Active Control of Vibration

Preface

Over the last decade there has been much work concemed with the active control of vibrations of flexible structures. The great majority of the work has been concerned with feedback control of large flexible systems at relatively low frequencies. The general topic of active vibration control in this context has been admirably treated in recent texts by Professors Meirovitch (1990) and Inman (1989). The subject matter of their texts has largely been devoted to modem control systems with an emphasis on multi-channel feedback control. More recently, as a result of a perceived need in the noise control community, advances have emerged in the active control of vibrations at audio frequencies and for steady state excitation. These advances have been largely due to the recent development of fast digital signal processors together with fast multichannel analogue-to-digital and digital-to-analogue converters. In addition, there have been significant advances in the development and use of control transducers which have enabled the realistic implementation of many active vibration control techniques. The overall aim of the book is to summarise these new advances in active vibration control with an emphasis on the fundamental scientific principles that form the basis of these techniques. In writing this book we have chosen to provide both a basic understanding of the subject and a research reference text. The book is thus aimed towards graduate students, researchers and engineers who have some knowledge of the theory of vibrations, mechanics and control. The book is written as a companion to the text by Nelson and Elliott (1992) which covers, in detail, the related area of active control of sound. In the interest of brevity, much of the material which is common to both fields has not been included in this text and references to Nelson and Elliott (1992) are provided where pertinent. However, when the material is essential to the understanding and continuity of the text, it is included in this book. In a similar way to the companion text, the book attempts to combine in a unified manner, material from mechanical vibrations, acoustics, signal processing, mechanics and control theory. Key new areas discussed in the text are the use of feedforward control, the modelling and use of distributed strain actuators and sensors, the control of waves in structures, the theory and implementation of active isolation of vibration and the active control of structurally radiated sound. Throughout the text considerable effort is directed towards highlighting and clarifying the dual nature of the 'wave' and 'mode' descriptions of the vibrations of structures. It is demonstrated that either form of description has its advantages, depending upon the type of application, understanding of the control problem and designing of the controller.

x

PREFACE

The book begins with a brief review of mechanical vibrations and wave propagation in structures. This material is well known, but it is necessary to review and introduce the material within the context of the subject matter of the book in order to develop a solid foundation of understanding. Effort is directed towards describing basic vibrations and wave propagation in order to understand the more advanced topics oriented towards active control that are described in later chapters. Chapter 1 is intended as an introduction to vibrations of lumped parameter systems and also includes a brief description of finite element analysis. It thus provides the basic equations for modelling the control of lightly damped structures with only a small number of modes. Chapter 2 summarises basic material describing longitudinal and flexural wave propagation in long slender beams and cylinders. These equations are then applied to modelling the motion of finite beams, plates and cylinders and a discussion is presented of the interpretation of the response in terms of either waves or modes. The equations describing the response of the above systems to various input force and moment configurations are developed and provide the basis of modelling the control of vibrations of such structures. Chapters 3 and 4 introduce some of the control concepts used in the book. In Chapter 3 feedback control is discussed, initially for a single-input single-output (SISO) system using a transfer function approach. The generalisation to multiple-input, multipleoutput (MIMO) feedback control systems is then described using state variables. The state variable formulation is a useful way of describing feedback controllers, and in particular it suggests a way of describing the independent control of the system's modes. Chapter 4 introduces feedforward control, again beginning with the SISO case analysed in the frequency domain. Adaptive digital filters are widely used for feedforward control and these are introduced in this chapter, and the generalisations required for multi-channel (MIMO) operation are described in some detail. The adaptivity of the feedforward controller ensures that it is not 'open loop', and a brief analysis is presented that shows how adaptive feedforward controllers can be represented as equivalent feedback systems. In Chapter 5 we describe material related to the use of actuators and sensors for active control of vibration. The chapter begins with a summary of recent work in modelling the use of distributed, piezoelectric strain actuators and sensors in various configurations. The use of point sensors configured in arrays in order to provide estimates of modal and wavenumber response is also discussed. Decomposition of wave fields into individual wave components is outlined. The chapter finishes with a brief description of advanced actuators such as those based on shape memory alloy which can be used for semi-active, adaptive, or steady state control of system parameters. Chapter 6 describes the active control of vibration in various distributed mechanical systems. Initially the active control of the mechanical response is analysed in terms of the structural modes of the complete system. The active suppression of the resonant response of these modes is then illustrated using both feedforward and feedback control methods. Alternatively, the motion of a system can be analysed in terms of the structural waves which propagate within it, and the active control of structural waves is described in the second half of this chapter. Particular attention is paid to the active control of flexural waves since they are dispersive and have near-field components, and both effects introduce their own complications into the active control problem. Chapter 7 deals specifically with the active isolation of vibrations. The first topic to

PREFACE

xi

be addressed is the isolation of a periodic source of vibrations from a resonant receiving structure. This problem is widely encountered in engineering practice and occurs whenever a rotating or reciprocating machine is mounted on a flexible structure. This problem is then generalised to include the isolation of transient machinery vibrations and also to deal with the case where one wishes to provide isolation for a system (a sensitive instrument for example) from externally generated vibrations. The considerable promise shown by the application of active techniques to these problems is clearly demonstrated. The final two chapters of the book deal with the new field of Active Structural Acoustic Control (ASAC) in which structurally radiated sound is directly controlled by active structural inputs. In Chapter 8 the concepts of ASAC are first outlined and then the mechanisms of sound radiation from vibrating structures are discussed. A review is presented of the application of ASAC to plate structures excited by various classes of disturbances and controlled using various transducer configurations. The behaviour and control of the system in the wavenumber domain is discussed. An example of an ASAC configuration and experimental results are described to illustrate the practical implementation of the approach. The use of multi-channel feedback control to implement ASAC is then outlined. Chapter 9 discusses the application of ASAC to cylindrical systems. The acoustic radiation and coupling with vibrating cylinders is briefly reviewed. The application of ASAC to the control of sound radiated by cylinders, interior cavity noise and vibrational power flow in fluid-filled cylinders is summarised. The chapter finishes with the description of the use of ASAC to control interior noise in an aircraft fuselage in order to illustrate a practical application of the technique. There are many well established topics in active control of vibration that have not been covered in this book and only those most appropriate to vibration control in the audio frequency range have been dealt with in depth. Although the theory of feedback control and its application to various structural systems are discussed briefly in the text, the reader is referred to the texts of Meirovitch (1990) and Inman (1989) for a more detailed description of this material. The main purpose of the material on feedback control included in this text has been to illustrate where it has been used recently at the higher audio frequencies and for the control of structurally radiated sound. It was also thought necessary to introduce this material in order to relate it to the newer field of feedforward control. Similarly, the control of vibrations in large flexible structures at very low frequencies has not been covered in this text. Throughout the book, numerous references to other books, research publications and text are provided. This list is not intended to be comprehensive but is intended to provide the reader with the information and guidance to find more detail on a particular subject. It is our hope that in this book we have described new material which will lead to the application, and stimulate research in the field, of active vibration control at audio frequencies. It is our view that the active control of vibrations shows much potential for solving many difficult noise and vibration problems. If the book provides the basis for guiding the reader towards using these new solutions then we believe it will have largely achieved its aim. In writing this book the authors have had the benefit of many valuable suggestions and criticisms from a large number of colleagues. In particular the authors would like to acknowledge the help given by Professor Ricardo Burdisso, Dr Gary Gibbs, Dr

xii

PREFACE

Cathy Guigou, Dr Bertrand Brevart, Dr Tao Song and Julien Maillard of Virginia Polytechnic Institute and State University. Professor Robert Clark of Duke University, Professor Jim Jones of Purdue University, Professor Peter Wang of National Pingtung Polytechnic Institute, Dr Andy von Flotow of Hood Technology Corporation, Dr Kam Ng of the Office of Naval Research, Dr Dean Thomas of the ISVR, University of Southampton and Drs Rich Silcox and Harold Lester of NASA Langley Research Center also provided very useful input. The authors are also indebted to NASA Langley Research Center, the U.S. Office of Naval Research and the U.K. Department of Trade and Industry for financially supporting much of the research which forms the basis of this book. We are also grateful to the Office of Naval Research for providing support for the preparation of the text. Many reviewers contributed their valuable time to reading and providing suggestions for improvement of the material and to Dr E. Anderson, Dr A. Baz, Dr M. Brennan, Professor J. Cuschieri, Dr C. Hansen, Professor M. Heckl, Professor D. Inman, M. Johnson, Professor C. Knight, Professor C. Liang, Dr B. Mace, Dr R. Pinnington, Professor W. Saunders, Dr J. Scheuren, Dr S. Snyder, Professor S. Sommerfeldt, and Dr T. Sutton, the authors wish to offer their thanks. Finally the authors are indebted to their families for unlimited tolerance during the difficult parts of the writing process, and to Dawn Williams, Crystal Carter, Maureen Strickland, Susan Hellon, Karl Estes and Cathy Gorman for their excellent typing and graphics skills.

Introduction to Mechanical Vibrations

1.1

Introduction

All mechanical systems composed of mass, stiffness and damping elements exhibit vibratory response when subject to time-varying disturbances. The prediction and control of these disturbances is fundamental to the design and operation of mechanical equipment. In particular, the use of secondary, active inputs to the system in order to modify the system response in a controllable way is the topic of this book. The analysis of controlled systems is founded on the same analytical approaches used to study the vibrations of elastic structures. A brief review of the main concepts of vibration analysis and the associated techniques of solution is necessary to set the foundation for the following chapters. In this chapter we begin by defining terminology and the mathematical methods for describing the linear response of vibrating systems. The equations of motion and linear behaviour of single-degree-of-freedom systems are outlined for both free and forced response. The use of the Laplace transform to solve for transient response is reviewed. The extension to multi-degree-of-freedom systems and then the use of finite element analysis are briefly introduced. These approaches are valid for lightly damped structures or elements that are small relative to the wavelength of motion. For more detail the reader is referred to the texts of Thomson (1993), Meirovitch (1967) and Inman (1994).

1.2 Terminology The following is a brief list of the main terminology and definitions used in analysing the vibratory response of mechanical systems. Mechanical system. A mechanical system is composed of distributed elements which exhibit characteristics of mass, elasticity and damping. Degrees of freedom. The number of degrees of freedom of a system is equal to the number of independent coordinate positions required to completely describe the motion of the system. System response. All mechanical systems exhibit some form of vibratory response when excited by either internal or external forces. This motion may be irregular or may repeat itself at regular intervals, in which case it is called periodic motion. Period. The period T is the time taken for one complete cycle of motion.

2

ACTIVE CONTROLOF VIBRATION

Harmonic motion is the simplest form of periodic motion whereby the actual or observed motion can be represented by oscillatory functions such as the sine and the cosine functions. Motion that can be described by a continuous sine or cosine function is called steady state. For example one may write the actual displacement w ( t ) in the form

w(t) = [A I cos(~ot + q~),

(1.2.1)

where w ( t ) and IAI are real, a~ is circular frequency in radians per second and q~ is an arbitrary phase angle in radians. Equation (1.2.1) can also be expressed as a superposition of a sine and a cosine function as w ( t ) - AR cos cot- At sin cot,

(1.2.2)

where AR and At are real numbers such that AR = I Alcos ¢,

(1.2.3a)

A, = I alsin ¢,

(1.2.3b)

where the phase angle is specified by qb = tan - ~(AJAR).

(1.2.4)

The constant I AI in equation (1.2.1) is related to the constants At and AR in equation (1.2.2) by

[al

= ( a z + a l z)1/2.

(1.2.5)

Frequency is the number of cycles per second (also called hertz) of the motion and is the reciprocal of the period. Therefore frequency is specified by

1 f= --

T

(1.2.6)

and circular frequency co (radians per second) is given by o9 = 2z~f. Amplitude is the measure of For example if the motion is corresponds to the amplitude of The mean square amplitude response. Thus, for example

(1.2.7)

the maximum response of the system during a period. specified by equation (1.2.1) then the constant I AI the motion. is defined as the time average of the square of the

= lim --1 [ r w2(t) dt. T--->~* Z J°

(1.2.8)

The root mean square (rms) amplitude is the positive square root of the mean square amplitude. For the harmonic oscillation of equation (1.2.1) the rms amplitude is independent of phase and is equal to Ial/~. Free vibrations are the motions of the system in the absence of external disturbances and as a result of some initial conditions. Forced vibrations are the motions of the system produced by external, persistently applied disturbances.

3

INTRODUCTION TO MECHANICAL VIBRATIONS

Natural frequencies are those frequencies at which response exists during free vibration. The lowest natural frequency is called the fundamental frequency. Transient motion is motion other than steady state response. If damping is present, transient response will decay with increasing time. Phasor. A phasor is a rotating vector representation of the harmonic motion of the system. The periodic motion of equations (1.2.1) and (1.2.2) can be represented in a complex form, which is more convenient for mathematical manipulations and is given by w(t) = A e j°~t, (1.2.9)

where A and w(t) are complex with the complex amplitude specified as (1.2.10)

A= AR + jAI,

The phasor representation of equation (1.2.9) is shown in Fig. 1.1. The length of the vector, I AI, is the real amplitude of motion. As the vector rotates with angular velocity to in a counterclockwise direction, its projection on the real and imaginary axis of the complex w plane varies harmonically with time t. A rotation of the vector through 360 ° corresponds to a cycle of motion. In this text the convention used is that the real component of the phasor or the complex description of the motion corresponds to the actual, observed or measured motion. Therefore the actual motion is given by w(t) = Re[A eJ°'t].

(1.2.11)

Using the relationship eJ°"=cos tot + j sin tot and substituting A = AR +jAI into the above expression yields equation (1.2.2). The phase of the motion, q~, is thus retained through the ratio of the imaginary and real components of the phasor as specified in +lm

AE J I I I I I

AR

+Re

ines phasor position at t = O)

Fig. 1.1 Phasor diagram representation of harmonic motion.

4

ACTIVE CONTROLOF VIBRATION

equation (1.2.4). Note that a negative sign has been used in equation (1.2.2) in contrast to many texts dealing with vibration. This choice ensures that the phase q~ is positive and since At is a constant to be determined by boundary conditions, the choice of negative sign does not affect the result. Since the phasor is a vector, any number of harmonic motions of the same frequency can be added vectorially. For linear motions written in complex notation, since the principle of superposition holds, this simply means separately summing the real and imaginary components of the individual motions. In this text the majority of the response equations are written using complex notation as this is the most convenient form for analysing systems where responses are superimposed (i.e. as is the case in active control simulations). The actual motion can be directly recovered by taking the real part of the complex description. Where the motion is directly described in the actual form, it is indicated in the text.

1.3 Single-degree-of-freedom (SDOF) systems Consider a mass M supported by a massless spring as shown in Fig. 1.2(a). As the displacement of the system can be completely specified at all times by a single variable w, the system is said to possess a single degree of freedom (SDOF). By appropriately w

T (a)

M

(b)

M

(d)

M

K

//

/

/ / / /

/

/ //'

1" (c)

M

K<

////I Fig. 1.2

G

// ///

SDOF systems and free body diagrams: (a) and (b), undamped; (c) and (d), damped.

INTRODUCTIONTO MECHANICALVIBRATIONS

5

choosing the origin of the coordinates at the rest position (i.e. in static equilibrium), the constant force due to gravity can be ignored. When the mass is displaced an amount w from its equilibrium position, the spring will exert a restoring force - K w due to being elongated (for positive w) as shown in the free body diagram of Fig. 1.2(b). On release of the mass, the spring will attempt to accelerate the mass and the restoring force and acceleration are related by Newton's second law of motion which shows that d2w M

dt 2

= -Kw.

(1.3.1)

Rearranging terms, we obtain the differential equation describing the motion of this simple SDOF system. This is given by dt 2 +

w-0.

(1.3.2)

Equation (1.3.2) is a second-order ordinary differential equation and therefore must have a solution which is specified in terms of two unknown constants or amplitudes of motion. Although the above analysis is straightforward, it does illustrate the basic process by which elastic systems are generally analysed. The system is first broken into elements (or blocks). For some initial conditions the restoring and inertial forces are balanced, thus providing the differential equation describing the motion of the system.

1.4

Free motion of SDOF systems

Based upon the observation that mechanical systems respond harmonically in free motion, the solution of equation (1.3.2)can be assumed to be of the form given by equation (1.2.2). Therefore we assume that the actual motion can be described as w ( t ) = AR cos oJt- AI sin tot.

(1.4.1)

where AR and A/are real amplitudes of motion. Substitution of equation (1.4.1) into equation (1.3.2), differentiating with respect to time and eliminating common terms, provides a relation for the frequency ton at which the system will naturally vibrate. This is given by oJn = I ~

(1.4.2)

and thus the solution of equation (1.3.2) becomes w ( t ) = AR cos to, t - A~ sin ~o,t.

(1.4.3)

In order to specify the motion completely, the unknown constants AR and A~ need to be determined and these are found by applying given boundary or initial conditions to the system. The frequency oJ, is called the natural or resonance frequency and is a very important characteristic of the system as will be shown in Section 1.6. Note that the natural frequency of the SDOF system increases with stiffness K and decreases with increasing mass M. These observations are in general true for all linear elastic systems.

6

ACTIVE CONTROL OF VIBRATION

To specify the motion of the system completely, one needs to apply initial conditions. For example, if at t = 0 the system has an initial real displacement w(0) and a real velocity ~¢(0) then the unknown constants in equation (1.4.3) can be determined from w(0) = AR,

(1.4.4a)

w(0) = - o9~A,,

(1.4.4b)

where the use of the overdot is a compact notation for differentiation with respect to time. The observed response can be obtained by solving for the constants AR and At from equations (1.4.4a) and (1.4.4b) and substituting these into equation (1.4.3). The actual response of the system to arbitrary initial conditions is then given by

w(t) = w(0) cos tOnt+

w(O)

sin (Ont.

(1.4.5)

This motion can also be written as w(t)= [A lcos(O)nt + ~9),

(1.4.6)

where the phase angle q~ is specified from equation (1.2.4) by q~= tan_l( -

tOnW(O)l~(O)

(1.4.7)

and the amplitude of motion that results from equation (1.2.5) is given by IAI=

[w(O)] ~ +

(1.4.8) (.On

Thus the response of the SDOF can be seen to be simple harmonic motion at the natural frequency ~o, with an amplitude [A[ and a phase angle q~ given by equation (1.4.7) and (1.4.8) respectively.

1.5 Damped motion of SDOF systems All vibrations in realistic systems occur with some form of damping mechanism, where the energy of vibration is dissipated during a cycle of motion. The simplest form of damping is when the resisting force associated with the damping is proportional to, and acts in an opposite direction to, the velocity of the element. Thus the damping force is specified by dw F~ - - C ~ ,

(1.5.1)

dt where C is the damping coefficient. Figure 1.2(c) shows an SDOF system with this form of damping which is called viscous damping. Including the additional damping force into the force balance of the new system, as shown in the free body diagram of

INTRODUCTION TO MECHANICALVIBRATIONS

7

Fig. 1.2 (d), leads to a new differential equation for an SDOF system given by

M

d2w

dt 2

+C

dw

dt

+ Kw = 0.

(1.5.2)

It is now more convenient to use a complex description of the motion. Thus a solution is assumed of the form

w(t) = A e "/',

(1.5.3)

where w(t) is now a complex variable. Substituting equation (1.5.3) into equation (1.5.2) provides the values of ~, for which a solution exists. These are given by 7=

2~/±

-

.

(1.5.4)

It is convenient to express C in terms of the critical damping Cc = 2M~on. The damping ratio is then defined by ~ = C/Cc. Equation (1.5.4) then reduces to 7 = -ogn~ ±jogJ 1 - ~2,

(1.5.5)

where co, is the undamped natural frequency given by equation (1.4.2). When ¢ > 1, both terms in equation (1.5.5) will be real and this implies a steadily decaying response with no oscillation. This is termed an overdamped system. When = 1, the system is said to be critically damped. This value of C represents the smallest possible damping required to prevent oscillatory motion and ensures that the system returns to its rest position in the shortest time as shown in Fig. 1.3. When ~ < 1 then the square root term will be real, positive and 7 in this case will be complex with a negative real part. Thus the response will oscillate at a damped natural frequency oJa = oJ~x/1 - ¢2

(1.5.6)

and decay in amplitude with increasing time. This is called light damping and it is exhibited by most structural systems which are thus described as underdamped. The observed response to the specified initial conditions defined in Section 1.4 is obtained by using the real part of equation (1.5.3) and the initial conditions to solve for the constants A ! and AR as described in Section 1.4. The actual displacement is then given by

w(t) = e -~,.¢t[w(0) cos COdt+ vi,(0) + ¢co,w(0)sin (Oat] .

(1.5.7)

(.,0d

This equation can also be written in simple harmonic form as -~o,¢t W(t) = IAle cos(~oat+ q~),

(1.5.8)

where the phase angle q~ is now given by q~= t a n _ l ( w ( 0 ) + ~ w ( 0 ) ) -

w(O)~oa

(1.5.9)

and the real amplitude by

I AI = {[w(0)]2 + [w(0)+ ~c0~w(0)]2/~03} 1/2.

(1.5.10)

8

ACTIVE CONTROL OF VIBRATION

Wo

light damping

~t)

~'<1

Wo

criticaldamping

w(t)

~r=l

Wo

~

heavydamping t

~'>1

Fig. 1.3 Response of an SDOF system with various values of damping. Initial displacement w(0) = w0 while initial velocity r0(0)= 0. Equation (1.5.8) reveals that the response consists of harmonic motion at a frequency of rod and with an amplitude given by IAle -~.~' which decays with increasing time. Note that from equation (1.5.6) it is apparent that the damped frequency is lower than the natural frequency. Obviously the inclusion and characteristics of damping are very important to active control methods since it represents a process by which the response of a system can also be reduced by passive means. Figure 1.3 shows typical response versus time curves for an SDOF system with light, critical and heavy damping. As predicted by the above equations, light damping leads to a response that oscillates at rod and is slowly decreased in amplitude with time. With critical damping the response moves towards the equilibrium position but does not cross it. Heavy damping leads to almost complete suppression of the oscillatory type motion; the damping force is such that it significantly slows the motion towards the equilibrium position and the system takes longer to return to the equilibrium position than with critical damping.

INTRODUCTIONTO MECHANICALVIBRATIONS

1.6

9

Forced response of SDOF systems

Many systems are excited by persistently applied disturbances rather than being initially excited as in the cases of free motion discussed above. Let us assume that the input disturbance is specified by a harmonic force of constant amplitude written in complex form and given by f ( t ) = F e TM,

(1.6.1)

where F is a complex number defining the amplitude and relative phase of the applied force. The homogeneous differential equation of equation (1.3.2) thus has to be modified to take account of the disturbance force and is written as d2w M ~

dt 2

+ K w = F e ~'.

(1.6.2)

Since it is assumed that the disturbance has been applied for all time, t, the transient response component is zero and it is reasonable to assume the steady state response will be again harmonic and specified in complex form by w ( t ) = A e TM,

(1.6.3)

where w ( t ) and A are again, in general, complex. On substitution of this assumed solution into equation (1.6.2) we obtain -~o 2 +

A = ~. M

(1.6.4)

The unknown complex response amplitude A can be obtained by rearranging equation (1.6.4) and is given by ElK

A=

.

(1.6.5)

1 - (O.)/O)n) 2

Equation (1.6.5) proviaes immediate insight into the response of elastic systems to a steafly state harmonic flisturbance. When ~o = a,. the flisplacement amplitufle will theoreticany be infinite for continuous steafly state excitation. Thus when systems are flriven at a frequency iclentical to or very close to their natural frequency a very large response will result. Under these conditions the system is ~escribefl as being flriven o n anti is a very important conflition from a control point of view. resonance

Extension of the above approach to more complex systems becomes increasingly difficult. In this case, it is often easier to use the impedance method. We define mechanical input impedance as the ratio of the complex amplitudes of input force to the velocity at the drive point. Thus for the harmonic displacement of equation (1.6.3) the velocity is given by vi,(t) and is specified by ¢v(t) = ja)A e TM.

(1.6.6)

The velocity thus has complex amplitude javA and input impedance is given by Z~ =

F jooA

(1.6.7)

10

ACTIVE CONTROL OF VIBRATION

evaluated at the disturbance location. The input impedance of the system in Fig. 1.2(a) can be shown to be

-jK[ 1 -

Z~ =

(co/co,) 2]

.

(1.6.8)

co

Once the input impedance of a system is determined or measured, the response of the system to a steady state harmonic disturbance force can be directly calculated using equation (1.6.7). Inclusion of damping into the relationship for the forced response modifies the resulting amplitude to

F/K

A= 1

-

.

(1.6.9)

(co/con)2 + j2~(CO/CO,)

(a) --

dctJ tO

~' = 0.05

6

~'= o.~

5

Orj

-tO

"~ t,-E

4

~= o.~5

3 ~"= 0.25

2

t.--

o

z

~'= 0.5 1 0.5

1.0 1.5 2.0 Frequency ratio, co~con

2.5

3.0

(b)

o~ o

d c'-

-90

03 t-

13_

~ ' ~ 0 ~ ~ -180

Fig. 1.4

0

1.0

2.0 3.0 4.0 Frequency ratio, o~/con

n 5.0

Forced response of an SDOF system: (a) magnitude, (b) phase.

11

INTRODUCTION TO MECHANICAL VIBRATIONS

Figure 1.4(a) presents the non-dimensional displacement response amplitude equal to

IAK/FI versus the non-dimensional input frequency, co/~o,, for various values of damping ratio ~. The first important feature apparent from Fig. 1.4(a) is that the inclusion of small amounts of damping has led to large reductions of response at or near resonance but has small effect away from the resonance condition. Secondly, damping leads to the response of the system being bounded at resonance. Increasing damping also causes the frequency of maximum response to move to lower values. The corresponding phase response is plotted in Fig. 1.4(b). For very light damping the phase of the displacement response relative to the excitation force of the system flips through nearly a 180 ° phase change as the excitation frequency is increased through the resonance frequency. Increasing the amount of damping leads to a decrease in sharpness of the phase transition. These observations will be seen in later sections to have important implications for the performance and stability of actively controlled elastic systems, particularly when they have many degrees of freedom.

1.7

Transient response of SDOF systems and the use of the Laplace transform

We first consider the response of an SDOF system to an impulse excitation as this case provides the basis for the study of more general forms of transient excitation. An impulse is generally thought of as a disturbance force acting on a system over a very short time. We define an impulse as the time integral of the force I = I ~f(t) dt, 0

( 1.7.1 )

where f(t) is a real function describing the variation of the force with time and r is the duration of the impulse. Since fdt = Mdw where w is the velocity of the mass we can rewrite equation (1.7.1) in the form I = I:(~)M dw.

(1.7.2)

If we make the further approximation that the impulse acts over such a small time that v~(r) -- w(0), or in other words the system reaches a velocity ~¢(0) instantaneously, then I = My0(0),

(1.7.3)

which is known as the impulse-momentum relation. Thus the initial conditions for the impulsively excited system are w(0) =0,

(1.7.4a)

¢v(O) = I/M.

(1.7.4b)

Applying these initial conditions to the damped SDOF system, the response to a unit impulse can be determined from equation (1.5.7) with I = 1 and is given by -~%t

w(t) =

e

Mood

sin cojt,

(1.7.5)

12

ACTIVE CONTROLOF VIBRATION

where w(t) is the actual motion of the mass. Equation (1.7.5) is known as the unit impulse response function of the system generally denoted h(t), and is a very useful parameter. Note that h(t) is a purely real function. The knowledge of the unit impulse response function allows us to derive expressions for the response of a mechanical system to any general excitation, including transients, and the response due to specified initial conditions using Duhamel's integral (Meirovitch, 1967; Newland, 1984). The actual response of the SDOF system to a general real forcing function f(t) is then given by -~tOn(t- r)

w(t) = I~f(r) e

MtOd

sin[tOd(t- r)] dr

[w w(O)+~tOnw(O) q sin rod • +e -¢'°"' (0) cos tOdt+ rod

(1.7.6)

A more specific form of equation (1.7.6) is called the convolution integral and is written as

w(t)= I~f(r)h(t- r) dr,

(1.7.7)

where h(t) is the unit impulse response function of the system. Equation (1.7.7) essentially expresses the response of the system to any general excitation f(t) by using the principle of superposition, that is, the total response consists of the sum of a series of impulses appropriately weighted and positioned with the magnitude of the forcing function f ( r ) at various times, r. We now examine the behaviour of the impulse function for decreasing time durations. Figure 1.5 shows a rectangular impulse function of length r, centred on t = 0 and of impulsive magnitude I. If we take I to be a constant, then by examining equation (1.7.1) it is apparent that as we progressively reduce r, then f(t) will increase. In the limiting case of r tending to zero we have a function which is infinite at t = 0 and has a value of zero at all other values of time t. If we further assume that the product f(t)

-d2

d2

Fig. 1.5 General impulse function.

t

INTRODUCTION TO MECHANICALVIBRATIONS

13

r F is equal to unity (i.e. I = 1) then we have defined the Dirac delta function described by the pair of relationships iS(t) = 0,

t ¢ 0,

I ~ d(t)dt = 1.

(1.7.8a) (1.7.8b)

If we assumed that the rectangular pulse was centred at time to then we could equivalently define the function in the form d(t - to) = 0,

t ¢ to,

I?= d(t- to)dt = 1.

(1.7.9a) (1.7.9b)

A very useful characteristic of the Dirac delta function is called the 'sifting' property given by

I? f(t)d(t- to) dt=f(to).

(1.7.10)

Other properties and uses of generalised functions such as the Dirac delta function are described by Farassat (1994). We now turn our attention to the use of the Laplace transform in analysing the response of mechanical systems to transient disturbances. The Laplace transform is a generalisation of the Fourier transform and is generally useful for disturbances which do not have a convergent Fourier transform (Nelson and Elliott, 1992). The Laplace transform can often be used to transform complicated disturbances associated with impulsive and transient excitation into a form which, in conjunction with the transformed equation for the response of the system, are more easily manipulated. The Laplace transform of f(t) is defined to be (see, e.g. Kuo, 1966; Meirovitch, 1967)

.~f(t) = F(s) = I : f ( t ) e -s' dt,

(1.7.11 )

where s is a complex variable defined by

s= cr + jw.

(1.7.12)

The inverse Laplace transform is defined by (Meirovitch, 1967) ~ - l F (s) = f ( t ) =

1 ~7-~ [,/+~ F(s) es' ds, 2Jrj

(1.7.13)

where the path of integration is a line parallel to the imaginary axis crossing the real axis at Re{ s} =~, and extending from -,,,, to +,,,,. As an example, the equation describing the motion of an SDOF system can be written in the form

MfC(t) + C~(t) + Kw(t)= f(t).

(1.7.14)

Applying the Laplace transform to both sides of the above equation, we obtain

M[sZW(s)- w(O)s- w(0)] + C[sW(s) - w(0)] + KW(s) = F(s).

(1.7.15)

14

ACTIVE CONTROL OF VIBRATION

Collecting terms and rearranging leads to the s-domain relation for the system response described by

F(s)

W(s) =

M s 2 + Cs +

K

+

(Ms + C)w(O) + MCv(O) Ms 2+

Cs + K

,

(1.7.16)

where the first term in equation (1.7.16) describes the forced response and the second term, the motion due to initial conditions w(0) and vi,(0). If we restrict our attention to the forced response, the system transfer function for an SDOF system can be written in the form

H(s) = M s 2 + Cs +

K

.

(1.7.17)

It is often useful in the analysis of active control systems to represent the input-output relationship using a block diagram as shown in Fig. 1.6. Note that if we choose to consider only the imaginary axis of the Laplace domain s, then the system transfer function H(s) is equivalent to a frequency domain transfer function H(jto), as discussed in Nelson and Elliott (1992). Meirovitch (1967) also demonstrates that the system transfer function can be obtained from the system impulse response function by the relationship

H(s)=~h(t).

(1.7.18)

We now illustrate the use of the Laplace transform by solving for the transient response of an SDOF system excited by a harmonic forcing function turned on at t =0. The input disturbance is specified by f(t) = O, f ( t ) = F sin tot,

t O.

(1.7.19)

Taking the Laplace transform of the damped inhomogeneous SDOF equation with the forcing function specified by equation (1.7.19) we obtain, with zero initial conditions

(Ms 2 + Cs + K)W(s)= F(s),

(1.7.20)

where F(s) is the Laplace transform of the input disturbance and is specified for the function described in equation (1.7.19) by 09

F(s) = ~ .

(1.7.21)

$ 2 + 0) 2

Transfer function Input

.I

-I

H(s)

Output

Fig. 1.6 System block diagram.

W(s)

15

INTRODUCTION TO MECHANICAL VIBRATIONS

Solving for W(s) we find

W(S)

"-"

M

S 2 + a)

2

X

$2

1)

22,

2 "

+ 2~a).s + o9~

Using the method of partial fractions and tables of known inverse Laplace transforms (Meirovitch, 1967), we find the solution for the actual response of the system to be given by

w(t)

=

1 M

)[

(a) 2 - 0)2) 2 - 4~2(.020) 2

-2~09 n COS a ) t -

(09 2 - 092n)+ 2~209n

-¢%t

2 ~ n COS OJat +

+e

sin cot O)

sin a~a •

(1.7.23)

O.)d

Hence the response of the system to a sine-wave forcing function turned on at t = 0 consists of a damped oscillation at the natural frequency of the system and a steady state response at the forcing frequency. If the damping is small or the excitation is close to the natural frequency of the system then the vibrations associated with the natural frequency of the system can extend for long times. This type of analysis and the results produced have important implications in terms of performance and stability of actively controlled mechanical systems.

1.8 Multi-degree-of-freedom (MDOF) systems When N independent coordinates are required to completely specify the system response, the system is said to have N degrees of freedom. Such a system is also said to have multi-degrees of freedom (MDOF). When a mechanical system has many degrees of freedom it is more convenient to use a matrix representation to describe and analyse the motion. In this section we formulate the equations of motion in matrix form. The reader is referred to the Appendix of the companion text (Nelson and Elliott, 1992) for a review of vector and matrix theory. Consider, for example, a simple N-degree-of-freedom system as shown in Fig. 1.7 (a) which is fixed at one end and free at the other. Excitation forces are directly applied to each mass. For a particular mass M,, the free body diagram is shown in Fig. 1.7 (b) and the corresponding equation of motion can be written as d2w M n

dt 2

=Kn(Wn_l-Wn)-Kn+l(Wn-Wn+l)Jrfn

.

(1.8.1)

This equation can then be manipulated to take the form

M~

d2wn

dt 2

+ K~w~_, + (K~ + K~+ ~)w~- K~+ ,w~+, =f~,

(1.8.2)

where fn is the forcing function applied to the nth mass. Similar equations for the motion of all the masses can be written except for n = 1, where w0 =0, and n = N,

16

ACTIVE CONTROL OF VIBRATION

where K u + 1=0, which correspond to the fixed and free boundary conditions of the system of Fig. 1.7 (a) respectively. The resulting simultaneous equations for each mass can be written in matrix form as M d2w

+ K w = f,

(1.8.3)

dt 2

where M and K are the mass and stiffness matrices. In the following text, matrices and vectors will be indicated by upper- and lower-case boldface symbols respectively. For typical linear structures, the matrices appearing in equation (1.8.3) are real and symmetric. The mass matrix M is specified by

M=

M1 0 0 0 M: 0 0 M~ 0

(1.8.4) 0

M~ which is a diagonal matrix. The stiffness matrix K is given by

(K, +K2)

K

._.

-K:

o

(/<2 + K3)

-K3

0

(1.8.5) 0

-K~_, (K~_,+ K~) 0 -KN which is a symmetric matrix. The vectors w and f are given by

1

w=

,

f=

Lw~(t)]

1.9

"

.

(1.8.6a,b)

f~(t)

Free motion of MDOF systems

If the forcing functions f are set to zero and the system is given some energy by an initial condition, then the MDOF system described by equation (1.8.3) will vibrate indefinitely due to lack of damping. The vector differential equation describing the motion can be written with matrix coefficients as d2w

M

dt 2

+ Kw = 0.

(1.9.1)

INTRODUCTION TO MECHANICALVIBRATIONS Wl I I

~

w2 I I

Wn

~

I I

17

WN- 1

,~ v

1 I

4

, i

WN

,.._ v

I I

'

(a)

Kn+l (Wn- Wn+l)

K n (wn_l- Wn)

~n (b) Fig. 1.7 Example MDOF system: (a) system configuration; (b) free body diagram of the nth mass. The solution of equation (1.9.1) will provide the undamped free vibrations of the system. We again assume a harmonic solution of the complex form w, (t) - A n

e TM,

n = 1,2 ..... N,

( 1.9.2)

where Wn is the displacement and A, is the complex amplitude (i.e. represents both amplitude and phase) of the motion of the nth mass. Substituting equation (1.9.2) into equation (1.9.1) we find the relation - 2 M A + Ka = 0,

(1.9.3)

where 2 = w 2 and the amplitude vector a is given by A1 A2 a=

AN.

(1.9.4)

By premultiplying equation (1.9.3) by M - ~ and rearranging we obtain the expression 2Ia - Ha = 0,

(1.9.5)

where I is the identity matrix and the dynamic matrix H is specified by H=M-~K.

(1.9.6)

Equation (1.9.5) can be further manipulated in the form [2I - H ] a = 0,

(1.9.7)

18

ACTIVE CONTROLOF VIBRATION

For a non-trivial solution a s 0 and setting the determinant of equation (1.9.7) to zero provides the characteristic equation of the MDOF system which can be written as 12I-

H I = 0,

(1.9.8)

The roots 2, of the characteristic equation are called the eigenvalues which provide the natural frequencies of the system. The expansion of equation (1.9.8) leads to an Nth order polynomial equation in 2. The solution of this polynomial equation provides N roots of 2 which are real and positive when the matrices M and K are positive definite (Cook, 1981). The positive square root of the N values of ;t will provide N resonance frequencies for the system with the lowest value, n = 1, corresponding to the fundamental resonance frequency, i.e. to, = a/r-~, n = 1, 2 . . . . . N.

(1.9.9)

If the number of degrees of freedom is small, the equation (1.9.8) can be solved algebraically. However, if the number of degrees of freedom of the system is large, then the solution of the polynomial obtained by expanding equation (1.9.8) is difficult and it is more appropriate to use numerical methods as discussed in Rao (1990) or Inman (1994). Once the eigenvalues are found, the mode shapes or modal eigenvector ¥, can be determined by substituting 2, into equation (1.9.7). The relative variation in the amplitude of motion of each element provides the mode shape ¥ , corresponding to the nth eigenvalue as [in I - H ] ¥ , = 0,

n = 1,2, ..., N,

(1.9.10)

where the mode shape for the nth mode is given by the vector of resulting amplitudes A1]

v.-

I .

.

(1.9.11)

[A~vJ The form of equation (1.9.10) is such that only the relative values of the mode vector components can be found. A constraint is imposed such as a unity value for one of the modal components, and then equation (1.9.10) is solved for the remaining components. Since the mode shape functions are arbitrary to a constant multiplier, it is convenient to normalise the modal vector. Two normalisation techniques are traditionally used in structural dynamics. The first technique is to set the maximum absolute component of the modal vector to one. That is max IV,, I = 1.

(1.9.12)

The second approach is to scale the mode shape with respect to the mass matrix as follows: ¥ ~ M ¥ , = 1.

(1.9.13)

The mode shapes also satisfy the following orthogonality conditions with respect to the mass and stiffness matrices

yTMym = 5,mM,,

(1.9.14a,b)

INTRODUCTION TO MECHANICAL VIBRATIONS

19

where Kn and M, are called the modal stiffness and mass coefficients, the indices m and n refer to the modal order and 6nm is the Kronecker delta function, defined as 6~m= 1 if n=mand6,m=0if n¢m. If the normalisation scheme of equation (1.9.13) is used, the coefficients in equation (1.9.14a,b) reduce to K~ = co,2 and M~ = 1. Also from the orthogonality conditions of equation (1.9.14a,b) we find 2

a)~ =

.

(1.9.15)

M.

The mode shapes form a set of linearly independent vectors, and thus they can be used as a basis in which to expand the solution vector w. Then, equation (1.9.2) which describes the individual motion of an element of the system, can be expanded in matrix form to describe the motion of the complete system by w(t) = ~ q ( t ) ,

(1.9.16)

where • is the modal matrix whose columns are the modal vectors, and q ( t ) = {ql(t) . . . . . qu(t)} T is referred as the modal displacement or generalised coordinate vector. Substituting equation (1.9.16) into (1.9.1), we obtain d2q MR/

dt 2

+ KRJq = 0.

(1.9.17)

We now premultiply equation (1.9.17) by the transpose of the mode shape matrix to obtain ~IJTM~I/ d2q + ~IJTK~IJq = 0. dt 2

(1.9.18)

Using equations (1.9.14a,b) and (1.9.15) we can then describe the original equation of motion of the MDOF system as a set of N second-order differential equations dZqn(t) dt 2

2

+ co~q,(t) = 0,

n = 1,2 . . . . . N.

(1.9.19)

This set of differential equations have the same form as an SDOF oscillator with a unit mass and a spring constant equal to ~on. 2 Thus the actual time history response of the nth modal coordinate to initial conditions can be written as (see equation (1.5.7) with

~=o) qn(t) = qn(0) cos %t +

4.(0)

sin %t,

(1.9.20)

(-On

where q,(0) and 4,(0) are the initial displacement and velocity of the nth modal coordinate. These are obtained from the initial conditions of the system w(0) and ";v(0) as follows"

q~ (0) = ~P~Mw (0),

O~(0) = ~P~ Mw (0).

(1.9.21)

20

ACTIVE CONTROL OF VIBRATION

Once the response of the modal coordinates are found, they are substituted into equation (1.9.16) to solve for the physical vector of coordinates w(t). Thus, the response of an MDOF system has been reduced to the linear combination of the response of a set of independent SDOF oscillators. Equation (1.9.20) can also be written in a form similar to equation (1.4.6) using equations (1.4.7) and (1.4.8).

1.10

Forced response of MDOF systems

The equation of motion of an MDOF system under an external force disturbance is given by M d2w ----- + Kw = fit). dt 2

(1 10.1)

To solve equation (1.10.1) it is first necessary to solve for the eigenvalues of the free system, as described in the previous section. This will provide a set of N natural frequencies, wl, w2 ... ton and associated mode shape vectors ¥1, Y2 .... YN- We can then write the solution vector w as a linear combination of modes by again separating the response vector w into spatial and time varying components:

w(t)=Wq(t),

(1.10.2)

where W is the mode matrix described previously. Substituting equation (1.10.2) into equation (1.10.1) and premultiplying the resulting equation by WT we obtain the following set of N uncoupled differential equations: d2wn dt 2

2

+ to,w~ = F~(t),

(1.10.3)

where F,(t) is the nth generalised force associated with the external force vector f(t) that is given by (1.10.4)

Fn(t)=wTf(t).

The solution of equation (1.10.3) for the actual motion can be obtained from Duhamel's integral as (Rao, 1990) 1 q,(O =

r| ' F,(r) sin [to,(t- r)] dr

Mn(.On J o

+q,(O) cos w,t +

q,(O)

sin to~t,

(1.10.5)

ton where F,(t) is the actual component of the generalised force.

1.11 Damped motion of MDOF systems With the inclusion of damping, equation (1.9.1) is modified to d2w M

dt 2

dw +C

, + K w = 0,

dt

(1.11.1)

INTRODUCTION TO MECHANICAL VIBRATIONS

21

where C is the damping matrix. We again replace the modal expansion of w(t) from equation (1.9.16) into (1.11.1) and premultiply by WT. In general, the matrix product w T c w , denoted as the modal damping matrix, will not result in a diagonal matrix. Thus, the matrix derivative equation in (1.11.1) will not be uncoupled by the modal matrix. The modal differential equations are said to be coupled by the damping matrix C. There are special cases that will result in a diagonal modal damping matrix, and therefore decouple the equations of motion. For example, this will occur when the damping matrix is given by (Thomson, 1993) C = a M + ilK,

(1.11.2)

which is a linear combination of mass and stiffness matrices. This condition leads to what is known as proportional damping. The purpose of choosing such damping is that the procedure outlined above will still produce a set of N uncoupled equations as in the undamped case. However, it should be remembered that this is not generally the case for real systems. Under these ideal circumstances it is straightforward to show that the differential equation of motion for the nth mode becomes (Rao, 1990)

Mn d2q"+ Cn dq. dt 2

- ~ t + K.x = F.(t),

(1.11.3)

where (1.11.4)

C. = a M . + ilK..

Equation (1.11.3) can also be written in the more convenient form as 2

F.(t)

tT.(t) + 2~.tO.q. + q.tO. = ~ ,

(1.11.5)

Mn where ~, is the damping ratio for the nth mode. The solution of this equation for the actual motion is again provided by Duhamel's integral and is given by

f

1 t F,(r) e q.(t) = M.tO. o +e

-~nwn(t-r) sin [tO,~(t- r)] dr

(0) cos to,~t +

~,to,~q,(0) sin to,~ , tOnd

(1.11.6)

where 09,d is the damped natural frequency of the nth mode defined as ~o~ = ~o,~/1 - ~2.

1.12

Finite element analysis of vibrating mechanical systems

The derivation of the equations of motion for the structural system of Fig. 1.7 was based on writing the equilibrium equation for each of the masses in the system using Newton's second law. This approach is applicable to simple one-dimensional systems. However, it quickly becomes cumbersome and impractical for complex structures. The finite element method (Zienkiewicz, 1977) is a technique developed in order to

ACTIVECONTROLOFVIBRATION

22

overcome these difficulties. The general steps involved in the finite element method (FEM) can be described as follows: (i) The basic concept is to subdivide the structure into a finite number of elements, a process which is known as the discretisation of the continuum. (ii) The displacement field of each element is then approximated by interpolation functions and the displacement is calculated at a reduced number of discrete points or nodes, N~. (iii) The equilibrium equations for each one of the nodes is derived to form the equations of motion for the free-free element that defines the element mass and stiffness matrices M e and K e, respectively. (iv) These element matrices are used to build the matrix differential equation for the complete structure. This process consists of assembling the global mass and stiffness matrices by bringing the contribution of each element mass and stiffness coefficients to the proper nodal point. (v) Finally, the boundary conditions are imposed on the system by either constraining the corresponding nodal displacements or applying nodal forces. The formulation described above will be illustrated for a simple one-dimensional uniform bar in longitudinal motion, as shown in Fig. 1.8(a). All motion will be described in terms of the actual or observed motion of the system. We first discretise the continuum into finite elements, as depicted in Fig. 1.8 (b). The elements will have a node at each end, and the axial displacement field will be approximated by linear interpolation functions as follows and as shown in Fig. 1.8 (c) (Zienkiewicz, 1977):

(112.1)

u(x,t) = Un(t)(l _ ~n) +Un+l(t) --x ln,

where x is a local coordinate, l~ is the element length, and u,(t) and u, +~(t) are the unknown actual axial displacements at the nodes n and n + 1, respectively. To derive the equations of motion of the element, it is convenient to use Lagrange's equation (Meirovitch, 1967) given by d

dr~ +

dt ~--~r]

r n,n + 1,

=0,

(1.12.2)

dur

where T n and V, are the kinetic and strain energy of the nth element, respectively. The kinetic energy, Tn, and strain energy, Vn, of the element are defined by 1

t,

T, = 2 Io pSu2(x) dx,

1

I,

V~= 2 Io ESu2(x) dx,

(1.12.3a,b)

where p is the mass density, E is the elastic modulus and S is the cross-sectional area which is assumed to be uniform. Substituting equation (1.12.1) into (1.12.3), performing the integrations and substituting the results into equation (1.12.2) yields

l, pS 6

]f } S[l l][U t f00/

1 an + 2 //,÷1 ~

=

-1

,

(1.12.4)

1 U,+l

which is the matrix differential equation for the nth element. The matrices in equation (1.12.4) are the element mass and stiffness matrices given by

/nf)S[2l l] Ue-T 2'

Ke WS[ 1 -1] ----~n--1 1 "

(1.12.5)

23

INTRODUCTION TO MECHANICAL VIBRATIONS

~1"- I

Bar

I -I~

(a)

Node

\

/-.I. v

A

,~

n

nth finite element A v

.L v

(b)

A v

n+l n

1- x/~n

(c)

X/~n~ / i

Fig. 1.8 Finite element analysis of the motion of a bar: (a) uniform bar in extension; (b) discretisation into nodes; (c) linear interpolation functions.

The global mass and stiffness matrices are next built by adding the influence of each element matrix coefficient. Then, the equation of motion of the complete structure is given by Mii + Ku = f,

(1.12.6)

where

M = p__S_S 6

211

l]

0

0

...

0

l]

2(/1 +/2)

12

0

...

0

0

12

2(/2+/3)

13

...

0

•

:

:

"..

0

0

0

In

0

...

0

(1.12.7)

2(In+In+]) ln+l ...

lu

0 2/u+ 1

24

ACTIVE CONTROLOF VIBRATION

is the mass matrix of the discretised system and

1/ll -I/I,

K = ES

I

-1/l, (lfl,) + (1/12)

0 -1/12

-1/12

(1//2) + (1//3/

0 0 -1/13

°,° °°° .°°

°°.

l

-1/l=

(1//,,) + (1//,,+,) -1//~+, °..

-1/lN

°°.

I/IN+, (1.12.8)

is the stiffness matrix, and U --'- { / , / 1 , / , / 2 ,

...,

U N } T,

f=

{f,,f2 . . . . .

fN} T

(1.12.9)

are the displacement and force vectors respectively. We will now, for illustrative purposes, assume that the beam is clamped at one end and driven by an oscillating in-plane force f = FR cos tot at the other end. This configuration would be implemented as follows. The clamped boundary condition is imposed by setting u~ = 0. In order to solve the linear system of equations represented by equation (1.12.6), the first row of vectors u and f is eliminated and the first row and column of matrices M and K are removed. The force boundary condition is imposed by setting at node N, fN = FR and f~ = 0 for n = 1 . . . . . N - 1. Since the in-plane motion of this one-dimensional beam model is governed by a second-order differential equation (see Section 2.2), the model will have N degrees of freedom for N nodes. This brief description of the FEM is only intended as an introduction for the reader not familiar with the technique. For a more detailed description of the methods for differing motions, structural shapes and boundary conditions the reader is referred to texts such as those by Zienkiewicz (1977), Cook (1981) and Petyt (1990).

2 Introduction to Waves in Structures

2.1

Introduction

Many practical structures are made up of long slender beam and shell elements. In these systems the response is often dominated by a wave type motion rather than a modal response, particularly when there is structural damping present. Thus a study of the nature and excitation of waves in simple long structural elements and their relation to modes is useful to understanding the behaviour of such systems. In particular it will be shown in Chapter 6 that it is sometimes more efficient to design a controller to observe and reduce waves in a system rather than use the more conventional method of controlling the modes of the structure. To undertake such an approach it will be necessary to be familiar with the basic principles of wave motion in elastic systems. In this chapter a brief review is presented of structural waves and their relation to the modal response of finite structures. For more detailed information the reader is referred to the texts of Cremer and Heckl (1988) and Brekhovskikh (1980). Figure 2.1 shows a string with a disturbance shape propagating at two time instants, tl and t2. We can describe the actual transverse displacement of the string using a general real function f, as

w ( x , t) = f

t -

,

(2.1.1)

where c is the velocity of propagation of the disturbance. At the time instant tl the displacement of the string at Xl will be given by Wl =f(tl---X~)c

(2.1.2)

and at t z the displacement at x 2 is (2.1.3)

26

ACTIVE CONTROL OF VIBRATION I

Time t 1

.........

i I I I

c(t2-tl)

I I I I I ' I I I I I

Time t 2

#1 I I I I I

I

Xl

x

x

x2

Fig. 2.1 Wave propagation on a string. If we assume that the pulse shape does not change as it travels down the string then Wl must equal WEand this implies that (2.1.4) or

(2.1.5)

x 2 - Xl = c ( t 2 - tl).

This demonstrates that the assumed waveform of equation (2.1.1) corresponds to a propagation of the wave in the positive x direction at the velocity of c. A similar analysis demonstrates that the function f ( t + ( x / c ) ) corresponds to wave propagation in the negative x direction. A general solution for the actual wave motion on the string can thus be written as

(c) (t

w(x,t) -- f t -

+g

+

,

(2.1.6)

which corresponds to propagation of waves in both the positive and negative x directions respectively. In equation (2.1.6) the real function g defines the shape of the negative travelling wave. For harmonic motion, the superposition of the two waveforms can be more conveniently written in complex notation (see Section 1.2) where the functions f ( t - ( x / c ) ) and g ( t + ( x / c ) ) are chosen respectively as A e j~('-(x/c)) and B e j~('÷(x/°) and therefore w(x, t) = A

e )~°t-jkx +

B e jc°t+jka,

(2.1.7)

INTRODUCTION TO WAVES IN STRUCTURES

27

where the variable k is known as the wavenumber and is defined as o)

k = --.

(2.1.8)

c

The amplitudes A and B are, in general, complex and the actual motion would again correspond to the real part of equation (2.1.7). Positive propagation can be seen to be determined by that part of the wave solution where the variables kx and cot have a negative relative sign, while a positive relative sign indicates negative propagation. By examining equation (2.1.7) for a fixed time t, it is apparent that wavenumber determines the spatial variation in phase. Different forms of waves can exist in structures. The waveform shown in Fig. 2.1 corresponds to a transverse wave where the string particle motion is perpendicular to the wave direction. Longitudinal wave motion corresponds to a particle motion which is parallel to the direction of propagation. Another important characteristic of waves in structures is the variation of the phase speed, c, with frequency. Waves whose phase speed are constant with frequency are termed non-dispersive while those whose phase speed varies with frequency are called dispersive. This characteristic is important as it affects a number of important parameters such as the rate of propagation of energy and the variation of the 'shape' of the disturbances propagating in structural systems. A pulse can be decomposed using Fourier analysis into many wave components at different frequencies (Nelson and Elliott, 1992). If the wave speed of the medium changes with frequency then components of the pulse will travel at different speeds and the pulse shape will change with increasing time (and thus location). The fact that the waveform of Fig. 2.1 was assumed to not change in time implies that the transmitting medium for this case was non-dispersive. As most realistic structural systems are characterised by the ability to support transverse shear as well as having internal stiffness the following discussion will focus on longitudinal (also called axial, extensional or in-plane) and flexural (also called bending) waves in thin beams. For the more advanced topics of 'thick' beams and torsional motion the reader is referred to Timoshenko and Goodier (1951) and Cremer and Heckl (1988). It should also be noted that systems whose response can be described by a sum of modes can be generally decomposed into elemental travelling waves and evanescent fields propagating in opposite directions whose relative magnitudes are determined by the system boundary conditions. The relationships between a wave and modal description of the response of a finite system is important to active vibration control since, as will be demonstrated in Chapter 6, effective control can often be achieved by actively changing the boundary conditions of the structure.

2.2

Longitudinal waves

Longitudinal waves are characterised by a particle motion which is parallel to the direction of propagation. Figure 2.2 shows an element of a slender bar undergoing what is called longitudinal, extensional, axial or in-plane motion and in the limit, the strain in the bar can be shown to be Ou/Ox where a partial derivative is used since u, the

28

ACTIVE CONTROL OF VIBRATION u

u+du

r---".--, v

v

v

v

v

w-

I

I

x

x+dx

Fig. 2.2 Longitudinal wave motion in a bar. internal axial displacement, is a function of time and position. By balancing the restoring force and inertial reaction of an element of the beam, the equation of motion for longitudinal waves can be written as ~2U

1 ~2U

~)X2

CL2 ~t 2

= O,

(2.2.1)

where the phase speed, CL, is given by CL= {E/p, and E is the Young's modulus of elasticity of the beam material while p is the density. A general solution of equation (2.2.1) for the complex in-plane displacement can be assumed to be u(x, t)= A e j~'-ja + B e j~'+ja,

(2.2.2)

where A and B are complex amplitudes whose values are given by the appropriate boundary conditions. Note that, since the differential equation (2.2.1) is of second order, there are only two wave solutions. Substituting the positive travelling wave solution of equation (2.2.2) into the wave equation (2.2.1), it is apparent that the wavenumber for longitudinal waves is specified by (2)

kL = - - .

(2.2.3)

cL

As a relevant example let us consider the response of a semi-infinite beam (existing for x~>0) to a harmonic axial force, f ( t ) = F e j°~t, located at x = 0. The internal axial force at x = 0 in the beam is given by ~u

f(x,t) = - S E ~ , Ox

(2.2.4)

where S is the cross-sectional area of the beam. The convention assumed in equation (2.2.4) is that a positive force is produced by a compression of the beam material. Since the beam is semi-infinite, only positive-going travelling waves will exist. Applying the appropriate boundary condition of continuity of force at x = 0 , the displacement amplitude of the positive travelling wave is found to be F A= ~ . j SEkL

(2.2.5)

I N T R O D U C T I O N TO W A V E S IN S T R U C T U R E S

29

If there were additional discontinuities in the beam at other axial locations then the negative wave component would have to be introduced and the appropriate boundary condition applied to obtain the magnitude of B, the unknown amplitude of the negative travelling wave. As described in Section 1.2 the actual displacement is given by the real part of equation (2.2.2). It is also of interest to evaluate the power carried by the beam for this form of excitation. Time-averaged power can be evaluated using the relationship in which instantaneous power is integrated over a period of motion. Thus HL = --lit Re[f] • Re[ti] dt, T 0

(2.2.6)

where we use the previously stated convention that the actual, observed motion corresponds to the real part of the complex motion and T is a period of harmonic motion. For time harmonic vibration, equation (2.2.6) can be shown to reduce to I-IL = ~1 Re[fti ],

(2.2.7)

where (*) denotes complex conjugate and () denotes differentiation with respect to time. Using the above relations, the power in the beam for longitudinal motion is given by

rI, =

1

2pc,S

IF 12.

(2.2.8)

Note that longitudinal waves are non-dispersive, that is, their phase speed does not vary with frequency. As a consequence of this, the power induced by a constant amplitude force is constant with frequency. Another important characteristic of longitudinal waves is that, since the beam is relatively stiff in extension, a high power is generated for relatively small particle displacements. The above relations for phase speed cL apply for what is called 'quasi longitudinal' motion by Cremer and Heckl (1988) when the lateral dimensions of the beam are much less than the wavelength of motion and are laterally unrestrained. In this case displacement also occurs in the transverse direction to the wave motion due to Poisson's coupling and leads to a slower wave speed than that in an infinite medium ('pure' longitudinal motion). For cases where one dimension is long relative to the wavelength of motion and thus contraction is effectively restrained in that direction (i.e. a plate) the effective modulus of elasticity is given by Eeff = J

V1

E

- v2

(2.2.9)

and the corresponding longitudinal wave velocity is CL=

where v is Poisson's ratio for the material.

p ( 1 - v e)

(2.2.10)

30

2.3

ACTIVE CONTROL OF VIBRATION

Flexural w a v e s

Many structures are excited by disturbances which are located off the central axis of the moment of inertia of the structure leading to wave motion which is transverse to the direction of propagation. In a relatively thin beam, for example, it is extremely difficult to excite purely longitudinal motion; the net result is a combination of both transverse and longitudinal displacements. The propagation and excitation of what are called f l e x u r a l , t r a n s v e r s e or b e n d i n g waves is thus very important. In addition, although it is well known, the derivation of the Euler-Bernoulli thin beam equation for flexural motion will be presented here since it outlines the basic procedure used in Chapter 5 for the analysis of distributed piezoelectric actuators and sensors attached to beams. Figure 2.3 shows an element of a thin beam excited in pure bending. For the following derivation it is assumed that the beam cross-section is symmetrical about the centreline and remains normal to the neutral surface. Under these assumptions and using static deflection relations from Timoshenko and Goodier (1951), the longitudinal displacement u is given by ~W u(x) = -z

(2.3.1)

,

Ox where w is out-of-plane displacement of the beam and z is the transverse coordinate of the beam section. Using Hooke's law in one dimension, the stress in the beam is given by /)2w (2.3.2)

a(z) = -zE ~ . Ox 2

q(x)Iw T[ + fl~. A x Ox

L.

AX

X

U

Ax

Fig. 2.3 Beam element in bending.

INTRODUCTION TO WAVES IN STRUCTURES

31

The resulting moment in the beam can then be obtained by integrating the stress distribution over the beam cross-section such that

Mx(x) = [

(z)z dz,

d-h/2

(2.3.3)

which then reduces to ~2W Mx(x) = - E l ~

~X 2 '

(2.3.4)

where I is the cross-sectional moment of inertia of the beam, I = b h 3 / 1 2 where b is the width and h is the height of the beam respectively. If we consider the beam element illustrated in Fig. 2.3 with the resultant shear forces and moments acting as shown, and neglecting rotary inertia of the element, the condition of moment equilibrium gives ~Mx

Ax = TiAx,

(2.3.5)

Ox where Mx and TI are the internal moment and transverse shear force acting upon the element. Applying Newton's law of motion in the vertical direction gives I

-q(x) +

(2.3.6)

= pS

i3x

igt2 '

where q ( x ) is an external load on the element and S is the cross-sectional area of the beam. Combining equations (2.3.6) and (2.3.5) we obtain

~X 2

- pS ~ + q(x). ~t 2

(2.3.7)

Finally, using the expression for bending moment from equation (2.3.4) we obtain the Euler-Bernoulli thin beam equation of motion given by

b4W E1 ~

bX 4

b2W + pS

~)t 2

= -q(x),

(2.3.8)

where q ( x ) is now the external load or forcing function (with units of force per unit length) on the beam. Equation (2.3.8) applies for thin beams relative to the wavelength of motion and its derivation ignores the variation in transverse shear and rotary inertia of the beam element. One solution of equation (2.3.8) can be written in complex form as w ( x , t) = A

eJ°~t-jklx.

(2.3.9)

On substituting equation (2.3.9), with q ( x ) set to zero (i.e. no external load) into the beam equation, the flexural wavenumber is found to be

I pSo) kI =

E1

(2.3.10)

32

ACTIVE CONTROLOF VIBRATION

As flexural wavenumber is also related to phase velocity by the standard relation, ki= ~o/cI, it is immediately apparent from equation (2.3.10) that flexural waves are dispersive, that is, their phase speed varies with frequency. Figure 2.4 shows typical dispersion curves (plots of wavenumber versus frequency) of a beam for both flexural and longitudinal motion. It is apparent that flexural waves have a relatively slow phase speed compared to longitudinal waves and this will be shown in Chapter 8 to have important implications in terms of sound radiation and control. Examining equation (2.3.10) and noting that the flexural wave equation (2.3.8) is fourth order, it can be shown that there are, in fact, four possible solutions with exponential dependence kI, - k I, jk I and - j k I. The assumed wave solution of equation (2.3.9) can now be written more explicitly as w(x, t) = A +n

e jc°t e-Jkl x + e ja't e

A, e j°~'e-kl x

(2.3.11)

+Jklx + nn ej°jt e +k/x,

where two additional waves are apparent, whose wavenumber solutions of equation (2.3.10) are purely imaginary. These latter components can be seen to decay with increasing x (for positive going disturbances, for example) and are thus known as evanescent or near-field solutions. Their presence is a result of the wave equation being of fourth order and the fact that a beam in flexure supports both bending and shear forces. Thus flexural wave motion in a beam is characterised by a positive-going and a negative-going travelling wave each of which can carry energy, and two decaying near fields which add to the transverse displacement near the disturbance point or boundary but do not carry any energy of vibration when propagating in an infinite, homogeneous beam (Fahy, 1985). Although the near-field components decay with increasing distance, their presence is important and, as discussed in Chapter 6, can strongly affect the behaviour of active control systems for flexural waves in beams. The above discussion pertains to beams of finite width and whose cross-sectional dimensions are small relative to the wavelength of motion. Occasionally it is useful to consider a 'beam' that is infinite in width (i.e. into the paper for the diagram of

kf

..Q

E t-.

1 1 kL

Frequency, Fig. 2.4 Typical dispersion curves for flexural and longitudinal waves in a beam.

33

INTRODUCTION TO WAVES IN STRUCTURES

Fig. 2.3). In this case the moment of inertia and the density of the beam are normalised to a unit width of beam. Thus a modified moment of inertia, I ' = h3/12 and density p'= p/b are used in the previous relationships. In addition since the beam displacement is now constrained in the width direction, the effective elastic modulus given by equation (2.2.9) is also used. Thus the thin beam equation can be written for the infinite width case as ~4w

EeffI'

OX4

~2w +

oh

= -p(x),

Ot2

(2.3.12)

where the external load p(x) now has units of force per unit area or pressure. The corresponding flexural wavenumber relationship is given by

~ pho)2 kf--

2.4

(2.3.13)

Eeff/'

Flexural response of an infinite thin beam to an oscillating point force

The flexural response of a semi-infinite or infinite beam to an oscillating point force is of interest and important to many of the analyses presented in later chapters. Assuming that the point force, f = F e j°~', is applied at x = 0 on an infinite beam, the flexural wave equation can be written in inhomogeneous, harmonic form as d4w E1 ~ - pSo)2w = dx 4

Fd(x),

(2.4.1)

where 6(-) is the Dirac delta function (see Section 1.7) and F has the units of force. It is useful to introduce the spatial Fourier transform, defined by the pair

W(k) = I:= w(x) ej'a dx,

(2.4.2a)

w(x) = 1 I'= j_= W(k) e-Jkx dk,

(2.4.2b)

2~'

which enables transformation between spatial functions w(x) and functions W(k) of wavenumber in a manner which is exactly analogous to Fourier transformation between functions of time w(t) and functions of frequency W(co). The wavenumber transform W(k) can thus be regarded as describing the spatial frequency or wavelength (since k = 2Jr/2) content of the function w(x). With application of the transform of equation (2.4.2a) to both sides of equation (2.4.1), the transformed or spectral displacement can be written as (Junger and Feit, 1986)

W(k) --

F El(k4-

. k:)

(2.4.3)

34

ACTIVE CONTROL OF VIBRATION

Taking the inverse transform of equation (2.4.3) and using the method of residues (Churchill et al., 1974) evaluated at the poles k = +k i, - j k I (chosen to ensure the system is causal) we find the response for x > 0 to be w(x)=

-iF (e-j~_je-krx). 4EIk}

(2.4.4)

Note that the solution is composed of both a travelling component and a near-field component which decays exponentially away from the drive point. A similar solution can be constructed for negative x. The motion of a semi-infinite beam subject to a point force has been derived by Cremer and Heckl (1988).

2.5

Flexural wave power flow*

The total instantaneous power of flexural waves is the sum of the components due to the shear force and moment contributions as (Cremer and Heckl, 1988)

(2.5.1) where the instantaneous transverse shear force contribution is II~= Re[rl]- R e [ , ]

(2.5.2)

and the instantaneous bending moment contribution is H ~ = Re[Mx]- Re[0],

(2.5.3)

where the variables involved are determined by the element relations described in Section 2.3 and are specified by:

angular velocity, ~2 W

O(x,t) = ~ , ~x~)t

(2.5.4)

bending moment, ~2 W Mx(x,t) = -EI

~)X2 '

(2.5.5)

and transverse force, ~Mx

Ty(x,t) = ~ . Ox

(2.5.6)

The instantaneous power flow in the positive x direction in the infinite beam for point

* Here we use the commonly accepted terminology of 'power flow' although 'power' is the correct description.

35

INTRODUCTION TO WAVES IN STRUCTURES

force excitation can be evaluated in the far field using the above relations and is given by

olFl:

I-IF = i 6 E / ~ sin2(°gt- kyx)

(2.5.7)

and ogIF] 2

1-I~ =

cos2(cot- kyx).

16Elk}

(2.5.8)

The total instantaneous power flow in the positive x direction is then 2

rI =

16Elk}

(2.5.9)

Note that the power flow II~B is both space and time invariant while the individual bending and shear components of the instantaneous power oscillate with space and time. Thus equation (2.5.9) is identical to time-averaged power. Physically, this implies that at a constant axial location, the energy of vibration of flexural waves is flowing to and fro between the motions associated with transverse displacement and rotation as the wave travels through the observation point. In addition the input power varies with frequency due to the dispersive nature of flexural waves. The total input power to the beam will be the sum of the power flowing in the positive and negative directions. Thus, by symmetry, the total input power to the infinite beam due to the oscillating point force is given by

IFI 2 1-I i =

2.6

~

.

8Elk}

(2.5.10)

Flexural response of an infinite thin beam to an oscillating line m o m e n t

In many applications the excitation of thin beams by oscillating moments is of interest. Chapter 5 will demonstrate that forcing functions associated with surface-mounted distributed strain actuators can be approximately modelled as line moments. Assuming that the forcing function is a line moment applied at x = 0 and specified by M(t)= M~6'(x)e j~'', the equation of flexural motio n of the beam can be written in moment form as

dx 2

~ o92p,~w = Mi6 ! (x),

(2.6.1)

where 6' (-) is the derivative of the Dirac delta function with respect to its argument, Mi has the units of moment per unit length and the time variation e j°~t is again omitted for brevity.

36

ACTIVE CONTROL OF VIBRATION

Taking the spatial Fourier transform of equation (2.6.1) gives

EIk4W(k)- o92pSW(k)=-jkMi.

(2.6.2)

Solving for W(k), applying the inverse Fourier transform and using the theory of residues as outlined previously in Section 2.4 gives the displacement of the infinite beam for x > 0 which is

M~

w(x) =

"~,e-jkrx - e

).

(2.6.3)

4Elk~ Note that in equation (2.6.3) the displacement field is again composed of a travelling wave component and a near-field component. The actual displacement is given by the real part of equation (2.6.3). The power flow in the positive x direction in the infinite beam can be evaluated using a similar procedure as outlined above and for moment excitation is given by (Gibbs and Fuller, 1992a) Fin = ~ . 16Elky

(2.6.4)

The total input power to the infinite beam system will be twice HB.

2.7

Free flexural motion of finite thin beams

In many control problems it is necessary to derive an expression for the response of finite beams to a disturbance. In order that the resulting motion be determined, it is first necessary to apply the appropriate boundary conditions in order to derive the free response. Commonly encountered simple boundary conditions are as follows. (a) Simply supported end. In this case the end of the beam is free to rotate but is constrained to have zero displacement and moment. The boundary condition is thus specified by

w(x) = 0

and

~2w(x) Ox2

= 0.

(2.7.1)

at the constraint location. (b) Clamped end. For a clamped boundary condition, both the displacement and rotation are constrained to zero. The boundary condition is thus specified by

w(x) = 0

and

bw(x)

= 0.

(2.7.2)

bx (c) Free end. For a free end, both the shear force and the internal bending moment must disappear. Thus the boundary condition is

~2w(x) ~x 2

~3w(x) =0

and

~x 3

= 0.

(2.7.3)

INTRODUCTION TO WAVES IN STRUCTURES

37

(d) General termination impedance. In many cases the beam is terminated by a known impedance. To completely specify the termination impedance both a bending termination impedance, Z'x and a transverse shear impedance Z) need to be specified. The appropriate boundary condition is then applied by matching the beam internal bending impedance to the termination bending impedance. Thus Ztx =

Mx(x) O(x)

(2.7.4)

and matching beam transverse shear impedance gives Z)= Ty(x)

(2.7.5)

w(x)

evaluated at the termination location. Simultaneously applying the above boundary conditions to an assumed wave field allows determination of both the travelling wave and near-field amplitudes. Let us apply the above approach to the case of a finite beam of length L, simply supported at each end. For the simply supported boundary conditions the flexural near fields can be shown to vanish. The beam response can be then written as the superposition of two travelling waves in the positive and negative directions with unknown coefficients A and B:

w(x, t)= A eJ~t-Jkl~ + B eJ°~t+Jklx.

(2.7.6)

Applying the boundary condition of (only) zero displacement at the constraints (since we do not have near-field components) we find at x = 0 that w(x) l x=O= O,

(2.7.7)

A = -B

(2.7.8)

w(x)[ x__L= O,

(2.7.9)

A e-JkIL + B eJ~;~L= 0.

(2.7.10)

which results in

and also at x = L, we can put

which therefore results in

Substituting equation (2.7.8) into equation (2.7.10) results in the displacement field being specified by

w(x, t)= - 2 A j sin klx e j~.

(2.7.11)

Applying the boundary condition of zero displacement at x = L to equation (2.7.11) allows the derivation of the system characteristic equation which is expressed in the form sin kiL = 0,

(2.7.12)

which therefore implies that the eigenvalues are given by k~= ~ ,

L

n = 1,2,3 .....

(2.7.13)

38

ACTIVE CONTROL OF VIBRATION

Thus unlike the infinite beam, a finite beam can only vibrate in free motion at discrete frequencies (i.e. resonate) such that the free flexural wavenumber equals the discrete values given by equation (2.7.13). We denote these discrete values of wavenumbers k,, n = 1,2,3, .... Figure 2.5 shows the dispersion curves of two beams of different thickness. Also shown in Fig. 2.5 are the eigenvalues given by equation (2.7.13) for a specified length, L. The resonance frequencies of the modes of the system are given by the intersection points of the eigenvalues and the dispersion curves. Figure 2.5 illustrates the duality in the wave and modal description of the vibration of finite elastic systems, since the beam motion can be thought of in terms of a standing wave (a mode) or two equal waves travelling in opposite directions in the beam. Using equation (2.7.13) and equation (2.3.10), the resonance frequencies of the simply supported beam can be calculated from n = 1,2, 3,...,

(2.7.14)

mI

where m' is mass per unit length of the beam. A more general solution of the free vibration of a finite beam with general impedances at each end would require equation (2.7.6) to be expanded to include terms describing the flexural near fields and the application of both boundary conditions of equations (2.7.4) and (2.7.5). The total normal response of the beam (i.e. without an external persistent disturbance) is given by the superposition of the individual eigensolutions, or modes, to give oo

w(x,t) = Z W~p, e

je)nt

,

(2.7.15)

n=l

where W~ is the modal amplitude, ~. is the mode shape of a simply supported beam given by ~. = sin(k.x) and co. is the resonance frequency of the nth mode. Beam (a)

..~

f,

J

2re

I ~-

Beam (b)

I--

-E

I== z //I

!

I

Frequency, co

Fig. 2.5 Relationship between eigenvalues and free wavenumbers for two differing finite beam systems.

39

I N T R O D U C T I O N TO W A V E S IN S T R U C T U R E S

Model

lo-2 V

Mode 3

Mode 1 A ~.

/ , " , ,'-2x

10-3 2

•~o

3 "

10 - 4

E ~ rn

0-5

~o-6

_ I

I

I

10

I

I

I

I

I[

I

100

I

t

I

t

I

t I

I

1000

Frequency (Hz)

Fig. 2.6

Frequency response function and mode shapes of a simply supported beam.

Figure 2.6 shows an example frequency response function (FRF) for a beam of length L = 0 . 3 8 m, bending stiffness E I = 5 . 3 2 9 N m 2 and mass per unit length, m' =0.6265 kgm -~ calculated using equation (2.8.12). Also shown are the mode shapes corresponding to the peaks, or resonance points in the FRF. The locations of the peaks are thus predicted by equation (2.7.14) while the associated mode shapes correspond to those of equation (2.7.15). An interesting characteristic of beam behaviour is that the resonance frequencies become spaced further apart with increasing mode number. This is also apparent from Fig. 2.5 and is due to the dispersive nature of flexural waves. Figure 2.6 will be further discussed in Section 2.8. Boundary conditions other than those for a simply supported beam will result in additional near-field motion located near the discontinuities. If the beam is long compared to the wavelength these near fields will decay before they reach the other constraint and thus will have little effect on the nature of the global response (i.e. they will not significantly influence the resonance frequencies and characteristic functions of the system).

2.8

Response of a finite thin beam to an arbitrary oscillating force distribution

We now study the flexural response of the finite beam to a harmonic force of arbitrary distribution f(x, t)= F(x)e j°'' where F(x) is considered positive in the upwards direction. The differential equation of motion can now be written in inhomogeneous frequency domain form as

F(x) E1

daw(x) - k~w(x)= ~ ,

dx 4

(2.8.1)

40

ACTIVE CONTROLOF VIBRATION

where w(x) describes the complex transverse displacement of the beam and e j'°' has again been omitted for brevity. If we further assume that the beam is finite and has simply supported boundary conditions then we can seek a solution of equation (2.8.1) in the form of a series composed from the eigenfunctions or free mode shapes found previously. We therefore assume that

w(x) = Z ~ sin knx,

(2.8.2)

n=l

where Wn are the unknown amplitudes of the response of the system. Substituting equation (2.8.2) into equation (2.8.1) we see that Z (k~ - ¢)Wn sin k n x

=

F(x)/EI.

(2.8.3)

n=l

In order to solve for the coefficients Wn, we use the orthogonality characteristic of the eigenfunctions (see Section 1.9) defined for the continuous beam system as if n , m,

(2.8.4a)

L o ~nWmdx = An if n = m,

(2.8.4b)

L

o

~)n~Imdx = 0

IIL

where ~Pn is the beam mode shape function and An is the mode normalisation constant 1 which is specified for the simply supported beam as An= ~. Note that equations (2.8.4a,b) are a specific form of the general orthogonality relations given in equations (1.9.14a,b) and are used when the beam is homogeneous and thus the mass and stiffness does not vary spatially. Thus to separate out individual modal contributions we multiply through both sides of equation (2.8.3) by ~Pn= sin knx and integrate over the length of the beam L. We then find an expression for the complex amplitudes Wn given by 2

I L F(x) sin k~x dx. IVn= EIL(k4-k}) o

(2.8.5)

Equation (2.8.5) demonstrates that in order to solve for the response of the system, we have conveniently expanded the forcing distribution into a series with the system free mode shapes as the basis functions; the magnitude of the modal amplitudes W, are dictated by the shape of the forcing function. Note that the choice of the mode shapes as the basis functions implies that the solution using equation (2.8.2) in conjunction with equation (2.8.5) is well-conditioned to determine the global system response and ill-conditioned to predict the near-field effects at say the drive point since the basis functions do not readily model the evanescent motions. In other words, at low frequencies the infinite sum in equation (2.8.2) can be truncated at a relatively small number of modes in order to obtain the global response of the beam but a relatively high number of modes is needed to accurately evaluate the near-field components. An alternative approach as used by Morse (1976), for example, is to use the normal modes of a free-free beam as the expansion functions. In this case the near fields are automatically included in the mode shapes.

