APPENDIX 1
PRINCIPAL COORDINATES
A system with N degrees of freedom has N principal modes of oscillations. It is possible to define generalized coordinates such that each of them can perform a harmonic oscillation only at one of the natural frequencies of the system. Such coordinates are called principal or normal coordinates.
* Consider first a system of two differential equations which is very often met in applied problems
ax + bx + ey = h, dy + ey + hx = 12,
(1)
where hand 12 are functions of coordinates and time, a, b, e, d, e and h are constants satisfied the conditions for the positiveness of O~, O~ (4).
Introduce now the principal coordinates 6, 6, ... , en connected with the primary coordinates x, y by the relations
(2)
where h aO~ - b
0"1 = 2 = dOl - e e
h aO~-b 0"2 = d02 = 2-e e
(3)
and 0 1 , O2 are the natw·al frequencies - the roots of the characteristic equation:
(b - a02)(e - d02) - he = 0,
1 O~ = -d [ae + bd - v(ae - bd)2 + 4aedh],
2a 1
O~ = -d [ae + bd + V (ae - bd)2 + 4aedh] . 2a
It is easy to verify that -ah
0"1·0"2 = dc . We have the following equations for the principal coordinates:
(4)
(5)
(6)
328 APPENDIX 1
where 0'2 1
k1 = = , a(0'2 - 0'1) a + to'~
k2 = 0'1 = 1 a(0'1 - 0'2) a + to'~
Note : The following inequalities are true for the case ad > 0, he > 0:
In fact, we have:
where
It is obvious that:
2 ~(e b)2 he ll. = - - - + 4- . d a ad
.: - 0 21 = !(': _ ! + ll.2) d 2 d a '
2 e 1 (b e 2) 02-d=i ~-d+ll. ,
! _ 0 2 = !(! _ .: + ll.2) a 1 2 ad'
0 2 _ !!. = ! (': _ !!. + ll. 2). 2 a 2 d a
Adding (9) and (10) we obtain:
(~- O~) + (O~ - ~) = O~ - O~ = ll.2 > O.
Multiplying (9) and (10) we have:
( e 2) (2 e ) 1 [4 ( e b ) 2] he d - 01 O2 - d =:4 ll. - d - ~ = ad > O.
From (13) and (14) it follows
e 2 - - 0 1 > 0 d '
Similarly, we have the inequalities (8):
b 2 - - 0 1 > 0, a
2 e O2 - d > O.
2 b O2 - - > O. a
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
(16)
Example 28
For equations:
we have
Hence,
PRINCIPAL COORDINATES
m1X1 + C1(X1 - X2) = Pcoswt,
m2 X2 + CdX2 - xd + C2 X2 = -af(x2),
and equations in the principal coordinates are
Xl = e1 + e2, X2 = 0'1e1 + 0'2e2,
329
(17)
- 2 P 0'2 a 1 e1 + 0 16 = --. coswt + -. /(0'16 + 0'26), (19) ml 0'2 - 0'1 ffl2 0'2 - 0'1
•• 2 P 0'1 a 1 e2 + 0 2 6 = --. coswt + -. f(O'lel + 0'2e2).
m1 0'1 - 0'2 m2 0'1 - 0'2
Example 29
For the system
we have
x + (1 + p) X - PY = e It, ny + p(y - x) = eh,
a = 1, b = 1 + p, C = -p, d = n, e = -h = p, -p -p
0'1 = .. 2 ,0'2 = .. 2 , nUl - p nU2 - p
k - 1 k _ __1----;0-1- l+nO'~' 2 - l+nO'~ ,
oi 2 = ~ [p + n{1 + p) T V'[p - n{1 + p)j2 + 4np2] , , 2n and the equations for 6 and 6 are
x=6+6, Y=0'16+0'26, •• 2 e 6 + 0 1 6 = 1 + 2 (It + 0'112),
nO'l
•• 2 e 6 + 0 2 6 = 1 + 2 (It + 0'212). n0'2
(20)
(21)
(22)
330 APPENDIX 1
* We consider now a system of n differential equations of second order
m1 X1 + C1 X1 + ... + CnXn = ft, 11l2 X2 + d1X1 + ... + dnxn = 12,
(23)
Suppose that, the characteristic determinant
C1 - m102 C2 Cn
D{(2) = d1 d2 - m2 02 dn
i1 i2 in - mn0 2
has n eigenvalues O~, O~, ... , O! - the real positive roots of the equation D{(2) = O. We have the following formulae, transforming (23) into the principal modes
n
Xj = L d!.k) ek, j = 1,2, ... , n, k=1
(24)
(k) ",(k) (k) where dj =:m and 0'; = Dj{On is the algebraic supplement of the element at
"'I
j-th column and last row of D(O~), i.e. O'}k) = (-1)iHDi "" where Di'" is obtained from D{(2) by striking out the last row and j-th column.
