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)]

REFERENCES

1. Alifov A.A., Frolov K.V.: Interaction 0/ Nonlinear O,ciliato'll SlIneml with Power SuPplll (in Rus­sian), Nauka, Moscow, (1985), 328p.

2. Andronav A.A., Vitt A.A. and Khaikin S.E.: The '1'IIeorJI O/O,cilIatiOfil (in Russian), second

edition, Fillmatgill, Moscow, (1959), 915p.

3. Babakov I.M.: The TheO'll 0/ O,cillatiOfil (in Russian), second edition, Nauka, Moscow, (1965), 625p.

4. Beletxkii V.V.: Motion 0/ an Artificial Satellite about it. Centre 0/ Mal, (in Russian), Moscow,

Nauka, (1965), 416p.

5. Blaquiere A.: Nonlinear SlInem Anallln,. Academic Press, (1966), 392p.

6. Bogoliubov N.N., Mitropolskii Yu.A.: A,ymptotic Methodl in the '1'IIeorJI 0/ Nonlinear O,cillatiofll.

(in Russian), fourth edition, Moscow, (1974), 410p.

7. Bolotin V.V.: DJlflGmic Stabilitll 0/ name SlIneml (in Russian), Gostekhilldat, Moscow, (1956),

600p.

8. Bulgakov B.V.: O,cillatiOfil (in Russian), Gostekhilldat, Moscow, (1954), 890p.

9. Chernousko F.L.: Re,onanee Phenomena in Motion 0/ Satellite about Centre 0/ Mal, (in Russian),

J. 0/ computational mathematic, and math. phllnc" Moscow, 3, No.3, (1963).

10. Chetaev N.G.: Stabi/ifJI 0/ Motion (in Russian), second edition, Gostekhilldat, Moscow, (1955),

207p.

11. Clendening W.R., Dubey R.N.: An AnallI"' 0/ Control Methodl/or Galloping SlInem. Thlfil.

ASME, series B, vo1.95, No.3, (1973).

12. Den Hartog J.P.: Mechanical Vibratiofll, Mc Graw-Hill, New York, (1947), 460p.

13. Frolov K.V.:

a) Decrealing the amplitude %,cillation 0/ re,onanee 'llneml bll control 0/ the pammeter,. COfiltruction

0/ Machine" Moscow, No.3, (1965).

b) Some nonlinear effect. in phllncal 'llneml with variable parameter,. Abh.. DAW, Abt. Math..

Naturwill. Thchn. Heft 5, (1977).

14. Ganiev R.F. and Kononenko V.O.: O,cillatiOfil 0/ rigid bodie, (in Russian), Moscow, Nauka,

(1976), 432p.

15. Hagedorn P.:

a) Nonlinear O,cilIatiOfil. Second edition, Oxford science publications, (1988), 311p.

b) Uber die Tilgung ,elbnerrergter Schwingungen. ZAMP., vo1.29, (1978) pp. 815-821.

16. Hale J .K.: O,cillatiofll in Nonlinear SlInllml, Mc Graw-Hill, New York, (1963), 229p.

17. Hayashi C.: Nonlinear O,cillatiofll in Phllncal SlIneml, Mc Mc Graw-Hill, New York, (1964),

204p.

18. Inoue J. and Others: On ,e/f·'JlflChroniz4tion 0/ Mechanical Vibrutor,. Bulletin 0/ the JSME vol.B,

2, (1970), No.56, pp. 258-263.

19. Kauderer H.: Nichtlineare Mechanill. Springer - Verlag Berlin (1958), 777p.

20. Kononenko V.O.: Vibrating SlInem with a Limited Power SuPplll, London, (1969), 236p.

REFERENCES 333

21. KrylotrN. and BogoliubotrN.: IntroductiontoNonlinearMechanic,. Pinceton, New York, (1949).

22. Lefschetz S.: Differential Equationl. Geametric Theo,.". Second Edition, N.Y. Intersci Pub!.

(1967), 364p.

23. Leimanis E. and Minorsky N.: DlJf&IJmic, and Nonlinear Mechanic,. John wiley & sons, Inc.

New York, (1968).

24. Magnus K.: Vibrationl. Blackie & son Limited, London, (1966), 299p.

26. Malkin J.G.: Some ProNe"" in tAe Theo,." 0/ Nonlinear O,cillationl (in Russian), Gostekhizdat,

Moscow (1966), 491p.

