Nonlinear normal modes of buckled beams: three-to-one and one-to-one internal resonances

Nonlinear Dynamics

Volumn 18, Issue 3, 1999, Pages 253-273

  Nayfeh, Ali H ^a   Lacarbonara, Walter ^a   Chin, Char Ming ^a  

^a State University   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BEAMS AND GIRDERS; BOUNDARY CONDITIONS; BUCKLING; INTEGRAL EQUATIONS; NONLINEAR SYSTEMS; PARTIAL DIFFERENTIAL EQUATIONS; PERTURBATION TECHNIQUES; PHASE SPACE METHODS; RESONANCE;

BUCKLED BEAMS; INTERNAL RESONANCE; METHOD OF MULTIPLE SCALES; NONLINEAR NORMAL MODES;

DYNAMICS;

EID: 0032655393     PISSN: 0924090X     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1008389024738     Document Type: Article

References (25)

1. Rosenberg, R. M., 'The normal modes of nonlinear n-degree-of-freedom systems', Journal of Applied Mechanics 30, 1962, 7-14.
2. Rosenberg, R. M., 'On nonlinear vibrations of systems with many degrees of freedom', Advances in Applied Mechanics 9, 1966, 155-242.
3. Rand, R. H., 'A direct method for non-linear normal modes', International Journal of Non-Linear Mechanics 9, 1974, 363-368.
4. Vakakis, A. F., 'Analysis and identification of linear and nonlinear normal modes in vibrating systems', Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 1990.
5. Rand, R. H., Pak, C. H., and Vakakis, A. F., 'Bifurcation of nonlinear normal modes in a class of two degree of freedom systems', Acta Mechanica 3, 1992, 129-146.
6. Vakakis, A. F. and Rand, R. H., 'Normal modes and global dynamics of a two-degree-of-freedom non-linear system-I. Low energies', International Journal of Non-Linear Mechanics 27, 1992, 861-874.
7. Vakakis, A. F., Mikhlin, L. I., Pilipchuk, Yu. V., and Zevin, A. A., Normal Modes and Localization in Nonlinear Systems, Wiley, New York, 1996.
8. Shaw, S. W. and Pierre, C., 'Non-linear normal modes and invariant manifolds', Journal of Sound and Vibration 150, 1991, 170-173.
9. Shaw, S. W. and Pierre, C., 'Normal modes for non-linear vibratory systems', Journal of Sound and Vibration 164, 1993, 85-124.
10. Boivin, N., Pierre, C., and Shaw, S. W., 'Non-linear modal analysis of structural systems featuring internal resonances', Journal of Sound and Vibration 182, 1995, 336-341.
11. Nayfeh, A. H. and Nayfeh, S. A., 'On nonlinear modes of continuous systems', Journal of Vibration and Acoustics 116, 1994, 129-136.
12. Nayfeh, A. H. and Nayfeh, S. A., 'Nonlinear normal modes of a continuous system with quadratic nonlinearities', Journal of Vibration and Acoustics 117, 1995, 199-205.
13. Nayfeh, A. H., Chin, C-M., and Nayfeh, S. A., 'On nonlinear normal modes of systems with internal resonance', Journal of Vibration and Acoustics 111, 1996, 340-345.
14. Benemar, R., Bennouna, M. M. K., and White, R. G., 'The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, Part I: Simply supported and clamped-clamped beams', Journal of Sound and Vibration 149, 1991, 179-195.
15. King, M. E. and Vakakis, A. F., 'An energy-based approach to computing resonant nonlinear normal modes', Journal of Applied Mechanics 63, 1996, 810-819.
16. Month, L. A. and Rand, R. H., 'An application of the Poincaré map to the stability of nonlinear normal modes', Journal of Applied Mechanics 47, 1980, 645-651.
17. Pak, C., Rand, R. H., and Moon, F. C., 'Free vibrations of a thin elastica by normal modes', Nonlinear Dynamics 3, 1992, 347-364.
18. King, M. E. and Vakakis, A. F., 'An energy-based formulation for computing nonlinear normal modes in undamped continuous systems', Journal of Vibration and Acoustics 116, 1993, 332-340.
19. Shaw, S. W. and Pierre, C., 'Normal modes of vibration for non-linear continuous systems', Journal of Sound and Vibration 169, 1994, 319-347.
20. Nayfeh, A. H., 'On direct methods for constructing nonlinear normal modes of continuous systems', Journal of Vibration and Control 1, 1995, 389-430.
21. Nayfeh, A. H., Kreider, W., and Anderson, T. J., 'Investigation of natural frequencies and mode shapes of buckled beams', AIAA Journal 33, 1995, 1121-1126.
22. Chin, C-M., Nayfeh, A. H., and Lacarbonara, W., 'Two-to-one internal resonances in parametrically excited buckled beams', AIAA Paper No. 97-1081, in 38th AIAA Structures, Structural Dynamics, and Materials Conference, Kissimmee, FL, April 7-10, 1997.
23. Kreider, W., Nayfeh, A. H., and Chin, C-M., 'Two-to-one resonances in buckled beams', Paper No. SYMP-95-50, in Proceedings of the 15th Biennial Conference on Mechanical Vibration and Noise, Boston, MA, September 17-21, 1995, pp. 449-462.
24. Nayfeh, A. H., Perturbation Methods, Wiley-Interscience, New York, 1973.
25. Nayfeh, A. H., Introduction to Perturbation Techniques, Wiley-Interscience, New York, 1981.