Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications

2 dL dcp

9

+ L dtdt + L

0,

(2.5.1)

where L(t) is the time-dependent length and 9 is the gravitational acceleration. We consider small changes of the length according to the formula

2.5 Nonstationary Nonlinear Systems

a)

43

E

a

.

C.. F\ B

H

..

'

.'

w

b)

a

c

w

= L o(1 + /-Lcoswt).

L

Fig. 2.4. Nonlinear jumps of amplitudes accompanying an increase (a) and a decrease (b) in frequency

(2.5.2)

Then, by introducing the new variable x = LIp and after some transformations, we obtain 2

d x dt 2

2 ) + (g L + /-Lw coswt

o

(2.5.3)

x = 0,

where 0 < /-L « 1 is a small perturbation parameter. When the fixed point zero (trivial solution) undergoes periodic movement (Fig. 2.6b), we get the following governing equation mLoep

= -mg sin

(2.5.4)

which leads to the Mathieu equation

X + (a

+ 16qcos27") X = 0,

(2.5.5)

where 27" = wt,

x

=
Yo q = 4£0'

(2.5.6)

2. Discrete Systems

44

3~----r-----.,......----.-----.,

a) a

2

h=O.5

n=l 1 - .-- ---- -------r-~______t~~;t__::71--1

0.5

1.5

1

2

w

b)

-1r/2

o ~h.....;;O"""'''''''

o

~_--t---l-_ _-----J 1 • 1.5 w 2

0.5

Fig. 2.5. Amplitude (a) and phase (b) versus w for different values of h

In the third example (Fig. 2.6c) the excitation y(t) can be defined as follows r2

y(t) = 4L

+ r coswt -

~2

4L cos 2wt,

(2.5.7)

and the problem is reduced to the Hill equation

x + [0" + J.L4'(r)] x

=

0,

(2.5.8)

where.

0" =

~,

2r = wt,

J.L =

~o'

4'(r) = w 2 cos 2r -

TW

2

cos 4r.

(2.5.9)

If 4'( r) = cos r, then we have the following equation d2 x dr 2

+ (0" +J.Lcosr)x =0.

(2.5.10)

The periodic solutions with periods 21r and 41r appear on the stability limits of the system governed by the Mathieu equation (2.5.10). We would like to

2.5 Nonstationary Nonlinear Systems

45

y(t)

I I I I

_.- .c)

b)

a)

Fig. 2.6. Mathematical pendulum with a periodically changed length L(t) (a), with the periodic movement of the fixed point 0 (b), and with anharmonic periodic excitation (c) estimate analytically the (a, ,) values for which the system possesses a 41rperiodic solution. We are looking for the following solution

x(r,l-t) = xo(r) + I-txI(r) + 1-t2x2(r) a(l-t) = ao + I-tal + 1-t2a2 + ....

+ ... , (2.5.11)

Substituting (2.5.11) into (2.5.10) and comparing the terms with the same power of the small parameter I-t «: 1, we get

d 2xo dr 2 d2xl dr 2 d2x2 dr 2

+ aoxo = + aOXl

0,

= -(al

+ aOX2 =

-(al

+ cos r)xo, + cos r)xl

(2.5.12)

- a2XO,

The first equation of (2.5.12) gives

xo(r)

=

A o cos Jl10r

+ B o sin Jl1Or.

(2.5.13)

Because we are looking for the 41r- periodic solution, therefore

k2

ao ="""4'

k=0,1,2, ...

(2.5.14)

2. Discrete Systems

46

and from (2.5.13) we obtain

kr

kr

xo(r) = A o cos 2

+ Bosin 2·

(2.5.15)

o. From (2.5.12) we find

We consider first the case when k =

Xl (r) = C l

+ C 2r

r2 - O"lA0 "2

+ Ao cosr,

(2.5.16)

where Ci (i = 1,2) are the integration constants. Taking into account the periodicity, we have xI(r) = C l

+ A o cos r.

(2.5.17)

From the second equation of (2.5.12) we get X2(T)

~ Cs + C4T - (<72+~) An ~2 + C1 COST + ~ COS2T.

Again, the 41T"-periodicity gives equation of (2.5.11), we obtain 0"

1-L

0"0

=

0"1

= 0,

0"2

= -~ and from the second

2

(2.5.19)

';!::j - - .

2

For n

(2.5.18)

= 1 we have 0"0 = ~,

xo(r) = A o cos ~

2

and

+ B o sin~.

(2.5.20)

2

The second equation of (2.5.12) has the Ikw form

::~ + ~X1 = - (<7

1

+

DAocos~

+ (-<7 1 +

D

Bosin

~

Ao 3 B . 3 - - cos -r - - o sm -r. (2.5.21) 2 2 2 2 In order to have a periodic solution, secular terms should vanish, which leads to the conditions

- (<71+ DAn =0, ( <71 + ~ )

B o = O.

(2.5.22)

There are three different solutions of (2.5.22), which are given below': 1. A o

# 0;

2. A o = 0; 3. A o = Bo

O"i l ) = -~; O"i

2

)

= +~;

Bo

= O.

B o # O.

= 0 with O"i 3 ) temporarily unknown.

(2.5.23)

2.5 Nonstationary Nonlinear Systems

47

The first two solutions of (2.5.23) give the following estimation of the stability limits u(1)

u(2)

~

! _ tt

~

! + tt.

4

2'

(2.5.24)

4 2 For n = 2 we have the following stability limits u(1)

~ 1 + ~H2

u(2)

~ 1 - -tt2 .

12"'" ,

1

(2.5.25)

2

In all these cases We introduced the small positive perturbation parameter tt, which characterizes the modulation depth of the parametric excitation. However, linear systems are an idealization of real systems which are nonlinear. Therefore, we consider the general form of the system of equations

{±}

= [[~o]

+ tt [~I(t)] + tt 2 [~2(t)] + ... J {x} +e [F(XI' ... ,xn )] .(2.5.26)

Above, we have two independent perturbation parameters tt and e. The first one defines the parametric modulation, whereas the second one characterizes the nonlinearity. Additionally, [~o] is the constant matrix, and [~i(t)] = [~i(t + T)], where T is the period. The solution of (2.5.26) is sought in the form {x(t, e, tt)}

= x(O,O) + ttX{I,O) + tt 2x{2,O) + .. , +e (X{O,I)

+ ttX{I,I) + tt 2x {2,1) +

+e 2 (x{O,2)

)

+ tt X{I,2) + tt2x {2,2) +

(2.5.27)

)

+ ... Substituting (2.5.27) into (2.5.26), and after comparing the expressions with the same power of ttkel, we get { ±(O,O) } -

[~o] { x{O,O)} = {O},

01

{ X(O,I) } _

[~ol { X(O,I) } = [~~]

ttl

{ ±(I,O) } -

[~o]

{ x(O,2) } -

[~ol { x(O,2)} = [~~]

tt 2

{ ±(2,O) } -

[~o] { x{2,O)}

OIL

{ X(l,l)}

[~oJ {X(l,l)} = [~ll { X(O,I) } + [~]

eO, ttO

0

2

-

{x{1,O)} =

=

{x(O,O)} ,

[~l] { x{O,O) }

[~2]

{

{X(O,I)}

x{O,O)}

(2.5.28)

,

+ [~O~]

{x(O,O)} ,

+ [~l] { X{I,O) }

,

{X(I,O)} ,

48

2. Discrete Systems

The first equation of the set (2.5.28) gives the fundamental part of the sought solution (2.5.27), which can be succesfully approximated by solving the recurrent set of the rest of equations (2.5.28). To achieve a complete ordering of all of the recurrent equations, we take the following additional condition e < 1-£. The tringle below gives the ordering from the smallest to the largest values asymptotically on each horizontal row, Le. defines the sequence of recurrent equations e

1-£ e el-£ 1-£2 e3 e21-£ 1-£2 e 1-£3 e4 e31-£ e21-£2 1-£3e 1-£4 ............................................................. 2

(2.5.29)

The question arises: which initial conditions should be taken for x(O,O) , X(l,O), X(O,l) and so on. The following condition should be fulfilled {x(to, e, I-£)} = {x(O,O) (to) } + e { X(O,l) (to) } + 1-£ { x(l,O) (to) } + ... ,(2.5.30)

and usually it is useful to take {x(O,O)(to)} = {x(to,e,I-£)} ,

(2.5.31)

which leads to the equations X(I,O)(tO)

= X(O,l)(tO) =

X(2,O) (to)

Examples of two calculation to considered now [28].

= ... = O.

supp~rt

(2.5.32)

the above consideration will be

Example 2.5.1. Consider the standard example of vibrations of a system with one degree of freedom, of the form

d2 x mcit

+ (ko + k l

cos 2t),x + k 2x 3 = 0,

(2.5.33)

where m is the mass of the vibrating body, and ko, k 1 and k2 are the stiffness coefficients. After assuming the parameters >..2 = ko/m, 1-£ = k1/ko, and e = k 2/m, we obtain the equation . d2 x

dt

+ >..2(1 + 1-£ cos 2t)x + ex3 = O.

(2.5.34)

For 1-£ = 0, we obtain the Duffing equation, and for e = 0, the Mathieu equation. Let us develop the quantities >.. 2 and x into a power series in the small parameters 1-£ and e:

2.5 Nonstationary Nonlinear Systems ..x. 2 = n 2

+ a(I,O) J-L +

+e 2 (a(0,2) x

49

J-L 2a(2,0) + ... + e ( a(O,I) + J-La(I,I) + ... )

+ J-La(I,2) + J-L 2a(2,2) + ... ) + ., .

(2.5.35)

= x(O,O) + J-Lx(1,O) + J-L 2 x(2,0) + ... +e (X(O,I) + J-LX(I,I) + J-L 2 x(2,1) +

+e 2 (x(0,2)

+ J-LX(I,2) + J-L 2 x(2,2) +

)

(2.5.36)

) + . . ..

Now, consider the first unstable zone (n = 1). After substituting (2.5.35) and (2.5.36) into (2.5.34), and equating the expressions representing the same power J-L and e and their combinations, we obtain the following recurrent system of linear differential equations

+ x(O,O) £(1,0) + x(1,O) £(2,0) + x(2,0) £(0,0)

= 0,

=

_x(O,O)

cos 2t

- a(I,O) x(O,O) ,

= _a(2,O) x(O,O) _ a(I,O) X(I,O) _ a(l,O) cos 2tx(0,0) - x(l,O) cos 2t,

£(0,1)

+ X(O,I)

£(1,1)

+ X(I,I) = -3 ( x(O,O») 2 x(1,O)

= (x(O,O») 3 _ a(O,I) x(O,O),

-a (1,1) x(O,O) £(2,1)

+ x(2,1) = _3x(0,0)

-

(2.5.37)

_ a(l,O) X(O,I) _ a(O,l) x(O,O)

X(O,I)

(X(I,O») 2 _

cos 2t

cos 2t, 3 ( x(O,O») 2 x(2,0)

_ a(2,1) x(O,O)

_a(1,I)x(I,O) _ a(0,l)x(2,0) _ a(O,I)x(1,O) cos2t _ a(2,0)x(0,1) _x(O,1)a(I,O)

cos 2t

- x(l,I)a(I,O) -

X(I,I)

cos 2t ,

Let us assume the initial conditions x(t = 0) = A o, x(t = 0) = B o. From the first equation of system (2.5.37) we obtain x(O,O) =

A o cost

+ B o sin t.

(2.5.38)

Substituting (2.5.38) into the second equation of system (2.5.37), from the condition of avoiding the secular terms, we get: lOa (1,0) = and B o = 0 0 and A o = O. Let us consider the first case: then X(I,O) _ or 2 a(I,O) = 1~ A o cos 3t and, taking into account the third equation, we have

-!

!

a (2,0)

=

2.

32'

x(2,0)

= - ~ A o cos 3t + A o cos 5t. 256 768

(2.5.39)

50

2. Discrete Systems

Acting analogously, we obtain

_~A02

...!:.-A30cos3t 32 ' 5 llA3 A3 a(l.l) = _A 2 X(l,l) = 0 cos3t + _ 0 cos5t 16 0' 256 384' 75 73 235 a(2,1) = --A~ x(2,1) = A~ cos3t A5 cos5t 1024 ' 24576 73728 16A4 + 1474~6 cas7t,

a (O,l) =

4'

X(O,l) =

(2.5.40)

(2.5.41)

3 3 3 a(O,2) = --A~, x(O,2) = --4Ag cos3t + 3072Ag cos5t. 128 102 Taking the above calculation into account in (2.5.35), we obtain 2 1 7 2 2 (3 A = 1 - '2J.L + 32J.L + ... + cAo -'4 +0

2

(

+

5 75 2 ) 16J.L + 1024J.L + ...

-1~8A~ + .. -) +....

(2.5.42)

For c = 0 we obtain one branch of the first limit of the stability loss of the Mathieu equation, and for J.L = 0, the dependence of the frequency on the amplitude for a conservative system with the Duffing characteristics.

Example 2.5.2. Consider an unsteady nonlinear system with two-degree-offreedom, the motion of which is governed by the following differential equat~ns

1

•

MZ 1 + C(ZI - Z2) + k(Zl - Z2) + k1(Zl - Z2)3 = 0, (2.5.43) mZ2 - C(ZI - Z2) - k(Zl - Z2) - k 1(Zl - Z2)3 = -(k3 - k o cos 2wt)Z2, where m and M are the masses, k, k 1 , k 3 and ko are the rigidities, and C is the damping coefficient. The frequencies of free vibrations of the conservative system are calculated after assuming C = k 1 = ko = 0 in (2.5.43). They amount to a21 2 = ! , 2

3 ]. [~ + k +m k3:r (~ + k +m k 3 ) 2 _ 4kk M M Mm

'

T

(2.5.44)

Our consideration is limited to calculating the first simple parametric resOnanCe around the frequency al. Equation (2.5.43) will be rearranged to the form

ih + A~Yl

-CMA~(cIY2 - c2yI)3 - J.LOIAl(cl'iJ2 - c2iJI) +J.LPIA~(YI - Y2) cos 2r, ih + A~v2Y2 = -CMA~v2(cIY2 - c2yI)3 - J.LOIAIV(cIY2 - c2iJI) +J.LP2A~v2(Yl - Y2) cos2r, =

(2.5.45)

2.5 Nonstationary Nonlinear Systems

51

where

1

1/J =

, 11 and the dot now denotes the differentiation with respect to r. We seek the solution of (2.5.45) of the form 2 YI = y1°'0) + /lYP'O) + /l2 y 1 ,0) + ... + cyiO,I)

-'2

2 (0,2) (1,1) +c YI + ... + /lcYI + ... (0,0) +

Y2 = Y2

+c

(1,0) +

/lY2

2 (0,2)

Y2

+ ... +

/l

2 (2,0) +

Y2

(1,1)

/lcY2

...

+

(0,1) cY2

(2.5.47)

+ ...

A~ = 1 + /laiI,O) + /l2 a 12 ,0) + ... + ca1o,I) + c2a1o,2) + ... + /lca1 I ,I) + ... Al

= 1

(1,0) (0,1) (1,1) (2,0) (0,2) a a a a1 4 + _I_/l + _ I _ c + _I_/l c + /l2_1_ + c _ _ +

a

.

2 2 2 4 4" After substituting (2.5.47) into (2.5.45), we obtain the following system of linear differential equations .. (0,0) + (0,0) YI YI

0

=,

.. (1,0) + (1,0) _ (1,0) (0,0) £ YI - UI YI YI - -a l

(

• (0,0) . (0,0») cIY2 - c2YI

+PI (y1°'0) - y~o,O») cos2r, .. (2,0) + (2,0) _ (1,0) (1,0) (2,0) (0,0) YI - al YI YI YI - -a l

-01 (CIy~I,O)

(1,0) al ( . (0,0) . (0,0») 1 -cIY2 - c2YI

0

2

I - c 2y 1 ,0») + PI aiI,O)

(y1°'0) - y~o,O») cos 2r

+PI (y1 I ,0) - y~I,O») cos 2r, .. (0,1) + (0,1) _ (0,1) (0,0) YI YI - -al YI -

M

..(0,2) Y1

a(0,2)y(0,0)

+

y(0,2) __ a(O,I)y(O,I) 1

-

1

1

_

1

(0,0) cIY2 -

1

-M { 3c~ (y~O,O) )2y~0,I) -

-

(0,0»)3 c2YI ,

(2.5.48)

Ma(O,I) (<=, y(O,O) 1

q

6c~c2y~0,I)y10'0)

2

(0,0»)3

- c2YI

52

2. Discrete Systems

.. (1,1) +

Y1

(1,1) _

Y1

- -

a(l,I)y(O,O) a(O,l)y(l,O) 1 1 1 1 -

M a (1,0) 1

a(l,O)y(O,l) 1

1

M{3 c13(y(O,O»)2 2 Y2(1,0)

(0,0) (0,0»)3 c1Y2 - c2Y1 -

2 (0,0) (1,0) (0,0) 2 (0,0) (1,0) (0,0) + 6 - 6 c1Y2 Y2 c2Y1 c1 c 2Y2 Y1 Y1 (0,1) ~ al (0,0) (0,0») ~ (.(0,1) .(0,1») - c1Y2 -VI- c2Y1 - vI c1Y2 - c2Y1

2

+P1 (y~O,l) _ y~O,l») cos2r

+ P1a~0,1)

(y~o,O)

_ y~o.o») COS 2T }, .. (0,0) +

Y2

.. (1,0) +

Y2

V

0 -,

2 (0,0) -

Y2

2 (1,0)

v Y2

+ P2 v2 .. (2,0) +

Y2

V

2 (2,0)

Y2

~

2 (1,0) (0,0)

= -val

=

Y2

- V2 V

2 (1,0) (1,0)

-

Y2

.(0,0») C2Y1 -

. (0,0) . (0,0») CIY2 - c2Y1

y~O,O) )

(yiO,O) -

-val

(

-

COS

2r,

2 (2,0) (0,0) V a1 Y2

t

~

.(1,0)

v2 V \'£lY2

P2v2a~1,0) (yiO,O)

(1,0) a1 (

8

-

2V -

2

-

.(0,0) c1Y2

.(1,0») c2Y1

-

y~O,O») cos2r

-

+P2 v2 (yp,O) _ y~l,O») cos2r, .. (0,1) +

Y2

2 (0,1) V

.. (0,2) +

Y2

V

Y2

=

-v 2 aiO,1)yiO,0) _ Mv 2

2 (0,2) _

(C1Y~0,0)

2 (0,1) (0,1) (0,2) 2 (0,0) - a1 al v Y2

- -v Y2

Y2

_ c 2y i O,0») 3 _

_ C2yiO ,0») 3 ,

-

V

2M

(2.5.49)

(0,1) ( (0,0) a1 c1Y2

Mv 2 (3c~ (y~O,O) )2y~O,l)

-6c?c2Y~O,O) y~O,I) y~O,O) + 6c 1 c~y~O,O) yiO,O) yiO' 1)

+3c1c~y~0,1) (yiO,O»)2 - 3c~(yiO,0»)2yiO,1)

-3e~(y~o.O) )2e2ylo.l») , .. (I,I) Y2

+ y(I,l) 2

_ _ v2a(I,l)y(0,0)

-

1

. (c I

y~O,O)

2

-

-

v2a(O,l)y(l,O) 1

c 2y iO,0») 3 _

2

-

v2a(1,0)y(0,1) 1 1

2

- V

v2 M ( 3c~ (y~O,O») 2y~ 1,0)

2M (1,0 a1

2.5 Nonstationary Nonlinear Systems

53

(2.5.50)

54

2. Discrete Systems

(2.5.51)

2.6 Parametric and Self-Excited Oscillation

_P1 2

+

(

(-v

2a O'1) Bo i

55

+ ~Mv2c~A5Bo + ~MV2c~B3) v 2 -1

~MV2c3A2BO - !Mv 2c3B 3) 4 2 4 2 2(v 2 - 9)

BO] ° _ P1a(O,l) 1 2

°

= O.

(2.5.52)

In order to avoid time-consuming calculations, we assume that B o = O. Then (1,1) 3- (1,0) 3 3 2 P2 v2 9 2 3- 2 P2 v2 A2 A a1 = '4Ma1 c2 O- MC1C22(v2 _ 1) . gAo - SMC 1C2 (v 2 _ 9)

°

2 (0,1)

2 3- 3 1 - 3 2 P1 va 1 - 64 MC2P1Ao - 64 MC2AO + 2(v2 - 1)

1 Mv2c~A5

+ -8

(v 2

-

9)

+

°

3M 2 3A2 VC2 8(v2 - 1)

p1aiO,l) 2

(2.5.53)

Substituting the calculated values ap,O), aiO,l), ap,l) into the third equation of (2.5.47), we obtain the analytic form of the limit of the loss of stability, which depends on J.L, c and the other parameters.