26. Mansour W.M.: Quenclai"., o/limit Cgcle, 0/ a Van der Pol O,cillator. J. Sound Vibration 26, (1972) pp.395-406.

27. Meirovitch L.: Element. 0/ Vibration Analy';'. Mc. Graw-Hill , Inc., (1976).

28. Minorsky N.: Nonlinear O,cillatiOnl. D. Van Nostrand Company, Inc., (1962), 714p.

29. Mitropolskii Yu.A.:

a) ProNe"" 0/ the A"mptotic Thea,." 0/ Nonnationa,." O,cillationl (in Russian), Nauka, Moscow,

(1964), 431p.

b) Averagifl{lMethodinNonlinearMechanic. (in Russian), Naukova Dumka, Kiev, (1971), 440p.

c) Certainl A,pectl de, Progre, de la Methode de Centrage, Edizioni cremonese, Roma (1973).

d) FUndamental 7hlndl in the Thea,." 0/ Nonlinear O.cillationl and tAeir Development (in Russian).

Proceedifl{l' o/ICNO, Kiev, Tom I, (1961), pp. 15-22.

e) The Development of Averaging Method. Proceedi"." o/ICNO IX., Kiev (1981).

30. Mitropolskii Yu.A. and Moseenkov B.I.: A.ymptotic SolutiOnl 0/ Equationl witA Partial Derivativel

(in Russian), Kiev, (1976), 690p.

31. Mitropolskii Yu.A. and Samoilenko A.M.: On A,ymptotic EzpaMonl in Nonlinear Mechanic. (in

Russian). U1crainian Math. Journal, Kiev, No.1, (1979), pp.42-53.

32. Mitropolskii Yu.A., Homa G .P.: Th.e Mathematical Foundation 0/ A,ymptotic Methode 0/ Nonlinear Mechanic. (in RUSSian). Naukova Dumka, Kiev, (1983), 215p.

33. Mitropolskii Yu.A., Lopatin A.K.: Theormco - Group Approach in the A,ymptotic Methode 0/ Nonlinear Mechanic, (in Ruissian). Naukova Dumka, Kiev, (1988), 272p.

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.

36. Nayfeh A.H. and Mook D.T.: Nonlinear O,cillationl. Wiley, (1979).

37. Nguyen Van Dao:

a) FUndamental methode o/the theo,." 0/ nonlinear o,cillationl. Education Publishers, Hanoi (1971), 193p.

b) Nonlinear O,cillationl 0/ Higher Order Syneml. NCSR Vietnam, Hanoi (1979), 64p.

c) . An Appro:amate Method lor lnvemgation 0/ the Stability 0/ Motion in the Critiml Cale. Bulletin de

L'Academie Polonaile de, Science" No.2, (1971), pp.17-21.

334 REFERENCES

d) On the Phenomenon 01 Parametric Be.onance 01 a Nonlinear Vibrator Under the Action 01 Electro­

magnetic Force. Proc. 01 Vibration ProNeml, Warsaw, Poland, No.3, (1972), pp.281-291.

e) Interaction between Parametric and Forced O.cillationl in Mvltidimennonal S"lfeml. J. 01 Thlhnieal

Ph"lie., Warsaw, Poland, No.3, (1975).

f) Interaction 01 Subharmonic O.cillationl. J. 01 7llehnieal Ph"lie., Warsaw, Poland, No.2, (1975), pp.227-237.

g) Some Properiie. 01 the Generolized Van tier Pol Equation. Journal 01 Thlhnical Ph"lie., Warsaw,

Poland No.2, (1976), pp.183-190.

h) Parametric O.cillationl 01 Dl/nGmieal S"lfeml Under the Influence 01 Nonlinear FnctiOnl. Acta

Mathematica Vietnam. Tom 3, No.2, (1978).

i) A'lIfflpiotie Method lor Stud" 01 Qualilinear Differential Equationl 01 Higher Order. A'lIfflptotie

Methotll 01 Mathematical Ph"lie., Kiev, Naukova Dumka, (1988), pp.191-203.

j) Lanchelfer Damper and Voigt Dl/nGmie Ab.orber lor Quenching Se/f-ezcited Vibratiom. Proceeding.

01 V111International Congre" on TheMl/ol Meehanilml8 Machine., Prague, vol.6, August (1991).

k) Dl/nGmie Ab.orber Effect lor Quenching Se/f-ezcited Vibration 01 Mechanical ."lfeml with Limit

Enern Re.ource. Nonlinear ProNeml 01 Machine Dl/nGmic., Moscow, Nauka, (1992), pp.63-69.

