Abstract
Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.
Similar content being viewed by others
References
Meirovitch, L., Analytical Methods in Vibrations, Macmillan, New York, 1967.
Rand, R. H., ‘A direct method for non-linear normal modes’, International Journal of Non-Linear Mechanics 9, 1974, 363–368.
Rosenberg, R. M., ‘On non-linear vibrations of systems with many degrees of freedom’, Advances in Applied Mechanics 9, 1966, 155–242.
Vakakis, A. F., ‘Analysis and identification of linear and non-linear normal modes in vibrating systems’, Ph.D. dissertation, California Institute of Technology, 1990.
Shaw, S. W. and Pierre, C., ‘Normal modes for nonlinear vibratory systems’, Journal of Sound and Vibration 164(1), 1993, 85–124.
Shaw, S. W. and Pierre, C., ‘Normal modes of vibration for nonlinear continuous systems’, Journal of Sound and Vibration 169(3), 1994, 319–347.
Shaw, S. W., ‘An invariant manifold approach to nonlinear normal mode of oscillation’, Journal of Nonlinear Science 4, 1994, 419–448.
Shaw, S. W. and Pierre, C., ‘On nonlinear normal modes’, Proceedings of the Winter Annual Meeting of the A.S.M.E., Anaheim, California, November 1992, DE-Vol. 50, AMD-Vol. 144.
Carr, J., Application of Centre Manifold Theory, Springer-Verlag, New York, 1981.
Vakakis, A. F., ‘Nonsimilar normal oscillations in a strongly nonlinear discrete system’, Journal of Sound and Vibration 158(2), 1992, 341–361.
Vakakis, A. F. and Cetinkaya, C., ‘Mode localization in a class of multi-degree-of-freedom systems with cyclic symmetry’, SIAM Journal on Applied Mathematics 53, 1993, 265–282.
Vakakis, A. F., ‘Fundamental and subharmonic resonances in a system with a "1–1" internal resonance’, Nonlinear Dynamics 3, 1992, 123–143.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Boivin, N., Pierre, C. & Shaw, S.W. Non-linear normal modes, invariance, and modal dynamics approximations of non-linear systems. Nonlinear Dyn 8, 315–346 (1995). https://doi.org/10.1007/BF00045620
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00045620