize easily for fa > ff, the specimen has other resonances at higher frequencies with substantial Q factors. In this case, all resonances are excited by shock and because of the frequency content particular to this kind of shock, the responses of the modes at high

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Mechanical shock

frequencies can be dominating. This process can thus lead to important undertesting; -by covering only the low frequencies, one can define an 'equivalent' shock of sufficiently low amplitude to be realizable on the drop testers. However, nothing is solved from the point of view of shock duration. The limitation of 2ms on the crusher programmers or 4ms approximately on the pneumatic programmers will not make it possible to carry out a sufficiently short shock. Its spectrum will in general envelop much too much of the pyrotechnical shock at low frequencies (Figure 8.7). Except for the intersection point of the spectra (f = f a ), simulation will then be incorrect over all the frequency band. Over-testing issometimes acceptable for f < fa, and under-test beyond. We tried to show in this chapter how mechanical shocks could be simulated on materials in the laboratory. The facilities described are the most current, but the list is far from being exhaustive. Many other processes were or are still used to satisfy particular needs [CON 76], [NEL 74], [POW 74] and [POW 76].

Chapter 9

Control of a shaker using a shock response spectrum

9.1. Principle of control by a shock response spectrum 9.1.1. Problems The response spectra of shocks measured in the real environment often have a complicated shape which is impossible to envelop by the spectrum of a shock of simple shape realizable with the usual test facilities of the drop table type. This problem arises in particular when the spectrum presents an important peak [SMA 73]. The spectrum of a shock of simple shape will be: - either an envelope of the peak, which will lead to significant over-testing compared with the other Frequencies; - or envelope of the spectrum except the peak with, consequently, under-testing at the frequencies close to the peak. The simulation of shocks of pyrotechnic origin leads to this kind of situation. Shock pulses of simple shape (half-sine, terminal peak saw tooth) have, in logarithmic scales, a slope of 6 dB/octave (i.e. 45°) at low frequencies incompatible with those larger ones, of spectra of pyrotechnic shocks (> 9 dB/octave). When the levels of acceleration do not exceed the possibilities of the shakers, simulation with control using spectra are of interest.

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Mechanical shock

Figure 9.1. Examples of SRS which are difficult to envelop with the SRS of a simple shock

The exciters are actually always controlled by a signal which is a function of time. The calculation of a shock spectrum is an unambiguous operation. There is an infinity of acceleration-time signals with a given spectrum. The general principle thus consists in searching out one of the signals x(t) having the specified spectrum. Historically, the simulation of shocks with spectrum control was first carried out using analogue and then digital methods [SMA 74a] [SMA 75].

9.1.2. Method of parallel filters The analogue method, suggested in 1964 by G.W. Painter and H.J. Parry ([PAI 64], [ROB 67], [SMA 74a], [SMA 75] and [VAN 72]) consists of using the responses of a series of filters placed simultaneously at the output of a generator of (rectangular) impulses. The filters, distributed into the third octave, are selected to cover the range of frequency of interest. Each filter output is a response impulse. If the filters are of narrow bands, each response resembles a narrow band signal which becomes established and then attenuates. If the filters are equivalent to one-degreeof-freedom systems, the response is of the decaying sinusoidal type and the reconstituted signal is oscillatory [USH 72]. Each filter is followed by an amplifier allowing regulation of the intensity of the response. All the responses are then added together and sent to the input of the amplifier which controls the shaker. One approaches the spectrum specified by modifying the gain of the amplifiers at the output of each filter. It is admitted that the output of a given filter affects only the point of the shock spectrum whose frequency is equal to the central frequency of the filter and to which the shock spectrum is insensitive with the dephasing caused by the filters or the shaker. The complete signal

Control of a shaker using a shock response spectrum

237

corresponding to a flat spectrum resembles a swept sine of initial frequency equal to the central frequency of the highest filter, whose frequency decrease logarithmically to the central frequency of the lower filter [BAR 74], [HUG] and [MET 67]. The disadvantage of this process is that one does not have practically any check on the characteristics of the total control signal (shape, amplitude and duration). According to the velocity of convergence towards the specified spectrum, the adjustment of the overall signals can be in addition be extensive and result in applying several shocks to the test item to develop the control signal [MET 67]. This method also was used digitally [SMA 75], the essential difference being a greater number of possible shapes of shocks. Thereafter, one benefited from the development of data processing tools to make numerical control systems which are easier to use and use elementary signals of various shapes (according to the manufacturer) to constitute the control signal [BAR 74].

9.1.3. Current numerical methods From the data of selected points on the shock spectrum to be simulated, the calculator of the control system uses an acceleration signal with a very tight spectrum. For that, the calculation software proceeds as follows: - At each frequency f0 of the reference shock spectrum, the software generates an elementary acceleration signal, for example a decaying sinusoid. Such a signal has the property of having a shock response spectrum presenting a peak of the frequency of the sinusoid whose amplitude is a function of the damping of the sinusoid. With an identical shock spectrum, this property makes it possible to realize on the shaker shocks which would be unrealizable with a control carried out by a temporal signal of simple shape (cf. Figure 9.2). For high frequencies, the spectrum of the sinusoid tends roughly towards the amplitude of the signal. - All the elementary signals are added by possibly introducing a given delay (and variable) between each one of them, in order to control to a certain extent the total duration of the shock (which is primarily due to the lower frequency components). - The total signal being thus made up, the software proceeds to processes correcting the amplitudes of each elementary signal so that the spectrum of the total signal converges towards the reference spectrum after some interations.

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Mechanical shock

Figure 9.2. Elementary shock (a) and its SRS (b)

Figure 9.3. SRS of the components of the required shock

The operator must provide to the software, at each frequency of the reference spectrum: -the frequency of the spectrum; - its amplitude; - a delay; - the damping of sinusoids or other parameters characterizing the number of oscillations of the signal. When a satisfactory spectrum time signal has been obtained, it remains to be checked that the maximum velocity and displacement during the shock are within

Control of a shaker using a shock response spectrum

239

the authorized limits of the test facility (by integration of the acceleration signal). Lastly, after measurement of the transfer function of the facility, one calculates the electric excitation which will make it possible to reproduce on the table the acceleration pulse with the desired spectrum (as in the case of control from a signal according to time) [FAV 74]. We propose to examine below the principal shapes of elementary signals used or usable.

9.2. Decaying sinusoid 9.2.1. Definition The shocks measured in the field environment are very often responses of structures to an excitation applied upstream and are thus composed of a damped sine type of the superposition of several modal responses of [BOI 81], [CRI 78], [SMA 75] and [SMA 85]. Electrodynamic shakers are completely adapted to the reproduction of this type of signals. According to this, one should be able to reconstitute a given SRS from such signals, of the form:

where: O=2nf f = frequency of the sinusoid n = damping factor NOTE: The constant A is not the amplitude of the sinusoid, which is actually equal to [CAR 74], [NEL 74], [SMA 73], [SMA 74a], [SMA 74b] and [SMA 75]:

9.2.2. Response spectrum This elementary signal a(t) has a shock spectrum which presents a more or less significant peak to the frequency f0 = f according to the value of n. This peak increases when n decreases. It can, for very weak n (about 10-3), reach an amplitude

240

Mechanical shock

exceeding by a factor 10 the amplitude of shock according to time [SMA 73]. It is an interesting property, since it allows, for equal SRS, reduction in the amplitude of the acceleration signal by an important factor and thus the ability to carry out shocks on a shaker which could not be carried out with simple shapes.

Figure 9.4. SRS of a decaying sinusoid for various values of n

Figure 9.5. SRS of a decaying sinusoid for various values of the Q factor

When n - 0.5, the SRS tends towards that of a half-sine pulse. One should not confuse the damping factor n, which characterizes the exponential decay of the

Control of a shaker using a shock response spectrum

241

acceleration signal a(t), and the damping factor E chosen for the plotting of the SRS. For given n, the SRS of the decaying sinusoid presents also a peak whose amplitude varies according to E or Q = 1/2 E (Figure 9.5). The ratio R of the peak of the spectrum to the value of the spectrum at the very high frequency is given in Figure 9.6 for various values of the damping factors n (sinusoid) and E (SRS) [SMA 75].

Figure 9.6. Amplitude of the peak of the SRS of a decaying sinusoid versus n and E

for approximated using the relation [GAL 73]:

the value of this ration can be

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Mechanical shock

Particular case where n = E

Let us set E = n + e. It becomes 2

yielding

If e is small, we have In

and if

9.2.3. Velocity and displacement With this type of signal, the velocity and the displacement are not zero at the end of the shock. The velocity, calculated by integration of a(t) = A e"11 equal to

If t - o

The displacement is given by:

If t - o, x (t) -

o (Figure 9.7).

sin Q t, is

Control of a shaker using a shock response spectrum 243

Figure 9.7. The velocity and the displacement are not zero at the end of the damped sine

These zero values of the velocity and the displacement at the end of the shock are very awkward for a test on shaker.

9.2.4. Constitution of the total signal The total control signal is made up initially of the sum (with or without delay) of elementary signals defined separately at frequencies at each point of the SRS, added to a compensation signal of the velocity and displacement. The first stage consists of determining the constants Ai and ni of the elementary decaying sinusoids. The procedure can be as follows [SMA 74b]: - Choice of a certain number of points of the spectrum of specified shock, sufficient for correctly describing the curve (couples frequency fi, value of the spectrum Si). - Choice of damping constant ni of the sinusoids, if possible close to actual values in the real environment. This choice can be guided by examination of the shock spectra of a decaying sinusoid in reduced coordinates (plotted with the same Q factor as that of the specified spectrum), for various values of n (Figure 9.4). These curves underline the influence of n on the magnitude of the peak of the spectrum and over its width. One can also rely on the curves of Figure 9.6. But in practice, one prefers to have a rule easier to introduce into the software. The value ni ~ 0.1 gives good results [CRI 78]. It is, however, preferable to choose a variable damping factor according to the frequency of the sinusoid, strong at the low

244

Mechanical shock

frequencies and weak at the high frequencies. It can, for example, decrease in a linear way from 0.3 to 0.01 between the two ends of the spectrum. NOTE: If we have acceleration signals which lead to the specified spectrum, we could use the Prony method to estimate the frequencies and damping factors [GAR 86]. 1 -n being chosen, we can calculate, using the relation [9.3], for Q = — given

(damping chosen for plotting of the shock spectrum of reference), the ratio R of the peak of the spectrum to the amplitude of the decaying sinusoid. This value of R makes it possible to determine the amplitude amax of the decaying sinusoid at the particular frequency. Knowing that the amplitude amax of the first peak of the decaying sinusoid is related to the constant A by the relation [9.2]:

we determine the value of A for each elementary sinusoid. For n small ( < 0.08 ), we have

9.2.5. Methods of signal compensation Compensation can be carried out in several ways. 1. By truncating the total signal until it is realizable on the shaker. This correction can, however, lead to an important degradation of the corresponding spectrum [SMA 73]. 2. By adding to the total signal (sum of all the elementary signals) a highly damped decaying sinusoid, shifted in time, defined to compensate for the velocity and the displacement [SMA 74b] [SMA 75] [SMA 85]. 3. By adding to each component two exponential compensation functions, with a phase in the sinusoid [NEL 74] [SMA 75]

Control of a shaker using a shock response spectrum

245

Compensation using a decaying sinusoid In order to calculate the characteristics of the compensating pulse, the complete acceleration signal used to simulate the specified spectrum can be written in the form:

where

0i is the delay applied to the ith elementary signal. Ac, wc, nc and 9 are the characteristics of the compensating signal (decaying sinusoid). The calculation of these constants is carried out by cancelling the expressions of the velocity x and of the displacement x obtained by integration of x.

246

Mechanical shock The cancellation of the velocity and displacement for t equal to infinity leads to:

yielding

Figure 9.8. Acceleration pulse compensated by a decaying sinusoid

Figure 9.9. Velocity associated with the signal compensated by a decaying sinusoid

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247

Figure 9.10. Displacement associated with the signal compensated by a decaying sinusoid

The constants Ac and 0 characterizing the compensating sinusoid are thus a function of the other parameters (Qc, nc). The frequency of the compensating

(
waveform fc = — should be about a half or third of the smallest frequency of the V 27iJ points selected to represent the specified spectrum. The damping nc is selected to be between 0.5 and 1 [SMA 74b]. The constants Ac and 0 can then be determined. Compensation using two exponential signals This method, suggested by Nelson and Prasthofer [NEL 74] [SMA 85], consists of adding to the decaying sinusoid two exponential signals and a phase shift 9. Each elementary waveform is of the form:

The exponential terms are defined in order to compensate the velocity and the displacement (which must be zero at the beginning and the end of the shock). The phase 0 is given to cancel acceleration at t = 0. With this method, each individual component is thus compensated. The choice of the parameters a, b and c is somewhat arbitrary. The idea being to create a signal resembling to a damped decaying sinusoid, one chooses:

248

Mechanical shock

where

The acceleration, velocity and displacement are zero at the beginning and the end of the shock if:

With these notations, the velocity and the displacement are given by:

Control of a shaker using a shock response spectrum

249

Figure 9.11 shows as an example a signal compensated according to this method. It is noted that the first negative peak is larger than the first positive peak. The waveform resembles overall a decaying sinusoid.

Figure 9.11. Waveform compensated by two exponential signals

Figures 9.12 and 9.13 give the corresponding velocity and displacement.

Figure 9.12. Velocity after compensation by two exponential signals

Figure 9.13. Displacement after compensation by two exponential signals

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Mechanical shock

NOTE: We also find the equivalent form

with

where k1 and k2 have the same definition as previously.

9.2.6. Iterations Once all these coefficients are determined, one calculates the spectrum of the signal thus made up. All this work was carried out up to now by supposing the influence of each decaying sine of frequency fi on the other points of the spectrum to be negligible. This assumption is actually too simplifying, and the spectrum obtained does not match the specified spectrum. It is thus necessary to proceed to successive iterations to refine the values of the amplitudes A i of the components of x(t):

The iterations can be carried out by correcting the amplitudes of the elementary waveforms by a simple rule of three. More elaborate relations were proposed, such as [BOI 81], [CRI 78]:

where - S c ^ n '(fj) is the amplitude of the spectrum calculated using the values Ai of the nth iteration at the frequency fi; - S(fi) is the value of the reference spectrum to the frequency fi; - Sc(fi) is the amplitude of the spectrum calculated with the values Ai of the nth iteration at the frequency fi, except for the coefficient Ai which is added with an increment AAi (0.05 gives satisfactory results);

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251

- p is a weighting factor (0.5 gives an acceptable velocity of convergence). If the procedure converges (and it is fortunately the case in general), the reconstituted spectrum and the specified spectrum are very close at the frequencies fi selected. The agreement can be worse between the frequencies fi. These intermediate values can be modified by changing the sign of the constants Ai (taken to be initially positive) [NEL 74]. One often retains the empirical rule which consists of alternating the signs of the components. If the levels are too high between the frequencies fi, one must decrease the values of ni. If, on the contrary, they are too weak, it is necessary to increase the values of ni or to add components between the fi frequencies. The calculated spectrum must match the specification as well as possible, as close to the peaks of the spectrum as to the troughs. One acts sometimes on the frequency fc to readjust to zero the residual velocity VR and displacement dR (to increase fc led to a reduction in VR and dR in general [SMA 74a]).

9.3. D.L. Kern and C.D. Hayes' function 9.3.1. Definition To reproduce a given spectrum frequency after frequency (and not simultaneously a whole spectrum), D.L. Kern and C.D. Hayes [KER 84] proposed to use a shock of the form:

where f= frequency n = damping A = amplitude of the shock e = Neper number

The signal resembles the response of one-degree-of-freedom system to a Dirac impulse function. It can seem interesting a priori to examine the potential use of this function for the synthesis of the spectra. The first parameters to be considered are residual velocity and displacement.

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Mechanical shock

9.3.2. Velocity and displacement Velocity By integration we obtain:

The velocity, different from zero for t = 0, can be cancelled by adding the term but then, the residual velocity is zero no longer.

Displacement The integration of v(t) gives:

One cannot cancel the velocity and the displacement at the same time for t = 0 and t large. As an example, the curves of Figures 9.14, 9.15 and 9.16 show the acceleration, the velocity and the displacement for f = 1 and n = 0.05.

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253

Figure 9.14. Example of D.L.Kern and C.D. Hayes' waveform

Figure 9.15. Velocity

Figure 9.16. Displacement

9.4. ZERD function 9.4.1. Definition 9.4.1.1. D.K. Fisher and M.R. Posehn expression The use of a decaying sinusoid to compose a shock of given SRS has the disadvantage of requiring the addition of a compensation waveform intended to reduce to zero the velocity and the displacement at the end of the shock. This signal modifies the response spectrum at the low frequencies and, in certain cases, can harm the quality of simulation. D.K. Fisher and M.R. Posehn [FIS 77] proposed using a waveform which they named ZERD (Zero Residual Displacement) defined by the expression:

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Mechanical shock

where

This function resembles a damped sinusoid and has the advantage of leading to zero velocity and displacement at the end of the shock.

Figure 9.17. ZERD waveform of D.K. Fisher and M.R. Posehn (example)

Gradually, the peaks of maximum amplitude come before it decreases at a regular rate. The positive and negative peaks are almost symmetrical. Therefore, it is well adapted to a reproduction on shaker.

9.4.1.2. D.O. Smallwood expression D.O. Smallwood [SMA 85] defined ZERD function by the relation:

where

It is this definition which we will use hereafter.

Control of a shaker using a shock response spectrum

9.4.2. Velocity and displacement By integration we obtain the velocity:

and the displacement

It can be interesting to consider the envelope of this signal. If we pose:

If n is small, T also is small, so that C(t) ~ Q t and q ( t ) = Q t cos [a t + 9(t)].

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256

Mechanical shock

Figure 9.18. ZERD waveform of D.O. Smallwood (example)

Acceleration becomes:

and for n small

The maximum of the envelope t e"11

1

1

is nQ r|Q the time-constant, the function reaches its maximum in time 1 / rj Q. The value of this maximum is

1

takes place for t =

. Since

, the maximum of cos [Q t + 9(t)] being equal to 1

Qne

A is thus the amplitude of a(t).

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257

9.4.3. Comparison of the ZERD waveform with a standard decaying sinusoid This comparison can be carried out through the envelopes,

and

where

Figure 9.19. Comparison of the ZERD and decaying sinusoid waveform envelopes

The plotting of these two curves shows that they have: - the same amplitude A; - the same slope (-1) in a semi-logarithmic scale for a t large; - a different decrement with

decreasing less quickly

9.4.4. Reduced response spectra 9.4.4.1. Influence of the damping n of the signal The response spectra of this shock are plotted in Figure 9.20 with a Q factor equal to 10; they correspond to signals according to the times defined for n = 0.01, 0.05 and 0.1 respectively.

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Mechanical shock

Figure 9.20. ZERD waveform - influence of damping n on the SRS

The peak of these spectra becomes increasingly narrow as n decreases. 9.4.4.2. Influence of the Q factor The response spectra of Figure 9.21 are plotted from a ZERD waveform (n = 0.05) for Q = 50, 10 and 5, respectively.

Figure 9.21. ZERD waveform - influence of the Q factor on the SRS

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259

9.5. WAVSIN waveform 9.5.1. Definition R.C. Yang ([SMA 74a], [SMA 75], [SMA 85], [YAN 70] and [YAN 72]) proposed (initially for the simulation of the earthquakes) a signal of the form:

where

where N is an integer (which, we will see it, must be odd and higher than 1). The first term of a(t) is a window of half-sine form of half-period i. The second describes N half-cycles of a sinusoid of greater frequency (f), modulated by the preceding window.

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Mechanical shock

Example Figure 9.22 shows an acceleration signal WAVSIN plotted for

Figure 9.22. Example of WAVSIN waveform

9.5.2. Velocity and displacement The function a(t) can also be written:

Velocity

At the end of the shock

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261

Whatever the value of N

Displacement

For

Figure 9.23. . Velocity corresponding to the Figure 9.24. . Displacement corresponding to waveform in Figure 9.22 the waveform in Figure 9.22

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Mechanical shock

If N is odd(N = 2n + l):

For the displacement to be zero at the end of the shock, it is thus necessary that N is an odd integer. The advantages of this waveform are: - The residual velocity and displacement associated with each elementary signal a(t) are zero. - With N being odd, the two functions sine intervening in a(t) are maximum for The maximum of a(t) is thus am. - The components have a finite duration, which makes it possible to avoid the problems involved in a possible truncation of the signals (the case of decaying sinusoids for example).

9.5.3. Response of a one-degree-of-freedom system Let us set f0 as the natural frequency of this system and Q as its quality factor. To simplify the writing, let us express a(t) in dimensionless coordinates, in the form:

where

Control of a shaker using a shock response spectrum

yielding

9.5.3.1. Relative response displacement

where:

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Mechanical shock

Particular cases

and

where:

Let us set, for

For

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265

9.5.3.2. Absolute response acceleration

Particular cases E = 0 and B = 1: the same relations as for the relative displacement E= l

For all the cases where 0 < 9 < 60, let us set:

It becomes, for 9 > 00:

9.5.4. Response spectrum The SRS of this waveform presents a peak whose amplitude varies with N with its frequency close to f. Figure 9.25 shows the spectra plotted in reduced coordinates for N = 3, 5, 7 and 9(Q = 10).

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Mechanical shock

Figure 9.25. WAVSIN - influence of the number of half-cycles s N on the SRS

Figure 9.26. WAVSIN - amplitude of the peak of the SRS versus N and Q

Figure 9.26 gives the value of the peak of the shock spectrum R(Q, N) standardized by the peak R(10, N) according to the half-cycle number N, for various values of Q [PET 81].

9.5.5. Time history synthesis from shock spectrum The process consists here of choosing a certain number n of points of the spectrum of reference and, at the frequency of each one of these points, choosing the parameters b, N and am to correspond the peak of the spectrum of the elementary waveform with the point of the reference spectrum. This operation being carried out for n points of the spectrum of reference, the total signal is obtained by making the sum:

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267

with 6j being a delay intended to constitute a signal x(t) resembling as well as possible the signal of the field environment to simulate (the amplitude and the duration being preserved if possible). The delay has little influence on the shock spectrum of x(t).

Choice of components The frequency range can correspond to the interval of definition of the shock spectrum (1/3 or 1/2 octave). Convergence is faster for the 1/2 octave. With 1/12 octave, the spectrum is smoother, without troughs or peaks. The amplitude of each component can be evaluated from the ratio of the value of the shock spectrum at the frequency considered and the number of half-cycles chosen for the signal [BAR 74]. ami allows a change of amplitude at all the points of the spectrum. Ni allows modification of the shape and the amplitude of the peak of the spectrum of the elementary waveform at the frequency fi. The errors between the specified spectrum and the realized spectrum are calculated from an average on all the points to arrive at a value of the 'total' error. If the error is unacceptable, one proceeds to other iterations. Four iterations are in general sufficient to reach an average error lower than 11% [FAV 74]. With the ZERD waveform, the WAVSIN pulse is that which gives the best results. It is finally necessary to check before the test that the maximum velocity and displacement corresponding to the drive acceleration signal remain within the limits of the test facility (by integration of x(t)).

9.6. SHOC waveform 9.6.1. Definition Method SHOC (SHaker Optimized Cosines) suggested by D.O. Smallwood [SMA 73], [SMA 74a], [SMA 75] is based on the elementary waveform defined by:

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Mechanical shock

The signal is oscillatory, of increasing amplitude according to time, and then decreasing (symmetry with respect to the ordinates). The duration t of the signal is selected to be rather long so that the signal can be t

regarded as zero when t > — and t < -—. The waveform is made up of a decaying 2 2 cosine and a function of the 'haversine' type, the latter being added to only be able to cancel the velocity and the displacement at the end of the shock. In theory, the added signal should modify as little as possible the initial signal.

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269

Figure 9.27. Composition of a SHOC waveform

The characteristics of this compensation function are given while equalizing, except for the sign of the area under the curve and the area under the decaying cosine. The expression of the haversine used by D.O. Smallwood can be written in the form:

This relation has two independent variables with which one can cancel the residual conditions [SMA 73].

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Mechanical shock

The velocity at the end of the shock is equal to 2AV, if AV is the velocity change created by the positive part (t > 0) of the shock.

being sufficiently large

AV is zero at the end of the shock if:

The largest value of a(t) occurs for t = 0 :

a(o) = am - 8

9.6.2. Velocity and displacement By integration of the acceleration:

we obtain the velocity

and the displacement

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Example SHOC waveform, f = 0.8 Hz and n = 0.065

Figure 9.28. Example of SHOC waveform

9.6.3. Response spectrum 9.6.3.1. Influence of damping n of the signal Figure 9.29 shows the response spectra of a SHOC waveform of frequency 1 Hz with damping factors n successively equal to 0.01, 0.02, 0.05 and 0.1. These spectra are plotted for Q = 10. We observe the presence of an important peak centered on D the frequency f = whose amplitude varies with n. 2n

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Mechanical shock

Figure 9.29. SHOC - influence of n on the SRS

9.6.3.2. Influence of the Q factor on the spectrum The Q factor has significance only in the range centered on the frequency of the signal. The spectrum presents a peak all the more significant since the Q factor is larger.

Figure 9.30. SHOC- influence of the Q factor on the SRS

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9.6.4. Time history synthesis from shock spectrum To approach a point of the shock spectrum to simulate, we have the following parameters: - damping n for the shape of the curve; - the frequency f, at the point of the spectrum to be reproduced; - the amplitude am, related to the amplitude of the spectrum (scale factor on the whole of the curve); - duration T, selected in order to limit the maximum displacement during the shock according to the possibilities of the test facility. In fact, n and T are dependent since one also requires that at the moment T / 2 the decaying cosine be near to zero. Considering the envelope, one can for example ask that with t / 2, the amplitude of the signal be lower than p% of the value with t = 0 yielding:

For f given, it is necessary thus that

The curve of Figure 9.27 is plotted, as an example, for which leads to the relation Examination of the dimensionless SRS shows that the advantages of the decaying sinusoid are preserved. If T decreases, the necessary displacement decreases and, as the low frequency energy decreases, the spectrum is modified at frequencies lower than approximately 2 / T . Each time that a correction proves to be necessary, a compromise must thus be carried out between the smallest frequency to which the shock spectrum must be correctly reproduced and the displacement available. If 1/T is small compared to the frequency of the lowest resonance of the system, the effect of the correction on the response of the structure is weak [SMA 73]. Due to symmetry around the y-axis t = 0, the shocks are added in the frequency domain (i.e. of the shock response spectra) as well as in the time domain. This

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Mechanical shock

simplifies the construction of complex spectra. Variations can, however, be observed between the specified and carried out the shock spectra, which had with nonlinearities of the assembly, with the noise. In general, these variations do not exceed 30%. The peaks of an acceleration signal built from SHOC functions are positive in a dominating way. For certain tests, one can carry out a shock which roughly has an equal number of positive peaks and negative peaks with a comparable amplitude. This can be accomplished by alternating the signs of the various components. This alternation can lead, in certain cases, to a reduction in the displacement necessary to carry out the specified spectrum.

9.7. Comparison of the WAVSIN, SHOC waveforms and decaying sinusoid The cases treated by D.O. Smallwood [SMA 74a] seem to show that these three methods give similar results. It is noted ,however, in practice that, according to the shape of the reference spectrum, one or other of these waveforms allows a better convergence. The ZERD waveform very often gives good results.

9.8. Use of a fast swept sine The specified shock response spectrum can also be restored by generation of a fast swept sine. It is pointed out that a swept sine can be described by a relation of the form (Volume 1, Chapter 7)

where for a linear sweep (f = b t + fj)

f| is the initial frequency of sweep and b the sweep rate. The number Nb of cycles f +f

carried out between f1 and f2 for the duration T is given by Nb =

l

2

T. The 2 signal describing this sweep presents the property of a Fourier transform of roughly constant amplitude in the swept frequency band, being represented by [REE 60]:

The first part of the response spectrum (SRS) consists of the residual spectrum (low frequencies). Knowing that, for zero damping, the residual shock response spectrum SR is related to the amplitude of the Fourier transform by

we have, by combining [9.72] and [9.73]

From these results, for the method which makes it possible to determine the characteristics of a fast swept sine from a response spectrum [LAL 92b]: - one fixes the number of points N of definition of the swept sine signal; - one is given a priori, to initialize calculation, a number of cycles (Nb =12 for example), from which one deduces the sweep duration

and the sweep rate

-between two successive points (fi, SRS,) and (fj+j, SRSj +1 ) from the specified spectrum, the frequency of the signal is obtained from the sweeping law f = bt + f i; - the amplitude of the sinusoid at the time t corresponding to the frequency f included between ^ and f i+ i is calculated by linear interpolation according to amp =

(the constant 1.3 makes it possible to hold account of the fact that the relation [9.73] is valid only for one zero damping whereas the spectra are in general plotted for a value equal to 0.05. This constant is not essential, but makes it possible to have a better result for the first calculation); - one deduces, starting from [9.71], the expression of the signal:

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Mechanical shock

- by integration of this signal of acceleration, one calculates the associated velocity change AV (by supposing the initial velocity equal to zero). By comparison with the velocity change AV0 read on the specified response spectrum (given by the slope at the origin of this spectrum, calculated from the first two points of the spectrum and divided by 2 TI), one determines the duration and the number of cycles Nb (up to now selected a priori) necessary to guarantee the same change velocity from

- with the same procedure as previously, one realizes a re-sampling of the signal x(t); - the response spectrum of this waveform is calculated and compared with the specified spectrum. From the noted variations of each N points, one readjusts the amplitudes by making three rules. One to two iterations are enough in general to obtain a signal of which: - the spectrum is very close to the specified spectrum; - the amplitude and the velocity change are of the same order of magnitude as those of the signal having been used to calculate the specified spectrum. This signal, to be realizable on the shaker, must be modified by the addition of a pre-shock and/or a post-shock ensuring an overall zero velocity change.

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277

Example Let us suppose that the specified spectrum is the shock spectrum of a half-sine waveform, of amplitude 500 m/s2 and of duration 10 ms (associated velocity change: 3.18 m/s). Figure 9.31 shows the signal obtained after three iterations carried out to readjust the amplitudes (without pre- or post-shock) and Figure 9.32 the corresponding response spectrum, superimposed on the specified spectrum.

Figure 9.31. . Example of fast swept sine

Figure 9.32. . SRS of the equivalent swept sine to the SRS of a half-sine shock

The velocity change is equal to 3 m/s, the amplitude is very close to 500 m/s2 and the duration of the first peak of the signal, dominating, is close to 10 ms.

This method is thus of the interest to coarsely represent the characteristics of amplitude, duration and velocity change of the signal at the origin of the specified spectrum. It has the disadvantage of not always converging according to the shape of the specified spectrum.

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Mechanical shock

9.9. Problems encountered during the synthesis of the waveforms The principal problems encountered are the following [SMA 85]: Problem

Possible remedy The step supposes that the elementary waveforms which constitute the shock of the specified spectrum are not too dependent on one another, i.e. the modification of the amplitude of the one of them only modifies slightly the other points of the spectrum. If the points chosen on the specified spectrum are too close to one another, if the damping is too large, it can be impossible to converge. The search for a solution can be based on the following: - the amplitude of a component cannot be reduced if the SRS is too high at this frequency: there is thus a limit with the possibilities of compensation with respect to the contribution of the near components;

- a small increase in the amplitude of one component can The iterations do sometimes reduce the shock spectrum to this frequency not converge. because of the interaction of the near components; - to change the sign of the amplitude of one component will not lower the SRS in general. It is noted that convergence is better if the signs of the components are alternated. If the SRS is definitely smaller at the high frequencies than in certain ranges of intermediate frequencies, there can not be a solution. It is known that any SRS tends at high frequencies towards the amplitude of the signal. The SRS limit at high frequencies of the components designed to reproduce a very large peak can sometimes be higher than the values of the reference SRS at high frequency. The SRS of the total signal is then always too large in this range. The solutions in the event of a convergence problem can be the following: - to give a high damping to the lowfrequencycomponents and decreasing it in a continuous way when the component frequency increases;

Control of a shaker using a shock response spectrum

The iterations do not converge (continuation).

279

- to change thefrequencyof the components; - to lower damping (each component); - to reduce the number of component; - to change the sign of certain components. This problem can be corrected by:

The spectrum is - increasing the number of components, while placing the well simulated at new ones close to the 'valleys' of the spectrum; the frequencies - increasing the damping of the components; chosen, but is too small between - changing the sign of the components; it should, however, these be known that the components interact in an unforeseeable way frequencies. when the sign of the one of the two near components is changed. The spectrum is well simulated at the frequencies chosen, but is too large between certain frequencies.

One can try to correct this defect: - by removing a component; - by reducing the damping of the near components; - by changing the sign of one of the near components.

If acceleration is too high, one can: The resulting - lower the damping to increase the ratio peak of the x(t) signal is not spectrum/amplitude of the signal; realizable (going - increase the delays between the components; beyond the - change the sign of certain components; limiting performances of -use another form of elementary waveform at each the shaker). frequency; - as a last resort, reduce the frequency range on which the specification is defined.

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Mechanical shock

If the velocity is too large: The resulting x(t) signal is not realizable (going beyond of the limiting performances of the shaker).

- The lowfrequencycomponents are usually at the origin of this problem and a compromise must be found at these frequencies, which can result in removing the first points of the reference spectrum (these components produce a large displacement also). It is also a means of reducing the duration of the signal when the specification imposes one duration maximum. This modification can be justified by showing that the test item does not have any resonance frequency in this domain. -If possible, one can try to change the elementary waveform, the displacement and the velocity associated with the new waveform being different. The ZERD waveform often gives better results. - If no compromise is satisfactory, it is necessary to change the shock test facility, with no certainty of obtaining better results.

9.10. Criticism of control by a shock response spectrum Whatever the method adopted, simulation on a shock test facility measured in the field requires the calculation of their response spectra and the search for an equivalent shock. If the specification must be presented in the form of a time-dependent shock pulse, the test requester must define the characteristics of shape, duration and amplitude of the signal, with the already quoted difficulties. If the specification is given in the form of a SRS, the operator inputs in the control system the given spectrum, but the shaker is always controlled by a signal according to the time calculated and according to procedures described in the preceding paragraphs. It is known that the transformation shock spectrum signal has an infinite number of solutions, and that very different signals can have identical response spectra. This phenomenon is related to the loss of most of the information initially contained in the signal x(t) during the calculation of the spectrum [MET 67]. It was also seen that the oscillatory shock pulses have a spectrum which presents an important peak to the frequency of the signal. This peak can, according to choice of parameters, exceed by a factor of 5 the amplitude of the same spectrum at the

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281

281

high frequencies, i.e. five times the amplitude of the signal itself. Being given a point of the specified spectrum of amplitude S, it is thus enough to have a signal versus time of amplitude S / 5 to reproduce the point. For a simple shaped shock, this factor does not exceed 2 in the most extreme case. All these remarks show that the determination of a signal of the same spectrum can lead to very diverse solutions, the validity of which one can question. This experiment makes it possible to note that, if any particular precaution is not taken, the signals created by these methods have in a general way one duration much larger and an amplitude much smaller than the shocks which were used to calculate the reference SRS (factor of about 10 in both cases). We saw in Chapter 4 that one can use a slowly swept sine to which the response spectrum is close to the specified shock spectrum [CUR 55], [DEC 76], [HOW 68]. In the face of such differences between the excitations, one can legitimately wonder whether the SRS is a sufficient condition to guarantee a representative test. It is necessary to remember that this equivalence is based on the behaviour of a linear system which one chooses a priori the Q factor. One must be aware that: - The behaviour of the structure is in practice far from linear and that the equivalence of the spectra does not lead to stresses of the same amplitude. Another effect of these non-linearities appears sometimes by the inaptitude of the system to correct the drive waveform to take account of the transfer function of the installation.

Figure 9.33. Example of shocks having spectra near the SRS

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Mechanical shock

- Even if the amplitudes of the peaks of acceleration and the maximum stresses of the resonant parts of the tested structure are identical, the damage by thefatigue generated by accumulation of the stress cycles is rather different when the number of shocks to be applied is significant. - The tests carried out by various laboratories do not have thesame severity. R.T. Fanrich [FAN 69] and K.J. Metzgar [MET 67] confronted this problem, based on the example of the signals of Figure 9.33. These signals A and B, with very different characteristics, have similar response spectra, within acceptable tolerances (Figure 9.34), but are they really equivalent? Does the response spectrum constitute a sufficient specification [SMA 74a], [SMA 75]? These questions did not receive a really satisfactory response. By prudence rather than by rigorous reasoning, many agree however on the need for placing parapets, while trying to supplement the specification defined by a spectrum with complementary data (AV, duration of the shock).

Figure 9.34.. SRS of the shocks shown in Figure 9.33 and their tolerances

9.11. Possible improvements To obtain a better specification one can, for example: -consider the signals of acceleration at the origin of the specified spectrum and specify if they are shocks with a velocity change or are oscillatory. The shock spectrum can, if it is sufficiently precise, give this information by its slope at very

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low frequencies. The choice of the type of simulation should be based on this information; -specify in addition to the spectrum other complementary data such as the duration of the signal time or the number of cycles (less easy) or one of the pre-set parameters in the following paragraphs, in order to deal with the spectrum and the couple amplitude/duration of the signal at the same time.

9.11.1. IES proposal To solve this problem, a commission of the IES (Institute of Environmental Sciences) proposed in 1973 four solutions consisting of specifying additional parameters [FAV 74], [SMA 74a], [SMA 75], [SMA 85] as follows: 1.

Limit the transient duration

This is a question of imposing minimum and maximum limits over the duration of the shock by considering that if the shock response spectrum is respected and if the duration is comparable, the damage should be roughly the same [FAN 69]. For complex shapes, one should pay much attention to how the duration is defined. 2. Require SRS at two different values of damping Damping is in general poorly known and has values different at each natural frequency of the structure. It can be supposed that if the SRS is respected for two different values of damping, for example E = 0.1 (Q = 5) and £ = 0.02 (Q = 25), the corresponding shock should be a reasonable simulation for any value of £. This approach also results in limiting the duration of the acceptable shocks. It is not certain in particular that a solution always exists, when the reference spectra come from smoothed spectra or an envelope of spectra of several different events. This approach intuitively remains attractive however; it is not much used except in the case of fast sweep sines. It deserves some atention being paid to it to evaluate its consequences over the duration of the drive waveform thus defined with shapes such as WAVSIN, SHOC, and a decaying sinusoid. 3. Specification of the allowable ratios between the peak of the SRS of each elementary waveform and the amplitude of the corresponding signal versus time The goal is here to prevent or encourage the use of an oscillatory type shock or a simple shape shock (with velocity change). It should, however, be recalled that if the shock spectrum is plotted at a sufficiently high frequency, the value of the spectrum reflects the amplitude of the shock in the time domain. This specification is thus redundant. It is, however, interesting, for it can be effective over the duration of the

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shock and thus lead to better being able to take account of the couple shock duration/amplitude. 4. To specifically exclude certain methods The test requester can give an opinion on the way of proceeding so that the test is correct. He/she can also exclude certain testing methods a priori when he/she knows that they cannot give good results. He/she can even remove the need of choice to the test laboratory and specify the method to be used, as well as the whole of the parameters that define the shock (for example, for components of damped sine type, frequencies, damping factors and amplitudes, etc.). Only laboratories equipped to use this method will be able to carry out this test, however [SMA 75] [SMA 85]. 9.11.2. Specification of a complementary parameter Several proposals have been made. The simplest suggests limiting arbitrarily the duration of the synthesized shock to 20 ms (with equal shock spectrum) [RUB 86]. Others introduce an additional parameter to better attempt to respect the amplitude shock duration. 9.11.2.1. Rms duration of the shock Let us set x(t) as a shock of Fourier transform of X(Q) . The rms duration is defined by [SMA 75]:

E is referred as the 'energy of the shock'. It is necessary that E is finite, ie that l

x(t) approaches zero more quickly than —, when t tends towards - o or + o.

r

In a general way, the rms duration of a transient is a function of the origin of times selected. To avoid this difficulty, one chooses the origin in order to minimize the rms duration. If another origin is considered, the rms duration can be minimized by introducing a time-constant T (shift) equal to [SMA 75]:

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285

The rms duration is a measure of the central tendency of a shock. Let us consider, for example a transient of a certain finite energy, composed of all the frequencies equal in quantity. An impulse (function 5) represents a shock of this type with one minimum duration. A low level random signal, long duration represents it with one maximum duration. The rms duration of most current transients is given in Table 9.1. This duration can also be calculated starting from [PAP 62]:

where

If the amplitude Xm(Q) of the Fourier transform of is specified, the rms d<j> duration minimum is given by — = 0, i.e. by <|)(Q) = constant. The constant can be dD zero. Equation [9.79] implies that the rms duration is related to the smoothness of the Fourier spectrum (amplitude and phase at the same time). The more the spectrum is smoothed, the shorter the rms duration.

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Mechanical shock Table 9.1. Rms duration of some often-used shocks [PAP 62] [SMA 75]

Function Rectangle

Equation x(t) = 1 for 0 < t < T x(t) = 0 elsewhere 7t t

Half-sine

TPS Triangle

Haversine

Decaying sinusoid

x(t) = sin —

Rms duration

0.29 T

for 0 < t < t

0.23 T

T

x(t) = 0 elsewhere x(t) = t / T f o r O < t < t x(t) = 0 elsewhere x(t) = l

2 |t|

T

0.19-c T

for-- 0 x(t) = 0 elsewhere for a « 1

0.16T

0.14 1 1 2aQ

9.11.2.2. Rms value of the signal T.J. Baca ([BAG 82] [BAC 83] [BAC 84] [BAC 86]) proposed characterizing the shock by its rms value, given by:

where T is the shock duration. The rms value is an "average" of the energy of the signal over the shock duration. NOTE: This method can be used to compare shocks by observing the variations of their rms value with time [CAN 80].

i.e., numerically

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287

where N = total number of points defining the signal.

9.11.2.3. Rms value in the frequency domain The rms value is a means of highlighting the contents of the frequency of a shock. It can also be calculated in the domain of the frequencies [BAC 82] [BAC 83] [BAC 84], by application of Parseval's theorem:

where the Fourier transform X(f) of x(t) is such that:

and where Fc is the cut-off frequency which limits the useful frequency range, taking into account the symmetry of |x(f )| , yielding:

NOTE: Also, according to the frequency:

where f is the current frequency at which the rms value is calculated. This expression thus makes it possible to highlight the contribution of all the frequencies (lower than Fc) for a shock of duration T.

288

Mechanical shock

9.11.2.4. Histogram of the peaks of the signal The shock spectrum does not give any direct information concerning the number of peaks of the signal x(t). The histogram of the peaks (a peak being the maximum or minimum between two zero passages) could constitute a complementary data (by possibly standardizing the ordinate of the curve by division by the amplitude of the largest peak). If this technique is used, it is important to specify the type of filtering which the signal underwent before establishment of the histogram, the comparison of two shocks making sense only if they were filtered under the same conditions (same frequency limits).

9.11.2.5. Use of the fatigue damage spectrum Being given a shock x(t), the fatigue damage spectrum D(f 0 ) (cf. Volume 5) is calculated starting from the response relative displacement z(t) of a linear onedegree-of-freedom mechanical system of natural frequency f0 and of quality factor Q:

The number N of cycles to the rupture is extracted from the Basquin law N a = C (b and C are constant functions of the material constituting the part). This yields, since a = K z

The damage at frequency f0 takes into account the number of peaks ni of amplitude Zj (histogram of the peaks of the response). The fatigue damage spectrum, which includes characteristics of the signal such as the duration and the histogram, is very complementary to the response spectrum and could thus be specified jointly with the shock response spectrum to define a test. 9.11.3. Remarks on the characteristics of response spectrum The need to specify a complementary parameter is much related to the fact that the specified spectrum is defined only for one limited frequency range. When it is

Control of a shaker using a shock response spectrum

289

calculated in a sufficiently large frequency interval, the response spectrum makes it possible to read of very useful information, such as: - At lowfrequencies,the velocity change AV associated with the shock (slope of the spectrum in the origin). This is not strictly exact if the damping is zero. However, for the usual values of £, the approximation is sufficient to determine if the shock is associated with a velocity change or not. - At highfrequencies,the magnitude of the signal varies with time. Consideration of these parameters should make it possible to obtain a simulation much more correctly akin to the real shock. It is thus desirable to specify the shocks with a spectrum calculated in a sufficiently broad frequency range to allow reading of these values on the curve. A signal which has a SRS very close to the SRS specified across all the frequency band has necessarily the same amplitude and the same velocity change as the origin of shock.

9.12. Estimate of the feasibility of a shock specified by its SRS Several methods were proposed to evaluate the feasibility of a shock specified using a shock response spectrum.

9.12.1. C.D. Robbins and E.P. Vaughan's method The shakers are limited with respect to force. Possible rms acceleration xrms is a function of the total mass M of the test package including the mounting fixture and attachments:

where Frms = maximum rms random force realizable on the shaker. The shaker can accept peak values equal to three times the rms force:

Each point of the SRS is simulated by an oscillatory signal having the frequency of the SRS at this point (Figure 9.35).

290

Mechanical shock

Figure 9.35. Simulation of a SRS point of reference

The spectrum of this elementary waveform has a peak at this frequency whose amplitude is R times its value at high frequencies, i.e. R tunes the amplitude of the signal in the time domain (R being a function of the type of signal used and of the number of oscillations). The possible maximum value SRSmax of the shock response spectrum is thus:

where R is equal to or higher than 2. If the penalizing value R = 2 is taken, we obtain

With the more realistic value R = 4, we have

K.J. Metzgar [MET 67], C.D. Robbins and P.E. Vaughan [ROB 67] checked by experiment that it was possible to reach spectrum values higher than l000g (they specified neither the mass nor the type of shaker). Tests confirming this value in addition were carried out with one 135kN shaker and a test item mass of 200kg.

Control of a shaker using a shock response spectrum

291

Problems of non-linearity After input of the specified spectrum on the control system, the calculation of the drive signal is normally carried out at low amplitude, for example 10% of the specification, to avoid damaging the test item while making it undergo all the shocks necessary to the development procedure. Once the spectrum obtained is considered to be satisfactory, one applies the shock to the test item. For small test items or larger test items of dead mass type, it can be agreed that the passage from level 1/10th to level 1 is effected linearly except for 10%. For heavier test resonant items, it is preferable if possible, to use a dummy item which is representative for the development of the test in order to guard against possible significant non-linearity, and to carry out intermediate level shocks. 9.12.2. Evaluation of the necessary force, power and stroke The dynamic force necessary to carry out a shock is related to acceleration on the table by the relation [DES 83]:

where - F(Q) and X(Q) are the Fourier transforms of the force F(t) and acceleration x(t) respectively; - m is the total mass of the test package (armature, table, fixture and test item); - H(O) is the transfer function of the test item. It is necessary to take account of the weight of the moving unit if the shaker, for a vertical configuration, supports this load directly. In most practical cases, the test item can be represented, approximately, by an only one-degree-of-freedom system, of natural frequency osp and damping factor £sp; this transfer function is then written:

where

292

Mechanical shock

Figure 9.36. One-degree-of-freedom model of an equipment tested on a shaker

Let us set SRS(f) as the SRS specified calculated for a damping equal to £. By definition, each point of this spectrum gives the largest response of a linear onedegree-of-freedom transfer function system:

For the usual values of damping factor £, we have, at the resonance (f = f0), H(f)«

. Knowing in addition that the response of a mechanical system is

2i$ related to the excitation by the relation R(f) = H(f) X(f), the maximum of the response being the point of the SRS, it comes, by noting a(Q) a function whose module is the SRS ( SRS(ft) = |o(Q)|):

yielding

The necessary maximum force is obtained in a conservative way for the maximum value of SRS.

Control of a shaker using a shock response spectrum

293

Let us set V(Q) as the Fourier spectrum of the velocity of the table during the shock movement. The power necessary is given by the real part of F V(Q): p = m[F V(Q)]

Knowing that X(Q) = iQV(Q), the necessary maximum power is estimated conservatively from [9.97] and [9.99] by

Conservatively,

Yielding

Taking into account [9.99], the Fourier transform of the displacement during the shock can be written:

294

Mechanical shock

This yields an estimate of the maximum stroke:

Relations [9.101], [9.102] and [9.103] show that, if n is the slope of the SRS, the power, the stroke and the force are respectively of the form pmax This 1 1 is a function increasing for n > — and decreasing for n < —. 2 2

Fm=k3 (where kj,k 2 and k3 are constants). Thus, in logarithmic scales: - the necessary power is given by the highest point of contact between the SRS and the line of slope !£; - the maximum stroke corresponds to the highest point of contact of the SRS with a line of slope 2; - the force at the highest point of contact between the SRS and a line of zero slope.

Figure 9.37. Calculation of the force, the power and the stroke necessary to carry out a shock of given SRS

The coordinates of these points of contact are used in relations [9.101], [9.102] and [9.103] to calculate the maximum force, the necessary power and the stroke.

Control of a shaker using a shock response spectrum 295 If the damping £sp of the test equipment is not known, one can use the approximate value of 0.05.

NOTES. 1. In the case of a specification defined by a velocity SRS, one will consider in the same way the points of intersection of the spectrum with [DES 83]: - a line of slope -1/2 for maximum power; - a line of slope 1 for the stroke; - a line of slope -1 for the force. Table 9.2. Slopes of the straight lines allowing evaluation of the possibility of carrying out a shock of a given SRS on a shaker

Quantity Power Stroke Force

Slope Relative displacement SRS (or absolute acceleration) 1

2 2 0

Velocity SRS

1 2 1 -1

2. If the dynamic behaviour of the material is known and if it is known that, in the frequency domain studied, the material does not have any resonance, one will be able to take £,sp = 0.5 for transfer function H(Q) equal to 1. 3. This method provides only one order of magnitude of the force, the power and the stroke necessary to carry out a shock specified from an SRS on shaker because: - the relations used are approximations; - according to the shape of the elementary waveform used to build the drive waveform, the actually generated waveform will require conditions slightly different from those estimated by this test.

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Appendix

Similitude in mechanics

In certain cases where the material is too heavy or bulky to be tested using the usual test facilities, it is possible to carry out tests on a specimen on a reduced scale. This method isoften used, for example in certification tests applied to containers of transport of nuclear matter which comprise, for example, of a test of free fall of 9m on a concrete flagstone (type B containers). The determination of the model can be made according to two different assumptions.

Al. Conservation of materials With this assumption, which most frequently used, the materials of the object on scale 1 and the model on a reduced scale are the same. The stresses and the velocities (in particular the velocity of sound in each material) are retained. Let us set L as a length on scale 1 and i the length corresponding to the reduced scale. The scale ratio [BAK 73], [BRI 31], [FOC 53], [LAN 56], [MUR 50], [PAS 67] and [SED 72] is:

L i The velocity being retained, i.e. V = — = v = -, the changing duration is given by T t

298

Mechanical shock

acceleration

becomes

In a similar way, analysis of the dimensions leads to the following relations:

Table A.I. Scale factor of various parameters

Surface

s = )?S

Volume

w = tf W

Density

Unchanged

Mass

m=X M

Frequency

3> cp = — X

Force

f=X2 F

Energy

e=X3 E

Power

p = X2 P

Pressure (stress)

Unchanged

Several requirements exist at the time of the definition of the model on a reduced scale, such as: - the clearance limits on the scale, the manufacturing tolerances, the states of surface; - for the screwed parts, the account taken, if possible, of the number of screws (dimensions on the scale). If the ratio of the scale leads to too small screws, one can replace them by screws of bigger size, of lower number, taking account of the total area of the cross-section of the whole of the screws; - in the case of stuck parts, taking account of the bonding strength in similitude;

Appendix

299

- in the case of measurements taken on the model by similarity, taking into account the similitude of the frequency bandwidth of the measuring equipment. This is not always easy to do, in particular in the case of impact tests where the shock to be measured already has on scale 1 high frequency contents; - the effects of gravity (when they are not negligible); - as far as possible the choice of sensors in similitude (dimensions, mass etc). It is advisable to also consider the scale factor with respect to their functional characteristics (acceleration and the frequency are 1 / X times on a reduced scale). Difficulties can arise whose importance need to be evaluated according to the case studied. For example, the gradient of the stresses in the part is not taken into account. If G is this gradient on scale 1 and if £ is the stress, we have on a reduced scale

i.e.

A2. Conservation of acceleration and stress We always have

Acceleration being preserved, it becomes, since

yielding

Table A.2 summarizes the main relations.

300

Mechanical shock Table A.2. Scale factors

Velocity

v = X3/2V

Surface

s = A2S

Volume

w = ?i3 W

Density

D d=— K

Mass

m=X M $ ~ /—

Frequency

(p

Force

f=X2 F

Energy

e = A? E

Power

p = X5/2P

Pressure (stress)

Unchanged

4\

Mechanical shock tests: a brief historical background

The very first tests were performed by the US Navy around 1917 [PUS 77] [WEL 46], Progress, which was slow up to World War II, accelerated as from the 1940s. The following list gives a few of the major dates. 1932

First publication on the shock response spectrum for the study of earthquakes.

1939

First high impact shock machine (pendular drop-hammer) for the simulation of the effects of submarine explosions on on-board equipment [CLE 72] [OLI 47].

1941

Development of a 10-foot free fall test method [DEV 47].

1945

Drafting of the first specifications for aircraft equipment [KEN 51] under various environmental conditions (A.F. Specification 410065).

1947

Environmental measurements on land vehicles for drafting specifications [|PRI 47].

1947

Use of an air gun to simulate shocks on electronic components (Naval Ordnance Laboratory) [DEV 47].

1948

Free fall machine on sand, with monitoring of the amplitude and duration of the shock (US Air Force) [BRO 61].

1955

Use of exciters for shock simulation (reproduction of simple shapeshocks) [WEL 61].

302

Mechanical shock

1964

Taking into account shocks of pyrotechnic origin, demonstrating the difficulty of simulating them with classic facilities. Development of special facilities [BLA 64].

1966

Initial research into shock simulation on exciters driven from a shock response spectrum [GAL 66]. These methods were only fully developed in the mid-1970s.

1984

The shock response spectrum becomes the benchmark in the MIL-STD 810 D standard for the definition of specifications.

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[BAC 84]

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[CAR 74] CARDEN J., DECLUE T.K., and KOEN P.A., 'A vibro-shock test system for testing large equipment items', The Shock and Vibration Bulletin, August 1974, Supplement 1, p. 1/26. [CAR 77] [CAV 64]

CARUSO H., 'Testing the Viking lander', The Journal of Environmental Sciences, March/April 1977, p. 11/17. CAVANAUGH R., 'Shock spectra', IESProceedings, April 1964, p. 89/95.

[CHA 94]

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[CLA 65]

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[CRI 78] [CRU 70]

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[CZE 67] [DAV 85]

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[DAV 92]

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[DEC 76]

DE CAPUA N.J., HETMAN M.G., LIu S.C., 'Earthquake test environment Simulation and procedure for communications equipment', The Shock and Vibration Bulletin, 46, Part 2, August 1976, p. 59/67. DE SILVA C.W., 'Selection of shaker specifications in seismic qualification tests', Journal of Sound and Vibration, Vol. 91, n° 1, p. 21/26, November 1983. 'Development of NOL shock and vibration testing equipment', The Shock and Vibration Bulletin, n° 3, May 1947.

[DES 83] [DEV 47] [DEW 84]

DE WINNE J., 'Etude de la validite' du critere de spectre de reponse au choc', CESTA/EXn° 040/84, 27/02/1984.

[DIN 64]

DINICOLA D.J., 'A method of producing high-intensity shock with an electrodynamic exciter', IES Proceedings, 1964, p. 253/256. DOKAINISH M.A. and SUBBARAJ K., 'A survey of direct time-integration methods in computational structural dynamics: I Explicit methods'; 'II Implicit methods', Computers & Structures, Vol. 32, n° 6, 1989, p. 1371/1389 and p. 1387/1401.

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HALE M.T. and ADHAMI R., 'Time-frequency analysis of shock data with application to shock response spectrum waveform synthesis', Proceedings of the IEEE, Southeastcon, 91 CH 2998-3, p. 213/217, April 1991. HAY W.A and OLIVA R.M., 'An improved method of shock testing on shakers', IES Proceedings, 1963, p 241/246. HlEBER G.M. and TUSTIN W., Understanding and measuring the shock response spectrum; Part 1: Sound and Vibration, March 1974, p. 42/49; Pan 2: Sound and Vibration, April 1975, p. 50/54. HOWLETT J.T. and MARTIN D.J., 'A sinusoidal pulse technique for environmental vibration testing', NASA-TM-X-61198, or The Shock and Vibration, n° 38, Part 3, 1968, p. 207/212.

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[HOW 68]

[HUG]

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chocs

au

moyen

de

vibrateurs

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[HUG 83a]

HUGHES T. J. R. and BELYTSCHKO T., 'A precis of developments in computational methods for transient analysis', Journal of Applied Mechanics, Vol. 50, December 1983, p. 1033/1041. [HUG 83b] HUGUES M.E., 'Pyrotechnic shock test and test simulation', The Shock and Vibration Bulletin, n° 53, Part 1, May 1983, p. 83/88. [IKO 64] IKOLA A.L., 'Simulation of the pyrotechnic shock environment', The Shock and Vibration Bulletin, n° 34, Part 3, December 1964, p. 267/274. [IMP] IMP AC 6060F - Shock test machine - Operating manual, MRL 335 Monterey Research Laboratory, Inc. [IRV 86] IRVINE M., 'Duhamel's integral and the numerical solution of the oscillator equation', Structural Dynamics for the Practising Engineer, Unwin Hyman, London, p. 114/153,1986. [JEN 58] JENNINGS R.L., 'The response of multi-storied structures to strong ground motion', MSc Thesis, University of Illinois, Urbana, 1958. [JOU 79] JOUSSET M., LALANNE C. et SGANDURA R., 'Elaboration des specifications de chocs mecaniques', Rapport CEA/DAM Z/EXDO 78086, 1979. [KEE73] KEEGAN W.B., 'Capabilities of electrodynamic shakers when used for mechanical shock testing', NASA Report N74 - 19083, July 1973. [KEE 74] KEEGAN W.B., 'A statistical approach to deriving subsystem specifications', IES Proceedings, 1974, p. 106/117. [KEL69] [KEN 51]

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KELLY R.D. and RICHMAN G., 'Principles and techniques of shock data analysis', The Shock and Vibration Information Center, SVM5, 1969. KENNARD D.C., 'Measured aircraft vibration as a guide to laboratory testing', W.A.D.C. A.F. Technical Report n° 6429, May 1951, or 'Vibration testing as a guide to equipment design for aircraft', The Shock and Vibration Bulletin, n°l I.February 1953. KERN D.L. and HAYES C.D., 'Transient vibration test criteria for spacecraft hardware', The Shock and Vibration Bulletin, n° 54, Part 3, June 1984, p. 99/109. KIRKLEY E.L., 'Limitations of the shock pulse as a design and test criterion', The Journal of Environmental Sciences, April 1969, p. 32/36. LALANNE C., 'Recueil de spectres de Fourier et de choc de quelques signaux de forme simple', CEA/CESTA/Z - SDA - EX DO 20, 22 August 1972. LALANNE C., 'La simulation des environnements de chocs mecaniques', Rapport CEA-R-4682 (1) et (2), 1975. LALANNE C., JOUSSET M., SGANDURA R., 'Elaboration des specifications de chocs mecaniques', CEA/CESTA Z/EX DO 78086, December 1978. LALANNE C., 'Realisation de chocs mecaniques sur excitateurs electrodynamiques et verins hydrauliques. Limitations des moyens d'essais', CESTA/EX/ME/ 787, 07/09/1983, Additif CESTA/EX/ME 1353, 28/10/1983, 2e additif: COUDARD B. CESTA/EX/ENV 1141, 16/12/1987. LALANNE C., 'La simulation des chocs mecaniques sur excitateurs', CESTA/DT/EX/EC DO n° 1187, December 1990.

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[PAP 62] [PAS 67]

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[POW 74]

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[USH 72]

USHER T., 'Reproduction of shock spectra with electrodynamic shakers', Sound and Vibration, January 1972, p. 21/25.

[VAN 72]

VANMARCKE E.H. and CORNELL C.A., 'Seismic risk and design response spectra', Safety and Reliability of Metal Structures, ASCE, 1972, p. 1/25. VIGNESS I., 'Some characteristics of Navy High Impact type shock machines', SESA Proceedings, Vol. 5, n° 1, 1947. VIGNESS I., 'Navy High Impact shock machines for high weight and medium weight equipment', US Naval Research Laboratory, Washington DC, NRL Report 5618, AD 260-008, June 1961.

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[VIG 63]

VIGNESS I. and CLEMENTS E.W., 'Sawtooth and half-sine shock impulses from the Navy shock machine for mediumweight equipment', US Naval Research Laboratory, NRL Report 5943, June 3, 1963. WALSH J.P. and BLAKE R.E., 'The equivalent static accelerations of shock motions', Naval Research Laboratory, NRL Report n° F 3302, June 21, 1948. WELCH W.P., 'Mechanical shock on naval vessels', NAVSHIPS 250 - 660 - 26, March 1946. WELLS R.H. and MAUER R.C., 'Shock testing with the electrodynamic shaker', The Shock and Vibration Bulletin, n° 29, Part 4, 1961, p. 96/105.

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Index

arbritary triangular pulse Fourier transform 12 background noise 90 bump 2 conservation of acceleration and stress 299 of materials 297 digitization frequency, choice of 52 DuhameFs integral 29

spectra 41 et seq haversine definition 136 shock 4 IBS proposal 283 initial peak saw tooth shock (IPS) definition 145 Fourier transform 10 shock motion 145 iterations 250 jerk 2

energy spectrum 5 extreme response spectrum 47 fast swept sine, use of 274 final peak saw tooth (FPS) 3 Fisher, D.K. & Posehn, M.R. expression 253 Fourier spectrum 5 transform 4 arbritary 12 definition 4 fast (FFT) 18 half-sine shock pulse 7 importance of 17 initial peak saw tooth pulse (IPS) 10 practical calculations (undigitized, digitized) 18 etseq rectangular pulse 14 reduced 6 simple shocks 6 et seq half-sine shock pulse 3 definition 121 Fourier transform 6

Kem, D.L. & Hayes, C.D. function 251 linear one-degree-of-freedom system response defined by acceleration 27, force 26 to simple shocks 33 arbitrary triangular 35 half-sine 33 initial peak saw tooth 34 rectangular pulse 34 trapezoidal 36 versed-sine 33 metal to metal impact, simulation by 230 pseudo-velocity 38, 62 et seq pulse, end of 69, 72 pyroshocks, simulation of 227 et seq barrel tester 228 conventional shock machine, by 232 electrodynamic shaker simulation 231 flower pot 233 metal to metal 229

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rectangular pulse Fourier transform 14 rectangular shock 3 definition 139 motion study 139 residual spectrum 74 Robbins, C.D. & Vaughan, E.P. method 289 shaker, shock generation 189 et seq advantages 190 control by using SRS 235 et seq numerical methods 237 parallel filter method 236 complementary parameter 284 decaying sinusoid 239 electrodynamic, limitations of 191 electrohydraulic, use of 193 Kern and Hayes function 251 SHOC waveforms 267 ZERD function 253 pre- and post-shocks 193 et seq abacuses 212 incidence 227 optimized 216 quality of simulation 220 shape, influence of 213 symmetrical, various 198 et seq time history response, one-degreeof-freedom system 220 principles 189 reduced response spectra 257 response spectrum 265 SHOC waveform 267 time history response 266 WAVSIN waveform 259 SHOC waveform 267 definition 267 response waveform 271 time history synthesis 272 velocity and displacement 270 shock 1 analysis 1 et seq defined by force 26 non-zero velocity change 60 pyrotechnic 88, 227 et seq severity 23 relative, and Fourier 86 simulation, use of fast swept sine 108 modulated random noise 112 modulated sine pulses, restitution by 115 random vibration 113 shaker 119 swept-sine 109

and stresses 23 in time domain 4 shock machines 149 et seq accelerated fall 158 electrodynamic exciter 151 exotic 151 free fall 151, 156 high impact 149, 160 lightweight 160 medium weight 162 impact 151 Collins 153 impulse 151 limitations 183 pendular 149, 154 pneumatic 153, 163 programmers 165 half-sine 165 limitations 180 rectangular-trapezoidal 180 TPS 173 steel punch method 179 propagation time, shock wave 168 rebound 169 sand-drop 149, 150 test facilities 164 universal 155, 181 shock motion 122 impact mode 126 perfect rebound 129 without rebound 128 impulse mode 124 shock response spectrum (SRS) calculation 48 et seq algorithms 47 domains 23 et seq, 59 and extreme response spectrum (ERS) 47 and Fourier spectrum 81 high frequency 75 use of 53 several-degrees-of-freedom systems 55 shock test specification 95 et seq measure signal, simplification 96 signal compensation 244 simple shocks, kinematics of 121 spectra, shock response (SRS) absolute acceleration 37 amplitude 101 calculation 90 correction factor 79 damping, choice of 78 influence 77 duration 101 frequency range, choice of 80 negative (or maximum negative) 39

Index positive (or maximum positive) 38 primary (initial) negative and positive 38, 68 relative displacement 37, 38 response 37 secondary or residual 38, 66 and Fourier transform 83 shape, choice of 100 specification 99 difficulties in 105 synthesis 98 use of 100 with various pulses 41 et seq spectral charts 81 standard response spectra 39 TSP pulse 43 stress strain curve 26 terminal peak saw tooth shock (IPS) 3 definition 142 pulse 65

transient signal 2 trapezoidal shock 3 Fourier transform 15 versed-sine pulse, definition 136 shock 4 Fourier transform 8 spectra 42 WAVSIN waveform 259 et seq ZERD function 253 and decaying sinusoid 257 zero shift 92

317

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Synopsis of five volume series: Mechanical Vibration and Shock

This is the second volume in this five volume series. Volume 1 is devoted to sinusoidal vibration. The responses, relative and absolute, of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms defined. By placing the properties of sinusoidal vibrations in the contexts of the environment and of laboratory tests, the transitory and steady state response of a single-degree-offreedom system with viscous and then with non-linear damping is evolved. The various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of the sweep rate are shown and a rule for choice of this rate deduced from it. Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be made with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution. Volume 3 examines the analysis of random vibration, which encompass the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing,

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overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). Having established the relationships which provide the response of a linear system with one degree of freedom to a random vibration, Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behaviour of a material subjected to fatigue, the laws of damage accumulation, together with the methods for counting the peaks of the response, used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal. The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non linear accumulation law, etc.). Volume 5 is more especially dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification as from the life cycle profile of the equipment is then detailed, taking account of an uncertainty factor, designed to cover the uncertainties related to the dispersion of the real environment and of the mechanical strength, and of another coefficient, the test factor, which takes into account the number of tests performed to demonstrate the resistance of the equipment. This work is intended first and foremost for engineers and technicians working in design teams, which are responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance, following the choice of the most suitable simulation means.

Random Vibration

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MechanicalVibration & Shock

Random Vibration Volume III

Christian Lalanne

HPS

HERMES PENTON SCIENCE

First published in 1999 by Hermes Science Publications, Paris First published in English in 2002 by Hermes Penton Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: Hermes Penton Science 120PentonvilleRoad London Nl 9JN © Hermes Science Publications, 1999 © English language edition Hermes Penton Ltd, 2002 The right of Christian Lalanne to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A CIP record for this book is available from the British Library. ISBN 1 9039 9605 8

Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn www.biddles.co.uk

Contents

Introduction

xiii

List of symbols

xv

1 Statistical properties of a random process

1

1.1. Definitions 1.1.1. Random variable 1.1.2. Random process 1.2. Random vibration in real environments 1.3. Random vibration in laboratory tests 1.4. Methods of analysis of random vibration 1.5. Distribution of instantaneous values 1.5.1. Probability density 1.5.2. Distribution function 1.6. Gaussian random process 1.7. Rayleigh distribution 1.8. Ensemble averages: ''through the process' 1.8.1. n order average 1.8.2. Central moments 1.8.3. Variance 1.8.4. Standard deviation 1.8.5. Autocorrelation function 1.8.6. Cross-correlation function 1.8.7. Autocovariance 1.8.8. Covariance 1.8.9. Stationarity 1.9. Temporal averages: 'along the process' 1.9.1. Mean 1.9.2. Quadratic mean - rms value 1.9.3. Moments of order n

1 1 1 2 2 3 4 4 5 6 7 8 8 9 9 10 12 12 12 13 13 15 15 16 18

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1.9.4. Variance - standard deviation 1.9.5. Skewness 1.9.6. Kurtosis 1.9.7. Temporal autocorrelation function 1.9.8. Properties of the autocorrelation function 1.9.9. Correlation duration 1.9.10. Cross-correlation 1.9.11. Cross-correlation coefficient 1.9.12. Ergodicity 1.10. Significance of the statistical analysis (ensemble or temporal) 1.11. Stationary and pseudo-stationary signals 1.12. Summary chart of main definitions (Table 1.2)

18 18 19 22 25 27 28 28 29 30 30 30.

2 Properties of random vibration in the frequency domain

33

2.1. Fourier transform 2.2. Power spectral density 2.2.1. Need 2.2.2. Definition 2.3. Cross-power spectral density 2.4. Power spectral density of a random process 2.5. Cross-power spectral density of two processes 2.6. Relation between PSD and correlation function of a process 2.7. Quadspectrum - cospectrum 2.8. Definitions 2.8.1. Broad-band process 2.8.2. White noise 2.8.3. Band-limited white noise 2.8.4. Narrow-band process 2.8.5. Pink noise 2.9. Autocorrelation function of white noise 2.10. Autocorrelation function of band-limited white noise 2.11. Peak factor 2.12. Standardized PSD/density of probability analogy 2.13. Spectral density as a function of time 2.14. Relation between PSD of excitation and response of a linear system 2.15. Relation between PSD of excitation and cross-power spectral density of response of a linear system 2.16. Coherence function 2.17. Effects of truncation of peaks of acceleration signal 2.17.1. Acceleration signal selected for study 2.17.2. Power spectral densities obtained

33 35 35 36 42 43 43 44 45 45 45 46 47 47 48 48 50 52 54 54 55 57 58 60 60 61

Contents

vii

3 Rms value of random vibration

63

3.1. Rms value of a signal as function of its PSD 3.2. Relations between PSD of acceleration, velocity and displacement 3.3. Graphical representation of PSD 3.4. Practical calculation of acceleration, velocity and displacement rms values 3.4.1. General expressions 3.4.2. Constant PSD in frequency interval 3.4.3. PSD comprising several horizontal straight line segments 3.4.4. PSD defined by a linear segment of arbitrary slope 3.4.5. PSD comprising several segments of arbitrary slopes 3.5. Case: periodic signals 3.6. Case: periodic signal superimposed onto random noise

63 66 69 70 70 71 72 73 81 82 84

4 Practical calculation of power spectral density

87

4.1 Sampling of the signal 4.2. Calculation of PSD from rms value of filtered signal 4.3. Calculation of PSD starting from Fourier transform 4.3.1. Maximum frequency 4.3.2. Extraction of sample of duration T 4.3.3. Averaging 4.3.4. Addition of zeros 4.4. FFT 4.5. Particular case of a periodic excitation 4.6. Statistical error 4.6.1. Origin 4.6.2. Definition 4.7. Statistical error calculation 4.7.1. Distribution of measured PSD 4.7.2. Variance of measured PSD 4.7.3. Statistical error 4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis 4.7.5. Confidence interval 4.7.6. Expression for statistical error in decibels 4.7.7. Statistical error calculation from digitized signal 4.8. Overlapping 4.8.1. Utility 4.8.2. Influence on number of degrees of freedom 4.8.3. Influence on statistical error 4.8.4. Choice of overlapping rate 4.9. Calculation of PSD for given statistical error 4.9.1. Case: digitalization of signal is to be carried out 4.9.2. Case: only one sample of an already digitized signal is available

87 90 91 92 92 98 98 100 101 102 102 103 104 104 106 107 108 111 124 126 128 128 129 130 132 134 134 135

viii

Random vibration

4.10. Choice of filter bandwidth 4.10.1. Rules 4.10.2. Bias error 4.10.3. Maximum statistical error 4.10.4. Optimum bandwidth 4.11. Probability that the measured PSD lies between ± one standard deviation 4.12. Statistical error: other quantities 4.13. Generation of random signal of given PSD 4.13.1. Method 4.13.2. Expression for phase 4.13.2. I.Gaussian law 4.13.2.2. Other laws

137 137 138 143 145 149 150 154 154 155 155 155

5 Properties of random vibration in the time domain

159

5.1. Averages 5.1.1. Mean value 5.1.2. Mean quadratic value; rms value 5.2. Statistical properties of instantaneous values of random signal 5.2.1. Distribution of instantaneous values 5.2.2. Properties of derivative process 5.2.3. Number of threshold crossings per unit time 5.2.4. Average frequency 5.2.5. Threshold level crossing curves 5.3. Moments 5.4. Average frequency of PSD defined by straight line segments 5.4.1. Linear-linear scales 5.4.2. Linear-logarithmic scales 5.4.3. Logarithmic-linear scales , 5.4.4. Logarithmic-logarithmic scales 5.5. Fourth moment of PSD defined by straight line segments 5.5.1. Linear-linear scales 5.5.2. Linear-logarithmic scales 5.5.3. Logarithmic-linear scales 5.5.4. Logarithmic-logarithmic scales 5.6. Generalization; moment of order n 5.6.1. Linear-linear scales 5.6.2. Linear-logarithmic scales 5.6.3. Logarithmic-linear scales 5.6.4. Logarithmic-logarithmic scales

159 159 160 162 162 163 167 171 174 181 184 184 186 187 188 189 189 190 191 192 193 193 193 193 194

:

Contents

ix

6 Probability distribution of maxima of random vibration

195

6.1. Probability density of maxima 6.2. Expected number of maxima per unit time 6.3. Average time interval between two successive maxima 6.4. Average correlation between two successive maxima 6.5. Properties of the irregularity factor 6.5.1. Variation interval 6.5.2. Calculation of irregularity factor for band-limited white noise 6.5.3. Calculation of irregularity factor for noise of form G = Const, f b.... 6.5.4. Study: variations of irregularity factor for two narrow band signals 6.6. Error related to the use of Rayleigh's law instead of complete probability density function 6.7. Peak distribution function 6.7.1. General case 6.7.2. Particular case of narrow band Gaussian process 6.8. Mean number of maxima greater than given threshold (by unit time) 6.9. Mean number of maxima above given threshold between two times 6.10. Mean time interval between two successive maxima 6.11. Mean number of maxima above given level reached by signal excursion above this threshold 6.12. Time during which signal is above a given value 6.13. Probability that a maximum is positive or negative 6.14. Probability density of positive maxima 6.15. Probability that positive maxima is lower than given threshold 6.16. Average number of positive maxima per unit time 6.17. Average amplitude jump between two successive extrema

195 203 206 207 208 208 212 215

7 Statistics of extreme values

241

7.1. Probability density of maxima greater than given value 7.2. Return period 7.3. Peak £p expected among Np peaks 7.4. Logarithmic rise 7.5. Average maximum of Np peaks 7.6. Variance of maximum 7.7. Mode (most probable maximum value) 7.8. Maximum value exceeded with risk a 7.9. Application to case of centred narrow band normal process 7.9.1. Distribution function of largest peaks over duration T 7.9.2. Probability that one peak at least exceeds a given threshold 7.9.3. Probability density of the largest maxima over duration T 7.9.4. Average of highest peaks 7.9.5. Standard deviation of highest peaks

241 242 242 243 243 243 244 244 244 244 247 247 250 252

218 220 222 222 224 226 230 230 231 234 235 236 236 236 237

x

Random vibration

7.9.6. Most probable value 7.9.7. Value of density at mode 7.9.8. Expected maximum 7.9.9. Average maximum 7.9.10. Maximum exceeded with given risk a 7.10. Wide-band centered normal process 7.10.1. Average of largest peaks 7.10.2. Variance of the largest peaks 7.11. Asymptotic laws 7.11.1. Gumbel asymptote 7.11.2. Case: Rayleigh peak distribution 7.11.3. Expressions for large values of Np 7.12. Choice of type of analysis 7.13. Study of envelope of narrow band process 7.13.1. Probability density of maxima of envelope 7.13.2. Distribution of maxima of envelope 7.13.3. Average frequency of envelope of narrow band noise

253 254 254 254 255 257 257 259 260 261 262 263 264 267 267 272 274

Appendices

279

Al. Laws of probability Al.l. Gauss's law A1.2. Log-normal law A1.3. Exponential law A1.4. Poisson's law A1.5. Chi-square law A1.6.Rayleigh'slaw Al.7. Weibull distribution A1.8. Normal Laplace-Gauss law with n variables A1.9. Student law A.2. 1/n* octave analysis A2.1. Center frequencies A2.1.1. Calculation of the limits in l/n* octave intervals A2.1.2. Width of the interval Af centered around f. A2.2. Ordinates A.3. Conversion of an acoustic spectrum into a power spectral density A3.1. Need A3.2. Calculation of the pressure spectral density A3.3. Response of an equivalent one-degree-of-freedom system A.4. Mathematical functions A4.1. Error function A4.1.1. First definition A4.1.2. Second definition A4.2. Calculation of the integral Je a x /x n dx

279 279 280 284 284 286 287 289 290 294 295 295 295 296 297 299 299 300 301 303 303 303 309 314

Contents

xi

A4.3. Euler's constant A.5. Complements to the transfer functions A5.1. Error related to digitalization of transfer function A5.2. Use of a fast swept sine for the measurement of transfer functions A5.3. Error of measurement of transfer function using a shock related to signal truncation A5.4. Error made during measurement of transfer functions in random vibration A5.5. Derivative of expression of transfer function of a one-degree-of-freedom linear system

314 315 315

325

Bibliography

327

Index

341

Synopsis of five volume series

345

319 321 323

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Introduction

The vibratory environment found in the majority of vehicles essentially consists of random vibrations. Each recording of the same phenomenon results in a signal different from the previous ones. Characterization of a random environment therefore requires an infinite number of measurements to cover all the possibilities. Such vibrations can only be analyzed statistically. The first stage consists of defining the properties of the processes comprising all the measurements, making it possible to reduce the study to the more realistic measurement of single or several short samples. This means evidencing the stationary character of the process, making it possible to demonstrate that its statistical properties are conserved in time, then its ergodicity, with each recording representative of the entire process. As a result, only a small sample consisting of one recording has to be analysed (Chapter 1). The value of this sample gives an overall idea of the severity of the vibration, but the vibration has a continuous frequency spectrum that must be determined in order to understand its effects on a structure. This frequency analysis is performed using the power spectral density (PSD) (Chapter 2) which is the ideal tool for describing random vibrations. This spectrum, a basic element for many other treatments, has numerous applications, the first being the calculation of the rms value of the vibration in a given frequency band (Chapter 3). The practical calculation of the PSD, completed on a small signal sample, provides only an estimate of its mean value, with a statistical error that must be evaluated. Chapter 4 shows how this error can be evaluated according to the analysis conditions, how it can be reduced, before providing rules for the determination of the PSD.

xiv Random vibration

The majority of signals measured in the real environment have a Gaussian distribution of instantaneous values. The study of the properties of such a signal is extremely rich in content (Chapter 5). For example, knowledge of the PSD alone gives access, without having to count the peaks, to the distribution of the maxima of a random signal (Chapter 6), and in particular to the response of a system with one degree-of-freedom, which is necessary to calculate the fatigue damage caused by the vibration in question (Volume 4). It is also used to determine the law of distribution of the largest peaks, in itself useful information for the pre-sizing of a structure (Chapter 7).

