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Journal of Applied Mathematics
Volume 2012, Article ID 958121, 9 pages
Research Article

An Analytical Approximation Method for Strongly Nonlinear Oscillators

School of Aeronautic Science and Engineering, Beihang University, 37 Xueyuan Road, Beijing 100191, China

Received 13 October 2011; Accepted 28 November 2011

Academic Editor: Wan-Tong Li

Copyright © 2012 Wang Shimin and Yang Lechang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


An analytical method is proposed to get the amplitude-frequency and the phase-frequency characteristics of free/forced oscillators with nonlinear restoring force. The nonlinear restoring force is expressed as a spring with varying stiffness that depends on the vibration amplitude. That is, for stationary vibration, the restoring force linearly depends on the displacement, but the stiffness of the spring varies with the vibration amplitude for nonstationary oscillations. The varied stiffness is constructed by means of the first and second averaged derivatives of the restoring force with respect to the displacement. Then, this stiffness gives the amplitude frequency and the phase frequency characteristics of the oscillator. Various examples show that this method can be applied extensively to oscillators with nonlinear restoring force, and that the solving process is extremely simple.