INTRODUCTION TO WAVES IN STRUCTURES

41

For illustrative purposes we study two example forcing conditions. If the forcing function is constant over the beam, i.e. F (x) = F then 2F L W~ = EIL(k 4 - k}) Io sin k~x dx,

(2.8.6)

and when the integral is evaluated, this can be written as

W~ = - EILk,(k 4 _ k} )

~ - - ~ ]Jo

which finally reduces to W~= -

4F nTtEZ(k 4 - k})

,

n = 1,3, 5 .....

(2.8.8)

Thus the response of the beam for a uniform (or by extension a symmetric) forcing function is only in the n = 1,3,5 .... or symmetric modes. This characteristic is also intuitively obvious when one examines the spatial phase distribution of the n= 2, 4, 6 .... or antisymmetric modes. For example the phase of the n = 2 mode changes through 180 ° at the mid point of the beam and thus the integral of equation (2.8.5), which can be split into two antisymmetric terms about the mid point, is zero for this mode. It is also interesting to note that the amplitude of the response of higher order modes decreases with increasing modal order. This characteristic will be found useful when controlling structural motion with point forces which tend to couple into many modes. If it is desired to excite only a particular mode in the beam corresponding to a given integer n, then it is apparent from equation (2.8.5) that the forcing function, for the case of a simply supported beam, takes the distribution F ( x ) = F sin knx. By the orthogonality relations of equations (2.8.4a,b) it is apparent that the response in all other modes except the nth mode will be zero. These observations generally apply to all structures whose vibratory motion can be described as a series of orthogonal modes. Characteristics such as these are important in the design of distributed actuators and sensors as outlined in Chapter 5. In general we desire to control a low number of modes of vibration and wish to achieve this without exciting other modes (an effect termed control spillover by Balas, 1978). In this case we can sometimes tailor the input control function F (x) to achieve the required modal excitation in the control field. A point force excitation is also of prime interest and is specified, using the Dirac delta function, by F ( x ) = F r ( x ) . Equation (2.8.5) then becomes

W~ = - E I L ( k 4 - k} ) I c 6(x - xi)sin knx dx.

(2.8.9)

Using the 'sifting' property of the Dirac delta function discussed in Section 1.7, we can evaluate the integral in equation (2.8.9) which leads to W~=

sin knxi ~ . ElL k 4 - k} 2F

(2.8.10)

42

ACTIVE CONTROLOF VIBRATION

The total response of the beam including the harmonic time component is then given by

w(x,t) =

2F ~

ElL ~=

sin k~xi sin k~x ejO~, .

k4 - k}

(2.8.11)

The response relations described by equations (2.8.5), (2.8.8) and (2.8.10) can easily be rewritten in terms of frequency by using equations (2.3.10) and (2.7.14) manipulated to give k 4 = m'to2JEl and k}= m'to2/El where m' is mass per unit length of the beam. For example the total response of the beam due to the point force can be written as a function of frequency in the form

w(x,t) =

- 2F ~ M

=

sin knX i sin knx _io~t to2_ to2

e ,

(2.8.12)

where ton is the natural frequency of the nth mode and M is the total mass of the beam. When kn= kI, or to = ton, the excitation frequency corresponds to the resonance frequency of the nth mode of the beam, and as expected the system response approaches infinity. This singularity in the response function can be overcome by the approximate method of introducing hysteretic damping into the beam. This involves expressing the Young's elastic modulus of the beam material as E ' = E(1 +jr/,) where JT, is the total loss factor (Cremer and Heckl, 1988) when evaluating kI, the free flexural wavenumber at frequency to. Figure 2.6 shows the velocity response of a simply supported beam (with material properties specified for the previous example of Section 2.7) to an oscillating point force of unit amplitude located at xi=O.1L. The beam is assumed to have a damping represented by a value of JTt= 0.001. It should be noted that the hysteretic model of damping is only strictly valid for steady state harmonic motion. Use of the hysteretic model for an impulsively excited structure leads to a non-causal response, i.e. the response of the system apparently anticipates the excitation. The poles of the system transfer functions which correspond to the roots of the system characteristic equation are found in both the positive and negative parts of the s-plane (see Section 1.7 for a brief description of the Laplace domain described by the complex variable s). As will be discussed in Chapter 3, these characteristics of the hysteretic damping model have important implications in active control in terms of controllability and stability. In these situations it may be advantageous to use a viscous model of damping, although this is certainly prohibitive when the system has many degrees of freedom. The reader is referred to the papers by Crandall (1970) and Scanlan (1970) for more details on this aspect of the hysteretic damping model. Equation (2.8.11) again illustrates the duality in the interpretation of the system response as either a sum of modes or as a superposition of travelling waves. As the system is being forced by a harmonic steady state input, the finite system will respond at the same frequency of excitation as the forcing function and not at the discrete wavenumbers defined by the eigenfrequencies. If the excitation frequency is away from the resonance frequencies of the beam, the response will be composed of free waves with wavenumbers +kI that travel backwards and forwards, without constructive reinforcement in the beam system. However, when the excitation frequency is such that ki= kn then the free wavenumber (or wavelength) also corresponds to the natural eigenvalue of the free system and the free waves reflect from the beam terminations

INTRODUCTION TO WAVES IN STRUCTURES

43

with constructive interference and reinforce the beam motion. The net result is a large response associated with the resulting resonance. Behaviour such as this will be shown later to be critical to the performance of active vibration control systems in terms of stability, power requirements and performance. The response in beam systems can also be seen to be critically dependent upon the location and distribution of the disturbance force. It has been demonstrated that a distributed input can be shaped in order to excite selected modes. In contrast, a point force input which is described using the Dirac delta function can be seen from the definition of the Dirac delta function (Section 1.7 and equation (2.4.2a)) to have a spatial Fourier transform value of unity. Thus a point force can also be thought of as being composed of a sum of all wavenumber components with equal amplitude, a characteristic termed spectrally white in a wavenumber sense. The point force will then couple into all modes if appropriately located. Equation (2.8.11) predicts, on the other hand, that if the point force is located on a nodal line of a particular mode then that mode will not be excited at all. Likewise the position of the force relative to the mode shape strongly influences the magnitude of the resultant modal amplitude. Obviously if control of a particular mode is required then the position of application of the point force control input is important. Finally equation (2.8.10) predicts that for a particular frequency, as the modal order increases, the contribution of the higher order modes to the total response will decrease. Thus structures such as beams act as low pass filters of a disturbance excitation, when considered in terms of their modal response. This effect can often work to advantage in curtailing the aforementioned control spillover problem.

2.9

Vibration of thin plates

The previous sections have been concerned with the vibration of thin beams. It is of interest to extend the scope of our discussions to consider elastic motion in two dimensions, i.e. the vibrations of thin plates or panels. The equation of motion of a plate can be written as a two-dimensional extension of that governing the motion of beams. Following Cremer and Heckl (1988) the equation of motion can be written as

E1( ~4w+ 2 ~4w + ~4wI + ph ~2w= -p(x, y, t), ~X 4

~X2c~y2

~y41

~t 2

(2.9.1)

where h is the thickness of the plate, I is the moment of inertia per unit width and p is the applied external pressure or load. For the in-vacuo case when considering the free motion of the plate, the applied pressure is set to zero. As discussed in Section 2.2 the value of modulus of elasticity given by equation (2.2.9) should be used when dealing with plates. Hence, in this case E l = h 3 E / [ 1 2 ( 1 - v2)]. Important approximations in deriving equation (2.9.1) are similar to those made in deriving the equations for beam motion. It is assumed that the plate is thin with respect to a wavelength of motion and transverse shear as well as rotary inertia of the plate motion are ignored. This effectively limits the mathematical description to low frequencies (Cremer and Heckl, 1988).

44

ACTIVE CONTROL OF VIBRATION

Waves in two-dimensional structures can be described in a variety of coordinate systems. A structural wave travelling in the plate at an angle a to the x axis (see Fig. 2.7) can be expressed as (2.9.2)

w(x, y, t) = A eJ~t-;kxX-jky y.

Substitution of equation (2.9.2) in equation (2.9.1) and using some algebraic manipulations in conjunction with equation (2.3.10) results in k~= k2 + k2.

(2.9.3)

It is apparent that equation (2.9.3) represents a vector relationship between the three wavenumber components and thus the x and y wavenumber components are related to the free wavenumber by kx = k I cos a,

(2.9.4)

k,= klsin a. Equation (2.9.3) demonstrates an important characteristic of wavenumber. The free wave travels at angle a (illustrated in Fig. 2.7) at the speed of flexural waves in plates. The free wavenumber kI can also be vectorially decomposed into x and y trace components as dictated by equation (2.9.4). Thus in two-dimensional systems whose boundaries are parallel to the coordinate axes, resonance will occur when the trace wavenumber components kx and ky in the x and y directions simultaneously equal an eigenvalue (see Section 2.10 for an example of this) in each direction. Free waves can exist in a variety of different forms expressed in terms of different coordinate systems. Choice of the appropriate coordinate system is dependent upon the system configuration. For example, the excitation of an infinite plate by a point force is more conveniently studied in cylindrical coordinates, as discussed in the text by Junger

/

Simply supportededges

a

y~ ~r v

L.,

X Vl

Fig. 2.7

Simply supported rectangular plate coordinate system.

INTRODUCTION TO WAVES IN STRUCTURES

45

and Feit (1986). In general, however, it is more appropriate to choose the coordinate system of the equation of motion based on the alignment of the system boundaries rather than the geometric characteristics of the forcing function.

2.10

Free vibration of thin plates

In many cases the system to be examined is such that it is either not possible to solve the system differential equation or the shape of the boundary cannot be easily described in terms of a coordinate system. In such cases, one may be able to pursue the analysis using approximate methods in which the continuous system is approximated by an N degree-of-freedom system as outlined in Sections 1.8-1.12. Such techniques are also well described by Meirovitch (1967). For the moment we will restrict ourselves to the study of the free vibrations of a rectangular thin plate which is simply supported along the edges. On the basis that a simply supported plate is a two-dimensional extension of a simply supported beam it is appropriate to choose a separable solution of the transverse modal displacement of the form Wmn(X ,

y, t)

= Wren s i n k m x sin

k,y

e )°Jt,

(2.10.1)

where Wm, is modal amplitude and m and n are modal indices. Applying the boundary conditions of zero transverse displacement at the plate edges, shown in Fig. 2.7, results in expressions for the wavenumber eigenvalues in each coordinate direction that are given by

met/a, k,, = net~b, km =

m = 1,2,3 .... , n = 1,2, 3 . . . . .

(2.10.2)

Substituting the eigenvalues associated with resonance of the trace wavenumber components kx and ky in the x and y direction into equation (2.9.1) and using the relation between frequency and free wavenumber given by equation (2.3.10) we can solve for the discrete frequencies at which the system resonates in two dimensions. These are given by

+

]

(1)ran ~-'-~/ L~-~] ~ b I J' where the subscripts m, n denote the (m, n)th mode of vibration. Examples of the shapes of different modes of vibration are shown in Fig. 2.8. The fundamental (1,1) mode has an associated motion with no phase change across the plate surface. The higher order modes are characterised by nodal lines through which the relative phase of the displacement function flips 180 °. The discussion presented in Chapter 8 will demonstrate that modal order and the corresponding mode shape function have a significant effect on sound radiation and control. Another important aspect of plate response is the modal density or number of modes within a frequency bandwidth. The modal density of thin plates is large and the 'modal overlap' increases with frequency (Cremer and Heckl, 1988). Thus modal control of two-dimensional systems is a far more difficult problem than the equivalent onedimensional problem since there is a much larger number of significant degrees of freedom in a given band of frequencies (see Chapter 6).

46

ACTIVE C O N T R O L O F VIBRATION

Mode (m,n) (1, I)

(2, 1)

+

+

f

Nodal line (3, 1)

(1,3)

+ +

+ 4-

Fig. 2.8 Selected mode shapes of a simply supported rectangular plate.

2.11

Response of a thin rectangular simply supported plate to an arbitrary oscillating force distribution

Following a similar procedure to that outlined in Section 2.8, the in-vacuo response of a rectangular plate to a two-dimensional forcing function F(x, y) e l~'t c a n be written as

EI(~)aw +2 O4W + ~4wI + ph OZw = -F(x,y) e_i,,,t, ~X4 ~x2~y2 ~y4 ] ~)t2

(2.11.1)

where for the plate, I = h3/[12(1- v2)] and F(x,y) has the units of pressure. We assume, as previously, that the forced response can be written in terms of a sum of modes of the free response of the plate vibrating at the forcing frequency, i.e. oo

oo

w(x,y,t) = Z Z Wrensin kmx sin knY ej~t.

(2.11.2)

m=l n=l

On substituting the assumed response into equation (2.11.1) and using the orthogonality property of the plate mode shapes in the x and y directions as discussed in Section 2.8, we obtain an expression of the plate response amplitudes given by

Wmn-- M(to24_(-Omn)2I aOI boF(x, y) sin kmx sin kny dx dy,

(2.11.3)

I N T R O D U C T I O N TO WAVES IN STRUCTURES

47

where M( = pshab) is the total mass of the plate. As before we have found a solution by expanding the input force into components with the system mode shapes as the basis functions. If the input forcing function is a point force, f ( x , t)= F 6 ( x - x i ) 6 ( y - y i ) e j'°', located at x/, y/, then, using the 'sifting' property of the Dirac delta function, the integrand has a value only at xi, y~ and the modal amplitudes are given by Wmn =

4F sin kmxi sin k~ Yi M(o)2

.

2 -- (Dmn)

(2.11.4)

Once again the plate modal response is observed to depend strongly upon the location of the input force, the modal input impedance and the input frequency. If the input frequency, co, equals a resonance frequency, com,, the (m, n) mode response will again approach infinity, due to lack of system damping. In order to evaluate the total plate response, the double summation of equation (2.11.2) is truncated at a finite number of modes chosen to ensure a satisfactory convergence of the series. In many cases of interest the plate is excited by input moments. It is then convenient to write the inhomogeneous plate equation in moment form as (Timoshenko and Woinowsky-Kreiger, 1984) c-:-:32Mx c32M~ ~2My ~2w _j~ot +2 +~ - ph ~ = M(x, y) c , ~x 2 axSy ay 2 at 2

(2.11.5)

where in equation (2.11.5), M ( x , y) the disturbance moment distribution has the units of moment per unit length per unit area of plate. The internal moments of the two-dimensional plate element are specified by (Ugural, 1981) M x = -EI.-z-7~ + v ----'S"

[ 02w Ox

02wI Oy l

(2.11.6a)

[ b2w 82w I My = -El~-~y 2 +V ~x'--S]'

(2.11.6b)

~2 W

M~y = -EI(1 - v)

.

(2.11.6c)

Oxay

The shear forces acting on the edges aligned along the x and y axes are respectively (Ugural, 1981) x Tf =

=

ay/xy

+ ~, Oy Ox

(2.11.7a)

" +~.

(2.11.7b)

by

Substitution of these expressions into equation (2.11.5) confirms its equivalence with equation (2.9.1).

48

ACTIVE C O N T R O L OF VIBRATION

An important forcing function associated with piezoelectric distributed actuators is a line moment around an axis located at Xl. In this case the plate equation is written in moment form as

Ox2

+2

~.

OxOy

.

Oy2

.ph .

Ot2

. Md'(x

X1)_jtat c ,

(2.11.8)

where t~' (-) is the derivative of the Dirac delta function with respect to its argument and the moment amplitude M has units of moment per unit length. This form and other two-dimensional moment configurations will be useful in Chapter 5 for analysing the response of plates to excitation by piezoelectric distributed elements.

2.12

Vibration of infinite thin cylinders

In the previous sections we have studied the vibrations of planar structures. In this section we will deal with a brief study of an important curved structure, the thin-walled shell. Shell theory is often used to model common structures such as the fuselages of aircraft or the hulls of submarines. Figure 2.9 shows the cylindrical coordinate system and the notation used in the analysis for the displacement in the radial, axial and torsional directions. Various theories describing the motion of the shell with different approximations have been derived and are summarised by Leissa (1973). The most significant aspect of the vibration of curved bodies is that the motion must be considered in three axes. Thus in thin-walled shell vibration, the equations are written in terms of the in-plane (axial) motion, u, the out-of-plate (radial) motion, w and the torsional motion, v. The simplest thin-walled shell equations are the Donnell-Mushtari equations which are written for in-vacuo motion as (Leissa, 1973) O2u

+

(1 - v) O2u

c)x2 (1 +v)

2a v Ou

l Ov

aOx

a 2 c)0

2a 2 O2u

i)xi)O w

+~+f12 a2

+

002 (l-v)

+

2

~x 4

O2v

2a

i)xi)O

O2v OX2

a

cL2

Ox

1 Ow

a 2 002

+ ~x2~02

//

I

~4W

+2

v Ow

1 O2v

~-t-

2

(a O4W

(1 + v)

a 2 t)04

iJ

=0,

~

2 CL pa(1- V2)

c t2

Eh

a 2 ~0

l ~4w )

=0,

+~=

(2.12.1a)

(2.12.1b)

. (2.12.1C)

In the above equations, the stiffness factor fl is given by fl= h2/(12a 2) and the longitudinal or axial phase speed CL is given by equation (2.2.10) and is the same as for plate motion. Setting f l - 0 results in the equations reducing to those describing the motion of a curved membrane. The variable Pa is again an external forcing function or load with the units of pressure. Unlike the wave equations used to describe the motion of beams and plates, the shell equations consist of three coupled equations for each axis of motion which must be solved simultaneously. Important assumptions used in deriving the Donnell-Mushtari equations are similar to those used previously for thin beams and plates with the additional assumption that the variation in transverse shear stress in the circumferential direction is ignored (Junger and Feit, 1986). Higher order

INTRODUCTION TO WAVES IN STRUCTURES

49

X

h

i Fig. 2.9

Cylindrical coordinate system for an infinite thin cylinder.

thin shell theories can be obtained by adding correction factors to the Donnell-Mushtari shell equations as described in Leissa (1973). In order to solve for free wave motion in an infinite cylindrical shell we first assume displacement distributions for the shell wall of the form jwt-jknsx + j:r/2

u(x, O,t)= Z Z U~scos nO e n=l

,

(2.12.2a)

s=l

jwt-jknsx

v(x, O,t) = Z Z V,s sin nO e

,

(2.12.2b)

n=Os=l oo

oo

w(x, O,t)= Z Z W,s cos nO e n=0

jwt-jknsx

.

(2.12.2c)

s= 1

The above assumed distributions have appropriately chosen angular and axial functions to ensure that the circumferential variation will be a stationary mode pattern. The subscripts n and s correspond to azimuthal modal order (n = 0, 1,2 .... ) and branch order (s = 1,2, 3 .... ) respectively. The modal order n can be seen to correspond to the number of radial nodal lines, while s indicates the order of particular eigensolutions

n=l

n=O

\

t

f"-~

/ t

Fig. 2.10

I

\ I

n=2

/

Xln=3

Circumferential mode shapes of an infinite thin cylinder.

50

ACTIVE CONTROL OF VIBRATION

for a fixed n. Typical lower order circumferential mode shapes are shown in Fig. 2.10. A complete solution should use a circumferential distribution with a form e ±j"° which can result in rotating angular distributions or 'spinning modes'. Substituting the above distributions into the shell equation of motion and setting the disturbance input pressure p~ to zero results in a system of equations which can be conveniently described in matrix form for a particular mode (n, s) as

where

Ll1

L12

~

Z31

I

Uns

0

/~. /~3

Vns =

O,

L32

Wns

o

L~ ._ _~-~2

+

L13

L33

(knsa) 2 + ~1 (1 - v)n 2 ,

1

L~2 = ~(1 + v)n(k,~a),

(2.12.3)

(2.12.4a) (2.12.4b)

L13 = v(k,~a),

(2.12.4c)

L21 = L12, L22 = _f~2 + ~1 (1 - v)(k,~a) 2 + n 2

(2.12.4d) (2.12.4e)

L23 = n,

(2.12.4f)

L31 = L13,

(2.12.4g)

L32 = L23, L3 3 = _~-~2 +

1 + fl2[(k,~a)2 + n2] 2.

(2.12.4h) (2.12.4i)

In the above equations (2.12.4a-i), f~ is the non-dimensional frequency, ~ = (Da/cL and c~. is longitudinal phase speed of the shell material with effective modulus of elasticity given by equation (2.2.9). A value of ~ = 1 corresponds to the ring frequency of the cylinder when the system resonates as a ring due to longitudinal waves travelling around the shell with a wavelength equal to the circumference of the shell. Thus f~ is alternatively defined as ~ = o.)/(.t) r where (.or the ring frequency of the cylinder is given by (D r - - C L / a . Expansion of the determinant of the amplitude coefficients in equation (2.12.3) provides the system characteristic equation from which dispersion plots can be generated which relate the non-dimensional axial wavenumber, k~sa to the nondimensional frequency, ~. A typical dispersion plot for free waves in a steel cylindrical shell with a thickness ratio of h/a=O.05 and Poisson's ratio v=0.31 is shown in Fig. 2.11 for a circumferential mode number n = 1. The results of Fig. 2.11 were calculated using the Fliigge shell equations in order to be accurate at very low values of ~. This is achieved by adding small correction terms to the Donnell-Mushtari equations as outlined in Leissa (1973). Dispersion curves for different values of n have shapes similar to those of Fig. 2.11, the main difference being that the waves cut-on at different frequencies. Unlike waves in plates, cylindrical shell waves can be seen to have three forms of roots; purely real roots which correspond to propagating axial, torsional and flexural type motion (for f~> 1), a purely imaginary root which asymptotes to the same value as a plate bending near field for ~ ~>1 and complex roots which when paired together correspond to an attenuated, near-field standing wave (see Fuller, 1981). The behaviour of the waves can generally be divided into two regions,

INTRODUCTIONTO WAVES IN STRUCTURES

Re knsa

51

+

5.0

/ o/

I

2.0

/i

! 4 / / ~ / , / \

-5.0 / f

Bending near field "\.

f

f f I

~"

(b)

Fig. 2.11 Dispersion plot for waves in an infinite thin-walled shell, v=0.31, h/a=O.05, n = 1; ~, purely real solutions; - - - - - , purely imaginary solutions, - - -, complex solution (after Fuller, 1981).

either below or above the ring frequency, ~ = 1. Above the ring frequency the behaviour of the shell is similar to a flat plate and most of the energy of vibration is in bending. Below the ring frequency, the behaviour is far more complex due to the increased relative curvature of the wall to the wavelength and most of the energy is in stretching. The solution denoted (a) in Fig. 2.11 corresponds to beam motion of a long slender rod at low frequencies ( ~ ~ 1) and approaches wave motion similar to that of flexural vibration of a flat plate at high frequencies (f~-> 1). The solution denoted (c) is purely imaginary at very low frequencies and cuts-on at f~ = 0.5 to approach torsional motion at higher frequencies. Solution (b) consists of two complex solutions at low frequencies which when combined together form an attenuated standing wave (or near field). As the frequency is increased the two solutions meet in the purely imaginary plane and then diverge to become either that of the bending near field on a fiat plate or, after cut-on, approach longitudinal motion in a fiat plate. More detailed discussion of the free-wave characteristics of cylinders is given by Fuller (1981). It should be noted that for some of the wave solutions, in particular the complex branches, the Donnell-Mushtari shell equations do not behave well at extremely low frequencies ( ~ ~ 0). In this case it is better to use a higher order shell theory such as the Fliigge shell equations.

52

2.13

ACTIVE CONTROL OF VIBRATION

Free vibration of finite thin cylinders

The above analysis can be modified to study the response of finite cylinders by using a similar procedure to that used for the study of the response of finite beams described in Section 2.7. We assume that the cylinder is of length L and has 'simply supported' boundary conditions (also called 'shear diaphragm', Leissa, 1973); that is the out-ofplane displacement w, the bending moment M x, the torsional displacement v and the shear force TxI are simultaneously constrained to be zero at the cylinder ends. Note that this 'simply supported' boundary condition is more complex than that used for thin beams due to the additional components of motions used in shell analysis. Assuming a wave field in the cylinder, for one circumferential mode of vibration, the radial displacement can be expressed as w,~(0, x, t) = (W/s cos nO e -Jk,~x + W,,r cos nO e.ik,~X)e j'°t

(2.13.1)

and applying the boundary conditions at the end of the cylinder we obtain a similar result for the transverse displacement distribution to that obtained in the beam analysis. This is given by w,~(x, O, t ) = - 2 j W i cos nO sin k,~x e j~'

(2.13.2)

and the corresponding eigenvalues are written as kr~ = s : r / L ,

s = 1,2, 3 ...

(2.13.3)

Resubstituting the eigenvalues back into the system characteristic equation results in a cubic equation in the squared non-dimensional frequency, f~2 (Junger and Feit, 1986). This is given by (~"~2)3 __ A2(~-~2)2 + A1 (~'~2) __ Ao = 0,

(2.13.4)

1 2 v){(l_

(2.13.5)

where Ao=

A1 -

{[(k,~a)2 + 2

+

v2)(knsa) 4 +

+ n214},

( )

+ (3 + 2v)(k,~a) 2 } + f12 3 - v [(k~,a)2 + n213' 2

a2 = 1 + ( 3 ) 2

[(k,~a) + n 2] + flZ[(k, sa)2 +

n212.

(2.13.6)

(2.13.7)

Finding the roots of the polynomial given by equation (2.13.4) results in the values of corresponding to the non-dimensional resonance frequencies of the system. Two points are important to note. First, the procedure for obtaining the resonance frequencies is identical to that used for beams, although it is more complicated due to the form of the three individual coupled shell equations. Second, the resonance frequencies could also be obtained, as in the case of the beam, by the intersection points of the eigenvalues specified by sz~/L with the free dispersion curves as shown in Fig. 2.12. For illustration purposes the location of the second flexural resonance frequency, ~2, and the first longitudinal resonance frequency, f~, are shown graphically.

53

INTRODUCTION TO W A V E S IN STRUCTURES

Flexural ./____ __/Torsional ~ ~

2~: L

/i I

.Q

/

I

E

-/

l>

.~ x

/Longitudinal

--!

L

!

. . . . .

<

Non-dimensionalfrequency,.Q Fig. 2.12

Relationship between eigenvalues and free axial wavenumbers for a finite cylinder.

A typical plot of resonance frequency versus circumferential mode number for a fixed value of knsa is given in Fig. 2.13. The purpose of presenting the resonance information in this form is that it illustrates the interesting result that for the s = 1 (flexural) branch which corresponds to predominantly radial motion, the resonance frequency initially falls and then increases with n. This behaviour is caused by the vibration energy being predominantly associated with stretching motion at low values of n. As n increases, the stretching energy decreases, resulting in lower resonance frequencies (Arnold and Warburton, 1949). At higher n, the energy increasingly shifts to flexural motion resulting in an increase in the resonance frequencies.

2.14

Harmonic forced vibration of infinite thin cylinders

The forced, harmonic response of cylinders is also of interest. We will consider, for example, the response of an infinite cylinder to a line force specified by

p(O, t) = F cos nO 6(x) e TM,

(2.14.1)

where F has units of force per unit length of circumference. In this case it is useful to express the shell displacements as spatial Fourier transforms such that 1 [~=~

U(X, O, t) =

2re '

v(x, O, t) = 2~

U,(k,) cos nO e

jwt-jk,,x+j:t/2

Z V~(k~)sin nO e/°~'-/k°x dk~, n=O

w(x, 0, t) =

W~(k~) cos nO 2:r

dk,,

(2.14.2a)

,=0

n=0

x

dkn,

(2.14.2b)

(2.14.2c)

54

ACTIVE CONTROL OF VIBRATION 10

s=3

0 tO

s=-2

0 tl:l tO 0~ tO

E

-9,t O

z

s=l

0.1

0

1

2

3

4

5

6

7

8

9

10

Circumferential mode number, n

Fig. 2.13

Typical resonance frequency plots for a simply supported cylinder,

k.~a= Jr/4 (after Junger and Feit, 1986).

h/a=O.05,

and applying a spatial Fourier transform to the forcing field of equation (2.14.1) we obtain P(k,, t)= F cos nO e j~'. (2.14.3) Note the subscript, ns, has been reduced to n since all wavenumbers (i.e. as related to s) are included in the integral over the k domain for each particular circumferential mode. Substituting equations (2.14.2a-c) and (2.14.3) into the original equations of motion of the shell, equation (2.12.1), gives the wavenumber domain (or spectral)shell equations of motion of the forced response of the system (for a particular value of n). These can be expressed as Lll

L12 G2

L13 Un(kn) V.(k.)

L32 1-,33 W.(k.)

0 =

0

f~2F/(p~ho02)

,

(2.14.4)

INTRODUCTIONTO WAVESIN STRUCTURES

55

where the elements of the matrix are as given in Section 2.12. Applying matrix theory (see the Appendix of Nelson and Elliott, 1992) we can solve for the spectral amplitudes and write Uo(k.,)

T,,

T,:

v.(k.)

T~, T~

T13

o

T~ =

0

W.(k,,) T31 T32 T33

,

(2.14.5)

~2F/(pshto2)

where the matrix T is the inverse of matrix L. Thus the spectral radial displacement amplitude (as a function of k,a) is given by

( ~')2F ) W.(~) =

p 2hto 2

T~.

(2.14.6)

Application of the inverse Fourier transform gives the radial displacement for a single circumferential mode which is therefore

w,(x, n) =

f~2F I?** -j(kna)(x/a) T33e dk~a, 2z~pshato 2

(2.14.7)

where the time variation has been suppressed. From matrix theory, T33 can be written in terms of the elements of matrix L as T33 = (L,,LE2- L12LE,)/IL[,

(2.14.8)

where ILl is the determinant of matrix L. Substituting equation (2.14.8) in equation (2.14.7) suggests that an appropriate method of solving the complex integral of equation (2.14.7) is by the method of residues (Churchill etal., 1974). The integration path chosen to ensure causality is shown in Fig. 2.14 and runs just below

] Rekna

/

I I I

x

\

®

\

\\

)

/I

®

)

/ / k n a plane /

\

)

z

/

/

f

Im kna

Fig. 2.14 Contour of integration for the cylinder input mobility calculation: ®, poles of the integrand function.

56

ACTIVE CONTROL OF VIBRATION

the real axis from mOOto 0 and just above the real axis from 0 to oo. To be compatible with the e j°~' time convention, the contour is closed by a semi-circle of infinite radius in the lower k,,a plane. The poles contained within the contour will be at the locations of the zeros of the determinant of matrix L" i.e. at the free wavenumbers solved for previously.

10 (a) 8,,..., II)

rr

.~ c~

6-

O

E

m

cO .m

4-

t--

E "9 t--

j

O

z

2

0 0.1

I

0.5

1

2

Non-dimensional frequency, ,.-c2

10 (b) ,,-.,.

>v

5

E H

E

0

O

t.-- -5 O Z

-10

0.1

I

I

,

0.5

1

2

3

Non-dimensional frequency, .(2

Fig. 2.15 Input mobility of a thin steel cylindrical shell, (b) imaginary part of y,,d (after Fuller, 1983).

h/a

= 0.05, n = 1" (a) real part of

ynd,

INTRODUCTION

TO WAVES IN STRUCTURES

57

By using the theorem of residues the radial response for a particular circumferential mode n can be written as ~,-22F

w,(x, n) =

0.

~

2Jrpshato 2 ~=

Res,

(2.14.9)

where Res~ =

(Lilt22

-

L12L21)/IL I'.

(2.14.10)

The prime in equation (2.4.10) denotes the derivative with respect to (k.~a). We have now returned to a full subscript notation of ns since the residues are evaluated at the poles which correspond to a particular branch solution s. We define the drive point mobility It. for a particular circumferential mode n as

Y,, =jcow,,/F.

(2.14.11)

Figure 2.15 presents a typical plot of non-dimensional drive point mobility, y,,d= Y,,(cLpsh) versus frequency for the n = 1 mode in a steel shell of thickness, h/a = 0.05. The mobility can be seen to be small at very low frequencies and peak near the ring frequency of the cylinder where the imaginary component changes sign. The peak in the real part of the mobility near f2 -- 1.5 is associated with the cutting-on of the longitudinal type shell wave. At high frequencies the mobility approaches that of a point-force-excited plate. In order to solve for the response of a point-driven cylinder one can take advantage of the periodic decomposition of the point force f = F r ( O ) into a series of circumferential modes. The input pressure distribution of the cylinder p(O, t) can then be written for an oscillating point force as t~

p(O, t) = Z F,, cos nO ~_jwt,

(2.14.12)

n=0

where/7, = (e,/2er)F and e, = 1 if n = 0 and e, = 2 if n ~ 0. The total response to the point force can then be obtained by summing the response to each modal component given by equation (2.14.9) as w (x, t) = Z w, cos nO e j'ot.

(2.14.13)

n=0

The above analysis outlines a brief description of the procedure used to derive the forced harmonic response of infinite thin shells. For further information the reader is referred to the texts of Leissa (1973) and Junger and Feit (1986).

3 Feedback Control

3.1

Introduction

The objective of active vibration control is to reduce the vibration of a mechanical system by automatic modification of the system's structural response. An active vibration control system can take many forms, but the important components of any such system are a sensor (to detect the vibration), an electronic controller (to suitably manipulate the signal from the detector) and an actuator (which influences the mechanical response of the system). The types of actuator used in active vibration control can be broadly classified into those which are fully-active and those which are semi-active. Fully-active actuators are able to supply mechanical power to the system. Examples are electromagnetic shakers, piezoelectric ceramics and films, magnetostrictive and electrohydraulic devices. Actuators such as these can be used to generate a secondary vibrational response in a linear mechanical system, which could reduce the overall response by destructive interference with the original response of the system, caused by the primary source of vibration. Semi-active actuators behave as essentially passive elements in that they can only store or dissipate energy. Their use in active control stems from the fact that their passive mechanical properties can be adjusted by the application of a control signal and such systems are thus sometimes called 'adaptive'. Semi-active actuators can be constructed using electrorheological fluids or shape memory alloys, for example, as discussed in Chapter 5. In this chapter we are concerned exclusively with active vibration control systems which employ fully-active actuators. We are particularly concerned with systems in which the original excitation of the structure, due to the primary source, cannot be directly observed and thus cannot be used as a feedforward control signal. Feedforward control systems will be discussed more fully in Chapter 4. The control systems discussed in this chapter will be those in which the control signal obtained from the sensor is affected by both the primary source and the secondary actuator over which we have control, and this is fed back directly to the actuator. Single-channel feedback control systems are discussed initially, for which the compromise between performance and stability can be most clearly seen. The general single-channel case is then illustrated using displacement, velocity and acceleration feedback applied to a mass-spring-damper system, and the effect of some important imperfections, such as delays in the feedback loop, are discussed. The conventional approach to the feedback control of systems with multiple control sensors and multiple

60

ACTIVE CONTROL OF VIBRATION

control actuators, using the state variable description, is then introduced and applied to a simple two-degree-of-freedom-system. This leads to a discussion of state estimation and optimal control. Finally, a feedback strategy called modal control is described, which arises directly out of the state variable formulation.

3.2

Single-channel feedback control

We will initially consider the behaviour of a system with a single control sensor and a single secondary actuator, for which only a single-channel feedback controller is required. Typically the control sensor will measure the total response of a mechanical system, which is then fed via the controller to the secondary actuator. The mechanical block diagram of such a feedback control system is shown in Fig. 3.1, in which it is assumed that the net excitation of the mechanical system is due to the difference between the primary and secondary excitations. The transfer function of the mechanical system is now defined to be the ratio of the Laplace transform of the response, W(s), to the Laplace transform of the net excitation, Fp(s)-F,(s), and is denoted G(s). The Laplace transform was introduced in Section 1.7 and is used here, with its associated complex frequency variable, s, so that both stable and unstable systems can be properly represented (see, for example, Richards, 1979; Meirovitch, 1990 and Franklin et al., 1994, which are good general references for the material covered in this chapter). The transfer function of the feedback controller is analogously defined as the ratio of the Laplace transform of the secondary excitation to the system, Fs(s ), to the Laplace transform of its response, W(s), and denoted H(s). The equivalent block diagram of the feedback control system shown in Fig. 3.1 can now be expressed in the form shown in Fig. 3.2, in which the differencing operation between the primary and secondary inputs to obtain the net excitation to the mechanical system is explicitly shown. Standard linear systems theory can now be used to derive the overall response of the system, including feedback control. From the definition of the transfer response of the original mechanical system G(s), we can express the Laplace transform of its response as

W(s) = G(s)[Fp(s)- Fs(s)].

(3.2.1)

Sensor detecting response of mechanical system

Primary excitation

Mechanical syste

T

Electrical feedback controller

t Secondary excitation via fully active actuator

Fig. 3.1 The components of a feedback control system.

FEEDBACK CONTROL

Primary excitation

61

Transfer function of mechanical system

Fp(S)~~

d G(s) I I"- I

~ • Response

.....

Fs(S[

H(s)

W(s)

I..,

I"

Transfer function of feedback controller

Fig. 3.2 Equivalent electrical block diagram of a feedback control system. The Laplace transform of the secondary excitation can also be expressed using the definition of the transfer function of the feedback controller, H(s), as

Fs(s ) = H(s)W(s).

(3.2.2)

Combining these equations we obtain

W(s) = G(s)[Fp(s)- H(s)W(s)].

(3.2.3)

After some simple algebraic manipulation, the transfer function of the mechanical system with feedback control can be written as

W(s) Fp(s)

=

G(s) 1 + G(s)H(s)

.

(3.2.4)

Equation (3.2.4) can be used to obtain the 'closed loop' characteristics of any given mechanical system and feedback controller. 'Classical' feedback control can be used to design such feedback controllers using either a pole-zero representation of the individual transfer functions (as in the root locus method, for example) or a frequency response representation (as, for example in the Nyquist method). The inherent compromise with any feedback system can be most easily illustrated using the frequency response approach, however (which assumes that the open loop system is stable). Transforming the transfer function of equation (3.2.4) into a frequency response by making the substitution s = j~o, we obtain

W(jw)

=

Fp(jW)

G(jw)

(3.2.5)

1 + G(jw)n(jw)

as the frequency response of the closed loop system. If we could arrange for the open loop frequency response, G(joo)H(jco), to have little phase shift in the frequency range of interest but simultaneously to have a gain of much greater than unity, then we can write ] 1 + G(joJ)H0"co) I ~>1

for o9 in the working range,

(3.2.6)

so that

W(jcv) F(j--p\-Co) "~ G(jog)

for co in the working range.

(3.2.7)

62

ACTIVE CONTROL OF VIBRATION

The response of the mechanical system is thus significantly reduced in this frequency region by the action of the feedback controller. Outside the working frequency range, however, it may not be possible to ensure that the phase shift of the open loop frequency response is small. In particular, any delays in the mechanical system or feedback path will inevitably give rise to an increasing phase shift at higher frequencies. If the open loop gain is unity at any frequency to~, where there is also 180 ° open loop phase shift, then G(jto)H(jto) = - 1

at

to =toc,

(3.2.8)

so that W(jto) ~ oo

at

to = to c,

(3.2.9)

and the system becomes unstable. More formal definitions of the conditions for stability are discussed in Section 3.3. The design of a practical feedback control system thus generally involves a compromise between having a high open loop gain to achieve good performance in the working frequency range, and having a lower open loop gain to ensure stability outside the working frequency range. It is possible to use electrical circuits in the feedback path which allow the open loop gain to be increased somewhat by increasing the frequency at which equation (3.2.8) is satisfied. Such 'compensators' are discussed, for example by Richards (1979) and Franklin et al. (1994).

3.3

Stability of a single-channel system

The stability of a linear single-channel feedback system can be most readily determined by an inspection of the position of its closed loop poles. For the feedback control system described by equation (3.2.4), the closed loop poles are determined by the roots of the characteristic equation given by 1 + G ( s ) H ( s ) = 0.

(3.3.1)

If we denote these poles by the complex numbers Pl, P2 . . . . . and the zeros of the system, i.e. the roots of G ( s ) = 0, by the complex numbers z~, z2. . . . . the closed loop transfer function equation (3.2.4) can be written in the factored form

W(s) Fp(s)

=

K(s- z~)(s- z2)...

= F(s),

(3.3.2)

(s - p l ) ( s - p2)...

where K is a constant gain factor. The poles and zeros are either real numbers or conjugate pairs of complex numbers. Many important properties of the system can be inferred from the positions of its poles and zeros in the complex s plane, as discussed, for example, by Nelson and Elliott (1992, Section 3.4). In particular, the impulse response of the system whose transfer function is defined in equation (3.3.2) can be written as f ( t ) , which is the waveform of the output of the system if the input waveform is a Dirac delta function, whose Laplace transform is unity. The impulse response can be derived by inverse Laplace transformation of the partial fraction expansion of equation (3.3.2) to give f ( t ) = al

e pat + a2 epzt + " " ,

(3.3.3)

FEEDBACK CONTROL

63

where al, a2, etc., are constants. If the pole Pi is real, then so is its corresponding constant ai in equation (3.3.3). In order for f(t) to decay to zero with increasing time, then clearly each real pole, pi, must be negative. For each pair of complex conjugate :¢ :¢ poles, pj, p j , the corresponding constants aj, aj are also complex conjugates so that the contributions of these poles to f(t) will be entirely real, and of the form

ajePJt+ aj:¢ ep~t = Aje ojt cos(wit+ Oj)

(3.3.4)

where crj and o)1.are the real and imaginary parts of pj, and Aj and 0j are real constants which depend on aj. Each of these terms in the expression for f(t) will also decay to zero with increasing time provided that crj is negative. The impulse response of the system, f(t), will thus decay to zero with increasing time only if the real parts of all the poles of F(s) are negative; in other words if all the poles of F(s) are to the left of the imaginary axis when plotted in the complex s plane. This property of the impulse response guarantees that it is absolutely integrable (see, e.g. Kuo, 1966, Ch. 10), i.e. that

I o l f ( t ) ldt < ,,o,

(3.3.5)

which, in turn, implies that any bounded input to the system produces a bounded output (as described, for example, by Franklin et al., 1994, Section 3.4.1). This provides a convenient and intuitive definition of stability which is sufficient for our purposes. It is worth noting that any system with a pole lying exactly on the imaginary s axis (i.e. having a zero real part) is not stable according to this definition. A pure integrator with a transfer function of the form 1/s will produce an unbounded ramp output for a bounded step input, for example. Such a transfer function could be used to describe the rigid body motion of a free mass. In some texts, such systems are also defined to be 'stable', with the more restricted definition of stability above being described as 'asymptotically stable' (see, for example, Meirovitch, 1990, Section 3.8), but this distinction is not necessary here. Provided the transfer functions of the plant, G(s), and controller, H(s), are available in closed form, the roots of the characteristic equation, equation (3.3.1), can be calculated analytically. This can be a tedious job if performed manually and although, these days, computers can be readily programmed to perform this task, a number of computational methods have historically been used to manually determine the signs of the real parts of all the roots of the characteristic equation. One such method, which uses the Routh Hurwitz criterion, is described, for example, by Franklin et al. (1994, Section 4.4.3). It is often the case in practice, however, that closed polynomial forms for H(s) and G(s) are not directly available, and stability must be assessed from input-output measurements, made on the system before control. The Nyquist stability criterion provides such a method, using the open loop frequency response, G(jo))H(jvo). This frequency response can generally be directly measured before the feedback loop is closed. The theoretical basis of the Nyquist criterion is described, for example by Franklin et al. (1994, Section 6.3) and has been reviewed in Nelson and Elliott (1992, Section 7.4). The outcome is that a closed loop system is stable only if the polar plot of the open loop frequency response does not enclose the point ( - 1 , 0 ) . This provides a convenient generalisation of the rather special condition described by equation (3.2.8),

64

ACTIVE CONTROLOF VIBRATION

which would correspond to the open loop frequency response passing through the point ( - 1 , 0 ) at the frequency co~. The Nyquist criterion is also extremely useful in defining not only the absolute stability of a system, but also its relative stability, which describes the proximity of the open loop frequency response to the point (-1,0). This is important when dealing with many practical systems which have some inherent uncertainty in their frequency response, because of changes with operating point or temperature, or over time, for example. A system is described as being robust if its stability and performance are relatively unaffected by such changes, and the Nyquist criterion provides an intuitively appealing way of expressing this robustness. An introductory description of these issues is to be found in Franklin et al. (1994, Section 6.9.2).

3.4

Modification of the response of an SDOF system

In this section we consider a particular example of an idealised single-channel feedback control system, which is applied to an SDOF lumped mass-spring-damper system. The dynamics of this system have been discussed in Chapter 1. The physical arrangement is illustrated in Fig. 3.3, in which the signal from the sensor, W, is assumed to be proportional to the downward displacement of the mass. The actuator generating the secondary input to the system is assumed to generate only a force, F,, proportional to the control signal. Any inherent stiffness or damping associated with the actuator can be lumped in with that of the mass-spring-damper system before control. Prior to active control, the control signal driving the secondary actuator will be zero and so the secondary force will also be zero. Under these conditions the dynamic response of the SDOF system can be deduced from its differential equation, derived in Chapter 1, which can be written in terms of the time histories of the primary force, fp(t), and the displacement of the mass, w(t), as fp(t) = MOO(t) + Cvg(t) + Kw(t).

Fp~ -Mass

Primaryforce excitation

M W

Sensor signal proportional to downward displacement of mass ,

K

(

Fs

,

,

L

F~ "-T-Feedback controller

/ / / / / / /

//

/ / / / / / / / / / / / /

Linear Viscous Secondary spring damper actuator

Fig. 3.3 Feedbackcontroller applied to a lumped mass-spring-damper system.

(3.4.1)

65

FEEDBACK CONTROL

The Laplace transform of each term of this differential equation can now be taken, and assuming zero initial conditions, i.e. w(0)= 0 and w(0)= 0, equation (3.4.1) can then be written as Fp(s) = Ms2W(s) + CsW(s) + ~ ( s ) , (3.4.2) in which Fp(s) and W ( s ) are the Laplace transforms of fp(t) and w(t). The transfer function defining the mechanical response of the system prior to control was defined above to be

W(s)

G(s) = ~ ,

(3.4.3)

Fp(s) which from equation (3.4.2) can be written, in this case, as 1

G(s) =

.

(3.4.4)

Ms 2 + Cs + K

In order to determine the closed loop response of the system shown in Fig. 3.3, we need to know the form of the feedback controller, H(s). We will assume that the electronic response of the controller is such that the secondary force has three components, which are proportional to the acceleration, velocity and displacement of the mass, with the gain constants ga, go and gal. The time history of the secondary force can thus be written as f~(t) = gjO(t) + govg(t) + gdW(t).

(3.4.5)

We would ideally use three separate transducers to measure the acceleration, velocity and displacement. In practice, often only a single transducer is available, to measure acceleration for example, and electronic integrators are used to derive signals proportional to velocity and displacement. Such an approach will only be valid over a certain frequency region, however, because practical electronic integrators do not have the infinite gain at zero frequency ('d.c.') which an ideal integrator should possess. The static position (or 'attitude') of the mass could thus not, in practice, be deduced from the output of an accelerometer alone. The philosophy here, however, is that we are seeking to control the vibration of the mass, and thus we confine our attention to the effects of feedback control at frequencies above d.c. Equation (3.4.5) can thus still provide a good model for feedback vibration control over the frequency range of importance in vibration problems, even with an accelerometer as the only sensor. If, alternatively, we had a single sensor available which measured the velocity of the mass, a feedback law of the form of equation (3.4.5) could still be realised with an electrical differentiator and an electrical integrator. The control strategy would then be that described in text books on control theory as 'proportional-integral-derivative', or PID control. Because we are not concerned with the zero-frequency response of the system, as discussed above, we can ignore the initial conditions associated with the feedback control law. The Laplace transform of equation (3.4.5) can then be taken term by term, and rearranged to give the transfer function of the feedback controller, H ( s ) , as H(s) =

Fs(s)

W(s)

2 = g~s + g~s + gal.

(3.4.6)

66

ACTIVE CONTROL OF VIBRATION

The two open loop transfer functions, equations (3.4.4) and (3.4.6), can now be used to deduce the closed loop response, given by equation (3.2.4), as W(s) Fp(s)

=

G(s)

1

=

1 + G(s)H(s)

.

(3.4.7)

( M + g~)s 2 + (C + go)s + (K + gd)

The effect of feeding back acceleration, velocity and displacement is clearly to modify the effective mass, damping and stiffness of the mechanical system, respectively. Equation (3.4.7) can also be expressed, by analogy with equation (3.4.4), as W(s)

1

=

Fp(s)

,

(3.4.8)

M's2 + C's + K'

where M' = M + ga is the modified mass, C' = C + go is the modified viscous damping term and K' = K + gd is the modified stiffness. In this idealised case the stability of the closed loop system is ensured provided M', C' and K' are all positive, as can be shown in this case using the Routh-Hurwitz criterion. In principle, then, the mass, damping and stiffness can be independently modified by the three gains in the feedback controller, such that each of these mechanical parameters can be set anywhere in the range from being zero to being arbitrarily large. In practice a number of effects act to reduce the range over which the properties of such a mechanical system can be modified by feedback control.

3.5

The effect of delays in the feedback loop

One of the most important effects which limits the performance of feedback controllers in practical mechanical systems is unmodelled phase shift. Such phase shift may arise because of the dynamic response of the sensors or actuators being used or may be due to time delays in the controller. Time delays are especially prevalent if a digital control system is being implemented, particularly if analogue anti-aliasing and reconstruction filters are used. The addition of such time delay in the feedback path causes the transfer function of the controller to be modified from equation (3.4.6), such that it can now be written H ( s ) = ~Fs(s) = e_S~(gas2 + gvs + gd),

(3.5.1)

W(s) where z is the assumed delay. If this delay is small, its frequency response can be expressed as e -j~ = 1 -jwr

for

wr ~ 1.

(3.5.2)

The closed loop frequency response of the feedback control system with delay is now more complicated than that deduced from equation (3.4.7), but can still be written in the form W(jw)

1

Fp(j~o)

jwC" + K" - o92M "

(3.5.3)

FEEDBACK CONTROL

67

The effective damping term can now be written as C " = C + g v - rgd + o92rga,

(3.5.4)

K" = K + gd

(3.5.5)

M" = M + ga - rgv,

(3.5.6)

the effective stiffness is now

and the effective mass is now

The effective stiffness has not, to a first approximation, been affected by the delay r. The effective mass is only marginally affected since the term rgv is found to be small in comparison to M for lightly damped systems. The effective damping term, however, is strongly affected by the delay, and affected in a frequency dependent manner as shown by the final term in equation (3.5.4). If displacement and acceleration feedback are assumed to be implemented in such a way that the effective stiffness and mass are, say, changed by a factor of two from their natural values, the relative magnitudes of the terms C, r g d and O)2•ga in equation (3.5.4) can be examined. It is found that for a lightly damped system, rgd is comparable with C when r is a small fraction of a period of the natural frequency of the system. Under similar conditions ~oZrga becomes comparable with C for frequencies above the natural frequency of the system. If either displacement or acceleration feedback are implemented in a control loop with a small delay, the effect of the delay is thus to significantly change the damping of the system. If the effective stiffness of the system is to be increased, or the effective mass decreased, then gd is positive and ga is negative, and the effect of both these terms, with a delay present, will be to reduce the effective damping. Even with modest delays present in the feedback loop, the effective damping can be reduced to the point where C" is negative, and the system becomes unstable. In contrast, velocity feedback, which increases the damping of the system, has only a small effect on the effective mass and stiffness. Velocity feedback is thus seen to be a more robust control strategy than displacement or acceleration feedback, as far as unmodelled phase shifts are concerned. This is in agreement with the experimental findings of Hodges (1989). Velocity feedback is very widely used to actively add damping to otherwise lightly damped mechanical systems. Velocity feedback can also be used on distributed mechanical systems and is guaranteed to be stable, provided the velocity sensor and force actuator are collocated (Balas, 1979). By placing the sensor and actuator at the same point on the structure the effect of velocity feedback is the same as if a passive damper had been attached between that point on the structure and an inertial reference: the so-called s k y h o o k damper (Kamopp et al., 1974). The use of velocity and displacement feedback in the active control of the vibrations of a circular saw has been described by Ellis and Mote (1979). They note that the thickness of a circular saw is a compromise between being thick enough to prevent vibration and being thin enough to minimise 'kerf' losses due to the blade thickness. These authors estimated that in 1979, U.S. lumber companies could save $4 million/ year by reducing kerf losses by one thousandth of an inch. The experimental system used to actively control the saw blade is illustrated in Fig. 3.4(a), in which an eddycurrent displacement sensor and an electromagnetic actuator are used. The block diagram of the control system is shown in Fig. 3.4(b). On a production saw these

68

ACTIVE CONTROLOF VIBRATION

Material

(a)

/~-- Feed I Electromagnet ,jx___

Saw blade

Vibration sensor

i

Electromagnets vibrationsensor

Disturbances from sawingprocess Referencet~Error.J Control H Electroposition signal"I algorithm magnets I

(b) Circular ~ saw

Lateral position

Electronicpositionsensor

Fig. 3.4 The mechanical arrangement (a) and electrical block diagram (b) of the feedback control system used by Ellis and Mote (1979) for the active control of circular saw vibrations. authors demonstrated a 100% increase in stiffness and 50% increase in damping using active displacement and velocity feedback control.

3.6

The state variable approach

Rather than directly transforming the differential equations which describe a dynamic system into the Laplace domain, an alternative approach is to recast the time domain equations into a standard form; in terms of the internal state variables of the system. It is then possible to manipulate this state variable representation, using wellestablished matrix methods, in order to derive a number of useful properties about the system, such as its stability, its controllability and observability and the effect of various forms of feedback control. These methods are the subject of 'modem' control theory and are described more fully by Richards (1979), Meirovitch (1990) and Franklin et al. (1994).

69

FEEDBACK CONTROL

In this section we will introduce the state variable representation by considering the dynamics of the SDOF, mass-spring-damper system discussed above. The real power of the representation, however, is that the same matrix manipulations can be performed on the state variable equations of arbitrary order. Provided the dynamics of multipleinput, multiple-output systems can be expressed in this form, standard procedures can again be used to establish the properties of the system and describe the effect of feedback control. Consider again the differential equation describing the SDOF system, which was previously expressed in the form (3.6.1)

f ( t ) = MOO(t) + Cvg(t) + Kw(t).

It should be noted that a matrix generalisation of equation (3.6.1) can be used to describe mechanical systems with many degrees of freedom (see Chapter 1). The process of transforming such a matrix equation into state space form is a straightforward extension of the scalar case considered here and is described, for example, by Meirovitch (1990) and Inman (1989). Equation (3.6.1) can be rewritten as ~(t) =

C

K 1 vi,(t) - ~ w(t) + ~ f(t). M M M

(3.6.2)

This second-order differential equation can be expressed as two simultaneous first-order differential equations. In doing so, it is convenient to define two variables which completely define the internal state of the system. These are termed state variables and are denoted x~(t) and x2(t). One choice of suitable state variables for this example is the displacement and velocity of the mass: x2(t) = w(t).

xl (t) = w(t),

(3.6.3a,b)

It is thus clear that the state variables are related by the first-order differential equation (3.6.4)

x2(t)=2~(t),

and by rewriting the differential equation of the SDOF system (equation (3.6.2)) in state variable form, we also obtain C ~2(t) = - ~ x2(t) M

K

1

x~(t) + ~ f ( t ) . M M

(3.6.5)

Equations (3.6.4) and (3.6.5) can be written as a single matrix equation of the form

[ ]

21(t) =

K

C

22(t)

M

M

0

lrxl t ] + 01

f(t)

(3.6.6)

M

or, more compactly, as :~(t) = Ax(t) + Bu(t),

(3.6.7)

70

ACTIVE CONTROLOF VIBRATION

where

~:(t)

=

"x2"t"

A=

0

1

K

C

M

M

and

x(t) = Xl(t) x2(t)J'

and in this case B is a vector and u(t) a scalar of the form 0 B-

1

,

u(t)-f(t).

Equation (3.6.7) describes the evolution of the internal states of the system when driven by an input u(t). A similar equation can also be used to describe the effect of the inputs and state variables on the output y(t), of the system, which can, in general, be written in the form y(t) = Cx(t) + Du(t).

(3.6.8)

If a velocity sensor is used on the SDOF system above, for example, then we would have C = [ 0]

and

D=0.

The state variable representation is indicated diagrammatically in Fig. 3.5. More complicated mechanical systems can be described using the same philosophy. In the case above, for example, the mass could be coupled to a second mass via separate springs and dampers, in which case four state variables would be required to define the internal state of the system. In general any linear mechanical system of arbitrary complexity can be completely described by the pair of matrix equations (3.6.7) and (3.6.8), as demonstrated, for example, by Meirovitch (1990, Section 3.7). By introducing the state variables which act as intermediate variables between the input signals and output signals, we admit to a broader class of behaviour than can be described by the simple input-output transfer function used above. The second mass mentioned above could, for example, be only weakly coupled to the driven mass, but have very little damping associated with its motion. The state variables associated with the second mass could thus become very large for certain excitation frequencies without significantly affecting the output of the system. It is partly this ability of the state variable method to account for the complete behaviour of a dynamic system which makes it so powerful. Some of the standard properties of the state variable method are reviewed in the Appendix, where it is shown that the transient response of the system is dependent on the eigenvalues of the system matrix A in equation (3.6.7). Also discussed in the Appendix are the conditions under which all the state variables affect at least one of the outputs (observability) and the conditions under which all the state variables are affected by at least one input (controllability). These are important considerations in determining the ability of feedback control to reduce various aspects of the response of systems with complicated dynamic behaviour.

FEEDBACK CONTROL

71

Initial conditions of states

t t t Dynamic system described by state variables which are elements of the vector x

Inputs, elements of the vector u

Fig. 3.5

Outputs, elements of the vector y

State variable representation of a dynamic system.

To illustrate the use of the state variable methods discussed in the Appendix, in the simplest case, we will consider the transient response of the SDOF system discussed above. The unforced solution to the state variable equations for the ith state variable can, in general, be written as

xi(t)=eil e a~' +ei2 e ~2t + ... +ein e ~.~,

(3.6.9)

where the constants eil, ei2, etc., depend on the initial conditions of the internal states and 21, 22, etc. are the eigenvalues of the A matrix. For the SDOF system, there are two state variables and thus two eigenvalues for the associated A matrix, and two terms in the transient response, equation (3.6.9). Following on from the discussion in Section 3.3, it can be seen from equation (3.6.9) that the free response of a system described in state variable form will decay to zero provided the real parts of all the eigenvalues of the A matrix have negative real parts (see also the Appendix). This is equivalent to the conditions for the stability of a single-channel system in Section 3.3, for which all the poles were required to have negative real parts. Since the eigenvalues of the A matrix can be computed for multiple-input, multiple-output systems described in state variable form, this condition on the eigenvalues of the system matrix provides a powerful generalisation of the stability condition discussed previously. The eigenvalues of A are the solutions of the characteristic equation resulting from setting the determinant of 2 1 - A to zero, which in this case, using the definition of A following equation (3.6.7), can be written as 2

-1

I;lI-Al= _g 2 + - CM

=0.

(3.6.10)

M

The characteristic equation is thus C K 22+--2+--=0, M M

(3.6.11)

which has the solutions

C ((~M)2 K) 1/2. ~1,2-" 2M ~

(3.6.12)

72

ACTIVE CONTROL OF VIBRATION

The mechanical parameters of the SDOF system can be expressed, as in Chapter 1, in terms of its natural frequency, too, and damping ratio, ~, such that K

2 = too

C

and

M

= ~o0.

(3.6.13a,b)

2M

If we now assume that K M

C

>

,

i.e. ~ < 1,

(3.6.14)

2M

the eigenvalues of the A matrix in this case can be written in the form 2,,2 = -~tOo +jtoo~/1 - ~2.

(3.6.15)

The transient response of the system, equation (3.6.9), thus takes the form expected from the conventional analysis (Chapter 1) of having a decaying oscillatory waveform. It is also clear that for this passive system the eigenvalues have negative real parts, as required for stability.

3.7

Example of a two-degree-of-freedom system

In order to illustrate the use of the state variable method on a more complicated system, but one which is still algebraically tractable, we will consider the mechanical system illustrated in Fig. 3.6. This figure shows a rigid body, of mass M and moment of inertia I about its centre of gravity, supported a horizontal distance l~ from its centre of gravity by a spring of stiffness K1 and a viscous damper with damping constant C1, and similarly by another spring and damper (/(2, C2) a horizontal distance l 2 from the centre of gravity on the other side. If the body is subject to an external force, Fw(t), and moment, Ho(t), about the centre of gravity, the equations of motion for this system, assuming the vertical

j~

..-*-"

J1

~J

J2

Fig. 3.6 A two-degree-of-freedom system which can have heave motion in the w direction from the center of gravity (~) and pitch motion about the centre of gravity associated with the rotation 0.

73

FEEDBACK CONTROL

displacement w and angular rotation 0 at the centre of gravity are small, may be written in the form (Porter, 1969)

MC¢(t) + (C~ + C2)w(t)+ (K1 + Kz)w(t) - (C~l~- Czl2)O(t)- ( K i l l - Kzlz)O(t)= Fw(t)

(3.7.1)

and

lO(t) + (C~l~ + Czl2)O(t) + (K,l 2+ K2lZ)O(t) - (C~l~- C212)w(t)- ( K i l l - Kzlz)w(t)= Ho(t).

(3.7.2)

Assuming the observed output signals from the system are the displacements at the mounting points, Wl(t) and Wz(t ) illustrated in Fig. 3.6, these are related to the displacement and rotation about the centre of gravity (again assuming small displacements) by the expressions Wl(t) = w ( t ) - llO(t),

Wz(t) = w(t)+ 120(t).

(3.7.3)

In order to describe this system in state space form, we first have to choose the state variables. One convenient choice for the four state variables needed in this case are the linear and angular displacements and velocities about the centre of gravity, i.e.

xl(t) = w(t), x3(t) = O(t),

Xz(t) = tO(t), x4(t) = O(t).

(3.7.4a,b,c,d)

The choice of state variables is not unique, and it is shown in the Appendix that any set of independent combinations of these variables could be used. The definition of equation (3.7.4) is convenient, however, because it allows the equations of motion, equations (3.7.1) and (3.7.2), to be easily expressed in standard form" :~(t) = Ax(t) + Bu(t)

(3.7.5)

as

0

1

0

0

K1 + K2

C1 + C2

Kill - K212

Clll - C212

M 0 K~ll - K212

M 0 Clt1-C212

M 0 K~12~+ K21~

M 1 Cl121+ C2122

I

I

I

I

Cy(t)

I

=

O(t) O(t)

•

•

I

o

0 1

+

w(t) ~v(t) O(t) O(t)

0 o

0

0

0

1 -I

Ho(t) '

(3.7.6)

.

from which the definitions of the matrices A and B in this case are clear. Similarly, the

ACTIVE CONTROL OF VIBRATION

74 output equation

(3.7.7)

y(t) = Cx(t) + Du (t), can, in this case, be written

WE(t)

[w(t)

-l,

o

)

+12 0 /

1 0

(3.7.8)

O(t)

'

LO(t) where the definition of C is clear, and D - - 0 in this case. We shall use this example to illustrate some of the properties of different forms of feedback control in the next section. For now we just draw attention to some aspects of behaviour of this two-degree-of-freedom system. For convenience, the variables in the system matrix in this case may be denoted 0

1

0

0

A= -Kw -Cw Kwo Cwo 0

0

0

1 '

(3.7.9)

Kow Cow -Ko -Co where the definition of the individual variables can be deduced from equation (3.7.6). Kw and Cw quantify the stiffness and damping in the 'heave' mode of the system, and K0 and C 0 quantify these quantities for the 'pitch' mode (although these variables have different dimensions to the stiffness and damping constants in Fig. 3.6). Kwo and Cwo quantify the coupling between the heave motion and the pitch motion. In general, a force acting at the centre of gravity, for example, will cause a rotation of the body shown in Fig. 3.6, as well as a translation. Under certain conditions, however, these cross-coupling terms are zero and the motion is said to be uncoupled. These conditions are when Kwo= Kow = 0 and Cwo= Cow= 0 and, from the definition of these variables, this condition can be written as

Kill =

Clll =

and

K212

(3.7.10)

C212 •

When the system is uncoupled, the state equation (3.7.6) can be written as

;v(t) ~(t) n

m

"O(t) O(t)

0

1 ~

-Kw -Cwl - -

0

0

0

0 I~_!t!

[w(t)

0

1

0

0

1 M

(3

0

0

0

1 I

Fw(t)] (3.7.11)

-1.

|0(t)

0

0

o

o ,, -Xo -Co [ O(t)

The dashed lines indicate a partitioning of this matrix equation into the two independent equations given by

(t)

[ o = -Kw

0 1

-Cw

(t)

+

1

Fw(t) ,

(3.7.12)

FEEDBACK

75

CONTROL

and 0

O(t) = - K o -Co

O(t)

I

which describe the heave and pitch motion. Each of these equations is of the same form as for the SDOF system, equation (3.6.6), indicating, as expected, that the system behaves as an SDOF system independently in each of these two modes. The natural frequencies and damping motions for these two uncoupled modes may thus be readily calculated. To consider a particular example: if the rigid body were a uniform beam, supported at the two ends by springs and dampers with identical characteristics, then l z - 1 2 - - l , K~ = K 2 -- K and C~ = C2 = C. The uncoupling conditions, equation (3.7.10), would clearly then be satisfied. For this example, the moment of inertia of the body is I = Ml2/3, and the natural frequency and damping ratio in the heave mode, Ww and ~w, and in the pitch mode, w0 and ~0 may be calculated as Wx=

2K ~,

M

~x =

C

42MK

(3.7.14a,b)

and ~/6K (DO -"

~ ,

M

C ~ 0 --

(3.7.15a,b)

~/2MK/3

Although the motions are uncoupled, we see that the natural frequencies and damping ratios of the two modes cannot be independently specified. In this particular case, their ratios are both equal to ~-. Although this ratio will vary depending on the geometry, for a given geometry the choice of springs and dampers to fulfil equation (3.7.10) inevitably leads to the natural frequency and damping ratio of one mode being related in a fixed way to those of the other mode. It should be noted that the eigenvalues of the equations of motion cast in the state space form, ~(t)= Ax(t) for free motion where x(t) is the vector of state variables, are not the same, but are clearly related to, the eigenvalues of the equations of motion when put in the standard mechanical engineering form of Chapter 1; M ~ ( t ) + K w ( t ) = 0 where w(t) is the vector of displacements of a number of points on the structure. One obvious reason for this difference is that there are generally twice as many state variables as elemental displacements, since in this case the elemental velocities must also be included as state variables. The difference may be explained in simple terms by noting that the equation M ~ ( t ) + K w ( t ) = 0 contains no damping terms. The Nth order equation M ~ ( t ) + Kw(t) = 0,

(3.7.16)

can be written, assuming M is not singular, in the form ~(t) = - M -1Kw(t),

(3.7.17)

whose characteristic equation (1.9.8) has N real eigenvalues, equal to the square of the natural frequencies (equation 1.9.9). By analogy with the SDOF case, equation (3.7.17) can be written in terms of the

76

ACTIVE CONTROL OF VIBRATION

state variables x ( t ) = [wT(t), ";VT(t)] as 0 -M-1K

~(t)

0

(3.7.18)

"iv(t)'

so that the system matrix in this case will have 2N imaginary eigenvalues, which occur in conjugate pairs, as in equation (3.6.12) with C=0, and are equal to ±j times the natural frequencies.

3.8 .Output feedback and state feedback The obvious multi-channel generalisation of the single-channel feedback control systems discussed in Section 3.2 is to feed each output back to each input via one element in a matrix of feedback gains. Such an arrangement is illustrated in Fig. 3.7, and we will begin this section by analysing the closed loop response of such a system using the state variable approach developed above. We assume that the mechanical system before control is governed by the state variable equations X(t) = Ax(t) + Bu(t),

(3.8.1)

y(t) = Cx(t) + Du(t).

(3.8.2)

and The input to the mechanical system is now the difference between the 'required' input to the system, which we denote r(t), and the signals fed back from the output, y(t), weighted by the elements of the gain matrix Go, so that

u(t)=r(t)-Goy(t).

(3.8.3)

Using equation (3.8.2) for y(t) in (3.8.3) allows the net input vector to the mechanical system to be expressed as (3.8.4)

u(t) = [I + GoD ] -~ [r(t) - GoCx(t)].

U\ Input

/>

Mechanical system

Go

Output y

(

Matrix of feedback gains

Fig. 3.7 Block diagram of a multi-channel mechanical system with output feedback.

FEEDBACK CONTROL

77

Substituting equation (3.8.4) into (3.8.1), we find that the system with output feedback is governed by the new state variable equation :t(t) = [ A - B[I + GoD]-~GoC]x(t) + B[I + GoD]-~r(t).

(3.8.5)

The dynamics of the original state variables of the mechanical system are now governed by the new system matrix A0= [ A - B[I + GoD]-IGoC].

(3.8.6)

For a given mechanical system, and matrix of feedback gains, the properties of the closed loop system could be calculated using equation (3.8.6). In particular, the stability of the closed loop system could be assessed by calculating the eigenvalues of A0. Various methods have also been developed for calculating the elements of the gain matrix Go to achieve a desired closed loop response. One widely discussed method is pole placement whose objective is to ensure that the eigenvalues of the closed loop system matrix are closer to those specified by the designer than those of the open loop system. For example, the closed loop eigenvalues may be moved further into the left half of the complex plane to improve stability robustness (see, for example, Meirovitch, 1990; Inman, 1989). Returning to our example of the two-degree-of-freedom system (Section 3.7), we have seen that in this case D = 0 and A, B and C are defined by equations (3.7.6), (3.7.8) and (3.7.9). Output feedback in this case would result in equation (3.8.3) having the form

Ho(t) = Hor(t)

g2~ g22 [Wz(t) '

(3.8.7)

where F~r(t) and Hor(t ) are the external force and moment applied to the closed loop system, so that F~(t) and Ho(t) remain the net force and moment on the physical system. The four feedback gains gll, g12, g21 and g22 can be independently chosen. Since D - 0 in this case, the new system matrix, equation (3.8.6), simplifies to A0 = [A - BGoC ].

(3.8.8)

Using the expressions for the A, B, C and Go matrices given above, the closed loop system matrix in this case can be written

Ao=

0 1 0 0 -Kw -Cw Kwo Cwo _ 0 0 0 1 Kow Cow -Ko -Co

0

0

0

g11+g12 M 0 g21 + g22

0

g2112-gllll

0 0

M 0 g2212- g211~

I

0

(3.8.9)

I

The effect of feeding back the displacements at the two mounting points is, as expected, to change the stiffness terms in the system matrix. The change in each stiffness term is, however, dependent on the value of two feedback gain constants, which makes this arrangement somewhat inconvenient for design purposes.

78

ACTIVE CONTROL OF VIBRATION

We now consider the effect on the response of the system of feeding back the state variables. This does, of course, assume that we have access to the state variables and, depending on the system under control, such access may be achieved in a variety of ways. The most direct and reliable method is to ensure that as many output signals are obtained from the system as there are states, and that these outputs are sufficiently independent that the state variables can be reconstructed from them. These conditions are equivalent to the assumption that the C matrix in equation (3.8.2) is invertible, as can be demonstrated by rearranging the output equation (3.8.2) in the form x(t) = C-1 [y(t) - Du(t)],

(3.8.10)

where C must be square and nonsingular for the expression to be valid. Having derived the waveforms of the state variables, these can be fed back via a gain matrix G x so that

u(t)=r(t)-Gxx(t),

(3.8.11)

where r(t) again denotes the input to the closed loop system shown in Fig. 3.8. By substituting equation (3.8.11) into equation (3.8.1), the state variable equation for the closed loop system with state feedback can be written as

(3.8.12)

k(t) = [ A - BGx]x(t) + Br(t). The resulting closed loop system matrix is thus A x = A - B G x.

(3.8.13)

To illustrate this control philosophy we again retum to the two-degree-of-freedom system from the previous section. In order to make the example more realistic, we now assume that the feedback affects the systems via two secondary force actuators acting in parallel with the passive springs and dampers, as illustrated in Fig. 3.9. The resultant secondary force and moment at the centre of gravity are related to these two secondary forces by the equation

Fw(t)]=[1 1][f,~(t)]=Tfs(t)" Ho(t) -ll 12 [f,2(t) Mechanical system

Input

(3.8.14)

Output y

x

Gx

C-1 (x I

Feedback gain matrix

Inverse outputmatrix

Fig. 3.8 Block diagram of a state variable feedback system using the output vector to calculate the state vector.

FEEDBACK CONTROL

w1

79

w2

ts2

tsl

Fig. 3.9 The two-degree-of-freedom system of Fig. 3.6 with the addition of two secondary force actuators, used for state variable feedback control. The output of the feedback controller could thus be weighted by the set of gains given by the elements of the matrix T-1, so that although the actuators in parallel with the springs and dampers are physically being driven, the effect was to independently influence Fw(t) and Ho(t). We also assume that both the displacement and velocity at the two mounting points can be measured, so that we have as many outputs as state variables. The output equation can now be written as

y(t)=

W ~

= 0

01 '0

W2

0

12

"2

1

0

i

-

= Cx(t),

(3.8.15)

t jt

where D = 0 again, and C is now invertible. The state variables of the system can thus be calculated from the system outputs using equation (3.8.10), which in this case takes the form x(t) = C-~y(t).

(3.8.16)

The state variables are fed back to the secondary actuators via the inverse transformation matrix T -1 and a 2 x 4 feedback gain matrix Gx, so that fs(t) = T-1GxX(t ).

(3.8.17)

The net input to the mechanical system is thus

u(t)=r(t)-Tfs(t),

(3.8.18)

and using equations (3.8.17) and (3.8.16) this can be written as

u(t)=r(t)-GxC-ly(t).

(3.8.19)

The block diagram of the complete control system is illustrated in Fig. 3.8, where in this case r(t) denotes the external force and moment acting on the mechanical system. Equation (3.8.16) reduces equation (3.8.19) to the original expression for state feedback control, equation (3.8.11), with the result that the system matrix is again given by equation (3.8.13). In the case being considered here, A is given by equation

80

ACTIVE CONTROL OF VIBRATION

(3.7.9) and B and G x are given by

B

0

0

1 M 0

0

0

I"

Gx=/g~l

[ g21

0 1 I

g12

g13

g]4

g22 g23 g24

l

(3.8.20a,b)

ip

so that the closed loop system matrix in this case is given by

0 1 0 0 -Kw -Cw Kwo Cwo _ Ax = A - BGx = 0 0 0 1 Kow Cow -Ko -Co

0

0

0

0

gll

g12

g13

g14

M 0

M 0

M 0

M 0

g21

g22

g23

g24

I

I

I

I

(3.8.21)

It is clear that each of the stiffness and damping terms can be independently adjusted by changing only one element of the state feedback gain matrix. In particular, if we set

g13 MKwo, g21 = IKow, =

g14 = M C w o , g22 = IC Ow,

(3.8.22a,b,c,d)

then we can decouple an otherwise coupled system by the use of active feedback control. We are then also free to independently specify the stiffness and damping ratio of the heave and pitch modes with the four remaining feedback gains (gll, g~2, g23 and g24). The use of state feedback thus allows us to synthesise a mechanical system which has dynamic properties which could not have been obtained by adjustment of the passive springs and dampers. This has been brought about by the ability of an active system to apply forces at one point on the structure in response to the motion at a physically separate point on the structure.

3.9

State estimation and observers

In the example above we assumed that all the state variables could be calculated directly by operations on the outputs. This is not always the case, particularly when the mechanical system being controlled has complicated internal dynamic behaviour which it is difficult to directly measure. A strategy which has been developed to estimate the state variables from a limited number of observations is the state estimator or observer (see, for example, Richards, 1979; Meirovitch, 1990). A state estimator is an electrical or digital system which models the intemal dynamics of the mechanical system being controlled. It is fed by the same input signals as the mechanical systems, u(t), and has its output, 5'(t), constantly compared with the output of the mechanical system, y(t). The objective is to ensure that the internal states of the electronic state estimator, which

FEEDBACK CONTROL

81

can of course be directly measured, will track the internal states of the mechanical system, which cannot be directly measured. The internal states of the state estimator are then used as estimates of the internal states of the mechanical system, and fed back to the inputs to implement state variable feedback control. The arrangement is illustrated in Fig. 3.10. One obvious way of implementing a state estimator would be to ensure that it had the same dynamics as the system under control, which are assumed known. The estimated states, ~, would thus be governed by the equation

~:(t) = A~(t) + Bu(t).

(3.9.1)

Assuming D = 0 in equation (3.6.8) for simplicity, the output of the state estimator would be, 5'(t) = C~,(t).

(3.9.2)

The vector of errors between the outputs of the plant and those of the state estimator is thus

ey(t) = y(t)

- ~,(t) = y(t) - C:~(t).

(3.9.3)

In order to adjust the estimated states to minimise the sum of the squares of these T errors, eyey, we could add another term to equation (3.9.1) which was proportional to the negative of the derivative of eyeyT with respect to ~,. This results in a form of gradient descent adaptive algorithm. Writing the sum of the squared errors as T eyey = yTy _

yTC~' _ ~TCTy + ~TcTc~,,

(3.9.4)

the required derivative can be written as (see the Appendix of Nelson and Elliott, 1992),

OT eyey = 2CTC ~ _ 2CTy = _2CT(y _ C~).