List of symbols

The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have another meaning which will be defined in the text to avoid any confusion.

a A A(t) b c Ej( ) E2( ) Erf E( ) f fsamp. fmax f0 g G G( ) G( )

Threshold value of 40 or maximum of 40 Maximum of A(t) Envelope of a signal Exponent Viscous damping constant First definition of error function Second definition of error function Error function Expected function of... Frequency of excitation Sampling frequency Maximum frequency Natural frequency Acceleration due to gravity Particular value of power spectral density Power spectral density for 0 < f
h h(t)

k K

t_ ~t

Cross-power spectral density Interval (f/f 0 ) or f 2 /fj Impulse response Transfer function Stiffness Number of subsamples Value of 40 Mean value of 41) Average maximum of Np peaks Rms value of 40

'4)

Rms value of C(t) Generalized excitation (displacement) First derivative of 40

L Lr

Second derivative of 40 Given value of 40 Rms value of filtered signal

L( Q)

Fourier transform of 4 0

e(t)

xvi

L( Q) m M Ma Mn n

Random vibration

Fourier transform of i( t) Mean Number of points of PSD Average number of maxima which exceeds threshold per unit time Moment of order n Order of moment or number of degrees of freedom

na

n^ no n

o

np N Np Na

NQ

Average number of crossings of threshold a per unit time Average number of crossings of threshold a with positive slope per unit time Average number of zerocrossings per unit time Average number of zerocrossings with positive slope per second (average frequency) Average number of maxima per unit time Number of curves or number of points of signal or numbers of dB Number of peaks Average numbers of crossings of threshold a with positive slope for given length of time Average number of zerocrossings with positive slope for given length of time

PN() P PSD

Q(u) r rms r(0 R

R*u

R(f) s Sn

Average number of positive maxima for given length of time Probability density Probability density of largest maximum over given duration Probability Power spectral density i < •"-"-' ' VI -r2 Probability that a maximum is positive Probability that a maximum is negative Probability density of maxima of £(t) Q factor (quality factor) Distribution function of maxma o Probability that a maximum is higher than given threshold Irregularity factor Root mean square (value) Temporal window Slope in dB/octave Cross-correlation function between ^(t) and u(t) Fourier transform of r(t) Auto-correlation function Standard deviation Value of constant PSD

List of symbols s( ) t T Ta u U

0 u(t) u(t) u(t) v

rms

x

rms

x(t)

XJTOS xm a Xn St §( )

Power spectral density for -oo < f < +00 Time Duration of sample of signal Average time between two successive maxima Ratio of threshold a to rms value /„„ of 40 Average of highest peaks Generalized response First derivative of u(t) Second derivative of u(t) Rms value of x(t) Rms value of x(t) Absolute acceleration of base of one-degree-offreedom system Rms value of x(t) Maximum value of x(t) Risk of up-crossing Variable of chi-square with n degrees of freedom Time step Dirac delta function

Af AF A£ At s y £U (p |i n ji'n n p T Tm co0 Q £ \l/

xvii

Frequency interval between half-power points or frequency step of the PSD Bandwidth of analysis filter Interval of amplitude of 40 Time interval Statistical error or Euler's constant (0.57721566490...) Coherence function between 40 and u(t) Phase Central moment of order n Reduced central moment of order n 3.14159265... Correlation coefficient Delay Average time between two successive maxima Natural pulsation (2 TT f0) Pulsation of excitation (2*f) Damping factor Phase

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Chapter 1

Statistical properties of a random process

1.1. Definitions 1.1.1. Random variable A random variable is a quantity whose instantaneous value cannot be predicted. Knowledge of the values of the variable before time t not does make it possible to deduce the value at the time t from it. Example: the Brownian movement of a particle.

1.1.2. Random process Let us consider, as an example, the acceleration recorded at a given point on the dial of a truck travelling on a good road between two cities A and B. For a journey, recorded acceleration obeys the definition of a random variable. The vibration characterized by this acceleration is said to be random or stochastic. If n journeys are performed, one obtains as many different V(t) curves, each recording having a random character. We define as a random process or stochastic process the ensemble of the time functions [^(t)) for t included between - <» and +00, this ensemble being able to be defined by statistical properties [JAM 47].

2

Random vibration

Random movements are not erratic in the common meaning of the term, but follow a well defined law. They have specific properties and can be described by a law of probability. The principal characteristic of a random vibration is simultaneously to excite all the frequencies of a structure [TUS 67]. In distinction from sinusoidal functions, random vibrations are made up of a continuous range of frequencies, the amplitude of the signal and its phase varying with respect to time in a random fashion [TIP 77], [TUS 79], So the random vibrations are also called noise. Random functions are sometimes defined as a continuous distribution of sinusoids of all frequencies whose amplitudes and phases vary randomly with time [CUR 64] [CUR 88].

1.2. Random vibration in real environments By its nature, the real vibratory environment is random [BEN 6la]. These vibrations are encountered: - on road vehicles (irregularities of the roads), - on aircraft (noise of the engines, aerodynamic turbulent flow around the wings and fuselage, creating non-stationary pressures etc) [PRE 56a], - on ships (engine, swell etc), - on missiles. The majority of vibrations encountered by military equipment, and in particular by the internal components of guided missiles, are random with respect to to time and have a continuous spectrum [MOR 55]: gas jet emitted with large velocity creates important turbulences resulting in acoustic noise which attacks the skin of the missile until its velocity exceeds Mach 1 approximately (or until it leaves the Earth's atmosphere) [ELD 61] [RUB 64] [TUS 79], - in mechanical assemblies (ball bearings, gears etc), etc.

1.3. Random vibration in laboratory tests Tests using random vibrations first appeared around 1955 as a result of the inability of sine tests to excite correctly equipment exhibiting several resonances [DUB 59] [TUS 73]. The tendency in standards is thus to replace the old swept sine tests which excite resonances one after the other by a random vibration whose effects are nearer to those of the real environment.

Statistical properties of a random process

3

Random vibration tests are also used in a much more marginal way: -to identify the structures (research of the resonance frequencies and measurement of Q factors), their advantage being that of shorter test duration, - to simulate the effects of shocks containing high frequencies and difficult to replace by shocks of simple form.

1.4. Methods of analysis of random vibration Taking into account their randomness and their frequency contents, these vibrations can be studied only using statistical methods applied to the signals with respect to time or using curves plotted in the frequency domain (spectra). Table 1.1. Analysis possibilities for random vibration

/

X

1

RANDOM MECHANICAL VIBRATION

STUDY IN THE TIME DOMAIN

~

\ \

STATISTICAL PROPERTIES OF THE PROCESS

\

STATISTICAL PROPERTIES OF THE SIGNAL STATISTICAL PROPERTIES OF THE RESPONSE OF A ONE D.O.F. SYSTEM

RESPONSE OF AN IDEAL RECTANGULAR FILTER

RESPONSE OF A ONE D.O.F. MECHANICAL SYSTEM

ENSEMBLE AVERAGES

/

f

TIME AVERAGES

INSTANTANEOUS VALUES DISTRIBUTION MAXIMA DISTRIBUTION

FOURIER TRANSFORM

/ \

POWER SPECTRAL DENSITY

EXTREME RESPONSE SPECTRUM

FATIGUE DAMAGE SPECTRUM

One can distinguish schematically four ways of approaching analysis of random vibrations [CUR 64] [RAP 69]: - analysis of the ensemble statistical properties of the process,

4

Random vibration

- methods of correlation, - spectral analysis, - analysis of statistical properties of the signal with respect to time. The block diagram (Table 1.1) summarizes the main possibilities which will be considered in turn in what follows. The parameters most frequently used in practice are: - the rms value of the signal and, if it is the case, its variation as a function of time, - the distribution of instantaneous accelerations of the signal with respect to time, - the power spectral density.

1.5. Distribution of instantaneous values 1.5.1. Probability density One of the objectives of the analysis of a random process is to determine the probability of finding extreme or peak values, or of determining the percentage of time that a random variable (acceleration, displacement etc) exceeds a given value [RUD 75]. Figure 1.1 shows a sample of a random signal with respect to time defined over duration T.

Figure 1.1. Sample of random signal

Statistical properties of a random process

5

The probability that this function £(t) is in the interval i, I + M is equal to the percentage of time during which it has values in this interval. This probability (or percentage of time) is expressed mathematically:

[1.1]

Prob|/

To precisely define p(^), it is necessary to consider very small intervals M and of very long duration T, so that mathematically, the probability density function is defined by: [1.4]

p(/) = limit limiti

1.5.2. Distribution function Owing to the fact that p(l) was given for the field of values of e(t), the probability that the signal is inside the limits a < e(t) < b is obtained by integration from [1.2]: Tb

Prob [a < 4t) < b] = J

d£

[1.5]

Since the probability that ^(t) within the limits -oo, + oo is equal to 1 (absolutely certain event), it follows that

[1.6] and the probability that t exceeds a given level L is simply Prob [L <

[1.7]

6

Random vibration

There exist electronic equipment and calculation programmes that make it possible to determine either the distribution function, or the probability density function of the instantaneous values of a real random signal £(t). Figure 1.2 shows how one passes from the signal ^(t) to the probability density and the distribution function.

Figure 1.2. Distribution of instantaneous values of the signal

Among the mathematical laws representing the most usual probability densities, one can distinguish two particularly important in the field of random vibrations: Gauss's law and Rayleigh's law.

1.6. Gaussian random process A Gaussian random process f.(i) is one such that the ensemble of the instantaneous values of i(t} obeys a law of the form:

expi-

[1.81

where m and s are constants. The utility of the Gaussian law lies in the central limit theorem, which establishes that the sum of independent random variables follows a roughly Gaussian distribution whatever the basic distribution. This the case for many physical phenomena, of parameters which result from a large number of independent and comparable fluctuating sources, and in particular the majority of vibratory random signals encountered in the real environment [BAN 78] [CRE 56] [PRE 56a].

Statistical properties of a random process

7

A Gaussian process is fully determined by knowledge of the mean value m (generally zero in the case of vibratory phenomena) and of the standard deviation s. Moreover, it is shown that: - if the excitation is a Gaussian process, the response of a linear time-invariant system is also a Gaussian process [CRA 83] [DER 80]; - the vibration in part excited at resonance tends to be Gaussian. For a strongly resonant system subjected to broad band excitation, the central limit theorem makes it possible to establish that the response tends to be Gaussian even if the input is not. This applies when the excitation is not a white noise, provided that it is a broad band process covering the resonance peak [NEW 75] (provided that the probability density of the instantaneous values of the excitation does not have too significant an asymmetry [MAZ 54] and that the structure is not very strongly damped [BAN 78] [MOR 55]). In many practical cases, one is thus led to conclude that the vibration is stationary and Gaussian, which simplifies the problem of calculation of the response of a mechanical system (Volume 4).

1.7. Rayleigh distribution Rayleigh distribution of which the probability probability has the form:

e2s2

[L9]

s

(t. > 0) is also an important law in the field of vibration for the representation of: -variations in the instantaneous value of the envelope of a narrow band Gaussian random process, - peak distribution in a narrow band Gaussian process. Because of its very nature, the study of vibration would be very difficult if one did not have tools permitting limitation of analysis of the complete process, which comprises a great number of signals varying with time and of very great duration, using a very restricted number of samples of reasonable duration. The study of statistical properties of the process will make it possible to define two very useful concepts with this objective in mind: stationarity and ergodicity.

8

Random vibration

1.8. Ensemble averages: 'through the process' 1.8.1. n order average

Figure 1.3. 'Through the process' study

Let us consider N recordings of a random phenomenon varying with time

l

l(t)

[i € (l, N)J for t varying from 0 to T (Figure 1.3). The ensemble of the curves ' t(i) constitutes the process | l ^(t)|. A first possibility may consist in studying the distribution of the values of i for t = tj given [JAM 47]. If we have (N) records of the phenomenon, we can calculate, for a given tl5 the mean [BEN 62] [BEN 63] [DAY 58] [JEN 68]: [1.10] N

If the values ^(t) belong to an infinite discrete ensemble, the moment of order n is defined by:

[1.11] (E[ ] = mathematical expectation). By considering the ensemble of the samples at the moment tj, the statistical nature of ^tj) can be specified by its probability density [LEL 76]:

Statistical properties of a random process

9

M and by the moments of the distribution:

if the density p t i ] exists and is continuous (or the distribution function). The moment of order 1 is the mean or expected value; the moment of order 2 is the quadratic mean.

For two random variables The joint probability density is written: P(/I» l l' ^2>t 2 J =

lun

and joint moments:

, J"J

1.8.2. Central moments The central moment of order n (with regard to the mean) is the quantity: E {[4t,)-m]

^

n

}= Urn -IP4)-ml" J

[1.16]

N->co N •_,

in the case of a discrete ensemble and, for p(^) continuous: [1.17]

1.8.3. Variance The vor/ance is the central moment of order 2

10

Random vibration

[U8] By definition:

S.f

-2 m

p[4t,)] d^t,) + m2 } "p[4t,)] d/

1.8.4. Standard deviation The quantity s^ \ is called the standard deviation. If the mean is zero, *

When the mean m is known, an absolutely unbiased estimator of s

2

is

2

. When m is unknown, the estimator of s is ^

N m' = — N

where

N-l

Statistical properties of a random process

11

Example Let us consider 5 samples of a random vibration given time t = tj (Figure 1.4).

and the values of i at a

Figure 1.4. Example of stochastic process

If the exact mean m is known (m = 4.2 m/s2 for example), the variance is estimated from: S2 =

(2 - 4.2)* + (5 - 4.2?

S2 =

+

(2 - 4.2? + (4 - 4.2? + (7 - 4.2f

= 3.64 (m/s2)2

If the mean m is unknown, it can be evaluated from

m =—

2+5+2+4+7

s2 = — = 4.50 (m/s2)2 4

5

20 2 = — = 4 m/s 5

2\2

12

Random vibration

1.8.5. Autocorrelation function Given a random process '^(t), the autocorrelation function is the function defined, in the discrete case, by:

R( t l ,t 1 + t)= lim -2/4',) I 4 t i + ^)

[L22]

[1-23] or, for a continuous process, by: [1-24]

1 .8.6. Cross-correlation function Given the two processes {t(t)} and |u(t)} (for example, the excitation and the response of a mechanical system), the cross-correlation function is the function: [1.25] or

R(T)= lim — Z / A ' i J - X t i 4 N-»°oN

j

The correlation is a number measuring the degree of resemblance or similarity between two functions of the same parameter (time generally) [BOD 72]. 1.8.7. Autocovariance

Autocovariance is the quantity:

C(t,, tl +T) = R(t,,t, -f t)-<(t,) ^

+ ,)

C(ti,t } +T) = R\t 1 ,t 1 + T) if the mean values are zero.

[1.28]

Statistical properties of a random process

13

We have in addition: R(t,,t 2 ) = R(t 2 ,t,)

[1.29]

1.8.8. Covariance One defines covariance as the quantity: [1.30]

1.8.9.Stationarity A phenomenon is strictly stationary if every moment of all orders and all the correlations are invariable with time t} [CRA 67] [JAM 47] [MIX 69] [PRE 90] [RAP 69] [STE 67].

Figure 1.5. Study of autostationarity The phenomenon is wide-sense for weakly) stationary if only the mean, the mean square value and the autocorrelation are independent of time tj [BEN 58] [BEN61b][SVE80]. If only one recording of the phenomenon ^(t) is available, one defines sometimes the autostationarity of the signal by studying the stationarity with n samples taken at various moments of the recording, by regarding them as samples obtained independently during n measurements (Figure 1 .5). One can also define strong autostationarity and weak autostationarity.

14

Random vibration For a stationary process, the autocorrelation function is written:

R(T)= lim - ( 0 ) N

[L31]

'*M

NOTES. Based on this assumption, we have:

R(-r) = R(-t) = R(-T) = R(T)

[1.32]

(R is an even function oft) [PRE 90], R(0) = E{/(0) ^0)} = E{^2(t)}

[1.33]

R(0) is ?/?^ ensemble mean square value at the arbitrary time t

We have

yielding

± 2 E40 ^T + ET

>0

R(0) ± 2 R(T) + R(0) > 0 and R(0)^|R(T)| As for the cross-correlation function, it becomes, for a stationary process,

[1.34]

Statistical properties of a random process

[1.35]

lim — N-»oo

15

N

Properties 1.

R#U

[1.36]

Indeed

2. Whatever T [1.37]

1.9. Temporal averages: 'along the process' 1.9.1. Mean

Figure 1.6. Sample of random signal

16

Random vibration

Let us consider a sample l(t) of duration T of a recording. It can be interesting to study the statistical properties of the instantaneous values of the function g(t). The first possibility is to consider the temporal mean of the instantaneous values of the recording. We have: = lim — P4t)dt T->oo2T ~ T

[1-38]

if this limit exists. This limit may very well not exist for some or for all the samples and, if it exists, it may depend on the selected sample ^(t); but it does not depend on time(l}. For practical reasons, one calculates in fact the mean value of the signal ^(t) over one finite duration T: ?(t)dt

[1.39]

1.9.2. Quadratic mean - rtns value The vibration t(i) results in general in an oscillation of the mechanical system around its equilibrium position, so that the arithmetic mean of the instantaneous values can be zero if the positive and negative values are compensated. The arithmetic mean represents the signal poorly [RAP 69] [STE 67]. Therefore it is sometimes preferred to calculate the mean value of the absolute value of the signal t)|dt

[1.40]

and much more generally, by analogy with the measurement of the rms value of an electrical quantity, the quadratic mean (or mean square value) of the instantaneous values of the signal of which the square root is the rms value. The rms value (root mean square value) If^ = \-c (t) is the simplest statistical characteristic to obtain. It is also most significant since it provides an order of magnitude of the intensity of the random variable. 1. One defines too x(t) from: 1

T

1

lim — f x(t) dt ou lim — f V

T^oo T *

'

T/0

T-voo T J-T/2

x(t) dt V '

Statistical properties of a random process

17

If one can analyse the curve l(t) by dividing the sample of duration T into N intervals of duration Atj (i e [l, N]), and if ^ is the value of the variable during the interval of time At i5 the mean quadratic value is written:

!?At,

*?Ati

4

At

N

[1.41]

with T = 2^i Atj • If the intervals of time are equal to (At) and if N is the number of points characterizing the signal, T = N At and:

Figure 1.7. Approximation to the signal

If all Atj tend towards zero and if N -> oo, the quadratic mean is defined by [BEN 63]: = - f/ 2 (t)dt T

(or by

_T

0

[1.42]

18

Random vibration

1.9.3. Moments of order n As in the preceding paragraph, one also defines: - moments of an order higher than 2; the moment of order n is expressed: « lim — f TV ( t ) d t 2T central moments: that of order n is defined by: fin = EJkt) -40]° j = lim — j ^kt) -^(t)]" dt 1. J T-»co 2 T ~!

[1.44]

For a signal made up of N points of mean i:

1N

i=l

1.9.4. Variance - standard deviation

2 The central moment of order 2 is the variance, denoted by s^: [1.45] is called the standard deviation. Signal made up of N points:

i=l

1.9.5. Skewness The central moment of order 3, denoted by [13, is sometimes reduced by division

[1.46]

Statistical properties of a random process

19

One can show [GMU 68] that fj/3 is characteristic of the symmetry of the probability density law p(^) with regard to the mean ^(t); for this reason, n'3 is sometimes called skewness.

Figure 1.8. Probability densities with non-zero skewness

Signal made up of N points:

(j,3 = 0 characterize a Gaussian process. For ji'3 > 0, the probability density curve presents a peak towards the left and for fi'3 < 0, the peak of the curve is shifted towards the right.

1.9.6. Kurtosis

4. The central moment of order 4, reduced by division by s^, is also sometimes considered, for it makes it possible estimation of flatness of the probability density curve. It is often termed kurtosis [GUE 80]. [1.47]

20

Random vibration Signal made up of N points: N

|x'4 = 3

For a Gaussian process.

fi'4 < 3

Characteristic of a truncated signal or existence of a sinusoidal component ( u'4 = 1.5 for a pure sine).

H'4 > 3

Presence of peaks of high value (more than in the Gaussian case).

Figure 1.9. Kurtosis influence on probability density

Statistical properties of a random process

21

Examples 1. Let us consider an acceleration signal sampled by a step of At = 0.01 s at 10 points (to facilitate calculation), each point representing the value of the signal for time interval At x

_ x 2 rms ~

+X+--.-H ~

T = N At = 10 . 0.01 s

0.1 3 8

2 2

*ms = - (m/s ) and

Xrms

=1 95

-

This signal has as a mean

m=

l + 3 + 2 + - - - + (-2)+0 n ^—'0.01 = -0.4 m/s2 0.1

22

Random vibration

And for standard deviation s such that 1

\2 At|

^("

T ^-l*i-mJ

s2 = -U(l + 0.4)2 + (3 + 0.4)2 + (2 + 0.4)2 + (0 + 0.4)2 + (-1 + 0.4)2 + (- 3 + 0.4)2

S 2 =3.64 (m/s2)2

s = 6.03 m/s2 2. Let us consider a sinusoid x(t) = x m sin(Q t + cp)

xm

|x(t)|=

2 .. x

=

(for a Gaussian distribution, |x(t)| » 0.798 s ).

1.9.7. Temporal autocorrelation function We define in the time domain the autocorrelation function R^(T) of the calculated signal, for a given T delay, of the product t(t) l(t + T) [BEA 72] [BEN 58] [BEN 63] [BEN 80] [BOD 72] [JAM 47] [MAX 65] [RAC 69] [SVE 80].

Figure 1.10. Sample of random signal

Statistical properties of a random process

23

[1.48] [1-49]

The result is independent of the selected signal sample i. The delay i being given, one thus creates, for each value of t, the product £(t) and ^(t + T) and one calculates the mean of all the products thus obtained. The function R^(t) indicates the influence of the value of t at time t on the value of the function i at time t + T . Indeed let us consider the mean square of the variation between ^(t) and ^(t + t),

One notes that the weaker the autocorrelation R^(T), the greater the mean square of the difference [^(t)-^(t + t)] and, consequently, the less ^(t) and 4t + t) resemble each other.

Figure 1.11. Examples of autocorrelation functions

The autocorrelation function measures the correlation between two values of considered at different times t. If R/ tends towards zero quickly when T

24

Random vibration

becomes large, the random signal probably fluctuates quickly and contains high frequency components. If R^ tends slowly towards zero, the changes in the random function are probably very slow [BEN 63] [BEN 80] [RAC 69]. R^ is thus a measurement of the degree of random fluctuation of a signal. Discrete form The autocorrelation function calculated for a sample of signal digitized with N points separated by At is equal, for T = m At, to [BEA 72]:

[1.51]

Catalogues of correlograms exist allowing typological study and facilitating the identification of the parameters characteristic of a vibratory phenomenon [VTN 72]. Their use makes it possible to analyse, with some care, the composition of a signal (white noise, narrow band noise, sinusoids etc).

Figure 1.12. Examples of autocorrelation functions

Statistical properties of a random process

25

Calculation of the autocorrelation function of a sinusoid 40 = tm sin(Q t) T) = — J *

sin Ot sin Q t + i dt

[1-52] The correlation function of a sinusoid of amplitude lm and angular frequency Q

4

is a cosine of amplitude — and pulsation H. The amplitude of the sinusoid thus 2 can, conversely, be deduced from the autocorrelation function: max

[1.53]

1.9.8. Properties of the autocorrelation function 1. Re(o)=Ep(t)]=£ 2 (t) = quadratic mean [1.54] For a centered signal (g = 0], the ordinate at the origin of the autocorrelation function is equal to the variance of the signal. 2. The autocorrelation function is even [BEN 63] [BEN 80] [RAC 69]: R,(T) =* R,(-r)

[1.55]

3.

R,M|
Vx

CL56]

If the signal is centered, R^(T) —» 0 when T —» QO. If the signal is not centered, R.,(T) ->• ~t when T -> QO.

26

Random vibration 4. It is shown that: [1.57]

[1.58]

Figure 1.13. Correlation coefficient

NOTES. 1. The autocorrelation function is sometimes expressed in the reduced form:

MO)

[1.59]

or the normalized autocorrelation function [BOD 72] or the correlation coefficient p^(i) varies between -1 and+l p^ = 1 if the signals are identical (superimposable) p^ = -1 if the signals are identical in absolute value and of opposite sign. 2. If the mean m is not zero, the correlation coefficient is given by

Statistical properties of a random process

27

1.9.9. Correlation duration Correlation duration is the term given to a signal the value T O oft for which the function of reduced autocorrelation p^ is always lower, in absolute value, than a certain value pfc . o

Correlation duration of: - a wide-band noise is weak, - a narrow band noise is large; in extreme cases, a sinusoidal signal, which is thus deterministic, has an infinite correlation duration. This last remark is sometimes used to detect in a signal ^(t) a sinusoidal wave s(t) = S sin Q t embedded in a random noise b(t) : [1.60] The autocorrelation is written: R,(T) = R s M + R b (T)

[1.61]

If the signal is centered, for T sufficiently large, R b (t) becomes negligible so that S2 R^(T) = R S (T) = — cos Q T

t1-62!

2

This calculation makes it possible to detect a sinusoidal wave of low amplitude embedded in a very significant noise [SHI 70a]. Examples of application of the correlation method [MAX 69]: - determination of the dynamic characteristics of a systems, - extraction of a periodic signal embedded in a noise, - detection of periodic vibrations of a vibratory phenomenon, - study of transmission of vibrations (cross-correlation between two points of a structure), - study of turbubences, - calculation of power spectral densities [FAU 69], - more generally, applications in the field of signal processing, in particular in medicine, astrophysics, geophysics etc [JEN 68].

28

Random vibration

1.9.10. Cross-correlation Let us consider two random functions ^(t) and u(t); the cross-correlation function is defined by: = lim

1 2T

fT I A t ) u ( t + T)dt -T

[1-63]

The cross-correlation function makes it possible to establish the degree of resemblance between two functions of the same variable (time in general). Discrete form [BEA 72] If N is the number of sampled points and T a delay such that T = m At , where At is the temporal step between two successive points, the cross-correlation between two signals t and u is given by

,

N-m

N-m 1.9.11. Cross-correlation coefficient Cross-correlation coefficient P^ U (T) or normalized cross-correlation function or normalized covariance is the quantity [JEN 68]

^/R,(0) R u (0) It is shown that:

If t(\) is a random signal input of a system and u(t) the signal response at a point of this system, p^ u (t) is characteristic of the degree of linear dependence of the signal u with respect to I. At the limit, if ^(t) and u(t) are independent,

If the joint probability density of the random variables /(t) and u(t) is equal to p(4 u), one can show that the cross-correlation coefficient p^ u can be written in the form:

Statistical properties of a random process

29

[1.66] where m^ m u s^ and s u are respectively the mean values and the standard deviations of ^(t) and u(t) .

NOTE. For a digitized signal, the cross-correlation function is calculated using the relation:

1

N

R, u (m At) = - £>(p At) u[(p-m) At]

[1-67]

1.9.12. Ergodicity A process is known as ergodic if all the temporal averages exist and have the same value as the corresponding ensemble averages calculated at an arbitrary given moment [BEN 58] [CRA 67] [JAM 47] [SVE 80]. A ergodic process is thus necessarily stationary. One disposes in general only of a very restricted number of records not permitting experimental evaluation of the ensemble averages. In practice, one simply calculates the temporal averages by making the assumption that the process is stationary and ergodic [ELD 61]. The concept of ergodicity is thus particularly important. Each particular realization of the random function makes it possible to consider the statistical properties of the whole ensemble of the particular realizations.

NOTE. A condition necessary and sufficient such that a stationary random vibration ^(t) is ergodic is that its correlation function satisfies the condition [SVE 80]. liml

frf,_llR(T)dt

T^«T M T;

=

o

[1-68]

-where R^(T) is the autocorrelation function calculated from the centered variable

30

Random vibration

1.10. Significance of the statistical analysis (ensemble or temporal) Checking of stationarity and ergodicity should in theory be carried out before any analysis of a vibratory mechanical environment, in order to be sure that consideration of only one sample is representative of the whole process. Very often, for lack of experimental data and to save time, one makes these assumptions without checking (which is regrettable) [MIX 69] [RAC 69] [SVE 80].

1.11. Stationary and pseudo-stationary signals We saw that the signal is known as stationary if the rms value as well as the other statistical properties remain constant over long periods of time. In the real environment, this is not the case. The rms value of the load varies in a continuous or discrete way and gives the shape of signal known as random pseudostationary. For a road vehicle for example, variations are due to the changes in road roughness, to changes of velocity of the vehicle, to mass transfers during turns, to wind effect etc. The temporal random function ^(t) is known as quasi-stationary if it can be divided into intervals of duration T sufficiently long compared with the characteristic correlation time, but sufficiently short to allow treatment in each interval as if the signal were stationary. Thus, the quasi-stationary random function is a function having characteristics which vary sufficiently slowly [BOL 84]. The study of the stationarity and ergodicity is an important stage in the analysis of vibration, but it is not in general sufficient; it in fact by itself alone does not make it possible to answer the most frequently encountered problems, for example the estimate of the severity of a vibration or the comparison of several stresses of this nature. 1.12 Summary chart of main definitions (Table 1.2) to be found on the next page.

Table 1.2

Main definitions

Through the process (ensemble averages)

Along the process (temporal averages)

Moment of order n

Central moment of order n

Variance

Autocorrelat ion

Crosscorrelation

Stationarity if all the averages of order n are Ergodicity if the temporal averages are equal to the ensemble averages. independent of the selected time tj.

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Chapter 2

Properties of random vibration in the frequency domain

The frequency content of the random signal must produce useful information by comparison with the natural frequencies of the mechanical system which undergoes the vibration. This chapter is concerned with power spectral density, with its properties, an estimate of statistical error necessarily introduced by its calculation and means of reducing it. Following chapters will show that this spectrum provides a powerful tool to enable description of random vibrations. It also provides basic data for many other analyses of signal properties.

2.1. Fourier transform The Fourier transform of a non-periodic ^(t) signal, having a finite total energy, is given by the relationship: (t) e-lQt dt

L(Q) =

[2-1]

-00

This expression is complex; it is therefore necessary in order to represent it graphically to plot: - either the real and the imaginary part versus the angular frequency Q, - or the amplitude and the phase, versus Q. Very often, one limits oneself to amplitude data. The curve thus obtained is called the Fourier spectrum [BEN 58].

34

Random vibration

The random signals are not of finite energy. One can thus calculate only the Fourier transform of a sample of signal of duration T by supposing this sample representative of the whole phenomenon. It is in addition possible, starting from the expression of L(n), to return to the temporal signal by calculation of the inverse transform.

[2.2]

One could envisage the comparison of two random vibrations (assumed to be ergodic) from their Fourier spectra calculated using samples of duration T. This work is difficult, for it supposes the comparison of four curves two by two, each transform being made up of a real part and an imaginary part (or amplitude and phase). One could however limit oneself to a comparison of the amplitudes of the transforms, by neglecting the phases. We will see in the following paragraphs that, for reasons related to the randomness of the signal and the miscalculation which results from it, it is preferable to proceed with an average of the modules of Fourier transforms calculated for several signal samples (more exactly, an average of the squares of the amplitudes). This is the idea behind power spectral density.

Figure 2.1. Example of Fourier transform

In an indirect way, the Fourier transform is thus very much used in the analysis of random vibrations.

Properties of random vibration in the frequency domain

35

2.2. Power spectral density 2.2.1. Need The search for a criterion for estimating the severity of a vibration naturally results in examination of the following characteristics: - The maximum acceleration of the signal: this parameter neglects the smaller amplitudes which can excite the system for a prolonged length of time, - The mean value of the signal: this parameter has only a little sense as a criterion of severity, because negative accelerations are subtractive and the mean value is in general zero. If that is not the case, it does not produce information sufficient to characterize the severity of the vibration, - The rms value: for a long time this was used to characterize the voltages in electrical circuits, the rms value is much more interesting data [MOR 55]: - if the mean is zero, the rms value is in fact the standard deviation of instantaneous acceleration and is thus one of the characteristics of the statistical distribution, -even if two or several signal samples have very different frequency contents, their rms values can be combined by using the square root of the sum of their squares. This quantity is thus often used as a relative instantaneous severity criterion of the vibrations [MAR 58]. It however has the disadvantage of being global data and of not revealing the distribution of levels according to frequency, nevertheless very important. For this purpose, a solution can be provided by [WIE 30]: - filtering the signal ^(t) using a series of rectangular filters of central frequency f and bandwidth Af (Figure 2.2), - calculating the rms value L^g of the signal collected at the output of each filter. The curve which would give Lrms with respect to f would be indeed a description of the spectrum of signal t(t), but the result would be different depending on the width Af derived from the filters chosen for the analysis. So, for a stationary noise, one filters the supposed broad band signal using a rectangular filter of filter width Af, centered around a central frequency f c , the obtained response having the aspect of a stable, permanent signal. Its rms value is more or less constant with time. If, by preserving its central frequency, one reduces the filter width Af, maintaining its gain, the output signal will seem unstable, fluctuating greatly with time (as well as its rms value), and more especially so if Af is weaker.

36

Random vibration

Figure 2.2. Filtering of the random signal

To obtain a characteristic value of the signal, it is thus necessary to calculate the mean over a much longer length of time, or to calculate the mean of several rms values for various samples of the signal. One in addition notes that the smaller Af is, the more the signal response at the filter output has a low rms value [TIP 77]. L2 To be liberated from the width Af, one considers rather the variations of — Af with f. The rms value is squared by analogy with electrical power. 2.2.2. Definition If one considers a tension u(t) applied to the terminals of a resistance R = 1 Q, passing current i(t), the energy dissipated (Joule effect) in the resistance during time dt is equal to: dE=Ri2(t)dt=i2(t)

[2.3]

Properties of random vibration in the frequency domain

37

Figure 2.3. Electrical circuit with source of tension and resistance

The instantaneous power of the signal is thus:

[2.4] and the energy dissipated during time T, between t and t + T, is written:

[2.5] P(t) depends on time t (if i varies with t). It is possible to calculate a mean power in the interval T using:

[2-6l The total energy of the signal is therefore:

[2.7] and total mean power:

[2.8]

By analogy with these calculations, one defines [BEN 58] [TUS 72] in vibration mechanics the mean power of an excitation ^(t) between -T/2 and +T/2 by:

[2.9]

38

Random vibration

where J.*T=*(t)

for|t
|^ T =0

for|t|>T/2

Let us suppose that the function ^ T (t) has as a Fourier transform L T (f). According to Parseval's equality, p.10]

yielding, since [JAM 47] *

Pm = lim - (""VrMl2 df = lim - J°° L T (f)P df T->«>T -°°

[2.11]

[2.12]

T-»ooT °

This relation gives the mean power contained in f(t) when all the frequencies are considered. Let us find the mean power contained in a frequency band Af . For that, let us suppose that the excitation ^(t) is applied to a linear system with constant parameters whose weighting function is h(t) and the transfer function H(f) . The response r T (t) is given by the convolution integral r1T ( t ) =J| h ( X ) M t-X)dX 1 Q

[2.13]

where X is a constant of integration. The mean power of the response is written: Pmresponse =^1(^(0^

[2.14]

i.e., according to Parseval's theorem: 2 fT,

, tf

Pmresponse = lim — 'p JUI I R-rlf 1 v /II df T

f) 11^1 L^-'J

If one takes the Fourier transform of the two members of [2.13], one can show that: R T (f) = H(f) L T (f) yielding

[2.16]

Properties of random vibration in the frequency domain

^response '^f f N'f

L

l(ff

df

39

P-171

Examples 1 . If H(f ) = 1 for any value of f, rP

mresponse

=lim

_

r 2 l L TOf

df = pr

. mmput

[2.18]

a result which a priori is obvious. Af

Af

2

2

2. IfH(f) = l f o r O < f - — < f < f + — H(f) = 0 elsewhere

In this last case, let us set: |2

GT(f)="LT(f)^

[2.20]

The mean power corresponding to the record ^j(t), finite length T, in the band Af centered on f, is written: T (f)df

[2.21]

and total mean power in all the record P(f,Af) = lim T 4 " 2 G T (f) df

[2.22]

One terms power spectral density the quantity: lim ^

T-»oo

Af

[2.23]

40

Random vibration

In what follows, we will call this function PSD.

Figure 2.4. Example of PSD (aircraft)

NOTE. By using the angular frequency Q, we would obtain:

[2.24] with

[2.25]

Taking into account the above relations, and [2.10] in particular, the PSD G(f) can be written [BEA 72] [BEN 63] [BEN 80]: [2.26]

where ^ T (t, Af) is the part of the signal ranging between the frequencies f - Af/2 and f + Af/2. This relation shows that the PSD can be obtained by filtering the signal using a narrow band filter of given width, by squaring the response and by taking the mean of the results for a given time interval [BEA 72]. This method is used for analog computations.

Properties of random vibration in the frequency domain

41

The expression [2.26] defines theoretically the PSD. In practice, this relation cannot be respected exactly since the calculation of G(f) would require an infinite integration time and an infinitely narrow bandwidth.

NOTES. - The function G(f) is positive or zero whatever the value off. - The PSD was defined above for f ranging between 0 and infinity, which corresponds to the practical case. In a more general way, one could define S(f) mathematically between - oo and + <x>, in such a way that S(-f) = S(f)

[2.27]

- The pulsation Q = 2 n f is sometimes used as variable instead off. 7/"Gn(Q) is the corresponding PSD, we have G(f) = 2 TC GQ(Q)

[2.28]

The relations between these various definitions of the PSD can be easily obtained starting from the expression of the rms value: = fG(f)df = «u

One then deduces: G(f) = 2 S(f)

[2.30]

'j — /"• ^o^

r^? ^ 11

G(f) = 4 n Sn(Q)

[2.32]

f(f\

NOTE. A sample of duration Tqfa stationary random signal can be represented by a Fourier series, the term aj of the development in an exponential Fourier series being equal to: sin

27ikt T

d; — 1

J T

-T/2

cos

2nkt

dt

T The signal t(t) can be written in complex form

42

Random vibration

P.34] 1 / v where ck = — ^ctj - Pj ij 2

77ze power spectral density can also be defined from this development in a Fourier series. It is shown that [PRE 54] [RAC 69] [SKO 59] [SVE 80]

T-»oo 2 Af The power spectral density is a curve very much used in the analysis of vibrations: - either in a direct way, to compare the frequency contents of several vibrations, to calculate, in a given frequency band, the rms value of the signal, to transfer a vibration from one point in a structure to another, ... - or as intermediate data, to evaluate certain statistical properties of the vibration (frequency expected, probability density of the peaks of the signal, number of peaks expected per unit time etc). NOTE. The function G(f), although termed power, does not have the dimension of it. This term is often used because the square of the fluctuating quantity appears often in the expression for the power, but it is unsuitable here [LAL 95]. So it is often preferred to name it 'acceleration spectral density' or 'acceleration density' [BOO 56] or 'power spectral density of acceleration' or 'intensity spectrum' [MAR 58].

2.3. Cross-power spectral density From two samples of random signal records ^j(t) and ^(t)' one defines the cross-power spectrum by

T-»oo

if the limit exists, Lj and L2 being respectively the Fourier transforms of ^(t) and ^ ( t ) calculated between 0 and T.

Properties of random vibration in the frequency domain

43

2.4. Power spectral density of a random process The PSD was defined above for only one function of time ^(t). Let us consider the case here where the function of time belongs to a random process, where each function will be noted J t(t). A sample of this signal of duration T will be denoted by 1 0 T (t), and its Fourier transform ^(f). Its PSD is

i

'L T (f) G T (f) =

[2.37]

By definition, the PSD of the random process is, over time T, equal to:

n being the number of functions ! i(\) and, for T infinite, G(f) = lim G T (f)

[2.39]

T->oo

If the process is stationary and ergodic, the PSD of the process can be calculated starting from several samples of one recording only.

2.5. Cross-power spectral density of two processes As previously, one defines the cross-power spectrum between two records of duration T each one taken in one of the processes by: [

GT(f) =

9 'T*l 'T 2 T

The cross-power spectrum of the two processes is, over T,

Z'G T (f) n

and, for T infinite,

[2-40]

44

Random vibration

G(f) = lim G T (f)

[2.42]

2.6. Relation between PSD and correlation function of a process It is shown that, for a stationary process [BEN 58] [BEN 80] [JAM 47] [LEY 65] [NEW 75]:

G(f)=2 r°R(T)

[2.43]

J-OO

R(T) being an even function of T, we have: G(f) = 4 J°° R(T) cos(2 it f t) dr

[2.44]

If we take the inverse transform of G(f) given in [2.43], it becomes: [2-45] i.e., since G(f) is an even function of f [LEY 65]: R(T)= J°°G(f)cos(27tf T)

[2.46]

R(Q) = r (t) = ff G(f) df = (rms value)^

[2.47]

and

NOTE. These relations, named Wiener-Khinchine relations', can be expressed in terms of the angular frequency Q in the form [BEN 58] [KOW 69] [MIX 69]: G(Q) = 71

R(T) cos(Q T) di

[2.48]

°

R(t)= j°°G(Q)cos(QT)dT

[2.49]

Properties of random vibration in the frequency domain

45

2.7. Quadspectrum - cospectrura The cross-power spectral density G^ u (f) can be written in the form [BEN 80]:

where the function C^ u (f) = 2 J R j u ( t ) cos(2 it f T) (h

[2-51]

—GO

is the cospectrum or coincident spectral density, and where Q^f) = 2 J^R^r) sin(2 w f T) dt

f2-52]

—CO

is the quadspectrum or quadrature spectral density function. We have:

R

fti W = Jr[ C ^u( f ) cos(2 * f T ) + Q*u( f ) sin(2 * f T)] df

f2>54]

[2.55]

[2.57]

2.8. Definitions 2.8.1. Broad-band process A broad-band process is a random stationary process whose power spectral density G(Q) has significant values hi a frequency band or a frequency domain which is rigorously of the same order of magnitude as the central frequency of the band[PRE56a].

46

Random vibration

Figure 2.5. Wide-band process Such processes appear in pressure fluctuations on the skin of a missile rocket (jet noise and turbulence of supersonic boundary layer).

2.8.2. White noise When carrying out analytical studies, it is now usual to idealize the wide-band process by considering a uniform spectral density G(f) = G 0 .

Figure 2.6. White noise A process having such a spectrum is named white noise by analogy with white light which contains the visible spectrum. An ideal white noise, which is supposed to have a uniform density at all frequencies, is a theoretical concept, physically unrealizable, since the area under the curve would be infinite (and therefore also the rms value). Nevertheless, model ideal white noise is often used to simplify calculations and to obtain suitable orders of magnitude of the solution, in particular for the evaluation of the response of a onedegree-of-freedom system to wide-band noise. This response is indeed primarily produced by the values of the PSD in the frequency band ranging between the halfpower points. If the PSD does not vary too much in this interval, one can compare it at a first approximation to that of a white noise of the same amplitude. It should however be ensured that the results of this simplified analysis do indeed provide a

Properties of random vibration in the frequency domain

47

correct approximation to that which would be obtained with physically attainable excitation.

2.8.3. Band-limited white noise One also uses in the calculations the spectra of band-limited white noises, such as that in Figure 2.7, which are correct approximations to many realizable random processes on exciters.

Figure 2.7. Band-limited white noise

2.8.4. Narrow-band process A narrow-band process is a random stationary process whose PSD has significant values in one frequency band only or a frequency domain whose width is small compared with the value of the central frequency of the band [FUL 62].

Figure 2.8. PSD of narrow-band noise

The signal as a function of time i(i) looks like a sinusoid of angular frequency Q0, with amplitude and phase varying randomly. There is only one peak between two zero crossings.

48

Random vibration

Figure 2.9. Narrow-band noise It is of interest to consider individual cycles and envelopes, whose significance we will note later on. If the process is Gaussian, it is possible to calculate from G(Q) the expected frequency of the cycles and the probability distribution of the points on the envelope These processes relate in particular to the response of low damped mechanical systems, when the excitation is a broad-band noise. 2.8.5. Pink noise A pink noise is a vibration of which the power spectral density amplitude is of inverse proportion to the frequency.

2.9. Autocorrelation function of white noise The relation [2.45] can be also written, since G(f) = 4 re S(n) [BEN 58] [CRA 63]: [2.58] If S(Q) is constant equal to S0 when Q varies, this expression becomes:

where the integral is the Dirac delta function S(T) , such as:

Properties of random vibration in the frequency domain

6(t) -> oo

when T -» 0

6(r) = 0

when T = 0

49

[2.60]

f 5(t) dt = 1

yielding R(T) = 2 TI S0 5(i)

[2.61]

NOTE. If the PSD is defined by G(Q) in (0, <x>), this expression becomes

R(T) = * ^- 8(1) 2

[2-62]

[2-63]

T = 0, R -> oo. Knowing that R(0) w e
Figure 2.10. Autocorrelation of a -white noise

It is noted in addition that the correlation is zero between two arbitrary times. An ideal white noise thus has an infinite intensity, but has no correlation whatever between past and present [CRA 63].

50

Random vibration

2.10. Autocorrelation function of band-limited white noise

J1

Figure 2.11. Band-limited white noise

Figure 2.12. Autocorrelation of band-limited white noise

From the definition [2.58], we have, if S(Q) = S0 [FUL 62], [2.64]

[2.65]

[2.66] R(T) can be also written [2.67]

The rms value, given by [BEN 6la] [2.68] is finite. If T tends towards zero, R(T) -» 2 S0 (^ ~^i) (scluare of the rms value). The correlation between the past and the present is nonzero, at least for small intervals. When the bandwidth is widened, the above case obtains.

Properties of random vibration in the frequency domain

51

NOTES.

1 . The result obtained for a -white noise process is demonstrated from this particular case when Qj = 0 andQ2 ~* °°' indeed, if £±2 = ft,

4S 0 Q 2 t . Q 2 t 2S 0 R(t) = - cos - sin - = - sin Q2 T T

2

2

T

7/Q2 -> oo [2.61]t

R(T) -> 2 7i S0 5(i) Conversely, //R(t) has this value,

S(Q) = — 271

2. ^Twe 5er Qj = Q0

AQ AQ . . -- and Q2 = Q0 + - , R(T) can be written [COU 70]: 2 2

/ x 4 S0 / v f T AQ R(T) = —5L cos (Q0 T) sid \

R(0) -^ 2 S0 AQ

RT 2 p = —7-7 = R(0) T A Q

cos

^OT SIn

T AQ

[2.70]

2

3. If in practice, the noise is defined only for the positive frequencies, the expressions [2.66] and [2.68] become

[2-72]

52

Random vibration

2.11. Peak factor

The peak factor or peak ratio or crest factor Fp of a signal can be defined as the ratio of its maximum value (positive or negative) to its standard deviation (or to its rms value). For a sinusoidal signal, this ratio is equal to -s/2 (» 1.414). For a signal made up of periodic rectangular waveforms, it equals 1 while for saw tooth waveforms, it is approximately equal to 1.73. In the case of a random signal, the probability of finding a peak of given amplitude is an increasing function of the duration of the signal. The peak factor is thus undefined and extremely large. Such a signal will thus necessarily have peaks which will be truncated because of the limitation of the dynamics of the analyser. From it will result an error in the PSD calculation. Let us consider a random signal ^(t) of rms value lrms . If the signal filtered in a filter of width Af has its values truncated higher than I Q , the calculated PSD is equal to

Af

Af

Let us set: Fp = -^-

[2-73]

The error will thus be, at frequency f, [2.74]

= 100

G(f)J with G'(f)_/%(f)/Af

4(f)

[2?5]

G(f) It is shown that [PIE 64], for a Gaussian signal, the error varies according to the peak factor Fp according to the law e = 100(l-p)

[2.76]

Properties of random vibration in the frequency domain

53

where p = 2 Fp2 |°p(x) dx + 2

P

p(x) dx - 2 Fp p(x)

[2.77]

and [2.78]

p(x) = -— exp /27l

The variations of the error e according to Fp are represented in Figure 2.13.

Figure 2.13. Error versus peak factor (according to [PIE 64])

The calculation of p can be simplified if it is noted that: _^_ ^_ jc_ JL f e ~ 2 dx+-4= f" e ~ 2 dx-f-4- f ° e ~ 2 dx = 1 J-« J-F -F Fp

and that the probability density is symmetrical about the y-axis: |p(x)

dx-!2

p is then written: *» r?2

2 £ P p(x)dx-2F p P (F p )

54

Random vibration

p = 2 (l - Fp2) £ p(x) dx + Fp2 - 2 Fp p(pp)

[2.79]

The integral I P p(x) dx can be calculated using the error function, knowing that this function can be defined in the form [LAL 94] (Appendix A4.1): [2.80]

2.12. Standardized PSD/density of probability analogy Standardized PSD is the term given to the quantity [WAN 45]:

It is noticed that the standardized PSD and the probability density function have common properties: - they are nonnegative functions, - they have an unit area under the curve, -if we set R(Q) = J G N (u) du, R(fl) increases in a monotonous way from zero (Q = 0) to 1 (for Q infinite). R(Q) can thus be regarded as the analogue of the distribution function of G(Q).

2.13. Spectral density as a function of time In practice, the majority of the physical processes are, to a certain degree, nonstationary, i.e. their statistical properties vary with time. Very often however, the excitation is clearly nonstationary over a long period of time, but, for small intervals, which are still long with respect to the time of response of the dynamic system, the excitation can be regarded as stationary. Such a process is known as '•quasistationary '. It can be analysed for two aspects [CRA 83]: - study of the stationary parts by calculation of PSD whose parameters are functions slowly variable with time, - study of the long-term behaviour, described for example by a cross probability distribution for the parameters slowly variable with PSD.

Properties of random vibration in the frequency domain

55

The non-stationary process can also be of short duration. This is particularly the case of a mechanical oscillator at rest suddenly exposed to a stationary random excitation; there is a phase of transitory response, therefore nonstationary. Many studies have been conducted on this last subject [CHA 72] [HAM 68] [PRI 67] [ROB 71] [SHI 70b]. Various solutions were obtained, among those of T.K. Caughey and H.J. Stumpf [CAU 61] (Volume 4), R.L. Barnoski and J.R. Maurer [BAR 69] and Y.K. Lin [LIN 67]. Other definitions also were proposed for PSD of nonstationary phenomena [MAR 70]. 2.14. Relation between PSD of excitation and response of a linear system One can easily show that [BEN 58] [BEN 62] [BEN 63] [BEN 80] [CRA 63]: - if the excitation is a random stationary process, the response of a linear system is itself stationary, - if the excitation is ergodic, the response is also ergodic. Let us consider one of the functions ' ^(t) of a process (whether stationary or not); the response of a linear system can be written: 'u(t) = J°°h(X) ^ ( t - X ) d X

[2.82]

yielding: 'uft,) 'u(t 2 )=J\(X) ^ ( t i - X J d X Jo°°h(u) M'l) %)=

J

4t2-M)dn

hWhGi) Xti-x) V ( t 2 - u ) d X d n

[2.83]

Ensemble average R u ( t l 5 t 2 ) = E[u(tj)

U (t 2 )j

M'l, 0 = I0°° J*MX) h(u) R,(tl - X, t2 - u) dX dn

[2.84] [2.85]

where - X, t2 - »i) = E[^t, - X) 4t 2 - u)]

[2.86]

56

Random vibration

Example of a stationary process In this case, R u (t l5 1 2 ) = R U (0, t2 - tj) = R u (t 2 - tj = R U (T) and R

- 1 f° u (W\ -

[2.87]

In addition, we have seen that [2.43]: G(f)=2 p°R(T)e- 27tifT dT J-oo

The PSD of the response can be calculated from this expression [CRA 63] [JEN 68]:

G u (f) = 2

R f (T + A. - n)dXdji
G u (f) = 2

H(f) G,(f) u (f)

= |H(f)| 2 G,(f)

[2.88]

Depending on the angular frequency, this expression becomes: G u (0)=

[2.89]

NOTE. This result can be found starting with a Fourier series development of the excitation ^(t). Let us set u(t) as the response at a point of the system. With each frequency fj, the response is equal to H j times the input (Hj = a real number). So

Properties of random vibration in the frequency domain

57

u(t) can also be expressed in the form of a Fourier series, each term o/u(t) being equal to the corresponding term of i(i) modified by the factor Hj and phase (pj: u j Hj sin| —^ t +
[2.90]

j

i.e. /\ ^-< [ . 2 TC j t Ult)= > U;H; COS (0: S1H V

'

J^J

J

J

rp

' J

I

. 2 7C J t h SU1 CO: COS T

J

rp

The rms value o/u(t) is equal to 2 *• • 2 C O S ^ C ; -t-— S1IT

When T -» oo, 2

fdf

[2-93]

Knowing that, if G u (f) w ^e PSD of the response, urms = fG u (f)df , #

u (f)

= H 2 (f)G,(f)

[2.94]

is method can be used for the measurement of the transfer function of a structure undergoing a pseudo-random vibration (random vibration of finite duration, possibly repeated several times). The method consists of applying white noise of duration T to the material, in measuring the response at a point and in determining the transfer function by term to term calculation of the ratio of the input and output coefficients of the Fourier series development.

2.15. Relation between PSD of the excitation and cross-power spectral density of the response of a linear system

'•><«) =,0 u(t) Vt + t)^ f°°hU) Xt) ^t + t - X ) »A

58

Random vibration

If the process is stationary, the ensemble average is:

and the cross-spectrum: G u (f) = 2 f Y 2 7 t i f t

Gft,(f) = = H(f)G,(f)

[2.95]

NOTE. If we set:

G f c (f)-|A(fJ «"« the transfer function H(f) can be written, knowing that the PSD G^(f) is a real Junction

G/(f)

[2.96]

2.16. Coherence function The coherence function between two signals ^(t) and u(t) is defined by [BEN 63] [BEN 78] [BEN 80] [ROT 70]:

tf\ G ( ( (( tf\) /"•Gm(t)

This function is a measure of the effect of input on response of a system. In an ideal case, G/u = H(f) G/>f

[2.97]

Properties of random vibration in the frequency domain

59

and

y^ u is in addition zero if the signals ^(t) and u(t) are completely uncorrelated. In general, y^ u lies between 0 and 1 for the following reasons: - presence of noise in measurements, - nonlinear relation between l(t) and u(t) , - the response u(t) is due to other inputs than Let us consider the case where noise exists only with the response. Setting G the PSD of the response without noise and G nn that of the noise alone, it becomes: G

uu( f ) = G w + G n n

where

G

[2.98] w =

yielding [2.99]

The quantity jfu G uu \sr\amedcoherentouputpowerspectrum. 2

G

uu ~ G nn

G

nn

[2.100]

G

If the noise is present only on the input, we set ^(t) = a(t) + m(t) where a(t) is the pure signal and m(t) the noise. We have in the same way, for the PSD,

60

Random vibration [2.101] [2.102]

2.17. Effects of truncation of peaks of acceleration signal The example below makes it possible to highlight the influence of a truncation of the peaks of a random acceleration signal on its power spectral density.

2.17.1. Acceleration signal selected for study The signal considered is a sample of duration 1 second of a white noise over a bandwidth 10 - 2500 Hz, of rms value x r m s = 49.9 m/s2 (PSD amplitude: 1 (m/s2)2/Hz, sampling frequency: 8192 Hz). This signal was truncated with various acceleration values: ±5 x rms ±4.5 x^ ..., until ±0.5 Xj

Figure 2.14. Truncated signals

Properties of random vibration in the frequency domain

61

2.17.2. Power spectral densities obtained The spectral densities of these signals were calculated between 10 Hz and 10000 Hz. We observe from the PSD (Figure 2.15) that: -truncation causes the amplitude of the PSD to decrease uniformly in the defined bandwidth (between 10 Hz and 2500 Hz); -this reduction is only sensitive if one clips the peaks below 2x^5 approximately; - truncation increases the amplitude of the PSD beyond its specified bandwidth (2500 Hz). This effect is related to the mode of truncation selected (clean cut-off at the peaks and no nonlinear attenuation, which would smooth out the signal in the zone concerned).

Figure 2.15. PSD of the truncated signals

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Chapter 3

Rms value of random vibration

3.1. Rms value of a signal as function of its PSD We saw that [2.26]: G(f) = lim - J T 4( t » Af ) dt

G(f) is the square of the rms value of the signal filtered by a filter Af whose width tends towards zero, centered around f. To obtain the total rms value 1^$ of the signal, taking into account all the frequencies, it is thus necessary to calculate

The notation 0 means that integration is carried out in a frequency interval covering f = 0, while 0+ indicates that the interval does not include the limiting case f = 0. In a given frequency band fj, f 2 ($2 >

The square of the rms value of the signal in a limited frequency interval f } , f 2 is equal to the area under the curve G(f) in this interval. In addition:

64

Random vibration

[3.3]

f>(f)df = where 2 .

is the variance of the signal without its continuous component and

^ is the standard deviation of the signal.

Figure 3.1. Non-zero lower limit of the PSD

In addition, [3.4]

is the mean value of the signal. We thus have: 4ns=sf+W2

[3.5]

Lastly, for f * 0, we have: f

G(f)df = 0

[3.6]

A purely random signal does not have a discrete frequential component.

NOTE. The mean value I corresponding to the continuous component of the signal can originate in: — shift due to the measuring equipment, the mean value of the signal being actually zero. This component can be eliminated, either by centring the signal I before the calculation of its PSD, or by calculating the PSD between fj = 0 + e and f 2 (E being a positive constant different from zero, arbitrarily small),

Rms value of random vibration

65

- permanent acceleration, constant or slowly variable, corresponding to a rigid body movement of the vehicle (for example, static acceleration in phase propulsion of a launcher using propellant). One often dissociates this by filtering such static acceleration of vibrations which are superimposed on it, the consideration of static and dynamic phenomena being carried out separately. It is however important to be able to identify and measure these two parameters, in order to be able to study the combined effects of them, for example during calculations of fatigue strength, if necessary (using the Goodman or Gerber rule, ... cf. Volume 4) or of reaction to extreme stress. Static and dynamic accelerations are often measured separately by different sensors, vibration measuring equipment not always covering DC the component. Except for particular cases, we will always consider in what follows the case of zero mean signals. We know that, in this case, the rms value of the signal is equal to its standard deviation. Obtained by calculation of a mean square value, the power spectral density is an incomplete description of the signal ^(t). There is loss of information on phase. Two signals of comparable nature and of different phases will have the same PSD. Example Let us consider a stationary random acceleration x(t) having an uniform power spectral density given by: G*(f) = 0.0025 (m/s2)2/Hz in the frequency domain ranging between fj = 10 Hz and f 2 = 1 000 Hz, and zero elsewhere.

Figure 3.2. PSD of a signal having a continuous component

66

Random vibration

Let us suppose in addition that the continuous component of the signal is equal to x = 2 m/s2. Let us calculate the rms value and the standard deviation of the signal. The mean square value of the signal is given by the relation [3.5]: 2 — 7

o

f°°

/ x

* =* +4=J0-Gx(f)df > ? = J>,(f)df /4000

I 0.0025 df 40 ? = (1000 - 10) 0.0025 = 2.475 (m/s2)2 x = 4 (m/sy

yielding the mean square value x 2 = 4 + 2.475 = 6.475 (m/s2)2 and the rms value Vx 2 « 2.545 m/s2 while the standard deviation is equal to s^ = 1.573 m/s The random signals are in general centered before the calculation of the 2

2

spectral density, so that x = s^.

3.2. Relations between PSD of acceleration, velocity and displacement Let us set as ^(t) a random signal with Fourier transform L(f) ; by definition, we have: (t)e-2*iftdt

[3-7]

*(t)= p L ( f ) e 2 7 t i f t d f

[3.8]

L(f)= J-OO

and

Rms value of random vibration

67

yielding

By identification, it becomes: [3.10] The conjugate expressions of L(f) and of L(f) are obtained by replacing i by -i. If Gf(f} and G-,(f) are respectively the PSD of <(t) and of )?(t), one thus obtains, since these quantities are functions of the products L*(f) L(f) and L*(f) L(f) [LEY 65] [LIN 67],

yielding [3.12]

[3.13] and, in the same way, [3.14]

NOTES. 1. These relations use an integral with respect to the frequency between 0 and + oo. In practice, the PSD is calculated only for one frequency interval (fj, f^. The initial frequency f| is a function of the duration of the sample selected; this duration being necessarily limited, f| cannot be always taken as low as would be desirable. The limit f 2 is if possible selected sufficiently large so that all the frequency content is described. It is not always possible for certain phenomena, if only because of the measuring equipment. A value often used is, for example, 2000 Hz. However,

68

Random vibration

the integral necessary for the evaluation of i^f^ includes a term in f which makes it very sensitive to the high frequencies. In the calculation of all the expressions utilizing ^nng, as will be the case of the irregularity parameter r which we will define later, the result could be spoilt having considerable error in the event of a inappropriate choice of the limits fj and f 2 . J. Schijve [SCH 63] considers that the high frequency/small amplitude peaks have little influence on the fatigue suffered by the materials and proposes to limit integration to approximately 1000 Hz (for vibratory environments of aircraft type). 2. It is known that the rms value of a sinusoidal acceleration signal is related to the corresponding velocity and the displacement by "l = 2 K f e = ( 2 n f ) 2 e

[3.15]

These relationships apply at first approximation to the rms values of a very narrow band random signal of central frequency f. This makes it possible to demonstrate differently the relations [3.13] and [3.14]. The PSD of a signal l(t) is indeed calculated while filtering £(t) using a filter of width Af whose central frequency varies in the definition interval of the PSD, the result being squared and divided by Af for each point of the PSD. One thus obtains [CUR 64] [DEE 71] [HIM 59]: G-?=(27if)2Ge /

r-\^ /-i

[3.16] { r\

f\^ /-<

f? 171

yielding [OSG 69] [OSG 82]:

and 2

, o,(f)

[3

One can deduce from these relations the rms value of the displacement of a very narrow band noise [BAN 78] [OSG 69]:

Rms value of random vibration

69

(-3 2Q1 l

]

3.3. Graphical representation of PSD We will consider here the most frequent case where the vibratory signal to analyse is an acceleration. The PSD is the subject of four general presentations: - the first with the frequency on the x axis (Hz), the amplitude of the PSD on the y axis [(m/s2)2/Hz], the points being regularly distributed by frequency (constant filter width Af throughout the whole range of analysis); 1 - the second, met primarily in acoustics problems, uses an analysis in the — * n octave, the filter width being thus variable with the frequency; one finds more often in this case the ordinates expressed in decibels (dB). The number of decibels is then given, by: /-!

n dB = 101og

[3.21] GO

where G is the amplitude of the measured PSD. G0 is a reference value, selected equal to 10~12 (m/s2)2/Hz in general, or, if we consider the rms value in each band of analysis, by n d B =201og — a

[3-22]

o

where a = rms value of the signal in the selected band of analysis, a 0 = reference value of (10"6 m/s2); 1 - sometimes, the analysis in — *" octave is carried out by indicating in ordinates n the rms value obtained in each filter. For a noise whose PSD varies little with the frequency (close to white noise), the rms value obtained varies with the bandwidth of the filter,

-the relationships [3.17] show that the PSD can also be plotted on a fourcoordinate nomographic grid on which can be directly read the PSD value for acceleration, velocity and displacement [HIM 59].

70

Random vibration

Figure 3.3. Four-coordinate representation [HIM 59]

3.4. Practical calculation of acceleration, velocity and displacement rms values 3.4.1. General expressions The rms values of acceleration, velocity and displacement are more particularly useful for evaluation of feasibility of a specified random vibration on a test facility (electrodynamic shaker or hydraulic vibration machine). Control in a general way being carried out from a PSD of acceleration, we will in this case temporarily abandon the generalized co-ordinates. We saw that the rms value x^s of a random vibration x(t) of PSD G(f) is equal to:

The rms velocity and displacement corresponding to this signal of acceleration are respectively given by: [3.23]

[3.24]

Rms value of random vibration

71

In the general case where the PSD G(f) is not constant, the calculation of these three parameters is made by numerical integration between the two limits f} and f 2 of the definition interval of G(f). When G(f) can be represented by a succession of horizontal or arbitrary slope straight line segments, it is possible to obtain analytical expressions.

3.4.2. Constant PSD in frequency interval

Figure 3.4. Constant PSD between two frequencies

In this very simple case where the PSD is constant between G(f) = G 0 , yielding:

and f 2 ,

[3.25]

[3-26]

[3.27]

NOTES. The rms displacement acceleration:

can be also written as a Junction of rms

72

Random vibration

x

[3.28]

[3.29]

3.4.3. PSD comprising several horizontal straight line segments

Figure 3.5. P5D comprising horizontal segments We then have [SAN 63]: [3.30]

[3.31]

[3.32]

Rms value of random vibration

73

3.4.4. PSD defined by a linear segment of arbitrary slope It is essential in this case to speciiy in which scales the segment of straight line is plotted. Linear-linear scales Between the frequencies f} and f 2 , the PSD G(f) obeys G(f) = a f + b, where a and b are constants such that, for f = f l 5 G = G } and for f = f 2 , G = G 2 , yielding

a=

G

2~G1

f2-f,

and b =

G2 - f 2

f,-f2

[3.33]

[3.34]

[3.35]

74

Random vibration

Linear-logarithmic scales

Figure 3.6. Segment of straight line in lin-log scales

The PSD can be expressed analytically in the form: lnG = a f + b

[3.36]

[3.37]

fe e fe" I —— df = +aJ df, this integral can be calculated 2 f f f by a development hi series (Appendix A4.2): aI f.e af af c a2^ fci a n fr-n J df = hi f I + — + +••• + +••• 1! 2 2! n n!

Knowuig that

In the same way,

Rms value of random vibration

75

[3.38]

The integral is calculated like above, from (Appendix A4.2):

e

af

e

— d f=

af 3

3f

f4

af -I3 f

ea f eaf — df = - 3 3f

j

3

df

a eaf

a2 e af

6 f:

6

f

+-

6

f—df f

Particular case where G2 = In this case, [3.39]

[3.40]

[3.41]

Logarithmic-linear scales

In these scales, the segment of straight line has as an analytical expression: G = alnf+b with a =

ms=a(f2lnf2-f1lnf1)+(f2-f1)(b

[3.42]

76

Random vibration

Figure 3.7. Segment of straight line in log-lin scales

a

Infii f

f

In f t

In f2 I

4*H l

2

a

Info o

1

1

a+b

[3.43]

2

a + 3b

1

Logarithmic-logarithmic scales

Figure 3.8. Segment of straight line in logarithmic scales

The PSD is such that: hiG(f) = l n G 1 + b ( l n f - l n f 1 ) whence

The constant b is calculated from the co-ordinates of the point f 2 , G 2

[3-44]

Rms value of random vibration

77

In b=

Rms acceleration [PRA 70]:

mic

—

f2G2 b +1

[3.45]

If b = - 1:

Adf *, 7

;:2

[3.46]

,2 _

G!

1

r> f b-2 d f

G,

-1

=

_L _L_ 5l_^il

[3.47]

Ifb = l: The parameter b can be equal to 1 only if — = — , the commonplace case = G 2 and fj = f 2 being excluded. On this assumption, G = Gj — and

78

Random vibration

1

G

' !h£ v m s = — .Piln^ 2* V f ,

2 x •'••rmc =

x

[3.48]

fj

, 16*4f4

nns ~

, GI \^r .1 [UJ J

1

G-

[3.49]

Ifb = 3:

x

_^ OLin*k ™"4^K

[3.50]

f,

In logarithmic scales, a straight line segment is sometimes defined by three of the four values corresponding to the co-ordinates of the first and the last point, supplemented by the slope of the segment. The slope R, expressed in dB/octave, can be calculated as follows: - the number N of dB is given by

G2 N = 10 Iog10 -+

[3.51]

- the number of octaves n between fj and f 2 is, by definition, such as — = 2 n , yielding:

n=

and R = 10 log,0(2)

Iog10

[3.52] f

f

2/ l

Rms value of random vibration

79

[3.53]

= 101og10(2)b«3.01b

R . Let us set a = 101ogio(2). It becomes, by replacing b by — in the preceding a expressions [CUR 71]: R+a If R * -a :

x

rms ~

[3.54]

-1

R+a

This can be also written [SAN 66]:

•A-rrr

h°2

f, G,

*+l

f2 G2

[3.55]

a

a

or [OSG 82]: aGa +R

R. f,V

[3-56]

Reference [SAN 66] gives this expression for an increasing slope and, for a decreasing slope,

fiG,

[3.57]

a if R * -a , or

1-i'

UJ

[3.58]

80

Random vibration

For R = -a : [3.59]

If R * a : R-a"

R-a v

Gj

a rms ~

2

4 7t (R - a) f{

UJ

Wtt

-'

a G2 2 4 7t (R - a) f 2

f

L

<-f ']a

UJ

[3.60]

For R = a :

v* 2

[3.61]

—

If R * 3 a : R-3a"

R-3a

G m

( c \ 2 I *

1

~\6n4 f, 3 (R-3a) U J

/"• 1

2

\ 16* 4 f 2 3 (R-3c0

1

2

r>> >;^v-i

U

L3.62J

UiJ

For R = 3 a :

xL

1

G} j/2 1671 ff " ^ 4

1 G 2 ^fj 167I4 f23 "f!

Figures 3 9 3 10 inrl 3 11 re^nprtivelv -hnw

[3.63]

2 xrms

i

2 fi ! v rms

G

J

—'4

f? x 2 * rms

Gj

versus —, for different values of R. Abacuses of this type can be used to calculate the rms value of x, v or x from a spectrum made up of straight line segments on logarithmic scales [HIM 64].

Rms value of random vibration

Figure 3.9. Reduced rms acceleration versus

Figure 3.10. Reduced rms velocity versus f 2 /f] andR

81

an

dR

Figure 3.11. Reduced rms Displacement versus f 2 /fj

3.4.5. PSD comprising several segments of arbitrary slopes Whatever the scales chosen, the rms value of a PSD made up of several straight line segments of arbitrary slope will be such as in [OSG 82] [SAN 63]:

82

Random vibration

x tms = Yx 2

[3-64]

JZ~i Urns

being calculated starting from the relations above. In the same way: v ™ = |>.vf_

[3.65]

and Xrmc

3.5. Case: periodic signals It is known that any periodic signal can be represented by a Fourier series in accordance with: = L0 +

Ln sin(2 7i n f,

Power spectral density [2,26]: G(f)= lira - j 4 ( t , Af)dt T-»ooTAf ° Af->0

is zero for f * f n (with f n = n f t ) and infinite for f = fn since the spectrum of is a discrete spectrum, in which each component Ln has zero width Af . If one wishes to standardize the representations and to be able to define the PSD f 00

of a periodic function, so that the integral J G(f) G(f) df is equal to the mean square value of ^(t), one must consider that each component is related to Dirac delta function, the area under the curve of this function being equal to the mean square value of the component. With this definition,

n=0

Rms value of random vibration

83

where ^Sn is the mean square value of the n* harmonic ^ n (t) defined by ,2 -rmsn

J(t)dt 1

n

[3.69]

[3.70]

~nf,

(n = 1,2, 3,...). ln(t) is the value of the n* component and

where T is arbitrary and i is the mean value of the signal l(t). The Dirac delta function §(f - n fj] at the frequency f n is such that: [3.71]

and [3.72]

S(f-f n ) =

for f *• f n (E = positive constant different from zero, arbitrarily small). The definition of the PSD in this particular case of a periodic signal does not require taking the limit for infinite T, since the mean square value of a periodic signal can be calculated over only one period or a whole number of periods.

Figure 3.12. PSD of a periodic signal The chart of the PSD of a periodic signal is that of a discrete spectrum, the amplitude of each component being proportional to the area representing its mean square value (and not its amplitude).

84

Random vibration

We have, with the preceding notations, relationships of the same form as those obtained for a random signal: [3.73] li

^

\

/

n=0

'o+ "

n=l

" [3.75]

and, between two frequencies fj and f; (fj = i fj - 8, f; = j fj + e, i and j integers,

[3.76]

Lastly, if for a random signal, we had: J f f _ + G(f)df = 0

[3-77]

we have here: fo(f)df = f n * [0

forf = n f i for f *n ^ et f * 0

[378]

The area under the PSD at a given frequency is either zero, or equal to the mean square value of the component if f = n fj (whereas, for a random signal, this area is always zero). 3.6. Case: periodic signal superimposed onto random noise Let us suppose that: 4t) = a(t) + P(t)

[3.79]

a(t) = random signal, of PSD G a (f) defined in [2.26] p(t)= periodic signal, of PSD G p (f) defined in the preceding paragraph.

