Normal vibrations of a general class of conservative oscillators

Nonlinear Dynamics

Volumn 11, Issue 1, 1996, Pages 1-15

  Mikhlin, Yu V ^a  

^a NATIONAL TECHNICAL UNIVERSITY   (Ukraine)

Author keywords

Internal resonances; Nonlinear normal modes of vibration; Nonlinear systems; Power series; Trajectories in configuration space

Indexed keywords

INTEGRAL EQUATIONS; LINEARIZATION; LYAPUNOV METHODS; MATRIX ALGEBRA; NATURAL FREQUENCIES; NONLINEAR SYSTEMS; NUMERICAL METHODS; VIBRATIONS (MECHANICAL);

ANALYSIS OF TRAJECTORIES; CONFIGURATION SPACE OF SYSTEMS; CURVILINEAR TRAJECTORIES; NONLINEAR NORMAL VIBRATION MODES;

OSCILLATORS (MECHANICAL);

EID: 0030230830     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF00045048     Document Type: Article

References (27)

1. Lyapunov, A. M., The General Problem of the Stability of Motion, Princeton University Press, Princeton, 1947.
2. Kauderer, H., Nichtlinear Mechanik, Springer-Verlag, Berlin, 1958.
3. Rosenberg, R. M. and Atkinson, C. P., 'On the natural modes and their stability in nonlinear two-degrees-of-freedom systems', Journal of Applied Mechanics 26, 1959, 377-385.
4. Rosenberg, R. M., The normal modes of nonlinear n-degrees-of-freedom systems', Journal of Applied Mechanics 30, 1962, 7-14.
5. Rosenberg, R. M., 'Nonlinear vibrations of systems with many degrees of freedom,', in Advances of Applied Mechanics, Academic Press, New York, Vol. 9, 1966, pp. 156-243.
6. Rosenberg, R. M. and Kuo, J. K., 'Nonsimilar normal mode vibrations of nonlinear systems having two degrees of freedom', Journal of Applied Mechanics 31, 1964, 283-290.
7. Rand, R., 'A higher order approximation for nonlinear normal modes in two degrees of freedom systems', International Journal of Non-Linear Mechanics 6, 1971, 545-547.
8. Manevich, L. I. and Mikhlin, Yu. V., 'Periodic solutions close to rectilinear normal vibration modes', PMM USSR 36(6), 1972, 1051-1058.
9. Vito, R., 'An approximate method for treating the nonlinear vibration of certain two degrees of freedom systems', Journal of Applied Mechanics 39, 1972, 620-621.
10. Mikhlin, Yu. V., 'The joining of local expansions in the theory of nonlinear oscillations', PMM USSR 49(5), 1985, 738-743.
11. Vedenova, E. G., Manevich, L. I., and Pilipchuk, V. N., 'Normal oscillations of a string with concentrated masses on non-linearly elastic support', PMM USSR 49(2), 1985, 203-211.
12. Vakakis, A. R, 'Non-similar normal oscillations in a strongly non-linear discrete system', Journal of Sound and Vibration 158(2), 1992, 341-361.
13. Vakakis, A. F. and Rand, R., 'Normal modes and global dynamics of a two-degree-of-freedom non-linear system. I. Low energies', International Journal of Non-Linear Mechanics 27(5), 1992, 861-874.
14. Normal modes and global dynamics of a two-degree-of-freedom non-linear system. II. High energies, International Journal of Non-Linear Mechanics 27(5), 1992, 875-888.
15. Shaw, S. W. and Pierre, C, 'Nonlinear normal modes and invariant manifolds', Journal of Sound and Vibration 150, 1991, 170-173.
16. Shaw, S. W. and Pierre, C., 'Normal modes for nonlinear vibratory systems', Journal of Sound and Vibration 164(1), 1993, 85-124.
17. Nayfeh, A. H. and Nayfeh, S. A., 'On nonlinear modes of continuous systems', Journal of Vibration and Acoustics 116, 1994, 129-136.
18. Synge, J. L., Classical Dynamics, Springer-Verlag, Berlin, 1960.
19. Wintner, A., The Analytical Foundations of Celestial Mechanics, Princeton University Press, Princeton, 1941.
20. Poincaré, H., Les méthodes nouvelles de la mécanique céleste. 3 Volumes, nouveau tirage, Gauthiers Villars, Paris, 1960.
21. Bellman, R., Introduction to Matrix Analysis, McGraw-Hill, New York, 1960.
22. Hsu, C. S., 'On a restricted class of coupled Hill's equations and some application', Journal of Applied Mechanics 29, 1962, 1-21.
23. Zhupiev, A. L. and Mikhlin, Yu. V, 'Stability and branching of normal oscillation modes of nonlinear systems', PMM USSR 45(3), 1981, 450-455.
24. Zhupiev, A. L. and Mikhlin, Yu. V, 'Conditions for finiteness of the number of instability zones in the problem of normal vibrations of non-linear systems', PMM USSR 48(4), 1984, 486-489.
25. Mikhlin, Yu. V, 'Matching of local expansions in the theory of nonlinear vibrations', Journal of Sound and Vibration 182(4), 1995, 577-588.
26. Nayfeh, A. H. and Mook, D., Nonlinear Oscillations, Wiley, New York, 1979.
27. Manevich, L. I., Mikhlin, Yu. V., and Pilipchuk, V. N., The Method of Normal Oscillation for Essentially Nonlinear Systems, Nauka, Moscow, 1989.