(3.9.5)

Mechanical system

Observer

Gx

1"

c,

A X

Feedback gain matrix

Error in observer output

Estimated states

Fig. 3.10 State variable feedback using an observer to obtain an estimate of the states of the mechanical system.

82

ACTIVE CONTROLOF VIBRATION

The observer equation, (3.9.1), thus becomes ~(t) = A~,(t) + Bu(t) + FCT(y(t) - C~,(t)),

(3.9.6)

where F is a matrix which determines the convergence properties of the algorithm. It is more conventional to write FC T as a single matrix, T, known as the observer gain matrix, in which case this method of estimating the states is known as a Luenberger observer and can be written as ~:(t) = A~,(t) + Bu(t) + T (y(t) - C~(t)).

(3.9.7)

The dynamic behaviour of the observer can now be expressed as a coupled set of first order differential equations, since from equations (3.8.1) and (3.9.7), we can write ~:(t) - ~(t) = Ax(t) - A~, (t) - T (y(t) - C~,(t)).

(3.9.8)

Using the fact that y ( t ) = Cx(t), and defining the error between the true and estimated states as ex(t) = x ( t ) - ~,(t), we can thus write equation (3.9.8) as

6x(t) = [ A - TC]ex(t).

(3.9:9)

If the estimated states are now used for state feedback control, so that u(t) = - G f i ( t ) ,

(3.9.10)

it is clear that both the feedback system and observer will have an inter-related dynamic behaviour. In particular, the dynamics of the control system can be described, using equations (3.8.1), (3.9.10) and the definition of ex, by the equation ~:(t) = [A - BGx]x(t) + BGxex(t).

(3.9.11)

The behaviour of the control system and the observer, equations (3.9.11) and (3.9.9), can now be combined into one matrix equation as [

] A-BGx, ~k(t) = 0 6.x(t)

' BGx x ,' A - TC

x(t)

ex(t)

.

(3.9.12)

The dynamics of the coupled controller and observer are determined by the eigenvalues of the square matrix in equation (3.9.12). These can be deduced using the identity for partitioned determinants given, for example, by Kailath (1980): MllM12 M2~M22 = I M ~ I l I M ~ - M21M1~M~21,

(3.9.13)

in which all of the sub-matrices are square and M~ is non-singular. In this case M2~ = 0, so that the eigenvalues of the coupled system are exactly equal to those of the control system with perfect feedback control, given by the solution to I M,, I = I 2,I - A + BGx [= 0, and those of the observer alone, given by the solution to I M2~[ = I Z ~ I - A + TC I=0. This important property is known as the deterministic separation principle, and implies that the characteristics of the observer can be chosen independently of those of the state feedback control system. It should be noted, however, that feeding back the estimated states, rather than true states, generally results in an increase in control effort, as discussed for example by Inman (1989) and Baz (1992).

FEEDBACK CONTROL

83

One is tempted to design the observer to respond very rapidly to any difference between its own output and that of the mechanical system. Unfortunately, such a strategy makes the state variable estimates very sensitive to any uncorrelated noise in the system, particularly 'sensor' noise at the observed output of the mechanical system. Knowing the statistical properties of the various sources of noise in the mechanical system allows the design of an 'optimal' state estimator which minimises the mean square difference between the state variables of the mechanical system and those of the estimator. Such an optimal state estimator is known as a Kalman filter. Even if the adaptive part of the state estimator can be optimally designed, it is still generally assumed that the dynamics of the estimator perfectly match those of the mechanical system. In other words it is assumed that the A, B, C and D matrices of the mechanical system are known perfectly. For this reason, state variable feedback using a state estimator is generally rather less robust to unmodelled plant dynamics than the direct method, equation (3.8.10), in which as many outputs as states are used.

3.10

Optimal control

In Section 3.7 the values of the feedback gains were chosen to achieve some prescribed change in dynamic properties of the mechanical system. The ultimate aim of feedback control, however, is often to reduce the motion of the mechanical system to the greatest possible extent. The choice of the prescribed change in the dynamic properties is generally motivated by this aim of reduced response. For example, the damping ratio of a system would typically be adjusted by feedback control so that the closed loop system was critically damped. There are more direct methods of designing feedback control systems which achieve the greatest possible reduction in response, and systems designed using these methods are known as optimal control systems (see, for example, Kwakernaak and Sivan, 1972). In optimal control the feedback control system is designed to minimise a cost function or performance index which is proportional to the required measure of the system's response. We assume here that the objective is to reduce the response to the greatest possible extent, in which case the control system is said to act as a regulator (control systems can also be formulated to optimally track some required system output which are then termed servos). It is algebraically very convenient to define a cost function which is quadratically dependent on the response, since this greatly simplifies the optimisation problem. One such cost function appropriate to a regulator would be

J= f~ [yT(t)Qy(t) + uT(t)Ru(t)] dt + yT(t/)Sy(t/),

(3.10.1)

where Q and R are positive-definite symmetric weighting matrices. An analogous cost function appropriate for a servo system can be obtained by replacing y(t) in equation (3.10.1) with ( y o ( t ) - y(t)), where yo(t) is a vector of desired output waveforms. The scalar quantity yT(t)Qy(t) is quadratically dependent on the outputs of the system under control, and uT(t)Ru(t) is quadratically dependent on the control inputs. The purpose of the second term in equation (3.10.1) is to account for the effort being expended by the control system, so that small reductions in the output are not obtained at the expense of physically unreasonable input levels. These two terms are integrated

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ACTIVE CONTROL OF VIBRATION

from time t = 0 to the final time tI in equation (3.10.1) and a terminal condition, yT (tl)Sy (tl) is also generally included to independently weight the output at this time. We assume that the output equation of the system under control can again be written in the reduced form y(t) = Cx(t).

(3.10.2)

The cost function (3.10.1) can then be expressed as J = I~S[xV(t)QlX(t)+ uT(t)Ru(t)] dt + xV(ty)Slx(ty),

(3.10.3)

in which Q~ = CTQC and S~ = CTSC are state variable weighting matrices which are now positive semi-definite, because in general there are more states than outputs. Equation (3.10.3) is the form of cost function generally considered in optimal control. Kwakemaak and Sivan (1972) show that the feedback control system which minimises this cost function for the linear, time-invariant system defined by the equation ~:(t) = Ax(t) + Bu(t)

(3.10.4)

uses state feedback with a time-varying feedback gain matrix G°(t), so that u(t) = G°(t)x(t).

(3.10.5)

The optimal time-varying feedback gain matrix is given by G O(t) = - R -1B Tp (t),

(3.10.6)

where P(t) is the solution of the matrix Riccati equation P(t) = - Q ~ - ATP(t) - P(t)A + P(t)BR-~BP(t).

(3.10.7)

This set of non-linear differential equations must be solved backwards in time with the initial condition P(tl)= S. A number of algorithms can be used to numerically solve the matrix Riccati equation, and these are reviewed, for example, by Meirovitch (1990). It has been noted by Meirovitch et al. (1983), however, that the convergence and stability of such algorithms cannot be guaranteed and the chances of obtaining a convergent solution decrease greatly as the order of the control system increases and as the number of actuators decreases. The feedback controller resulting from minimising equation (3.10.1) is said to be globally optimal. If the final time in the cost function (tI in (3.10.1)) tends to infinity, then the optimal gain matrix becomes time invariant and can be calculated by setting the time differential of P(t) (equation (3.10.7)) to zero. The feedback controller is then

asymptotically optimal. This optimal solution does assume that the state variables are available to be fed back, as in equation (3.10.5). Kwakemaak and Sivan (1972, Section 5.3) also show that if only an incomplete set of measurements, corrupted by white Gaussian noise, are available, the optimal feedback controller still consists of a state feedback controller with gains defined by equation (3.10.6), but in this case estimates of the state variables are used. These can be obtained via a Kalman filter, as described in Section 3.9. Specifically, the state estimates are assumed given by the solution of the equation ~:(t) = A~,(t) + Bu(t) + K°(t) [y(t) - C~(t)],

(3.10.8)

which comprises a model of the system under control (A~,(t)+ Bu(t)), and an update

FEEDBACK CONTROL

85

term proportional to the difference between the true output of the system y(t) and the estimated output S'(t)=C~(t), weighted by the Kalman gain matrix K°(t). This equation for the Kalman filter is clearly similar to that for the Luenberger observer, equation (3.9.7), except that in the observer the gain matrix, T, was not time varying and was determined by prior design. The Kalman gain matrix depends upon the noise levels present in the system. It is assumed that noise is present both within the system itself and also at the system output, so the state equation becomes x(t) = Ax(t) + Bu(t) + v(t),

(3.10.9)

and the output equation becomes y(t) = Cx(t) + w(t),

(3.10.10)

in which v(t) and w(t) are vectors of random signals which are assumed to be white, have a Gaussian probability density function, and are uncorrelated with the input signals u(t). The Kalman gain matrix which leads to the optimal feedback controller is then shown by Kwakemaak and Sivan (1972) to be of the form K°(t) = M (t)C'rW -1,

(3.10.11)

in which M (t) is the solution to another matrix Riccati equation, given by 1~1(t) = V + AM(t) + M(t)A ~ - M(t)CTW-~CM(t),

(3.10.12)

which is solved with the initial condition M ( 0 ) = E[x(0)x~(0)] where E denotes the expectation operator. The correlation properties of the state and observation noise vectors are quantified in equation (3.10.12) by the correlation marices

and

E[v(t)vT(t)] = V,

(3.10.13)

E[w(t)w~(t)] = W.

(3.10.14)

It can be difficult to use the optimal control strategy in practice, since it assumes perfect knowledge of the system under control and of the noise processes, and also requires numerical solution of the two Riccati equations. The power of the approach often comes from simulations of the optimal control method, which provide a benchmark by which to judge other more practical control schemes. The approach also demonstrates that the structure of the optimal control consists of a device to estimate the system states, whose design depends on the noise in the system but not on the cost function being minimised, and a feedback controller, whose design depends on the cost function but not the noise. This is another statement of the separation principle referred to in Section 3.8. These results depend upon the system under control being linear and the cost function (3.10.1) being quadratic. Because we must also assume that the random perturbations in equations (3.10.9) and (3.10.10) are Gaussian, this control philosophy is called Linear, Quadratic and Gaussian, or LQG, control. For the regulator problem discussed above, it is specifically called Linear Quadratic Regulator, or LQR, control. The analogous formulation for discrete-time systems is discussed, for example, by ,~strom and Wittenmark (1984) and Furuta et al. (1988). Porter (1969) has considered the application of optimal control theory in choosing the values of the springs and dampers in the two-degree-of-freedom system introduced

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ACTIVE CONTROL OF VIBRATION

in Section 3.7. He shows that in order to minimise a cost function of the form

J = Io alw(t)2 + fll~i'(t)2 +/~lfP(t)2 + a20(t)2 + f12/)(t)2 +/z20(t)2 dt,

(3.10.15)

then the equations of motion must be uncoupled. Furthermore, expressions for the values of the springs and dampers of the system which optimally minimise equation (3.10.15) are given in terms of the weighting parameters a~, fl~,/Zl, a2, f12,/z2, which could be chosen, for example, according to some subjective acceptability criterion if the two-degree-of-freedom system was a simplified model of a vehicle's suspension. More generally, such techniques could also be used to design the optimal feedback gain matrix which minimises the cost function of equation (3.10.15) for an active vibration control system such as the one illustrated in Fig. 3.9.

3.11

Modal control

Modal control is a term used to describe a wide variety of control techniques. In general, modal control is an approach to control system design in which the state variables are chosen such that the unforced behaviour of each state variable is relatively independent of the behaviour of the other state variables (Porter and Crossley, 1972). As noted by Inman (1989), for example, modal control can be cast either in 'state space' form or 'physical space' form, i.e., in terms of the physical modes of the mechanical system. Because of the difficulties in defining such physical modes for mechanical systems with general viscous damping (see Section 1.11 and Meirovitch, 1990) we choose to work in state space form. In order to understand the approach, however, the 'modes' derived from the state space analysis can be thought of as being similar to those of the equivalent undamped mechanical system, subject to the discussion at the end of Section 3.7. We have seen that the equations of motion of an MDOF mechanical system can be expressed in the state space form i:(0 = Ax(t) + Bu(t),

(3.11.1)

where x(t) is a vector of state variables, usually containing the generalised displacements and velocities associated with the mechanical system. It is shown in the Appendix that the choice of state variables is not unique, and any reversible transformation of x(t) will give another equally valid set of state variables. In particular, we consider the set of variables, z(t), obtained by transforming x(t) with the inverse matrix of eigenvectors of the system matrix A, such that z(t) = E-ix(t).

(3.11.2)

The matrix E is defined by writing the system matrix in the 'normal' form: A =EAE -1,

(3.11.3)

so that E is the matrix of eigenvectors of A and A the diagonal matrix of its eigenvalues. The state equation can now be written in terms of z(t) by substituting x(t) = Ez(t) in equation (3.11.1) to obtain E~(t) = AEz(t) + Bu(t),

(3.11.4)

FEEDBACK CONTROL

87

so that ~(t) - E-1AEz(t) + E-1Bu(t).

(3.11.5)

It can be seen from (3.11.3), however, that E-~AE can be written as A, and so equation (3.11.5) becomes ~(t) = Az(t) + E-~Bu(t).

(3.11.6)

In the absence of any input, u ( t ) = 0, the natural response of each element of z decays independently of all the others, since A is diagonal. The elements of z are known as the modal coordinates of the control system. It is intuitively appealing to consider the dynamics of a mechanical system in terms of its modal response. This motivates the desire to design a control system which does not affect the form of these modes (i.e., does not change the eigenvectors of A) but provides independent control over the natural frequencies and damping of these modes (i.e., allows modification of the eigenvalues of A). For simplicity, we will assume that there are as many measured responses from the system as there are generalised displacements and velocities, and hence modal coordinates, and that D = 0. The output equation may thus be written, using equation (3.11.2), as y(t) = Cx(t) = CEz(t)

(3.11.7)

in which case C, as well as E, is now a square matrix. Assuming that no column of the matrix CE has all zero elements the system is said to be observable (see the Appendix) and if CE can be inverted z (t) can be directly calculated from y (t), z(t) = E-1C-ly(t).

(3.11.8)

Similarly, we assume that there are as many inputs to the system as state variables, so that the matrix B in equation (3.11.1) is square, and that none of the rows of the matrix E-1B in equation (3.11.6) has all zero elements, so that the system is controllable. The matrix E-~B can then be inverted so that each mode can be independently controlled. We assume now a modal feedback controller, as illustrated in Fig. 3.11, of the form u(t) = r(t) - [B-~E]Gzz(t)

(3.11.9)

so that the driving term in the modal response, equation (3.11.6), has the form E-1Bu(t) = E-~Br(t) - Gzz(t),

(3.11.10)

where r(t) is the input vector to the closed loop system. The matrix B-~E in equation (3.11.9) can be called the 'mode synthesiser' and E-~C -~ in equation (3.11.8) the 'mode analyser'. Substituting equation (3.11.10) into equation (3.11.6), the closed loop system equation can be written ~(t) = [ A - Gz]z(t) + E-~Br(t).

(3.11.11)

If G~ is diagonal we can thus achieve our objective of independently changing the eigenvalues of the system, via the diagonal elements of G z, without changing the eigenvectors, since [ A - G~] is also diagonal. Because feedback is performed in the space of modal amplitudes, this strategy is called independent modal-space control, or

88

ACTWE CONTROLOF VIBRATION

Input

>

Mechanical system

°zt:

Mode Modal synthesiser gainmatrix

~~> Output

Mode analyser

Fig. 3.11 Block diagram of a modal feedback controller, in which the modal coordinates z, are directly measured from the output, y, and can be fed back to independently influence the modal amplitudes. IMSC (see Meirovitch, 1990, for example). Although the implementation of the mode analyser and mode synthesiser in Fig. 3.11 would require full matrix multiplications, their design is fixed, and independent of the control objectives. The modal gain matrix, however, could be 'tuned' to give different design objectives. The number of non-zero elements in the diagonal modal gain matrix is equal to the number of state variables, which is in contrast to the full state variable feedback gain matrix (Fig. 3.8) in which the number of non-zero elements is, in general, equal to the number of state variables multiplied by the number of inputs to the system. The discussion above began with the assumption of an MDOF, lumped parameter mechanical system. The dynamics of such a system can be described exactly by an ordinary differential equation, and the system has a finite number of modes. The dynamics of a mechanical system with distributed parameters can only be described using partial differential equations, and such a system has, in principle, an infinite number of modes. Distributed parameter systems can be exactly described using modal equations with infinite-dimensional state vectors; see, for example, Meirovitch (1990). In practice, however, only a finite number of modes significantly contribute to the response, and a finite-dimensional approximation to the modal response of a distributed parameter system can be made arbitrarily accurate by increasing the number of modes included. Concepts of modal control similar to those outlined above can thus be carried over into the active control of distributed parameter systems. In this case, however, often equation (3.11.8) cannot be used to directly calculate the modal amplitudes from a reasonable number of discrete point measurements of the system's response, because of the large number of significantly excited modes. One alternative to this direct modal analyser approach is to use a modal estimator, such as an observer or Kalman filter, to estimate the mode amplitudes by constructing an electronic model of the mechanical system, which is driven by the same inputs as the physical system. It is not, however, possible to accurately estimate large numbers of modal amplitudes from small numbers of measured outputs using an estimator, because of inevitable inaccuracies in the A and B matrices used in the observer. The implementation of modal control on distributed parameter systems thus presents difficulties if discrete sensors are used. An interesting altemative, discussed for example by Burke and Hubbard (1987) and Lee

FEEDBACKCONTROL

89

and Moon (1990), is to design distributed sensors which can be made sensitive to only one mode of the system's motion. These concepts will be further discussed in Chapter 5. Another aspect of modal control which is particularly important for distributed parameter systems is the effect of unmodelled modes. Suppose the vector of modal coordinates is partitioned into those which are accounted for by the feedback control system, Zc(t), and those which are not accounted for by the control system and are termed residual, ZR(t). The equations describing the mechanical system thus contain both these sets of modes and may be written as [Zc(t)]ZR(t)= [Ac0 AR0][Zc(t)]ZR(t)+ [Bc] u(t)'BR

(3.11.12)

y(t) - [CcCR] ZR(t) '

(3.11.13)

where the matrices [B~ B~] T and [Cc CR] are equal to E-~B in equation (3.11.6) and CE in equation (3.11.7), respectively. The output of the system, y, is seen to be affected by at least some of the residual modes via the matrix CR in equation (3.11.13), an effect called observation spillover, and the inputs to the system can excite at least some of the residual modes via the matrix BR in equation (3.11.12), an effect called control spillover (Balas, 1978). We now assume that a modal feedback control system is designed which only takes account of the controlled modes, such that equations (3.11.8) and (3.11.9) can in this case be written as u(t) = r(t) - Bc~GcCc~y(t),

(3.11.14)

where Gc is a diagonal gain matrix. Substituting this feedback law back into equation (3.11.12) and using equation (3.11.13), we obtain the closed loop system equation

r ]:[[: ][ ~,c(t) LzR(t)

0 _ Gc GcCclCR AR BRBc~Gc BRBclGcCc~CR

]]r ]

Zc(t) + LzR(t) B

r(t).

(3.11.15)

Although the diagonal elements of G c would affect the eigenvalues of the controlled modes, A c, in the way expected if only those modes were present, a number of other effects now also occur because of the residual modes. It is clear that the system matrix is now no longer diagonal, which implies that the elements of the vector [zT zT] T no longer represent the modal coordinates of the system, and that the eigenvectors of the closed loop system are generally no longer equal to those before control. As well as changing the eigenvectors of the closed loop system, all the eigenvalues are also changed. In the worst case, the real parts of some of these eigenvalues may be increased to the extent that they are no longer negative, resulting in an unstable closed loop system. It should be noted that both observation spillover (non-zero CR) and control spillover (non-zero BR) must be present to destabilise the system. Both of these effects will inevitably be present when a distributed parameter system is controlled using a modal control system which assumes a finite number of modes. It has been shown, however, that a small amount of damping greatly reduces the possibility of instability due to spillover (Meirovitch et al., 1983).

4 Feedforward Control

4.1

Introduction

In this chapter we will continue to make the assumptions made in the previous chapter: that the system under control is linear and that the secondary actuator is fully active. A secondary disturbance can thus be generated which destructively interferes with that due to the primary source, as a result of superposition. In contrast to the previous chapter, however, we will now begin to relax the assumption that we have no information about the original excitation of the mechanical system, due to the primary source. This allows us to use such information as the basis of a feedforward control approach. There are two important examples of where some prior knowledge of the excitation due to the primary source can be obtained. The first of these is where the disturbance is deterministic. In principle the future behaviour of such a disturbance can be perfectly predicted from its previous behaviour. In practice, a reference signal is usually derived from the primary source of the disturbance, and used to maintain the synchronisation of the secondary excitation. For example, in the case of disturbances caused by reciprocating machines such as internal combustion engines, a tachometer signal related to the crankshaft rotation is often used to generate a reference signal. The second example of where prior knowledge of the primary disturbance is available is when the vibrational disturbance is propagating through a mechanical structure, and a sensor can be used to detect this incident disturbance. We should make clear that this 'detection' sensor is not the same as the 'response' sensor discussed in the previous chapter, which was used to generate the excitation signal for feedback control. Such a response sensor may still be used in feedforward control, but is not used to directly drive the electronic controller in this case. In feedforward control systems the response sensor is only used to monitor the performance of the controller. The frequency response or impulse response of the electrical controller may be adjusted or 'tuned' in response to the output of this sensor in order to make the feedforward control system adaptive. We shall see that feedforward control relies on a delicate balance between the effects of the primary source, and those of the secondary input, and so the amplitude and phase characteristics of the feedforward controller must be adjusted very carefully. This is in contrast to the requirements for the response of the single-channel feedback controller for example, for which, provided the open loop gain is high and the open loop phase shift is not too great, reasonable reductions in the primary disturbance will be achieved with a range of controller responses, as discussed

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in Section 3.2. The need for an adaptive algorithm to adjust the response of the feedforward controller is thus seen to be greater than in the case of a feedback controller. In this chapter we will consider the feedforward control of disturbances which are either deterministic or random. The deterministic disturbances will be considered as being decomposed into their constituent frequency components; the harmonics if the disturbance is periodic. The effect of feedforward control on each of these discrete frequency components can then be considered independently, because the system under control is assumed linear and so superposition applies. Random disturbances represent a more severe problem, as we shall see, because of the requirement that the feedforward controller has a prescribed response at all frequencies in the range of interest, rather than just at a number of discrete frequencies, and so there is an additional constraint of causality for a realisable controller. We begin the discussion with single-channel feedforward systems for the control of harmonic, and then random disturbances, in which the performance limits of such controllers are discussed. The filtered-x LMS algorithm is then introduced for the automatic adjustment of an FIR digital filter used as the controller in a single-channel feedforward control system, and some of the properties of this algorithm are discussed. Adaptive controllers are then described, in the frequency and time domains, for the feedforward control of systems with multiple sensors and actuators. Finally, an interesting interpretation of adaptive feedforward controllers as equivalent fixed feedback controllers is described, which relates to the material presented in Chapter 3.

4.2 Single-channel feedforward control The physical components of a single-channel feedforward control system are illustrated in Fig. 4.1. The difference between this diagram and that for the feedback controller, shown in Fig. 3.1, is that the electrical controller, H, is now driven by an estimate of

Excitation

Primary force

fp

r

~e

Response Mechanical system

Secondary force

Electrical feedfoward controller

fs T

Fig. 4.1 The components of a feedforward control system.

FEEDFORWARDCONTROL

93

the original excitation of the mechanical system due to the primary source x. The signal proportional to the response of the mechanical system e, plays no direct part in the control path, but could be used to adapt the response of the controller, as discussed above. The original excitation is assumed to influence the mechanical system via the primary force, fp, which is related to the original excitation via the primary transmission path P. The net excitation of the mechanical system is again assumed to be proportional to the difference between the primary and secondary forces ( f p - f s ) , and the response of the system is related to this excitation via the response of the mechanical system, G. Figure 4.2 thus shows the equivalent block diagram of the feedforward control system, in which the signals are all represented by their Laplace transforms, and the responses of the various components by their transfer functions. One potentially complicating feature of feedforward control systems, which is often present when the excitation is random and a detection sensor is used to obtain an estimate of the original excitation, is feedback from the secondary input back to the detection sensor. This feedback path is generally due to mechanical disturbances, caused by the secondary force, finding their way back to the detection sensor through the primary path, which in practice is not just the unidirectional transfer function represented in Fig. 4.2. A more complete block diagram which includes this feedback path, and also includes measurement noise signals in the outputs from the detection and response, is given in Fig. 6.9 of Nelson and Elliott (1992). In order to analyse the behaviour and limitations of the feedforward controller, such a feedback path can be lumped into an overall controller response and does not affect the conclusions of the analysis, as discussed, for example, in Section 6.6 of Nelson and Elliott (1992). In practical controller implementations the effect of the feedback path can be removed by having a separate electrical feedback filter in the controller, adjusted to have a response equal to that of the feedback path, whose output is subtracted from the detected excitation signal. In telecommunications such a technique is known as 'echo cancellation' (Sondhi and Berkley, 1980) and similar 'feedback cancellation' architectures have been used in active sound control (Poole et al., 1984) and active vibration control (Elliott and Billet, 1993). It is clear from Fig. 4.2 that the Laplace transform of the response of the mechanical system can be expressed in this case as

E(s) = G ( s ) [ P ( s ) - H(s)]X(s).

Mechanical system

Primary path Excitation X(s)

v

v

P(s)

(4.2.1)

Fp(S)~

G(s)

~esponseE(s)

Fs(s)

H(s) Electrical controller Fig. 4.2 Equivalentblock diagram of a feedforward control system.

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ACTIVE CONTROL OF VIBRATION

In this simple case we have assumed that the response of the mechanical system is due only to the primary excitation and the secondary source. In particular it is assumed that no other uncorrelated signals are present, which would act as noise in the system. The effects of such measurement noise will be considered in the following section, but for now we will examine the consequences of equation (4.2.1) without this complication. In the absence of noise we could, in principle, drive the response of the system to zero using feedforward control, i.e., we could perfectly cancel the response of the system at all times by exactly balancing Fp with F s. The Laplace transform of the response would also be zero under these conditions and from equation (4.2.1) this could clearly be achieved if the controller response exactly matched that of the primary path, i.e., if H ( s ) - P(s) then E ( s ) - 0.

(4.2.2)

Assuming the original disturbance is random, equation (4.2.2) must be satisfied for all values of the complex frequency, s. This means that the magnitude and phase of the feedforward controller's frequency response must exactly match that of the primary path for all frequencies, so that the impulse responses of these systems must also be identical. In principle this task is just a matter of electrical filter design. In practice, a number of problems arise, particularly when the feedforward controller is implemented digitally, as is usually the case when the controller is made adaptive. Electrical filters implemented digitally have an inevitable delay associated with them. This is partly due to the processing time of the digital system but, more importantly, is also due to the phase shift of the analogue low pass filters which are generally used to prevent aliasing, and reconstruct the output waveform (Nelson and Elliott, 1992, Section 6.13). A consequence of this inherent delay is that the initial part of the impulse response of the primary path cannot be modelled by a digital controller. This does not represent too great a problem in the active control of sound, since the propagation of acoustic waves in air is non-dispersive and the propagation speed is relatively low. In structures, however, the wavespeed of compressional disturbances is generally considerably larger than that for acoustic waves in air, and the high frequency components of flexural waves can also propagate very rapidly. The delay associated with a digital implementation of the controller for the feedforward control of random vibration may thus have serious consequences on the extent of the cancellation achieved. If the disturbance is deterministic, however, such causality issues are not a problem because the future behaviour of the disturbance is, in principle, perfectly predictable from its previous behaviour. As an example of the control of such a signal we consider the control of a sinusoidal disturbance of angular frequency too, and choose to work in terms of complex frequency domain variables. The reference signal in this case is taken to be a unit complex sinusoid (X(jtoo)=e j'°ot) s o that the complex response of the mechanical system can be written as

E(jwo) = G(jwo)[P(jwo)- n(jwo)].

(4.2.3)

In order to exactly cancel the response at this frequency we only have to arrange that the amplitude and phase response of the controller at the frequency too are equal to those of the primary path. In terms of the complex responses at too: if H(jwo)= P(jwo) then E(jwo)= 0.

(4.2.4)

This condition, for a single value of too, is obviously a far less stringent condition to

FEEDFORWARDCONTROL

95

meet than that in the case of random excitation (equation (4.2.2)), since the frequency response of the controller only has to match that of the primary path at a single frequency. It should be noted, however, that to attenuate E(joo) by 20 dB for example, then the complex response of the controller, H(j~o), must match that of the primary path, P(jog) to within about +0.6 dB in amplitude and about +4 ° in phase (Angevine, 1992).

4.3

The effect of measurement noise

In order to examine the effects of measurement noise, it is convenient to redraw the block diagram of the feedforward controller in Fig. 4.2, as in Fig. 4.3(a). The transfer functions of the controller and mechanical systems have also been replaced by their frequency responses, and the spectra of the various signals are shown rather than their Laplace transform. In this diagram the notional summing junction has also been moved to the other side of the mechanical system and the subtraction introduced in Fig. 4.2, by the positioning of F, in Fig. 4.1, has been absorbed into G. The spectrum of the output of the response sensor before control is now denoted D(j~o) and this has contributions from the primary source but is also now assumed to be contaminated with uncorrelated contributions from other sources. Similarly the estimate of the primary excitation used to drive the feedforward controller, whose spectrum is X(j~o), is also assumed to be contaminated with measurement noise which is uncorrelated with the disturbance from the primary source. The path from the excitation, X(jog), through the controller, H(jw), to the input of the mechanical system, U(jco), and then through the mechanical system, G(jco), to the response, E(j~o), remains unchanged. It should, however, be noted that with the block diagram in the form of Fig. 4.3(a), any dynamic behaviour of the secondary actuator can also be accounted for by considering it as part of the response of the mechanical system, G(jco), in which case U(j~o) and E(j~o) (a)

Disturbance Controller Excitation

_1

xq~) -I

H(j(~)

Mechanical syste

_1 u(j(~) -I

G(/o~)

E(jog) Net response

(b) Disturbance Controller Filtered excitation R(jog)

H(jog)

~N)

Eqo,) .,)

response

Fig. 4.3 Alternative block diagrams of the feedforward control system.

96

ACTIVE CONTROL OF VIBRATION

can be interpreted as the electrical voltages applied to the secondary actuator and measured at the response sensor, respectively. Using this interpretation it is clear that the block diagram of Fig. 4.3(a) is exactly valid, provided the transducers and physical system are linear, no matter what the mechanical response of the secondary actuator, or its internal mechanical impedance. Figure 4.3(a) can be further redrawn as in Fig. 4.3(b) in which the spectrum of the filtered excitation signal is now used, which is defined to be

R(jo9) = G(ja~)X(j¢o).

(4.3.1)

The block diagram of Fig. 4.3(b) is exactly equivalent to that shown in Fig. 4.3(a), provided the controller and mechanical system are linear and time-invariant, as can be seen by transposing the order of these two elements in Fig. 4.3(a). The spectrum of the net disturbance can now be written, using Fig. 4.3 (b), as

E(jo9) = D ( j ¢ o ) - H(j~o)R(j¢o).

(4.3.2)

Even though measurement noise is present in the excitation and response signals, some degree of feedforward control can generally still be achieved. In order to reduce E(jto) to the greatest possible extent, a compromise has to be drawn in deciding the response of the controller. Perfect cancellation of the primary disturbance could be achieved if the controller were designed as if there were no noise present (equation (4.2.2)), but this would cause amplification of the measurement noise present in the excitation signal. No such amplification would occur if the response of the controller were set to zero, but clearly no cancellation of the primary disturbance would then be achieved either. This compromise in the design of the active controller is similar to that encountered in deriving the optimal estimate of the frequency response which is assumed to relate two observed signals, as discussed for example by Bendat and Piersol (1986). The frequency response of the optimal controller, nopt(jto), which achieves this compromise can be derived using the orthogonality principle (Papoulis, 1981). This states that the mean square error is reduced to the greatest possible extent when the residual error is completely uncorrelated with the signal driving the filter. Referring to Fig. 4.3 (b) we can see that the optimality criterion is satisfied in this case if

E[R* (jto)E (jto) ] = 0,

(4.3.3)

in which E[ ] denotes the expectation operator and * denotes complex conjugation. If the error signal is expanded out using equation (4.3.2), and we note that the expectation operation and multiplication by H(jco) are both linear operations, and thus commute, equation (4.3.3) can be written as

E[R* (j to )D (j to ) ] - Hopt(j to )E[R* (j to )R (j to ) ] = 0,

(4.3.4)

where Hopt(jto) is the value of H(jto) required to satisfy the optimality condition, equation (4.3.3). This can be expressed as Srd( (.O) - Hopt (Jo) )Srr( O.)) = 0,

(4.3.5)

where the cross-spectral density between R(jto) and D(jto) and the power-spectral density of R(jto) are defined as

and

Srd(tO) = E[R* (jto)O(jto) ],

(4.3.6)

Srr((.o) = E[R* (jto)R(jto) ].

(4.3.7)

FEEDFORWARD CONTROL

97

Equation (4.3.5) can be used to derive an expression for the frequency response of the optimal controller in terms of these cross- and power-spectral densities as

Srd(~) Srr(~O)

Hopt(j¢o ) = - ~ .

(4.3.8)

Equation (4.3.8) can be used as the basis for the practical design of the controller, as discussed by Roure (1985) and Nelson and Elliott (1992). We limit the discussion here to an examination of the effect of implementing such a controller on the performance of the active control system. By substituting equation (4.3.8) into (4.3.2), the powerspectral density of the residual response can be expressed as See(O))min =

E[ I O(j~o) - Hopt(jog)R(j(o) [ 2],

(4.3.9)

and using equations (4.3.6) and (4.3.7) this can be written as

ISrd(O) 12 See(aO~n = Saa(oJ) - ~ .

Srr(O0)

(4.3.10)

Using the definition of the filtered excitation signal (equation (4.3.1)), the cross- and power-spectral densities (4.3.6) and (4.3.7), can be expressed as

Sra(O) = G(ja))Sxa(a~)

(4.3.11)

and Srr(O)) =

I G(j~o)[2Sxx(Oj),

(4.3.12)

where S xa(a)) is the cross-spectral density between the observed excitation signal and the observed response of the system in the absence of control, and S=(o9) is the powerspectral density of the observed excitation. Recognising that Saa(~o) is the power-spectral density of the response before control, the fractional change in the power-spectral density of the response can now be written using equations (4.3.11) and (4.3.12), as See((.D)nfm

= 1-

] Sxd((l) ) ]2

.

(4.3.13)

The final term in this expression is the coherence between the observed excitation signal and the primary response signal and equation (4.3.13) provides a very convenient method of establishing the best possible reduction in the response which could be achieved with a linear time-invariant controller (Ross, 1980; Nelson and Elliott, 1992).

4.4 Adaptive digital controllers Although it is possible to derive the frequency response of the optimal feedforward controller using the methods outlined in the previous sections, the problem of designing a practical filter which implements this frequency response still remains. This problem is compounded by the fact that the optimal controller depends upon the statistical properties of the excitation signal, and on the frequency response of the primary path. In practice both the excitation signal and primary path will change slowly with time, and to maintain the delicate balance required for feedforward control, the response of the controller must also change with time. It is relatively easy to change the

98

ACTIVE CONTROL OF VIBRATION

characteristics of a digital filter by adjustment of its coefficients, but it is generally difficult to change the response of a complicated analogue filter. For this reason most practical implementations of active systems for the feedforward control of vibrations use adaptive digital filters. These controllers operate on sampled versions of the signals and are thus often referred to as 'time domain' controllers. This section is concerned with an algorithm which can be used to automatically adjust the coefficients of such a filter to achieve a desired objective and so implement an adaptive digital controller. The properties and implementation of digital filters are described in numerous textbooks (for example, Oppenheim and Shafer, 1975; Rabiner and Gold, 1975; Bozic, 1979; Lynn, 1982; BeUanger, 1984), and have been summarised in Chapter 4 of Nelson and Elliott (1992). We shall concentrate here on the adaptation of digital filters whose outputs are formed from the weighted sum of previous inputs (Widrow and Steams, 1985). Such digital filters have an impulse response which is of finite duration and are known as Finite Impulse Response, or FIR, filters. If the excitation signal derived from the primary source is sampled at a fixed rate to produce the sequence x(n), and this is used as the input signal for an FIR filter which acts as the controller in a feedforward vibration control system, the output sequence of the controller can be written as 1-1

u(n) = Z hi x ( n - i).

(4.4.1)

i=0

In this expression, n denotes the sample number, which may only take integer values, n = . . . - 3 , - 2 , - 1,0, 1,2, 3, etc, and the variables h/denote the filter coefficients which weight the current and previous I - 1 input samples of the signal x(n). For notational convenience we shall use the operator notation described, for example, by Astr6m and Wittenmark (1984), Goodwin and Sin (1984) and Johnson (1988), to indicate the digital filtering operation. In this notation q-l denotes the unit delay operator, which, when it operates on the sequence x(n), transforms it into the same sequence delayed by one sample. This operation is conventionally written as

q-ix(n) = x ( n - 1).

(4.4.2)

The action of the FIR filter can also be represented as an operator, which we can denote as

H(q) = ho + hlq -l + hzq -2 +

...

+

hi_ lq t- l,

(4.4.3)

so that equation (4.4.1) can now be written in the operator form

u(n) = H(q)x(n).

(4.4.4)

In order to use the output sequence defined by equation (4.4.1) to drive the secondary actuator, it must be converted into an analogue voltage using a digital-to-analogue converter, whose waveform must be smoothed using an analogue low pass filter. Similarly, before the net response of the mechanical system can be used to adjust the coefficients of the digital controller, it too must be converted into a sequence, sampled at the same rate, by passing the analogue signal through an analogue low pass filter, to prevent aliasing, and then through an analogue-to-digital converter. The overall signal path from the output sequence of the controller to the sequence representing the net

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100

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many spectral components as there are controller coefficients, so that the control filter is persistently excited (see, for example, Goodwin and Sin, 1984 or Johnson, 1988), this quadratic function has a unique global minimum. A simple gradient descent algorithm is thus guaranteed to converge to the globally optimal solution for this problem. Such an adaptive algorithm can be written as

3J , hi(n + 1) = hi(n) - It ~3hi(n)

(4.4.11)

where It is a convergence coefficient and hi(n) is the ith controller coefficient at the nth sample time. From the definition of the cost function J, given in equation (4.4.10), the derivative in equation (4.4.11) can be written as

3J - 2e(n) 3e(n) 3hi(n) ~~i(n) "

(4.4.12)

From equation (4.4.9) the derivative of e(n) with respect to h~ can also be seen to be simply r ( n - / ) . W e assume that the coefficients of the control filter are only changing slowly so that the fact that hi is in fact also a function of time, hi(n), does not significantly alter equation (4.4.9). The steepest descent algorithm required to adapt the coefficients of the digital controller, given by equation (4.4.11), can thus be written as

hi(n + 1)= hi(n)- ae(n) r ( n - i),

(4.4.13)

in which a = 2It is another convergence coefficient. This algorithm is known as the filtered-x LMS algorithm, since r(n) is obtained by filtering the reference signal x(n) with G(q) in equation (4.4.8). The algorithm was first proposed by Morgan (1980) and independently for feedforward control by Widrow et al. (1981) and for active sound control by Burgess (1981). The convergence properties of the filtered-x LMS algorithm are similar to those of the normal LMS algorithm, whose properties are described in detail by Widrow and Steams (1985), for example. The difference between the two cases is that the convergence behaviour of the filtered-x LMS algorithm is determined by the eigenvalues of the autocorrelation matrix of the filtered reference signal, r(n), rather than those of the reference signal itself, x(n), which determine the behaviour of the normal LMS algorithm. In particular, the maximum convergence coefficient for the filtered-x LMS algorithm is related to the mean square value of the filtered reference signal ~ in the same way as it is to the mean square value of the reference signal in the normal LMS algorithm (Widrow and Steams, 1985, Chapter 6), so that the largest stable value of a is approximately 1

amax

~I'

(4.4.14)

where I is the number of coefficients in the adaptive filter equation (4.4.1). When delays are present in the loop this convergence coefficient must be reduced somewhat and some simulations presented by Elliott and Nelson (1989) suggested that amax-- 7 ( / + d) ,

(4.4.15)

FEEDFORWARD CONTROL

101

may be a more accurate estimate of the maximum convergence coefficient in a system with a pure delay of 6 samples. In practice, a separate digital filter which approximates the true response G(q) must be used to generate the filtered reference signal. The convergence of the filtered-x LMS algorithm has been found to be very robust to differences between the response of this model filter and the true path (Morgan, 1980; Elliott etal., 1987; Boucher etal., 1991). The block diagram of a practical implementation of the single-channel filtered-x LMS algorithm is shown in Fig. 4.5, in which the filtered reference signal is obtained using a digital filter, G(q), whose response is an approximation to the true secondary path, G(q). In Fig. 4.5, H~ (q) is a dummy adaptive filter, driven by r(n) and adapted to minimise e2(n), whose coefficients are copied into the controller, H(q). The variation in the optimum convergence coefficient with the phase difference between the true response G(q) and the model G(q) for a pure tone reference signal is shown in Fig. 4.6, together with the resulting convergence time (from Boucher et al., 1991). It can be seen that for phase errors of less than about +45 ° , the effect on the convergence time is relatively small. An example of the use of the filtered-x LMS algorithm in the active control of broadband flexural disturbances propagating in a beam (Elliott and Billet, 1993), will be described in Chapter 6. As discussed in Chapter 2, the modal density of structures can be relatively high. For broadband excitation the frequency response of the system is likely to contain multiple poles associated with these resonances and this creates difficulties in building an FIR filter estimate of G(q) which is relatively compact (a reasonable number of coefficients) and accurate. In this case it is sometimes more efficient to use Infinite Impulse Response (IIR) filters which can represent such frequency responses in a much more compact form, although stability of the filter is an important issue. Vipperman et al. (1993) have studied the design and use of IIR filters for G(q) in the control of broadband, random disturbances on a beam. The use of IIR filters as the adaptive filter H(q) is far more difficult since stability of the filter is affected at each update point of the LMS algorithm, but has been used for acoustic control by Eriksson et al. (1987) and for structural control by Elliott and Billet (1993).

Reference sequence

v

H(q)

G(q)

)i ^

Hl(q)

G(q)

~ )-F- e(n)

/

~

I Observed error Isequence

/ Fig. 4.5 A block diagram of a practical implementation of the filtered-x LMS algorithm in which the normal LMS algorithm is used to update a slave filter Hi(q) which is driven by the filtered reference sequence r(n) obtained by passing the reference sequence x(n) through an estimate (~(q) of the secondary path G(q). The coefficients of H~(q) are then copied into the FIR control filter H (q).

102

ACTIVE CONTROL OF VIBRATION

0.2 0.18

~

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o

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=>

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E 2500

1

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t 12

>

c 1000

o o

o

E = E

500

~

c--

N

o -80

|

-60

-40

i

-20 0 20 Phase error (deg)

40

60

80

Fig. 4.6 The convergence coefficients (a) required to give the fastest convergence time in a simulation of the filtered-x LMS algorithm with a sinuisoidal reference signal having four samples per cycle with various phase errors between the estimate of the secondary path G(q) and the true secondary path G(q). The three graphs correspond to pure delays in the secondary path G(q) of 4, 8 and 12 samples. Also shown in (b) is the convergence time resulting from the use of these optimum convergence coefficients under the same conditions.

4.5

Multichannel feedforward control

A single-channel active control system is, in principle, able to completely control the vibration in one direction, at a single point on a structure. It is often found, however, that what needs to be controlled is either the vibration in several directions, or the vibration at several points on a structure. Multiple control actuators must then be used to achieve active control and the performance of these actuators is generally sensed by multiple response (or error) sensors. We will leave aside the design and positioning of such actuators and sensors for the moment and concentrate on the feedforward control problem posed by such multi-channel systems.

103

FEEDFORWARD CONTROL

Vector of primary onse signals u

.

Vector of reference signals

Matrix of control filters

Matrix of mechanical paths

Vector f response (error) signals

Fig. 4.7 Block diagram of a multi-channel feedforward control system. The block diagram of a multi-channel feedforward control system is illustrated in Fig. 4.7, which is a generalisation of that shown in Fig. 4.3. Note that multiple excitation signals, x, have also been assumed, which may, for example, be harmonic reference signals at different frequencies, or estimates of independent random excitations. The controller, H, consists of a matrix of electronic filters which drive each of the actuators with the sum of the filtered versions of each of the excitation signals. The response of the mechanical system, G, is also assumed, in the most general case, to be fully coupled, with the output to each actuator, u, affecting every response signal, e. As in the single-channel case, feedback paths may again exist from the controller output, u, back to the excitation signals, x. It is still possible, however, to lump these feedback paths into the response of an effective controller response, without affecting the analysis of the system performance, as described by Nelson and Elliott (1992). In the following sections the behaviour of such a multi-channel feedforward controller will be considered in both the frequency domain and the sampled time domain, and adaptive algorithms for the adjustment of the controller will be discussed.

4.6 Adaptive frequency domain controllers Assume that the set of excitation signals; depicted in the general multi-channel block diagram of Fig. 4.7, consists of a set of sinusoids. These may be the harmonics of a periodic primary disturbance, for example. The spectrum of each of the elements of the response vector, e, will thus contain tonal components at these frequencies. Providing each of the mechanical paths from actuator to sensor is linear, however, and the system is in the steady state, the action of the set of filters in the controller which affect one frequency will have no effect on the response at any other frequency. The analysis of the control problem is considerably simplified in this case because the adjustment of each of the sets of filters affecting each reference frequency can be considered independently. The analysis of the performance of such a multi-channel system thus has to be performed only at a single frequency. It is convenient to adopt complex notation to denote the amplitude and phase of the various signals, and of the frequency responses of the mechanical paths at the reference frequency. We do not need to explicitly include the reference signal in the analysis, however, since by assuming it takes the form of a complex exponential at

104

ACTIVE CONTROLOF VIBRATION

the reference frequency tOo, it disappears from the analysis. The vector of complex response signals may now be expressed as (4.6.1)

e(flOo) = d (flOo) + G (jto0)u (jto0),

where u(jto0) denotes the vector of contributions to each of the actuators at the frequency too. The block diagram of the multbchannel feedforward controller with frequency domain variables is shown in Fig. 4.8. Note that the amplitude and phase of the controller at too (H in Fig. 4.7) have been absorbed into the definition of u(jto0) and it is now the real and imaginary parts of the vector u(jto0) which are the variables which can be adjusted by the controller. The general properties of several different algorithms which could be used to adjust the components of u(jto0) to minimise the sum of the squares of the error signals, e(jto0), have been discussed in Chapter 12 of Nelson and Elliott (1992). In this section we will concentrate on the minimisation of a cost function which is consistent with that used in optimal feedback control, as discussed in the previous chapter, and has particular application to active vibration control. In the remainder of this section we will drop the explicit dependence of the variables on too for notational convenience. Equation (4.6.1) can thus be expressed as e = d + Gu.

(4.6.2)

The cost function we seek to minimise by the adjustment of the real and imaginary parts of the components of u can now be defined as J = eHQe + uHR U,

(4.6.3)

in which the superscript H denotes the Hermitian (conjugate transpose) of the vectors, and Q and R are positive definite, but not necessarily diagonal, Hermitian weighting matrices (so that QH= Q, RH= R) and J is a real scalar. Note the similarity between this cost function and that minimised in optimal feedback control (equation (3.10.1)). The first term in equation (4.6.3) depends on the response of the system under control. The use of the general weighting matrix Q, however, allows particular aspects of the response to be emphasised, such as that corresponding to the sound power radiated by the mechanical system, for example, as described by Elliott and Rex (1992) and in Chapter 8. The second term in equation (4.6.3) depends on the 'effort' expended by the actuators and the weighting matrix R allows the effort of some actuators to be discriminated against more than others, for example. By suitable choice of the matrix R, the effort term could also be made proportional to the mean square excitation of a set of structural modes not detected by the error sensors (Elliott and Rex, 1992).

u

(jco o)~

G

(j~o o)

)~

e q,oo)

Fig. 4.8 Block diagram of the steady state behaviour of the multi-channel feedforward controller at the reference frequency tOo.

FEEDFORWARDCONTROL

105

Substituting equation (4.6.2) into (4.6.3), the cost function can be expressed as J = uH[GHQG + R ]u + uHGHQd + dHQGu + dnQd,

(4.6.4)

which can also be written in the standard Hermitian quadratic form (Nelson and Elliott, 1991, Section A.5) as J = uHAu + uHb + bHu + C,

(4.6.5)

where the definitions of A, b and c are obvious from equation (4.6.4), and it should be noted that the matrix A in equation (4.6.5) is not the same as the state variable system matrix used in Section 3.6. The vector of control variables which minimise the cost function in equation (4.6.3), u0, and the resulting minimum value of J(Jmin) can then be immediately identified as being u0 = A-~b and J m i n = c - bHA-lb (Nelson and Elliott, 1992, Section A.5). In this case the optimal set of actuator signals can be written as Uo = - [GHQG + R ] -1GHQd.

(4.6.6)

The complex Hessian matrix [A = GHQG + R] associated with this cost function is guaranteed to be positive definite. In particular, the assumed positive definiteness of the effort weighting R ensures this condition even if G HQG is ill-conditioned, or is rank deficient, as would be the case if there were fewer response sensors than secondary actuators, for example. The cost function is thus guaranteed to have a unique global minimum value for u = u0 as given above. One of the standard methods of adjusting the control variables, when the cost function is a quadratic function of these variables with a guaranteed global minimum, is the method of steepest descent. Some care needs to be taken in the development of this algorithm for complex variables, but it is shown in Section 12.4 of Nelson and Elliott (1992), that such an algorithm can be expressed in the standard form u(k + 1) - u(k) - a [Au(k) + b],

(4.6.7)

in which u(k) denotes the vector of control variables at the kth iteration, A and b are the terms defined by equation (4.6.5) and a is a convergence coefficient. Substituting the expressions for these terms (deduced from equation (4.6.4)) into equation (4.6.7) gives the steepest descent algorithm which minimises the cost function defined by equation (4.6.3). This adaptive algorithm can then be written, using equation (4.6.2), as u(k + 1)= [ I - a R ] u ( k ) - aGHQe(k),

(4.6.8)

where e(k) is the vector of complex response signals measured in the steady state after the application of the control variables u(k). The convergence behaviour of a gradient descent algorithm such as this is described, for example, by Widrow and Steams (1985) and Nelson and Elliott (1992). In particular, the convergence behaviour of the cost function can be described in terms of the decay of a number of independent 'modes' of convergence, leaving a residual level which is equal to the exact least squares solution, J m i n = c - b HA-~b. The decay rates of these modes of convergence are determined by the eigenvalues of the matrix [GHQG + R l, and the level to which they are initially excited depends upon the primary disturbance vector, d. Figure 4.9 shows an example of the overall convergence measured for a 32-sensor, 16-actuator control system operating at 88 Hz (Elliott et al., 1992) together with the calculated decay curves of the individual modes

106

ACTIVE CONTROL OF VIBRATION

-10

II

-20 A

1:13 "o

--~ -30 ID

._1

-40 -50 _60

1H, ! 1i I , 0 2

\

,

4

, 6

k,, 8

ll0

, 12

, 14

, -.. , 16 18

I 20

Sample number (thousands)

Fig. 4.9 The convergence of the sum of the squared outputs of the 32 error microphones predicted from equation (4.6.8), - - -, together with the convergence of each of the individual modes of the control system. of this control system. One problem with such an algorithm may be the slow convergence of control 'modes' associated with small eigenvalues of the matrix [GHQG + R ]. It should be noted, however, that the effect of the effort weighting term in the cost function is to increase the value of these small eigenvalues and so reduce this problem (Elliott et al., 1992). In addition, the effort weighting term also makes the algorithm more robust to errors in the measurement of the matrix of responses of the mechanical system at the reference signal, G, which are used in equation (4.6.8) (Boucher et al., 1991). Figure 4.10, for example, shows the measured convergence of a 32-sensor, 16actuator acoustic control system at a frequency of 88 Hz, in which the estimate of the transfer responses from each actuator to each sensor was corrupted by random errors (Elliott et al., 1992). With no effort weighting, the sum of the squared sensor outputs begins to rise after about 15 000 samples due to the presence of slow unstable modes. With a small effort weighting (dashed line), however, the control is stabilised, but achieves a lower level of control (22 dB) than that achieved with a control system with an exact model of the transfer responses (33 dB). To further reduce the problems associated with the slow 'modes' of the steepest descent algorithm, other algorithms may be used which 'rotate' the direction of the gradient algorithm. One such algorithm uses the Gauss-Newton method which, in general, can be written (Nelson and Elliott, 1992, Section 12.5) as u(k + 1) = u(k) - aA-1 [Au(k) + b].

(4.6.9)

in which A and b are again defined by equation (4.6.5). In the case being considered here, equation (4.6.8) reduces to u(k + 1) = u ( k ) - a[GHQG + R]-~ [GHQe(k) + Ru(k)].

(4.6.10)

FEEDFORWARD CONTROL

107

-5

-10 v

-~-15

-20 -25 0

,

,

10

20

,

,

I

l

30 40 50 60 Sample number (thousands)

,

70

80

Fig. 4.10 An example of the behaviour of the control system when the estimate of the transfer response (from each secondary source to each microphone) used in the update equation is corrupted by random errors with a variance which is approximately equal to the real and imaginary parts of the true response. The solid curve is with no effort weighting; the dashed curve is with a small effort weighting.

4.7

Adaptive time domain controllers

Instead of assuming that the set of reference signals (x in Fig. 4.7) are continuous-time sinusoids, as in the previous section, we now assume that they are sampled sequences. It may be that these sequences still represent sinusoids at the excitation frequencies of the primary source, but more generally they could also now represent sampled estimates of a number of random primary excitations. In this section, we will consider the adjustment of the coefficients of an array of digital FIR filters whose inputs are these K reference sequences, xk(n), and whose outputs Um(rl), drive the M secondary actuators. We denote the ith coefficient of the filter driving the mth actuator from the kth excitation signal as h mki, SO that the output of this filter can be generally represented as K

1-1

Um(n)=ZZhmkiXk(rl--i),

(4.7.1)

k=l i=0

which may be regarded as a generalisation of equation (4.4.1), such that there are now MK control filters which each have I coefficients. Again using the operator notation introduced in Section 4.4, equation (4.7.1) can be rewritten as K

Um(n) = Z Hmk(q)xk(n), k=l

(4.7.2)

108

ACTIVE

CONTROL

OF VIBRATION

in which the operator Hmk(q) is defined to be I-1

Hm~(q)= Z hmkiq-i.

(4.7.3)

i=O

Again adopting the philosophy developed in the single-channel case, we assume that the overall response, including the analogue filters, data converters, actuator response and mechanical system, in the path from the mth output of the controller to the /th sampled error (response) signal, is represented by a fixed digital filter whose response is denoted a s G tin(q). The lth error sequence can thus be written as M

et(n) = dr(n) + Z Gtm(q)um(n),

(4.7.4)

m=l

where dr(n) is the lth error sequence in the absence of control. Using equation (4.7.2) this error sequence can also be written explicitly in terms of the controller response as M

K

et(n) = dr(n)+ Z Z Gtm(q)Hmkt(q)xk(n).

(4.7.5)

m=lk=l

This can in turn be expressed as M

K

et(n) = dr(n) + Z Z H,~(q)rtmk(n),

(4.7.6)

m=lk=l

where the filtered reference signals are now defined to be

rtm~(n) = Gtm(q)xk(n ).

(4.7.7)

The expression for the lth error signal can now be explicitly expressed in terms of the coefficients of the filters in the digital controller, using equation (4.7.6), as M

K l-1

el(n)- dr(n) + Z Z Z h,~i rt,,~(n- i),

(4.7.8)

m = l k = l i=O

which can be written in vector notation as

et(n) = di(n) + r~(n)h,

(4.7.9)

where rt(n) = [rlll(n ) rill(n- 1)... rlMx(n-- I+ 1)] w and •

]T.

h = [hi10 h111 .. hMKI_ 1

Defining the vectors of all the error signals, and primary signals as e ( n ) = [el(n) e2(n).., eL(n)] T, d ( n ) = [d~(n) d2(n).., dL(n)] ~,

(4.7.10)

we can now express the steady state sampled response of the multi-channel

FEEDFORWARDCONTROL

109

feedforward control problem in the matrix form (Elliott et al., 1987) as e(n) = d(n) + T (n)h,

(4.7.11)

where T ( n ) = [rl(n) r2(n) ... rL(n)] T. Returning to equation (4.7.1), we can also express this sequence feeding the mth actuator as T(n)h, Um(Fl)'- X m

(4.7.12)

where xT= [0 0... xl(n) x l ( n - 1)... 0 ... x2(r/) x2(r/- 1) ... 0], so that the vector of signals driving the actuators can be written as (Elliott and Nelson, 1988) u(n) = X(n)h,

(4.7.13)

where X ( n ) = [xl(r/) x2(r/) ... XM(r/)]T. We now define a generalised cost function (Elliott, 1993), similar to that used in optimal feedback control theory, and in the previous section, which includes both error and 'effort' terms, as

J= E[eH(n)Qe(n) + un(n)Ru(n)],

(4.7.14)

in which the superscript H denotes the Hermitian (complex conjugate transpose) and E denotes an expectation operator. The Hermitian transpose is retained here to allow for the possibility that the sampled signals may be complex, and could represent transformed variables, for example. Q is an error weighting matrix, which is Hermitian and positive definite but not necessarily diagonal, and R is an effort weighting matrix which is also Hermitian and positive definite but not necessarily diagonal. Using the equations for e(n) and u(n) above, this cost function can be expressed in the complex quadratic form J = hnAh + hUb + bnh + c,

(4.7.15)

in which A = E[TH(n)QT (n) + XH(n)RX(n)],

b=E[TH(n)Qd(n)], and

c = E[dH(n)Qd(n) ].

This equation has a unique global minimum, assuming A is positive definite, for a set of control filter coefficients given by hopt = - A - l b ,

(4.7.16)

which result in the least squares value of the cost function Jmin

-" c -

bHA - lb.

(4.7.17)

110

ACTIVE CONTROLOF VIBRATION

The vector of derivatives of the real and imaginary components of the vector of control filter coefficients, hR and hi, can be written as (Haykin, 1987; Nelson and Elliott, 1992) =

~J

g ~hR

+j

aJ

-~i

= 2[Ah + b]

(4.7.18)

which, in this case, can be written as g = 2E[TU(n)Qe(n) + XU(n)Ru(n)].

(4.7.19)

In practice only an approximation to each of the paths from secondary source to error sensor can be measured and used to generate the practically implemented filtered reference signals, the matrix of which may be denoted T (n). Using the instantaneous estimate of g, with 1" (n), to update all the control filter coefficients at every sample, yields the algorithm (Elliott, 1993): h(n + 1) = h(n) - a[qf"(n)Qe(n) + X"(n)Ru(n)].

(4.7.20)

If all the error signals are equally weighted (Q = I) and no effort term is used (R =0), this algorithm reduces to the Multiple Error LMS algorithm (Elliott and Nelson, 1985). In this simplified case, equation (4.7.20) can be written in terms of the adaptation of the individual coefficients of the controller as L

hmki(rt +

1) =

hmki(n) - a

(4.7.21)

~ ftmk(n)el(n - i), 1=1

where f~mkis the filtered reference signal obtained by passing the reference signal xk (n) through an estimate of the path from the mth actuator to the lth error sensor, (~ lm(q)" The algorithm with diagonal weighting matrices for error and effort has also been discussed by Elliott et al. (1987, 1992). The Multiple Error LMS algorithm has been used by Jenkins et al. (1993) for the feedforward control of the harmonic signals fed to four active mounts, to minimise the sum of the squared signals from eight accelerometers on the receiving structure. It has also been used by Fuller et al. (1990a) to minimise the sum of the squared harmonic outputs from two piezoelectric sensors on a beam in a study of the simultaneous control of flexural and extensional waves, and in investigations by Fuller et al. (1989a) and Thomas et al. (1990) of active control of sound transmission through panels. A convergence analysis of this algorithm can be performed in a similar manner to that generally used for the LMS algorithm (Widrow and Steams, 1985). One difference in this case is that the algorithm, if stable, is found to converge to the solution (4.7.22)

hoo= - E [ T ( n ) Q T ( n ) + X H ( n ) R X ( n ) ] - ~ E [ t H ( n ) Q d ( n ) ] ,

which is not, in general, equal to the optimal solution, hop t above, since t (n)¢ T (n). Using this expression for h=, substituting for e ( n ) = d ( n ) + T (n)h(n), and making the usual assumption that the filter weight vector is statistically independent of the reference signals, the update equation can be written as E[h(n + 1) - hoo] = [I - aE[TH(n)QT (n) + XH(n)RX(n)]]E[h(n) - ho.],

(4.7.23)

the convergence of which depends on whether the real parts of the eigenvalues of the generalised autocorrelation matrix, E[I"H(n)QT (n) + XH(n)RX(n) ], are positive. Note ,

FEEDFORWARD CONTROL

111

that the eigenvalues of ~'H(n)QT(n) are, in general, complex since l"(n) is not necessarily equal to T(n), and the real parts of these eigenvalues are also not guaranteed positive (as they would be in the normal LMS analysis) for the same reason. The effort term in this expression, XH(n)RX(n), is guaranteed to be positive definite, however (assuming the control filters are persistently excited), and thus will have positive real eigenvalues which can have the effect of stabilising an otherwise unstable system (Elliott et al., 1992).

4.8 Equivalent feedback controller interpretation When the reference signal is a sinusoid, the adaptive feedforward controller has an interesting interpretation as an equivalent fixed feedback control system. This interpretation follows from an analysis of electrical adaptive cancellers presented by Glover (1977), which was originally used to analyse the behaviour of time domain digital feedforward controllers by Elliott et al. (1987) and Darlington (1987). A similar approach has also been used to analyse both analogue and digital time domain and frequency domain feedforward controllers by Sievers and von Flotow (1992), and by Morgan and Sanford (1992), who also present an interesting explanation of the analysis in terms of the manipulation of block diagrams. In this section we present the results of such an analysis for a single-channel digital feedforward controller which employs the filtered reference LMS algorithm, as discussed in Section 4.4. The block diagram of such a controller is shown in Fig. 4.11 (a) in which the reference signal is a synchronously sampled sinusoid so that x ( n ) = cos o90n,

(4.8.1)

where o90 = 2 e r / N and N is the number of samples per period. The mechanical system being controlled is represented by G, and the adaptive feedforward controller has coefficients updated according to equation (4.4.13). By taking the single-channel case of the result derived by Elliott et al. (1987), it can be shown that the relationship between the z transform of the control filter output, U ( z ) , and that of the error signal, E ( z ) for this algorithm, is given exactly by the response of a linear time-invariant system whose z transform is H(z) = U(z) _ E(z)

aAJ

2

[

]

z cos(og0 - q~) - cos q~

1 - 2z cos o90+ z 2

(4.8.2)

"

In this expression a is the convergence coefficient of the adaptive algorithm, the estimated response of the mechanical system at the frequency of the reference signal is given by (-~(Jog0) = A e j~, and I is the number of coefficients in the control filter. The implication of this analysis is that the behaviour of the adaptive feedforward controller illustrated in Fig. 4.11 (a) is exactly the same as the fixed feedback controller shown in Fig. 4.11 (b), in which H represents the system whose z transform is given by equation (4.8.2). By way of illustration, the modulus of the frequency response of this equivalent linear feedback controller is shown in Fig. 4.12(a), for the case in which too = er/2. Notice that the response of this equivalent linear system tends to infinity at the frequency of the reference signal. As the convergence coefficient of the adaptive

112

ACTIVE CONTROLOF VIBRATION

x(n)

t "~1 hi(n)

u(n)

/

_~n)

G

._e(n)

(a)

~e(n)

u(n)

(b) Fig. 4.11 Block diagram of an adaptive feedforward controller (a) and the equivalent fixed feedback control system (b).

I

I

(a)

IGc(j )l

(b)

1

v

o~o

~o

to

Fig. 4.12 The modulus of the frequency response of (a) the equivalent fixed feedback controller (equation 4.8.2) and (b) the net closed loop response obtained from equation (4.8.3). feedforward algorithm increases, and hence its speed of response becomes more rapid, the bandwidth of the significant response in Fig. 4.12(b) is increased and a wider range of frequencies near to the reference frequency is affected by the controller. The overall frequency response of the control system, from original disturbances to residual error, is given, from the closed loop response of the block diagram in Fig. 4.11 (b), as

E(jto) 1 = • Gc(j~) = ~ O(jto) 1 - G(ja~)H(jog)

(4.8.3)

FEEDFORWARD CONTROL

113

For the case in which the response of the mechanical system corresponds only to a pure delay, of duration equal to one sample of the reference frequency, the modulus of the closed loop frequency response is shown in Fig. 4.12 (b). When the frequency of the disturbance corresponds exactly to the reference frequency, ~o0, the overall closed loop response, equation (4.8.3), is zero, because the response of the equivalent feedback controller, equation (4.8.2), tends to infinity. For a sinusoidal disturbance with a frequency which is far removed from ~o0, the residual signal is approximately equal to the disturbance, since the closed loop response is unity at these frequencies for which the equivalent feedback controller has a very small response. The closed loop response of the control system away from the reference signal may have a magnitude which becomes significantly greater than unity, however, if the mechanical system has resonances in this frequency region. In extreme cases this can lead to instability of the controller at these frequencies, as described by Morgan and Sanford (1992). The equivalent linear system approach has been used to analyse the variation of the maximum convergence coefficient with delay in the secondary path by Elliott et al. (1987) and Morgan and Sanford (1992). The effect of errors in the model of the secondary path has been investigated using the equivalent linear feedback system by Boucher et al. (1991) and Darlington (1991). Darlington has shown that the relative heights of the two peaks in the frequency response of the system, on either side of the reference frequency, depend on the phase error of the secondary path model, and suggests that this asymmetry could be exploited as a diagnostic tool to detect such phase errors. Further parallels between the behaviour of harmonic adaptive feedforward and linear feedback systems have been discussed by Sievers and von Flotow (1992), who point out that a similar technique has been used to analyse algorithms for the higher harmonic control of helicopter vibration, by Hall and Wereky (1989). Because of the requirement for an infinite gain at the reference frequency in the feedback controller, the direct feedback control method illustrated in Fig. 4.11 (b) is probably not a practical method of implementing a narrow-band active control system. Its equivalence to the adaptive feedforward controller of Fig. 4.11 (a) can, however, provide a useful way of analysing the feedforward case, and may allow the extensive array of analytical tools developed for feedback control to be brought to bear on the feedforward control problem.

5 Distributed Transducers for Active Control of Vibration

5.1

Introduction

An important element of any practical control system are the transducers used for implementation of the control. Sensors are needed for measurements which can be used to estimate important disturbance and system variables. Actuators are used to apply control signals to the system in order to change the system response in the required manner. In general sensors provide information to the controller to determine the performance of the control system or to provide control signals related to the system response. Thus sensors and actuators provide the link between the controller and the physical system to be controlled and their design and implementation is of prime importance. In general, control transducers come in three main forms; point transducers, arrays of point transducers or continuously distributed transducers. Each particular format has its advantages and disadvantages and these are outlined in this chapter. Types of actuators range from point force actuators, electrostrictive and piezoelectric distributed actuators, to those based on the materials of shape memory alloy (SMA) and magnetostrictive systems. The choice of actuator is dependent upon system requirements such as required control authority (amount of control force, moment, strain or displacement, etc.), power consumption, frequency response, and physical constraints such as size and mounting requirements, etc. Sensors also range from conventional transducers such as accelerometers, strain gauges and proximity detectors to the systems based on piezoelectric material, optical fibres and shape memory alloy as well as advanced non-contacting sensors such as laser vibrometers. Choice of the particular sensor configuration is dependent upon the system variable to the observed, and to some degree, the form of signal processing to be used. The following sections are a brief discussion of the basic theory and issues regarding the implementation of distributed actuators and sensors commonly encountered in active vibration control systems. Electrodynamic shakers can be readily modelled as point force inputs and the response of structures to these types of forcing functions have been outlined in Chapters 1 and 2. For more information on these devices the reader is referred to the text of Cremer and Heckl (1988) and the handbook by Broch (1984). This chapter is mainly concerned with the relatively new theory of distributed piezoelectric actuators and sensors. The analysis presented here is limited to a static approach where inertial effects associated with the actuator itself are ignored. However,

116

ACTIVE CONTROL OF VIBRATION

the static approach is shown to be satisfactory in many cases and more recent work on advanced theories which include material dynamic terms, actuator-structure coupling, impedance effects etc. as well as the dynamics of the associated electrical circuitry, is briefly reviewed. Some brief discussion is also given of actuators and sensors based on advanced new materials such as shape memory alloy. It should be noted that although the theory is derived in terms of piezoelectric material, it is generally applicable to all distributed strain-inducing actuators and sensors.

5.2

Piezoelectric material and definitions

The piezoelectric effect was first discovered in 1880 by Pierre and Jacques Curie who demonstrated that when a stress field was applied to certain crystalline materials, an electrical charge was produced on the material surface. It was subsequently demonstrated that the converse effect is also true; when an electric field is applied to a piezoelectric material it changes its shape and size. This effect was found to be due to the electrical dipoles of the material spontaneously aligning in the electrical field. Due to the internal stiffness of the material, piezoelectric elements were also found to generate relatively large forces when their natural expansion was constrained. This observation ultimately has led to their use as actuators in many applications. Likewise if electrodes were attached to the material then the charge generated by straining the material could be collected and measured. Thus piezoelectric materials can also be used as sensors to measure structural motion by directly attaching them to the structure. Most contemporary applications of piezoelectricity use polycrystalline ceramics instead of naturally occurring piezoelectric crystals. The ceramic materials afford a number of advantages; they are hard, dense and can be manufactured to almost any shape or size. Most importantly the electrical properties of the ceramics can be precisely oriented relative to their geometry by poling the material as described below. The relationship between the applied forces and resultant responses of piezoelectric material depend upon a number of parameters such as the piezoelectric properties of the material, its size and shape and the direction in which forces or electrical fields are applied relative to the material axis. Figure 5.1 shows an element of piezoelectric material. Three axes are used to identify directions in the piezoelectric element termed 1, 2 and 3 in respective correspondence with the x, y and z axes of rectangular coordinates. These axes are set during the poling process, which induces the piezoelectric properties of the material by applying a large d.c. voltage to the element for an extended period. The z axis is taken parallel to the direction of polarisation and this is represented in Fig. 5.1 by the vector p pointing by convention from the positive to negative poling electrode (shown in the figure) or in the negative z direction. Piezoelectric coefficients, usually written in a form with double subscripts, provide the relationship between electrical and mechanical quantities. The first subscript gives the direction of the electrical field associated with the voltage applied or the charge produced. The second subscript gives the direction of the mechanical strain of the material. Several piezoelectric constants are used and the interested reader is referred to the IEEE Standards on Piezoelectricity for a full definition of these (IEEE, 1988). Anderson (1989) has also provided a good introduction to piezoelectric materials and their associated definitions.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

117

3(z) E,ectro

+1 V

Applied field, E

I

P

,,,,,--

e

#2(y)

l(x) Fig. 5.1

Piezoelectric element and notation.

The constitutive equations for a linear piezoelectric material when the applied electric field and the generated stress are not large can be written as (Uchino, 1994) ~i

E

= Sijcr~ + dmiEm,

(5.2.1a)

D m = dmio i + ~ikEk,

(5.2.1b)

where the indices i, j = 1 , 2 , . . . , 6 and m, k = 1, 2, 3 refer to different directions within the material coordinate system. In equations (5.2.1a) and (5.2.1b) e, o, D and E are respectively the strain, stress, electrical displacement (charge per unit area) and the electrical field (volts per unit length). In addition S e, d and ~ are the elastic compliance (the inverse of elastic modulus), the piezoelectric strain constant and the permittivity of the material respectively. The piezoelectric strain constant d is defined to be the ratio of developed free strain to the applied electric field E. In particular the strain constants d33, d31 and d32 are of major interest. The subscript 33 implies that the voltage is applied or charge is collected in the 3 direction for a displacement or force in the same direction. The subscript 31 implies that the voltage is applied or charge is collected in the 3 direction while the displacement or force occurs in the 1 direction. For much of the following analyses, while values of d32 and d31 are often significantly different in real materials, it is assumed that the piezoelectric material behaves identically in the 1 and 2 directions. The d32 and d31 constants are related to d33 by a Poisson effect with a negative sign. The above relations also imply the use of electrodes to apply or collect the electrical field and for the type of motion discussed here, these are shown as a shaded region on the top and bottom of the element depicted in Fig. 5.1. The strain of the piezoelectric element in the z direction can be simplified, for onedimensional motion, in the absence of an applied stress, to the relation l~.p e3

=

d3 3V~ h a,

(5.2.2)

where V is the applied voltage (note E V/ha) and h~ is the element thickness in the z direction. For the following discussion we restrict the voltage V to relatively small values so that the piezoelectric material can be considered relatively linear. The subscript pe now denotes strain of the piezoelectric element. For the same applied field =

118

ACTIVE CONTROLOF VIBRATION

the actuator will also deflect in the x and y directions and the resultant strains are in the x direction 1 = d31 V / h a Epe

(5.2.3)

2 = d32 V/ha. Epe

(5.2.4)

and in the y direction

By convention when a field (relatively small in value compared to the poling field) is applied to the piezoelectric element in the same direction as the poling vector as shown in Fig. 5.1, the element will expand in the z direction. At the same time, due to Poisson coupling, the element will contract in the x and y directions. Thus the d33 constant is typically specified as a positive value while the d31 and d32 are negative for piezoelectric ceramics. The above simplified definitions are sufficient to perform the analyses required in this chapter. For a more detailed description of piezoelectric terminology and behaviour the reader is referred to the text of Moulson and Herbert (1990), the work by Anderson (1989) and the IEEE Standards on Piezoelectricity (IEEE, 1988).

5.3

Piezoelectric stack actuators

The following analysis of the actuation of elastic structures is based upon what is known as the static approach. The static response of an interaction between a piezoelectric element and a structure is first determined by coupling the constitutive relations of the piezoelectric element and structure with their equilibrium and compatability equations. Once the equivalent static force or moment due to the actuator is obtained it is then used as a frequency-independent amplitude for a harmonically varying input to the system. This approximate approach has been found to provide reasonable results for relatively lightweight piezoelectric elements driven well below their internal resonance frequency. Most importantly, the static approach includes the distributed forcing function effects of the piezoelectric elements which will be shown to be a very important attribute for selective control of the states of the structural system. The first configuration of piezoelectric material we consider is the stack arrangement shown in Fig. 5.2 which is working against an applied external force F and an external stiffness represented by a spring. A stack is defined to be a single or multi-layered piezoelectric element which is relatively long in the z direction. This configuration is intended to induce motion in the 3 direction by applying voltages over electrodes at the top and bottom of the element. Two configurations of the actuator are shown. In Fig. 5.2(a) the actuator is working against an external spring stiffness arranged in parallel with the actuator while in Fig. 5.2 (b) the stiffness is positioned in series. In both cases for zero voltage the extemal spring is in equilibrium and applies no stiffness force. Note that the actuator also has an internal stiffness associated with its material Young's modulus of elasticity. The objective is to find the resultant displacement of the actuator and thus the effective stiffness when a voltage is applied to the actuator. The following static analysis of the parallel configuration of Fig. 5.2(a), although simple, does illustrate the basic technique for solving for coupled response of piezoelectric-structural systems. The

DISTRIBUTEDTRANSDUCERSFORACTIVECONTROLOFVIBRATION

119

F

tW

l

:t W •:_ ":_ .:•:- .:_ .:•:_ .:- .:. ,.- ..- .":-':-':- l .:- .:- .." !i- .'.":-:.":'. La

K

Piezoelectric ~ stack actuator

External .l--e" spring

~i~j

load

///////

// (a)

(b)

Fig. 5.2 Piezoelectric stack actuator working against an external stiffness load in (a) parallel, (b) series with the load.

piezoelectric material is of area A a, length La and is assumed to have a Young's elastic modulus o f E~. (E~ is assumed to be the ' short-circuited' modulus.) The unconstrained strain (i.e. when no external resisting stiffness is present) of the actuator in the z (or 3) direction is given by

F_,pe:

(5.3.1)

d33 V/La

and thus the unconstrained displacement of the actuator is

(5.3.2)

W a = d33V.

When a stiffness resists the motion of the actuator to w as illustrated in Fig. 5.2(a) then the internal force F a that the actuator exerts in the positive z direction is related to the constrained motion of the actuator by Ea(w a - w)

F~ =

(5.3.3)

Aa. L~

Applying a force balance to the arrangement of Fig. 5.2(a) gives an expression for the external force F in terms of the actuator and external spring constants and their deflections. This is given by F = Kw -

E~(w~-w) L~

Ao,

(5.3.4)

where K is the constant of the external spring. Solving this equation for the displacement of the actuator gives d33V + F/K~ w =

,

1 + K/Ka

(5.3.5)

120

ACTIVE CONTROL OF VIBRATION

where K~ is the actuator equivalent spring constant given by K~ = E~Aa/L a. Setting the applied external force F to zero results in an expression for the deflection when a voltage is applied given by w=

d33V

(5.3.6)

1 + X/X,, and the actuator force is then given by F~ =

d33VK

.

(5.3.7)

1 + K/K~ When the actuator displacement is constrained to zero, an important quantity called the blocked force of the actuator is given by F~

.