Rms value of random vibration

85

The PSD of *(t) is equal to: G,(f) = G a (f) + G p (f)

[3.80]

G,(f) = G a ( f ) + Z 4 6 ( f - f n )

P.81]

n=0

where fn = « f,

n = integer e (0, °o) fj= fundamental frequency of the periodic signal in = mean square value of the n* component ^ n (t) of ^(t) The rms value of this composite signal is, as previously, equal to the square root of the area under G «(f).

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Chapter 4

Practical calculation of power spectral density

Analysis of random vibration is carried out most of the time by supposing that it is stationary and ergodic. This assumption makes it possible to replace a study based on the statistical properties of a great number of signals by that of only one sample of finite duration T. Several approaches are possible for the calculation of the PSD of such a sample.

4.1. Sampling of the signal Sampling consists in transforming a vibratory signal continuous at the outset by a succession of sample points regularly distributed in time. If 5t is the time interval separating two successive points, the sampling frequency is equal to fsamp = 1 / 8t. So that the digitized signal is correctly represented, it is necessary that the sampling frequency is sufficiently high compared to the largest frequency of the signal to be analysed. A too low sampling frequency can thus lead to an aliasing phenomenon, characterized by the appearance of frequency components having no physical reality.

88

Random vibration

Example Figure 4.1 thus shows a component of frequency 70 Hz artificially created by the sampling of 200 points/s of a sinusoidal signal of frequency 350 Hz.

Figure 4.1. Highlighting of the aliasing phenomenon due to under-sampling

Shannon's theorem indicates that if a function contains no frequencies higher than fmax Hz, it is completely determined by its ordinates at a series of points spaced 1/2 fmax seconds apart [SHA 49]. Given a signal which one wishes to analyse up to the frequency fmax , it is thus appropriate, to avoid aliasing - to filter it using a low-pass filter in order to eliminate frequencies higher than fmax (the high frequency part of the spectrum which can have a physical reality or noise), -to sample it with a frequency at least equal to 2f max [CUR87] [GIL88] [PRE 90] [ROT 70].

NOTE. f

Nyquist = fsamp. / 2 » called Nyquist frequency.

In practice however, the low-pass filters are imperfect and filter incompletely the frequencies higher than the wanted value. Let us consider a low-pass filter having a decrease of 120 dB per octave beyond the desired cut-out frequency (f max ). It is

Practical calculation of power spectral density

89

considered that the signal is sufficiently attenuated at - 40 dB. It thus should be considered that the true contents of the filtered signal extend to the frequency corresponding to this attenuation (f_ 40 ), calculated as follows.

Figure 4.2. Taking account of the real characteristics of the low-pass filter for the determination of the sampling frequency

An attenuation of 120 dB per octave means that

-120 =

where AQ and Aj are respectively the amplitudes of the signal not attenuated (frequency fmax ) and attenuated at -40 dB (frequency f_ 4 Q ). Yielding

1*2

-120 =

and log 2 f

-40 = 10 3

«L26

90

Random vibration

The frequency f_40 being the greatest frequency of the signal requires that, according to Shannon's theorem, fsamp = 2 f_4o , yielding f

samp. _

The frequency f_4Q is also the Nyquist frequency This result often resulted in the belief that Shannon's theorem imposes a sampling frequency at least equal to 2.6 times the highest frequency of the signal to be analysed. We will use this value in the followings paragraphs.

4.2. Calculation of PSD from rms value of filtered signal The theoretical relation [2.26] which would assume one infinite duration T and a zero analysis bandwidth Af is replaced by the approximate relation [KEL 67]:

where '(.^ is the mean square value of the sample of finite duration T, calculated at the output of a filter of central frequency f and non zero width Af [MOR 56]. NOTE. Given a random vibration l(t) of white noise type and a perfect rectangular filter, the result of filtering is a signal having a constant spectrum over the width of the filter, zero elsewhere [CUR 64]. The result can be obtained by multiplying the PSD G0 of the input l(t) by the square of the transmission characteristic of the filter (frequency-response characteristic) at each frequency (transfer function, defined as the ratio of the amplitude of the filter response to the amplitude of the sinewave excitation as a function of the frequency. If this ratio is independent of the excitation amplitude, the filter is said to be linear). In practice, the filters are not perfectly rectangular. The mean square value of the response is equal to G0 multiplied by the area squared under the transfer function of the filter. This surface is defined as the 'rms bandwidth of the filter '.

Practical calculation of power spectral density

91

If the PSD of the signal to be analysed varies with the frequency, the mean square response of a perfect filter divided by the -width Af of the filter gives a point on the PSD (mean value of the PSD over the width of the filter). With a real filter, this approximate value of the PSD is obtained by considering the ratio of the mean square value of the response to the rms bandwidth of the filter Af, defined by [BEN 62], [GOL 53] and [PIE 64]:

Af =

H(f)

[4.2]

dt

H,

•where H(f) is the frequency response function of the (narrow) band-pass filter used and Hmax its maximum value.

4.3. Calculation of PSD starting from Fourier transform The most used method consists in considering expression [2.39]: ^

r~

~i

[4.3]

G M (f) = lim - d |L(f, T)P L

-

J

NOTES. 1. Knowing that the discrete Fourier transform can be written [KAY81] L

( m T ) - T y i c::f t 2 * j m 1X1 ^ J v1 N N N

[4.4]

j=o

r/ze expression of the PSD can be expressed for calculation in the form [BEN 71] [ROT 70]: N-l

G(mAf) = N

m

i exp -1

j=o

[4.5]

N

where 0 < m < M and Cj = j 5t. 2. The calculation of the PSD can also be carried out by using relation [2.26], by evaluating the correlation in the time domain and by carrying out a Fourier transformation (Wiener-Khintchine method) (correlation analysers) [MAX 86].

92

Random vibration

The calculation data are in general the following: - the maximumfrequencyof the spectrum, - the number of points of the PSD (or the frequency step Af ), - the maximum statistical error tolerated.

4.3.1. Maximum frequency Given an already sampled signal (frequency fsamp. ) and taking into account the elements of paragraph 4.1, the PSD will be correct only for frequencies lower than ^ max = * ' 2-

4.3.2. Extraction of sample of duration T Two approaches are possible for the calculation of the PSD: - to suppose that the signal is periodic and composed of the repetition of the sample of duration T, - to suppose that the signal has zero values at all the points outside the time time corresponding to the sample. These two approaches are equivalent [BEN 75]. In both cases, one is led to isolate by truncation a part of the signal, which amounts to applying to it a rectangular temporal window r(t) of amplitude 1 for 0 < t < T and zero elsewhere. If l(i) is the signal to be analysed, the Fourier transform is thus calculated in practice with f(t) = ^(t) r(t).

Practical calculation of power spectral density

93

Figure 4.3. Application of a temporal window

In the frequency domain, the transform of a product is equal to the convolution of the Fourier transforms L(Q) and R(£i) of each term:

F(Q) = J L(o>) R(Q - G>) dco (CQ is a variable of integration).

Figure 4.4. Fourier transform of a rectangular waveform

[4.6]

94

Random vibration

The Fourier transform of a rectangular temporal window appears as a principal central lobe surrounded by small lobes of decreasing amplitude (cf. Volume 2, Chapter 1). The transform cancels out regularly for Q a multiple of

(i.e. a T

1 frequency f multiple of —). The effect of the convolution is to widen the peaks of T the spectrum, the resolution, consequence of the width of the central lobe, not being 1

able to better Af = —. T The expression [4.6] shows that, for each point of the spectrum of frequency Q 2n (multiple of ), the side lobes have a parasitic influence on the calculated value of T the transform (leakage). To reduce this influence and to improve the precision of calculation, their amplitude needs to be reduced.

t

Figure 4.5. The Manning window

This result can be obtained by considering a modified window which removes discontinuities of the beginning and end of the rectangular window in the time domain. Many shapes of temporal windows are used [BLA 91] [DAS 89] [JEN 68] [NUT 81]. One of best known and the most used is the Manning window, which is represented by a versed sine function (Figure 4.5): / x

if

r(t) = - 1-cos

27C

0

[4.7]

Practical calculation of power spectral density

95

Figure 4.6. The Bingham -window [BIN 67]

This shape is only sometimes used to constitute the rising and decaying parts of the window (Bingham window, Figure 4.6). Weighting coefficient of the window is the term given to the percentage of rise time (equal to the decay time) of the total length T of the window. This ratio cannot naturally exceed 0.5, corresponding to the case of the previously defined Harming window. Examples of windows The advantages of the various have been discussed in the literature [BIN 67] [NUT 81]. These advantages are related to the nature of the signal to be analysed. Actually, the most important point in the analysis is not the type of window, but rather the choice of the bandwidth [JEN 68]. The Harming window is nevertheless recommended. The replacement of the rectangular window by a more smoothed shape modifies the signal actually treated through attenuation of its ends, which results in a reduction of the rms duration of the sample and in consequence in a reduction of the resolution, depending on the width of the central lobe. One should not forget to correct the result of the calculation of the PSD to compensate for the difference in area related to the shape of the new window. Given a temporal window defined by r(t), having R(f) for Fourier transform, the area intervening in the calculation of the PSD is equal to: Q = J4* |R(f)|2 df

[4.8]

96

Random vibration

From Parse val's theorem, this expression can be written in a form utilizing the N points of the digitized signal:

[4.9]

Practical calculation of power spectral density

97

Table 4.1. The principal windows Window type

Compensation factor

Definition

/x i!

[ io

7t(t-9T/lo)~|{

ft f 1 _L J~/-»C T\l) — ..' , .' 4*\L1 + COS

Bingham (Figure 4.6)

j

1/0.875

1 T JJ T 9T for 0 < t < — and
r(t\ ft fl8 4. fi A.&

Hamming

1/0.3974

T J for 0 < t < T

/r . ] F

(11; t 1— ... . .. 1 1

Hanning

(2*tyi

/-rvc COS

—.

1/0.375

2L V T J, for 0 < t < T f 2t 3 (t) = l 6 ^ — -1 +6 1 r J T T 3T for — < t 4 4

Parzen

2t

r(t) = 2 1-

1/0.269643

T

T J T 3T for 0 < t < — anci — < t < T 4 4 \

(}

(

9

^«J 1 f 2 7 1 t j1- r, x1^10£ u ,^

COS^

T

^ t1 ,J

Flat top

1/3.7709265 - 0.388 cosf 3 — 1 1 + 0.032 cosf 4 — t j I T J V T j for 0 < t < T r(t)= 1-1.24 cosf — 1 1 + 0.244 cosf 2 — 1|

KaiserBessel

IT J

- 0.00305 cosf 3 — t ) 1 T J for 0 < t < T

\ T;

1/1.798573

98

Random vibration

The multiplicative compensation factor to apply in order to take account of the difference between this area and the unit area of a rectangular window are thus equal tol/Q[DUR72].

4.3.3. Averaging We attempt in the calculations to obtain the best possible resolution with the data at our disposal, which results in trying to plot the PSD with the smallest possible frequency step. For a sample of duration T, this step cannot be lower than 1/T. With this resolution, the precision obtained is unacceptable. Several solutions are possible: - to carry out several measurements of the phenomenon, to calculate the PSD of each sample of duration T and to proceed to an average of the obtained spectra, - if only one sample of duration T is available, to voluntarily limit the resolution by accepting an analysis step Af larger than 1/T and to carry out an averaging [BEN 71]: - either by calculating the average of several frequential components close to the considered spectrum component, separated by intervals 1/T, when the noise to be analysed can be comparable to a white noise. If the average is carried out on K PSD, the average obtained is assigned to the central frequency of an interval of width equal to K/T (which characterizes the effective resolution of the PSD thus calculated), - or by dividing up the initial sample of duration T into K subsamples (or blocks) of duration AT = T/K which will be used to calculate K spectra of resolution I/AT and their average [BAR 55] [MAX 81]:

The results of these two approaches are identical for given duration T and given resolution [BEN 75]. It is the last procedure which is the most used. The window, rectangular or not, is applied to each block.

4.3.4. Addition of zeros The smallest interval Af between two points of the PSD is related to the duration 1 of the block considered by at least Af = —. The calculation of the PSD is carried AT out at M points with distances of Af between 0 and fsamp./2 (fsamp. = sampling

Practical calculation of power spectral density

99

frequency of the signal). As long as this condition is observed, it is said that the components of the spectrum are statistically independent.

Figure 4.7. Addition of zeros at end of the signal sample

One can however add components to the spectrum to obtain a more smoothed curve by artificially increasing the number of points using zeros placed at the end of the block (leading to a new duration AT > AT).

Figure 4.8. The addition of zeros increases the number of points of the PSD

Although the components added are no longer statistically independent, the validity of each individual component remains whole. The additional points of the PSD thus obtained lie between the original points corresponding to the duration AT and are on the continuous theoretical curve. The resolution is unchanged. All the components have an equal validity in the analysis [ENO 69]. One should attach no importance particularly to the components 1 spaced out at —, except that they constitute an ensemble of independent AT components. An equivalent unit could be selected by considering the points at the

100

Random vibration

frequencies (l + 8f )/T, (2 4- 5f )/T etc where 8f is an increment ranging between 0 and 1 [BEN 75].

4.4. FFT J.W. Cooley and J. Tukey [COO 65] developed in 1965 a method named Fast Fourier Transform or FFT making it possible to reduce considerably the computing time of the Fourier transforms. A FFT analyser functions with a number of points which is [MAX 86]: -a power of 2 for the Cooley-Tukey algorithms and those which derive from them, - a product of integral powers of prime numbers (Vinograd's algorithm). With the Cooley-Tukey's algorithm, the computing time of the transform of a signal defined by N points is proportional to N Iog2 N instead of the theoretically necessary value N . Table 4.2. Speed ratio for FFT computation

Number of Points

Number of points of the Fourier transform

N

Speed ratio •

Iog2 N

256

128

32

512

256

56.9

1024

512

102.4

2048

1024

186.2

4096

2048

341.3

Calculations of PSD are done today primarily using the FFT, which also has applications for the calculation of coherence functions (square of the amplitude) [CAR 73] and of convolutions. This algorithm, which is based in practice on the discrete Fourier transform, leads to a frequency sampling of the Fourier transform and thus of the PSD.

Practical calculation of power spectral density

101

NOTES.

1. Whilst in theory equivalent, the FFT and the method using the correlation can lead in practice to different results, -which can be explained by the non cognisance of the theoretical assumptions due to the difficulties of producing the analysers [MAX 86]. J. Max, M. Diot and R. Bigret showed that a correlation analyser presents a certain number of advantages such as: - a greater flexibility in the choice of the frequency sampling step, facilitating the analysis of the periodic signals, - a choice more adapted to the conditions of analysis of the signal. 2. When these algorithms are used to calculate the Fourier transform of a shock, one should not forget to multiply the result by the duration T of the treated signal.

4.5. Particular case of a periodic excitation The PSD of a periodic excitation was defined by [3.68]:

n=0

The PSD of such an excitation being characterized by very narrow bands centered on the frequencies f n , the calculation of G'(f) supposes that l(t] is analysed in sufficiently narrow filters Af. The PSD is approximated by: Af

YT/ n=0

^

rn can be obtained either by direct calculation of:

T

n

o

2

1 k with T = — or T = —, i.e. by calculation of the mean value: f

*n

n

7; = - PL n sin27if n dt = -L n

T °

n

T having the same definition. It is noted that:

[4.13]

102

Random vibration

[4 M]

'

1

T must be multiple of — .If it is not the case, the error is all the weaker since the fn

number of selected periods is larger. For a periodic excitation, the measurment or calculation accuracy is only related to the selected width Af of the chosen filter (the signal being periodic and thus determinist, there is no error of statistical origin related to the choice of T).

4.6. Statistical error 4.6.1. Origin Let us consider a stationary random signal whose PSD we wish to determine. The characteristic of such a signal being precisely to vary in a random way, the PSD obtained is different according to the moment at which it is calculated.

Figure 4.9. Estimates of the PSD for various signal samples

Let us consider the PSD G(f) evaluated at frequency f starting from a sample of duration T chosen successively between the times t0 and t0 + T, then t0 + T and t 0 + 2 Tetc. The values of G(f ) thus calculated are all different from each other and different also from the exact value G(f). We have:

G(f)= l i m -

Practical calculation of power spectral density

103

The true PSD is thus the mean value of the quantities G(f) estimated at various times, when their number tends towards the infinity. One could also define the standard deviation s of G(f). For N values, 12

s= [4.16]

N-l

and, for N -> <x>

s= lim s

[4.17]

N-»°o

s being the true standard deviation for a measurement G(f), is a description of uncertainty of this measure. In practice, one will make only one calculation of G(f) at the frequency f and one will try to estimate the error carried out according to the conditions of the analysis.

4.6.2. Definition

Statistical error or normalized rms error is the quantity defined by the ratio: £_

S

_4

[4.18]

(variation coefficient) where t ^ is the mean square value of the signal filtered in the filter of width Af (quantity proportional to G(f)) and s 2 is the standard *

deviation of the measurement of finite duration T.

related to the error introduced by taking a

NOTE. We are interested here in the statistical error related to calculation of the PSD. One makes also an error of comparable nature during the calculation of other quantities such as coherence, transfer function etc (cf. paragraph 4. 12).

104

Random vibration

4.7. Statistical error calculation 4.7.1. Distribution of measured PSD

V

If the ratio s = •=£=• is small, one can ensure with a high confidence level that a

4

measurement of the PSD is close to the true average [NEW 75]. If on the contrary s is large, the confidence level is small. We propose below to calculate the confidence level which can be associated with a measurement of the PSD when s is known. The analysis is based on an assumption concerning the distribution of the measured values of the PSD. 2

The measured value of the mean square z of the response of a filter Af to a random vibration is itself a random variable. It is supposed in what follows that z can be expressed as the sum of the squares of a certain number of Gaussian random variables statistically independent, zero average and of the same variance:

p /n x 2 (t)d t+ /

J

J

'2T/n ..2/ 'T/n X ( t ' d t + ""' JT(l-l/n)

[4.19]

One can indeed think that z satisfies this assumption, but one cannot prove that these terms have an equal weight or that they are statistically independent. One notes however in experiments [KOR 66] that the measured values of z roughly have the distribution which would be obtained if these assumptions were checked, namely a chi-square law, of the form: X2 = X2 + X2 + X32 + -+Xn

[4.20]

If it can be considered that the random signal follows a Gaussian law, it can be shown ([BEN 71] [BLA 58] [DEN 62] [GOL 53] [JEN 68] [NEW 75]) that 2

measurements G(f) of the true PSD G(f) are distributed as G(f) — where x n *s n the chi-square law with n degrees of freedom, mean n and variance 2 n (if the mean value of each independent variable is zero and their variance equal to 1 [BLA 58] [PIE 64]). Figure 4.10 shows some curves of the probability density of this law for various values of n. One notices that, when n grows, the density approaches that of a normal law (consequence of the central limit theorem).

Practical calculation of power spectral density

105

Figure 4.10. Probability density: the chi-square law

NOTE. Some authors [OSG 69] consider that measurements G(f) are distributed more Y

2

like G(f)

, basing themselves on the following reasoning. From the values n-1

Xj, X 2 , X3, • • • , X n of a normally distributed population, of mean m (unknown value) and standard deviation s, one can calculate —.\2 (x 1 -x) +(x 2 -x) +(x 3 -x) + ... + (x n -x)

[4.21]

-where [4.22] (mean of the various values taken by variable X by each of the n elements). Let us consider the reduced variable X: -X

[4.23]

106

Random vibration

The variables Uj are no longer independent, since there is a relationship between them: according to a property of the arithmetic mean, the algebraic sum of the deviations with respect to the mean is zero, therefore ^(Xj - X] = 0, and = 0 yielding:

consequently,

In the sample of size n, only n -1 data are really independent, for if n — 1 variations are known, the last results from this. If there is n -1 independent data, there are also n -1 degrees of freedom. The majority of authors however agree to consider that it is necessary to use a law with n degrees of freedom. This dissension has little incidence in practice, the number of degrees of freedom to be taken into account being necessarily higher than 90 so that the statistical error remains, according to the rules of the art, lower than 15% approximately.

4.7.2. Variance of measured PSD The variance of G(f) is given by: G(O Xn S

G(f) ~

S

G(f) ~

[4.24]

var . n.

However the variance of a chi-square law is equal to twice the number of degrees of freedom: [4.25]

Var

yielding 2

o1" S

G(f)~

G 2 (f)

*) ______ n 2 n

The mean of this law is equal to n.

_2

n

~

O 2

G 2 (f)

n

[4.26]

Practical calculation of power spectral density

107

4.7.3. Statistical error

Figure 4.11. Statistical error as function of the number ofdof

G(f) =

G(f)

G(f) = G(f)

[4.27]

The statistical error is thus such as:

£

2

S

G(f)

— .. _- . <•>

G(f) [4.28] s is also termed 'standarderror'.

108

Random vibration

4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis This relation can be obtained,

either using

a series expansion

of

E^ [G(f ) - G( f ) j r , or starting from the autocorrelation function.

From a series expansion: It is shown that [BEN 61b] [BEN 62]:

TAf

[4.29]

576 bias

variability

Except when the slope of the PSD varies greatly with Af, the bias is in general negligible. Then 2 8 =

E{[6(f)-G(f)f

G 2 (f)

1 TAf

This relation is a good approximation as long as e is lower than approximately 0.2 (i.e. for T Af > 25). [4.30]

The error is thus only a function of the duration T of the sample and of the width Af of the analysis filter (always assumed ideal [BEA 72] [BEN 63] [NEW 75]).

Figure 4.12. Statistical error

Practical calculation of power spectral density

109

Figure 4.12 shows the variations of this quantity with the product TAf. The number of events n represented by a record of white noise type signal, duration T, filtered by a filter of width A f , is thus, starting from [4.28]: n = 2AfT

[4.31]

Definition The quantity n = 2 Af T is called number of degrees of freedom (dof). From the autocorrelation function Let us consider ^(t) a vibratory signal response collected at the output of a filter of width Af. The mean square value of i(i) is given by [COO 65]:

2

2

Setting t^f the measured value of \^, we have, by definition: ft/2 ,2

,2

[4.32] e=

"Af

.4 A Af ~

8 = :

Af

However, we can write: 9

dv

i.e., while setting t = u and t = v-u = v - t ,

110

Random vibration

yielding

dt where D{T) is the autocorrelation coefficient. Given a narrow band random signal, we saw that the coefficient p is symmetrical with regard to the axis i = 0 and that p decrease when T| becomes large. If T is sufficiently large, as well as the majority of the values oft: 2 fT f+oo 2 / x — j dt J P2(l) dT

2 S =

T^

0

-00

•Af

yielding the standardized variance 8 [BEN 62]: 2

_ 2 f+oo

[4.33]

T ' .

[4-34]

*

Particular cases 1. Rectangular band-pass filter We saw [2.70] that in this case [MOR 58]:

cos 2 7t fft T sin 7i Af T P(T) = TC T Af

yielding 2

2

2 4 foo cos 2 7i f 0 T sin rc Af T s » —J T—T rn ^o T u

2 2 Ar.2 71 T Af

Practical calculation of power spectral density

e

2

111

1 *

TAf and [BEN 62] [KOR 66] [MOR 63]:

[4.35]

Example For e to be lower than 0.1, it is necessary that the product T Af be greater than 100, which can be achieved, for example, either with T = 1 s and Af = 100 Hz, or with T = 100 s and Af = 1 Hz. We will see, later on, the incidence of these choices on the calculation of the PSD.

2. Resonant circuit For a resonant circuit: /

\

^

r-

P(T) = cos 2 7t f 0 T e

-ft t Af

yielding 2

4

f°°

2 ~

-,

-2 it T Af ,

E « — J cos 2 7i f 0 T e 00

dt

-27iiAf

[4.36]

4.7.5. Confidence interval Uncertainty concerning G(f) can also be expressed in term of confidence interval. If the signal ^(t) has.a roughly Gaussian probability density function, the distribution of

X2 -/ v , for any f, is the same as —. Given an estimate G(f) obtained G(f) n

G(f)

112

Random vibration

from a signal sample, for n = 2 Af T events, the confidence interval in which the true PSD G(f) is located is, on the confidence level (l - a):

n G(f)

[4.37]

2 Cn, l-oc/2

2

, a/2 2

where x n a/2 and % n i_ ot /2 ^ave n degrees of freedom. Table 4.3 gives some values 2 of % n a according to the number of degrees of freedom n for various values of a.

•/ Figure 4.13. Values of%n

a with

respect to the number of degrees of freedom and ofo.

Figure 4.13 represents graphically the function parameterized by the probability a.

xa with respect to n,

Example 99% of the values lie between 0.995 and 0.005. One reads from Figure 4.13, for n = 10, that the limits are %2 = 25.2 and 2.16.

Practical calculation of power spectral density

113

Figure 4.14. Example of use of the curves X n a (n)

Example Figure 4.14 shows how in a particular case these curves can be used to evaluate numerically the limits of the confidence interval defined by the relation [4.37]. Let us set n = 10. One notes from this Figure that 80% of the values are within the interval 4.87 and 15.99 with mean value m = 10. If the true value of the mean of the calculated PSD S0 is m, it cannot be determined exactly, nevertheless it is known that 4.87

10

S

15.99 10

2.05 S0 > m > 0.625 S0

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20 21 22 23 24

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37.6 38.9 40.3 41.6 43.0

34.2 35.5 36.8 38.1 39.4

31.4 32.7 33.9 35.2 36.4

28.4 29.6 30.8 32.0 33.2

23.8 24.9 26.0 27.1 28.2

19.3 20.3 21.3 22.3 23.3

15.5 16.3 17.2 18.1 19.0

12.4 13.2 14.0 14.8 15.7

10.9 11.6 12.3 13.1 13.8

9.59 10.3 11.0 11.7 12.4

8.26 8.90 9.54 10.2 10.9

7.43 8.03 8.64 9.26 9.89

25 26 27 28 29

46.9 48.3 49.6 51.0 52.3

44.3 45.6 47.0 48.3 49.6

40.6 41.9 43.2 44.5 45.7

37.7 38.9 40.1 41.3 42.6

34.4 35.6 36.7 37.9 39.1

29.3 30.4 31.5 32.6 33.7

24.3 25.3 26.3 27.3 28.3

19.9 20.8 21.7 22.7 23.6

16.5 17.3 18.1 18.9 19.8

14.6 15.4 16.2 16.9 17.7

13.1 13.8 14.6 15.3 16.0

11.5 12.2 12.9 13.6 14.3

10.5 11.2 11.8 12.5 13.1

30 40 50 60

53.7 66.8 79.5 92.0

50.9 63.7 76.2 88.4

47.0 59.3 71.4 83.3

43.8 55.8 67.5 79.1

40.3 51.8 63.2 74.4

34.8 45.6 56.3 67.0

29.3 39.3 49.3 59.3

24.5 33.7 42.9 52.3

20.6 29.1 37.7 46.5

18.5 26.5 34.8 43.2

16.8 24.4 32.4 40.5

15.0 22.2 29.7 37.5

13.8 20.7 28.0 35.5

70 80 90 100

104.2 116.3 128.3 140.2

100.4 112.3 124.1 135.8

95.0 106.6 118.1 129.6

90.5 101.9 113.1 124.3

85.5 96.6 107.6 118.5

77.6 88.1 98.6 109.1

69.3 79.3 89.3 99.3

61.7 71.1 80.6 90.1

55.3 64.3 73.3 82.4

51.7 60.4 69.1 77.9

48.8 57.2 65.6 74.2

45.4 53.5 61.8 70.1

43.3 51.2 59.2 67.3

116

Random vibration

More specific tables or curves were published to provide directly the value of the limits [DAR 72] [MOO 61] [PIE 64]. For example, Table 4.4 gives the confidence interval defined in [4.37] for three values of 1 - a [PIE 64], Table 4.4. Confidence limits for the calculation of a PSD [PIE 64] Degrees of freedom

Confidence interval limits relating to a measured power spectral density G(f) = l

(l- a) = 0.90

(l - a) = 0.95

(l - a) = 0.99

n

Lower limit

Higher limit

Lower limit

Higher limit

Lower limit

Higher limit

10 15 20

0.546 0.599 0.637

2.54

0.483 0.546 0.585

3.03 2.39 2.08

0.397 0.457 0.500

4.63 3.26 2.69

25

0.662

0.615

1.90

0.532

2.38

30

0.685

1.71 1.62

0.637

0.559

2.17

40 50

0.719 0.741

1.51 1.44

0.676 0.699

1.78 1.64

0.599 0.629

1.93

75

0.781 0.806

1.34 1.28

0.743

1.42

0.769

1.35

0.680 0.714

1.59

100 150 200

0.833 0.855

1.22 1.19

0.806 0.826

1.27 1.23

0.758 0.781

1.49 1.37 1.31

250

0.870 0.877

1.16 1.15

0.847 0.855

1.20

300

1.18

0.800 0.820

1.25

400

0.893

0.877

1.15

0.840

1.21

500

0.901

1.13 1.11

0.885

1.14

0.855

1.18

750 1000 5000

0.917

1.09 1.08 1.03

0.909 0.917 0.962

1.11 1.09 1.04

0.877

1.15 1.12

0.934 0.971

2.07 1.84

1.54

0.893 0.952

Multiply the lower and higher limits in the table by the measured value G(f ) to obtain the limits of the confidence interval of the true value G(f).

1.78

1.27

1.05

Practical calculation of power spectral density

117

NOTE. When n > 30, -^2 x,n follows a law close to a Gaussian law of mean ^2 n -1 and standard deviation 1 (Fisher's law). Let \ be a normal reduced variable and a a value of the probability such that

Probjx
[4.38]

where k is a constant function of the probability a. For example:

a

k(a)

90%

95%

99%

1.645

1.960

2.58

We have [4.39]

Prob^/2n-l -k(a)< && < J2*-1 + k(a)j yielding the approximate value of the limits 0/Xn

Prob

L/2n-l -k(a)|2

_vy_i_..

\ /J

< V

2

L/2n-l + k(a

^ -?-"----_--- .---

and that of the confidence interval limits of

__.!--

[4.40]

_,--

G(f)

G (since the probability of— is the G(f) G

V

same as r/rar of—):

n

Prob

2n 2n ^G(f)^ [v/2-n-TT + k(a)]2 ~ G(f ) " [fi^i _ k(a)]2

— 1

n

[4.41]

For large values qfn [n > 120], i.e. for E small, it is shown that the chi-square law tends towards the normal law and that the distribution of the values o/G(f) can itself be approximated by a normal law of mean n and standard deviation ^2 n (law of large numbers). In this case,

Practical calculation of power spectral density

119

Figures 4.15 to 4.18 provide, for a confidence level of 99%, and then 90%: — variations in the confidence interval limits depending to the number of degrees of freedom n, obtained using an exact calculation (chi-square law), by considering the Fisher and Gauss assumptions, - the error made using each one of these simplifying assumptions. These curves show that the Fisher assumption constitutes an approximation acceptable for n greater than 30 approximately (according to the confidence level), •with relatively simple analytical expressions for the limits.

Figure 4.19. Confidence limits (G/G)

Figure 4.20. Confidence limits (G/G) [MOO 61]

120

Random vibration

G G The ratio — (or —, depending on the case) is plotted in Figures 4.19 and 4.20 G G with respect to n, for various values of the confidence level.

Example Let us suppose that a PSD level G = 2 has been measured with a filter of width Af = 2.5 Hz and from a signal sample of duration T= 10 s. The number of degrees of freedom is n = 2 T Af = 50 (yielding s = ..•

= 0.2). Table 4.4

gives, for 1 - a = 0.90 : 0.741G
— < 0.69 on the confidence level 5%, G

s~< — < 1.35 on the confidence level 95%. G With a confidence level of 90%, we thus have:

0.69 < — < 1.35 G i.e.

1.35

0.69

0.74G
Practical calculation of power spectral density

121

[4.44] From this inequality, can be written [PIE 64]: [4.45]

The confidence limits on the 68% level are plotted in Figure 4.21 for n ranging between 2 and 1000, then ranging between 20 and 1000.

IB3

Figure 4.21. Confidence limits at the 68% level

NOTE. At confidence level 1 - a = 68%, the expressions [4.37] and[4.45] show that

[4.46]

122

Random vibration

yielding S =

2 X

[4.47]

•where, if E < 0.2, 8 »

, one deduces: [4.48]

77»s expression is applicable for any nfor confidence level 68% and any a -when n w large.

Figure 4.22. G/G as function of frequency of filter and length of analysis [CUR 64]

Figure 4.22 shows the variations of: G _ true PSD (large T) G measured PSD with respect to the central frequency of the filter, for various lengths of analysis, at central frequency the confidence level 80% and for a ratio = 10 [CUR 64]. Af

Practical calculation of power spectral density

123

Central frequency of the filter (Hz)

Figure 4.23. G/G as function of frequency of the filter and probability

Figure 4.23 is parameterized, in the same axes, by the probability. Figures 4.22 and 4.23 are deduced from Figure 4.21 as follows: for a given f, f G G Af = — is calculated, then, for a given T, n = 2 Af T, yielding — and —. 10 G G Example

We want to calculate a PSD with a statistical error less than 17.5% at confidence level 95%. At this level 95%, we have ±1.96 times the standard error. The standard error should thus not exceed: e = — = 8.94% 1.96

Knowing e, the calculation conditions can be chosen from

124

Random vibration

4.7.6. Expression for statistical error in decibels

2 2 While dividing, in [4.40], <^2 x n by its mean value, i.e. x n

2 n-1

becomes 2"

2

Prob

[

[ V 2 n - l - k ( a ) f ^ v"-n ^, 1LA* " 2n-l " 2n-l

2

2

2n-l o Z 2

[4.49]

The error can be evaluated from —, i.e. 2-^—^-y, in the form n G ~]

[4.50]

= 10 log 10 It is raised, according to n, by

»10

=

10

2n-l

1+

2 k(a)

k 2 (a)

12 n -1

2 n -1

[4.51]

Figure 4.24 shows the variations of sdB with the number of degrees of freedom

n. If k(a) = 1, there is a 68.27% chance that the measured value is in the interval ±1 SA/ f \ and an 84.13% chance that it is lower than 1 S£/ f \. Then:

8dB = 10 log10

2

/2 n -1

1 2 n -1

Practical calculation of power spectral density

125

Figure 4.24. Statistical error in dB

d.o.f. number

Figure 4.25. Statistical error approximation

The curves in Figure 4.25 allow comparison of the exact relation [4.50] with the approximate relation [4.52]: the approximation is good for n > 50.

126

Random vibration

Example If it is required that s = ±0.5 dB , i.e. that s = ± 12.2 % , it is necessary, at confidence level 84%, that TAf = 67.17, or that the number of degrees of freedom is equal to n = 2 T Af « 135. If A f = 24 Hz, T1

i1Un4

- ° ^..ij5 s. At confidence level 90%, the (0.122fAf variatioiis of the PSI) are, in the interval [BAN 78]:

n 50 100 250

-

Lower limit (dB)

Upper limit (dB)

-1.570 -1.077 -0.665

1.329 0.958 0.617

4.7.7. Statistical error calculation from digitized signal Let N be the number of sampling points of the signal x(t) of duration T, M the number of points in frequency of the PSD fsamp the sampling frequency of the signal f fmax the maximum frequency of the PSD, lower or equal to -samp 2.6 (modified Shannon's theorem, paragraph 4.1)

5t the time interval between two points. We have: T=N5t

[4.53] [4.54]

2M

Practical calculation of power spectral density

127

NOTE. M points separated by an interval Af lead to a maximum frequency f = M Af = -samp' . To fulfill the condition of paragraph 4.3.1, it is necessary l

to limit in practice the useful field of the PSD to f m a x *

samp. 2.6

If we need a PSD calculated based on M points, we need at least AN = 2 M points per block. Since the signal is composed of N points, we will cut up it into K=

N

T

blocks of duration AT = —. 2M K

Knowing that fsanm = —: St

Af = 2M8t yielding

i.e. [4.55]

128

Random vibration

Example N = 32 768 points

M = 512 points

T = 64 s

yielding 2M = 1 024 points per sample N

K=

= 32 samples (of 2 s)

2M

K 32 Af = — = — = 0.5 Hz T 64

N 32768 fsamp = — = = 512 points/s T 64

32768 Even if M Af = 512 0.5 Hz = 256 Hz, we must have, in practice, 2.6

2.6

4.8. Overlapping 4.8.1. Utility One can carry out an overlapping of blocks for three reasons: - to limit the loss of information related to the use of a window on sequential blocks, which results in ignorance of a significant part of the signal because of the low values of the window at its ends [GAD 87]; - to reduce the length of analysis (interesting for real time analyses) [CON 95]; - to reduce the statistical error when the duration T of the signal sample cannot be increased. We saw that this error is related to the number of blocks taken in the sample of duration T. If all the blocks are sequential, the maximum number K of blocks of fixed duration AT (arising from the frequency resolution desired) is equal to the integer part of T/AT [WEL 67]. An overlapping makes it possible to increase this number of blocks whilst preserving their size AT.

Practical calculation of power spectral density

129

Overlapping rate The overlapping rate R is the ratio of the duration of the block overlapped by the following block over the total duration of the block. This rate is in general limited to the interval between 0 and 0.75.

Figure 4.26. Overlapping of blocks

Overlapping in addition makes it possible to minimize the influence of the side lobes of the windows [CAR 80] [NUT 71] [NUT 76].

4.8.2. Influence on number of degrees of freedom Let N be the number of points of the signal sample, N' (> N) the number of points necessary to respect the desired statistical error with K blocks of size A N ( N ' = K A N ) . The difference N'-N must be distributed over K - l possible overlappings [NUT 71]:

N'-N = ( K - l ) R A N yielding R=

N'-N AN ( K - l )

=

N'-N

.. ... [4.56]

N'-AN

For R to be equal to 0.5 for example, it is necessary that N'= 2 N - AN. Overlapping modifies the number of degrees of freedom of the analysis since the blocks cannot be regarded any more as independent and noncorrelated. The

130

Random vibration

estimated value of the PSD no longer obeys a one chi-square law. The variance of the PSD measured from an overlapping is less than that calculated from contiguous blocks [WEL 67]. R. Potter and J. Lortscher [POT 78] showed however that, when K is sufficiently large, the calculation could still be carried out on the assumption of non overlapping, on the condition the result could still be corrected by a reduction factor depending on the type of window and the selected overlapping rate. The correlation as a function of overlapping can be estimated using the coefficient: r(t)r[t + ( l - R ) A T ] d t [4.57]

Table 4.5. Reduction factor Window

Correlation coefficient C

Coefficient \i

R = 25%

R = 50%

R = 75%

R = 50%

R = 75%

Rectangle

0.25000

0.50000

0.75000

0.66667

0.36364

Bingham

0.17143

0.45714

0.74286

0.70524

0.38754

Hamming

0.02685

0.23377

0.70692

0.90147

0.47389

Hanning

0.00751

0.16667

0.65915

0.94737

0.51958

Parzen

0.00041

0.04967

0.49296

0.999509

0.67071

Flat signal

0.00051

-0.01539

0.04553

0.99953

0.99540

Kaiser-Bessel

0.00121

0.07255

0.53823

0.98958

0.62896

4.8.3. Influence on statistical error When the blocks are statistically independent, the number of degrees of freedom is equal t o n = 2K = 2 T A f whatever the window. With overlappings of K blocks, the effective number of blocks to consider in order to calculate the statistical error is given [HAR 78] [WEL 67]: - f o r R = 50%by: K

1 50 =

2

2

CSQ%

K

K 2

2 c50%

K

'50%

[4.58]

Practical calculation of power spectral density

131

-for R = 75% by: K

75%

* ~ ~,1 + T~2 2C

T~2 ^ ~ 2 _ c2 o + c2 T~2 75% + 2 c50o/0 + 2 c2s% 75 /0 50% + 3 C25o/0 K K2

1

+

2 75% +2 C50% + 2 C25%

= u« K

[4.59]

2C

(the approximation being acceptable for K > 10). Under these conditions, the statistical error is no longer equal to 1/vK , but to: e

[4.60]

=

The coefficient |i being less than 1, the statistical error is, for a given K, all the larger as overlapping is greater. But with an overlapping, the total duration of the treated signal is smaller, which makes it possible to carry out more quickly the analyses in real time (control of the test facilities). The time saving can be calculated from [4.56]: n _

N'-N

T-T

N'-AN

T-AT

(AT =duration of a block). To avoid a confusion of notations, we will let To be the duration of the signal to be treated with an overlapping and T the duration without overlapping. We then have:

N'-N

T-TO

N'-AN

T-AT

yielding

TO = T ( I - R ) + R A T

[4.6i]

Since R < 1 and AT « T, we have in general To « T (l - R) . The time saving T is thus approximately equal to — « (l - R) .

132

Random vibration

Example T = 25 s, Af = 4 Hz (i.e. K = T Af = 100), s0 = 0.1 (without overlapping). With R = 0.75 and a Hanning window, p,«0.52; yielding E = 1/^/0.52 x 25 x 4 » 0.139. But this result is obtained for a signal of duration T0 « (l - 0.75) 25 « 6.25 s.

If we consider now a sample of given duration T, overlapping makes it possible to define a greater number of blocks. This K1 number can be deducted from [4.56]: N'=

N-RAN 1-R

yielding, if N ' = K ' A N K-R

[4.62]

K' = 1-R

The increase in the number of blocks makes it possible to reduce the statistical error which becomes equal to:

1-R

1 8 =

= S0

K-R

[4.63]

1-R

Example With the data of the above example, the statistical error would be equal to 1 - 0.75 S«80

* 0.693 80 = 0.0693.

4.8.4. Choice of overlapping rate The calculation of the PSD uses the square of the signal values to be analysed. In this calculation the square of the function describing the window for each block thus intervenes in an indirect way, by taking account of the selected overlapping rate R. For a linear average, this leads to an effective weighting function r^^t) such as [GAD 87]:

Practical calculation of power spectral density

133

[4.64] where T is the duration of the window used (duration of the block), i is the number of the window in the sum, K is the number of windows at time t

Figure 4.27. Ripple on the Manning -window (R = 0.58;

Figure 4.28. Manning \vindo\vfor R = 0.75

Figure 4.29. Ripple amplitude versus 1 - R

With the Harming window, one of the most used, it can be observed (Figure 4.27) that there is a ripple on r^^t), except when 1 - R is of the form 1 / p where p is an integer equal to or higher than 3 (Figure 4.28). The ripple has a negligible amplitude when 1-R is small (lower than 1/3) [CON 95] [GAD 87]. This property can be observed in Figure 4.29, which represents the variations of the ratio of the maximum and minimum amplitudes of the ripple (in dB) with respect to 1-R.

134

Random vibration

This remark makes it possible to justify the use, in practice, of an overlapping equal to 0.75 which guarantees a constant weighting on a broad part of the window (the other possible values, 2/3, 3/4, 4/5, etc..., are less used, because they do not lead as 3/4 to a integer number of points when the block size is a power of two).

4.9. Calculation of PSD for given statistical error 4.9.1. Case: digitalization of signal is to be carried out Given a vibration ^(t), one sets out to calculate its power spectral density between 0 and fmax with M points (M must be a power of 2), for a statistical error not exceeding a selected value 8. The procedure is summarized in Table 4.6 [BEA 72] [LEL 73] [NUT 80]. Table 4.6. Computing process of a PSD starting from a non-digitized signal The signal of total duration T (to be defined) will be cut out in K blocks of unit duration AT, under the following conditions: f samp. — 2-6 tmax _ *• Nyquist

*samp.

Condition to avoid the aliasing phenomenon (modified Shannon's theorem). Nyquist frequency [PRE 90].

£r

1f

Af

Nyquist M

st

* *samp.

Interval between two points of the PSD (this interval limits the possible precision of the analysis starting from the PSD). Temporal step (time interval between two points of the signal), if the preceding condition is observed.

AN=2M

Number of points per block.

N -2M N "s2

Minimum number of signal points to analyse hi order to respect the statistical error.

T = N8t

Minimum total duration of the sample to be treated.

Y

N

2M

AT = — ( = 2 M 6 t = — ) K Af

Number of blocks. Duration of one block.

Practical calculation of power spectral density

135

Calculation of — |L/f )| for

AT '

^

Calculation from the FFT of each block.

each point of the PSD, where f = mAf ( 0 < m < M )

-t-^f fcf * K

AT

1V

Averaging of the spectra obtained for each of the K blocks (stationary and ergodic process)

With these conditions, the maximum frequency of the PSD computed is equal to f'max = ^Nyquist • But it is preferable to consider the PSD only in the interval (0, fmax).

NOTE. It is supposed here that the signal has frequency components greater than fmax and that it was thus filtered by a low-pass filter to avoid aliasing. If it is known that the signal has no frequency beyond f max , this filtering is not necessary and 1 f' max = fLmax -

4.9.2. Case: only one sample of an already digitized signal is available If the signal sample of duration T has already been digitized with N points, one can use the value of the statistical error to calculate the number of points M of the PSD (i.e. the frequency interval Af), which is thus no longer to be freely selected (but it is nevertheless possible to increase the number of points of the PSD by overlapping and/or addition of zeros). Table 4.7. Computing process of a PSD starting from an already digitized signal

Data: The digitized signal, fmax and 8. p 1

Af

max

samp.

Lf

rifnaX

5t-

Theoretical maximum frequency of the PSD (see preceding note).

samp. 2.6

Practical maximum frequency.

l

Temporal step (time interval between two points of the signal).

*samp.

136

Random vibration

T N =— 8t

Number of signal points of duration T. Number of points of the PSD necessary to respect the statistical error (one will take the number immediately beneath that equal to the power of 2).

M~Ns2 2 ,, * Nyquist Af

*samp.

Nyquist frequency.

Nyquist

Interval between two points of the PSD.

M AN = 2 M K-

Number of points per block.

N

Number of blocks.

2M

Etc

If the number of points M of the PSD to be plotted is itself imposed, it would be necessary to have a signal defined by N' points instead of N given points (N < N'). One can avoid this difficulty in two complementary ways: - either by using an overlapping of the blocks (of 2 M points). One will set the overlapping rate R equal to 0.5 and 0.75 while taking smallest of these two values (for a Harming window) which satisfies the inequality: -R 2M <8

u.

N

When it is possible, overlapping chosen in this manner makes it possible to use N' N-2MR K' blocks with K' = , where [4.56] N'= ; 2M 1-R - or, if overlapping does not sufficiently reduce the statistical error, by fixing this rate at 0.75 to benefit as much as possible from its effect and then to evaluate the size of the blocks which would make it possible, with this rate, to respect the statistical error, using:

[1-R AN <E

U

N

The value AN thus obtained is lower than the number 2 M necessary to obtain the desired resolution on the PSD. Under these conditions, the number of items used for the calculation of the PSD is equal to:

Practical calculation of power spectral density

N' =

137

N - 0.75 AN 1 - 0.75

and the numbers of blocks to K'= N'/AN. One can then add zeros to each block to increase the number of calculation points of the PSD and to make it equal to 2 M. 2 M K'-N' For each block, this number is equal to . This is however only an K' artifice, the information contained in the initial signal not evidently increasing with the addition of zeros. 4.10. Choice of filter bandwidth 4.10.1. Rules It is important to recall that the precision of calculation of the PSD depends, for given T, on the width Af of the filter used [RUD 75]. The larger the width Af of the filter is, the smaller the statistical error s and the better the precision of calculation of G(f). However, this width cannot be increased limitlessly [MOO 61]. The larger Af is, the less the details on the curve obtained, which is smoothed. The resolution being weaker, the narrow peaks of the spectrum are not shown any more [BEN 63]. A compromise must thus be found. Figure 4.30 shows as an example three spectral curves obtained starting from the same vibratory signal with three widths of filter (3.9 Hz, 15.625 Hz and 31.25 Hz). These curves were plotted without the amplitude being divided by Af, as is normally the case for a PSD.

5 Frequency (Hz)

6 xi82

Figure 4.30. Influence of width of filter

13 8

Random vibration

One observes in these conditions, the area under the curve calculated for Af = 15.625 Hz is approximately half of that obtained for Af = 31.25 Hz. In the case of a true PSD, division by Af gives the same area for whatever the value of Af. We note in addition on these curves that the spectrum obtained for Af = 15.6 Hz is very much smoothed; in particular, the peak observed for Af = 3.9 Hz has disappeared. To choose the value of Af, it would be necessary to satisfy two requirements: 1. The filter should not be broader than aquarter of the width of the narrowest resonance peak expected [BEN 61b] [BEN 63] [FOR 64] [MOO 61] [WAL 81]; 2. The statistical error should remain small, with a value not exceeding approximately 15%. If the first condition is observed, the precision of the PSD calculation is proportional to the width of the filter. If, on the contrary, resonances are narrower than the filter, the precision of the estimated PSD is proportional to the width of the resonance of the specimen and not to the width of the filter. To solve this problem, C.T. Morrow [MOR 58], and then R.C. Moody [MOO 61] suggested making two analyses, by using the narrowest filter first of all to emphasize resonances, then by making a second analysis with a broader filter in order to improve the precision of the PSD estimate. Other more complicated techniques have been proposed (H. Press and J.W. Tukey for example [BLA 58] [NEW 75]).

4.10.2. Bias error Let us consider a random signal ^(t) with a constant PSD (white noise) G^(f) = G^Q applied to a linear system with transfer function of one-degree-offreedom [PIE 93] [WAL 81]:

0 being the natural frequency and Q the quality factor of the system). The response u(t) of this system has the following PSD:

G

Practical calculation of power spectral density

139

[4.66]

G

Let us analyse this PSD, which presents a peak at f = f 0 , using a rectangular filter of width AF centered on f c , with transfer function [FOR 64]: [4.67]

We propose to calculate the bias error made over the width between the halfpower points of the peak of the PSD response and on the amplitude of this peak when using an analysis filter AF of nonzero width. For given f c , the PSD calculated with this filter has a value of: [4.68]

df

[4.69]

It is known [LAL 94] (Volume 4, Appendix A3) that the integral f dh

Consequently,

140

Random vibration

In

AF

[4.72] i.e. Arc tan

[4.73]

AF However, by definition, the bandwidth between the half-power points is equal to fn

Af = —, yielding: Q [4.74]

At the half-power points, the calculated spectrum has a value: [4.75]

where f c = f 0 +

. One deduce that:

Practical calculation of power spectral density GF

G^

_Q

o Arc tan AfF + AF Arc tan AfF - AF Af 2 AF Af f

141

[4.76]

From [4.74], [4.75] and [4.76], it becomes:

AF Af

Afp + AF

Af

L

AfF-AF]

Af

[4.77]

J

Figure 4.31. Width of the peak at half-power versus width of the analysis/liter (according to [FOR 64])

Af,F Af AF The curve in Figure 4.31 gives the variations of - against — after numerical Af Af resolution. It is noted that the measured value Aff of the width of the peak at halfpower is obtained with an error lower than 10% so long as the width of the analysis filter is less than half the true value Af . AF AfFF Setting x = — and y = - . Af

'

Af

Arc tan x = Arctan(y + x) - Arctan(y-x) Knowing that Arc tan a - Arc tan b = Arc tan

a-b 1 + ab

Arc tan x = Arc tan

2x

, we have:

142

Random vibration

This yields x = AfF

Af

2X

r- anc* v

1 + y -x

"(AF?

2= x 2+

^ i-e<

"

UfJ _ " +1

[4.78]

In addition, the peak of the PSD occurs for f = f 0 : [4.79]

yielding the relationship between the measured value of the peak and the true value: G F V(fu; Af AF 0) p = — Arc tan — P G AF Af

[4.80]

Figure 4.32 shows the variations of this ratio versus AF/Af. If AF = — Af 4 according to the rule previously suggested,

^--Jl + p-l -1-
Figure 4.32. Amplitude of peak versus filter width

Practical calculation of power spectral density

143

Under these conditions, the error is about 3% of Af and 2% of the peak. Example Let us consider a one-degree-of-freedom system of natural frequency fo =100 Hz and quality factor Q = 10 , excited by a white noise. The error of measure of the PSD response peak is given by the curve in Figure 4.32. If AF = 5 Hz:

0

100

^ fo

yielding: GF = 0.92 P G Q 10 For f 0 = 50 Hz and Q = 10, one would obtain similarly —AF = — f0 50

G

= 0.78 .

G

4.10.3. Maximum statistical error When the phenomenon to be analysed is of short duration, it can be difficult to obtain a good resolution (small AF ) whilst preserving an acceptable statistical error.

144

Random vibration

Example

Frequency (Hz)

Frequency (Hz)

Figure 4.33. Influence of analysis filter width for a sample of given duration

Figure 4.34. Influence of analysis filter •width for constant statistical error

Figure 4.33 shows, as an example, the PSDs of same signal duration 22.22 seconds, calculated respectively with AF equal to 4.69 Hz; 2.34 Hz and 1.17 Hz (i.e. with a statistical error equal to 0.098, 0.139 and 0.196). We observe that, the more detailed the curve (AF small), the larger the statistical error. Although the duration of the sample is longer than 20 s, a resolution of the order of one Hertz can be obtained only with an error close to 20%. A constant statistical error with different durations T and widths AF can lead to appreciably different results. Figure 4.34 shows three calculated PSD of the same signal all three for e = 19.6 %, with respectively: T = 22.22 s

and

AF = 1.17 Hz

7 = 11.11 s

and

AF = 2.34 Hz

T = 5.555 s

and

AF = 4.69 Hz

The choice of AF must thus be a compromise between the resolution and the precision. In practice, one tries to comply with the two following rules: AF less than a quarter of the width of the narrowest peak of the PSD, which limits the width measurement error of the peak and its amplitude to less than 3%, and a statistical

Practical calculation of power spectral density

145

error less than 15% (which corresponds to a number of degrees of freedom n equal to approximately 90). Certain applications (calculation of random transfer functions for example) can justify a lower value of the statistical error. Taking into account the importance of these parameters, the filter width used for the analysis and the statistical error should always be specified on the power spectral density curves.

4.10.4. Optimum bandwidth A.G. Piersol [PIE 93] defines the optimum bandwidth AFop as the value of AF minimizing the total mean square error, sum of the squares of the bias error and of the statistical error: e

~ 8bias + sstat

The bias error calculated from [4.80] is equal to:

Af

[4.82]

Figure 4.35. Total mean square error

where Af is the width between the half-power points of the peak. Whence:

146

Random vibration

Af

AF

AF

Af,

[4.83]

— Arc tan — -1

AFT

Figure 4.35 shows the variations of bias error, statistical error and e with AF. Error s has a minimum at AF = AFop. The optimum bandwidth AFop is thus obtained by cancelling the derivative of E carried out numerically.

with respect to AF. This research is

Figure 4.36. Optimum bandwidth versus peak frequency and duration of the sample

The curves in Figure 4.36 show AFop versus f 0 , for

= 0.05 and for some

values of duration T. If AF/Af < 0.4, the bias error can be approximated by: 1 fAF N 2

[4.84]

Then,

AF

1

9 Af4

T AF

[4.85]

Practical calculation of power spectral density

147

Whence, by cancelling the derivative, 9 F4 4T

1/5

4/5

2

[4.86] rl/5

Figure 4.37 shows the error made versus the natural frequency f0, for various values of T.

Figure 4.37. Comparison of approximate and exact relations for calculation of optimum bandwidth

Figure 4.38. Comparison of the optimum bandwidth with the standard rules

It can be interesting to compare the values resulting from these calculations with the standard rules which require four points in the half-power interval (Figure 4.38). It is noted that this rule of four points leads generally to a smaller bandwidth in general. The method of calculation of optimum width must be used with prudence, for it can lead to a much too large statistical error (Figure 4.39, plotted for | = 0.05). To confine this error to the low resonance frequencies, A.G. Piersol [PIE 93] suggested limiting the optimum band to 2.5 Hz, which leads to the curves in Figure 4.40.

148

Random vibration

Figure 4.39. Statistical error obtained using Figure 4.40. Statistical error obtained using the optimum bandwidth the optimum bandwidth limited to 2.5 Hz

By plotting the variations of Af/AF op , one can also evaluate, with respect to f0, the number of points in Af which determines this choice of AF0_, in order to compare this number with the four points of the empirical rule. Figures 4.41 and 4.42 show the results obtained, for several values of T, with and without limitation of the AFopband.

Figure 4.41. Number of points in Af resulting from choice of optimum bandwidth

Figure 4.42. Number of points in Af resulting from the choice of optimum bandwidth limited to 2.5 Hz

Practical calculation of power spectral density

149

4.11. Probability that the measured PSD lies between ± one standard deviation We saw that the approximate relation s =

1

is acceptable as long as

TAf 8 < 0.20. In this same range, the error on the measured PSD G (or on G/G) has a roughly Gaussian distribution [MOO 61] [PRE 56a]. Let us set s = SA to simplify the notations. The probability that the measured PSD is false by a quantity greater than 0 s (error in the positive sense) is [MOR 58]: 1

, da

P=

[4.87]

If we set v = —, P takes the form: s

P=

dv A

Knowing that:

2 Erf(x) = —

fx _ 2 I e-t dt

[4.88]

Figure 4.43. Probability that the measured PSD lies between ± 1 standard deviation

P can be also written, to facilitate its numerical calculation (starting from the approximate expressions given in Appendix A4):

150

Random vibration [4.89] p =-

1-Er — IV2,

The probability of a negative error is identical. The probability of an error outside the range ± 0 s is thus equal to: [4.90] I

Example

6=1

P = 68.26 %

e=2

P = 95%

4.12. Statistical error: other quantities The statistical error related to the estimate of the mean and mean square value is, according to the case, given by [BEN 80]: Table 4.8. Statistical error of the mean and the mean square value Mean estimation Ensemble averages Temporal averages

NH

Error

^xV^

Estimate of tiie mean square value

Error

IN 1

^__^_^__

Calculations of the quantities defined in this chapter are carried out in practice on samples of short duration T, subdivided into K blocks of duration AT [BEN 71] [BEN 80], by using filters of non-zero width Af. These approximations lead to the errors in Table 4.9.

Practical calculation of power spectral density

151

Table 4.9. Other statistical errors Quantity Z Direct PSD

Error s

yy

Cross PSD

Coherence

Pxy (f)

Yxy xW

Pxy(f)

Gyy(f)

Px xy(f)

TA

Transfer function Pxy (f)

P xy x(f)

These expressions can be used with an estimated value P of the correlation coefficient instead of P (unknown); one then obtains approximate values of er, when er is small (i.e. er < 0.20 ), which can be limited at the 95% confidence level using: z ( l - 2 e r ) < Z < z ( l + 28 r ) where Z is the true value of the parameter and Z its estimated value. Figure 4.44 shows the variations of the error made during the calculation of the transfer function H w (f) , given by:

152

Random vibration 1/2

[4.91] 'xy

2TAf

for various values of nd = T Af. xy'

H(f) =

H(f)|

G x (f) H(f) = measured transfer function H(f) = exact function [BEN 63] [GOO 57].

Figure 4.44. Statistical error related to the calculation of the transfer function

It is shown that, if P = Prob

H(f)-H(f)

H(f)

P = l-

< sin er and

[4.92] -v^(f)cos 2 e

where n is the number of degrees of freedom, equal to 2 T Af.

Practical calculation of power spectral density

153

In(l-P)

n=

In

[4.93]

1-vxy

Figure 4.45. Number of dof necessary for the Figure 4.46. Number of dof necessary for statistical error on the transfer function to be the statistical error on the transfer function to lower than 0.10 with probability P be lower than 0.05 with probability P

The statistical error resulting from the calculation of the autocorrelation Rx is given by [VIN 72]:

A reasonable value of T for the calculation of Rv is T =

1

— . For the crossAfsJ

correlation R v : [4.95] TAf

154

Random vibration

4.13. Generation of random signal of given PSD 4.13.1. Method The method of generation of a random signal varying with time of given duration T from a PSD of maximum frequency f,^ includes the following stages: - calculation of the temporal step St = *samp.

^-" ^max

- choice of the number M of points of definition of the PSD (power of two), T - calculation of the number of signal points: N = —, 5t

- possibly, modification of N (and thus of the duration) and/or of M in order to respect a maximum statistical error e0 (for a future PSD calculation of the generated M eg signal), starting from the relation — < —, maintaining M equal to a power of two, N 2

- calculation of the frequency interval between 2 points of the PSD Af =

samp

,

2M

- for each M points of the PSD, calculation at every time t = k 8t (k = constant integer between 1 and N) of a 'sinusoid' - of the form:

m

x(t)= m x max sin(2 n f t + cp m ),

- of duration T, - of frequency fm = m df (m integral such that 1 < m < M ), - of amplitude ^2 G(f m ) Af [where G(f m ) is the value of the given PSD at the frequency fm, the amplitude of a sinusoid being equal to twice its rms value], -of random phase cp m , whose expression is a function of the specified distribution law for the instantaneous values of the signal, - sum of the M sinusoids at each time.

Practical calculation of power spectral density

155

4.13.2. Expression for phase 4.13.2.1. Gaussian law It is shown that one can obtain a normal distribution of the signal's instantaneous values when the phase is equal to [KNU 98] 9m = 27i^-21nr1cos(27ir2 )

I4-96!

or
'

'

"

[4-97]

In these expressions, rt and r2 are two random numbers obeying a rectangular distribution in the interval [0, 1]. Definition A random variable r has an uniform or rectangular distribution in the interval [a, b] if its probability density obeys p(r)-

1 bll [0

"•"•*" for r < a or r > b

[4.98]

If a random variable is uniformly distributed about [0, 1], the variable a+b y = a + ( b - a ) r is uniformly distributed about [a, b], having a mean of and 2 b-a standard deviation s =2 V3

4.13.2.2. Other laws

We want here to create a signal whose instantaneous values obey a given distribution law F(x). This function being nondecreasing, the probability that x < X is equal to [DAH 74]: P(x < X) = P[F(x) < F(X)]

[4.99]

156

Random vibration

Let us set F(x) = r where r is a random variable uniformly distributed about [0,1]. It then becomes: p[F(x)
[4.100]

From definition of the uniform distribution, P(r < R) = R where R is an arbitrary number between 0 and 1, yielding P(x < X) = P[r < F(X)] = F(X). To create a signal of distribution F(x), it is necessary thus that:

F(x) = r

[4.101]

The problem can also be solved by setting:

F(x) = l-r

[4.102]

Examples 1. Signal of (Appendix Al)

exponential

F(X) = l-

distribution: the

distribution

-XX

is

defined by [4.103]

From [4.102], l- e - Xx

[4.104]

=l-r

yielding

Inr

[4.105]

X — ~~"

and cpm = -2 TT

Inr

[4.106]

2. Signal with Weibull distribution: from (Appendix A.I)

1- exp

X-s

\<x

X>s

-e 0

X<8

it is shown in a similar way that we must have:

[4.107]

Practical calculation of power spectral density

(v-s)(-lnr) 1/a

157

[4.108] [4.109]

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Chapter 5

Properties of random vibration in the time domain

5.1. Averages 5.1.1. Mean value The mean value of a random vibration l(i) calculated over duration T,

m

= -5(t)dt

is related to the difference between the positive and negative areas ranging between the curve e(t) and the time axis [ORE 81]. The mean m of a centered signal is zero, so this parameter cannot be used by itself alone to correctly evaluate the severity of the excitation.

Figure 5.1. Random vibration -with non-zero mean

160

Random vibration

The mean value is equal to the absolute value of the parallel shift of the Ot axis necessary to cancel out this difference. A signal ^(t) of mean m, can be written:

t5-2]

*(t) = m + /(t)

where £ (t) is a centered signal. This mean value is in general a static component which can be due to the weight of the structure, to the manoeuvrings of an aircraft, to the thrust of a missile in phase propulsion etc. In practice, one often considers this mean to be zero. 5.1.2. Mean quadratic value; rms value The rms value is calculated from the mean quadratic value of the instantaneous values of the signal. The dispersion of the signal around its mean is characterized by:

dt = = -- J

0

c.2

s

2

(t) dt

[5.3]

=

2

It is pointed out that s is the variance of the distribution of the instantaneous values of ^(t) and that s is the standard deviation. Two signals having very different frequency contents, corresponding to very dissimilar temporal forms, can have the same mean quadratic value. In this expression, the rms value takes into account the totality of the frequencies of the signal. If the mean m is zero, the standard deviation s is equal to the rms value of the signal NOTE. On the assumption of zero mean, one can however note a difference between the standard deviation and the rms value -when the latter is calculated starting from the power spectral density, which does not necessarily cover the -whole ofthefrequential contents of the signal, in particular beyond 2000 Hz (value often selected as upper limit of the analysis band). The rms value is then lower than the standard deviation. The comparison of the two values makes it possible to evaluate the importance of the neglected range.

Properties of random vibration in the time domain

161

Example Vibratory environment on an aircraft, represented by acceleration as function of time:

1. Taxi 7. Maximum velocity at low altitude 2. Takeoff 8. Climbing turn 3. Climb 9. Deceleration 4. Cruise at high altitude 10. Landing approach 5. Maximum velocity at high altitude 11. Touchdown 6. Cruise at low altitude Figure 5.2. Rms acceleration recorded on a aircraft during flight [KAT 65]

Figure 5.3. Rms value of vibrations measured on a satellite during launch

162

Random vibration

It is very useful to plot the variations of the rms value against time (sliding mean on n points), hi order to: -choose the time intervals over which the rms value varies little: each corresponding phase can then be characterized by a PSD, - study the very short duration phenomena (nonstationary phenomena). The analysis for example measure the number of times that the rms value crosses a given threshold with respect to the amplitude of this threshold (rms value of the total signal or of the response of a one-degree-of-freedom mechanical system of constant Q factor, generally equal to 10, whose natural frequency varies in the useful frequency band) [KEL 61]. The variation of the rms value with time has also been used as a monitoring tool the correct operation of rotating machinery based on a statistical study of their vibratory behaviour [ALL 82] [PAR 82].

5.2. Statistical properties of instantaneous values of random signal The analysis of the statistical properties of the instantaneous values of a random signal i(t) is based primarily on the work of S.O. Rice [RIC 44] and of S.H. Crandall [CRA 63]. One is more particularly interested in the study of the probability density of the instantaneous values of the signal and in that of the peaks (positive and negative maximum amplitude). This study results in considering simultaneously at a given time f(t) and its derivatives J?(t) and J?(t) which respectively represent the value of the signal, its slope and its curvature at the time t. These parameters are in particular associated with a multidimensional normal probability density function of the form [BEN 58]: [5.4]

for the research of the distribution law of the peak values. 5.2.1. Distribution of instantaneous values The distribution of the instantaneous values of the parameter describing the random phenomenon can very often be represented by a Gaussian law [MOR 75]. There can of course be particular cases where this assumption is not justified, for example, for vibrations measured on the axle of a vehicle whose suspension has just

Properties of random vibration in the time domain

163

compressed an elastic thrust after deflection of the dampers (non linear behaviour in compression only).

5.2.2. Properties of derivative process Let us consider a stationary random vibration ^(t) and its derivative J?(t), defined by: >(t)= lim

4t + At)-^(t)

[5.5]

At

At-»0

With the condition that At) -

lim

=0

[5.6]

At

At-»0

Average value of the derivative process This is [5.7]

= lim E At->0

At

If the process is stationary,

yielding [5.8]

t =0

NOTE. The autocorrelation R^(t) presents an absolute maximum for T = 0. One thus has R ( 0 ) < 0. [5.9]

= lim E At-»0

= R'^(O)

At

[5.10]

164

Random vibration

The derivative of the autocorrelation function of a derivable process is: - continuous and derivable at any point, - even. It is thus cancelled for t = 0, yielding

E* e = o

[5.11]

There is no correlation between a stationary process £(t) and the derivative process J?(t) (whatever the distribution law). Mean square of the derivative T2~

+ At) - t(t)

=2 (At)

At

[5.12] A stationary process ^(t) is thus derivable in the mean square sense if and only if its correlation function R ( T ) contains a continuous second derivative. Correlation function of the process and its derivative 1. By definition [1.48], R#M = EJJ?(t) J?(t + T)]

R,,(T)= lim lO"™ 1 '

At,

R f t (t)= lim

-"vt

+ At

At2 - At|) - R^(T - At]) - R^(T + At2) -

At At,->0

R - > ( T ) = lim At,->0

2)-

At-

At,

Properties of random vibration in the time domain

R»M = -—T—=-R;W

R./T) = lim d <(t) At->0

165

[5.13]

—^1 At

L

«R (/T \) = lim ,. \ */ At->OL

R

(i)

At

_dR^)

[5.14]

dt

In the same way: [5.15]

dt In a more general way, if r and u are the m* derivative processes of and n* of u(t), one has, if the successive derivatives exist,

R. m >..( n >=t-i) r

[5.16]

Variance of the derivative process [5.17] Since E[^J = 0 , the variance s-^ is equal to [5.18]

Power spectral density of the derivative process By definition [2.45]:

166

Random vibration

R,(t) = C

S,(Q) *—QO

iQt

edQ

*

Knowing that:

i Q) S,(Q) e' n T dQ

[5.19]

This yields [5.20] 2

2 E\t ] = tJ'_oo StV(Q) dQ = wJ__oo Q I J

dQ

[5.21]

and of the same way [MOR 56] [NEW 75] [SVE 80]: E[?! = J •'-

S-,(Q) dQ = J <•

=Q

Q4 S,(Q) dQ

[5.22]

-

-

[5.23]

[5.24]

NOTES. The autocorrelation functions of the derivative processes oft(t) depend only on T. The derivatives of a stationary process are stationary functions. However, the integral of a stationary function is not necessarily stationary. The result obtained shows the existence of a transfer function H(Q) between l(t) and its derivatives: S^(Q) = |H(Q)|2 S,(Q)

[5.25]

S-^(Q) = |H(Q)|4 S,(n)

[5.26]

where

H(Q) = i Q

Properties of random vibration in the time domain

167

5.2.3. Number of threshold crossings per unit time Let us consider a stationary and ergodic random vibration £(t); and p(£), the probability density function of the instantaneous values of ^(t). Let us seek to determine the number of times per unit time na the signal crosses a threshold chosen a priori with a positive slope. Let us set na the number of occasions per unit time that the signal crosses the interval a, a + da with a positive or negative arbitrary slope, da being an very small interval corresponding to the time increment dt. We have, on average, n; = -*2

[5-27]

Let us set nQ the number of occasions per unit time that the signal crosses the threshold a = 0 with a positive slope (n^ gives an indication of the average frequency of the signal). Let us set finally 40 the derivative of the process 40 and b the value of 40 when i = a. Let us suppose that the time interval dt is sufficiently small that the variation of the signals between t and t + dt is linear. To a-40 cross the threshold a, the process must have a velocity 40 greater than dt The probability of crossing is related to the joint probability density p(l,l) between t and 't . Given a threshold a, the probability that: a < 4 0 £a + da , and

[5.28]

b < l(t) < b + db is thus, in a time unit, p(a,b) da db = PJa < f(t) < a + da, b < i?(t) < b + db]

[5.29]

Setting ta the time spent in the interval da:

da
[5.30]

PI

(ta being a primarily positive quantity). The number of passages per unit tune in the interval a, a + da for J?(t) = b is thus:

168

Random vibration

p(a, b) da db

= |b| p(a, b) da

[5.31]

and the average total number of crossings of the threshold a, per unit time, for all the possible values of
n a = f~|b|p(a,b)db=:2n;

[5.32]

where f oo

/

.\

I p/,/

[5.33]

This expression is sometimes called the Rice formula. The only assumption considered is that of the stationarity. One deduces some, for a = 0, [5.34] and n II

n

rt

o

„+ II r»

n

+ o

f

•?

b p(a, b) db

f+°°. i J |b|p(0,b)db

[5.35]

These expressions can be simplified since the signals ^(t) and l(t) are statistically independent: [536]

Then, db

na = p(a)

[5.37]

and n

n0

" nj

[5.38]

p(0)

Lastly, if 7r(J?) is an even function of b, 7l(b)

= 7t(-b)

Properties of random vibration in the time domain

169

yielding na = 2 p(a) f° b 7i(b) db

[5.39]

Particular case

If the function l(t) has instantaneous values distributed according to a Gaussian law, zero mean and variance l^ms, such that

e

2i?

[5-40]

™

it comes, starting from [5.37]: __b

na = p(a)

or snce n

et-<x>

b b

" 2

2* - »

[5.41]

s even, 21

p(a)

[5.42]

If the instantaneous acceleration is itself distributed according to a Gaussian law (0, ^rms): [5.43]

2

1

e

^-

?

-J

[5.44]

and [LEY 65] [LIN 67] [NEW 75] [PRE 56a] [THR 64] [VAN 75]: a

[5.45]

170

Random vibration

[5.46]

a2

na =n e

2

[5.47]

<-

Figure 5.4. Probability density of instantaneous values of a random signal

Since [5.48]

•rtns = f°G(n) dO = R(0)

= f°Q 2 G(Q) dO = -R*(o

[5.49]

[5.50] there results [DEE 71] [VAN 75]:

l 2

nn a

zn 2 an + - ' K

*fa G(Q) dn Jo(Q) dn

a2

>j

exp . ^2z ) / I ^' c rms /

na is the mean number of crossings of the threshold a per unit time.

[5.51]

Properties of random vibration in the time domain

171

n^ is the mean number of crossings of the threshold a with positive slope and per unit time.

5.2.4. Average frequency Let us set [PAP 65] [PRE 56b]:

n0 =

2 1 J Q G(Q) dQ

2

Too

J G(Q)dQ . Jo

1 1 R (2) (0) 7i V

[5.52]

R(0)

Depending on f, n0 becomes [BEN 58] [BOL 84] [CRA 63] [FUL 61] [HUS 56] [LIN 67] [POW 58] [RIC 64] [SJO 61] [SWA 63]: 1 f°°

J

9

IT" 2

r G(f ) df

[5.53]

+

0 n O n0 - 2 n 0 -2 n

£o
J

The quantity n0 {average or expected frequency) can be regarded as the frequency at which energy is most concentrated in the spectrum (apparent frequency of the spectrum). Band-limited white noise

6(f)

f2

f

Figure 5.5. PSD of a band-limited white noise

172

Random vibration If the PSD is defined by G(f) = G0

for f] < f < f2

G(f) = 0

elsewhere

we have

[5.54]

[5.55]

Ideal low-pass filter

Iff, = 0, 2

[5.56]

V3

a narrow band noise

6(f)'

Figure 5.6. F5D o/a narrow band noise

Let us consider a random vibration of constant PSD G(Q) = G0 in the interval , zero elsewhere. We have [COU 70] [NEW 75]:

Properties of random vibration in the time domain

173

[5.57]

[5.58]

Figure 5.7. PSD of a narrow band noise

Let us set Af = 2 s . We have fj = f0 - e and f2 = f0 - s, yielding

and [5.59]

174

Random vibration

HO tends towards f0 when Af tends towards zero. For whatever f0, n0 is equal to or higher than f0. In the case of the response of a linear slightly damped one-degree-of-freedom system, n^ will be thus in general close to the natural frequency f0 of the system.

5.2.5. Threshold level crossing curves Threshold level crossing curves give, depending on the threshold a, the number of crossings of this threshold with positive slope. These curves can be plotted: - either from the time history signal by effective counting of the crossings with positive slope over a duration T. For a given signal, the result is deterministic, - or from the power spectral density of the vibration, by supposing that the distribution of the instantaneous values of the signal follows a Gaussian law to zero mean. One obtains here the expected value of the number of threshold crossings a over the duration T [LEA 69] [RIC 64]:

[5.60] with n^= expected frequency defined in [5.53]:

Properties of random vibration in the time domain foo +2

Ilrt =

1

7

f

°(f>

175

df

J o °°G(f)df The knowledge of G(f) makes it possible to calculate n0 and ^nns, then to plot Na as a function of the threshold value a. In practice, one generally represents a with respect to Na, the first value of Na being higher or equal to 1. For N^ = 1, 0

[5.61]

=

a0 is, on average, the strongest value of the signal observed over a duration T.