2.6 Parametric and Self-Excited Oscillation in a Three-Degree-of-Freedom Mechanical System 2.6.1 Analysed System and Equation of Motion We consider a mechanical system with friction induced self-excited oscillations and parametric oscillations [14d, 17d]. The latter come from a rotor with rectangular cross-sections, which has a mass concentrated eccentrically at its centre. The rotor is fixed on the base placed on the belt moving with a constant velocity. At a certain value of the belt velocity and the frequency of rotor revolutions, parametric and self-excited'oscillations appear in addition to the forced oscillations caused by the unbalanced rotor. The diagram of the system is presented in Fig. 2.7. A weightless shaft with a rectangular cross-section and with a cylinder-like mass concentrated at its centre is supported by the base placed on the belt moving with constant velocity va. The friction coefficient between the belt and the base depends on their relative velocity. The character of this dependence (Fig. 2.8) causes self-excited vibrations. The effect is described in basic works on nonlinear vibrations. On the other hand, considering the nonidentical cross-section of the rotor at various values of its rotational speed, parametric vibrations occur. The vibrations cause a change to the normal force holding down the base to the belt in the vertical direction, and hence they cause changes to the friction force. It is assumed that despite the vibration of the rotor, the contact between the base and the belt is maintained.

56

2. Discrete Systems

m

k'l

.....-Vo Fig. 2.7. Diagram of the system

The calculation model of the system is presented in Fig. 2.9. The equations of motion of the system have the form mXc = -ewke cos

Iz"ep

= -Mo + a(-ewke cos 'Po

(2.6.1)

+ 71wk17 sin <po),

where

a,
coordinates of the centre of mass of the cylinder, mass moment of inertia of the cylider with mass m in relation to the axis z" of the system 0"x" y" z" moving with translatory motion in relation to Oxyz, coordinates of the point of puncture by the shaft in the coordinate system 0'e71, coordinate system whose axes are parallel to the main, central inertia axes of the cross-section of the shaft, shaft rigidities in the direction of the axes e and 71, driving torque reduced by all the resistance torques, parameters characterizing the position of the centre of mass of the disk C in relation to the point of puncture by the shaft.

For the states near the steady states, the torque M o is very small. Let I zIt =

·2 m'l.s'

(2.6.2)

where is is the inertia radius. Then the third equation of (2.6.1) will assume the form . c
+ 71wk 17 sin 'Po ) .

(2.6.3)

As the eccentricity a and the shaft deflection ew and 71w are small if compared to the inertia radius, the following can be assumed

2.6 Parametric and Self-Excited Oscillation

57

J.L

Fig. 2.8. Dependence of the friction coefficient on the relative velocity

ep = 0,


= w = const,

(2.6.4)

The following geometric dependences result from Fig. 2.9:

ew =

(x w - x) cos

Tlw = (x w - x) sin
+ Yw cos
Yc = Yw

+ acos(

=

+ asin(

Xc

Xw

(2.6.5)

where x w , Yw are the coordinates of the point of puncture by the shaft W in the system Oxyz. In order to write down the equation of motion of the mass M, it is necessary to determine the dynamic reactions on the shaft at its points of support. They are determined from the equations

Xl + X 2 + ew k { coswt + TlwkT/ sinwt = 0, Yl + Y2 - ewk{ sinwt + TlwkT/ coswt = 0,

(2.6.6)

where Xl, Yl and X 2 , Y2 denote the support reactions on the left and right end of the shaft, respectively. The rotor reactions on the supports are

Rx

= -X l -X2 ,

Ry = -Yl

-

Y2 .

(2.6.7)

The equation of motion of a body with mass M, on the assumption that Mg + R y > 0, has the form

Mi = -kx - eX + Rx w = Vo -

+ (Mg + Ry)J.L(w) ,

x.

(2.6.8)

The dependence of the friction coefficient on the relative velocity w can be described by the polynomial J.L(w) = e sgn w - ow

+ j3w 3 .

(2.6.9)

58

2. Discrete Systems

I·

x

I Xl x O~---4------~:""----.t-T----r---\-----r-----a--~ X2 x'

,

,

y.,=y.,

Yc=Yc -,::y(/"

x"

0" C mg

x.,

y"

Y'

Y

Fig. 2.9. Calculation model of the system

Finally, the equations of motion of the system, after assuming that x = Xl, Xw = X2, Yw = Xa, have the form t 2 Xl = -Xl (n + nl + n~ + (nl- n~) cos2wt) - HXI

-X2 (-(nl

+ n~)

- (nl- n~) cos2wt) - xa(nl- n~) sin2wt

+[g - X2(nl- n~)sin2wt - Xa (-(nl + n~) + (nl- n~) cos2f.Vt) + XI(nl- n~) sin 2wt](csgn (vo -

xd + {3( Vo - xd 3 ) , Xl (w~ + w~ + (wl- w~) cos2wt) -X2 (wl + w~ + (wl- w~) cos 2wt) + x3(wl- w~) sin·2wt - 0:( Vo

X2 =

xd

-

(2.6.10)

+a: 2 w sin(wt + CPo), x~ = -xI(wl- w~) sin 2wt + x2(wl- w~) sin 2wt +X3 (-(wl + w~) + (wl- w~) . cos 2wt) + OtW 2 cos(wt + \00) + g, 2 where: n = kiM, nl = k~/(2M), n~ = k."/(2M) , H = elM, wl k~/(2m), w; = k."/(2m).

2.6 Parametric and Self-Excited Oscillation

59

2.6.2 Transformation of the Equations of Motion to the Main Coordinates Let us introduce the following notation n~

=

nl +

2 2 WI =W~ ..

n;,

+Wq2 ,

a cos
ni = nl- n~, W22 =We2 -WTJ2 , a sin
H = J1.H 1, a: X = -, 6

{3 p= -,

( ) 2.6.11

6

J1.G = g,

where J1. = w~ / w~ = n~ / n? = (k e - kTJ) / (k e + kTJ) is the perturbation parameter. After accounting for (2.6.11) in the equation system (2.6.10), it will assume the form Xl = -Xl n 2 - Xl n~(3 + J1. cos 2wt) - J1.Hl I l +X2 n~ (1 + J1. cos 2wt) + X3n~ J1. sin 2wt +6(g - X2n~ J1. sin 2wt + Xl n~ J1. sin 2wt +X3n~( -J1. cos 2wt) )(sgn(vo - It} - X(vo - It}

+ p(vo -

X2 = -xlw~(1 + J1. cos 2wt) - x2w~(1 + J1. cos 2wt) +X3W~J1. sin 2wt + J1.(P sin wt + Q sinwt)w 2 , X3 = -xlw~J1.sin2wt + x2w~J1.sin2wt - x3w~(1- J1. cos 2wt) +J1.(Pcoswt - Qsinwt)w 2 + J1.G.

It}3), (2.6.12)

If we introduce J1. = 6 = 0 into the equation system (2.6.12), we obtain the homogeneous Linear differential equation system

Xl + Xl(n 2 + n~) - X2n~ = 0, X2 + W~(X2 - xt} = 0, X3 +W~X3 = O.

(2.6.13)

When we assume the solution of (2.6.13) in the form we find the following frequencies

Xi

=

Ai

cos pt, i = 1,2,3,

vb = ~ (n2+ nf +wf ± JU12 + nf +w?)2 - 4n2w~) , p~ = wr

(2.6.14)

Let us introduce the main coordinates el, for which at J1. = 6 = 0 the uncoupling of the linear part of the first two equations of the system (2.6.12) will occur. Let us now multiply these equations by el and e2, respectively, and add the sides. The result will be as follows 2 x16 + xt{n + n~)6 - x2n~6 + X2e2 + x2w~6 - xlw~6 = J1. [ - Xl n?6 cos 2wt - HlIlel + 6X2n~ cos 2wt -6X3n~ sin 2wt

+ 6XlW? cos 2wt -

6X2W~ cos 2wt

(2.6.15)

60

2. Discrete Systems

+6X3W? sin 2wt

+ w2e2 (P sin wt + Q cos wt) ]

+€6 [g - x2n?tLsin2wt + x3w?(1- tL cos 2wt) +xln?tLsin2wt] [sgn(vo By denoting el e 2 = (n 2 + n~)el

xd -

X(vo -

xd + p(vo - xd 3] .

- w?6,

6 e2 = -n~el + w~6

(2.6.16)

we find

(n 2 + n~ - (P)6 -w~6 = -n?el + (w? - (}2)6 = O.

0, (2.6.17)

In order for (2.6.17) to be fulfilled for following dependence must occur

n2 +-n? n? - e 2

-w? w? _ e2

6

=

=

different from zero, the

(2.6.18)

(}l = Pl' From the second equation

"Yle~,

(2.6.19)

where

"Yl

6

= o.

Hence, e? = p? and e~ = p~. Let el = e~ and 6 = e2 be denoted for of the system (2.6.16) we find e~

and

.

n?

(

•

2 2' WI - PI

Making use of the dependences (2.6.16) and (2.6.19), (2.6.15) is transformed to

fit + XlP~ + X2"Y1 + X2P~"Y1

=

tL[XlP~"Yl cos 2wt - HlXl

-P~"YlX2cos2wt +P?"YlX3sin2wt + "YlW2(P sin wt + Qcoswt)]

+€ [g - X2n~tL sin 2wt + x3n~(1 - tL cos 2wt) + xln?tL sin 2wt] . [sgn (vo - xI) - X(vo - xd + {3(vo - xd 3]. (2.6.20)

e2

Analogously, for e2 = P2 the following notation is used: = ~~, while e~ = "Y2e~,

where

e

1

= e~ and

(2.6.21)

2.6 Parametric and Self-Excited Oscillation

61

If we take (2.6.16) and (2.6.21) into account in (2.6.15), the equation will

assume the form

Xl

+ XlP~ + X2'Y2 + X2P~'Y2 = J.L[XlP~'Y2 cos 2wt - HlXl -P~'Y2X2 cos2wt + P~'Y2X3 sin 2wt + 'Y2w2(Psinwt + Q coswt)] +e[g - x2I1;tJsin 2wt + x 3I1;(l - J.Lcos 2wt) + x l I1; sin 2wt] . [sgn (vQ - xd - X(vQ - xd + p(vQ - xd 3]. (2.6.22)

Let us denote Yl = Xl

+ 'YlX2, (2.6.23)

Y2 = Xl + 'Y2 X2· The reverse dependences can be determined from (2.6.23)

Xl = IhYl - IhY2, X2 = t/J(YI - Y2),

(2.6.24)

where 1 'Y2 'Yl ,132 = - - t/J= - - 'Y2 - 'Yl 'Y2 - 'Yl 'Yl - 'Y2 Let us additionally assume that X3 = Y3. Taking (2.6.23) and (2.6.24) into account in (2.6.22), (2.6.20) and (2.6.12), we obtain the following differential equation system

131 =

iii + P~Yl

=

J.L [p~'Yl (13lYl - 132Y2) cos 2wt - HI (13dll - 132Y2) -P~'Yl t/J . (Yl - Y2) cos 2wt + P~'YlY3 sin 2wt +w2'Y1 (P sinwt + Q coswt)] +e [g - J.LI1~t/J(Yl - Y2) sin 2wt + J.LI1~(13lYl - 132Y2) sin 2wt

+I1;Y3(1 - J.L cos 2wt)][sgn(vQ - 13lYl + 132Y2) -X(vo - 132Yl

ih + P~Y2

=

+ 132Y2) + p( VQ -

13lYl

+ 132Y2) 3] ;

J.L[p~'Y2(13lYl - 132Y2) cos2wt - HI (13lYl - 132Y2) -P~'Y2t/J(Yl - Y2) cos 2wt + P~'Y2Y3 sin 2wt +w2'Y2(Psinwt + Qcoswt)]

(2.6.25)

+e[g -.J.LI1;t/J(Yl - Y2) sin 2wt + J.LI1~(13lYl - 132Y2) sin 2wt

+ 132Y2) 13lYl + 132Y2) 3] ;

+I1;Y3(1 - J.L cos 2wt)][sgn(vQ - 13lYl

+ 132Y2) + p(vQ P~(13lYl - 132Y2) sin 2wt + P~t/J(YI

-X(vo - 13lYl

ih + P~Y3

=

J.L[ - Y2) sin 2wt 2 +P~Y3 cos2wt +w (Pcoswt - Qsinwt) + G].

2. Discrete Systems

62

= wt, we obtain

After introducing the dimensionless time r

iit + A~Yl

2 = /1[A("r'I(€IYl - €2Y2) cos2r - AI H- l(1hYII - {32Y2' )

+A~"YIY3 sin 2r

+ "Yl (P sin 7" + Q cos r)] +€[g + /1[}~(€IYl - €2Y2) sin2r + [}~Y3(1 -/1cos2r)]

. [A~ sgn(Vo - {31WY~ - {32WY;) PI +wp(AI V~ - {31Y~ + {32y~)3];

AIXdAIV~ - {31Y~ + {32Y;)

ih + A~Y2 = /1[A~"Y2(€IYl - €2Y2) cos 2r - A2 f lt ({31Y~ - {32Y;) +A~"Y2Y3 sin 2r + "Y2(Psin r

(2.6.26)

+ Q cos r)]

+€ [g + /1[}~ (€1 Yl - €2Y2) sin 2r + [}~Y3 (1 - /1 cos 2r)] . [A~sgn(vo -

{31WY~ -

PI +wp(A2V~ - {31Y~

{32WY;) -

A2X2(A2V~ - {31Y~ + {32Y;)

+ {32y~)3];

ih + A~Y3 = /1[ - A~(€IYl - €2Y2) sin 27" 2 -] +A~Y3 cos 2r + P cos r + Q sin r + A3 G ,

where

Yi

dYi, =dr

€k

= 13k -1/;,

-

G

G=2' P3

~ , 1 ,• = 1, 2 , 3,

\2, -_

A

w2

1

HI

Xk

=

= X,

HI -, PI

k = 1,2,

Pk

1 I

Vo

=

Vo -, PI

•

II

Vo

Vo = -, P2

2.6.3 Zones of Instability of the First Order The procedure of solving the system of equations (2.6.26) consists in assuming two perturbation parameters /1 and € connected with the parametric excitation and friction, respectively. The sought periodic solutions of Yi(r) are presented in the form of the double power series (i) (i) 2 (i) ( (i) (i) 2 (i) ) Yi () r , Yo,O+/1YO,I+/1 YO,2+"'+€ Yl,O+/1Yl,I+/1 Yl,2+'"

(26 27) +.....

where yi~L k, l = 0,1,2, ... must fulfil the condition of periodicity. Periodic solutions are only possible for certain values of the parameters A~ presented in the form of the analogous series . A~ = n 2+/1ao,1 +/12ao ,2+' .. +€ (0:1,0 + /1al,1 + /120:2,2 + ...) + ... (2.6.28) where ak,l k, l = 0,1,2, ... are unknown coefficients, which are determined from the condition of periodicity, avoiding in the solution terms unrestrictedly growing in time. For a first-order resonance n 2 = 1 we shall determine

2.6 Parametric and Self-Excited Oscillation

63

the parametric unstability zones, for which the frequency of parameter modulation fulfils, consecutively, the dependences w P1, W P2, and w P3. In series (2.6.27) and (2.6.28) for w P1 and w P2 we shall limit our considerations to the first powers of the small parameters J.L and €. On the other hand, for w P3 we shall limit ourselves in the calculations to the second approximation. In all three cases we shall assume that sgn(vo - (31 wYl - (32WY~) = 1. Let "'J

"'J

"'J

"'J

"'J

"'J

\2 ""2 \2 ""3

2

\2

2

\2

= V2 ,1""1'

(2.6.29)

= v3 ,l""1'

where P2 V21 =-, , P1 P3 V31 = - , , P1

and let us assume that V2,1 and V3 ,1 are not integers. Let us first consider the case w ~ P1, assuming that

ya~6(r) = ya~6(r)

=

O.

(2.6.30)

The assumption is accounted for by a weak conjugation of (2.6.26) for € « 1 and J.L « 1. For J.L = € = 0 we shall obtain an unlinked system of three linear differential equations. For the resonance coordinate Y~'T), the magnitude of the oscillation of the other two main coordinates should be of the order of the small parameters J.L and €. Let us substitute series (2.6.27) and (2.6.27) into the differential equation (2.6.26), taking into consideration the dependences (2.6.19) and (2.6.30) and the expansion 0:0,1 a1,0 ( ""1 = 1 + J.L-- + €-2- + .. .. 2.6.31) 2 After equating to zero the coefficients of the same powers € and J.L, we obtain the system of recurrent differential equations \

"'J

y(1)" 0,0

+ y(1) 0,0

-

(1)"

(1) + Y1,0

= -a1,OYo,0

Y1,0

O., (1)

9 + P~

- 1 - gX1 vo

-3gwp(v1)2{31y(1)1 o 0 ,0 (1)"

YO,l

(1)1 + gX1{31Yo,0 + gwp (vo1)3

+ 3gwpv1{32 (y(1)/)2 + gwp{331 (y(1)/)3. 0 1 0,0 0 ,0 ,

(1) (1) (1) + YO,l = -ao,lYo,o + 1'l€lYo,O cos 2r -fI1{32ya~6' + 1'l Psin r + 1'1 Q cos r;

64

2. Discrete Systems

(2)11 (2) (2) Y10 + V2,1 Y1,0 ,

2

=

V2,1 - {3 (1)1 gp,2 gv 2 v V" + gV2,1X2 1Y'0,0 2 2,1/'\.2 0 2 (V,,)2 {3 y(1)1 +gWpv32,1 (V,,)3 + 3gwv2,1 0 0 1 0,0 +3gwPV2, 1V~ {3?

(2.6.32)

(Y~~6 ,) 2 + gw p{3~ (Y~~6 ,) 3 ;

2 H- {3 (1)1 cos T - V2,1 2 1YO,0 +"Y2Psin T + "Y2Q COST; (3)II + v(2) y(3) - O. 3,1 1,0 , Y1,0 (3)11 2 (3) (1) (1). 2 P Q . 2 GYO,1 + V3,1YO,1 = -V3,1 c1Yo,0 sm T + COS T sm T + V3,1 .

(2)11 (2) (2) YO,1 + v 2,1YO,1 =

2 (1) "Y2v2,1c1Yo,0

Assuming the solution of the first equation of the system (2.6.32) in the form y~16 , = a1 cos T + b1 sin T we obtain from the second equation: p2

Y~~6" + Y~~6

=

;

(2.6.33) 3

gX1v~ + gwp(v~) + 2gwP(v~){3?(a~ + b~)

-

+COST[ - a1,Oa1 + gX{31b1 -

+~ gwp{3~b1an + sin T [ -

3gwp(v~)2{31b1 - ~gwp{3~b~

0:1,ob 1 -

+3gwp ( Vo') 2 {31 a 1 + 43 gW:P{313 b21a1

gX1{31 a 1

+ 43 gwP{313 a13]

(2.6.34)

<

+~gW{3V~{3;(b~ - a~) CO~2T + 3gwpv~{3?a1b1 sin 2T 1 +COS3T[ - 4gwp{3~b~

3

+ 4gwp{3~b1a~] 3

. [-14gwp{31a1 33 32] +sm3T + 4gwp{31b1a1 . From the condition of periodicity we obtain two algebraic equations -a1,Oa1

3gwp(v~)2{31 - ~gwp{3~ADb1 =

0,

3gwp(v~)2{31-~gwp{3~ADa1-a1,ob1=

0,

+ (gX1{31

-(gX1{31-

-

(2.6.35)

where A~ = a~ + b~. For non-zero a1 and b1 , the following relation must occur -a1,0 gX1{31 - 3gwp(v~)2{31 - ~gwp{3fA~ gX1{31 - 3gwp(vO)2{31 - ~gwp{3fA~ - 0:1,0

= o.

(2.6.36)

Hence, (2.6.37)

2.6 Parametric and Self-Excited Oscillation

65

The only real solution of (2.6.37) is 0::1,0 =

0,

A21 = X

-33wP(v~)2

(2.6.38)

4 w p(3r The following function is the solution of (2.6.34) (1)

Y1,0 =

-, (')3 3 '(32A 2 p?9 - gX1 Vo + gwp Vo + "2gwpvo 1 1

+~gwpv~(3i(b~ - a~) cos 2r + gwpv~(3~a1b1 sin 2r + (3129WPi3:b~ + 3329WPi3~bla~)

. cOS 3T

(2.6.39)

1 gwP (331a13 - 32 3 gwP(331 b2 )' 3 1a1 sm r. + ( 32 The solution omits the general integral of the homogeneous equation by as1 sociating it with ya J. When we substitute (2.6.33) into the fourth and sixth equation of the equation system (2.6.32), after transformations, we obtain (2)"

Y10 ,

+ V22"1Y1(2)0

2

9

2

-"

= V2, 1-2 - V2 19X2 Vo P2'

+ gWP23, 1 (Vo")3

2 3 "(32A - (31b1 +"2V2,lgwpvo 1 1 + cos r [ v2,lgX2

-3gwPv~,1 (v~)2(31b1 + ~9Wp(3~A~b1] +sinr[ - V2,lgX2(310::1 + 3gwpv~,dv~)2(31a1 2 ] 3 " 2 2 +"43 gwp(33A 1 1a1 + "2V2,lgWPVO (3db 1 +3V2,lgwpv~(3ra1b1sin 2r

(3)"

Y1,0

2 (3) + v3,lY1,0

2) a1 cos

2r

2)

+ (

-41 gwP(313b31 +

1 3 "2gwP(31 b1a 1 cos 3r

+ (

-41 gwP(331a13 + "43 gwP(313b2)' 1a1 sm 3r;

= O.