The equations for ek are:
(25)
where n
Mk = L t7li[d!k)j2, k= 1,2, ... ,n. (26) i=1
APPENDIX 2
SOME TRIGONOMETRIC FORMULAE OFTEN USED IN THE AVERAGING METHOD
• 2 1 ( ) 1. sm a = 2" 1 - cos 2a
1 2. cos2 a = 2(1 + cos 2a)
· 1 . 3. Slnacosa = 2"sm2a
• 3 1 (. .) 4. sm a = 4 3 sm a - sm 3a
1 5. cos3 a = 4(3 cos a + cos 3a)
• 2 1 ( . .) 6. SIn a COS a = 4 sm a + sm 3a
7. cos a sin2 a = ~(cosa - cos3a)
8. sin4 a = ~(3 - 4cos 2a + cos4a)
1 9. cos4 a = 8(3 + 4 (:os 2a + cos 4a)
10. sin3 a cos a = ~(2 sin 2a - sin 4a)
• 3 1 (. .) 11. smacos a = 8 2sm2a+sm4a
12. sin2 acos2 a = ~ (1- cos4a)
13. sin5 a = 1~ (sin 5a - 5 sin 3a + 10sin a)
1 14. cos5 a = -(cos 5a + 5cos 3a + lOcos a)
16
15. sin a sin,8 = ~[cos(a -,8) - cos(a + ,8)1
16. sin a cos,8 = ~[sin(a -,8) + sin(a + ,8)1
1 17. cos a cos,8= 2"[cos(a-,8)+cos(a+,8)]
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29. Mitropolskii Yu.A.:
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d) FUndamental 7hlndl in the Thea,." 0/ Nonlinear O.cillationl and tAeir Development (in Russian).
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34. Mitropolskii Yu.A., Nguyen Van Dao, Nguyen Dong Anh.: Nonlinear O,cillationl in the Syne"" 0/ Arbitra,." Order (in Russian). Naukova Dumka, Kiev, (1992), 329p.
35. Moiseev N.N.: A,ymptotic Methode 0/ Nonlinear Mechanic, (in Russian). Nauka, Moscow, (1969), 380p.
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b) Nonlinear O,cillationl 0/ Higher Order Syneml. NCSR Vietnam, Hanoi (1979), 64p.
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g) Some Properiie. 01 the Generolized Van tier Pol Equation. Journal 01 Thlhnical Ph"lie., Warsaw,
Poland No.2, (1976), pp.183-190.
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k) Dl/nGmie Ab.orber Effect lor Quenching Se/f-ezcited Vibration 01 Mechanical ."lfeml with Limit
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pp.251-260.
b) Quenching of Self-Excited Vibration: One and two frequencll vibration. JoumoJ Sound and
Vibration 42 (1975) pp. 261-271.
c) Quenching of Self-excited Vibrations: Effect of D'1I Friction. JoumoJ Sound and Vibration 45,
(1976) pp.285-294.