Figure 5.9. Example of threshold level crossing curve for a Gaussian signal

The curve in Figure 5.9 shows the variations of

with respect to A

plotted starting from the expression:

N; -* T T

N

,

176

Random vibration

Figure 5.10. Largest peak, on average, over a given duration

The variations of

as function of the product n0 T are represented in +

rms

Figure 5.10:

It is observed that it is possible to obtain, in very realistic situations, combinations of n^ and T such that the ratio —— is equal to or higher than 5. For this, it is necessary that: + ~ . 25/2 n0 T > e

njT>2.7105 For

T = 600 s

it is necessary that

n+

> 447 j^z

T = 3600 s

n

J > 74.5 Hz

T = 4 hours

n£>18.6Hz

Properties of random vibration in the time domain

177

x

\

h\

x

4

:

i

\^

;^\

;

19-* 8.8

1.8

2.8

3.8

4.8

5.8

a It rms Figure 5.11. Time necessary to obtain, on average, a given maximum level, versus the average frequency

Figure 5.12. Probability of crossing a given threshold, versus the threshold value

Figure 5.11 indicates the duration T necessary to obtain a given ratio aQ/^, as function of n.

u+ "a

Figure 5.13. Noteworthy points on the threshold level crossings curve

For a = 0,

Nj = N£ = nj T The probability that the signal crosses the level a with a positive slope and that a < t < a + Aa is equal to Nj Aa/Nj £^5 .

178

Random vibration

Figure 5.14. Values of the signal in the interval a, a + da, after crossing threshold a with positive slope

The probability that ^(t) is higher than a is equal to:

da

P=

[5.62]

Ll

™ da

e

Knowing that the error function can be written:

J u 2 fu -T" Erfl— = - J e 2 du v v/T^ yJ

V «r

i TT

[5.63]

0

and that:

resulting in, if u =

P=

e

2

du

[5.64]

Properties of random vibration in the time domain

179

This yields, after standardization: [5.65]

1-Erf

P(u> rms

rms

Figure 5.12 shows the variations of P(U > Oand5.

for

ranging between

Figure 5.15. Example of threshold level crossing curve

Example Let us consider a random acceleration defined over a duration T = 1 hr by its PSDG(f): G(f) = G! = 0.1 mV/Hz from 10 Hz to 50 Hz G(f) = G2 = 0.2 mV/Hz from 50 Hz to 100 Hz G(f) = 0

elsewhere

xLs = (50 +10) 0.1-K100 - 50) 0.2= 14 (m / s2)2

180

Random vibration

*nns=3.74m/S 2

From [5.53]: +2

0.1 (50 -10) + 0.2 (100

-50)

3 x™

2

Hz

nj = 66.8 Hz

and [5.60]: NT = 66.8 3600 e

z2 1
= 2.4

a 28

Figure 5.16. Example of curve threshold level crossings

The threshold which is only exceeded once on average over the duration T has an amplitude a0 = 3.74^/2 In 66.8 3600 = 18.62 m/s2 These threshold level crossings curves were used to compare the severity of several random vibrations [KAZ 70], to evaluate their damage potential or to reduce the test duration. This method can be justified if the treated signal is the stress applied to a part of a structure, with just one reserve, which is the non immediate relation between the number of peaks and the number of threshold level crossings; it is not, on the other hand, usable starting from the input signal of acceleration. The

Properties of random vibration in the time domain

181

threshold crossings curve of the excitation x(t) is not representative of the damage undergone by a part which responds at its natural frequency with its Q factor. In random mode, one cannot directly associate a peak of the excitation with a peak of the response.

NOTES. \.All the relations of the preceding paragraphs can be applied either to the vibration input on the specimen, or to the response of the specimen. 2. ONERA proposed, in 1961 [COU 66], a method of calculation of the PSD G(f) of a stationary and Gaussian random signal starting from the average number of zero level crossings, its derivative and the rms value of the signal. The process can be extended to non Gaussian processes.

5.3. Moments Many important statistical properties of the signal considered (excitation or response) can be obtained directly from the power spectral density G(Q) and in particular the moments [VAN 79]. Definition Given a random signal ^(t), the moment of order n (close to the origin) is the quantity: 1

dt

n/2

f+T

= lim

2T

dt n/2 J

dt

[5.66]

(if the derivative exists). The moment of order zero is none other than the square of the rms value t^: Mn = M0 =

G(f) df

The moment of order two is equal to:

Mo =

[5.67]

182

Random vibration However, by definition,

R(t) =

G(Q) cos QT dQ

[5.68]

If we set:

S(i) = E[#t)£(t + T)]

[5.69]

, . fw 7 d R(T) S(t) = I Q2 G(Q) cos QT dQ = ' 0

2

r,

_„, [5-70]

dT

(if R exists). We have, for i = 0 . S(T) = J°° Q2 G(Q) dn

[5.71]

In the same way, if: [5.72]

, .

d 4 R(t)

T(T) = -

fco

4

— = I Q4 G(Q) cos QT

4

[5.73]

it becomes, for T = 0, T(0) = J°° Q4 G(Q) dQ

[5.74]

yielding [KOW 69]: M2 =-R(o)= r°Q 2 G(Q)dQ = (27i)2 f°f 2 G(f)df = ^27ns

M =-

G(Q)dQ = (27t)4 |°f4 G(f)df = /^

[5.75]

[5-76]

More generally, the n* moment can be defined as [CHA 72] [CHA 85] [DEE 71] [PAR 64] [SHE 83] [SWA 63] [VAN 72] [VAN 75] [VAN 79]:

Properties of random vibration in the time domain 183 Mn = J°° Qn G(Q) da = R (n) (0) [5.77]

or n

M =

fn G(f) df

j

(n integer) where R (2n) (0) = (-!)"

J

Q 2 n G(Q) dO

[5.78]

Mn are the moments of the PSD G(Q) with respect to the vertical axis f = 0. Application One deduces from the preceding relations [CRA 68] [CHA 72] [LEY 65] [PAP 65] [SHE 83]: 1

n

o

1

/

Y-

M2 2

[5.79]

271 , M oJ

[5.80]

NOTES. Some authors [CHA 85] [FUL 61] [KOW 69] [VAN 79] [WIR 73] [WIR 83] define Mn by:

M n = f" f" 0(f) df

[5.81]

which leads to [BEN 58] [CHA 85]:

nn =

M

(sometimes noted Q^) [VAN 79],

[5.82]

184

Random vibration

5.4. Average frequency of PSD defined by straight line segments 5.4.1. Linear-linear scales

fST with

M O = rI? o(f)df where

G(f) = af + b

[5.83]

2

'2 =

f2 G(f) df

M = (2n

Figure 5.17. PSD defined by a straight line segment on linear axes

[5.84]

Properties of random vibration in the time domain

185

This yields, after having replaced a and b by their value according to fj, f2, Gj and G2 [BEN 62],

tf-fj1 n +"

o - fj - c

4

3 2

- fl )

+

[5.85]

(f2G, - f,G 2 ) (f2 - f,)

Particular cases = G2 = G0 = constant

fo-f

fl + f l f 2 + f2

[5.86]

If fj = 0 and if G = GQ until f2, there results:

nf = -*-

[5.87]

nj = 0.577 f2

[5.88]

i.e.:

£

If the PSD is a narrow band noise centered around f0, we can set f{ = f0 — and 2 fj = f0 + - [BEN 62], yielding: 2 n:2 = f. + —

12 8->0,

If

n

o -^

[5-89]

186

Random vibration

5.4.2. Linear-logarithmic scales In this case, the PSD is represented by: lnG = a f + b 2

M0=

[5.90]

edf

=

e

[5.91]

f2eaf+bdf After integration by parts, it becomes, if a * 0, af+b M2 = (2*) i-y(a2 f2 -2 a f + 2) 2

[5-92]

yielding af,+b/ 2 f2 ~ ~ ~\ af.+b/ 2 f2 ~ f ~ e 2 la f2 - 2 a f2 + 2! - e ' la fj -2 a ft + 2

+
0

[5.93]

2 /la af,+b af.+bl af.+b\ le 22 -e ~ 1' I

the constants a and b being calculated starting from the co-ordinates of the points fls Gj,I' £> and G->. 2, £ Particular case

G2 = Gl5 (a = 0) [5.94] [5-95]

and +2 n

_

f

2 ~fi

_ fi

° ~3(f2-fi)~

+ f f

1 2 + f2 3

[5 96]

Properties of random vibration in the time domain

187

5.4.3. Logarithmic-linear scales

G(f) = a l n f + b M,0 =

[5.97]

f 2 ( a l n f + b)df a(ftaf)J|+(f2-f1)(b-a)

[5.98]

(2rc)2 JP 2 f 2 (va l n f + b)df f, ' a|lnf2--

f 3 a In f 2 — + b -fl 3 a f l n f 1

a Infi -- +b

_r

H]

3y

[5.99]

[5.100]

a (f2 In f2 - fi In fi)+ (f2 - fj)(b - a)

Particular case Gj = G2 = G0 = constant In this case, a = 0 and b = G0, yielding:

and +2

n0 =

f3

f3

*2 ~ M

f2

M

[5.101]

188

Random vibration

5.4.4. Logarithmic-logarithmic scales [5.102]

G(f) = G, -

the constant b being such that b =

In G 2 /Gj

In ff

Mn= r G

—

df = M

(ifb*-l): M2 = (27t)2 JJ2 f2 G(f) df = (27t)2 ^±

(if b* -3). It yields:

n

[5.103]

= ° "b + 3 f b + 1 - f b + 1

Ifb = -l: In — and f2_f2

M2 = (27i)2 G! f,

n

=

r 2 -f 2 2 In f 2 /f,

L

[5.104]

Properties of random vibration in the time domain

189

Ifb = -3:

f

1

1

\

VM

M2 = (27t)2 G! ff In —

2 ta f

+2 ,2 f2 O = 2 fl f2

f2 r

2/ f l f2 ~r

[5.105]

NOTE. If the PSD is made up o/n straight line segments, the average frequency n0 is obtained from:

n+ n 2

1

1=1

o =7~-r~fT

[5.106]

5.5. Fourth moment of PSD defined by straight line segments The interest of this parameter lies in its participation, with M0 and M2 already studied, in the calculation of n and r.

5.5.1. Linear-linear scales By definition.

MA=(2nr I rG(f)df G(f) = a f+ b yielding:

190

Random vibration

MA = UTI

where a =

[5.107]

G2 — GI

f*2 ~ M f

and b =

f 2 Gj - fjG 2

hf ~ Mf

Particular cases 1. Gj = G2 = G0 =constant, i.e. a = 0 and b = G0: M 4 =(27t) 4

[5.108]

2. fj = 0 4 Fa 4

3.

6+

b

[5.109]

2

L6 5 = 0 and G0 = constant [5.110]

5.5.2. Linear-logarithmic scales

a=

In G 2 /G, f2-fL

b=

f2 In Gj - fj In G2

U=

4

f G ( f ) d f =(2*>4 jf f e

af+b

After several integrations by parts, we obtain, if a * 0,

df

[5.111]

Properties of random vibration in the time domain

M4 =

(2*)'

af 2 +b

a

—p af,+b '

4

12 ff 24 f, M + —y— a a

4ff

fj

12 K

24 f,

24

a

a

a

24

191

[5.112]

Particular cases 1. Gj = G2 = G0 =constant Then, a = 0 and b = In G0 f5 ~

2

1

f5

[5.113]

~ *

2. f = 0 and a = 0 12 ,24 241 24 e - | I 2 — f 2 J + — f|-— f 2 + — |a a a a y a

[5-114]

and, if Gt = G2 = G0 M4 = (2;t)4 GO —

[5.115]

5.5.3. Logarithmic-linear scales

G = aln f + b 4

= (27r)4 J 2 f4 ( a l n f + b ) d f *i

M4 = (2n)4

a taf!-- +b

a In f> - —

where GO — GiL

a =~

In f 2 /f,

Go

and b = -^-

Inf

[5.116]

192

Random vibration

Particular cases = G2 = G^ = constant, i.e. a = 0 and b = G( M4=(27r)4-

[5.117]

f 11 2

Iff, =0: I

M4 =

G

0

i

5.5.4. Logarithmic-logarithmic scales

yielding, if b * -5, 5 Gi ff -^ 5 J

M4 = (2*)T-£ ff

b+5

—

or M4 =

(271)

[5.118]

Ifb = -5:

f

M4 = (27t)1 ff

b—

Particular case If Gj = G2 = GJQ = constant and if b * -5

[5.119]

Properties of random vibration in the time domain

M)

b+5

193

[5.120]

NOTE. If the PSD is made up ofn horizontal segments, the value 0/"M4 is obtained by calculating the sum: [5.121]

5.6. Generalization; moment of order n In a more general way, the moment Mn is given, depending on the case, by the following relations.

5.6.1. Linear-linear scales The order n being positive or zero, IV! n

- \^t

n+1

1

1

Ln + 2

v\

2

1+

M

'

n+P

\12

h

[5.122]

]

'_

5.6.2. Linear-logarithmic scales

Mn

( ^nl c af,+b[

^Ttj 1C

[ L a

-e af.+bLn |rl La

n

n n

t2

I2

,

+

n(n - 1) 2

a

I2

2

"+

n a

n! .n-I n(n ~ ]) rn-2 ]l + M + l "" n ! 2 a aJ J

[5.123]

5.6.3. Logarithmic-linear scales \ ^n+l

Mn = (2;t)n (n>0)

mf,n+l

dlnfj V

+b n+ l

[5.124]

194

Random vibrition

5.6.4. Logarithmic-logarithmic scales

.

^ Mi+1 ^ r-n+1 .„ n U') to — Oi it

.

^ r.b+n+1 ^ ._ n(ji to - Oi it }

Mn = (27i)l -^-^ —— = (27i) -g- -—b+n+1 fj b+ n +1

[5.125]

Ifb = -(n + l): Mn=(2jnf1n+1G1ln-^-

[5.126]

Chapter 6

Probability distribution of maxima of random vibration

6.1. Probability density of maxima It can be useful, in particular for calculations of damage by fatigue, to know a vibration's average number of peaks per unit time, occurring between two close levels a and a + da as well as the average total number of peaks per unit time.

NOTE. One is interested here in the maxima of the curve which can be positive or negative (Figure 6.1).

Figure 6.1. Positive and negative peaks of a random signal

196

Random vibrjttion

For a fatigue Analysis, it -would of course be necessary to also count the minima. One can acknowledge that the average number of minima per unit time of a Gaussian random signal is equal to the average number the maxima per unit time, the distributions of the minima and maxima being symmetrical [CAR 68]. A maximum occurs when the velocity (derivative of the signal) cancels out with negative acceleration (second derivative of signal). This remark leads one to think that the joint probability density between the processes ^(t), J?(t) and £(t) can be used to describe the maxima of t(t}. This supposes that ^(t)l is derivable twice. S.O. Rice [RIC 39] [RIC 44] showed that, if p(a, b, c) is the probability density so that ^(t), i?(t) and £(t) respectively lie between a and a + da, b and b + db, c and c + dc, a maxiimum being defined by a zero derivative and a negative curvature, the average number the maxima located between levels a and a + da in the time interval t, t + dt (Window a, a + da, t, t + dt) is: = -dt

CO

[6.1]

c p(a, 0, c) dc

where, for a Gausldan signal as well as for its first and second derivatives [CRA 67] [KOW63]: p(a, 0, c) i (27i)~3/2 |M ~1/2 exp

a" + n33 c + 2 u13 a c

2|M|

[6.2] J

with

-t^2 2

M = -t rms

Let us recall trjat:

The determinatnt JMJ is written:

[6.3]

Probability distribution of maxima 2 2 "t ^rms (v l - r* )/

2 2 2 4 c 2 M |m| = £ rms (t V-rms c"t rms -ic rms;\= rms

|M = M 0 M 2 M 4 ( l - r 2 )

197 [6.4] [6.5]

if

I^2rms

M

[6.6]

R is an important parameter named irregularity factor. |M| is always positive. The cofactors y.i\ are equal, respectively to: [6.7] [6.8]

^13 = ^rms = M2 ^33 = -ftns t2m* = M0

[6.9]

M

2

yielding va = -da dt

f

(2s)-3/2 ,/M0M2M4(l-r2) 2

2

2

M4 a + M 0 M 2 c + 2 M 2 a c

exp

2 M0 M 2 M 4 ( l - r 2 )

da dt (27i)"3/2

VIt) '0 -oc

M

[6.10]

dc

~2M 0 (l-r 2 ) 2

2 M 4 ( l-r )

c exp

r |

1 /

2\

C

2 M4(l-r2)t

\ 2 M2 ac 1

2 +

M0

J_

dc

198

Random vibration

2M 0 (l-r 2 )

M

c exp"] — c + —-a M 2 M, II -r 2 ) v o

2M0(l-r2)

dli dt

v, = -

dc

2M0(l-r2)

M4 ( l - r 2 )

M; Mo J a 1 expj 2 M4|lM0 J

M

c+ V

2

M, c + —-a. dc-

M2 a fo M

J

dc

ex

Pl

2 1VU1-

M

Let us set v =

and w = Vv . It becomes:

2 M,

v. = 2

e2M<]M4

^M0M2M4(l-r ) 12 a/M 0 0

After integration![BEN 58] [RIC 64],

v =

(27i)~3/i da dt

M0

IIV1

2

a2r2/2M0(l-r2) —00

dw

_v

e

dv

Probability distribution of maxima

+ firMff

\

(2

199

a2 2

e

ar

2M f t

° 1 + Erf

'A/MO"

[6-11]

i.e.

[6.12]

vr aa = «n "p q(a) da dt

where [6.13]

(average number of maxima per second), n^can be also written: R (4) (0)

[6.14]

NOTE. va can be written in the form [RIC 64]: -R(2)(0)

a

2R(0) If

2R(0))/R^ 4 ) (0)R (0)

J2 k R(0) r~ (2)

( V(2 [R

^

")

)

[ V2kR(0)J_

\2ll

[6.15] [(

2 k:R(0)

where = R(0)R (4) (0)-[R (2) (0)]"

[6.16]

The probability density of maxima per unit time of a Gaussian signal whose amplitude lies between a and a + da is thus [BRO 63] [CAR 56] [LEL 73] [LIN 72]:

200

Random vibration

q(a) =

XI7

ra

ar

1 + Erf

[6.17]

2 fx _X2 { where Erf(x) = —p J e dX (Appendix A4.1). The probability so that a •Jit ° maximum taken ramdomly is, per unit time, contained in the interval a, a + da is

q(a)da . If we set m =

, it becomes:

va I a 1 da — = q(a) da = q(u) du = d dt VrmsJ^rms

[6.18]

yielding [BER 77] [CHA 85] [COU 70] [KOW 63] [LEL 73] [LIN 67] [RAV 70] [SCH 63]: u2

U

^-r

2

" 2(i-r<) 6

&

r

ue

2

r—

1 + Erf

2

ru

Ij'MJj

[6.19]

The statistical distribution of the minima follows the same law. The probabilty density q(u) is thus the weighted sum of a Gaussian law and Rayleigh's law, with coefficients function of parameter r. This expression can be written in various more or less practical foims according to its application. Since: foo

I

J

-oo

where

e

_X2 A

fX

dl = -Jn = 2 JJ

0

2

2 r°° -x Erf(x) = l--—J e* dA,

[6.20]

it becomes: 2

q(u) = —±r-

-

e

21-r 2

1

ue

foo

-—r J_ru

[6.21]

Probability distribution of maxima

201

Setting A, = — in this relation, we obtain [BEN 61b] [BEN 64] [HIL 70] V2 [HUS 56] [PER 74]:

x

q( u) =

VI-r

2

-

e

2 ) 2 (l-r 7

+ru

1-

foo

J

ru

e

2

dt

[6.22]

One also finds the equivalent expression [BAR 78] [CAR 56] [CLO 75] [CRA 68] [DAY 64] [KAC 76] [KOW 69] [KRE 83] [UDW 73]: 2

-

[6.23]

q(u) = —-——e where t

2 dt

<**)=-£=/*' ru V =

q(u) = V l - r 2 e

ru

2

/ -

V27i

v2

u2 ri /,,\

mu^ — v i ~ -r2 e 2

or

A/i

I-r

e

2

— + v O(v) V27i

[6.24]

202

Random vibration

q(u) = V l - r 2 e

d<E(v)

2

L

dv

[6.25]

q(u) = Vl-

dv

Particular cases 1. Let us suppose that the parameter r is equal to 1, q(u)then becomes, starting from [6.19], knowiiitg that Erf(co) = 1,

q(u) = ue

[6.26]

2

which is the probability density of Rayleigh's law of standard deviation equal to 1. a Since u = anid:

q(a) da = q^u) du =

a

da

[6.27]

it becomes q( a)=*)=

4- .'

[6.28]

2.1fr = 0,

q(u) = -—re

2

[6-29]

V27T

(probability density of a normal ie Gaussian law). There exists in this (theoretical) case an infinite number of local maxima between two zero crossings with positive slope. We will reconsider these particular cases.

Probability distribution of maxima

203

6.2. Expected number of maxima per unit time It was seen that the average number of maxima per second (frequency of maxima) can be written [6.13]:

n

PL p = J_ 27i WM 2

Taking into account the preceding definitions, the expected maxima frequency is also equal to [CRA 67] [HUS 56] [LIN 67] [PAP 65] [PRE 56a] [RJC 64] [SJO 61]:

1

R<4)(0)

rms

[6.30]

rms

n

:

G(Q)dQ 2

—oo

P=

f+oo

J

271

£

22

n G(Q) dQ

r G(f) df f+oo

I

2 2

[6.31]

f G(f)df

In the case of a narrow band noise such as that of Figure 5.6, we have:

1 /

1

[6.32]

i.e. co0 [6 331

'

n_ is thus approximately equal to n0: there is approximately 1 peak per zero crossing; the signal resembles a sinusoid with modulated amplitude.

NOTE. Using the definition of expression [5.81], [CHA 85] : „+ "p

M =

4

would be written [BEN 58]

204

Random vibration

Starting from the number of maxima va lying between a and a + da in the time interval t, t -t- dt, olie can calculate, by integration between ^ and t2 for time, and between - oo and + o> for the levels, the average total number of maxima between tj and t2: = — r—

[6.34]

q(a)dadt

Per second, n;=—

1

27i y M2

n =

P

r

fjvl. f-t M-J q(a)da -co

V

dt,

i FM, i y M2

and, between tj and t2,

[6.35]

Application to the case of a noise with constant PSD between two frequencies Let us consider a vibratory signal l(t) whose PSD is constant and equal to G0 between two frequencies fj and f2 (and zero elsewhere) [COU 70]. We have:

Probability distribution of maxima

205

This yields

1 n+

p-

f5 2 J1 If2^ _ Ij ^

[6.36]

3

_5f2 -d

Iff

[6.37]

If £ = f0

Af

Af

and f2 = fo H

2

(narrow band noise Af small).

2

Af

2; f

1+2 n

+^ ,2 p =fO

f

\2

Af

L2f0J

1

Af

5 l2f(J

[6.38]

Af

IfAf

np Af Figure 6.2 shows the variations of— versus —.

fo

f

o

206

Random vibration

Fligure 6.2. /ivera^e number of maxima per second of a narrow band noise versus its width

6.3. Average time interval between two successive maxima This average time is calculated directly starting from rip [COU 70]: [6.39]

In the case of a narrow band noise, centered on frequency f0:

xm -> — when Af -> 0.

Probability distribution of maxima

207

Figure 6.3. Average time interval between two successive maxima of a narrow band noise versus its width

6.4. Average correlation between two successive maxima This correlation coefficient [p(t m )] is obtained by replacing T by Tm in equation [2.70] previously established [COU 70]. If we set:

g

Af

it becomes:

Figure 6.4 shows the variations of jpj versus 5. The correlation coefficient does not exceed 0.2 when 8 is greater than 0.4. We can thus consider the amplitudes of two successive maxima of a wide-band process as independent random variables [COU 70].

208

Random vibration

Figure 6.4. Average correlation between two successive maxima of a narrow band noise versus its bandwidth

6.5. Properties of the irregularity factor 6.5.1. Variation interval The irregularity factor:

iLs

_ -R(2)(o)

can vary in the interval [0,1]. We have indeed [PRE 56b]: Mo

JV "0

G(Q) dQ [6.42]

According to Sch ware's inequality,

J

Q2 G(iQ) dn < J J G(Q) dD J J°° Q4 G(Q) dQ

i.e. [6.43] Since M2 ^ 0, it becomes:

Probability distribution of maxima

0<

M-

<1

209

[6.44]

Another definition The irregularity factor r can also be defined like the ratio of the average number of zero crossings per unit time with positive slope to the average number of positive and negative maxima (or minima) per unit time. Indeed, r=

M

[6.45]

Example Let us consider the sample of acceleration signal as a function of time represented in Figure 6.5 (with few peaks to facilitate calculations).

Figure 6.5. Example of peaks of a random signal

The number of maxima in the considered time interval At is equal to 8, the number of zero-crossing with positive slope to 4 yielding:

The parameter r is a measure of the width of the noise: - for a broad band process, the number of maxima is much higher than the number of zeros. This case corresponds to the limiting case where r = 0. The

210

Random vibration

maxima occur above or below the zero line with an equal probability [CAR 68]. We saw that the probability density of the peaks then tends towards that of a Gaussian law [6.29]:

q(u) =

:-e

2

- when the number of passages through zero is equal to the number of peaks, r is equal to 1 and the signal appears as a sinusoidal wave, of about constant frequency and slowly modulated amplitude passing successively through a zero, one peak (positive or negative), a zero, and so on. We are dealing with what is called a narrow band signal, obtained in response to a narrow rectangular filter or in response of a one-degree-of-freedom system of rather high Q factor (higher than 10 for example).

Figure 6.6. Narrow band signal

All the maxima are positive and the minima negative. For this value of r, q(u) tends towards Rayleigh's law [6.26]:

q(u) = u e

2

The value of the parameter r depends on the PSD of the noise via n0 and n_ (or the moments M0, M2 and M4). Figure 6.7 shows the variations of q(u) for r varying from 0 to 1 per step Ar = 0.15 .

Probability distribution of maxima

211

Figure 6.7. Probability density function of peaks for various values ofr

Example The probability that u0 < u < u0 + Au is defined by: nu K+AU — =J q(u)du nR -o where nR is the total number of occurrences. For example, the probability that a peak exceeds the rms value is approximately 60.65%. the probability of exceeding 3 times the rms value is only approximately 1.11%[CLY64]. NOTES. 1. Some authors prefer to use the parameter k = 1/r [SCH 63] instead of r. Others, more numerous, prefer the quantity [CAR 56] [KRE 83]: [6.46]

(sometimes noted z) whose properties are similar: - since r varies between 0 and 1, q lies between 0 and 1, — q is close to 0 for a band narrow process and close to 1 for a wide-band process,

212

Random vibrtition

- q = 0 for a pure sinusoid with random phase [UDW 73]. One should not confuse this parameter q with the quantity r.m.s. value of the slope of the envelope of the process q= :? often noted using the r. m.s. value of the slope of the process same letter; this spectral parameter also varies between 0 and 1 (according to the Schwartz inequality) and is function of the form of the PSD [VAN 70] [VAN 72] [VAN 75] [VAN 79]. It is shown that it is equal to the ratio of the rms value of the envelope of the signal to that of the slope of the signal itself. To avoid any confusion, it will hereafter be noted qg (Volume 5). 2. The parameter r depends on the form of the PSD and there is only one probability density of maxima for a given r. But PSD of different forms can have the same r. 3. A measuring instrument for the parameter r f"R meter") has been developed by the Bruel andKjaer Company [CAR 68] . 6.5.2. Calculation of Irregularity factor for band-limited white noise The following definition can be used: r 2_

™t M0M4 -f

M2 = (2 nf J]2 G f2 df = (2 n) G f,

[6-47]

L

[6-48]

3

M4 = (2 n)4 Jff2 G f4 df = (2 n)4 G f-^-

[6-49]

yielding .

5

(rf-^f

[6.50]

Probability distribution of maxima

213

i.e., if h = —,

V5 If f2 -> fj, h -> 1 and r -» 1. If f2 -» oo, h -» oo and r -> —.

When the bandwidth tends towards the infinite, the parameter r tends towards — = 0.7454 . This is also true if ft -> 0 whatever value f2 [PRE 56b]. The limiting case r = 0 can be obtained only if the number of peaks between two zero-crossings is very large, infinite at the limit. That is for example the case for a composite signal made up of the sum of a harmonic process of low frequency f2 and of a band-limited process at very high frequency and of low amplitude compared with the harmonic movement.

214

Random vibration

L.P. Pook [POO 76] uses as an analogy the rectangular filter - a one-degree-off0 freedom mechanical filter in which Af = — = 2 £ f0 to demonstrate, by considering Q that the band-limited PSD is the response of the system (f 0 , Q) to a white noise, that:

[6.52]

r=

8 90

«_

6 75

8.78

0.

Figure 6.9. Irregularity factor of band-limited-white noise versus damping factor

It is noted that r -> 1 if £ -» 0.

NOTE. 77ze parameter i of a narrow band noise centered on frequency f0, whose PSD has a width Af, is written, from the above expressions [COU 70] [RUD 75]:

Probability distribution of maxima

1 1+-

n r= o

Af

3

f 1+ 2

Af

215

V

[6.53]

4

1 Af +-

6.5.3. Calculation of irregularity factor for noise of form G = Const, f

Figure 6.10. PSD of a noise defined by a straight line segment in logarithmic scales

The moments are expressed

[6.54]

b+3

;f

[6.55] M2 = (2 Tif f{ G! In-^

if b = -3

216

Random vibration

G,

b+5 2

-f!b+5)

ifb*-5 [6.56]

4

M 4 =(27i) ff G! I n - -

Case:

if b = -5

-l, b * -3, b * -5

Let us set h = -- . Then:

(h b + 3 -l)

2 _

(b + 3)2

[6.57]

(hb+I-l)(hb+5-l)

The curves of Figures 6.11 and 6.12 show the variations of r(h) for various values of b (b < 0 and b > 0).

Figure 6.11. Irregularity factor versus h, for Figure 6.12. Irregularity factor versus h, for various values of the negative exponent b various values of the positive exponent b

For b < 0, we note (Figure 6.11) that, when b varies from 0 to - 25, the curve, always issuing from the point r = 1 for h = 1 goes down to b = -3, then rises; the curves for b = -2 and b = -4 are thus superimposed, just as those for b = -1 and b = -5. This behaviour can be highlighted in a more detailed way while plotting, for given h, the variations of r with respect to b (Figure 6.13) [BRO 63].

Probability distribution of maxima

217

Figure 6.13. Irregularity factor versus the exponent b

We observe, moreover, that, for b = 0, the curve r(h) tends, for large h, towards r0 = — = 0.7454. All occurs then as if f, were zero (signal filtered by a low-pass 3 filter). Case: b = -1

MA = f, G, In —

yielding r=

h 2 -!

[6.58]

218

Random vibration

Case: b = -3 /

i

1

1

\

M2 = (2 TC) ' fj3 Gt In —

r=

2h Inh

[6.59]

h 2 -!

This curve gives., for given h, the lowest value of r. Case: b = -5 /

M0 =

1

1

\

/

1

M2 = -(2 7t)

4

~ , M4 = (2 n) r-5 ff G! In

1

\

f

2

r=

[6.60]

6.5.4. Study: variations of irregularity factor for two narrow band signals Let us set Af = f2 - fj in the case of a single narrow band noise. The expressions [6.47], [6.48] and [6.49] can be approximated by supposing that Af being small, the

Probability distribution of maxima f

frequencies f,and f2 are close to the central frequency of the band f0 =

219

l+f2 . We 2

have then:

M0 = G Af M2 « (2 7i)2 G Af and M4 * (2 7i)4 G Af Now let us apply the same process to a two narrow bands noise whose central frequencies and widths are respectively equal to f0, Af0 and f], Afj. B(f)

Bl

Figure 6.14. Random noise composed of two narrow bands

With the same procedure, the factor r obtained is roughly given by [BRO 63]: \2

(27r)4(Af0f02G0+Af1f12G1)^

2

r —

(Af0 G0 + Af, Gj) (2 7t)4 (Af0 f04 G0 + Afj f,4 Gj

' Af

f 1 l*

i^ 1 1 _1_

Af, f2 G! Af0 f02 GO \f 4 Af, i GI

1+

G

0> V

Af0 f b

[6.61] ^

G

oJ

220

Random vibration f Af G l l l . It Figures 6.15 and 6.16 show the variations of r with — and of

IS

Af0 G0

observed that if — = 1, r is equal to 1 for whatever Af0 G0

Figure 6.15. Irregularity factor of a two narrow band noise

Figure 6.16. Irregularity factor of a two narrow band noise

These results can be useful to interpret the response of a two-degrees-of-freedom linear system to a white noise, each of the two peaks of the PSD response being able to be compared to a rectangle of amplitude equal to Qj times the PSD of the n ffl excitation, and of width At = — — [BRO 63]. 2 Q

6.6. Error related to the use of Rayleigh's law instead of complete probability density function This error can be evaluated by plotting, for various values of r, variations of the ratio [BRO 63]:

q(u) P r (u) where q(u) is given by [6.19] and where p r (u) is the probability density from Rayleigh's law (Figure 6.17):

Probability distribution of maxima

221

2

p r (u) = u e

When u becomes large, these curves tend towards a limit equal to r. This result can be easily shown from the above ratio, which can be written:

q(u)

\l-r

p r (u)

fin

e

' u

r

1+Erf

2

ru

[6.62]

v \ ~r /

Figure 6.17. Error related to the approximation of the peak distribution by Rayleigh 's law

It is verified that, when u becomes large,

q(u)

-> r. One notes in addition from

Pr(u) these curves that: - this ratio is closer to 1 the larger is r, - the greatest maxima tend to obey a law close to Rayleigh's, the difference being related to the value of r (which characterizes the number of maxima which occur in alternating between two zero-crossings).

222

Random vibration

6.7. Peak distribution function 6.7.1. General case From the probability density q(u), one can calculate by integration the probability that a peak (maximum) randomly selected among all the maxima of a random process be higher than a given value (per unit time) [CAR 56] [LEY 65]:

Q (u) = J°° q(u) du =

re

1-

ru

[6.63]

where

P(X O ) is the probability that the normal random variable x exceeds a given threshold x0. If u-»oo, P(X O ) -> 1 and Q p (u) -> 0. Figure 6.18 shows the variations of Q P (u) for r = 0; 0.25; 0.5; 0.75 and 1.

Figure 6.18. Probability that apeak is higher than a given value u

NOTES. 1. The distribution function of the peaks is obtained by calculating 1 - Qp(u). 2. The function Q p (u) can also be -written in several forms.

Probability distribution of maxima

223

Knowing that: 1

A=

1

= -——

1

fXoA/2 _x>

I

e

1

0

e

L2

j_

tf*ere results [HEA 56] [KOW 63]:

Q (u) = -il-Erf)

+ —e

2

2

ru

1 + Erf

[6.64]

or

Q D (u) = - Erfc

Erf

ru

[6.65]

This form is most convenient to use, the error Junction Erf being able to be approximated by a series expansion with very high precision (cf. Appendix A4.1). One also sometimes encounters the following expression:

Qn(u)

if

= l-

2

rr~ J ^ 2 71

fu /^ e " -^

3. For /arge u [HEA 56],

Qp(u)*re

_^_ 2 .

yielding the average amplitude of the maximum (or minimum):

[6.66]

224

Random vibration

171

[6.67]

2 6.7.2. Particular case of narrow band Gaussian process For a narrow band Gaussian process (r = 1), we saw that [6.28]: / \ a 2 f2 q(aj= -^— e ""* "rms

The probability so that a maximum is greater than a given threshold a is then: a2 Q ( a ) =e ~~2~7L

[6.68]

It is observed that, in this case [5.38],

yielding q( a ) =

d[p(a)]/da — p(0)

[6.69]

These two last relationships suppose that the functions f.(i) and J?(t) are independent. If this is not the case, in particular if p(/)is not Gaussian, J.S. Bendat [BEN 64] notes that these relationships nonetheless give acceptable results in the majority of practical cases.

NOTE. The relationship [6.28] can also be established as follows [CRA 63] [FUL 61] [POW 58]. We showed that the number of threshold level crossings -with positive slope, per unit time, na is, for a Gaussian stationary noise [5.47]:

•where

Probability distribution of maxima

225

_

n0 = — 271

The average number of maxima per unit time between two neighbouring levels a and a + da must be equal, for a narrow band process, to: _ a ~ n a+da ~

da da yielding, by definition of q(a), n

nD q(a) da = - —- da da The signal being assumed narrow band, rip = rig. This yields

1 dn+ q(a) = —da and

<•)= /f w shown that the calculation of the number of peaks from the number of threshold crossings using the difference na - na+da is correct only for one perfectly narrow band process [LAL 92]. In the general case, this method can lead to errors.

Figure 6.19. Threshold crossings of a narrow Figure 6.20. Threshold crossings of a wideband noise band noise

226

Random vibration

Particular case -where f} -> 0 We saw that, for a band-limited noise, r

when fj -» 0. Figures 6.21 and -X

6.22

respectively

P(a < u tms)

show

the

variations

of

the

density

q(u)

and

of

= 1 - Qp(u) versus u, for r = —.

Figure 6.21. Peak probability density of a band-limited noise with zero initial frequency

Figure 6.22. Peaks distribution function of a band-limited noise with zero initial frequency

6.8. Mean number of maxima greater than given threshold (by unit time) The mean number of maxima which, per unit time, exceeds a given level a = u .^5 is equal to: M

i u If a is large and positive, the functions P , fl-r zero; yielding:

Qp * r e"

and

[6.70]

a = n p Qp( u >

I ur and P •,

tend towards

fl-r [6-71]

Probability distribution of maxima

Ma * np r e

227

[6.72]

n o i.e. [RAC 69], since r = -J, n p

[6.73] This expression gives acceptable results for u>2 [PRE 56b]. For u < 2 , it results in underestimating the number of maxima. To evaluate this error, we have exact value plotted in Figure 6.23 variations of the ratio of M0: approximate value

P QpOO Qp(») e"2/2 - "V2 '

with respect to u, for various values of r. This ratio is equal to 1 when r = 1 (narrow band process).

Figure 6.23. Error related to the use of the approximate expression of the average number of maxima greater than a given threshold

— 2A> -V/2 This yields Q p (u) « * r e " '* ' and Ma « e u ' (same result as for large a). In these two particular cases, the average number per second of the maxima located above a threshold a is thus equal to the average number of times per second which

228

Random vibration

€(t) crosses the threshold a with a positive slope; this equivalent to saying that there is only one maximum between two successive threshold crossings (with positive slope). For a narrow band noise, one thus has: Ma = n! Q D (a)

[6.74]

NOTE. ,

4 - 7

The expression [5.47] (n& = HQ e

/

""*) is an asymptotic expression for large a

[PRE 56b]. The average frequency n0 =

Q2 G(Q) dQ

1

is independent

27C

V

"0

of noise intensity and depends only on the form of the PSD. In logarithmic scales, [5.47] becomes:

In na = In HQ In na iy thus a linear function of a , the corresponding straight line having a slope ~—. One often observes this property in practice. Sometimes however, the curve (in na, a 1 resembles that in Figure 6.24. It is in particular the case for turbulence phenomena. One then carries out a combination of Gaussian processes [PRE 56b] -when calculating: k^

^ Pi n+(a) where Pi is a coefficient

[6.75] characterizing the contribution brought by the P

component and naj is the number of crossings per second for this f1 component. If it is supposed that the shape of the atmospheric turbulence spectrum is invariant and that only the intensity varies, n0 is constant. A few components then often suffice to representing the curve correctly.

Probability distribution of maxima

229

Figure 6.24. Decomposition of the number of threshold crossings into Gaussian components

One can for example proceed according to the following (arbitrary) steps: -plot the tangent at the tail of the observed distribution ©; - plot the straight line (D starting from the point of the straight line 1 which underestimates the distribution observed by a factor 2, and tangent to the higher part of the distribution; —plot straight line ® from @ in the same way. The sum of these three lines gives a good enough approximation of the initial curve. The slopes of these lines allow the calculation of the squares of the rms values of each component. The coefficients Pj are obtained from:

Mi(a) = P 1 nJe

[6.76]

for each component. Each term Mj can be evaluated directly by reading the ordinate at the beginning of each line (for a = Q), yielding [6.77]

230

Random vibration

6.9. Mean number of maxima above given threshold between two times If a is the threshold, and tj and t2 the two times, this number is given by [CRA 67] [PAP 65]:

[6.78]

6.10. Mean time interval between two successive maxima Let T be the duration of the sample. The average number of positive maxima which exceeds the level a in time T is: [6.79]

MaT=njQ(a)T and the average time between positive peaks above a is:

a

1

1

Ma

np +Q(a)

[6.80]

For a narrow band noise, T 1 = a

1 M

1

1

n Q(a)

n Q(a)

[6.81]

Ta =

M2 M

or

[6.82]

Probability distribution of maxima

231

6.11. Mean number of maxima above given level reached by signal excursion above this threshold n

o

The parameter r = —— makes it possible to compare the number of zerocrossings and the number of peaks of the signal. Another interesting parameter can be the ratio Nm of the mean number, per unit time, of maxima which occur above a level a0 to the mean number, per unit time, of crossings of the same level a0 with positive slope [CRA 68]. The mean number, per unit time, of maxima which occur above a level a0 is equal to: Ma = np 0

q(u) du u

[6.83]

o

where u0 = —— and q(u) is given by [6.19]. The mean number, per unit time, of *rms crossings of the level a0 with positive slope is [5.47]:

n+a = n+0 e

2

This yields M

[6.84]

[6.85]

Figure 6.25 shows the variations of Nm versus u0, for various values of r. It is noted that Nm is large for small u0 and r: there are several peaks of amplitude greater than u0 for only one crossing of this u0 threshold.

232

Random vibration

Figure 6.25. Average number of maxima above a given level through excursion of the signal above this threshold

For large u0, Nm decreases quickly and tends towards unity for whatever r. In this case, there is on average only one peak per level crossing. During a time interval tj -10, the average number of maxima which exceed level a is: [6.86]

Let us replace the rms value lrms by CTms and seek the rms value another random vibration which has the same number n_ of peaks so that, over time t3 - 12 = tj - 10, we have [BEN 61b] [BEN 64]: [6.87]

It is thus necessary that:

[6.88]

Probability distribution of maxima

23 3

If the two vibrations follow each other, applied successively over tj -10 and t2 -tj, the equivalent stationary noise of rms value ^5 applied over:

which has the same number of maxima np exceeding the threshold a as the two vibrations lms I illdi and £Ims HIS-> , is such that: 7

MaT=Ma(t1-t0)

and

f

\ a

M,

[6.90]

This yields

a

a

p

T=

\

1

[6.91]

0

a

.

a

L and [6.92]

This expression makes it possible to calculate the value of ^nnSea (for a *• 0).

234

Random vibration

6.12. Time during which the signal is above a given value

Figure 6.26. Time during which signal is above a given value

Let a be the sel<jcted threshold; the time during which ^(t) is greater than a is a random variable [RAC 69]. The problem of research of the statistical distribution of this time is not yet solved. One can however consider the average value of this time for a stationary random process. The average time during which one has a < ^(t) < b is equal to:

T

[6.93]

and, if b -» oo, the time for which £(t) > a is given by:

[6.94] (^rms = rms value of ^t)). This result is a consequence of the theorem of ergodicity. It should be noted that this average time does not describe in any way how time is spent above the selected threshold. For high frequency vibrations, the response of the structure can have many excursions above the threshold with a relatively small average time between two excursions. For low frequency vibrations, having the same probability density p as for the preceding high frequencies, there would be fewer excursions above the threshold, but longer, with the excursions being more spaced.

Probability distribution of maxima

235

Proportion of time during -which l(i) > a Given a process ^(t) defined in [0, T] and a threshold a, let us set [CRA 67]: ri(t)=l r|(t) = 0

ifal \ elsewhere j

[6.951

and t = mT

[6.96]

the proportion of time during which l(i) > a , the average of Z0 is: m

1 fi = ~ J0 mT!

dt

= mn

mz = p[/(t) > a]

( a ^ mz =!_((,_!_ UimsJ where ^s

[6.97]

= MQ = R(0) and ((>( ) is the Gaussian law. The variance of Z0 is of the a*

A -7- InT form — e rm when T —> oo. K T f

6.13. Probability that a maximum is positive or negative These probabilities, respectively qj^ and q^, are obtained directly from the expression of Q p (u). If we set u = 0, it becomes [CAR 56] [COU 70] [KRE 83]:

, 1+ r q;ax=— ^max = 1 ~ ( lma X yielding

since

'

for u ec ual to

l

~ °°> Qp(u)=

[6-99]

236

Random vibration q^ax is me percentage of positive maxima (number of positive maxima divided

by the total number of maxima), q^ the percentage of negative maxima [CAR 56]. These relations can be used to estimate r by simply counting the number of positive and negative maxima over a rather long time. J_

For a wide-band process, r = 0 and

2'

For a narrow band process, r = 1 and q^

= 1, q^

= 0.

6.14. Probability density of positive maxima This density has the expression [BAR 78] [COU 70]: = -- q(u) 1+ r

[6.100]

6.15. Probability that positive maxima is lower than given threshold Let u be this threshold. This probability is given by [COU 70]: P(u) = l

1+r

[6.101]

Q p (u)

yielding

1+ r

1+r

-P 2 1 + Erf

ur

[6.102]

6.16. Average number of positive maxima per unit time The average number of maxima per unit time is equal to [BAR 78]:

+ = r0 P

i.e. [6.13]

*—00

[6.103] '—OC

Probability distribution of maxima

237

(the notation + means that it is a maximum, which is not necessarily positive). The average number of positive maxima per unit time is written:

n

p>o=f

J-O

l

p>0

1

[6.104]

471

6.17. Average amplitude jump between two successive extrema Being given a random signal l(t), the total height swept in a time interval (- T, T) is [RIC 64]:

F J

-T

d/(t)

dt

dt

Let dn(t) be the random function which has the value 1 when an extremum occurs and 0 at all the other times. The number of extrema in (- T, T) is J

dn(t).

Figure 6.27. Amplitude jump between two successive extrema

The average height hT between two successive extrema (maximum - minimum) in (- T, T) is the total distance divided by the number of extrema:

238

Random vibration

p —

-T dt ^

f dn(t) ~T

_L F

dt

2T ~

[6.105]

-1 f dn(t) 2T ~

If the temporal averages are identical to the ensemble averages, the average height h is:

= rlim r~ hT=

lim — F dt 2T ~T dt

[6.106]

lim — I T dn(t) 2T ~

where np is the number of extrema per unit time. For a Gaussian process, the average height h of the rises or falls is equal to [KOW 69] [LEL 73] [RIC 65] [SWA 68]: [6.107]

or [6.108] For a narrow band process, r = 1 and: h=

J2n

[6.109]

This value constitutes an upper limit when r varies [RIC 64]. NOTE. Calculation ofh can be also carried out starting from the average number of crossings per second of the threshold [KOW 69]. For a Gaussian signal, this number is equal to [5.47]:

The total rise or fall (per second) is written:

Probability distribution of maxima

£

°0

l+«>

jf2

na da = n0 I e 30 J-00

r-

"•* da = J2n n0 lt

239

[6.110]

This yields the average rise or fall [PAR 62]:

h-*nn

n.

•rms

[6.111]

Example Let us consider a stationary random process defined by [RIC 65]:

<*»)-

-rms

for (3 G>0 < Q, < (00 [6.112]

G(Q) = 0

elsewhere

J.R. Rice and P.P. Beer [RIC 65] show that: h

I

10~Ji

[6.113]

For P = 0 (perfect low-pass filter),

h

[6.114]

If J3 -> 1 (narrow band process), [6.115]

This page intentionally left blank

Chapter 7

Statistics of extreme values

7.1. Probability density of maxima greater than given value Let us consider a signal ^(t) having a distribution of instantaneous values of probability density p(/)and distribution function

Let A,N be a new random variable such thatX N = max^peakl. . XN is the largest i=l,n peak obtained among the Np peaks of the signal ^(t) over a given duration. The distribution function of XN is equal to:

and the probability density function to:

[7.2]

If the probability Q that a maximum is higher than a given value is used,

242

Random vibration

we have .

N -1

. ,XI

,

where Np [l -Q(^')]

p

~1

[7.3]

is the probability of having \Np -l) peaks less than a

value t among the Np peaks.

7.2. Return period The return period T(X) is the number of peaks necessary such that, on average, there is a peak equal to or higher than X. T(x) is a monotonous increasing function ofX.

T(X) = —^— l-P(X)

[7.4]

where P(X) is related to the distribution of t. It becomes: T(X) [l - P(X)] = T(X) Prob(x > X) = 1

[7.5]

7.3. Peak I expected among N_ peaks is the value exceeded once on average in a sample containing Np peaks. We

[7.6]

) = !- — ' P

and Np

ft*

The return period of ^ is equal to: P.7]

Statistics of extreme values

243

7.4. Logarithmic rise The logarithmic rise ctN characterizes the increase in the expected maximum £p in accordance with the Napierian logarithm of the sample size:

[7.8] <*N

^

N

p)

From [7.6], we have

p yielding

and

d(ln N p ) N p p(/ p ) =

i.e. aN =Np

7.5. Average maximum of N_ peaks t

7.6. Variance of maximum

[7.10]

244

Random vibration

7.7. Mode (most probable maximum value) / \ Let us set t M such that PN(^M j *s maximum. The calculation of

=0

gives:

(<M)

=0

[7.12]

7.8. Maximum value exceeded with risk a This value, noted ^N , is defined by: [7.13] a is the probability of recording a maximum value higher than peaks.

among Np

7.9. Application to case of centred narrow band normal process 7.9.1. Distribution function of largest peaks over duration T If it be considered that the maxima are distributed according to a Rayleigh density law f

i "N

£

= -j s f

e;X

P|

2 si

and if it be supposed that the peaks of the narrow band random signal are themselves randomly distributed (a broad assumption in a strict sense, because such a signal may have a correlation between consecutive peaks), the probability that an arbitrary peak ^eafc is lower than a given value i is equal to: exp -

2 si

Statistics of extreme values

245

We obtain, from the above relationships, the distribution function of the largest peaks

(l < i < Np I. PN is the probability that each of the Np peaks is lower than £, if the peaks are independent [KOW 69]. Figure 7.1 shows this probability for some values of no T (equal to Np since, for a narrow band noise, np = HQ), plotted versus

u=

Figure 7.1. Distribution function of largest peaks of a narrow band noise

Figure 7.2 presents the variations of the function QN = 1 - PN , QN being the probability so that the largest peak is higher than a given value U during a length of time T.

246

Random vibration

NOTES. 1. For N_ large (i.e., in practice, for sf/l
Figure 7.2. Probability that the largest peak is higher than a given value

2. This relation can be written in the form: t

= J-21ri{l-exp[(ln P N ) / N

[7.16]

"« Figure 7.3 shows the variations ofl/s^ versus Np,for various values o/PN.

Figure 7.3. Amplitude of the largest peak against number of peaks, for a given probability

Statistics of extreme values

247

7.9.2. Probability that one peak at least exceeds a given threshold The probability that one peak at least exceeds the threshold t is equal to:

1-e

[7.17]

2s]

V

/

where (1 < i < N 1, yielding the probability so that a maximum ^peak. lies between

i.e.

[7.18]

1- 1 - e 2 5 '

7.9.3. Probability density of the largest maxima over duration T The probability density of the largest maxima is thus

1- exp

r.

\

r.

f2

[7.19]

or, while noting \ = j

[7.20]

Over time T, the number of maxima higher than u =

is I, -rms

v=Q(u)N,

[7.21]

248

Random vibration

where v is such that 0 < v < Np. N -1

p N ( u ) d u = -N P N (U)

dQ - Q(u)]

du = N

p N (u) du = d

[7-22] For large u, we can accept that Q(u) can be approximated by [CAR 56]: [7.23]

Q(u) « r e (Rayleigh's law). For large Np, we have, on average, for a given duration T,

In addition, we still have v = ND Q(U), yielding, since r = 1,

v « n0 T e +

T

[7.24]

z2

and

[7.25]

p N (u)du = d exp

From the relationship [7.25], we can express, by integration, this density in the form: ( u2 u 2Y1] = n , T u expj — + n T exi 0 PN( ) ( 2 2 U

1

1—Jjj

[7.26]

Statistics of extreme values

249

Figure 7.4 shows the variations of PN(U) for various values of n0 T between 10 2 andl0 8 .

Figure 7.4. Probability density of the largest maximum over duration T

Each one of these curves gives the distribution law of the largest maximum over duration T of n signal samples to be studied (Figure 7.5).

Figure 7.5. Largest peak of a sample of given duration

Figure 7.6 shows this same probability density for nj T = 3.6 10 to 3.6 10 , superimposed over the probability density curve of the instantaneous values of the random signal (Gauss's law).

250

Random vibration

Figure 7.6. Probability densities of peaks and highest maxima

7.9.4. Average of highest peaks u

The relation [7.24] makes it possible to express u according to v:

On the assumption that In(n 0 T| is large compared to In(v), A.G. Davenport [DAV 64] deduces the average value of £Q:

2s]

[7.27]

i.e., after a MacLaurin series development and an integration by parts [KOW 69] [LON 52]:

Statistics of extreme values

251

[7.28]

For large values of Np, M.S. Longuet-Higgins [KRE 83] [LON 52] shows that one can use the asymptotic expression [7.29]

where e is the Euler's constant equal to 0.577 215 664 90 ... (cf. Appendix A4.3), the /

\~3/2

difference with the whole expression being about I In N_ I

Figure 7.7. Comparison of the approximate average value of the distribution of the highest peaks to the exact value

[UDW 73].

Figure 7.8. Average of the highest peaks

The approximation is very good [CAR 56] , even for small Np (error less than 3% for all Np > 2 and less than 1% if N > 50).

252

Random vibration

NOTE. On the assumption ln(n0T]» ln(v). The ratio

+ In H T

In v

+1

In H T

In n0 T

[7.30]

21nnjT

is small with regard to the unit if\\ « 2 In n0 T. The approximation [7.29] is very acceptable for a narrow band process, i.e. for r close to 1 [CAR 56] [POO 76].

7.9.5. Standard deviation of highest peaks On the same assumptions, the standard deviation of the largest peak distribution is calculated from

X = V u o~( u o)' Tt

~"D

1

V^ln^l)

5

Statistics of extreme values

253

Figures 7.8 and 7.9 respectively show the average u0 and the standard deviation su as a function of n0 T. We note on these curves that, when n^ T increases, the average increases and the standard deviation decreases very quickly. We notice in Figure 7.10 that the slope of the curve P N (u) increases with n^ T, result in conformity with the decrease of su.

Figure 7.10. Probability density of the largest peaks close to unity

7.9.6. Most probable value The most probable value of t corresponds to the peak of the probability density lm curve defined by [7.19], i.e. to the mode £m (or to the reduced mode m = ——). If

we let v =

, it occurs when ¥

v " "e/

d

1/2 -v v' e =0

dv i.e. when [PRA 70] [UDW 73]

v = In N_ -hi 1-

If N

is large

[7.32]

254

Random vibration v*lnN,

[7.33]

yielding the most probable value m=

[7.34]

m=

[7.35]

7.9.7. Value of density at mode [7.36] A typical example of the use of the preceding relations relates to the study of the distribution of the wave heights, starting from an empirical relationship of the acceleration spectral density [PIE 63]. 7.9.8. Expected maximum The expected maximum ^ is such that

N,

%=

[7.37]

= 1-exri — 2 si

[7.38]

In N

7.9.9. Average maximum

(

*]

f'-'/

( 2s2t)

(

r2 \

^

2S{J

ex 2 s

^

P

[7.39]

Statistics of extreme values

255

7.9.10. Maximum exceeded with given risk a IN.

fI 2sjj_ ^4

) !

[7.40]

[7.41]

N« = .2 In

i.e., for a « 1, [7.42]

One finds in Table 7.1 the value of the parameters above defined from the relationships [7.29] [7.31] [7.35] [7.36] for some values of nj T. Table 7.1. Examples of values of parameters from the highest peaks distribution law u

-r n+ 0 T

3.6 102 3

3.6 10

4

3.6 10

5

3.6 10

6

3.6 10

o

3.5993 4.1895 4.7067 5.1725 5.5999

S

u0

X/UQ

m

PN m

0.3738

10.386 10-2

3.4311

1.2622

0.3169

2

4.0469

1.4888

2

4.5807

1.6851

2

5.0584

1.8609

2

5.4948

2.0214

0.2800 0.2535 0.2334

7.565 10' 5.949 10'

4.901 10"

4.168 1Q-

It is noted that su /u0 is always very small and tends to decrease when n0 T increases. Table 7.2 gives, with respect to n^ T, the values of Q = 1 - P for u = u0 and u = m.

256

Random vibration

Table 7.2. Examples of values of parameters from the highest peaks distribution law

m

Q(m)

0.4239

3.0349

0.6321

3.8722

0.4258

3.7169

0.6321

4.4264

0.4267

4.2919

0.6321

4.7067

0.4271

4.5807

0.6321

4.9188

0.4273

4.7985

0.6321

5.1725

0.4275

5.0584

0.6321

5.3663

0.4277

5.2565

0.6321

3.6 106 107

5.5999

0.4279

5.4948

0.6321

5.7794

0.4280

5.6777

0.6321

8

6.1648

0.4282

6.0697

0.6321

HOT

u

o

Q(«O)

102 103 104 3.6 104 105 3.6 105 106

3.2250

10

It is noticed that, for whatever n^ T, Q(U O ) and Q(m) are practically constant. In many problems, one can suppose that with slight error the highest value is equal to the average value u0 . It is also noted that the average is higher than the mode, but the deviation decreases when n0 T increases. Over one hour of vibrations and for an average frequency n0 of the signal varying between 10 Hz and 1000 Hz, one notes that the average u0 varies between 4.7 and 5.6 times the rms value lims (Figure 7.8 and Table 7.2). The amplitude of the largest peak therefore remains lower than 5.6 1^$. The amplitude of the probability density to the mode increases with respect to T (Figure 7. 11).

Statistics of extreme values

257

Figure 7.11. Value of density of largest peak at the mode 7.10. Wide-band centered normal process 7.10.1. Average of largest peaks The preceding calculations were carried out on the assumption of a narrow band noise (r « l). For a wide-band noise (r * 1), D.E. Cartwright and M.S. LonguetHiggins [CAR 56] show that the average value of the largest peak in a sample of Np peaks is equal to: [7.43] (e = 0.57721566490...= Euler's constant). One obtains the relationship [7.29] for r= 1, Np being then equal to n0 T. Let us set ^m2 as the rms value of the peak distribution, where = +r

2

[7.44]

u

Figure 7.12 shows the variations of —pzr with respect to r, for various values of Vm2

Np. u

o

For large Np, ~p=r is a decreasing function of r. When the spectrum widens, the average value of the highest peak decreases. When r -> 0 (Gaussian case), the

258

Random vibration

expression [7.43] cannot be used any more, the quantity r Np becoming small compared to unity. The general expression is complicated and without much interest. R.A. Fisher and L.H.C. Tippett [CAR 56] [FIS 28] [TIP 25] propose an asymptotic expression of the form,

sm

[7.45]

UQ =

+n where m is the mode of the distribution of maxima, given in this case by m

[7.46]

me

Figure 7.12. Average value of the highest peak of a wide-band process versus the irregularity factor

Figure 7.13. Average value of the highest peak of a wide-band process versus the number of peaks

The distribution [7.19] is thus centered around this mode for large ND. From [7.46], it becomes:

yielding

Statistics of extreme values

259

One can show that u0 converges only very slowly towards this limit.

Figure 7.14. Mode of the distribution law of the highest peaks of a wide-band noise

Figure 7.15. Average value of the highest peaks Ofa wide-band noise over duration T

7.10.2. Variance of the largest peaks The variance is given by [FIS 28]

The standard deviation is plotted against N_ in Figure 7.16.

260

Random vibration

8.68 8.95 8.58 8.45

:'"" S, =\

g \ :

8.48 8.35 8.36 8.25 8.28

E E [ -

\s ^\ 1 II III

]

Figure 7.16. Standard deviation of the distribution law of the highest peaks of a wide-band noise

calculated from

Table 7.3 makes it possible to compare the values of

[7.28] with those given exactly by L.H.C. Tippet for some values of N

[TIP 25].

Table 7.3. Comparison of exact and approximate values of

N

P

100

200

500

1000

2.37533

2.61467

2.90799

3.11528

Relation [7.28]

1.70263 1.99302 2.58173

2.80726

3.08549

3.28326

L.H.C. Tippett

1.53875 1.86747 2.50758

2.74604

3.03670

3.24138

M

E(/) s

<

10

20

1.43165 1.74393

7.11. Asymptotic laws The use of exact laws of probability for extreme values, established from the initial distribution law of the instantaneous values or from the distribution law of the maxima, leads to calculations which quickly become very complicated. They can be simplified by treating only the tail of the initial law, but with many precautions because, as one well imagines, several asymptotic laws can be used in

Statistics of extreme values

26 1

this domain. Moreover, the values contained therein, of weak probability, appear only occasionally and the real law is not well known. 7.11.1. Gumbel asymptote This approximation is used for the distribution functions of the exponential type, which tend towards 1 at least as quickly as exponential for the great values of the variable [GUM 54]. This asymptotic law applies in particular to the normal and lognormal laws. Let us consider a distribution function which, for x large, is of the form P(x) = l-aexp(-bx)

[7.49]

The constants a and b are selected according to the law being simulated. If, for example, we want to respect the values of the expected maximum xp and the logarithmic increase aN : [7-50]

(x

In comparing these expressions with those derived from the P(X) law, it becomes: -

= ae~bXp

aN=Npabe"bXp yielding r,b = a N I

[

_La* 3

V

The adjusted distribution function around x_ is thus

[7.52]

f 7 - 53 ]

262

Random vibration

P(x) = 1 - — exp[- ctN (x -

[7.54]

For large Np, one obtains an approximate value of the distribution function of the extreme values making use of the relationship

which yields P N (x) * exp|-exp[-aN (x - x p ) j

[7.55]

7.11.2. Case: Rayleigh peak distribution We have

If we set x = u sx and if the reduced variable TJ = aN I x - x_ I is considered, we have = ./21nNp(vu->/21nNp

[7.56]

The distribution function is expressed as P N (x) = exp[-exp(-T|)]

[7.57]

while the probability density is written: = exp[-Ti-exp(-Ti)]

[7.58]

Statistics of extreme values

Figure 7.17. Probability density of extreme values for a Rayleigh peak distribution

263

Figure 7.18. Distribution function of extreme values for a Rayleigh peak distribution

7.11.3. Expressions for large values 0/N p Average maximum [7.59]

where e = 0.57722 ... (Euler's constant). Standard deviation of maxima S

K

N=

1

aN

[7.60]

Probability of an extreme value less than x_ 1

PN(X P )*-* 0.36788

[7.61]

264

Random vibration

7.12. Choice of type of analysis The prime objective is to simplify the analysis by reducing the number and duration of the signals studied. The starting datum is in general composed of one or more records of an acceleration time history. If there are several records, the first step is to carry out check of the stationarity of the process and, if it is the case, its ergodicity. If one has only one record, one checks the autostationarity of the signal and its ergodicity. These properties make it possible to reduce the analysis of the whole of the process to that of only one signal sample of short duration (a few tens of seconds for example). This procedure is not always followed and one often prefers to plot the rms value of the record with respect to time (sliding average on a few tens of points). In a complementary way, one can add the time variations of skewness and kurtosis. This work makes it possible to identify the various events characteristic of the phenomenon, to isolate the shocks, the transitional phases and the time intervals when, the rms value varying little, the signal can be analysed from a sample of short duration. It also makes it possible to make sure that the signal is Gaussian. The rms value irms of the signal gives an overall picture of the excitation intensity. It can be useful to calculate the average E(l) = m . If it differs from zero, one can either centre the signal, if it is estimated that the physical phenomenon has really zero average and that the DC component is due to an imperfection of measurement, or calculate the rms value of the total signal and the standard deviation s = ^l^s - m . In order to have a precise idea of the frequential content of the vibration, it is also important to calculate the power spectral density of the signal in a sufficiently broad range not to truncate its frequency contents. If one has measurements carried out at several points of a structure, the PSDs can be used to calculate the transfer functions between these various points. The PSDs are in addition very often used as source data for other more specific analyses. The test facilities are controlled starting from the PSD and it is still from the PSD that one can most easily evaluate the test feasibility on a given facility: calculation of the rms value of acceleration (on all the whole frequency band or a given band), of the velocity and displacement, average frequency etc. The autocorrelation function is a little more specific mode of analysis . We saw that this function is the inverse Fourier transform of the PSD. Strictly, there is no more information in the autocorrelation than in the PSD. These two functions however underline different properties of the signal. The autocorrelation makes it possible in particular to identify more easily the periodic signals which can be

Statistics of extreme values

265

superimposed on the random vibration (measurement of the periods of the periodic components, measurement of coherence time etc) [VIN 72]. The identification of the nature of the probability density law of the instantaneous values of the signal is seldom carried out, for two essential reasons [BEN61b]: - this analysis is very long if one wants points representative of the density around 3 to 4 times that of the rms value lnns (a recording lasting 18.5 minutes is necessary to estimate the probability density to 4 £ms of a normal law with an error of 30%); - the tendency is generally, and sometimes wrongly, to consider a priori that the signal studied is Gaussian. Skewness and kurtosis are however simple indicators to use. Peak value distribution The distribution of the peak values is especially useful to know when one wishes to make a study of the fatigue damage. The parameter as function of time to study must be, in this case, not acceleration at the input or in a point of the specimen, but rather the relative displacement between two given points (or better, directly strains or stresses in the part). The maxima of this displacement are proportional to the maximum stresses in the part on the assumption of linearity. We saw that if the signal is Gaussian, the probability density of the distribution of the peak values follows a law made up of the sum of a Gaussian law and Rayleigh law. Extreme values analyis This type of analysis can also be interesting, either for studies of fatigue damage, or for studies of damage due to crossing a threshold stress, while working under the same conditions as above. It can also be useful to determine these values directly on the acceleration signal to anticipate possible disjunctions of the test facility as a result of going beyond its possibilities.

266

Random vibration Table 7.4. Possibilities of analysis of random vibrations

Threshold level crossings The study of threshold crossings of a random signal can have some interest in certain cases:

Statistics of extreme values

267

- to reduce the test duration by preserving the shape of the PSD and that of the threshold level crossings curve (by rotation of this last curve) [HOR 75] [LAL 81]. This method is little used; - to predict collisions between parts of a structure or to choose the dimension of the clearance between parts (the signal being a relative displacement); - to anticipate disjunctions of the test facility.

7.13. Study of envelope of narrow band process 7.13.1. Probability density of maxima of envelope It was previously shown how one can estimate the maxima distribution of a random vibration. Another method of analysing the properties of the maxima can consist in studying the smoothed curve connecting all the peaks of the signal [BEN 58] [BEN 64] [CRA 63] [CRA 67] [RJC 44].

Figure 7.19. Narrow band vibration and its envelope

Given a random vibration ^(t), one can use a diagram giving t(t) with respect to l(l). For a sinusoidal movement, one would have:

= A sin o)0 t

[7.62]

j?(t) = A (DO cos o>0 t

[7.63]

and the diagram according to £(t) would be a circle of radius A, since: co0

268

Random vibration

r(0+

[7.64]

Figure 7.20. Study of the envelope of a sinusoidal signal and of a narrow band signal

The envelope of this sinusoid is made up of two straight lines: ± A. In the case of a narrow band random signal, envelope A is a time function and can be regarded as the amplitude of a function of the form [DEE 71]: u(t) = A(t) sin[co0 t + 6(t)]

[7.65]

in which A(t) and the phase 0(t) are random functions that are supposed to be slowly variable with co0. There are in reality two symmetrical curves with respect to the time axis which are envelopes of the curve i(i). By analogy with the case of a pure sinusoid, A(t) can be considered the radius of the image point in the diagram l(t), l(t):

(A > 0), where sin[6(t)] co0 cos[6(t)]

Statistics of extreme values

269

The probability that the envelope lies between A and A + dA is equal to the £ joint probability that the curves i and — are located in the hatched field ranging co0

between the two circles of radius A and A + dA (Figure 7.21).

Figure 7.21. Probability that the envelope lies between A and A + dA

Consider the corresponding two dimensional probability density p I, — |. We

I <°o,

have: d x d — =co 0 p(t,t)dld l^o ) t} Idtdl ('<} t sin 0,A co cos 0)A dA d0 I, — — = p(A 0

= q(A, 0) dA d6

[7.66]

q(A, 0) = A p(A sin 0, A co0 cos 0)

[7.67]

did where

The probability density function q(A) of the envelope A(t) is obtained by making the sum of all the angles 0:

270

Random vibration

[7.68]

d0

Let us suppose now that the random vibration ^(t) and its derivative i?(t) are statistically independent, with zero averages and equal variances sf = s? (=

^s),

according to a two dimensional Gaussian law:

[7.69]

[7.70]

2sJ_ and [7.71]

q(A) = — exp 2 si

(A > 0). The probability density of the envelope A(t) follows Rayleigh's law.

Figure 7.22. Probability density of envelope A(t)

NOTES. 1. The probability density q(A), calculated at a given time t, is independent oft, the process being supposed stationary.

Statistics of extreme values

271

One could calculate this density from an arbitrary signal £(i) . The result would be independent of the sample chosen in t(\) if the process is ergodic [CRA 63] . 2. The density q(A) has the same form as the probability density q(a) of maxima [CRA 63]. It is a consequence of the assumption of a Gaussian law for i(i) and l(t). In the case of a narrow band noise for which this assumption would not be observed, or if the system were nonlinear, the densities q(A) and q(a) would have different forms [BEN 64] [CRA 61]. When the process has only one maximum per cycle, the maxima have the same distribution as its envelope (this remark is strictly true when r = 1). When the number of maxima per second increases and tends towards infinity, it has been seen that the distribution of maxima becomes identical to that of the instantaneous values of the signal (Gaussian law) [CRA 68].

Figure 7.23. Probability that the envelope exceeds a given threshold AQ

The probability that the envelope exceeds a certain given value A0 is obtained by integrating q(A) between A0 and infinity. jQQ

P(Envelope> A 0 )= | q(A)dA

P(Envelope

1= exp

2s?

272

Random vibration Table 7.5. Examples of probabilities of threshold crossings

AO^

P 0.8825 0.6065 0.1353 0.0111

0.5 1 2 3

7.13.2. Distribution of maxima of envelope S.O. Rice [RIC 44] showed that the average number of maxima (per second) of the envelope of a white noise between two frequencies fa and fb is: [7.72]

N« 0.64110 ( f b - f a )

A Let us set v = —

. If v is large (superior to 2.5), the probability density q(v)

can be approximated by: v2

E q(v)~ MV/

6

* (v 0.641 10V

2

[7.73]

lie 2

and the corresponding distribution function by:

=Q( A

rnax

ve

[7.74]

0.64110

QA is the probability that a maximum of the envelope chosen randomly is lower than a given value A = v s^. The functions q(v) and QA are respectively plotted in the general case (arbitrary v) on Figures 7.24 and 7.25.

Statistics of extreme values

Figure 7.24. Probability density of envelope maxima

273

Figure 7.25. Distribution Junction of envelope maxima

Figure 7.26. Comparison of distribution functions

Figure 7.26 shows, by way of comparison, the distribution functions of: -the

1 P =2

instantaneous

values

of

the

signal

(Gaussian

law)

(A):

Erflv

- the maxima of the signal [6.64] (B),

- the instantaneous values of the envelope (Rayleigh's law) (C): P = 1 - e

2

,

274

Random vibration

- the maxima of the envelope (curve given by S.O. Rice [RJC 44]) (D). 7.13.3. Average frequency of envelope of narrow band noise It is shown that [BOL 84]: [7.75]


-rms

where ^rms

= rms va ue

l

of the noise l(t)

0 = average pulsation of the noise \2 n f0) f0 = average frequency o For a signal t(t) whose PSD G(f) is constant between frequencies fj and f2 and centered on f0, this relationship leads to:

i.e. to: 1/2 [7.76]

Statistics of extreme values

275

Summary tables of the main results Table 7.6 (a). Main results

Parameter Number of crossings of a threshold a with positive slope per unit time

Relation

Expression a2

[5.47] a — ~ n"0~

*» e

2£ 2nns

1 [5.53] Average frequency

Moments

C

[J0 °f G(f)df"p

+

[5.79]

n

O ~

[5.77]

Probability density of the maxima

[6.6]

foo

[6.13] [6.31]

[6.40]

J

2n\MQ

'L*L

n(n\

R<2)(0)

VMoM4

^/R(0)R(4>(0)

u2

Vl-r z

-

2(l-rM

r

"

' 4-

r

11 p

P

m

1

J4* f2 G(f ) df

f

L

2« l( M2 i

r Af v1 i+-13(2— fj /-

O

1 _1_ O 1 2

\2 Af

t 2

/_L

1

\4 Af

UfoJ s U f o J J +

)

^(..^J

i"ir ( > ? i pu 1

r

ru

1 4- Frf

^

f4G f df

a+

f

"2F

-~T2

2

r Average time between two successive maxima (narrow band noise)

^

M2

/Ls

^

Average number of maxima per second

~

J G(f)df

/

[6.19]

!

Mn = (27t)n 1°° fn G(f) df r

Irregularity factor

2

276

Random vibration

5 4lV2

Average correlation between two successive maxima

^

(

1 + 262 +[O.HlJ

P-

'

2*5

L

tf 1+ —

1+

y/2

f

T

2

3

1+2S + —

J

^

y2

1+ —

v.

5 )

v.

+

S4

+

TJ

Table 7.6 (b). Main results Parameter

Relation

Expression

Distribution function of the peaks

[6.64]

/ * 1 J u r ~, i" u Q ( u ) = - 1-Erd . +-e 2 1 + Erf . 2 [^(l-r 2 )] 2 [^2(l-r 2 )

Average number of maxima greater than a threshold a per unit time

r

i

r

.'

a2

[6-74]

Ma

a2

] I4 e 2^e 27iVM2

l |M2 e 27t\M 0

/

Average number of positive maxima per second

[6.104]

Average time interval between the maxima

[6.80]

Average time interval between the maxima (narrow band noise) Probability so that a maximum is positive or negative

1

+

Rp

N

JM^

IM^

47i^M0+iyM2J np

1

.

1

.

nj Q(a)

Ma

a:

[6.82]

T

_ O = / TT 7t

MH 2M0 n 0 e

M2 [6.98] [6.99]

+

1+r

^max ~

_

1-r

'Imax ~~

2

2

t1

Time during which the signal is above a given value

[6.93]

Average amplitude jump between two successive maxima

[6.107]

—

T 1 .

ab

f>

1

1 / e ^nnsV 2 7 C

1

r

9 C#2

* rm S fl/?

4

h = V2^ r ^rms

"*•

2Mft

Statistics of extreme values

Probability density of the largest peaks

[7.19]

N p -l

( t\

XT

/

I 2sU

cxri m-r-J

1

, >

i cx \ r

f * }}

"

PNUJ- Np 1 n

277

...

2 "7

n 2 1 2sJ,

Table 7.6 (c). Parameter

Relation

Probability density of the largest maximum over duration T

[7.26]

Average for the large values of the number of peaks (narrow band noise)

[7.29]

Expression

fr2

/ \ = n+T J T p N (u) 0 Tuexpl- — + n 0 T e x p U

+

[h

— u n

/

/ ,

-.-

\") Inl n

£

\

Tl -L

} V ^("o T) 1

[7.31]

"•• " v? JT.^)

Most probable value (mode)

[7.35]

m = ^21n(njT)

Maximum exceeded with a risk a.

[7.41]

Standard deviation

[7.48]

a

N

|o In

[7.71]

c

' «T Vo-ar-'' T7 u

0

/ Inl / N p i\1 _Li , e «~ J^ /O ~ inirr "NI ^21n(rN p ) 2

7t

2

m

2

6 ( ra 2 + 1)2

(wide-band noise) Probability density of the envelope of a narrow band Gaussian process

2

° - r Ullno Tj ' / 2 / K

[7.43]

U

A

q( ) -

A exp| 2 2

S^C

1

\ JJj

Standard deviation (narrow band noise)

Average for the great values of the number of peaks (wide-band noise)

( 2\ii\

L.

