Nonlinear Normal Modes of a Continuous System With Quadratic Nonlinearities

[+] Author and Article Information
A. H. Nayfeh, S. A. Nayfeh

Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219

J. Vib. Acoust 117(2), 199-205 (Apr 01, 1995) (7 pages) doi:10.1115/1.2873898 History: Received March 01, 1993; Revised October 01, 1993; Online February 26, 2008


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.

Copyright © 1995 by The American Society of Mechanical Engineers
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