(2.6.40) (2.6.41

)

The following functions are the particular solutions of the above equations (2) 9 - 2Vo" Y1,0 = 2'X P2

2 3 "(32A + gwpv 2,1 (Vo")3 + -2-gwpvo 1 1 V2,1

+ V?, 11-

- (31 - 3gwpv2,1 2 (")3(3 1 [ v2,lgX2 Vo 1

+ v 2. 1-

1 [ - V2,lgX2(31 a 1 +

2 1

-

"43 gwp(313 A 3] 1 b1 cosr

3gwPv~,1(v~)2(31a1

(2.6.42)

66

2. Discrete Systems

and (3)

(2.6.43)

YI0 , = O.

By substituting (2.6.39) into the third equation of the system (2.6.32), we obtain

y~~I" + y~~1 = ( -CtD,lal + ~'l'1olal + ( -CtD,lb l + ~1'lolbl

HilMI + 'nQ) - Hlihal

+ 1'I

cOST

P)

sin T

(2.6.44)

+ ~"YIClal cos 3r + ~"YIClbl sin 3r. We avoid terms unrestrictedly growing in time in its solution if the following equations are fulfilled (CtD,1 -

-

~1'101) al + H l l3l bl = 1'IQ,

-Hl{31 a l

+

(

aO,1

1)

+ "2"YI CI

.

\

(2.6.45)

b1 = "YIP.

For the case P = Q, after transformations, we obtain from (2.6.45): 2

P 2 aO,1 = ± ( Ai"Y1

±

1

2 2

+ 4"Yl cl

P (AI "Yd

4

- 2 2

- HI {31

P2)!

'2 P

+ "Yl C1 A~ - 2H- 1{31 A~ "Yl3 Cl 4 2

(2.6.46)

The particular solution of (2.6.44) is (1) 1 YO,1 = -16 "Yl c l(al cos3r

.

+ bl sm3r).

(2.6.47)

Taking (2.6.33) into consideration in the fifth and seventh equation of the system (2.6.32), we find the particular solutions.

y~~1 =

1 V?1 _ 1 GV?'I1'20 I a l -1I2,IH213Ibl

+ vl

1 _ 1 (l

+ 1'2Q) COST

~V~,I1'20 1bl + V2, I H2131 a I + ')'2P)

sin T

2.6 Parametric and Self-Excited Oscillation 2

2

V2,11'2 Cl V2,11'2 c l . + (2 9) al cos 31' + ( 2 ) bl sm3T; 2 v 2,1 2 v2 ,l - 9

y~~l =

G+

v 2 1_1 a,l

(

67

2.6.48)

(-~Vj'lclbl + p) COST

1 + 2 1_ (-2 vj,lclal + va,l 1

Q) sinT

Va2 lCl Va2 lCl + (2' )blcos3T+ (2' )alsin3T. 2 va,l - 9 2 va,l - 9

(2.6.49)

We have thus determined the particular terms of the series (2.6.27) and (2.6.28), limiting the calculations to the first approximation. Let us now concentrate on the analysis of the case w rov P2. The solutions will be sought, as has been done previously, in the form of the series (2.6.27) and (2.6.28) for i = 2. From (2.6.28) we obtain al,O

.A2 = 1 + c-2 Let us denote \2 "'1

=

0:0,1

+ J.L-- + . . . . 2

(2.6.50)

2 \2 VI ,2"'2,

\2 2 \2 "'a = va,2"'2'

(2.6.51)

where Vl,2 = Pl/P2' Va,2 = Pa/P2' and Vl,2 and Va,2 are not integers. Analogously to (2.6.30), we have (1)

Yo,o( 1')

(a) = Yo,o

= O.

(2.6.52)

Substituting (2.6.27) into (2.6.26), with (2.6.50), (2.6.3) and (2.6.52) taken into account, after equating to zero the coefficients of the same powers of J.L and c, we obtain (1)" 2 (1) 9 - 2' {3 (2), a ( ,)a Yl,O + Vl ,2Yl,O = p~ - gxv l ,2 vO- gX Vl,2 2Yo,o + gwpv l ,2 Vo 2 (V')2 Yo,o (2), + 3gWPVl,2 {322VO '( Yo,o (2),)2 +3 gwp{32Vl,2 o

+gwp{3~ (Y~~6,)a j (1)" 2 (1) 2 (2) - {3 (2), YO,l + Vl,2YO,1 = -1'lVl,2c2YO,O cos 21' + Vl,2 H l 2Yo,o +(3l Psin l' + 1'1 Q cos Tj y(2)" + y(2) - O., 0,0 0,0 -

(2.6.53)

(2)" + (2) (2) + 9 -" - {3 (2), ( ,,)a Yl,O Yl,O = -O:l,OYO,o P~ - gX2Vo - gX2 2Yo,O + gwp Vo

+3gWp(v~)2{32y~~J' + 3gwpv~{3~(Y~~6,)2 + gwp{3~(y~~J,)a; (2)" (2) YO,l + YO,l

(2)

(2)

-

(2),

= -ao,lYo,o - 1'2c 2Yo,O cos 21' + H2{32YO,0 +1'2PsinT

+ 1'2Q COST;

68

2. Discrete Systems Yl,O

(3)11

2 y(3) + V3,2 1,0

- O.

(3)11 YO,l

3 (3) + V3,2YO,l

= V3,2€2YO,0

-

, 2

sm 2T + P COS7

(2).

-

Q' 2 G sm T + V3,2 .

After substituting the following expression in the fourth equation of the system (2.6.53) y~2J ,

= a2 cos T + b2 sin T

(2.6.54)

and using the trigonometric relations, we obtain (2)"

Y1,O

+ Y1,0 (2) _ .!!.... _ 9X v" + gwp(v")3 + ~gwp(vll)2(a2 + b2) - p~ 2 0 0 2 0 2 2 +( -

al,Oa2 -

gX2j32b2 +

3gwp(v~)2j31b1 + ~gwpj3~b~

+~gWpj3~b2a~) cos T + ( -3gwp (vo")2j3 2a2 3 gwPVo"j32(b2 +2" 2 2

al,ob 2 + gX2j32a2

. T 2 sm + "43 gwp j323b22a 2 - "43 gwp j332a3)

2) - a2

cos 2T

+3gwpv~ j3ia2b2 sin 2T + ~(b~ - 3a~)gwpj3~b2 cos 3T +41 (2 a2 -

3 . 3T. 3b2) 2 gwpj32a2 sm

(2.6.55)

From the condition of periodicity of the S9lution, we get

-"I,oa. + ( - g5l..130 + 3gwp( v~)' (3. (gX.{3. - 3gwp(';,{)' (3. -

\

+ ~ gw p~A~ )

~ gw p~A~ ) ao -

"I

b. = 0,

,Db. = O.

(2.6.56)

where A~ = a~ + b~. For the non-zero a2 and b2 the main determinant of equation system (2.6.56) must equal zero. From this condition we obtain a1,O = 0, A 2 _ X2 - 3wp(v~)2 2 ~wpj3~

(2.6.57)

The particular solution of (2.6.55) is

y~~J

=

g2 P2

gX2v~ + gwp(v~)3 + ~2gwpv~j3iA~

-2"1 gwpvo"j322 (b22

2) - a2

cos 2T + gwPVo"j32. 2a2b2sm2T

( 2.6.58)

+ 32 (3a~ - b~)gwpj3~b2 cos 3T + ;2 (3b~ - a~)gwpj3~a2 sin 3T. 1

2.6 Parametric and Self-Excited Oscillation

69

Making use of (2.6.54) in the first and sixth equation of system (2.6.53), we obtain their particular integrals 3 '{32A2 + gwp v 1,2 (')3 Vo + -2-gwfJVo 2 2 V1,2

(1) 9 -, Y10 = ""2 - gXl Vo , P1

+ II~ 2 _ 1

1 (-9;\1111,2130

,

+ 3gwPI30Il~,2(V~)2 + ~gwP{j~A~) b2 COS T

{33 2)

.

(gX1 - V1,2{32 - 3gwp{32V1,2 2 (Vo')2 + -43 gwP 2A 2 a2 sm r + 21 V12 1 , 3V1 2 2, 2 2 + 2( 2' ) gwp{32 v o(b2 - a 2) cos 2r v 1,2

+ 2(

-

3V1

2 2 '_ v 1,2

(2.6.59)

1

2 ,

)

1

•

gwP{32 v oa 2b2 sm 2r

1 2 2 gwp{3~ 1 2 2 gwp{3~ . +-4 (b 2 - 3a2) 2 9 b2 cos 3r + -4(a2 - 3b 2) 2 a2 sm 3r V 1 ,2 -

V 1 ,2 -

9

and (3) -

Y1,0

-

0

(2.6.60)

.

The substitution of (2.6.54) into the fifth equation of system (2.6.53) gives (2) (2) YO,l + YO,l

=

(

-QO,l a2

1 - '212c2a2 + H- 2{32 b2 + 12Q ) cos r

+ ( -aD, 1b2 + ~'Y2g2b2 -

H2{j2a2

+ 'Y2P) sin T

+~12c2a2cos3r - ~12c2b2sin3r.

(2.6.61)

The following expression is obtained from the condition of periodicity after transformations and after assuming that P = Q: 4 QO,l

2 (- 2 2 1 2 2 p2 2) ( - 2 2 1 2 2) 2 + 20: 0 ,1 H 2{32 - 412c2 - A~ 12 + H 2{32 - 412C2 1 4 2 p2 2 _ p2 ( +"212c2 A2 - 212 {32 H 2A2 H 2{32 - 12c 2) = 2

o.

(2.6.62)

2

Hence

(2.6.63)

70

2. Discrete Systems

The particular integral of (2.6.61) is the following

Yo(2)1 ,

12 e ). =-2 ( a2 cos 3'T + b 2 sm'3'T 16

(2.6.64)

On the other hand, after substituting (2.6.54) into the second and seventh ,equation of (2.6.53), we find the particular solutions (1) YOI = ,

2

1 (1 1 (1 1

V 1 ,2 -

2 ) --2Vl,2/le2a2+Vl,2Hl{32b2+/lQ COS'T

2 -VI2 2/l e 2b2 - VI 2H- l{32 a 2 + lIP) sin'T 2v 1,2 - 1 2 ' , Vl,2/le2 Vl,2/l e 2 . . - ( 2 )a2cos3'T - 2( 2 _ 9)b 2 sm3'T, 2 v l ,2 - 9 v l ,2

+

(a) YOI =G+ ,

1 (1 1 (1 2

Va,2 -

+ 2 va,l - 1

1

(2.6.65)

2 e -Va2 2b2+ P ) COS'T 2 '

2 -2 va,2e2a2 -

Q) sm'T .

(2.6.66)

2 2 Va,2e2 Va,2 e2 . + ( 2 ) b2 cos 3'T - 2( 2 ) a2 sm 3'T. 2 va ,2 - 9 va ,2 - 9

Finally, let us consider the case of W ~ Pa. Periodic solutions are possible for the particular value of the parameter A,a: A, ~ 1 + e al,O + aO,l + 2 a2 ,O + 2 aO,2 + al,l + (2.6.67) J-L 2 e 2 J-L: 2 eJ-L 2 ... a2

.

I

Let us denote \2 2 \2 Al = vl,aAa, A,22 = V2,a 2 A,2a,

( 2.6.68 )

where Vl,a = PI/Pa, V2,a = p2/pa, and Vl,a and V2,a are assumed not to be integers. Similarly to the previous considerations, assuming that

y~~J('T) = Y~~6('T)

= 0

(2.6.69)

we obtain the following recurrent differential equation system from equation system (2.6.26) (1)" 2 (1) 2 (a) Yo,o - + Vl,aYl,o = (g + (}l Yo,o)

(1)"

2

(1)

2

(a).

[

Vl,a 2 -, a, a 2 p~ - Vl,aXl Vo + WPVl,a(Vo) ] ; .

YO,l +Vl,aYo,l = IlVl,aYo,osm2'T+/l(PSm'T+Qcos'T); (1)" Y2,O

2 (1) 2 (1) ( r'l2 (a)) [al,o 2 ~ , + Vl ,aY2,o = -Vl,aal,OYl,O + 9 + J&lYO,O p~ - Vl,aXlal,OVO - ( {3 (I), {3 (2), 2 (Vo')2 (al,O -Vl,aXl - lYl,O + 2Yl,O + WPVl,a -2-Vl,aVo,

2.6 Parametric and Self-Excited Oscillation

71

(2)')] + J'I £"}2 YI,O (a) [ P~ 1 - VI,aXI 2 - Vo , + WPVI,a a (Vo, )a] ; + {3 IYI,O (I)" 2 (l) _ 2 (I) 2 ((I) (2») 2 YO,2 + v I ,aYo,2 - -vI,aQo,IYO,I + VI ,a'1'I €IYO,I - €2Yo,I cos r (I) - {32YO,I (2)') +VI,a H- 1 ({3 IYO,I (a) + YO,I (a»). +VI2,a'1'I ( ao,IYo,o sm 2r; (I)" 2 (l) 2 ({I) (I») 2 ( (I) YI,I + VI,aYI,I = -VI,a aO,IYo,O + aI,oYO,I + VI ,a'1'I €IYI,O -

€2yi~J) cos 2r +

VI,aHI

({3Iyi~J' - {32yi~J')

0 +V~,a'1'1 (QI 'Oy~aJ + y~aJ) sin 2r + (g + {}~y~aJ) [a 2,1 " , Pa - a O,I Vo, - VI,aXI - ( - {3IYO,I (I), + {32YO,I (2»)] -VI2,2Xl 2 ( ')2 (2 V, QO,I {3 (I), {3 (2)') +WPVI,a Vo VI ,a o-2- - IYO,I + 2YO,I

(2.6.70)

£"}2( (a) (a) )( 1 2 -, a ( , )a) +Jel YO,I + Yo,o cos2r P~ - vI,aXIVo +WpvI,a Vo ; (2)" 2 (2) 2 (a) V2,a -" 2" a 2 YIO + V2 aYI 0 = (g + {}IYO 0) -2 - V2,aX2 Vo + WPV2 a(Vo) ] ; , J' ' P[ 2 ' (2)" 2 (2) 2 (a). 2 (P . Q ) YO,I + v2,aYo,I = v2,a'1'2Yo,0 sm r + '1'2 sm r + cos r ; (2)" 2 (2) 2 (2) ( £"}2 (a») [QI,O 2 - " Y2,0 + V2,aY2,0 = -V2,aQI,oYI,0 + 9 + JeIYO,O P~ - V2,aX 2Vo QI,O - ( - {3 IYI,O (l), (2)') + WPV 2,a ( Vo")2 -V22,aX2 + {32YI,0 2 . (aI,o 2 v2,a v" 0 - {31y{l), 1,0 + {32y(2)') 1,0 ] £"}2 YI(a)0 [ -2 1 2 -V " V2 ( II)a] ; +Jel , P-aV2' aX2 o + WP 2,a Vo (2)" 2 (2) _ 2 (2) 2 ((I) (2») YO,2 + V2,aYo,2 - -V2,aaO,IYo,I + V2,a'1'2 €IYO,I - €2YO,I cos 2r - ({3 (I), {3 (2)') +V2,a H 2 IYI,O - 2YO,I 2 ( (a) (a»). +V2,a'1'2 ao,IYo,o + YO,I sm 2r; (2)" 2 (2) 2 ((2) (2») 2 (l) YI,I + V2,aYI,I = -V2,a aO,IYI,O + aI,OYO,I + V2,a'1'2(€IYI,0

+€2y~~J) cos 2r - v2,aH2({3Iy~~J' - {32yi~J) 2

(

(a)

(a).

+V2,a'1'2 aI,oYo,o + YI,O) sm 2r £"}2 (a») [aO,I 2 " - ( {3 (2), +( 9 + JeIYO,O P~ - V2,aX2 a O,I VO - V2,aX2 - IYO,I

7'2

2. Discrete Systems + -

(2),) (32YO,1

2 ( ")2 ( ,,00,1 + WPV2 ,3 V o l.I2,3 V O

2

(l), (2)') ] 2( (3) (3) {3lYO,l + {32YO,0 + [11 YO,l - Yo,o

2-

cos 2) T

2 ( ")3] .,

1 - V2,3X2VO"+ WPV2 ,3 Vo . [ p~

o·

(3)" + y(3) 0,0 , Y 0,0 (3)" (3) _ (3). Yl,O + Yl,O - -Ol,OYO,O'

y~~l" + y~~l

=

-00,lY~~6

(3)" (3) Y2,0 + Y2,0

= -Ol,OYl,O - 02,OYO,0'

+

Y~~6 COS2T +

(3)

QsinT

PCQ;ST -

+

OJ

(3).

(3)" (3) (3) (3) (I) . 2 YO,2 + YO,2 = -ao,lYo,l - 00,2Yo,0 - clYO,l sm T (2) . (3) +c2YO,1 sm 2T + OO,lYo,o

cos 2T

(3) + YO,l

cos 2T

+ 00,1

a-.,

(3)" (3) (3) (3) (3) (3) Yl,l + Yl,l = -Ol,lYO,O - Ol,OYO,l - 00,lYl,o - clYl,O (I) . 2 + (3) -ClYl,O sm T OO,lYo,o

cos 2T

+

(3)" (3) (3) (3) (3) Yl,l + Yl,l = -Ol,lYo,o - Ol,OYO,l - OO,lYl,o (2) . (3) +c2Yl,0 sm 2T + Ol,OYo,o

cos 2T

00,1

G~',

+ 01,0

a- .

cos 2T + (I) . 2 ClYl,o sm T (3) YO,l

(3) + Yl,O

cos 2T

After substituting

y~36 ,

= a3

cos T

.

,

+ b3 sin T

(2.6.71)

into the twelfth equation of system (2.6.70) we get (3)" (3) Yl,O + Yl,O

For non-zero 01,0

a3

=

( -01,0 a3 COST +

and

b3

b3 sm . T ).

(2.6.72)

from the condition of periodicity we get

o.

(2.6.73)

= O.

(2.6.74)

=

Hence, (3) Yl,O

Making use of (2.6.71) in the first and sixth equation of (2.6.70), we obtain their particular solutions (l) Yl,O =

[g n? -2- + 2 Vl ,3

1 (a3

COST + b3

. T) ] sm

Vl,3 -

2 - + WP V l,3 2 (Vo')3] ,

. [ V?,3 p~ - Vl,3Xl VO

(2.6.75)

2.6 Parametric and Self-Excited Oscillation

73

2

n

9 + 2 1 (aa cos 'T + ba sin 'T) ] yi~J = -2[ v2 ,a v2 ,a - 1 .

[v~,a

2 -" p~ - V2,aX2Vo

" + WPV22 ,a(V o)

a]

(6

2..76)

.

After substituting (2.6.71) into the thirteenth equation in system (2.6.70), we have

y~~I" + y~~1 ~ {; + ( -aa,la3 + ~a3 + + ( -aa,lb3 - ~b3 -

Q)

p)

cosr

sin r

(2.6.77)

+ ~ (a3 cos3r + b3 sin 3r).

The condition of periodicity gives ~ 2 1

",,(1) _

'-L0,1 -

(2) _

a O,l -

+

P

,

a3

Q b '

-"2 -

(2.6.78)

3

The particular solution of this equation is the following

y~al,

= (; -

1 (aa cos 3'T + ba sin 3'T)

16

(2.6.79)

•

When substituting (2.6.71) into the second and seventh equation of system (2.6.70), we obtain their particular integrals (1)

YO,l

=

"Y1 2 v 1,a -1

v2

+ 2( ~,a

(v?,a) - ba + Q 2 "y

1 )

v1a , - 9

(2) _ YO,l -

"Y2 2 v2 ,a -

v2

1

(v~,a) -2 ba + Q

+ v 2 a"Y11,

-

ba cos 3'T)

cos 'T

1

(2.6.80)

,

+ v 2,a"Y2-

1

2

"V

+ 2( V 22,aa -12 9) 2,

(aa sin 3'T

COS'T

(

• 3 aa sm 'T

-

ba cos 3) 'T .

(2.6.81)

After substituting (2.6.71) and (2.6.73) into the fourteenth equation of system (2.6.70), we obtain the following from the condition of the existence of periodic solutions: a2,0

= O.

(2.6.82)

Analogously, taking (2.6.71), (2.6.79), (2.6.80), (2.6.81) into account in the fifteenth equation of system (2.6.70), we obtain equations which, after transformations, will assume the form

74

2. Discrete Systems

1 1) 1'2C2V~,a (1 1)

(1) _ 1'l c 1V?,a (v?,a 0:0,1 4 2

+

+ v?,a - 9

~22,a -

4

1+

~22,a - 9

1 (1) 1 P (C11'1 C21'2) +"2 00 ,1 - 32 - aa 2(v?,a - 1) - 2(v~,a - 1) ,

c V o (2) - - 1'l 1 ?,a 0,2 -

4

1'2 C2v~ a + 4'

~0(2)

_

- 2 0,1

(1 (1 v~,a

v?,a - 1

~_

1) 1) v~,a

+--=-2-v 1,a - 9

- 1+

- 1

Q ( C11'1 + c21'2 ). ba 2(v?,a - 1) 2(v~,a - 1)

32

(2.6.83)

(2.6.84)

The following algebraic equation system will be obtained from the condition of periodicity of the solutions of equation system (2.6.70) after substituting (2.6.71), (2.6.74), (2.6.75), (2.6.76) and (2.6.79) into the sixteenth equation of system (2.6.70): c2 c 2!1? c1c1!1? -O:l,l a a - 2(v?,a _ 1) ba + 2(vl _ 1) ba = 0, a C1 c1!1? c2c2!1~; -O:l,l ba - 2(v?,a _ 1) aa + 2(v~,a _ 1) da = 0,

(2.6.85)

where C1 =

2 v1,a

-2 -

2 - VI + 2 ( ,)a v 1a X1 O wpv1a Vo ,

PI'

,

2

v2,a

2 -

I

C2 = - 2 - v2 aX2 Vo P2'

+ wpv ' 2 a( V ,)a . 2 o ,

(2.6.86)

From the condition of a non-zero solution of equation system (2.6.85) in relation to aa and ba, we obtain

0:(1,~) = 1,1

± [!1? ( 2C2 C2 _ 2

v 2,a

-

1

C1 C 1 )]. 2 - 1 v 1,a

(2.6.87)

The coefficients of the series (2.6.67) are determined by expressions (2.6.73), (2.6.78), (2.6.83), (2.6.84) and (2.6.87).