2 S^v

-1

A2]

278

Random vibration

Distribution of maxima of the envelope of a narrow band process Average frequency of the envelope of a narrow band noise of constant PSD

[7.73]

[7.76]

(v}~~ a <4\yj

I* V6 (2 iv 0.641 10 v

fj + f ! f 2 + f 2

°PM

«) lie 2 '

.

v

1/2 2

ID IM +12/ + 1o

Appendices

Al. Laws of probability Al.l. Gauss's law This law is also called the Laplace-Gauss or normal law.

Probability density

r^-L-l

PW -

1 fx-mY 1

1 .

c

r

s yJ2 n

2^-

S

m = mean s - sianuam ueviauon

J

The law is referred to as reduced centered normal if m = 0 and s = l. Distribution function

1 fx-mY

T<( ~y\ -- p/v I^A; i ^x ^- Y^ A; —

X

2

— jf 00 e ^ s /27i -

Reduced varial.1^. *

s

^ dx ax

x -m s

If E] is the error function

nu TT^T\

2L

T7 i1 +_L iii _

where T —

IV2 J.

Mean

M

Variance (central moment of order 2)

s

X-m s

280

Random vibration

: Central moments

CT i,

— , k

keven(k = 2 r):

4

f * t k r> ~T , ^ rit

1 J

" "V^ -

2r!

-.

*

2"^ r

^'Vr!

k odd (k = 2 r + 1) ^2 r+1 = °

Moment of order 3 and skewness

0

,1' ^ 1

Kurtosis

M-4- ~ T ~ 3 s

Median and mode

0

A1.2. Log-normal law

Probability density

p(x) = m y =mean s = standard deviation of the normal random variable y = In x

dy

>(y)dy =

The log-normal law is thus obtained from the normal law by the change of variable x = e y . 1 In x-nr

Distribution function

F(X) = P(x < X) = (

X

1

2

x s,

Mean

m = E(X) = e

2

s

dx

Appendices

sW m ' +S > (e S '-l)=m 2 (e S '-l) Variance (central moment of order 2)

s2 = [ E (x)] 2 v 2

v = V eS> - 1

, -o s

+ y

2 ( s2 ~} m y T s =| e y - l l e

Expressions for

m y and, s2v with respect to E(X) and s

281

v = variation coefficient

2

i1 f

r / M

i

s

2

m In T7f v l .— in In 1 m i -L + v — mi Jivxn — 2 L E 2 (x)J

m y = ln[E2(x)] --

ln[s2 + E 2 (x)]

r

2

i

s

sc v — In in 1i -i. -t- ^ E 2 (x)J u s y = -lnJE 2 (x)J + ln[s2 + E 2 (x)j

It is noted that the transformation x = ey applies neither to E(X), nor to s . 1

/

2\

m v = In x = In m - — In 1 + v 2 V '

f7 v = Ve y -1

s y = ln(l + v 2 )

Variation coefficient

If v is the variation coefficient of x, we have also m = x Vl + v If two log-normal distributions have the same variation coefficient, they have equal values of s (and conversely). Moment of order j Central moment of order 3

V. j-- ee A / 2

^=(e '-lj

Median

\2

,

X i - ^ . f e ^ - l ¥ e ^ H - 2 T - v 3 + 3» ^

V

)

J\

4 ( 12 , 10 ,^

A4 = m l v

.

(e s ' + 2j = m 3 ( v 6 + 3 v 4 )

^

s

Kurtosis

2

\3/2 -f 2 m + s ^ ) / 2

s

Skewness Central moment of order 4

i , , j m j + - j' s;

+6v

8

6

, 4\

+ I 5 v + 1£ l6v+3vl

-i = v8 + 6 v 6 + 15 v 4 + 16 v 2 + 3 s

x=e

my

282

Random vibration

-s"

For this value, the probability density has a maximum equal to

Mode

s!

[AIT 81] [CAL 69] [KOZ 64] [PAR 59] [WIR 81] [WIR 83].

NOTES. 1. Another definition can be: a random variable x follows a log-normal law if and only ify = \nxis normally distributed, with average mJv and variance st,. * 2. This law has several names: the Galton, Me Alister, Kapteyn, Gibrat law or the logarithmic-noiTnal or logarithmo-normal law. 3. The definition of the log-normal law can be given starting from base 10 logarithms (y = Iog10 x): log,0 x-m y

p(x) =

n fin

xs

[Al.l]

In 10

With this definition for base 10 logarithms, we have: m y = Iog10 x

[A1.2]

my4log10(s^/0.434)

m = 10L

V = 'ViU

2

s;/0.434 y/

, -1

[A1.3]

[A1.4]

m =log 1 0 x--log 1 0 (l + v 2 ) v 2 '

[A1.5]

s y =0.434 Iog10(l + v 2 )

[A1.6]

Hereafter, we will consider only the definition based on Napierian logarithms.

Appendices

283

4. Some authors make the variable change defined by y = 20 log x, y being expressed in decibels. We then have:

[ALT] f 20

since - ««75.44 .

5. Depending on the values of the parameters m y and s y , it can sometimes be difficult to imagine a priori which is the law which is best adjusted to an enseble of experimental values. A method allowing for choosing between the normal law and the lognormal law consists in calculating:

s - the variation coefficient v = — , m X3 - the skewness —, s X4

- the kurtosis —r, 4 S

knowing that

n

m) [A1.9]

n \3

n

and

284

Random vibration

Ifskewness is close to zero and kurtosis close to 3, the normal law is that which *3 15 best adjusted. If v < 0.2 and — « 3, the log-normal law is preferable. v

A1.3. Exponential law This law is often used with reliability where it expresses the time expired up to failure (or the time 'interval between two consecutive failures). Probability density

p(x) = A e - X x

Distribution function

F(X) = P ( x < X ) = l - e ~ X X

Mean

mj = E(x) = — A

m

Moments Variance (central moment of order 2)

n!

n

2

1

'™I m -' S =

n^

Central moments

^n = l + -" n -l A

Variation coefficient

v=l

Moment of order 3 (skewness)

, ^3 ,3 . u3 = — = A +3

Kurtosis

^ = ^4 + 4 A3 + 12

s

A1.4. Poisson's law It is said that a random variable X is a Poisson variable if its possible values are countable to infinity x 0 , x l ? x2..., xk..., the probability that X = x k being given by:

Pk =

r_ k!

where X, is an arbitrary positive number.

[A1.12]

Appendices

285

One can also define it in a similar way as a variable able to take all on the integer values, a countable infinity, 0, 1,2, 3..., k..., the value k having the probability: [A1.13]

k! The random variable is here a number of events (we saw that, with an exponential law, the variable is the time interval between two events). The set of possible values n and their probability p k constitutes the Poisson law for parameter A.. This law is a discrete law. It is shown that the Poisson law is the A. limit of the binomial distribution when the probability p of this last law is equal to — k and when k tends towards infinity.

Distribution function

Mean

v~> ?k F(X) = P ( 0 < x < X ) = X 6 " ^ —- ( n < X < n + l) k! k=0 oo mj = E(X) = ^ k e

,k — = A. k!

k=0

Moment of order 2

m 2 = A. (A. + 1)

Variance (central moment of order 2)

2 s = iA,

u 3 = A. Central moments Variation coefficient

Skewness

Kurtosis

H4 = X + 3),2

v=

1

vf

1 VA

^3 = ~F

..' M-4

1 + 3A-

A

NOTES. 1. Skewness u/3 being always positive, the Poisson distribution is dissymmetrical, more spread out on the right.

286

Random vibration

2,Ifk tends towards infinity, u'3 tends towards zero and u'4 tends towards 3. There is convergence from Poisson 's law towards the Gaussian law. When X, is large, the Poisson distribution is very close to a normal distribution.

A1.5. Chi-square law Given v random variables Uj, u2..., u v , supposed independent reduced normal, i.e. such that: du.

one calls chi-square law with v degrees of freedom (v independent variables) the 2

probability law of the variable x v defined by:

[AL15] 2

The variables U; being continuous, the variable % is continuous in (0,<x>). NOTES.

2 1 . The variable % can also be defined starting from v independent non reduced normal random variables Xj whose averages are respectively equal to nij = E x and the standard deviations Sj, while referring back to the preceding definition with

Xj -nij 2 V"1 2 the reduced variables u j = - and the sum x v = / . ui •

2. The sum of the squares of independent non reduced normal random variables does not follow a chi-square law.

Probability density v= number of degrees of freedom r= Euler function-second kind (gamma function)

Appendices

287

V

Mean Moment of order 2

m 2 = v(2 + v)

Standard deviation

s = y2v

Central moments

u2 = 2 v

u 3 = 8 v u 4 = 12 v (v + 4)

v=

Variation coefficient

J!

Skewness

fj

\\ v

Kurtosis

t

*> v-f-4 v

Mode

M = v-2

This law is comparable to a normal law when v is greater than 30 approximately. Al .6. Rayleigh 's law X" X

Probability density

( \ p(x) =— e v

2v2

v is a constant X2

Distribution function F(X) = P ( x < X ) = l - e

2y2

|7t

Mean m=

^~V

Median

X = v^2 In 2 » 1.1774v

Rms value

,g?) = vV2

288

Random vibration

2

Variance

(

f4

^1 2

/"L-

( k - .Ori - l1\)

TYI

If k is even (k = 2r)

m

m^.!

2rr!

r

r! v2 r

2r

=2

H0 = 1

Central moments

">

( 2 r ) » HT

If k is odd

Moment of order k

^

s = 2-- v = - - l i r a ' \ 2) U )

^ I| -2 v\;

""

jij - 0

r ^2

.2=l2--jv2

f^" /

•*

,3 = J- ( j [ - 3 )v 3

/4

Variation coefficient

fit

a- ilF a

Skewness

7l

~ 3 1/1 -06311 v-oj i i

'2f2-«f I 2)

( i ^ L 3* 4 ^4=|8-— 4- v

I

ICurtosis

b_32-3u

;

2

(4-^ Mode

M=v

Reduced law If we set u = —, it becomes, knowing that p( x) = — e s s

, . 1 x 2s 2 ! 9 ] /v p(x} = e = - u e z =-p(u) s s

[A1.16]

Appendices

[x i dx p(u) du = p(u) J — = s p(x) — = p(x) dx \sJ s

[A1.17]

Table Al.l. Particular values of the Rayleigh distribution

X

(x

X]

Vv

\)

Probj - > — V

1

0.60653

1.5

0.32465

2

0.13534

2.5

4.3937 10'2

3

1.1109 10"2

3.5

2.1875 10"3

4

3.355 10-4

4.5

4.01 10'5

5

3.7 10"6

A1.7. Weibull distribution

Probability density

a ( x -8^1 i \ p(xj = j v - a U - e j

( x -z\ P

U-sy

[0

x<s a and v= positive constants

Distribution function

CA.LJ

M

.

289

y\. ^ o

U-8J

j

0

X<8

Mean

m = 8 + (v-e) I|l + ~

Median

X = s + (v-e)(ln2) 1/a

V

a/

290

Random vibration

s2 = (v-s) 2

Variance

Mode


M = e + (v-e)|l--| V oJ

[KOZ 64] [PAR 59]. NOTE. One sometimes uses the constant r\ = v - s in the above expressions. A1.8. Normal Laplace-Gauss law with n variables Let us set x1? ^2—> x n n random variables with zero average. The normal law with n variable Xj is defined by its probability density:

x

2|M| u

i xj

[A1.18]

where |M| is the determinant of the square matrix:

[A1.19] U

n2

U:; = El X:, X; I =moments of second order of the random variables y \ * j> M y = cofactor of |iy in |M|.

Examples 1. n =

Appendices

291

with

MH=1 -E(xf)yielding

/(x,)\ =

2s 2

[A1.20]

which is the probability density of a one dimensional normal law as defined previously.

2. n = 2 exp

P(X!>

L 2|M|

[A1.21]

+M ]2 Xj x2 + M 21 x2 x, + M22 x 2 with

M-22

) = E(x2 X L ) = ^1 = Psl S2

M

12 = -

= M 21

p is the coefficient of linear correlation between the variables Xj and x2 , defined by: COV p= s x

^X^ S X

( l) ( 2)

where cov^Xt ,^2 ) is the covariance between the two variables Xj and x 2 :

[A1.22]

292

Random vibration

cov(X b X 2 ) = J r°[x, -E(X 1 )][x 2 -E(X 2 )]p(x 1 ,x 2 )dx 1 dx2 J

[A1.23]

J-OO

Covariance can be negative, zero or positive. Covariance is zero when x} and x 2 are completely independent variables. Conversely, a zero covariance is not a sufficient condition that Xj and x2 be independent. It is shown that p is included in the interval [-1, l]. p = 1 is a necessary and sufficient condition of linear dependence between Xj and x 2 . Yielding )(xl5x2) =

I .

/ 7 cxp l 27tSls2Vl-p L

2

f

x \2

/

l

/ 2\ 1 1 - P ) Vs! )

*1*2 "P

s

\2~] *2 1

\S2J J

l S2

[A1.24]

NOTE. If the averages were not zero, we would have

1

1

\2

> V

*

Y

'

V

Tpi Y. 1

S

l

/

v Pi v \ P/v 1 1 Xj - tf (l X- v{ )l - X |1 Xv2 ~ *\X2) | 2 - i^X 2 ; + l 1 s S S l 2 V 2 J J v

-2p

If p = 0, we can write p(x, ,x 2 j= p(x, Jbvx? /

[A 1.25]

L

( \ P(X!) =

z=re Sj

X,

I

\

and p(x 2 | =

1

—— e

2s;2 2s"

. Xj and x2 are independent random variables.

71

2sr

Appendices

It is easily shown, by using the reduced centered variables t, =

and s

to =

293

l

x -

[A 1.26]

dx = 1

Indeed, with these variables, 1

exp

[A1.27]

2 71 Sj S2

and

[A1.28] L

Let us set u = \ 1

f

- p t2

l*-L/v,

and calculate

"> /

~> I

e

7~ JJe" 271

~*

dt du

[A 1.29]

z

i.e. e~ l 2 / ^ dt,

[A1.30]

J2n

We thus have f-HJO

J

—oo

f+00

p(u) du J

—oo

/

v

p(t 2 ) dt2 =

[A1.31]

NOTE. // is shown that, if the terms u^ are zero when i * j, /.e if all the correlation coefficients of the variables X| and X: are zero (\ * j), we have:

294

Random vibration

|in

0 -

0

0

'-. •••

0

0

••• '-.

0

o

o o un

[A1.32]

Ml =

and [A1.33]

[A1.34] For normally distributed random variables, it is sufficient that the crosscorrelation functions are zero for these variables to be independent. ^1.9. Student law The Student law with n degrees of freedom of the random variable x whose probable value would be zero for probability density:

n +1

p(x) =

n+i [A1.35]

Appendices

295

A2. l/nth octave analysis Some analysers make it possible to express the power spectral densities calculated in dB from an analysis into 1/3 octave. We propose here to give the relations which make it possible to go from such a representation to the traditional representation. We will place ourselves in the more general case of a distribution of the points in l/nth octave.

A2.1. Center frequencies A2.1.1. Calculation of the limits in l/nth octave intervals By definition, an octave is the interval between two frequencies fj and f 2 such f that — = 2. In l/nth octave, we have

[A2.1]

i.e. Iogf 2 =logf 1 +

log 2

[A2.2]

296

Random vibration

Example Analysis in 1/3 octave between fj = 5 Hz and f 2 = 10 Hz. log f a = log 5 + -

= 0.7993

f a = 6.3 Hz log f b = log f a

= 0.8997

f b = 7.937 Hz

fc=f2 =

A2.1.2. Width of the interval Af centered around'f The width of this interval is equal to Af = upper limit - lower limit

Figure A2.1. Frequency interval

Let a be a constant characteristic of width Af (Figure A2.1) such that:

(log f + log a) - (log f - log a) =

[A2.3]

Appendices

297

yielding [A2.4] One deduces f Af = a f - a

[A2.5]

[A2.6]

This value of Af is particularly useful for the calculation of the rms value of a vibration defined by a PSD expressed in dB. Example For n = 3 , it becomes a « 1.122462...and Af « 0.231563...f . At 5 Hz, we that have Af = 1.15 Hz.

A2.2. Ordinates We propose here to convert the decibels into unit of amplitude [(m/s2)2/Hzj. We have, if x^g is the rms value of the signal filtered by the filter (f, Af ) defined above: [A2.7]

*ref

xref is a reference value. If the parameter studied is an acceleration, the reference value is by convention equal to 1 um/s = 10"6 m/s (one finds sometimes 10"5 m/s in certain publications). Table A2.1 lists the reference values quoted by Standard ISO/DIS 1683.2.

298

Random vibration

Table A2.1. Values of reference (Standardise 1683 [ISO 94]) Formulate (dB)

Reference level

201og(p/p0)

20 uPa in air 1 uPa in other media

Acceleration level

201og(x/x0)

1 fim/s

Velocity

201og(v/v0)

1 nm/s

Force level

201og(F/F0)

1 uN

Power level

101og(p/P0)

IpW

Intensity level

10 log(l/I0)

1 pW/m2

Energy density level

101og(w/W0)

1 pJ/m3

Energy

10 log(E/E0)

Parameter Sound level

pressure

IpJ

Yielding [A2.8]

*rmS = xref 1020 The amplitude of the corresponding PSD is equal to N x

/-. _

ref

[A2.9]

Af

Af

[A2.10]

G=

or /n

log

-l fG

[A2.ll]

Appendices

299

Example If xref = 10~5 m/s2 N -- 10

1010 Af and if n = 3 N -- 10

I/6

1010 /-i _ 2 21/3-! f N N

10

10

10

0.23 f If, at 5 Hz, the spectrum gives N = 50 dB, 1/6

2

50 —10 1010

G * 8.6369 10"6 (m/s2)2/Hz

A3. Conversion of an acoustic spectrum into a power spectral density A3.1. Need When the real environment is an acoustic noise, it is possible to evaluate the vibratory levels induced by this noise in a structure and the stresses which result from it using finite element computation software. At the stage of writing of specifications, one does not normally have such a model of the structure. It is nevertheless very necessary to obtain an evaluation of the vibratory levels for the dimensioning of the material.

300

Random vibration

To carry out this estimate, F Spann and P. Patt [SPA 84] propose an approximate method based once again on calculation of the response of a one-degree-of-freedom system (Figure A3.1).

Figure A3.1. Model for the evaluation of the effects of acoustic pressure

Let us set:

P = acoustic pressure GP = power spectral density of the pressure A = area exposed to the pressure P - effectiveness vibroacoustic factor M = mass of the specimen and support unit. The method consists of: - transforming the spectrum of the pressure expressed into dB into a PSD G p expressed in (N/m2)2/Hz, -calculating, in each frequency interval (in general in the third octave), the response of an equivalent one-degree-of-freedom system from the value of the PSD pressure, the area A exposed to the pressure P and the effective mass M, - smoothing the spectrum obtained. A3.2. Calculation of the pressure spectral density By definition, the number N of dB is given by P

P A T H

N = 201o g]0 —

[A3.1] —5

P0 = reference pressure = 210

N/m"

Appendices P = rms pressure =

301

G P Af

For a 1/n* octave jfilter centered on the frequency f c , we have [A3.2]

Af= 2

yielding

P0 10"'" I

[A3.3]

Af In the particular case of an analysis in third octave, we would have 2l/6

i

[A3.4] 4.32

and P0 10N/20) [A3.5]

A3.3. Response of an equivalent one-degree-of-freedom system

Figure A3.2. One-degree-of-freedom system subjected to a force

302

Random vibration

Let us consider the one-degree-of-freedom linear system in the Figure A3..2, excited by a force F applied to mass m. The transfer function of this system is equal to: [A3.6]

F

F

2 2

m (l-h )

+ hVQ

2

|

y and z being respectively the absolute response and the relative response of the mass m, and

At resonance, h = 1 and H=— m

[A3.7]

The power spectral density GF of the transmitted force is given by: GF=(pA)2GP

[A3.8]

(F = P A P) and the PSD of the response y to the force F applied to the one-degreeof-freedom system is equal, at resonance, to: [A3.9]

G

[A3.10]

m

- Q

(Po

[A3.ll]

,l/2n ,l/2n

In the case of third octave analysis,

[A3.12]

mJ

2V6_.

,1/6

Appendices

303

F. Spann and P. Part set Q = 4.5 and £ = 2.5 ; yielding , = 126.6 - G P m

[A3-13]

A4. Mathematical functions The object of this appendix is to provide tools facilitating the evaluation of some mathematical expressions, primarily integrals, intervening very frequently in calculations related to the analysis of random vibrations and their effect on a onedegree-of-freedom mechanical system. A4.1. Error function This function, also named probability integral, is the subject of two definitions. A4. 1 . 1 . First definition The error function is expressed: E,(x)

-^fV ./7 o

[A4.1]

J

If x -> oo, Ej(x) tends towards Eloo which is equal to '!d, = l

[A4.2]

u and if x = 0, Ej(0) = 0. If we set t = —— , it becomes V2

\

x I

u

2 fx — du

u" 2

du

[A4.3]

304

Random vibration

Figure A4.1. Error function E](x)

One can express a series development of Ej(x) by integrating the series development of e

E 1 (x) =

2

between 0 and x: 2n+l

3 X

X

5

1!3

215

[A4.4]

x-

n!(2n

This series converges for any x. For large x, one can obtain the asymptotic development according to [ANG 61] [CRA 63]: 1

E,(x)*l xVTt

2x

1.3 2

2

2 x

4

1.3.5 3 6

2 x

K~l)

,U5...(2n-3) 1 •—x— 1~n-l 2n-2 2

x

[A4.5] For sufficiently large x. we have -X"

[A4.6]

If x = 1.6, E :I (X)= 0.976, whilst the value approximated by the expression above is * 0.973.

Appendices

305

For x = 1.8 , Ej(x)« 0.9877 instead of 0.890. The ratio of two successive terms, equal to

2n-l —, is close to 1 when n is close x

2

to x . This remark makes it possible to limit the calculation by minimizing the error on E)(x).

NOTE. E(x) is the error function C complementary error function'). foO

2

ERFC(x) = — J

2

f

Function Ej(x) = -— J e

and [l-E(x)], noted

ERFC(x), is the

2

e

dt

_t _

dt

[A4.7]

306

Random vibration Table A4.1. Error function E|(x) (continuation)

X

EjU)

0.025

0.02820

0.030

0.05637

0.075

0.08447

0.100

0.11246

0.125

0.14032

0.150

0.16800

0.175

0.19547

0.200

0.22270

0.225

0.24967

0.250

0.27633

0.275

0.30266

0.300

0.32863

0.325

0.35421

0.350

0.37938

0.375

0.40412

0.400

0.42839

AE,

0.02820 0.02817 0.02810 0.02799 0.02786 0.02768 0.02747 0.02723 0.02697 0.02666 0.02633 0.02597 0.02558 0.02517 0.02474 0.02427

X

E!(X)

0.425

0.45219

0.450

0.47548

0.475

0.49826

0.500 0.520500 0.525

0.54219

0.550 0.56332 0.575

0.58388

0.600

0.60386

0.625

0.62324

0.650 0.64203 0.675

0.66022

0.700

0.67780

0.725

0.69478

0.750 0.711156

AE1

0.2380 0.02329 0.02278 0.02224 0.02169 0.02113 0.02056 0.01998 0.01938 0.01879 0.01819 0.01758 0.01698 0.01638 0.01577

0.775

0.72693

0.800

0.74210

X

Et(x)

0.825

0.75668

0.850

0.77067

0.875

0.78408

0.900

0.79691

0.925

0.80918

0.950

0.82089

0.975

0.83206

1.000

0.84270

1.025

0.85282

1.050

0.86244

1.075

0.87156

1.100

0.88021

1.125

0.88839

1.150

0.89612

1.175

0.90343

1.200

0.91031

0.01517

AE!

0.01458 0.01399 0.01341 0.01283 0.01227 0.01171 0.01117 0.01064 0.01012 0.00962 0.00912 0.00865 0.00818 0.00773 0.00731 0.00688

Appendices

1.225 0.91680 1.250 0.92290 1.275 0.929863 1.300 0.93401 1.325 0.93905 1.350 0.094376 1.375 0.94817 1.400 0.95229 1.425 0.95612 1.450 0.95970 1.475 0.96302 1.500 0.9661 1 1.525 0.96897 1.550 0.97162 1.575 0.97408 1.600 0.97635 1.625 0.97844

0.00649 0.00610 0.00573 0.00538 0.00504 0.00472 0.00441 0.00412 0.00383 0.00356 0.00332 0.00309 0.00286 0.00265 0.00246 0.00227 0.00209

1.650 0.98038 1.675 0.98215 1.700 0.98379 1.725 0.98529 1.750 0.98667 1.775 0.98793

0.00194 0.00177 0.00164 0.00150 0.00138 0.00126

0.00116 1.800 0.989090 0.00106 1.825 0.99015 0.00096

2.075

0.99666

2.100

0.99702

2.125

0.99735

2.150

0.99764

2.175

0.99790

2.200

0.99814

2.225

0.99835

2.250

0.99854

2.275

0.99871

1.850 0.99111

0.00088

1.875 0.99199

0.00080

2.300

0.99886

1.900 0.99279

0.00073

2.325

0.99899

1.925 0.99352

0.00066

2.350

0.9991 1

1.950 0.99418

0.00060

2.375

0.99922

1.975 0.99478

0.00054

2.400

0.99931

2.000 0.99532

0.00049

2.425

0.99940

2.025 0.99781

0.00045

2.450

0.99947

2.475

0.99954

2.500

0.99959

2.050 0.99626

307

0.00040 0.00036 0.00033 0.00029 0.00026 0.00024 0.00021 0.00019 0.00017 0.00015 0.00013 0.00012 0.00011 0.00009 0.00009 0.00007 0.00007 0.00005

Approximate calculation ofE^ The error function can be estimated using the following approximate relationships [ABR 70] [HAS 55]: E 1 (x) = l-(a 1 t + a 2 t 2 + a 3 t 3 + a4 t 4 + a 5 t 5 ) e where (0 < x < oo)

t=

1 + px |s(x)|< 1.5 10~ 7

x

+e(x)

308

Random vibration p = 0.327 5911 a, = 0.254 829 592 a2 = -0.284 496 736

a3 =1.421 413 741 a4 = - 1.453 152 027 a5 = 1.061 495 429

E^x) = 1 - (a, t + a 2 t2 + a3 t 3 ) e"x" + e(x) t=

[A4.9]

s(\\<2.5 10~5

1 + px

p - 0.470 47 a, = 0.348 024 2

a2 = - 0.095 879 a3 = 0.747 855 6

Other approximate relationships of this type have been proposed [HAS 55] [SPA 87], with developments of 3rd, 4th and 5th order. C. Hastings also suggests the expression 1 E

l (

X

)

=

1

" ~ 2

3

4

5

606"

(1 + aj x + a2 x + a3 x + a 4 x + a5 x + a6 x )

a, = 0.070 523 078 4 a2 = 0.042 282 012 3 a3 = 0.009 270 527 2

[A4.10]

a4 = 0.000 152 014 3 a5 = 0.000 276 5672 a6 = 0.000 043 063 8

(0<x
Derivatives dEj(x) 2 -= dx VTC d 2 E,(x)J 4 _X2 -Y -=—p x e x dx VTI

[A4.ll]

[A4.12]

Appendices

309

Approximate formula The approximate relationship [DEV 62] f 4x2^

[A4.13]

= Jl-exd-

1-Tj

gives results of a sufficient precision for many applications (error lower than some thousandths for whatever x). A4.1.2. Second definition The error function is often defined by [PAP 65] [PIE 70]: t2 1 E 2 (x) = —zr e V27r °

2

dt

[A4.14]

With this definition

EiM E 2 (x)=

b

W2y

[A4.15]

9

yielding [A4.16] Applications

- = f ' e'^dx . a jt i

[A4.17]

a

a.

where a and (3 are two arbitrary constants [PIE 70] and

310

Random vibration

Properties of E2 ( x ) E 2 (x) tends towards 0.5 when x -> <x> :

E =

2

I e *

[A4.19]

dt = 0.5

E 2 (0)= 0 E 2 (-x) = -E 2 (x) Function E 2 (x) =

1

fx

dt

J

Table A4.2. Error Junction E2(x) X

E2(x)

0.05

0.01994

0.10

0.03983

0.15

0.05962

0.20

0.07926

0.25

0.09871

0.30

0.11791

0.35

0.13683

0.40

0.15542

0.45

0.17364

0.50

0.19146

0.55

0.20884

0.60 0.65

0.22575

0.70

0.25804

0.75

0.27337

0.80

0.28814

0.24215

AE2 0.01994 0.01989 0.01979 0.01964 0.01945 0.01920 0.01892

X 0.85

0.30234

0.90

0.31594

0.95

0.32894

1.00

0.34134

1.05

0.35314

1.10

0.36433

1.15

0.37493

1.20 1.25

0.38493

1.30

0.40320

1.35

0.41 149

1.40

0.41924

1.45

0.42647

1.50

0.43319

1.55

0.43943

1.60

0.44520

0.01859 0.01822 0.01782 0.01738 0.01691 0.01640 0.01589 0.01533 0.01477

E2(x)

0.39435

AE2 0.01420 0.01360 0.01300 0.01240 0.01180 0.01119 0.01060 0.01000 0.00942 0.00885 0.00829 0.00775 0.00723 0.00672 0.00624 0.00577

x

E2(x)

1.65

0.45053

1.70

0.45543

1.75

0.45994

1.80

0.46407

1.85

0.46784

1.90

0.47128

1.95

0.47441

2.00

0.47725

2.05

0.47982

2.10

0.48214

2.15

0.48422

2.20

0.48610

2.25

0.48778

2.30

0.48928

2.35

0.49061

2.40

0.49180

AE2 0.00533 0.00490 0.00451 0.00413 0.00377 0.00344 0.00313 0.00284 0.00257 0.00232 0.00208 0.00188 0.00168 0.00150 0.00133 0.00119

Appendices

2.45

0.49286

2.50

0.49379

2.55

0.49461

2.60

0.49534

2.65

0.49598

2.70

0.49653

2.75

0.49702

2.80

0.49744

2.85

0.49781

2.90

0.49813

2.95

0.49981

0.00106 0.00093 0.00082 0.00072 0.00064 0.00055 0.00049 0.00042 0.00037 0.00032 0.00028

3.00

0.49865

3.05

0.49886

3.10

0.49903

3.15

0.49918

3.20

0.49931

3.25

0.49942

3.30

0.49952

3.35

0.49960

3.40

0.49966

3.45

0.49972

3.50

0.49977

0.00024 0.00021 0.00017 0.00015 0.00013 0.0001 1 0.00010 0.00008 0.00006 0.00006 0.00005

Figure A4.2. Error function E2(x)

3.55

0.49841

3.60

0.49984

3.65

0.49987

3.70

0.49989

3.75

0.49991

3.80

0.49993

3.85

0.49994

3.90

0.49995

3.95

0.049996

4.00

0.49997

311

0.00004 0.00003 0.00003 0.00002 0.00002 0.00002 0.00001 0.00001 0.00001 0.00001

312

Random vibration

Approximate calculation 0/E 2 (x) The function E 2 (x) can be approximated, for x > 0, by the expression defined as follows [LAM 76] [PAP 65]: 5 4 E 2 (x)*i 1 - (a t + b t + c t + d t + e t )e

[A4.20]

where t=

1

1 + 0.2316418 x

c=1.421 413741

a = 0.254 829 592

b = -0.284 496 736

d = -1.453 152027

e = 1.061 405 429

The approximation is very good (at least 5 decimal points).

NOTE. With these notations, the function E 2 (x) is none other than the integral of the Gauss function:

G(x) = — r e

2

V27I

Figure A43. Comparison of the error function E2(x) andofG(\)

Appendices

313

Figure A4.3 shows the variations ofG(x) and o/E 2 (x) for 0 < x < 3. We thus have: TC

exp 2a

tA4-21]

d u = - a E, -— V2

Calculation of x for E 2 (x) = E0 The method below applies if x is positive and where 0 < EQ < 0.5 [LAM 80]. One calculates successively: z = V-21n(l-2E 0 ) and

x = g0 + g, z + g7 z2 + • • • + g10 z10

[A4.22]

where g 0 = 6.55864 10"4

g6 = -1.17213 10~2

g,=-0.02069

g 7 = 2.10941 10'3

g2 = 0.737563

gg =

g 3 =-0.207071

g9

g4 = -2.06851 10~2

g10 = -2.93138 10~7

_2.18541 10"4

= 1.23163 10-5

g5 =0.03444 For negative values, one will use the property E 2 (-x) = -E 2 (x)

NOTE. Ej

X

To calculate x from given Ei set E2 = —, calculate x, and then ——. /O 21 V2

314

Random vibration

A4.2. Calculation of the integral J ea / xn dx One has [DWI 66]:

feax , , a x a 2 x 2 a3 x 3 a11 x n I - dx = ln|x| + - + -- + -- + •-• + -x 1! 2 2! 3 3! n n!

+-.

[A4.23]

yielding, since /f .ea * f. .ea x ea x a I - dx = -- + - I --dx J n / ,\ n-1 , •> n-1 x (n -1) x n-1 x fe~

JI

x

e"~ dx = - r ,\ n-1r (n-l)x

n

ae~ /

,\ /

»\

(n-l)(n-2)x

n-27

a" * e /

,\ ,

(n-l)!x

[A4.24]

L

+

J

a" *

fe — (n-l)! x /

,\ ,

J

[A4.25]

A4.3. Euler's constant Definition 1 1 e = lim 1 + — H— + — - In n 2 n

[A4.26]

8 « 0.577 215 66490... An approximate value is given by [ANG 61]: 1 2

i.e. s « 0.577 2173... Applications It is shown that [DAY 64]: I Jr\

In X e

X

dX = -e

[A4.27]

Appendices

315

and that

1 (ln X) foo

2

e

dX = - + 6

[A4.28]

AS. Complements to the transfer functions A5.1. Error related to digitalization of transfer function The transfer function is defined by a certain number of points. According to this number, the peak of this function can be more or less truncated and the measurement of the resonance frequency and Q factor distorted [NEU 70]. Any complex system with separate modes is comparable in the vicinity of a resonance frequency to a one-degree-of-freedom system of quality factor Q.

Figure A5.1. Transfer Junction of a one dof system close to resonance

Let us set y as the value of the quality factor read on the curve, Q being the true value. X f read resonance frequency , M . ,.„ Let us set P = — and a = — = . When a is different Q f0 true resonance frequency from unity, one can set a = 1 - 8, if 5 is the relative deviation on the value of the resonance frequency. For 5 = 0, one has a = 1 and (5 = 1. For 8 * 0, P is less than

31 6

Random vibration

one. The resolution error is equal to e R = 1~(3. The amplitude of the transfer function away from resonance is given by y such that: 2

y2

=

<

+«

— ~^ a ~T/ n l— i _ , v^ ) —2 2

2

[A5J]

2

+a

Q

(1-a) + 2

a 1"+ —r 2

'1

Q2(l-aT+a2

Q V

1 r

2

For large Q, we have

Q2(l-a2) 2

i.e., replacing a by 1 - 8 and supposing Q large compared to 1, 1 - 2 8 + 4 Q 2 S2

[A5.4]

and

[A5.5] •/I - 2 6 + 4 Q

Appendices

Figure A5.2. Digitalization ofn points of the transfer function between the half-power points

317

Figure A5.3. Effect of a too low a sampling rate

)2 » 2 8, i.e. if Q2 26 -2\-V2

[A5.6] [A5.7]

Let us suppose that there are n points in the interval Af0 between the half-power points, i.e. n -1 intervals. We have: 5 = l-a = l - —

[A5.8]

Afn

[A5.9]

yielding

6=

M*

1

2(n-l)f0

2(n-l)Q

-> a o s , i.e., since 8 R « 2

[A5.10]

318

Random vibration

[A5.ll]

Figure A5.4 shows variations of the error S R versus the number of points n in Af 0 . To measure the Q factor with an error less than 2%, it is necessary for n to be greater than 6 points.

Figure A5.4. Error of resolution versus number of points in Af

NOTE. In the case of random vibrations, the frequency increment Af is related to the sampling frequency f s by the relationship [A5.12]

Af = 2M

where M is the total number of points representing the spectrum. Ideally, the increment Af should be a very small fraction of the bandwidth Af0 around resonance. The number of points M is limited by the memory size of the calculator and the frequency f s should be at least twice as large as the highest frequency of the analysed signal, to avoid aliasing errors (Shannon's theorem). A too large Af leads to a small value of n and therefore to an error to the Q factor measurement. Decreasing f s to reduce Af (with M constant) can lead to poor representation of the temporal signal and thus to an inaccuracy in the amplitude of the spectrum at high frequencies. It is recommended to choose a sampling frequency greater than 6 times the largest frequency to be analysed [TAY 75J.

Appendices

319

A5.2. Use of a fast swept sine for the measurement of transfer functions The measurement of a transfer function starting from a traditional swept sine test leads to a test of relatively long duration and requires in addition material having a great measurement dynamics. Transfer functions can also be measured from random vibration tests or by using shocks, the test duration being obviously in this latter case very short. On this assumption, the choice of the form of shock to use is important, because the transfer function being calculated from the ratio of the Fourier transforms of the response (in a point of the structure) and excitation, it is necessary that this latter transform does not present one zero or too small an amplitude in a certain range of frequency. In the presence of noise, the low levels in the denominator lead to uncertainties in the transfer function [WHI 69]. The interest of the fast linear swept sine resides in two points: - the Fourier transform of a linear swept sine has a roughly constant amplitude in the swept frequency range. W.H. Reed, A.W. Hall, L.E. Barker [REE 60], then R.G. White [WHI 72] and R.J. White and R.J. Pinnington [WHI 82] showed that the average module of the Fourier transform of a linear swept sine is equal to: X(o>) = ~^b 2Vb

[A5.13]

where x m = amplitude of acceleration defining the swept sine

f2-f, b = - = sweep rate and that, more generally, x

X =

where f is the sweep rate for an arbitrary law.

[A5.14]

320

Random vibration

Example Linear sweep: 10 Hz to 200 Hz Durations: 1 s - 0.5 s - 0.1 s and 10 ms x m =10ms' 2 Depending on the case, the relationship [A5.14] gives 0.3627 - 0.256 - 0.1147 or 0.03627 (m/s).

Figure A5.5. Example of fast swept sine

Figure AS .6. Examples of fast swept sine Fourier transforms

Appendices

321

- sweeping being fast (a few seconds or a fraction of a second, depending on the studied frequency band), the mechanical system responds as to a shock and does not have time to reach the response which it would have in steady state operation or with a slow sweep (Q time the excitation). Accordingly, the dynamics of the necessary instrumentation is less constraining and measurement is taken in a domain where the non linearities of the structure are less important. The Fourier transform of the response must be calculated over the whole duration of the response, including the residual signal after the end of sweep. A5.3. Error of measurement of transfer function using a shock related to signal truncation With a transient excitation, of shock type or fast swept sine, the transfer function is calculated from the ratio of the Fourier transforms of response and excitation: HO 0) -^4 X(ifl)

[A5..5]

X(i Q) = \**x(t) e~ i Q t dt

[A5.16]

Y(in)=

[A5.17]

where

yWe'^'dt —OO

If x(t) is an impulse unit applied to the time t = 0, we have X(i Q) = 1 for whatever Q and (Volume 1, expression [3.115]): H(ifl) = f ° h ( t ) e ~ i n t d t

[A5.18]

where h(t) is the impulse response. For a single-degree-of-freedom system of natural frequency f 0 (Volume 1, relationship [3.114]), i f.\

h(t) = ..

U

e

~~S Wfi

l

•

/1

i-£

sin V I - ^
[A5.19]

322

Random vibration

yielding the complex transfer function ,

x

1

CO 1-

[A5.20]

2

(0

T

CO,

V

The relationship [A5.18] could be used in theory to determine H\i Q) from the response to an impulse, but, in practice, a truncation of the response is difficult to avoid, either because the decreasing signal becomes non measurable, or because the time of analysis is limited to a value t m [WHI 69]. The effects of truncation have been analysed by B.L. Clarkson and A.C. Mercer [CLA 65] who showed: - that the resonance frequency can still be identified from the diagram vector as ds the frequency to which the rate of variation in the length of arc with frequency, —, df is maximum, -but that the damping measured from such a diagram (established with a truncated signal) is larger than the true value. These authors established by theoretical analysis that the error (in %) introduced by truncation is equal to: ]

e (%) = 100

1 -£co t m ^1 2 2 2 ^1 -e ° l + ^ a ) 0 T m + - ^ <°0 T mJ

1 1

1-e

°

ra

[A5.21]

^1 + q O)Q T m J

Figure A5.7. Error of measured value of£, due to truncation of the signal

Appendices

323

It is seen that if | co0 T m > 1, the error of the measured value of £ is lower than 5%. It is noted that one can obtain a very good precision without needing to analyse extremely long records. For example, for f 0 = 100 Hz and £ = 0.005 , a duration of 2 s led to an error less than 5% (£ co0 i m =1).

A5.4. Error made during measurement of transfer functions in random vibration 2.

The function of coherence y is a measurement of the precision of the calculated value of the transfer function H(f) and is equal to [2.97]: 2

Figure A5.8. Error of measurement of the transfer function in random excitation versus y, for K = 20

[A5.22]

Figure A5.9. Error of measurement of the transfer function in random excitation versus the probability, for K = 20

324

Random vibration

Figure A5.10. £ra>r of measurement of the Figure A5.11. Error of measurement of the transfer function in random excitation transfer function in random excitation versus y, for P = 0.90 versus K, for P = 0.90

If the system is linear and if there is no interference, y = 1 and the calculated value of H(f) is correct. If y < 1, the error in the estimate of H(f) is provided with a probability P by:

[A5.23]

where K is the number of spectra (blocks) used to calculate each PSD [WEL 70].

Figure A5.12. Error of measurement of the transfer function in random excitation versus K, for y = 0.5

Appendices

325

A5.5. Derivative of expression of transfer function of a one-degree-of-freedom linear system Let us consider the transfer function:

1 H((D) =

[A5.24]

l - h2 + 2i

where (0

<x>0

through multiplication of the denominator's conjugate quantity, we obtain H(co) = ( l - h J A - 2 i ^ h A

if we set A=

1

yielding

dH dh

A / OA ? \\ d -2hA-2i4A+l-h2 ; dh

hi

dA

[A5.25]

dh

with

dA

[A5.26]

dh

dH

f -> 1 dA OA = -2(h + i 4) A + L 1 - h^ - 2 ^ h Ji — dh dh

[A5.27]

This page intentionally left blank

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Index

acceleration 70 et seq practical calculation 70 rms values 70 acoustic spectrum conversion 299 along the process 15,31 temporal averages 31 amplitude jumps 237 successive extrema 237 analysis, types of 264 autocorrelation function 264 extreme values 265 peak value distribution 265 power spectral density 264 rms value 264 threshold level crossings 266 asymptote, Gumbel 261 asymptotic laws 260 autocorrelation function 12,264 properties of 25 autocovariance 12 autostationarity 13 strong 13 weak 13 band-limited white noise 50,212 irregularity factor 212 bandwidth, optimization 145 bias error 138 Bingham window 95 broad-band process 45 center frequencies 295 central moments 9 standard deviation 10 variance 9 chi-square law 105,286 coherence function 58 coherent output power spectrum 59

coincident spectral density 45 confidence interval 111, 118 confidence limits 119 et seq correlation 26 coefficient 26 duration 27 function 27 and PSD 44 cospectrum 45 coincident spectral density 45 covariance 13 crest factor 52 cross-correlation 12, 28 coefficient 28 function 12,28 • normalized covariance 28 normalized function 28 cross-power spectral density 42 two processes 43 displacement 70 et seq practical calculation 70 rms values 70 distribution of maxima 195etseq average correlations 207 average numbers per unit time 236 average time intervals 206 expected number per unit 203 fatique analysis 196 irregularity factor 197 mean number 226,231 mean time intervals 230 negative 235 positive 235,236 density 236 probability 195 distribution function 6

342

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ensemble averages 8 et seq central moments 9 n order average 8 'through the process' 8 ergodicity 29 error normalised rms 103 PSD 102 statistical 103, 126 error Junction 303, 306 Euler's constant 314 exponential law 284 extrema 237 extreme values 241 et seq analysis 265 maximum exceeded with 244 mode 244 probability density of maxima 241 statistics 241 variance of maximum 243 Fast Fourier Transform (FFT) 100 filter bandwidth 137 choice of 137 maximum statistical error 143 Fourier leakage 94 spectrum 33 transform 34 Gaussian process 6 narrow band 224 random 6 Gauss's law 279 normal law 279,290 Gumbel asymptote 261 Hanning window 94 highest peak, average 250 standard deviation 252 instantaneous values 4 distribution 4 function 5,6 probability density 4 function 6 irregularity factor 197, 208 et seq band limited white noise 212 variation interval 208 kurtosis 19 Laplace-Gauss law 279,290 laws of probability 279 chi-square law 105, 286 exponential law 284

Gauss's law 279 log-normal law 280 normal Laplace-Gauss law 290 Poisson's law 284 Rayleigh's law 220,287 Student law 294 Weibull distribution 289 leakage 94 linear-logarithm scale 186,190,194 logarithmic rise 243 log-linear scale 187,192, 193 log-log scale 188, 192, 193 log-normal law 280 mathematical functions 303 error function 303,306 Euler's constant 314 maxima average correlation 207 average time interval 206, 230 envelope distribution 272 expected number 203 number greater than given threshold 226,230,231,241 probability density 247 probability distribution 195 et seq, 235 variance 243 mean quadratic value 160 rms value 160 mode 244 expected maximum 254 probable maximum value 244 moments 181 central moments 18 of order n 18, 193 n order average 8 narrow band envelopes 48,267 distribution of maxima 272 probability density of maxima 267 individual cycles 48 irregularity factor 218 process 47 normal 244 signal 210 noise 2 band limited 50 of form 21 irregularity factor 215 pink 48 white 46 autocorrelation function of 48 Nyquist frequency 88

Index octave analysis 295 center frequencies 295 one-degree-of-freedom system 301 linear 325 transfer function 325 response of equivalent 301 optimum bandwidth 145 ordinates 297 overlapping 128etseq anddof 129 rate 128 choice of 132 and statistical error 130 peak distribution function 222 peak factor 52 peak ratio 52 crest factor 52 peak value distribution 265 peaks 243 average maximum 243 average of highest 250 standard of highest 252 truncation of 60 periodic excitation 101 periodic signals 82 random noise 84 pink noise 48 Poisson's law 284 power spectral density (PSD) 35 et seq, 39, 61,264 acceleration 66 average frequency 184 calculation from Fourier transform 91 et seq calculation from rms value of filtered signal 90 constant in frequency interval 71 continuous component signal 65 conversion of an acoustic spectrum 299 correlation function 44 definition 36,39,42,264 displacement 66 distribution of measured 104 excitation of 55 fourth moment 189 graphical representation 69 linear-logarithmic scales 186 logarithmic-linear scales 187 logarithmic-logarithmic scales 187 normalized rms error 103 periodic excitation 101 periodic signals 83 onto random noise 84 practical calculation of, 87 et seq random process 43 random signal 154

signal sampling 87 standardized 54 statistical error 102,103,134 et seq variance of measured 106 velocity 66 pressure spectral density 300 probability density 4, 220 function 220 laws 279 et seq pseudo-stationary signals 30 quadratic mean 16 rms value 16 quadrature spectral density 45 quadspectrum 45 random process 1 et seq Gaussian 6 PSD of 43 statistical properties 1 stochastic vibration 1 variable 1 random signal 162 generation 154 instantaneous values 162 moments 181 probability density 170 statistical properties 162 threshold crossings 167 threshold level crossing curves 174 random vibration 2 average frequency 171 in the frequency domain 33 properties of 33 instantaneous values 4 et seq mean value 159 et seq laboratory tests 2 maxima probability distribution 195 methods of analysis 3 noise 2 real environments 2 rms value 63 et seq, 160 in the time domain 33 et seq, 159 et seq Rayleigh distribution 7 peak 262 Rayleigh's law 220,287 error related 220 return period 242 rms value 264 random vibration 63 et seq and PSD 63 sampling 87 of signal 87 Shannon' s theorem 315 skewness 18

343

344

Random vibration

spectral density 54 function of time 54 standard deviation 10 standardized variance 110 statistical error 150etseq calculation 104 calculation of PSD 134 et seq digitized signal 126 maximum 145 quantities 150 'standard error' 107 stationarity 13 stationary signals 30 stochastic 1 process 1 vibration 1 Student law 294 temporal autocorrelation 22 temporal averages 15 et seq 'along the process' 15 mean 15 skewness 18 standard deviation 18 variance 18 threshold crossing 266 curves 174 et seq number 167 'through the process' 31 ensemble averages 31 time 234 signal 234

transfer function 315 complements 315 digitization of 315 related error 315 measurement of 319 errors 321,323 one-degree-of-freedom linear system 325 truncation of peaks 60 signals 60 variance 9,18 standard deviation 18 standardized 110 variation interval 208 velocity rms value 70 et seq practical caculation 70 Weibull distribution 289 white noise 46 autocorrelation function of 48 band-limited 47,212 autocorrelation function of, 50 wide-band centered normal process 257 Wiener-Khinchine relations 44 windows 97 Bingham 95 Manning 94 principal 97 weighting coefficient 95

Synopsis of five volume series: Mechanical Vibration and Shock

This is the third volume in this five volume series. Volume 1 is devoted to sinusoidal vibration. The responses, relative and absolute, of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms defined. By placing the properties of sinusoidal vibrations in the contexts of the environment and of laboratory tests, the transitory and steady state response of a single-degree-offreedom system with viscous and then with non-linear damping is evolved. The various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of the sweep rate are shown and a rule for choice of this rate deduced from it Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be made with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution. Volume 3 examines the analysis of random vibration, which encompass the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing,

346

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overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). Having established the relationships which provide the response of a linear system with one degree of freedom to a random vibration, Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behaviour of a material subjected to fatigue, the laws of damage accumulation, together with the methods for counting the peaks of the response, used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal. The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non linear accumulation law, etc.). Volume 5 is more especially dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification as from the life cycle profile of the equipment is then detailed, taking account of an uncertainty factor, designed to cover the uncertainties related to the dispersion of the real environment and of the mechanical strength, and of another coefficient, the test factor, which takes into account the number of tests performed to demonstrate the resistance of the equipment. This work is intended first and foremost for engineers and technicians working in design teams, which are responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance, following the choice of the most suitable simulation means.

Fatigue Damage

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Mechanical Vibration & Shock

Fatigue Damage Volume IV

Christian Lalanne

HPS

HERMES PENTON SCIENCE

First published in 1999 by Hermes Science Publications, Paris First published in English in 2002 by Hermes Penton Ltd Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: Hermes Penton Science 120 Pentonville Road London Nl 9JN © Hermes Science Publications, 1999 © English language edition Hermes Penton Ltd, 2002 The right of Christian Lalanne to be identified as the author of this work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. British Library Cataloguing in Publication Data A CIP record for this book is available from the British Library. ISBN 1 9039 9606 6

Printed and bound in Great Britain by Biddies Ltd, www.biddles.co.uk

Contents

Introduction List of symbols 1

Response of a linear one degree-of-freedom linear system to random vibration

1.1. Average value of response of a linear system 1.2. Response of perfect bandpass filter to random vibration 1.3. PSD of response of a single-degree-of-freedom linear system 1.4. Rms value of response to white noise 1.5. Rms value of response of a linear one-degree-of-freedom system subjected to bands of random noise 1.5.1. Case: excitation is a power spectral density defined by a straight line segment in logarithmic scales 1.5.2. Case: vibration has a power spectral density defined by a straight line segment of arbitrary slope in linear scales 1.5.3. Case: vibration has a constant power spectral density between two frequencies 1.5.3.1. Power spectral density defined in a frequency interval of arbitrary width 1.5.3.2. Case: narrow band noise of width Af = ^fo/Q 1.5.4. Excitation defined by an absolute displacement 1.5.5. Case: excitation defined by power spectral density comprising n straight line segments 1.6. Rms value of the absolute acceleration of the response 1.7. Transitory response of dynamic system under stationary random excitation 1.8. Transitory response of dynamic system under amplitude modulated white noise excitation

xi xiii

1 1 2 4 5 11 11 17 19 19 22 25 27 30 31 38

vi

2

Fatigue damage

Characteristics of the response of a one-degree-of-freedom linear system to random vibration

2.1. Moments of response of a one-degree-of-freedom linear system: irregularity factor of response 2.1.1. Moments 2.1.2. Irregularity factor of response to noise of constant PSD 2.1.3. Characteristics of irregularity factor of response 2.1.4. Case: band-limited noise 2.2. Autocorrelation function of response displacement 2.3. Average numbers of maxima and minima per second 2.4. Equivalence between transfer functions of bandpass filter and one-degree-of-freedom linear system 2.4.1. Equivalence suggested by D.M. Aspinwall 2.4.2. Equivalence suggested by K.W. Smith 2.4.3. Rms value of signal filtered by the equivalent bandpass 3

41 41 41 45 47 58 60 60

filter

Concepts of material fatigue

3.1. Introduction 3.2. Damage arising from fatigue 3.3. Characterization of endurance of materials 3.3.1.S-Ncurve 3.3.2. Statistical aspect 3.3.3. Distribution laws of endurance 3.3.4. Distribution laws of fatigue strength 3.3.5. Relation between fatigue limit and static properties of materials 3.3.6. Analytical representations of S-N curve 3.3.6.1. Wohler relation (1870) 3.3.6.2. Basquin relation 3.3.6.3. Some other laws 3.4. Factors of influence 3.4.1. General 3.4.2. Scale 3.4.3. Overloads 3.4.4. Frequency of stresses 3.4.5. Types of stresses 3.4.6. Non zero mean stress 3.5. Other representations of S-N curves 3.5.1.Haighdiagram 3.5.2. Statistical representation of Haigh diagram 3.6. Prediction of fatigue life of complex structures 3.7. Fatigue in composite materials

63 63 66 67 69 69 70 73 73 76 77 81 82 85 85 86 92 94 94 95 95 97 98 98 100 100 107 108 109

Contents

4 Accumulation of fatigue damage

vii

Ill

4.1. Evolution of fatigue damage 4.2. Classification of various laws of accumulation 4.3. Miner's method 4.3.1. Miner's rule 4.3.2. Scatter of damage to failure as evaluated by Miner 4.3.3. Validity of Miner's law of accumulation of damage in case of random stress 4.4. Modified Miner's theory 4.5. Henry's method 4.6. Modified Henry's method 4.7. H. Corten and T. Dolan's method 4.8. Other theories

Ill 112 113 113 117 121 122 127 128 129 131

5 Counting methods for analysing random time history

135

5.1. General 5.2. Peak count method 5.2.1. Presentation of method 5.2.2. Derived methods 5.2.3. Range-restricted peak count method 5.2.4. Level-restricted peak count method 5.3. Peak between mean-crossing count method 5.3.1. Presentation of method 5.3.2. Elimination of small variations 5.4. Range count method 5.4.1. Presentation of method 5.4.2. Elimination of small variations 5.5. Range-mean count method 5.5.1. Presentation of method 5.5.2. Elimination of small variations 5.6. Range-pair count method 5.7. Hayes counting method 5.8. Ordered overall range counting method 5.9. Level-crossing count method 5.10. Peak valley peak counting method 5.11. Fatigue-meter counting method 5.12. Rainflow counting method 5.12.1. Principle of method 5.12.2. Subroutine for rainflow counting 5.13. NRL counting method (National Luchtvaart Laboratorium) 5.14. Evaluation of time spent at given level 5.15. Influence of levels of load below fatigue limit on fatigue life 5.16. Test acceleration 5.17. Presentation of fatigue curves determined by random vibration tests

135 139 139 141 142 143 145 145 147 148 148 150 151 151 154 156 160 162 164 168 173 175 175 181 184 187 188 188 190

viii

6

Fatigue damage

Damage by fatigue undergone by a one-degree-of-freedom mechanical system

193

6.1. Introduction 6.2. Calculation of fatigue damage due to signal versus time 6.3. Calculation of fatigue damage due to acceleration spectral density 6.3.1. General case 6.3.2. Particular case of a wide-band response, e.g., at the limit, r = 0 6.3.3. Particular case of narrow band response 6.3.3.1. Expression for expected damage 6.3.3.2. Notes 6.3.3.3. Calculation of gamma function 6.3.4. Rms response to narrow band noise G0 of width Af when GO Af = constant 6.4. Equivalent narrow band noise 6.4.1. Use of relation established for narrow band response 6.4.2. Alternative: use of mean number of maxima per second 6.4.3. P.H. Wirsching's approach 6.4.4. G.K. Chaudhury and W.D. Dover's approach 6.4.5. Approximation to real maxima distribution using a modified Rayleigh distribution 6.5. Comparison of S-N curves established under sinusoidal and random loads 6.6. Comparison of theory and experiment 6.7. Influence of shape of power spectral density and value of irregularity factor 6.8. Effects of peak truncation 6.9. Truncation of stress peaks

193 194 196 196 200 201 201 205 210

7

253

Standard deviation of fatigue damage

212 213 213 216 217 220 224 228 232 237 237 238

7.1. Calculation of standard deviation of damage: J.S. Bendat's method 7.2. Calculation of standard deviation of damage: S.H. Crandall, W.D. Mark and G.R. Khabbaz method 7.3. Comparison of W.D. Mark and J.S. Bendat's results 7.4. Statistical S-N curves 7.4.1. Definition of statistical curves 7.4.2. J.S. Bendat's formulation 7.4.3. W.D. Mark's formulation

253

8

277

Fatigue damage using other assumptions for calculation

8.1. S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit) 8.2. S-N curve defined by two segments of straight line on log-lin scales 8.3. Hypothesis of non-linear accumulation of damage

258 263 269 269 270 273

277 280 283

Contents

ix

8.3.1. Corten-Dolan's accumulation law 8.3.2. J.D. Morrow's accumulation model 8.4. Random vibration with non zero mean: use of modified Goodman diagram 8.5. Non Gaussian distribution of instantaneous values of signal 8.5.1. Influence of distribution law of instantaneous values 8.5.2. Influence of peak distribution 8.5.3. Calculation of damage using Weibull distribution 8.5.4. Comparison of Rayleigh assumption/peak counting 8.6. Non-linear mechanical system

283 284 287 289 289 290 291 294 295

Appendices

297

Al. Gamma function A 1.1. Definition A1.2. Properties A1.3. Approximations for arbitrary x A2. Incomplete gamma function A2.1. Definition A2.2. Relation between complete gamma function and incomplete gamma function A2.3. Pearson form of incomplete gamma function A3. Various integrals

297 297 297 300 301 301

Bibliography

323

Index

347

Synopsis of five volume series

351

302 303 303

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Introduction

Fatigue damage to a system with one-degree-of-freedom is one of the two criteria adopted for comparing the severity of different vibratory environments, the second being the maximum response of the system. This criterion is also used to create a specification reproducing on the equipment the same effects as all the vibrations to which it will be subjected in its useful lifetime. This volume focuses on calculation of damage caused by random vibration This requires: - having the relative response of the system under vibration represented by its PSD or by its rms value. Chapter 1 establishes all the useful relationships for that calculation. In Chapter 2 is described the main characteristics of the response. - knowledge of the fatigue behaviour of the materials, characterized by S-N curve, which gives the number of cycles to failure of a specimen, depending on the amplitude of the stress applied. In Chapter 3 are quoted the main laws used to represent the curve, emphasizing the random nature of fatigue phenomena, followed by some measured values of the variation coefficients of the numbers of cycles to failure. - determination of the histogram of the peaks of the response stress, supposed here to be proportional to the relative displacement. When the signal is Gaussian stationary, as was seen in Volume 3 the probability density of its peaks can easily be obtained from the PSD alone of the signal. When this is not the case, the response of the given one-degree-of-freedom system must be calculated digitally, and the peaks then counted directly. Numerous methods, ranging from the simplest (counting of the peaks) to the most complex (rainflow) have been proposed and are presented, with their disadvantages, in Chapter 5.

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Fatigue damage

- choice of a law of accumulation of the damage caused by all the stress cycles thus identified. In Chapter 4 are described the most common laws with their limitations. All these data are used to estimate the damage, characterized statistically if the probability density of the peaks is available, and deterministically otherwise (Chapter 6), and its standard deviation (Chapter 7). Finally, in Chapter 8 are provided a few elements for damage estimation from other hypotheses concerning the shape of the S-N curve, the existence of an endurance limit, the non linear accumulation of damage, the law of distribution of peaks, and the existence of a non-zero mean value.

List of symbols

The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have locally another meaning which will be defined in the text to avoid any confusion. a b

c C d

dof D

Dt erf e(t) E( ) f f0

Threshold value of z(t) Exponent in Basquin's relation (N a = C) or Exponent Viscous damping constant Constant in Basquin's relation (No = C) Damage associated with one half-cycle or Exponent in Corten-Dolan law or Plastic work exponent Degree-of-freedom Damage by fatigue or Damping capacity (D = J < j n ) Fatigue damage calculated using truncated signal Error function Narrow band white noise Expectation of... Frequency Natural frequency

G( ) h h(t) H H( )

i

J k K lms ^(t) m Mn n

Power spectral density for 0 < f < oo Interval (f/f 0 ) Impulse response Drop height Transfer function

^\

Damping constant Stiffness Constant of proportionality between stress and deformation Rms value of ^(t) Generalized excitation (displacement) Mass or Mean or Number of decibels Moment of order n Number of cycles undergone by test-bar or a material or Order of moment Exponent of D = J an or

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n0 nQ

np N

N_ p( ) PSD

Number of constant levels of a PSD Mean number of zerocrossings per second Mean number of zerocrossings with positive slope per second (expected frequency) Mean number of maxima per second Number of cycles to failure Number of dB per octave Number of peaks over duration T Probability density Power spectral density

S( )

Vl -r Probability density of maxima Reduced response First derivative of q (0) Second derivative of q(0) Q factor (factor of quality) Irregularity factor Root mean square (value) Yield stress Ultimate tensile strength Correlation function Standard deviation Standard deviation of damage Power spectral density for

t

—oo < f < +00 Time

T u

Duration of vibration Ratio of peak a to rms

q q(u) q (0) q (0) q(0) Q r rms Re Rm R z (t) s SD

value lms Ujms

O

ofl(t)

Rms value of u(t)

u0 UQ u(t) u(t) ii(t) v VN

zrms zrms

Initial value of u(t) Initial value of u(t) Generalized response First derivative of u(t) Second derivative of u(t) Variation coefficient Variation coefficient of number of cycles to failure by fatigue Rms value of x(t) Rms value of x(t) Rms value of x(t) Absolute displacement of the base of a one-degreeof-freedom system Absolute velocity of the base of a one-degree-offreedom system Absolute acceleration of the base of a one-degreeof-freedom system Rms value of y(t) Rms value of y(t) Rms value of y(t) Absolute response displacement of the mass of a one-degree-offreedom system Absolute response velocity of the mass of a onedegree-of-freedom system Absolute response acceleration of the mass of a one-degree-of-freedom system Rms value of z(t) Rms value of z(t)

zrms zp

Rms value of z(t) Amplitude of peak of z(t)

\Tms xms xms x(t)

x(t)

x(t)

yrms yrms yrms y(t)

y(t)

y(t)

List of symbols

z(t)

z(t) z(t)

a B ^ A Af

AT

Relative response displacement of mass of one-degree-of-freedom system with respect to its base Relative response velocity Relative response acceleration CT 2 A/1 -E2 a / -> \ 2 1- 2 E ] \ ^ I Index of damage to failure Frequency interval between half-power points

Width of narrow band noise Effective duration or Time of correlation

y( ) p(

v

) '

T|
G)0 Q

£

xv

Incomplete gamma function Gamma function Exponent in Weibull's law Signal of simple form Reduced excitation 3.14159265... Reduced time (co0 t) Stress Alternating stress Fatigue limit stress Rms value of stress Mean stress

Truncation level of stress , . Natural pulsation (2 TT f0) Pulsation of excitation (2 TI f) Damping factor

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Chapter 1

Response of a one-degree-of-freedom linear system to random vibration

1.1. Average value of response of a linear system The response of a linear system to an excitation assumed to be random stationary is given by the general relation

where ^(6) is the excitation, a is a variable of integration, and h(9) is the impulse response of the system. The ensemble average of q(0) ('through the process') is

The operators E and sum being linear, we can, by permuting them, write,

Since, by assumption, the process X(Q) is stationary, the ensemble averages are independent of the time at which they are calculated:

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This relation links the average value of the response to that of the excitation. We saw that the Fourier transform of h(6) is equal to:

and

If the process X(0) is zero average, the response is also zero average [CRA 63]. 1.2. Response of perfect bandpass filter to random vibration Let us consider a perfect bandpass filter defined by:

Figure 1.1. Bandpass filter

The response of this filter to the excitation ^(t) is given by the convolution integral

Response of a one-degree-of-freedom linear system

3

where h(t) is the impulse response of the system. In the frequency domain, one saw that, if G^(f) is the input PSD of the signal ^(t) and G u (f) that of the response u(t):

yielding the rms value urms of the response

If the PSD of the input is a white noise, G^(f) = G^Q = constant:

If the transfer function of the filter is such that H0 = 1:

The rms value of the response signal is equal to the rms value of the input in the bandfa,fb. NOTE. - The PSD symbol is modified by an index indicating on which signal it is calculated (l(t), x(t),...,). The symbol G is followed by Q or f between brackets to specify the nature of the variable, pulsation or frequency. If the PSD is constant, one adds the index 0 (eg Ge (&)).

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1.3. PSD of response of a single-degree-of-freedom linear system Consider a mass-spring-damping linear system of natural frequency f0 = co0/2 TT and quality factor Q = 1/2 £. As above, the response u(t) of the system to the input l(i) has as a general expression (Volume 1):

where the impulse response of the system h(t) is equal to

For a physical system,h(t) = 0 for t<0 domain, the PSD are related by

(causal system). In the frequency

For a relative response the function H(Q)is given by

and for an absolute response by

In a more general way, one can write the complex transfer in the form [1.13] where Bj = 0 or 1 in terms of case selected.

Response of a one-degree-of-freedom linear system

5

1.4. Rms value of response to white noise We will suppose initially that the power spectral densities S(Q) are defined in the most general case, for Q between - oo and + oo. The rms value of the response, u

r m s » i s given by

Let us examine the theoretical particular case of a white noise where S^(Q)= constant = S^Q from - oo to + oo. Then

It can be shown [LAL 94] that the integral

has as a value

Assumption

n° 1 (relative response)

n° 2 (absolute response)

B0

1

1

B,

0

l/Qw 0

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1

1

A1

l/Qw0

1/Qcoo

A2

V«0

l/a>o

I

7tQco 0

A0

This yields assumption n° 1 (relative response)

assumption n° 2 (absolute response)

Particular cases Case 1.

and [CRA 63] [CRA 66]

Case 2. The PSD is defined, in terms of Q, in (0, oo) [GAL 57]:

G^(Q) = 2 S,(Q)

Response of a one-degree-of-freedom linear system

Case 3. From case

Case 4. The PSD is defined in terms of fin (- oo, + oo). From [1.14], since S^ 0 (f) = 2 7t S^(Q)

Case 5. From case n° 4, if t(t)

Case 6. The PSD is defined in terms of fin (0, <x>):

and [1.19] [BEN 61] [MOR63] [MOR75] [PIE 64]:

x(t) Case 7. From case n° 6, if *(t) = —— [BEN 62] [BEN 64] [CRA 63] [PIE 64], o0

7

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NOTE. 1. One could find, in the same way, the results obtained on assumption n° 2 while replacing, in the relations above, Q by

1 + Q2

Q

or by

1 + Q2

Q

4

co0 (in terms of whether

it is about G^Q or G^J. 2. The response of a linear one-degree-of-freedom system subjected to a sine wave excitation is proportional to Q. It is proportional to -^/Q if the excitation is random [CRE 56a]. Table 1.1 lists various expressions of the rms value in terms of the definition of the PSD. If the PSD is defined in (0, oo), consider now various inputs (acceleration, velocity, displacement or force). The transfer function of a linear one-degree-offreedom system can be written:

The rms value of the response is calculated, for an excitation defined by a displacement, a velocity or an acceleration, respectively using the following integrals:

Table 1.2 lists the rms responses of a linear one-degree-of-freedom system for these excitations (of which the PSD is constant) [PIE 64].

Response of a ONE-DEGREE-OF-FREEDOM

linear system

9

Summary chart Table 1.1. Expressions of the rms values of the response in tems of the definition of the PSD

Type of transfer function

Definition of PSD

PSD of the reduced variable in terms of Q in (—00,00) PSD of x(t) in terms of Q in (-00,00) PSD of ^(t) in terms of Q in (0,») PSD of x(t) in terms of Q in (0,«)) PSD of ^(t) in terms of f in (—00,00) PSD of x(t) in terms of f in (-00,00) PSD of ^(t) in terms of f in (0,oo) PSD of x(t) in terms of f in (0,00)

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Fatigue damage Table 1.2. Rms values of the response in terms of the definition of excitation

Excitation

Parameter response

Absolute displacement G x in (0,oo)

Absolute displacement

Absolute velocity G^

Absolute velocity

Absolute acceleration

Absolute acceleration

GX

Force G X

Rms value

Relative displacement

GF k2

Relative or absolute displacement

If the noise is not white, one obtains the relationships by taking the value of G for Q = 0)0 and by supposing that the value of G varies little with Q between the half-power points. The approximation is much better when £ is smaller (0.005 < £ < 0.05). For a weak hysteretic damping, one would set r| = 2 £ in these relations. These relations are thus applicable only: - if the spectrum has a constant amplitude, - if not, with slightly damped systems (for which the transfer function presents a narrow peak), with the condition that the PSD varies little around the natural frequency of the system. In all cases, the natural frequency must be in the frequency range of the PSD.

Response of a one-degree-of-freedom linear system

11

1.5. Rms value of response of a linear one-degree-of-freedom system subjected to bands of random noise 1.5.1. Case: excitation is a power spectral density defined by a straight line segment in logarithmic scales

Figure 1.2. PSD defined by a segment of straight line in logarithmic scales

We propose to examine the case of signals whose spectrum can be broken up into several bands, each one having a constant or linearly variable amplitude in logarithmic scales; the PSD will be defined in a band fj, f2(or Ql5 Q2) by:

where co0 is the natural pulsation of the one-degree-of-freedom system and b is a constant whose value characterizes the slope of the segment defined in logarithmic scales (h = Q/co0). NOTE. - We saw that the number of octaves n which separate two frequencies f1 and f2 is equal to [LAL 82]

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In addition, one defines the number of decibels m between two levels of the PSD by

yielding, from

the number of dB per octave,

The mean square value of the response is

where Gl(Q) is the PSD of the input ^(t), defined from 0 to infinity, and H(Q) is Q the transfer function of the system. With the notation h = —, the function H can be co0

written as follows (Table 1.3). In a general way,

As in case n° 1,

a

where hj

Table 1.3. Transfer functions relating to the various values of the generalized variables

Case 1

2

3

Transfer function

Excitation and response

14

Fatigue damage For case n° 2,

For case n° 3,

If we set

it is shown (Appendix A3.1) that, for b * 3,

and for b = 3,

This recurrence formula makes it possible to calculate Ib for arbitrary b [PUL 68]. Table A3.1 in Appendix A3 lists expressions for Ib.

Response of a one-degree-of-freedom linear system

15

Particular cases It can be useful to use a series expansion in terms of h around zero to improve the precision of the calculation of the rms values for larger values of f0, or an asymptotic development in terms of 1/h for the small values of f0. 1. For h small, lower than 0.02, a development in limited series shows that, to the fifth order,

2. For h large, greater than 40, we have, to the fifth order,

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NOTE. - For large values of h (>100) and more particularly -with I 0 (h), it can be interesting to evaluate directly AI = l(hj + Ah) -l(hi) starting from the latter relationships when the -width Ah = h2 - hj is small. We then have

where the terms

can be replaced by series developments.

On the most current assumption where the excitation is an acceleration [^(t) = -x(t)/co 0 ] ar>d the response is relative displacement z(t), the relation , GX(Q) Gko(Q) . . [1.33], transformed while replacing G^ by G^(Q) = — = —^4—, and GJJ(^) co0

by

G s (f) 27T

, then becomes:

o)0

Response of a one-degree-of-freedom linear system

Since G^ = (2 n f) in the same way by:

17

Gz and G± = (2 TI f) G^, the rms velocity zrms is given

and rms acceleration by:

Figure 1.3. PSD defined by a straight line segment in linear scales

Let us suppose that the vibration is characterized by a signal ^(t) of which the power spectral density can be represented by an expression of the form

where

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In case n° 1,

This expression can be written, with the notations of the above paragraphs:

i.e., by replacing A and B by their expressions,

If the vibration is defined by an acceleration [^(t) = -x(t)/o)Q] and if the PSD is C (f) defined in terms off, we have G^(Q) = —-— and 271

Likewise it can be shown that

Response of a one-degree-of-freedom linear system

19

and that

The functions I 0 (h) to I 5 (h) are defined by the expressions [A3.20] to [A3.25] in Appendix A3.

1.5.3. Case: vibration has a constant power spectral density between two frequencies 1.5.3.1. Power spectral density defined in a frequency interval of arbitrary width For a white noise of constant level G^ Q (Q) between two frequencies Qj and Q2, [CRA 63], by making G^ = G(2 = G(Q in [1.56]:

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NOTES. — I f ^ i —> 0, the term between square brackets tends towards zero. It tends towards unity when h2 —> °°. This term is thus the corrective factor to apply to the corresponding expression of paragraph 1.4 to take account of the limits fj and f2 respectively non zero and finite.

Figure 1.4. Constant PSD between two frequencies

More simply, if the excitation x(t) is a noise of constant power spectral density GXO between two arbitrary frequencies f, and f2, we have, since ^(t) = -x(t)/co 0 , G f (Q) = G^(Q)/coo and G x (f) = 2 n G x (o>),

The rms values of the velocity and acceleration have the following expressions:

Response of a one-degree-of-freedom linear system

21

Example

Rms values of the response

Average frequency

Number of maxima per second

Parameter r

Particular case where £, = 0 In this case

It is supposed here that G^Q is expressed in terms of Q. If G^Q is defined in terms of f, we have, as previously,

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Let us set h = th x

and

1.5.3.2. Case: narrow band noise of width Af - ^f 0 /Q It is assumed that

where K is a constant. From [1.61], let us calculate I0 for

and

Response of a one-degree-of-freedom linear system

23

Figure 1.5. Narrow band noise of width Af = X f0 / Q

If G is the value of the PSD and supposing Q sufficiently large so that V4Q2-1«2Q,

i.e., at first approximation

and, if

Equivalence with white noise There will be equivalence (same rms response) with white noise G^Q > if

i.e. if

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Particular cases

Figure 1.6. Determination of an equivalence with white noise

There is the same rms response if

i.e. if

If

Response of a one-degree-of-freedom linear system

25

Equivalence with a white noise G^Q exists if

1.5.4. Excitation defined by an absolute displacement In this case, the relative displacement rms response is given by:

where

In terms of the expressions listed in Table 3.1 of Volume 1:

If the PSD is defined by a straight line segment of arbitrary slope (in linear scales), and with the notations of paragraph 1.5.2,

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yielding Ttf,

^{Afo^^)-^)]^!^)-!^)]}

or

In the same way, the rms velocity and acceleration are given by:

and

I6, I7, I8 and I9 being respectively given by [A3.26], [A3.27], [A3.28] and [A3.29]. If the PSD is constant in the interval (hj, h 2 ), the rms values result from the above relationships while setting Gj = G2 = G X Q .