=

(5.3.8)

L~

These relationships demonstrate some fundamental aspects of coupled piezoelectric behaviour. Increasing the equivalent stiffness of the actuator relative to the system stiffness that the actuator is working upon increases the displacement of the actuator. However, increasing the stiffness of the piezoelectric material does not have such a significant effect on the maximum force exerted; the maximum force will be exerted when the actuator is working against a very stiff material. Although the above simple analysis is for a static situation it does illustrate an important point regarding piezoelectric actuators; the best configuration of actuator will depend upon the impedance of the system to be driven. The above analysis could be extended to a dynamic formulation by including material inertial effects. In this case different conclusions will be drawn, particularly when the coupled system is being excited near resonance. The coupled displacement of the series configuration of Fig. 5.2(b) can be found by using a similar procedure as outlined above. Use of piezoelectric stack actuators in a series configuration has been reported by Scribner et al. (1993) and the reader is referred to this reference for more details of their implementation.

5.4 Piezoelectric one-dimensional asymmetric wafer actuators The other common form of arrangement of a piezoelectric actuator is the asymmetric, wafer configuration shown in Fig. 5.3. In this arrangement the actuator is bonded to the surface of the structure and when a voltage is applied across the electrodes (in the direction of polarisation) the actuator induces surface strains to the beam through the d31 and d32 mode of the piezoelectric material response. We term this configuration a wafer arrangement since the piezoelectric element is very long (in the x and y directions) compared to thickness (in the z direction) through which it is polarised. For the 1-D analysis we shall follow the early work of Bailey and Hubbard (1985), Fanson and Chen (1986) and Crawley and de Luis (1987). Important assumptions are that the beam is covered by a layer of thin piezoelectric material of thickness, h~ which is perfectly bonded to the beam and strains only in the x direction. The following

121

DISTRIBUTED T R A N S D U C E R S FOR ACTIVE C O N T R O L OF VIBRATION

Z

,._

Piezoelectric actuator ~ Beam

Fig. 5.3 Piezoelectric asymmetric wafer configuration and associated strain distribution. derivation is an approximation using a static approach; inertial effects of the piezoelectric element are ignored, which is valid if the element is thin and lightweight compared to the beam system. In connection with this approximation, work by Pan et al. (1992a) has shown little difference in behaviour of a dynamic and static model for the geometries and frequencies of interest studied here. When a voltage is applied across the unconstrained (i.e. not attached) piezoelectric element the actuator will strain by an amount e, pe in the direction 1 which is parallel with the x axis as dictated by

d31V

~'pe = ~ ,

ha

(5.4.1)

where V is the applied voltage in the direction of polarisation, ha is the actuator thickness and d31 is the piezoelectric material strain constant When a voltage is applied across the bonded piezoelectric element it will attempt to expand but will be constrained somewhat due to the stiffness of the beam. Due to the symmetric nature of the load the beam will both bend and stretch, leading to an asymmetric strain distribution as shown in Fig. 5.3, where the origin of the z axis lies on the centre of the beam. The method of analysis outlined here follows previous work by Gibbs and Fuller (1992a). We assume that the strain distribution is linear as a result of Kirchoff's hypothesis of laminate plate theory (Jones, 1975) and thus can be written e(z) = Cz + e0,

(5.4.2)

where C is the slope and e0 is the z intercept. Equation (5.4.2) can be decomposed into the sum of an antisymmetric distribution Cz (i.e. flexural component) about the centre of the beam and a uniform strain distribution e0 (i.e. longitudinal component) as shown in Fig. 5.4. Using the strain distribution of Fig. 5.3 and Hooke's law, the stress distribution within the beam is given by Ob(Z) = Eo(Cz + Co),

where

E b

(5.4.3)

is Young's elastic modulus of the beam material. The stress distribution

122

ACTIVE CONTROL OF VIBRATION

I ,~X

m m Z

x

-I- -

Flexural

/

~X

Longitudinal

Fig. 5.4

Decomposition of asymmetric strain distribution.

within the piezoelectric a c t u a t o r ape(Z) is a function of the unconstrained piezoelectric actuator strain, the Young's elastic modulus of the actuator material Epe and the strain distribution shown in Fig. 5.3. This stress distribution can be written as a p e ( Z ) -- E p e ( C z "Jr"E 0 -- Epe)"

(5.4.4)

Applying moment equilibrium about the centre of the beam produces the relation

I'-hb ao(z)z dz + Ihb'+' (Tpe(Z)Z dz = 0,

(5.4.5)

where ho is the half-thickness of the beam. Next we apply the condition of force equilibrium in the x direction which shows that hb

I-hb ao(z)z dz + I hohb+h°%e(Z) dz = 0.

(5.4.6)

After integration, equations (5.4.5) and (5.4.6) can be solved for the unknowns C and e0, which, after some algebra, are given by

eo = K Lepe,

(5.4.7)

where the material-geometric constant is specified by

KL =

Epeha( 8 Eb h ~ -I-Epeh]) 2 4 16E2h 4 + EoEpe(32h3bha + 24h2h2a + 8hoh 3) + Epeha

(5.4.8)

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

123

and the slope is given by (5.4.9)

C---U~_,pe.

In this expression the material-geometric constant is specified by

Kf =

12E~pehbha(2hb + hu) 2 2 + 8hbh 3)+Epeha 2 4 16E2h~ + EbEpe(32h3h,~ + 24hbha

(5.4.10)

The induced moment distribution, mx, in the beam beneath the actuator is given by

mx(x) = Et, IKfepe •

(5.4.11)

The uniform strain component across the beam cross-section is 6 , ( X ) - " 6,0 -" KLr, pe,

(5.4.12)

where ~pe is related to the applied voltage by equation (5.4.1). Thus the response of the beam to the asymmetric actuator consists of a moment distribution mx(x), specified by equation (5.4.11) and a longitudinal strain distribution e(x), specified by equation (5.4.12). Both of these fields will exist at every point beneath the infinite piezoelectric element. We now examine the excitation of an infinite thin beam by an asymmetric piezoelectric wafer element of finite extent. In order to induce motion, a harmonically oscillating voltage v(t) = V e j°~' is applied to the electrodes of the piezoelectric element. For the following derivation we make further assumptions. We assume that the piezoelectric element is very long and thin and hence end effects (such as where the stress field vanishes) are ignored. This assumption is supported by the work of Liang and Rogers (1989) and Anderson (1989) who showed that the actuator strain field for a distributed actuator is unaffected by a free edge beyond approximately four actuator thicknesses distance from the boundary. Therefore for actuators that are large with respect to their thickness, the strain distribution of Fig. 5.3 can be taken to exist in the actuator-beam system. The other important assumptions are that the actuator is perfectly bonded and inertial effects of the actuator material are ignored. First we solve for the flexural response of the beam associated with the induced moment field. The Bernoulli-Euler equation of motion of the thin beam has been derived in Chapter 2. Written in moment form and including the actuator induced moments the beam-actuator equation of motion is given by 02[Mx(x) - mx(x)] Ox2

- o)2pSw = O,

(5.4.13)

where M~ is the internal beam bending moment and mx(X) is the actuator-induced bending moment, while p and S denote the density and cross sectional area of the beam. Following the approach of Crawley and de Luis (1987), equation (5.4.11) can be modified for a finite patch of L~ and then substituted into equation (5.4.13). For a finite length element, equation (5.4.11) is written as

m~(x)

= Corpe[H(x

) -

H ( x - L~) ],

(5.4.14)

where H(.) is the unit Heaviside step function defined as

H(x) = I i' x > O, [ 0, x < 0 .

(5.4.15)

124

ACTIVE CONTROL OF VIBRATION

Equation (5.4.14) implies that the induced moment only exists at every point under the location of the finite actuator. Substituting the moment distribution into equation (5.4.13), taking the second partial derivative with respect to x, and moving the actuator terms to the fight hand side results in ~2Mx(x) - t o 2 D S w = Col~pe[6'(x ) - 6 ' ( x OX2

L~)]

(5.4.16)

where Co = EDIKI and 6' (.) represents the derivative of the Dirac delta function with respect to its argument. Substituting the relation for bending moment, M~(x) =-EIO2w/Ox 2, it can be shown that equation (5.4.16) can be written in similar form to that of equation (2.3.8). Equation (5.4.16) demonstrates the classic result, pointed out by Fanson and Chen (1986) and Crawley and de Luis (1987), that the induced bending of the actuator can be represented as an external load consisting of a pair of line moments of opposite sign located at the actuator edges. As shown previously, the magnitude of these line moments is proportional to the applied voltage. We next consider the longitudinal motion of the beam. The equation of motion in this case (see equation (2.2.1)), written in terms of displacement and including a forcing term, is given by

to2pu

d2u +

d,x 2

E

de(x) =

dx

,

(5.4.17)

where u is in-plane displacement and e(x) is the applied strain distribution due to the actuator. Equation (5.4.12) can be modified for a finite length of patch La by assuming excitation strain only under the element such that ~,(X) = K L r p e [ n ( x ) -

H ( x - La) ].

(5.4.18)

Equation (5.4.18) is then substituted into equation (5.4.17) which produces the inhomogeneous longitudinal equation of motion given by 1

dZu -t-

2 tO U = KLgpe[6(X) -- d ( X --

La)],

(5.4.19)

where c~ = E/p. For the longitudinal motion, the finite piezoelectric element can be seen to be equivalent to an external load of two equal and opposite line forces acting in the x direction, longitudinally at the element edges and along the central axis of the beam. Simultaneously controlling both flexural and extensional motion in beams has been shown to be an important problem if vibration in beams are to be controlled effectively by active means (Fuller et al., 1990b). In order to efficiently control these types of vibration it is important to excite these wave forms to a varying relative degree. In order to demonstrate this we analyse an infinite thin beam excited by a pair of independently driven, but symmetrically located wafer actuators as shown in Fig. 5.5. The flexural equation of motion, equation (5.4.16), is solved using the procedure outlined in Chapter 2 for each individual element (with complex excitation voltages V~

125

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

Z

a

?

i

_~~_ Beam

Actuator, V1

I I I

I I

~X

N\\\\\\\'q ,

I

Actuator, V2 Fig. 5.5

Infinite thin beam excited by two co-located piezoelectric wafer elements.

and V2). The total motion w,(x) of the beam for the two actuators is found by superposition to be (Gibbs and Fuller, 1992a)

wt(x, t) = d31KI(V1 - V2) [(1 - e ~TL°)e -~Tx- (1 - e -jk~L°)e -jkzx] c"J~", 4k~h~

(5.4.20)

where 09 is the frequency of oscillation of the voltages and kI is the flexural wavenumber. The power flow associated with this motion is derived using the method outlined in Chapter 2 and is given by (Gibbs and Fuller, 1992a) 1-IB , -

8hike

IV1 - V2i [ 1 - cos kfta].

(5.4.21)

The longitudinal equation of motion, equation (5.4.19), is solved as demonstrated in Chapter 2 (see Section 2.2) and the solution is (Gibbs and Fuller, 1992a)

ut(x, t) = jKtd31(V1 + V2) (1 - e -jk'~L°)e j°~'-jk'x 2kLha ,

(5.4.22)

where kL is the longitudinal wavenumber. The power flow associated with this wave motion as outlined in Chapter 2 is (Gibbs and Fuller, 1992a) 1-I~ =

(KL )2d231t°ES 4kLh ]

12 IV1 + V2 [1 - cos kLLa].

(5.4.23)

As an illustrative example, an aluminium beam of thickness 2h b equal to 3.175 mm and width 7.62 cm is considered. The piezoelectric actuators are assumed to be of G1195 ceramic material with properties given in Table 5.1. The excitation frequency is 800 Hz. For the first result the voltage amplitude input to both actuators is fixed at V~- V: = 400 V p.-p. and the length is fixed at 3.81 cm. Figure 5.6 presents the flexural and longitudinal power flow (dB relative to 10 -~2 W) plotted as a function of the relative phase between V~ and V2. When the actuators are perfectly in phase, only longitudinal waves are generated and conversely when the actuators are 180 ° out of phase only flexural waves are generated. Variation in the relative magnitudes of flexural and longitudinal wave power flow can be achieved by choosing the phase between 0 and 180 °.

126

ACTIVE CONTROL OF VIBRATION Table 5.1

Typical piezoelectric ceramic actuator properties (PZT, G1195).

Vpe = 0.30 ha = 0.1905

gpe = 6.3 x 101° N/m 2 P pe =

7650 kg/m 3

mm

d31 =

d32 =

d36 =

0

-

166 x 10 -12 m/V

120

~

100

A

~ ' x Bending wave

80

133 "0 v

Longitudinal

% % % %

_o

60

o a.

40-

200

0

I

I

I

I

I

I

I

I

20

40

60

80

100

120

140

160

180

Relative phase, V 1-V 2 (deg)

Fig. 5.6 Power flow as a function of actuator relative phase, f = 800 Hz, La= 3.81 cm (after Gibbs and Fuller, 1992a). In the second illustrative result, the phase of V~ is chosen to be 90 ° in advance of V2. The power flow for both longitudinal and flexural waves as a function of actuator length is presented in Fig. 5.7. It can be seen that as the actuator length approaches zero, the power flow for both wave types, as expected, also approaches zero. When the actuator length is equal to 9.6 cm (corresponding to half a flexural wavelength at 800 Hz), the flexural wave power flow is at a maximum, since, as shown previously, an individual piezoelectric actuator effectively acts as two line moment sources in antiphase that are externally applied to the beam at the ends of the element. The waves generated from each end in this case, are thus perfectly in phase. Conversely when the actuator is 19.2 cm long, the element length corresponds to a flexural wavelength and the flexural power flow drops to zero. It is interesting to view the above wave based results in terms of the excitation of finite beams. For example, if a simply supported beam was completely covered with a piezoelectric element, then the element length would be equal to a complete wavelength corresponding to the second mode of motion of the finite beam. As discussed above, from a wave point of view it is apparent that the second mode would not be excited. Likewise from a modal point of view using the orthogonality principle outlined in Chapter 2, it is also apparent that a symmetric actuator cannot couple into an antisymmetric beam motion. The above observation is another useful illustration of the wave-mode duality exhibited by vibrations of extended systems.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

127

110

100

nn 1:} v _o

o 13.

90

80

_ i////

Longitudinal w

/I

/ 70 -I

I I

60

50

I

I

I

I

I

I

I

I

I

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Actuator length,

0.2

La (m)

Fig. 5.7 Power flow as a function of actuator length, f=800 Hz, (/)12 "-90 ° (after Gibbs and Fuller, 1992a). Finally, due to the higher wave speeds of longitudinal waves, the longitudinal power flow does not have a maximum power flow until the actuator is relatively much longer. At 800 Hz the longitudinal half-wavelength in aluminium is approximately 3.1 m. Thus, the results indicate that in order to dominantly drive longitudinal motion with this configuration, longer, extended arrays of actuators are needed.

5.5

Piezoelectric one-dimensional anti-symmetric wafer actuators

A simplification of the asymmetric wafer configuration is when two, identical wafer piezoelectric elements are located symmetrically about the beam and driven 180 ° out of phase with the same signal as shown in Fig. 5.8. Due to the 180 ° phase difference of each element we call this an anti-symmetric configuration. The analysis presented here essentially follows that of Section 5.4 except that, because the system is being excited in an anti-symmetric fashion, pure bending of the beam will occur without any excitation of longitudinal waves. By applying moment equilibrium about the centre of the beam we find that

I

-h -h b - ha

~ ~pe(Z)Z dz

+

I hb I hb + ha (9 OpeZ d z = 0, Oo(Z)Zdz + -h b hb

(5.5.1)

where superscripts ® and ® denote wafer elements shown in Fig. 5.8. Writing the stress in the piezoelectric elements and beam in terms of the Young's elastic modulus of the material, the strain slope C and the unconstrained strain of the piezoelectric elements can be deduced in the manner used above. We can then solve

128

ACTIVE CONTROL OF VIBRATION Z

l

I

S

Piezoelectrielcement(~ be/beam 01 Piezoelectrielc ement(~)

Fig. 5.8 Piezoelectric anti-symmetric wafer actuator configuration and associated strain distribution. equation (5.5.1) for the strain slope which gives c=

where the material-geometric constant is now specified by Kf=

3Epe[(hb+

ha)z- h 21

2{Epe[(hb + ha)3 - h 31 + Ebh3}

•

(5.5.2)

The moment surface density mx induced in the beam by the actuator is again given by (5.5.3)

mx(x) = EblKf epe•

It is interesting to note that the value of mx for anti-symmetric excitation is not exactly twice that given for a single anti-symmetric wafer which is specified by equation (5.4.11). This result arises because the expression for the asymmetric actuator does not include the stiffness of the bottom-located actuator. If this stiffness is included in the analysis (with applied voltage set to zero on that element) then it can be shown that the anti-symmetric actuator provides exactly twice the input moment. As an example application of the above analysis we will consider excitation of a simply supported Euler-Bemoulli beam by multiple anti-symmetric piezoelectric actuators depicted in Fig. 5.9 (Clark et al., 1991). In this case we again assume that actuators will be excited by an oscillating voltage, V e jc°t, and their inertial effects are ignored. Since each actuator is now finite, we can again assume that the induced moments of the actuator(s) are only present directly under the actuator(s) location. Following the analysis of the previous section, the equation of motion for the beam-actuator system becomes "-'-'----~2 X -- o')2~Sw "- Z C°EP i=1

'(X - x1)i _

6'(x

-

x2)],~

(5.5.4)

where 6' (') again represents the derivative of the Dirac delta function with respect to its argument.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION f!!iiiiiii]

tiii!!!iiit

-76mm-~"

129

40~mm

i 266 mm

I"

380 mm I ....

':1

I: ....

I

Simply supported beam Piezoelectric ceramic element

Fig. 5.9 Excitation of a simply supported beam by multiple anti-symmetric piezoelectric actuators. In equation (5.5.4), x~ and x2 are the location of the edges of the ith actuator for a series of N~ anti-symmetric actuators. We find a solution of equation (5.5.4) by expanding the response of the simply supported beam in terms of its basis mode shapes such that oo

w(x, t) = Z W~ sin(mrx/L) e j~'.

(5.5.5)

n--1

Using equation (5.5.3), substituting equation (5.5.5) into equation (5.5.4) and using the orthogonality property of the modes shapes outlined in Chapter 2, we find an expression for the modal amplitudes of the beam response given by

IV, = i ~1 2Coep cos "=

e

- cos

L

(to~ - toZ)LZm"

,

(5.5.6)

where the unconstrained strain of the ith actuator is given by i ~ - d31V i ej~, +j~' Epe

(5.5.7)

and where V i is the voltage amplitude and q~i is the phase applied to the ith actuator. In equation (5.5.6), m" is the mass per unit area of the beam and to, is the resonant frequency of the nth mode of vibration of the beam. Equation (5.5.6) can now be used to evaluate the modal amplitudes of a typical beam responding to an array of anti-symmetric actuators. The example results given below are for a steel beam of 380 mm length, 40 mm width and 2 mm thickness. Figure 5.10 presents modal amplitudes of the beam, plotted as acceleration amplitude, for two identical piezoelectric anti-symmetric actuators (note, here 'actuator' implies two antisymmetric elements positioned as in Fig. 5.9 and driven out of phase with the same voltage magnitude). The excitation frequency is 200 Hz which is an off-resonance excitation case for the beam system. The piezoelectric elements were of length 38.1 mm, 15.8 mm width and 0.2 mm thickness and were constructed from a ceramic material, G1195 with properties specified in Table 5.1. The actuators were each driven with a voltage of 60 V p.-p.

130

ACTIVE CONTROLOF VIBRATION 1.8

r---1 1.6

Theoretical

Experimental

1.4 E

1.2

t-

.9

1

"~ 0.8 o o

0.6

-

"o o

0.4 0.2 1

2

3

4

5

6

7

8

Modal number

Fig. S.10 Modal amplitudes of beam response, f=200 Hz, both actuators out of phase (after Clark et al., 1991). In the analytical predictions the moments along the edges of the actuators parallel with the beam edges were ignored due to the extremely high stiffness of the beam about its width. In addition, the magnitude of the moments were scaled by the ratio of the width of the actuator to the width of the beam, since the actuator did not completely cover the beam across its width. Also shown in Figs 5.10 and 5.11 are experimentally measured values of amplitude taken from the work of Clark et al. (1991). 2

r-'n

1.8 ¢kl

E tO .m

Theoretical

Experimental

1.6 1.4 1.2 1

o o "o o

0.8 0.6 0.4 0.2 0

1

2

3

4

5

6

7

8

Modal number

Fig. S.ll Modal amplitudes of beam response, f = 200 Hz, both actuators in phase (after Clark et al., 1991).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

131

As can be seen in Fig. 5.10, the dominant response is in the n = 2 mode and this is expected as the driving frequency is near the corresponding n = 2 resonance frequency of 124 Hz. Figure 5.11 presents the same configuration, except now the actuators are driven in phase. It is apparent that the n = 2 mode has now largely been suppressed and the dominant response is that given by the n = 3 mode. This behaviour can be understood by studying the phasing of the actuators relative to the beam normal mode shapes. For example, by considering the orthogonality property of the modes of the beam, it would be theoretically impossible for two actuators symmetrically positioned about the beam mid point as shown in Fig. 5.9 and driven in phase, to couple into the n = 2 mode which has a 180 ° phase change in response through the beam mid point. The above example is important as it outlines the fundamental basis of distributed actuators. In order to selectively control required modes, without exciting unwanted modes (the control spiUover phenomenon discussed in Chapter 3), it is necessary to create a distributed actuator configuration with the required amplitude and phase distribution. Obviously the more independent actuators that are employed, the larger the success in achieving this goal. It is also apparent from the agreement with experimental results shown in Fig. 5.10 and 5.11, that the above static theory provides a reasonable model of the excitation of one-dimensional structures by piezoelectric ceramic wafer elements and to a large degree validates the approximations made in the theory for the geometries and frequencies considered here. Such one-dimensional anti-symmetric wafer elements have also been used to excite and control bending waves in thin beams (Gibbs and Fuller, 1992b).

5.6

Piezoelectric two-dimensional anti-symmetric wafer actuators

The above discussions relate to the excitation of one-dimensional structures. We now turn our attention to the excitation of two-dimensional plates in pure bending produced by a piezoelectric patch configuration in an anti-symmetric arrangement. The analysis is a logical extension of the one-dimensional case.

Fig. 5.12

Two-dimensional piezoelectric wafer element and structure.

132

ACTIVE CONTROLOF VIBRATION

Figure 5.12 shows a piezoelectric patch element located on a flat plate and defines the coordinate system used in the analysis. Figures 5.13(a) and 5.13(b) show the assumed strain distribution resulting from two identical piezoelectric elements located synunetricaUy on the plate and driven 180° out of phase with the same signal. The analysis presented below essentially follows that of Dimitriadis et al. (1991) except that the strain slope is assumed to ~ continuous through all laminae in correspondence to the previous analyses. Due to the anti-symmetric nature of the piezoelectric wafer excitation the strain distributions in the x and y direction of the plate are given by ex = Cx z,

(5.6.1a)

~,y-- CyZ,

(5.6.1b)

where Cx and Cy a r e the slopes of the strain distribution for the x - z and y - z planes respectively. For the 2-D case we assume that the piezoelectric material has similar properties in the 1 and 2 directions such that d31 = d32. The unconstrained strain of the actuator in both the x and y direction is then given by e,pe =

d31V / h a.

(5.6.2)

Piezoelectric element C) Plate

i

I ,

I

J

/x

i

Piezoelectric element (~ (a)

Piezoelectric element (~ Plate

I ~Piezoelectric element (~

(b) Fig. 5.13 Piezoelectric anti-symmetric wafer element two-dimensional strain distribution: (a) x - z plane; (b) y - z plane.

133

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

Following the approaches of the previous sections, the stresses in the plate for the x and y direction are given respectively by =

Ep (ex + Vpey) l_v 2

(5.6.3a)

op =

Ep. (ey + Vpex), l_v 2

(5.6.3b)

off and

where Vp is Poisson's ratio of the plate material. The stresses in the top piezoelectric element are given by OPe(l)--"

Epe

[~-'x "~ "12pe~'y- (1 + 'Vpe)l?,pe]

(5.6,4a)

E~y "st"'llpel?,x - - ( 1

(5.6.4b)

1 -

and 0 pe~-"

Epe

+ 'Ppe)l?,pe],

1 - Vp2e where 'l,'pe is Poisson's ratio of the piezoelectric element. The stresses in the bottom piezoelectric element are given by pe®__

Epe

[l?,x .-1-llpel?,y "4" (1 q- llpe)E, pe ]

(5.6.5a)

1 - 'pp2e and pe®

o~

=

Epe

1 -"fft2e [ey-1" 'llpel?.x "l- (1 + "llpe)~.pe].

(5.6.5b)

Since F_,pe is the same in both axes and the plate is assumed homogenous then Ex-- Ey-- E. Therefore the strain can be written as

(5.6.6)

~, -- CxZ-- C y z - " Cz.

As previously, we apply the condition of moment equilibrium about the x and y axis by integrating over the stress distributions in the x - y and y - z planes. After much algebra we can solve for the strain distribution in both the x and y directions (which are equal) yielding the result (5.6.7)

C = Kff, pe,

where the material-geometric constant is given by Kf =

3Epe[(h o + h,~)2- h2](1 -

vp)

2Epe[(ho + ha) 3 - h31(1 - vp) + 2Eph3(1 - "Vpe) '

where the plate is of thickness 2h b and the actuator elements are of thickness ho.

(5.6.8)

134

ACTIVE CONTROLOF VIBRATION

As an example of the use of the above two-dimensional analysis we now consider harmonic excitation of a simply supported thin plate by a single anti-symmetric actuator. As previously, we assume that the piezoelectric actuator will induce internal moments in both the x and y direction which are only present under the piezoelectric patch extent. As shown above, these moments will be equal and are thus specified by

ms = my-- CoEpe[n(x- x1)- H ( x - x 2 ) ] [ H ( y - y ~ ) - H ( y - Y2)],

(5.6.9)

where (Xl, Yl) and (x2, Y2) are the coordinates of the patch comers and Co = EIK I. We also assume that the actuator is driven by a harmonically oscillating voltage and ignore inertial and end effects of the actuator. The actuator is also assumed to be perfectly bonded to the plate. Using the classical thin plate theory outlined in Chapter 2, substituting the moment distribution of equation (5.6.9) into the 2-D thin plate equation, evaluating the differential operators and moving the piezoelectric terms to the fight hand side (as in the 1-D case) we derive the inhomogeneous plate equation given by ~2w

E1 V4w + m"

c3t2

= Coepe[b'(x - Xl) - 6'(x - Xz)][H(y - Yl) - H(y - Y2)]

+ Co~.pe[H(x- x 1 ) - H(x- x2)][6' ( y - Y l ) - 6' (Y- Y2)], (5.6.10) where E1 denotes the bending stiffness of the plate and V 2 is the Laplacian operator. Equation (5.6.10) implies that the 2-D anti-symmetric piezoelectric actuator effectively applies line moments of amplitude CoEpet O the plate at the location of the actuator edges. The solution of equation (5.6.10) can be found by using the modal expansion of the plate response w(x, y) as described in Chapter 2, which is given by oo

w(x, y, t) = Z

oo

Z wmnsin kmx sin k~y e j~'t,

(5.6.11)

m=ln=l

where Wmn is the plate displacement modal amplitudes which can be calculated by substituting w(x, y) back into the equation of motion, and using the orthogonality property of the plate modes. The modal amplitudes are found to be given by

Wmn =

,,,~4C°epe,2 [ - (k~'2'+ k2n) (cOs kmxl - cOs kmx~)(cOs knyl - cOs kny2)]' m ~pt~Om,,-092) kmkn

(5.6.12)

where m" is the mass per unit area of the plate, Sp is the plate area, (.Dmnare the plate natural frequencies and k m and kn are the plate eigenvalues given in Chapter 2. Example results are presented for a steel plate whose dimensions are: width a = 0 . 3 8 m, height b = 0 . 3 0 m and thickness h = 1.588 mm. Two different configurations of piezoelectric ceramic actuators were considered, as shown in Fig. 5.14. In configuration (a), the element is long in the y direction, narrow in the x direction and symmetric about the b/2 line with x~ = 0.32 m, x2 = 0.36 m, Yl = 0.04 m and Y2 = 0.26 m. In the second case, configuration (b), the actuator was rotated such that it was long in the x direction and narrow in the y direction, i.e. x~ =0.04 m, x2=0.34, y~ =0.23 m and y2=0.27 m. Thus in configuration (b) the actuator is symmetric about the line corresponding to x = a/2.

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

135

Actuator

Simply supported plate .. .°%%%° °.Oo. %%° °°°°° °°o° ..... %%. ...°. .°-. -°.. .... ..°° ...° .°.°. ... .°°. °.. %%.........

X

X

(a)

Fig. 5.14

(b)

Piezoelectric actuator test configurations.

Figure 5.15 illustrates the plate normalized displacement distribution along the line corresponding to y = b/2 at a frequency of f = 9 5 . 5 Hz for configuration (a). The displacement distribution shows evidence of multi-modal excitation due to the excitation frequency being between the (1, 1) and (2, 1) resonance frequencies. It is also apparent from Fig. 5.15 that for this case there is a nodal line being excited close to the x = x~, actuator boundary. This behaviour is associated with the forcing function of the piezoelectric element which was shown to be a line moment along its edge. The piezoelectric element thus tends to induce a rotation at its edge rather than out-of-plane displacement. Tables 5.2 and 5.3 present modal amplitudes of plate response (normalised to the largest value in each table) for an excitation of f = 148 Hz in configurations (a) and (b) respectively. In configuration (a), the (2, 1) mode is dominant as to be expected since the excitation frequency is close to the (2, 1) resonance frequency of 149.8 Hz. Note that the response in the anti-symmetric y distributions (even n indices) is effectively zero. This result is due to the orthogonality between the symmetric actuator forcing

133 "lD v

-o m

-10

c~

E

-20

E

o ..i...,

.Q >

= E o Z

-30 -40 _50

I 0.0

! 0.2

!

I 0.4

I

I 0.6

I

I 0.8

I 1.0

Axial location, x/a

Fig. 5.15 Plate normalised displacement, f=95.5 Hz, configuration (a) (after Dimitriadis et al., 1991).

136 Table 5.2 1991).

m

ACTIVE CONTROL OF VIBRATION Plate displacement amplitudes (dB), f = 148 Hz, case (a) (after Dimitriadis et al.,

1

2

Plate displacement amplitudes (dB), f = 148 Hz, case (b) (after Dimitriadis et al.,

1 2 3 4 5 6

0.0 -345.8 -55.4 -358.5 - 84.7 -357.4

- 32.8 -348.1 -54.3 -356.1 - 81.6 -352.2

3 - 39.1 -352.5 -57.2 -357.9 - 82.6 -355.2

4 -45.7 -358.5 -62.4 -362.3 - 86.5 -348.9

- 74.2 -69.4 -67.9 - 68.2 -72.1

6

Table 5.3 1991).

2

- 670.3 -665.9 -664.8 - 665.6 -670.3

5

-42.1 0.0 -32.7 - 38.1 -46.6

1

- 64.4 -60.9 -60.8 - 62.4 -68.2

4

1 2 3 4 5

m

- 648.3 -650.0 -652.6 - 655.6 -622.8

3

5 - 54.9 -367.3 -70.8 -370.3 -94.1 -349.1

- 668.5 -663.6 -661.8 - 661.8 -665.1

6 - 343.4 -655.6 -358.8 -658.0 - 381.5 -350.6

function and the anti-symmetric response in these distributions. This behaviour is further illustrated in the results of Table 5.3 which is for configuration (b) with the actuator rotated through 90 °. In this configuration the dominant modes are the (1, 1) and (1,2) modes and the (2, 1) mode is now not excited even though it is being driven close to its resonance frequency. The above results illustrate an important feature of the behaviour of piezoelectric actuators. Since they are of a distributed nature, the shape and location of the actuator can be chosen to excite a required mode or modal distribution. This observation has important considerations in terms of reducing control spillover. In addition, experiments carried out by Clark et al. (1993) have demonstrated that the above static model provides a reasonable approximation to the dynamic excitation of thin plates by wafer type piezoelectric actuators. The above analyses have provided the equivalent forcing functions for 1-D asymmetric and anti-symmetric actuator configurations as well as a 2-D anti-symmetric configuration. The results show that the actuator can be replaced by equivalent line moments acting along the edges of the actuator whose magnitude is given by

m ( x ) = m ( y ) - Coep,,

(5.6.13)

where the m ( y ) value is used in the 2-D case. Relations for Co~E1 = K I are summarised in Table 5.4 for the three actuator configurations where E1 is bending stiffness. The previous formulations are based upon simplified static models to estimate the piezoelectric induced strains. Although experiments have shown that these models

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

137

Table 5.4 Summaryof strain constants. Excitation condtion 1-D asymmetrict 1-D anti-symmetric 2-D anti-symmetric

Geometric constant, KI = ColE1 12EbEp~hbha(2hb + ha) 2 2 2 4 16E~h4~ + EbEpe(32h3bh~ + 24hbha + 8hbh3) + Epeha

3Epe[(hb + ha)z - h 2] 2Epe[(hb + ha) 3 - ha] + 2Ebh3b 3Epe[(hb + ha)2 - h2](1 - Vp)

2Epe[(hb+ ha)3- h3](1 - v,,) + 2Eoh3(1 - 'llpe)

~f For the asymmetric configuration there will also be in-plane excitation.

provide reasonable predictions of dynamic forcing functions there are many, more accurate, extensions of the static models. Crawley and Anderson (1989) have developed detailed models for piezoceramic actuator effects including the effects of dynamics. Hagood et al. (1990) have developed models for the dynamics of piezoelectric actuators for structural control and have included the dynamic influence of the electrical power circuit through the piezoelectric effect. Since the analysis of Haygood et al. (1990) is based upon a variational approach it is also very useful for analysing complex piezoelectric-structure systems. More recently Stein e t a l . (1993) have derived expressions for a coupled piezoelectric-structure system including the impedance of the electrical network. These expressions allow derivation and design of the actuator in terms of power consumption and impedance matching. The single-layered actuator analysis has been extended to multi-layered actuators by Cudney et al. (1990) while Jia and Rogers (1989a) have developed models for embedded distributed actuators in structural systems using classical laminate plate theory. Numerical models have been developed by a number of workers. Ha et al. (1992) have presented a finite element formulation for modelling the dynamic as well as static response of laminated composites containing distributed piezoceramics subjected to both mechanical and electrical loading. Work has also been carried out for actuators on curved surfaces such as cylinders. Dimitriadis and Fuller (1992) developed expressions for a piezoelectric actuator in 2-D cylindrical coordinates. Banks et al. (1995) have developed a model for actuation of cylinders by finite 2-D patches. Tzou and Gadre (1989) investigated the axisymmetric excitation of multi-layered cylinders by embedded piezoelectric layers. Lester and Lefebvre (1991) analytically and experimentally studied harmonic excitation of a finite composite cylinder by piezoceramic patches. The above brief review indicates that there has been much work carried out in the study of piezoelectric actuation. The analyses developed in Sections 5.3-5.6 are valid for all forms of distributed induced strain actuators, i.e. those forms of actuators that apply distributed tractions to a structure due to their internal expansion or contractions. The use of piezoelectric actuators in sound radiation control from structures will be illustrated in Chapter 8. Other forms of advanced actuator are based on magnetostrictive materials which are similar to piezoelectric materials except that they are activated by a magnetic rather than an electrical field. They show much potential for their use in active vibration control due to their extremely high strain rates (Goodfriend and Shoop, 1991; Hiller et al., 1989).

138

5.7

ACTIVE CONTROL OF VIBRATION

Piezoelectric distributed sensors

The previous sections have dealt with the use of piezoelectric elements as control actuators. We now turn to a related problem; the use of distributed piezoelectric elements as sensors in active control. Much previous work has concentrated on the use of discrete point sensors in controlling vibrations in distributed structures. The reader is referred to Beranek (1988) and Broch (1984) for information on conventional point transducers. Point sensors are usually employed in arrays of transducers whose outputs are processed to obtain some estimate of a required variable or state of the system to be controlled (see Chapter 3 and the next section for a discussion on some of these techniques). The basis of this approach is that in order for the control to be effective without 'observation spillover', then the controller has to be designed to observe only those motions which are required to be reduced. The main disadvantage of this approach is due to the signal processing requirements necessary to process the transducer outputs and thus obtain estimates of the required variables. Distributed piezoelectric sensors show potential to overcome this disadvantage in that they can be shaped so as to act as spatial filters which only observe certain motions. As the piezoelectric sensors are continuous, this spatial filtering is achieved by what is effectively a continuous analogue integration of the measured variable over the sensor surface and thus does not require any signal processing. The main disadvantage of these types of sensor are that they are fixed in shape and are thus fixed in terms of the characteristic which they observe, unlike the point array sensor which can be reconfigured through the use of different signal processing techniques. In addition the output of the sensor is sensitive to the accuracy of the shape of the sensor as well as its positioning on the structure. Figure 5.16 shows a one-dimensional thin beam covered with a thin layer of piezoelectric material. Additional assumptions to those made in the previous sections are that the piezoelectric material is mechanically isotropic, and that the sensor has constant properties along its length and is also thin compared to its length. We also assume that the sensor has no effect on the motion of the beam, i.e. the stiffness and inertial force of the sensor are very small compared to the beam.

Z

l

/ Piezoelectsensor ric

L ~ : : J . ' : : : ~ ' . ~

~' Beam

Fig. 5.16 Piezoelectric distributed sensor-beam distribution.

configuration and

associated

strain

DISTRIBUTEDTRANSDUCERSFOR ACTIVECONTROLOF VIBRATION

139

The strain in the x direction at the surface of the sensor can be written as ~2w

e(x, t)= ~

~X 2

(hb + hs)

(5.7 1)

'

where hb is half thickness of the beam and hs is the thickness of the piezoelectric sensor. Due to the reciprocity of the piezoelectric effect, deformation of the sensor will produce a charge across the sensor electrodes. As discussed by Lee and Moon (1990) the sensor output is a function of the effective electrode width F(x,y) and the polarisation of the piezoelectric material P(x, y). Following Collins (1990) we define an arbitrary sensor shape as +b/2F(x, y)P(x, y) dy, f(x) = [J-b/2

(5.7.2)

where +b define the transverse limits of the sensor. In general P will be +1 for piezoelectric material poled through its thickness and F(x, y) denotes the spatial pattern of the electrodes, i.e. F(x, y) will be either 1 or 0 at coordinate x, y depending upon whether that point is covered by the electrode or not. For predicting the electrical output of piezoelectric sensors it is more convenient to use the piezoelectric stress constant e~q. The stress constant e~q is directly related to the strain constant d~q introduced in Section 5.2 by the Young's elastic modulus and the Poisson's ratio of the piezoelectric material as discussed in Auld (1990). The stress constant has the units of coulombs per square metre in the metric system. The total charge q(t) generated by the piezoelectric sensor when it is deformed can then be calculated by integrating the local beam surface strain multiplied by the piezoelectric material stress constant e31 and weighted with the sensor shape to obtain the one-dimensional sensor relationship given by q ( t ) = - JL ° ~e3d°(x) Ox2 (hb + h,)dx,

(5.7.3)

where L s is the total length of the sensor. The strain variation through the sensor is assumed negligible (due to its thinness). Lee and Moon (1990) have also derived an expression for the charge generated by a two-dimensional distributed sensor which is given by

[

~2W

~2W

~2W ]

q( t) = - j[ s f (x, y) e31 ~~X 2 + e32 ~y2 + 2e36 ~x ~y (hb + hs) dx dy,

(5.7.4)

where S is the area covered by the sensor and e36 is the stress constant in the shear direction. When a sensor is used with no skew angle then e36 = 0 (Lee and Moon, 1990). As an example of a one-dimensional sensor we consider a simply supported beam model. We desire to develop a modal sensor, that is a distributed piezoelectric sensor whose output is only related to the motion of one particular mode of the beam. As described in Chapter 2 the motion of a simply supported beam can be written as ¢,o

Ajwt

w(x, t) = ,2~ W~q~n(x)e ,

n=l

(5.7.5)

140

ACTIVE CONTROL OF VIBRATION

where Wn are the modal amplitudes and ~pn are the modes shape functions given by ~p, = sin k,x,

k n =

ner/L.

(5.7.6)

Substituting equation (5.7.5) into the one-dimensional sensor equation (5.7.3) yields an expression for the charge output given by oo

q(t) = Z W,Bn,

(5.7.7)

n=l

where

B~ = -(hb + h~)e311~"f(x) ~2p~ dx. ~x 2

(5.7.8)

As is well known, the modes of a self-adjoint system such as the one-dimensional simply supported beam are orthogonal (see Chapter 2). We can thus take advantage of this property to design a sensor shape in order to observe only the required modes. From equation (5.7.8), it is apparent that if we choose f ( x ) to be proportional to the second derivative of a mode shape then q(t) will be proportional only to the amplitude of the corresponding mode, i.e. we have constructed a modal sensor. In this case the sensor charge output will be given by

q. = - (hb + h~)e3~K.A.L~W.,

(5.7.9)

where K~ is a constant related to the sensor gain and A. is the mode normalisation constant specified by

1 L, A~ = ~ I0 ~p~dx.

(5.7.10)

Figures 5.17(a) and (b) show two sensor shapes designed to observe the first and second mode of a simply supported beam respectively. Note that for the first mode, the sensor weighting (i.e. width) is largest where the strain of the first mode is largest. For the second mode, the polarisation factor P is + 1 in the left half and - 1 in the fight half of the beam in accordance with the response of the second mode which flips through 180 ° in phase over the beam mid point. Other important work in shaped distributed sensors has been performed by Collins et al. (1991) who designed shaped piezoelectric sensors based on the sinc function in order to respond with unity gain to all structural vibrations below a cut-off frequency o)~ and not to observe motions above w ~. In the investigations of Collins the piezoelectric material used was the polymer polyvinylidene fluoride (PVDF). As discussed by Collins and von Flotow (1991), PVDF material makes an excellent sensor. It is lightweight and flexible and thus causes little change to the system response. It has a high piezoelectric charge constant and can readily be shaped into complicated forms either by cutting or etching the electrodes. Changing the polarity of the sensor is achieved by simply flipping the sensor material over in the required areas so that the polarity direction is reversed relative to the system coordinate axis. Another important aspect of the implementation of piezoelectric sensors is the development of the necessary high input impedance electronics (from the sensor side) to measure the

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

141

(a)

(b) Fig. 5.17 Distributed sensor shapes for a one-dimensional simply supported beam modal observer: (a) n = 1 mode; (b) n = 2 mode.

charge over the sensor without causing 'leakage'. Circuit arrangements to provide high input impedance voltage amplification etc. are discussed by Collins (1990) and Clark et al. (1992a). As mentioned previously, care must be taken in accurately shaping and positioning the distributed sensors. Errors in the charge output due to these aspects have been investigated by Gu et al. (1994). Lee and Moon (1990) and Burke and Hubbard (1991) have also considered the design of two-dimensional modal sensors. In this case, it is more difficult to shape the sensor in order to obtain the necessary sensor weighting since variation in the vibration profile occurs in both the x and y directions. In order to solve this problem, efforts have been directed towards locally varying the piezoelectric material properties or thickness in order to obtain the necessary integrated rejection of the structural motions that are not required to be sensed. Chapter 8 will also illustrate the use of two-dimensional piezoelectric sensors in the control of sound radiation from structures. Typical properties of PVDF can also be found in Table 8.3.

5.8

Modal estimation with arrays of point sensors

In many situations it is advantageous to use arrays of point sensors and to electrically combine their outputs in order to construct with point sampling an output that is equivalent to a distributed sensor. Here a point sensor is defined as one whose size is small compared to the wavelength of structural motion. As the method of combining the individual sensor outputs is usually electronically based, it is possible to construct a sensor that is adaptive by changing the configuration of the electronics. A point sensor

142

ACTIVE CONTROLOF VIBRATION

can of course be seen to be a particular case of a distributed sensor, with its spatial weighting factor represented by a Dirac delta function. It can then be shown that the single point sensor observes all modally weighted motions equally at that point. By combining the outputs of individual sensors, as shown in the simple example of Fig. 5.18 we can construct an array of sensors whose combined shape function has reduced modal observability. We first discuss an array of point sensors which can be used to estimate modal amplitudes of known systems. Let us assume that the system is defined by a series of response characteristics such as mode shapes, ~Pm,-Thus the out-of-plane motion of the system could be described by Ajo)t

w(x,y,t)= Z Z AmJpmn(x'y) c ' m=ln=l

(5.8.1)

where Amn are modal amplitudes. The above relationship could easily be written for other variables such as velocity, acceleration, strain etc. We desire to measure or estimate a mode or state of the system. In order to do this we can sample the structural response at J positions and represent their values as a vector ws. Assuming that the structural response is dominated by J modes, the vector w, is then related to the system modal amplitudes and known mode shape functions by

I

wi

°

~P 1ll

~

~P~N 1

A11 ,

°,,

(5.8.2)

LWs where the elements of Ws comprise the measured complex displacements and M + N = J. The e j~' time factor has been omitted for convenience. By solving the above system of equations, we can obtain the modal amplitudes as a = W-~w ~,

(5.8.3)

where a [All A12 ... A12 ... AMN] T, W s = [W] W s2 ... WsJ] T and W is a matrix of the modal contributions at the sample points. Thus if we position an array of J sensors on a structure, we can process the output of the sensors using equation (5.8.3) in order to obtain information related to individual modes. Several points are important. The individual =

Sensor output

Summer

Gain

Point sensor Structure

Fig. 5.18

Array of point sensors configured as a general distributed observer.

DISTRIBUTED TRANSDUCERSFOR ACTIVECONTROLOF VIBRATION

143

sensors need to provide estimates of absolute amplitudes and phase at all points simultaneously. This necessitates the use of a multi-channel data acquisition system and the processing of the information in the frequency domain in order to resolve the relative phases. If the system is being driven at a steady frequency then a reference and a roving sensor can be used with the phases measured relative to the reference sensor. The above approach also requires an accurate knowledge of the system mode shapes or characteristic functions and this often is not readily available. Finally, spatial aliasing will occur if the individual modes are not sampled with sufficient spatial resolution. In general, two sampling points are needed in a wavelength of a motion of the structure to be observed. If modes of significant amplitude occur at wavelengths shorter than the Nyquist wavelength (equal to twice the spacing of the sensors) then the information from these modes will fold back around the Nyquist value and corrupt the estimates of the magnitudes of lower order modes in a manner analogous to discrete time-frequency processing of signals. The ad hoc approach is to increase the number of sample points until modal amplitude is observed to roll-off at some set value. Errors in the modal estimation associated with noise and positioning of the sensors are described by Clark et al. (1993). The technique is, of course, aided by the fact that, as discussed in Chapter 2, structures act as low pass filters to broadband disturbances in terms of modal response. The method can also be made more robust by using more measurement points than required modal amplitudes. In other words, a system of overdetermined simultaneous equations is formed. In this case it is appropriate to use a pseudo inverse or least mean squares technique to solve for the modal amplitudes as a = [WTW]-~WT W~.

(5.8.4)

The above method can be applied to systems described in terms of waves as well as modal response functions. Fuller et al. (1990c) have used the procedure to estimate complex amplitudes of travelling and near-field waves in vibrating thin beams. Arrays of sensors are also used in state feedback control of systems to estimate the states of the system. Such an estimate is usually in the form of a Kalman filter and this technique is described in Chapter 3.

5.9

Wavenumber estimation with arrays of point sensors

In many cases it is advantageous to estimate directly (or sense) the wavenumber components of a structural motion. As discussed in Chapter 8, sound radiation is directly related to structural wavenumber components having a supersonic phase speed. Thus a sensor that provides wavenumber information from structural measurements may in principle be used to infer properties of the sound radiation. If a onedimensional structure has a motion described by

w(x, t) = w(x) e j°~

(5.9.1)

we can apply a one-dimensional spatial Fourier transform given by

W(kx) = I = w ( x ) ~ + dx,

(5.9.2)

where W(k,) are the spectral wavenumber amplitudes at a particular wavenumber value

144

ACTIVE CONTROL O F VIBRATION

kx. In effect, since kx = 2z~/2, we have decomposed F(x) into its Fourier wavenumber components of different wavelengths when equation (5.9.2) is applied to a spatial response for a particular frequency as discussed in Section 2.4. In practice it is impossible to obtain a continuous estimate of F(x); hence we can reduce equation (5.9.2) to a discrete Fourier transform (DFT) for a finite record length (again analogous to time-frequency manipulations) which can be written as I

g'(kx) = x~ w(xi) eft~i Ax,

(5.9.3)

i-1

where if' are the spectral estimates and I discrete samples are taken over equal spacings of width Ax (Maillard and Fuller, 1994). Equation (5.9.3) thus enables an estimate of the amplitude of a particular wavenumber component to be made for particular frequencies. It is also possible to evaluate the discrete wavenumber transform using fast Fourier transform (FFT) algorithms as demonstrated by Williams and Maynard (1982) and Wahl and Bolton (1992). Note that the usual sampling requirements in terms of finite record length and bandwidth apply as discussed by Maillard and Fuller (1994) and Nelson and Elliott (1992). In order to build a wavenumber sensor, we can measure the motion at a required number of points and apply the relationship given by equation (5.9.3). Approaches of this type as well as finite difference techniques for separating wave components have been pursued by Pines and von Flotow (1990a). Different wavenumber components can then be directly used as sensor information. Note that this method applies at a particular frequency and is thus a frequency domain method, and that the equations can readily be extended to two-dimensional systems. Often it is required to estimate the wavenumber components in the time domain. Maillard and Fuller (1994) have developed a system which estimates time domain, structural wavenumber information from an array of accelerometers whose outputs are passed through a bank of digital FIR filters and then summed to provide an estimate of the wavenumber components of the structural response.

5.10 Wave vector filtering with arrays of point sensors In many active control problems the vibrational field to be attenuated consists of multiple waves. For example the vibrational field can consist of waves travelling in opposite directions (i.e. a standing wave) due to scattering from a discontinuity located at the end of the structural element. As discussed in Chapter 6, an effective control strategy in this case is to minimise the power flow travelling towards the discontinuity. In order to implement such a control system is necessary to use a sensor which provides error information proportional to the positive travelling wave component. On the other hand, if one wished to make the beam discontinuity act like a perfect absorber then the sensor would provide error information proportional to the reflected wave. The following analysis outlines a procedure based upon an approach developed by Elliott (1981) which filters out positive and negative wave components from a standing wave field in a non-dispersive medium (see Chapter 2).

DISTRIBUTED

TRANSDUCERS

FOR ACTIVE CONTROL

OF VIBRATION

145

Let us assume that we can measure the displacement field of a beam vibrating in longitudinal motion at two points. If the wavelength of motion is long compared to the spacing between the sensors, Ax, then the total displacement ut is given by ut(x, t) = [ul(x, t) + u2(x, 0 ] / 2 ,

(5.10.1)

where the subscripts 1 and 2 refer to the sensor positions. If the beam is carrying both positive and negative travelling waves then the total displacement field is also given by ut(x, t) = ui(x, t) + Ur(X, t)

(5.10.2)

2 U i + 2/,/r = U 1 q- U2,

(5.10.3)

and thus where the arguments of the displacement u have been eliminated for the sake of simplicity. For longitudinal wave motion, as discussed in Chapter 2, the in-plane force f ( x , t) is given by

0u (5.10.4)

f(x,t) = -SE ~,

i)x

with S and E denoting area and modulus of elasticity of beam respectively. Using the finite difference method as outlined by Elliott (1981) this force can be estimated from the two sensor positions by evaluating the expression f, .~ SE u 2 - u_______L1"

(5.10.5)

Ax We also know that for a single longitudinal wave the internal force is related to the particle displacement by (5.10.6)

f = + pLcLjtOu

for positive and negative propagation respectively, where P L is mass per unit length and CLis longitudinal wave speed given by c~ =~-ET-~L. Hence the total force in the beam in terms of the assumed wave field is ft = p l c L j o o ( u i - Ur).

(5.10.7)

This total force should equal that estimated from the finite difference expression given by equation (5.10.5), i.e. /62 -- U 1

pr.CLflO(Ui - Ur) -- - S E ~ . Ax

(5.10.8)

Manipulating this equation we see that 2U~ -- 2Ur =

2SCL

(Ul -- U2).

(5.10.9)

joAx

If we now sum equations (5.10.3) and (5.10.9) we can solve for the incident or positive travelling wave field such that 4u~ = (Ul + u2) +

2Sct. floAx

(Ul - u2).

(5.10.1 O)

146

ACTIVE CONTROLOF VIBRATION

If we assume that both sensors are of equal sensitivity and have a gain G then 1 u = -- V G

(5.10.11)

and we can re-write equation (5.10.10) in terms of V + which is the sum of the voltage outputs of sensors 1 and 2 and V- which is the difference between sensors 1 and 2, i.e. 4Gu/= V÷ +

2Scz.

V-.

(5.10.12)

jcoAx Similarly, if we take the difference between equations (5.10.3) and (5.10.9) we obtain an estimate of the negative travelling wave component that can be written as

4Gur = V +

2SCL V-. j~oAx

(5.10.13)

Figure 5.19 shows a block diagram of a circuit designed to perform the above functions in order to produce signals proportional to u~ and Ur. The outputs of both sensors are fed into the circuit via buffer amplifiers, one of which has a variable gain so that the apparent sensitivities of both sensors may be equalised. The signals are then summed and differenced and then passed through high pass filters in order to remove spurious low frequency noise which could be passed through the integrator and thus overload the circuit. In order to implement the above expression the signal of the bottom channel of the circuit in Fig. 5.19 is passed through an integrator with gain 2ScL/Axjw where S is the cross-sectional area of the beam, cL is phase speed and Ax is the sensor spacing. The above method can be used to resolve wave fields in various non-dispersive media, by calculating different gains for the integrator depending upon the wave type and material properties (for example, Elliott, 1981, discusses separation of acoustic plane waves). An important aspect of the technique is that it can be implemented so as to provide time domain information of the wave components which is compatible with the time domain implementation of the adaptive LMS algorithm (see Chapter 4). Practical implementation aspects of the above method and a discussion on the accuracy can be found in the work of Elliott (1981).

-..t Sensor 1 ~

G

4Ku i

Sensor 2

4Ku r 2Sc L Axj~

Fig. 5.19 Circuit diagram for an analogue wave vector filter in a one-dimensional, nondispersive medium (after Elliott, 1981).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROL OF VIBRATION

147

Many structural systems carry dispersive waves (such as beams and plates in flexure) and in this case it is impracticable to use the above analogue method. Gibbs et al. (1993) have developed and demonstrated a method for separating out positive and negative travelling waves in a thin beam excited by broad band noise. The procedure is very similar to the above method except that the differenced output of two piezoelectric sensors designed to observe flexural strain is passed through an IIR filter designed to give the correct transfer functions which are causal over a band of frequencies. Similar to the above approach, the method of Gibbs et al. (1993) provides time domain signals which are proportional to both positive and negative travelling flexural waves. The interested reader is referred to the work of Gibbs et al. for more details on implementation of wave vector filtering in dispersive media. The method could also be extended to two-dimensional systems. In this case the vector component of the resolved wave field will point in the same direction as the finite difference array. Pines and von Flotow (1990a) discuss in some detail the theory and difficulties associated with wave separation from sampled information in a dispersive medium. Chapter 6 will present more discussion of the use and performance of various configurations of sensor arrays designed to separate out travelling flexural waves and flexural near fields in the active control of vibration in long beams.

5.11 Shape memory alloy actuators and sensors The previous sections have been concemed with actuators that apply an oscillating control input to the system usually at the same frequency (or frequencies) as the disturbance. These forms of control inputs have been previously defined in Chapter 3 as 'fully active' actuators. We now consider the use of an actuator which applies relatively steady-state or slowly time-varying control inputs which tend to change the system response by altering the system characteristics itself. As these types of inputs are not vibratory in nature and do not add energy to the dynamic system we call these systems 'semi-active' or 'adaptive' as defined in Chapter 3. The following discussion will centre on the use of filaments or wires of shape memory alloy (SMA) embedded in composite panels. However, the description could be applied to any actuator capable of inducing static strain that is attached to, or embedded in, a structure. The shape memory effect can be described very basically as follows: a material in the low-temperature martensitic condition, when plastically deformed and with the external stresses removed, will regain its original shape when heated. The material phenomenon that 'memorises' its original shape is the result of the reverse transformation of the deformed martensitic phase to the higher temperature austenite phase (Jackson et al., 1972). The solid-solid phase transformation also yields other useful characteristics such as the ability to reversibly and reliably control the material properties such as Young's elastic modulus and the yield strength. Many materials are known to exhibit the shape memory effect. They include the copper alloy systems of Cu-Zn, Cu-Zn-A1, C u - Z n - G a , C u - Z n - S n , C u - Z n - S i , Cu-A1-Ni, C u - A u - Z n , Cu-Sn, and alloys of Au-Cd, Ni-A1, Fe-Pt and others (Jackson et al., 1972). The most common of the shape memory alloys or transformation metals is a nickel-titanium alloy known as Nitinol. Nitinol is the SMA of choice for use as an embedded distributed actuator because of its unusually high resistivity which allows for resistive heating through the passage of an electrical current.

148

ACTIVE CONTROLOF VIBRATION

Nickel-titanium alloys (Ni-Ti) of proper composition exhibit unique mechanical 'memory' or restoration force characteristics and the ability to provide reversible changes of the material properties. The material can be plastically deformed in its lowtemperature martensite phase and then restored to the original configuration or shape by heating it above the characteristic transition temperature. This behaviour is limited to Ni-Ti alloys having near-equiatomic composition. Plastic strains of typically 6-8% may be completely recovered by heating the material in order to transform it to its austenite phase. Figures 5.20 and 5.21 show typical non-linear mechanical properties of Nitinol SMA as a function of temperature (Jackson et al., 1972). It can be seen that the Young's elastic modulus of Nitinol can increase by three to four times and the 0.2% yield strength increases from about 83 MPa to about 620 MPa; the recovery stress, which is the stress caused by the restoration tendency when the edges of the Nitinol are fully restrained when activated, changes as a function of temperature and initial strain. The class of the material referred to as SMA hybrid composite is simply a composite material that contains SMA fibres (or films) in such a way that the material can be stiffened or controlled by the addition of heat to the SMA. One of the many possible configurations of the SMA hybrid composite material is one in which the SMA fibres are plastically elongated before embedding and constrained from contracting to their 'normal' of 'memorised' length upon curing the composite material with high temperature. The plastically deformed fibres are therefore an integral part of the composite material and the structure. When the fibres are heated, generally by passing a current through the fibre shape memory alloy material, they 'try' to contract to their 'normal' or 'memorised' length and therefore generate a predictable in-plane force and

80 70 A

co

10% 7% 6% 5%

60

o x

-~

Q.

50 4%

oO t_

~"

40

3%

30

-.-

2%

•- -

1%

o

~ n-.

20

Initial strain

10 I.

0 ~J.....__._.a...._...__t......._..__t.~_--.-.------

60

100

140

180

220

260

300

340

380

Temperature (°F)

Fig. 5.20 Recovery stress versus temperature for Nitinol shape memory alloy. Heat rate equals 40 ° per minute (after Jackson et al., 1972).

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION 13

149

130

-

l

oo

70 50 ~'-

3

,--

x

1

x •~ 10

(D O

30

(a) Cooling

¢'~

I

vco

v

Z3

"o

o

11

Mf 1

~O

13

M

I

Md

S

m 130 -

i

_~ 110

-

90

-

70

-

50

-

30

-

10

O0

-20

j

s ~

60

140

~(Hb)atlng 220

300

Temperature (°F)

Fig. 5.21 Young's elastic modulus of Nitinol shape memory alloy versus temperature: r-l, modulus; ©, 0.2% yield stress; (a) cooling; (b) heating (after Jackson et al., 1972).

moment which can be used to provide active control of the dynamic and static response of the composite. This technique is referred to as active strain energy tuning (ASET) (Rogers et al., 1989a). SMA fibres can also be embedded into a composite material without prestrain. When the embedded SMA fibres are electrically activated, the Young's elastic modulus of the SMA fibres increases three to four times, as illustrated in Fig. 5.21, the overall stiffness of the composite structure will be increased, and therefore the response of the composite structure will be modified. In this case, significant numbers of SMA fibres relative to the structural volume is needed to achieve a usable tuning effect. This technique is referred to as active property tuning (APT) (Rogers et al., 1989a). To provide adaptive control of the response of the SMA hybrid composites, one can either 'tune' the mechanical properties of the embedded actuator fibres (APT) or impart large internal restoring stresses throughout the structure (ASET). Tuning the mechanical properties means that the SMA fibres or films are embedded without being plastically deformed, i.e. there is no shape memory effect, and it may avoid introducing large internal loads and stress concentrations in the structures. Active strain energy tuning is accomplished with SMA fibres (or actuators) that are embedded with a certain

150

ACTWE CONTROLOF VIBRATION

amount of initial strain. Once the fibres are actuated, the overall stiffness of the structure changes and an internal force and restoring stress, is created because of the shape memory effect. This allows control of various material properties of the structure. Liang et al. (1991) have investigated analytically the structural acoustic behaviour of adaptive shape memory alloy reinforced laminate plates in terms of their structural and acoustic response characteristics. Liang et al. use a Rayleigh-Ritz approach to find the normal mode response of a simply supported rectangular plate with embedded SMA fibres by including the recovery stress due to activation of the SMA fibres directly into the governing laminate equations (the ASET principle). The acoustic field radiated is then coupled to the plate motion using a procedure similar to that outlined in Chapter 8. The analysis is reported in full detail in Liang et al. (1991). Here we discuss an illustrative example of the use of adaptive SMA fibres. For the results discussed, a composite plate of dimensions 1.1 m by 0.8 m by 8 mm was studied. The stacking sequence of the composite plate layup was [0/-45/45/90 ° ] resulting in a quasi-isotropic plate (Jones, 1975). The plate was assumed to be made of equal thickness graphite/epoxy and Nitinol/epoxy laminae. Nitinol/epoxy obeys the rule of mixture and detailed information about the constitutive relations of SMA hybrid composites is discussed in detail in the work by Jia and Rogers (1989b) and material properties are available in the work of Rogers et al. (1989b). The Nitinol/ epoxy laminae are considered to be the top and bottom laminate, which has a 40% Nitinol fibre volume fraction which yields a SMA fibre volume fraction of 10% for the entire plate. It is assumed that the recovery stress of the embedded SMA fibres upon activation is 280 MPa. The recovery stress is related to the initial strain, the recovery stress-strain relation of the SMA fibre, the curing process of the composite plate which could cause stress relaxation, and the boundary conditions of the plate (Rogers and Barker, 1990). The plate is assumed to have a damping coefficient of 0.01. Table 5.5 contains the calculated natural frequencies of the first ten modes of the plate when inactivated and activated (i.e. with heating of the SMA fibres). The results show that, for example, the first natural frequency has been increased by approximately 70% upon actuation of the embedded SMA fibres. Typical changes in mode shapes for the third to the sixth mode of the free plate vibration are given in Fig. 5.22. The result of both Table 5.5 and Fig. 5.22 indicate significant changes in the plate natural frequencies and associated mode shapes upon activation of the embedded SMA fibres. As discussed in Chapter 8, sound radiation from a harmonically vibrating plate is strongly influenced by the value of the plate resonance frequencies (relative to the drive frequency) and plate mode shapes. Figure 5.23 shows sound intensity radiation Table 5.5 Changeof the first ten natural frequencies (after Liang et al., 1991) Natural frequencies (Hz) Mode

1

Inactive 41.3 Active 71.5

2

3

4

5

6

7

8

9

10

82.8 114.8~ 144.4 166.9 224.0 233.7 245.5 2 9 0 . 7 317.9 129.7 146.6 203.4 239.5 246.9 296.5 3 2 2 . 4 355.4 403.4

DISTRIBUTED TRANSDUCERS FOR ACTIVE CONTROLOF VIBRATION

151

Activated

Inactivated

,

.

3rd

3rd

4th

4th

.

.

,

.

,/c 5ih

5th

'6th

'

"

6th

Fig. 5.22 Change in mode shapes for an SMA composite, simply supported plate (after Liang et al., 1991).

0 i

4.5

/

2.25

xq~,,-

0

i

2.25

I

I

4.5

Intensity (W/m2x 106)

Fig. 5.23 Radiation directivity pattern of an SMA composite plate, f=220 Hz: inactivated; - - - , activated (after Liang et al., 1991). directivity patterns for the same SMA composite panel located in a rigid baffle and excited on one side by a sound wave incident at 45 ° with an amplitude of 1 Pa. The frequency of the wave is 220 Hz and the far-field directivity patterns are in the y = b/2 mid plane of the plate (see Fig. 8.10(b) for the radiation coordinate system). The results show a significant change in level and shape of the radiation pattern when the SMA is activated. This is due to two mechanisms. Firstly, as the plate resonance frequencies are changed, its response amplitude is modified (see Chapter 2), leading to

152

ACTIVE CONTROLOF VIBRATION

an overall change in the radiation level. Secondly, the coupling between the plate motion and the sound radiation is altered due to the change in activated mode shapes (see Chapter 8). This results in a drastically different sound radiation pattern. The work discussed in this section illustrates the use of SMA actuators that apply relatively steady state loads to the system or change the overall system properties. Fluids can also be used as a form of actuation in the sense that they are employed to change the system properties similarly to the SMA implementation termed APT and discussed above. Electrorheological (ER) fluids are suspensions of highly polarised fine particles dispersed in an insulating oil (Stangroom, 1983). When an electric field is applied to the ER fluid the particles form chains which lead to changes in viscosity of the medium of several orders of magnitude, as well as alteration of elasticity. The ER fluid can be thus embedded in a structure, for example, and used to tune its overall mechanical properties such as damping and stiffness by electrical actuation (Gandhi and Thompson, 1989). It is worthwhile to note that SMA can also be used as a 'fully active' actuator to control vibrations in dynamic systems. Baz et al. (1990) have analytically and experimentally demonstrated the use of SMA fibres to control transient motion in a cantilevered beam. The results show that the use of an SMA-based active system significantly increased the damping of the system. In general SMA is limited to control of very low frequencies due to its large thermal time constant and provision must be made for quickly dissipating the thermal output of the SMA fibres in order to quicken the cooling phase. In the work of Baz et al. (1990) the fibres were located outside of the material of the beam to facilitate cooling. Shape memory alloy can also be used as a distributed sensor. Work reported by Fuller et al. (1989b, 1992) demonstrated that an embedded SMA fibre when used in a Wheatstone bridge configuration can give accurate estimations of oscillating strains in a cantilevered beam. As SMA material can be manufactured to be super-elastic, very large strains can be measured before the failure of the sensor. This technique has been extended by Baz et al. (1993) who used arrays of different lengths of SMA fibres embedded in a cantilevered beam in conjunction with a matrix technique (similar to that of Section 5.8) to provide estimates of modal amplitudes of response. The technique shows much promise where large strains and hostile environments are encountered. The use of multiple embedded actuators and sensors in structures is part of a rapidly growing field termed adaptive, intelligent or smart structures. For more information on this field the reader is referred to the review paper by Wada et al. (1990).

6 Active Control of Vibration in Structures

6.1

Introduction

In this chapter we review a number of different approaches to the active control of mechanical vibration in structures. These approaches are distinguished from those discussed in Chapters 3 and 4 in that the structure is now assumed to be governed by a partial differential equation rather than an ordinary differential equation. In other words, the structure is assumed to be distributed rather than having 'lumped' springs, masses and dampers. There are a number of ways of describing the motion of such a structure, each of which is consistent with the governing partial differential equation. One way of expressing the velocity distribution over an entire structure, for example, is in terms of the sum of the contributions from a number of structural modes. Another approach is to describe the motion in terms of the amplitudes of a number of different types of mechanical w a v e s in the structure. Both of these representations and their relation to each other were discussed in Chapter 2. The most 'efficient' description, which requires the fewest parameters to describe the motion of the structure, will depend very much on the geometry of the structure, its boundary conditions and the frequency of excitation. These two descriptions of the motion of a distributed structure lead to two rather different approaches to active control. The first concentrates on controlling the m o d e s of a structure. By actively reducing the amplitudes of these structural modes the spaceaverage mean square velocity over the whole structure is reduced, and the control can be said to be 'global'. It should be emphasised that reducing the total vibrational energy of a distributed structural system, for example, does not guarantee that the s o u n d radiated by the structure will be correspondingly reduced (due to the nature of the structure-acoustic coupling). The active reduction of sound radiation from a structure will be discussed in more detail in Chapters 8 and 9, and for now we restrict ourselves to considering only the reduction of vibration p e r se. This is still an important area of practical application, however; for example in order to improve the positioning or pointing accuracy of an antenna or robot arm, or to reduce fatigue in highly-stressed structures. Whereas the control of structural modes tends to imply the global control of vibration throughout the structure, the active control of structural w a v e s is normally used when the flow of vibrational energy from one part of a structure to another is important. This would be true, for example, where there was a concentrated source of

154

ACTIVE CONTROLOF VIBRATION

vibration, and a particularly sensitive component was located at another point on the structure, connected via a relatively long structural component in which only a reasonable number of structural waves can transmit power. In the active control of structural waves we thus tend to be concerned with the suppression of vibration transmission rather than global control of the entire structure. The active control of vibrations transmitted through vibration isolation mounts could be said to also fall into this classification, but because of their practical importance they are described separately, in Chapter 7. It should be noted that suppressing vibration transmission into one part of the structure may well increase the vibrational energy in another part of the structure, and will generally not achieve global control. In this chapter we will first consider the modal approach to global active vibration control using both feedforward and feedback techniques. The feedforward approach, in which we assume a single-frequency excitation, is reasonably straightforward to describe analytically and it allows model problems to be solved which illustrate the ultimate performance limits of any active control system. Feedback control is a more practical strategy when knowledge of the primary disturbance is limited. The active control of structural waves is similarly treated, except that in this case feedforward control can be applied to a wider class of disturbances. This is because wave control implies that something is known about the direction of structural power flow, so that it is sometimes possible to position a sensory array between the primary and secondary sources of excitation which can detect the incident structural wave, whatever its waveform (see Nelson and Elliott, 1992, Chapters 5 and 6, for a discussion of the equivalent acoustical problem). We particularly concentrate on the active control of flexural waves, because of their importance in practice. A brief discussion of feedback control of flexural waves is also included, in which it is shown that the most successful strategy, of displacement feedback, is equivalent to a conventional method of wave control using a linear spring.

6.2

Feedforward control of finite structures

We will assume that the structure to be controlled is excited by a primary excitation at a single frequency and that all altemating quantities are proportional to e j~'t. The excitation is described by a force distribution f(x, y, to) acting over the structure, which for convenience is assumed to be a plate extending in the two directions x and y. The distribution of transverse displacement over the plate, w(x, y, to) is expressed, as in Sections 2.10 and 2.11, as the sum of the contributions from a finite number of natural modes of the structure, N, so that N

w(x, y, a~) - Z A~(a~)~Pn(X,y),

(6.2.1)

n=0

in which An(to) is the complex amplitude of the nth natural mode at frequency to. ~Pn(x, y) is the spatial distribution of the nth natural mode shape, which is orthogonal with respect to any other mode shape, Is ~'n(x,Y)~'m(X,Y) dx dy

0

if n

:#

m,

(6.2.2)

155

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

and is assumed normalised such that 1

I ~n2(x'y) dx dy S s

1,

if n = m,

(6.2.3)

where S is the area of the plate. In principle an infinite number of modes must be used in the summation of equation (6.2.1), but in practice the distribution of displacement can be described to any arbitrary accuracy by a finite number of modes. For excitation frequencies in the range of the natural frequency of the first few structural modes the number of modes required for an accurate description is typically fairly small, indeed close to the natural frequency of one of these lower order modes in a lightly damped structure it is often only necessary to account for the contribution of one 'dominant' mode. The amplitudes of each of the structural modes in equation (6.2.1) can be expressed as

An(a)) = An(a)) Isf(X, y, (.O)t~n(X,y) dx dy.

(6.2.4)

The second-order resonance term, with damping, can be written A'(a)) =

1

M(w 2 - 092 + j2~nOgnO))

,

(6.2.5)

where M is the total mass of the plate, w, is the natural frequency of the nth mode and ~, the damping ratio associated with this mode. The total time-averaged vibrational kinetic energy of the plate when driven by a pure tone force distribution can be written as Mo) 2 g

Ek(w) = ~

4S

J w(x, y, w)w* (x, y, w) dx dy. s

(6.2.6)

Using the expansion in terms of the normal modes given by equation (6.2.1), and noting that the mode amplitudes are not dependent on the position on the plate, the total kinetic energy can be written as Ek(~o) =

Mw 2 4 Zn Zm A"(o)A*m(m) -S1 I s

t~n(X' Y)lPm(X'

y) dx dy.

(6.2.7)

The orthonormality of the natural modes described by equations (6.2.2) and (6.2.3), can now be used, so that the total kinetic energy can be written as M~o2 N Ek(m) =

(6.2.8)

[, 4

n--0

i.e., proportional to the sum of the modulus squared mode amplitudes. The total vibrational potential (or strain) energy of the plate can also be written as

Ep((,o)=

4

s

(I2w2 ~~x 2

+ ~~y2

)

dxdy,

(6.2.9)

which can itself be expressed in terms of the amplitudes of the natural modes. For

156

ACTIVE cONTROLOF VIBRATION

many common boundary conditions, however, including free-free, simply supported and clamped-clamped, the kinetic and potential energies are equal (Morse and Ingard, 1968). We thus concentrate on the problem of the feedforward active control of the kinetic energy in a harmonically excited plate. The analysis closely follows the analogous problem of minimising the total acoustic potential energy in an enclosure, considered in detail in Chapter 10 of Nelson and Elliott (1992). The modal expansion of the transverse displacement at some point on the plate (equation (6.2.1)) is first expressed as a vector inner product

w(x, y, o))= aT(to)~(x, y),

(6.2.10)

where aT(to) = [Al(to) A2(to)...AN(to)], gtT(x, Y) = [~Pl(X,Y) ~P2(X,y)... ~PN(x,Y)].

(6.2.11)

The total vibrational kinetic energy of the plate given by equation (6.2.8), can then be written as Ek(co) =

Mto 2

aH(o))a(to).

(6.2.12)

4 The vector of structural mode amplitudes is itself the superposition of components due to the primary source ap(to) and those due to an array of M secondary force distributions, whose complex amplitudes are contained in the vector f s, coupled to the structural mode amplitudes via a matrix of modal coupling coefficients B, so that in general a = ap + Bf~.

(6.2.13)

The total vibrational kinetic energy can thus be written in the standard Hermitian quadratic form as Me92 Ek-

4

H

H

H

H

(fsHBHBfs+ fs B ap + ap Bfs + apap).

(6.2.14)

E k is thus a quadratic function of each component in fs which is guaranteed to have a minimum value, since the matrix BHB is positive definite, for a value of f~ given by (Nelson and Elliott, 1992, Appendix)

f~o - - [BHB ] -~BHap

(6.2.15)

with a corresponding minimum value of E k given by Ek(m~n)-

Mto 2

a n [ l - B[BHB]-~BH]ap.

(6.2.16)

4 To illustrate these general principles, several authors have taken specific examples of structure and primary and secondary force distribution. Curtis (1988), for example, performed numerical simulations of the minimisation of the total vibrational strain energy in an undamped beam of length L. Some typical results from these simulations are presented in Fig. 6.1.

157

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

/ t"-

g

t-

09 0

.¢ ""

I

0

I

I

I

100

200 300 Frequency (Hz)

400

500

Fig. 6.1 Vibrational strain energy as a function of pure tone excitation frequency for a beam excited by a primary point force at Xp -0.3L ( ~ ) and after the energy has been minimised by the action of a secondary point force at Xs- 0.1L (- - -) (after Curtis, 1988). The formulation for global control presented above was generalised somewhat by Post (1990) and Post and Silcox (1990), who considered the minimisation of the mean square transverse displacement over some part of the length of a beam, say from x = xl to x:. The cost function being minimised is thus of the form

j =

l dx.

Xl

(6.2.17)

This results in a similar quadratic optimisation problem to that considered above, which can be solved at a number of excitation frequencies to give the minimum cost function. The control strategy is still, however, entirely feedforward and is illustrated Primary frequency generator (o~)

J Hqco) J ~ "-I

wI P,I,

Fig. 6.2

xp

xs

Primary force

Secondary force

Block diagram of the feedforward control system on a beam (after Post, 1990).

158

ACTIVE CONTROLOF VIBRATION

in Fig. 6.2, in which H ( j t o ) is the complex number which describes the amplitude and phase of the secondary source relative to that of the primary source. As another example of global control, Fig. 6.3 shows the results computed by Post (1990) and Post and S ilcox (1990) for a simply supported beam with a damping factor ~, of 1%, in which a cost function equal to the total kinetic energy on the beam was minimised. The physical parameters in this simulation were normalised so that the first mode had a normalised natural frequency of to = 10. From equation (6.2.6) it is clear that the total kinetic energy in this case is proportional to the mean square transverse displacement of the beam, averaged over its length. Significant reductions in cost function can be achieved near the natural frequencies of the beam, and almost no reductions can be achieved at the frequencies in between these resonances. The distributions of transverse displacement on the beam with and without active control are shown in Fig. 6.4 for excitation frequencies corresponding to the second normalised natural frequency ( t o - 40) and half way between the first and second normalised natural frequencies (to = 25). As an example of the minimisation of a local cost function, Fig. 6.5 shows the cost function given by the mean square displacement averaged only over the final quarter of the beam length, with and without active control. This cost function is even more significantly reduced than the global cost function for excitation frequencies close to the natural frequencies of the beam. The distributions of transverse displacement on the beam before and after local control are shown in Fig. 6.6 for the same on-resonance and off-resonance excitation frequencies as were used in Fig. 6.4. The residual displacement distribution for the on-resonance excitation frequency, Fig. 6.6(a) is similar to that obtained by global control, Fig. 6.4(a). The effect of local control offresonance, however, is that although some reductions have been obtained in the final quarter of the beam, significant increases in the displacement are created elsewhere on the beam. -10

-20

. uncontrolled

-30

"0 v

3

------ c o n t r o l l e d

-40 -50

-60 -70

8°o J

I

2'o

I

I

do

I

do

I

J

Normalised frequency, to Fig. 6.3 The global cost function, proportional to the total kinetic energy on the beam, due to the primary force alone, at xp= L/6, (uncontrolled) and (controlled) with the effects of an optimally adjusted secondary force at Xs = 3L/4 (after Post, 1990).