2.6.4 Calculation Examples The analytically obtained diagrams of parametric instability zones are presented below in order to illustrate the influence of particular parameters of

2.6 Parametric and Self-Excited Oscillation

75

the system on their magnitude and position. The physical parameters of the system are given in the form in which they occur in differential equation (2.6.12). Figures 2.10 and 2.11 present the influence of the unbalance J.LP, the damping (J.LHt) , and the shape of the friction characteristic (0:/{3) on the magnitude of the parametric instability zones for PI and P2, for the following data: [J2 = 900 s-2, = 480 s-2, = 4800 s-2, 9 = J.LG = 9.81 m s-2, VQ = l O.4ms- , e = 0.2. On the basis of (2.6.14), PI = 73.32s- 1, P2 = 28.35s- 1, and P3 = 69.288- 1 have been obtained. The adequate coefficients assume the form ')'1 = 0.833, ')'2 = 0.12, {31 = 0.126, {32 = -0.674, el = 1.176, e2 = 0.376. The other quantities characterizing the system have been marked in the figures (when 0:/ (3 = 0:/ p). The parametric instability zones presented in Figs 2.10 (for pd and 2.11 (for P2) expand with the increase of the unbalance J.LP, while, depending on the value of the quotient 0:/ {3, this tendency can have different intensity. In the case of 0:/ {3 = 0.5 m 2s- 2, the doubling of the unbalance has caused the instability to be expanded twice for the zones corresponding to PI and P2. For 0:/ {3 = 1 m 2s- 2, the unbalance has been increased three times, which has brought about a comparatively small expansion of the instability zones for PI, while for P2 the expansion is still almost doubled. In the case of large unbalance of the rotor, the changes of the quotient 0:/{3 do not influence the magnitude of the parametric instability zones. The influence of damping on the magnitude of the instability zones corresponding to the frequencies PI and P2 is also very different. Small damping (J.LH I = 0.05 s-l) causes considerable shift of the zone for P2 in the direction of the growing value of the modulation depth J.L (J.L > 0.15). In the case of the double increase of damping the zone will not occur for J.L ~ 0.3. The magnitude and position of the instability zones for PI are not sensitive to the changes of the damping coefficient. In the case of J.LH I = 10s- 1 the proper zone of the frequency PI exists for J.L > 0.034. When J.LHI = 20s- 1 damping is doubled, and the lower border of the occurrence of the zone is shifted to the value of J.L = 0.07. The parametric instability zones for P3 are presented in Fig. 2.12. The magnitude of the zones depends on the initial conditions of the system motion. The diagrams have been prepared on the assumption that a3 = b3 = 0.01 m, where a3 = Y3(0), b3 = Y3(0). The calculations, in the case of the resonance coordinate Y:3, have been performed with with a precision of up to second order, hence the inclination of the unstability zones in the direction of the growing values of the parameter ..x~ has appeared. For the first approximation, the zones remain symmetrical in relation to the straight line ..x~ = 1. As in the cases considered above, the increase of the unbalance considerably expands the instability zone. The changes of the value of the quotient 0:/ {3 and damping have a negligible influence on the magnitude of the zone. Figure 2.13 presents the parametric instability zones for various

n?

w?

76

2. Discrete Systems

a)

0.7 0

b)

0.1

0.2

0.3

1.3 - r - - - - - - - - - - - - - - - - - - - - - , H1=0, /-LP=21·1Q-4 m .x 21 HI =0, /-Lp=7·1Q-4 m 1.2 HI=P=O

1.1 1.0

0.9 0.8 0.7+------.....--------,r---------l

o

c)

0.1

0.2

0.3

1.3,...--------------------,

.x~ 1.2

0.8 0.7+-------.--------,.--------l

o

0.1

Fig. 2.10a-c. Instability zones for PI

0.2

0.3

2.6 Parametric and Self-Excited Oscillation

a)

1.02 rHl=O, ttp=7·1Q-4 m

,\~

rHl=O, ttP=3.5·1Q-4 m

1.01

r

HI - P - O

c.

1.00

'-P:

O. ttHI 0.05s- 1

0.99 I

0.98

b)

o

0://J=0.5m2s- 2

1

0.2

0.1

J-L

0.3

1.02

,\~

rHl=O, ttP=14.1Q-4 m

1.01

rHl=O, ttP=7·10-4 m

r Hl =P=O ~

1.00

""'C.p 0, ttHI 0.05 S-1

0.99

10://J= 0.65 m 2s- 2 1 0.98

c)

o

0.2

0.1

J-L

0.3

1.02 ,\2

rHl=O, ttP=21.1Q-4 m

2

1.01

rHl=O, ttP=7·1Q-4 m , H1

1.00

p 0 ~ '-p 0, ttH} 0.05s- 1

0.99

0.98

2 2 I 0://J= 1 m s- ]

o

0.1

Fig. 2.11a-c. Instability zones for P2

0.2

J-L

0.3

77

78

2. Discrete Systems

2.0.......----------------~

\2 "'3

ttP=ttQ=21.1Q-4 m ttP=ttQ =7.10- 4 m

1.5

l.0t========:::::::::::=;;;;~

0.1

0.2

J-L

0.3

Fig. 2.12. Instability zones for 2 2 P3 (0./(3 = 0.5m s- , 0./(3 = 0.65m2 s- 2 , 0./(3 = 1 m 2 s- 2 )

values of the parameters {}2, {}? and w? For the zones denoted by 1 we get {}2 = 14400s- 2 , {}? = 1920s- 2 , w? = 19200s- 2 ; for the zones denoted by 2 we have {}2 = 3600s- 2 , {}? = 480s- 2 , w? = 4800s- 2 , and for the zones denoted by 3: {}2 = 900s- 2 , {}? = 120s- 2 ,"w? = 1200s- 2 . In all the cases the magnitudes of the other parameters are as follows: J-LHI = lOs-I, e = 0.2, o/f3 = 0.5m2 s- 2 , VQ = 0.5ms- 1 , J-LP = 0.0015m. As shown in Fig. 2.13a,b the growth of the squares of frequencies {}2, {}?, w? (resulting from the increase in rigidity of the elastic elements in the system, or from the decrease in the values of the masses) causes the instability zones for PI and P2 to expand. For example, when the parameters {}2, {}?, and w? increase by four times, it brings about an approximately doubled expansion of the zones. The unstability zone for pa is not influenced by the frequency changes in the system (Fig. 2.13c). Figure 2.14a, b presents the influence of the velocity changes of the belt VQ on the magnitude of the instability zones for PI and P2. Calculations have been performed for the data denoted by 1, except for the velocity VQ, whose value has been changed. In each case the increase in the belt velocity causes the expansion of the instability zones. For VQ < 0.3 JIlS- 1 these changes are less evident. The influence of the velocity changes VQ on the parametric instability zone for Pa is practically negligible. We can summarize the obtained results as follows. (1) The method of seeking a solution as a power series of the two perturbation parameters J-L and e used in the considerations makes it possible to

2.6 Parametric and Self-Excited Oscillation

a)

79

1.5 1

2

c

1.0

3

0.5J------..---------,..----------l o 0.1 0.2 {L 0.3

b)

1.5

\2 ""2

1 2

0.5

c)

3C

C

1.0

o

0.1

0.2

0.3

1.5

1.0

0.5+--------..---------,..-------~

o

0.1

0.2

{L

0.3

Fig. 2.13. Influence of parameter changes on the instability zones for: (a) Pl; (b) P2; (c) P3

80

2. Discrete Systems

investigate the single resonances of any order for the systems with weak nonlinearity and weakly modulated systems (J.L « 1). When we perform calculations with a p,recision of up to second order, it turns out that the limits of instability zones incline in the direction of the growing values of the parameter >.~ (Fig. 1.7). For the first approximation, the limits remain symmetrical in relation to the straight line >.~ = 1. (2) The parametric instability zones for Pll and P2 expand with the increase in the rotor unbalance. Depending on the value of the quotient alp, this tendency has different intensity. In the case of a/ (3 = 0.5 m 2 s-2, the double increase in the unbalance has brought about a considerable expansion of the unstability zones, for PI as well as for P2. For a/(3 = 1 m 2 s-2 the unbalance, causes a rather small expansion of the instability zones for PI, while for P2 the expansion is still almost doubled. In the case of a lack

a)

1.5

r

tb=O.4ms- I

; ; '\)=0.35ms- 1 I Vo 0.25ms-

r

"- tb-0.15ms-I

1.0

! \

0.5

b)

0

0.1

1.5

\2

A

2

c

1.0

0.5

o

c:

0.1

0.2

0.3

,;;=r

'\)=0.4 ms-

/r

vo=0.25ms- I

~

I

tb= 0.35 ms- I

tb°.15ms-I

0.2

Fig. 2.14. Influence of the belt velocity changes PI; (b) 112

J.L Vo

0.3

on the insta.bility zones for: (a)

2.7 Modified Poincare Method

81

of rotor unbalance the changes of the quotient a/{3 do not influence the magnitude of the parametric instability zones. The influence of damping on the magnitude of the instability zones corresponding to PI and P2 is also very different. The minimum damping (J.LHI = 0.05s- 1 ) causes a considerable shift of the zone for P2 in the direction of the growing values of the modulat,ion depth (J.LHI = 0.15s- 1 ). The magnitude and position of the instability zones for PI are not so sensitive to the damping coefficient changes. The regularities indicated here are the more clear, the greater the difference between the values of the frequency PI and P2 (i.e. for PI » P2) is. The increase in the unbalance also produces a considerable expansion of the instability zone for P3; however the changes of the parameter and of damping have no essential influence on the magnitude of the zone. The growth of the frequency squares n2 , n? and w~ causes the expansion of the instability zones for PI and P2. The parametric instability zone for the frequency P3 is not sensitive to the frequency changes in the system. In the case of the belt velocity increase (vo), the instability zones for PI and P2 are expanded. This property is noticeable within the range of great velocities (vo > 0.4ms- 1 ). The influence of the velocity changes Vo on the parametric instability zone for P3 is practically negligible. (3) For the frequencies PI and P2, the position of the instability zone limits does not depend in the first approximation on the initial conditions of the system motion. The magnitude of the instability zone limits for P3 depends on these conditions.

2.7 Modified Poincare Method 2.7.1 One-Degree-of-Freedom System Consider a one-degree-of-freedom nonlinear system governed by the equation [33, 117]

ddt + 2

y

2

W

2

Y = eQ

(d y, dt ,e

Y )

.

(2.7.1)

The above e > 0 is a small perturbation parameter and the function Q is analytical with regard to its arguments y, dy/dt and e for 0 < e < co. We are going to find a periodic solution to (2.7.1) depending on the perturbation parameter e. The system under consideration is autonomous, therefore we can arbitrarily take

dY(O) = 0 (2.7.2) dt ' because starting with the initial conditions for to, (2.7.1) will not change with a shift of time.

82

2. Discrete Systems

For c = 0 we have d2 y _ + w2 Yo - 0, Yo(O) = 0, dt and a solution to (2.7.3) is given by -2

(2.7.3)

yo(t) = A o cos wt,

(2.7.4)

where the amplitude A o is not yet defined. However, contrary to the previous investigations and following Proskuriakov [65d] , we are going to find an invariant orbit depending also on the second parameter b = b(c), where b(O) = O. Therefore, we have

y(O) = A o + b(c).

(2.7.5)

On the basis of the assumption of analycity of F, this function will be analytical also with respect to A o + b. We are focused on finding a periodic solution

y(t) = y(t + T), y(t) = y(t + T),

(2.7.6) (2.7.7)

and the period T is defined as T = To

+ a(c),

(2.7.8)

where

To = 21rw- 1 , a(O) = O.

(2.7.9) (2.7.10)

From (2.7.6) and (2.7.7) we get

y(To + a, A o + b, c) = y(O, A o + b, c) = A o + b, y(To + a, A o + b, c) = y(O, A o + b, c) = O.

(2.7.11) (2.7.12)

A general solution form of (2.7.1) is a function of the two parameters b and c: K

y(t, Ao + b, c) = (A o + b) coswt

+ LYk(t, A o + b)c k,

(2.7.13)

k=l

and K

y(t, Ao + b, c) = -w(A o + b) sin wt + LYk(t, A o + b)c k .

(2.7.14)

k=l

From (2.7.13) for t

Yk(O, Ao + b) = 0, Yk(O, Ao + b) = O.

=

0 and taking into account (2.7.2) and (2.7.5) we get (2.7.15) (2.7.16)

2.7 Modified Poincare Method

83

We develop the right-hand side of (2.7.1) into a power series of e in the neighbourhood of e = 0 putting fJ = 0 into series (2.7.13) and (2.7.14). We obtain (8 Q dy 8Q dy ad + F de e=O=O y e e=o=O y e=o=O 2 y + 8Q ) + e2(~ 8 Q (d )2 8e e=o=O 2 8 y2 de e=O=O 2 2Q +! 8 Q2 Y) + ~ (8 2 ) 2 8y de e=o=O 2 8e e=O=O 2 2 8 Q dy dy 8 Q dy 8 2Q dy +---+ --+ --8y8y de de e=o=O 8y8e de e=O=O 8i/8e de e=o=O 2 2 y 8Q d 8Q d y ) + O( e + - -2 +-2 8y de e=o=O 8y de e=O=O

. {. eQ(y,y,e) = e Q(yo,yo,O)

+e

(d

3)}

= e{ Q{yo,Vo)o +e (( ~~). YI + (~~)o VI + (~~)J

+e

2

[~ (~:~) °y? + ~ (~:~) °V? + ~ (~:~)

2 82Q ) . ( 8 Q) + ( 8y8i/ Yl Yl + 8y8e

2Q ( 8 ) 0 Yl + 8fj8e

+ (~~) °Yl + (~~) °V2] + 0{e

3 )}.

0

0

. Yl (2.7.17)

Now we demonstrate that using the periodicity conditions we are able to solve the problem, Le. to find A o, o:(e), and then fJ(e). To show this, let us begin with (2.7.12). From (2.7.14) and (2.7.2) for T = To + 0: we obtain y(To +

0:,

A o + fJ,e) = -w(A o + fJ) sinw(To +

0:)

(2.7.18)

K

+ LYk(To + 0:, Ao + fJ)e k = 0, k=l

which leads to the equation !<

- w(A o + fJ) sinwo:+ LYk(To +

0:,

A o + fJ)e k = O.

(2.7.19)

k=l

0:

The above solution allows us to find o:(e). For this purpose, we first express in an analytical form of e (2.7.20) o:(e) = eO:l(Ao + fJ) + e2O:2(Ao + fJ) + ... + ekO:k(Ao + fJ).

Each of O:k depends on A o,To and fJ and can be developed into the Maclaurin series in the neighbourhood of fJ = 0:

84

2. Discrete Systems

8Cik 1 82ak '2 ( ) 8A b + 2 8Ag b + ... , k = 1, ... , K. 2.7.21 o The corresponding ak can be obtained by differentiating (2.7.19) with respect to e and putting a = b = e = O. We find Cik(A o + b)

= ak(Ao) +

e

-w' An (~;) 0 + Yl(To,Aol = 0,

e'

-w'Ao

(~:~)o +2 [Yl (~;)o +y.] =0,

3

-w' An

(~:~)

e

+21

0

(2.7.22) (2.7.23)

+ 6{!i3 + (~~) 0 Yl + (~;)/.

(8a) 2 [oo. 8e 0 Y I

+"31 W 2 YI ] } ,

(2.7.24)

where 8 ka/8e k = Cik. If Yk are known then we can find Cik according to the subsequent equations (2.7.22), (2.7.23) and (2.7.24). Now we illustrate how we obtain Yk' For this purpose we have to reconsider the left-hand side of (2.7.1). Taking into account (2.7.14), we obtain

y(t, A o + b, e) = -w 2(A o + b) coswt +

K

L Yk(t, A o + b)e k.

(2.7.25)

k=l

Finally, from (2.7.1), (2.7.17) and (2.7.25) we get the following recurent set of linear differential equations (for b =r= 0) YI + W 2YI = Q(yo, Yo), . (2.7.26) Q Q Q .. (2.7.27) Y2 + w2 Y2 = (8fJy ) 0 YI + (88y )0.YI + (88e ) 0'

2Q 2Q 2Q 1 (8 ) 2 1 (8 ) .2 1 (8 ) Y3 + w Ya - 2 8y2 0 YI + "2 8 y 2 0 YI + 2 8e 2 0 2 2 82 Q ) . ( 8 Q) ( 8 Q) . + ( 8y8y 0 YI YI + 8y8e 0 YI + 8y8e 0 YI ..

2

_

+ (~~) /2+

(~~)

0

y.,

(2.7.28)

and so on, which allows us to find Yk(A o,To). To conclude, using condition (2.7.12) we have found ak, and therefore the unknown period T = To + a(e) is defined. Using condition (2.7.11) we illustrate how to find A o and b. For this purpose let us develop (2.7.11) into the Maclaurin series with respect to Ci

y(To,Ao +b,e) + Ciy(To,Ao +b,e) +

~a2Y(To,Ao +b,e)

1 a'" a Y (To, Ao + b, e) + ... - A o - b = O. +6

(2.7.29)

2.7 Modified Poincare Method

85

The left-hand side L of (2.7.29) can be described by the series

L = eLl (To, A o + b)

+ e 2 L2(To, A o + b) + ... + e k Lk(To, A o + b).(2.7.30)

Each of the Lk(To, Ao + b) term can be developed into the Maclaurin series with respect of b

8Lk 182Lk 2 Lk(To, Ao+b) = Lk(To,A o)+ 8A b+2" 8Ag b +... , k = 1 ... , K.(2.7.31) o Taking into account (2.7.31) in (2.7.30) we obtain the following evident form

~ (L k (rr' A) L = e L...J oLO, 0

o, A o)b + ~ 8 Lk(To, Ao) b2 + + 8 L k(T 8A 2 8A2 .. , 2

k=l

0 2

+

~ 8 Lk(To, A o)bk) k!

&A~

e

0

0 -.

k-l -

(2.7.32)

From (2.7.29) for e = 0 we get y(To, A o + b) = A o + b. Each of the terms y(m) = dmy/dt m depends on A o, To and b, and can be developed into the Maclaurin series in the neighbourhood of b = 0: (m) _ (m) 8y(m) ~ 8 2y(m) 2 y (To,Ao + b,e) - y (To,Ao) + 8A b + 2 8Ag b + ... o k 1 8 y(m) k + k! 8A~ b (2.7.33) Now for b = 0 we compare the terms standing by the same powers of e in (2.7.29), taking into account (2.7.20) and (2.7.30) and (2.7.32). First, we take b = 0 and from (2.7.29) we get

y(To, A o, c)

1

+ oy(To, A o,c) + "20?y(To, A o, c) + 6"1 a 2'"Y (To, A o, c) + ... - A o =

(2.7.34) O.

Comparing the coefficients of the same powers of e in (2.7.32) and (2.7.34), we obtain cO

el e2

Ll(To,A o) = yt{To,Ao) = 0, L2(To,Ao ) = Y2(To,Ao) + 0lYJ (To, A o ) - ~ll:~Aow2, . 2 L3(To, A o) = Y3(To, A o) + ll:2Yl (To, A o) + ll:lY2(To, A o) -Aow2ll:lll:2 - ~ll:~Yl(To,Ao).

(2.7.35) (2.7.36)

(2.7.37)

Equation (2.7.35), further called the amplitude equation, can possess two solutions: either yt{To, A o) = 0 or yt{To, A o) = O. In the first case this equation allows us to find the amplitude A o. If A o is real and not a multiplied root of the above equation, then it corresponds to the only real solution of (2.7.1).

86

2. Discrete Systems

In the latter case we divide (2.7.32) bye and L2(To, A o) = 0 will serve as the amplitude equation, and so on. We discuss further the first case. From (2.7.35) we calculate A o, and then (2.7.32) defines the following implicit fUl1ction of the two parameters band e

8L l 2 8L 2 1 282L l L(e, b) = eL 2 + b 8A + e L 3 + eb 8A + 2b 8A5 o o 3 2 8L3 1 2 82 L2 1 3 83 L l eb 4 +e L + e b 8A + 2 8A~ + '2 b 8A~ +.... o

(2.7.38)

According to the implicit function theorem, the number of branches of the implicit function b(e) is defined by the smallest power of bk in the following series 8L l 1 282 L l 1 383 L l 1 k 8k L l _ b b L(O, b) = b 8A + 2 8A5 + '6 8A~ + ... + k! b 8~ - O. (2.7.39) o According to (2.7.35) Ll (A o) = ydA o) and if all the derivatives up to the kth order 8LI/8Ao, 8 2 LI/8A~, ... , 8 k LI/8A~ vanish, then A o has kmultiplicity, Le. A~. In what follows k-multiplicity of Ao defines the number of branches of b(e). Each of the k-branches can be described by the 'following series b(e)

= Al/me l / m + A 2 / m (e l / m) 2 + A 3 / m (el/m) 3 + ... ,

(2.7.40)

where m equals one of the numbers 1,2, ... , k. In the case of fractional powers, the sum of the denominators cannot be greater than k. · For m = 1 the problem is reduced to the well-known situation of the classical series \

(2.7.41)

2.7.2 General Nonlinear Systems 2.7.2.1 Introduction. Let us consider an autonomous nonlinear system in the general form

dX -at =

F(X),

(2.7.42)

where:

10 F is continuous and X

E

IRn ;

2° F fulfills the Lipschitz condition: IIF(X) - F(Y)II ::; CIIX - YII, where C is constant; 3° a solution X is defined as: X = X(t, XCO)), where XCO) = X(to) (further we take to = 0).