38. Nguyen Van Dao, Nguyen Van Dinh:

a) Dl/nGmie Ab.orber EJlectlor Se/f-ezcitetl S"lfeml. Advance. in Mechanic., Warsaw, vol. 14, No.1, (1991).

b) Dl/nGmic Ab.orber lor S"lferm with Di8fributed Parameter •. Proceeding. 01 NCSR Vietnam, vol. 2 ,

(1990), pp.3-23.

c)· A'lIfflptotie Method lor Invemgation 01 Multifrequenc" O.cillatiom 01 Qualilinear S"lfem 01 Second

Order. Proceeding. 01 NCSR Vietnam, Vol. 1, No 1, (1987). pp.lO-15.

39. Nguyen Van Dinh:

a) The 7Uned Ab.orber in Se/f-ezcited S"lfem. Journal 01 Mechanic •• Hanoi, No. 3-4, (1979), pp.21-

30.

b) The Dl/nGmie. Ab.orber in Qualilinear S"lfeml. Dissertation, Hanoi, (1980), 62p.

40. Osinski Z.:

a) Vibration Theo'l/. Warsaw, Poland (1978).

b) Damping 01 Mechanical Vibratiom. Warsaw, Poland (1979).

41. Panovko Ya.G.: Introduction to the TheO'l/ 01 Mechanical O.cillatiom (in Russian). Nauka,

Moscow, (1977), 233p;

42. Pisarenko G.S.: Enern Dillipation in Mechanical Vibrationl (in Russian). Kiev (1962), 436p.

43. Pontryagin L.S.: Ortlina'l/ Differential Equation. (in Russian). Fizmatgiz, Moscow, (1961),

3llp.

44. Roseau M.: Vibrationl in Mechanical S,,8feml. Springer Verlag, (1989). 515p.

45. Rubanik V.P. O.cillatiom in Qualilinear S"lfeml with Time Lag. (in Russian) Nauka, Moscow,

(1969), 287p.

46. Schmidt G.: Parametererregte Schwingungen. Berlin, (1975), 3l3p.

REFERENCES 335

47. Schmidt G. and Tondl A.: Nonlinear Vibratiom. Cambridge Univ. Press. and Akademic-Verlag

(1986).

48. Starzhinskii V.M.: Applied MethoiU in the Th.eo'1l of Nonlinear O,cillation.. Mir Publishers, Moscow, (1980), 264p.

49. Stoker J.J.: Nonlinear Vibrationl. Interscience Publishers LTD, London. (1957).

50. Tondl A.:

a) Quenching of SeI/-E%l!ited Vibration.. Equi/ibrium A,pect,. JoumoJ Sound and Vibration 4' (1975)

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.

51. Tran Kim Chi:

a) A'IIfflptotic Solution. of QUlJlilinear Differential Equation. of Third Order with Delall. Ucrainian

Math. JoumoJ (in Russian). No.3. (1983), pp.392-397.

b) A'IIfflptotic Method for Comtruction of Differential Equation. of N - Order with Slowlll Va'1/ing Parameter, (autonomous .:ase) (in Russian), Ucrainian Math. Journal, T. 33, No 3, (1981),

pp.427-429.

c) A'IIfflptotic Method for Coutruction of Differential Equatiom of N - Order with Slowlll Va'1/ing

Parameter, in Non-autonomoUl CGle (in Russian), Ucrainian Math. JoumoJ, T. 33. No 4, (1981),

pp.567-570.

d) Parametric O,cil/ation with High Degree 71Irm at Modu/ation Depth. Proc. of 11th International

Conference on Nonlinear O,cillation., Budapest, (1987), pp.498-501.

52. Tran Kim Chi, Nguyen Van Dao:

a) Coutruction of the Solution, of Nonlinear Higher Order Differential Equation.. Acta 71Ichnica CSA V, Prague. No.3, (1980), pp.358-368.

b) Nonlinear DiJJerential Equation with SelJ-ezcited and Parametric Ezcitation.. The Colloquium on

Differential Equatiom and Application.. Budapest, Hungary, (1991). pp.65-73.

c) A'IIfflptotic Methodfor Stud"ing of the QUIJIilinear O,cillation. of DlIft4I'I1ic S",tema 0/ Higher Order.

Advance,inMIlCMnic,. Warsaw. vol. 3, No4, (1980), pp.3-21.

53. Volosov V.M. and Morgunov V.I.: The Method 0/ Averaging in the TheorII 0/ Nonlinear SlIltema.

(in Russian), Izd-vo, MGU, Moscow, (1971).

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