0.028 0.024

159

a)

0.02 0.016 0.012 0.008 0.004 00

0.2

0.4

0.6

0.8

1

x/L 0.016

(

0.014 0.012 0.001 0.0008 0.0006 0.0004 0.0002

f

k3(/ --con,ro,,e

oo

0.2

0.4

0.6

0.8

t ffL

Fig. 6.4 Transverse displacement distributions on the beam after minimisation of the global cost function at (a) an on-resonance excitation frequency (~ =40) and (b) an off-resonance excitation frequency (co = 25) (after Post, 1990). -20 -30

uncontrolled controlled

-40 -50 1:13 -.o v -60 ~" -70 -80 -90 -100 0

20

40

60

80

100

Normalised frequency, o9

Fig. 6.5 The local cost function, proportional to the kinetic energy from x = 3L/6 to x = L on the beam, due to a primary force at Xp = L/6 alone (uncontrolled) and controlled with the effects of an optimally adjusted secondary force at Xs = 3L/4 (after Post, 1990).

ACTIVECONTROLOFVIBRATION

160 0.028[

~

0.024f

/

o.o~ol/ * °°'°I /

k

~" 0"014I /

\

\ \

o.oo.I/ o oo;v

/

/

/

/

~

\

(a)

\

\ \ / --unc?n][oile<J\ oo \

0.2

0.4

0.6 x/L

0.008 (b)

0.007

uncontrolled

0.006 0.005 "~" 0.004 0.003 0.002 0.001 0

0.2

0.4

0.6

t

F/////..,!

t

Xp

xs x/L

Fig. 6.6 Transverse displacement distributions on the beam after minimisation of the local cost function (proportional to the kinetic energy over the shaded region on the x axis) when excited (a) on-resonance (to = 100) and (b) off-resonance (to = 250) (after Post, 1990). Experimental approximations to the space-averaged mean square displacement minimised in the numerical simulations described above could, for example, be obtained from the sum of the squared outputs of a number of accelerometers on the beam. This is analogous to minimising the sum of the squared outputs from a number of microphones to obtain an approximation to the total acoustic potential energy. The positioning of such discrete error sensors, and the errors involved in this approximation to the true space-averaged cost function, are described in Section 10.11 of Nelson and Elliott (1992). These errors are minimised if the sensors are positioned so that they are able to observe each of the structural modes which are significantly excited. An alternative approach to the practical measurement of the residual vibrations is to use spatially distributed sensors which, for example, are preferentially sensitive to certain structural modes. Gu et al. (1994) discuss some of the practical problems of implementing such distributed sensors on two-dimensional structures, and show that such sensors

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

161

must be very accurately shaped if they are not to be sensitive to structural modes other than that for which they were designed. Nevertheless, Gu et al. (1994) experimentally observed significantly better attenuation of the (3, 1) structural mode on a plate using two specially shaped distributed sensors than when using two point accelerometers. Broadband excitation of structures is also an important problem. Vipperman et al. (1993) and Burdisso et al. (1993c) have analytically and experimentally studied the SISO feedforward control of transverse displacement at a point on a simply supported beam. In their work they investigated the use of IIR fixed filters as plant models (Vipperman et al., 1993) and studied the effects of non-causality due to delays through the control hardware, as also discussed in Section 6.13 of Nelson and Elliott. Their results demonstrate that as the control system becomes increasingly non-causal, the amount of achievable attenuation across the band is reduced. However, this can be somewhat compensated for if the number of coefficients in the adaptive filter is increased (Burdisso et al., 1993c).

6.3

Feedback control of finite structures

The various approaches to the feedback control of distributed structures have been described by Meirovich (1990), for example, and were also briefly discussed in Chapter 3. There has been a great deal of theoretical work in this area, with experimental investigations being rather thin on the ground. For this reason we concentrate on two examples of practical implementations, reported by Hodges (1989) and Rubenstein et al. (1991). The block diagram of an experimental arrangement used by Hodges (1989) is shown in Fig. 6.7. The objective of the experiment was to independently control the two lowest

F Cantileverbeam Accelerometers | Primary Secondary_l ~..~l~l, Force transducer voltageVp(t) shaker I~ "~lh' IFI Primary I 1]1 slaker Sec°ndary i~ Dual I voltage Vs( t ) i~I channelI , Current \\\\\\\\\\\" analyserII ~xamplifier IChargelf(t) I amp. I ~2(t) I Charge Reconstruction filter amp. I~ 1 t) ] Charge ] amp. I Micro-, I A ,,I DAC processor ADC , TMS 320C25+ host PC 4

I I

I I I

N I

I I I

Fig. 6.7 Block diagram of the modal feedback experiment on a cantilever beam described by Hodges (1989).

162

ACTIVE CONTROLOF VIBRATION

natural modes of a cantilever beam, which was 0.63 m long, 50 mm wide and 5 mm thick. A primary force was provided by a shaker which was connected to the beam via a force transducer, and driven by random noise, low pass filtered so as to have a bandwidth of 100 Hz. The secondary force acted at the same position on the beam, on which were also mounted two accelerometers, one close to the force inputs, and one close to the tip. The force inputs were arranged to be close to the node of the third structural mode on the beam, so that this mode was not significantly excited by either the primary or secondary force input. The outputs from the charge amplifiers driven by the two accelerometers were fed to a pair of analogue-to-digital converters. No anti-aliasing filters were necessary because the input force was band limited. The absence of antialiasing filters reduced the delay in the feedback loop, which was shown to be detrimental to the performance in Section 3.5. The sampled accelerometer signals were passed to a Texas Instruments TMS 320C25 processor mounted on a DSP (Digital Signal Processing) card in a host personal computer. The sample rate of the system was 50 kHz, which was chosen to minimise the processing time, and again reduce the loop delay. The processor implemented a pair of digital integrators (Hodges et al., 1990) and 55

"-" 45 "7t ~ -~ 35

uncontrolled

(a)

o

"" 25

-

t__

m 15 "o r-

5 -5

-15 -25

I

0

100

50 Frequency (Hz)

55 A

45

uncontrolled ........ controlled

"7o') -~ 35

~

~

k

(b)

o

"

25

rn 15 "o

5

-'~

-5

t~

-15 -25

I

0

50

100

Frequency (Hz) Fig. 6.8 Transfer inertances measured on the cantilever beam with and without (a) feedback of the velocity associated with the first structural mode; (b) feedback of the velocity associated with the second structural mode (after Hodges, 1989).

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

163

a modal analyser, which gave two outputs, proportional to the velocities of the lowest two structural modes on the beam. These signals were weighted by two modal velocity gains and fed back to the secondary shaker via a digital-to-analogue converter, reconstruction filter and current amplifier. Figure 6.8 shows the transfer inertance (acceleration per unit force) measured between the primary force and the tip acceleration for two different modal velocity feedback gain settings. Figure 6.8(a) shows the effect of feeding back only the velocity of the lowest order structural mode. It can be seen that the response of this mode at its natural frequency has been reduced by more than 20 dB because of the additional damping due to the velocity feedback (the damping ratio of this mode was increased from 0.005 to 0.158). The response due to the second mode, however, has hardly been affected by this feedback strategy. In contrast, Fig. 6.8(b) shows the effect of driving the secondary force with a signal proportional to the velocity of the second structural mode alone. In this case the amplitude of the second modal response has now been reduced by more than 15 dB at its natural frequency, but the response due to the first mode is largely unaffected. In fact, the natural frequency of the second mode has been slightly increased by this feedback strategy, which is probably the result of a small decrease in the modal mass due to delays in the feedback loop, as predicted by equation (3.5.6) in Section 3.5. In the investigation of Rubenstein et al. (1991), state feedback control was used to optimally control one structural mode on a 3 mm thick steel plate of dimensions 0.6 m x 0.5 m. The experimental arrangement is indicated schematically in Fig. 6.9. Twelve accelerometers were used on the panel whose outputs were passed through a 'modal filter' which used the assumed mode shapes of the plate to compute the modal accelerations of the first two structural modes. These were then used to update a Kalman filter (Section 3.9), which optimally estimated the modal velocities and

7 Panel • in frame -q

Primary shaker generating disturbance

-1

Twelve accelerometers

v

;construction filter riving secondary shaker

A

-kc

C

" -~

I~,

Kalman filter -:

Modal-filter -~

A D : C ~

4

Digital-toanalogue converters

Fig. 6.9 (1991).

LQR feedback gains

L_

Y

J

Analogue-to-digitalconverters

State estimator

Block diagram of LQR state feedback experiments described by Rubenstein et al.

164

ACTIVE CONTROLOF VIBRATION

accelerations of the first two structural modes, which, together with the states of the reconstruction filter and the disturbance model, were the eight state variables used to describe the system. The use of a disturbance model to augment the vector of state variables is described by Johnson (1976), and had been previously used in the active control of a torsional system, for example, by Burdess and Metcalfe (1985). These estimates of the state variables are then fed back to an electromagnetic shaker acting as the secondary source, via a set of eight feedback gains. The complete feedback system was implemented digitally on a transputer-based system operating at a sample rate of 3.3 kHz. A second-order reconstruction filter was used before driving the shaker, but again no anti-aliasing filters were used before sampling the inputs from the accelerometers. The LQR feedback gains were chosen to minimise a quadratic cost function of the sort described in Section 3.10, but which included only the amplitude of the first structural mode of the plate. These gains depend upon the assumed form of the disturbance. The disturbance was assumed to be either a pure tone at 60 Hz, close to the natural frequency of the first structural mode at about 50 Hz, or narrow-band random noise. In both cases the disturbance was modelled as white noise passed through a second-order resonant system whose damping was chosen so that its output was either harmonic or narrow-band random. It should be noted that the centre frequency of the disturbance is assumed known, so that the output of the disturbance model can accurately model the physical disturbance. In the case of a pure tone disturbance the disturbance observer will have no damping and will become an oscillator whose frequency is assumed to be known exactly, and whose states are steady state sinusoids. Under these conditions the controller designed using LQR feedback methods becomes 10 5

(a)

0

~-5 !

-100

0.1

0.2

!

0.3 0.4 Time (secs)

0'.5

O'.6

0.7

4 t,"-

o

.m

2 (b)

0 0


!

0.1

012

I

I

0.3 0.4 Time (secs)

15

O.

0.6

0.7

Fig. 6.10 The measured waveforms of the acceleration in (a) the first, and (b) the second structural modes of the plate used in the experiments by Rubenstein et al. (1991). The LQG controller, which was adjusted to optimally control the 60 Hz disturbance of the first mode only, is switched on after 0.2 s.

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

165

very similar to the feedforward methods described above, which assume a pure tone reference signal. The experimentally measured waveforms of the acceleration in the first two structural modes are shown in Fig. 6.10. The feedback controller is switched on after 0.2 s and the amplitude of the first mode is rapidly suppressed. The amplitude of the second mode, which is not included in the cost function being minimised, is, however, somewhat increased by the action of the feedback control.

6.4

Feedforward control of wave transmission

Practical engineering structures are often fabricated from a number of components, held together with relatively long, simple connecting elements. Examples of such constructions are large space structures, connected together with truss-work beams, large buildings, connected together with concrete beams, and the rotor and gearbox of a helicopter, connected to the fuselage with a small number of cylindrical struts. Rather than try to explicitly control the vibrations of the whole structure, as was described above, another approach to active control, which may be more appropriate at higher frequencies, is to prevent the transfer of vibrational energy between the components of the structure, along the connecting elements. The connecting elements must thus be characterised by an input/output relationship which describes this energy flow in terms of the internal mechanical state variables of the connection. A convenient way of describing this process is in terms of the propagation of a number of uncoupled structural waves along the connecting component. Common types of structural wave motion in components such as beams and struts are longitudinal, flexural and torsional. Longitudinal and flexural waves were introduced in Chapter 2, and for a more complete description of structural wave types, the reader is referred, for example, to the books by Graft (1975) and Cremer and Heckl (1988). Another justification for breaking up the problem of actively controlling the whole structure into individual wave control systems is provided by the inherent limitations of a global control system at high frequencies. Von Flotow (1988a), for example, discusses controlling the individual modes of a structure at excitation frequencies for which a large number of structural modes are significantly excited, which he called the 'acoustic limit'. Von Flotow demonstrates the extreme sensitivity of the eigenfunctions, or mode shapes, to small modelling errors in this frequency range and the difficulty in designing a stable system to control these ill-defined structural modes. Figure 6.11 illustrates the general form of an active system for the control of vibrational waves in one direction along a structural component. The vector a~ denotes the amplitudes of the incident components of the different wave types, which impinge on the sensor array being used to detect the waves. The signals from the sensor array are fed to the controller, which then drives the actuator array. The actuator array, in general, causes waves of each of these different types to propagate in both the fighthand and left-hand directions. The left-hand going waves make up the elements in the vector of reflected components aR, and the interference between the right-hand going waves generated by the actuator array and the original incident waves results in the elements of the vector of transmitted wave components aT. Although the sensor and actuator array are shown as being distinct in Fig. 6.11, implying a feedforward control strategy, they could also overlap, or be co-located, as in a feedback control system.

166

ACTIVE CONTROLOF VIBRATION

f az

Structural element

aR v

I

Detection sensor array

I

Controller

1F

I

Secondary actuator array

Fig. 6.11 General form of an active system for the control of a number of wave types on a connecting element joining parts of a larger structure. Clearly the objective is to design the controller so that the transmitted components are suppressed and a t = 0 . This turns out to be possible if the wave amplitudes are measured at a single plane to the fight of the secondary source, in contrast to the measurements which would ordinarily have to be made at multiple points along the structure to measure the total energy, for example. Under these conditions the parts of the structure on the fight-hand side of the control system would be effectively isolated, as far as vibration were concerned, from those on the left-hand side. In order that the control system least disturb the components on the left-hand side of the connecting component, it would also be desirable that the reflected wave components were also zero, i.e., a R = 0. This condition is only possible whilst fulfilling the condition that a r = 0, if the actuator array is perfectly directional for each of the wave types. We now consider the detailed design of two different structures for a feedforward controller in the general multi-channel case. We shall return to a more detailed consideration of a single-channel system for the control of flexural waves in beams in Section 6.7. From what has been described in the previous section, the most obvious structure of a feedforward system for the control of a number of different wave types is illustrated in Fig. 6.12. In this figure the signals from the detection sensor array are decomposed into the amplitudes of the incident components of each of the various

Detection aI sensor array

I

aR 4__

Secondary Error actuator a T sensor array __=,. array

I L~

I

I

il

Wave analyser Wave analyser K

Fig. 6.12 The structure of an adaptive feedforward system for the active control of multiple wave types, which uses individual controllers for each type of wave, adapted independently.

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

167

wave types using a wave analyser, or w a v e vector filter, as introduced in Section 5.10. These incident wave amplitudes are then fed forward by individual, single-channel, control filters and drive the inputs to a w a v e synthesiser which combines these signals together into a set of drive signals for the actuator array. The control system is made adaptive, in a similar way to that described in Chapter 4, by adjusting the controller to minimise the outputs of an error sensor array, which provides an estimate of the net transmitted wave amplitudes. Notice that the output of the error sensor array is also decomposed into individual wave components, so that each control filter is adjusted independently to minimise only one error signal. It is interesting to compare the control structure illustrated in Fig. 6.12 with that for modal f e e d b a c k control shown in Fig. 3.11, in which case the modes are those of the whole vibrating system, rather than wave components on only one connecting part. When the waveform of the disturbance being controlled is deterministic rather than random, the detection sensor array can often be dispensed with. If the original, primary, source of vibration is a rotating or reciprocating machine, for example, a tachometer signal can be used to provide a reference signal to drive the individual controllers, which has the same frequency content as the incident waves. The error and associated wave analyser must still be retained, however, if the controller is to be made independently adaptive. An arrangement using multiple independent controllers with a deterministic reference signal, has been described theoretically by Pan and Hansen (1991) and experimentally by Clark et al. (1992b), in which the longitudinal wave and the two flexural wave components were actively controlled on a 25 mm x 50 mm thick aluminium beam. The wave analysers and synthesisers illustrated in Fig. 6.12 assume a detailed knowledge of the dynamics of the connecting component within which the waves are propagating, and generally require actuators and sensors whose sensitivity are closely matched in amplitude and phase, as will be described in Sections 6.5 and 6.6. An alternative approach to the control problem is to dispense with these wave analysers and synthesisers, to connect each actuator to each detection sensor via an array of control filters, and to adjust each of the control filters to minimise the sum of the squared output signals directly observed by the error sensor array. The adjustment of each of the control filters in this case is no longer independent, and a change in one control filter now requires compensating changes in the other control filters, and so a fully coupled multi-channel control algorithm must be used, such as that described in Sections 4.5, 4.6 and 4.7. The advantage is that the need to implement wave analysers and synthesisers, and the requirement for matched transducers, is removed. Fuller et al. (1990a), for example, describe experiments in which the amplitudes and phases of the inputs to two piezoceramic actuators on either side of a beam were adjusted to cancel both the flexural and longitudinal motion, as described in more detail in Chapter 5. Such a fully coupled approach has been used for the simultaneous control of longitudinal waves and the two flexural wave components using three magnetostrictive actuators inside a cylindrical strut structure by Brennan et al. (1992a) and Elliott et al. (1994). In these experiments, the amplitudes and phases of the signals driving the three actuators were adjusted to minimise the sum of the squared outputs from six accelerometers on the receiving mass. It is difficult to proceed any further in such a general discussion without some knowledge of the propagating wave types and the properties of the sensor and actuator

168

ACTIVE CONTROL OF VIBRATION

arrays. Arrays of sensors for the decomposition of waves in structures have been considered in a general sense in Chapter 5. The properties of sensor and actuator arrays will now be considered in the following two sections for the special case of flexural waves. We consider the detection and generation of flexural waves in particular for a number of reasons. Firstly, because they are often the most important wave type in many kinds of structure, in that they can have large out-of-plane displacements and can readily cause the radiation of sound. Secondly, flexural waves are dispersive, and have a nearfield component associated with their generation, as described in Chapter 2, which complicates their behaviour somewhat. If the detection and generation of flexural waves are understood, the behaviour of other wave types can be deduced by deleting the nearfield terms in the more general expression. Lastly, the detection and generation of longitudinal waves in structures has many analogies with the detection and generation of plane acoustic waves in a duct (since both obey a second order wave equation and are non-dispersive) a subject which was considered in detail by Nelson and Elliott (1992).

6.5

Actuator arrays for control of flexural waves

Following on from the analysis presented in Section 2.4, in which the propagating and near-field components of the flexural waves generated on either side of a point force acting on a Euler-Bemoulli beam was considered, in this section we consider the result of driving an array of point forces on an infinite beam. Figure 6.13 shows the general arrangement, in which up to four secondary forces may be used to control the propagating and near-field components on either side of the array. We know from Section 2.4 that the complex transverse displacement of the infinite beam on the fight hand side of a point force, Fs], at the origin can be written mj~ 1 "(e-jk~x-je

w(x)= 4Elk 3

),

x>0,

(6.51)

where the implicit e TM time dependence has been omitted, k/ is the flexural wavenumber, given by k/= to~/2(pA/EI) ~/4, and x is the position along the beam.

Ap

en

An beam

Fsl

Fs3 Fs4 IP.X

Fig. 6.13 The positions of the four point forces which make up the most general case of the actuator array for flexural waves on a beam and the definition of the propagating and near-field components generated on either side of the array.

169

ACTIVE CONTROL OF VIBRATIONIN STRUCTURES Equation (6.5.1) can generally be written as

w ( x ) = Ape -&x + A,, e -k/,.

for

x>0,

(6.5.2)

where in this case,

Ap=

-jFsl

-Fsl A,, = ~ .

and

4Elk~

(6.5.3a,b)

4Elk~

Similarly, on the left hand side of the beam, we can, in general, express the complex transverse displacement as

w ( x ) = Bp e & x + B n ekl x,

for

x<0

(6.5.4)

where, again in this case,

jFsl

Bp=

and

4EIk~

-Fsl

Bn = ~ .

(6.5.5a,b)

4Elk~

Figure 6.14(a) shows the modulus of the complex transverse displacement on either side of a single point force which is driven to cancel an incident propagating flexural wave of unit amplitude (so that Ap = - 1 ) as a function of the non-dimensional distance x/2. The incident wave could have been generated, for example, by a primary point force a long distance away on the infinite beam. Notice the evanescent near field on the 2.5

1.2

(a)

I w/Ai] 1

[ w/Ai[ 0.6 0.4

0.5

0.2

0

-2 -1.5 -1

1

0.8

1.5

i

-0.5

0

,

0.5

1

1.5

2

i

02-~.~-~-o.5

x/X

1.2

(b)

q

1

2

o 0.5

ff

i

1.2 1

0.8

0.8

I w/A,l 0.6

I .,/~,108

0.4

0.4

0.2

0.2

0 -2 -1.5 -1

-0.5

0

0.5

1

1.5

2

x/X

0 -2 -1.5 -1 -0.5

2

(O)

(c) 1

11~

0

0.5

1

1.5

2

xD.

Fig. 6.14 Modulus of the transverse displacement of a beam as an incident flexural wave is controlled using an array of: (a) one point force; (b) two point forces; (c) three point forces; (d) four point forces.

170

ACTIVE CONTROL OF VIBRATION

fight hand side of the actuator, and the standing wave on the left hand side of the actuator caused by the interference between the incident wave and the reflected propagating wave. It is clear that using a single force actuator as a secondary source, we can only cancel an incident flexural wave by at the same time generating a near field on either side of the secondary actuator, and also generating a reflected propagating wave. By using two secondary forces it is possible to generate a propagating wave on the fight of the array without a corresponding propagating wave being generated on the left. This can be understood by first writing the amplitudes of the propagating and nearfield wave components, on the fight and left hand sides of the actuator array, due only to the secondary force F~2 at x - 12 as

Ap= -iF,2 4Elk}

e

An =

'

-F,2 4Elk/

e

(6.5.6a,b)

'

and

-jFs2

Bp=

e

-jk/

-F,2

, .4,=

4EIk:

e

-k/,

,

(6.5.7a,b)

4Elk/

Notice that Ap is advanced by a phase angle of k/lz, compared with the case where Ap is generated by F,~ at x = 0, and Bp is now similarly delayed by a phase angle of k/12, compared with the case where Bp is generated by F,~. The net amplitudes of the two propagating waves due to the action of both F,z and Fs2 can now be written in matrix form as

Bp

[1 eJ"lrl

= 4~-k}

1 e

-JkA" .. JLFs2j

If we wish to drive Ap to some desired level Apo, and suppress action of F~I and F~2, we can use equation (6.5.8) to obtain

F,2

=j4

1

] 1[o]

e -'i~t2

.

(6.5.8)

Bp by

the combined

(6.5.9)

This matrix equation can be solved to show that 2Elk 3

a,,o

(6.5.10)

-::12F~2.

(6.5.11)

sin/9/2 and Fsl "-

-e

This combination of secondary forces is analogous to the strengths of a pair of acoustic sources arranged to cancel downstream radiation with no reflection in a duct (Nelson and Elliott, 1992, Section 6.7). Figure 6.14(b) shows the modulus of the transverse velocity due to two point forces, driven according to equations (6.5.10) and (6.5.11), for a separation 12= 2/10, together with that of an incident wave of unit amplitude (Brennan et al., 1992b). The standing wave on the left hand side of the

171

A C T I V E C O N T R O L O F V I B R A T I O N IN S T R U C T U R E S

actuator array has clearly been suppressed at the cost of a slightly more intense near field. Two control forces could alternatively be used to suppress both the propagating and near-field components on the fight hand side of the array (Mace, 1987; McKinnell, 1989), in which case a propagating wave would again be generated on the left hand side by the array, causing a standing wave. It is possible to suppress the near-field contribution to the fight of the actuator array, as well as suppressing the transmitted and reflected travelling waves, by using three point forces. This may be important if the beam connecting two parts of a structure is short and the distance from the actuator array to the structure on the fight hand side is small compared to the flexural wavelength. The near-field component A n under these conditions could combine with the reflected near-field component generated by the structure on the fight hand side of the array, and allow the transmission of energy into the structure, as described, for example, by von Flotow (1988a). The contributions to the two propagating and fight hand near-field amplitudes from Fsl a t x = 0, F~2 at x = l 2 and Fs3 a t x = l 3 can be written in matrix form as Ap

1

An =

-j 4Elk}

Bp

-j 1

e ~k/~

e jk/~

F~I

. jk/2 e jk~t3 F~2 -J e -j e -i~'~t: e -j~'/~ Fs3

(6.5.12)

in which equations (6.5.3), (6.5.5), (6.5.6) and (6.5.7) have been used, together with the corresponding equations for Fs3. The required values of Fs~, Fs2 and F,3 can be found by solving equation (6.2.12) for [ApAnBp]=[ApoO0]. The displacement distribution on the beam resulting from such an actuator array, with 12= 0.1). and l 3 =0.22, used to cancel an incident wave of unit amplitude is shown in Fig. 6.14(c) (Brennan et al., 1992b). It is also possible to conceive of an array of four point secondary sources which suppress the near-field components on both sides of the actuator array as well as the left hand going travelling wave (Scheuren, 1985 and 1990). Extending the matrix formulation above to four forces, we can then write "

.

Ap "

1

e jk/2

e j~t~

An

-j

-j

- j e ik/2

- j •e jk/3

Bp

4EIk}

1

e -jk/2

e -j~/3

Bn.

-J

_je-~k/2 _je-~3

e J,/4

-je

jkfl4

Fsl" Fs 2

(6.5.13)

e -jkfl4

Fs3

.-jkz~

Fs4.

-je

This equation can be solved to give F~I, F~2, Fs3 and Fs4 for any combination of generated wave components. In particular, Fig. 6.14(d) shows the resulting transverse displacement of the beam with an incident wave of unit amplitude, and if [ApA~BpB~] is set equal to [-1 0 0 0 ] with/2 = 0.12,/3 = 0.22 and/4 = 0.32. It should be noted that a progressively larger control effort is required as more secondary sources are introduced to control a larger number of wave types. If the separation between each of the secondary forces is 0.12, the sum of the squared forces required to achieve the result shown in Fig. 6.14(b) with two secondary forces is 1.45 times that required when a single secondary force is used, Fig. 6.14(a). When three

172

ACTIVE CONTROLOF VIBRATION

secondary forces are used, Fig. 6.14(c), the sum of the squared forces is nearly six times that required with a single force, and with four secondary forces, Fig. 6.14(d), the sum of the squared forces is about 27 times that required with a single force (Brennan et al., 1992(b)). This increase in control effort with the more complicated actuator arrays becomes more pronounced at lower frequencies, for which the actuator separation is an even smaller part of a flexural wavelength.

6.6

Sensor arrays for control of flexural waves

We can use matrix methods similar to those used in the previous section to decompose the outputs of an array of sensors into a set of constituent flexural wave amplitudes. This method is an extension of that outlined in Section 5.10 for non-dispersive media. Consider first the most general case in which both propagating and near-field flexural wave components are present in both the left hand going and fight hand going directions. An array of four sensors is required to independently measure these four components, and we will assume, for the sake of convenience, that an array of accelerometers is used, spaced along the beam as indicated in Fig. 6.15(a). It is further assumed that the accelerometers have negligible mass and rotary inertia, a condition discussed, for example, by Mace (1991). The general expression for the transverse acceleration of the beam, with all four wave types present, as illustrated in

Bp~ ~ Ap BnW-......~_ ~~An //////////////////////////////////////////////////////

Y

Y

(a)

beam

(b)

/H//////////////////////////////////////////////////

beam

Y Fig. 6.15 (a) A four-sensor array used to decompose the flexural wave field into two propagating and two near-field components; (b) a two sensor array used to decompose the flexural wave field into two propagating wave components.

173

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

Fig. 6.15(a), is then

~(x) = -o92(Ape-J~sx + A~e-kI x + BpeJtq x + B, eklX).

(6.6.1)

The vector of accelerations measured at four sensors located at x = 0, 12, 13 and l 4 is thus

"f+(O) i

1

1

1

1 ]'ALp"

(12) = --(2)2 e -i~/2 e -~2 eJkJt: e ~t: An #(/3) e-J~'/3 e-~/3 J'/~ e St3[ Bp

(6.6.2)

"

e-Jkfl4 e-~/4 ej~:II4 e kfl4] .B,,.

.W (/4).

It is clear that, provided the square matrix in equation (6.6.2) is not singular, the vector of wave amplitudes, [ApA,,BpB,,] "r, can, in principle, be calculated from the vector of accelerations at the sensors by inversion of the square matrix. In practice, however, the matrix inversion will be prone to errors due to ill-conditioning as discussed, for example, by Halkyard and Mace (1995). If the sensor array is more than a flexural wavelength away from any sources or discontinuities on the beam, only the two propagating waves will then have a significant amplitude. Under these conditions, only two sensors need be used to decompose the wave field into the two propagating components, as illustrated in Fig. 6.15 (b). The accelerations at these two sensor positions, chosen to be symmetrically positioned about the origin, at x = - 1 / 2 and l/2 in this case, may be written in terms of the propagating wave amplitudes, Ap and B p, as

f/,(l/2)

e -jk//2

e jk//2

Bp

The vector of propagating wave amplitudes may be readily obtained in this case, by inversion of the 2 x 2 matrix in equation (6.6.3), which gives

Ap = j e j~1/2 - e Bp 2(2)2 sin kfl _e-J~?t/2 eJ~?/2 ~(1/2) "

(6.6.4)

The wave component propagating in the fight hand direction, for example, is thus given by

A~, =

J

[eJ~A/zfi,(-l/2) - e-i~~'/2~(l/2)].

(6.6.5)

2o92 sin kfl Further simplification is possible if it is assumed that the spacing between the sensors is small compared with the flexural wavelength, i.e., kll ~ 1. Under these conditions, sin kil~. kil, e±J~'/2 = 1 + jkil/2, and equation (6.6.5) can be written as

Ap= 2o92

jkI

l

,

(6.6.6)

174

ACTIVE CONTROLOF VIBRATION

in which the two terms in the inner brackets can be recognised as the finite difference approximations to the acceleration and the spatial derivative of the acceleration at the position x = 0 on the beam. Equation (6.6.6) is thus the structural equivalent to the method proposed by Elliott (1981) for decomposing a one-dimensional acoustic field into its two constituent plane wave components using the outputs of two closely spaced microphones (see Section 5.10 for the method applied to longitudinal waves). The implementation of such a wave sensor for flexural vibrations is more complicated than the equivalent acoustical or longitudinal case, however, since the waves are dispersive and the factor 1/jk I appearing in equation (6.6.6) does not correspond, within a constant of proportionality, to a pure integration, as it does in the non-dispersive (longitudinal wave) case. Further practical difficulties arise if any differences in the response of the two accelerometers are not carefully compensated for, particularly the phase response, and due to finite difference errors and the presence of near field, as discussed, for example, by Redman-White (1984). In related work, Gibbs et al. (1993) have demonstrated the separation of broadband travelling bending waves on a finite beam in which the matrix equation (6.6.4) is solved using summing and differencing circuits in conjunction with IIR filters to model the system broad band transfer functions.

6.7

Feedforward control of flexural waves

In this section we will look in more detail at the form of feedforward controller required for the suppression of flexural waves with random waveforms propagating along a beam. The individual elements of detection sensor, controller, secondary actuator and error sensor are shown in Fig. 6.16. We will assume, however, that only the simplest form of point acceleration sensor and point force actuator are to be used, which detect or generate flexural waves propagating equally well in either direction. The feedback path from secondary actuator to detection sensor, mentioned in Section 4.2, must thus be explicitly considered in this case. The acceleration at the detection sensor can be written as the sum of the contributions from the primary source, ¢0p(-l~), and the secondary source, calculated using the results of Chapter 2, so that •

jo9

2

.

fO(-la) = ~ ( - l l ) + 4Elk~ (e-jk/' - j e

-kA

)F~,

(6.7.1)

Similarly, the acceleration measured at the error sensor can be written as •

¢0(/2) = ~(/2) +

jo9

2

4Elk~

(e -jk/~ - j e-~t~)F~.

(6.7.2)

Assuming the primary source is some distance away from the control system, and that no scattering of the primary wave occurs due to the presence of the secondary source or sensors, the acceleration due to the primary source at the error sensor can be written in terms of that at the detection sensor as ffp(12) = w p ( - / 1 ) e-Jk;(t~ ÷ t2),

(6.7.3)

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

175

Incident flexural wave on beam

l

l

at x = 0

"1"-I /H

Excitation (detection) accelerometer at x = - ~ 1

?

Secondary actuator l

Response error

~~cce~erometer

" [I- at x=~ 2

Adaptive digital controller

Fig. 6.16 An adaptive system for the active control of broadband flexural disturbances on a beam.

The controller, H, is used to drive the secondary source using the signal from the detection sensor, so that /7,= Hff(-/1).

(6.7.4)

Assuming there is no measurement noise, the controller can, in principle, drive to zero the acceleration at the error sensor given by equation (6.7.2). This condition, together with equations (6.7.3) and (6.7.4), can be used in equation (6.7.1) to give an expression for the ideal controller, Go, which drives ff(12) to zero, in terms of the physical properties of the beam (Elliott and Billet, 1993). This expression is given by Ho

4Elk~

=

jw 2

- e -jki(t~ +t2)

(e-JkI~_ j e -k~) _

e-Y~(/~

+/2)(e-Jkf/l_ j e -~/l)

.

(6.7.5)

Notice that the objective of the controller is to drive the total acceleration at the error sensor position to zero. Since, in general, the acceleration at this position has contributions from the propagating flexural waves due to the primary and secondary sources, a n d the near-field component of the response of the secondary actuator, driving the signal to zero will not necessarily suppress the propagating part of the flexural wave. The attenuation of the p r o p a g a t i n g component can be analysed under these conditions by expressing the net acceleration at the error sensor, assuming it is remote from any discontinuity, as w(12) = A e -jkt2 + B(e-J*: t2 - j e-k:t2),

(6.7.6)

where A is the incident propagating wave amplitude as it passes the secondary actuator, and B the contribution from the secondary source. If the secondary source is adjusted to drive if(12) to zero, then -jk~ -Ae B = - j k ~ e-,r~" e -j

(6.7.7)

176

ACTIVE CONTROL OF VIBRATION

The residual propagating wave amplitude will be equal to A + B and the ratio of this to the original incident wave amplitude can be deduced, following Elliott and Billet (1993) and using equation (6.7.7), to give

A +B

- j e -kA -

A

. e -jk~t2_ j e -k~t2

(6.7.8)

Providing e-kl t2 is reasonably small, the attenuation level (in dB) is an approximately linear function of kll 2 (Pan and Hansen, 1993). The attenuation level achieved by driving the acceleration to zero is plotted against frequency in Fig. 6.17, for the case of a 6 mm thick steel beam with 12 = 0.7 m. It can be seen that in this case, provided attenuations of no more than 20 dB are anticipated, the effect of the near field of the secondary actuator at the error sensor is negligible above about 12 Hz. At this frequency the distance from the secondary source to the error sensor is about 3/8 of a flexural wavelength. Provided the control system only operates above this frequency, then we can make the approximation e-kZt2~ 1 in equation (6.7.5) for the ideal controller, which then becomes

Ho

=

4EIk~ -e , jw 2 1 -- e-J~ll(e -j~ll - j e -~/l)

(6.7.9)

which can also be written (Tartakovskii and Knyazev, 1965; Scheuren, 1985) as

Ho

=

4EIk~

1

o92

2 sin

.

kill

+

(6.7.10)

e -kd'

Under these conditions the response of the ideal feedforward controller only depends on the 'local' response of the section of beam under control, from detection sensor to 60

50 A

I:I)

40 ,4,-'

.E_ ,'-

30

:::}

"

20

<

10

0

I

I

I

10

20

30

I

I

I

I

40 50 60 70 Frequency (Hz)

I

I

80

90

100

Fig. 6.17 Maximum attenuation of a propagating flexural wave in a 6 mm steel beam using a control system with a single error sensor a distance of 0.7 m from the secondary source. The reductions are limited by the near field of the point secondary force.

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

177

secondary source (von Flotow, 1988a), an observation made previously in the case of feedforward systems for acoustic control by Roure (1985). Whereas the near-field components of the response of the secondary actuator provide a fundamental low frequency limit on the performance of such a control system, the dispersive nature of flexural wave propagation generally leads to a fundamental high frequency limit. This can be demonstrated by calculating the group delay from detection sensor to secondary source;

.gg= ll Okf _

ll ( pA ) 1/4

Oco - 2 - - ~ ~ - H ]

'

(6.7.11)

which is plotted as a function of frequency in Fig. 6.18 for the case of a 6 mm thick steel beam with l l - 1 m. The electrical controller has to model a rather complicated frequency response (equation (6.7.9)) and the most accurate and flexible way of implementing such a controller is to use digital filters. Such digital controllers have an inherent delay associated with their response, however, due to the processing time of the digital device and the phase lag in the analogue anti-aliasing and reconstruction filters which have to be used (see, for example, Nelson and Elliott, 1992, Section 6.13). It is clear from Fig. 6.18 that at some upper limiting frequency the group delay along the beam will become less than the inherent delay in the controller, and active suppression of flexural waves with random waveforms will not be possible. In the experiments reported by Elliott and Billet (1993), for example, the delay in the processor was about 2.4 ms and, from Fig. 6.18, broadband control was limited to frequencies below about 800 Hz for the 6 mm steel beam used in the experiments. It is interesting to note that if broadband feedforward control of longitudinal waves on a steel beam was being contemplated using a controller with an inherent delay of 2.4 ms, a distance of greater than 12 m would be required from detection sensor to secondary actuator. This is because although longitudinal waves are not dispersive, they have a high wave speed; about 5000 meter/sec in a steel beam. As the beam becomes thinner,

20 18 16 ~'14 E >,12 '~ "O 10 8 0

(.9 6 4

2 0

i

0

i

I

i

I

!

i

i

i

100 200 300 400 500 600 700 800 900 1000 Frequency (Hz)

Fig. 6.18 The group delay of a flexural wave propagating a distance of 1.0 m from the detection sensor to the secondary source for a 6 mm steel beam.

178

ACTIVE CONTROL OF VIBRATION

-35 -40 x~-45

8 -5o e.

....

a~ -60

|

O.

i70

-75

: . , :.."";" : / ' ; " t '

. ': .'"

P

0

(

L 100

|:

• -,Z-"

.. : ~v_

i:

.-

.,,,~.

~ ~'.'L~-

,...,;,:i . i ~ 200 300 400 500 Frequency (Hz)

t

600

700

800

Fig. 6.19 The power-spectral density at the response error sensor on the beam illustrated in Fig. 6.16 before control ( ) and after the adaptive controller has converged (- - -). the upper frequency limit for the control of flexural waves (due to wave dispersion) is increased and the lower frequency limit (due to the near field of the secondary actuator) is decreased, so that for a given controller geometry, the bandwidth of control is very much greater for a thin beam than it is for a thicker one. Figure 6.19 shows the practical results from a feedforward system for the active control of flexural waves on a 6 mm thick steel beam using an adaptive controller (Elliott and Billet, 1993). In this study, accelerometers were used as detection and error sensors, and a coil suspended in a fixed magnetic field was used as the secondary actuator. The beam was terminated in sand boxes which provided anechoic terminations for flexural waves above a few hundred hertz. The acceleration at the error sensor is plotted in Fig. 6.19 before and after active control, and it can be seen that reductions in level of up to 30 dB have been achieved from about 100 Hz to 800 Hz. The reductions in the transverse acceleration were also measured at a monitoring sensor position, further along the beam than the error sensor, and the attenuations observed were similar to those at the error sensor, which are shown in Fig. 6.19. At frequencies below 100 Hz, external sources of vibration contribute significantly to the output of both detection and error sensors, with a resultant loss in the coherence between their outputs. This limits the performance available from any control system, as was described in Section 4.3. Above about 800 Hz, the delay associated with the propagation of flexural waves from detection sensor to secondary actuator becomes less than the electrical delay in the controller, resulting in the performance limitations due to causality described in Section 4.2.

6.8 " F e e d b a c k c o n t r o l o f f l e x u r a l w a v e s In contrast to feedforward control, in which the detection sensor is placed upstream of the secondary actuator, in this section we will consider the case of a co-located sensor

ACTIVE CONTROLOF VIBRATIONIN STRUCTURES

179

and actuator. The control strategy is thus one of feedback control, as discussed in Chapter 3. A simple, single-channel, case of such a feedback strategy applied to the control of flexural waves on a beam is illustrated in Fig. 6.20. In this example, the transverse acceleration of the beam is detected and fed back, via a gain -ga, to the secondary force actuator. The magnitude of this point force is thus

Fs = -ga~(O),

(6.8.1)

where it has been assumed that the secondary actuator is at the origin of the longitudinal (x) axis. The transverse displacement of the beam at x = 0 is the superposition of the displacement due to the incident flexural wave, A t, and that due to the secondary force, F~. The latter contribution can be obtained from equation (6.5.1), for the transverse displacement of a beam subject to a point force at the origin, by setting the position, x, at which the displacement is measured to zero. The net displacement at the origin is thus (1 +j)F~ w(0) = ,4/- ~ .

4Elk~

(6.8.2)

The amplitude of the flexural wave propagating on the upstream side of the secondary force, which is equal to the transmitted wave, is given by

ar = A~ - 4EI-----~"

(6.8.3)

In order to calculate the effect on the transmitted wave of the feedback control strategy illustrated in Fig. 6.20, we must solve equations (6.8.1) and (6.8.2) for F~, and substitute this into equation (6.8.3), which yields

Ar = i l +~-t_o~+'-4-EIk} At.

(6.8.4)

It has been shown in Chapter 3 that feeding back a force proportional to acceleration was equivalent to adding mass to the mechanical system. The feedback strategy illustrated in Fig. 6.20 is thus equivalent to attaching a point mass (m = ga) to the beam. The effect of such a point mass on the propagation of flexural waves has been analysed by Mead (1982), for example, who gives the equation for the transmitted wave as being

A r = [ -AJ I( '4p+- jl (O4]+ p ) I

(6.8.5)

x

Fig. 6.20 Acceleration feedback control system, using a co-located accelerometer and a point force actuator, for control of the propagation of flexural waves on a beam.

180

ACTIVE CONTROLOF VIBRATION

in which p = ~o2m/EIk}=mkJpA, which is a more elegant version of equation (6.8.4). The variable p is equal to the added mass divided by that of a length of beam equal to about 2/6. The power transmission coefficient is defined to be the ratio of the power in the transmitted flexural wave to the power in the incident flexural wave, which in this case is given by [Ar[ 2 a T -"

IA/ 12

(4 + p)2 -"

(4 +/.t) 2 +

(6.8.6)

~U 2 ,

where p = gakJpA, by analogy with the definition above. The variation of this transmission coefficient with the parameter p is plotted in Fig. 6.21(a). It can be seen that the flexural wave is progressively blocked by the feedback system as the gain ga, and hence p is increased, until the power transmission coefficient reaches a limiting value of 1/2 for p ~ oo, which corresponds to a very high value of point mass imposed on the beam. It was shown in Section 2.5 that the power flow associated with a flexural wave in a beam is equipartitioned between contributions from the transverse force and velocity, and from the moment and angular velocity. The point mass is able to reflect the contribution from the transverse motion but will have no effect on the contribution from the angular motion, so that as p ~ oo, aT--> 1/2, as observed. The case in which the discontinuity on the beam has both linear and rotational inertia has been analysed by Cremer and Heckl (1988). They show that this combination of discontinuities can completely suppress a propagating flexural wave at a single critical (a) Acceleration feedback t°O m if}

1.0

E l'-- . E

O 13..

(b) Velocity feedback

L_

0.0

i

0

J

~

1.0

c °O ~ if}

t

i

i

~

,

Normalised gain,/t

O t'l

,

0.0

~

100

i

0

i

~

i

~

~

Normalised gain, v

,

J

100

(c) Displacement feedback 1.0 c°O ~ 0~

i,tl:i

°~ l~::

11)

O

O

a..

I

0.0

I

0

I

I

I

I

Normalised gain, e

I

I

100

Fig. 6.21 The effect on the power transmission coefficient of a beam of varying degrees of: (a) acceleration feedback control; (b) velocity feedback control; (c) displacement feedback control.

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

181

frequency. Clearly both mechanisms by which energy can be transmitted by flexural waves in the beam are affected by such an arrangement. If the controller is designed to feed back the velocity of the beam so that F~ = -g~w(0),

(6.8.7)

then by a similar line of reasoning to that used above, the resultant propagating transmitted wave is given by (j -- S ~ -

4-EI/93 ,41.

(6.8.8)

This may be expressed as

jr-4 ] Ar = - j r - (4 + v) A~, where v=~og,,/Elk}=g,,k:/copa. feedback controller is thus

at=

(6.8.9)

The power transmission coefficient for such a [at[ 2 .2 =

[A, [

v 2 +16

v2 + (4 + 1,,)2

'

(6.8.10)

which is plotted in Fig. 6.21 (b). The power transmission coefficient falls more rapidly with feedback gain in this case, reaching a value of a r=0.4 for v = 8 , but then increases for higher values of feedback gain. As the feedback gain becomes very large, the beam is effectively pinned, or simply supported, at the point of application of the secondary force. This boundary condition again reflects the power in the flexural motion associated with the transverse motion, but cannot affect that associated with the rotational motion, so that again the power transmission coefficient tends to the value of 1/2 as the feedback gain becomes very large. Finally, we will consider what turns out to be the most interesting case of feedback control; when the secondary force is proportional to the transverse displacement of the beam, so that

F~ = - gdw(O).

(6.8.11)

Following the same steps as above, we obtain the equation for the transmitted propagating wave which in this case is given by

]

(1 +J)ga + 4EIk~ A~,

(6.8.12)

which can be written, by analogy with the result of Mead (1982) for a spring attached to the beam, as AT =

j ( 4 - e)

(6.8.13)

e + j ( 4 - e) in which e= ga/Elk}= gak:/w2pA. The corresponding value of power transmission

182

ACTIVE CONTROL OF VIBRATION

coefficient becomes ( 4 - e) 2 ar = (4 - E) 2 - E2

(6.8.14)

which is plotted in Fig. 6.21 (c). It can be seen that as gd and hence e, become large, which again corresponds to the case of a pinned or simply supported discontinuity on the beam, then again a r tends to the value of 1/2. The most interesting aspect of Fig. 6.20(c), however, is that aT has a minimum which, from equation (6.8.14), is seen to be given by ar=0

if

e=4.

(6.8.15)

The value of the ideal feedback gain corresponding to e = 4 is thus g d = 4 Elk},

(6.8.16)

so for a given frequency of excitation there is a single value of displacement feedback gain which completely suppresses flexural waves. In conventional terms, this result could also be achieved by a linear spring, having an appropriate spring constant, acting between the beam and an inertial reference position. Unfortunately because of the frequency dependence of kI in equation (6.8.16), this critical value of feedback gain will change if the excitation frequency is changed, and broadband control cannot be achieved with a single gain setting. If the gain of the ideal feedforward controller, equation (6.7.9), is considered in the limit as the distance between sensor and actuator, l l, tends to zero, we obtain lim Go =

l,---)0

F~ = 4Elkf

W(0)

O.)2 '

(6.8.17)

which thus results in F~ = - (4Elk})w(O),

(6.8.18)

exactly as obtained for the displacement feedback controller in equation (6.8.16). It is interesting to note that to suppress flexural waves on a beam, the ideal feedback controller should have a finite gain, whereas the ideal feedback controller for suppressing acoustic plane waves propagating in a duct should have an infinite gain (Nelson and Elliott, 1992, Section 7.8). This difference is due to the near-field contribution to the response in the case of flexural waves on a beam. The simple feedback controller is not as useful as it first appears, however, since the optimal feedback gain, given in equation (6.6.16), has a response which is real, but proportional t o 0.) 3/2 (since klo~ 0.)1/2). Since the frequency response of the feedback controller is real, it must have an impulse response which is symmetric about the time origin. Because the frequency response increases according to to 3/2, the impulse response is not confined to the origin of time and so must be non-causal. The ideal controller is thus not realisable for broadband excitation, as discussed by von Flotow (1988) and Miller et al. (1990). Alternative feedback strategies are also possible. For example, Miller et al. (1990) and von Flotow (1988a) describe the suppression of the reflected flexural wave at the free end of a beam by applying a moment at the end proportional to the tip velocity.

ACTIVE CONTROL OF VIBRATION IN STRUCTURES

183

Similar wave absorbing terminations for the free end of a beam using feedforward control have been described by Scheuren (1990) and Pines and von Flotow (1990b). Miller et al. (1990) also discuss the strategy of maximising the power absorption of a set of secondary sources, an idea investigated experimentally on an infinite beam by Redman-White et al. (1987). Such a control strategy should only be exercised with some caution, however, since it was shown for the analogous acoustical case by Elliott et al. (1991), that the power output of the primary source can be significantly increased by maximising the power absorption of the secondary sources in a finite system with reflections from the ends. This can result in dramatic increases in the stored energy in the system, a result which has been confirmed in the vibrational case by Brennan et al. (1993). Von Flotow (1988a) points out that the near-field contributions in flexural wave propagation can also provide the coupling necessary to increase the power output of the primary, and concludes that 'it is not yet clear whether this type of "energy vacuuming" will be useful in application'. Although power absorption does not appear to be a practical control strategy in finite systems, the use of feedback control for wave suppression does appear promising. It should be emphasised that this feedback control strategy is different from that of damping structural modes, as described in Section 6.3. Whereas the practicalities of modal control have been widely investigated, the implementation problems involved in wave control have only recently been considered, and are still under investigation.

7 Active Isolation of Vibrations

7.1

Introduction

This chapter will deal with the application of active techniques to problems in vibration isolation. Broadly speaking, there are two classes of such problem; instances where we wish to isolate a vibrating body (such as a machine of some kind) from a 'receiving structure' (such as a car body, ship hull, aircraft fuselage or building) and instances where we wish to isolate a body (such as sensitive equipment or a railway car) from vibrations imposed by another source (such as ground vibrations or railway track unevenness). In both classes of problem, the source of the vibrations may be either deterministic (i.e. having a perfectly predictable waveform) or random (i.e. having a waveform that is not perfectly predictable). However, it is generally true to say that most problems of the first kind, which involve isolating machinery vibrations from a receiving structure, have a deterministic source of vibrations. This is the case whenever the source of vibrations is a rotating or reciprocating machine. In these cases we can adopt what is essentially a feedforward control approach to the problem. Thus one only needs to know (or be able to detect) the frequency of the vibration source and the necessary control forces can be synthesised using the adaptive feedforward techniques described in detail in Chapter 4. It is also true to say that the second class of problem, which involves isolating a vibrating body from external sources of vibrations, is mostly dealt with using feedback techniques of the type described in Chapter 3. For example, one of the most commonly addressed problems of this type is the design of active vehicle suspension systems. Thus the body to be isolated is the passenger cabin of the vehicle and the source of vibrations is the variable height of the road surface, the latter being a random process. Although isolation problems associated with a deterministic excitation are mostly treated using feedforward techniques and those associated with a random process are mostly treated using feedback, feedforward control can be applied to problems with random disturbances and feedback can be applied to problems with deterministic disturbances. Many of these possibilities will be considered here. The intention of this chapter is to give an introductory survey of these types of vibration isolation problem. First, the isolation of periodic vibrations from a flexible receiving structure will be described, since this is one of the most commonly encountered problems faced by the vibration control engineer. Basic guidelines for the positioning and requirements of secondary actuators will be given and some of the performance limits of such systems will be discussed. Examples will be described of

186

ACTIVE CONTROL OF VIBRATION

the practical application of the technique to automobiles and helicopters. Some discussion will also be presented of such vibration isolation problems when the vibration source is not deterministic and feedback control techniques must be used. Finally, an introduction will be given to the application of feedback control for the active isolation of bodies from random vibrations. Much of the work in this area has been undertaken within the framework of linear quadratic Gaussian (LQG) optimal control theory and an introductory discussion of these techniques will be presented.

7.2

Isolation of periodic vibrations of an SDOF system

First we will follow Nelson et al. (1987a) and consider the isolation of an SDOF system from a flexible substructure. The vibration of SDOF systems has been considered in Chapter 1. A similar analysis to that given below has been presented by von Flotow (1988b). The problem studied represents the simplest possible model of a mounted machine whose vibrations we wish to isolate from a receiving structure. It will also be assumed that the single degree of freedom system is harmonically excited, such that in practice, an adaptive feedforward control system could be used. This in turn assumes that the primary excitation forces in the machine are deterministic (perfectly predictable) which is of course the case for a large class of problems arising in practice; the isolation of machines such as engines, pumps and compressors from flexible structures such as the hulls of ships and submarines or the bodies of automobiles. As illustrated in Fig. 7.1, a secondary force can be applied at three possible locations associated with the mass-spring-damper system. Here we have assumed that the primary complex excitation force fe (representing the unsteady forces generated by the machine) is applied to the mass M of the system. Thus an obvious approach to the problem is that illustrated in Fig. 7.1 (a) where the secondary force is also applied to the mass and that to achieve zero response of the system we require simply that f~ = -fp.

(7.2.1)

In practice therefore, this implies that the secondary force would have to be applied to the body of the vibrating machine via an inertial exciter of some kind (e.g. an electrodynamic exciter which is provided with a mass against which it can react). A second approach to the problem is that illustrated in Fig. 7.1 (b). Here the secondary force is applied directly to the receiving structure with the objective of reducing the response of the receiving structure to zero. Again the secondary force would have to be applied by using some form of inertial exciter. It is important to calculate the magnitude of the secondary force required relative to that of the primary force applied to the system. Since we are dealing with periodic vibrations it will suffice to work at a single frequency. First assume that the receiver can be characterised by a complex input receptance R (jto), such that its complex displacement wn is related to the applied force f by WR= R(jto)f. (Each of these terms is a complex number which varies with to; see Chapter 1.) The force applied to the receiver is the sum of the secondary force and the forces applied via the spring and viscous damper. Thus we can write WR = R(jco)[fs + K ( w s - WR) + j t o C ( w s - wR)],

(7.2.2)

ACTIVE ISOLATION OF VIBRATIONS

187

to

M

M

//////////

(a)

[•C

1"////////

(b)

M

fJ///'ffff~/'/f~//fff (c) Fig. 7.1 Active isolation of a harmonically excited SDOF system from a receiving structure. Three arrangements are shown for the application of a secondary force (a) directly to the mass of the system, (b) directly to the receiver and (c) directly to the receiver with reaction against the mass of the system. where Ws has been used to denote the complex displacement of the mass M, the subscript s implying the displacement of the 'source'. This equation implies that for wR = 0 we require f , = - (K + je)C)ws

(7.2.3)

and the secondary force required is that necessary to cancel the sum of the stiffness and damping forces applied to a rigid foundation by a complex displacement Ws of the mass. Note that this result holds irrespective of the receptance R(jco) of the receiver. We can deduce fs in terms of fp by examining the equation of motion of the mass given by -ogZMws + j~oC(ws - wR) + K ( w s - wR)= fp

(7.2.4)

and it therefore follows that when wR = 0, we have ( - ~oZM + jogC + K)ws = f p.

(7.2.5)

188

ACTIVE CONTROLOF VIBRATION

Substitution of this result into equation (7.2.3) therefore shows that - ( K + floC) fs =

-o92M + flo C + K

(7.2.6)

fp.

This result, which relates the required secondary force to the primary force, can be expressed non-dimensionally in terms of the natural frequency co,,='qK/M and damping ratio ~ = C / 2 M r o , , of the SDOF system (these parameters have been previously discussed in Chapter 1). First note that

-[(K/M) + (floC/M)] (7.2.7)

f~ = -09 2 + (floC/M) + ( K / M ) fp

and therefore +

f,

=

2j o¢co.)

-co 2 + 2jog~tOn + tO,,2

(7.2.8)

fp.

This expression can also be written in terms of non-dimensional frequency f~= co/~o, such that 1 +2j~

f, =

~'-

1 - 2j~O

(7.2.9)

fp.

The frequency dependence of this relationship is illustrated in Fig. 7.2 which shows a plot of the ratio of the modulus of the secondary force to that of the primary source for a damping ratio ~ = 0.01. This plot emphasises that, as one would expect, the secondary force required to cancel the vibration of the receiver is much larger than the primary force at the natural frequency of the mounted system. Thus, in the region of this natural frequency, primary force fluctuations are effectively amplified by the 100

10

(b) (a)

I

fso I

0.1

0.01

I

0

,

,O=l

i

t

J

2

3

4

Non-dimensional frequency, ,(2

Fig. 7.2 The magnitude of the secondary force required relative to that of the primary force for the three arrangements shown in Fig. 7.1: (a) cancellation of the force applied to the mass; (b) cancellation at the receiver; (c) cancellation at the receiver with reaction against the mass.

ACTIVE ISOLATION OF VIBRATIONS

189

response of the system and result in large forces transmitted to the receiver by the spring and damper. Naturally, in conventional passive isolation systems, this is the situation that one seeks to avoid by designing the mounted natural frequency of the machine to be well below the lowest excitation frequency generated by, for example, the rotational speed of the machine. A further approach to the problem, which as we shall show below, may have some distinct advantages, is that illustrated in Fig. 7.1 (c). Here the secondary force is applied in parallel with the spring and damper such that it acts on the receiver with a reaction against the source. Such a situation would arise if, for example, an electrodynamic exciter were used with the body of the exciter rigidly fixed to the source and the excitation applied to the receiver; such an arrangement generates equal and opposite forces applied to both source and receiver. Under these circumstances, equation (7.2.3) for the secondary force required still holds, but equation (7.2.4) describing the motion of the mass becomes - c o 2 M w s + jcoC(ws-

wR) + K(ws + wR) = f p - fs"

(7.2.10)

Again setting wR = 0 and using equation (7.2.3) for f, shows that WS --

-fP

(7.2.11)

o) 2M

This shows that the displacement of the mass is exactly as if it were freely suspended; the dynamic displacement of the mass is determined only by its inertia. This results because the secondary force cancels the contribution of the stiffness and damping forces which act on the mass, in addition to those that act on the receiver. The net result is that the dynamic displacement of the mass is as if it were 'floating' in free space. Combining equations (7.2.3) and (7.2.11) then shows that

fs =

(K + jcoC) ooZM fp.

(7.2.12)

This in turn can be written non-dimensionally as fs =

1 +2j~ ~2

fp"

(7.2.13)

The modulus of the ratio fs and fp is plotted in Fig. 7.2. The clear difference between the magnitude of the secondary force required in this case and that required when cancelling at the receiver is that in the region of the mounted natural frequency I L l ' - I L l . Thus the required secondary force is much less than that necessary to cancel the receiver motion at the natural frequency of the system. As far as the practical application of active control is concemed, the use of an actuator in parallel with a passive isolation stage could have distinct advantages. In a given application, if an actuator can be found that provides a secondary force of the order of the primary force, then it may be possible to use a much higher mounted natural frequency associated with the passive isolation stage than would otherwise be possible. This in turn has advantages with regard to the stability of the mounted machine; 'soft' passive isolators designed to give a low mounted natural frequency cause problems with, for

190

ACTIVE CONTROL OF VIBRATION

example, the alignment of shafts with other machines. Such stability requirements are usually met by increasing the mass of the machine with a supplementary 'inertia base'; this is very often an undesirable increase in mass.

7.3

Vibration isolation from a flexible receiver; the effects of secondary force location

One of the inadequacies of the simple models presented in the last section is that the secondary force applied to the receiver is assumed to be coincident with the point of application of the force applied via the spring and damper. In real active isolation systems, there will inevitably be some mismatch between the point (or area) of application of the primary force (via a passive isolator for example) and the point (or area) of application of the secondary force. As a first step towards evaluating the effect of this, here we examine the problem of a single point secondary force fs separated by a distance d from a point primary force fp, both forces being applied transverse to a thin infinite plate (Fig. 7.3). We calculate the secondary force necessary to ensure that the total vibrational power input to the plate is minimised. This analysis shows that the separation distance between the two forces determines a high frequency limit above which active control cannot be globally effective. Here we follow the analysis presented by Jenkins (1989) and Jenkins et al. (1993). First note that the complex transfer mobility relating the transverse velocity of an infinite thin plate to the transverse applied force at a frequency to can be written as o)

(7.3.1)

Y(r) = 8Dk} [H(°Z)(klr)- H(°Z)(-jklr)]

which is the result given by Cremer and Heckl (1988), where H(02)denotes the Hankel function of the second kind. Here k/is the wavenumber and D is the plate bending

OO

OO

OO

Fig. 7.3 A point primary force and a point secondary force applied to an infinite thin elastic plate.

ACTIVE ISOLATIONOF VIBRATIONS

191

stiffness given by Eh3/12(1 - v:), where E is Young's modulus, h is the plate thickness and v is Poisson's ratio. The wavenumber kI of flexural waves propagating in the plate is given by k}= ~ : p h / D , where p is the plate density as derived in Chapter 2. The power input to the plate by a harmonic point force f can be written as II = ~1 Re{f*ja~wR},

(7.3.2)

where joowR is the complex transverse velocity of the plate at the position of application of the force. The total power output of the force pair can be written as the sum of the power outputs of the primary and secondary forces. Each force will produce a velocity at its own point of application (determined by the input mobility of the plate at that point) and at the point of application of the other force. Thus we can write for the total power input 1

H = 7 R e { f * [ f p Y ( O ) + f s Y ( d ) l + f * [fsY(0)+fpY(d)l},

(7.3.3)

where Y(0) is the input mobility of the plate and Y(d) is the transfer mobility which determines the complex velocity produced at a radial distance d from a point force. Of course equation (7.3.3) assumes that the structure is linear such that the velocity fields produced by the two forces can simply be superposed. Equation (7.3.3) can be written as

ri = Ifsl 2'7 Re{ Y(0) } + 71Re{f*fsY(d) +fs*fpY(d)} + Ifpl 2'~ Re{ Y(0)}. (7.3.4) Since Re{~f,Y(d) } = ½ rearranged to show that

ri =

1

fsr(a) +

(a)l, the second term in this equation can be

rlfsl 2 Re{Y(0)} +fs* R e l Y ( d ) } f p + f * Re{Y(d)}fs + ILl 2 Re{Y(0)}l,

(7.3.5)

which is a quadratic function of the complex force fs having the form (7.3.6)

rI = A lf s l2 + f * b + b *f s + c.

The

parameters

in

this

equation

are

1

A=sRe{Y(0)},

1

b=sRe{Y(d)}fp

and

c = =1l L I 2Re{Y(0)}. This expression for the total power output has a well-defined minimum value II 0 = c - I b l 2 / A that is associated with an optimal secondary force given by f,o = - b / A . (See Nelson et al., 1987b, or Nelson and Elliott, 1992, for a full description.) Since it follows from equation (7.3.1) that Re{Y(0)} =oo/8Bk 3 and Re{ Y(d)} = Y(O)Jo(kfl) where J0 is a zeroth-order Bessel function of the first kind (Abramowitz and Stegun, 1972), then the expressions for the minimum power output and optimal secondary force input become 1-I0

(7.3.7a)

- 1 - j2o(kld ),

lip

Lo - - = -Jo(kld ).

.~

(7.3.7b)

The results have been expressed non-dimensionally where lip = ILI2,o/leOk 2 is the power input to the plate by the primary force in the absence of the secondary force. These results are depicted graphically in Fig. 7.4. This shows that appreciable

192

ACTIVE CONTROLOF VIBRATION I I I I I

fso

d=3Z/8

,,l

2~

d-- 3~,/8

Z

2~,

10 log 10 //o" (dB)

0

Radial distance, d

Fig. 7.4 (a) The optimal secondary force input required to minimise the total power output of the force pair illustrated in Fig. 7.3. (b) The maximum possible reduction in power output as a function of the distance d relative to the flexural wavelength in the plate at the frequency of excitation. reductions in total power output can only be achieved if the secondary force is applied at a distance well within 3;t/8 from the primary force, where 2 is the flexural wavelength in the receiving structure at the frequency of interest. This result follows since the first zero of the Bessel function depicted in Fig. 7.4 occurs at kid ~ 3 Jr~4. As an illustration of the practical implications of this result, Table 7.1 shows the primary/secondary force separation distances necessary to achieve given levels of power input reduction when two point forces are applied to an (infinite) steel plate of 2 mm thickness. This shows, for example that 10 dB reduction in power output would only be achieved for frequencies less than 100 Hz if the two forces are separated by about 30 mm. Thus the misalignment between the points of application of the primary and secondary forces imposes quite severe limitations on the frequency range of applicability of active techniques. Inevitably success in this type of problem is restricted to low frequencies.

193

ACTIVE ISOLATION OF VIBRATIONS

Table 7.1 Primary/secondary force separations for a specified input power reduction. Primary/secondary force separation (mm) for a reduction in power flow of Frequency (Hz) 50 100 200 500 1000

2 = 2zt/k I (mm)

10 dB

5 dB

0 dB

625 442 313 197 139

45 32 22 14 10

84 59 42 26 19

239 169 119 75 53

7.4 Active isolation of periodic vibrations using multiple secondary force inputs It is also inevitable in practical installations that rotating or reciprocating machinery will be mounted off flexible receiving structures via a multiplicity of mounting points. Machinery of this type, depending upon the application, is usually mounted via typically four, six or eight passive isolators. The analysis presented in Section 7.2 gives some indication of the benefits and disadvantages of applying secondary control force inputs at various locations in such an installation. Here we introduce the analytical framework appropriate to the analysis of active isolation systems where multiple secondary forces are used in order to apply control to the system. The general problem is illustrated in Fig. 7.5. The typical mounted machine constitutes an MDOF system, potentially having three translational and three rotational modes of motion. The vibration of MDOF systems has been discussed in Chapter 1. Thus, for example, as illustrated in Fig. 7.6, control could be applied 'at the source' by using six appropriately oriented inertial actuators in order to synthesise three translational and three rotational dynamic inputs. These would directly apply the requisite secondary forces and moments in order to exactly cancel the primary forces and moments generated by the machine itself. A similar problem in principle exists if control is to be applied at the position of the mounts; as illustrated in Fig. 7.5(b), at any given mounting point the machine is capable of applying primary excitation to the receiving structure via all six forms of input. One approach to this problem is that suggested by Ross et al. (1988) who advocate the use of 'an intermediate structure' in a two-stage isolator, all degrees of freedom of the intermediate structure being controlled by secondary inertial inputs (Fig. 7.7a). Another approach is that suggested by Smith and Chaplin (1983) who used passive isolators, which were very compliant to sheafing motions, in parallel with electrodynamic actuators at each mounting point, which acted to control only vertical motion. A further approach, which was used with some success by Jenkins et al. (1991), employs a pneumatic isolator which can in principle only transmit a normal stress to the receiving structure. The pneumatic isolator is in turn controlled by only a single translational actuator which is capable of controlling the pressure fluctuations

194

ACTIVE CONTROL OF VIBRATION

I Vertical ~.~

(a)

Twisting

i

I

i

Machine /

,

fo fo

f

f

of°

f o r ° f o f ° - -

"

Raft

Z Longitudinal

Pitching ,

(b)

& />---"l

Lateral

|

i Fig. 7.5 (a) The six modes of motion that can be excited by a vibrating machine. (b) The same types of motion are possible in an isolator that is compliant in all three directions.

Actuators

Fig. 7.6 Active control of primary excitation by acting on the machine to cancel excitation in six degrees of freedom. The use of inertial actuators is assumed.

ACTIVE ISOLATIONOF VIBRATIONS

195

within the isolator and through this controlling the transmitted normal stress. This is illustrated in Fig. 7.7(b). Finally, Fig. 7.7(c) shows the arrangement presented by Staple (1989) for use on helicopter vibration control systems. Irrespective of the mode of application and number of secondary force inputs, systems designed to control periodic machinery vibrations most conveniently employ a feedforward control approach as discussed in detail in Chapter 4. It is interesting to note however, that early approaches to this type of problem (see Calcaterra and Schubert, 1968) used a feedback approach with narrow-band filters in the control loop, the filters being tuned to the excitation frequency and its harmonics. Sievers and von Flotow (1990) have discussed the relationships between the various approaches to controlling periodic vibrations. However, in using a feedforward approach, one is generally faced with adjusting the magnitude and phase of a multiplicity of secondary force inputs in order to minimise some appropriate quadratic cost function. One such cost function is the total vibrational energy (kinetic or strain) in the receiving structure. Inevitably such a function must be approximated by undertaking measurements at discrete locations distributed over the receiver (a car body or helicopter passenger cabin, for example). The analysis of such systems is straightforward, again assuming linear behaviour of the structural response and the applicability of the superposition principle. At a given frequency, the displacement of the receiving structure can be characterised by the complex vector w of order N whose elements are the complex displacement at N points on the receiving structure. Clearly as N becomes larger one obtains a better approximation to the global response of the structure as long as the sensors are appropriately distributed. The displacement at each point can be considered to consist of a superposition of the displacements produced by the primary excitation and those produced by the secondary actuators. One can thus write W = Wp "+"

Rf,,

(7.4.1)

where Wp is the vector of complex displacements produced by the primary excitation and the product Rf, defines the complex vector of displacements produced by M secondary force inputs. Thus R is the N x M complex receptance matrix characterising the response at the N points considered due to the application of the M secondary forces, these in turn being characterised by the complex vector Is. If we now choose the cost function for minimisation to be the sum of the squared displacements at N positions defined by N

J-

Iw~ - w w,

(7.4.2)

n=l

then substitution of equation (7.4.1) shows that J may be written as J = fHRHRf~ + fHRHWp+ wpHRf~+ WpHWp.

(7.4.3)

This function can be written as the Hermitian quadratic form J - f~Af~ + fnb + bnf~ + c,

(7.4.4)

where the matrix A - R HR, the complex vector b= RHWp and the scalar constant c - WHWpis the sum of the squared displacements due to the primary excitation alone. As described by Nelson et al. (1987b) and dealt with in Chapter 4, a function of this

196

ACTWE CONTROLOF VIBRATION

type has a unique global minimum Jo associated with an optimal vector of secondary force inputs f~o. These are defined by J0 = c - bHA-lb,

f,o = - A - l b

(7.4.5a,b)

The unique minimum in the function is assured provided that the matrix A is positive definite. This is ensured in this case since f~Af, defines the sum of the squared displacements due to the secondary force inputs alone. This must therefore be greater

Vibrating I-1

assiveisol.ator (a)

Secondary actuator

,P' .eceiver Electrodynamic secondary force actuator

Intermediate plate Pneumatic mount

K

J

Source raft (b)

• Passive isolation

Receiver

Fig. 7.7 (a) The active mounting configuration suggested by Ross et al. (1988) for control of transmission in multiple degrees of freedom. (b) The active mounting configuration used by Jenkins et al. (1991) to enable control to be applied in only the vertical direction at the machinery mounting point. (c) The mounting arrangement described by Staple (1989) for the application of control to the interface between the fuselage of a helicopter and the raft supporting the engine and gearbox. Note that the elastomer (shown hatched) provides a stiffness giving passive isolation between raft and fuselage. The additional oscillatory input provided by the hydraulic actuator acts in parallel with this stiffness to provide isolation in the manner illustrated in Fig. 7.1 (c).

ACTIVE ISOLATIONOF VIBRATIONS

~

~

197

Raft

attachment

Primary loadpath (c)

Elastomer Fuselage~ attachment " ~

I

I_oscillatory Additional input

Actuator

Fig. 7.7 Continued. than zero for all non-zero f r Note that equation (7.4.4) is the vector equivalent of the scalar quadratic function of a complex variable given by equation (7.3.6). The objective of a feedforward controller in this case is therefore to adjust the complex secondary force inputs in order to minimise this function. As dealt with in Chapter 4, the case with which this can be achieved will to a large extent be determined by the conditioning of the matrix A. In addition to this, the cost function defined above can be made more sophisticated and include terms which also penalise the 'effort' (fHf s for example) used in achieving the minimisation. The inclusion of such terms may help in making the problem better conditioned. Bound up with the conditioning problem is the number and location of secondary force inputs used and the number of sensors used on the receiving structure in order to adequately represent its response. As a first step towards evaluating the influence of these factors on the success of control, a finite element analysis of active control applied to an MDOF system will be described in the next section. An introduction to finite element analysis of distributed elastic systems can be found in Chapter 1.

7.5

Finite element analysis of an active system for the isolation of periodic vibrations

Jenkins et al. (1993, see also Jenkins, 1989) have undertaken a detailed investigation of a specific active isolation system designed to be representative of a typical

198

ACTIVE CONTROLOF VIBRATION

machinery isolation problem. The problem analysed is one of a rigid thick plate mounted via four passive isolators onto a flexible receiving structure in the form of a clamped thin plate. Laboratory experiments were undertaken on the system depicted in Figs 7.8(a) and (b). Note that the passive isolators (in the form of foam rubber cylinders) are supplemented by electrodynamically generated secondary forces which effectively act in parallel with the passive isolators (as illustrated schematically in Fig. 7.1 (c)). The equivalent finite element model is illustrated in Fig. 7.9. In the real system the plate and raft have dimensions of 0.99 m x0.66 m x0.002 m and 0.22 m x 0.22 m x 0.01 m respectively. The foam rubber isolators are of outer radius 0.04 m and inner radius 0.025 m and each have a vertical stiffness of approximately 14.4 x 103 N m -~. The fundamental vertical mass/spring resonance of the system occurs at approximately 14 Hz. The thin receiver plate was constructed from mild steel whilst the raft was aluminium.

Primaryforcegeneratingunit (coil and magnet) Force signal / Acceleration ~ - ~~ s~ ~i ~g. ~n a l ~._~ SecondaryforceacutaO tS ras) g

Clamping

'

It

ii

I

Receiving structure Foam"cylinder"isolator Sensorinputs(accelerometers) to the controlsystem "Measuring" accelerometer (a)

Sensorinputs / ...~ Clamping

Primaryforce actuator

I~")

Q

Plate Secondaryforce actuator

(b) Fig. 7.8 The experimental active isolation system described by Jenkins view; (b) top view.

et al.

(1993): (a) side

ACTIVE ISOLATION OF VIBRATIONS

Central primarye x c ~

Spring/damperelements

199

Off-centralprimaryexcitation ,,

,,

<

-i Four nodesdefininga shell element ~ Position of the secondary actuators Fig. 7.9 The finite element model described by Jenkins et al. (1993). The points marked (o) indicate positions at which the fluctuating displacement is minimised by the action of the secondary force inputs. Those marked (o) were used when cancelling the vibrations at only four locations. Secondary forces are shown being applied directly to the receiver although for the results presented in Fig. 7.10, they were applied in parallel with the spring/damper elements representing the passive isolator. The finite element modelling of the raft and receiver was relatively straightforward and was undertaken by using a set of finite plate elements as illustrated in Fig. 7.9. The isolators were e a c h modelled as four spring/damper elements acting in parallel. This was an attempt to model the effect of the finite size of the isolators, since as demonstrated in Section 7.3, the effect of a finite separation distance between the points of application of the primary and secondary forces is likely to set an upper frequency limit on the performance of active systems. Further details of the modelling procedure are presented by Jenkins et al. (1993) and by Jenkins (1989). One of the advantages of the approach adopted is that a very good approximation can be made to the total energy in the receiver by evaluating the sum of the squared displacements at all N nodes in the analysis of the receiver. Thus the optimal secondary force inputs are deduced by first evaluating the elements of the complex receptance matrix R relating the four secondary force inputs to the displacement at each point on the receiver. The optimal secondary force input is then found by inversion of the matrix RHR in accordance with equation (7.4.5b). The primary force exciting the raft into vibration was always assumed to be applied vertically to the centre of the raft, the finite element model then being used to evaluate the complex vector w p of displacements due to the primary force. The procedure of evaluating the optimal secondary force input was then repeated at 1 Hz intervals in the range from 0 to 200 Hz. In addition to deducing the secondary forces necessary to

200

ACTIVE CONTROLOF VIBRATION

minimise the displacements at all (81) nodes of the receiver plate, the same process could be adopted when only a restricted number of nodal measurements of displacement were used. This simulated the use of a realistic number of sensors (either four or eight in the cases considered) and the receiver plate energy could then be evaluated to measure the effectiveness of a feedforward controller. The results of the finite element analysis are illustrated in Fig. 7.10. This shows the isolation efficiency defined by

N ~-~ ] Wn ]2gidlink EN

--

"=~

N

(7.5.1)

°

~"~ I Wn 12ithisolation n=l The numerator of this expression is the sum of the square displacements in the receiving structure when the active/passive isolators are replaced by rigid connections between the raft and plate. The denominator is the same quantity evaluated after the introduction of isolation, whether passive, active or a combination of both. This measure of the effectiveness of isolation follows from that used by Ungar and Dietrich (1966) and Soliman and Hallam (1968). It expresses the increase (or decrease) in performance produced by the introduction of isolation. The results shown in Fig. 7.10 were evaluated from summing the squared displacements over 35 equispaced nodes on the receiver, the position of the nodes being chosen to facilitate direct comparison with experimental results. First note that the effectiveness of passive isolation was found to be negative in some specific

70 60 50

L(iii)

~ 40 v

o

~ 3o g 2O m 10 0 -10 First mass-spring mode at 14 Hz

100

200

Frequency (Hz)

Fig. 7.10 The results of the isolator effectiveness predicted by the finite element model of Jenkins et al. (1993) for: (i) active control using four sensors; (ii) active control using eight sensors; (iii) active control using 81 sensors; (iv) passive isolation only.