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c) A'IIfflptotic Method for Coutruction of Differential Equatiom of N - Order with Slowlll Va'1/ing
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INDEX
Absorber 76,86
- dynamic 75, 76, 82, 92, 96
- strong 80, 82, 84
- weak 78, 82, 88
Amplitude 3,27,243
- stationary 237, 247
Aperiodic 66
Approximation 2, 24, 56, 61, 99, 224, 236
- first 7, 10
- refinement of the first 8, 44, 286
- second 8, 18, 53, 61, 115, 290
- third 204 Average
Averaging
Balance
115, 134, 295
10,293,302
- dynamical 76
- harmonic 124 Beam
Beat
Case
94
128, 254
- critical 65
Circuit - electrical 16, 243
Coefficient
- damping 99, 101
- Fourier 296 - of distribution 179
Computer
- analog 46, 67, 260
Condition
- boundary 94,97, 102, 104, 198
Coordinate
- generalized 178, 327
- principal 327
Criterion
- Lienard 68
- Routh - Hurwitz 68,68, 171,221
Curve
- amplitude 266
- integral 302
A
B
c
- resonance 248, 249, 279
Cycle
- limit 58, 67
Damping 12
- coefficient 25, 30
- linear 18
- nonlinear 18
Decrement 115
Degree of freedom 85, 92, 226
Delay 45,49
Determinant 68, 131
- characteristic 90, 330
Detuning 252
Development
- Fourier 6
Deviation 118, 123, 212, 297
Dissipation 230
Energy
- kinetic 33, 291
- potential 33, 291
Entrainment 107
Equation
- averaged 129, 258, 283, 292
INDEX
D
E
- characteristic 68, 179, 181, 212,258,322
- degenerated 97, 102
- Duffing 12, 129, 132 - Hamilton 34
- Hill 267
- Lienard 58
- Mathieu 197,204
- Newton 73
- quasi - linear 45
- Rayleigh 61
- Van der Pol 66, 245, 257
- Variational 221, 258, 266, 278
Equilibrium 64, 65, 79, 81, 88, 100, 180
Excitation 196
- external 109, 115, 235, 247, 249
- harmonic 129
- nonlinear 1
- self 22, 116
337
338 INDEX
Expansion 2
-asymptotic 2, 3, 12
- Fourier 19
Force
- damping 82, 88, 106
- elastic 17
- electromagnetic 240
- exciting 109, 114 - external 76, 179, 181, 234
- frictional 217,222
- generalized 88
- impulsive 293
- restoring 24, 26, 146 Form
- standard 95, 283, 291
Frequency 98, 102, 254
- combination 107, 108, 115
- exciting 108, 206,247
- instantaneous 146
- momentary 1
- natural 103,108, 179,198,247
Friction 76, 219, 266
- combination 218
- dry 215 - linear 214, 224 - nonlinear 17
- turbulent 217,223 Function
- Bessel 14 - Dirac 293, 297
- Delta 293
Hamiltonian 34
Harmonic
- fundamental 3, 6, 27, 43, 124, 146
Impulse 297
Instability 66, 234
Interaction 245, 319
Invariant
- adiabatic 25, 32, 39, 40
F
H
I
INDEX
Jump
- in amplitude 107, 136
- phenomenon 129, 280
Linearization 22, 26
Matrix 67, 68, 286
Method
- asymptotic 69, 146, 235,322
- averaging 282, 297, 304, 319
- Van der Pol 282
Mode
- principal 80, 82, 92, 320,330
Moment
- of inertia 14, 83, 94, 199
Motion
- aperiodic 316
- periodic 68
- steady state 78
Operator
- averaging 43, 117, 140, 231, 284
- integrating 284, 286
Orthogonality 40, 97, 181, 291
Oscillation
- asynchronous 128
- combination 262
- forced 107, 196, 246, 247, 262
- free 13, 46, 311
- harmonic 78, 87, 107, 180, 243, 262
- heteroperiodic 118
- multi - frequency 166, 180
- non-linear 246, 282
- nonstationary 146
- parametric 234, 267, 264, 268
- self-excited 68, 72, 76, 82, 247, 262
- self-sustained 245
- single frequency 178
- stationary 21, 79, 234, 239, 267, 276
- subharmonic 137, 241, 263, 268
- synchronized 247, 249, 252
J
L
M
o
339
340
Oscillator 176, 206
