Affiliation:
1. Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219
Abstract
We use two approaches to determine the nonlinear modes and natural frequencies of a simply supported Euler-Bernoulli beam resting on an elastic foundation with distributed quadratic and cubic nonlinearities. In the first approach, we use the method of multiple scales to treat the governing partial-differential equation and boundary conditions directly. In the second approach, we use a Galerkin procedure to discretize the system and then determine the normal modes from the discretized equations by using the method of multiple scales and the invariant manifold approach. Whereas one- and two-mode discretizations produce erroneous results for continuous systems with quadratic and cubic nonlinearities, all methods, in the present case, produce the same results because the discretization is carried out by using a complete set of basis functions that satisfy the boundary conditions.
Reference25 articles.
1. Benamar R. , BennounaM. M. K., and WhiteR. G., 1991, “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, Vol. 149, pp. 179–195.
2. Carr, J., 1981, Application to Centre Manifold Theory, Springer-Verlag, New York.
3. Hsieh, S.-R., Shaw, S. W., and Pierre, C., 1993, “Normal Modes for Large Amplitude Vibration of a Cantilever Beam,” International Journal of Solids and Structures, to appear.
4. King, M. E., and Vakakis, A. F., 1993, “Nonlinear Normal Modes in a Class of Nonlinear Continuous Systems,” Nonlinear Vibrations, R. A. Ibrahim, A. K. Bajaj, and L. A. Bergman, eds., DE-Vol. 54, pp. 33–41.
5. Nayfeh, A. H., 1973, Perturbation Methods, Wiley, New York.
Cited by
83 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献