ACTIVE ISOLATIONOF VIBRATIONS

201

frequency ranges. Thus the introduction of passive isolation increased the vibration of the receiver at these frequencies. That this should occur at the lowest natural frequency of the system (= 14 Hz) is not surprising, but a similar effect also occurs in the region of 30 Hz which reference to Table 7.2 shows is the natural frequency of the first (0, 0) bending mode of the receiver plate. This is a classic example of the effectiveness of a passive isolator being severely degraded by the presence of a foundation resonance. (For more details see Snowdon (1968) and for a practical example see Nelson (1982).) Above this frequency, however, the effectiveness of the passive isolation increases substantially. (Note that the isolation efficiency of the passive system exhibits spurious numerical results in the region of 75 Hz; the frequency at which there is a degenerate mode of the plate as shown by Table 7.2. See Jenkins (1989) for more details.) Figure 7.10 shows clearly the benefits to be obtained by application of the 'parallel' active system. Substantial increases in isolation efficiency are available at low frequencies, although these benefits reduce rapidly at frequencies below the fundamental mass/spring resonance of the system (14 Hz). This results from the increasingly large secondary force inputs required below this frequency, as illustrated by Fig. 7.2. Thus, as one would expect with this arrangement, infinite secondary force would be required to prevent any motion of the receiver in the low frequency limit. Above the fundamental resonance, however, the finite element analysis predicts considerable gains in performance, especially in the frequency range up to 76 Hz, with the performance benefits of the active system deteriorating above this frequency, especially in the case where only four 'sensor' outputs were used in order to determine the optimal secondary forces. The use of eight rather than four outputs is clearly beneficial over the whole frequency range and illustrates the advantages in using more sensors than secondary force inputs; a 'square' feedforward control system with equal numbers of inputs and sensors is always guaranteed to produce zero displacement at the positions of the sensors. However, as discussed in detail by Nelson and Elliott (1992) within the context of the active control of acoustic fields, it is often more beneficial to use an 'overdetermined' system if global benefits are to be obtained. The results shown in Fig. 7.10 also indicate that this applies here. In fact the use of eight sensors produces results which are not dissimilar to those produced by using 81 sensors, indicating that the sum of the squared displacements measured by the eight sensors used gives a reasonable approximation to the energy of the system. This is to be anticipated for the low modal densities of the receiving structure examined in this case. From the evidence Table 7.2

Predicted and measured plate natural frequencies. Natural frequency (Hz)

Plate mode (m, n) 0, 0 1,0 0, 1 2, 0 1, 1 3,0

Predicted

Measured

30.7 47.4 75.3 75.7 90.7 115

30 48 f Combined degenerate ] mode at 77 Hz 93 116

202

ACTIVE CONTROLOF VIBRATION

presented here, and from that deduced in examining the control of acoustic fields at low modal densities, it is to be anticipated that 'overdetermined' feedforward active controllers can be used successfully by employing this simple strategy provided the receiver has a low density of modes.

7.6

Practical examples of multi-channel feedforward control for the isolation of periodic vibrations

The general strategy described in the last two sections has already found its way into practical use; two specific examples being the control of the periodic vibrations produced in automobiles and helicopters. The application of active engine mounts to automotive vibration problems is described in detail by McDonald e t a l . (1991). Figure 7.11 shows the details of an active engine mount developed by Freudenberg (1986) and used by Lotus Engineering in a series of trials on a Volkswagen Golf GTI 16 valve vehicle. This is a high torque, front wheel drive vehicle, and conventional engine mounts have to be of relatively low stiffness to provide adequate noise and vibration isolation. This in turn can produce a static implementation problem for the engine and the vehicle may suffer from 'secondary ride shake' or other transient problems especially on start-up or run-down of the engine. The active engine mounts used consist of a conventional hydraulic mount in which is embedded an electromagnetic actuator. Below 25 Hz, the response is dominated by the hydraulic fluid with its motion between the internal reservoirs of the mount providing appropriate damping. At frequencies above this however, the fluid effectively provides a rigid link between the electromagnetic actuator and the body of the automobile, allowing the secondary force applied by the actuator to be transmitted and thus cancel the primary excitation provided by engine vibrations. Figure 7.12 shows the details of the implementation of the system. Accelerometers are located on the chassis side of

Actuator

Diaphragm " ~ ~ , ~ . ,

, ~ ~ f [ . ~ Workin.g reservoir

reservoir Rubber element Balance reservoir Bellows Fig. 7.11 Details of the active engine mount developed by Freudenberg (1986) (after McDonald et al. , 1991).

ACTIVE ISOLATION OF VIBRATIONS

203

Sound measurement Micr°ph°ne zone ~ Speakers below as,

,tive ,r

/<-

Freudenberg \ f ",,~'~ ~ active engine \ ~ "-~%'~'~ mount (M1) ~ ~ /~A~ (,,,j) \\i Acceler°m ~ ' , . . . , , . . ~ ~ ~ 1 Rubber mount-~~~---~: Accelerometer Freuden active en mount (M3)

,=~...,-~:~,~....

~ i

\'" measurement ~ ........... zone Freudenberg active engine mount (M2)

/ ~ Accelerometer

Fig. 7.12 The installation of the active mounts in a Volkswagen Golf GTI 16 valve vehicle (after McDonald et al., 1991).

20

A

13. 13.. ¢-

oL _

10 t.-

E

°

~

121

"i.".

"..i !

1000

,

"'i F

.....-.-"

. •

".

I

2000

3000

4000

5000

6000

7000

Engine speed (RPM) Fig. 7.13 Experimental results illustrating the effect of active control of engine vibrations produced in the Volkswagen Golf GTI 16 valve vehicle shown in Fig. 7.12. Three active mounts were used and the results show the displacements measured in the driver's footwell produced by second order engine vibrations: ~ , before active control; - - - , after active control (after McDonald et al., 1991).

204

ACTIVE CONTROLOF VIBRATION

the engine mounts, with two accelerometers placed on either side of the front mounts since the cross-member supporting the front mount was quite lightly damped. The multiple error LMS algorithm described in Section 4.7 was then used to determine the voltage inputs to the three engine mounts in order to minimise the sum of the squared accelerations measured by the four accelerometers, a reference signal being generated for the control system at the engine firing frequency. Figure 7.13 shows the reduction in 'second-order' (twice rotational frequency) vibration levels produced when the control is applied as a function of the engine speed. Significant reductions in level have clearly been achieved. McDonald et al. (1991) also emphasise that when used in conjunction with loudspeakers for controlling the interior noise (see Fig. 7.12) significant reductions in the noise levels produced can also be achieved. A further benefit emphasised by McDonald et al. is the improvement in ride and handling qualities of a vehicle whose engine mounts can be made much stiffer than those used conventionally. More recent work presented by Ushijima and Kumakawa (1993) describes a similar approach to automotive engine vibration control, but in this case the mounts used include a piezoelectric actuator embedded in a conventional passive isolator.

(a)

(b)

(c) Fig. 7.14 Possible techniques for applying secondary forces in the active control of helicopter vibrations described by Staple (1990)" (a) raft installation using dual point actuators; (b) actuator/struts acting across structural compliance; (c) single point actuators reacting against seismic masses.

ACTIVE ISOLATIONOF VIBRATIONS

205

The application of active techniques to helicopter vibration problems is described by Staple (1990). Helicopters generate vibrations directly as a result of the aerodynamic imbalance on the advancing and retreating sectors of the rotor disc. Vibrations are transmitted to the airframe at the blade passing frequency and its higher harmonics. Several possible techniques exist for applying secondary force inputs to a helicopter airframe and those described by Staple are illustrated in Fig. 7.14. The system used by Westland Helicopters was of the type illustrated in Fig. 7.14(a) and incorporated the mounting arrangement illustrated in Fig. 7.7 (c) which acts between the airframe and the raft supporting the engine and gearbox. Alternative approaches suggested by Staple are shown in Fig. 7.14(b) and (c). A system of this type was the subject of successful flight evaluation on a Westland 30 helicopter in early 1987. The four elastometric mounts at the airframe/raft interface were supplemented with hydraulic actuators acting in parallel, as shown in Fig. 7.7 (c). These four secondary force inputs were used to minimise a cost function consisting of the weighted sum of squares of the outputs of 12 accelerometers plus the weighted sum of squared actuator inputs. The control system (which was essentially feedforward in nature) operated in the frequency domain, using a discrete Fourier transform of the measured accelerations, essentially to establish the complex matrix relating the secondary force inputs to the secondary acceleration outputs and the vector of complex accelerations produced by the primary excitation. The solution to the least squares problem can then be found by a number of techniques (see Nelson and Elliott 1992, Chapter 12, for full details). The hydraulic actuators were capable of producing a maximum force of 2000 lbf and displacement of 0.01 in at the blade passage frequency of 21.7 Hz. The results of the flight trials i!i!iiiiiiii!iiii Baseline helicopter Head absorber Active control of structural response

. . . . . . . . . ..........

0.5 v

~" 0.4 o o ~

> 0.a

o ~

-

iii!!ii!i!il

iiiilaiiii!!!ii!

i i !i i i i i!i

i i i i i i i !i

:.:.:.:.:.:.:.:.

-ii i i i i i i ::::::::::::::::

o o o

"

. m

..Q

i:i:i:i:!:!:!:!: ................

0.2

m

O

i,.,.

0.1

40

60

80

100

Forward Speed, Knots

Fig. 7.15 Results of a flight test of an active vibration control system implemented on a Westland 30 Helicopter. The results show the comparison of the baseline helicopter vibrations with those produced when a conventional dynamic vibration absorber and with 'active control of structural response' (ACSR) (after Staple, 1990).

206

ACTIVE CONTROL OF VIBRATION

(Fig. 7.15) demonstrated that average reductions of cockpit vibrations between 72% and 82% could be produced. This was much more favourable than the best reductions that could be produced by a rotor mounted (head) vibration absorber which was the best passive device available. Finally, it is worth pointing out that similar feedforward strategies to those used for the control of periodic vibrations are being applied to the reduction of noise and vibration produced by mass imbalances in jet engine turbofans. Such out-of-balance forces can cause problems with high levels of low frequency tonal noise in the passenger cabins of commercial aircraft. Swanson and Miller (1993) for example, describe the design of an active engine mounting suitable for such applications. The mount is essentially electromagnetic in its operation; the basic actuator is used to produce an alternating differential pressure across the passive mounting of the engine to the airframe.

7.7

Isolation of unpredictable vibrations from a receiving structure

Hitherto in this chapter, we have dealt only with problems of isolating deterministic vibrations from a receiving structure and of course this covers a wide class of problems of practical interest. There are, however, also many instances where it is desirable to isolate the vibrations produced by either a transient source (a punch press for example) or indeed a source of random vibrations (such as those produced by turbulent fluid flowing in a pipe). In either case, the vibrations are not perfectly predictable and a feedforward control strategy is difficult to use, unless of course there is either a sufficient propagation delay between the source of the vibration and the point or points at which the vibration enters the receiver, or the disturbances are narrow-band and have a high degree of predictability. However, one is generally faced with using a feedback controller in order to reduce the transmission of such random vibrations. The principle of operation of such a feedback controller is sketched in Fig. 7.16(a). An SDOF model is again used to describe the basic dynamics of a machine mounted via a passive isolation stage onto a receiving structure. An unpredictable primary force fp is shown applied to the system, and also shown is a transducer for directly measuring the force f, transmitted to the receiving structure. The objective of the controller is simply to minimise the value of the transmitted force. First consider the transfer function of the system relating fp to ft in the absence of control. Laplace transformation of the equation of motion of the mass (see Chapter 1) gives

S 2 ~ s ( S ) + sCWs(s ) + KW's(S ) = Fp(s),

(7.7.1)

where Ws(s) is the Laplace transform of the displacement Ws of the mass. Assuming that the foundation is perfectly rigid, the force transmitted to the structure is simply the force applied by the spring and damper and is given by

F,(s) = sCWs(s) + KW~(s).

(7.7.2)

The ratio of the force transmitted to the primary force defines the transmissibility which can therefore be written as

F,(s)

Fp(s)

=

sC+K s2M + sC + K

.

(7.7.3)

ACTIVE ISOLATION OF VIBRATIONS

I1 (a)

207

-g

M

, iC I / / / / / / / / / / / / / / / / /

H(s)

(b)

_-tt

Fig. 7.16 The principle of operation of a feedback system for the reduction of transmitted vibrations from an unpredictable source. (a) Schematic representation of a possible system and (b) its representation in block-diagram form.

Making the substitutions to,2 = K/M, ~ = C/2Mto n, as defined in Chapter 1 and s = jto enables the frequency response of the transmissibility to be written as

F~(jto) Fp(jto)

2

=

ton + 2jto~tOn to 2 - co2 + 2jto~ton

.

(7.7.4)

This expression can in turn be written as a function of the non-dimensional frequency ratio ~ = tO/tOn of the excitation frequency to to the natural frequency On of the system such that F,(jto) F~(j~o)

=

1 + 2jCf~ 1- ~

.

(7.7.5)

+ 2j~f~

This is the classical expression for the force transmissibility associated with a simple SDOF passive isolation system. A plot of the modulus of this function is shown in Fig. 7.17(a) for different values of damping ratio ~ as a function of the nondimensional frequency f2. As emphasised in Section 7.2 (see equation (7.2.9)) the objective of a passive isolation system is to ensure that the transmissibility is minimised by choosing the natural frequency to, of the mounted system to be much less than the excitation frequency ~o. Now consider the effect on the transmissibility of feeding back a secondary force input to the mounted mass that is proportional to the force transmitted. This is illustrated in Fig. 7.16(a) together with an equivalent block diagram shown in

208

ACTIVE CONTROL OF VIBRATION

10 2 0.01

~. 101

(a)

Damping ratio

0.05 0.2

•~ ._

10 0

E e-

;

0

10-~

0 U..

10-2

101

Lt.~

l loop g=l \...= g=2

10 0

(b)

g=5 g =10

u." ._z:B 1gl o_ ._

g =50

E e-

~ 10-2

0L 0 U--

10-3 0

t 1

l t 2 3 Non-dimensional frequency, t3

t 4

5

Fig. 7.17 Force transmissibility for the system sketched in Fig. 7.16: (a) with no feedback and for different values of passive damping ratio; (b) as a function of feedback gain g for a passive damping ratio of 0.05. Fig. 7.16(b). With the secondary force applied, the Laplace-transformed equation of motion of the mounted mass becomes

s2MW,(s) + sCW,(s) + KW,(s) = Fp(S) + F,(s).

(7.7.6)

Assuming that the feedback gain is g such that

F~(s) = -gF,(s)

(7.7.7)

209

ACTIVE ISOLATION OF VIBRATIONS

and since F,(s) is given by equation (7.7.2), the equation of motion can be written as (sZM + sC + K ) W , ( s ) = Fp(s) - g(sC + K)W,(s).

(7.7.8)

This can be rearranged to give [sZM + sC(1 + g) + K(1 + g)]W,(s) = Fp(s),

(7.7.9)

such that the ratio of the transmitted to primary forces becomes F,(s)

Fp(s)

sC+K

=

.

(7.7.10)

s2M + sC(1 + g) + K(1 + g)

This can in turn be expressed as the frequency response function Ft(jw )

1 + 2j~ =

Fp(jw)

.

(7.7.11)

(1 + g ) - f~2 + 2 j ~ ( 1 + g)

Figure 7.17(b) shows the modulus of this function as the gain g is varied. Clearly for very large values of gain g, the reduction in the transmissibility is substantial. At lower values of g, however, it is interesting to note that there is an effective shift in the natural frequency and damping of the mounted system as both the stiffness and damping terms in the denominator of equation (7.7.10) become modified by the action of feedback. Note that the poles of the closed loop system defined by the transfer function of equation (7.7.10) are simply those of a second-order system with damping C(1 + g) and stiffness K(1 + g). The closed loop system will thus remain stable provided that the gain g remains positive (and thus the secondary force Fs(s)= -gFt(s)). However, the practical realisation of this approach is quite far removed from this idealised model, since in a real system there are clearly additional transfer functions associated with the application of the secondary force and the sensing of receiver vibrations. Nevertheless, work by Tanaka and Kikushima (1988) has demonstrated the practical feasibility of what is essentially this approach. An outline sketch of the experimental system employed is shown in Fig. 7.18(a). The secondary force is applied via hydraulic actuators capable of producing forces of up to 4 kN and quartz force transducers were used to sense directly the force applied to the concrete receiving structure. This signal was then fed back to the hydraulic actuator via an analogue computer which was used to implement a compensating filter which enabled the highest values of loop gain to be used whilst still maintaining system stability. Full details of the compensator design are given by Tanaka and Kikushima (1988). One of the filters used had a transfer function of the form G(s) = g

(s/o),

)2 + 2~s/o),

(s/o)o + 1) 2

+1

.

(7.7.12)

The results for the experimental reductions in force transmissibility are illustrated in Fig. 7.18(b) as a function of different values of o)~. The system designed was clearly successful in producing reductions in the transmitted force in excess of 10 dB over a frequency range from 2 to 20 Hz. It is interesting that Tanaka and Kikushima chose to use the force applied to the foundation as the variable to be fed back rather than a direct measurement of the

210

ACTIVE CONTROLOF VIBRATION • . I Active isolator

Exciter

:

:

=

~ .

II

~

I

I

l~,~,~~,vo~

l~ol',

~ .

~ / t ' ~ . ~ , . . Servovalve

I~ Bed

'Servoamp ~ < I I " ~

I

. i ~erv°ampl -fp__f~t_s[ . ' Control input . Charge amp

~', .' Foundation Load cell ,"

....:.~......

....

<

o

i1"' I .....

~,o

XY-Plotter I I

b I I,,,,,,Dn I oI / iol _1! ~ IJ Anal°guec°mpute'

o oolj

, i Servoanalyse r

(a) 20

'01 -

.-

0

._~ E -10

/ . ~" k

,,-,,-

"Y'. ....."

e-

~ -20 O

" -30

° t 1

I

10 Frequency (Hz)

!

100

(b)

Fig. 7.18 The approach used by Tanaka and Kikushima (1988) to reduce the transmission of machinery vibrations to the machinery foundations. (a) The experimental arrangement used and (b) the force transmissibility with control applied for a range of compensation filters used in the feedback loop as specified in equation (7.7.12) with co~ (Hz) set at ~ , 40; - - -, 60; m _ m , 80; ,100. displacement response. Work presented by Watters et al. (1988) used a similar strategy in experiments on the active isolation of diesel engine vibration using active control. These authors chose to use a feedback system (rather than a feedforward system), even though the basic excitation was predominantly periodic, in view of the large number of harmonics in the spectrum of the excitation. They also chose to use a secondary force in parallel with the passive isolation stage. Watters et al. pointed out, however, that a measurement of the foundation displacement would be highly influenced by the details of the receptance of the foundation. Thus since We(s)= R(s)F(s), if the function R(s) describing the response of the foundation exhibits a highly resonant behaviour, then the open loop transfer function relating the control force input to the displacement output will also contain this dynamics. This makes the design of the compensation filter

ACTIVE ISOLATIONOF VIBRATIONS

211

in the feedback loop much more difficult and may result in unnecessarily conservative selections of loop gain in order to provide an adequate stability margin (see the discussion presented by Nelson and Elliott, 1992, Ch. 7). If, on the other hand, the force applied to the foundation is chosen as the variable to be fed back, the open loop transfer function will not be influenced by R(s). A more comprehensive discussion of this point has been presented by Blackwood and von Flotow (1993) who develop a general analysis of the active isolation problem when both the foundation and the mass to be isolated have finite frequency-dependent receptances (analogous to the analysis presented by Sykes (1958) for the passive isolation case). Further evidence for the effectiveness of using an active system built on the use of feedback of transmitted force measurement has been presented by Spanos et al. (1993). These authors describe an electrodynamic actuator which is again used in parallel with a passive isolator together with a control loop which was a direct measurement of the total force applied to a flexible structure from which the isolation of a broadband disturbance source is required. Very large (40 dB) improvements in force transmissibility are demonstrated at the natural frequency of the passive isolation system. The high level of performance is achieved by using a sophisticated compensation network in the feedback loop which introduces in excess of 20 dB of gain over approximately a decade of frequencies. The compensation design was implemented as a parallel bank of nine second-order filters on a 50 MHz type 68030 processor running at 4 kHz and achieved outstanding performance over a frequency range from about 5 Hz to 100 Hz.

7.8

Isolation of vibrating systems from random external excitation; the possibilities for feedforward control

The previous sections in this chapter have largely dealt with cases which involve the isolation of vibrations from a source (such as a vibrating machine) from a flexible receiving structure. The other major class of vibration isolation problem includes those cases where we wish to isolate a vibrating system from vibrations due to some external source. Examples of problems of this type include the isolation of buildings and sensitive equipment from ground vibrations and the isolation of the passenger cabins of road vehicles from vibrations induced by road undulations and roughness. Most approaches to problems of this type have involved the application of feedback control, since disturbances causing the vibration (especially in the case of road vehicles) are generally random. The topic of feedback control in these cases will be discussed in the next sections. However, in some instances, the possibility exists for adopting a feedforward approach and although hitherto not widely used, some discussion is worthwhile of the potential of the technique in the current context. The application of feedforward control, for example in the case of sensitive equipment to be isolated from ground vibrations, is illustrated in Fig. 7.19. The structure of the control system in this instance can be described exactly by the block diagram of Fig. 4.4. Thus vibrations travelling through the ground can be detected prior to their arrival at the base of the equipment to be isolated. These signals are detected and processed via a digital filter before being used to drive a secondary force input to the equipment. The filter is then adapted in order to minimise (in this case) some mean

212

ACTIVE CONTROLOF VIBRATION

Controller

Primary source of random vibrations

M

Fig. 7.19 Feedforward control for the improved isolation of sensitive equipment from random foundation vibrations. square measure of the systems response (displacement, velocity or acceleration). All of the features and performance limits associated with such a control approach are described fully in Chapter 4. One of the principle requirements for successful operation of such a controller is that the detected signal (effectively containing prior information regarding the vibrations about to be received by the controlled system) is well correlated with the vibrations of the controlled system. This requirement is formally expressed in terms of the coherence function which determines the performance limits of such a control approach (see equation 4.3.13). It should also be noted that a similar criterion applies in the case where multiple sources of differing vibrations are present. It will also often be necessary to apply multiple secondary force inputs to the vibrating system in order to control the various rigid body (or even flexible) modes of the controlled system's response. In such cases N detected signals could be processed via a matrix of N x M adaptive filters in order to minimise the response measured at L vibration sensors on the controlled system. The performance limits in such cases are again determined by multiple coherence functions and full details are given by Nelson and Elliott (1992, see Chapter 12, Section 12.7). The means of adapting the controller in such cases is also described in detail in Section 4.7 of this text. Although this feedforward approach to equipment isolation has hitherto not been widely used, its application in the field of vehicle suspension design has at least been considered in principle. Early work by Bender (1968) has examined in detail 'preview control'; a control scheme in which an input is sensed before it reaches the controlled plant. See also the work of Balzer (1981). This approach is identical to feedforward control in the sense discussed here. Bender in fact clearly recognised that such a preview controller would essentially consist of a Wiener filter which operates on a signal derived from a direct measurement of the road surface irregularities measured ahead of the vehicle. The system thus 'previews' the vibrations about to be applied to the vehicle suspension and thus can be thought of as a controller with the structure illustrated in Fig. 7.19 and shown in the block diagram of Fig. 4.4. The design procedure adopted by Bender was to synthesise the transfer function of a causally constrained (and therefore realisable) filter which operated on the signal detected ahead

ACTIVE ISOLATION OF VIBRATIONS

213

of the vehicle in order to minimise a quadratic cost consisting of a sum of two terms; the weighted mean square acceleration response of the vehicle and the mean square value of the relative displacement between the vehicle and the road. Classical Wiener filter theory was used (see Nelson and Elliott, 1992, Chapter 3, Section 3.8 for a brief description) and it is interesting to note that the adaptive feedforward approach described in Chapter 4 of this text essentially solves the same filter design problem automatically, although with a filter which is constrained to have a finite (as well as causal) impulse response. Bender compared the preview control approach with an optimised system that did not include preview information and concluded that the addition of preview control could account for rms vibration reductions by as much as a factor of 16 at constant clearance space (relative vehicle-road displacement). The computations were undertaken for a roadway displacement spectral density which was proportional to 1/o9 2. Although there are clearly advantages available with this approach it has, to the authors' knowledge, yet to find practical application. This is manifestly as a result of the difficulties associated with developing a sensor system which is capable of producing adequate accurate preview of the road disturbances ahead of the vehicle. Another field in which the use of adaptive feedforward methods has been considered is the active control of seismically induced building vibrations. In a series of papers presented by Burdisso et al. (1992, 1993a,b) an investigation has been carried out of the potential for detecting ground vibrations and then passing this signal, representative of typical seismic disturbances, through an adaptive filter whose output drives a force actuator used to control the vibrations of a small scale model of a flexible building. The experiments described show considerable promise in the reductions produced in the building response. Certainly this area of work shows potential for further development.

7.9

Isolation of vibrating systems from random external excitation; analysis of feedback control strategies

In the absence of any prior knowledge of the random disturbances impinging on a vibrating system, one is then faced with using a feedback strategy in order to implement control. Thus for example, in the case of vehicle suspension systems referred to above, one has to use detected system variables and feed back their values via some gain (or gains) in order to determine the secondary (controlling) forces required to act upon the system. The particular problem of feedback control applied to vehicle suspension systems has received considerable attention and this work will be reviewed in Section 7.11. Another area in which active techniques have been considered is in the isolation of sensitive equipment carried by spacecraft. In particular the isolation of telescopes from unwanted vibrations is especially critical (see the review by Collins and von Flotow, 1991). For example, the Hubble Space Telescope is equipped with a precision pointing control system required to meet remarkably stringent line of sight requirements. As described by Rodden et al. (1986), an accuracy of 0.01 arcsec (rms) is required together with a steady state stability of 0.007 arcsec (rms). Active approaches to improving the isolation of a telescope structure from an associated vibrating equipment module have been studied analytically by Kaplow and Velman (1980) who examine the possibility of using 'skyhook damping' for improving

214

ACTIVE C O N T R O L OF VIBRATION

the isolation effectiveness of a passive mounting system. The principle of such a system has already been described in Section 3.5; in this context an active force generating element acting in parallel with a passive mounting produces a force proportional to the velocity of the sensitive item to be isolated. The components of such a system are illustrated in Fig. 7.20. The equation of motion of the mass M (to be isolated) when Laplace transformed can be written as

sZMWR(s) + sC[WR(s)- Ws(s)] + K[WR(s) - W~(s)] = F,(s),

(7.9.1)

where W,(s) is the Laplace transform of the displacement w,(t) of the source of vibrations. If it is assumed that the secondary force fs(t) is made equal to a gain g multiplied by the velocity dwR(t)/dt of the mass (receiver) to be isolated, then

(sZM + sC + K)WR(s)- (sC + K)W,(s)= sgWR(s).

(7.9.2)

This can be rearranged to show that the displacement transmissibility is given by

WR(s) Ws(s)

=

sC + K

,

(7.9.3)

sZM + s(C + g) + K

which can be written as a function of frequency by putting s = jw such that

wRCjw) jwC + K = . W:(jw) -wZM +jw(C + g) + K

(7.9.4)

2 This in turn can be expressed as a function of the natural frequency defined by to,= K/M and the damping ratios ~pa:~= C/2Mwn and ~a~t= g/2Mw,,, such that

WR(jw)

2jW~passWn+ Oo~

(7.9.5) 2

~

Ws(jOo)

o

- 0 ) 2 + 2jf.o¢-On(~pass + ~act) + ¢.on

Finally this ratio can be expressed in terms of the non-dimensional excitation frequency = co/ w,, which gives WR(j(.o )

Ws(jW)

=

1 + 2j~pass~'~

1 -- ~,~2 +

2j(~pass +

.

(7.9.6)

~act) ~'~

This expression is shown graphically in Fig. 7.21 for a range of values of ~a~t. In the case of ~act= 0, the expression reduces to that given by equation (7.7.5) for the force transmissibility associated with passive isolation of a receiver from an SDOF system

-t

/

9;-'

/

/

/

/

Fig. 7.20 Active isolation of an SDOF system by using a secondary force applied in parallel with passive isolation.

215

ACTIVE ISOLATION OF VIBRATIONS

_

10 2

_

~"

lol

m

Increasing feedback

° _

•E ffl . _

00

t'2_ t_

f--

•

E

10-1 "Ideal" response

. _

a

10-2 0

1 Non-dimensional frequency, .Q

10

Fig. 7.21 The effect of feedback gain as the displacement transmissibility of the system sketched in Fig. 7.20 which implements a 'skyhook damper'. Note that there is no degradation in the high frequency isolation as the 'active damping' is increased. excited by a fluctuating force. This is depicted in Fig. 7.17(a). This simple graph demonstrates the difficulty with the use of only passive isolation; there is the possibility of resonant amplification of the disturbances when ~pas~is chosen to be low, whilst high frequency transmissibility is increased with high values of ~pas~-Active isolation, with the gain g chosen such that ~act- 1, can however, produce a net result which is a reasonable approximation to the 'ideal' transmissibility response illustrated in Fig. 7.21; that is a unit value of transmissibility below f~= 1 and a zero value above this frequency. Such a transmissibility function ensures that 'low frequency' attitude control commands are faithfully transmitted to the isolated item whilst high frequency vibrations are not. In fact, the low and high frequency asymptotes associated with the active system are precisely those associated with a pure stiffness, but the active system introduces damping into the response curve in the region ~ - 1. Whilst the study presented by Kaplow and Velman (1980) includes an analytically based simulation of the dynamics of a real space telescope, more recent work presented by Schubert (1991) and Edberg and von Flotow (1992) clearly demonstrates the practical effectiveness of this simple technique for enhancing the performance of passive isolation systems. Schubert, for example, describes a conventional general purpose isolation system based on pneumatic 'air spring' isolators. These passive isolators are enhanced with electrodynamic active elements which generate an additional force proportional to the velocity of the mounted payload. The latter is measured electrodynamically using a 'geophone'; a magnet supported off the mounted mass moves relative to a coil whose inertia tends to hold it fixed in space. The coil is mounted via springs off the magnet and the device therefore has mechanical dynamics

216

ACTIVE CONTROL OF VIBRATION

which effectively act to produce an output which is a high pass filtered version of the velocity signal. Experimental results are presented which demonstrate a displacement transmissibility that is improved by a factor of 100 in the region of the 4 Hz resonance of the passive air spring isolation system. A similar order of magnitude of improvement in transmissibility has been demonstrated in the practical system described by Edberg and von Flotow (1992). This work, stimulated by the need for a working micro-gravity isolation system required for a number of scientific experiments to be conducted in space, demonstrated the practicability of an active/passive isolation system in which the isolators were fabricated from a number of layers of etched polyvinylidene fluoride (PVDF) piezoelectric material. This enabled the isolators to provide a very low passive stiffness (with a correspondingly low mounted natural frequency) whilst also being capable of providing a secondary force input through the piezoelectric effect when the control voltage was applied. Similar requirements for space based experiments have also led to the development of active magnetic isolation systems. One such system is described by Allen et al. (1986). In this case, the isolation is provided entirely by the active force input generated by passing a measure of the relative displacement of the mounted mass via a compensation network G(s) to a magnetic actuator. Thus no passive isolation stage is included and the equation of motion of the mounted mass, when Laplace transformed, is given by s2MWR(S) =

G(s)[W,(s)- WR(S)],

(7.9.7)

which leads to the transmissibility function defined by

WR(s) G(s) = . Ws(s) s2M-G(s)

(7.9.8)

The authors claim that appropriate choice of G(s) can then give a transmissibility function which rolls off very rapidly with increasing frequency (100 dB/decade) which is greater than the high frequency asymptotes associated with either 'skyhook' active isolators (40 dB/decade) or passive isolators alone (20 dB /decade ).

7.10

Isolation of vibrating systems from random external excitation; formulation in terms of modern control theory

The simple analyses presented above are strictly speaking restricted to SDOF systems, even though in the practical examples cited, the principles have been applied in practice to MDOF isolation systems. However, there are methods for analysing MDOF isolation systems and we will give a brief introduction here. By far the most prevalent method of analysis has been undertaken within the framework of the linear quadratic Gaussian (LQG) regulator theory introduced in Section 3.8. A brief account of the basis for this approach in the context of active isolation of vibrations will be presented here. In Section 3.8 the solution is given for the problem of finding a control u(t) which acts on a system described by

~(t)=Ax(t)+Bu(t)

(7.10.1)

ACTIVE ISOLATION OF VIBRATIONS

217

such that u (t) = - G O(t)x(t),

(7.10.2)

where G°(t) is a matrix of time variable feedback gains, this matrix being that which ensures the minimisation of the quadratic cost function described by equation (3.10.1). The optimal gain matrix G°(t) is determined from the solution of the matrix Riccati equation given by equation (3.10.7). Now it turns out that for a stochastically excited system described by ~:(t) = Ax(t) + Bu(t) + v(t),

(7.10.3)

where v is a vector of external inputs each component consisting of zero mean Gaussian white noise, then the control u (t) which minimises the cost function J= E[xV(t)Qx(t) + uT(t)Ru(t)],

(7.10.4)

where E[ ] denotes the expectation operator, is given by u (t) = - R -1BTpx (t),

(7.10.5)

where the matrix P is a symmetric positive semi-definite matrix which satisfies the steady state matrix Riccati equation - P A - ATp + PBR-1BTp- Q =0.

(7.10.6)

Thus the stochastic control problem of minimising a performance index of the type described by equation (7.10.4) has an identical solution to the deterministic optimal regulator problem described in Section 3.8. A lucid description of this principle and its connection to problems in vehicle suspension design is given by Kamopp (1973). For a more rigorous description of the relationship between the deterministic and stochastic regulator problems, the reader is referred to more advanced texts on modem control systems theory (for example Stengel, 1986). The usefulness of the theory within the current context is of course that it is ideally suited to problems involving the active control of random vibration. Thus for example, in the case of a system describable by two state variables, x~ and x2 with two control inputs ul and u2, the performance index (7.10.4) with Q = ql and R = rl reduces to J= E[q(x 2+ x~) + r(u~ + u~)].

(7.10.7)

Thus the optimal control is synthesised in order to minimise the 'mean square' (equivalent to 'time averaged' for a stationary ergodic process) response as measured by x 2 and x22whilst penalising the mean square expenditure of effort as described by Ul2and u~.

7.11

Active isolation of vehicle vibrations from road and track irregularities

The theoretical approach outlined in the last section has been used by a number of authors in the analysis of vehicle suspension systems. Karnopp (1973) presents a number of simple examples where the order of the state space description of the dynamic system (equation 7.10.1) is sufficiently small to enable the algebraic Riccati equation (7.10.6) to be solved relatively easily and the optimal feedback gains to be

ACTIVE CONTROLOF VIBRATION

2 18

deduced. Perhaps one of the simplest systems amenable to analysis by such techniques is the base excited mass-spring system illustrated in Fig. 7.20 with the damper excluded. The state variables describing the system can be chosen to be the velocity wR of the receiver (mass) and the relative displacement wR- w, between the receiver and source (moving foundation). The equation of motion of the mass can be written as MffJR(t) + K(wR(t) - w~(t)) = f,(t),

(7.1 1.1)

where f , ( t ) is the control input to the mass and the dots symbolise differentiation with respect to time. This equation can be written in matrix form as the first-order differential equations

[

f v R ( t ) - fv,(t)

] [0 1

fvR(t) wR(t) - w,(t)

0

]

+

f,(t)

0 ]. -w~(t)

(7.11.2)

This equation thus has the form R(t) = Ax(t) + Bu(t) + v(t),

(7.11.3)

where the input vector u (t) is deafly the scalar f~(t) and the excitation v (t) is provided by the foundation velocity w~(t). Let us assume that we wish to find the optimal feedback gain that minimises the cost function J = E[qrOE(t) + ruE(t)],

(7.11.4)

where q is a weighting on the mean square velocity of the mass and r is a weighting and the mean square control 'effort' applied. Thus in terms of the more general cost function defined by equation (7.10.4), this amounts to choosing R - rI, where I is the identity matrix and

[~ o0]"

Q=

~11~

The algebraic Riccati equation that has to be solved (equation 7.10.6) thus reduces to PA

ATP _ _1 PBBT P + Q = o.

+

(7.11.6)

r It is useful to write this equation out in full. Since the matrix P is symmetric, and using the relation co,2 = K / M , equation (7.11.6) can be written as

[,11

PiE P22] 1

-to.

r

0

P12 P22

PlZ P22

[ ][: 0]

0] Pll P12 + P 12 P22

0

=0.

(7.11.7)

0

Multiplication of the matrices involved yields

f ][

11

PIE -(onP + [P22 to~P12

P12 2

-°)nPll

P22

1

092.p12]-rM2 -

r

PI1 LPllP12

PllP12 + P~2

= 0.

(7.11.8)

219

ACTIVE ISOLATION OF VIBRATIONS

Equating the sum of the appropriate elements of each of the matrices to zero yields four independent equations, two of which are identical. Three equations in the unknown elements Pll, P12 and P22 thus result and are given by 2P~2- (I/rME)p2~ + q = 0 , P22

(7.11.9)

2 _ (1/rM2)PllP12 = 0, - tOnPll

(7.11.10)

a)nPl2 (I/rM2)P22=O.

(7.11.11)

--2

2

_

Now note that the feedback law given by u(t) = -R-~BTPx(t)

(7.11.12)

can in this case be written as the scalar

fs(t)

=

1,1,o, o[;11

12 e 2 2 J [ w R ( t )

r

- Ws(t)

]

'

which therefore reduces to fs(t) =

1

rM

{PllWR(t) + Plz(wR(t)- w~(t)}.

(7.11.14)

The terms P~I and P12 can be deduced directly from equations (7.11.11) and (7.11.9). There are two solutions to equations (7.11.11) given by

Pl2 __ O,

P12_. -2rMZt°n2

(7.11.15a,b)

and the solution to equation (7.11.9) can be written as

PII = a/rMZ(q + 2P12).

(7.11.16)

Finally note that equation (7.11.10) shows that P22

= c°2~PI1 + (1/rMZ)Pll P 12.

(7.11 17)

These equations demonstrate that there are a large range of possible solutions for the elements of the matrix P. There is only one solution, however, that ensures that the matrix P is positive semi-definite, i.e. that iTPi~>0 for any choice of the vector i. Positive semi-definiteness of the matrix P ensures that the control is stabilising; thus the eigenvalues of the closed loop loop system matrix given by [A-R-~BTP] will all lie in the left half of the s-plane. In this case, assuming that i is the unit vector leads directly to the condition P~ + 2P12+

P22 >~ 0 .

(7.11.18)

It now becomes clear that we must choose P12 = 0,

PI~ = x/rMZq,

P22 = ~/to,rMZq

(7.11.19a,b,c)

if the positive semi-definiteness condition on P is to be satisfied. The feedback control law given by equation (7.11.14) thus reduces simply to

f s( t) = -~/q/rfv R( t).

(7.11.20)

220

ACTIVE CONTROL OF VIBRATION

This is precisely the feedback law discussed in Section 7.9 which results in 'skyhook damping' of the controlled system. This analysis demonstrates that skyhook damping constitutes an optimal strategy in the sense that it minimises the mean square velocity of the mass to be isolated. The feedback gain g introduced in Section 7.9 is shown here to be given by ~/q/r and is thus seen to be a simple function of the ratio of the relative penalty on minimising mean square error and mean square effort; the smaller r (the effort weighting) the higher the feedback gain. An analysis of a very similar type of problem is given by Hrovat and Hubbard (1981) who formulate the state space description of a two mass system separated by a spring with an active force in parallel. The time derivative of the acceleration of one of the masses and the relative displacement between them are included in the performance index for minimisation. They conclude that the optimal controller in this case also includes a 'skyhook spring' in addition to a skyhook damper. Charts are presented which enable preliminary design calculations to be undertaken graphically. In addition to these relatively simple analyses, the fact that the principles outlined in the last section can be readily applied to systems of much higher order, has enabled analyses to be undertaken of much more complex dynamic models. A typical example is the work presented by Ha6 (1986) (see also Ha6 1985) who considers the active control of a flexible beam system mounted via two suspensions (one at each end of the beam), each suspension consisting of a two-stage dynamic system. The inclusion of a flexible beam model was of course to enable the effects of real vehicle flexibility to be introduced into the dynamic analysis of the active system. Again the LQG formalism was applied to the problem with feedback gains chosen to minimise a performance index consisting of the sum of the squared accelerations at a number of positions in the system, the mean square values of the primary and secondary suspension deflections and the active control forces. In addition a Kalman filter was included to produce optimal estimates of the 'noisy' state variables (see Chapter 3). Ha6 concluded that the flexibility of the body of the vehicle could be neglected if the lowest natural frequency of bending is more than ten times larger than the lowest natural frequency corresponding to rigid body modes. A similarly sophisticated analysis has been presented by Sinha et al. (1978) who used two rigid truck models each having two degrees of freedom together with a threedegree-of-freedom rigid car body model in an analysis of the potential of active control as a technique for improving rail vehicle performance at high speeds. Operation of conventional rail vehicles is limited by a number of dynamic problems including ride quality and curve negotiation. The seven-degree-of-freedom model included active control actuators which applied additional steering torques and lateral forces between the car body and truck. The LQG control problem was solved with a performance index including a weighted sum of mean square tracking errors, car body acceleration and control effort. The case studies presented suggested that the controller had the potential for producing reductions in rms acceleration and tracking errors by factors of about five from the baseline vehicle. Other studies of similar problems have been undertaken by Yoshimura et al. (1986), Hall and Gill (1987), Celniker and Hedrick (1982) and E1 Madany and E1Razaz (1988). Whilst the analytical studies referred to above make good use of the optimal control formalism and theoretical studies can give clear guidelines for the synthesis of real controllers, practical implementations rarely correspond exactly to these idealised

ACTIVE ISOLATION OF VIBRATIONS

221

dynamic models. Nevertheless, considerable progress has been made in practical active suspension design. Much of this work was stimulated by the use of the aerodynamic 'ground effect' in Formula one Grand Prix racing cars (Wright, 1978). The ground effect cars introduced by the Lotus team in 1978-79 could generate total downforces of up to 25 kN at velocities of 75 m s -~ enabling cornering accelerations near to 4g. The conventional suspensions of the cars, consisting of hydraulic dampers and coil springs, were made very stiff to ensure that the vehicle skirts were kept on the ground. This made the cars particularly uncomfortable to drive. Active suspension was introduced for both ride quality improvement and to maintain ground clearance as the aerodynamic downforce was increased (see Dominy and Dominy, 1984; Wright, 1978; Dominy and Bulman, 1985). These developments are now finding their way towards more widespread applications on road vehicles, although problems such as excessive power consumption by hydraulic actuators in such 'fully active' systems have still to be overcome before mass-production becomes likely.

8 Active Structural Acoustic Control. I Plate Systems

8.1

Introduction

In numerous industrial applications, structurally radiated noise is a persistent problem which is often poorly alleviated by passive means, particularly at low frequencies. The possibility of active noise control was suggested over 50 years ago (Lueg, 1936) but its implementation is a relatively recent development due to advances in fast microprocessors for digital signal processing. In this chapter we discuss the active control of sound radiation from distributed vibrating structures. Two forms of control sources are generally available. The use of acoustic control sources have been investigated by previous workers (Deffayet and Nelson, 1988 and Fuller et al., 1991). In general it has been shown that when the sound source is complex or distributed over multiple surfaces, many acoustic control sources are required in order to provide global control. An alternative approach as embodied in the research of Fuller and his co-workers (Fuller, 1985a, 1987, 1988), is to use control inputs applied directly to the structure in order to reduce or change the vibration distribution with the objective of reducing the overall sound radiation. This technique has been termed Active Structural Acoustic Control (ASAC), an abbreviation that conforms with the generally accepted terminologies of Active Noise Control (ANC) and Active Vibration Control (AVC). Figure 8.1 shows a genetic arrangement of a distributed elastic system excited by an oscillating disturbance. Sound radiation occurs as a result of the continuity of particle displacement at the interface between the structure and the surrounding compressible medium. The objective is to reduce the sound radiation. Obviously completely reducing the overall structural response with active vibration control would lead to an attenuation of the sound radiation. However, as shown later, various modes of vibration have differing radiation efficiencies and some are better coupled to the radiation field than others. This suggests that in order to reduce sound radiation, only selected modes need to be controlled, rather than the whole response. In addition, if the relative phases and amplitudes of a multi-modal response can be adjusted so that they destructively interfere in terms of radiated sound, then the radiation field may be attenuated with little change in the overall response amplitude of the system. In other words, the controlled system will have an overall lower radiation efficiency than the uncontrolled system. The above observations suggest the general arrangement of ASAC; control inputs are applied to the structure while minimising radiated pressure or pressure related variables.

224

ACTIVE CONTROL OF VIBRATION

Sound radiation

I

----

--

---

"-

,,, ,, •

/,

Distr!buted

~ ~

Disturbance

\

-

I

fp"

',

',

Radiation

: +senors

,, ,

.

error

/

elastic system

,

',

\

~ ~ , ~ ' w N ~ ~ ~ s \

"~ w

Normal surface velocity

f

Control forces

Fig. 8.1 Generalarrangement for Active Structural Acoustic Control (ASAC). It should be emphasised that ASAC is not simply a matter of applying AVC. Knyazev and Tartakovskii (1967) have demonstrated that AVC can lead to an increase in radiated sound levels. Conversely as will be seen later, reduction of radiated sound is sometimes accompanied by an increase in structural vibration levels. The advantages of ASAC can be seen to be a marked reduction in the number of control channels and the control power consumed in particular applications. In addition when ASAC is implemented using piezoelectric or other induced strain transducers, a very compact and lightweight control configuration is obtained. These advantages have stimulated research in ASAC by a number of workers such as Vyalyshev et al. (1986), Meirovitch and Thangjitham (1990), Hansen and Snyder (1991), Thi et al. (1991), Naghshineh and Koopman (1991) and Thomas et al. (1990). Before we begin an outline of the application of ASAC to plate systems, it is advantageous to review some of the basic theory of structural acoustics, or how vibrating structures couple to their radiated sound fields. We then apply ASAC to harmonic radiation from plates and finish by considering transient and broadband disturbances and the active control of their associated sound radiation.

8.2

Sound

radiation

by planar

vibrating

surfaces;

the Rayleigh integral

One of the most important problems dealt with by researchers in the field of active control in recent years is that of the active suppression of sound radiation from a vibrating plane surface. The reason for the preoccupation with this topic is that it

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS

225

represents the simplest idealisation of a whole class of problems of practical interest involving the radiation from, or transmission through, some form of structure, whether it is the radiation from the hull of a submarine or the transmission of sound through the fuselage of an aircraft. Before describing details of recent research into this problem, it will be useful to review briefly some of the basic techniques involved in calculating the sound field radiated by vibrating surfaces. Here we will give an outline of a subject which is dealt with at length and in considerably more depth by authors such as Junger and Feit (1986) or Fahy (1985). The analysis presented in this section is generally for harmonic motion of one mode of plate vibration. The total radiation consists of a superposition of radiation from individual modes that 'interfere' in the radiation field and this will be described in Section 8.4. The evaluation of the Rayleigh integral is probably the conceptually simplest approach to calculating the sound field radiated by an area of vibrating surface that is surrounded by an otherwise rigid infinite plane (see Fig. 8.2). The Rayleigh integral gives the complex pressure (associated with an e j~' time dependence) at a given field point p(r) in terms of the complex velocities v0(r,) associated with an elemental sources at points r s on the surface S. Thus

I ja)p°fv(rs)e-J~ dS, p(r)= s

(8.2.1)

2erR

where R = I r- r, I and v~(r,) is the component of the complex velocity normal to the surface S, while P0 is the density of the acoustic medium. This equation can be derived

p(r) \ \ \ \ \

Elemental source

~fJf I~

)<.

f f

,. f

f

Y

f \

Fig. 8.2 A rectangular plate in an infinite baffle showing nodal lines, coordinate system and an elemental sound source associated with the plate motion.

226

ACTIVE CONTROL OF VIBRATION

from the more general solution to the wave equation presented by, for example Pierce (1981) or Nelson and Elliott (1992). Essemially the integral evaluates the sum of the fields of a distribution of elemental sources, each having a complex volume velocity rO(r~) dS. As an example of the application of this theoretical approach, consider the sound field radiated by the rectangular plate illustrated in Fig. 8.2. The analysis is assisted by expressing the far-field pressure in the spherical coordinates (r, 0, q~). The classical assumption made in order to evaluate the far-field pressure is that the value of R in the exponential term e-ju~ in equation (8.2.1) is approximated by R .~ r - x sin 0 cos q~- y sin 0 sin q~,

(8.2.2)

where x and y define the coordinate position on the plate and (r, 0, q~) are the coordinates of the field point. This assumption is valid provided R~> a, b, the plate dimensions (see Junger and Feit, 1986, for a full discussion). In addition, the term R in the denominator of equation (8.2.1) can be approximated by R--r, a less stringent approximation being required for this term than for the exponential term which determines the relative phase of the contributions to the pressure at the field point from the different elemental sources. A particular form of out-of-plane vibration of a rectangular plate which, with the above assumptions, leads to an analytically tractable form of equation (8.2.1), is given by ~'(r') = W"~ sin(mZ~X)sin(-~)a

{a0}, 0 x, y, , b "

(8.2.3)

This corresponds to the complex velocity distribution associated with a simply supported plate vibrating in its (m, n)th mode as discussed in Chapter 2. Equation (8.2.1) then reduces to P(r'O'~)=jogp°~Vmne-ikr

Io Iao sin ----~-](mz~X/sin(nZCY)eJ(ax/a+~y/b)dxdy---ff-, (8.2.4)

where a = kasin0cosq~ and fl = kbsinOsincp. This integral has been evaluated by Wallace (1972) who gives the solution

p(r,O, qb)=joopo(Vmne-Jk~ ab [(-a)me-~a-a][(-1)~e-Jt~-l] 2z~r mnz~ 2 -~a'~m~'~---1 (fl/nz02-1 ' '

(8.2.5)

The far-field pressure is clearly a complicated function of the geometry and the modal integers (m, n). In the discussions of this section we will initially restrict attention to a single global measure of the far-field sound; that is the total radiated acoustic power. This can be found by integrating the far-field acoustic intensity over a hemisphere surrounding the plate. This intensity is given by

= 2pocol W,,~12 kab 2p0c0

2

COS

COS

sin

sin

z~3rmn [(a/mzO 2- 1][(fl/nzO z- 1

•

(8.2.6)

This expression is in the form given by Wallace (1972), where cos (a/2) is used when

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

227

m is an odd integer and sin (a/2) is used when m is even. Also, cos (fl/2) is used when n is odd and sin (fl/2) is used when n is even. Note that P0 and Co are respectively the density and sound speed of the medium and k - tO~Co is the acoustic wavenumber. An important feature of this result has been pointed out by Fahy (1985). If one evaluates the maximum value of this intensity, when the wavelength 2 of the sound is such that it exceeds both the 'structural trace wavelengths' of the plate vibration (i.e. such that ka ~ m~ and kb~ n~), it can be shown that the intensity produced by the vibrating plate is never greater than that which would be produced by a single 'cell' of vibration acting alone. (The cells are illustrated in Fig. 8.2; a single cell corresponds to an area bounded by nodal lines shown as dashes.) This is one of the most significant features of plate radiation; it is the interference produced by different areas of the vibrating plate which largely characterises the low frequency radiation of most interest in studies of active control. The destructive interference produced by contributions from neighbouring cells has a profound influence on the radiation characteristics, and as we shall see, it is the contribution from 'uncancelled' cells which dominate the radiation characteristics of a given mode of vibration (Maidanik, 1962). In order to quantify these effects we can evaluate the total power radiated by the (m, n)th mode which is given by integrating the far-field intensity over a hemisphere surrounding the plate. Thus the sound power is given by

0, 0

r sin 0 dO dq~.

(8.2.7)

2p0c0

Wallace (1972) has undertaken this integral numerically using equation (8.2.6) for the acoustic intensity. In order to better compare the results for different mode orders, Wallace defines a modal radiation efficiency given by am, =

H

(l fi,'mn[2)lOocoab

,

(8.2.8)

where (I Wreni2) is the temporal and spatial average modal velocity of the plate, which in this case is simply given by ll;Vm, 12/8. Some examples of the radiation efficiency curves computed by Wallace are shown in Fig. 8.3. Note the widely differing form of the radiation efficiency curves for the different modes at low frequency. These radiation efficiencies are applicable when the plate response is dominated by one mode, i.e. at resonance. Off-resonance, when more than one mode is significant, the radiated power cannot be simply calculated using individual radiation efficiencies. The results shown in Fig. 8.3 are plotted as a function of the dimensionless ratio k/k b, where kb is the structural wavenumber given by +

.

(8.2.9)

~at Thus once k/kb,> 1, the radiation efficiency of all modes becomes unity; this corresponds to the condition that the structural wavelength exceeds the acoustic wavelength and under these conditions there is no appreciable interference between the contributions from neighbouring 'cells'.

228

ACTIVE CONTROL OF VIBRATION

10-I

(m, n)

o>~ 1 0-2 ,-.

.

_

o _

t-

.~

-3

-~ 10

.

_

"o rr

10-4

165

(2, 2)

0.1

1

Non-dimensional wavenumber, k~ kb

Fig. 8.3 Radiation efficiency curves for a number of modes of a square plate (after Wallace, 1972).

The frequency at which the structural wavelength equals the acoustic wavelength in the surrounding medium is known as the critical frequency. In terms of free wavenumber it is thus defined as U = k b = ki,

(8.2.10)

where kI is the flexural wavenumber defined in Chapter 2. For a thin plate the critical frequency f~ is given by (Cremer and Heckl, 1988) 2

fc - ~ CO , 1.8CLh

(8.2.11)

where Co is sound velocity, CL is the longitudinal plate wave phase speed given in Chapter 2 and h is the thickness of the plate. The critical frequency is thus an important characteristic of structural response in terms of sound radiation. Below the critical frequency interference effects between neighboring cells are the dominant factor in determining radiation efficiency. Well above the critical frequency there is no significant interference between the radiation from cells and the radiation efficiency is thus largely independent of modal order. Wallace gives the following expressions for the radiation efficiency of different mode types in the low frequency limit when the acoustic wavelength is much greater than either of the plate structural wavelengths ( o r f ~ f~).

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS

229

For m, n both odd,

Omn

'~'

2 5 m 2ner

12

(m~) 2 b + 1

20

(mzO2 b + 1

(ny~) 2

.

(n"~'~)2

•

(8.2.12)

For m odd, n even, Omn

~"

8(ka)(kb)3 f ~ - - 1 3mZn2~ 5

For m, n both even, (~mn

~

15m2n2er 5

( 4)a (

f

1

1- ~ + 1 (met) 2 b

64

(~-~)2

(8.2.13)

•

(8.2.14)

When m is even and n is odd, an identical result to equation (8.2.13) follows with m and n interchanged. Still more instructive forms of these expressions result when ka, kb e 1 (i.e. the acoustic wavelength is much larger than the plate dimensions). In that case, assuming for the moment that b = a, we have the following. For m, n both odd, 32(ka) 2 Om n ~

~

m 2n 2y./~5

(8.2.15)

.

For m odd, n even, 8(ka) 4 ~mn

~

3m2n27g5

•

(8.2.16)

For m, n both even,

2(ka) 6 am,, --

15m2n2er 5

•

(8.2.17)

The dependence of the radiation efficiency of these three mode classes on increasingly high powers of ka shows that the three classes exhibit radiation efficiencies which are respectively characteristic of monopole, dipole and quadrupole type sources. This is shown in Fig. 8.4 which illustrates the 'comer monopole' model of low frequency plate radiation. This is based on the notion of perfect cancellation by neighboring 'half-cells' of vibration, such that the only uncancelled cells appear at the comers of the plate. The relative phase between these effective 'comer monopole sources' then determines whether the new source is of monopole, dipole or quadrupole type (Maidanik, 1962). This model was used by Deffayet and Nelson (1988) to describe the effectiveness of using acoustic secondary sources to control the low frequency radiation of a rectangular plate. It was shown that the field of a monopole type mode could be adequately controlled globally using a single secondary monopole source, whereas a dipole type mode required two correctly oriented secondary sources for global control to be achieved. For control of radiation from quadrupole type modes, four appropriately located and phased secondary acoustic sources were required.

ACTIVECONTROLOFVIBRATION

230 Y +

+

+

+

~X

Y +~,,

=

+

m odd, n odd (monopole)

,~X

m even, n odd (dipole)

Y

y i

,

+

I I I . . . . . . .

I. . . . . . . I I I

~X

+

+ m odd, n even (dipole)

+

I

=

~X

m even, n even (quadrupole)

Fig. 8.4 The comer monopole model of low frequency plate radiation.

8.3

The calculation of radiated sound fields by using wavenumber Fourier transforms

Another approach for the calculation of sound fields radiated by vibrating surfaces involves working with the spatial Fourier transforms of the variables involved. The transforms used in dealing with, for example, radiation from a two-dimensional surface in the x - y plane have the form

F(kx' kY) = I~ I~ f(X' y) eJ(kxX+kyY)dx dy,

(8.3.1)

1 -.i(kxx+ kyy) f(x, y)= (270 2 I?**I?= F(kx, ky) e dkx dky.

(8.3.2)

Thus, as discussed in Section 5.9, the Fourier transforms used, rather than transforming from say time to frequency, transform from the spatial variables x and y to the wavenumber variables kx, ky (and vice-versa). The utility of these expressions can be demonstrated by first applying the transform given by equation (8.3.1) to the Helmholtz equation which governs the form of the complex pressure in a three-dimensional harmonic sound field. This is given by (V 2+ kZ)p(x, y, z) = O,

(8.3.3)

where ~r2 is the Laplacian operator. The transformed equation can thus be written as

~92 ~)2 f-,.r-,, (-~X2+ by 2

~92 2) bz 2

Y, z) e j(kxx+kyy)ax

y=o.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

231

The derivatives with respect to x and y integrate respectively to -k2p(kx, ky, z) and -k2yP(kx, ky, z) while the derivative with respect to z can be taken outside the integral. The transformed equation then becomes (see, for example, Junger and Feit, 1986) 2

2-k2-ky

2

z) = o.

(8.3.5)

This equation has the solution

P (kx, ky, z) = A e -jk~z,

(8.3.6)

where the wavenumber kz is given by

kz = x/k 2 - k 2 - k 2

(8.3.7)

and A is an arbitrary constant. The solution to the transformed equation thus has the form of a simple plane acoustic wave propagating in the positive z-direction with the wavenumber kz. Note that if (k 2 + k2) > k 2, then kz will be imaginary and equation (8.3.6) will represent an exponentially decaying solution in the positive z direction. (The real part of the exponent in equation (8.3.6) must always be negative in order to satisfy the Sommerfeld radiation condition; see Junger and Feit, 1986, Ch. 5.) The transformation of the complex acoustic pressure by using equation (8.3.1) thus effectively decomposes the field into a sum of plane waves. The formal approach adopted in the solution of radiation problems using this technique is to also transform the boundary conditions. Thus, for example, in the case of acoustic radiation from a plane surface, the linearised equation of conservation of momentum requires that on the surface

jwpofv(x, y) +

i)p(x, y, z)

= 0,

(8.3.8)

bz

where w(x, y) is the complex velocity of the surface in the positive z direction (normal to the x - y plane) and P0 is density of the acoustic medium. The transformed boundary condition is then simply, at z = 0

jwpoW(kx, ky) + ~ P(kx, ky, z) = 0.

(8.3.9)

Thus we can use the general solution of equation (8.3.6) and determine the constant A in terms of the transformed surface velocity distribution W(kx, ky). Substitution of equation (8.3.6) into equation (8.3.9) gives

A = wpoW(kx, ky)/kz

(8.3.10)

and therefore the transformed pressure field is related to the transformed velocity field by

P(k x, ky,

z') =

wpo(V(kx, ky) -~k=z e

kz

.

(8.3.11)

232

ACTIVE CONTROL OF VIBRATION

The resulting complex pressure field can then be calculated by an inverse double Fourier transformation of this result. Thus

W( x, y)e p(x, Y, z) = J_oo J_oo kz (2Jr) 2

dkx dky.

(8.3.12)

Integrals of this type can be evaluated relatively easily by using the method of stationary phase. A full account of the use of this technique in solving planar radiation problems is given by Junger and Feit (1986). An important observation given by Junger and Feit is that radiation at a particular angle corresponds directly to a particular vector wavenumber quantity of the planar structural motion. Another approach which has been used more recently is to use the Fast Fourier Transform algorithm for efficient numerical evaluation of the integral transforms. Thus for example, given a complex surface velocity distribution, the discrete Fourier transform corresponding to equation (8.3.1) is evaluated numerically in order to approximate W(k~, ky) at a series of discrete wavenumber values. Equation (8.3.11) is then used to evaluate P(kx, ky, z) at a given value of the coordinate z and the inverse transform is evaluated numerically in order to recover p(x, y, z). This is essentially the approach adopted in generalised acoustic holography (Maynard et al., 1985; Veronesi and Maynard, 1987). In that case it is also sometimes useful to make measurements of surface pressure on a plane (at z = zH say) just above the source. The general solution, equation (8.3.6), to the transformed Helmholtz equation can be used to show that if P(kx, ky, zH) is the measured wavenumber transform at z = zH, then the wavenumber transform at any other plane z is simply given by

P(kx, ky, z) = P(kx, ky, ZH) e -ik:(z - z,).

(8.3.13)

A description of the numerical implications of the use of this technique is presented by Veronesi and Maynard (1987). A further use of the wavenumber transform is that it can be used to give expressions for the total sound power radiated by a vibrating planar surface in terms of acoustical and vibrational variables measured on the plate surface. The sound power radiated by harmonic vibrations of a surface whose complex normal velocity is ~i,(x, y) and on which the complex acoustic pressure is given by p(x, y) is given by FI =

g Re

p(x, y)

_oo

(x, y) dx dy ,

(8.3.14)

which is the integral over the surface of the time averaged normal acoustic intensity. Note that Re denotes the operation of taking the real part and (*) denotes the complex conjugate. Clearly if ~i,(x, y) is non-zero over some finite region of the surface, then the limits on the integral become finite. However, in order to derive expressions for the acoustic power radiated in terms of wavenumber transforms, it is helpful to use Parseval's formula which, in this context, can be expressed as

I?ooI?oop(x, y)Cv* (x, y)dx dy= ~

4:r2

I?o. I?o. P(kx ky)(~/r~(k x, ky) dk x dky.

(8.3.15)

Using this relationship together with equation (8.3.11) to relate P(kx, k,) to IiV(k~, ky)

233

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

shows that the acoustic power output can be expressed in terms of the surface velocity transform as I W(kx, k s) 12 dkx dky . kz

H = Wpo Re 8:r 2

(8.3.16)

We now note that 1/kz is only real for acoustic wavenumbers k = vo/c that satisfy k i> a/k 2 + k~

(8.3.17)

and thus equation (8.3.16) reduces to

n OPo8

l '(kx, ~22

-

ks)I.= dkx dk s,

(8.3.18)

+k~ ~<

where the range of integration is only over those wavenumbers satisfying the inequality given by equation (8.3.17). Alternatively one may write the expression for the acoustic power in terms of the wavenumber transform of the acoustic pressure. Again using equations (8.3.15) and (8.3.11) shows that n =

1

II

2 dxx d k y ,

IP(kx) k y ) 1 2 4 k 2 - k 2 - ky

(8.3.19)

8~2wP° k~+~y -< 2 k2 where again the integration is undertaken only over those wavenumbers for which k2 + k2 ~
¢v(x)={ Wsin(mzex/a)O

0<x<;}0>x>"

(8.3.20)

The wavenumber transform is given by W(kx) = WIo sin(merx/a)e '~'xxdx,

(8.3.21)

which results in m -jkxa

-1] W(kx) = (V (mJr/a)[(-1) e k 2 _ (mzr/a) 2

(8.3.22)

This has the modulus squared given by [W(kx) 12 --" [ W I

2

2m~r/a k ~ - (mzr/a) 2

sin2 kx a -2 mJr).

(8.3.23)

The resulting wavenumber spectrum is sketched in Fig. 8.5. Note that since the displacement is uniform in the y direction then the ky spectrum is effectively a Dirac

234

ACTIVE CONTROL O F VIBRATION

Radiating wavenumber components /a) increasing~ ,

-m]z"

-(.z)

a)

a

cO

cO

/>,

m_...~ Wavenumber,kx 8

Fig. 8.5 A typical wavenumber spectrum of a vibrating plate and the identification of radiating wavenumber components (after Fahy, 1985). delta function at the origin. This reduces the condition of equation (8.3.17) to k ~ +kx. The peak in the spectrum is given by kx = m~r/a. However, as we have seen in equation (8.3.18), only those values of I kxl k, the acoustic wavenumber, will contribute to acoustic power radiation (for the one-dimensional case). This range of wavenumbers is shown cross-hatched, for example, in Fig. 8.5. Physically speaking, if one associates with the wavenumber kx, a phase speed of propagation Cx, such that kx = cO/Cx then the condition that ] kx[ >-k amounts to requiring that ] c~ ] ~
ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS

235

field will occur, since the wavenumber transform will consist of two Dirac delta functions as shown in Fig. 8.6(a), outside the radiation circle. On the other hand, if the same standing wave exists only over a finite region, as for a finite plate, then the wavenumber spectrum will be a result of the convolution of the wavenumber spectrum of the infinite system and that of the 'window' associated with the finite size and shape of the structure. In this case, the wavenumber transform will have the approximate form of Fig. 8.6(b) and now radiation will occur due to the presence of supersonic components. In other words the discontinuities associated with the plate edges have led to a scattering of energy from subsonic to supersonic wavenumber motion. Once again this provides an example of the dual interpretation of structural response in terms of waves or modes. Radiation from higher order modes can be either thought of as resulting from uncancelled modal cells as shown from Fig. 8.2 or due to supersonic wave components. Both of these behaviours, due to the edge scattering, lead

v (kx)l I

I I I I I I I I I I I I I I

-kf

(a)

-k

+k

+kf

Wavenumber, kx

Supersonic region

W(kx)I

(b)

I

I

I I I

I I I

I I I I

I I I I

-kf

-k

+k

+kf

,,

Wavenumber,kx

Fig. 8.6 Schematic illustration of the wavenumber distribution on (a) an infinite onedimensional plate with subsonic bending waves, (b) a finite one-dimensional plate with subsonic bending waves.

236

ACTIVE CONTROL OF VIBRATION

to identical radiation in the spatial domain. Observations such as this, aid in understanding the behaviour of active controllers and sometimes lead to alternative approaches to the control problem. This concept will be briefly expanded upon next.

8.4

Sound power radiation from structures in terms of their multi-modal response

In Section 8.2 we have described the acoustic radiation from individual modes of a vibrating plate. In reality, however, at a given frequency of excitation, a plate will exhibit a response which is a superposition of a number of individual modal responses. Strictly speaking, the response can only be fully described by an infinite series of modes, although in practice a finite series will give a good representation; the number of modes used being dependent on the damping of the plate, the frequency of interest and the convergence of the truncated series. Thus the complex velocity of the plate surface excited at a frequency to can be described by

m=M n=N

~jcot

¢v(x, y) = Z Z (Vm~Pmn(X,y)~ ,

(8.4.1)

m=O, n=O

where Wm~ is the complex amplitude of the (m, n)th mode and ~'m~(X,y) is the mode shape function which describes the modal space dependence. Clearly in the case of the simply supported plate described in Section 8.2, ~mn(X,Y) can be expanded into separable functions such that lffmn(X, y ) = lpm(X)lpn(y), where ~m(X) and ~p~(y) are respectively given by sin(mJrx/a) and sin(nzry/b) and a and b are the plate dimensions in the x and y directions, respectively. Equation (8.4.1) can also be written in the form W(x, y ) = vgT~3(X, y),

(8.4.2)

where the vectors ,iv and lp(x, y) are defined by (8.4.3a,b) "/vT= [W01W10W~... WMN], ~pT(x, y)= [~Po(X)~Pl(y) ~P~(X)~Po(y) ~Pl(X)~Pl(Y)... ~PM(X)~pN(Y)]. An expression for the power output due to the plate response at a given frequency can be found by evaluating equation (8.3.18) which expresses the sound power output of a planar vibrating surface in terms of the wavenumber transform of the surface velocity field. Thus if we define the wavenumber transformed vector ~P(kx,ky) by

~p(kx, ky) = I~_~I?o.lp(x, y)e j(kxx+kyy) dx dy,

(8.4.4)

then the modulus squared of the wavenumber-transformed velocity field can be written as

[ W(kx, ky)[ 2= [,~¢T~)(kx, ky)12=~cH~p*(kx, ky)~T(kx, ky)*W.

(8.4.5)

Substitution of this result into equation (8.3.18) then shows that the expression for the sound power output of the plate can be written as II = ~¢HM~¢,

(8.4.6)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

237

where the matrix M is given by

M=

~op0 Re {I °°

8yg2

I;-

~P*

T (k"m' k'-')IP(kx~' kY)dkx dky}. a/k - kx - ky 2

-oo

2

(8.4.7)

2

The matrix M thus has diagonal terms which quantify the 'self' radiation resistances of the individual modes when acting in isolation, whereas the off-diagonal terms quantify the 'mutual' radiation resistances which arise as a direct result of the interference of the fields of two different modes. Thus if we characterise one mode by the integers (m, n) and another mode by the integers (m', n' ), the corresponding entry in the matrix M is given by

Mmn'm'n'

=("0/902 Rell °° I;oo

8 yg

_oo

lP*m(kx)~;(2Y21P--m'('kx)lPn'(kY)5--;5---75 ] ~fk - kx - ky dkx dky .

(8.4.8)

Using the results of equation (8.3.22) which gives the wavenumber transform of sin(mzcx/a), shows that for a simply supported plate, we can write

~ *m(kx)~m'(kx) =

tam'jr 2 fmm,(kxa), aZ[k~- (mz~/a)Z] [k 2 -(m'yc/a) 2]

(8.4.9)

where the function fmm' (k~a) is given by 2 (1 - cos kxa) 2 (1 + cos kxa) f mm,(kxa) = 2j sinxa - 2 j sin kxa

m even, m' even m odd, m' odd m odd, m' even m even, m' odd

(8.4.10)

Exactly analogous results follow for the product ~p* (ky)~pn,(ky). The expression for the entry in the matrix M due to the interference of the (m, n)th and (m', n' )th modes can thus be written as

ogpo mm'nn'275 8yg2 aZb 2

4

Mmm'nn' =

× Re

"

(8.4.11) This integral can be simplified considerably in the low frequency limit. Thus, if one assumes that kxa~mzr and kyb~ nzc, terms like (k 2- ( m ~ / a ) 2) c a n be replaced by (mzr/a) 2 in the denominator of the integrand. In addition, for small values of k,a, series expansion of the expressions given in equation (8.4.10)shows that, to leading order,

fmm,(k,:a) =

(kxa) 2...

m even, m even

4 - (kxa)2 ...

m odd, m odd

2j(kxa)...

m odd, m' even

- 2 j (kxa).. •

m even, m' odd

(8.4.12)

238

ACTIVE CONTROLOF VIBRATION

Exactly analagous results are again produced in the low frequency limit (when kyb ,~. 1) for f,,,,.(kyb). This demonstrates that the only modal interactions of significance at low frequencies are due to the interference of the radiated fields of modes for which m and m' are odd and for which n and n' are odd. In other words it is only the mutual interference of monopole type modes that significantly modifies the acoustic power output of a simply supported plate at low frequencies. This conclusion was reached by Thomas (1992) in studying the interaction of plate modes in contributing to sound power radiation by a plate under active control. This interaction can be illustrated by plotting the self-radiation efficiencies of the lower order modes of a plate, which can be written as 0"mn,mn

--

2Mmn,mn ~

(8.4.13)

,

pocoab

together with the mutual radiation efficiencies between these modes, which are defined to be

Mmnm'n' 0"mn,m'n'

"--

'

(8.4.14)

.

pocoab

The variation of these radiation efficiencies with non-dimensional frequency is shown in Fig. 8.7 for a plate with b/a =0.57 (Elliott and Johnson, 1993). At low frequencies, ka ~ 1, the plate modes with a net monopole component have the highest radiation efficiencies, a~.~ and O'31,31, although the mutual radiation term 0"1~,3~is more important than the self-radiation term 0"31,31" At higher frequencies, ka,> 1, the effect of the mutual radiation term decreases since each of the modes radiates independently and with approximately equal efficiency. 101 100

°11,11

10-1 _ >:,

o

¢-.

1

0.2

-

°11,31

(D

i_~

10 -3 -

,-.

10-4 _

o

1 0-5 tr'

°31,31 °21,21

10_ 6 10-7 _ 10-8 10-1

. . . .

I

100

.

.

.

.

.

.

,,I

101

i

|

|

,

,

,

,,

103

Non-dimensional frequency, ka

Fig. 8.7 The radiation efficiencies of selected structural modes on a rectangular plate. The self-radiation efficiency of the (1, 1), (1,2) and (3, 1) modes are designated al~.~, crzl.z~, cr3~.31 and mutual radiation efficiency of the (1, 1) and the (3, 1) modes is designated 0~.3~.

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATESYSTEMS

239

It is possible to define a set of velocity distributions on the plate which do radiate sound independently at any one frequency. Consider the eigenvalue/eigenvector decomposition of the matrix M in equation (8.4.6): M = pT~p,

(8.4.15)

where, since M is symmetric and positive definite, P is a real unitary matrix of eigenvectors and f2 is a diagonal matrix of positive real eigenvalues. The power radiated by the plate, equation (8.4.6), can now be written as II = ";vHPTf2P-;v= bHf~b,

(8.4.16)

where b = Pw is the set of structural mode amplitudes transformed by the eigenvectors of the radiation resistance matrix. Since fl is diagonal, equation (8.4.16) can be written as N

I-[ = ~

(8.4.17)

f ~ . l b . I 2,

n=0

so that the velocity distributions corresponding to this transformed set of velocity distributions radiate sound independently. The existence of this set of velocity distributions has been described by Borgiotti (1990) and Cunefare and Koopman (1991), and is implicit in the work of Baumann et al. (1991). They have been termed radiation modes by Elliott and Johnson (1993), who have plotted out their radiation efficiencies, proportional to the f2,'s, for the plate described above, as shown in Fig. 8.8. At low frequencies, ka ~ 1, the first-order (1) radiation mode is much more efficient than those labelled (2) and (3), and generally the variation of their radiation efficiency with frequency is simpler than that for the radiation efficiency of the structural modes (Fig. 8.7). The velocity distributions corresponding to these theoretical radiation modes are weak functions of the excitation frequencies. At low frequencies, ka ~ 1, however, the velocity distributions are almost independent of excitation frequency, and for the plate considered here are plotted in Fig. 8.9 (Elliott and Johnson, 1993). 101 10 0 I~

10-1

i

I

i

I

i

i

i

i

i

i

i

I

i

i

i

i

i i

i

Radiatiom n o d e ~

10"2

~

0.3

.E_O

10 .4

.(1:1 "0

10-5

rr

10..6

(3) (4)

10 .7 10 .8 10-1

i

10 °

I

I

I

I

I

I II

I

101

i

i J J i ii102

Non-dimensional frequency, ka

Fig. 8.8 The radiation efficiencies of the first six radiation modes of the rectangular plate.

240

ACTIVE CONTROL OF VIBRATION

velocity, ~,

(1)

(2)

(3)

(4)

(5)

(6) Fig. 8.9 The velocity distributions corresponding to the first six radiation modes of the rectangular plate at an excitation frequency corresponding to k a = O. 1. The most efficiently radiating velocity distribution (1) clearly corresponds to the net volume displacement of the plate, as expected. The advantage of the radiation mode approach is that, by using Fig. 8.8, it allows one to quantify the extent to which other velocity distributions are also significantly radiating at any one excitation frequency. It also suggests an efficient method of sensing the velocity distributions of the plate which are most important in radiating sound, as will be discussed in Section 8.8.