1.5.5. Case: excitation defined by power spectral density comprising n straight line segments For a power spectral density made up of n segments of straight line, we have

each term z,^

being calculated using the preceding relations:

- either by using abacuses or curves giving Ib in terms of h, for various values of b [PUL 68], - or on a computer, by programming the expression for Ib. When the number of bands is very large, it can be quicker to calculate directly [1.31] through numerical integration than to use these analytical expressions. In the usual case where the excitation is an acceleration, one determines zrms, z.rms and zrms starting from the relations [1.57], [1.58] and [1.59]. The rms displacement is thus equal to

where, for

with

and,

Similarly

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and

Figure 1.7. PSD made up of horizontal straight line segments

When the PSD is made up only of horizontal straight line segments, these relations are simplified in terms of

Response of a one-degree-of-freedom linear system

29

For example,

Particular case where the n levels all have constant width Af On this assumption, the excitation being an acceleration, the rms values zrms, zrms and zms can be obtained from:

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Fatigue damage

1.6. Rms value of the absolute acceleration of the response If the excitation is an acceleration x(t),

f Let us set h = —. If the PSD is defined by straight line segments of arbitrary fo slope in linear scales, G k (f) = A f + B and

where

For a noise of constant PSD between fj and f2, constant,

Example Under the conditions of the example quoted in paragraph 1.5.3.1, we have Yms

= 39 82

-

m/s2 (to be compared with 4 7t2 f^ zms = 39.62 m/s2).

Response of a one-degree-of-freedom linear system

31

1.7. Transitory response of dynamic system under stationary random excitation T.K. Caughey and H.J. Stumpf [CAU 61] analysed, for applications related to the study of structures subjected to earthquakes, the transitory response of a linear one-degree-of-freedom mechanical system subjected to a random excitation: - stationary, - Gaussian, - with zero mean, - of power spectral density G/Q). The solution of the differential equation for the movement

can be written in the general form

The input £(t) being Gaussian and the system linear, it was seen that the response u(t) is also Gaussian, with a distribution of the form

which requires knowledge of the mean and the standard deviation. The mean is given by

where ^(A,) = 0 since l(t) was defined as a function with zero average, yielding

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Fatigue damage

m(t) depends on u0 and u0. In addition,

T.K. Caughey and H.J. Stumpf [CAU 61] show that

where

If the function G(Q) is smoothed, presents no acute peak, and if ^ is small, s (t) can approach

Response of a one-degree-of-freedom linear system

33

where 9 = co0 t. It is noticed that, when 0 is very large, s (0) -> — o)0 Q G^(co 0 ) 2 which is the same as the expression [1.17]. For £, = Q, this expression becomes

The variance s (0) is independent of the initial conditions. Variations of the 1 2

quantity

2s

—- in terms of 0 are represented in Figure 1.8 for several values TKOO G^(co 0 )

of£.

Figure 1.8. Variance of the transitory response of a one dof system to a random vibration

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Fatigue damage

It is noted that, when £ increases, the curve tends more and more quickly towards a horizontal asymptote (lower because £ is larger). This asymptote corresponds to the steady state response, which is obtained during some cycles. NOTES. — The results are theoretically exact for a vibration of white noise type. They constitute a good approximation for a system slightly damped subjected to an excitation of which the spectral density is slightly variable close to the natural frequency of the system [LIN 67]. Particular cases < A

*(t)

,

.

G£

1. The excitation is an acceleration, ^(t) = -—— and G^(Q) = ——; yielding C00

OD0

and, for £ = 0

Being a Gaussian distribution, the probability that the response u(t) exceeds a given value, k s, at the time t is

where

Response of a one-degree-of-freedom linear system

35

while the probability that

du

du

[1.105]

(zero initial conditions). On this assumption, m = 0 and [1.106]

Figure 1.9 shows the variations of P with k.

Figure 1.9. Probability that the response exceeds k times the standard deviation

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Fatigue damage

Example Let us consider a random acceleration having a PSD of constant level GX0 = 1 (m/s2)2/Hz applied at time t = 0 to a simple system having as initial conditions

The simple system is linear, with one-degree-of-freedom, natural frequency f0 = 10 Hz and factor quality Q = 10. The mean m is stabilized to zero after approximately 1.5 s (Figure 1.10). The standard deviation s(t), which starts from zero at t = 0, tends towards a limit equal to the rms value z,rms (m is then equal to zero) of the stationary response z(t) of the simple system (Figure 1.11):

Figure 1.10. Mean of response versus time

Figure 1.11. Standard deviation versus time

The time pattern of probability for z(t) higher than 1.5 zrmstends in stationary mode towards a constant value P0 approximately equal to 6.54 10

Figure 1.12).

Response of a one-degree-of-freedom linear system

37

Figure 1.13 shows the variations with time of the probability that z(t) is higher than 1.5 s(t), a probability which tends towards the same limit P0 for sufficiently large t. Figures 1.14 and the 1.15 show the same functions as in Figures 1.12 and 1.13, plotted for k = 1, 2, 3 and 4.

Figure 1.12. Probability that the response is higher than 1.5 times its stationary rms value

Figure 1.13. Probability that the response is higher than 1.5 times its standard deviation

Figure 1.14. Probability that the response is higher than k times its standard deviation

Figure 1.15. Probability that the response is higher than k times its stationary rms value

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Fatigue damage

1.8. Transitory response of dynamic system under amplitude modulated white noise excitation Let us consider a non stationary random acceleration of the form [BAR 68] [1.107] where e(t) is a wide-band white noise, (t) is a signal of simple shape (rectangular wave, then half-sine wave) intended to modulate e(t) (envelope function); (t) is of the form 9(t) = 0 ( t ) with

elsewhere

elsewhere

The study of the response of a one-degree-of-freedom linear system to such a signal is concerned mostly with mechanical shock (the durations are sufficiently short that one can neglect the effects of fatigue). These transients are also used to simulate real environments of similar shape, such as earthquakes, blast of explosions, launching of missiles, etc. The fact of modulating the amplitude of the random signal led to reduction of the energy transmitted to the mechanical system for length of time T. R.L. Barnoski * f() At [BAR 65] [NBA 66] proposed use of the dimensionless time parameter 0 =

Q in which intervenes AT , the effective time interval in which the energy contained in the modulated pulse is equal to that of the stationary signal over the duration T. This parameter 6* is a function of the time necessary for the amplitude response of a 1 Q simple resonator to decrease to — times its steady state value, namely T0 = : e f0 [1.108]

Response of a one-degree-of-freedom linear system

[1.109]

dt

For a rectangular wave,

39

and for a half-sine wave,

From an analogue simulation, R.L. Barnoski and R.H. Mac Neal plotted the ratio P of the maximum response to the rms value of the response versus 0 , for values of the probability PM (probability that all the maxima of the response are lower than P times the rms value of the response). The oscillator is supposed to be at rest at the initial time.

Figure 1.16. Reduced maximum response versus the dimensionless time parameter

The curves obtained have all the same characteristic appearance: fast rise of p from the origin to 0* 1, then slow increase in P in terms of 0* (Figure 1.16). The curves plotted in the stationary case constitute a upper limit of the transitory case when the duration of the impulse increases. These results can thus make it possible to estimate the time necessary to effectively achieve stationarity in its response. For 0 > 1 (impulse duration long compared to the natural period of the system), the results of the stationary case can constitute a conservative envelope of P. For 0 large and a great number of response cycles, the response tends to becoming independent of Q factor. One can note in addition that, when the duration of the impulse increases and becomes long compared with the natural period of the resonator, the response peak tends to becoming independent of the shape of the modulation.

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Fatigue damage

Figure 1.17. Reduced maximum response to stationary and pulsed random excitation versus the dimensionless time parameter (rectangular envelope, Q = 20, f0 = 159 Hz) [BAR 65]

Chapter 2

Characteristics of the response of a one-degree-of-freedom linear system to random vibration

2.1. Moments of response of irregularity factor of response 2.1.1. Moments By definition,

with

Setting

we can write:

a

one-degree-of-freedom

linear

system:

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Fatigue damage

i.e.

or

[2.1]

For white noise,

[2.2]

The main contribution to the integral comes from the area around the natural frequency : me results obtained for white noise are a good approximation to actual cases where Gl varies little around 0. For noise with constant PSD between two frequencies f1 and f2,

[2.3] In being defined in Appendix A3 [LAL 94]. The most useful moments are M0, M2 and M 4 . Moment of order 0

[2.4]

If the excitation is a white noise, Gl(h) =constant, [ l o ( ) - lo^)] = 1 and

Characteristics of the response to random vibration

43

[2.5]

If the excitation is an acceleration,

and

yielding

[KRE 83] [LEY 65]: [2.6]

Moment of order 1 [2.7]

constant,

[2.8]

Knowing [LAL 94] that

it becomes, in the most usual case

[2.9]

or

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Fatigue damage

[2.10]

The relationship [2.9] can also be written, since arc tan x + arc tan

[CHA 72], [2.11]

[2.12]

The ratio

M1 — varies little with , so long as E, 0.1 and is then close to 1 Urms

(Figure 2.1).

Figure 2.1. First moment of the response of a one dof system versus its damping

The expression [2.12] can be written in the equivalent form [DER 79]:

Characteristics of the response to random vibration

45

[2.13]

Moment of order 2 [2.14]

If Gl= constant,

[2.15] or

[2.16] We find again the relationship

2.1.2. Irregularity factor of response to noise of constant PSD By definition, [2.17]

According to [2.3], we have, for noise of constant PSD,

46

Fatigue damage

[2.18]

since [2.19]

It was seen in addition that [2.5] yielding

and that

[2.20]

For white noise, M4 is infinite since its calculation assumes integration between zero and infinity of the quantity

which tends towards a constant (equal to 1) when function plotted for

Figure 2.2 shows this

Figure 2.2. Function to be integrated for calculation of fourth moment versus h

Characteristics of the response to random vibration

47

The integral thus does not converge and the parameter r is zero. The ratio h being also large possibly when f0 is small, the parameter r of the response tends in the same way towards zero when the natural frequency of the system decreases. NOTE. - The curves I 0 (h), I 2 (h) and I 4 ( h ) show that [CHA 72]: — if is small, these functions increase very quickly when h is close to 1, — I0 and I2 tend towards 1 when h becomes large, while I4 continues to increase. So ideal white noise can give a good approximation to wide-band noise for the calculation of zrms and zrms, but it cannot be used to approximate zrms. If is small and if h is large compared to unity, the functions I0 and I2 are roughly equal to 1.

Figure 2.3. Integrals I0, I2 and I4 versus h,

Figure 2.4. Integrals I0, I2 and I4 versus h,

2.1.3. Characteristics of irregularity factor of response If the PSD is constant between two frequencies f1 and f2, the parameter r varies with and f0 (via h). Let us take again the example of paragraph 1.5.3.1. The variations of the rms values zrms, zrms and zrms according to f0 (for = 0.01, 0.05 and 0.1) and according to , (for f0 = 10 Hz, 100 Hz and 1000 Hz) are shown on Figures 2.5 to 2.10.

48

Fatigue damage

Figure 2.5. Rms relative displacement of the response of a one dof system versus its natural frequency

Figure 2.6. Rms relative displacement of the response of a one dof system versus its damping factor

Figure 2.7. Rms relative velocity of response of a one dof system versus its natural frequency

Figure 2.8. Rms relative velocity of response of a one dof system versus its damping

Characteristics of the response to random vibration

Figure 2.9. Rms relative acceleration of response of a one dof system versus its natural frequency

49

Figure 2.10. Rms relative acceleration of response of a one dof system versus its damping

Figure 2.11. Rms absolute acceleration of Figure 2.12. Rms absolute acceleration of response of a one dof system versus its response of a one dof system versus natural frequency its damping

Figures 2.13 and 2.14 show the variations of the parameter r under the same conditions. It is observed from Figure 2.13 that, whatever f0, r —> 1 when —> 0. The distribution of maxima of the response thus tends, when tends towards zero, towards a Rayleigh distribution. When increases, r decreases with f0. Figure 2.14 underlines the existence of a limit independent of frequencies f0 < f1 and f0 > f2.

for the

50

Fatigue damage

Figure 2.13. Irregularity factor of the response of a one dof system versus its damping

Figure 2.14. Irregularity factor of the response of a one dof system versus its natural frequency

This example shows that: - r is closer to 1 when

is smaller and f0 is larger,

- when f0 becomes large compared to f2, r tends, whatever towards the same limiting value (0.749). When f0 -> the PSD input is completely transmitted and the signal response has the same characteristics as the input signal (input x, f response z). This result can be shown as follows. We saw that, when h = — is small (i.e. when f0 is large), we have

Characteristics of the response to random vibration

51

and 2 rms

In the same way, rms value of the first derivative of x rms value of th

value of the second derivative of x

Figure 2.15. Natural frequency greater than the upper limit of the noise

It is thus normal that r has as a limit the value

calculated from the PSD of the excitation. From the expression [6.50], Volume 3, we obtain

In addition, when f0 is small compared to f1, r tends towards a constant value r0 equal, whatever , to 0.17234 (for the values of f1 and f2 selected in this example). Indeed, for h large, i.e. for f0 small, we have

52

Fatigue damage

Figure 2.16. Natural frequency lower than the lower limit of the noise

Knowing that

it becomes

yielding

i.e., at first approximation,

Characteristics of the response to random vibration

53

[2.21] n = 0 since the excitation is a PSD of constant level G0 between f1 and f2. Thus

and

where xms

is the rms displacement of the excitation x(t), namely

It would be shown in the same way that zrms = xrms and zrms = xrms if xrms and x rms are the rms velocity and acceleration of the excitation

yielding, according to the above relationships: [2.22]

54

Fatigue damage

Example If f1 = 10 Hz and f2 =1000 Hz, we obtain r = 0.17234 Figure 2.17 summarizes these results.

Figure 2.17. Irregularity factor of response of a one dof system versus its natural frequency

Particular case where In this case, and for white noise, [2.23]

Let us set

[2.24]

Characteristics of the response to random vibration

55

Let us calculate the first terms: dh

While setting h = tg

we obtain

[2.25]

yielding [2.26]

[2.27]

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Fatigue damage

Example f1 = 10 Hz f2 = 1000 Hz

Figure 2.18 shows the variations of r with f0 for

= 1.

Figure 2.18. Irregularity factor of the response of a one dof system versus its natural frequency, for a damping factor equal to 1

Particular case where

= 0 (for a white noise) dh

[2.28]

Let us set [LAL 94]

dh

Appendix A3.2 gives the expressions for J0 and J2, as well as the formula of recurrence which allows J4 to be calculated.

Characteristics of the response to random vibration

57

NOTE. - Approximate calculation of r for a power spectral density with several peaks (for example, the case of a PSD obtained in the response of a several-degreesof-freedom system). G(f)

Figure 2.19. PSD with several peaks

G(f)

Figure 2.20. Equivalent 'box' spectrum

J. T. Broch [BRO 70a] transforms this power spectral density into equivalent boxes (the box spectrum containing the same amount of energy centered around the resonance frequencies, cf. paragraph 2.4.1) and shows that r can be approximated by:

[2.29]

where: fj = frequency of the first resonance fn = frequency ofn'h resonance

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Fatigue damage

energy ratio of the maximum responses to the resonances width of peak n at — 3 dB.

2.1.4. Case: band-limited noise If f1 = 0, the relationship [2.18] is written [CHA 72]: [2.30]

If

is small and if h2 is large compared to 1, we have I0

I2

1 yielding [2.31]

Figure 2.21. Comparison of the exact and approximate expressions of the irregularity factor of the response to a narrow band noise for a damping ratio equal to 0.01

Figure 2.22. Comparison of the exact and approximate expressions of the irregularity factor of the response to a narrow band noise for a damping ratio equal to 0.1

Characteristics of the response to random vibration

59

NOTE. - Under these conditions, we have [2.32] where

[2.33] and, if the noise is defined by an acceleration,

Figure 2.23. Constant PSD between two frequencies

[2.34] yielding, for [LEY 65]

with

in this particular case.

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Fatigue damage

2.2. Autocorrelation function of response displacement It is shown that this function is equal to [BAR 68]:

[2.35]

[2.36]

The duration (time or correlation interval) AT necessary that the envelope of Rz (T) decreases 1/e times its initial amplitude is given by [2.37]

2.3. Average numbers of maxima and minima per second If E[n + (a)l is the average number per second of crossings of a given threshold a with positive slope, the average number of peaks per second lying between a and a + da can be approximate by using the difference [POW 58]:

Figure 2.24. Minimum with positive amplitude

This method can lead to errors, since it supposes that the average number of minima per second (such as M in Figure 2.24) taking place with a positive amplitude

Characteristics of the response to random vibration

61

is negligible compared to the average number of peaks per second [LAL 92]. It is so for narrow band noise. To evaluate the validity of this approximation in the case of band-limited noise, J.B. Roberts [ROB 66] calculates, using the formulation of S.O. Rice: - the average number of peaks per second between a and a + da:

the average number of minima per second between a and a + da:

where f(a,b, c) is the joint probability density function of the random process. The function f(a, b, c)dadbdc is the probability that, at time t, the signal z(t) lies between a and a + da, its first derivative z(t) between b and b + db and its second derivative z(t) between c and c + dc, a maximum being defined by a zero derivative (Volume 3, paragraph 6.1).

[2.40]

[2.41] The ratio of the number of minima to the number of peaks is equal to [RIC 39]: [2.42]

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Fatigue damage

Figure 2.25. Ratio of the number of minima to the number of peaks

where

R is thus a function of the only parameter X, which depends on the reduced level and of parameter r. One notes in Figure 2.25 that R decreases when increases. If R is small, the process is within a narrow band. R is thus a measurement of the regularity of the oscillations. Figure 2.26 shows R varying with r, for

varying from 1 to 5.

Characteristics of the response to random vibration

Figure 2.26. Ratio of the number of minima to the number of peaks versus the irregularity factor

63

Figure 2.27. Ratio of the number of minima to the number of peaks, versus the threshold a

is larger. This

The ratio R decreases when r increases, and this faster when tendency is underlined in another form by the curves

for variable r.

2.4. Equivalence between transfer functions of bandpass filter and one-degreeof-freedom linear system There are several types of equivalences based on different criteria.

2.4.1. Equivalence suggested by D.M. Asplnwall The selected assumptions are as follows [ASP 63] [BAR 65] [SMI 64a]: - the bandpass filter and the linear one-degree-of-freedom system are supposed to let pass the same quantity of power in response to white noise excitation: the two responses must thus have same rms value, - the two filters have same amplification (respectively at the central frequency and the natural frequency).

64

Fatigue damage

Figure 2.28. Bandpass filter equivalent to the transfer function of a one dof system

The rms value of the response u(t) to the excitation l(t) is given by

For the mechanical system, we have for example

and for the bandpass filter:

The first assumption imposes PSD of the white noise

or

Characteristics of the response to random vibration

65

and the second

yielding

[2.43]

or [2.44]

This interval Af is sometimes named [NEW 75] mean square bandwidth of the mode. If the mechanical system has as a transfer function

we have, with the same assumptions

and

yielding

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Fatigue damage

[2.45] and, if Q is large, [2.46]

The error is lower than 10% for Q > 4.

2.4.2. Equivalence suggested by K. W. Smith We search width B of a rectangular filter centered on f0, of height Q, such that the area under the curve H2 (Q) is equal to that of the rectangle Am B (Am = Q) [SMI 64a].

Figure 2.29. Bandpass filter equivalent to the transfer function of a one dof system according to K. W. Smith

Knowing that the PSD Gu of the response is related to that of the excitation by we have, for white noise,

Characteristics of the response to random vibration

67

However

yielding [2.47] and, since we must have Am = Q, [2.48]

2.4.3. Rms value of signal filtered by the equivalent bandpass filter This is equal to

i.e.

If the vibration is an acceleration, we must replace

It then

becomes:

[2.49]

68

Fatigue damage

Example Let us consider a one-degree-of-freedom linear mechanical filter of natural frequency f0 =120 Hz and quality factor Q = 22, subjected to a white random noise of spectral density of acceleration G0 = 10 (m/s2)2 Hz between 1 Hz and 1000 Hz (rms value: x from [2.49] is equal to

100 m/s2). The rms value of the response calculated

It can be checked that the rms displacement zms obtained from relations [1.33] and [1.34] is in conformity with the value extracted from this result (i.e. z rms = 0-358 mm).

Chapter 3

Concepts of material fatigue

3.1. Introduction Fatigue phenomena, with formation and growth of cracks in machine elements subjected to repeated loads below ultimate strength was discovered during the XIXth century with the arrival of machines and freight vehicles functioning under dynamic loads larger than those encountered before [NEL 78]. According to H.F. Moore and J.B. Kommers [MOO 27], the first work published on failure by fatigue was by W Albert, a German mining engineer, who carried out in 1829 repeated loading tests on welded chains of mine winches. S.P. Poncelet was perhaps the first to use the term of fatigue in 1839 [TIM 53]. The most important problems of failure by fatigue were found about 1850 during the development of the European railroad (axes of car wheels). A first explanation was that metal crystallizes under the action of the repeated loads, until failure. This idea has as a source the coarsely crystalline appearance of many surfaces of parts broken by fatigue. This theory was disparaged by W.J. Rankine [RAN 43] in 1843. The first tests were carried out by Wohler between 1852 and 1869 [WOE 60]. The dimensioning of a structure to fatigue is more difficult than with static loads [LAV 69], because ruptures by fatigue depend on localised stresses. Since the fatigue stresses are in general too low to produce a local plastic deformation and a redistribution associated with the stresses, it is necessary to carry out a detailed analysis which takes into account at the same time the total model of the stresses and the strong localised stresses due to the concentrations.

70

Fatigue damage

On the other hand, analysis of static stresses requires only the definition of the total stress field, the high localised stresses being redistributed by local deformation. Three fundamental steps are necessary: - definition of the loads, - detailed analysis of the stresses, - consideration of the statistical variability of the loads and the properties of materials. Fatigue damage depends strongly on the oscillatory components of the load, its static component and the order of application of the loads. Fatigue can be approached in several ways and in particular by: -the study of the Wohler's curves (S-N curves), - study of cyclic work hardening (low cycle fatigue), - study of the crack propagation rate (fracture mechanics). The first of these approaches is the most used. We will present some aspects of them in this chapter. 3.2. Damage arising from fatigue One defines as fatigue damage the modification of the characteristics of a material, primarily due to the formation of cracks and resulting from the repeated application of stress cycles. This change can lead to a failure. We will not consider here the mechanisms of nucleation and growth of the cracks. We will say simply that fatigue starts with a plastic deformation first very much localized around certain macroscopic defects (inclusions, cracks of manufacture etc) under total stresses which can be lower than the yield stress of material. The effect is extremely weak and negligible for only one cycle. If the stress is repeated, each cycle creates a new localised plasticity. After a number of variable cycles, depending on the level of the applied stress, ultra-microscopic cracks can be formed in the newly plastic area. The plastic deformation extends then from the ends of the cracks which increase until becoming visible with the naked eye and lead to failure of the part. Fatigue damage is a cumulative phenomenon.

Concepts of material fatigue

71

Figure 3.1. Unclosed hysteresis loop

If the stress-strain cycle is plotted, the hysteresis loop thus obtained is an open curve whose form evolves depending on the number of applied cycles [FEO]. Each cycle of stress produces a certain damage and the succession of the cycles results in a cumulative effect. The damage is accompanied by modifications of the mechanical properties and in particular of a reduction of the static ultimate tensile strength Rm and of the fatigue limit strength. It is in general localised at the place of a geometrical discontinuity or a metallurgical defect. The fatigue damage is related at the same time to metallurgical and mechanical phenomena, with appearance and cracks growth depending on the microstructural evolution and mechanical parameters (with possibly the effects of the environment). The damage can be characterized by: - evolution of a crack and energy absorption of plastic deformation in the plastic zone which exists at the ends of the crack, - loss of strength in static tension, - reduction of the fatigue limit stress up to a critical value corresponding to the failure, - variation of the plastic deformation which increases with the number of cycles up to a critical value. One can suppose that fatigue is [COS 69]: - the result of a dynamic variation of the conditions of load in a material,

72

Fatigue damage

- a statistical phenomenon, - a cumulative phenomenon, - a function of material and of amplitude of the alternating stresses imposed on material. There are several approaches to the problem of fatigue: - establishment of empirical relations taking account of experimental results, - by expressing in an equation the physical phenomena in the material, beginning with microscopic cracks starting from intrusion defects, then propagation of these microscopic cracks until macro-cracks and failure are obtained. R. Cazaud, G. Pomey, P. Rabbe and C. Janssen [CAZ 69] quote several theories concerning fatigue, the principal ones being: - mechanical theories: - theory of the secondary effects (consideration of the homogeneity of the material, regularity of the distribution of the effort), - theory of hysteresis of pseudo-elastic deformations (discussion based on the Hooke's law), - theory of molecular slip, - theory of work hardening, - theory of crack propagation, - theory of internal damping, - physical theories, which consider the formation and propagation of the cracks using models of dislocation starting from extrusions and intrusions, - static theories, in which the stochastic character of the results is explained by the heterogeneity of materials, the distribution of the stress levels, cyclic character of loading etc, - theories of damage, - low cycle plastic fatigue, in the case of failures caused by cycles N less in number than 104 approximately. The estimate of the lifetime of a test bar is carried out from: - a curve characteristic of the material, which gives the number of cycles to failure according to the amplitude of stress, in general sinusoidal with zero mean (SN curve), - of a law of accumulation of damages.

Concepts of material fatigue

73

The various elaborated theories are distinguished by the selected analytical expression to represent the curve of damage and by the way of cumulating the damages. To avoid failures of parts by fatigue dimensioned in statics and subjected to variable loads, one was first of all tempted to adopt arbitrary safety factors, badly selected, insufficient or too large, leading to excessive dimensions and masses. An ideal design would require use of materials in the elastic range. Unfortunately, the plastic deformations always exist at points of strong stress concentration. The nominal deformations and stresses are elastic and linearly related to the applied loads. This is not the case for stresses and local deformations which exist in metal at critical points and which control resistance to fatigue of the whole structure. It thus proved necessary to carry out tests on test bars for better apprehension of the resistance to fatigue, under dynamic load, while starting with the simplest, the sinusoidal load. We will see in the following chapters how the effects of random vibrations, most frequently met in practice, can be evaluated. 3.3. Characterization of endurance of materials 3.3.1. S-N curve The endurance of materials is studied in laboratory by subjecting to failure test bars cut in the material to be studied to stresses of amplitude (or deformations), generally sinusoidal with zero mean. Following the work of W6hler(1) (1860/1870) [WOE 60] [WOH 70] carried out on axes of trucks subjected to rotary bending stresses, one notes, for each test bar, depending on a, the number N of cycles to failure (endurance of the part or fatigue life). The curve obtained while plottingCTagainst N is termed the S-N curve (stressnumber of cycles) or Wohler 's curve or endurance curve. The endurance is thus the aptitude of a machine part to resist fatigue. Taking into account the great variations of N with a, it is usual to plot log N (decimal logarithm in general) on abscissae. Logarithmic scales on abscissae and on ordinates are also sometimes used.

1. Sometimes written Woehler

74

Fatigue damage

Figure 3.2. Main zones of the S-N curve

This curve is generally composed of three zones [FAC 72] [RAB 80] (Figure 3.2): - zone AB, corresponding to low cycle fatigue, which corresponds to the largest stresses, higher than the yield stress of material, where N varies from a quarter of cycle with approximately 104 to 105 cycles (for mild steels). In this zone, one very quickly observes significant plastic deformation followed by failure of the test bar. The plastic deformation _ can be related here to the number of cycles to the failure by a simple relationship of the form

[3.1] where the exponent k is near to 0.5 for common metals (steels, light alloys) (L.F. Coffin [COF 62]), - zone BC, which is often close to a straight line on log-lin scales (or sometimes on log-log scales), in which the fracture certainly appears under a stress lower than previously, without appearance of measurable plastic deformation. There are very many relationships proposed between a and N to represent the phenomenon in this domain where N increases and when cr decreases. This zone, known as zone of limited endurance, is included between approximately 104 cycles and 106 to 107 cycles, - zone CD, where D is a point which, for ferrous metals, is ad infmitum. The SN curve in general presents a significant variation of slope around 106 to 107 cycles, followed in a way more or less marked and quickly by a zone (CD) where the curve tends towards a limit parallel with the N axis. On this side of this limit the value of a, noted D, there is never failure by fatigue whatever the number of cycles applied.

Concepts of material fatigue

75

called fatigue limit. D is thus the stress with zero mean of greater amplitude Dis for which one does not observe failure by fatigue after an infinite number of cycles. This stress limit cannot exist or can be badly defined for certain materials [NEL 78] (high-strength steels, non-ferrous metals). For sufficiently resistant metals for which it is not possible to evaluate the number of cycles which the test bar would support without damage [CAZ 69] (too large test duration) and to take account of the scatter of the results, the concept of conventional fatigue limit or endurance limit is introduced. It is about the greatest amplitude of stressCTfor which 50% of failures after N cycles of stress is observed. It is noted, for m = 0, D (N). N can vary between 106 and 108 cycles [BRA 80a] [BRA 80b]. For steels, N = 10 and

D (l0

I

D.

The notation

D

is in this case

used.

NOTES.Brittle materials do not have a well defined fatigue limit [BRA 81] [FID 75]. For extra hardened tempered steels, certainly for titanium, copper or aluminium alloys, or when there is corrosion, this limit remains theoretical and without interest since the fatigue life is never infinite. When the mean stress m is different from zero, it is important to associate m with the amplitude of the alternating stress. The fatigue limit can be written a or in this case. aD

Figure 3.3. Sinusoidal stress with zero (a) and non zero mean (b)

Definition The endurance ratio is the ratio of the fatigue limit to the ultimate tensile strength R of material:

D

(normally at 107 cycles)

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Fatigue damage

[3.2]

NOTE. - The S-N curve is sometimes plotted on reduced scales in axes /R m , N), in order to be able to proceed more easily to comparisons between different materials.

Figure 3.4. S-N curve on reduced axes

3.3.2. Statistical aspect The S-N curve of a material is plotted by successively subjecting ten (or more) test bars to sinusoidal stresses of various amplitudes. The results show that there is a considerable scatter in the results, in particular for the long fatigue lives. For a given stress level, the relationship between the maximum value and the minimal value of the number of cycles to failure can exceed 10 [LAV 69] [NEL 78]. The dispersion of the results is related on the heterogeneity of materials, the surface defects, the machining tolerances and especially to metallurgical factors. Among these factors, inclusions are most important. Scatter is due in fact because the action of fatigue in a metal is in general strongly localised. Contrary to the case of static loads, only a small volume of material is concerned. The rate of fatigue depends on the size, on the orientation and on the chemical composition of some material grains which are located in a critical zone [BRA 80b] [LEV 55] [WIR 76]. In practice, it is not thus realistic to want to characterize the resistance to fatigue of a material by a S-N curve plotted by carrying out only one fatigue test at each stress level. It is more correct to describe this behaviour by a curve in a statistical manner, the abscissas giving the endurance Np for a survival of p percent of the test bars [BAS 75] [COS 69].

Concepts of material fatigue

77

The median endurance curve (or equiprobability curve) (N 50 , i.e. survival of 50% of the test bars), or sometimes the median curves ±1 to 3 standard deviations or other isoprobability curves is in general given [ING 27].

Figure 3.5. Isoprobability S-N curves

Without other indication, the S-N curve is the median curve. NOTE. - The scatter of fatigue life of non-ferrous metals (aluminium, copper etc) is less than that of steels, probably because these metals have fewer inclusions and inhomogeniuties.

3.3.3. Distribution laws of endurance For high stress levels, endurance N follows a log-normal law [DOL 59] [IMP 65]. In other words, in scales where the abscissa carry log N, the distribution of log N follows, in this stress domain, a roughly normal law (nearer to the normal law when a is higher) with a scatter which decreases when a increases. M. Matolcsy [MAT 69] considers that the standard deviation s can be related to the fatigue life at 50% by an expression of the form

[3.3] where A and (3 are constants functions of material.

78

Fatigue damage

Example Table 3.1. Examples of values of the exponent P

Aluminium alloys

1.125

Steels

1.114to 1.155

Copper wires

1.160

Rubber

1.125

G.M. Sinclair and T.J. Dolan [SIN 53] observed that the statistical law describing the fatigue lives is roughly log-normal and that the standard deviation of the variable (log N) varies according to the amplitude of the applied stress according to an exponential law.

Figure 3.6. Log-normal distribution of the fatigue lives

In the endurance zone, close to D, F. Bastenaire [BAS 75] showed that the inverse 1/N of endurance follows a modified normal law (with truncated tail). Other statistical models were proposed, such as, for example [YAO 72] [YAO 74]: -

the normal law [AST 63], extreme value distribution, Weibull's law [FRE 53], the gamma law [EUG 65].

Concepts of material fatigue Table 3.2. Examples of values of the variation coefficient of the number of cycles to failure

Authors I.C. Whittaker P.M. Besuner

[WHI 69a]

T. Endo [END 67] J.D. Morrow [WIR 82]

Materials

Conditions

Steels Rm < 1650 MPa(240ksi)

36

Steels Rm > 1650MPa (240 ksi)

48

Aluminium alloy

27

Alloy titanium

36 (Lognormal) 14.7

Steel 4340 7075-T6 2024-T4

Low cycle fatigue (N < 103)

Steel SAE 1006 Maraging steel 200 grade Maraging steel Nickel 18%

T.R. Gurney [GUR 68]

Welded structures

17.6 19.7 65.8

Titanium 8 1 1 S.R. Swanson [SWA 68]

V N (%)

Fatigue under narrow band random noise

25.1

Mean

52

38.6 69.0

79

80

Fatigue damage Table 3.3. Examples of values of the variation coefficient of the number of cycles to failure Authors

Materials

Conditions

V N (%)

R. Blake W.S. Baird[BLA69]

Aerospace components

Random loads

3 to 30

Epremian Mehl [EPR 52]

Steels Log-normal law

A.H.S. Ang [Ang 75] W.H. Munse

Welding

52

I.C. Whittaker [WHI 72]

Steel UTS < 1650 MPa Steel UTS > 1650 MPa

36 48

Aluminium alloys Titanium alloys

22 36

P.H. Wirsching [WIR 83b]

Welding (tubes)

70 to 150

P.H. Wirsching Y.T. Wu [WIR 83c]

RQC - 100 Q

Waspaloy B Super alloys Containing Nickel

s

m

log/ log

2.04 to 8.81 Lognormal law

Plastic strain

15 to 30

Elastic strain

55

Plastic strain

42

Elastic strain

55

VN , often about 30% to 40%, can reach 75% and even exceed 100%. P.H. Wirsching [WIR 83a]

Low cycle fatigue field: 20% to 40% for the majority of metal alloys. For N large, VN can exceed 1 00%.

T Yokobori [YOK 65]

Steel Rotational bending or traction compression

28 to 130

T.J. Dolan [DOL 52] H.F. Brown

Aluminium alloy 7075. T6 Rotational bending

44 to 80

G.M. Sinclair T.J. Dolan [SIN 53]

Aluminium alloy 75.S-T Rotational bending

10 to 100

J.C. Levy [LEV 55]

Mild steel Rotational bending

43 to 75

Concepts of material fatigue I. Konishi [KON 56] M. Shinozuka

Notched plates - Steel SS41 Alternate traction

18 to 43

M. Matolcsy [MAT 69]

Synthesis of various test results

20 to 90

S. Tanaka S. Akita [TAN 72]

Silver/nickel wires Alternating bending

16 to 21

81

P.H.Wirshing [WIR 81] however checked, from a compilation of various experimental results that, for welded tubular parts, the log-normal law is that which adapts best. It is this law which is most often used [WIR 81]; it has the following advantages in addition: - well defined statistical properties, - easy to use, - adapts to great coefficient variations. Tables 3.2 and 3.3 give as an indication values of the variation coefficient of the number of cycles to failure of some materials noted in the literature [LAL 87]. Value 0.2 of the standard deviation (on log N) is often used for the calculation of the fatigue lives (for notched or other parts) [FOR 61] [LIG 80] [LUN 64] [MEH 53].

3.3.4. Distribution laws of fatigue strength Another way of getting to the problem can consist of studying the fatigue strength of the material [SCH 74], i.e. the stress which the material can resist during N cycles. This strength has also a statistical character and strength to p percent of survival and a median strength are defined here too.

Figure 3.7. Gaussian distribution of fatigue strength

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Fatigue damage

The curve representing the probability of failure during a test for which duration is limited to N cycles depending on the stress a is named response curve [CAZ 69] [ING 27]. The experiment shows that the fatigue strength follows a roughly normal law whatever the value of N and is fairly independent of N [BAR 77]. This constancy is masked on the S-N diagrams by the choice of the log-lin or log-log scales, scatter appearing to increase with N. Some values of the variation coefficient of this law for various materials, extracted from the literature, are given in reference [LAL 87]. Table 3.4. Examples of values of the variation coefficient of the endurance strength for given N Authors

Materials

Parameter

V N (%)

J.C. Ligeron [LIG 80]

Steels Various alloys

Fatigue limit stress

4.4 to 9.4

T Yokobori [YOK 65]

Mild steel

Fatigue limit stress

2.5 to 11.3

E Mehle [MEH 53]

Steel SAE 4340

R.F. Epremian [WIR 83a]

Large variety of metallic materials

20 to 95

Endurance stress (failure for given N )

5 to 15

For all the metals, J.E. Shigley [SHI 72] proposes a variation coefficient D (ratio of the standard deviation to the mean) equal to 0.08 [LIG 80], a value which can be reduced to 0.06 for steels [RAN 49].

3.3.5. Relation between fatigue limit and static properties of materials Some authors tried to establish empirical formulae relating the fatigue limit D and its standard deviation to the mechanical characteristics of the material (Poisson coefficient, Young's modulus etc). For example, for steels, the following relations were proposed [CAZ 69] [LIE 80] (Table 3.5). A. Brand and R. Sutterlin [BRA 80a] noted after exploitation of a great number of fatigue tests (rotational bending, on not notched test bars) that the best correlation between D and a mechanical strength parameter is that obtained with the ultimate strength Rm (tension):

Concepts of material fatigue

83

All these relations correctly represent only the results of the experiments which made it possible to establish them and are therefore not general. A. Brand and R. Sutterlin [BRA 80a] tried however to determine a more general relation, independent of the size of the test bars and stress, of the form:

where a and b are related to is the real fatigue limit related to the nominal fatigue limit

stress concentration factor, X is the stress gradient, defined as the value of the slope of the tangent of the stress field at the notch root divided by the maximum value of the stress at the same point

84

Fatigue damage Table 3.5. Examples of relation between the fatigue limit and the static properties of materials

Houdremont and Mailander

Re= yield stress Rm= ultimate stress

Lequis, Buchholtz and Schultz

A% = lengthening, in percent. a proportional to Rm and (3 inversely proportional

Fry, Kessner and Ottel Heywood

Brand (to 15% near)

Lieurade and Buthod [LIE 82]

S = striction, expressed in%

Juger Rogers Mailander

Stribeck In all the above relations, Steel, bending: Feodossiev

Very resistant steels: Non-ferrous metals:

and

are expressed in

Concepts of material fatigue

85

Variation coefficient

v is independent of Rm. A. Brand et al [BRA 80a] advises to take 10 %.

3.3.6. Analytical representations ofS-N curve Various expressions have been proposed to describe the S-N curve representative of the fatigue strength of a material, often in the limited endurance domain (the definition of this curve having besides evolved over the years from a deterministic curve to a curve of statistical character).

Figure 3.8. Representation of the S-N curve in semi-logarithmic scales

The S-N curve is in general plotted in semi-logarithmic scales log N, a, in which it presents a roughly linear part (around an inflection point), variable according to material (BC), followed by an asymptote to the straight line a = aD. Among the many more or less complicated representations and of which none is really general, the following can be found [BAS 75] [DEN 71] [LIE 80].

3.3.6.1. Wohler relation (1870)

This relation does not describe the totality of the curve since a does not tend towards a limit GD when N -> oo [HAI 78]. It represents only the part BC. It can be also written in the form:

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Fatigue damage

3.3.6.2. Basquin relation The relation suggested by Basquin in 1910 is of the form or [BAS 10]

while setting:

The parameter b is sometimes named index of the fatigue curve [BOL 84].

Figure 3.9. Significance of the parameter b of Basquin's relation

In these scales, the curve can be entirely linearized (upwards) by considering the amplitudes of the true stresses (and neither nominal). The expression [3.7] can also be written:

or

(TRF being the fatigue strength coefficient. This expression is generally valid for 4 high values of N (> 10 ). If there is a non zero c0 mean stress, constant C must be replaced by:

Concepts of material fatigue

87

where C is the constant used when a0 = 0 and Rm is the ultimate strength of the material [WIR 83a]. In the expression N CT = C, the stress tends towards zero when N tends towards the infinite. This relation is thus representative of the S-N curve only in its part BC. In addition, it represents a straight-line in logarithmic scales and not in semilogarithmic scales (log-lin). A certain number of authors presented the results of the fatigue tests in these scales (log-log) and showed that part BC is close to a straightline [MUR 52]. F.R. Shanley [SHA 52] considers in particular that it is preferable to choose these scales. H.P. Lieurade [LIE 80] note as for him that the representation of Basquin is appropriate less well that the relation of Wohler in the intermediate zone and that it is not better around the fatigue limit. It however is very much used. To take account of the stochastic nature of this curve, P.H. Wirsching [WIR 79] proposes to treat constant C of this relation like a log-normal random variable of mean C and standard deviation CTC and gives the following values, in the domain of the great numbers of cycles: -median: 1.55 1012 (ksi)(2), - variation coefficient: 1.36 (statistical study of S-N curves relating to connections between tubes).

Some numerical values of the parameter b In Basquin's relationship Metals The range of variation of b is between 3 and 25. The most common values are between 3 and 10 [LEN 68]. M. Gertel [GER 61] [GER 72] and C.E. Crede, E.J. Lunney [CRE 56b] consider a value of 9 to be representative of most materials. It is probably a consideration of this order that led to the choice of 9 by such standards as MIL-STD-810, AIR, etc. This choice is satisfactory for most light alloys and copper but may be unsuitable for other materials. For instance, for steel, the value of b varies between 10 and 14 depending on the alloy. D.S. Steinberg [STE 73] mentions the case of 6144-T4 aluminum alloy for which b = 14 (Na 1 4 =2.26 1078). b is near to 9 for ductile materials and near to 20 for brittle materials, whatever the ultimate strength of the material [LAM 80].

2. 1 ksi = 6.8947 MPa

88

Fatigue damage Table 3.6. Examples of values of the parameter b [DEI 72] a

Material

Type of fatigue test

min

b

CT

max

2024-T3 aluminium 2024-T4 aluminium 7075-T6 aluminium 6061-T6 aluminium ZK-60 magnesium BK31XA-T6 magnesium QE 22-T6 magnesium 4 130 steel Standardized Hardened 6A1-4V Ti Beryllium Hot pressed Block

Axial load Rotating beam Axial loading Rotating beam

-1

Axial load Rotating beam Wohler

0.25

Axial load Axial load Axial load

-1

Axial load

-1 -1 _]

-1 -1

-1 -1

4.9

0

10.8 8.7

0.2

Anneal copper 1S1 fiberglass

Axial load Axial load

7.0 4.8 8.5 5.8 3.1 4.5 4.1

-1

Cross Rol Sheat Invar

5.6 6.4 5.5

0.2

12.6 9.4 4.6 11.2 6.7

The lowest values indicate that the fatigue strength drops faster when the number of cycles is increased, which is generally the case for the most severe geometric shapes. The lower the stress concentration, the higher the value of parameter b. Table 3.6 gives the value of b for a few materials according to the type of load applied: tension-compression, torsion, etc. and the value of the mean stress, i.e. the

Concepts of material fatigue

89

A few other values are given by R.G. Lambert [LAM 80] with no indication of the test conditions. Table 3.7. Examples of values of the parameter b [LAM80]

Material

b

Copper wire

9.28

Aluminium alloy 6061-T6

8.92

7075-T6

9.65

Soft solder (63-37 Tin - Lead)

9.85

4340 (BHN 243)

10.5

4340 (BHN 350)

13.2

AZ3 1 B Magnesium alloy

22.4

Figure 3.10. Examples of values of the parameter b [CAR 74]

It is therefore necessary to be very cautious in choosing the value of this parameter, especially when reducing the test times for constant fatigue damage testing. Case of electronic components The failures observed in electronic components follow the conventional fatigue failure model [HAS 64]. The equations established for structures are therefore applicable [BLA 78]. During initial tests on components such as capacitors, vacuum tubes, resistors, etc. and on equipment, it was observed that the failures (lead

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Fatigue damage

breakage) generally occurred near the frame resonance frequencies, generally below 500 Hz [JAC 56]. The analysis of tests conducted on components by D.L.Wrisley and W.S. Knowles [WRI] tends to confirm the existence of a fatigue limit. Electronic components could be expected a priori to be characterized by a parameter b of around 8 or 9 for fatigue strength, at least in the case of discrete components with copper or light alloy leads. That is moreover the value chosen by some authors [CZE 78]. Few data have been published on the fatigue strength of electronic components. C.A. Golueke [GOL 58] gives S-N curves plotted from the results of fatigue testing conducted at resonance on resistors, for setups such that the resonance frequency is between 120 Hz and 690 Hz. Its results show that the S-N curves obtained for each resonance are roughly parallel. On log N - log x scales (acceleration), parameter b is very close to 2. Components with the highest resonance frequency have the longest life expectancy, which demonstrates the interest of decreasing the component lead length to a minimum.

Figure 3.11. Examples of S-N curves of electronic components [GOL 58]

This work, already old (1958), also reports that the most fragile parts as regards fatigue strength are the soldered joints and interconnections, followed by capacitors, vacuum tubes, relays to a much lesser extent, transformers and switches. M. Gertel [GER 61] [GER 62] writes the Basquin relation N ab = C in the form

where aD is the fatigue limit. If the excitation is sinusoidal [GER 61] and if the structure, comparable to a one-degree-of-freedom system, is subjected to tensioncompression, the movement of mass m is such that

Concepts of material fatigue

91

a = stress in the part with section S. If the structure is excited at resonance, we have: and

Knowing that the specific damping energy D is related to the stress by

and that the Q factor can be considered as the product

dimensionless factor of the material, where E is Young's modulus, and of Kv, the dimensionless volumetric stress factor, we obtain

yielding, if

and

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Fatigue damage

we obtain

yielding the value of the parameter b of resistors in N - a axes (instead of N, x): b = 2 (n-1)

[3.18]

which, for n = 2.4 , is equal to 2 x 1.4 = 2.8. These low values of b are confirmed by other authors [CRE 56b] [CRE 57] [LUN 58]. Some relate to different component technologies [DEW 86]. Among the published values are, for instance, C.E. Crede's [GER 61] [GER 62]: Resistors: b = 2.4 to 5.8 Vacuum tubes: b = 0.6 and those of E.J. Lunney, C.E. Crede [CRE 56b]: Capacitors: b = 3.6 (leads) Vacuum tubes: b = 2.83 to 2.13. 3.3.6.3. Some other laws Stromeyer (1914)

or

or

Here, a tends towards CTD when N tends towards the infinite.

Concepts of material fatigue

93

Palmgren (1924) [PAL 24]

or

a relation which is better adjusted using experimental curves than with Stromeyer's relation. Weibull (1949)

where Rm is the ultimate strength of studied material. improve the preceding one. It can be also written:

This relation does not

where F is a constant and A is the number of cycles (different from 1/4) corresponding to the ultimate stress [WEI 52]. It was used in other forms, such as:

with n = 1 (Prot [PRO 48]) n = 2 (Ferro and Rossetti [PER 55]) and

where a and b are constants (Stussi [FUL 63]). Corson (1949) [MIL 82]

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Fatigue damage

Bastenaire [BAS 75]

3.4. Factors of influence 3.4.1. General There is a great number of parameters have a bearing on fatigue strength and thus the S-N curve. The fatigue limit of a test bar can thus be expressed in the form [SHI 72]: a D = K sc K s K e K f K r K v o' D

[3.30]

where CT'D is the fatigue limit of a smooth test bar and where the other factors make it possible to take into account the main following effects: KoovP

scale effect

Ks Ke

surface effect temperature effect

Kf

form effect (notches, holes etc)

Kr

reliability effect

K

various effects (loading rate, type of load, corrosion, residual stresses, stress frequency etc)

These factors can be classified as follows [MIL 82]: - factors depending on the conditions of load (type of loads: tension/compression, alternating bending, rotational bending, alternating torsion etc), - geometrical factors (scale effect, shape etc), - factors depending on the conditions of surface, - factors of a metallurgical nature, - factors of environment (temperature, corrosion etc). We examine below some of these parameters.

Concepts of material fatigue

95

3.4.2. Scale For rather obvious reasons of simplicity and cost, the tests of characterization of strength to fatigue are carried out on small test bars. The tacit and fundamental assumption is that the damage processes apply both to the test bars and the complete structure. The use of the constants determined with test bars for the calculation of large-sized parts supposes that the scale factor has little influence. A scale effect can appear when the diameter of the test bar is increased, involving an increase in the volume of metal and the surface of the part concerned, and thus an increase in the probability of cracking. This scale effect has as origins: - mechanics: existence of a stress gradient in the surface layers of the part, variable with its dimensions, weaker for the large parts (case of the nonuniform loads, such as torsion or alternating bending), - statistics: larger probability of existence of defects being able to start microscopic cracks in the large parts, - technological: surface quality, material heterogeneity. It is noted in practice that the fatigue limit is smaller when the test bar used is larger. With equal nominal stress, the more dimensions of a part increase, the more its fatigue strength decreases [BRA 80b] [BRA 81] [EPR 52]. B.N. Leis [LEI 78], then B.N. Leis and D. Broek [LEI 81] showed that, on the condition of taking some precautions to make sure that similarity is respected strictly at the critical points (notch root, crack edges etc), precise structure fatigue life predictions can nonetheless be made from laboratory test results. However, respect of this similarity is sometimes difficult to obtain, for lack of correct comprehension of the factors controlling the process of damage rate. 3.4.3. Overloads We will see that the order of application of loads of various amplitudes is an important parameter. It is observed in practice that: 1. For a smooth test bar, the effect of an overload leads to a reduction in the fatigue life. J Kommers [KOM 45] shows that a material which was submitted to significant over-stress, then to under-stress, can break even if the final stress is lower than the initial fatigue limit, because the over-stress produces a reduction in the initial fatigue limit. By contrast, an initial under-stress increases the fatigue limit [GOU 24]. J.R. Fuller [FUL 63] noted that the S-N curve of a material having undergone an overload turns in the clockwise direction with respect to the initial S-N curve,

96

Fatigue damage

around a point located on the curve having for ordinate the amplitude GJ of the overload.

Figure 3.12. Rotation of the S-N curve of a material which underwent an overload (J.R. Fuller [FUL 63J)

The fatigue limit is reduced. If after nl cycles on the level GJ , n2 cycles are carried out on the level a2, the new S-N curve takes the position (D. n i Rotation is quantitatively related to the value of the ratio — on the over-stress NI level GJ. J.R. Fuller defines a factor of distribution which can be written, for two load levels,

where q is a constant, equal to 3 in general (notch sensitivity of material to fatigue to the high loads) NA = number of cycles on the highest level CTA Na = number of cycles on the lower level aa If P = 1, all the stress cycles are carried out at the higher stress level (Na = 0). This factor P allows characterizing of the distribution of the peaks between the two limits CJA and cra and is used to correct the fatigue life of the test bars calculated under this type of load. It can be used for a narrow band random loading.

Concepts of material fatigue

97

2. For a notched test-bar, of which most of the fatigue life is devoted to the propagation of the cracks by fatigue, this same effect led to an increase in the fatigue life [MAT 71]. Conversely, an initial under-load accelerates cracking. This acceleration is all the more significant since the ensuing loads are larger. In the case of random vibrations, they are statistically not very frequent, and of short duration and the under-load effect can be neglected [WEI 78]. 3.4.4. Frequency of stresses The frequency, within reasonable limits of variation, is not important [DOL 57]. It is considered in general that this parameter has little influence so long as the heat created in the part can be dissipated and a heating does not occur which would affect the mechanical characteristics (stresses are considered here as directly applied to the part with a given frequency. It is different when the stresses are due to the total response of a structure involving several modes [GRE 81]). An assessment of the influence of the frequency showed that [HON 83]: - the results published are not always coherent, particularly because of corrosion effects, - for certain materials, the frequency can be a significant factor when it varies greatly, acting differently depending on materials and load amplitude, - its effect is much more significant at high frequencies. For the majority of steels and alloys, it is negligible for f < 117 Hz. In the low number of cycles fatigue domain, there is a linear relation between the fatigue life and the frequency on logarithmic scales [ECK 51]. Generally observed are: - an increase in the fatigue limit when the frequency increases, - a maximum value of fatigue limit at a certain frequency. For specific treatment of materials, unusual effects can be noted [BOO 70] [BRA 80b] [BRA 81] [ECK 51] [FOR 62] [FUL 63] [GUR 48] [HAR 61] [JEN 25] [KEN 82] [LOM 56] [MAS 66a] [MAT 69] [WAD 56] [WEB 66] [WHI 61]. I. Palfalvi [PAL 65] showed theoretically the existence of a limiting frequency, beyond which the thermal release creates additional stresses and changes of state. The effect of frequency seems more marked with the large numbers of cycles and decreases when the stress tends towards the fatigue limit [HAR 61]. It becomes paramount in the presence of a hostile environment (for example, corrosive medium, temperature) [LIE 91].

98

Fatigue damage

3.4.5. Types of stresses The plots of the S-N curves are in general carried out by subjecting test bars to sinusoidal loads (tension and compression, torsion etc) with zero mean. It is also possible to plot these curves for random stress or even by applying repeated shocks.

3.4.6. Non zero mean stress Unless otherwise specified, it will be supposed in what follows that the S-N curve is defined by the median curve. The presence of a non zero mean stress modifies the fatigue life of the test bar, in particular when this mean stress is relatively large compared to the alternating stress. A tensile mean stress decreases the fatigue life, a compressive stress increases it. Since the amplitudes of the alternating stresses are relatively small in the fatigue tests with a great number of cycles, the effects of the mean stress are more important than in the tests with a low number of cycles [SHI 83]. If the stresses are enough large to produce significant repeated plastic strains, as in the case of fatigue with a small number of cycles, the mean strain is quickly released and its effect can be weak [TOP 69] [VAN 72].

Figure 3.13. Sinusoidal stress with non zero mean

When the mean stress am is different from zero, the sinusoidal stress is generally characterized by two parameters chosen from aa, crmax, crmin and R

= a min/ C T max-

Concepts of material fatigue

Figure 3.14. Representation of the S-N curves with non zero mean versus R

99

Figure 3.15. Representation of the S-N curves with non zero mean versus the mean stress

Although this representation is little used, it is possible to use the traditional representation of the S-N curves with on the abscissa the logarithm of the number of cycles to failure and on the ordinate the stress crmax, the curves being plotted for different values of am or R [FID 75] [SCH 74]. Other authors plot S-N curves giving aa versus N, for various values of am and propose empirical relations between constants C and b of the Basquin's relation ( N < y b = C ) a n d a m [SEW 72]:

Figure 3.16. Example of S-N curves with non zero mean

100

Fatigue damage

Example Aluminium alloy: log,0 C - 9.45982 - 2.37677 am +1.18776CT^- 0.25697 a^ b = 3.96687 - 0.213676 am - 0.04786 a^ + 0.00657 G3m (am in units of 10 ksi)(^)

It is generally agreed to use material below its yield stress (a max < Re) only, which limits the influence of am on the life. The application of static stress led to a reduction in aa (for a material, a stress mode and a given fatigue life). It is thus interesting to know the variations of aa with am. Several relations or diagrams were proposed to this end. For tests with given am, one can make correspond a value of the fatigue limit CTD to each value of am. All of these values aD are represented on various diagrams known as 'endurance diagrams' which, as with the S-N curves, can be drawn for given probabilities [ATL 86].

3.5. Other representations of S-N curves 3.5.1. Haigh diagram The Haigh diagram is made up by plotting the stress amplitude cra against the mean stress at which the fatigue test was carried out, for a given number N of cycles to failure [BRA 80b] [BRA 81] [LIE 82]. Tensions are considered as positive, compressions as negative.

3. 1 ksi = 6.8947 MPa.

Concepts of material fatigue

101

Figure 3.17. Haigh diagram

We set crm the mean stress, aa the alternating stress superimposed on am, a'a the purely alternating stress (zero mean) which, applied alone, would lead to the same lifetime and aD the fatigue limit. Point A represents the fatigue limit CTD in purely alternating stress, the point B corresponds to the ultimate stress during a static test (aa =0). The half straight lines starting from the origin (radii) represent couples aa, am. They can be aa parameterized according to the values of the ratio R = . The co-ordinates of a °m

point on the straight line of slope equal to 1 are (cr m , am) [SHI 72]).

(repeated stress

The locus of the fatigue limits observed under test for various values of the couple (a m , aa) is an arc of curve crossing by A and B. The domain delimited by arc AB and the two axes represents the couples (a m , aa) for which the fatigue life of the test bars is higher than the fatigue life corresponding to aD. So long as amax (= am + cr a ) remains lower than the yield stress Re, the curve representing the variations of aa with am is roughly a straight line. For crmax < Re, we have, at the limit, CT

max = Re = a m+ CT a

CT

a = Re-CTin

102

Fatigue damage

This line crosses the axis Oam at a point P on abscissa Re and Oaa at a point Q on ordinate Re. Let us set C the point of Oaa having as ordinate amax (< R e ). The arc CB is the locus of the points (cr m , aa) leading to the same fatigue life. This arc of curve crosses the straight line PQ at T. Only the arc (appreciably linear) crossing by T on the left of PQ is representative of the variations of cra varying with am for amax < Re. On the right, the arc is no longer linear [SCH 74]. Curve AB has been represented by several analytical approximations, starting from the value of aD (for am =0), and of aa and am, used to build this diagram a priori in an approximate way [BRA 80a] [GER 74] [GOO 30] [OSG 82]. Modified Goodman line

Soderberg line

Gerber parabola

Figure 3.18. Haigh, Gerber,Goodman and Soderberg representations

Concepts of material fatigue

103

The Haigh diagram is plotted for a given endurance N0, in general fixed at 107 cycles, but it can also be established for any number of cycles. In this case, curve CTB, in a similar way, can be represented depending on case by: Modified Goodman

Soderberg

Gerber

These models make it possible to calculate the equivalent stress range Aa eq . taking into account the non zero mean stress, using the relation [SHI 83]:

where Aa is the total stress range (modified Goodman) om = mean stress Rm= ultimate tensile strength. The relationships [3.35] to [3.37] can be written in the form Modified Goodman

Soderberg

Gerber

Depending on materials, one or the other of the representations is best suited. The modified Goodman line leads to conservative results (and thus also for oversizing) [HAU 69] [OSG 82], except close to the points am = 0 and Rm = 0. It is good for brittle materials, conservative for ductile materials. The Gerber representation was proposed to correct this conservatism; it adapts better to the experimental data for aa > am. The case am » aa can correspond to

104

Fatigue damage

plastic deformations. The model is worse for am < 0 (compression). It is satisfactory for ductile materials. The Soderberg model eliminates this last problem, but it is more conservative than that of Goodman. E.B. Haugen and J.A. Hritz [HAU 69] observe that: -the modifications made by Langer (which excludes the area where the sum CTa + cm is higher than Re) and by Sines are not significant, - it is desirable to replace in this diagram the static yield stress by the dynamic yield stress, -the curves are not deterministic, one should better use a Gerber parabola in statistical matter, of the form:

where crD and Rm are mean values, as

where Sa

and SR are respectively the standard deviations of CTD and Rm

[BAH 78].

Figure 3.19. Haigh diagram. Langer and Sines modifications

Concepts of material fatigue

105

NOTE. — The Haigh diagram can be built from the S-N curves plotted for several values of the mean stress am (Figures 3.20 and 3.21).

Figure 3.20. S-N curves with non zero mean stress, for construction of the Haigh diagram

Figure 3.21. Construction of the Haigh diagram

A static test makes it possible to evaluate Rm. A test with zero mean stress gives CTa. For given N, the curves am. have an ordinate equal to <J D (N)J. Other relations Von Settings-Hencky ellipse or J. Marin ellipse [MAR 56]:

a'a = allowable stress when am = 0 aa = allowable stress (for the same fatigue life N) for given am * 0 ml = constant. The case of ml = 1 (Goodman) is conservative. The experiment shows that ml < 2. Value 1.5 is considered correct for the majority of steels [DBS 75].

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Fatigue damage

J. Bahuaud [MAR 56]

where Rt = true ultimate tensile and compressive strength. If strength Rt is unknown, it can be approximated using

Zu = striction coefficient Rm = conventional ultimate strength Dietmann

All these relations can be gathered in the more general form

where kl5 k2, rj and r2 are constant functions of the chosen law.

Concepts of material fatigue

107

Table 3.8. Values of the constants of the general law (Haigh diagram) r

i

2

k,

k2 Re

r

Soderberg

1

1

1

Modified Goodman Gerber Von Mises-Hencky J. Marin

1 1 2 1 2

1

1 1 1 1 1 1

Dietmann Kececioglu

(

b *>

2 2 ml 1 2

Rm i i i i i i

( ) b = factor function of the nature of material (close to 1)

NOTE. - In rotational bending, the following relation can be used, in the absence of other data [BRA 80bJ: _ aD tension-compression D rotative bending ~ TT

3.5.2. Statistical representation of Haigh diagram We saw that in practice, the phenomena of fatigue are to be represented statistically. The Haigh diagram can take such a form. If the relation Gerber is chosen for example, parabolic curves are plotted, on the axes aa, cr m , to describe the variations of a given stress cr a , corresponding to a given number of cycles to failure, with given probability.

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Fatigue damage

Figure 3.22. Statistical representation of the Haigh diagram

It is shown that the distribution of the alternating stresses obtained while crossing the arcs of parabola by a straight line emanating from the origin O (slope aa ) is roughly Gaussian [ANG 75]. CT

m

3.6. Prediction of fatigue life of complex structures A very difficult problem in the calculation of the fatigue life of a structure is the multiplicity of the sites of initiation of cracks and the mechanisms which determine the life resulting from fatigue of the structure. It could be observed that these sites and mechanisms depend on the environment of service, the amplitude and the nature of the loads. B.N. LEIS [LEI 78] classes methods of analysis of fatigue in two principal categories: - indirect approach, in which one tries to predict the fatigue life (estimate and accumulation of damage) on the basis of deformation and stress acting far from the potential areas of initiation of the cracks by fatigue, depending on the external displacements and forces (black box approach), - the direct approach, in which one tries to predict the fatigue life on the basis of stress and deformations acting on the potential sites of initiation. These stresses and deformations are local. The first approach cannot take into account the local inelastic action at the site of initiation of fatigue, whereas the direct approaches can introduce this non linearity.

Concepts of material fatigue

109

The direct approach allows correct predictions of initiation of cracks in a structure on the condition of correctly taking into account the multiplicity of the sites of initiation and mechanisms which control the life in fatigue. 3.7. Fatigue in composite materials An essential difference between metals and composites lies in their respective fatigue behaviour. Metals usually break by initiation and propagation of crack in a manner which can be predicted by the fracture mechanics. The composites present several modes of degradation such as the delamination, failure of fibres, disturbance of the matrix, presence of vacuums, failure of the matrix, failure of the composite. A structure can present one or several of these modes and it is difficult a priori to say that which will prevail and will produce the failure [SAL 71]. Another difference with metals relates to behaviour as a result of low frequency fatigue. It is often admitted that metals follow, for a low number of cycles, a law known as Coffin-Manson 's relating the number of cycles to failure N to the strain range, of the form

(where P » 0.5). The composites are more sensitive to the strain range and more resistant to fatigue when undergoing large numbers of cycles than with a low number of cycles. A structure can break owing to the part of the load spectrum relating to small stresses if it is metallic, whereas the same structure in composite would break because of high loads. The fatigue strength of composite materials is affected by various parameters which can be classified as follows [COP 80]: — factors specific to the material: low thermal conductivity, leading to an important increase in the temperature if the frequency is too high, defects related on the heterogeneous structure of material and its implementation (bubbles etc), natural aging related to its conditions of storage etc. - factors related to the geometry of the test bars: - shape, holes, notches (stress concentration factors), as with metals, - irregularities of surface being able to modify the thickness of the test bars in a significant way (important in the case of nonuniform stresses (torsion, bending etc), - stress and environment conditions: - mode of application of the stresses (torsion etc), - frequency,

110

Fatigue damage

- mean stress, - hygrothermic environment, - corrosion (surface deterioration of polymer comparable with corrosion). R. Cope and A. Balme [COP 80] show for example that a resin polyesterfiberglass composite or laminate obeys a fatigue degradation model because of: - degradation of the interface fibre-resin, - degradation by cracking and loss of the resin polyester, - progressive damage of the reinforcement. These use an index of reference for damage evolution depending on number n of G cycles the ratio (G: rigidity modulus in torsion after n cycles, G0: the same GO modulus at the test commencement) and observe a threshold ns beyond which the material starts to adapt in an irreversible way, and thus to undergo damage. The effect of temperature results in a decrease of the threshold ns for weak deformations and an increase in the number of cycles for failure (for t > 60°C). The presence of mean stress amplifies in an important way the fatigue damage. In addition, it is often considered that Miner's rule strongly over-estimates the fatigue life of the structures in composites [GER 82].

Chapter 4

Accumulation of fatigue damage

4.1. Evolution of fatigue damage From a fundamental point of view, the concept of fatigue damage is not well defined because of a lack of comprehension of the physical phenomena related to the material. The constituent material of any part subjected to alternating stresses undergoes a deterioration of its properties. Depending on the stress level and the number of cycles carried out, this deterioration is partial or can continue until failure.

Figure 4.1. Accumulation of fatigue damage up to failure

In order to follow the evolution of deterioration, one defines the concept of damage undergone by the test bar, which can rise from zero to 100% at the instant of failure. To follow the progress of the damage over time, a first solution could consist for example [LLO 63] of studying the evolution of specific physical properties of the test bar.

112

Fatigue damage

Strength in (static) tension decreases gradually as the number of stress cycles increases. Measurement of the evolution of these parameters encounters two difficulties: its variation is very weak for most of the life of the test bar and particularly if the test to be carried out is destructive. During the fatigue test, cracks appear which are propagated until failure of the test bar. Unfortunately, cracks become detectable and measurable only towards the end of the fatigue life of the test bar. Additionally, the studies conducted to elucidate evolution of fatigue as a function of the characteristics of elasticity, plasticity, viscoelasticity and buckling showed that these parameters do not evolve during the application of alternating stress cycles [LEM 70]. With regard to this impossibility, one tried to imagine laws of damage accumulation by fatigue being based on other criteria. Many laws of this type were established, in general starting from experimental results. 4.2. Classification of various laws of accumulation Some laws of accumulation of damage are independent of the stress level a. Others depend on it: the D(n) curve is different according to the value of c [BUI 80] [LLO 63] [PRO 48]. For others, the D(n) curve varies not only with a, but also with the preceding series of alternating stresses applied to the material (damage with interaction).

Figure 4.2. Various processes of stress accumulation

Accumulation of fatigue damage

113

In general, all these laws of damage accumulation relate to alternating loads with zero average applied to ordinary temperature. There are however publications treating of these two particular problems (strong temperature or non zero mean stress). 4.3. Miner's method 4.3.1. Miner's rule One of the oldest rules, simplest and the most used, is that known as Miner's rule [MIN 45] (or Palmgren - Miner, because this rule was actually proposed initially by Palmgren [PAL 24] in 1924, then by Langer in 1937 [LAN 37]). It is based on the following assumptions: - the damage accumulated by material during each cycle is a function only of the stress level a; for n cycles, one defines as damage, fraction of fatigue life at the level of sinusoidal stress a the quantity:

where N is the number of cycles to the failure at level a. If W is the stored energy at failure after N cycles at the stress level a and w that absorbed after the application of n cycles at this same level, we have:

- the appearance of a crack, if it is observed, is regarded as a fracture [MIN 45], - at this level of stress a, failure intervenes (in a deterministic way) when n = N, i.e. when d = 1 (endurance); we will see that N could also correspond to the number of cycles with a failure rate of 50% (number of cycles for which half of the test bars tested are broken). This number N is given by the S-N curves previously defined (which are, in general, plotted starting from tests under sinusoidal stresses with constant amplitude). The stresses lower than the fatigue limit stress are not taken into account (N is infinite, therefore d is zero), - the damages d are added linearly. If k sinusoidal stresses Oj of equal or different amplitudes are applied successively in n; cycles, the total damage undergone by the test bar is written:

114

Fatigue damage

It is supposed finally that the damage accumulates without there being influence of one level on the other. In this case, the failure occurs for

NOTE. - The assumption of linearity is not necessary to obtain

to the failure. It is enough to suppose that the rate of damage is a function of n/N independent of the amplitude of the cyclic stress [BLA 46]. Miner's rule thus consists of proceeding to a sum of the fractions of fatigue life of the specimen consumed with each cycle and each stress level, the damage corresponding to a cycle of stress a being equal to 1/N. It is thus a linear law independent of the stress level and without interaction. We will see that Miner's rule can also be used for a random excitation or even for shocks. In this case, it is necessary to establish a histogram of the peaks of stress, a table indicating, for the period of validity of the stress, the number of peaks having a given amplitude (or included in a range of amplitudes). For a given amplitude GJ, the partial damage relating to a peak (half-cycle) is:

yielding total damage for HJ peaks of amplitude c^:

Accumulation of fatigue damage

115

Example Let us consider a test in which each test bar is subjected successively to the 3 following tests: - a sinusoidal stress of amplitude 15 daN/mm2, frequency 10 Hz, duration 1 hour, - a sinusoidal stress of amplitude 20 daN/mm2, frequency 20 Hz, duration 45 min, - a sinusoidal stress of amplitude 12 daN/mm2, frequency 15 Hz, duration 1 hour 30 min. The supposed S-N curve of the material considered is plotted in Figure 4.3.

Figure 4.3. Example of S-N curve

For cjj = 15 daN/mm2, the number of cycles to failure is equal to Nt = 3 10 For a2 = 20 daN/mm2, N2 = 6 105 and for a3 = 12 daN/mm2, N3 = 107. The numbers of cycles carried out at each level are respectively equal to n, = ft

Tj =103600 = 3.6 104

(Hz) (s)

n2 =20x2700 = 5.4 104

116

Fatigue damage

and

n3 =15 5400 = 8.1 104 yielding

D = 0.1101 If after application of these 3 tests, one continues the test until failure under the conditions of the third test, it takes one duration of complementary test Tc such that:

where nc is given by

n c = ( l - D ) . N 3 =(l-0.110l)l0 7 =8.899 10 6 cycles yielding

Application If it is considered that the S-N curve can be described analytically by Basquin's relationship N CT = C, we can write:

Accumulation of fatigue damage

117

yielding, for example, the expression of an equivalent alternating stress ae which would produce the same fatigue damage if it were applied during Ne cycles:

In this relation, the only characteristic parameter of the S-N curve is the exponent b. Miner's rule for the estimate of the fatigue life under stresses with variable amplitude is easy to implement; although much criticized, it is thus largely used in the design calculations. Fatigue being a complex process, implying many factors, this rule, a simplified description of fatigue, does not provide very precise estimates. It seems however acquired that it leads to results acceptable for the evaluation of the initiation of the cracks under random stresses [WIR 83b]. 4.3.2. Scatter of damage to failure as evaluated by Miner Many tests were carried out to confirm this law by experiment. They showed that actually, one observes a significant scatter in the value of D at the instant of failure [CUR 71] [GER 61]. M.A. Miner specifies besides that the value 1 is only an average. The study of the scatter of the sum 2l< n i/^i to failure was the object of multiple works of which one can find a synthesis in the book by P.O. Forrest [FOR 74]. All these works showed that the order of application of the sinusoidal stresses is not without influence over the fatigue life of the test bars and that the rate of cracking does not depend only on the amplitude of the loading at a given time, but also of the amplitudes during preceding cycles [BEN 46] [DOL 49] [KOM 45]. For decreasing levels (a2 < 1 when one applies the levels increasingly (cjj < a2) (weaker cracking rate due to the

118

Fatigue damage

under-load). This last phenomenon, much less significant than the previous one, is in general neglected [NEL 78] [WEI 78]. This difference in behaviour can be explained by considering that cycles of great amplitude, when they are applied at first, create damage in the form of intergranular microscopic cracks that the following cycles at low level will continue to propagate. Alternatively, the first cycles, at low levels, cannot create microscopic cracks [NEU 91]. In certain cases however, and in particular for notched samples, one observes the opposite effect [HAR 60] [NAU 59]. D.V. Nelson [NEL 78] considers that the variations between real fatigue life and fatigue life estimated using Miner's law, are related to the order of application of the loads, and are, for the smooth parts, lower than or equal to ±50%. Calculations are often carried using this law by supposing that any stress lower than the fatigue limit stress aD of material has no effect on the material, whatever the number N of cycles applied. However the tests showed that this stress limit can be lowered when the test bar underwent cycles of fatigue at a higher level than aD, lowering which is not taken into account by the Miner's rule [HAI 78]. The damage DF obtained with failure can, in practice, depending on the material and the stress type, vary over a broad range. One finds, in the literature, values of about 0.1 to 10 [HAR 63] [HEA 56] [TAN 70], or 0.03 to 2.31, and even sometimes 0.18 to 23 [DOL 49]. In general however, the range observed is narrower, from 0.5 to 2 for example [CZE 78] [FID 75]. These values are only examples among many of other results [BUG 77] [BUG 78] [COR 59] [DOL 49] [FOR 61] [GAS 65] [GER 61] [JAG 68] [JAG 69] [KLI 81] [LAV 69] [MIN 45] [STE 73] [STR 73] [WIR 76] [WIR 77]. Compilations of work of various authors showed that: - Depending on the materials and test sequences, Dp does not deviate, in general, strongly from 1 and is most frequently larger than 0.3 when all the stress levels are greater than the fatigue limit stress [BUG 78]. The variation in the nonconservative direction is larger when the amplitude of the stresses close or lower than the fatigue limit stress is larger than that of the stresses higher than this limit. In addition, it is noted that Dp is practically independent of the size of the specimens. - With a cumulative damage less than 0.3, the probability of non-failure is equal to 95% (560 test results, without reference to the various types of loads and for a constant average stress, on various parts and materials) [BRA 80b] [JAG 68]. In practice, it would be more correct to set, at failure:

Accumulation of fatigue damage

119

where C is a constant whose value is a function of the order of application of the stress levels. This approach supposes however that the damage on the same stress level is deterministic. However the experiment shows that the test results carried out under the same conditions are scattered. P.H. Wirsching [WIR 79] [WIR 83b] proposes a statistical formulation of the form:

where A, index of damage to failure, is a random variable of median value near to 1, with a scatter characterized by the coefficient of variation VA , the distribution being regarded as log-normal. The purpose of this formulation is to quantify uncertainties associated with the use of a simple model to describe a complicated physical phenomenon. The author proposes, to separate the factors:

where VN = variation coefficient for cycles with constant amplitude until failure. V0= variation coefficient related to the other effects (sequence,...). The failure takes place for D > A , with a probability given by: Pf = P ( D > A )

[4.12]

One will find in Table 4.1 some values recorded in the literature by P.H. Wirsching.