2.7 Modified Poincare Method

87

Now we consider a set of first-order equations, which can govern not only mechanical, but also biological or chemical nonlinear dynamical systems. Of course, the set of second-order differential equations obtained using Newton's laws or the Lagrange equations can be easily reduced to (2.7.42). We are going to find only the periodic solutions to (2.7.42) with a period T > 0 taking the initial conditions X(O) in such a way that Z(T, X(O») = X(T, X(O»).

(2.7.43)

X(T,X(O») is the solution to (2.7.42), and X(O,X(O») = X(O). If we find the real X and T which fulfil the equation Z(T,X) = X,

(2.7.44)

the above map reduces the problem of finding the periodic solution of (2.7.42) to finding the fixed points of the map (2.7.44). Such an idea is often used during an application of numerical techniques to exhibit periodic solutions. 2.7.2.2 Autonomous System. Consider now the case where the righthand side of (2.7.42) depends on a small parameter e

s

= 1, ... ,n,

(2.7.45)

where the same assumptions as for (2.7.42) are valid and 0 ~ e < eo. Because (2.7.42) is autonomous, then for an arbitrary to if we take t + to instead of t in (2.7.45) this equation will not be changed. It implies that a solution to (2.7.45) does not depend on to and we can take to = O. Let us assume that for e = 0 the linear system (2.7.46)

has a family of periodic orbits with a period To. Let us assume that it corresponds to the existence of K pairs of V s eigenvalues ±ri/To, r = 0,1,2, ... , i 2 = -1. Even if they are multiple values but with simple elementary divisors, system (2.7.45) can be reduced to the form dxs

dt

=

-vsYs

+ eips(Xl,'"

,xk, Yl,···, Yk, Zl,···

,ZM, e),

dys . dt = vsxs + e1/Js(Xl,"" Xk, Yl,···, Yk, Zl,···, ZM, e),

d: = ~

dz·

L.- PijZi

+ ehj(Xl""

,Xk, Yb" . ,Yk, Zl,· .. , ZM, e),

i=l

s = 1,2, ... , K,

j = 1, 2, ... , M,

M

+ 2K =

n.

(2.7.47)

88

2. Discrete Systems For e

= 0 the linear system

dx s = -VsYs, dt

-

dys = VsX s ,

(2.7.48)

dt dz M - = LPijZi dt i=l

has a family of To periodic solutions depending on 2K arbitrary constants C s and D s of the form

Xs = Cs cos vst - D s sin vst, Ys = Cs sin vst + D s cosvst, Zj

(2.7.49)

= 0,

s = 1, 2, ... , K,

j

= 1, ... , n -

2K.

Because the system under consideration is autonomous, we can arbitrarily take one of Ys = 0 for to = O. For s = 1 we get (2.7.50)

Y1 = D1 = O. This means that instead of 2K variables we have to obtain 2K - 1. The solutions to system (2.7.47) have the following form

! ! t

Xl = (C 1 + Ci) cos v1t

+e

[CP1 COS)!"l (t - r) - t/J1 sin V1 (t - r)] dr,

o

t

Y1 = (C1 + Ci) sin V1t +e

[CP1sinv1(t - r) - tP1COSV1(t - r)] dr,

o

Xs = (C s + C:) cosvst - (D s + D:) sin vst t

+e

!

.

[CPs cos vs(t - r) - tPs sin vs(t - r)] dr,

o

Ys

= (Cs + C:) sin vst +

(D s + D:) cos vst

t

+e

J

[CPs sin vs(t - r) + t/Js cosvs(t - r)] dr,

s

=

2, ... , K,

o

! t

{z} = e[P]t{z·}

+e

o

e[P](t-r) {h}dr.

(2.7.51)

2.7 Modified Poincare Method

89

The above solutions for t = 0 have the following initial values

Xl(O) = C l +Ci(e), Yl(O) =0, xs(O) = Cs + C;(e), Ys(O) == D s + D;(e), {z} = {z·(e)},

s

= 2, ... ,K, (2.7.52)

where {z} = col(Zll ... ,Zn-2K), {z·} = col(zi,···,z~_2K)' [P] {h} =col(h l , ... , hn - 2K ), and these solutions possess a new period

+ a(e).

T = To

= [Pij], (2.7.53)

For e ---. 0 we have a(e) ---.0, C: ---.0, D: ---.0, {z·} ---. O. Now we show that solutions (2.7.51) fulfil (2.7.47). For this purpose we put (2.7.51) into (2.7.47) and we get

:t

{!

['P. cos 1I.(t - r) - ,p. sin 1I.(t - r)] dr } t

=

~

J

-vs

{!

[CPs sin vs(t - r)

+ 'l/Js cosvs(t - r)] dr + CPs,

o

['P. sin 1I,(t - r) +,p, cos 1I,(t - r)] dr } t

=

V

s / [CPs cos vs(t - r) - T/Js sin vs(t - r)] dr o

s= 1, ... ,K,

~

[j

e[P)(t-T) {h}

dr] = [P]

!

e[P)(t-T){h}

dr +

+ T/Js'

{h}.

(2.7.54)

'Ib prove the identity of the above equations, we use the Laplace transformation, remembering that t

.

Xl (r) * X2(r) == / Xl (r)X2(t - r)dr, o L[Xl(t) * X2(t)] = Xl (s)X 2(s), L[cos at] = L[sin at] =

s

(2.7.56)

2

+a

2'

(2.7.57)

2

+a

a

2'

(2.7.58)

S S

(2.7.55)

90

2. Discrete Systems

Therefore, we find that in fact the following equations are fulfilled 82 82

+ V;L(cps) -

8Vs 82

+ l/;L(tP s )

= - v [ 2 Vs 2 L (cp s) 8

+ Vs

+ 8 2 +8 V 2 L (tP s)] + L (CPs), s

2

8VS 8 L (CPS) + 2+ 2 L (tPS) 2 2+ 8 V 8 V

s

s

= Vs [ 8 2: V 2L (CPS) s

88 _

1 [p]L(h)

= [P] 8 _

8

2~

1 [p]L(h)

2 L (tP s )]

+ L(tPs),

Vs

+ L(h).

(2.7.59)

The temporarily unknown quantities C;, D: and z·, which depend on c, should be obtained in order to fulfil the periodicity conditions. As system (2.7.47) is analytical, therefore also its solutions, as well as the period, are analytical because of the initial conditions and the parameter c. The periodicity condition applied to solutions (2.7.51) will take the form

J T

Xl

(T) -

Xl

(0) = c

[CP1 cos V1 (T - r) -tP1 sin V1 (T - r)] dr

o +(C1 + C;)[cosv1T - 1] = 0,

J T

Y1 (T) - Y1 (0) = c

[CP1 sin V1 (T -17) - tP1 cos V1 (T - r)] dr

o +(C1 + C;) sin V1T = 0, T

xs(T) - xs(O) = c

J

[CPs cosvs(T - r) - 'l/Js sin vs(T - r)] dr

o +(Cs + C;)[cos vsT - 1]

= 0,

T

Ys (T) - Ys(O) = c

J

[CPs sin vs(T - r) + tPs cos vs(T - r)] dr

o

.

+(Ds + D:) [cos vsT - 1] = 0,

J T

{z(T)} - {z(O)}

= (e[P]T -

[I]) {z·} +c

eIP](T-r){h}dr = O. (2.7.60)

o According to the above equations, we have n equations with the unknowns

Ci, T, C:, D: (8 = 2, ... , K). It is easy to show that sin vsT = sin VsD:, cosvsT = COSVsD:,

(2.7.61)

2.7 Modified Poincare Method

91

and from the second equations of (2.7.60) it follows that

a(e)

=

ea:1(e).

(2.7.62)

For Ci = C: = D; = e = 0 (8 = 2, ... ,K) we get from (2.7.60) n equations with the unknowns 2K - 1, the initial conditions Cs , D s (D1 = 0), and the period To. In order to simplify further considerations we denote the left-hand side of (2.7.60) by F i (i = 1, ... , n), and additionally we introduce the new variables

XK = XK,

Y1

= XK+1,

To = aK+1, D2 = aK+2,

(2.7.63)

DK = a2K, •K+11 zl• = a2

Zn-2K =

Xn ,

The fundamental idea of the Poincare method is to find n parameters ai (i = 1, ... , n) and n functions ai(e) (i = 1, ... , n) from the equations

Fi(a1 +ai,· .. , a2K +a2K, a2K+1"'" a~, e)

= 0,

i = 1,2, ... , n.(2.7.64)

Because the general form of solutions (2.7.51) can not always be expressed by elementary functions (after necessary integrations), therefore to omit the potential difficulties, we look for the solution as a power series of the perturbation parameter e of the form co

Xi = ~,o

+

L Ri,l(a1 + ai, . ..

I

a2K

+ a2K, a2K+1"'" a~)el,

1=1

where

R1,O = (a1 R 2,o = (a2

+ an cos v1t, + a2) cos V2 t -

(aK +2

+ ai< +:G) sin V2 t ,

RK,O = (aK +ai<)cosvKt- (a2K +a2K)sinvKt, RK+1,O = (a1 + ai)sinv1t, R K+2,O = (a2 + a2) sinv2t + (aK+2 + ai<+2) cosv2t,

(2.7.65)

92

2. Discrete Systems

R2K,O

=

(aK

+ ai<) sin vKt + (a2K + a~K) cos vKt,

{:~:.o } e(P]'{a2~~+1 =

}.

(2.7.66)

For t = 0 from (2.7.65) and (2.7.66) we obtain

Rl,O(t = 0) = al + ai(c), R 2,o(t = 0) = a2 + ai(c), RK,O(t = 0) = aK + ai«c), RK+l,O(t = 0) = 0, RK+2,O(t = 0) = aK+2 + ai<+2(c), R2K,O(t = 0) = a2K + aiK(c), R 2K+l,O(t = 0) = a2K+l(c), Rn,o(t = 0) =

(2.7.67)

a~(c),

and ~,l(t = 0) = 0 for i = 1, ... ,no In general it can happen that some of the functions r i have the following structure F i = cf'i. The necessary periodicity condition is then equivalent to either c = 0 or f'i = O. The first assumption implies that the function Ri,O(t) is periodic for all parameters ai. In the second case we get <Xl

f'i

=L

R i ,l(al

+ ai,·.'·, T, ... ,a2K + aiK' aiK+l' a~)c'-l = O.

1=1

For c = 0 we have

(2.7.68)

a: (0) = 0, and from (2.7.68) we obtain

f'i(al, ... ,a2K,O, ... ,O,0) =0,

i = 1, ... ,2K.

(2.7.69)

Let us suppose that the algebraic nonlinear set of equations possesses one or a few solutions which we denote by

(2.7.70) and which can be multiplied. Then, we constrain such a solution to the periodicity condition

2.8 Hopf Bifurcation

Fi(iil +ai, ... ,li2K+a2K,a2K+I, ... ,a~,c)=0 i=l, ... ,n.

93

(2.7.71)

When we introduce ai = ... = a~ = c = 0, then we have the lefthand side of (2.7.71) equal to zero. We are going to find the functions ai(c) (i = 1, ... , n) on the basis of the theory of the implicit functions with the condition ai (0) = O. The functional determinant of (2.7.64) for (2.7.70) and for ai a~ = c = 0 has the form

an aal

Zr

1

an

i1 0 =

=0. ~rn al

(2.7.72)

~rn

an

If for iiI, ... ,iin we get i1 0 #- 0, then we have a simple solution, and in this case the unknown functions can be expressed by a~(c) = cAil +c 2A i2 +c 3A i3 + ... , (2.7.73)

which fulfils the condition ai (0) = O. If we get i1 0 = 0, then the solution is multiple and if the rank of the matrix built on the determinant is equal to n -1, then from (2.7.71) we exclude n - 1 quantities. We finally obtain a new nonlinear algebraic equation of the form

+ ai, c) = 0, where for 2K + 1 < i ~ F(iii

i

= 1, ... ,n,

(2.7.74)

n we have iii = 0, and i is equal to one of the numbers

from 1 to n. For the unknown iii the problem reduces to finding ai(c) as the implicit function of c in the form of the power series of either the total or fractional powers of the parameter c. The number of the solution branches as well as their series representations depend on the multiplicity solutions of (2.7.69). The latter is defined by the first nonvanishing term of 8 k F/ 8ai . In the case when the rank of the matrix is equal to n - 1, n is large, or when the rank of that matrix is less than n - 1, there is still a method proposed by McMillan [64d].

2.8 Hopf Bifurcation Bifurcation from equilibrium to dynamics is associated with the so-called Hopf bifurcation [116]. Up to now there have been many papers addressing this phenomenon, which is encountered in many branches of science [26, 27, 89, 91, 98, 15d, 16d, 18d-20d, 45d, 46d]. Here, we would like to present some examples of the analytical approach based on both the perturbation and harmonic balance methods similar to those obtained by Huseyin and his co-workers.

94

2. Discrete Systems

We begin with an analytical method of determining the post-critical family of the periodic solutions of some autonomous ordinary nonlinear differential equations dependent on one bifurcation parameter. It is known [116] that when the bifurcation parameter reaches the critical point, and when the system of linearized differential equations originating from the initial system of equations has complex conjugate eigenvalues of the Jacobian (the other values have non-zero real terms), and after fulfilling additionally the Hopf conditions, equilibrium path stability is lost and a family of periodic solutions appears. Denoting the bifurcation parameter as "T/", the following case has been considered: for T/ < T/c (where 1]c is a critical point) the system has a stable equilibrium path. For T/ = T/c the complex conjugate eigenvalues of the Jacobian of the characteristic equation occur (the others have negative non-zero terms), and together with an increase in T/ (T/ > T/c) the real terms of the eigenvalues become positive. The Hopf conditions ensure that the intersection of the imaginary axis with the complex conjugate eigenvalues occurs with non-zero velocity. A method for searching for bifurcation solutions was presented in [91,98]. Nevertheless, it is troublesome to use because of the time-consuming calculations for equations of dimension bigger than two. As the Hopf bifurcation occurs in nonlinear systems, it would be desirable to use for their analysis the analytical methods widely known in the field of nonlinear vibrations. The most popular and effective are the methods of harmonic balancing and the perturbation method. The main defect of the harmonic balancing method, which makes it useless for Hopf bifurcation analysis, is the necessity of a priori knowledge about the solution. An adv..antage of the perturbation method lies in constructing the solution by the subsequent solving of the perturbation equations of the linear differential equations, when it is only necessary to know the solutions of the undisturbed differential equation system. The combination of the two methods makes it possible to solve the Hopf problem (the method of harmonic balancing is used to solve each of the perturbation equations of the linear, differential equations). We consider the system of differential equations, whose characteristic equation is of the form (u - ut)(u - (2)P(U) = 0,

(2.8.1)

where

Ul.2 = e(T/) ± iW(T/),

w(T/c)

i= 0,

e(T/c) = 0,

°

a~c) i= 0,

and the roots of the polynomial P (u) = have negative real parts. It results from the centre manifold theorem [88] that the critical subsystem is mainly responsible for the bifurcation and bifurcated solution, and for the qualitative assesment of the bifurcated solution it is possible to limit oneself only to the solution of the two-dimensional critical differential equation. Here this solution serves as the initial approximate solution of the full nonlinear differential equation system, and the "detailed'· solution is determined by

2.8 Hopf Bifurcation

95

the method of slllccessive approximations. The latter also makes it possible to solve the problem where there are nonanalytical nonlinearities. Let us consider the differential equation system having the form

d dt (x) = F(l1, x),

n

x E IR

(2.8.2)

,

where T/ is the parameter vector and F (T/, x) is a nonlinear function, analytical in the state variables T/ and x. For the purpose of the further analysis it has been assumed that T/ is a one-dimensional bifurcation parameter. Let XQ fulfil the equation F(T/, XQ) =

o.

(2.8.3)

Examination of the stability of the equilibrium pathxQ is known to be limited to the determination of the eigenvalues of the Jacobian Fx(T/, xQ), where Fx(T/,xQ)

(~:~)

=

]

, (i,j = 1, ... ,n).

(2.8.4)

x=xo

Let the equilibrium path XQ for T/ < T/e (the critical value of the parameter) be the stable solution of system (2.8.2). On the other hand, when T/ = T/e the two complex conjugate eigenvalues cross the imaginary axis with non zero velocity, Le. let 0"1

= e(T/) + iW(T/),

w(T/e) = We

i= 0,

(2.8.5)

For T/ > T/e, the real parts 0"1, 0"2 become positive. A family of periodic solutions is created at the critical point. Let us assume that equation system (2.8.2) can be presented in the form .

Ku(T/)u

+ Kv(T/)v + K(T/,u,v),

U

=

iJ

= Su(T/)U + Sv(T/)V + S(T/,u,v),

2

IR , v E IRn - 2 , U

E

(2.8.6)

where x = colon(u,v), and the matrices KC*)(T/), and SC*)(T/) are the linear parts of the expansion of F(T/, x) into the Taylor series in the equilibrium path XQ. Let the characteristic equation (2.8.6) have the form of (2.8.1), while p ((j) = 0 has roots with negative real parts, and 0"1 and 0"2 are the eigenvalues of the two-dimensional matrix Ku(T/). The matrix K u, known also as the critical matrix, decides about the Hopf bifurcation. From the centre manifold theorem it follows that in the neighbourhood of the equilibrium path XQ = 0 there exists a function v = f(u) which in a sufficiently close neighbourhood XQ = 0 has the property 8f/8u = O. This allows us to assume to a first approximation that

v = f(u) =

o.

(2.8.7)

96

2. Discrete Systems

Taking into account (2.8.7) in (2.8.6), we obtain the following .

U

= Ku(Tf)u

+ K(Tf,u,O),

U

E

2

lR .

(2.8.8)

Later we shall assume that K(Tf, u,O) = K(Tf, u). Let us develop the matrix K(Tf) into a Taylor series in the neighbourhood of the critical point Ku(Tf)

= Ku(Tfc) +

1 2 K U17 (Tf - Tfc) + "2KUT1T1(Tf - Tfc) + ... ,

(2.8.9)

where etc. Let u(Tf) = 0 be the solution of (2.8.8) for 7J < Tfc, and for Tf = Tfc the periodic solution u(t; e) = u(t + T; e) of the period T bifurcates, which is dependent on one formally assumed small parameter e connected with the amplitude. After the transformations this parameter can be arbitrarily assumed to be e = 1. In order to obtain the period T = 211", we shall introduce the dimensionless time r = wt and, as a result, we will obtain the following expression from (2.8.8): d W(e) dt u(r;e) = K(Tf)u(r;e) + K(Tf, u(r;e)).

(2.8.10)

The periodic solution u( r; e) will be sou~ht in the form of a certain Fourier series, where the amplitudes and the freqttencies depend on the parameter e: K

ui(rj e) = 2:)Pik(e) cos kr

+ Tik(e) sin kr).

(2.8.11)

k=O

Because the system (2.8.10) is autonomous, then Tl1(e) = O. Moreover, pik(e), Tik(e), Tf(e) and W(e) are developed into a power series of the parameter e of the following form .()_ ' +1,,2 P,k e - Piko +Pike "2Pike

.()_ T,k e - Tiko Tf (e") -_ Tfc

W(e) =

We

+ ... ,

+' 1"2 + ... , Tik e + "2Tike

+' Tf e + "21 Tf " e 2 +

(2.8.12)

... ,

1 + W'e + "2w"e2 + ... ,

where Piko = Tiko = 0, because ui(rj 0) = 0 at the critical point. The solution of ui(rj e) is also sought in the form of a power series ui(rje) =

u~(r)e) + ~u~'(r)e2 + ... ,

(2.8.13)

2.8 Hopf Bifurcation

97

where K

up (r) = 2:)p~~ cos kr + T~J sin kr).

(2.8.14)

k=O

We can now proceed in two ways. We can either introduce relations (2.8.11)-(2.8.14) into (2.8.10) and by comparing terms in the same power c obtain the perturbation equations of the linear differential equations, or obtain these equations by means of successive differentiation of (2.8.10) with respect to c. As an example, let us consider the mechanical system with 1 ~ degrees of freedom, presented in Fig. 2.15. The vibration equations of the system have the form

fil Ultt = -k l Ul - k(Ul - ua)a + (au~ CUat = -kaua - k(ua - ut}a.