8.5

General analysis of Active Structural Acoustic Control (ASAC) for plate systems

In this section we outline the basic analysis of ASAC applied to plate systems. The system studied consists of a simply supported rectangular plate positioned in an infinite baffle as shown in Fig. 8.10. Two different disturbances to the system are considered as

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS

Incident pressure,Pi

241

/,,

Iii

// ~ t e d ~

pressure Pr

Y

,J v!

b2f - I - Uniforml_~ pressure

Point liD'-force

I I I I

b

y

Piezoelectricpatch

I I I I I

-

--~Ye

i

a2

x2

(c)

Fig. 8.10 Coordinate system and arrangement of a baffled rectangular plate with inputs (a) on the incident wave side, (b) on the radiated wave side and (c) configurations of the input forces. examples. Firstly, an acoustic plane wave incident on the plate at an oblique angle is taken as the noise input shown in Fig. 8.10(a). Secondly, a forcing function over a small localised area on the plate is introduced that is considered to be representative of a structural disturbance (in the limit this forcing function will be a point force) shown in Fig. 8.10(c). Both of these disturbances will excite the plate into motion resulting in radiation on the transmitted half plane of the radiated field shown in Fig. 8.10(b). As discussed in Chapter 1 and Section 8.2 the form of the disturbance has an important influence on the resultant plate motion and thus the sound field to which it is coupled. When the disturbance is a low frequency plane wave, the input phase distribution is relatively constant over the plate surface and the result is that the plate response is dominated by lower order modes that are often excited well above their resonance

242

ACTIVE CONTROLOF VIBRATION

frequencies. This behaviour leads to an overall vibration pattern with fewer nodal lines and a high radiation efficiency. For a localised structural input, the plate responds with a much richer modal distribution and higher order modes tend to dominate near their resonance frequencies. In this case at higher frequencies (however, still below the plate critical frequency) the overall radiation efficiency is lower. Thus, as demonstrated by McGary (1988), airborne disturbance inputs to structural systems will tend to radiate/ transmit higher levels of sound than structural inputs for the same frequency and total magnitude of load. In the following analysis a general procedure is outlined. The inputs are assumed to be at single frequencies and all systems (structural, acoustic and electrical) are assumed linear so that superposition of response holds. The type, number and location of the control transducers is assumed known. In this section the fluid loading is assumed small (e.g. as in air) hence the plate response can be determined using the inv a c u o equations described in Chapter 2. The following steps are taken in the general ASAC analysis: (1) An expression is derived for the response of the plate to the input or primary disturbance(s). (2) An expression is derived for the coupled radiation to the far-field from the plate due to the disturbance. (3) An expression is derived for the response of the plate to the multiple structural control or secondary inputs. (4) An expression is derived for the far-field radiation from the plate due to the structural control inputs. (5) The total far-field pressure response is found by superimposing the disturbance and control fields. (6) A quadratic cost function is formed that is based on the required observed radiated pressure field variables.t (7) Quadratic optimisation theory is used to find the optimal control inputs that minimise the cost function as outlined in Section 4.6. (The input disturbance(s) is assumed constant and known.) (8) The optimal control inputs are substituted into the relations for the total field response (i.e. far-field pressure, plate out-of-plane displacement etc.) in order to evaluate the control performance. (Note that the reduction in the cost function also provides a measure of the overall control performance in terms of the observed error variables.) Before we carry out the above steps, we first derive the basic system response equations. Note that as outlined by Pan et al. (1992b) the method can also be formulated by using transfer matrices written in terms of input and radiation impedances. In addition, as described in Section 8.8, design techniques for optimally shaping and locating the control actuators and sensors are available. Figure 8.10 shows the arrangement and coordinate system of the baffled, simply supported, rectangular thin plate excited by a harmonic disturbance pressure acting over an area of the plate. To calculate the radiated sound field, a description is required of

t One could also minimise a pressure related variable such as supersonic wavenumbercomponents.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

243

the plate complex vibration distribution. For the simply supported thin plate, the displacement distribution is given by equation (2.11.2) in modal form as oo

oo

W(X,y, t) "- Z Z wmn sin kmx sin kny e i~t,

(8.5.1)

m=ln=l

where the eigenvalues are given by km

met =

~ ,

a

kn =

nJr b

.

(8.5.2a,b)

The plate modal amplitudes for various forms and distributions of input forcing functions have been calculated by Wang and Fuller (1991) and are given by Wren "-

Pro. ph(ooZm~-co 2)

,

(8.5.3)

where 09 are the natural frequencies, p the plate density and h is the plate thickness. The modal force, Pmn, due to the input disturbance depends upon an exact description of the external load. Gu and Fuller (1993) have studied active control of sound radiation from a plate in the presence of a heavy fluid. However, for the present discussion we limit the analysis to a light fluid loading such as air and thus radiation loading effects are ignored. Modal forces for various input disturbances acting upon an in-vacuo simply supported plate have been derived by Wang and Fuller (1991) as follows. (1) Uniformly distributed pressure. For a uniformly distributed pressure with amplitude, Q, located between coordinates a~, a2 and b~, b2, as shown in Fig. 8.10(c), the modal force is given by mn

Pmd =

4Q

m nx~ 2

(cos kma 1 -

cos

kma2)(cos knbl -

cos

knb2),

(8.5.4)

where the superscript d will hereafter denote the input disturbance. (2) Obliquely incident plane wave. An obliquely incident plane wave as shown in Fig. 8.10(a) can be described by pi(x, y, t) = Pie j°~t-jksin °i

cos q,,-jksin 0, sin ~i.

(8.5.5)

Again assuming that the fluid loading is light we can calculate the total input pressure at the plate surface from the plate blocked pressure. That is, we assume total reflection of incident waves and thus the total input pressure at the plate surface is twice the incident pressure. Using such an approach the modal force for an oblique incident plane wave has been calculated first by Roussos (1985) and later by Wang and Fuller (1991) and is given by Pare.= 8Pilmln where the coupling constants are given by

-j [m = 2

sgn(sin Oi cos q~i)

if (met)2 = [sin 0icos ~ b i ( o ) a / c ) ] 2,

(8.5.6)

244

ACTIVE CONTROL OF VIBRATION

or

mn{ 1 - ( - 1 ) m exp[-j sin Oi cos ~i(toa/c)} I n --

(mn) 2 - [sin Oi cos ~i(wa/c)] 2 if (mn) 2~ [sin Oi cos q~i(toa/c)] 2

(8.5.7a,b)

and -J sgn(sin Oi sin (])i)

rn- 2

if (n~:)2 = [sin Oi sin q~i(~ob/c)] 2, or

L-

nn{ 1 - (-1)n exp[-j sin 0 i sin qbi(wb/c)] } (net) 2- [sin Oi sin ~ i ( t o b / c ) ]

2

if (nn)2~ [sin Oi sin ~i(tob/c)] 2.

(8.5.Sa,b)

(3) Piezoelectric actuator. As discussed in Chapter 5 a rectangular piezoelectric actuator configured for pure bending is considered to consist of two wafer elements located symmetrically on each side of the plate and driven 180 ° out of phase in the d31 mode. The modal force for such a piezoelectric actuator as shown in Fig. 8.10(c) is given by Wang and Fuller (1991) as P C m = 4C°l~Pe mn~2

(kZm+

k2n)(cos k m x 1 - c o s k m x z ) ( C O S

knYl -

cos knY2)

(8.5.9)

where x~, x2, and Yl, Y2 are the coordinates of the actuator edges. The parameter Co is a constant that is function of the piezoelectric actuator/plate properties and geometries specified by C o - E b I K I, where K I is given by equation (5.6.8). The unconstrained strain, epe, of the piezoelectric element is defined by d31V l?,pe -- ~ ,

(8.5.10)

ha where d31 is the piezoelectric transverse strain constant and h a is the piezoelectric element thickness. Most importantly Epe and thus Pm~ is seen to be linearly related to V, the input complex voltage to the actuator. (4) Point force. The modal force associated with excitation by a harmonic point force of amplitude F located at (Xr, Yr) as shown in Fig. 8.10(c) is pc _

4F

ab

sin k mXf sin k,y I,

(8.5.11 )

where the superscript c denotes control force. The sound radiation caused by the vibration due to the above inputs is related to the plate velocity distribution. As outlined in Sections 8.2 and 8.3 the radiated pressure can

ACTIVI~ STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTtIM$

245

be evaluated by using the Rayleigh integral or spatially Fourier transformed variables in conjunction with the method of stationary phase. As demonstrated by Junger and Feit (1986), both mathods yield identical solutions. Roussos (1985) has derived e×pressions for the radiation from a rectangular plate using the Rayleigh integral. The radiated field shown in Fig. 8.10 (b) is given by eo

e@

p(R, 0, q~)- K Z Z

(8.5.12)

Wm, lml, e m,

m-'-I n=-I

where the radiation constant K is defined as

1[

K - =topoab exp ]to t . . . .

2r~R

c

]}

(a cos ~ + b sin ~) . 2c

(8.5.13)

The coupling constants I,, and I, for the radiated field ere identical to equations (8.5.7a,b) and (8.5.8a,b) except that the coordinate angles (0i, 'hi) am replaced by (0, ,)) which define the coordinate of the observation point in the radiated fieldand the sign in the argument of the exponential function is changed to minus. We now have all the necessary components in hand to derive the response of the total radiated field,i.e.the disturbance field (also called primary) plus the control field (also called secondary). For a feedforward control arrangement the disturbance source and the control actuators act simultaneously and am assumed Imrfectly coherent. By superposition the totalcomplex pressure can be written as p, = pp + p,,

(8.5.14)

where the subscripts t, p and s refer to total, disturbance and control pressure respectively. Using the previous relations the total pressure in the far field can be written for the following configurations. Case I. Plane wave disturbance = point control forces u~ p, = P~B + ~ F;Cj,

(8.5.15)

/-=1

where N~ is the total number of control forces, while P~ and Fj are the complex amplitudes of the disturbance wave and control forces respectively. Case I1. Localised structural disturbance = point control forces #,

p, = O.B + Z r;cj,

(s.5.16)

/=-1

where Q is the amplitude of the disturbance pressure acting over the small area specified previously. Case III. Plane wave disturbance -- piezoelectric actuators N, P,

= PIB + ~ VjCj, /=-1

where Vj is the complex control voltage applied to the jth actuator.

(8.5.17)

246

ACTIVE CONTROLOF VIBRATION

Case IV. Localised structural disturbance - piezoelectric actuators

s pt= QB + Z V~Cj, j=l

(8.5.18)

where Q is the amplitude of the disturbance pressure. In the above equations the transfer functions are defined as: plane wave disturbance, B - K

-~

ImI~,

(8.5.19a)

Imln,

(8.5.19b)

localised structural disturbance, B =K

= = y

where the appropriate form of WPmnis calculated using equation (8.5.3) with either equation (8.5.6) or equation (8.5.4) respectively; point control force, =

= -~j

Iml.,

(8.5.20a)

=

__ - ~ j

ImIn'

(8.5.20b)

piezoelectric control actuator, Cj=K

where the appropriate form of Wm~njis calculated using equation (8.5.3) with equation (8.5.11) and equation (8.5.9) respectively. In the above transfer functions, B and C are normalised with respect to their appropriate forcing function amplitudes in order to put the relationships for total response in a form which can more readily be manipulated in terms of control amplitudes. Thus in effect, B, for example, will represent a complex transfer function between the specified input pressure of an obliquely incident wave and its corresponding radiation pressure at some observation angle in the far field. Control of the disturbance field can be achieved by appropriately choosing the vector of point force amplitudes f~= [F~ F~ F~ ...]T or piezoelectric voltages v~= [V~ V~ V~... ]T in order to minimise a chosen cost function. Choice of the cost function is defined by the form of control performance required. The ideal cost function for globalt control of sound radiation is the total acoustic power radiated by the plate since this is the variable that we seek to reduce. In this case the cost function is defined as the integral of intensity flowing through a hemisphere surrounding the plate which is given by J =

1 f

Ip, 12dS

2p0c0 ' s t Here global means through an extended area or volume.

(8.5.21)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

247

and which can be described in polar coordinates as J =

2p0c0 ' 0 J0

IP,

sin 0 dO dq~.

(8.5.22)

If directional control is required then equation (8.5.22) can be modified to only attenuate power over a required directional sector and is thus written J =

1

fo2f¢2 ]p 12R2 sin 0 dO d~b, 2 p o c o ' o, ~,, '

(8.5.23)

where 01, 02, q~l, q~2 define the limits of the sector. Usually it is not possible to design a sensor such that a cost function of the form of equation (8.5.21) can be measured, and in practice microphones are often used as point error sensors. In this case the cost function is written in discrete form as Ne

1

J=

~ [pI(R~, 0 i, ~)i)12,

(8.5.24)

2poco i= 1 where N e is the number of error sensors and (Ri, Oi, ~)i) is the location of the ith error sensor microphone. Equation (8.5.24) can be written in a more accurate discrete form as 1

J =

N~

~ Ipll = A S i,

(8.5.25)

2p0c0 i= 1 where A S i is the projected area associated with the jth sensor on the hemispherical surface. In practice, however equation (8.5.24) provides a cost function which is a reasonable estimate proportional to the total radiated power as long as a suitable geometry of error sensors are chosen. Choice of number and location of error sensors is an important topic of research and will be briefly discussed later. In general, for global control, the number of point error sensors required is equal to the number of modes contributing to the overall sound power radiation. The error sensors, by analogy with time sampling theory (see, for example, Nelson and Elliott, 1992, Ch. 2), should be spaced in sectors of the radiated field defined by regions of 180 ° phase change through nodal lines in the radiation field associated with that mode. For example, Fig. 8.11 shows a hypothetical low frequency, rectangular plate, radiation pattern which is a combination of radiation from the (1, 1) and (2, 1) modes. Positioning a single error sensor at location (a) will only lead to control of the (1, 1) mode since this angle corresponds to a node in the (2, 1) radiation pattern. Positioning two sensors at locations (b) will not lead to global control since the (1, 1) and (2, 1) modes will cancel at the error sensors and reinforce in the other sector. Locations (c) shows a situation where two sensors are positioned in two sectors which are 180 ° out of phase for the (2, 1) mode. When used with two actuators, the only way the controller can attenuate the error signals is by completely attenuating each mode, thus leading to global control. Another important observation regarding the choice of point sensors is that generally the number of sensors should be equal to or greater than the number of control inputs used. If the number of error sensors equals the number of actuators then total

248

ACTIVE CONTROL OF VIBRATION

~mode

1

mode 2 + (a)

0 ensor

(b)

e error sensor

+ (c)

Fig. 8.11 Influence of error sensor position on modal radiation control. attenuation is theoretically achieved at each error sensor since the system is perfectly determined. This situation generally leads to poor attenuations at other positions especially if the number of degrees of freedom in the system exceeds the number of error sensors. If the number of control actuators used is greater than the number of error sensors then the system is underdetermined. Note, however, that Elliott and Rex (1992) have presented a methodology to ensure well-conditioned relations for underdetermined systems by introducing a control effort term into the cost function. It should again be stressed here that in ASAC, although the control action is applied directly to the structure, the cost function is derived from the far-field radiated pressure (or far-field radiated pressure-related variables). Thus, inherent in the definition of the cost function, is the natural structural acoustic coupling that relates the plate vibration to the radiated sound. This arrangement should be contrasted to the more obvious

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

249

approach in which plate vibration is directly observed and minimised. Naturally, completely reducing overall plate vibration will lead to a reduction in sound radiation. However, as will be demonstrated, this latter approach generally requires many more channels of control (i.e. number of sensors and actuators) and a much more subtle and efficient control paradigm can be implemented when far-field pressure is used as an error variable. One is thus directly observing the field variable to be controlled, which in this case is radiated sound. The following analysis for derivation of the vector of optimal control inputs is for the ideal case of minimising total radiated power and thus equation (8.5.21) is used to define the cost function. Although the following derivation is formulated for the above Case IV, it can be written in exactly the same form for the other three cases with appropriate substitution of variables. When the expression for total pressure Pt from equation (8.5.18) is substituted into the cost function definition, equation (8.5.21), it can be demonstrated that the cost function is a scalar which is quadratic in the vector of complex control voltages v, or control force amplitudes, f,. The reader is referred to the Appendix of Nelson and Elliott (1992), as well as Chapter 4 of this text, for a proof and discussion on the nature and minimum value of the quadratic cost function. It can be shown that this cost function will possess a unique minimum which will define the optimal control voltages. The minimisation procedure is based upon the setting the gradient of the cost function J with respect to the control vector v, to zero in order to find the stationary point of the quadratic form. The total complex pressure can be expressed in vector form as p, = hTq + cTv ~,

(8.5.26)

where q=[Q~Q2Q3 ...]v is the vector of complex input disturbances and h = [H~ H2 H3 ...IV is the vector of aforementioned transfer functions associated with those disturbances. For cases in which there is a single input disturbance these vectors reduce to scalars such that q=Q

(8.5.27)

h = H~.

(8.5.28)

and

In equation (8.5.26) the control transfer function vector is defined by

c = [Ci C2... Cu,]"r,

(8.5.29)

while the input control voltages or control forces can also be written in vector form as

v,=

T.

(8.5.30)

Note that a similar vector expression as equations (8.5.26) and (8.5.30) could be used for f,, the vector of complex control force amplitudes. As outlined by Nelson and Elliott (1992) it is convenient to write the cost function using matrices in a Hermitian quadratic form. For a single input disturbance the squared modulus of the total complex pressure can be written IP, [2= p,p,= v~Cv, + v~x + XHV,+

QH,H*~Q*.

(8.5.31)

250

ACTIVE

CONTROL

OF VIBRATION

In equation (8.5.31) the Hermitian matrix C is defined by C = c*c T

(8.5.32)

x = QHlC.

(8.5.33)

and the vector x is given by

The cost function as defined by equation (8.5.22) which is written in terms of total radiated acoustic power can now be specified as J = v HAv, + Vnb + bHv s +

(8.5.34)

C,

where the individual terms are now given by AN, ×U,

=

1 [2~ [,q2 [c.cT]R 2 sin 0 dO dq~ 2poco ' o Jo

1 I2, [,#2 bN'×l= 2p0c0,0 J0

c=

1012

r/o rjo,

(8.5.35)

QHlcR2 sin 0 dO dq~

(8.5.36)

sin 0 d0 d~.

(8.5.37)

2poco '

As discussed in Nelson and Elliott (1992), all the combined terms of equation (8.5.34) are scalars and as long as A is positive definite then there is a unique minimum value of J. A typical element of matrix A is defined by A~=

~. kli~ mnjI kliI mnj1" sin 0 d 0 d q~ RoCo "

k = l I=1 m--1 n--1

i = 1 , N , ; j = 1,N,.

(8.5.38)

A typical element of vector b is oo

oo

oo

1 [2=Ij/2 K1K* Z ~ Z Z QP~tlQP,~,'PktalS£nsR2sin OdO dq), s= 1,Ns.

pOCo '

k = l !=1 m = l n = l

(8.5.39) Finally the single entry of h for a single input disturbance is defined as n 1-

1 f 2=[zr/2K1K~ Z 0 dO poCo ' k=l

Z

Z

P* IP lP* D 2 sin0d0dq~ QpkllQmnl*kll*mnl*~

(8.5.40)

1=1 m = l n = l

where

w;%

QS,,j = - - , V~

wL1

Q~l = ~ . Q

(8.5.41a,b)

Note in the above equations that the quadruple infinite sums in (k, l) and (m, n) result from the multiplication of two double modal series. As discussed in Section 8.4 the cross terms are important and the additional modal indices (k, l) are employed. The

ACTIVE STRUCTURALACOUSTICCONTROL. I PLATE SYSTEMS

251

optimal solution of control strengths to minimise the cost function of equation (8.5.34) can be found using the result derived in the Appendix of Nelson and Elliott (1992) and Chapter 4 which shows that V~o= - A - l b -

(8.5.42)

Equation (8.5.42) thus defines the vector of optimal control voltages V,o for the case of piezoelectric actuators. On obtaining V,o, the minimised far-field radiated pressure can then be calculated using equation (8.5.26). The minimum of the cost function can also be calculated from Jmin = c - bHA-lb.

(8.5.43)

Equation (8.5.43) can be used to calculate the attenuation in total radiated power obtained when the control is invoked as long as enough error sensors are used to provide a reasonable estimate. Note that if there are more control actuators than error sensors then matrix A will be singular and the procedure developed by Elliott and Rex (1992) for underdetermined systems should be used. The reader is also referred to the Appendix of Nelson and Elliott (1992) for discussions on this and other aspects of finding the minimum of quadratic functions. The previous analysis enabling the derivation of the optimal control voltages can readily be applied to the different cases I, II and III outlined above with use of the appropriate variables and using similar methodology. In the next two sections, results from example applications using this analysis will be discussed.

8.6

Active control of sound transmission through a rectangular plate using point force actuators

In this section we study the active control of sound transmission through a rectangular, baffled, simply supported plate using point force actuators. Systems similar to these have been previously investigated by Fuller (1990) and Wang and Fuller (1991). All disturbance frequencies are well below the coincidence frequency of the plate which is fc_~6300 Hz. Table 8.1 presents the specifications of the steel plate, while the corresponding natural frequencies for the simply supported boundary conditions, computed using equation (2.10.3), are given in Table 8.2. The acoustic medium is assumed to be air with P0 = 1.21 kgm -3 and Co = 343 ms -~. To calculate the plate response and the corresponding radiated field it is necessary to truncate the infinite summations. In the following examples, truncating the indices k, l, m and n at a value of five (i.e. 25 modes are considered in the doubly infinite sums) was found to provide close to 0.01% error in the radiated pressure amplitude at the highest frequency considered. This choice of truncation can be seen from Table 8.2 to effectively limit the input disturbance frequency such that f < 1750 Hz. In addition, it Table 8.1

E = 207 x 1 0 9 N m pp = 7870 kg m - 3

Plate specifications. v = 0.292 h = 2 mm

a = 0.38 m b = 0.30 m

252

ACTIVI~CONTROLOF VIBRATION Table 8.2 Naturalfrequencies of the plate (Hz).

m

1

2

3

4

5

1 2 3 4 5

87.71 188.74 357.13 592.88 895.98

249.81 350.85 519.23 754.98 1058.08

519.98 621.02 789.40 1025.15 1328.25

898.22 999.25 1167.64 1403.39 1706.48

1384.53 1485.56 1653.95 1889.69 2192.79

was necessary to calculate the integrals of equations (8.5.35) to (S.5.37) and this was carried out numerically using Simpson's rule. In order to use the above equations for this case, the input control terms are replaced with those for point control forces. The plate response results presented in Sections 8.6 and 8.7 consist of the distribution of plate vibrational amplitude plotted along the ) , - b / 2 horizontal plate mid plane (see Fig. 8.10). The results, presented in decibels (dB), were normalised to the largest amplitude obtained in each figure. Radiation directivity patterns are also presented along the y-b/2 axis at a distance of R - 2 m. Although this observation point is relatively close to the plate, far-field radiation equations were used for simplicity, and thus the results also reflect the behaviour at large distances from the plate. For convenience, angular positions with a negative sign of 0 in the figures correspond to the coordinate ~ - ~ positions. For the following example the input disturbance was assumed to be a plane wave with amplitude P~-1 N m =2 incident at angles of 0~ - 45 ° , ~ - 0 e. Figure 8.12 presents the radiation directivity patterns with and without control when the excitation frequency was set to 186 Hz. Note that negative values of sound pressure level correspond to pressure magnitudes less than the reference pressure of 2 x 10 =s N m =2. From Table 8.2 it is apparent that this frequency is near the resonance frequency of the (2, 1) mode, and correspondingly, the uncontrolled radiation field appears to have the distorted version of the radiation pattern associated with a (2, 1) mode. An examination of the modal contributions confirms that the radiation field is mostly due to the (1, 1), (2, 1) and (3, 1) modes. The slight offset of the node in the radiation field is due to the presence of a monopole type radiator such as the (1, 1) mode. Applying one control force, as shown in the schematic diagram at the top of Fig. 8.12 as a black dot, leads to little reduction in radiated power. Positions of the control forces are shown to scale in the schematic figures. However, it appears that the (1, 1) radiation contribution has been controlled as the residual radiation pattern has now the characteristic dipole shape associated with the (2, 1) mode and this is confirmed by examining the modal contributions. The single, centrally located control force is unable to couple into the (2, 1) mode of the plate as shown in equation (2.11.4). Using two forces leads to a large reduction in radiated power as now the (2, 1) mode is controlled and residual field is now largely due to the (3, 1) mode contribution. On using three point force actuators positioned as shown in the schematics of Fig. 8.12, control is achievable over the (1, 1), (2, 1) and (3, 1) modes, and power reductions of the order of 67 dB are predicted.

ACTIVI~STRUCTURALACOUSTICCONTROL,I PLATI~$YSTISMS Unoontrollti¢l

I Foroti

a

llllllllll=

i1= ,

Forolill

253

3 Forollit

=III=I=I=IIIIUlIIII=I

I

48,7 (dB)

80,8 (¢IB)

I

I

i

01=_48" 0l =- O*

Power rttduotlon

o,e (aa)

5O ....... -46°

0--'0°

""':::,

I

/

/ 0

F

=50

7/

=100

.............

....... ":1:::~ ~

............ I ............. 48'1 .:=;:'"

,.................. ~

(

Illl $

/

",...,' . . . .

~

I

.::;:"......

........................ i)N /

F........"

",~:~2 ... \

[ %1

L( ( ;-; !....]i......... li '2........... (

-50

=50 =I00 -80 Sound prnmum Iovel (d8)

0

0

60

Fig, 8,12 Radiationdirectivity for different numbers of point for, o actuators, f=- 186 Hz,

Curves of the power transmission loss versus frequency for the same situations as above are plotted in Fi$, 8,13, Her~ power transmission loss is defined as (Wang, 1991) Transmission loss-~ 10 log(lli/l=I~),

(8,6,1)

where the incident power is given for an obliquely incident plane wave by the relation

hi--- Ipil~ ab cos 0~

(8.6.2)

2pete

and the radiated power is given by Jo

~,, Ip(r, 0, O)[~ r ~ sin 0 dO dO, o 2pete

(8.6.3)

whore p ( r , 0, 0) is evaluated in the far field usin8 equation (8,$,26) for the cases with and without control. Note that negative values of tr~amission lo~a are a numodoal artifact of the computer calculations; in practice transmission loss cannot be loss than zero, For the primary or uncontrolled case, the transmissiort loss cu~e ha~ a number of dips a~sociatod with the plate modal resontm,os, When one ,ontrol fo~o is u~od, it ,an be soon from Fig, 8,13 that the fall in transmission loss at the (1,1) re~onanco frequency has been eliminated as discussed above, Similarly, use of two control forces eliminates the dips at the (1, 1) and (2, 1) resonance frequencies, The figure shows that attenuation greater than 50 dB can M theoretically obtained over a frequency range of

254

ACTIVE

Uncontrolled

CONTROL

OF VIBRATION

1 Force

2 Forces

3 Forces

.................................

0i==45°~i 0° i 200

I

~'~"1

|

150

~,,,',,,

A

rn "o

o (0

100

m

tO .B

50

.B

.'~..

E

,,.

...........~~:...~ ~

\ :~"'.....

,.,,..~..-..-..-:....-.,

e--

-50

0

I

I

I

I

I

I

I

1O0

200

300

400

500

600

700

800

Frequency (Hz)

Fig. 8.13 Plate transmission loss for different numbers of point force actuators. 0 to 450 Hz with three control forces. Whether this is achieved in practice for broadband, random disturbances depends upon a number of issues such as causality, filter size, etc. as described in Chapter 4. However, these frequency domain results do define the ultimately achievable performance with the limited number of actuators used in the locations specified.

8.7

Active control of structurally radiated sound using multiple piezoelectric actuators; interpretation of behaviour in terms of the spatial wavenumber spectrum

The previous section briefly discussed results of using point force actuators as control inputs. However, there are disadvantages to using point force actuators such as their size and the need for a back reaction support. In this section we discuss a few representative examples of control of structurally radiated noise (i.e. a structural rather than an airborne input) with arrays of independently controlled piezoelectric actuators. The piezoelectric actuators were assumed to be manufactured from ceramic material with typical properties given in Table 5.1 (corresponding to G1195 material). As discussed in Chapter 5 the actuators were configured to produce pure bending in the plate. The system used for the analysis is exactly the same as that presented in the previous section except that in this case the input disturbance is assumed to act over a very small area of 40 x 40 mm approximating a localised structural input. For this case the input

255

ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS

disturbance amplitude was set to Q = 7.9 x 103 N m -~ giving an input force of 12.65 N. In the following figures the prescribed disturbance source and actuator locations and size are shown in schematics at the top of each figure, drawn to scale looking into the plate surface from the radiated field. The black rectangle represents the size and position of the disturbance source while the clear rectangles represent the size and position of the two-dimensional piezoelectric actuators. At this stage no attempt is made to optimally configure the control actuators; their selection is made on an ad hoc basis, linked with a knowledge of the acoustically significant plate modes and their response shapes. For the first case the disturbance frequency is set to 85 Hz. An examination of Table 8.2 reveals that this frequency is close to the resonance frequency of the (1, 1) mode.t Hence this case corresponds to an 'on-resonance' excitation. Figure 8.14 presents the normalised vibration amplitude distribution with and without control for four different configurations of piezoelectric actuators. In Fig. 8.14 the solid line depicts the displacement distribution of the plate only under the influence of the disturbance, and as expected is close to a (1, 1) mode shape. When the various configurations of control (1)

(2)

(3)

(4)

I

I

I

I

rn I1) "O

,m=

-50

..../.--:':~'"~.L.. -'~.-...

E

.:----, :,,

tl:l ¢...

•~

-~00 -

,.,.~,-~... -~

.dI3 >

/

t;:',, '(" #

"

.,

:

',.k/. "-

" ~1

"-,,

&, "':~, . . . . i':_..-:...... - .

"13

I

-150

i"

I,

\

|".f

',

E 0

z

-200

I

I

I

I

I

I

0

0.2

0.4

0.6

0.8

1

x/a

.............

Uncontrolled 1 Piezo

.......................... 2 P i e z o s 3 Piezos .......

4 Piezos

Fig. 8.14 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 85 Hz. 1 Since no damping is included in the model, the disturbance frequency is not set exactly to the resonance frequency.

256

ACTIVE CONTROLOF VIBRATION

actuators are applied, the vibration amplitudes are significantly reduced and the (1, 1) mode is well controlled. However, increasing the number of actuators does not lead to a significant reduction in vibration. (Note the we are using an acoustic cost function in this example, in contrast to the configurations in Chapter 6.) This effect will be discussed below. Figure 8.15 presents the radiation directivities corresponding to the cases shown in Fig. 8.14. Also presented is the total reduction in radiated acoustic power calculated from equation (8.2.7) for each case. As expected the uncontrolled field is uniform and corresponds to the monopole like radiation of the (1, 1) mode. When one control actuator is applied, reductions in radiated power of the order of 60 dB are obtained. Use of two control actuators brings a further 10 dB reduction. Further increasing the number of actuators has little effect on the radiated power until for case (4) when actuators are located off the y = b/2 line (see Fig. 8.10) and a further 30 dB of power reduction is obtained. From these cases, it is apparent that good sound control is achievable with a single actuator and increasing the number of control channels has no significant practical advantage. In general this observation is true for systems on or near resonance of an efficiently radiating structural mode such as the (1, 1) mode considered here. Figure 8.16 presents the wavenumber transform of the plate vibration calculated (1)

(2)

(3)

~5°1 ,,

(4)

I po.0°

i,,.,~° ....

................... <' -50

~ "1

,0o ............

i

0

50

!

1O0

/

/

150

Sound pressure level (dB) Power reduction (riB) Uncontrolled ............. 1 Piezo .......................... 2 Piezos ..... 3 Piezos ....... 4 Piezos

0.0

60.6 70.0 70.0

93.4

Fig. 8.1$ Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 85 Hz.

ACTIVE S T R U C T U R A L ACOUSTIC CONTROL. I PLATE SYSTEMS

(1)

(4)

(3)

(2)

257

100 A

e~

80

"7

6O

o

~

40

o=1==,====== ` ='1'= 1' I'

==================================

(N

~

20

-20 ~ / 0 k

i

I

I

10

2O kx(m'l )

30

.............

40

Uncontrolled

1 .......................... 2 . . . . . 3 ....... 4

Piezo Piezos Piezos Piezos

Fig. 8.16 Wavenumbcr spectrum of plate response for different numbers of piezoelectric ceramic actuators, f = 85 Hz or k = 1.556 m- ~. Note values are only shown for positive kx.

using equation (8.3.1) applied to the plate out-of-plane vibration distribution (see also Wang, 1991). The free wavenumber k of the radiation field is also shown. Although the wavenumber distribution exists in two dimensions for k,. and ky, values are only presented here for the positive k~. components. As discussed in Section 8.3 only supersonic wavenumber components, where (k~ + k~)~/2~ k will radiate to the far field. When control is applied, the wavenumber spectrum of Fig. 8.16 is attenuated across the complete wavenumber range including the supersonic components. A modal decomposition of the response also shows that all of the amplitudes of modes which are well coupled to the sound radiation are reduced. Thus radiation control has been achieved by suppressing the supersonic wavenumber components, and vibration of the mode(s) which are dominantly coupled to the radiated far field by virtue of their high radiation efficiency. We term this mechanism of control modal suppression (Fuller et al. , 1991). The disturbance frequency is now increased to 128 Hz which is an off-resonance frequency located between the (1, 1) and (3, 1) modes. Figures 8.17 and 8.18 present the plate displacement distributions and corresponding radiation directivity patterns. In this case it is apparent that increasing the number of control actuators leads to a realistic increase in control performance. At this off-resonance frequency the modal density is

258

ACTIVE CONTROLOF VIBRATION

higher; therefore more modes are now contributing significantly to the cost function, thus requiting multiple actuators for high reduction in sound power. The displacement distributions of Fig. 8.17 are also interesting. Using one control actuator leads to very little overall attenuation of the plate vibration; in fact, it is increased in some locations. As the number of actuators is increased, there is also seen to be a lack of significant change in the overall amplitude of plate motion. However, examining Fig. 8.17 it can be seen that the vibration distribution under multiple control action is quite complex with many nodal lines across the plate surface. A calculation of the overall plate radiation efficiency demonstrates that it is considerably lower when control is applied (Wang et al., 1991). Thus in this case the sound attenuation shown in Fig. 8.18 is not achieved by reducing the overall plate vibration, rather the controlled response of the plate exhibits a lower overall radiation efficiency. Figure 8.19 gives the corresponding positive wavenumber distributions computed along the x axis. In this case it can be seen that when control is applied, the supersonic wavenumber components (where kx <~k) are attenuated while the subsonic, non-radiating components are increased, except for case (3). In this off-resonance case, a multiplicity of modes are contributing significantly to the radiation field. The controller modifies the dynamics of the system so that the uncontrolled modes of significance (in terms of radiation) now have differing phases

(1)

(2)

I

I

0

.."

"0

/

.X

: ..~.-.~-"~..

..?,

-40

(4)

I

./"...-S. ........ -",:,.- .......

.. . . . . . . . . . .

133

(3)

"..

,...-

-, :./......

.

E t.--

.£ ~

-80

;

"0

•

..

._-,

I

~N

\1

\

'

li!I

!i~

~ -~2o

o Z

I

I

I

I

I

0

0.2

0.4

0.6

0.8

.............

1

1

x/a Uncontrolled ............. 1 Piezo ..........................2 Piezos ..... 3 Piezos ....... 4 Piezos

Fig. 8.17 Variation in normalised vibration amplitude along the y= b/2 axis for different numbers of piezoelectric ceramic actuators, f = 128 Hz.

ACTIVE STRUCTURALACOUSTIC CONTROL. I PLATE SYSTEMS (1)

(2)

,4-1 I

100

..........

(3)

259

(4)

I

I

-- 0 °

"',,.. ........_45 °

,,,,'""" 45 ° .......

rn

-~ >

50-

"'"x

, ,. ,,, , ,,.

o/ / ////"''"

i,... co

co

t,..

Q. "0

e-

O-

o

-90 -5010 0

90 °

50

0

-50

0

50

100

Sound pressure level (dB) Power reduction (dB) Uncontrolled ............. 1 Piezo ..........................2 Piezos 3 Piezos 4 Piezos

0.0 18.9 30.1 30.1 51.4

Fig. 8.18 Radiation directivity for different numbers of piezoelectric ceramic actuators, f = 128 Hz.

and amplitudes which interact to cancel the overall radiation in the far field. The closed loop structural response now has lower overall radiation efficiency even though its overall vibrational amplitude has not been attenuated (in some cases it may also increase!). A modal decomposition of the response shows that the modal amplitudes of the dominant sound radiators are not significantly changed. This second mechanism of control, where the overall plate vibration amplitude is not significantly attenuated or sometimes increased while supersonic wavenumber components, and the associated sound radiation are reduced, we term control by modal restructuring (Fuller et al., 1991). An alternative point of view of modal restructuring proposed and analysed by Burdisso and Fuller (1992a) is that the controlled structural response has new eigenvalues and eigenfunctions which have lower radiation efficiency. Burdisso and Fuller (1992a) also demonstrate that these controlled eigenfunctions are nonvolumetric. The above work has shown that global control of planar far-field sound radiation always corresponds to a fall in the supersonic components of the plate structural wavenumber spectrum. Furthermore, as discussed by Junger and Feit (1986), the radiation pressure from a plate at a particular angle can be shown to be directly proportional to a particular plate wavenumber component. This suggests that an

260

ACTIV]~ CONTROL OF

VIBRATION

(~)

(a)

(4)

+'4:=1:: ii :--1: llllllllllllll lllllllllllll

60

~i

~,

.lilt

40 ("

~,~,~.,,=,,=~,=~,=~...... Ill/

~=

"="= == =,.,,,,=

A

tiig~qa,

\

I

__O"i o

k

I

I

I

lo

2O

30

40

kxlm =1) Urtoontrollocl ............. 1 Pie=to ..........................a Plozoe . . . . . 3 Plozoo ....... 4 Piozoo

Fig. 8.19 Wavenumber spectrum of plato response for diffaront numbors of piezoelectric ceramic actuators, f-~ 1:28Hz or k -=2.345 m =t, Note values are only shown for positive k~:. alternative control formulation for minimising sound radiation would b~ to e×press the cost function in the wavenumber domain and suppress discrete (or bands of) wavonumbors, Fuller and Burdisso (1992) have analytically formulated a wavonumber control technique that suppresses radiation towards particular anglos. Clark and Fuller (1992a) have o×porimontally and theoretically demonstrated a wavenumber domain controller that is designed to minimiso the band of wavonumbors enclosed within the supersonic circle. This approach shows much potential. However, its practical implementation is dependent upon development of realistic, time domain, wavenumbor structural sensors that work over a broad frequency range. Work of this nature has been carried out by Maillard and Fuller (1994). Finally, Fig. 8.20 presents the radiated power with and without control over a range of frequencies from 0 < 800 Hz, Peaks in the uncontrolled power are associated with modal resonances. It is apparent that good control is achieved with a single actuator for very low frequencies, where f-~ 150 Hz. However, in order to provide good broadband power reduction it is necessm~ to use an array of appropriately positioned actuators. Oood attenuation is obtained over a frequency range of 0 ~f~650 with four actuators with the exception of the peak near 330 Hz, Table 8,2 indicates that this range encompasses the resonance frequencies of 13 structural modes and thus the result demonstrates the efficiency of ASAC.

261

ACTIVE STRUCTURAL ACOUSTIC CONTROL, IPLATE SYSTEMS

(~,)

(~)

(?)

(4)

180

~=

~=0

6O

o I I

=60

I

I

I

I

I

I

I

1O0

200

300

400

BOO

600

700

800

Fmqu#noy (Hz) Unoontroll~d ............. t Plezo ..........................2 Pl~zom

..... .......

Fig, 8,20

8.8

3 Pl~zo~ 4 Pl~zo~

Radiated ~ound power for different numbor~ of piezoelectric ceramic actuator~,

The use of plezoele~ric distributed structural error sensors in ASAC

In many applicationsof ASAC the use of microphone error sensors is impracticable.In this case it is desirable to configure error sensors which provide estimates of far-field pressure when located on, or very close to, the radiatingstructure.Various desisns hays been suggested in order to implemem this concepL Baumann ct el. (1991) have simulated the use of radiationfilterswith good success to modify the structuralstates of systems m radiationstatesin feedback implememations of ASAC. Clark and Fuller (1992b) have used a model reference implementation of the feedforward LMS all~orithmin which the vibrations of the structureare driven to a reference vector (i,e. not zero) corresponding to minimum acoustic power radiation. Fuller and Burdisso (1992) have also cortsidereda paradigm based upon a wavenumber domain approach in which Sul~rsonic structural wavenumber componems am comrolled, loading to minimisation of radiated pressure at selected anglos, In thi~ ~oction we di~cu~ the u~e of rectangular piezoelectric distributed ~tructural sensors as appropriate error sensors in ASAC. In particular we are interested in implementing structural error sensors for control of sound radiation from two-

262

ACTIVE CONTROL OF VIBRATION

dimensional plates. The use of distributed strain actuators and sensors integrated directly on or into the structure in conjunction with a 'learning' type adaptive controller implies that this configuration falls into a class of systems which are part of a rapidly expanding, related field known as adaptive, smart or intelligent structures. The reader is referred to the review paper by Fuller et al. (1992) for more information on the use of adaptive structures for controlling sound radiation. Recalling equation (5.7.4) which represents the charge output of a two-dimensional element of piezoelectric material bonded to the surface of the plate, we must simply define the area of application, F(x, y), to obtain the sensor electrical response as a function of the plate response. In the case of a rectangular sensor element this function is defined as F(x, y) = [ H ( x - Xel)- H ( x - x g ) ] [ H ( y - ye) _ H ( y - y~)],

(8.8.1)

where H(-) is the Heaviside unit step function of the spatial coordinates. Substituting this expression into equation (5.7.4) and integrating over the area of the sensor yields the charge output of the rectangular sensor given by

qe(t)

= (h + She) Mm~l = Nn~lWren[e31 ma nb + e32 × (COSkny e2 - c o s k , yl) e ~_jot,

(COS

kmx2- c o s kmx~) (8.8.2)

where (x~,y~) are the coordinates of the lower left comer of the rectangular piezoelectric element (viewed from the front of the plate), (x~, y~) are the coordinates of the upper fight comer of the sensor and h e is the sensor thickness. An examination of equation (8.8.2) reveals that the sensor charge output is composed of summation over the modal contribution. The degree to which a given mode contributes to the total sensor output is proportional to modal amplitude Wm, and is also related to modal order (m, n) as well as the sensor size and position. Thus different sensor shapes and positions can be used to selectively observe modes or combinations of modes. In order to modify the quadratic optimisation procedure, the output of the sensor is used as an error signal. Thus the control cost function is modified to be Ne g =

Z Iv; I

(8.8.3)

i=1

where it is assumed that voltage output of the sensor Ve is proportional to the charge generated by the s e n s o r qe and N e is the number of independent sensors used. As a preliminary example, we study the use of two distributed PVDF structural sensors shown approximately to scale in Fig. 8.21. The shape of these sensors is based upon the observation that, in general, odd-odd plate modes (e.g. (1, 1), (3, 1), etc.) are more efficient radiators than the even-even modes (e.g. (2, 2), (4, 4) etc.) at frequencies well below the critical frequency of the plate (see Section 8.2). Substituting the form of the sensor shapes of Fig. 8.21 in equation (8.8.2) reveals that the sensor denoted PVDF2 will only observe the odd modes in the x direction, while PVDF1 will only observe the odd modes in the y-direction. Thus if the outputs of both sensors are minimised then the structural motion of the efficiently radiating modes should be minimised (rather than minimising the total response) and this should lead to a fall in radiated sound. Figure

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

263

Plate

J

::::::::

iiiiiiii

iiii iiiii

piezo 1

~iliii

~

iiii

/ PVDF2

300 mm NJNI --- Disturbance

!i!iiiiii :i:i:i$

PVDFI

iiiii!iii iiiil/

Piezo 2

iiiii --------~ X

380 mm

P,

Fig. 8.21 Simply supported rectangular plate with piezoelectric ceramic actuators and piezoelectric PVDF sensors.

8.22 shows a typical analytical result for a f r e q u e n c y o f 349 Hz for the plate system considered in the previous sections. In this e x a m p l e the plate is excited by a point force located at x = 240 m m and y = 130 m m and two piezoelectric actuators were used for control, as shown in Fig. 8.21 (see Fig. 8.10 for coordinate specifications). The piezoelectric sensors were assumed to be m a n u f a c t u r e d f r o m polyvinylidene fluoride ( P V D F ) material with typical material properties given in Table 8.3. A p p r o x i m a t e dimensions and locations o f the sensors are s h o w n to scale in Fig. 8.21. Also s h o w n in Fig. 8.22 are the controlled results w h e n three error microphones were used that were located at mid plane (i.e. y = b/2) radiation angles o f 0 = +45 °, 0 ° and - 4 5 ° and at 0

0o

/

//

/ /

,,",-" ...... , ",, ,, .... ° , -,

,,'" /

k\

\

"'""

X

"

I

-90 ° 80

90 ° 60

40

20

0

20

40

60

80

Radiated sound pressure level (dB)

Fig. 8.22 Radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).

264

ACTIVE CONTROL OF VIBRATION Table 8.3 Typical piezoelectric polymer sensor properties (PVDF). E, = 2 x 109 N m -2 p,= 1.78 x 103 kgm "3 h,. = 0.16 mm

e3~= 65.3 x I0 "3 C m e32= 38.7 x I0 -3 C m e36=0

-2 -2

R = 2 m. The results show that the PVDF structural error sensors provide of the order of 10-15 dB sound reduction. It should be noted here that simply minimising the vibration at two points often leads to sound radiation increases, as demonstrated in the experiments of Metcalf et al. (1992). The control performance can be seen to be not as good as using microphones and this suggests that the P V D F sensors were not correctly placed to properly observe the structuralmotion associated with radiation. Various strategies have been developed for designing P V D F structural sensors in order to correctlyweight the structuralmodes for sound radiationcontrol (see the work of Clark et al., 1992a). The work by Wang et al. (1991) and Clark and Fuller (1992c) has demonstrated that for higher modal densities of response, optimising the shape and location of the control transducersis of the same order of importance as increasingthe number of channels of controlin terms of obtaining global reduction.In one particular example examined by Clark and Fuller (I992c) for a SISO system with relativelyhigh modal density,the optimal actuatorlocationwas found to be near the comer of the plate where itcould effectivelycouple into multiple structuralmodes. The optimal sensor was a long narrow strip located at the bottom edge of the plate where it could observe structuralmotion associatedwith the edge radiationphenomena discussed in Section 8.2. The resultsof both Wang et al. (199I) and Clark and Fuller (I992c) suggest that for more complex, realisticsystems the actuator and sensors will have to be optimally located to obtain reasonable performance with a low number of transducers.The work of Clark et al. (1992a) also demonstrates that extreme care must be taken in accurately manufacturing and positioningthe distributedsensor shape. Section 8.4 has demonstrated how the sound power radiation of a plate can be described in terms of a set of velocitydistributionson the structurewhose sound power radiation is independent of the amplitudes of the other velocity distributions.These velocity distributionsare termed 'radiationmodes' and are obtained using an eigenvalue decomposition as described in Section 8.4. The form of these radiationmodes suggest an approach for designing distributedstructuralsensors which are shaped (as described in Chapter 5) to respond only to the velocity distributionscorresponding to these radiation modes. Such an approach is only strictlyvalid in the low frequency region ka,t I since at higher frequencies the shapes of the radiation modes depend more strongly on the excitation frequency. Nevertheless both Snyder et al. (1993) and Johnson and Elliott (1993) have developed and successfully tested distributedP V D F sensors designed to observe these radiationmodes for the active control of structurally radiated sound. As noted in Section 8.4, by far the most important radiation mode at low frequencies corresponds to the net volume displacement of the plate (see mode (I) of Fig. 8.9). Differentiatingthe signal from the sensor which detected this distribution would give the net volume velocity of the surface, and a P V D F distributed sensor which measures the volume velocity was originallysuggested for A S A C by Rex and Elliott (1992). The shape of the resultantsensor is seen to be almost identicalto that obtained by the design procedures of Clark et al. (I992a).

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

265

Burdisso and Fuller have pursued a different but related approach to the design problem for distributed structural sensors and actuators in ASAC. Burdisso and Fuller (1992a,b) demonstrated that feedforward-controlled radiation from a structure can be described in terms of new eigenvalues and eigenfunctions for the controlled system. Using this formulation Burdisso and Fuller (1994b) developed an eigenassignment design technique by which the actuator and sensor shapes can be designed so as to cause the closed loop system to behave in a controllable manner. In order to minimise sound radiation from a structure excited by multiple frequencies, Burdisso and Fuller (1993, 1994a) used the eigenassignment procedure to design actuator and sensor locations and shapes so that controlled structural modes were radiators of low acoustic power (and thus had a very low radiation efficiency at frequencies well below the plate critical frequency) and the closed loop resonances (i.e. at the closed loop eigenfrequencies) were detuned from the excitation frequencies. The controlled structural mode shapes were shown to be independent of frequency. The procedure enables simultaneous design of the sensor and actuator shapes and position. Analysis and experiments with this procedure demonstrated high, global attenuation of structurally radiated sound over a wide band of frequencies.

8.9 An example of the implementation of feedforward ASAC In this section we briefly discuss a typical arrangement of ASAC in order to illustrate how such systems are implemented in practice. Figure 8.23 shows a schematic layout of the experimental fig and associated control system. The structural system consists of a baffled simply supported plate located in an anechoic chamber. The disturbance to the plate is provided by a point force actuator mounted on the plate through a force transducer and driven by a steady state single frequency. Control inputs are achieved by two piezoelectric ceramic actuators bonded to the plate in the required positions (actuator in this case implies two symmetrically located piezoelectric wafer elements driven 180° out of phase, as discussed in Chapter 5). Two different sensor arrangements are employed; either three microphone error sensors located at (R, 0, ¢~) coordinates of (2m, +45 °, 0°), (2m,-45 °, 180°) and (2m, 0 °, 0 °) or two PVDF strip sensors attached to one side of the plate are used. Figure 8.21 shows the locations and relative size of the piezoelectric actuators and sensors. The configuration corresponds to the analysis of the Section 8.8. Note that for graphical convenience we use 0 = ±45 ° to indicate ¢~= 0, 180°. A signal generator was used to create the harmonic disturbance and the signal was amplified to drive the shaker. The same signal was passed through two adaptive filters to provide the control signals to the piezoelectric actuators. The coefficients of the adaptive filters were updated at each time step using the multi-channel version of the filtered-x LMS algorithm outlined in Chapter 4, in order to drive the error signals to a minimum. Note that, as discussed in Chapter 4, the reference signal has to be prefiltered with estimates of the transfer functions (G~, G~2 etc.) from each actuator to each sensor. These transfer functions are measured at the frequency of interest before carrying out the experiments. As the tests are for a single frequency of excitation only two coefficients are needed in the adaptive and fixed FIR filters. The instantaneous values of the error signals are also required in the update equation to compute the instantaneous gradient estimate. These variables are used to adapt the filter coefficients in the filtered-x LMS update equation presented in Section 4.7.

266

ACTIVE CONTROL OF VIBRATION ) Reference input Disturbance ~__ Simply supported plate shaker 1 - ~ ~ ' - "

1~ ~

PZTactuat°r II

/

r Adaptive filterA

l error sensors I .

.

.

.

.

.

[~apt,ve,,,ter Fixed filters •.-,"~' Gll I ~~1

~i ,MS h/ q algorithmP= I P--q G21 ~ i G22 I - ~ ~-~ G2z

,~11[

LMS E I 1algorithm IZ i r~-

Fixed filters

Fig. 8.23

Schematic layout of controller and test rig for an ASAC experiment.

In order to evaluate the control performance, the radiated field was measured with a movable microphone traverse centred on the plate and with a radius of 2m (the radius being limited by the dimensions of the anechoic chamber). The output of the traverse thus provides radiation directivity plots in the b/2 mid plane of the plate. The magnitude of the disturbance force was also measured in order to provide an absolute comparison between theory and experiment.

0

0o

/')
-9oo 80

i, 60

40

X\

"

, 20

\

/, 0

";,i ,,X 9oo 20

40

60

80

Radiated sound pressure level (dB)

Fig. 8.24 Measured radiation directivity, f = 349 Hz: ~ , uncontrolled; - - - - , controlled with two PVDF sensors; . . . . . , controlled with three error microphones (after Clark and Fuller, 1992a).

267

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

Figure 8.24 shows experimental results for a frequency of 349 Hz. When the results are compared to the theoretical predictions of Fig. 8.22 (with a disturbance amplitude set to that measured in the experiment) reasonable agreement is apparent in terms of shape, absolute values and trends for the different types of sensor. This brief example is intended to introduce the reader to the practical implementation of ASAC. Further details of such experiments can be found in the work of Clark and Fuller (1992a). Smith et al. (1993) have also extended the above application to control of broadband plate radiation using integrated piezoelectric actuators and sensors.

8.10

Feedback control of sound radiation from a vibrating baffled piston

Hitherto in this chapter, attention has been restricted to feedforward control of sound radiation from structures where there is a reference signal available which characterises the primary disturbance. Here we begin a discussion of the potential for feedback control of sound radiation from structures where the excitation is neither deterministic nor can be detected at some time well before it excites the structure (a turbulent boundary layer for example). An important contribution to this field of study is that made by Baumann et al. (1991) who introduced the concept of a radiation filter which enables the evaluation of the sound power radiated from a vibrating surface directly from the properties of the surface vibration. The concept is most easily explained with reference to a simple SDOF model of a vibrating piston in an infinite baffle that is illustrated in Fig. 8.25. This model has been used by Thomas and Nelson (1994a) in order to investigate the potential for feedback control of stiff lightweight structures when used as a secondary 'trim' in passenger aircraft. Such stiff light panels can be designed in order to have elastic modes with a relatively high natural frequency such that the control problem is considerably simplified. The use of this type of structure has already shown some benefits in feedforward control of low frequency sound transmission (Thomas et al., 1993a; Cameal and Fuller, 1993) and shows promise when feedback control is applied (Thomas and Nelson, 1994b). Using the model illustrated in Fig. 8.25, the equation of motion of the rigid circular piston can be written as

Mff~(t) + Cvg(t) + Kw(t) = fp(t) + fs(t),

(8.10.1)

where M is the mass of the piston, C and K are the damping and stiffness of the suspension and fp(t) and fs(t) are the 'primary' disturbance and 'secondary' control force respectively. This equation can be written in matrix form as [ ~"= ('!( t]) [-C/M1

-K~M ] (,)LW(,) r ~i, ]+[1/oM]fs(,)+ [l/oM]fp(,) ,

(8.10.2,

which is thus in the state space form (see Section 3.6) ~(t) = Ax(t) + Bu(t) + Av(t).

(8.10.3)

The equation of motion is thus 'forced' by a vector of external inputs v(t) which, in this case, is simply the scalar primary force input fp (t). If it is assumed that this input is simply Gaussian white noise, then we can again apply LQG regulator theory (outlined

268

ACTIVI~CONTROLOF VIBRATION

Random oxoltmtlon, v (t)

Radiation fllt,r ~

Control foroe, u (t )

g(t)

Diaplaooment, w (t)

Fig, 8,25 A circular, baffled piston mountod on a spring and a damper, The radiation filter has an input given by the piston velocity and an output whoso moan ~quarod output is the acoustic power radiated, in Section 7,10) and find the control input that, for o×ample, minimizes a quadratic cost function such as that specifi¢d in ~quation (7,10,4), As an e×ample, Thomas and N~l~on (1994a) cho~o to minimize tho cost function y-- ~[¢~(t) + a~(t)], (s,~o,4) which is thu~ tho ~um of tho moan squared piston voloeity and tho moan ~quamd ~eondary for¢o woightod by tho factor a, Thi~ factor ponali~ the control 'effort' u~od, Tho r~ult~ (Fig, 8,26), which chow the apaetral dandify of the piston volocity when control is appliod, iUuatrat~ the dop~nd~ne~ on th~ ehoic~ of the w~ighting factor a, Clearly, th~ lower the value of a, the higher the value of the control gain (~e Seetion~ 7,10 and 7,11), Minimization of moan ~quaro velocity, however, does not imply minimization of acoustic power radiated*, A direct measure of the acoustic power can b~ d~dueod from a knowledge of th~ piston velocity by pa~ing thi~ signal through a radiation filter who~o timo avoragod ~quar¢d output i~ o×ttctly equal to the acoustic power radiated, Defining the output of the filt~r to b~ ~(t), the acoustic power can b~ written as

l~ = E[~(t)],

(s, ~o,~)

If the filter i~ a~sumod to havo a frequency r¢~pon~o function density of the acoustic power radiatod i~ given by

s.cw)--- s..(,o)-- I

I

G(jto),

then th~ ~p¢ctral

(s.ao.6)

t Minimizationof pistonaccelorationwouldof cour~oprovid~~ b~tter~pproxim~tionto the minimizationof ~ou~fl~ poworr~di~ted,but ~h~u~eof th~ volo~ity~ign~lher~help~explainth~ ~on~optof thQr~diationfiltor,

ACTIVE STRUCTURAL ACOUSTIC CONTROL, I PLATE SYSTI~MS

269

=10 open loop 1 xlO =3

=16 t

=20

~

=25

1 xlO =6

=30

1 xlO 4

~=35 =40

1 xlO =8

=45 0

0,2

0,4

0,8

0,8

1

1,2

1,4

1.8

1,8

2

Fill, 8.26 Optimal reduction in piston velocity for values of a between 10 =-~and 10 =~, Results show the ratio of the spectral density of the piston velocity to the spectral density of the input disturbance force. where S,o~(co) is the spectral density of the piston velocity signal. The spectral density of the acoustic power is a real, even and positive function of frequency. This implies that we can find an approximation to S,,(~o) that is in the form of a rational spectrum. That is, we can write

N(coa) S~(co) - DCwa) ,

(8.10.7)

where N(co a) is a polynomial of order q in ¢o~ and D(co 2) is a polynomial of order p in ma. A simple example of a rational spectrum is given by S.(co)

1

-

~ ,

(8.10.8)

o)4+a

where a must be real since S, (co) is real. Now note that if we assume the piston velocity signal to be white noise of unit spectral density (i.e. S,~,~(¢a)= 1) then S,(w)---I G(jco)I a, the modulus squared of the radiation filter. We now write S,(co) as the product of two factors such that

S,,(o)) = O(jo))O* (jo))-- O(joa)O(-jo)).

(8.10.9)

Thus, for example, we could express equation (8.10.8) in the form I

I

S~(oa),, (b+jco) (b-jm)

I

I

(b*+ja~) (b*-jm)

(810.101

where b is the complex number given by b= (a/4) ~14+j(a/4) ~14.The spectral factor G(jco) is simply l/[(b+jco)(b* +jco)] which thus specifies the frequency response function of the radiation filter. This also implies that we could express the transfer

270

ACTIVE CONTROLOF VIBRATION

function of the radiation filter as the more general Laplace transform G(s)= 1/[(b + s) (b* + s)] where the Laplace variable s = cr +jw. It turns out that this type of decomposition, known as spectral factorisation, is generally applicable to rational spectra since the power spectrum must be a real, even, non-negative function of o9. This subject is dealt with in more detail by Van Trees (1968) who points out that the pole-zero plot of G(s) G(-s) must have the following properties. (1) Symmetry about the jw axis (otherwise S~(w) would not be real). (2) Symmetry about the cr axis (otherwise S,(w) would not also be even). (3) Any zeros on the jw axis occur in pairs (otherwise S~(w) would be negative for some value of co). (4) There are no poles on the j to axis. As a direct consequence of these properties we can, in general, find G(s) such that all its poles and zeros are in the left half of the s plane (or <0), while G(-s) has all its poles and zeros in the fight half of the s plane (or>0). Since G(s) has all its poles and zeros in the left half of the s plane, it is guaranteed to be a stable, causal filter. Furthermore, it is minimum phase and will have a stable inverse (see Nelson and Elliott, 1992, Chapter 3). Thomas and Nelson (1994a) used a least squares method to adjust the coefficients of a rational Laplace transform with sixth-order numerator and denominator in order to match the well known variation with frequency of the acoustic radiation resistance of a rigid circular piston. The radiation resistance curve (which exhibits the frequency dependence of S~(w) when the piston velocity is a white noise signal) is shown in Fig. 8.27. The pole-zero plot of S~(s)= G(s) G(-s) shown in Fig. 8.28 provides a fit to this curve to within 0.01% up to ka = 2.5, where a is the piston radius and k = tO~Co. Note that the symmetry of the pole-zero plot complies with conditions (1) and (2) given above. Taking the left half plane poles and zeros shown in Fig. 8.26 to define G(s) one can use one of several techniques to find a state space realisation of the radiation filter with this transfer function (see, for example, Maciejowski, 1989). Such a realisation can be written as il(t) = Asrl(t) + B sw(t),

(8.10.11)

z(t) = Cg11(t ) + Dg'/v(t).

(8.10.12)

Thus rl(t) is the vector describing the 'states' of the radiation filter while z ( t ) = z(t), the scalar output of the radiation filter and "/v(t)= W(t) is its input. This realisation can then be combined with the state space model of the piston dynamics given by equation (8.10.3) such that

[A 0][x,t,] ["0] [o]

il(t ) = Bs),

Ag

n(t)

z ( t ) = [Dg¥ where ¥= [1

+

u(t) +

x(t)] ' Cg] ~l(t)

v(t),

(8.10.13)

(8.10.14)

0]. This state space model has an output z(t) whose time-averaged

271

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

1.2

I

I

I

I

I

I

I

I

I

I 0.4

I

I

I

I

I

I

I

0.2

0.6

0.8

1 ka

1.2

1.4

1.6

1.8

0.8 0 tO O

Q. 0.6

a:

0.4 0.2 00

Fig. 8.27

2

The radiation resistance of a plane circular baffled piston of radius a. I

I

I

I

I

I

o

I

I

I

I 3

! 4

o

21

#0

-

X

-1 -2

--

-3--

-4

o

--

-5 -5

I -4

I -3

I -2

o I -1

I 0

I 1

I 2

a a/c O

Fig. 8.28 The pole-zero diagram for the rational Laplace transform approximation found for the radiation resistance of a plane circular baffled piston. Poles are represented by x and zeros by O. Note that there are a pair of zeros placed close to the origin. squared value is equal to the acoustic power radiated. Therefore the optimal control that minimises the cost function

J= E[zT(t)z(t) + auX(t)u(t)]

(8.10.15)

will minimise acoustic power radiated plus a factor a times the mean square effort used in the control. Thomas and Nelson (1994a) have computed the solution to this problem and the results are illustrated in Fig. 8.29. These clearly show the difference between the

272

ACTIVE CONTROL OF VIBRATION

130

I

I

~2ol

I

loop

110 Vibration control

~100

Radiation control

,o "0

8o

7O 8o

5O

40 30

0

0.5

1.O /ca

1.5

2.0

Fig. 8.29 Soundpower radiated by a plane circular piston without feedback (open loop), with optimal feedback minimising the squared velocity of the piston (vibration control) and with optimal feedback minimising sound power radiated (radiation control). The mean squared velocity of the primary excitation is 0.083 m s "2. results produced by choosing the control that simply minimises vibration and those produced by choosing the control that minimises acoustic power radiation. The results shown in Fig. 8.29 are for the same value of time-averaged control power used in the two cases (which used two different values of a). The frequency weighting provided by the radiation filter's representation of the radiation resistance curve is clearly evident.

8.11

Feedback control of sound radiation from distributed elastic structures

In this section we discuss the active control of sound radiated from a structure exhibiting multi-modal response using feedback control, following the approach of Baumarm et al. (1991) introduced in the previous section for an SDOF system. It was shown in Section 8.4 that the acoustic power radiated by a harmonically excited structure, whose velocity distribution is described by the sum of contributions from N structural modes, can be written in frequency domain form as (8.11.1) l-IC~) = ~'"C~)MC~)~(~), where ~ ( ~ ) is the vector of N complex structural mode velocity amplitudes, and M(o~) is a matrix of self and mutual radiation resistances (which is real and symmetric) while superscript H denotes the Hermitian operator. We can factorise M (~) into the form G" (~)G (~), so that the radiated power can also be written as (8.11.2)

273

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

Defining the vector of transformed modal amplitudes G(co),~(to) to be z(to), equation (8.11.2) can be written as FI (oj) -- ZH(CO)Z(0~).

(8.11.3)

The acoustic power radiation due to any one element of z(to) is thus independent of that due to each of the other elements of z (to) (Borgiotti, 1990; Cunefare, 199 I). This is in contrast to the power radiation due to each structural mode amplitude, which is generally dependent on the amplitudes of many other structural modes, as accounted for by the off-diagonal mutual radiation terms in M(~). This procedure has been discussed in more detail in Section 8.4. If the structure is subject to a transient excitation, the total radiated acoustic energy can then be written as E = [" ",VH(~)M(w) W(co) dto ,/ 0

~

(8 11.4) '

where . ( t o ) is the Fourier transform of the vector of the waveforms of the structural mode amplitudes ",V(t). The evolution of these time histories can be described in state space form by defining a state vector x(t) (as described in Section 3.6) which includes the velocities ~(t), and, for example, the displacements w(t) associated with each of these velocities, which then obey the state space equation :¢(t) = Ax(t) + Bu(t),

(8.11.5)

where A is the state matrix and u(t) is the vector of inputs to the system due to the secondary forces. The radiated energy equation (8.11.4) can also be written, using equation (8.10.3) as E

= I"0

ZH(~0)Z(0J)do~P

(8.11.6)

where at each frequency we again have z(m) - G (m),,V(m)

(8.11.7)

and G(to) can now be identified as the matrix of frequency response of the 'radiation filters' which operate on the structural mode amplitudes to give each of the elements in z(to). Baumann et al. (1991) demonstrate that these 'radiation filters' can be chosen to be causal, and thus realisable (to arbitrary accuracy) in state space form. The procedure to be adopted first requires finding a rational Laplace transform approximation to the elements of M(to). The algorithm provided by Francis (1986) can be used to carry out the necessary spectral factorisation. Note, however, that the algorithm can only be used provided that the rational Laplace transform approximation to M(to) is also positive definite. The state equations of this array of radiation filters can be written as t'(t) = Acr(t ) + Bcx(t ),

(8.11.8)

z(t) = Ccr(t) + Dcx(t ),

(8.11.9)

in which r(t) is the vector of state variables of the radiation filters, and x(t) is the vector of state variables for the structure, as defined above. These two sets of state

274

ACTIVE CONTROL OF VIBRATION

variables can be combined to give an overall state variable model for the radiating system -

[A 0][x]+[.] Ba

Aa

r

0

u,

(8.11.10)

with a corresponding output equation z=[Dc

CG][X].

(8.11.11)

Now that the complete system, including sound radiation, has been cast in state space form, the standard tools of 'modem' control theory can be used to calculate the optimal LQG feedback gains, as described in Section 3.10. The cost function which must be minimised is of the form Jrad = Io [zT(t)Z(t) + auT(t)u(t)] dt,

(8.11.12)

where the first term is the total sound energy radiated by the structure expressed as an integral over time, which must be equal to the frequency domain form, equation (8.11.6), by Parseval's theorem. The second term in equation (8.11.12) must be introduced to make the equations soluble, and may be physically interpreted as being proportional to the control effort, i.e., the sum of the squared inputs to the secondary sources. The optimal set of full state feedback gains which minimises this cost function is the solution of a steady state Riccati equation (see Section 3.10), and the resulting feedback control law can be expressed in the form u(t) = K ~ x ( t ) ,

(8.11.13)

where x = [x TrT] T is the full state vector, and K rad is the optimal feedback gain matrix. Baumann et al. (1991) present some numerical results of the effect of such optimal feedback control of sound radiation on a beam excited by a short duration pulse. The 1 m x 0.125 m steel beam was assumed undamped, clamped at both ends and supported in an infinite baffle. For simplicity it was assumed that the structure and its sound radiation could be accurately modelled by using the first three structural modes only. A single secondary actuator was located half way along the beam, and driven by each of the full state variables via the feedback gain matrix defined by equation (8.11.13). The resulting velocities of the three structural modes are shown in Fig. 8.30(a) (Baumann et al., 1991, Fig. 4). It can be seen that the first (volumetric) mode, mode 1 and the third mode, mode 3, are most strongly controlled using this feedback strategy. This is because the second mode, mode 2, acts as an acoustic dipole at frequencies close to its natural frequency and is thus not an efficient sound radiator. Controlling this mode will thus have little influence on the total acoustic radiated energy. In contrast to this radiation control strategy, the optimal feedback problem can also be solved to suppress the mechanical vibration of the beam. This can be achieved by using the LQG theory described above to minimise a cost function of the form Io [xT(t)X(t) + a2uT(t)U(t)] dt,

(8.11.14)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. I PLATE SYSTEMS

275

0.025 O.02

(a)

0.015 ~,

,, :':

0.01!

/I

"~ o.oo5i

i~/: !

~o

i~.

0 i -0.005

'..\~

! ! i /i

i

.--.

, i

..

"..v

" ""--- ' " -

' '....

: :

"~:............ v

"

/

:~ -o.ol i 'i~ -0.015 -

0 •02 ] /

-0.025 /

i

0.02 ~

.

0.015 t [

i~/(~i

i

i

.

i

.

t

.

t

.

~ Mode 2 ..... Mode 3 t i

I

/ / J

. (b)

o0 ii ii i i i

>"

!t i i :

i I. i/!

o ~ ii £,

-ooo5

i i

.~ :. :~!

i i

'i

!'" !

:

.z i

. :: "

~i

i

i ii'ii

il

i

~..

!,~i

"~

!/i

i i

ii

!

-0.015 ~ 1 / ii ! :'

V

-001:

:-

~,:

i

"

:i

~!

ii

:.

.".,

"~ ---

l

I 0.1

""'.

"

"--

""

- -

-0.02 / 0

~--

..

"'~

Mode 1

~

Mode 2 ..... Mode 3

t 0.2

i 0.3

i 0.4

t 0.5

i 0.6

i 0.7

i 0.8

t 0.9

1

Time (sec)

Fig. 8.30 Velocities of the three structural modes of the beam modelled by Baumann et al. (1991) to demonstrate state feedback control of (a) sound radiation and (b) structural vibration. which results in a feedback law of the form u(t) = KV~bx(t).

(8.11.15)

In the results presented by Baumann et al. (1991), the weighting on the effort term in equation (8.11.14), a z, was adjusted to make the total energy used by the controller

276

ACTIVI~ CONTROL OF VIBRATION 70

~

I

I

I

I

00

=

80

=

40

=

3° /

.10 0

~,

....

i 50

.....

i 100

....

:7

i 160

-,.,, .......

J 200

,

260

Frequenoy (Hz) Fig. 8.31 Totalradiatedpower without statefeedback (====) and with statefeedback (= ==) two control forces placed at x - 0.1 m, y -- 0.1 m and x - 0,25 m, y - 0.125 m with a - 10=°. Results presented from Thomas and Nelson (1993) for the predicted reduction in sound power radiated from an aluminium plate excited by a turbulent boundary layer, The plate was simply supported, measured 0.5 x 0.25 m and was 1 mm thick. A structural damping factor of 0.01 was assumed.

equal to that used in the simulation above. Figure 8.30(b) shows the velocities of the three modes modelled in the computer simulation when implementing feedback control of vibration (equation (8.11.15)). In this case, all three structural modes have been controlled to approximately the same degree, with the result that the third mode, in particular, rings for considerably longer than in Fig. 8.30(a). Baumann et al. (1991) state that the total acoustic energy radiated when using feedback control to suppress vibrations (Fig. 8.30(b)) was 38% greater in these simulations than when feedback control was used explicitlyto suppress radiation (Fig. 8.30(a)). A similar formulation has also been used to analyse the feedback control of sound radiation from a structure excited by random disturbances (Baumann et al., 1992). Thomas and Nelson (1993) have also used Baumann's theory to examine the feasibility of providing active control of the sound power radiated from a simply supported flexible plate excited by a turbulent boundary layer. An example of the results they derived is shown in Fig. 8.31 which demonstrates the reduction of sound power radiated that can be in principle achieved with optimal feedback control for an aluminium panel typical of those used in aircraft fuselage construction. The results presented are for a specificchoice of 'effortweighting' a in the cost function used; the reductions produced were found to be crucially dependent on this choice and thus the control gains used, More detailsare presented by Thomas and Nelson (1995).

9 Active Structural Acoustic Control. II Cylinder systems

9.1

Introduction

The previous chapter considered Active Structural Acoustic Control (ASAC) applied to sound radiation and transmission of two-dimensional plate systems. We now turn our attention to the application of ASAC to sound transmission through, and radiation from, cylindrical structures. Such systems are representative of many applications. The control of sound transmission into aircraft is important, and in the low frequency region the fuselage can be adequately modelled as a cylinder (Koval, 1976; Fuller, 1986b; Bullmore et al., 1990). Piping systems are common in many applications and often carry unwanted vibrations (White and Sawley, 1972) and radiate noise (Holmer and Heymann, 1980). Submarine hulls can also be approximated as cylinders and active control shows promise for the reduction of low frequency structurally radiated sound (Clark and Fuller, 1993). In the following sections we will consider ASAC applied to various cylindrical structural systems which can be considered to form the basis of the above applications. Before we do this, it is necessary to extend the previously described cylinder equations of Sections 2.12 and 2.14 to allow for coupled interior and radiated acoustic fields, various forcing functions and the effects of finiteness of the cylinder.

9.2

Coupled cylinder acoustic fields

Consider the infinite cylinder discussed in Chapter 2 and shown in Fig. 9.1. If the cylinder is harmonically vibrating and is totally immersed in a compressible fluid then the interior and exterior acoustic fields will be excited into motion by the radial vibration of the cylinder wall (ignoring the effects of viscosity). The interior acoustic field can be modelled by a double modal series as (Morse and Ingard, 1968) oo

pi(r, O, x, t)

oo

ZZ = n=l

pins cos nOJ,(k~ir) e

jsx,

(9.2.1)

s=l

where k~~is the internal radial wavenumber which is related to the axial wavenumber k~s by the vector relationship k~~= +~/(ki) 2 - kZ,s,

(9.2.2)

278

ACTIVE CONTROL OF VIBRATION

i

~

, ,,jln=o

0=1 I

!

~~"

n=2

Fig. 9.1 Coordinate system and modal shapes for an infinite thin cylinder. where k ~ is the wavenumber of the interior acoustic field. Note that J,(-) denotes a Bessel function of the first kind as described by Abramowitz and Stegun (1972). The radiated field only experiences outward radiation and can be written as (Junger and Feit, 1986)

ZZ pO cos nOH,(k~ r) e oo

pr(r, O, x, t) =

oo

to

,

(9.2.3)

n=0s=l

where H , ( k , r ) is the Hankel function of the first kind (Abramowitz and Stegun, 1972) and a similar relation to equation (9.2.2) is obtained for k r° in terms of outside field properties and wavenumbers. In equations (9.2.1) and (9.2.3) the index n corresponds to the circumferential modal order (n = 0, 1,2, 3 . . . . ) and is related to the number of diagonal nodal lines across the mode shape as shown in Fig. 2.10. The subscript s indicates a branch or wave solution (s = 1,2, 3, ...) for a particular n and does not relate to any particular characteristic of the mode shape except that increasing s implies increasing modal complexity. The superscripts o and i correspond to the 'outside' and 'inside' acoustic fields respectively. Application of the boundary condition of continuity of displacement at the shell wall in the three media results in results in P / a n d Pn° being written in terms of W,, as (Fuller and Fahy, 1982) P/=

~-P~ ri

t

W,~

(9.2.4a)

ri

ksJn(ksa) and

[ 20] P__z

rO

l

rO

k, Jn(k~ a)

(9.2.4b)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

279

The free equations of motion for the shell (i.e. without external disturbances) are subsequently modified by the fluid loading due to radiation such that all the matrix terms specified previously by equation (2.12.4) are similar except the L33 term which is now written as (Fuller and Fahy, 1982) L33 = -~'-'2 2 +

1 + fl2[(k,sa)2 + n2] 2 - FL,

where the fluid loading term FL is given by (Fuller, 1986a) FL = ~"22

Jn ( k ;ia ) ¢

ri

ri

i_\Psl Jn(ksa)ksa

n

i o,l

(9.2.5)

ro]

Hn(k s a) t

FO

,

(9.2.6)

ro

~ Ps I Hn(ks a)ks a

The system of equations represented by the modified shell equations (2.12.4) including the fluid loading term of equations (9.2.5) and (9.2.6) thus represents coupled equations of motion for the vibration of the shell and fluid media. In equation (9.2.6), P s represents the shell density, while P li and pOy are the densities of the 'inside' and 'outside' acoustic media respectively and (') denotes differentiation with respect to the argument. The symbol f~ indicates non-dimensional frequency such that ~ = wa/cL. By setting p} and/or p~ to zero the equations for the free response of the cylinder can be reduced to any of the three in-vacuo cases, with only a contained fluid, with only outward radiation or totally in-vacuo. Thus under the action of a disturbance, on solving the system of equations for Wns, the response of the interior and radiated acoustic fields can also be obtained, as described in the next section.

9.3

Response of an infinite cylinder to a harmonic forcing function

In practical problems structural systems are excited by various forms of forcing functions whether they are disturbance or control inputs. In the following analyses we will consider two forms of forcing function which are representative of those found in a number of practical situations; a point force which is applied to the structure or a monopole source which is located in the fluid medium.

Point force input The response of a fluid-filled cylinder to line and point forces has been derived by Fuller (1983). As discussed in Section 2.14, a point force input f = F e j°~' applied at x = 0, 0 = 0 can be expressed in cylindrical coordinates as

p(x, t) = 1 ~o enF cos nO 6(x) e_riot, 2Jr =

(9.3.1)

where en = 1 if n = 0 and e , = 2 if n~0. Following the analysis of Section 2.14 the response of the system is described by

1 i5~ Wn(kn) e -j(k,.)(x/a) d(kna), w(x)= ~o = 2Jra

(9.3.2)

280

ACTIVE CONTROLOF VIBRATION

where the spectral wavenumber displacement is

Wn(kn)= ( pshe)2 ~2F II ] 33

(9.3.3)

and •33 =

(LllL22- L,2L21)/ILI,

(9.3.4)

where the time variation e i~'' has been suppressed. In equation (9.3.4), for a fluid-loaded shell, the L33 term of the determinant of matrix L is given by equation (9.2.6), and the result of setting ILl to zero also provides the axial wavenumbers, k,sa, for the fluid-loaded system. Equation (9.3.3) in conjunction with equations (9.2.4a,b) enables evaluation of the interior and exterior acoustic fields. Spectral quantities can also be seen to only have one subscript n since the contributions from all waves for a particular n are included in the integral of the inverse Fourier transform.