- harmonic 1 - nonlinear 129, 155, 245, 319
- parametrically-excited 198 - self-excited 245, 247, 254
- self-sustained 254
- Van der Pol 143,247, 251
Pendulum 32, 196, 282 - mathematical 32, 40, 307
Perturbation 213, 221
Phase I, 31, 93, 109, 135
Phase plane 62,65 Plate 100 Point
- central 65
- critical 62 - elementary 62
- equilibrium 64, 66, 75
- focal 64
- nodal 63
- saddle 64, 66, 251 - singular 62, 64, 139
Quench 76, 78 Quenching 76, 80, 103, 116
- asynchronous 116
Regime - heteroperiodic 116
- oscillatory 99, 100, 311
- stationary 87
- steady state 81
- synchronous 127
Resonance 108, 146, 211, 274, 316
- fundamental 108
- internal 180
- parametric 280
- principal 205, 206, 313
- subharmonic 108
- superharmonic 108
Rigidity 101
INDEX
p
Q
R
Root
- characteristic 62,65, 320
Rotation 303, 318
Series
- asymptotic 164
- Fourier lll, ll5, 121, 160
- Taylor 130
Skeleton 133, 135
Solution
- asymptotic 309
- harmonic 109
- periodic 264
- stationary 78, 144, 219, 261
- synchronous 128
Spring 41, 73, 78
Stability 65, 171, 234,249
- asymptotic 65, 140, 324
- mere 65 Stiffness 78, 88, 198
Synchronization 245, 252
System
- autonomous 62
- averaged 301, 302, 312
- conservative 10
- degenerated 181, 300
- dynamical 145, 180
- electromechanical 239
- Hamilton 34
- mechanical 72, 75 - multidimensional 272
- nonlinear 76, 108, 178, 219,234
- oscillatory 22, 25, 76, 251, 291
- quasi-linear 179
- self-excited 76, 248
- unexcited 179, 180, 302
Trajectory
- phase 62,67, 305
- point 62
INDEX 341
s
T
Mechanics SOUD MECHANICS AND ITS APPLICATIONS
Series Editor: G.M.L. Gladwell
44. D.A. Hills, P.A. Kelly, D.N. Dai and A.M. Korsunsky: Solution of Crack Problems. The Distributed Dislocation Technique. 1996 ISBN 0-7923-3848-0
45. V.A. Squire, RJ. Hosking, A.D. Kerr and PJ. Langhorne: Moving Loads on Ice Plates. 1996 ISBN 0-7923-3953-3
46. A. Pineau and A. Zaoui (eds.): IUTAM Symposium on Micromechanics of Plasticity and Damage of Multiphase Materials. Proceedings of the IUTAM Symposium held in Sevres, Paris, France. 1996 ISBN 0-7923-4188-0
47. A. Naess and S. Krenk (eds.): IUTAM Symposium on Advances in Nonlinear Stochastic Mechanics. Proceedings of the IUTAM Symposium held in Trondheim, Norway. 1996
ISBN 0-7923-4193-7 48. D. Ie§aD and A. Scalia: Thermoelastic Deformations. 1996 ISBN 0-7923-4230-5 49. J. R. Willis (ed.): IUTAM Symposium on Nonlinear Analysis of Fracture. Proceedings of the
IUT AM Symposium held in Cambridge, U.K. 1997 ISBN 0-7923-4378-6 50. A. Preumont Vibration Control of Active Structures. An Introduction. 1997
ISBN 0-7923-4392-1 51. G.P. Cherepanov: Methods of Fracture Mechanics: Solid Matter Physics. 1997
ISBN 0-7923-4408-1 52. D.H. van Campen (ed.): IUTAM Symposium on Interaction between Dynamics and Control in
Advanced Mechanical Systems. Proceedings of the IUTAM Symposium held in Eindhoven, The Netherlands. 1997 ISBN 0-7923-4429-4
53. N.A. Fleck and A.C.F. Cocks (eds.): /uTAM Symposium on Mechanics of Granular and Porous Materials. Proceedings of the IUTAM Symposium held in Cambridge, U.K. 1997
ISBN 0-7923-4553-3 54. J. Roorda and N.K. Srivastava (eds.): Trends in Structural Mechanics. Theory, Practice,
Education. 1997 ISBN 0-7923-4603-3 55. Yu. A. Mitropolskii and N. Van Dao: Applied Asymptotic Methods in Nonlinear Oscillations.
1997 ISBN 0-7923-4605-X
Kluwer Academic Publishers - Dordrecht / Boston / London