1J)Ult,

(2.8.15)

c Fig. 2.15. Mechanical system with 1 ~ degrees of freedom

The bifurcation parameter 1J is related to damping, and the other coefficients in (2.8.15) are positive. After applying the first step of the perturbations, we obtain

1 1 a 1 -3- aul + -1JUl! fil fil fil = -k l Ul - k(Ul - ua)a, ka k a = --Ua - -(ua - ut} . c c

WUl T = -U2 WU2T WUaT

(2.8.16)

98

2. Discrete Systems

The roots of the characteristic equations (2.8.16) are 0"12 ,

~2 (..2L.. ± (..2L..) 2 _ ml ml

=

4k l ), ml

k3

(2.8.17)

= --.

0"3

c

= TIc = 0 we have

For TI

±iJ

l (2.8.18) k . • ml The first two equations of (2.8.16), after the assumption of U3 = 0, have the form 1 1 a 3 WUl r = -TlUl + - U 2 - -3-Ul' ml ml ml (2.8.19) WU2r = -k l Ul - ku~, <112

= ±iwc =

and we get

fff fff

1, ml

l'

- u lr = -u2'

ml

' (2.8.20) -l u' = - klUI' ml 2r Having taken into account Til = 0, we will obtain the following solution of Eqs. (2.8.20)

,

,

u l = Pu cos T, U2 = -.JklmlP~l sin T.

(2.8.21)

From the second system of perturbation equations, we obtain " P2l

" = T2l " = T22 " = TI , = W ' = 0. = P22

(2 .8.22)

Finally, the third syste~ of perturbation equations, after taking into account (2.8.22), has the form 1 1 1 1 a -w u'" + -w"u' = --u'" + --n"u' - __ (u'}3 6 c lr 2 lr 6ml 2 2ml 'f 1 3ml 1 , 1

'"

6WcU2r

W U2r + 21"

=

-61 k'" lUI -

k(u ,l )3 .

(2.8.23)

Comparing the terms in sin T and cos T in (2.8.23), we obtain 1",

- 6 WcPll

1",

1",

= '2 w Pll + 6ml T2l ,

1 '" 1" , 9a ( , }3 + -2TI Pll - -4- Pll , ml ml ml w c P2l '" = k lTll' '" 1 ", kl '" 3 k( ') 1" ~ , 6WcT2l = -"6Pu - 4 Pu + 2w V 1\;1mlPUl 1

",

-6 WcTll = -6-P2l

(2.8.24)

2.8 Hopf Bifurcation

99

and therefore we have '" = T fI' = P21 21 = 0 ,

'" TH

'"

9 k ( , )3 -4:k PH , 1

"

="43 Vkk1 ml (')2 P11

P11 =

W

(2.8.25)

I

, ):2 . 7J " = '920: (PH

Comparing the terms in sin 3r and cos 3r, we obtain 1

'"

-'2WCP13

1

1 III 6ml T 231

=

'"

1

III

-2WcT13 =

-6- P23

1

'"

k IT IN 131

1

'"

'2WCP23 = '2WcT23 =

ml

kIlO

-(fP13

-

30: ( , )3 - 4 PH'

(2.8.26)

ml

)3 + 4:1 k(p'11·

and then we get ", 3 k ')3 P13 = -16 k (P11 , 1 '"

T 13 = III

P23

27 0: (')3 16 Vm1k 1 PH ,

(2.8.27)

9 (')3 = 160: Pu I

III

T 23 =

9

16k

Vff!!!(')3 k; P11 •

From the third equation of (2.8.16), we obtain k3 k 3 + -U 1 · e e After equating the terms in sin r and cos r of (2.8.28), we obtain WU3T

", T 31 -

= --U3

(2.8.28)

3k(p'11)3

---::----~~--__:_-

4C( ~+ (~r~)' (2.8.29)

On the other hand, after equating the terms in sin 3r and cos 3r of (2.8.28), we obtain

100

2. Discrete Systems rill -

k(pll)3

-~-_-.::::....=..::~---:=---~

33- 4C(3~+~(!;r~)' k 3k(pll)3

pili _

(2.8.30)

!;)

33 - 12c2 (3~ + ~ ( 2) .

We shall limit ourselves to terms in (Pll)3 in calculations. The periodic bifurcation solution has the form 3k ( I )3 , 1 (PH ')3 cos 3T UI = Pl1 cOST - 8 kl Pl1 cOST - 32 a (p/)3' (2831) r=-r:11 sm 3T, •. 32 ymlk l ~ I 3 I 3 3 k §I( I )3 . 3 u2=-ymlkIPl1sinT+ 16 o (Pl1) COS3T+ 16 VIC; Pl1 sm T,

9 +-

U3

=

3k3k(pll)3

4c2( ~ + (~ r) +

cOST

+

3k(pll)3

4c( ~ + (!; r~)

k(pl1 )3

4C(3~+~(!;r~)

sin T

cos 3T

i I

+

kk3 (pld

3

12c2(3~ + ~(!;

sin 3T

r) ,

Finally, we obtain the amplitude parameter-frequency relations

(k;

3

k

"

W=V~+8~(Pll)

2

(2.8.32)

and the parameter-amplitude relations 1]

=

9 2 0 (Pll) . 4 I

(2.8.33)

2.9 Stability Control of Vibro-Impact Periodic Orbit 2.9.1 Introduction It is well known that mechanical vibro-impact systems have been widely employed in both theoretical and applied mechanics for a long time. Vibroimpact dynamics can be observed in many real engineering systems, such

2.9 Stability Control of Vibro-Impact Periodic Orbit

101

as hammer-like devices, ball-and-race dynamics in a ball bearing assembly, wheel-rail impact dynamics, etc. [41d, 63d]. Nowadays again this field of research has attracted strong interest, but in the framework of theories of modern dynamical systems. Recent industrial examples (tube fretting wear through vibro-impact behaviour in nuclear reactors or impacts between old and high buildings excited by earthquakes) belong to additional but not satisfactorily solved questions of discontinuous dynamical systems. There are two parallel branches of investigations in the framework of vibro-impact dynamics. The first one is based on a better approximation of laws for impact motion and restitution coefficients, and it is more involved in the physics of materials. The second branch includes control of steadystate vibro-impact motion with the possibility of stability changes (either to destabilize or to st,abilize the vibro-impact attractor). Recently many papers have appeared, which are devoted to control of nonlinear oscillators, including also control of chaotic orbits [66d, 145]. In general, these methods could be devided for feedback control with a time delay [71d], sliding mode control [146], repetitive control [43d] , iterative learning control [25], adaptive control [146], and so on. The main purpose of these methods is to control complicated systems, even with imprecise knowledge of their mathematical models. However, the control of the attractor or repeller is based on the numerical observations of the results by the introduction of "helping" control coefficients. Theoretical prediction are rather not given. Here we address one, not yet satisfactorily solved problem of vibro-impact dynamics control with delay feedback and we give an analytical prediction of the proper choice of control parameters.

2.9.2 Control of Vibro-Impact Periodic Orbits We analyse the following one-degree-of-freedom vibro-impact system with one clearance presented in Fig. 2.16. The equation of dynamics is as follows:

x + ci + o:2 x = Po coswt + A [x(t) +B [x(t) - x(t - T)] and

x+ = x_,

x+ = -Rrx_

for

x(t - T)] + for x < x>5

5,

(2.9.1)

where: Po = Yok 2 /m, c = cI/m, 0: 2 = (kl + k 2 )/m, A = k 2 aI/m, B = k 2 bI/m, and T = 21f / w is the period of the periodic orbit being stabilized. A key point of this control is that a periodic solution possesses the same period as the excitation, Le. Xo = xo(t - T) and Xo is a particular solution of both the controlled and uncontrolled system [52d]. The delay loop is switched off where perturbations are not present. In the case of perturbations the controller causes the perturbation to vanish more quickly than in

102

2. Discrete Systems

x

a)

s

b)

Y

I

X ...

,(" Y - mX+Ct X + (k t +"=2)x= k 2 y I '< ~

~

~

'I'

Ys memory storage

-

.-

--

x

Ys = at[x(t) -x(t-T))+ f+-

+ bt[x(t) -x(t -T))

---

Fig. 2.16. One-degree-of-freedom kinematid,lly excited vibro-impact system with one clearance (a) and its control diagram (b) (8 denotes the clearance)

the case without control. The problem of analytical estimation of the influence of control coefficients for periodic orbit stability cannot be solved in a standard way. Here we propose the following approach. Because in practise the differences x(t)-x(t -T) and x(t) -x(t-T) are small we express them by introducing the small parameter c, which allows us then to apply the KBM method formally and next to take c = 1 [27d]. We assume damping of the same order as c and from (2.9.1), we obtain

x + Q2 X = Po coswt + cA [x(t) -

x(t - T)]

+cB [(1 - ~) x(t) - x(t - T)] .

(2.9.2)

Po 2 cos wt, -w

(2.9.3)

Introducing

x = z+

a

2

2.9 Stability Control of Vibro-Impact Periodic Orbit

103

we get Z + a 2z

= ef(a, 7], 1/J),

(2.9.4)

where ef(a, 7], t/J) = eA [z

-

+ 0: 2 Po 2 coswt -w

a2~w2COSW(t-T)] HB[(l- ~)

.(z 7] =

wt,

z(t - T)

a

[ow 2 sinwt) - i(t - T) + -w

a

[ow 2 sinw(t - T)], -w

't/J = at.

Using the KBM method we have truncated the e series up to order O(e) and we have 0 btained da 1 Aa . Ba dt = "2(B - c)a + 2a slnaT - """'2 coso:T,

dt/J A AI. = 0: - - + -cosaT+ -Bsmo:T. (2.9.5) dt 20: 20: 2 For A = B = 0 we get the uncontrolled solution, which supperts the validity of our approach. Therefore, we analyse the following equivalent solution -

x =

a

2 Po 2 cos(wt - w

+ rp) + eRt(G cos o:ot + D sin aot),

(2.9.6)

where: G = s-Pocosrp/J(0:~-w2)2+c2w2, D = (G/sin2/3'x)(e13c cos 2/3'x) , ,X2 = 0:2 - w 2, /3 = rrk/w, cosrp = (J(0:2 -w 2)2 +c2w 2/PO)[s(R r + l)x_ sin 2/3,X/'x(2 cos 2/3,X - e- 13c - e13C )]. After integration of (2.9.5), we get a(t) = Goe Rt , t/J(t) = aot + ()o, 1

A

B

= "2(B - c) + 2a sin aT - 2" coso:T, A A 1 ao = a - 2a + 2a cos aT + "2 B sin aT,

R

(2.9.7)

and according to (2.9.7) and (2.9.6) one obtains G

= Gocos()o,

D

= -Gosin()o,

Go

= JG2 + D2.

2.9.3 Stability Control From (2.9.6) it is seen that when R < 0 the assumed solution is stabilised more quickly in comparison to the case of R = O. However, the problem of the stability investigation of the vibro-impact state is much more subtle.

104

2. Discrete Systems

Before impact number l, the mass possesses the velocity Xl-. This causes the following perturbation solution to occur

+ bXl = e RTI

x

[(C + bC,) cosaO'Tl

+F COS(W'Tl

+ cP + bcpl),

+ (D + bD,) sinao'T,] F

=

Po 2' a -w

(2.9.8)

2

A new time 'T is measured from the l-th ilmpact 'Tl = 'T + b'Tl. For example, the next impact occurs for 'Tl+l = 21r /w + b7" where b7i denotes the perturbation period T = 21r /w. After some calculations we get

+ bC, cos ao'T + Do:obn cos ao'T +bD, sin ao'T + IM'TlC cos ao'T + R8'TlD sinao'T) -Fbcpl sin(w'T + cp) - Fwb'Tl sin(w'T + cp), e RT [2Ra ob'Tl (D cos 0:0'T - C sin aoT) + mC, cos 0:0'T

bXl = e RT (-Ca:ob'Tl sin ao'T

bIl =

+IMD, sin 0:0'T + R 2 b'TlC cos 0:0'T + R 2 b'TlD sin 0:0'T -a5b'Tl (CcosO:O'T + Dsinao'T) - bC,o:osino:o'T + bD,o:O cosao'T]- Fwbcpl cos(W'T + cp) - Fw 2 b'Tl cos(W'T

+ cp).

The following boundary conditions are introduced l: l

'T

+ 1:

= 0, 'T

=

bTl = 0, bXl = 0, bIl = bXl+, 21rk - + b'Tl, b'Tl = b7;, bXl = 0, w

(2.9.9) bXl

= bI(l+I)-'

After some calculations we obtain the loequations bC, - Fbcpl sin cP = 0,

e 2f3R

(2.9.10)

. 2,8ao + w {Cv C l cos 2,80:0 + bD, sm 1 (6cpl+1I. -

bcpl)

. [(ao D + RC) cos 2,6<>0 + (RD - aoC) sin 2,6aol } - 6CI +1 = 0, R r e 2f3R { 6C, (Rcos 2{3ao - 0:0 sin 2,80:0) + bD, (Rsin 2,8ao 1 + ao cos 2,80:0) + - (bcpl+lI. - bcpl) [(R 2 C - 0:5C w

+ 2Ro:oD)cos2{3ao + (R 2 D - 0:5D - 2RaoC) sin2,8ao] } +R8C +1 + ao bD,+lI. - (R r + 1) Fwbcpl+l coscp = 0, ' where ,8 = 1rk/w and Rr < 1 as usual denotes the restitution coefficient. Assumming that l

bCPl = bcpo

+

E wbT

i ,

i=1

(2.9.11 )

2.9 Stability Control of Vibro-Impact Periodic Orbit

105

and introducing ~D, = a21',

bG, = a1 1',

b'{J, = a31'

(2.9.12)

we get the following characteristic equation

b2 1 2 + bll. 1 + bo = 0,

(2.9.13)

where

b2 = aD { F sin", -

+ (RD -

~ e2J'R [(aD D + RC) cos 213ao

aoG) sin 2}3a:o] },

bl = e 2J'R { ~ "'" [( RC + aDD)

+ (RD -

(cos 213a o - R,e2J'R)

aoG) sin 2}3ao] - (R r + I) Fw cos '{J sin 2}3a:o

+F sin", I( R, bo = R,aoe4J'R

1) aD cos 213ao + (R, + 1) Rsin 213ao I },

[~ (Re + aDD) -

(2.9.14)

F sin "'] .

Note that G(s) could be obtained using a similar approach but without the perturbations. Therefore, the problem of stability is reduced to consideration of the second-order characteristic equation (2.9.12). If the roots of (2.9.13) are 111,21 < 1, then according to the assummed solutions (2.9.12) bG" bD, and b'{J, approach zero for l ---+ 00, and the solutions will be asymptotically stable. We can easily estimate the stability regions, which are defined by the following inequalities

b2 < 1 bo

and

b1

bo + b2

<1.

(2.9.15)

Taking into account (2.9.14) it is now easy to find parameters of the system (or a delay loop) which fulfil inequalities (2.9.15). Additionally, for mechanical reasons, we have x( t} < s. 2.9.4 Simulation Results During numerical simulations we used the MATLAB-Simulink package and the MATLAB-model for (2.9.1) with the boundary conditions (Fig. 2.17). We have taken the following parameters: m = l[kg], k 1 = 3[N/m], k2 = I[N/m], C1 = 0 [Ns/m], Xo = l[m], T = O.47l'[s], I4 = 0.65, s = O.OI[m], = -O.OI[N/m], b, = -0.045[Ns/m]. For these parameters according to (2.9.7) we get R = -0.04, which shows that the delay loop control coefficients A and B allow us to obtain quicker

a,

106

2. Discrete Systems

Pocos(wt} Object

Delay x

o +....- - - - - - - - - 1 L:J-_........_~

Clock

"'-~~"-_--I Delay x'

Fig. 2.11. MATLAB-Simulink model of the vibro-impact system presented in Fig. 2.16

damping of free oscillations in the solution (2.9.6) than without control (Fig. 2.18). Additionally, for the given parameters we have found from Eq. (2.9.13) that 1"Y1,21 lie closer to the origin for the system with the control coefficients than without control (with the delay loop; 1"Y1,21 = 0.62, whereas without the loop 1'1'1,21 = 0.65). " For the given parameters numerical simulations confirm the analytical predictions. It can be seen in Fig. 2.18 that with control the transients vanish more quickly than in the case without control. In the case presented above the periodic orbit is achieved after about 24.1 seconds for the system analysed without the delay loop and after 21.8 seconds for the system analysed with the delay loop (IJ.LI < 10- 4 ), respectively.

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom 2.10.1 Definition The normal mode of a finite-dimensional system behaves like a conservative system having a single degree of freedom. In this case all position coordinates can be well defined by

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

107

3,-------.---------r-----r------r-----, x

10

i

4

~-.l . .~- . ---.\, ,

2

,I

, I

\

I

\

I

\

I

1

\

- - - - ,\ - -

,

---1

- ---

--

1 1 1

.--.

h

".

.~-~---I-~~

_/ without control ...,,

~

1.---.. . .

0

I

~

--

--+-11---'"'"'=1:=-1

'-'

H I

-

with control

.--. -1

----~----~-l-

~

'-'

H

-2

-4 L...20

--'-

-'-

24

22

J...-

26

----'-

28

--'

t[sec} 30

Fig. 2.18. Difference between two transients x(t) - x(t - T) approaching a periodic orbit for the system (al = -0.01, bt = -0.045) Xi

= Pi(X),

X -

Xl,

i

= 2,3, ... , n,

(2.10.1)

Pi (x)

being analytical functions. Rosenberg [135] gets credit for being the first to introduce broad classes of essentially nonlinear conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space (2.10.2)

"A phenomenon like 'vibration in normal modes' exists in nonlinear systems; in a purely verbal manner, this motion may be described as vibration in which all masses move periodically "in unison", among all the properties of normal vibrations we consider asYmptotic the basic one that the normal coordinates decouple the equation of motion" [135]. For instance, homogenous systems whose potential is an even homogenous function of the variables refer to such a class. It is interesting to note that the number of modes of normal vibrations in the nonlinear case can exceed the number of degrees of freedom. This remarkable property has no analogy in the linear (non-degenerate) case. Naturally this definition may now be called "naive", and the problem of a more accurate definition was studied by many authors [55d]. But it is very clear from the intuitive point of view and we accept Rosenberg's definition form the asymptotic standpoint in our further investigations.

2. Discrete Systems

108

In systems of a more general type, trajectories of normal vibrations are curvilinear. Lyapunov [54d] showed that solutions of this kind exist in nonlinear finite-dimensional systems with an analytical first integral, which are close to generating linear systems.

2.10.2 Free Oscillations and Close Natural Frequencies! For systems with two degrees of freedom which have quadratic nonlinearities, the internal resonance at the frequency ratio 1:2 has been studied, along with the internal resonance for systems with cubic nonlinearities and the frequency ratio 1:3 [119]. In recent years attention has turned to mode interactions (of an internal resonanse type) for close natural frequencies. Experimantal observations and solutions of particular problems show that this effect is relevant to the description of oscillatory processes in suspension bridges [1], cylindrical shells and other constructions [41-52,54-77, 28d-35d]. However, the literature does not contain any general analysis of mode interactions of free oscillations in nonlinear systems with close natural frequencies. In particular, we do not know what types of oscillation modulation are possible, what determines the degree and period of energy exchange in a system, what is the number of steady-state regimes (without modulation), which of them are stable, etc. Consider a nonlinear oscillatory system (initially, in general, with damping), described by the equations

.. Uk

+ 2J.L.Uk . + W.2 Uk = bkkUk3 + b12UIkU~, 3-k

k =, 1 2.

(2 .10.3 )

The frequencies WI and W2 are assumed to be close, and the damping factors for the two modes are taken to be the same. Equations (2.10.3) represent the general case of a system with symmetric potentials (when J.L. = 0) that include terms of the second and fourth degree. They are similar to the Duffing equation for systems with one degree of freedom and describe a broad class of mechanical systems (in general, no restrictions are imposed on the coefficients bij ). In accordance with the method of multiple scales we introduce the "fast" and "slow" times To = t, TI = eTo, T 2 = e2TO (the time TI will not be necessary below). We will seek a solution of system (2.10.3) in .the form of an expansion Uk = eUkl(To, TI, T 2, ... )

+ e3uk3(To, T 1, T 2, ...) +...

(2.10.4)

(terms of order e 2 vanish for a system with cubic nonlinearities). The smallness of J.L. and the difference in frequency are introduced through the conditions (2.10.5) 1

By courtesy Ye.V. Ladygina, A.I. Manevitch [1061

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

109

Taking into account that

d 2 a a D a Cit = Do + eD l + e D 2 + ... ; Do = 8T ' Dl = aT ' 2 = aT ' l 2 o 2 d 2 2 2 dt 2 = Do + 2eDoDl + e ( 2DoD2 + D l ) + ... , we obtain the following systems of equations for the two approximations

D5ukl + w2Ukl D5uk3 + w2Uk3

= =

0,

(2.10.6)

-2D o(D 2Ukl + J.Lukd + bkkU~l +b12utlu~lk - c52kO'ukl,

(2.10.7)

where c5ij is the Kronecker delta. The solution of system (2.10.6) is written in the form

Ulk = Ak(T2) exp(iwTo) +

Ak(T2) exp( -iwTo),

(2.10.8)

where the bars denote complex conjugats. Substituting (2.10.8) into system (2.10.7), from the condition that there are no secular terms in the resulting equations, we have ,

2 -

-

-2iw(A k + J.LA k ) + 3bkk A kA k + 2b1 2Ak A 3-k A 3-k +b12AkA~_k - c5 2k O'A k = 0,

(2.10.9)

where the prime denotes differentiation with respect to T 2 . Putting the complex amplitude in the exponential form A k = 1/2ak exp(iOk) (k = 1,2), we separate (2.10.9) into real and imaginary parts and obtain a system of equations governing the amplitude and phase modulation of both modes:

(a%)' + 2J.La% = (-I)kb12(4w)-la~a~sin2')', 8w8~ = -3bkka~ - b12a~_k(2 + cos 2')') + 4c5 2k O',

(2.10.10) ')' = 82 - 81. (2.10.11)

Having eliminated sin 2')', we obtain from (2.10.10) the integral (2.10.12) a~ +a~ = Eexp(-2J.LT2) = Eexp(-2e 2J.Lt), where the arbitrary constant E is proportional to the energy of the system (in the first approximation). In particular, for a conservative system (J.L = 0) a~ + a~

= E.