Interior monopole source A simple form of acoustic excitation inside piping systems investigated by Fuller (1986a) is a point source of sound located at rp. The free-field pressure resulting from this arrangement is given by

p(r,O,z) = Po eJkir'/r',

(9.3.5)

where r' is the distance to the observation point from the monopole location (expressed in cylindrical coordinates) and P0 is the monopole source strength in units of force per unit length. Skelton (1982) and James (1982) have derived the expression for the interior pressure field associated with the monopole positioned on the interior of an elastic cylinder which can be written in spectral form in cylindrical coordinates as

{ri]

i P(r, n, k ri) = pi~oZaWn(kn)Jn(krir)[kriaJ'n(kria)] -1

Jn(k Fp) + 2poe,, J',,(kr'a) ' ria)Y,,(k rir)] - [J,,(krir)y',,(kria)- J,,(k

for r > rp, (9.3.6)

[ Jn(krir) ]_ + 2poe~ J'n(kr'a)

_ , Jn(kria)Yn(krirp)]

[Jn(krirp)Y'n(kria)

for r < rp, (9.3.7)

where the monopole is located at r = rp, 0 = 0, and x = 0. The factor e n = 1 if n = 0 and e, = 2 if n > 0 and radial wavenumber k ri is given by equation (9.2.2) with suppressed subscript s. Equations (9.3.6) and (9.3.7) were obtained by assuming that the interior field consisted of a direct term from the monopole source and a scattered term associated with the forced vibration of the cylinder wall. The unknown amplitude of the scattered term was obtained by applying the Euler boundary condition at the wall as

ACTIVESTRUCTURALACOUSTICCONTROL.II CYLINDERSYSTEMS

281

described by James (1982) and Fuller (1984). Note that this implies, at large values of the distance of x, the interior pressure field will be dominated by the term associated with the vibrations of the cylinder, Wn(k~), since the direct terms will decay as 1/r'. Substituting the forcing functions into the equations of motion of the cylinder provides a system of equations which can be expressed in matrix form as

L2! L22 L23 Vn(kn) = 0 , Lal L32 L33 Wn(kn) T31

(9.3.8)

where the elements of L are given by equations (2.12.4a-i) and include the fluid loading effects of FL, given by equation (9.2.6), in term L33. The source term T31 is given by (Fuller, 1984)

Jn( rp

T31(kn, n)= 2enPo J'n(k ia)k ia

(9.3.9)

p,~oZh '

where (') implies differentiation with respect to the argument. The response of the system to the interior monopole can be again found by solving for Wn(k~) and applying the inverse Fourier transform. This gives the radial displacement relationship

(

~-~2 )n~0

W(rp, x, O) = Po ~pshaco2 =

cos

too [kri]_j(kna)(x/a)J..2n(rp.....~) e d

J'n(k ia)kria

na

(9.3.10) By applying inverse transforms to equation (9.3.6) and (9.3.7), similar relations for the interior pressure response can be obtained. The radiated field can be found directly from Wns using equations (9.2.3) and (9.2.4b).

Exterior monopole source James (1982) has used a procedure similar to that described above to find the pressure and response of a cylinder excited by an exterior monopole. The spectral form of the monopole, given by equation (9.3.5), in the presence of the cylinder can be written

as P(n k r°) ,

ro rp) H'n(kr°a)kr°a

2 2

2poenHn(k =

-

ro

pf(.O Wn(kn)Hn(k a) +

H'n(kr°a)kr°a

,

(9.3.11)

where in equation (9.3.11) the pressure is evaluated at the shell surface, r = a. Using a similar procedure to that employed in the interior monopole problem, the radial displacement response of the shell to the exterior monopole is found to be given by

w(x/a, O) = Po[

~,-~2 ]~0 f~-oo[ Hn(kr°rp)] 133 e -j(kna)(x/a)d(k,a) pszcha(.o2 = e n cos nO H,n(kroa)kroa (9.3.12)

282

ACTIVE CONTROL OF VIBRATION

and the interior pressure field is given by i

pi(r/a, x/a, O) = Po pserhaw2 PlW

2a

Z en cos nO

n=O

Hn(k rp) H'n(kr°a)kr°a

[Jn(krir)]-i(k.a)(x/a) x ..... 133 e d(kna). J',(kria)kria

(9.3.13)

The integrals obtained by applying the inverse Fourier transform could be solved by contour integration (including branch cuts if necessary) and the method of residues as described in Churchill et al. (1974). Such approaches have been used in Chapter 2. However, it is often more convenient to evaluate the integral numerically by directly integrating along the kna axis as described by Fuller (1986a). Care must be taken to choose sufficiently large values of +kna to ensure that the integral converges satisfactorily. In addition, a small amount of damping is usually added to the shell system to move the poles (associated with the eigenvalues) off the real axis thus avoiding the problem of numerical instability. As discussed in Chapter 2 this form of hysteretic damping is valid for harmonic motion. A more detailed discussion of this technique has been presented by Fuller (1986a).

9.4

Active control of cylinder interior acoustic fields using point forces

The control of sound transmission through cylindrical structures to the contained interior space is an important problem in many applications. For example, much work has been carried out on controlling aircraft interior acoustic fields using arrays of active acoustic sources located in the interior space (Elliott et al., 1989; Lester and Fuller, 1990; Silcox et al., 1990). In this section we study an alternative technique based on ASAC whereby active forcesare applied to the structure and are optimised to minimise the interior acoustic field. The analysis essentially follows work by Jones and Fuller (1990). Figure 9.2 shows an axial view of the system used for this study which is similar to the interior noise model developed by Fuller (1986b) where the aircraft fuselage is modelled by an infinitely long shell with coupled interior and exterior acoustic fields. The rationale behind using the infinite shell model is that mid to large size aircraft have appreciable structural and acoustical damping. This has the effect of damping axial waves, with the consequence that the response behaviour is dominated by a wave rather than a modal behaviour in the axial direction. Likewise inhomogeneities such as ribs and stringers are assumed 'smeared' while other asymmetries such as the floor etc. are ignored. The acoustic source due to an exterior propulsion unit such as a propeller is modelled as a monopole which radiates towards the fuselage, drives it into motion, which in turn, forces the interior acoustic field to respond. The control inputs are modelled by normal point forces arranged around the cylinder circumference in the source plane as shown in Fig. 9.2. Excitation and control is assumed to occur for harmonic oscillations which is representative of the dominant noise inside the cabin of propeller aircraft.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

283

Oi Po

0=0

a

Fig. 9.2 Coordinate system and input locations for active control of cylinder interior noise (x axis is into the page). The total complex interior acoustic field is by superposition, the sum of the contributions due to the exterior monopole disturbance (also called the primary source), pp, and the control force (also called the secondary sources) inputs, Ps, respectively and thus

p,(r/a,x/a,O)=pp+ps.

(9.4.1)

The interior field due to the monopole disturbance is specified by expressing equation (9.3.13) as a function of angular position such that

pp(rla, xlo. o) poAo(rla, xla. O..ola). =

(9.4.2)

where P0 is the monopole source strength (specified), and the internal and external media are assumed identical. The time variation e j~'' has been suppressed for conciseness. The transfer function, A0, in equation (9.4.2), is given by oo

A0 = Z ~PPcos(n0- nOo),

(9.4.3)

n=O where

/pP-- [En~a2][~sf][h]-I

H~(krr°) Jn(krr) 133 e H'n(kra)kra J'n(kra)kra

d(kna). (9.4.4)

In equations (9.4.2)-(9.4.4) r 0 is the radial location of the external source, 00 is the angular location of the source and r is the coordinate of the internal radial observation point.

284

ACTIVE CONTROL OF VIBRATION

The interior pressure response due to N, control forces located at x/a = O, 0 = 0 i is given by N,

ps(r/a, x/a, O)= Z FiBi(r/a' x/a, O, 0~),

(9.4.5)

i=1

where F i are complex force amplitudes (unknown) and the transfer function B~ is specified by oo

ni = Z ~Pn(r/a' x/a) cos (nO- nOi),

(9.4.6)

n=0

where

I?~ ~s=

Jn(krr) ] 133e j(kna)(x/a)d(k,,a).

J'n(kra)kra

era JLPsJ

(9.4.7)

Using the approach outlined in Chapter 4 we define a quadratic cost function given by

J(Fi) = --1 ~s ~r Ip,(r/a, x/a, S

o) I

dS,

(9.4.8)

which is the square of the modulus of interior pressure integrated across an axial plane of the interior acoustic field. For the following results we will simplify equation (9.4.8) by noting that for a rigid cylinder, all acoustic modes will have a pressure maximum at the wall. Thus, we can reduce equation (9.4.8) to a line integral around the circumference at the axial location of interest, and approximate the cost function as

i yg 0 Ip,(1, x/a, O) dO.

J(F3 --- --1

(9.4.9)

This simplified cost function results in substantial savings in computer time and has demonstrated very good control characteristics. The purpose of the analysis is to solve equation (9.4.9) such that the cost function J(Fi) is a minimum. The procedure for obtaining the optimal solution is similar to that outlined in Chapter 4 and is described briefly in this section. For further details the reader is referred to Jones and Fuller (1990). To obtain the optimal solution, the cost function is best expressed in matrix form by substituting equation (9.4.1) into equation (9.4.9) and performing the required integration using the orthogonality characteristics of the circumferential modes on the interval (0, 2zl). The resulting cost function is a real scalar quantity expressed in terms of unknown control amplitudes described by the vector f , = [F1 F2 F3 ... ]T, with all other variables specified or assumed known. To solve for the vector of control amplitudes which minimises J(fs), the cost function is differentiated with respect to f, and equated to zero. The resulting optimal solution f~o is given by, in matrix form (see the Appendix of Nelson and Elliott, 1992, and Chapter 4)

fso = -A-~b,

(9.4.10)

where b = xp0. For vector x of size [N, x 1 ] a typical element in the jth row is oo

= n~0 __ 2E

s

p:~

o,)],

(9.4.11)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. 11CYLINDER SYSTEMS

285

where 0] is the angle of the jth control force and 0 p is the angular location of the noise source. For the Hermitian matrix A of size [N~ x Ns] a typical element in the ith row and jth column is oo

A~j = = 2 ~ ( ~ ' ~ ) * c o s [ n ( 0 ~ - 0~)l,

(9.4.12)

where the factor e = 2 if n = 0 or e = 1 if n > 0. For the following illustrative results the cylindrical shell was assumed to be of 0.254 m radius, 1.63 mm thickness and constructed from aluminium with properties given in Table 9.1. The non-dimensional excitation frequency was set at ~ - 0.193. This value of ~ would correspond to, for example, 166 Hz in an aluminium cylinder of 2 m diameter. The objective was to minimise the interior acoustic field in the source plane, thus the integral of equation (9.4.9) was evaluated at x / a - O . Note that, as discussed in Section 2.8, a hysteretic damping q, as specified in Table 9.1, was introduced into the shell and fluid media to condition the inverse Fourier integrals for numerical evaluation and model the damping of a realistic aircraft fuselage. As shown in Fig. 9.3(a) the monopole is offset slightly from 0 = 0 ° by an angle of 0 d = - 8 ° to introduce some asymmetry into the model, corresponding to more practical situations. Figure 9.3(b) presents the interior sound pressure level distribution due to the disturbance alone, evaluated in the source plane. It can be seen that the interior pressure field is dominated by a slightly rotated cos 20 mode, with some small contribution from the other circumferential modes. In order to interpret the results it is convenient to describe the shell and interior pressure field response as an azimuthal series such that p or w = Z An cos nO + B n sin nO.

(9.4.13)

n=0

Figure 9.4 presents the shell response decomposed into modal amplitudes]" I An l a n d I Bn I for the conditions leading to the results shown in Fig. 9.3 (b). For the disturbance alone, the shell response is dominated by many modes such as A2, A3, A4, A5 and n 3 (note that the subscript refers to circumferential modal order). Figures 9.3(b) and 9.4 reveal an important characteristic of the coupled sound field. Out of all the strongly responding shell modes only the n = 2 motion is well coupled to the interior acoustic field (Fuller, 1985b; Thomas, 1992). Thus in order to control the interior acoustic field for this system, it is only necessary to control the n = 2 structural motion. Table 9.1 Medium

Aluminium Air

Youngs modulus E(N m -2) 71

×

m

10 9

Material properties

Poisson's ratio

Density p(kg m -3)

Free wave speed cL, cr (m s-')

Damping ratio

v

0.33 --

2700 1.21

5432 343

0.2 0.001

t In effect a wavenumber transform has been applied to the response in the angular direction.

rls, rlt

286

ACTIVE CONTROL OF VIBRATION

!

Pointcontrol \ ~ forces

~

I '

/

~~ Acoustic

Jsource

I i

(a)

(b)

(c)

(d) 50

70 90 SPL (dB)

110

Fig. 9.3 Interior sound pressure level distributions, x/a = 0, ~ = 0.193: (a) test configuration; (b) uncontrolled; (c) controlled with one force; (d) controlled with two forces (after Jones and Fuller, 1990). Figure 9.3(c) presents the controlled field when one control force located at 01 = 180 ° is employed. It is apparent that reductions o f the order of 10 dB have been obtained. However, the residual field now appears to have the shape of the sin 20 mode, suggesting control spillover has occurred. For the next test we apply a second control force at 0~= 45 ° and the total field given in Fig. 9.3(d) now shows reductions of the order of 50 dB with no discernible residual mode shape. In effect the use of two control forces has resulted in a distributed control input which has reduced coupling into the sin 2 0 mode, thus limiting the control spillover. The corresponding shell modal amplitudes for the case of two simultaneous control forces are also plotted in Fig. 9.4. The results show that the A2 and B2 shell amplitudes have been reduced leading to global interior noise reduction. Figure 9.4 also demonstrates significant control spillover into the higher order structural modes. As these modes are not well coupled to the interior acoustic field, the control spillover is constrained to the structure and does not lead to performance degradation in terms of

ACTIVE STRUCTURALACOUSTICCONTROL.II CYLINDERSYSTEMS Noise source o Control source A Noise and control source o

~50[

[] A O

I

100

x

E E v

287

,< -

-

50

t-

0

O ¢::

0

0 0

2

4

6

8

10

12

14

16

Circumferential mode number, n

C

150

E

~100 .~

g

50 "0 n"

0

2

4

6

8

10

12

14

16

Circumferential mode number, n

Fig. 9.4 Circumferential modal amplitudes of shell radial displacement response, x/a=O, = 0.193 (after Jones and Fuller, 1990). controlling the interior acoustic field. These results again illustrate the basic concept behind ASAC; it is only necessary to control or modify those structural motions which are associated with significant sound radiation. The results of Fig. 9.4 are also analogous to those presented in Chapter 8 which were discussed with regard to their wavenumber content. As can be seen from Figs 9.3 and 9.4, only the low angular wavenumbers (i.e. low values of n) are well coupled to the interior field, while the high wavenumbers are not well coupled and are thus not as important in terms of interior radiation. If the increase of structural response is unacceptable then shaped actuators could be used as in the work described by Dimitriadis et al. (1991), or a cost function could be used which includes a weighted contribution from the shell response. Both of these techniques can be employed in order to keep the structural vibration bounded to acceptable levels.

9.5

Active control of vibration and acoustic transmission in fluid-filled piping systems

Piping systems are common in many industrial situations, and they often can transmit unwanted vibrational energy to points away from the excitation source. Vibration transmission and energy distributions in fluid filled elastic cylindrical shells have been investigated in detail by Fuller and Fahy (1982). Due to the mixed media of the transmission path (i.e. the structural and fluid path) the passive control of vibrations in fluid filled pipes can be difficult. For example, the work of Fuller (1983) demonstrates that the internal fluid can cause a significant flanking of the energy of predominantly structural waves around a radial line constraint applied to the cylinder wall.

288

ACTIVE CONTROL OF VIBRATION

In this section we study active control of wave propagation in a fluid-filled elastic infinite shell. The work discussed is a summary of previous investigations reported by BrEvart and Fuller (1993). The configuration of the system to be analysed is shown in Fig. 9.5. The most convenient location to apply the control forces is directly to the shell wall, even though the object may be to minimise either structural power flow, fluid power flow or the sum of both. For the following analysis we restrict ourselves to axisymmetric (n = 0) and beam type (n = 1) wave motion and choose to minimise the radial displacement of the cylinder wall at up to two axial locations as shown in Fig. 9.5. The rationale again is that it is easiest to observe the structural motion; direct control of the interior field would require obtrusive arrays of control sources and sensors in the acoustic field which may impede flow of the internal fluid. Using the coordinate system of Fig. 9.5 where the incident wave has complex amplitude Wff,at x = 0, the complex disturbance displacement can be written as

Wp(x) = W~ cos n 0 e -jk~x,

(9.5.1)

where k,p, is the axial wavenumber of the chosen disturbance wave type. As discussed by Fuller and Fahy (1982), k~ is strongly dependent upon frequency and type of the wave in the shell-fluid system. The time variation e j°'' has again been suppressed, since we perform the analysis for a single frequency of motion. The control or secondary forces are considered to be axisymmetric (n = 0) or beam type (n = 1) radial line forces (corresponding to circumferential distribution of the disturbance) applied to the pipe wall as in Fig. 9.5. The azimuthal distributions of these line forces are given in Fig. 9.1, and can be written as

p,(x, O)= F, cos nO 6 ( x - x~),

n = 0, 1,

(9.5.2)

where x,i is is the axial point of application. The radial response of the shell fluid system to these line forces has been derived previously in Section 9.3 and is given by Ns

i

w~(.,x, n) = i=~1 2:rpsc2h/a _F~ cos nO I:** 133 e -jk.(x-x',) dkna,

for n = 0, 1,

(9.5.3)

where N, is the total number of control forces. In the following analysis we restrict ourselves to a maximum of two control forces, N, = 2. Input disturbance wave ~ W ° hs

Control force(s)

Error sensor(s) 7

Fluid

I I x=O

Fig. 9.5 System arrangement and coordinate system for active control of vibrations in fluidfilled cylinders.

289

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

The total displacement field in the shell system is, by superposition, the sum of the disturbance and control fields. Thus the complex radial displacements at the two error 1 Xe 2 for the case of two control forces is given by locations, x = Xe,

d WPs+ E f s,

W'=

(9.5.4)

where the vectors d, f s and w' and the matrix E are specified by -jknsxe T d = [e_J~sX~ e , 2] •

COS

1

•

dkna

133 e

f?~133 e -j~o~x~ x~, dkna

i_= 133 e

133 e

nO

E =

1

(9.5.5) 1

2

dkna

'

2Jrpsc2h/a

-

fs = [F~ w ' = [w'(x~)

(9.5.6)

F 2,T

s],

w'(x~)] T.

- y k . ( x e - x s)

~ ~

dkna

'

(9.5.7) (9.5.8)

Note that the subscript s is retained on the disturbance amplitude WPs and in the terms of vector d since the disturbance is that corresponding to a particular wave (n, s). On the other hand, the subscript s is suppressed in the integrals of matrix E as discussed in Section 9.3. For the following control strategy we choose to minimise the cost function defined by ue

J= Z I~Ztlwt(xie)12dO,

(9.5.9)

j=l

which is proportional to the out-of-plane vibrational energy per unit length of the shell at the error sensor locations. The cost function of equation (9.5.9) can be written in matrix form as J(fs) = g[fsH Afs + f~ b + bHfs+ c],

(9.5.10)

where the superscript H denotes the Hermitian transpose operator and e = 2 for n = 0 or e = 1 for n > 0. The vector b is given by b = [EHd]WPs.

(9.5.11)

A = EHE,

(9.5.12)

The Hermitian matrix A is given by

while the constant c is specified by the expression c = dHd]

WPs12.

(9.5.13)

The optimal control force vector that minimises the cost function has been previously derived in Chapter 4 and is given by

fso =

-A-lb.

(9.5.14)

In order to evaluate the performance of the active control, we now need to derive expressions for the total power flow in the shell-fluid system based upon previous work

290

ACTWECONTROLOF VIBRATION

of Fuller and Fahy (1982). We express the shell variables for a particular circumferential mode n as a series of complex wave solutions given by

u(x, O, t) =

Z

jwt-jknsx +jzt/2

U,, cos nO e

,

(9.5.15)

s=l jwt -jknsx

v(x, O, t) = Z Vr~ sin nO e

,

(9.5.16)

s=l oo

jcot-jk.sx w(x, O, t) = Z W,~ cos nO e

(9.5.17)

s=l

and the corresponding, coupled interior complex pressure as oo

p(x, r, O, t) =

Z

j~ot-jk.sx P,~ cos nO J,,(k~r) e .

(9.5.18)

s=l

Power flows in the axial direction in the fluid field, H I, and in the shell wall, H~, have already been derived in several previous references (e.g. Fuller and Fahy, 1982) and are given by oo

YL'

3,--x3

oo

i

1-If=2CL~,zpfeZZWnse

-jknsxT ~g

~ f

Wni[e -jk"ix] Fnsi,

(9.5.19)

s=l i - 1

where the fluid power factor Ffi is given by

1[ 1 ][

Ffi = -'~

kraJ,n(kra)

,ia

kraj,(Uia)

r ,

o J.(k~r)J n (k~r)r dr

(9.5.20)

and oo

i-is

oo

3 -jknsxT *r-jknix-,* f = Y('psCL~"~E Z Z wns e Wni[e J ansi, s=l i=l

(9.5.21)

where the shell power factor Sfnsiis given by

sfi = [(h/a)3/12][(k,~a)2(k.~a) * + vn2(k.ia) * + R~(k,~a)(knia)* + nRts(knia)*]

+[(h/a)/2]t(k,~a)R,,,R*. + vnRt~R,,* + vR,,*] + [ ( h / a ) / 4 ] ( 1 - v)[nRasR~ + kn~aRtsR~].

(9.5.22)

In equation (9.5.22), R~, and Rts are the ratios of axial to radial and axial to torsional amplitudes respectively, obtained by resubstituting the derived axial wavenumber k,, back into the shell equations of motion, stated in equations (2.12.3) and (2.12.4) including fluid loading if necessary, as discussed in Section 9.2. In equations (9.5.19) and (9.5.21), the radial displacement amplitude is found from the relation for the total response of the system given by equation (9.5.4) with the optimal control force value.

ACTIVE STRUCTURALACOUSTIC CONTROL. II CYLINDERSYSTEMS

291

The performance of the discontinuity caused by the active control system is then evaluated by use of the power transmission coefficient T~ defined as T~ =

Total transmitted power flow

(9.5.23)

Total incident power flow ( r I f + ]'-[ ~transmitted ~'SIX > X e (rIf.+.

(9.5.24)

l'-I ~incident xxslx

<xc

The power transmission loss is then defined as TL = 10 log~0(T~).

(9.5.25)

The above performance criteria assumes that the source strength is independent of the reflected wave which is reasonable for very long piping systems. For the following results we consider a steel shell of thickness h/a = 0.05 filled with water. Results are given for two cases of control; when one control force is applied and the radial shell displacement is minimised at one location, and when two control forces spaced a distance AXc/a=O.1 are used to minimise the radial displacement at two different axial locations closely spaced at a distance Axe/a = 0.05 apart. In the second case the two control forces have been spaced a distance of about 10 times smaller than the minimum wavelength at the highest frequency of interest (f~= 3.0). In this situation, the control inputs effectively introduce a line moment component into the shell system as well as a radial force. It was also desired to keep the control hardware reasonably compact and thus the spacing between the control forces and the error point was chosen as three radii. The inverse integrals of equation (9.5.6) were evaluated using the residue theorem as outlined in Section 2.14. This enabled the total system response to be broken down into contributions from each wave (n, s). The power flow relations of equations (9.5.19)-(9.5.22) could then be used. The infinite sums were truncated at eight roots which was found to provide sufficient accuracy in the solution. Material properties for the system are given in Table 9.2. Note that a hysteretic damping r/s = 0.02 has been added to the shell material, making E, the Young's modulus of elasticity, complex and equal to E' = E(1 - j r / s ) as discussed in Chapter 2. For the first case we consider an axisymmetric (n = 0) disturbance and control forces. Figure 9.6 presents the transmission loss when the incident wave is the branch denoted s = 2 in the notation of the work by Fuller and Fahy (1982), which behaves dominantly like a structural wave at low frequencies. For the case of one control force, even though the radial displacement is minimised at the error point, the active input can be seen to be ineffective in controlling total power flow in the system. The reason for this behaviour is Table 9.2 Materials

Steel Water

Material properties

Young's modulus E(N m -2)

Poisson's ratio

Density p (kg m -a)

Free wave speed cL, c: (m s -1)

Damping ratio

v

19.2 x 101° --

0.3 --

7800 1000

5200 1500

0.02 0.0

rls, fly

292

ACTIVE

CONTROL

OF VIBRATION

10 ~" "o v _J

o

5 |

°.,.

0 -5

tO

._~ -10 E

!."....... -. "

e--

o

:o,

~ -15 ;

"200~

0.5

1

1.i 5

2.L

215

Non-dimensional frequency, ~ Fig. 9.6 Transmission loss of the active control system shown in Fig. 9.5, s = 2 wave, n = 0: ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). that the fluid field leads to flanking of energy around the discontinuity provided by the control input. Using two control forces leads to attenuation of between 10 to 17 dB at low frequencies except around the frequency ~ =0.85 where the shell fluid systems are strongly coupled through coincidence behaviour. At very low frequencies, since there are only two waves cut-on, one would expect total attenuation of the power (TL---> -oo) with two control forces. However, due to the compact arrangement of the control transducers, near-fields of cut-off waves generated by the control inputs are observed by the error sensors. As discussed in Chapter 6 this leads to a reduction in the attenuation achieved. Design of the system is thus a compromise between the compactness of the transducers (in terms of their axial spacing) and desired reduction in power. At higher frequencies, I> 1.3, the incident, s = 2, wave becomes a dominantly fluid type wave and most of the energy propagates via the fluid path and the attenuations achieved are negligible. Figure 9.7 shows the transmission loss when the incident disturbance is the s = 1 wave (Fuller and Fahy, 1982) which has predominantly fluid wave type characteristics. As discussed by Fuller and Fahy (1982), this wave in the fluid field is subsonic and consists of a pressure near field located near the shell wall. However, Fuller and Fahy (1982) also demonstrate that below the ring frequency ~ = 1, the power flow of the s = 1 branch is dominantly in the fluid field. Applying one control force, Fig. 9.7 shows the surprising result that the power flow is almost uniformly reduced across the frequency range by around 10 dB. Due to the resonances of the coupled system associated with cut-on of higher order duct modes, an increased power flow occurs at a few discrete frequencies. This aspect of the results illustrates a potential drawback of active control. In contrast to passive control, active approaches can lead to an increase in power carried by the system when they act by introducing energy. When two control forces are used, higher attenuations of the order of 20 to 50 dB are obtained. The explanation for this good control performance is associated with the nature of this particular wave; the near field closely hugging the shell wall behaves like an attached mass loading and is thus strongly affected by structural forces. We now turn our attention to beam like (n = 1) motion of the system. Figure 9.8 shows the transmission loss with the s = 1 branch incident. As described in Fuller and

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

293

10 0 13 v

-10 o -20

i

tO .m (/)

i:', Y'..

u) -30 -i .i : E " (/)

: i

i" .........."

\:',1

" -40 -50 ~ 0

~

0.5

1

1.'5

2'.5

3

N0n-dimensi0nal frequency, .O

Fig. 9.7 Transmission loss of the active control system shown in Fig. 9.5, s = 1 wave, n = 0: ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). 20 ~" ...I

I0 tO

0 -20 -40

-60

~E -80

:'",.- -i

-100 -120~

0'5

'

'

'

'

1 1.5 2 2.5 Non-dimensional frequency, ~

3.0

Fig. 9.8 Transmission loss of the active control system shown in Fig. 9.5, s = 1 wave, n = 1" ~, one control force; - - - , two control forces (after Br6vart and Fuller, 1993). Fahy, (1982) this wave behaves like a slender beam at low frequencies where the presence of the enclosed fluid simply acts as an additional mass. The acoustic response consists of a forced near field located near the shell wall. At low frequencies, below the first acoustic wave cut-off frequency, high attenuations of the order of 10 to 60 dB are observed due to the dominantly structural motion of the system. The performance is again limited by the presence of control-generated near fields observed at the error sensors. However, above the first cut-off frequency, as higher order acoustic modes begin to propagate the control performance is severely decreased due to energy now propagating in the fluid path. Increasing the number of control forces to two leads to significantly better control performance particularly at low frequencies. At higher frequencies the attenuations are maintained at around 15 dB except near the cut-on frequencies of higher order modes. In this case it appears that the controller can reduce both the structural and fluid path simultaneously.

294

ACTIVE CONTROL OF VIBRATION

The preceding analysis demonstrates the potential for active control of total energy flow in fluid-filled piping systems. The results indicate that due to the coupled nature of the system it is possible in some situations to control the interior fluid power flow by applying structural inputs to the piping wall. The structural inputs effectively change the wall impedance seen by the fluid field making it more resistive and thus energy absorbing. This has obvious benefits in realistic applications such as ease of implementation and nonobtrusive (in the flow field) control hardware. It is apparent from the results that the controller needs to be of higher order, i.e. multiple input and output channels. This observation is undoubtedly related to the complex nature of the waves in fluid filled cylinders, in particular that there are many propagation paths and thus degrees of freedom in the system. In addition the performance would likely be improved by use of wave vector type sensors (as outlined in Section 5.10 and Chapter 6) which separate out positive travelling waves of a particular type and uses this information as error signals.

9.6

Active control of sound radiation from vibrating cylinders

The radiation of sound from cylindrical structures occurs in many important applications. Pertinent examples are the radiation of sound from submarine hulls, 'break out' of sound from circular air conditioning ducts and radiation of noise from piping systems. Radiation from cylinders excited by an internal flow has been studied previously by Holmer and Heymann (1980). Fuller (1986a) has analytically studied the related but idealised problem of sound radiation from an infinite cylinder excited by an internal monopole source. In this section, we study the active control of sound radiation from long cylindrical structures by radial forces applied to the wall of the structure. As an illustrative example we will consider an infinite cylindrical shell excited by an internal monopole source representative of, for example, a simplified model of an acoustic source due to a flow disturbance. Control is in the form of point forces applied radially to the shell wall. As most realistic sources are asymmetric we will locate the monopole source off the centreline at rp/a = 0.9 in order to induce asymmetry in the system response. As discussed by Kuhn and Morfey (1976) most realistic piping and duct systems carry higher order circumferential waves (n > 0) due to the source and duct asymmetries. The geometry and coordinate system used in the analysis is given in Fig. 9.1. Following the derivation presented in Section 9.1 the spectral response of a fluidfilled cylinder to a point force and an internal monopole respectively can be written in the wavenumber domain as

W~n(kn) =

a/33

p,c (h/a

F

(9.6.1)

and a Jn(krirp) W~(k,) = 2en133 p~ C2 L(h/a) (kria)J'n(k'~a) Po,

(9.6.2)

where rp is the location of the monopole and equations (9.6.1) and (9.6.2) hold for a particular circumferential mode n. All terms in the equations have been defined previously in Section 9.1.

295

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

By superposition the total spectral wavenumber displacement of the system consisting of the disturbance (primary) and control (secondary) forces is then given by t Wn(k,)

Us Z =

si Wn(kn) + WPn(kn),

(9.6.3)

i--1

where Ns is the number of control forces. The spectral radiated pressure is found by applying the boundary condition of continuity of radial displacement (similar to that carried out previously with an interior acoustic field). This leads to or-x2 2

ro

Pex(kn) = Pf~ CL Hn(krr) Wn(kn), a (kr°a)H'n(kr°a)

(9.6.4)

where r is the radial observation point. Applying an inverse Fourier transform to equation (9.6.4) gives the total radiated pressure in the spatial domain as p tex(r,

O, X) = i~1= ~a "4- --P~ £ a

cos n(O

COS ?'/(0-

n--o

foo

-

Oi)J_oo

02

nn(kr°r) -jk.x e dkna 2Z~ps(h/a) kr°an'n(kr°a) pf£) 133

}

-jknx Om) I?oo enpff2° 2133 .... Jn(k'-rir "p) nn(kr°F) e dk, a, erp~(h/a) (kria)J'n(kria) (kr°a)H'n(kr°a)

(9.6.5) where Oi, i = 1 to Ns and 0 m a r e the angular locations of the control force(s) and disturbance monopole respectively. Equation (9.6.5) assumes that the control forces, Fi, a r e applied in the source plane, x = 0. The disturbance source has amplitude p~. However the relationships could be easily modified to account for control forces out of the source plane. We now desire to minimise the sound field at a particular circular location in the far field. In the same manner as before, we form a cost function as an integral of the square of the total far-field pressure amplitude defined by J(fs) =

i IP'exl R dO, o 2p;c;

(9.6.6)

where P'e.~is evaluated at a particular axial location, x, and radial location, R. The cost function can again be written in matrix form as J = fsHAfs + where the control force vector fs = [F~

c

=

(

~-" IpPl2

fHb + bnfs + c,

F 2 . . . ]T

(9.6.7)

and b = p~0x. The scalar c is specified by

)

2nOm + --1 sinZnOm

e COS

n=0

E

I+°° 6nP;~'~2133

Jn(krirp)

'r_non(kr°?') -jk.a 12. _oo 2Ztps(h/a) (k ia)J'n(k~ria) (k ma)H~n(kr°a) e dkna

(9.6.8)

296

ACTIVE CONTROL OF VIBRATION

The vector x is of size N= x 1 and a typical element is given by

( [i_+:

1

m/

X~I = ~~o--~ e cos nOj cos nOm + --e sin nOj sin nO

x

x

pz~ 133 H.(kr°n) -ik.x 2~p=(h/a) (kr°a)g~(U°a) e dkna

gnpf~'~ 133 Jn(krirp) [I5 2zcp=(h/a) o: (kria)Jn(kria)

]"

Hn(kr°r) -jk.x ] e dk,,a . (kr°a)n'(kr°a)

(9.6.9)

The Hermitian matrix A is of size N, x Ns and a typical element is given by cos nOi cos n0j + --e sin nOi sin n

A/j = ~~o--~

o

j2.

2

I +00 PI ~ 133 Hn(kr°r) e jk,,x dk, a -00 4~ps(h/a) (kr°a)H'(kr°a)

x

(9.6.10)

In the above equations again e = 2 if n = 0, e = 1 if n ~ 0. The optimal force vector fso to minimize J is then given by

f=o = - A - ~ b .

(9.6.11)

The system used in generating the following example results consists of a steel shell of thickness h/a = 0.05 filled with water. Material properties are as used in Section 9.5. The monopole disturbance source is assumed to be located at rp/a = 0.9, 0 = 0 and the 90 ° 120 o ~

L3_O

~

60 ° ...,/

]o'~-~./

150 °

/'

0

"'~ ,,

..:'C"

"'"

~8o I'~',

..'"

.............

\

~ ,

i

!

"

" ...... /'.',

...............

.....

".',,

i,

"

",,

"~*

',

"

, ,,

:

-~

".. i ' ..-.>'.~ z .......... ,., ]"-,. ', i \ ,. . . . . . . . ,,, z ', ." ""--,~ ." ". . . . '~"~:. ""'),/' 'ix...... .i:i

'.,. ....."'.. o

"'""

"

"

\'... " ~

% '.. % "....

Z

...................

~ ~ "'"" " "....

/ -~(.

/i

24o ° ~

% ~

............... ~

".

"... ""-'"" ..'¢

.."

,J,l°

:

,.,................ ir"iSnk! i

0

"

: , L,!

...... i

3o°

"...,......" ....... ~,

)<¢ •.,I

. .......... ............

...... /" o

",,

~,'.~

............ /'

i

o

I I

i

i

........ ." "'..~"

.." I | I

... "-.....

"" .-"

""

o

~,.s ..~, .;:.-

,, 3oo °

270 ° Fig. 9.9 Radiation directivity as the result of the active control of sound radiation from a cylinder using point forces, x/a = 20, R/a = 20, f2 = 0.9: ~ , uncontrolled; - - - , controlled.

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

297

excitation frequency was ~ = 0 . 9 . The far-field pressure was minimised at a circumferential location of x/a= 20 and R/a = 20, i.e. well away from the source plane. The inverse Fourier transform integrals were evaluated numerically as detailed in work presented by Fuller (1986a). Parametric studies showed that the best location of the control forces was in the source plane and the control forces should be relatively closely spaced in order to exert some additional form of moment input. Figure 9.9 shows the uncontrolled and controlled radiation directivities at x/a = 20, R/a=20 with three control forces located at 0 s = 0 °, 37 °, 66 °. The uncontrolled directivity pattern exhibits a many lobed pattern in one half of the radiation field which is significantly distorted on the source side of the system. This indicates that a number of higher order modes are contributing significantly to the radiation field. When the control is applied, reductions of close to 10 dB are observed at nearly all angles and the radiation pattern is much smoother, indicating that the contributions of the higher order modes have been reduced. Figure 9.10 presents radiation directivities for the same case, evaluated at x/a = 30, R/a = 20, i.e. out of the plane of minimisation. The results indicate poor control except on the source side where around 5 dB of attenuation is achieved. The results indicate two major conclusions. Firstly, it is possible to control radiation from piping systems with control forces applied to the shell well away from points at which the error is minimised. The control forces in effect create a structural control field which propagates down the shell system and radiates to the minimisation plane in order to cause destructive interference. The best location for the control forces is in (or near) the source plane even if the minimisation plane is located well away from it. Secondly, the error sensors need to be located in the acoustic space where control of the sound field is desired. Due to the fact that radiation from the shell occurs as waves leaving at angles from the wall (see Junger and Feit, 1986, for a discussion of this phenomenon) 90 °

150

•

~-"....

,.,"

a'

,,, .....................

,:" "...... . ..'"

',,,

i

,."

/"

,,

30 °

", ,,.':.:-.

"",,, ....... "" "-

i' 8

80

o

~

~ , ,~,

-

.

',i

i

"

:

~

,:

!

~

!

,...:..:...,

,

,,

~,

'............................... . ............... /

.......... I ,,,~-,, .......... "'., .,.,, ..-'"'"" ""',. "-?". :"'ii.......11. ~

:,

"~:"....

~ ~

:,~1.-! .....

0

o

SPL

"- "'-~ t ...", : '",, :'.,-'" ,.""'"'"-.. .," ,:/

"---._..s_~__'.~

270

,

--'~'."" -.

"

.'"

""

o

°

Fig. 9.10 Radiation directivity as the result of the active control of sound radiation from a cylinder using point forces, x/a-30, R/a - 20, ~ = 0.9: - - , uncontrolled; - - - , controlled.

298

ACTIVE CONTROLOF VIBRATION

minimisation at a particular axial location does not guarantee control at other axial locations. In practice many cylinder systems are finite (e.g. the hull of a submarine) and Clark and Fuller (1993) as well as Ruckman and Fuller (1993) have respectively investigated experimentally and numerically the active control of radiation from such finite shell systems. In this case the cost function is usually the integral of acoustic intensity over a sphere surrounding the cylinder. The next section will consider active control of sound inside finite cylinders.

9.7

Active control of sound in finite cylinder systems

The previous sections in this chapter have all considered infinite cylinders for modelling the system of interest. In some applications it is more representative to model the structural and acoustic systems using a finite shell and enclosed volume. Examples occur in the study of active control of interior noise in small propeller aircraft or automobiles. Silcox and Lester (1989), Thomas (1992) and Thomas et al. (1993b) have all developed analytical models which use a finite shell section with rigid end caps, in order to study ASAC of the interior noise in aircraft. In general the procedure is similar to that described in Section 9.4 apart from a number of important differences. In order to find the coupled interior acoustic response, Silcox and Lester assume an interior modal solution of the complex form

pi(x,r, O)= Z Z P~Jm(krr)cos

cos nO

m=0n=0

q- Z Z eLJm(krf)cos

sin

nO,

(9.7.1)

m=0n=l

where kr =

I

k2 - ~

.

(9.7.2)

2

Note that equation (9.7.1) only expands the three-dimensional acoustic field into coefficients of a two-dimensional description of the pressure field at the shell wall. Thus P mnis the total response of all radial modes for a particular (m, n). The harmonic time component e i°'' has been omitted for convenience. The complex shell radial displacement w(x, 0), obtained from the response of the shell system to a disturbance due to an external pressure or control forces is then expanded using the acoustic mode shapes as basis functions such that

w(x,o= Z Z

Wren c o s

c o s r/O

W,,~ cos

sin nO.

m=On=O

+

ZZ

m=On=l

(9.7.3)

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

299

Using orthogonality of the basis functions, the modal amplitudes in the expansion are determined by o,o

~o =ZCm

forn

,m%,~

0,1 , 2, ....

-

(9.7.4)

m r-. 1

where the coupling coefficients Gin,m are specified by Cm

[A (-~ +A (+) m'm

tm

m'm

0

if

m',m,

if

m'=m,

(9.7.5)

where A(~:)= 1 [ 1 - c o s ( m ' + m ) z c ] m'n

m' + -m

~e

"

(9.7.6)

By applying the boundary condition of continuity of displacement at the wall, the pressure coefficients of equation (9.7.1) are now specified by 2

Pro,, =

(9.7.7)

PIO0 W,,,n. krJ'(kra)

The interior acoustic field for an input disturbance and a number of control forces can then be obtained by superposition which shows that (9.7.8)

P,=Pp+Ps,

where the control field Ps is a function of fs, the vector of control strengths. The quadratic cost function to be minimised is now defined as =

I0IoI?

r dr dO dx,

(9.7.9)

which is proportional to an integral of the acoustic energy density throughout the interior volume. Expressing this cost function as a quadratic function of the control force vector f~, as previously (see, for example, Section 9.5), enables the optimal control force vector fso to be determined in order to minimise J(f~). The optimal vector f~o is then resubstituted into equation (9.7.8) in order to evaluate the controlled or residual interior field as in previous sections. Point forces are used as control inputs. The response of the shell to the disturbance (primary) or control (secondary) forces is obtained by expanding the input function in terms of the shell response functions given by equation (9.7.3). The shell modal amplitudes WSn and Wm~nare then obtained by using the orthogonality property of the normal modes as described in a similar procedure to that outlined in Chapter 2 for the forced response of beams. For example, if the input pressure distribution is p ( x , O) then the expansion coefficients can be written as =

z _..---:

+Z Z m=ln=l

( )

sin ___

cos nO

'L

s s'nt

in.0 t

(9.7.10)

300

ACTIVE CONTROLOF VIBRATION

The modal coefficients are then obtained by use of orthogonality of the basis functions and are given by erL o

o P(X'O) sin

(mx) L

cos nO d x dO.

(9.7.11)

A similar relation can be derived for FSmn. Once the forcing pressure field has been expanded into the appropriate functions (i.e. the fight hand side of equation (9.7.10)), the functions can be substituted into the fight hand side of the radial equation of the shell equations which in this case are the Donnell-Mushtari relations (see equation (2.12.1c)). With the shell response now written in the appropriate form for a simply supported shell (see Section 2.13) the amplitude of the shell radial response can be solved by matrix inversion, which results in Wm~= (1-v2)a 2 (LllL22-L12L21) Fm~,

(9.7.12)

ILl

Eh

where equation (9.7.12) applies to both the cos nO and sin nO azimuthal distributions. The response of the shell and the coupled interior acoustic field can be obtained by using this procedure for any arbitrary forcing pressure field at the exterior shell wall surface (for example a pressure distribution obtained by measurement). The work by Thomas et al. (1993b) follows a similar procedure except that the interior acoustic field is found by using a Green's function for the interior field. An interesting aspect of Thomas's work is that restricted segments of the shell wall can be considered as sources, representative of real aircraft interiors. This aspect leads to coupling between non-identical order circumferential modes in the shell and acoustic space. On obtaining the shell response to the disturbance and control forces the corresponding interior acoustic fields can be evaluated as described above. The optimal control forces can be obtained by minimising J. For more explicit details the reader is referred to Silcox and Lester (1989) and Thomas et al. (1993b). The following example results are calculated for a simply supported aluminium cylinder which is 1.68 m in diameter, 3.66 m long and has a thickness of 1.7 mm. The

Pointforce

,/~ ~

.,~.,,~,~ x

/~//

Monopole

Fig. 9.11 Systemarrangement and coordinate system for the finite cylinder model.

ACTIVE STRUCTURALACOUSTIC CONTROL. II CYLINDERSYSTEMS

301

cylinder is closed with rigid end caps which are assumed not to contribute to the interior field. The disturbance is due to a harmonic monopole source located at cylindrical coordinates (rp, Xp, Op)= (1.5a, L/2, 0 °) where a and L are the radius and length of the cylinder. Due to the source location, only symmetric axial modes and cos n0 circumferential modes appear in the primary response function. Point force active inputs are applied to the shell wall in the source plane. The configuration is shown in Fig. 9.11. Figure 9.12 shows interior pressure distributions evaluated in the source plane, x = L/2, for the cases of uncontrolled and with two control forces located at 0 = 0 °, 180 ° also in the source plane. The excitation frequency is 100 Hz which closely corresponds to the resonance of the ( m = 1, n = 2 ) cylinder mode. Figure 9.12(a) presents the uncontrolled interior response and a modal breakdown reveals that two interior modes, the (m = 0, n = 2) and the (m = 2, n = 2) are contributing strongly to the response. Both of these modes are excited, since the axial distribution function, sin(nztx/L) couples into all symmetric acoustic modes, while the circumferential continuity dictates that there is a one to one coupling between azimuthal modes in the shell and acoustic space. Figure 9.12(b) presents the minimised pressure distribution using the two point forces as control inputs. The average sound pressure level reduction throughout the interior volume is 17.1 dB. The main effect of the controller is to attenuate the response of the (m = 1, n = 2) cylinder mode which leads to a corresponding decrease of the coupled sound field (i.e. what was previously referred to as modal suppression). However, the overall vibration levels of the cylinder with control is increased by approximately 5 dB due to control spillover into the (m = 2; n = 6, 8, 10, 12) higher order shell modes. As these modes are not well coupled to the acoustic field the spillover is constrained to the shell system and an overall sound attenuation is achieved in a similar manner to the results of the infinite cylinder discussed in Section 9.4. This phenomenon has been called modal restructuring and is similar to the behaviour of ASAC applied to plates that has been described in Section 8.7.

Sound pressure level (dB) -50

0

/ (a)

(b)

Fig. 9.12 Interior sound pressure levels using the active control system of Fig. 9.11 with two control forces, x = L/2, f= 100 Hz: (a) uncontrolled; (b) controlled (after Silcox and Lester, 1989).

302

ACTIVE CONTROLOF VIBRATION

The results demonstrate that ASAC is equally applicable to finite shell systems in addition to those that are very long and are better represented by infinite systems. For more detail on ASAC in finite systems the reader is referred to the works of Silcox and Lester (1989) and Thomas et al. (1993b).

9.8

Control of interior noise in a full scale jet aircraft fuselage

The previous sections have discussed ASAC applied to simplified systems represented by homogeneous cylinder structures. In this section we discuss a practical example in which the ASAC technique is applied to a DC-9 fuselage in order to reduce interior noise. This work was performed by Simpson et al. (1991). Figure 9.13 shows a diagrammatic representation of a plan view of the rear of the DC-9 test fuselage. In the experimental tests reported here, the fuselage was fully furnished with interior trim, seats etc. Figure 9.13 also shows the location of microphones which were arranged to be at head height for the tests. Selected microphones were used as error sensors for the adaptive controller. A steady sinusoidal disturbance representative of engine out-of-balance forces was applied to either of the fore or aft engine pylons by an external shaker mounted at an angle to excite multiple degrees of motion. Up to two control forces were applied using internally mounted shakers attached to fuselage frames. The forward control shaker was placed near the front pylon attachment point and attached to a longeron at the same axial location as the disturbance shaker. The aft control shaker was opposite the rear engine mount, behind the pressure bulkhead. The control shakers acted inertially and were applied at an angle in order to clear structural fuselage components, as shown in Fig. 9.13.

)ntrol shaker ,ourceshaker d controlshaker wardsourceshaker 20 ,ee, Microphone'~ I ~ ~ 0 locations L t0 ~ L~ ~ ~ ~

0 I

iO,,O

U ~ ~1

Accelerometer locations (A, B, D, E, at Iongeron10, Catlongeron15)

Vvo

Fig. 9.13 Test arrangement for active control of aircraft interior noise using point forces (after Simpson et al., 1991).

ACTIVE STRUCTURALACOUSTICCONTROL.II CYLINDERSYSTEMS

303

A schematic of the control system and the experimental set up is shown in Fig. 9.14. The control implemented was based upon a frequency domain based least mean squares algorithm with a gradient descent minimisation technique of the type outlined in Chapter 4. Further details are given in the work by Simpson et al. (1991). Figure 9.15 shows the uncontrolled and controlled sound pressure levels when the excitation was at the forward pylon point and one error sensor was used (microphone number 2). The excitation frequency was 170 Hz. The results show high attenuation at the error microphone while increased levels (control spillover) are observed at microphone position numbers 3, 4 and 7. Increasing the number of error microphones to four, such that some are located at regions where levels increased, tends to reduce the control spillover while maintaining 9.1 dB of averaged sound reduction. The use of multiple error microphones leads to a more smoothed residual pressure field in the cabin. For comparison the single microphone error sensor was replaced with a single accelerometer located immediately adjacent to the forward control point (accelerometer C). Figure 9.16 presents the uncontrolled and controlled pressure distributions similar to the previous test except now with the use of a structural error sensor. The results show that the global reduction has been reduced to 4.8 dB. A measurement using five accelerometers located on the fuselage near the pylon attachment point show averaged reductions of 13.3 dB over roughly a 2 m diameter area. This result highlights the main aspect of ASAC, in that minimisation of vibration at selected points does not necessarily guarantee reduction (in this case largest reduction) of radiated sound. In fact minimising local vibration often leads to i n c r e a s e d sound levels. For interior noise reductions, the error sensors need to observe the interior sound pressure levels, even though the noise is structure-borne. In the final test the excitation was maintained at 170 Hz and moved to the rear mount point. The rear mount point is characterised by a lack of carry through structure to the fuselage from the engine attachment and thus the excitation is far more distributed than the forward pylon point. Therefore in order to obtain reasonable global reductions it was necessary to use two control shakers in conjunction with four error sensors as ~ Noise signal

Error rl microphones f Noise signal II Q (3 [Oscillator t--- I -[ i Case (,21) .... ~ " I ! IL ,-.--. control "~,, ~..-z,""! Controller I I Case (2)",--IA/DI-- ~ and IIC°ntr°l "-J~ ~ T ~ data I I signal J[~ |

I,I,, Amp ,11 ~-L

. LIP::::::I[I-II Ihaker Vibration source Engine pylon -x

<

//acqu'siti°n

Error

Irl IT-I accelerometers

II i"-11 i~

,.. , A ' IJ"cont;ol t-use~age- / shaker

I arnp [~--I , , ! amp I~

--~Laboratorv eauiDment -" -' '-

Fig. 9.14 Control system and test equipment arrangement for the system of Fig. 9.13 (after Simpson et al., 1991).

304

ACTIVE CONTROL OF VIBRATION r-n Source levels Minimised levels

70

Noise

m "O

reduction

60

o

50

if)

~

40

/., ,.-I i., i..

30

is i.,

Q.

~

;g

20 rs .-s

0

co

t6 !

~

c !

C '

•

O

n

0

10

Js f~ /.,

1

;g

7

f J l J i`" f J f J

2*

f.. / J

I.,.,

/ J f.. / J I I

3

f`" /`" f`" /`" i.. f`" /`"

.~j f`" f`" f j f j / J .~j

/.,~ i~. /`" ....~ i`" .~`" /`" f.~

/`" / J f J f J / J f J / J f j

I`" /`" /.. i..

/ J / J / J f /

i..

/..

i.. /.. i.. /.. i f /~. i f i ~

/ / / J / / / J / / / / /`" / /

f J f J

O u")

3

4

O

/ J / J i J

O') cxl

.-..,I / . ,

,,-.,,I ,.-A f A ...,,I

....,I

5

Source, 170 Hz

`"J I J i J i i i i

r..., i... i.,

Control

6

/ J i i i.. i J i J i.. i / i i i / i i i / i J f / / i i J u

7

Microphone number

Fig. 9.15 Performance for the active control system of Fig. 9.13, forward pylon excitation, single microphone error sensor (* denotes the error sensor), f = 170 Hz (after Simpson et al., 1991). Source levels Minimised levels

70

_ o~

~" 60

"~~

m

-~

• co

~

50

30

~t.-- 20 0

co 10

-

i

m

5

u£,

-

t-.

-

l J

f j

i`" i.. f J

i.. i..

f J f J

r.,

f.,,4 f ~

.-.,,i

f J f J ,-j

2

f J

.-j ,..,

f J f J f J

,-~

i..

r., ,-., .-..

f j f j f j

3

4

O4 CO I.,,4 f.,,t

f J

f / / / / / / / / / / / / / / / / / / /

1

uO

reduction •

/ / f J / J f / /

40

Noise

/`" F J

/.. / J

5

1

6

f.,,,t f.,,4 1.,,'I f.,,4 f.,,4 f.,,I f..4

f ~ f.,'l f .,,,I I".,,4 1.,,4

f..,I

7

Microphone number

Fig. 9.16 Performance for the active system of Fig. 9.13, forward pylon excitation, single accelerometer error sensor ( * denotes the error sensor), f = 170 Hz (after Simpson et al., 1991). shown in Fig. 9.17 (microphone numbers 2, 3, 6 and 7). The results of the tests also given in Fig. 9.17 demonstrate an overall reduction of the order of 10 dB. In this case it was believed that two control forces were necessary due to the structure-bome sound travelling to the interior by two paths; one transmitting into the rear acoustic space and then through the rear pressure bulkhead, the other travelling down the fuselage structure and then radiating to the interior. The experiments demonstrate that ASAC works effectively in realistic structures. In particular the advantage was shown of using error transducers which sense and

ACTIVE STRUCTURAL ACOUSTIC CONTROL. II CYLINDER SYSTEMS

305

r-n Source levels

Minimised levels

70 m -~

60 50

Noise reduction

> t'O

m

40

--e5 cO

~

d

I

f-, fl,

m

"o

30

~

r,.: I

20

-~

/11

:'~

.-I,

/ -

/ J i J t J /.. f J

f ~ f -

,.-j f J f J r l

f A

i.,,,

O3, OO i s / J f J

f ~ i.,,,

Control Source, 170 Hz Control

..-j r l

it,

//i

1

2*

3*

4

r.,

r..

f A / / i

5

-j.

r l

6*

7*

Microphone number

Fig. 9.17 Performance for the active control system of Fig. 9.13, aft pylon excitation, four microphone error sensors (* denotes the error sensor), two control forces, f = 170 Hz (after Simpson et al., 1991). minimise the interior acoustic field rather than structural error sensors. This arrangement is a key aspect of ASAC. More recent work by Fuller et al. (1990d) and Silcox et al. (1992) has demonstrated that the point force actuators can be successfully replaced with surface mounted, distributed, ceramic piezoelectric actuators. These forms of actuators have been discussed in Chapter 5.

Appendix. State Variables

A.1

Introduction

The state variable philosophy can be expressed in terms of the block diagram illustrated in Fig. 3.5, in which the state variables are described by the equations :~(t) = Ax(t) + Bu(t)

(A.I.1)

y(t) = Cx(t) + Du(t).

(A.1.2)

and

These equations are represented diagrammatically in Fig. A.1. We will assume that the system being described is linear and time-invariant so that A, B, C and D are matrices of constant coefficients. Some of the consequences of this formulation are, very briefly, set out below. For a more detailed treatment the reader is referred to the books by Kailath (1980), Richards (1979) and Meirovitch (1990), for example. The state equations may also be cast in discrete time instead of continuous time, as described, for example, by Astrrm and Wittenmark (1984). o

A.2

General solution to the state variable equation

Rather than directly manipulate the time domain equations (A.I.1) and (A.1.2), it is convenient to take their Laplace transforms, which yields s x ( s ) - x(0)= Ax(s) + Bu(s),

(A.2.1)

y(s) = Cx(s) + Du (s),

(A.2.2)

where x(s) is the Laplace transform of the vector x(t), and x(0) is the vector of initial values of the state variables. These equations can be solved algebraically to give y(s) = C[sl - A]-~x(0) + [C[sl - A]-~B + D]u(s)

(A.2.3)

in which the first term on the fight hand side of equation (A.2.3) corresponds to the free response of the system, and the second to the forced response. For zero initial conditions, H ( s ) = C [ s I - A ] - ~ B + D is clearly the matrix of transfer functions relating the outputs to the inputs of the system. An advantage of the state variable

308

ACTIVE CONTROLOF VIBRATION

•

u(t)_~ Input

|

B

Xo I Initial states

j" dt

~.,~(t) C

put

A

I°l ,,

Fig. A.1 Block diagram of the state variable equations.

formulation is that provided the matrix A can be identified, a common set of manipulations can be performed to compute the response of the system. If the numerical values of the elements of A, B, C and D can be deduced, the matrix of frequency response functions, for example, can be easily calculated by computer for various values of co using the equation H ( j w ) = C [ j w l - A]-IB + D

(A.2.4)

so that if u(w) and y(to) are the Fourier transforms of the vector of inputs and outputs of the system, these can be related via the equation y(w) = H(jw)u(w).

A.3

(A.2.5)

The transient response

Inverse Laplace transformation of equation (A.2.3) can be used to obtain the time history of the output, given the input time history and the initial conditions: t

y(t) = CO(t)x(0) + C

I O ( t - r)Bu(r) dr + Du(t).

(A.3.1)

0

where the matrix ~ ( t ) is the inverse Laplace transform of the matrix [ s l - A] -1. The matrix of impulse responses relating output waveforms to input waveforms can be deduced from the final terms in equation (A.3.1) as being h (t) = C ~ (t)B + D. Note that the unforced, natural response of the

states

k(t) = Ax(t).

(A.3.2) is given by (A.3.3)

By taking the Laplace transform, gathering terms and taking the inverse Laplace transform, as was done above, the time histories of the natural response of the states can be expressed in the form x(t) = • (t) x (0).

(A.3.4)

309

APPENDIX. STATE VARIABLES

The matrix • (t) is thus called the state transition matrix. The state transition matrix can be written in the form ~(t) = 57- l [ s l _ A]_I = 5e_l(adj(slA))[sI_A] '

(A.3.5)

where 5C-~ denotes inverse Laplace transformation and adj ( s I - A) denotes the adjoint of the matrix ( s I - A ) . The determinant [ s I - A I can be expanded to give the characteristic polynomial of A, which can be factorised and written as [ s l - A [ = (s - / ~ l ) ( S - / ~ 2 ) ... (s - ~n) "~ 0,

(A.3.6)

where/1,/are the eigenvalues of A (which is assumed to be of order n and full rank). Each term of • (t) will thus be of the form dPo(t) =

~_,

~(s)

,

(A.3.7)

(S -- 21)(S --/~2)...(S --/~n)

where fij(s) is the relevant term in adj ( s l - A). Equation (A.3.7) can itself be expanded by partial fractions to give q)o(t)_~-i

aij~ +

s-2~

a°2

+...

s-,t2

,

(A.3.8)

s-L,

where each aijk is a constant. This expression can be inverse Laplace transformed by inspection to give (Pij(t) = aijl e;tlt + aij2ea2t + ... aijn e'~"t.

(A.3.9)

Since x ( t ) = ~(t)x(0), for some arbitrary initial conditions x(0), the unforced solution for any state can be written xi(t) = eile~lt + ei2e~2t + ... eine ~,,t,

(A.3.10)

where, for example, eil

=

£ aolxi(O).

(A.3.11)

j=l

The system will only have a decaying natural response, i.e. be stable, if each 2;has a negative real part.

A.4

Transformation of variables

The full solution for the natural response of the states, equation (A.3.10), i.e. the solution to x ( t ) = • (t)x(0), may be written as x(t) = el e~l' + e2e~2t + ... + ene~.'

(A.4.1)

in which the vector ej = [e~j ezj.., enj] v. Substituting each term into equation (A.3.3) gives 2~ei = Ae~

so that

( 2 i i - A)e~ = 0.

(A.4.2a,b)

310

ACTIVE CONTROL OF VIBRATION

From equation (A4.2) it is clear that the vectors e i are the eigenvectors of A. Equation (A.4.2a) is valid for i from 1 to n, i.e. le2 = ~,2e2 =

Ae 1 Ae2 (A.4.3)

°

2 . e . = Ae., which may all be written in matrix form as (A.4.4)

EA = AE, where A=

22

(A.4.5)

°1

2, and

(A.4.6)

E = [ele2...e.].

The system matrix can thus be written in 'normal' form as A = E A E -1

(A.4.7)

and the diagonal matrix of eigenvalues is equal to A = E-1AE.

(A.4.8)

It should be noted that the choice of state variables for any problem is not unique. For the two-degree-of-freedom system considered in Section 3.7, for example, we could have chosen the state variables to be x ' = [21 21 x2 22] T for example, which are related to the original state variables, x = [x 2 0 0 ]T, by °

12

x

0

ll

11 + 12

ll + 12

0

0 11+/2

-1

0

Xl

11+12

21

(A.4.9)

x2

+1

ll + 12

11+ 12 0

ll

-1

0

ll + 12

+1

11 + 12

or, generally, x(t) = Tx' (t).

(A.4.10)

Substituting this into the general state variable equation (A.1.1) we obtain T~:' (t) = ATx' (t) + Bu (t)

(A.4.11)

APPENDIX. STATE VARIABLES

311

so that ~:' (t)= T-~ATx' (t) + T-~Bu (t).

(A.4.12)

The eigenvalues of the system matrix applicable to this new state variable equation are given by the roots of ] 2 I - T-1ATI =0.

(A.4.13)

We note that 2 I - T-~AT is equal to T-~(2I - A)T so that equation (A.4.13) can be written as I T-~ ]l 21 - A [I T I = 0.

(A.4.14)

Provided T is not singular, ITI and I T- I will be finite, and equation (A.4.14)thus reduces to equation (A.3.6). The eigenvalues of the system matrix, and hence the poles of the transfer function matrix, are thus unaffected by a linear transformation of state variables.

A.5

Modal coordinates

Consider the new state vector, z, obtained by setting the transformation matrix, T, equal to the matrix of eigenvectors, E, so that from equation (A.4.10), we have x(t) = Ez(t).

(A.5.1)

The new state variable equation (A.I.1) then becomes E~(t) = AEz(t) + Bu(t)

(A.5.2)

~(t) = E-~AEz(t) + E-~Bu(t)

(A.5.3)

~,(t) = Az(t) + Ru(t),

(A.5.4)

so that

or, using equation (A.4.7),

where A is the diagonal matrix of eigenvalues and R = E-lB. It is clear that because A is diagonal, the unforced response of each element of z is independent or uncoupled from all the others and may be written

Zi(t) =/],iZi(t)

(A.5.5)

zi( t) = zi(O ) e ~,'.

(A.5.6)

so that

The coordinates z(t) are referred to as the modal coordinates of the system. The state equations of the system (A.I.1) and (A.1.2) can now be written in terms of the modal coordinates as ~(t) = Az(t) + Ru(t), where R = E-~B

(A.5.7)

y(t) = Sz(t) + Du(t), where S = CE.

(A.5.8)

and

312

ACTIVE CONTROLOF VIBRATION

If all the elements of any row of R are zero, then there will be an element of z which is unaffected by any of the inputs, u, so there will be state variables which cannot be controlled by the inputs and such state variables are said to be not controllable. Controllability is assured if B is square and non-singular. Similarly, if all the elements of any column of S are zero, then there will be elements of z which do not affect any element of the output vector y, and there are state variables which are said to be not observable. Observability is ensured if C is square and non-singular.

Index

Acceleration feedback 65 Acceleration feedback on a beam 179 Acoustic field, internal 277 Acoustic field radiated from cylinders 278 Acoustic power, broad band 268 Active control of enclosed sound fields 282 Active control of interior noise in a jet aircraft 302 Active control of radiated sound from a plate 254 Active control of sound inside finite cylinders 298 Active control of sound radiation due to transient vibration of a beam 274 Active control of sound radiation due to turbulent boundary layer excitation of a plate 276 Active control of sound radiation from cylinders 294 Active control of sound transmission through a plate 251 Active control of vibrations in piping systems 287 Active control of wave transmission 154 Active engine mounts 202 Active property tuning (APT) 149 Active strain energy tuning (ASET) 149 Active structural acoustic control (ASAC) 223 Active structural acoustic control, general formulation 240 Active suspension 185, 221 Actuator arrays 168 Actuator, fully-active vs semi-active 59 Adaptive controller for aircraft application 303 Adaptive controllers 103, 107 Adaptive structure 152,262 Aircraft trim panel 267 Amplitude 2 Analogue wave vector filter 146 Angular velocity 34

Anti-symmetric piezoelectric one-dimensional actuator 127 Anti-symmetric piezoelectric two-dimensional actuator 131 ASAC applied to a plate system 265 Asymmetric piezoelectric one-dimensional actuator 120 Asymptotic and global optimality 84 Axial motion of a cylinder 48 Axial waves 27 Beam response to a point force 41 Bending moment, beam 34 Bending moment, plate 47 Bending stiffness 190 Bending waves 30 Bessel function 191,278 Block diagram 14 Block diagram of state variable equation 308 Blocked force, piezoelectric stack actuator 120 Bounded-input, bounded-output stability 63 Branch number 278 Broad band sound power reduction 260 Causal filter 270 Cell of vibration 227 Cell radiation interference 227 Characteristic equation 18, 62, 71 Charge output of piezoelectric sensor 139,262 Circular saw vibration control 67 Circumferential modal order 278 Circumferential mode shapes for a cylinder 49 Clamped boundary condition 36 Closed loop transfer function 61 Coherence function 212 Coherence limit on attenuation 97 Complex description of vibration 3 Continuity of displacement 278

328

INDEX

Control effort 268 Control effort weighting 276 Control spillover 41 Control spillover in cylinders 286 Control vector 249 Controllability 312 Controlled eigenfunctions 259 Convolution integral 12 Corner monopole sources 229 Cost function 83,104, 109, 217,262 Cost function for an enclosed field 284 Cost function for external cylinder radiation 295 Cost function for interior field in a finite cylinder 299 Cost function for piping systems 289 Cost function for radiation control 246, 247 Cost function minimum 251 Cost function with control effort 268 Critical damping 7 Critical frequency 228 Critically damped system 7 Curved piezoelectric actuators 137 Cylinder coupling coefficients for the shellinterior field 299 Cylinder interior acoustic field 277 Cylinder response to a point force 57 Cylinder response to an intemal monopole 294 Cylinders 48 Cylindrical coordinate system 49 Damped natural frequency 7, 21 Damped motion of SDOF systems 6, 20 Damping 42 Damping matrix 21 Damping ratio 7, 21, 188,207 Decaying response 309 Degrees of freedom 1 Delay in the control loop 66 Delays due to digital controller 66 Design of piezoelectric structural sensors 264 Diagonal nodal lines 278 Digital feedforward control 97 Dipole 229 Dirac delta functionl3 Dispersion limits on attenuation 177 Dispersive waves 27 Displacement feedback 65 Displacement feedback on a beam 181 Displacement transmissibility 214 Distributed actuators for control of interior noise 305 Donnell-Mushtari shell equations 48 Drive point mobility for a cylinder 57 Drive point mobility, definition 57

Dynamic effects of piezoelectric actuators 137 Dynamic matrix 17 Edge scattering 235 Effective modulus of elasticity 29 Effort weighting 83, 104, 109 Eigenassignment control design approach 265 Eigenvalues 18,309 Eigenvalues of mass and stiffness matrices 17, 75 Eigenvalues of state-space system matrix 71, 76, 309 Eigenvectors 18, 310 Elastomer 196 Electrodynamic and electromagnetic actuators 115,202 Electrorheological (ER) fluid 152 Elemental source 224 Engine pylon for a jet engine 302 ER semi-active actuator 152 Estimation of modal amplitude 142 Euler-Bernoulli thin beam theory 30 Evanescent flexural waves 32 Expansion coefficients for external forcing function of cylinders 299 Extensional waves 27 Exterior monopole source 281 Far-field acoustic intensity for plate radiation 226 Fast Fourier transform (FFT) 232 Feedback control 59,206 Feedback control of sound radiation from structures 267, 272 Feedback interpretation of feedforward control 111 Feedforward control 91 Filtered-x LMS algorithm 100 Finite cylinders 52 Finite difference method 145 Finite element analysis 21 Finite element model 199 Finite piezoelectric actuator 123, 128, 134 Finite thin beam 36 FIR filters 98 Flanking of energy around control system in pipes 292 Flanking of energy in piping systems 287 Flexible receiving structures 190 Flexural wavelength 192 Flexural wavenumber 31 Flexural waves 30,190 Flexural wave control, feedback 178 Flexural wave control, feedforward 174

INDEX Flfigge shell equations 50 Fluid loading term 279 Fluid power factor 290 Force transmissibility 207 Forced response of MDOF systems 20 Forced response of SDOF systems 9 Forced vibrations 2 Fourier transform, spatial 33, 53,143,230 Fourier transform, spatial, discrete form 144 Free boundary condition 36 Free motion of MDOF systems 16 Free motion of SDOF systems 5 Free vibration 2 Frequency 2 Frequency response function (FRF) 39 Fully active actuator 59, 147 Gauss-Newton algorithm 106 General solution to the state variable equation 307 General state variable equation 307 General termination impedance 37 Generalised coordinate vector 19 Generalised force 20 Geophone 215 Global active control 153 Global minimisation of beam response 156 Hankel function 190, 278 Harmonic motion 2 Heave motion of two-degree-of-freedom system 72 Heaviside step function 123 Helicopter vibration 195,204 Helmholtz equation 230 Hermitian matrix 250 Hermitian quadratic form 195,249 Holography, generalised acoustic 232 Hydraulic actuator 196, 205, 221 Hysteretic damping 42,285 Hysteretic damping for piping systems 291 IIR filters 101 Impedance of a piezoelectric actuator 137 Impulse 11 Impulse-momentum relation 11 In-plane motion 27 In-plane motion of a cylinder 48 Independent modal-space control (IMSC) 87 Inertial actuators 186, 193 Infinite baffle 241 Influence of sensor position and number on control performance 247

329

Input impedance 10 Input mobility 191 Input power reduction 193 Instantaneous flexural power flow 35 Instantaneous power, definition 34 Intelligent structure 152, 262 Interior monopole source 280 Interior pressure field 282 Interior pressure field for finite cylinders 298 Isolation efficiency 200, 201 Kalman filter 83, 85 Kinetic energy of a plate 155 Kronecker delta function 19 Laplace transform 13,206, 307 Laplace variable, s 270 Light damping 7 Line force 53 Line force in cylindrical coordinates 288 Line moment 35 LMS algorithm 100 Local minimisation of beam response 158 Longitudinal phase speed 27 Longitudinal waves 27 LQG control 85 LQG regulator 216 Luenberger observer 82 Magnetostrictive material 137 Mass matrix 16 Material-geometric constant, flexural excitation 137 Material-geometric constant, longitudinal excitation 122 Matrix Riccati equation 217 MDOF systems 15,193 Measurement noise in feedforward control 95 Mechanical system 1 Method of residues 34, 55, 57 Minimum power output 191 Modal control 87, 161 Modal coordinates 311 Modal decomposition of cylinder vibration 285 Modal displacement vector 19 Modal estimation 141 Modal matrix 19 Modal overlap 45 Modal radiation efficiency 238 Modal restructuring 259 Modal sensor, one-dimensional 139 Modal sensor, two-dimensional 141

330 Modal suppression 257 Mode normalisation 18 Mode normalisation constant 40 Mode shape 18 Mode shapes, simply supported beam 38 Mode shapes, simply supported plate 45 Mode shapes, SMA composite plate 151 Model state-space co-ordinates 311 Modes of feedforward control system 106 Moment of inertia, beam 31 Monopole 229 Multi-degree-of-freedom systems 15, 193 Multi-layer piezoelectric actuators 137 Multi-modal response 236 Multi-channel feedback control 76 Multiple coherence function 212 Multiple Error LMS algorithm 110, 204 Mutual radiation efficiencies 238 Mutual radiation resistance 237 Natural frequency 3, 5, 188,207 Natural response of states 308 Near-field effects on attenuation of flexural waves 176 Near-field flexural waves 32 Nitinol 147 Non-dimensional drive point mobility 57 Non-dimensional frequency, cylinder 50, 279 Non-dispersive waves 27 Normal matrix 310 Nyquist spatial sampling criterion 143 Nyquist stability criterion 63 Observability 312 Open loop transfer function 61 Optimal control 83 Optimal control vector 251 Optimal secondary force 191 Optimisation of sensors and actuators 264 Orthogonality 18 Orthogonality of mode shapes 40 Out-of-plane motion of a cylinder 48 Output feedback 76 Overdamped system 7 Parallel secondary actuator 189 Parallel stack actuator 119 Passive isolation systems 189 Patch, piezoelectric 131 Perfectly determined control system 247 Performance index 217 Period 1

INDEX Phase angle 2 Phase speed or velocity 27 Phasor 3 PID control 65 Piezoelectric axes 116 Piezoelectric ceramic (PZT) actuator, typical properties 126 Piezoelectric constitutive equations 117 Piezoelectric distributed error sensor, use in ASAC 261 Piezoelectric effect 116 Piezoelectric material, definitions 116 Piezoelectric stack actuator 118 Piezoelectric strain constant 117 Piezoelectric stress constant 139 Piezoelectric unconstrained strain 117, 118, 119 Piping systems, noise 277 Piping transmission paths 287 Piston, baffled 267 Piston state space model 270 Pitch motion of two-degree-of-freedom system 72 Plate equation, moment form 47 Plate input disturbance, oblique plane wave 243 Plate input disturbance, piezoelectric actuator 244 Plate input disturbance, point force 244 Plate input disturbance, uniform pressure 243 Plate modal radiation efficiency 229 Plate response to a line moment 48 Plate response to a point force 47 Plate vibrations 190 Pneumatic isolator 193 Point force 41 Point force, cylindrical coordinates 57 Point force excitation of a cylinder 279 Pole-zero plot 270 Poles 34, 55 Poling of piezoelectric material 116 Polyvinylidene fluoride (PVDF) 140, 216 Power flow in fluid of piping system 290 Power flow in shell of piping system 290 Power, longitudinal waves 29 Power transmission loss for piping systems with active control 291 Pressure bulkhead 304 Preview control 212 Proportional damping 21 Q operator 98 Quadratic form 191 Quadrupole 229 Quasi longitudinal motion 29

INDEX Radial motion of a cylinder 48 Radial wavenumber for interior field 298 Radial boundary condition at the wall 278 Radiation efficiency 227 Radiation control strategy 274 Radiation filters 267,268,273 Radiation filter transfer function 269,270 Radiation from planar surfaces 224 Radiation filter frequency response function 269 Radiation control for structures 223 Radiation circle 234 Radiation constant 245 Radiation modes 239, 264 Rational spectrum 269 Rayleigh integral 224, 225 Receptance 186 Receptance matrix 195 Recovery stress of Nitinol 148 Reference signal 91 Regulator 83 Relative stability 64 Residual or unmodelled modes 89 Resonance frequency 5 Resonance frequencies of cylinders 52 Resonance frequencies of plates 45 Resonance frequencies of beams 38 Riccati equation 84, 217 Ring frequency of cylinders 50 RMS amplitude 2 Robustness 64 S-domain 13 SDOF system 4, 186, 206 Seismic vibrations 213 Self radiation efficiencies 238 Self radiation resistance 237 Semi-active actuator 59, 147 Sensor arrays 141, 172 Sensor, point 138 Sensor shape function 139 Sensor, shape memory alloy 152 Sensor, piezoelectric 138 Sensor, wavenumber 143 Separation principle 82, 85 Series stack actuator 119 Servo system 83 Shape memory alloy(SMA) 147 Shape memory effect 147 Shaped actuators for cylinder control 287 Shear diaphragm boundary condition 52 Shell power factor 290 Sifting property 13 Single-degree-of-freedom systems 4, 186,206 Simply supported boundary condition 36

3 31

Simply supported boundary condition for cylinders 52 Skyhook damper 213,219 SMA adaptive structure 149 SMA fully active actuator 152 SMA hybrid composite 149 Smart structure 152,262 Sound power radiation for a plate 232 Sound power radiation from planar structures 236 Sound power transmission loss 253 Sound radiation from an SMA composite plate 151 Spatial aliasing 143 Spectral factorisation 270 Spectral displacement 33 Spectral wavenumber components 143 Spectrally white 43 Spillover 89 Spillover in finite cylinder systems 301 Spinning modes 50 Stability of feedback systems 62 Stable filter 270 Stable system 309 State estimation 80 State feedback 78 State space form 267 State transition matrix 309 State variables 68,307 States 308 Static approach for piezoelectric actuator analysis 118 Stiffness factor for a cylinder 48 Stiffness matrix 16 Strain (potential) energy of a plate 155 Structural trace wavelengths 227 Structural wavenumber 227 Subsonic wavenumbers 234 Superposition principle 191, 195 Supersonic wavenumbers 234 Suspension systems 86 Symmetric forcing function for a beam 41 System matrix 70, 307, 310 System response 1 Telescope vibrations 213 Thin plate 43 Time averaged flexural power flow 35 Time averaged power, definition 29 Total loss factor 42 Trace wavenumber components 44 Transducers 115 Transfer function 14 Transfer functions, matrix 307 Transformation of variables 309

332

INDEX

Transient excitation 273 Transient motion 3 Transient response in state variable description 308 Transient response of SDOF systems 11 Transient response of state variable equations 308 Transmissibility 206 Transverse shear force of a beam 34 Transverse shear force of a plate 47 Transverse waves 30 Turbofan engine vibrations 206 Turbulent boundary layer disturbance 267 Two-degree-of-freedom system 72

Uncoupled motion of two-degree-of-freedom system 74 Uncoupled response 311 Underdamped 7 Underdetermined control system 248 Uniform forcing function for a beam 41 Unit impulse response function 12 Velocity feedback 65

Velocity feedback on a beam 181 Vibration control strategy 274 Vibration isolation 185 Vibration power input 191 Viscous damping 6 Volume velocity structural sensor 264 Wafer piezoelectric element 120 Wave motion 25 Wave propagation, general form 26 Wave speed 26 Wave transmission control 165 Wave vector filter 144 Wave vector filter, broad band 147 Wavenumber 27 Wavenumber control approach 260 Wavenumber estimation 143 Wavenumber Fourier transform 33, 53, 143, 230 Wavenumber of an internal acoustic field 277 Wavenumber plane 56 Wavenumber spectrum 254 Wavenumber spectrum of a vibrating plate 234 Wiener filter 212

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