120

Fatigue damage

Table 4.1. Examples of statistical parameters characterizing the law of damage to failure

Variation coefficient

Mean of A

Median of A

1.33

1.11

0.65

1.62 1.47 1.39 1.22

1.29 1.34 1.15 0.98

0.76 0.45 0.68 0.73

Aluminium (n = 389 specimens) Steel (n = 90) Steel (n = 87) Composite (n = 479) Composite

VA

P.H. Wirsching [WIR 79] recommends, with a log-normal distribution, a median value equal to 1 and a variation coefficient equal to 0.70. He quotes however some other results deviating from these values: 7075 alloy: VA = 0.98 2024 T4 aluminium alloy: VA =0.161 For sinusoidal loads or random sequences, W.T. Kirkby [KIR 72] deduced from a compilation of test results on aluminium alloys: A = 2.42

VA = 0.98

Standard deviation = 2.38 Other authors [BIR 68], [SAU 69], [SHI 80] and [TAN 75] proposed to ascribe a statistical character to the Miner's rule by considering in particular that: - the number N4 of cycles to failure is a random variable with average Nj at the stress level QJ, - the expected value of the failure is given by:

- the probability density of the damage to failure

Accumulation of fatigue damage

121

follows a log-normal distribution [SHI 80] [WIR 76] [WIR 77] with mean HD and standard deviation equal to 1. NOTE. - All that precedes -was defined for the case of stresses with zero average. If it is not the case, the method continues overall to apply, by using the S-N curve plotted for the value of the mean stress considered. 4.3.3. Validity of Miner's law of accumulation of damage in case of random stress A.K. Head and F.H. Hooke [HEA 56] were the first to be published on tests where the amplitude of the stress varies in a completely random way. Miner's rule was sometimes regarded as inadequate and dangerous to apply for loads of random amplitude [OSG 69]. This opinion however is not commonly held and the majority of authors consider that this rule is even better for these loads: the order of application of the stresses being random, its influence is much weaker than in sinusoidal mode. The results are often regarded as resolutely optimistic. Much work carried out on aluminium alloy test bars show that the Miner assumption led to predictions of fatigue lives larger than those observed in experiments, with variations in the ratio from 1 to 1.3 or 1 to 20 according to case [CLE 65] [EXP 59] [FRA 61] [FUL 63] [HEA 56] [HIL 70] [JAC 68] [PLU 66] [SMI 63] [SWA 63]. There is thus failure noted for

Other studies show on the contrary that the Miner's rule is pessimistic [EXP 59] [KIR 65a] [LOW 62]. This seems to be in fact the case for loads with average tensile stress associated with an axial loading [MAR 66] or when there are frictional phenomena [KIR 65a]. The precision of the estimates however is considered to be acceptable in the majority of the cases [BOO 69] [BOO 70] [BOO 76] [KOW 59] [NEL 77] [TRO 58]. Although the Miner's rule leads to coarse results, varying considerably, many authors consider that there is no rule more applicable than Miner's, the regularity of the results obtained with the other methods being not significantly better with regard to their complexity [EST 62] [FRO 75] [ING 27] [NEL 78] [WIR 77]. The Miner's hypothesis remains a good first approximation confirmed by experiment. The error depends on the rule itself, but also on the precision of the S-N curve used [SCH 72a].

122

Fatigue damage

All other rules developed to make it more precise, to be quoted later, brought an additional complexity while not always using well-known constants; they lead to better results only in certain specific cases. Miner's rule is very much used for mechanical structures and even for the calculation of strength to fatigue of electronic equipments. D.S. Steinberg [STE 73] suggests for these materials calculation of fatigue life from:

instead of 1 (and not 0.3 as sometimes proposed, a value which can lead to an =

increase in superfluous mass). W. Schutz [SCH 74] proposed /^—

N

high temperatures, when the yield stress is definitely lower, 2_.~~ Nj statistical studies [OSG 69]).

0.6 and, at

i

=

0.3 (result of

Miner's rule can also be used to make comparisons of severity of several vibrations (the criterion being the fatigue damage). This use, which does not require that V—-

N

is equal to unity at failure, gives good results [KIR 72] [SCH 72b]

i

[SCH 74]. NOTE. - Alternatives very similar to Miner's rule were evaluated [BUC 78]: n = 1, where N min is a value corresponding to a certain N min probability of survival (95% for example), is not always conservative,

Z

— the sum \ fN mean = mean number of cycles to failure at a given ^"^ N ^mean stress level) does not give either a precise estimate of the fatigue life (sequence effect, non linear accumulation law etc). 4.4. Modified Miner's theory To try to correct the variations observed between the calculated fatigue lives starting from Miner's theory and the results of the tests, in particular during the application of sinusoidal stresses at several levels ai? it was planned to replace the linear accumulation law by a non-linear law of the form:

Accumulation of fatigue damage

123

[4.15]

where x is a constant presumably higher than 1.

Figure 4.4. Damage versus the number of cycles, as function of stress amplitude Figure 4.4 shows the variations of the damage D versus the number of cycles carried out at two stress levels at and a2. For given a, one supposes that the test bar breaks when D = 1. The number of cycles to failure is greater since CT is smaller. If the damage is linearly cumulative (Miner's hypothesis), the curve D(n) is a straight line. With the modified law, the curve D x (n) is, at level a, of the form r~v D = a n*

a being a constant such that D = 1 when n - N:

For x > 1, D x (n) is below the straight line of the linear case (dotted curve). Accumulation of damage using modified Miner's rule Two methods can be considered [BAH 78] [ORE 81]: - method of equivalent cycles, - method of equivalent stresses,

124

Fatigue damage

according to whether the damage is expressed as a function of the number of cycles brought back to only one level of stress or the damage dj is calculated independently at each stress level a, before determining the total damage

The reasoning is based on the damage curves (damage D undergone by the test bar according to the number of cycles n) for a given stress level a. As an example, we will consider two levels below
Method of equivalent cycles One carries out: - nj cycles at level al5 failure occurring on this level for n = Nj cycles, - n2 cycles at level a2 (failure for N2 cycles).

Figure 4.5. Determination of equivalent number of cycles

The damage d2 created by these n2 cycles at level a2 could be generated by nt cycles at level CTI? nj" being such that

i.e., whatever x, for

Accumulation of fatigue damage

125

If it is supposed that the block made up of these DJ and n2 cycles of stress of amplitudes respectively equal to G{ and a2 is repeated p times until failure, one will have at this time carried out nlR cycles at level
If we set

this relationship can be also written

i.e.

while setting N

R

= n

lR + n2R = p ( n l

+ n 2)

126

Fatigue damage

The method can be extended to the study of a loading made up of an arbitrary number of levels. Calculation is facilitated if one uses an analytical form of the damage curves. Method of equivalent stresses Instead of considering a number of cycles equivalent to the single level a1? one calculates the partial damage created by each sequence [BAH 78] [GRE 81]; with the same notations, one has, to failure:

yielding

and

Miner's theory is a particular case of this method, with linear damage curves. On this assumption, the two approaches (equivalent cycles and equivalent stresses) lead to the same results. Modified Miner's rule thus envisages a fatigue life longer than the rule of origin, nearer consequently to the experiment when c^ < a2 (first stress level lower than the second). The improvement is however only partial, since it does not bring anything when QJ > a2. The modified Miner's rule is also independent of the level of stress and without interaction.

Accumulation of fatigue damage

127

4.5. Henry's method D.L. Henry (1955) [HEN 55] supposes that: -the S-N curve of a steel specimen can be described by a relation of a hyperbolic type of the form N(a-aD) = C

[4.26]

where C is a constant and CJD is the fatigue limit stress of material, - the two parameters C andCTDare modified as the fatigue damage accumulates. The endurance to fatigue decreases and C varies proportionately to aD. The author thinks that the relation [4.26] is correct so long as a remains lower than 1.5 CTD. One of the interests of this representation is related to the introduction of a fatigue limit. With these assumptions, D.L. Henry shows that the damage D stored by a test bar, defined here like the relative variation of the fatigue limit

is written

or, if we set y

stress ratio and P

It is noted that forCT'D= 0, we have D = 1 and that, when a -> aD, y —> 0 and D-»0.

128

Fatigue damage

Figure 4.6. Accumulation of damage according to Henry's hypothesis

The curve D((3) plotted for several values of y shows that, when y increases (i.e. for the strong values of the stress), the law tends to becoming linear. For a given (3 ratio, the damage is larger sinceCTis higher. This theory thus supposes that fatigue damage is a function of applied stress level and of the order of application of the stresses (law dependent on the level of the stresses and with interaction).

4.6. Modified Henry's method The relation of Henry was modified by addition of a term D [EST 62] 60]:

[INV

where Dc is critical fatigue damage (damage at which the part breaks completely), in order to take into account the application of a load which exceeds the residual resistance of the item to the test. Calculations of fatigue life are carried out according to same process as that used with method EFD [EST 62]. The increments

Accumulation of fatigue damage

129

n D — are added along the curve of damage; the failure takes place when — = 1. This N Dc method requires the knowledge of an additional parameter, Dc.

Henry's rule and this modified rule have a limited use, the materials having in general a badly definite fatigue limit stress (aluminium and light alloys in particular).

4.7. H. Corten and T. Dolan's method These authors propose a non-linear theory of cumulative damage, based on the number m of nuclei likely to be damaged and on the velocity r of propagation of the cracks [COR 56] [COR 59] [DOL 49] [DOL 57] [HIL 70] [LIU 59] [LIU 60]. The accumulated damage on the stress level i is written:

where: aj is an experimentally determined constant, m and r are constants for a given stress level. Damage to failure occurs for D = 1. It is expressed as a function of the number of cycles according to

It is about a law depending on the stress level, with interaction. The effects of interaction are included in the concept of propagation of cracks. In a test, the cycle of maximum load is decisive for the initial damage since it determines the number of points where cracks will be formed. Once this number is established, one supposes that the propagation is carried out according to a cumulative process without interaction. With the following notations: cCj= percentage of cycles carried out at level aa. d = constant GJ= maximum amplitude of the sinusoidal stress, for a fatigue life N, N o = total number of cycles of the test program,

130

Fatigue damage

we have

yielding

or

This relationship, comparable with that of A.M. Freudenthal and R.A. Heller, can be reduced to that of Miner by using a modified fatigue curve which, on log a, log N scales, intersects the y-axis atCTJand has a negative slope equal to 1/d. The equation of this curve is then

where N' is the fatigue life given by this modified curve. From [4.33], ctj can be substituted in [4.35], by taking account of [4.37], to calculate c^ and c^:

where

Accumulation of fatigue damage

131

yielding

B.M. Hillberry [HIL 70] shows that the Corten-Dolan theory, like that of Palmgren-Miner, over-estimates the fatigue lives under random stress (by a factor varying from 1.5 to 5 for Miner and 2.5 approximately for Corten-Dolan). This result is compatible with those of other authors. It can be explained by noticing that, according to these two theories, for random loads, the damage accumulate under a certain weighted average stress and that this same stress produces the failure. With such a load, however, the failure can take place earlier under one of the largest peaks present in a statistical way. This point of view [NEL 77] is confirmed by the test results carried out by Gerks. 4.8. Other theories Taking into account their relatively significant number, we will limit ourselves within the framework of this book to quote the authors of these theories and to give the references of the papers in which they are developed. This list does not claim to be exhaustive, from the point of view of methods as well as references. NOTE.- Among all the models suggested, of -which we will quote only a small number, little have found practical application. Miner's rule, with its imperfections, remains simplest, most general and the most used by far [BLA 78], and often sufficiently accurate. The other rules in general use several constants whose value is difficult to find and which complicate the analysis in a way not justified by the result. One can find, for example in the references [FA T 62] [FUL 61] [RIC 65b], comparisons of these various methods.

132

Fatigue damage Table 4.2. Other theories of fatigue damage accumulation

Date

References

Miner modified by E. Haibach

1970

[HAI 70] [STR 73]

S.M. Marco and W.L. Starkey

1954

[MAR 54]

F.R. Shanley

1952/1953

[MAS 66b] [SHA 52]

B.F. Langer

1937

[LAN 37]

J. Kommers

1945

[KOM 45]

F.E. Richart and N.M. Newmark

1948

[COR 59][RIC 48]

E.S. Machlin

1949

[MAC 49]

B. Lunberg

1955

[LUN 55]

A.K. Head and F.H. Hooke

1956

[HEA 56]

J.C. Levy

1957

[LEV 57]

A.M. Freudenthal and R.A. Heller

1956-19581959 - 1960

[FRE 55] [FRE 56] [FRE 58] [FRE 60] [FRE 61]

C.R. Smith

1958-1964

[SMI 58] [SMI 64b]

H. Grover

1959

[GRO 59] [GRO 60]

A.L. Eshleman, J.D. Van Dyke and P.M. Belcher

1959

[ESH 59]

E. Parzen

1959

[PAR 59]

R.R. Gatts

1961 - 1962

[GAT61][GAT62a] [GAT 62b]

S.R. Valluri

1961 -19631964

[VAL61a][VAL61b] [VAL 63] [VAL 64]

E. Poppleton

1962

[POP 62]

Method of equivalent fatigue damage (EFD Method)

1962

[EST 62]

S.V. Serensen

1964

[SER 64]

R.W. Lardner

1966

[LAR 66]

1968-1969

[BIR 68] [BIR 69a] [BIR 69b]

1968

[ESI 68]

Theory

Z.W. Birnbaum and S.C. Saunders A. Esin

Accumulation of fatigue damage

G. Filipino, H.T. Topper and H.H.E. Leipholz

133

1976

[PHI 76]

1964 - 1968

[ESI 68] [KOZ 68]

K.J. Marsh

1965

[MAR 65]

R.A. Heller and A.S. Heller

1965

[HEL 65]

S. Manson, J. Freche and C. Ensign

1967

[MAN 67]

A. Sorensen

1968

[SOR 68]

J.L. Caboche

1974

[CHA 74]

1971 - 1982

[BUI 71] [BUI 82] [DUB 71]

Z. Hashin and A. Rotom

1977

[HAS 77]

S. Tanaka and S. Akita

1975 - 1980

[TAN 75] [TAN 80]

J.L. Bogdanoff

1978 to 1981

[BOG 78a] [BOG 78b] [BOG 78c] [BOG 80] [BOG 81]

P.H. Wirsching

1979

F. Kozin and A.L. Sweet

J. Dubuc, T. Bui-Quoc, A. Bazergui and A. Biron

[WIR 79]

83b][

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Chapter 5

Counting methods for analysing random time history

5.1. General Counting methods were initially developed for the study of fatigue damage generated in aeronautical structures. The signal measured, in general a random stress O(t), is not always made up of a peak alone between two passages by zero. On the contrary, several peaks often appear, which makes difficult the determination of the number of cycles absorbed by the structure.

Figure 5.1. Random stress

The counting of peaks makes it possible to constitute a histogram of the peaks of the signal (number of events counted on the abscissa whose amplitude is shown on the ordinate) which can then be transformed into a stress spectrum (cumulative frequency distribution) giving the number of events (on abscissae) for lower than a given stress value (on ordinates), i.e. the number of peaks lower than a given threshold level versus the value of this threshold. The stress spectrum is thus a

136

Fatigue damage

representation of the statistical distribution of the characteristic amplitudes of the signal as function of time [SCH 72a]; it is used in two applications: - to carry out estimated fatigue life calculations on the parts subjected to stresses measured, - to transform a complex random stress into a simpler test specification, made up for example of several blocks of sinusoidal vibrations of constant amplitude. Several test strategies are possible, such as the realization of sinusoidal cycles with constant amplitude in random sequence or the application of individual cycles in randomized sequence. The tests are carried out on material test bars with a control in force. The importance of this analysis of the signal in these studies and its difficulty is easily understood, which explains the significant number of methods suggested. The analysed signal is a stress varying with time. Within the framework of this work, these same methods can be used to establish a peak histogram of the relative response displacement z(t) of the mass of a system with one-degree-of-freedom subjected to a vibration and to calculate the fatigue damage which results from it (the stress being supposed proportional to this relative displacement). All the methods assume that the result of counting always includes an assumption of damage, which postulates that the same fatigue life is obtained with the original signal and the signal reconstituted from counting, or at least that starting from the two signals, there exists a constant ratio of the fatigue lives for all the possible material combinations, stress concentrations, stress ratio, treatment of surfaces etc. Beyond this, it is certain that the evaluation of a result of counting is also much related the procedure of a specific test or to a theoretical method of fatigue life prediction (modification in the order of application of the stress levels can for example change the fatigue life) [BUX 73]. The evaluation of the fatigue damage is thus carried out in three steps: - counting of the cycles, - choice of a relation cycle - generated damage, - summation of the damage generated by each cycle. Various methods of counting were proposed, leading to different results and thus, for some, to errors in the calculation of the fatigue lives:

Counting methods for analysing random time history

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.

137

Peak count Range-restricted peak count Level-restricted peak count Mean-crossing peak count Range count Range-mean count Range-pair count Ordered overall range method Racetrack method Level crossing count Modified Level crossing count Peak valley pair (PVP) counting Fatiguemeter count Rainflow count NRL counting Time spent at a given count level

Schematically, these counting methods can be classified in two principal categories [WAT 76]: - methods in which a counting of a signal characteristic not necessarily connected in a simple way to damage accumulation is carried out (peaks, level crossings etc), - methods using the stress or deformation ranges (range counting, range-mean counting, range-pair counting, rainflow). They in general are preferred, for they are stated in parameters more directly connected to the fatigue damage. We will see that two among them, the range-mean counting and the rainflow counting, have the additional advantage of taking into account the mean stress level (or deformation) for each range. In all these methods, it is necessary to be able to eliminate the small variations. This correction, initially intended to remove the background noise of the measuring equipment, also aims at the transformation of long duration signals into signals easier to use [CON 78]. This simplification results in defining a threshold value in on the side which the peaks or ranges are not counted. As an example, let us consider the signal sample O(t) in Figure 5.2. The signal has peaks A, B, ..., F from which ranges can be defined: AB, BC, CD, ..., EF, with amplitudes respectively equal to r1, r2, r3, r4, r5 and with means m1, ..., m5.

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A

O(t)

Figure 5.2. Elimination of small variation ranges

The small variations such as CD can be neglected by imposing a threshold for counting. In this case, the ranges BC, CD, and DE are replaced by the single range BE(r' 2 ). Taking into account the expression for the fatigue damage, in which the peak amplitude intervenes to the power b, if the linear part of the S-N curve is represented by the Basquin's relation: N ab = Const. it is seen easily that the total damage is much smaller when it is calculated by taking account of the small variations than when they are removed:

(independently of the mean stress), would be this only because the small variations can be for certain below the fatigue limit and thus be neglected [DOW 72] [WAT 76]. A reduction of the counting threshold, which results in considering more ranges of small amplitude, can thus lead to a weaker expected damage and vice versa. This phenomenon is not systematic and can be negligible depending on the shape of the

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signal (for example when the small cycles are dominating). P. Watson and B.J. Dabell [WAT 76] observe that, to guard against this problem, a solution can consist in successively calculating the damage with several thresholds and to retain only the greatest value. The effect of omission of small variations has been studied by several authors [CON 78] [KID 77] [POT 73]. It appears often that the cycles of loads below the fatigue limit can in general be ignored if the load peak levels of the remainder of the curve versus time (e(t) or O(t)) are not too large. A study by A. Conle and H.T. Topper [CON 78] shows that, even with the rainflow method, the damage actually omitted with a signal without small variations is much larger than that predicted. When the signal has very high levels (load or deformation), it is necessary to be very careful during the elimination of the smallest variations. J.M. Potter [POT 73] observes in experiments that the suppression of the levels which, according to a conventional analysis of fatigue damage do not give any damage, leads (in experiments) to an increase in the fatigue life. We will describe in the following paragraphs the methods quoted. 5.2. Peak count method 5.2.1. Presentation of method This method is undoubtedly the simplest and oldest. The values considered as significant here are the maxima and the minima, which are observed over the duration T of the signal [BUX 66] [DEJ 70] [NEL 78] [RAV 70] [SCH 63] [SCH 72a] [STR 73]. A peak located above the mean is regarded as positive; below, it is negative. The ordinate axis can be divided so as to define classes and to build a histogram.

Figure 5.3. Stress time history peaks

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One can choose to count only the peaks above a certain threshold value (in absolute value) or to decide that small stress variations lower than a given value are negligible, these small variations being defined like the variation between a maximum and a minimum (or the reverse). Figure 5.3 for example, the peaks marked by the symbol I I are not counted [HAA 62]. The results can be expressed either in the form of the maximum (or minimum) number: - having a given amplitude, - or located above a given threshold, this threshold being variable to cover all the range of the amplitudes, - or by curves giving the peak occurrence frequency versus their amplitude. The curves giving the maxima and the minima are in general symmetrical (but not necessarily). It is the case for a Gaussian process [RAV 70]. Occurrence frequency

ji

Minima

Maxima

Amplitude Figure 5.4. Occurrence frequency of maxima and minima

With this method, any information on the sequence of load, on the order of appearance of the peaks is completely lost [VAN 71]. After counting, it is not possible to know if a counted peak is associated with small or with great load variation. The interpretation of the load spectrum made up with this method can thus be disturbed by small load variations. Counting being finished, a signal is reconstituted from this peak histogram by associating a positive peak with a negative peak of the same amplitude to constitute a complete cycle [HAA 62]. Consequently, it can been seen that the real total length of the excursion of the signal is in general less than that obtained by reconstituting the signal from the histogram.

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Peaks such as A and B in Figure 5.5 for example are treated both as starting from zero and returning to zero.

t Figure 5.5. Peaks counted as complete half-cycles

The reconstituted signal is thus more severe than the real signal [WEB 66]. So this method of counting is often named total peak count as opposed to the net peak count method in which the peaks are defined as the algebraic sum of the extrema, the valleys (minima) being counted negatively. The lengths of excursions tend in this case to be equal. Another way of counting the extrema can consist in retaining only the positive maxima and the negative minima. An alternative to this process is frequently used for the study of the vibrations measured during fighter aircraft operations [LEY 63]. The sorting of the extrema can be carried out from the results of the above method. The peaks count method is also sometimes named simple peak count method [GOO 73] as opposed to the peak count method in which the extrema between two passages by the mean value are taken (further described). Each peak is noted. It is the least restrictive of all the methods. It can be used as means of characterization of the signal, by comparing the extrema distribution with a given distribution (Rayleigh's law for example).

5.2.2. Derived methods The present maxima in each class of amplitude are counted (Figure 5.6). The reconstituted signal does not account for the peak order nor of the frequency. The small strain variations are amplified. To reduce the influence of it, a secondary condition can be introduced, intended to neglect the extrema which are associated with a load variation lower than a given value (hatched zone / / / / in Figure 5.6).

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Another possibility is to count the minimal values in addition to the maximal values, but that led to a very great number of combinations to reconstitute the signal and it is not a suitable method [WEB 66]. It is also sometimes chosen to count only the maxima above a specified level (or minima below this level). This modification does not improve the validity of the results.

Figure 5.6. Omission of small variations below a given threshold

O. Buxbaum [BUX 66] suggests neglecting all the signal variations lower than 5% of the maximum value, which do not have any influence on the fatigue life. F.E. Kiddle and J. Darts [KID 77] observed that, on a specimen with bolted connection, the progressive omission of the peaks of small amplitudes above the fatigue limit does not have an appreciable effect on the mean endurance. This effect is independent of the maximum load present in the spectrum. The endurance of this same specimen is on the other hand sensitive to the greatest loads.

5.2.3. Range-restricted peak count method The objective is here to count only the most significant peaks associated with the greatest load variations [MOR 67] [WEL 65]. Counts are limited to the peaks beyond mean thresholds (for example, if it is an acceleration, the minima which are below 0 g and the maxima above 2 g) and which are at the same time preceded and followed by a minimal variation of the amplitude (1 g for example) or which some exceed a given percentage of the incremental peak value (for example 50%) (the largest of the two results is chosen). Here the incremental peak value is defined as the difference between the peak value and the mean load level [GOO 73] [MOR 67] [WEL 65].

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Figure 5.7. Counting of ranges higher than a given value

In Figure 5.7 for example, the ranges R1 and R2 are higher than the threshold R0 - 1 g or 50% of AL. The intermediate fluctuations are neglected with this method, as well as some fluctuations which are not really of no importance. However, countings relating to largest maxima and the smallest minima assume more importance.

5.2.4. Level-restricted peak count method After having defined classes, the procedure is as follows: - a maximum (primary peak) is accepted only if the signal goes again below a larger value than a specified threshold [VAN 71] [WEL 65] (all crossings prior to this primary level being forgotten), - the value attributed to the peak is that of the threshold crossed just before the peak.

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Figure 5.8. Elimination of peaks not followed by sufficient amplitude variation

Figure 5.9. Counting of all threshold crossings (fatiguemeter method)

For example, peak A (Figure 5.8) is counted, because the signal crosses threshold 2' after having crossed the larger threshold level 2 before passing through the maximum value at A. This method uses the same type of threshold specification as the fatiguemeter method. Only the peaks which are preceded and followed by the necessary threshold are considered [GOO 73]. But in contrast to the fatiguemeter method, all the other passages across the threshold below 2 are not considered. As an example, for the same signal, the fatiguemeter method would lead to the result in Figure 5.9.

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5.3. Peak between mean-crossing count method This method may be called mean-crossing peak count method or maximum peaks between zero crossing, or peak-count method [GOO 73] [SEW 72] (the last being an ambiguous with the above method).

5.3.1. Presentation of method

Figure 5.10. Counting of largest peaks between two passages through mean value

Only the largest maximum or minimum between two passages through average value is counted, which is equivalent to completely neglecting the stress variations between two passages through zero, even if they are important [SCH 72a] [WEB 66]. The signal is thus reduced to only one peak between two passages through zero. With an identical signal, the number of peaks counted is therefore smaller than with the first method [BUX 66] [HAA 62] [STR 73] [VAN 71] and the total duration of the reconstituted signal from this histogram of peaks is shorter than that of the initial real signal.

°±

Figure 5.11. Suppression of all small variations not crossing mean

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The values of this count can be deduced from the results of the first method (with elimination of approximately 25% of the initially counted peaks) [LEY 63] [RAV 70]. This method is similar to that sometimes used to count the turbulence loads observed by transport aircraft [DEJ 70]. This type of counting can lead to incorrect results. If the signals in Figure 5.12 are considered for example, the same count is obtained in both cases whereas signal (b) is certainly more severe than (a).

( a)

t

'

( b)

t

Figure 5.12. Example of two different signals leading to the same counting result

All the secondary variations of the load between two zeros and the maximum level are ignored by assumption. Some of these secondary load variations can however be very significant with respect to their contribution to the total fatigue damage by [SEW 72]. NOTE. - A specific measuring instrument — the VGH-Recorder (N.A.C.A.) — has been developed to carry out these counts [WAL 58] [WEB 66]. It is equivalent to count the number of up-crossings through the mean value of the signal, which allows, for narrow band signals, calculation of the mean frequency of the signal (the narrow band signals have the same number of zero threshold crossings and of peaks. It is thus preferable to use this method only in this case). If the distribution of the instantaneous values of the signal is, in addition, Gaussian, this measurement of the mean frequency gives access to the peak distribution or to the curve of level crossings. It is shown indeed that in this case, the number of crossings of a level a with a positive slope is given by (Volume 3, eq [5.60]):

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with

T = signal duration, mean frequency.

5.3.2. Elimination of small variations

Figure 5.13. Elimination of small variations around the mean

Two mean threshold levels of reference are fixed, one positive, the other negative. One retains the largest maximum between two arbitrary successive crossings of the higher threshold and the smallest minimum between two crossings of the lower threshold (Figure 5.13) [DON 67]. With this modification, the peaks associated with small variations around the static value are neglected. In addition, smaller maxima are neglected and smaller minima will be neglected.

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5.4. Range count method Or amplitudes regardless of mean. 5.4.1. Presentation of method The range is defined as the difference between two extreme values of the stress (or the deformation) with a sign according to direction of the transition (minimum -> maximum: +, maximum —» minimum: -) [BUX 66] [DEJ 70] [GRE 81] [NEL 78] [RAV 70] [SCH 63] [SCH 72a] [STR 73] [VAN 71] [WEB 66].

Figure 5.14. Range counting

As opposed to the above methods, we obtain here information on the real variations of the stress, on the sequence (but not on the maximum amplitude) [LEY 63] [WEB 66]. All the ranges counted are considered as symmetrical with respect to the zero axis [HAA 62]. The damage is then calculated from these ranges by considering them as half-cycles of sinusoidal stress using a S-N curve or its representation (Basquin for example), and with a accumulation rule (Miner, ...), by

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considering (or not) the effect of the mean stress of each half-cycle (Goodman, Gerber etc [STA 57]). Several methods were proposed to take into account the effect of non zero mean stress. For example, curves stress / number of cycles to the rupture are given, where Aa / 2 the y-axis carries, instead of a or Aa, the quantity [SMI 42] or 1

-°m / Rm

l-°m/Rf

Aa = Stress range am = Mean stress Rm = Ultimate tensile strength Rf = True ultimate tensile strength (load to the fracture divided by the minimal cross-section after fracture) The calculation of damage by this process is relatively easy when the signal is periodic, the counting of the ranges being carried out over one period. If the signal is random, calculations are more difficult and it can be worthwhile to consider the statistical properties of the signal. For a Gaussian signal for example, it can be shown that the mean value Aam of the ranges Aa is equal to:

[5.3] O

rms

CT

= rms value

rms

of the stress a

= rms value

of the first derivative of a

Aam - fin ams r

r = irregularity factor n0 = number of zero crossings (centered signal) per unit time np = number of extrema per unit time.

[5.4]

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5.4.2. Elimination of small variations The method makes it not possible to take into account the stress variations lower than a given threshold and considered as not very important for the fatigue process. J.B. de Jonge [DEJ 70] note that this method presents a serious disadvantage. Being given the signal of Figure 5.15, three ranges can be counted:

and

Figure 5.15. Elimination of small variations If in consequence of an unspecified filtering, the small variations such as a 2 —0 3 are removed, counting will give only o 4 -a 1 instead of the first three already quoted ranges. The result of counting depends much on the amplitude of the small load variation considered. In addition, there is always the possibility of not being able to pair the positive and negative ranges if both are counted.

Figure 5.16. Example of range counting

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The range counting method detects variations which actually take place, but neglects the variations of the mean load. By centring all the ranges on the zero value, it tends without reason to support the number of the smallest loads in the spectrum [RAV 70] [SCH 63]. A characterization of the endurance to fatigue requires knowledge of the range and its mean value (or the maximum and minimum values) [WEB 66]. NOTE. — In the case of a Gaussian process, the plotting of the load spectrum can be carried out, rather than by counting, from an approximation of the distribution law of the ranges (increasing or decreasing direction). The exact determination of this distribution in a continuous random process is very difficult [RIC 65a]. There are however techniques making it possible to obtain an approximate relation, among which are those of: - H.P. Schjelderup and A.E. Galef [SCH 61a], which treat the particular case of Gaussian processes of which the PSD is made up of narrow peaks of very remote frequencies; -J. Kowaleski [KOW 61a], which gives a distribution law for a Gaussian process, without demonstration (but this law was considered to be incorrect by J.R. Rice, F.P. Beer and P.C. Paris [RIC 65a]); — J.R. Rice and F.P. Beer [RIC 64] [RIC 65a], which proposes a relation calling upon the autocorrelation function of the signal and its first four derivatives.

5.5. Range-mean count method Or range-mean pair count method, or means and amplitudes.

5.5.1. Presentation of method This method was proposed in 1939 by A. Teichmann [TEI 41]. A gap in the range counting method, the absence of measurement of the mean value of the signal, is eliminated here. The ranges are counted as with the above method and the mean value of the range is noted complementary to Aa [BUX 66] [FAT 77a] [NEL 78] [RAV 70] [SCH 63] [VAN 71].

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Fatigue damage Table 5.1. Example of statement of ranges and means

AB BC CD OF EF FG

i

Ad;

1 2 3 4 5 6

8 -3.5 5.5 -7 6 -10.5

CT

mi

1

3.25 4.25 3.25 3 0.75

Figure 5.17. Counting of ranges and means

Figure 5.18. Example of range and mean counting

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Some authors prefer to consider Aa1 / 2. These values make it possible to constitute a histogram with two dimensions (for each mean corresponds to a list of ranges) [STR 73]. The information collected being rich, the results of some of the other methods can result from the results of this one [WEB 66]: the range counting method is a particular case of the range-mean count method. In the same way, the results of this counting can be transformed to find those of the peaks counting method or the level-crossing count method. The mean is defined as the algebraic half-sum of two successive extrema:

The amplitude is the algebraic half-difference between these extrema.

Figure 5.19. Example of presentation of results of a cow

The results can be presented in the shape of curves giving, for a given mean a'mi , the occurrence frequency versus the range amplitude (or the half-amplitude). mi The distributions thus obtained have the same form whatever the mean a m i This method is generally regarded as most significant in fatigue [LEY 63]. It takes into account the assumption that the fatigue behaviour is mainly a function of the amplitude of the stresses around the mean value [GRE 81] [RAV 70]. The reason for the non-utilization of this method is two-fold [HAA 62]: -the instrumentation necessary is complicated. An equipment however was developed for this use (strain analyser); Verhagen and De Docs [VER 56] [WEB 66]), - reproduction of the loads counted is not easy.

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5.5.2. Elimination of small variations As with the above method, the suppression of small variations of a is not simple, since it results in modifying the preceding and following ranges. It is the weak point of these methods, sensitive to the smallest range neglected [WEB 66]. Elimination is carried out by removing all the variations of less than a given value. On the curve where these small variations take place, the mean and the interval between the peaks are calculated immediately before and after these small fluctuations.

Figure 5.20. Elimination of small variations

Range amplitude

Figure 5.21. Frequency of event after elimination of small variations

In general, this treatment especially results in removing the small variations close to the mean, which have little effect on fatigue. From these results, it is possible to carry out a calculation of fatigue damage as follows [STA 57]. Let us consider, as an example, the signal in Figure 5.22 (a). This signal can be smoothed to eliminate the small irregularities (Figure 5.22 (b)).

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Figure 5.22 (a) to (d). Decomposition in ranges

The signal is then broken up into 4 ranges AB, BC, CD and DE which are used to define cycles ABB', BCC etc (Figure 5.22 (c)). The damage due to the signal (a) will be half of that due to the complete cycles of (c). The cycles are centred with respect to the zero mean value, by modifying the amplitudes obeying a rule such as Gerber or modified Goodman (Figure 5.22 (d)). By translation according to the time axis, a continuous signal is reconstituted (Figure 5.23) producing a damage double of (a), which can then be evaluated using Miner's rule (for example).

Figure 5.23. Process of cycle reconstitution

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5.6. Range-pair count method Or range-pair cycles counting or range-pair excedance count method. This method, which was the object of several variants [BUR 56] [BUX 66] [FUC 80] [GRO 60] [JAC 72] [RAV 70] [SCH 63] [SCH 72a] [VAN 71] consists in counting a stress (or deformation) range as a cycle if it can be paired with a subsequent stress (or deformation) range of equal amplitude in the opposite direction.

Figure 5.24. Counting of range-pairs The ranges are thus counted per pair. A range is defined here as a load variation starting from an extremum. A counting is carried out if the positive range exceeds a certain value r and if it is followed by a negative range exceeding the same value r [STR 73] [TEI 55] [VAN 71]. The intermediate variations (r e ) are neglected.

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Figure 5.25. Process of counting of ranges by pairs

The procedure of counting is repeated for other values of r.

Figure 5.26. Example of counting of ranges by pairs

The result of counting can be expressed in the form of the number of range pairs exceeding a given value r. This method counts mainly the greatest fluctuations. The small fluctuations are regarded as superimposed on the largest. If the pairs of ranges are counted for |r| > r2, the intermediate positive or negative ranges, for which r < r2, do not affect the result of counting. They can thus be eliminated.

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Fatigue damage

This method takes into account the sequence of load and is insensitive to the amplitude of the smallest load ranges considered (counting thresholds). The small ranges can thus be counted or neglected at will, without any effect on the large ranges [DEJ 70] [WEB 66]. The method combines the greatest load increments with the smallest following decrements, which is important from the damage point of view. On the other hand, any information is stored in the mean value of the counted cycles. The ranges are paired without respect to their occurrence time. Each counted element is insensitive to the ranges other than those of its own previously chosen level. All the load excursions are not fully counted.

Figure 5.27. Combination of stress increments

This method can be used in the plastic range by associating the pairs of plastic deformation ranges with the pairs of stress ranges [KIK 71]. Measuring instrument: Strain Range Counter [TEI 55] [WEB 66].

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Figure 5.28. Association of deformation and stress ranges

NOTES. 1. N.E. Dowling [DOW 72] proposes to carry out counting according to the procedure described below, being based on Figure 5.29.

Figure 5.29. Counting according to N.E. Dowling [DOW 72]

Starting from a minimum 1, the signal increases to the largest peak 6, which is for the moment the only one selected. With this rise 1 - 6 the descent 6 - 1' of the

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Fatigue damage

same amplitude is associated. When we start from peak n° 2, the signal decreases according to 2 - 3 and we associate 3 - 2' with this variation, etc. The result is close to that of the rainflow counting method (paragraph 5.12). 2. O. Buxbaum [BLJX 66] (or T.J. Ravishankar [RA V 70]) define classes of range ri, r1 being the smallest amplitude chosen for the counting, the lower values being neglected.

Figure 5.30. Counting according to O. Buxbaum [BUX 66]

Starting from minimum 1, we will have a range each time the signal crosses a limit of class with positive slope. The range is closed when a negative variation of the same amplitude is crossed. For example, we open range AA' in A, BB' in B, CC' in C and we close these same ranges in A', B' and C'... These cuttings per pair of ranges are however artificial and, if the stresses are in the plastic range, can lead to an inaccurate description of the stress-strain cycles, i. e. to an error in the calculated damage [WAT 76]. 5.7. Hayes counting method This concerns a similar method to that of the rainflow, but easier to visualize [FUC 77] [FUC 80] [NEL 78]. One starts here by counting the cycles of smaller amplitude, then they are removed from the curve to count the cycles of greater amplitude. Let us consider as an example the signal in Figure 5.31 (a).

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Figure 5.31. Hayes counting procedure

Hayes first identifies small ranges such as BC and DE, which are interruptions of a larger range (AF). There is interruption: - for a peak-valley pair such as BC when the following peak, D, is higher than the peak B; - for a valley-peak pair, such as GH, when the next valley, I, is of smaller amplitude than the valley G. The hatched surface corresponds to the cycles relating to the ranges thus selected. In Figure 5.31, 20 alternations (20 peaks) are counted. Among these peaks, 14 are retained to constitute 7 pairs materialized by the hatchings, then eliminated from the curve (Figure 5.31 (b)). There remain 6 alternations (6 extrema) with 3 peaks and 3 valleys. The two hatched cycles I - L and R - U are counted, then are removed to obtain the curve in Figure 5.31 (c) in which there remains only the peak F and the valley M. The result of this counting can be expressed in the shape of a table of occurrence of the ranges with, possibly, their mean value.

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5.8. Ordered overall range counting method Or racetrack counting method. Let us consider, to illustrate the method, the signal in Figure 5.32 (a) which will be treated to obtain that in Figure 5.32 (c). The elimination of the smallest ranges is carried out as indicated in Figure 5.32 (b) [FUC 80] [NEL 77].

Figure 5.32. Racetrack counting method

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The method consists of defining a track of width S, limited by barriers which have the same profile as the signal. One retains as extremum only those for which a racing driver would have to change direction from top to bottom, as in f and n, or from bottom to top, as in m and o. The width S of the track determines the number of alternations which will be counted. This method was originally named ordered overall range method [FUC 77] [NEL 78]. Its aim is to condense a long and complex signal into a signal simpler to use. The procedure used is based on the assumption that the most significant part of the signal is the maximum range, i.e. the interval between the largest peak and the lowest valley. The distance between the second lowest valley and the second largest peak is the next most important element, provided that this second range crosses the first (between the largest peak and the smallest valley) or that it is outside the interval of time defined by the extreme values of the first range. By continuing the process in this manner, one can either exhaust the counting of all alternations, or to stop counting to a selected threshold and to regard all the smaller peaks and valleys as negligible. The results can be presented in two ways: - in the form of a histogram, of a spectrum or a list giving the range amplitudes and their occurrence frequency. If counting is carried out until exhaustion of the peaks and valleys, this method leads to the same result as a rainflow type counting and to a result very close to the range-pair count method . The racetrack method makes it possible to stop counting before exhaustion of the extrema and to take into account only a fraction of the ranges. In many cases, that led to the same result as an exhaustive counting of alternations, - in the form of a synthesized curve in which the essential peaks and valleys are the subject of a list with their origin sequence. If the procedure is continued until exhaustion of all the extrema, one then reproduces the initial curve in all its details. If counting is limited, small variations of the curve are omitted, but one retains alternations which produce the greatest ranges. The condensed curve contains more information than the spectrum. It includes the sequence of events, which can be important if the deformation produces residual stresses which remain active when following alternations. This method chooses initially the greatest ranges and preserves them in a correct temporal sequence. It is useful to condense curves by preserving only a few events, perhaps 10%, which produce most of the damage, usually more than 90% [FUC 77]. These condensed curves make it possible to reduce the duration of calculations and the tests and to focus the attention on a reduced number of the most significant events.

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5.9. Level-crossing count method Or simple level-crossing method.

Figure 5.33. Counting of level crossings

One counts here the number of times that the signal crosses a given level ao with a positive slope, depending on CT0 [BUX 66] [GOO 73] [NEL 78] [RAV 70] [SCH 63] [SEW 72] [VAN 71].

Figure 5.34. Example of level-crossing counting

The instrument developed for this measurement is the contact extensometer (Svenson) [SVE 52] [WEB 66]. One can expect that the mean level (zero in general) is the most often crossed, since the number of crossings of threshold falls when the

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threshold increases. Counts can be carried out with positive slope or negative slope above and below the mean (the results being the same) [GOO 73]. If the signal has a symmetrical distribution, it is enough to choose a direction and to count only above the mean [STR 73]. The passages below a certain threshold can be eliminated. If they are very few, the small load variations can be neglected without appreciable effect on the number of up-crossings. Very different models of load can lead to the same counting result [GOO 73]. The insufficiency of the method can be shown from Figure 5.35 [VAN 71].

Figure 5.35. Different signals having the same number of level crossings

Although very different, the two signals above lead to the same number of crossings of each level. The small load variations, which are of less importance in the fatigue process, increase counting. To compensate (at a certain point) for this source of error, level crossings associated with load variations lower than a given threshold can be neglected [GOO 73] (see the fatigumeter method). This is equivalent to defining classes of amplitudeCTJ,CTJ+ ACT of width ACT such that any oscillation included inside a class is not counted, by suitable choice of ACT [BUX 66]. All the small oscillations do not disappear. One continues to take into account those which cross the limit between two classes. The neglect of small alternations can however have an influence on the results [WEB 66].

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Figure 5.36. Elimination of small alternations

The level-crossing counting method can be used to compare the signal distribution with that of a Gaussian signal [BUX 66] [ORE 81]. This counting is sometimes used to calculate the number of peaks between two levels, by difference of the up-crossings of these two levels.

Figure 5.37. Example of error in the calculation of the number of peaks from a level-crossing counting

This step can lead to errors [SCH 63] [SCH 72a] [SVE 52] [VAN 71]. H.A. Leybold and E.C. Naumann [LEY 63] show that in a given interval of amplitude, the calculation of the number of peaks carried out from the up-crossings of levels can be false because of a certain compensation of maxima and minima. An example is given from the signal in Figure 5.37; no peak is detected by difference of crossings of thresholds 2 and 1 whereas there is actually one. As a consequence, one notes that the number of peaks calculated in the interval close to the mean is very weak.

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This method of calculation of the number of the peaks can however be used provided that the signal is stationary, Gaussian, with narrow band, and that the threshold is sufficiently large compared with the rms value.

Figure 5.38. Frequency of event of peaks deduced from counting of crossings of threshold

It is also sometimes used to plot curve occurrence frequency - peak amplitude [LEY 63]. Modified level-crossing method

Figure 5.39. Level-crossing counting method modified to eliminate small variations

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This method, sometimes used in the car industry, consists of: - choosing load levels (stress, deformation etc), - between several consecutive zero-crossings, recording only one counting each time that a level is exceeded (a level can be exceeded several times between two passages by zero, but only the first level up-crossing is counted).

Figure 5.40. Example of very different signals leading to the same result from using the modified level-crossing counting method

This method has it also its limits. It will produce for example same counts for signals (a) and (b) in Figure 5.40. In this case, (a) is probably much more damaging than (b). This method is not appropriate to detect the important differences between such loads. 5.10. Peak valley peak counting method OrP.y.P. method. The direct reproduction of the signal (stress or deformation) is not very practical, because it is too long to be able to expect reliable statistics [HOL 73].

Figure 5.41. Definitions

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The methods related to the use of a stress spectrum in general result in carrying out, during the fatigue tests of materials on test-bars, blocks of sinusoids not very representative of reality (fluctuations of amplitudes and of half-cycle durations). The histograms do not give information on the sequence or on the mean-range interactions, which have an effect on the fatigue life. The PVP method takes account of these elements and makes it possible to reduce the test duration in a notable manner. It consists of: - to analyse the signal measured and to plot: - a histogram of the range mean (row), - a histogram of the ranges,

Figure 5.42. Histogram of the ranges

Figure 5.43. Histogram of the means

Figure 5.44. Histogram of the ranges after suppression of events below fatigue limit

- to modify this last histogram by eliminating levels lower than the fatigue limit and considered this as without effect, - to reconstitute a signal from the remaining blocks. The method which would result in considering blocks of sinusoids by respecting the levels and numbers of cycles is not used, because it is considered non representative here.

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Figure 5.45. Constitution of the peak and valley matrix

Instead, the PVP method uses a matrix such as that in Figure 5.45, including, according to each axis, classes located by their medium level, with the values of the peaks on the vertical axis and those of the valleys on the horizontal axis. Each event is placed in the appropriate cell. This matrix thus makes it possible to describe the signal using the two parameter range - mean value since diagonal T is at constant mean, and diagonal '2' is at constant range.

Figure 5.46. Example of matrix

Figure 5.47. Diagonals of the matrix [HOL 73]

By using the properties of materials under fatigue, the authors propose a method of elimination of the levels lower than the fatigue limit, leading to a very important reduction of time (90% in the given example). The procedure is:

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— to note the fatigue limit of materials used and its ultimate strength («D-^X - to plot the Goodman diagram (with the range on ordinates and the mean on abscissae), in order to determine the boxes of the matrix which are below aD. This results in eliminating at the same time values in the histogram of ranges and of averages, -to generate a random signal. One starts in general from a valley and one chooses a valley-peak range in the matrix: 25 and 75 for example (Figure 5.50). One misses by 1 the number of occurrences indicated in the box.

Figure 5.48. Goodman diagram

It is then necessary to search a range starting from a peak of amplitude 75; this choice can be carried out in one of hatched boxes in Figure 5.49. In the example treated, the possible valleys following +75 are -75, -25 and +25 (five ranges on the whole), the probabilities being thus 1/5, 2/5 and 2 / 5. One thus chooses the following range taking into account these last values. Let us suppose that the selected transition is (+75, -25). The following transition will go towards a peak and will have to be selected in the third column (Figure 5.52). There are two possibilities, 125 and 75, with the same probability.

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Figure 5.49. Example of matrix

Figure 5.50. Choice of first range

Figure 5.51. Constitution of a signal

Figure 5.52. Choice of second range

Figure 5.53. Addition of range on the signal

If 125 is chosen, the next valley will have to be taken in the hatched line, and one proceeds thus until the table is empty. Towards the end of the signal, it will be

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possible that direct connections are impossible, some boxes being already empty. In this case, one will connect the arcs continuously. Frequencies We saw that the life in fatigue is, to a certain extent, independent of the frequency. This parameter is thus not taken into account. The authors use rather the velocity limitations of the actuators to determine the duration of each event.

Figure 5.54. Connections of unjoined arcs

5.11. Fatigue-meter counting method Or level crossings eliminating small fluctuations or restricted level-crossing count method or variable reset method. The fatigumeter was developed especially in aeronautics to measure and record vertical accelerations in the centre of gravity of aircraft in flight, for fatigue studies [LAM 73] [MEA 54] [RID 77]. Use of this equipment, such as the VGH recorder for example, or Fatigue-meter (RAE) [TAY 53], is limited apart from this particular application [HAA 62]. This material uses a level-crossings counting method with elimination of the small load variations.

Figure 5.55. Fatigumeter counting

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For that, a count of a level up-crossing above the mean stress is retained only if the stress is crossed before at least one interval A previously chosen with a positive slope [GOO 73] [NEL 78] [RAV 70] [SEW 72] [SCH 63] [SCH 72a] [SIR 73] [TAY 50]. This increment A corresponds to a stress interval below the fatigue limit stress and thus negligible from the point of view of fatigue [LEY 63]. It is equivalent to saying that a rearmament threshold is used as when: — for the levels higher than the mean, counting is started at the time of the passage of a given level; - a new passage on this same level gives place to a counting only if, meanwhile, the signal has descended rather low and crossed at a lower level fixed in advance with a negative slope before ascending [NEL 78] [RAV 70].

Figure 5.56. Counting with elimination of small variations

Counting is similar for the levels lower than the mean. This method thus introduces a new parameter, the release threshold, which can vary according to the level (all the larger since the level is higher) [GRE 81]. It has the advantage of recording the majority of the secondary load variations of some importance [SEW 72]. It results in neglect of certain small variations. There is loss of information on the sequences. But the small variations do not influence counting. The number of peaks calculated in each class by subtracting the number of upcrossing of a level i from the number of up-crossings of the following threshold i +1 is not exact, as with the preceding level-crossing count method.

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With this method, very different signals can still lead to the same result [VAN 71]. Example

Figure 5.57. Example of signals leading to the same number of crossings

This problem is here also related to small variations in stress. Example of threshold levels [GOO 73]: Table 5.2. Examples of primary and secondary threshold values

Acceleration levels in g Primary 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 Secondary 0.00 0.05 0.05 0.10 0.10 0.15 0.15 0.20 0.20 0.25 0.25 It is seen that the higher the primary level, the higher the secondary level (and thus the threshold). 5.12. Rain flow counting method Or Pagoda roof method. 5.12.1. Principle of method This method [DOW 72] [DOW 87] [FAT 93] [FUC 80] [KRE 83] [LIN 87] [NEL 78] [RYC 87] [SHE 82] [SOC 83] [WIR 77] was initially proposed by

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M. Matsuiski and T. Endo [END 74] [MAT 68] to count the cycles or the half-cycles of strain - time signals. Counting is carried out on the basis of the stress-strain behaviour of the material considered [STR 73]. The extracted cycles are in conformity with those of the tests of constant amplitude from which the predictions of fatigue life are generally carried out.

Figure 5.58. Stress-strain cycles

This is illustrated in Figure 5.58. In this example, the response e(t) can be divided into two half-cycles ad and de and into a complete cycle bcb' which can be considered as an interruption of a larger half-cycle. Let us imagine, to describe this method and to explain the origin of its name, that the time axis is vertical and that the signal a(t) represents a series of roofs on which water falls. The rules of the flow are the following ones:

Figure 5.59. The drop is released from the largest peak

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The origin of the signal is placed on the axis at the abscissa of the largest peak of the signal (this condition is not necessary with certain algorithms derived from this method) [FUC 80]. Water drops are sequentially released at each extrema. It can be agreed that the tops of the roofs are on the right of the axis, bottoms of the roofs on the left.

Figure 5.60. Flow rule of the drop from apeak

If the fall starts from a peak: a) the drop will stop if it meets an opposing peak larger than that of departure, b) it will also stop if it meets the path traversed by another drop, previously determined, c) the drop can fall on another roof and to continue to slip according to rules a andb. NOTE. - A new path cannot start before its predecessor is not finished If the fall begins from a valley [END 74] [NEL 78]: d) the fall will stop if the drop meets a valley deeper than that of departure, e) the fall will stop if it crosses the path of a drop coming from a preceding valley, f) the drop can fall on another roof and continue according to rules d and e. The horizontal length of each rainflow defines a range which can be regarded as equivalent to a half-cycle of a constant amplitude load.

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Figure 5.61. Drop departure from a valley

Figure 5.62. Flow rule of the drop from a valley

Example Let us consider the signal in Figure 5.63, the time axis being plotted vertically. On this Figure, pairs of ranges can be identified which, taken together, are equivalent to whole cycles: a) (1 - 8) + (8 - 11): b)(2-5) + (5-C) c) (6 - 7) + (7 - B) d)(3-4) + (4-A) (9-10) + (10-D)

largest cycle of the sequence \ ) cycles of intermediate size ) I the smallest cycles

The damage can be calculated from these cycles. Intermediate ranges (2 - 5) and (5 - C) are simply interruptions of a larger range (1 - 8). Small ranges (3 - 4) and (4 - A) are, in their turn, of the interruptions of the intermediate range (2 - 5), etc.

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Figure 5.63. Example of signal for a rainflow count

This method lends itself with difficulty to an analytical formulation [KRE 83]. Several algorithms were developed to carry out counting (cf. paragraph 5.12.2) [DOW 82] [DOW 87] [FAT 93] [RIC 74] [SOC 77]. It is currently rather well established that this method gives the best estimates of fatigue life in the case of cycles of strain under load of variable amplitude [DOW 72] [FAT 77b].

Another example Figure 5.64 (a) shows a signal and the part of this signal retained in a first passage. The left ranges are those of Figure 5.64 (b). The procedure is still applied to the remaining peaks and there remains then only the peak of Figure 5.64 (c). The result of this counting gives the half-cycles (25;-14), (14; 5), (16;-12) and (7; 2). The rainflow counting gives here the same result as the range-pair count method, except the small additional half-cycle (7; 2) counted here.

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Figure 5.64. Example of application of the rainflow counting process

Figure 5.65. Rainflow method combined with strain analysis [MAR 65]

The rainflow method is interesting when it is combined with a strain analysis [FUC 80]. Let us consider for example the case in Figure 5.65. The damage can be calculated for each cycle as soon as it is identified in the counting procedure.

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Each part of strain time history e(t) is counted only once. The damage created by a great variation of the strain is not affected by an interruption which produces a small cycle. For each cycle of stress, one follows a closed hysteresis loop returning at the initial point of the beginning of cycle, the preceding curve continuing as if this cycle had not existed. There is simply addition of the damage. The signal e(t) can thus be broken up into half-cycles whose terminals are known and which one can calculate the mean value. These data make it possible to calculate the damage undergone with an accumulation law [WAT 76].

NOTES. 1. There are methods very similar to this one (the 'maximum — minimum procedure', the 'pattern classification procedure') [END 74] which give the same results. 2. The rainflow method is very similar to the range-pair counting method. If we consider again the Figure 5.29, the difference between these two methods can be explained by considering the half-cycle 1 -6. In the case of the range-pair counting, the half-cycle which is complementary for it is 6 - 7' and the part 1' - 7 is lost. With the present method, when the range which could constitute the second half-cycle has an range larger than 6-1' (which is the case here, with 6 - 7), counting 1 - 6 is suspended (on standby of another later complementary half-cycle) and a new counting 6—7 is opened. In addition, the residual half-cycles are preserved here (those which could not be paired), whereas the range-pair counting method is unaware of them. 5.12.2. Subroutine for rainflow counting The routine below makes it possible to determine the ranges of a signal to use for the calculation of the fatigue damage according to this method. The signal must be first modified in order to start and to finish by the largest peak. The total number of peaks and valleys must be even and an array of the peaks and valleys (Extrema()) must be made up from the signal thus prepared. The boundaries of the ranges from the peaks are provided in arrays Peak Max() and Peak_Min() and those of the ranges resulting from the valleys in Valley_Max () and Valley_Min(). These values make it possible to calculate the two types of ranges Range_Peak( ) and Range_Valley( ), as well as their mean value Peak_Min(i) + Peak Max(i) , Valley_Min(i) + Valley_Max(i) and .

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' Procedure RAINFLOW of peaks counting (GFA-BASIC) 'According to D. V. NELSON [NEL 78] ' The procedure uses as input/output data: ' Extremum(Nbr_Extrema%+2)=array giving the list of Nbr_Extrema% extrema ' successive and starting from the largest peak ' At output, we obtain: ' Peak_Max(Nbr_Peaks%) and Peak_Min(Nbr_Peaks%)= limits of the ranges of the peaks ' Valley_Max(Nbr_Peaks%) and Valley_Min(Nbr_Peaks%)= limits of the valley ranges ' These values make it possible to calculate the ranges and their mean value. ' Range_Peak(Nbr_Peaks%)= array giving the listed ranges relating to the peaks 'Range_Valley(Nbr_Peaks)= array of the ranges relating to the valleys i PROCEDURE rainflow (Nbr_Extrema%,VAR ExtremumQ) LOCAL i%,n%,Q%,Output&,m%j%,k% ' Separation of peaks and valleys Nbr_Peaks% = (Nbr_Extrema% + 1) / 2 FOR i% = 1 TO Nbr_Peaks% Peak(i%) = Extremum(i% * 2 - 1) NEXT i% FOR i% = 2 TO Nbr_Peaks% Valley(i%) = Extremum(i% * 2 - 2) NEXT i% ' Research of deepest valley Valley_Min = Valley(2) FOR i% = 2 TO Nbr_Peaks% IF Valley(i%) < Valley_Min Valley_Min = Valley(i%) ENDIF NEXT i% Valley(Nbr_Peaks% + 1) = 1.01 * Valley_Min ' Treatment of valleys FOR i% = 2 TO Nbr_Peaks% // Initialization of the tables with Peak(l) L(i%) = Peak(l) LL(i%) = Peak(l) NEXT i% FOR i% = 2 TO Nbr_Peaks% n% = 0 Q% = i% Output& = 0 DO // Calculation of the Ranges relating to the Valleys IF LL(i% + n%) < Peak(i% + n%)

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Range_Valley(i%) - ABS(LL(i% + n%) - Valley(i%)) // Array of the Valleys Ranges Valley_Max(i%) = LL(i% + n%) //Array of the Maximum of the Ranges of the Valleys Valley_Min(i%) = Valley(i%) // Array of the Minimum of the Ranges of the Valleys Output& = 1 ELSE IF Valley(i% + n% + 1) < Valley(i%) Range_Valley(i%) - ABS(Peak(Q%) - Valley(i%)) Valley_Max(i%) = Peak(Q%) Valley_Min(i%) - Valley(i%) Output& = 1 ELSE IF Peak(i% + n% + I) < Peak(Q%) L(i% + n% + 1) = Peak(Q%) n% - n% + 1 ELSE L(i% + n% + 1) = Peak(Q%) Q% - i% + n% + 1 n% = n% + 1 ENDIF ENDIF ENDIF LOOP UNTIL Output& = 1 m% = i% + 1 IF m% < = Q% FORj% = m%TOQ% LLG%) = LO'%) NEXTj% ENDIF NEXT i% ' Treatment of peaks FOR i% - 2 TO Nbr_Peaks% + 1 // Initialization of the arrays with Valley JMin L(i%) = Valley_Min LL(i%) = Valley_Min NEXT i% FOR i% = 1 TO Nbr_Peaks% n% - 0 k% - i% + 1 Q% = k% Output& - 0 DO IF LL(k% + n%) > Valley(k% + n%)

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Range_Peak(i%) - ABS(Peak(i%) - LL(k% + n%)) //Array of the Ranges of the Peaks Peak_Max(i%) = Peak(i%) //Array of the Maximum of the Ranges of the Peaks Peak_Min(i%) = LL(k% + n%) // Array of the Minimum of the Ranges of the Peaks Output& = 1 ELSE IF Peak(k% + n%) > Peak(i%) Range_Peak(i%) = ABS(Peak(i%) - Valley(Q%)) Peak_Max(i%) - Peak(i%) Peak_Min(i%) - Valley(Q%) Output& = 1 ELSE IF Valley(k% + n% + 1) > Valley(Q%) L(k% + n% + 1) = Valley(Q%) n% = n% + 1 ELSE L(k% + n% + 1) = Valley(Q%) Q% = k% + n% + 1 n% = n% + 1 ENDIF ENDIF ENDIF LOOP UNTIL Output& = 1 m%-k% + 1 IF m% < = Q% FORj% = m%TOQ% LL(j%) = L0%) NEXTj% ENDIF NEXT i% RETURN

5.13. NRL counting method (National Luchtvaart Laboratorium) Or range-pair-range count method This concerns an extension of the range-pair counting method taking into account information on the mean value and thus making it possible to characterize the extreme levels correctly [SCH The procedure of counting is applied in two steps. In the first, all the intermediate cycles of load are detected and counted by noting the associated mean values. Each cycle of intermediate load counted is then eliminated from the signal

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varying with time [VAN 71]. The procedure is repeated until the signal does not present any more cycle of intermediate load. The signal obtained at the end of this first step is of oscillatory type with narrow band, with cycles of variable amplitude, each extremum being preceded and being followed by a zero-crossing. In the second step, the residual load is analysed according to the range-mean count method. Example Let us consider the sample of signal in Figure 5.66 (a).

Figure 5.66. Principle of the NRL counting method (step n° 1)

Step n° 1 Four successive peaks i to i + 3 are considered. If x i+1 and x i+2 are in the interval xi, xi +3 , two half-cycles of amplitude ——

— and mean —^

—

are counted. The values x i+1 and x i + 2 are then removed from signal (b). The procedure is repeated for the peaks Xi_ 2 , Xi_ l Xi and x i + 3 , and led, after counting and removal of the range Xi_1, xi to the signal (c) [DEJ 70].

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The repetition of this procedure on all the signal led to a new signal of the form of that of Figure 5.67 at the end of step n° 1.

Figure 5.67. Signal obtained at end of step n° 1

Step n° 2 Counting according to the range-mean method consist of noting the couples

These countings are added to those of step n° 1.

The total result can be described as the number of cycles (range Aa, mean am) for each combination of ACT and cr m . This representation is not very convenient because of its two-dimensional character. As fatigue is primarily due to the amplitude of the load (range) and, with the second order, to the mean, one decides here to calculate a mean value CTm(Ac) for each value of amplitude ACT and to preserve the standard deviation of the mean as information on the variations of the value CTm, i.e.

This description makes it possible to represent the result in a monodimensional form. For each amplitude of range ACT, one gives [DEJ 70]:

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- the number of cycles, -the mean value of these cycles a m (Aa), -the standard deviation of the mean sm(Aa). It is considered that this method gives more significant information from the fatigue point of view than the other methods described (except for the rainflow, to which it is close [VAN 71]). It has the same advantages as the range-pair counting method without having their already quoted disadvantages [SCH 72a].

5.14. Evaluation of time spent at given level The range of variation of the signal is broken up into sections of small width ACT,. Being given one of these sections included between the levels a, and G{+\, one counts the times At; during which the signal remains inside this section. The sum of these Atj is a fraction of total time T of the signal studied.

Figure 5.68. Time of staying at given level

This fraction is all the larger since the section is broader. To eliminate this disadvantage, one weights it by the width A<JJ. The results so obtained make it possible to calculate the probability density.

[5.7]

Very often, this density can be approached by that of a Gaussian law.

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This method, which is based on the time spent in a section ACT, is thus very dependent on the first derivative of the signal. The result will be very different according to whether the signal varies quickly or slowly. It represents poorly the frequency of appearance of the levels. It is a method little used [BUX 66]. 5.15. Influence of levels of load below fatigue limit on fatigue life From a synthesis of much work carried out on alloys 2024-T3 and 7075-T6, T.J. Ravishankar [RAV 70] showed that the low levels, lower than the fatigue limit, produce damage while contributing to the propagation of the fissure when this was started. Their omission thus increases the life time. 5.16. Test acceleration One can be led by this objective to increase the amplitudes of the sequence. It is advisable to be careful and to increase only those of the cycles of average size, an increase in the amplitude of the strongest levels being able to modify the fatigue life of the test-bar [SCH 74]. Let us consider a signal whose instantaneous values are distributed according to a Gaussian law and peaks according to Rayleigh's law. 1. The spectrum is maintained linear by increasing the probability uniformly.

Figure 5.69. Reduction of the duration of a test by increasing all the stresses

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The highest load levels occur more frequently while the smallest loads are less affected. This case is similar to the in situ situation where the segments of the worst road are traversed with a frequency higher than the normal [BUS 72]. W. Schtitz [SCH 72b] considers that if a test must be accelerated, that should not be carried out by increasing the stresses, but rather by neglecting the stresses of low amplitude. So that a test is representative, it is necessary to simulate the real stresses as accurately as possible with respect to the sequences and the amplitudes. The reduction of the duration of a test by increasing the frequency (central frequency of the narrow band noise) does not have a great beneficial effect, for there is an accompanying increase in the fatigue life of the material [KEN 82]. 2. Another method can consist in neglecting the low levels [SCH 74] (loads lower than a certain percentage of the static ultimate load, 2% for example). J.D. Tedford, A.M. Carse and B. Grassland [TED 73] note that the levels of stress lower than 1.75 times the rms stress do not have any significant effect on the fatigue damage. Their elimination makes it possible to reduce the duration of test considerably (it is to be noted that M.N. Kenefeck [KEN 82] gives as a limit 2.5 times the rms value). It is on the other hand important to have a good knowledge of the highest levels of the environment since they produce the damage.

Figure 5.70. Reduction of duration of a test by elimination of the lowest stresses

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The method consists in distorting the low part of the load spectrum by considering that the low levels do not contribute to fatigue. This is equivalent to considering that the vehicle does not run on the best parts of the road. NOTE. - A combination of the two methods (rotation of the spectrum and suppression of the low levels) is possible and is analogous to bumping along roads where potholes and paving stones are present [BUS 72].

Figure 5.71. Reduction of duration of a test by suppression of the lowest stresses and by increase in other stresses Although one can think a priori that such a suppression has little consequence, this step must however be applied with prudence [NAU 64]. A synthesis of various publications shows indeed that the low levels, lower than the fatigue limit, contribute to the damage. Their omission increases the fatigue life. They take part in the propagation of the fissure (dominating phenomenon under random loading) when this is started [JAC 66] [RAV 70]. 5.17. Presentation of fatigue curves determined by random vibration tests The S-N curves plotted for sinusoidal loads give the number N of cycles to rupture (on the abscissae) against amplitude of the stress a (on the ordinates). If the load is a random vibration, the load amplitude is not constant any more and the most significant parameter (for constant PSD shape) is the rms value [BOO 70] [BOO 76] [KEN 82] [PER 74], for a given value of the mean stress.

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The number of cycles is in general badly defined (except for the narrow band noises for which there is one peak between two through-zero passages). Two possibilities exist and are used: - the number of passages through zero with positive slope [BOO 66] [BOO 70] [BOO 76] [KEN 82] [RAV 70], when this parameter is representative of the number of cycles (r close to 1), - the number of peaks above the mean value (if the signal has symmetrical positive and negative distributions) [BOO 66] [HIL 70] [PER 74] [RAV 70]. This number of peaks can be evaluated either from the counting methods, or from a statistical calculation using its PSD if the signal is Gaussian. In the case of a sinusoidal signal, these two definitions are equivalent. One notes in these tests that, contrary to the case of a sinusoidal load, a fatigue limit is not observed for steels. This result is explained easily since, for a given rms value, there is, in the signal, peaks of amplitude equal to several times this rms value. According to the characteristics of the test control system, the peak factor (ratio of the largest peak to the rms value) can be about 4 to 4.5. The comparison of the results obtained under sinusoidal and random loads can thus be carried out only by modifying the representation of the curves plotted in sinusoidal mode on the stress axis. Several representations exist, among which [PER 74]: - instead of aa (amplitude of the sinusoidal alternating stress), one uses

[5.8] (am= mean stress), - one uses —^ [RUD 75] in sine and arms calculated on the centered signal, the V2 two values for am being given (the most usual method).

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Chapter 6

Damage by fatigue undergone by a one-degree-of-freedom mechanical system

6.1. Introduction We propose in this Chapter to consider the damage by fatigue created by a random vibration on a one-degree-of-freedom linear system, of natural frequency f0, quality factor Q, and to thus provide all the relations necessary to the layout of the fatigue damage spectra (Volume 5). This calculation can be carried out directly from a sample of the vibratory signal defined in the time domain or in a statistical manner (mean and standard deviation of the damage) from an acceleration spectral density of the vibration. Unless otherwise specified, one will make the following assumptions: - S-N curve represented by Basquin's relation (of the form N CT = C), - Linear accumulation law of the fatigue damage (Miner's rule), - Vibration of zero mean, - Gaussian distribution of the instantaneous values of the vibratory signal. We will examine however at the end of this volume some different assumptions: -

S-N curve having an ultimate endurance stress, Linear S-N curve in logarithmic-linear scales, Truncated signal, Damage accumulation law of Corten-Dolan, Non zero mean stress,

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— Some other probability distribution laws governing instantaneous values of excitation.

6.2. Calculation of fatigue damage due to signal versus time Let us consider a random vibration with zero mean applied to a one-degree-offreedom linear mechanical system of natural frequency f0 and damping ratio £. If the excitation is defined by acceleration x(t) in the time domain, in analog or numerical form, it is possible to calculate, over duration T corresponding to the duration of the signal, the response of the one-degree-of-freedom mechanical system, characterized by the relative displacement z(t) between the mass and its support. If the damping ratio £, is sufficiently small, the response is appeared as an oscillation around the zero average value, passing in general successively through positive and negative peaks and of which the amplitude varies in a random way. The frequency of the response is very close to the natural frequency f0 of the system (narrow band noise). In this simple case, the response can be broken up into half-cycles which are classified and counted according to their amplitude (histogram). If the response has a more complex form, the histogram of the peaks must be established using a peak counting method such as the rainflow method [LAL 92] (cf. Chapter 5). The system being linear, the stress created in the elastic element is proportional to the relative displacement zp corresponding to each extremum of z(t) (z = 0 at these points):

If the S-N curve of the material can be treated using Basquin's analytical expression [LAL 92],

where N is the number of cycles to failure of a test bar under sinusoidal stress of amplitude a and b and C are constants characteristic of material (Chapter 3 gives some values of the b parameter), we have:

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Figure 6.1. Representation of the S-N curve

The damage undergone by the system during the application of a half-cycle of stress oi is:

and, for ni half-cycles on the stress level ai,

In this relation, Ni is the number of cycles to failure at level ai, ni is the number of half-cycles counted at this level CTi (what explains factor 2). If one defined m classes of levels zpi, the generated total damage can be written, according to the Miner's linear rule of accumulation:

NOTE. - J. W. Miles [MIL 54] shows that, in the case of a structure random vibrations, Miner's hypothesis can be regarded as correct for of fatigue life. A non-linear hypothesis of damage accumulation unimportant variation of the fatigue life when the amplitudes of the distributed continuously over a broad range.

subjected to the estimate leads to an stresses are

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6.3. Calculation of fatigue damage due to acceleration spectral density 6.3.1. General case Let us consider a record x(t) of a random vibration (acceleration) representative of a stationary ergodic process with mean zero, duration T and PSD G(f). Let us set q(zp) as the peak probability density of the stress response of the system to this vibration. The mean number of maxima (positive or negative) is equal to:

and the mean number of peaks (maxima + minima) is equal to 2 Np . The number of peaks dn of the stress response whose amplitude is included, in absolute value, between crp and ap + dap is, over duration T,

The amplitude of the stress is supposed to be proportional to relative strain level z:

The damage by fatigue dD generated by these dn peaks is, according to Miner's rule[CRA63]:

The mean damage D (or expected damage, also noted E(D) in the following pages) is the sum of the partial damages for all the positive values of a [BRO 68a] [HIL 70] [LIN 72] (one is not interested in negative maxima, nor the positive minima):

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NOTES. 1. The mean damage per peak is equal to

2. The expected fatigue life T is obtained by making D = 1, yielding:

If the S-N curve of the material can be approached using Basquin's relation N a = C ,it can be written [PER 74]: [6.15] Lastly, if the relation between the stress and the strain is linear (a = K zp ), it becomes

If we set finally

where zrms is the rms value of z(t) and G,^ that of cy(t), we have, knowing that Q(z)= }

and

q ( z ) d z = Q ( u ) = J" q(u) du

t6-18]

198

Fatigue damage

or

The calculation of the mean damage D uses the rms value of the relative displacement z rms (or the rms stress) and an integral. We saw in Chapter 1 how 7rms can be evaluated starting from a PSD defined by straight line segments. Let us suppose that the distribution of the instantaneous values of the input signal x(t) (applied to the base of the one dof system) is Gaussian. The response of the system is then itself Gaussian and the probability density p(zp) of the maxima zp of the response can be expressed analytically in the form of the sum of a Gaussian distribution and Rayleigh's distribution (Volume 3, Chapter 6) [LAL 92]:

(r = irregularity factor = n^/n£ , ratio of the expected frequency to the mean number of maxima per second). This expression can be also written in the form

where

or

Damage by fatigue

199

From [6.15] and [6.24]

Knowing thatCT= K z and that

it becomes

NOTES. 1. Depending on the value of r, the response has peaks which follow a Gauss or Rayleigh distribution or intermediary. In the case of the Rayleigh distribution, corresponding to a low damping, the response is of narrow band type and resembles a sinusoid of randomly modulated amplitude. Each positive peak is followed by a negative peak, itself followed by a positive peak, ... All the authors agree that, for a given rms stress level and a given expected frequency, the most severe loading case (one-degree-of-freedom system) is that for which the peaks of the response have a Rayleigh distribution [BRO 70a] [SCH 61b].

200

Fatigue damage

Figure 6.2. Variations of damage versus irregularity factor of response

Figure 6.2 shows the variations of the integral of the relation [6.21] versus parameter r, for b = 10. It is checked that for a constant number n_ T of peaks, the damage increases with r, i. e. when the peak distribution tends towards Rayleigh 's law. 2. The calculation of D requires the evaluation of integrals of the form

There are available approximate expressions facilitating [SYL 81] (cf. Appendices A3.8).

this calculation

6.3.2. Particular case of a wide-band response, e.g., at the limit, r = 0 In this case [CHA 85],

with

Damage by fatigue

it becomes

Let us set

is the gamma function (Appendix A1)

The integral

with

yielding

6.3.3. Particular case of narrow band response 6.3.3.1. Expression for expected damage From the expression for the damage [6.25], we can write:

It can be shown that, if x is large,

201

202

Fatigue damage

The development in series of the error function can be limited to two terms without too much error only if x >= 0.9 , the approximation being then greater than 90% of the true value. From this, the condition results

It is shown in addition [PUL 67] that 90% of the damage is produced by the reduced stresses cr/a,^ when higher than 1.85 when the signal presents a Rayleigh peak distribution, yielding

i.e.

NOTE. -Ifr real value.