(2.10.13)

From (2.10.11) we obtain the equation for the phase difference ')': Bw')" = (3bl1 -2b12)a~ + (2b12 -3b22)a~ +b 12 (a~ -a~) cos 2')' +40'.(2.10.14)

For further analysis it is convenient [1] to introduce the new variable = a?/ E (0 < < 1). Then, from (2.10.11) with k = 1 and (2.10.14) we obtain a system of equations governing the amplitude-frequency modulation in variables:

e

e

e, ')'

110

2. Discrete Systems ~' = -2J1.e ')" =

+ r oe(1- e) sin 2')',

er1 + r2

1

+ r o( 2 -

(2.10.15 )

e) cos 2')'.

Here

ro = r2

=

b12E

r _ (3bll - 4b12

4w'

8w

1-

(2b12 - 3b22)E 8w

+ 40-

+ 3b 22 )E '

(2.10.16)

.

Without loss of generality we assume ro :/= 0, because otherwise b12 = 0 and system (2.10.3) decomposes into two decoupled equations. We will perform the further analysis for the case of a conservative system (J1. = 0). Dividing the second equation of (2.10.15) by the first one, we obtain d')' e (rl - r o cos 2')') + r2 + 1/2ro cos 2')' (2.10.17) de r oe(1 - e) sin 2')' The solution of this ordinary differential equation is roe(1 - e) cos 2')' + r 1e 2 + 2r2e = C, (2.10.18) where C is a constant ofintegration and determines the trajectory in the (e, ')') plane of the "amplitude-phase portrait" of the system (the AP-portrait). Eliminating')' from the first equation of (2.10.15) and (2.10.18), we obtain 1 (de) r~ dT 2 = F12(e) - F22(e),

2

(2.10.19)

2 rle +1,2r2e - c. ro The form of this equation is identical with that derived in [1] for the case when W2 = 3wl, but the functions Fl(e) and F2(e) are of a different form. The condition Fl (e)

= e(1 -

e),

F2(e)

=

(2.10.20) for the solution of (2.10.19) to exist means that solutions correspond to the parabolic segments F2(e) inside the domain bounded by the parabolic arcs ±F1(e) over the interval [0,1] (Fig. 2.19). The points of intersection of the parabolas F2(e) with the axis e, as can be seen from (2.10.18), correspond to the condition cos 2')' = 0, Le. ')' = ±(2n+ 1}rr/4 (n = 0,1,2, ... ), or the values ~ = 0 and e = 1. The points of intersection of the parabolas F2(e) and F 1 (e) correspond to extremal values of the functions e(T2 ) and ak(T2 ), respectively (k = 1,2). It follows from (2.10.15) (for J1. = 0) that at these points sin 2')' = 0 (if e:/= 0 and e:/= 1), Le. ')' = ±n1r/2 (n = 0,1,2, ...), and that the points on the lower curve (Fl < 0) correspond to the even values of n and those on the upper curve to the odd values. Consequently, the minimum and maximum values of e determining the amplitude modulations and the degree of energy exchange between the modes are equal to the roots of the equations F 1(e) = ±F2(e), Le. the equations

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

111

(2.10.21 ) where the upper and lower signs correspond to the points of intersection of the parabola F 2 (e) with the upper and lower curves ±Fl (e), respectively, while the value of C is governed by the initial values eo and /0. 0.4 .,----,......----., ~~

F1

A

!

,

.-

,~,.,

"

,

.~-.

,

~--r-------,

3\.

o ~._-~_. ~j"..-:~._. ~

,

0.4

0.4 .,...---'.,•.1-----, Fl

o

~._-j------

-~/~.

.. k"·~"

2"><~--~ F 1

-0.4 +--.--...-------1 o 0.5 ~ 1

-0.4 - f - - - - t - - - - - - i 0 0.5 1

a)

e

1/'

-0.4 +--"'---1------1 o 0.5 e 1

b)

c)

Fig. 2.19a·c. Graphical representation of the oscillatory regime

The solutions of (2.10.21) and the construction of a "characteristic graph" (Fig. 2.19) give a graphical representation of the oscillatory regime. There are two basic ways in which the curves Fl(e) and ±F2(e) can intersect in a "coarse" system, corresponding to the two basic oscillatory regimes: 1. both the points of intersection lie on the same parabola +Fl (e) or - Fl (e) (curves 1 and 2 in Fig. 2.19a); 2. the parabola F 2(e) intersects both the parabolas +Fl(e) and -Fl(e) in the interval [0, 1] (curve 3). In the first case, the phase difference / oscillates about the value / = ±mr/2. Synchronization of the oscillations proceeds "on average" over the modulation period: at the times when the extrema al and a2 are achieved the oscillations on both degrees of freedom proceed either in phase or in antiphase, if both the points of intersection lie on the lower branch, or the phase difference in these instants is equal to rr/2, (3rr /2) if both the points lie on the F l > 0 branch. In the second case the phase difference increases monotonically, running through the sequential values nrr /2 (n = 0, 1, ... ) at the extremal times. These two types of oscillatory regime with oscillating and monotonically increasing phase differences will respectively be called modulations of the first and second type. The solution of (2.10.19) has the form

± Irol J( ~

1

Fl2 (e) - F22 (e)) -1/2 de

= T2 -

T20,

eo = e(T20).

(2.10.22)

~o

Suppose el,'" ,e4 are the roots of the fourth-degree polynomial Fr(e)arranged in increasing order, with 6 and 6 lying inside the domain

Fi (e),

112

2. Discrete Systems

bounded by the parabolas ±Fl(e). The modulation semiperiod (for the oscillating phase case) corresponds to e varying over the interval (6, e3), and so the modulation period is equal to {3

T· = IFJ _ 2Frll/2

f ((e - 6)(e - 6)(e - 6)(e -

e4))-l/2 de· (2.10.23)

{2

For "non-coarse" systems one must consider singular cases for the position of the F 2 (e) curve (Figs 2.19b and c): the passage of F 2 (e) through the points e = 0 or e = 1 (Fig. 2.19b), and "external" touching of the parabolas F l (e) and F 2 (e) (corresponding to lines 1 and 2 in Fig. 2.19c). In these cases two of the roots ej coincide: in the first case 6 = 6 = 0 or 6 = e4 = 1, and in cases 2 and 3 6 = 6. Because the improper integral in (2.10.23) diverges when two of the roots ej coincide, the modulation period tends to infinity as these regimes are approached. These are "boundary" regimes separating the modulations of the two types distinguished above (lines 1 and 3) and associated with separatrices in the plane (e, 1), or regimes of stationary oscillations without modulation (curve 2). We remark that the" aperiodic" oscillations described in [ld] correspond to these boundary regimes. Consider the possible AP-portraits in the plane (e,1) which are given by integral (2.10.18) and which graphically describe the oscillatory modes of the system. Stationary points corresponding to oscillations with no modulation are found using (2.10.15) from the system of equations

.

I \

eo - e) sin 21 = 0, eFl

+ F2 + F2(1/2 -

e) cos 21 = 0,

which can have the solutions 2F2 = 0, cos 21 = - - , Fo

e

e= 1

1,

cos 21 =

= ± ~1r (n =

(2.10.25)

2(F~ + F2) , Fo

0, 1,2, ...),

(2.10.24)

e= e: = ±~/~ ~oF2.

(2.10.26)

(2.10.27)

These solutions exist when the following conditions are satisfied: (1) 12F21 < IFol, (3) 0 ~ e-:- < 1,

(2) 21Fl + F21 < IFol, (4) 0 < e: ~ 1.

(2.10.28)

Using the periodicity with respect to 1, we confine ourselves to the plane rectangle (0 < e < 1; 0 < 1 < 1r). The stationary points (2.10.25)-(2.10.27) can be positioned on the boundary lines of this rectangle and on the mid-line 1 = 1r /2 (with not more than one point on aline). It is easiest to investigate the nature of a stationary point with the help of (2.10.18), considering the

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

113

form of the integral curves in the neighbourhood of the stationary point. The stationary points at ( = 0 and = 1 are saddle points and therefore unstable. From this it follows that the presence of a second degree of freedom makes oscillations along the first generalized coordinate unstable if the stationary points (2.10.25) or (2.10.26) exist. In the neighbourhoods of the stationary points on the lines 1 = ±n1l" /2 the trajectories can be of either elliptic or hyperbolic type, and consequently these stationary points can be stable or unstable. The stability conditions for odd and even n, respectively, have the forms

e

(5) Fo(Fo + FI) > 0,

(6) Fo(Fo - FI) < O.

(2.10.29)

On the characteristic graph (Fig. 2.19c) the "externally" touching hyperbolas (curve 1) correspond to the stable stationary points and the "internally" touching ones (curve 2) to the unstable points. The stationary points on the lines 1 = ±n1l" /2 correspond to synchronous single-frequency modes, Le. normal oscillations of the nonlinear system. It follows from (2.10.3) and (2.10.8) that the points on the lines 1 = 0 and 1 = 11" correspond in the (U1, U2) configuration space to the two straight lines U2 = ±hU1' where

h

=

/ a2 = (1_e;)1/2 = (Fo/2 - F1 - F2)1 2 a1 Fo/2 + F2

e;

The stationary points on the lines 1 = 11"/2 and 311" /2 correspond to the ellipses

u2

1

et

+

u2 2 1- et

= Eg2,

e

e

and the points on the lines = 0 and = 1 correspond to straight lines along the axes OU2 and Ou 1. The separatrices pass through the possible unstable stationary points. For separatrices passing through the "left" points (2.10.25) one should put C = 0 in (2.10.18). We obtain equations for two branches: (2) cos 21 = -

eF1 + 2F2 no(1- ),

e

(2.10.30)

which exist when condition 1 of (2.10.28) is satisfied. The "right" separatrix, passing through the stationary points (2.10.26), exists when condition 2 is satisfied. The equations of the branches of this separatrix are obtained from (2.10.18) with C = F 1 + 2F2 (1)

e=

1,

(2) cos 21 =

(e + 1~~ + 2F2 .

(2.10.31)

The central separatrix (CS) passing through the stationary points (2.10.27) for odd (or even) n exists when condition 3 (condition 4) is satisfied and condition 5 (condition 6) is violated. Substituting the coordinates of points

114

2. Discrete Systems

(2.10.27) into (2.10.18), we obtain C = (-F2 ± Fo/2)2 /C~Fo - Fd and the equations for the branches of the central separatrix

e= B± VB2_D, Fo cos 21' + F2 - Fo cos 21' - Fl'

B -

D -_

(F2 =F Fo/2)2 . (Fl ± Fo)(Fl - Fo cos 21')

(2.10.32)

The stationary points and separatrices possess the following properties. 1. If the "left" stationary points (2.10.25) exist (i.e. condition 1 is satisfied), then in the rectangle (0 < ~ < 1, 0 ~ l' < 'IT) there is at least one "intermediate" stationary point (2.10.27) on the line l' = 1r /2 or l' = 0, and this point

is stable. Indeed, when condition 1 is satisfied the sign of the numerators in condition 3, 4 is given by the sign of their first term, and for their moduli we have I ± Fo/2 - F21 < IFol. If the signs of Fl and Fo are the same, then the sign of the denominator in condition 3 is the same as the sign of the numerator, and because we then have IFl + Fol > IFol, condition 3 is satisfied and, clearly, condition 5. In the case of opposite signs for Fl and Fo, the signs of the numerator and denominator in condition 4 are the same and IFl - Fal > IFo!' so that conditions 4 and 6 are satisfied. A similar assertion holds for the "right" stationary points (2.10.26).

2. If one stationary unstable point (2.10.27) exists on the line l' = 1r /2 (or l' = 0), then a stable stationary point exists on the line l' = 0 (or l' = 1r /2); and there are no separatrices (2.10.30) ati.d (2.10.31). Suppose condition 3 is satisfied and condition 5 is not satisfied (Le. the stationary point at l' = 1r /2 is unstable). Then Fo and Fl have opposite signs and IFll > IFol. It follows from condition 3 (because the sign of the denominator is governed by the sign of F 1 and is opposite to the sign of Fo) that the signs of Fo and F2 are the same and IF21 > IFl l/2. Condition 1 is therefore violated. Considering the case F l > 0 and F l < 0 separately, and taking into account that the sign of F 1 is opposite to the signs of Fo and F2 and that IFll > IFol, we find that in both cases condition 2 is violated, and the (right) inequality in condition 4 is also violated, which proves the assertion. These properties enable us to describe the various possible AP-portraits in the 'plane (e,1'). Each side separatrix (SS) joins two unstable stationary points at = 0 or ~ = 1. The branches of these separatrices surround a single stable stationary point at l' = 1r /2 or l' = 0 (0 < e< 1). One can verify that if, for example, between the "left" separatrices there is a point on the line l' = 0, then the abscissa of the point of intersection of the separatrix with the line l' = 0 is twice the abscissa of the stationary point it is obvious that ~ < 1/2. A similar property is satisfied by the right separatrix: here it is necessary for the stationary point surrounded by its branches to be in

e

e;

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

115

e

the right half of the rectangle. The separatrix originating from = 0 cannot intersect the line ~ = 1, and conversely. The branches of the CS join the two unstable stationary points (2.10.27), corresponding to even or odd values of n, and surrounding the stable stationary points. The CS cannot intersect the lines = 0 or = 1. Inside the domains surrounded by the SS or CS a modulation regime of the first type exists, and outside these domains, there is a regime of the second type. Thus, four qualitatively different types of the AP-portrait, governed by conditions 1-6, are possible and they are shown in Fig. 2.20

e

1 0

a)

b)

e

1 0

c)

1

d)

Fig. 2.20a-d. Four qualitatively different types of the AP-portrait

1. Conditions 1 and 2 are satisfied. There are stable stationary points at "y = n1l" /2 (2.10.27) for even and odd n, in the left section (e < 1/2) and right section (e > 1/2) of the rectangle, Le. three stable normal modes exist (and two trivial unstable ones Uk = 0, k = 1,2). Each of the stationary points is surrounded by the corresponding SS; there are no CS (Fig. 2.20a). 2. Only one of conditions 1 and 2 is satisfied. There is a stable stationary point (2.10.27) only for an odd or even n, and only one SS (on the left if condition 1 is satisfied, and on the right if condition 2 is satisfied); there are no CS (Fig. 2.20b). All the normal modes, apart from the single Uk = 0, k = 1 or k = 2 mode, exist and stable modes also exist that are either rectilinear (if condition 4 is satisfied), or elliptic (when condition 3 is satisfied). 3. Neither condition 1 nor 2 is satisfied, but condition 3 is satisfied. The stable and unstable stationary points (2.10.27) alternate (with the point for odd n being stable if condition 5 is satisfied). There is a CS, but no SS (Fig. 2.20c). Three normal modes exist, where either the rectilinear one is stable (when conditior.. 6 is satisfied), or the elliptic one is stable (when condition 5 holds). 4. Conditions 1-3 are not satisfied. There are no stationary points (normal modes) or separatrices. All oscillatory modes are of modulation type 2, with the modulation being relatively small if compared with cases 1-3 (Fig. 2.20d).

116

2. Discrete Systems

In cases 1-3 one can distinguish subcases. In. case 1 there are two subcases distinguished by the position of the left stationary point: on the line '"Y = 0 or on '"Y = 11"/2. Similarly, in case 3 the stationary point can be stable at '"Y = 0 or at '"Y = 11"/2. Four subcases are possible for case 2: a left or right separatrix, and a stationary point at '"Y = 0 or '"Y = 11"/2. The corresponding AP-portraits can be obtained from those shown in Fig. 2.20. We introduce the parameters

°

bu

b22 40' 02 = b12' 0' = b 2E ' 1 Then, conditions 1-6 can be represented in the form 01

= b12'

(1) (2) (3) (4) (5) (6)

302 - 3 < 0'0 - 301 + 1 < 0'0 - 301 + 1 < 0'0 30 2 - 1 < 0'0 302 - 3 < 0'0 - 301 + 3 < 0'0 01 + 02 > ~, 01+02<2.

< 302 ~ -301 < 302 < -301 < -301 ~ 302 -

1, + 3, 1 + 1 +3 3

for for for for

01 01 01 01

+ + + +

02 > ~, 02 < 3' 02 < 2, 02 > 2,

(2.10.33)

(2.10.34)

Unlike 0' and E, the dimensionless frequency detuning parameter 0'0 does not depend on the choice of c and can be written in the following form 0'

°

-

40'. --"....------,,,..--

- bI2(U~(0)

+ u~(O))'

(0'.

= c 20' = w22 -

2)

WI .

(2.10.35)

As can be seen from (2.10.34), the ty~e of AP-portrait is determined by the relative positions of the points Cl = 302 - 3, C2 = 302 - 1, d1 = -301 + 1, d2 = -301 + 3

(2.10.36)

and the quantity 0'0. Four possible positions of the intervals (Cl' C2) and (dl,d 2) are shown in Fig. 2.21 (C2 < d 1,Cl < d 1 < C2,Cl < d 2 < c2,d2 < C2)' The type of the AP-portrait (easily determined from (2.10.34)) is shown above the intervals. In case (a) the interval (C2' dd contains the stable stationary point at '"Y = 0 (11"), Le. the rectilinear normal mode is stable, and the unstable one is at '"Y = 11"/2 (311"/2) (i.e. elliptic). In case (d) these points (and normal oscillations) "exchange" stability. Figure 2.21 graphically demonstrates the influence of the parameter 0'0 on the'system behaviour. If 0'0 lies in the interval

01 <

0'0

< 02,

01 = min(cl' dd,

02 = max(C2' d 2),

(2.10.37)

then we have the AP-portraits of types 1-3 with stationary points and pronounced modulation of the amplitude and phase (energy exchange). If 0'0 lies outside this interval, the AP-portait of type 4 is indicated with relatively small modulation. Thus, condition (2.10.37) allows one to specify the smallness of the frequency detuning parameter. The minimum width of interval (2.10.37) is 2. The centre of the interval is the point

2.10 Normal Modes of Nonlinear Systems with n Degrees of Freedom

a)

ci

2°

3°

2°

4°

~

1°

2°

'"'"

d)