^

= 0.16, the series limited to two terms gives approximately half the

If this condition is observed, D can be written, after replacement of the error function by its approximate value and after simplification,

Let us set

We have

yielding [HAL 78]

b The integral of this relation has the form of the gamma function with x = 1 + —, 2 yielding [MIL 53]:

Damage by fatigue

203

Knowing that n^ = r n* (relation [6.45] Volume 3), it becomes [BRO 68a] [CRA63] [LIN 67]:

Since

If the excitation has a PSD made up of straight line segments, the damage D can be written [1.79]:

If the straight line segments are horizontal, we have, starting from [6.35] and from [1.86],

NOTES. 1. Relationships [1.57], [1.58] and [1.59] are used to calculate zrms, zrms and zrms and then nQ and r, if the PSD is composed of broken straight lines or [1.61], [1.63] and [1.64] if the PSD of excitation is constant in the frequency interval considered. 2. Calculation of the true value of the damage D is made difficult by ignorance of the coefficients K and C. They are therefore set in practice equal to I when calculating D which is thus obtained to a near arbitrary multiplicative coefficient. This choice is in general without consequence since this damage is evaluated in order to compare the severity of several stresses on the same structure, therefore for given K and C. 3. Basquin's relation can be also written:

204

Fatigue damage

The mean fatigue damage becomes then, on the assumptions selected:

a relation in which n0 T is the number N of alternate loadings to failure. The mean time to reach failure is such that D = 1, yielding:

or

Equation [6.40] is Basquin 's relation for random fatigue of a weakly damped system with one-degree-of-freedom.m On logarithmic scales, the S-N curve for random fatigue is a line with the same slope -1 / b as for alternating fatigue and A' can be interpreted as the ultimate stress (N = 1). Particular case where the input noise is white noise The rms displacement response to a vibration l(t) of constant power spectral density varying with the pulsation Q between zero and infinite, is (Chapter 1):

If the PSD is defined with respect to frequency f,

In this last case, if the input is an acceleration,

Damage by fatigue

205

So finally one can consider that the response is narrow band (r = 1):

Fatigue life The failure by fatigue takes place, according to Miner's assumption, when D = 1, yielding the expression for the fatigue life:

If the response can be regarded as a narrow band noise,

and if moreover the input is a white noise,

6.3.3.2. Notes /. On the Rayleigh hypothesis, the damage can also be calculated from the expression [6.3]

relating to a half-cycle using

[6.48]

206

Fatigue damage

where NT is the total number of half-cycles. If NT is large, the expected value E(a ) o/a

is equal to

yielding [WIR 83b]:

2. The relationship [6.35] is exact when r = 1, the stresses being then distributed according to Rayleigh's law. We will see that for r close to unity, one obtains with this expression a very satisfactory approximation of damage. 3. When r = 1, i.e. for a narrow band stress response, the relation [6.35] can be demonstrated more directly from the probability density of the Rayleigh distribution [BEN 64] [BRO 68a]: [6.51]

The narrow band response of a mechanical system to one-degree-of-freedom is carried out at a mean frequency close to the natural frequency of the system

fno * f o/ There is only one peak associated with each crossing of the mean value with positive slope [BER 77]. The mean damage per maximum (or half-cycle, since is [POW58]:

and, for n0 T cycles,

Damage by fatigue

where

207

yielding:

which leads, after transformation, to the relation [6.35]. 4. If N* is the number of cycles to failure at the rms stress level [POW 58], the expected damage [6.53]

can be also written, while setting

Let us set

The damage to failure is, assuming

yielding another expression of time-to-failure

Gms

208

Fatigue damage

where Ne = r| N is the number of cycles to failure corresponding to the stress G rms • This formulation makes it possible to build the S-N curves for random stresses starting from traditional S-N curves. 5. The quantity 7i(a) =

q(<0

can be regarded as a probability density:

N(a)

Figure 6.3. Density of probability of damage

Figure 6.3 shows variation of n(G) for b = 10, 8 and 6, C = 1 and a,-^ = 1. This law has the following characteristics:

Damage by fatigue

209

The mode, the mean and the standard deviation are linear functions of oms functions of parameter b [LAM 76].

and

Figure 6.4. Mode, mean and standard deviation of the damage

Figure 6.4 shows the variations of these three quantities with b for Gms = 1. 6. The expression of E(D) for r= 1 is also sometimes written in the form [LIN 67] [SHE 83]:

where M0 and M2 are respectively the moments of order 0 and 2 of the excitation, <jj and N j are the co-ordinates at a particular point of the curve N a = C. 7. The method of calculation ofD leading to the expression [6.34]

210

Fatigue damage

is sometimes named Crandall 's methodl [TAN 70]. Other methods were proposed later, such as those of M. Shinozuka [SHI 66] and D. Karnopp and T.D. Scharton [KAR 66]. 8. It is noted that the damage D varies with T and cjrms. A small variation of a rms , i.e. of the amplitude of the excitation, is thus much more sensitive than a variation of the duration T. In addition, since the stress level depends on power b, the damage is primarily created by the highest levels (in practice about four times the rms value [POW 58]). 6.3.3.3. Calculation of gamma function If b is an even positive integer number, it is shown that

If b is odd integer,

If b is arbitrary, we have:

lying between 1 and 2. Tables give

for

e[l,2][ABR64].

Damage by fatigue

211

Table 6.1. Values of the function r(l + b/2)

b 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25

r(i+b/2) 0.8966 0.9191 0.9534 .0000 .0595 .1330 .2223 .3293 .4569 .6084 .7877 2.0000 2.2514 2.5493 2.9029 3.3234 3.8245 4.4230 5.1397 6.0000 7.0355 8.2851 9.7971 11.6317 13.8636 16.5862 19.9162 24.0000 29.0214

b 8.50 8.75 9.00 9.25 9.50 9.75 10.00 10.25 10.50 10.75 11.00 11.25 11.50 11.75 12.00 12.25 12.50 12.75 13.00 13.25 13.50 13.75 14.00 14.25 14.50 14.75 15.00 15.25 15.50

r(i + b/2) 35.2116 42.8625 52.3428 64.1193 78.7845 97.0916 120.0000 148.7344 184.8610 230.3860 287.8853 360.6710 453.0107 570.4130 720.0000 910.9983 1155.3813 1468.7106 1871.2545 2389.4457 3057.8220 3921.5895 5040.0000 64.90 103 8.38 103 1.08 104 1.40 104 1.82 104 2.37 104

b 15.75 16.00 16.25 16.50 16.75 17.00 17.25 17.50 17.75 18.00 18.25 18.50 18.75 19.00 19.25 19.50 19.75 20.00 20.25 20.50 20.75 21.00 21.25 21.50 21.75 22.00 22.25 22.50 22.75

Example r(6.44) = 5.44 r(5.44) r(6.44) = 5.44 .4.44 3.44. 2.44 .1.44 l(\A4) with r(l.44) =0.8858. Yielding r(6.44)« 258.6

r(1+b/2) 3.09 4.03 5.27 6.91 9.07 1.19 1.57 2.07 2.74 3.63 4.81 6.39 8.50 1.13 1.51 2.02 2.71 3.63 4.87 6.55 8.82 1.19 1.61 2.17 2.94 3.99 5.42 7.37 1.00

4

10 104 104 104 104 105 105 105 105 105 105 105 105 106 106 106 106 106 106 106 106 107 107 107 107 107 107 107 108

b

r(1+ b/2)

23.00 23.25 23.50 23.75 24.00 24.25 24.50 24.75 25.00 25.25 25.50 25.75 26.00 26.25 26.50 26.75 27.00 27.25 27.50 27.75 28.00 28.25 28.50 28.75 29.00 29.25 29.50 29.75 30.00

1.37 108 1.87 108 2.55 108 3.50 108 4.79 108 6.57 108 9.03 108 1.24 109 1.71 109 2.36 109 3.26 109 4.50 109 6.23 109 8.63 109 1.20 1010 1.66 101° 2.31 1010 3.21 101° 4.48 10'° 6.24 1010 8.72 1010 1.22 10" 1.70 1011 2.39 10" 3.35 1011 4.70 10" 6.60 10" 9.29 10" 1.31 1012

212

Fatigue damage

The gamma function can also be approximated from a series development (cf. Appendix Al). In Table 6.1 are listed values of J I11 +bl — as function of b V 2) varying between 1.25 and 30.

6.3.4. Rms response to narrow band noise G0 of width Mwhen G0 Af = constant Let us consider a noise of constant power spectral density G0 and of width Af centered around a frequency fm limited by the frequencies f1 and f2 such that and

i.e., with the reduced co-ordinate

and

The relationships [1.61] and [A3.20] make it

possible to calculate the rms response zrms. If Ah is small, we obtain

or

It is noted that z^s varies with the product G0 Ah. When fm is small with respect to f0 and for the usual values of damping, it is seen that the quantity COQ zrms is close to the rms value xrms of the excitation. We saw that, for a narrow band noise,

In addition, all things being equal, the damage D is thus a function of the product (GQ Ah)

. If this product is constant, z^

is constant as well as D. If hm = 1,

Damage by fatigue

213

and

Lastly, if hm = 1 and Q = l,

6.4. Equivalent narrow band noise The estimate of the fatigue damage requires the calculation of a histogram of the peaks of the response. In the preceding paragraphs, the probability density of the peaks of the response was estimated from the statistical properties of the signal. One can also use one of the many methods of direct peak counting, in more general use, the most current being the rainflow method. Another approach is that of the equivalent narrow band noise. It is based on the simplified relation established on the assumption of a Rayleigh peak distribution (r » 1), with several alternatives: - Use of the relation [6.35] obtained for a Rayleigh peak distribution (r being supposed equal to 1), whatever the real distribution, - Same assumption, by replacing the expected frequency n0 by the mean number of peaks per unit time np , - Relation [6.35] corrected by a factor X (P.H. Wirsching), - Relation [6.35] corrected by a term function of b and r (G.K. Chaudhury and W.D. Dover), - Transformation of the real law of peak probability density into a Rayleigh distribution.

6.4.1. Use of relation established for narrow band response Even when the distribution of the peaks does not follow exactly a Rayleigh distribution, current practice is, to simplify calculation, to assume that r is equal to 1. The response z(t) is thus supposed to be a narrow band noise [CHA 85], which makes it possible to apply the relation [6.35]

214

Fatigue damage

by considering that it is still correct whatever the true value of r. It is thus supposed that a narrow band stress which has at the same time the same rms value and the same number of zero crossings (n 0 )as the real broad band stress provides an acceptable estimate of the damage. The interest of this assumption, if it is verified, lies in the facility of use of the analytical relation [6.35], which makes it possible to avoid rainflow type counting and heavier numerical calculations of integration. Error due to the Rayleigh distribution approximation The relation [6.35] is a good approximation of the damage for 0.567 < r < 1 (paragraph 6.3.3.1). We propose to consider, depending on the parameter r, up to what point and with which error one can use in all the cases (r arbitrary) the Rayleigh's law to calculate the fatigue damage. For that, let us calculate the ratio p of the damage E(D) obtained with a Rayleigh peak probability density q(u) and the damage D deduced from the most general law q(u):

Damage by fatigue

215

Figure 6.5. Ratio of the damage calculated using Rayleigh 's law and the complete formulation

Figure 6.5 shows that p tends towards 1 when r tends towards 1, more quickly for large values of b. The error arising, assuming Rayleigh's law, remains lower than 4% for b e[3, 30] and r > 0.6 (Figure 6.6), thus confirming M. Bernstein's calculation, already quoted [BER 77].

Figure 6.6. Error due to use of Rayleigh's law

Figure 6.7. Range in which the approximation using Rayleigh's law is acceptable

The damage D calculated using Rayleigh's law is always lower than the damage obtained with the more general formulation (whatever b and r).

216

Fatigue damage

NOTE. - L.P. Pook [POO 78] considers that for problems of fatigue, a process can be considered as with a narrow band if q < 0.14 where q = \\ - r , i.e. for r > 0.99, a condition which seems very severe from the above curves.

6.4.2. Alternative: use of mean number of maxima per second The comparison complete formulation / Rayleigh formulation was carried out above starting from the expression of the damage [6.35]:

in which the expected frequency n0, equals, for r = 1, the mean number of maxima ttp per unit times. J.T. Broch [BRO 68b] [SCH 61b] showed by experiment that, although n0 and ams (or z,^) are the same, two tests carried out with different values of r lead to different fatigue lives. This tendency is confirmed by other authors [BER 77] [FUL 62]. If r is different from 1, it is justified to replace n^ by np in the relation [6.35], the damage being related to the number of peaks. The comparison of the expression thus obtained

with that resulting from a complete formulation, carried out under the same conditions as previously, led to the curves of Figure 6.8. Thus the damage calculated simply is larger than that obtained with the complete formulation. This result is logical since the selected assumption results in regarding each peak as a well formed half-cycle, which is obviously not the case for r < 1. Approximation of the narrow band type has the principal disadvantage of overestimating the probability of occurrence of the largest peaks and thus leads to a conservative evaluation of the damage [BER 77] [CHA 85] [EST 62]; when the distribution of the peaks is close to a Gaussian law (r small), its use is debatable.

Damage by fatigue

217

Figure 6.8. Ratio of damage calculated-with a mean number of peaks and the Rayleigh law hypothesis with the complete formulation

In many practical cases, the results obtained are however sufficiently accurate [PHI65][RUD75][SCH61a].

6.4.3. P.H. Wirsching's approach After a study intended to compare several counting methods, P.H. Wirsching [WIR 80a] [WIR 80b] [WIR 80c] [WIR 83b] defined the parameter AR =

fatigue damage calculated from the " x" method fatigue damage calculated from the " equivalent narrow band" method

and plots curves giving AR against r (or q = V1 - r ) for various values of b and using several methods: X R : Rainflow type counting Ap: Counting of the peaks of the response A. z : Counting of zero crossings ^N: Simplified relation (r = 1). The curves obtained are given in Figure 6.9. They are plotted for b = 3, but their shape aspect is the same whatever b (the variation between the curves increases, but their relative position remains the same one).

218

Fatigue damage

Figure 6.9. Ratios of damage obtained with various peak counting methods

Whatever b, and whatever the method, A,x -» 1 when r -> 1. The reference being the rainflow method, considered in general to be the most precise, P.H. Wirsching shows that the method of peak counting and the equivalent narrow band method are conservative. The zero crossing counting method gives non conservative results. From four particular forms of power spectral densities, an empirical relation from this study is deduced allowing, starting from the simplified narrow band relation, calculation of the damage which would be obtained with rainflow counting. If DNB is the narrow band damage and D that given by rainflow counting,

the empirical relation between A, b and r being

where

If r = 1, A. = 1 and if r = 0, A. = A(b). Figures 6.10 and 6.11 show the variations of A, versus r and b. It is noticed that for b given, A. varies little with r if r < 0.9 .

Damage by fatigue

Figure 6.10. Factor of correction of damage calculated on a narrow band assumption (versus r)

219

Figure 6.11. Factor of correction of damage calculated on a narrow band assumption (versus b)

The study was carried out for loads resulting from the action of waves on offshore structures, for which q > 0.50 (r < 0.866) and b « 3 (welded joints). In this range, A. varies little and only b is influential. For small b, the narrow band model provides a reasonably conservative estimate of the lifetime. The accuracy of this approximation can also be evaluated by considering the ratio damage according to Wirsching damage from the general relation q(u)

If r -> 0, the damage calculated from the most general expression of q(u) is equal to

220

Fatigue damage

and

Figure 6.12. Error related to the use of the P.M. Wirsching relation Figure 6.12 shows the variations of 1 - p with r, for b = 3, 6, 10 and 15.

6.4.4. G.K. Chaudhury and W.D. Dover's approach These authors [CHA 85] propose a method derived from the general expression of the peak probability [6.24]:

We obtain, while replacing a by

The error fianction erf (x) (which varies between 0 and 1 according to x) can be approximated to 1/2:

yielding mean damage:

tends towards

(relation already obtained in [6.29]) and if r -» 1 (q -> 0), D tends towards

(np = no ). This relation gives a value lower by 25% than that of a theoretical narrow band process. The difference comes from the choice of the value of the error function. Calculations seem to follow better the experimental results in the case treated by the authors (stresses in tubular offshore structures) with peak counting using the rainflow method. To take account of the results of these countings, they correct and readjust the relation [6.34] established using Rayleigh's hypothesis

222

Fatigue damage

or

where

if the peak distribution is Rayleigh in form,

if this distribution is Gaussian and

in the general case, for 0.5 < r < 1. The relationship between the damage calculated by G.K. Chaudhury and W.D. Dover and that estimated on Rayleigh's hypothesis makes it possible to compare these two approaches (Figure 6.13).

Damage by fatigue

Figure 6.13. Comparison of damage calculated on the G.K. Chaudhury and Rayleigh assumptions

Figure 6.14. Comparison of damage calculated on the G.K. Chaudhury assumption and with complete formulation

223

224

Fatigue damage

The value of this method can also be evaluated by taking as reference the damage calculated using the general formulation of q(u) (Figure 6.14):

6.4.5. Approximation to real maxima distribution using a modified Rayleigh distribution The response of a linear one-degree-of-freedom system to a Gaussian random vibration is itself Gaussian, with a probability density of maxima [6.23]:

The parameter r of a narrow band response (or of a stress directly in a structure) is close to 1 and q(u) tends towards a Rayleigh distribution which, we saw, led to much simpler calculations. In the general case, q(u) is represented versus u by a curve which has an arc for u < 0 and, depending on r, an arc relatively close to a Rayleigh curve for u > 0. To approach Rayleigh's law, it would first be necessary to remove the negative part and, so that the new probability density q*(u) so defined has an area under the curve still equal to unity, to multiply q(u) by a standardization factor. We established that, if Q(u O ) is the probability such that u > u0,

Damage by fatigue

Figure 6.15. Probability density of the maxima

For

equal to

corresponds to area under the curve q(u) for u > 0, yielding

which is represented by the area on the left of the vertical axis (Figure 6.16).

Figure 6.16. Suppression of negative part of probability density of maxima

225

226

Fatigue damage

Since

So that the new density q (u) is standardized, it is necessary that [HIL 70] [KAC 76]:

Figure 6.17 shows q*(u) for r = 0.6 ; 0.75; 0.8; 0.9 and 1, plotted in its definition interval (0, QO). Calculation of the fatigue damage requires that the number of maxima dn lying between the stress levels cr and a + da:

where n is the mean number of positive maxima per second. When the response is of narrow band, r= 1 and np = n0. One prefers in this case to use n0 whose calculation assumes only that of zTms and zrms (that of n requires evaluation of the parameters zms and zrms). To preserve this facility, we can write

Figure 6.18 shows, for the same values of r as above, the variations of q(u) q (u) = . After these two corrections, we see that the maxima lower than r z rms are more numerous than those expected from Rayleigh's law. The taking into account of a Rayleigh distribution could thus seem slightly conservative, if it were not known that the low level maxima have a quasi-negligible contribution to the total damage. „„,

Damage by fatigue

Figure 6.17. Truncated and standardized probability density of maxima

227

Figure 6.18. Truncated, standardized and modified probability density of maxima for using expected frequency

When z > 2 zrms, all the distribution laws are almost confused with Rayleigh's law for 0.6 < r < 1, an interval which includes almost all the values found in practice in structures with one or several degrees of freedom [KAC 76]. NOTE. - B.M. Hillberry [HIL 70] proposes another approach based on the relation [6.21]:

where u =

. When r is arbitrary between 0 and 1, the curve q(u) presents rms values for u < 0 (Figure 6.19). CT

Figure 6.19. Suppression of negative maxima

228

Fatigue damage

To standardize the new peak probability density obtained after suppression of negative maxima, B.M. Hillberry writes the damage D in the form

6.5. Comparison of S-N curves established under sinusoidal and random loads Analytical expression selected to describe the S-N curve in its linear part (logarithmic scales) is that of Basquin (N a = C) which can be written, if the stress is characterized by its rms value instead of its amplitude,

The above results makes it possible to write, in the case of a test bar excited by a narrow band stress [6.35]:

There is failure by fatigue, on average, when D = 1. Then [ROO 64]:

where N = n0 T (mean number of cycles to failure), yielding:

The S-N curve thus evaluated for a random loading is thus a line having the same slope as that obtained under sinusoidal stress, with a smaller ordinate at the origin (the ordinate at the origin being here defined as the ordinate of the point N = 1, i.e. the theoretical ultimate stress of the material [ROO 64]: it is known that the approximation of the S-N curve by a line in this zone of low numbers of cycles, is not correct).

Damage by fatigue

229

All the parameters appearing in the right hand side of relation [6.94] are known from the results of the fatigue tests in sinusoidal mode: this relation allows the extrapolation of the curves SN obtained with sine loads to the case of random loads. NOTE. - If it is supposed that the peak distribution of the response follows a Gaussian distribution (instead of a Rayleigh distribution), i.e. that r = 0, we obtain in the same way:

We set here

Example 7075-T6 aluminium alloy

b = 9.65 C = 5.56 1087 (SI unit)

Figure 6.20. Comparison ofS-N curves

Figure 6.20 shows the curves a(N) plotted under the following conditions: (cr = maximum stress in sinusoidal mode)

230

Fatigue damage

random, Rayleigh's distribution [6.94] random, Gaussian distribution [6.96] The variation between the initial S-N curve (in sinusoidal mode) and the curve relating to random (Rayleigh's assumption) is a function of the value of the parameter b.

Figure 6.21. Variation between the S-N curve in sinusoidal mode and random (according to L. W. Root [ROO 64])

Figure 6.21 shows the variations of the ratio

rms random CT

= S versus

max sine

parameter b:

NOTE. - The fatigue calculations carried out for a peak distributions close to a Gaussian law are in general nonconservative [r small, therefore many cycles of low amplitude (under-cycles) superimposed on the cycles between two zero crossings]. R.G. Lambert [LAM 93 ] proposes to modify the relation [6.95] to take account of this phenomenon and to correct it simply as follows:

Damage by fatigue

231

More general case We saw that the expected damage by fatigue is given by [6.21]:

where q(u) is the probability density function of maxima, represented by [6.22] if the distribution of the instantaneous values of the random excitation is Gaussian. If we set N = n0 T, we obtain, for D = 1, a relation between arms and N of the form

If we preserve

we obtain:

Figures 6.22 and 6.23 show the curves plotted for r = 0.25 ; 0.50; 0.75 and 1, 80 with reference to the curve established in sine mode (b = 10, C = 10 ), when one carries on the abscissa n0 T or np T. It is noted that these curves are (into random) very tight and therefore not very sensitive to the variations of r. According to choice of the definition of N and according to the value of r, a calculation carried out using Rayleigh's assumption can thus be slightly optimistic (N = n0 T) or pessimistic (N = n T).

232

Fatigue damage

Figure 6.22. S-N curve into random plotted as a function of the number of zerocrossing with positive slope, for various values of the irregularity factor

Figure 6.23. S-N curve into random plotted as a function of the peak number, for various values of the irregularity factor

6.6. Comparison of theory and experiment One finds in the bibliography test results carried out in various configurations, under stationary random stresses with or without zero average, used for the layout of S-N curves, and for fatigue life calculations [GAS 65] [KIR 65a]. The conclusions are not always homogeneous and vary sometimes according to authors. Generally, it seems that the parameter considered as most representative of fatigue under random stress is the rms value of the stress (confirmation of the theory) [CLE 66] [CLE 77] [KIR65a]. S-N curves The Miner/Rayleigh method for the determination of S-N curves was checked experimentally, and demonstrated a reasonable scatter of fatigue damage [CLY 64] [ESH 64]. The rms stress vs number of cycles to failure curves are straight lines on logarithmic scales (note that one finds the two definitions of N, i.e. n0 T and n* T, perhaps allowing one to explain some differences in the results). The majority of the authors agree to locate the S-N curve for random loads on the left of the curve obtained for sine [BRO 70b] [CLE 66] [CLE 77] [CLY 64] [ELD 61] [HEA 56] [LAM 76] [PER 74] [MAR 68] [ROO 64] [SMI 63]; the fatigue

Damage by fatigue

233

lives are shorter for random than for sine, for equal rms stress values. This result is logical since the damage by fatigue is primarily produced by large peaks which can exceed 5 times the rms value for random.

Figure 6.24. Comparison between the S-N curves determined for sinusoidal and random stress

The difference observed is sometimes larger at low levels [PER 74] [SMI 63]. R.G. Lambert [LAM 76] gives some experimental results in Table 6.2, which we reproduce as an indication. The values of the ultimate stress ratio

Rs

(stresses for N = 1 in the Basquin

RA relation) respectively obtained for a sinusoidal stress and a random stress are close to 2 for ductile materials and 3 for relatively brittle materials (the materials are more brittle when their b parameter is larger). Some authors obtain however a longer fatigue life for high random stress levels and shorter for low levels [KIR 65b] [FRA 59] [LOW 62]. J.R. Fuller [FUL 61], S.R. Swanson [SWA 63], H.C. Schelderup and A.E. Galef [SCH 6la] have noted that the S-N curves plotted on logarithmic scales with sine stresses and narrow band random stresses (distribution of the peaks close to Rayleigh's law) are roughly straight lines of different slopes, the curve in random mode being deduced from that in sine mode by a clockwise rotation centered on the axis N = 1 (Figure 6.25). To take account of this remark, L. Fiderer [FID 75] proposes to represent the S-N curve by a relation derived from Basquin's (N a = C = I.CTJ , where a, is a fictitious stress level, since G} > Rm, stress to rupture for 1 cycle, i.e. in statics) of the form

234

Fatigue damage

the constant A taking account of observed rotation. The relative position of the theoretical S-N curves thus obtained and of the experimental curves is not so clearly established.

Figure 6.25. Transformation of the S-N curve in sine mode to the S-N curve into random mode

Table 6.2. Examples of constants in Basquin 's relation for sinusoidal and random stresses [LAM 76]

Material

Aluminium alloy 6061-T6 Copper Wire Aluminium alloy 7075-T6 Gl0 Epoxy Fiberglass

b

Random stress Constant Ratio Ultimate A stress RA RS (Pa) (Pa) RA

Sinusoidal stress Constant Ultimate C stress Rs (SI unit) (Pa) 1

8.92

7.56 108

1.57 1079

3.43 108

1.36 1076

2.20

9.28

5.65 108

1.66 1081

2.54 108

9.93 1077

2.22

9.65

12.38 108

5.56 1087

5.51 108

2.25 1084

2.25

12.08

5.58 108

2.28 108

9.20 10100

2.45

4.56 10105

Damage by fatigue

Wrought Steel SAE 4130 BHN267 Magnesium alloy AZ31B

235

17.54

12.13 108

2.14 10159

4.27 108

2.39 10151

2.84

22.37

2.99 108

3.99 10189

9.38 107

2.18 10178

3.19

Work carried out by C. Perruchet and P. Vimont [PER 74] shows an important diversity in conclusions: - The tests confirm relatively well the estimated lifetimes: - Whatever the stress level [ELD 61] [KOW 61b] [MAR 68], - Only at low stress levels, real fatigue lives being longer than those estimated for high levels [ELD 61] [FRA 59] [KIR 65b] [LOW 62], - Only at high levels, real fatigue lives being shorter than the expectation for low levels [CLE 66] [CLE 77], -The experimental fatigue lives are smaller by a factor of 2 to 10 [BOO 69] [BRO 70b] [CLE 66] [FUL 62] [HEA 56] [MAR 68] [PER 74] [SMI 63] whatever the stress level, - The experimental fatigue lives are longer by a factor of 2 to 3 approximately (between 104 and 107 cycles) whatever the stress level [ELD 61] [ROO 64]. Generally, it is considered that the estimates of fatigue life carried out using Miner's rule are rather good for random, the shape of the estimated S-N curve being correct [BRO 70b] [MAR 68]. This rule, relatively simple to use, provides adequate evaluations when an approximate result is required [BOO 69]. NOTE. - Mc Clymonds and J.K. Ganoug [CLY 64] proceeded to a comparison of the SN curves established for random, sine and swept sine (for Miner's and Rayleigh 's assumptions). Swept sine produces a significant damage only during the time for which the excitation frequency crosses the frequency interval between the halfpower points at each system resonance. Random loading creates damage by fatigue for each resonance over all the test duration. By referring the S-N curves to the load corresponding to the resonance peak for swept sine, to the rms load for random, they found, at a given stress level, a longer fatigue life for swept sine than for sine, and longer for sine than random.

236

Fatigue damage

Figure 6.26. Comparison of the S-N curves established experimentally under sine and random stresses

This tendency is confirmed by calculation. Figure 6.26 shows the S-N curves plotted using the following relations: sine [6.93]:

random [6.94]:

For the swept sine, the relation is established using the approximation suggested by M. Gertel [GER 61] for the calculation of fatigue damage undergone by a onedegree-of-freedom system excited by a linear swept sine. It is shown that [LAL 82] (Volume 5, Chapter 4):

where AN is the number of cycles performed between the half-power (function of the sweep rate)

points

Damage by fatigue

237

yielding, for D = 1,

6.7. Influence of shape of power spectral density and value of irregularity factor The shape of the PSD and the bandwidth are generally regarded as not very influential parameters [BRO 68b] [HIL 70] [LIN 72]. Studies carried out to evaluate the importance of the factor r lead however to more moderate conclusions: - The parameter r has little influence for 0.63 < r < 0.96 (it was seen that the narrow band approximation is acceptable in this range) [CLE 66] [CLE 77] [FUL 62]; -The parameter r has little influence, except when it is very weak, about 0.3. One then observes an increase in the fatigue lives for notched samples [GAS 72] [GAS 77]; - The parameter r has a notable influence (except if the probability density functions of the peaks are comparable) [BRO 68b] [LIN 72] [NAU 65]; - The fatigue life is larger when r is smaller, both from the point of view of crack initiation and their propagation [GAS 76]. For S.L.Bussa [BUS 67], this effect is particularly sensitive for r < 0,8 ; - S.R. Swanson [SWA 69] and J Kowalewski [KOW 63] arrive at qualitatively similar results with regard to the influence of the parameter r, without a law binding r and the fatigue life being able to be truly established.

6.8. Effects of peak truncation The fatigue damage being narrowly dependent on the number and the amplitude of the peaks of stress undergone by the part, it is interesting to speculate about the effect of a truncation of the peaks. This truncation can have several origins. In theory, calculations take into account a law of peak distribution q(u) without limitation, thus being able statistically to include very large peaks. In practice, it is rare to observe peaks higher than 6 standard deviations (rms value) in the real environment. During simulation tests in laboratory, the control system often cut the peaks higher than 4.5 times the rms value. Before anticipating the analysis, it is wise to specify the nature of the truncated control signal. This signal can be:

238

Fatigue damage

1. An acceleration, a force or any other quantity imposed on a structure, the stress at a point being the result of this excitation only. 2. A stress or a deformation directly measured on the part or the result of force applied directly to the part (case of tests on test bars). Depending on the case, the results are different. We will consider in the following paragraph only this last case. The influence of a truncation of the acceleration signal will be examined in Chapter 5 of Volume 5. 6.9. Truncation of stress peaks Let us suppose that the S-N curve can be represented by the analytical Basquin expression N eb = C and that we use Miner's rule. We saw that, on these assumptions, the fatigue damage can be written, without truncation,

If the peaks are truncated at stress level at, the damage becomes:

Figure 6.27. Truncation level of peaks on the S-N curve

Damage by fatigue

239

Particular case of a narrow band noise If the peak distribution of a(t) can be represented by Rayleigh's law (r = 1), the expression [6.105] of the damage can be written [POO 78]

(If we take into account a fatigue limit

aD,

this damage

becomes

Figure 6.28. Truncated peak

Since the peaks are truncated at value at, the amplitudes higher than this value are fixed at a = at for calculation of the damage when N ab = C [POO 78]. This damage

thus becomes

yielding

240

Fatigue damage

NOTE. - This result can be arrived at by considering that truncation to at is equivalent to the addition of a Dirac delta function at a = at which restores to a value 1 the truncated area under the curve q(a). This area is equal to [LAM 76]:

Figure 6.29. Addition of a Dirac delta function to the probability density of peaks to compensate for the area removed by truncation

The truncated probability density then becomes

where 0 < a < at, and damage by fatigue relating to larger peaks than at :

The total damage is thus, in the presence of a fatigue limit stress:

Damage by fatigue

241

Knowing that the term

can be written in the form

where y[B, i] is the incomplete gamma function, it becomes, with B = 1 + — and 2

and

242

Fatigue damage

The function y is tabulated and can be calculated from approximate relations. If

NOTES. 1. The expression [6. 111] can be also written [WHI69]:

(the function P is tabulated). 2. If, in the relationship [6.113], we set aD = 0 (no fatigue limit), we find the Crandall expression[6.34]. The effect of truncation is to reduce the damage in the ratio related to b.

,which

is

Damage by fatigue

Figure 6.30. Effect of truncation on the damage in the absence of fatigue limit

243

Figure 6.31. Effect of truncation on the damage for a fatigue limit equal to 3 times the rms value

Figures 6.30 and 6.31 show the variations of this ratio for aD = 0 and versus

parameterized by b (variable between 2 and 10 per step of 1).

For CD = 0, one notes that, if b < 7, truncation for

> 4 has negligible

effect. But if b > 7, 50% at least of the total damage by fatigue is produced by the peaks of the response exceeding 3 arms, in spite of their infrequent occurrence [SMA 65]. Although the damage created by the peaks between 2 arms and 3 arms is important [POO 78], it appears clearly that a truncation of the response signal to 3 arms modifies much the resulting damage. In certain cases (observed for 'missile' environments), the responses greater than 3 arms can have an occurrence much more frequent than for a Gaussian noise, with peaks higher than 5 arms very probable. The damage produced is then more important than that envisaged with a Gaussian assumption. To avoid this problem, a method could consist in adjusting the total amplitude of the random noise created during the test laboratory so that the response peaks produced are similar to those in the real environment.

244

Fatigue damage

For b < 7 and aD = 0, truncation has little effect if and 2 [WHI 69]. For b < 10 and

long as

> 4 (Figure 6.32), the reduction of the damage is weak so The level ag at which the greatest damage takes place is

obtained while searching ag such that

Figure 6.32. Effect of truncation on the damage as function of fatigue limit

is maximum, i.e. for

yielding

Figure 6.33. Stress leading to greatest damage

Damage by fatigue

If

For

level

245

it becomes

small (Figure 6.33), there is a field, for each value of b, in which the is equal to

These curves are valid only if aD < ag (there is no

damage if aD > at) [WHI 69].

Figure 6.34. Stress leading to greatest damage in the absence of truncation

These results show that, generally, the effect of truncation cannot be neglected a priori. It is thus important to specify, by test, which was the value of at. The truncation of the peaks larger than 2.5 arms has a notable influence, leading to longer fatigue lives. The truncation of the peaks above ±3.5 arms has little influence on the experimental results [BOO 69].

246

Fatigue damage

The predictions of fatigue lives carried out using Miner's hypothesis and a truncated signal are good with a truncation between 40 arms and 5.5 arms, worse with 3.5 arms and bad with 2.5 arms [BOO 69]. The relation [6.116] established in the absence of truncation shows in addition that the maximum damage is brought about by the peaks located between 2 and 4.5 times the rms value, this maximum being a function of the parameter b (Figure 6.34). NOTE. - A. J. Curtis [CUR 82] presents the calculation in a slightly different way. Being given a Rayleigh distribution truncated for a > at, and if there is a fatigue limit aD, it states the damage by fatigue D in the form

yielding, while setting

or

and

,

Damage by fatigue

247

Table 6.3. Values of the incomplete gamma function [CUR 82]

u t

b = 3.0

0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 .00 .10 .20 .30 .40 .50 .60 .70 .80 .90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00

0.0000 0.0000 0.0002 0.0007 0.0020 0.0048 0.0100 0.0185 0.0314 0.0498 0.0746 0.1065 0.1460 0.1932 0.2478 0.3092 0.3763 0.4479 0.5227 0.5990 0.6753 0.7501 0.8222 0.8903 0.9536 .0115 .0635 .1096 .1498 .1844 .2136 .2380 .2581 .2744 .2874 .2978 .3058 .3120 .3167 .3202

b = 6.5 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.004 0.008 0.017 0.033 0.059 0.099 0.159 0.244 0.359 0.510 0.702 0.937 1.217 1.542 1.908 2.310 2.744 3.199 3.667 4.140 4.606 5.058 5.488 5.890 6.259 6.592 6.888 7.146 7.368 7.556 7.712 7.841

b = 9.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.001 0.003 0.007 0.016 0.035 0.072 0.136 0.243 0.413 0.672 1.048 1.575 2.284 3.210 4.380 5.816 7.528 9.517 11.770 14.260 16.950 19.791 22.729 25.704 28.656 31.531 34.277 36.852 39.224 41.372 43.283 44.956

b = 25.0 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 256 256 512 1280 3328 8448 19968 45312 98304 204288 406528 776448 1427456 2530560 4333056 7179264 11527168 17960704 27195392 40066816 57502976 80484608

2 2 - e-u 11 e-ut/2t/ 0.005 0.020 0.044 0.077 0.118 0.165 0.217 0.274 0.333 0.393 0.454 0.513 0.570 0.625 0.675 0.722 0.764 0.802 0.836 0.865 0.890 0.911 0.929 0.944 0.956 0.966 0.974 0.980 0.985 0.989 0.992 0.994 0.996 0.997 0.998 0.998 0.999 0.999 1.000 1.000

r(1+b/2) b = 3.0

b = 6.5

b = 9.0

b = 25.0

1.3293

8.285

52.343

1.711 109

248

Fatigue damage

with

A.J. Curtis [CUR 82] shows that the fatigue life is more modified when b is larger (for small UD and

Table 6.3 gives, for ut ranging between 1.10

and 4.00, the values of:

— The function

for b respectively equal to 3.0- 6.5 - 9.0 and 25.0,

- The expression Layout of the S-N curve for a truncated distribution We have, if aD = 0 and at arbitrary, while setting Nt = n0+ T

The ratio of fatigue life with and without truncation is thus [LAM 76]:

This ratio depends only on

and on the parameter b. To compensate for the CT

rms

truncation and to produce the same damage, it would be necessary to multiply the rms value of the stress by the factor

Damage by fatigue

249

(and the amplitude of the PSD by A2)

Table 6.4 gives the values of A, for different values of b and

Table 6.4. Values of the compensation factor of truncation A

b

<*t CT

rms

3 4 5 6 7 8 9 10 11 12

3 .01 .0159 .02336 .032285 .04248 1.05377 1.066 1.0789 1.0924 1.10644

3.5 1.0022 1.0039348 1.000643 1.00976 1.01398 1.0191 1.0250 1.03178 1.0392 1.04732

4 1.000377 1.000756 1.001375 1.00231 1.00364 1.00543 1.00772 1.01055 1.01393 1.01784

4.5 1.000048 1.00011146 1.000224 1.000417 1 .000724 1.001185 1.0018425 1.002738 1.00391 1.00539

5 1.00000142 1.0000126 1.0000279 1.0000569 1.000108 1.000194 1.000331 1.000536 1.000833 1.001245

250

Fatigue damage

Figure 6.35. Compensation factor X versus the parameter b

Figure 6.36. Compensation factor X versus the truncation threshold

Figures 6.35 and 6.36 show the variations of A. with b and

It is noted that

X increases with b and decreases when at increases.

Figure 6.37. Ratio of damage of truncated and untruncated signals versus the truncation threshold

Figure 6.38. Ratio of the damage of the truncated and untruncated signals versus the parameter b

NOTE. - One could also consider the ratio of the damage of the truncated signal Dt to the damage of complete signal D. This ratio is equal to

Damage by fatigue

251

It is important to recall here that the truncation considered in this paragraph relates to the stress and not the acceleration signal applied to the base of the reference one-degree-of-freedom system. Some examples show that in this last case, truncation has little effect when it is greater than 3 times the rms value of the signal (Volume 5).

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Chapter 7

Standard deviation of fatigue damage

In previous chapters is was seen how fatigue damage undergone by a onedegree-of-freedom system can be calculated starting from the characteristics of random stress, Miner's rule and parameters of the S-N curve. When the peak distribution of the response can be represented analytically, one can obtain damage expectation E(D) . To this mean damage can be applied a standard deviation SD (or variance SD) characterizing scatter related to the random aspect of the excitation. Several methods have been proposed to calculate SD approximately on the assumption of a Rayleigh peak distribution.

7.1. Calculation of standard deviation of damage: J.S. Bendat's method Assuming the signal to be stationary and ergodic and by making an assumption concerning the form of the autocovariance function of the damage created by each half-cycle

of 2

the

response

(considered

equal

to

sd2

e-2

,

where

2

sd = E ( d ) -E (d), (d being the damage associated with the half-cycle k), J.S. Bendat [BEN 64] showed that the variance of the damage undergone by a singledegree-of-freedom linear system can be written

where

254

Fatigue damage

and

with

where

The variation coefficient v is given by:

Standard deviation of fatigue damage

255

Example

Figure 7.1. Variation coefficient of damage versus number of cycles (J.S. Bendat)

Figure 7.2. Variation coefficient of damage versus natural frequency (J.S. Bendat)

256

Fatigue damage

Figure 7.3. Variation coefficient of damage Figure 7.4. Variation coefficient of damage versus damping (J.S. Bendat) versus parameter b (J.S. Bendat) Figures 7.1 to 7.4 show the variations of v with b, f0, n0+ T and £ in the case of a one-degree-of-freedom system of natural frequency 100 Hz subjected to a random noise of constant PSD equal to 1.65 (m/s2)2/Hz between 1 Hz and 2000 Hz, with duration T = 3600 s, under the following conditions: b = 10, C = 1080, K = 6.3 1010 (SI units). Since the peak distribution follows, by assumption, a Rayleigh distribution,

yielding

Standard deviation of fatigue damage

257

Lastly, if 2 np+ T £, is large, we have

yielding

and

Table 7.1 gives, as an example, some values of v calculated as function of b and the product 2 n Z, np+ T (np+ = n0+ since r = 1).

258

Fatigue damage

Table 7.1. Values of the variation coefficient of the damage (J.S. Bendat) b 2 n ^ n0+ T

6

4

2

10 3.162 10-1 -1

25 2.000 10

-1

7.071 10-1

8

1.378

4.472 10-1 8.718 10-1 -1

-1

14

12

10

2.627

5.010

9.601

18.523

1.661

3.169

6.076

11.715

3.162 10

6. 164 10

1.175

2.241

4.297

8.284

75 1.155 10-1

2.582 10-1

5.033 10-1

9.592 10-'

1.829

3.508

6.764

-1

100 1.000 10

-1

2.236 10

4.359 10-1

8.307 10-1

1.584

3.038

5.858

250 6.325 10-2

1.414 10-1

2.757 10'1

5.254 10-1

1.002

1.922

3.705

500 4.472 10-2

1.000 10-1

1.95010'1

3.715 10-1

7.085 10'1

1.359

2.620

-2

-2

1.592 10-1

3.033 10-1 5.785 10-1

1.109

-2

1

1.378 10'

2.627 10-1

5.01010'

2500 2.000 10-2 4.472 10-2

8.718 10'2

1.661 10-1

5000 1.414 10-2

3.162 10-2

6.164 10-2

-2

-2

2

50 1.414 10

750 3.652 10

1000 3.16210-2

7500

1.155 10

8.165 10

7 071 10

2.582 10

1

2.139

9.607 10'

1

1.852

3.167 10-1

6.075 10'1

1.172

1.175 10'1

2.241 10-'

4.297 10-'

8.284 10-1

5.033 10-

9.592 10'

2

1.83010'1

3.508 10-1

6.764 10-1

10000 1.000 10-2

2.236 10-2

4.359 10'2

8.307 10'2

1.584 10-'

3.038 10-1

5.858 10-1

-3

-2

2

2.757 10'

5.254 10'2

1.00210-1

1.922 10-1

3.705 10-1 2.620 10-1

25000 6.325 10

1.414 10

50000 4.472 10-3

1.000 10-2

1.949 10'2

3.715 10'2

7.085 10'2

1.359 10''

-3

-3

2

2

2

2

100000 3.162 10

7.071 10

1.378 10'

2.627 10'

5.010 10-

9.607 10-

1.852 10-1

500 000 1.414 10-3

3.162 10-3

6.164 10'3

1.175 10'2

2.241 10'2

4.297 10-2

8.284 10'2

1 000 000 1.000 10-3

2.236 10-3

4.359 10'3

8.307 10-3 1.584 10-2

3.038 10-2

5.858 10'2

7.2. Calculation of standard deviation of damage: S.H. Crandall, W.D. Mark and G.R. Khabbaz method S.H. Crandall, W.D. Mark and G.R. Khabbaz calculate the variance of the damage in the two following ways [CRA 62] [MAR 61] [YAO 72]: a) By supposing random the amplitude of the stress cycles and the number of cycles in (the interval) [0, T], b) Starting from an approximation considering only the random aspect of the stress amplitude. These two approaches converge in extreme cases when the bandwidth tends towards zero. The problem in both cases is related to strong correlation in the incremental damage of successive cycles. It is necessary to carry out approximations which are valid only when T is large compared to the decrease in time of correlation.

Standard deviation of fatigue damage

259

NOTE. - There is a modified method of calculation of the variance (M. Shinozuka [SHI 66]) which assumes a normally distributed load (mathematically simpler than another distribution law). With the following assumptions: - The excitation is of white noise type and the peaks of the response are distributed according to Rayleigh's law, - The excited system is linear, with only one-degree-of-freedom, with a damping ratio very much lower than unity (in practice, less than or equal to 0.05), - The mean number of cycles of the response for the length of time T is very large compared to 1/2£. The approximate analytical expressions of the variance established in both cases (a) and (b) converge towards the same expression when damping £, tends towards zero [the mean damage E(D) being the same on the two assumptions, insofar as

The variance can be written, by supposing the amplitude of the half-cycles and their number to be random,

where

260

Fatigue damage

Figure 7.5. Variations of functions f1(b), f2(b) and f 3 (b)

Figure 7.5 shows the variations of f1(b), f 2 (b) and f 3 (b) versus b. Table 7.2 gives some numerical values of these three functions for b varying between 3 and 20.

Standard deviation of fatigue damage

261

Table 7.2. Values of functions f,(b), f 2 (b) and f 3 (b)

b 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0

10.5 11.0 11.5

f 2 (b)

f 3 (b)

b

0.370 0.522 0.716 0.964 1.280 1.684 2.202

0.029 0.040 0.054 0.070 0.090

0.212

12.0

0.281 0.375

12.5 13.0 13.5

2.868 3.730 4.846 6.300 8.198 10.686 13.956 18.268 23.971

0.179

f1(b)

31.533 41.586

0.115 0.144

0.223 0.275 0.340 0.420 0.518 0.641

0.794 0.987 1.230 1.537

0.503 0.679 0.920 1.250 1.704 2.328 3.188 4.375 6.013 8.278 11.411 15.750 21.763 30.102 41.676

14.0 14.5 15.0 15.5 16.0 16.5 17.0 17.5 18.0 18.5 19.0 19.5 20.0

f1(b)

f 2 (b)

54.980 72.866 96.799 128.886 171.982 229.961 308.087 413.520 556.003 748.811 1010.046 1364.413 1845.653 2499.884

1.928 2.427 3.067 3.890 4.951 6.326 8.110 10.433 13.466 17.435 22.641 29.485 38.502 50.405 66.146 87.000

3390.189 4602.919 6256.346

114.676

f 3 (b)

57.750 80.087 111.146 154.356 214.500 298.256 414.946 577.590 804.375 1120.719 1562.149 2178.339 3038.750 4240.558 5919.723 8266.515 11547.250

We deduce variation coefficient [CRA 63]

It should be noted that, while E(D) varies linearly with duration T of the vibration, the standard deviation SD varies as with VT .

262

Fatigue damage

Example Figures 7.6 and 7.7 show the variations of v with £, b or f0 in the particular case where the input is a white noise G = 1 (m/s2)2/Hz between 1 Hz and 2000 Hz, applied during 3600 s.

Figure 7.6. Variation coefficient of damage Figure 7.7. Variation coefficient of damage versus parameter b (W.D. Mark) versus natural frequency (W.D. Mark)

Figure 7.8. Variation coefficient of damage Figure 7.9. Variation coefficient of damage versus damping (W.D. Mark) versus number of cycles (W.D. Mark)

Standard deviation of fatigue damage

263

These curves, plotted starting from expression [7.20], show that v: - is an increasing function of b, - decreases when f0 increases, - decreases when £, increases. Figure 7.9 shows that v decreases when time T (and thus the number of cycles N = n0+ T) increases. In many practical cases, it can be considered that £, n0+ T is large compared to unity. The last two terms of the bracket are then negligible and we have

The variation coefficient (or relative dispersion) is then written:

This ratio is then larger since [CRA 63]: - e is smaller, is smaller (i.e. since

half of the interval between

the half-power points, is smaller), - T is smaller. It can be seen that E(D) is related to T and that SD varies as with VT .

7.3. Comparison of W.D. Mark and J.S. Bendat's results We studied the variations of the ratio

V(Bendat) v

with f0, b, £ and n0+ T in the

(Mark)

case of the example already treated. This ratio is written, for a narrow band response:

264

Fatigue damage

Figure 7.10. Ratio of variation coefficients calculated by J.S. Bendat and W.D. Mark versus natural frequency

Figure 7.11. Ratio of variation coefficients calculated by J.S. Bendat and W.D. Mark versus damping ratio

One notes in the figures: -7.11 and 7.12 that p does not in practice vary with £; it is higher since b is bigger, -7.10 and 7.13 that p tends towards a constant value when f0 increases from 10 Hz to 2000 Hz, all more quickly because £ is larger. This limiting value is obtained by neglecting the terms of the denominator in n0+ (n 0 + = f0) yielding:

Standard deviation of fatigue damage

265

Example

For b = 10:

In the same way, p tends towards constant p^ when n0 T tends towards infinity (Figure 7.13).

Figure 7.12. Ratio of variation coefficients calculated by J.S. Bendat and W.D. Mark versus parameter b

Figure 7.13. Ratio of variation coefficients calculated by J.S. Bendat and W.D. Mark versus number of cycles, for b = 10

- 7.14 that the ratio p varies very little with n0+ T. It is larger since b is larger.

266

Fatigue damage

Figure 7.14. Ratio of variation coefficients calculated by J.S. Bendat and W.D. Mark versus number of cycles for £, = 0.05

NOTE. - J.P. Tang [TAN 78] shows that a limit of the failure probability is given by:

where D is the damage E(D) is the mean damage SD is the standard deviation of the damage. Example Let us consider a steel structure working in tension compression and which can be comparable with a one-degree-of-freedom linear system, natural frequency f0 = 100 Hz and relative damping £ = 0.05 . Let us suppose that the stress cr is related to the peak relative displacement zp

by

with K = 6.3 1010 Pa/m and that steel has a S-N curve characterized by a 80 parameter b equal to 10, with a constant C = 10 (SI unit).

Standard deviation of fatigue damage

267

Let us suppose finally that this structure is subjected to a random acceleration applied to its base defined by a constant acceleration spectral density of amplitude GX = 1.65 (m/s2)2/Hz between 1 Hz and 2000 Hz. We propose to calculate: - the fatigue damage D after 50 hours of vibrations, - the standard deviation of the damage SD. For a system of natural frequency 100 Hz, the excitation can, at first approximation, being regarded as a white noise, yielding, using the approximate relations mentioned above: zms = 1.29 10-4 m r = 0.664 TO,,,, = 0.0809 m/s Zrms = 76.48 m/s2

yms = 51.147 m/s2

np+ n0+

[=

= 150.47 = 99.87

(2 TT f0)2 zrms]

D: Relation [6.26]: 0.86270 Rayleigh: 0.86196 SD: Relation [7.6]: 0.00577 Bendat relation: 0.00575 Mark relation: 0.003886 v: s D Mark / E(D) Rayleigh: 0.004508 SD Bendat/ E(D) Rayleigh: 0.0066666 Relation [7.6]/ D relation [6.26]: 0.006693 P(D > 1): < 2.48 10-7 for SD Bendat <8.83 10-11for Mark < 2.93 10-7 for SD Bendat [7.6] and [6.26] The mean damage calculated using Rayleigh's assumption is thus slightly lower than the value obtained with relation [6.26]. The approximate Bendat expression gives a value of SD very close to that deduced from relation [7.6]. The Mark relation leads to a definitely lower result. This variation is not ascribable to ignoring the assumptions selected to establish these expressions (r = 0.664 instead of 1), a difference being still observed if the coefficient r is very near to unity.

268

Fatigue damage

Mean fatigue life

Mean numbers of cycles to failure (under random vibrations) n0+ T = 99.87 57.9 3600 * 2.08 107 cycles Rms values

Number of cycles to failure under a sinusoidal stress having the same rms value and a frequency equal to 80 Hz: N ( K z m ) b = 1080

N = 34 694 cycles Now let us calculate the number of cycles to failure under sinusoidal stress which creates the same rms response as the random vibration. The random vibration generates at 100 Hz an rms response of (for Q = 10): zrms = 1.29 10-4 m

Standard deviation of fatigue damage

269

To obtain this response with a sinewave excitation of frequency 80 Hz (for example), an amplitude would be needed equal to

yielding cycles.

7.4. Statistical S-N curves 7.4.1. Definition of statistical curves We showed that, for Gaussian random excitation and a narrow band response (r = 1), the damage can be expressed in the form [6.35]:

and the standard deviation SD by the approximate relations:

J.S. Bendat[7.15]:

W.D. Mark [7.20]:

270

Fatigue damage

We will consider in what follows the curves defined by the fatigue damage E(D)-oc SD and E(D) + a SD where the constant a is a function of the selected degree of confidence, for a given distribution law of the damage (the most often used law being the log-normal distribution). For a given stress level, these values of the damage allow to calculate the number of cycles n0+ T for which failure can take place. These curves consider only the random aspect of the stress and do not take into account scatter of the fatigue strength of material, more important and studied elsewhere (Chapter 3). 7.4.2. J.S. Bendat's formulation The equation of the first curve, defined by

for D = 1 (failure) can be also written

i.e.

Let us set N = n0+ T

Standard deviation of fatigue damage

271

It becomes

with

The second equation [E(D)(l + o c v ) = l] leads to the same result. Example Let us consider a random noise of constant PSD between 1 Hz and 2000 Hz with an amplitude G such that 1 < G < 9, then 2 < G < 30 (to explore two ranges of stresses). Let us suppose that:

b = 10 5 = 0.05

f 0 =100 Hz K = 6.3 1010 Pa/m 80

C = 10

(SI units)

Figure 7.15. Mean ±3 standard deviations S-N curves for high values of N

Figure 7.16. Mean ±3 standard deviations S-N curves for low values of N

272

Fatigue damage

Figures 7.15 and 7.16 show the s(N) curves obtained starting from these data for E(D) = 1, E(D)-3s D = 1 and E(D) + 3sD = 1. It is seen that, when the rms stress arms is large, the curves corresponding to E(D) + 3 SD tend towards a limit, which can be calculated as follows. When G is large, A becomes large and

whereas N [E(D)-a s D ] — > 0. For a = 3, ^ = 0.05 and b = 10, we have B = 251 and the limit is equal to 7191 cycles. This limit increases very quickly with b and decreases when E, increases (Figure 7.17). One must however point out one of the assumptions in calculating SD which supposes that 2 rr n0+ T £, is large. For N = 1000, we have 2 7 i N £ = 314. It is necessary at least that +

4

4

2 re n0+ T £, = 10 , i.e. that N > 3 10 , because it eliminates this limit, which does not have any physical reality. Figures 7.15 and 7.16 show that dispersion is stronger for the high levels of excitation (contrary to dispersion related to the fatigue strength of material).

Figure 7.17. Limit of number of cycles to failure for a = 3

Standard deviation of fatigue damage

273

7.4.3. W.D. Mark's formulation As previously, let us write that, to failure

yielding

Let us also set

It then becomes:

provided that the discriminant is positive or zero, or that:

It can be shown that this expression is always positive. The condition E(D) + a SD = 1 leads to the same expression of N. Example With the numerical conditions of the previous example, we obtain, for 1 < G < 9 , the curves in Figure 7.18, which show the same tendency of the scatter to increase with a rms . Use of Figure 7.19 makes it possible to evaluate the influence of damping.

274

Fatigue damage

Figure 7.18. Mean ±3 standard deviations S-N curves for b = 10 and £,= 0.05 (W.D. Mark)

Figure 7.19. Mean ±3 standard deviations S-N curves for b = 10 and % = 0.01 (W.D. Mark)

With the same calculation criteria, but with £, = 0.01 (instead of 0.05), it is necessary, to obtain the same curve E(D), that 0 . 2 < G < 1 . 8 , i.e. a weaker excitation. Scatter on the other hand tends to increase.

Figure 7.20. Mean ±3 standard deviations S-N curves, for b = 8 and £ = 0.05 (W.D. Mark)

Lastly, the same study carried out with b = 8 instead of 10 shows that, to obtain the same damage, the amplitude G of the PSD of the excitation must be greater (Figure 7.20, for £ = 0.05 ).

Standard deviation of fatigue damage

275

NOTE. — As previously, all these curves should not be plotted (with this formulation) for too large a G, i.e. for a low numbers of cycles.

Figure 7.21. S-N curves for low numbers of cycles (W.D. Mark)

By taking again the first example (b = 10, £ = 0.05,), one notes that, if G is too large (between 2 and 10), the number of cycles corresponding to E(D) ± 3 SQ = 1 also tends towards a (notphysical) limit equal to (Figure 7.21): N[E(D)-3s D ]-»0.65 N [E(D) + 3 SD] -> 3288 cycles This limit can be found by calculation, starting from the relation [7.33] (for a =3):

When G becomes large, and thus also A, we have

The upper limit increases with b; it is weaker when damping is larger. The lower limit decreases with b. Curves in Figure 7.21 are plotted out of the range of

276

Fatigue damage

approximations used for calculation of sD. It was seen that one needed i.e., in our example, that N » 10.

Figure 7.22. Limit of number of cycles relating to mean damage minus 3 standard deviations

Figure 7.23. Limit of number of cycles relating to mean damage plus 3 standard deviations

In Figures 7.22 and 7.23, we note that N must be somewhat larger than 104 cycles for the upper limit.

Chapter 8

Fatigue damage using other assumptions for calculation

8.1. S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit) It is supposed that the S-N curve is defined by:

Figure 8.1. S-N curve with fatigue limit (logarithmic scales)

278

Fatigue damage

The damage is equal to

the probability density q(a) being described by [6.24] for a Gaussian excitation. If we can consider that the peak distribution of the response follows a Rayleigh distribution:

Let us set

with

The integral

Fatigue damage using other assumptions for calculation

279

b is the incomplete gamma function (with x = 1 + —). 2

While setting a = K z ,

It is noted that if TD tends towards zero, the expression [8.7] tends towards relation [6.35]. J.W. Miles [MIL 54] estimates that it is possible to neglect the fatigue limit when the stresses are distributed over a broad range. Figure 8.2 shows the variations of the ratio

plotted versus the ratio TD for various values of b. It is noted that the damage decreases when the endurance limit increases, more quickly when b is smaller.

280

Fatigue damage

Figure 8.2. Ratio of the damage calculated with and without taking into account fatigue limit

8.2. S-N curve defined by two segments of straight line on log-lin scales The S-N curves published in the literature are often plotted in logarithmic-linear scales and are, in these scales, comparable with two segments of straight line, the horizontal part corresponding to the fatigue limit.

Figure 8.3. S-N curve with fatigue limit in semi-logarithmic scales

With this representation, this curve can be described analytically by [MUS 60]:

Fatigue damage using other assumptions for calculation

281

A and B being constant characteristics of material. The damage is then equal to

q(a) being the probability density function of stress peaks, given by [6.24]. While comparing here this distribution with Rayleigh's law, it becomes:

Let us set

with

However

Let us set

It comes

yielding

282

Fatigue damage

In addition,

Fatigue damage using other assumptions for calculation

283

This yields

If

8.3. Hypothesis of non-linear accumulation of damage 8.3.1. Corten-Dolan's accumulation law Miner's rule is most frequently retained for the calculation of fatigue damage. Some attempts [SYL 81] were made to use other assumptions, such as CortenDolan's [COR 56] [COR 59], which supposes that the damage is added in a nonlinear way according to

(where the ai and d are constants). Knowing that the S-N curve can be represented here by

we have, for random stress,

284

Fatigue damage

H. Corten and T. Dolan showed that d is a constant when a1, varies. It is thus the constant A which is related toCT1. For untruncated random loadings or if the largest peaks do not exceed the yield stress of material, the value of
and

can be identified for a = a'j; yielding

Let us substitute this value in [8.16]:

where

Since u'1 and the integral are constants for a given rms level, this

expression differs from that obtained with Miner's rule only by one constant. If the stress is truncated, or if the peaks of stress exceed the yield stress (a form of truncation), GJ1 is fixed for the various levels of rms stress and the expression [8.16] is applicable with A constant such that

When there is truncation, the random fatigue curve is thus parallel to the modified curve. If there is no truncation, it is parallel to that established in sinusoidal mode.

8.3.2. J.D. Morrow's accumulation model R.G. Lambert [LAM 88] uses the accumulation damage model developed by J.D. Morrow [MOR 83] in the form

Fatigue damage using other assumptions for calculation

285

a = amplitude of the stress a

max

=

value of the greatest stress

nCT = number of cycles of stress at level a Nc

= number of cycles to failure at level CT

d = plastic work exponent We saw [6.11] that, in the linear case, the elementary damage dD could be written

where

a

Using the non-linear accumulation law [8.20], we have:

rms being the rms stress, let us note

as the crest factor:

286

Fatigue damage

Let us set, as previously, u =

(or integral between 0 and cimax). If a = K z,

The parameter d varies between - 0.25 and - 0.20. Integration can be carried out in practice between 0 and 8. The damage calculated under these conditions is higher than that obtained on a linear assumption of approximately 10% to 15%. Suppose than q(u) is the probability density of a Rayleigh distribution.

Figure 8.4. Probability density of damage using non-linear damage accumulation law of Corten-Dolan

Figure 8.4 shows the variations of the probability density u b+d +1 / eu /2 for d = 0 (linear case) and d = - 0.207 (non-linear case). It is noted that the damage is primarily created by the stress peaks between 2 and 5 times the rms value arms. The peaks larger than 5 crms are extremely damaging when they take place. The peaks lower than 2 Gms are very frequent, but contribute little to the damage.

Fatigue damage using other assumptions for calculation

287

8.4. Random vibration with non zero mean: use of modified Goodman diagram It was assumed until now that the mean stress was zero. When it is not the case, H.C. Schjelderup [SCH 59] proposes to use the modified diagram of Goodman or that of Gerber [LAL 92] (Chapter 3). These diagrams can be represented analytically by

with the following notations R m = ultimate stress Aa0=

amplitude of the zero mean sinusoidal fluctuation

AG= amplitude of the non zero mean sinusoidal stress fluctuation am = mean stress (> 0 if it is about a tension, < 0 if it is a compression) n = constant

n = 1 in the relation of Goodman n = 2 in that of Gerber

This relation can be written

If N(ACTQ) represents the S-N curve with zero mean stress, the S-N curve with mean stress crm is such that

This result can be applied to the problem of random vibrations, by making the following assumptions: - The distribution of the peaks of stress is not modified by the variations of the mean stress. - The Miner's rule constitutes a satisfactory tool for the calculation of the fatigue life for a mean stress.

288

Fatigue damage

In this case, the relation [8.24] can be transformed for application to the random vibrations according to

Gms = rms stress for a non zero mean GO

= rms stress for a zero mean.

The effects of the non zero mean stress are taken into account while replacing We can thus expect similar tendencies in fatigue with random vibrations and sinusoidal vibrations. The restrictions imposed by the selected assumptions limit this result to the modes of fatigue at long lifetime, low stress levels. An experimental study by S.L. Bussa [BUS 67], carried out on notched samples subjected to random vibrations having various spectral shapes, shows that the effect of the mean stress is in concord with the Goodman relation. This result is confirmed by work of R.G. Lambert [LAM 93]. NOTE. - Another way of taking into account the influence of the mean stress can consist, by using the same principles, to modify the constant C1 in relation [6.95] according to [LAM 93]:

or to directly correct the damage by fatigue as in

In these relations, crm must be lower than a'f, if not the failure takes place statically before the application of the alternate stress. a'f is the true ultimate stress, stress causing the failure for dynamics cycle.

Fatigue damage using other assumptions for calculation

Figure 8.5. Ratio of the damage calculated with a non zero mean stress and with a zero mean stress, for n = 1

289

Figure 8.6. Ratio of the damage calculated with a non zero mean stress and with a zero mean stress, for n = 2

8.5. Non Gaussian distribution of instantaneous values of signal The assumption is generally made that the distribution of the instantaneous values of the signal of excitation (and thus of the response) follows a Gaussian law and, consequently, that the distribution of the peaks of the response of the excited system follows a Rayleigh distribution law [BEL 59] [MIL 54] [SCH 58]. 8.5.1. Influence of distribution law of instantaneous values A study by R.T. Booth and M.N. Kenefeck [BOO 76], carried out on steel testbars, made it possible to compare the results obtained with three distributions, characterized by their crossings curve of the form:

where N0 = n0 T. For a Gaussian distribution, A = 2 and B = 0.5 and, for an exponential distribution, A = 1 and C = v2. The three laws considered are characterized as follows: - Gaussian distribution truncated to 4 times the rms value (a ms ). -Distribution close to the distribution experimentally observed from measurements in the real environment on a vehicle suspension device (A = 1.384 , B = 0.955) with truncation to 5.5 arms.

290

Fatigue damage

- Distribution obtained in the real environment measured on dampers (A = 1.19 , B = 1.098 ), with a truncation at 6.5 a^ . The irregularity coefficient is identical in the three cases, about 0.96 (PSD of the same shapes) and the S-N curve is plotted with rms on ordinates: Nofcrms-OD^C

[8.31]

(a D = fatigue limit stress). These authors note that: - The changes of distribution law have a significant influence on the fatigue strength: the distribution of the instantaneous values of the signal is thus an important factor. The assumption of a Gaussian signal can lead to significant errors if it is not checked. - The results obtained with the Miner rule are always optimistic (there is failure under test below

but not unreasonable taking into account the scatter

observed in the fatigue tests. -The calculations carried out on the Gaussian assumption differ from the experimental results by a factor of about 2, if the distribution is Gaussian. If the distribution is not Gaussian, the error can reach a factor equal to 9, when the real distribution contains a more significant number of stress peaks of greater amplitude.

8.5.2. Influence of peak distribution A study carried out by R.G. Lambert [LAM 82] on the influence of the distribution law of the peaks of the response of the system (i.e. of the stress) on fatigue damage shows that: — In a general way, the damage is primarily produced by the peaks of amplitude higher than twice the rms value crms. For a Rayleigh distribution, they are the peaks between 2 a,^ and 4 Gms, while for an exponential law, they are the peaks of between 5 Gms and 15 crms. - Because of the larger presence of peaks of great amplitude, the exponential law is more severe than a Rayleigh distribution.

Fatigue damage using other assumptions for calculation

291

8.5.3. Calculation of damage using Weibull distribution The fatigue damage can be written, according to Miner' rule [NOL 76] and [WIR 77]:

Knowing that N CT = C and that cr = K zg if the stress-strain relationship is not linear, it becomes

i.e., in the continuous case,

with

The integral can be regarded as the (bg)th moment of q(z). Let us set Q(Z) the distribution function of z,

If the distribution of the peaks follows a Weibull distribution,

where

is a constant characterizing the form of the law:

exponential distribution Rayleigh distribution

292

Fatigue damage

yielding

it becomes, with a

and

knowing that

NOTE. - K.G. Nolte and J.E. Hansford [NOL 76] established this relation to study fatigue of immersed structures subjected to waves of height H. It is sufficient to replace z by H to treat this particular case. The distribution wave height is in general approximated by a Rayleigh distribution for which r\ = 2 and 8 =

Hs

V2

(Hs = wave height indicative of the state of the sea). This yields

where

T is the total number of waves.

Fatigue damage using other assumptions for calculation

Figure 8.7. Reduced damage for Weibull peak distribution

293

Figure 8.8. Reduced damage for Weibull peak distribution

b parameter Figure 8.9. Damage reduced for Weibull peak distribution versus parameter b

Figures 8.7 to 8.9 show the variations in the quantity

depending on b or g, for various values oft], in order to highlight the influence of non linearity (for ^2 zrms lower, then higher, than 1).

294

Fatigue damage

8.5.4. Comparison of Rayleigh assumption /peak counting H.C. Schjelderup [SCH 61b] proposes to use a method close to the range-mean counting method (Chapter 5) and assumes that: - the mean stress follows a normal distribution

- the alternate stress around this mean follows also a normal distribution:

a where a is the standardized stress — a

aM = standard deviation of the standardized mean stress aA= standard deviation of the standardized alternating stress aa= mean of the standardized alternating stress

Figure 8.10. Alternating stress around the mean The probability of occurrence of a mean stress am and an alternating stress aa is

The number dn of cycles (a m , aa) among N cycles is equal to

yielding damage dD created by these dn cycles:

Fatigue damage using other assumptions for calculation

295

where N(a m ,a a ) is the number of cycles to failure for (a m , aa). The total number of cycles to failure N is that for which D = 1; yielding:

The function N(cr m ,cr a ) can be obtained starting from the modified Goodman diagram:

ultimate strength. From this formulation, H.C. Schjelderup shows, for an example, that the Rayleigh distribution gives more severe results and that this method seems to better adapt to the experimental results. 8.6. Non-linear mechanical system It is supposed that Basquin's distribution applies (N cr = C) and that the non linearity is of the form a = K zg. Then,

and, in the continuous case,

where

296

Fatigue damage

If

Lastly, if the peak distribution follows a Rayleigh distribution,

NOTE. - D. Karnopp and T.D. Scharton method. These authors approach the response of a non-linear hysteretic system slightly damped subjected to a random excitation by an artificial linear process [KAR 66] [TAN 70]. They calculate, starting from this process, the mean value of the displacement higher than a specific limit as an approximation to plastic deformation. They finally use the Coffin criterion to evaluate the mean fatigue life of the system under loads belonging to the low cycle fatigue domain.

Appendices

Al. Gamma function Al.1. Definition The gamma Junction (or factorial function or Euler function-second kind), F(X), is defined by [ANG 611:

A1.2. Properties Whatever value of x, integral or not,

If x is a positive integer:

298

Fatigue damage

[CHE

66],yielding,

for x a positive integer:

If x is arbitrary much higher than 1, we have:

For x an integer, one uses the relation

to calculate

relation

Figure A1.1. Gamma function

or the

Appendices

299

For arbitrary x, the relation [A1.2] and Table A1.l allow determination of

r(i + x). Example T(4.34) = 3.34 F(3.34) r(4.34) = 3.34 x 2.34 x 1.34 T(l .34) r(l .34) being given in Table A l . l . T(1.34) = T(l + 0.34) = 0.8922 yielding T(4.34) = 9.34

Table A1.1. Values of the gamma function [HAS 55]

r(1+ x) X

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

0.9514 0.9182 0.8975 0.8873 0.8862 0.8935 0.9086 0.9314 0.9618 1.0 1.0000

1

2

3

4

5

6

7

8

0.9943 0.9474 0.9156 0.8960 0.8868 0.8866 0.8947 0.9106 0.9341 0.9652 1.0043

0.9888 0.9436 0.9131 0.8946 0.8864 0.8870 0.8959 0.9126 0.9368 0.9688 1.0086

0.9835 0.9399 0.9108 0.8934 0.8860 0.8876 0.8972 0.9147 0.9397 0.9724 1.0131

0.9784 0.9364 0.9085 0.8922 0.8858 0.8882 0.8986 0.9168 0.9426 0.9761 1.0176

0.9735 0.9330 0.9064 0.8912 0.8857 0.8889 0.9001 0.9191 0.9456 0.9799 1.0222

0.9687 0.9298 0.9044 0.8902 0.8856 0.8896 0.9017 0.9214 0.9487 0.9837 1.0269

0.9642 0.9267 0.9025 0.8893 0.8856 0.8905 0.9033 0.9238 0.9518 0.9877 1.0316

0.9597 0.9237 0.9007 0.8887 0.857 0.8914 0.9050 0.9262 0.9551 0.9917 1.0365

9 0.9555 0.9209 0.8990 0.8879 0.8859 0.8924 0.9068 0.9288 0.9584 0.9958 1.0415

300

Fatigue damage

Example

A1.3. Approximations for arbitrary \ For arbitrary x (integer or not)

[LAM 76],

This relation is rather precise for x > 2 (better than 0.5%). For x integer positive, we have, starting from the above relations,

x! can be approximated to the Stirling formula

or, better, to:

can be calculated with an error lower than polynomial [HAS 55]

where

usine me

Appendices

301

A2. Incomplete gamma function A2.1. Definition The incomplete gamma function is defined by [ABR 70] [LAM 76]:

This function is tabulated in various published works [ABR 70] [PIE 48]. Figures A2.1 and A2.2. show the variations of y with x, for given T, then with T, for given x. We set:

It is shown that

can be written

where

(chi-square probability distribution). The function P is tabulated also [ABR 70].

302

Fatigue damage

A2.2. Relation between complete gamma function function

and Incomplete

We have

yielding

Figure A2.1. Incomplete gamma function versus x

Figure A2.2. Incomplete gamma function versus T

gamma

Appendices

303

A2.3. Pearson form of incomplete gamma function

Tables or abacuses give I varying with u for various values of p [ABR 70] [CRA 63] [FID 75].

A3. Various integrals A3.1.

A3.1.1. Demonstration From

if we set

By division, we obtain

304

Fatigue damage

yielding the recurrence relation [A3.1]. The use of these expressions requires knowledge of I b ( h ) for some values from b, for example b = 0, 1 and 2.

A3.1.2. Calculation o/I 0 (h) and I 2 (h) The calculation of I 0 (h) and I 2 (h) can be carried out easily by writing the functions to be integrated respectively in the forms

and

Let us set

Knowing what

can be put in the form with

it becomes

Appendices

In the same way,

With a change of variable such that:

we obtain

yielding

NOTES. 1. Knowing that

305

306

Fatigue damage

can be also written

The expression [A3.6] must however be used with caution, [A3.5] being correct only under precise conditions. The relation [A3.5], established from

is exact only if

From these first two conditions, if tan a

it is necessary that

Taking into account these inequalities, a + b can thus vary between -re and TT 71

71

(limits not included). The second member imposes in addition - — < a + b < — . Let 2 2 us set 0 = a + b. If

one must thus take as the value arc tan

of the angle

Appendices

307

If

one considers 2. It is easy to show that we are in the situation of an inequality [A3.7] as soon as we have x + y > 2. Indeed let us search for the minimal value of the sum

i.e. of

While setting X = tan x, one finds that the derivative

is cancelled when

yielding z = 2 (condition [A3.7]) or z = -2 (condition [A3.8]). Following what precedes,

and

yielding

308

Fatigue damage

which can be written, with the same reservations as above,

NOTES. - When h

and yielding, while setting

A3.1.3. Calculation of

Appendices

309

yielding

Knowing that

can be also written (always with the same reservations):

Application Let us consider the integral

The relations below give the expressions of [PUL 681:

for some values of b

310

Fatigue damage Table A3.1. Integrals I b (h) for b ranging between - 6 and 9

Appendices

311

A3.1.4. Approximations When the resonance frequency is within the frequency range of the excitation and when the influence of damping is small, it is possible to neglect the term in £ and to write Ib in the form [MIL 61]:

provided that

The condition is then written

312

Fatigue damage

Figure A3.1. Validity domain of the approximation [A3.30]

yielding

Knowing that h is always positive or zero and that £ varies between 0 and 1, these conditions become

For

these conditions lead for example to:

The smaller damping is, the larger the useful range.

Appendices

313

If

One increases, for safety, these values by approximately 40%:

For h small and less than 1:

For h large and higher than 1:

While replacing

by its expression in

and while integrating, it

becomes

for

for as long as no denominator is zero. If one of them is zero, the corresponding term is to be replaced by a logarithm. Thus, if b ± m = 0, the term

must be written

314

Fatigue damage

A3.1.5. Particular cases

where

These results can be deduced directly from [A3.22] and [A3.20] [WAT 62]. NOTE.- More generally, if b = 0 [DWI 66]:

where

and a, c and K > 0.

A3.2.

From the above relations, we deduce

Appendices

315

The expressions of J b (h) for the other values of b can be calculated starting from these relations using the recurrence formula.

A3.3.

where

the roots of

assumed located in the upper half plane for

Figure A3.2. Real and imaginary axes

The first values of In, extracted from the work of H.M. James, N.B. Nichols and R.S. Philipps [JAM 47], are the following:

316

Fatigue damage

Application From these expressions, S.H. Crandall, W.D. Mark [CRA 63] and D.E. Newland [NEW 75] deduce the value of the integral

where

For

Appendices

317

etc. A3.4.

These two integrals are calculated simultaneously while multiplying I2 by i in order to constitute the integral:

yielding

A3.5,

and

by separating the real and imaginary parts.

318

Fatigue damage

yielding

Applications

[PAP 65]

[CRA 63] A3.6.

This result is demonstrated by setting [ANG 61]:

Appendices

yielding

and

One obtains an Euler integral-first kind of the form

which can be expressed using gamma functions.

A3.7.

If

(n even),

319

320

Fatigue damage

If

A3.8.

Approximations Integral A can be approximated by the expression:

One obtains, with this relation, values a little higher than the true values. The relative error is equal to 6.4% for k = 2, to 2.5% for k = 5 and to 1.3% for k = 10 [DAY 64]. It can be reduced while evaluating (k - l)! using the Stirling formula [A 1.13]:

This error gives a value of the factorial smaller than the true value. For k = 2, the relative error is equal to - 7.8 %. It is equal to -2.1 % for k = 5 and to - 0.9 % if k = 10. Integral A can then be written, for (k > 2),

The relative error is here equal to -1.94% for k = 2, close to +0.4% for 5 < k < 10 and lower than 0.4% if k > 10.

aAppendices ces

•

£

l

2 6

1 8

Figure A3.3. Relative error made using the approximate expression for integral A

321

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