V~

~

3°

2° ci

~~M ~~

~~~

~

dl

2°

4°

~

ci

dl

Fig. 2.21a-d. Four possible positions of the intervals

4° ~

ttW#A

~

(Cl l C2)

and (d l , d 2 )

3 b22 - bll

3

a. = 2(a 2

~

dl

4°

2°

4°

2°

W/~

~ "'~~

dl 4°

1°

2°

4° ci

W§Aa

c)

b)

4°

117

-al)

=2

b12

.

In the case when b u = b22 we have al = a2: a. = 0, Le. interval (2.10.37) is symmetric in relation to the origin. If al 1:- a2 the interval is displaced relative to the origin and for sufficiently large la2 - all (or Ib22 - buD the point crO = 0 can turn out to lie outside the interval. One must also take into account that the sign of crO is governed by the sign of b12 (one can always put cr. > 0, Le. W2 > WI)' The signconstancy condition on crO indicates either the positive or negative part of interval (2.10.37) (if it exists). Two conclusions follow from this: 1. it is not necessary for the large modulation to correspond to the smaller value of cro: combinations of the coefficients bij are possible with the AP-portraits of types 1-3 for intervals with crO far from the point OJ 2. for certain combinations of bij only the type 4 AP-portraits are possible, irrespective of the energy and frequency separation. In the above analysis there is a natural separation of the influence of the oscillation energy and the ratio of the initial amplitudes of the two modes on the energy exchange. The quantity E acts on crO according to (2.10.35) (increasing E is equivalent to decreasing cr), and together with cr. it therefore determines the type of AP-portrait. The initial amplitude ratio ~o determines the phase trajectory in a given AP-portrait. Consider the special case when bu = b22 = 0, b1 2 1:- O. Then al = a2 = 0, CI = -3, C2 = -1, d l = 1, d 2 = 3, Le. we have Fig. 2.21, case (a). Condition 5 is not satisfied, while condition 6 is satisfied. For -3 < crO < -I' we have the type 2 AP-portrait with a left separatrix and stable stationary point at 1 = 0 (1r), Le. with rectilinear normal oscillations. For -1 < crO < 1 the AP-portrait is of type 3 and has a stable stationary point at 1 = 0 (1r) and an unstable one at 1 = 1r /2 (31r /2), Le. with stable rectilinear normal modes and an unstable elliptic mode. When 1 < crO < 3 the AP-portrait is of type 2 with a right separatrix and stable rectilinear normal modes. Finally, for crO < -3 and crO > 3 the AP-portrait is of type 4.

118

2. Discrete Systems

In conclusion we note that the numerical integrations of (2.10.3) performed for the purpose of estimating the accurancy of the solution obtained by the multiple scales method demonstrated almost complete agreement between the analytic and numerical solutions in all the cases considered with an arbitrary choice of e ~ 0.1 and amplitudes of up to 0.5 (the error in determining the amplitude was of the order of 0.1%). But when e was increased beyond 0.1, the error increased rapidly. For example, when e = 0.15 the error in the amplitude computation reached 30%.

2.11 Nontraditional Asymptotic Approaches 2.11.1 Choice of Asymptotic Expansion Parameters Introducing a small parameter into the nonlinear problems is a very delicate and nontrivial matter. We consider in this section an elementary illustrative problem: finding the roots of a fifth-degree polynomial [49, 10d]. We are concerned here with finding the real root Xo of the polynomial equation

+x =

x5

1.

(2.11.1)

We have chosen the degree of this polynomial to be 5 because it is high enough to be sure that there is no quadrature formula for the roots. However, one can be sure that there is a unique real r09t Xo and that this root is positive because the function x 5 + x is monotone fucreasing. Using Newton's method we compute Xo = 0.75487767 ...

(2.11.2)

There are several conventional perturbative approaches that we could use to find xo. One such approach, which we will call the weak-coupling perturbation theory, requires that we int[loduce a perturbative parameter e in front of the x 5 term [49]: ex5

+x

= 1.

(2.11.3)

Now, x depends on e and we assume that x(e) has a formal power series expansion in e: '

X ( c) =

ao + ale + a2e 2 + a3e3 +....

(2.11.4)

To find the coefficients we substitute (2.11.4) into (2.11.3) and expand the result in an asymptotic series in powers of e. We ,find that the coefficients an are integers which oscillate in sign and grow rapidly as n increases: ao = 1, al = -1, a2 = 5, a3 = -35, a4 = 285, a5 = -2530, as = 23751,

(2.11.5)

etc. In fact we can find a closed-form expression of an valid for all n,

2.11 Nontraditional Asymptotic Approaches an

= [( -1)n(5n)!] [n!(4n + I)!],

119

(2.11.6)

from which we can determine the radius of convergence R for the series in (2.11.4):

44

R = 55 = 0.08192.

(2.11.7)

Evidently, if we try to use the weak-coupling series in (2.11.4) directly to calculate x(I), we will fail miserably. Indeed, using the seven coefficients in (2.11.5) for e = 1 gives 6

x(l) =

2: an =

21476,

n=O

which is a poor approximation to the true value of x(l) in (2.11.2)! Of course, we can improve the prediction enormously first by coupling the [3/3] Pade approximants and then by evaluating the result at e = 1. Now we obtain the result x(l)

= 0.76369,

(2.11.8)

which differs from the correct answer in (2.11.2) by 1.2%. A second conventional perturbative approach is to use a strong-coupling expansion. Here, we introduce a perturbative parameter e in front of the term x in (2.11.1) [49]:

x 5 + ex = 1.

(2.11.9)

As before, x depends on e and we assume that x(e) has a formal series expansion in powers of e: x(e) = bo + b1e + b2e 2 + b3e 3 + .... (2.11.10) Determining the coefficients of this series is routine and we find that 111 bo = 1, bl = -"5' b2 = - 25' b3 = -125' _ 21 _ 78 b4 = 0, b5 - 15625' b6 - 78125' (2.11.11) etc. Again, we can find a closed-form expression for bn valid for all n,

r (4n-l) :

bn

= - 5r (¥) n!

(2.11.12)

from which we can determine the radius of convergence R of the series in (2.11.10) 5 R = 44 / 5 = 1.64938. .. . (2.11.13) Now, e = 1 lies inside the circle of convergence so it is easy to compute x(l) by summing the series (2.11.10) directly. Using the coefficients listed in (2.11.11), we have

120

2. Discrete Systems 6

x(l)

=L

bn

(2.11.14)

= 0.75434,

n=Q

which differs from the true result in (2.11.2) hy 0.07%, a vast improvement over the weak-coupling approach. Now we use the o-expansion method to find the root XQ. We introduce a small parameter 0 in the exponent of the nonlinear term in (2.11.1) [49],

+x

x 1+6

1,

=

(2.11.15)

and seek an expansion for x(o) as a series of powers of 0:

+ CIO + C202 + C303 + ... .

x(o) = CO

(2.11.16)

The coefficients of this series may be computed easily. The first few are 1 1 1 CO = 2' Cl = 4 1n2 , C2 = -gln2, 1

1

3

1

C3

2 = - 48 In 2 + 32 In 2 + 16 ln 2,

C4

= 32 In 2 - 64 In 2 - 32 In 2,

C5

= 480 In 2 - 768 In 2 - 128 In 2 + 64 In 2 + 64 In 2,

1

3

3

1

7

5

1

5

C6 = -192 In 2

1

2

3

4

35

+ 1536 In

4

3

3

2

1

5 3 5 2 1 2 + 768 In 2 - 128 In 2 - 128 In 2,

etc. The radius of convergence of the 0 series in (2.11.16) is 1. A heuristic argument for this conclusion is as follows. The radius of convergence is determined by the location of the nearest singularity of x(o) in the complex -0 plane. To find this singularity we differentiate (2.11.15) with respect to 0 and solve the resulting equation for x'(o): x 1+6 1n x

x'(o)

=

-1

+ x 6 (1 + 0)"

Since x( 0) is singular where its derivative ceases to exist we look for zeroes of the denominator 1 + x 6 (1

+ 0)

= O.

We solve this equation simultaneously with (2.11.15) to eliminate 0 and obtain a single equation satisfied by x:

o=

x In x

+ (1 -

x) In (1 - x).

The solution to this equation corresponding to the smallest value of 101 is x = O. From (2.11.15) we therefore see that 0 = -1 is the location of the nearest singularity in the complex-o plane. In fact, as 0 decreases below -1, (2.11.15) abruptly ceases to have a real root. This abrupt transition accounts for the singularity in the function x(o).

2.11 Nontraditional Asymptotic Approaches

121

Clearly, to compute Xo it is necessary to evaluate series (2.11.16) at fJ = 4. For this large value of fJ we use the coefficients in (2.11.1) and convert the Taylor series to [3/3] Pade approximants. Evaluating the Pade approximant at fJ = 4 gives x(fJ

= 4) = 0.75448,

(2.11.17)

which differs from the exact answer in (2.11.2) by 0.05%. The fJ series continues to provide excellent numerical results as we increase the order of the perturbation theory. If we compute all the coefficients up to C12 and then convert (2.11.16) to a [6/6] Pade approximant, we obtain x( fJ

= 4) = 0.75487654,

(2.11.18)

which differs from XQ in (2.11.2) by 0.00015%. Last but not least, we may introduce a small parameter in our equation in the following way [10d]:

xe

-1

+X

c « 1.

= 1,

After substituting x = ye one obtains y + ye

= 1.

Taking into account the relation c e = 1 + c In c

+ ...

we may represent y in the form y = c

+ o(clnc).

Then, we have

and for c = 1/5, x 3.9%).

~

0.724780 (the error of the first approximation is only

2.11.2 6-Expansions in Nonlinear Mechanics (49] Let us start with the solution of a simple nonlinear differential equation. Consider the nonline,ar ordinary differential equation problem

y'(x) = [y(x)]n, y(O) = 1.

(2.11.19)

The exact solution of this problem is

y(x) = [1 - (n - l)x]-l/(n-l).

(2.11.20)

To solve (2.11.19) approximately using the fJ expansion, we let n = 1 + fJ and solve

y'(x) = [y(x)P+6.

(2.11.21)

122

2. Discrete Systems

To solve (2.11.21) perturbatively, we can seek a solution y(x) in the form of a. series in the powers of 8: (2.11.22) For example, Yo(x) satisfies the linear differential equation problem

Yo = Yo(x),

Yo(O) = 1,

whose solution is x

Yo = e .

Indeed, all functions Yn (x) satisfy linear differential equations which are easy to solve. We find that 1 x x, 2 Y2(X) = e x [1"3x3+ gX 1 4] , Yl(X) = 2'e etc. The reason for using a perturbative approach if that, in general, even when one cannot solve the nonlinear differential equation, the differential equation for the perturbation coefficients Yo(x), Yl(X), Y2(X), ... are always linear and therefore can be solved in quadrature form. . For the particular problem (2.11.21) a closed-form solution exists. Therefore, we can determine the radius of convergence R of series (2.11.22): 1 R=~.

We have computed the series in (2.11.22) up to the 8 10 term. Let us examine the numerical accuracy of the 8 s~ries. The exact value of y(x) at x = 1/4 for the case n = 4 (8 = 3) is y

n)

= 1.587401.

(2.11.23)

Directly summing the 8 series 2:~ 8nYn(1/4) gives 1.284 when n = 0 (19% error), 1.404 when n = 1 (11.5% error), 1.470 when n = 2 (7.4% error), 1.5099 when n = 3 (4.9% error), 1.5626 when n = 6 (1.6% error), and 1.58128 when n = 10 (0.39% error). We can also compute a Pade approximant from the 8 series and then set 8 = 3. The [3/3] Pade approximant gives 1.58692 (0.03% error) and the [5/5] Pade approximant gives 1.587395 (3.7 x 10- 4 % error). It is numerical results such as these that encourage us to use the 8 expansion to solve difficult nonlinear differential equations. Now we turn to a more complicated problem. The classical anharmonic oscillator is defined, by the nonlinear ordinary differential equation d2y dt 2

3

+ Y + ey = O.

(2.11.24)

2.11 Nontraditional Asymptotic Approaches

123

We impose the conventional initial conditons

y(O) = 1,

(2.11.25)

y'(O) = O.

Our objective here will be to find the period of the anharmonic oscillator. It is well known that the initial-value problem (2.11.24) can be solved exactly in terms of elliptic functions and the period T can be expressed exactly as an elliptic integral ,..

J ~

T = 4

dO [1 + ~ (1 + sin 2 0)] -

1

~.

(2.11.26)

o

The integral in (2.11.26) can be expanded as a series in the powers of e 21 ] 3 T = 21T" [ 1 + -e + _ e 2 + ... 8 256

-I

(2.11.27)

One cannot use the conventional perturbation theory to find the period T for small lei. It is true that when lei is small the exact solution y(t) approximates the motion of a harmonic oscillator of period 21T". However, solving the Duffing equation perturbatively requires some subtlety. If we seek a conventional perturbative solution for y(t) as a series in powers of e, we find that there is a resonant coupling between successive orders in the perturbation theory. As a result, the coefficient of e in the pertubation series for y(t) grows linearly with t, the coefficient of e 2 grows quadratically with t, the coefficient of e 3 grows like t 3 , etc. Thus the perturbative solution is only valid for times t which are small compared with 1Ie. At such short times we cannot use the perturbation expansion for y(t) to obtain the series expansion in (2.11.27). More sophisticated perturbative methods have been devised which enable us to calculate y(t) perturbatively for times t "'" lie and thus to obtain the series in (2.11.27). One such method is called multiple-scale perturbation theory. We will attack (2.11.24) using the 6 expansion and will find that here, too, the methods of multiple scale perturbation theory must be used. To use the 6 expansion we replace y3 by yl+26 and consider the differential equation d2 dt 2

+ Y + (w 2 -

1)yl+26 = 0,

y(O) = 1,

y'(O) = O.

(2.11.28)

In (2.11.28) we have found that it is convenient to set e = w2

-

1,

(2.11.29)

so that when 6 = 0, (2.11.28) describes a classical harmonic oscillator whose frequency is w. Note, also, that y26 is to be interpreted as the positive quantity (y2)6. Thus when we expand y26 as a series in powers of 6 we obtain

y26 = 1 + 6In(y2)

+

62

63 2" [In(y2)]2 + "6[ln(y3)]3

+"',

(2.11.30)

124

2. Discrete Systems

in which the argument of the logarithm is always positive and no complex numbers appear. Let us begin by trying to find a conventional perturbative solution to (2.11.28) as a series in powers of 6 00

y(t)

=

L 6nYn(t).

(2.11.31)

n=O

Substituting (2.11.31) into (2.11.28) and using (2.11.30) we obtain a sequence of linear equations and associated initial conditions which must be solved. The first few read as

~~o + w2yO = ~~l

+W

2

2

y~(O) = 0,

Yl = _(w 2 -1)Yoln(Y5),

~;'2 + W 2 Y2 = ddtY3 2

Yo(O) = 1,

0,

+ W 2Y2

1) {Ylln(y~)

_(w 2

-

Y2(0)

= Y2(0) = 0,

(2 ) { Y2 1n (Yo2) - 1

= - W

Yl(O)

(2.11.32)

= y~(O) = 0,

+ 2Yl + ~Yo[ln(y~)]2 } ,

(2.11.34)

y~ + "2Y1 1 [1n (Yo2)]2 + 2Y2 + Yo

2 + "61Yo [ln(yo)] 23} , + 2y 1 ln(yo) Y3(0) =

(2.11.33)

(2.11.35)

y~(O) = O.

The solution to (2.11.32) is

Yo = cos(wt).

(2.11.36)

We can solve (2.11.33) using the method of order. We let

Yl(t) = cos(wt)u - 1(t).

(2.11.37)

The equation satisfied by Ul (t) is d 2 ul cos(wt) dt 2

dUl

-

2w sin (wt) dt = _(w 2 - 1) cos(wt) In [cos2 (wt)], (2.11.38)

which has cos(wt) as its integrating factor d -[cos 2 (wt) dUl] dt dt = -(w 2 - 1) cos 2(wt) In[cos 2 (wt)].

(2.11.39)

Two integrations of (2.11.39) give from (2.11.37)

J

J

o

0

t

Yl (t) = - cos(wt)(w 2-1)

~() cos ws

8

dr cos 2(wr) In [cos 2 (wr)].(2.11.40)

2.11 Nontraditional Asymptotic Approaches

125

The integral with respect to s in (2.11.40) can be performed by interchanging the orders of integration: 2

- cos (wt) (w - 1) Yl = w

f t

2

dr cos2 (wr) In[cos (wr)][tan(wt) - tan(wt)]

o =

cos(wt)

w

2

1 ~ {cos 2 (wt) - 1 -In[cos 2 (wt)] cos 2 (wt)} 2w wt

-Sin(wt)W~~1 f
(2.11.41)

o Note that the integral in (2.11.41) grows linearly with t for large t because the integrand is a positive periodic function. However, we know that the exact solution to (2.11.28) is a bounded function. Hence, (2.11.41) can only be valid for times that are short compared with I/o. This problem appears because each successive order in the perturbation theory is resonantly coupled to the previous orders. To see this, note that (2.11.33) is the differential equation for a driven harmonic oscillator of the natural frequency w. The driving term (the inhomogeneous part of the differential equation) also has the frequency w because it is a functional of Yo. Thus the oscillator described by (2.11.33) is driven on resonance and the solution exhibits secular behaviour (it grows with t). We can still try to use the expression in (2.11.41) to infer a value for the period of the oscillator accurate to order O. We assume that the period T of (2.11.28) is itself a series in powers of 0: 21r a (2.11.42) w w To determine the coefficient a in (2.11.42) we require that after a quarterperiod the amplitude of the oscillator T

= - + -0 + .. '.

y(t) = yo(t)

+ 0Yl (t) = 1 at t = 0 to Y = 0 at t = T /4.

will fall from y Evaluating the expressions for Yo in (2.11.36) and Yl in (2.11.41) we obtain, to order 0,

aD ~ 6W~~

,..

1 ]
a

= -4

w

f i

2

w -1 2

o

dx cos2 x In (cos 2 x)

= 1r 1-w 2 w

2

(1 ~ 2ln 2).

(2.11.43)

126

2. Discrete Systems

Thus, to the leading order in 6, the period of the oscillator is T = -21r [ 1 + 6 w

w

2 -

2

1 (21n 2 - 1)] .

(2.11.44)

2w Let us reexamine the problem in (2.11.28) llsing the methods of multiplescale perturbation theory. We assume that there are two time scales in the problem: a short-time scale described by the variable t and a long-time scale described by the variable 7"

=6t.

We then seek a solution to (2.11.24) of the form

y(t) = Yo(t, 7")

+ 6Yl (t, 7") + ... ,

(2.11.45)

where the initial conditions in (2.11.25) become 8Yo

at (0,0)

Yo(O,O) = 1,

8Yo

Y1 (0, 0) = 0,

87" (0,0)

=

0,

8Yl

+ at

(2.11.46)

(0,0) = 0,

etc. H we substitute (2.11.45) into (2.11.28), we obtain

82 at 2Yo(t, 7")

+ W 2 Yo(t,7")

=

°

(2.11.47)

to the zeroth order in 6 and

82 at 2Yl(t,7")

8 2 Yo 2 + W Y l (t, 7") = -2 at87"

"

+ (1 - w2)yo In(y5)

(2.11.48)

to the first order in 6. The most general real solution to (2.11.47) is Yo(t,7") = A(7")eiwt + A*(7")e- iwt , (2.11.49) where A(7") is to be determined. Using (2.11.49) we can e~aluate the right-hand side of (2.11.48):

(1 _w2)[A(7")eiwt + A*(7")e- iwt ]1n [(2IAI2) (1 + A2e2iwt + A*2 e -2iwt)] 2 21AI 21AI 2 -2[iweiwt A'(7") - iwe- iwt A*'(7").

(2.11.50)

Nov..> we expa~d the logarithm in (2.11.50) to identify all terms proportional to e1wt and e 21wt • Such terms oscillate at the frequency wand thus give rise to the seculiar behavior in Yl . The coefficient of e iwt is -2iwA'(T)

+ (1

- w 2 )A(T) In(2IAI 2 )

1- w ~ 1 4 + 2 A(7") 2k + 1 2

6

k [

2

_

1 -2 w A(T)

2k + 1 ] k+1 .

~ ~4-1 [

2; ]

(2.11.51)

2.11 Nontraditional Asymptotic Approaches

127

Evaluating the sums in (2.11.51) gives

-2iwA'(r)

w 2 )A(r) In(2IAI 2 )

+ (1 -

- (1 - w 2 )A(r) In 2 + (1 - w2 )A(r).

(2.11.52)

Thus, the condition that there is no secular behaviour in Y1(t, r) is that the expression in (2.11.52) (as well as its complex conjugate) vanishes:

- 2iwA'(r)

+ (1 -

w 2 )A(r)[1 -In(IA2 1)]

= O.

(2.11.53)

To solve (2.11.53) we let

A( r) = R( r )ei9 ('T"),

(2.11.54)

substitute (2.11.54) into (2.11.53), and decompose the result into its real and imaginary parts: R'(r) = 0, (J' ( r)

=

w 2 -1

2w

(1

+ 2ln R).

(2.11.55)

Hence, R(r) is a constant R(r) =

and

(J ( r)

Flo,

(2.11.56)

is a linear function of r,

(J(r) =

w 2 -1

2w

(1

+ 2ln Flo)r + (Jo.

(2.11.57)

The initial conditions in (2.11.46) imply that Flo our final result for To(t, r) is

¥Ott, r)

= cos [wt + r

w2 2W

= 1/2 and

1 (1- 21n 2)] .

(Jo

= 0,

thus

(2.11.58)

Finaliy, we eliminate r in favour of 8t to obtain the MSA result TMSA =

w - 6w

2: ~1r, (21n 2 - 1)

(2.11.59)

:1

which we expand to order 6:

TMSA=~ [1+6w~~1(2ln2_1)]

+0(6 2 ).

(2.11.60)

To our surprise, (2.11.60) agrees exactly with the order-6 result we obtained in (2.11.44) using the 6-perturbation method at a quarter-period. It is a long but routine calculation to carry out the 6-perturbation series to order 62 . Using the quarter-period method we find that at 6 = 1,

T =

~ [2'-+ 0.5238 w~~ 1 + 0.6041 (w~~ 1)'] .

(2.11.61)

128

2. Discrete Systems

Table 2.1. Comparison of the exact value of the period of the anharmonic oscillator with the period calculated from the order-8 quarter-period method (same 88 MSA) and the order-8 2 quarter-period method f:

1 3 8

W

J2 2 3

T( exact) 4.76802 3.52114 2.41289

T( order 82 ) 4.73488 3.50794 2.40871

T(order 8) 4.87195 3.59669 2.45397

In Table 2.1 we compare three results: the exact numerical calculation of the period T; the order-8 quarter-period calculation, which is the same as the order-8 MSA result in (2.11.60); and the order-8 2 quarter-period calculation in (2.11.61). We set 8 = 1 and look at three values of c = w 2 -1. As expected, the MSA and order-8 results are excellent, having an accuracy of about 2%. The order-8 2 results are even better, having a relative error of less than 0.5%.

2.11.3 Asymptotic Solutions for Nonlinear Systems with High Degrees of Nonlinearity As an example, one can consider the equation

x.. + xn= 0,

n = 2k

+ 1,

k = 1,2, ... ,

for which we will seek a single-parameter family of periodic solutions which are skew-symmetric with respect to the origin of coordinates in the limit as n

--+ 00.

e

.

Let us introduce the function = xj!A (A is the amplitude) for which the inequality 0 < lei < 1 holds. Note that the function e is continuous and periodic. The initial equation can then be represented as follows:

e.. + An-len = O.

(2.11.62)

en

We will expand the function in a series in 1I n as n do this, we first transform the function

_{en,