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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 531480, 22 pages
Research Article

Approximate Super- and Sub-harmonic Response of a Multi-DOFs System with Local Cubic Nonlinearities under Resonance

1Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
2Traction Power State Key Laboratory, Southwest Jiaotong University, Chengdu 610031, China

Received 15 June 2012; Accepted 1 October 2012

Academic Editor: Livija Cveticanin

Copyright © 2012 Yang CaiJin. 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.


A multi-degree-of-freedom dynamical system with local cubic nonlinearities subjected to super/subharmonic excitation is considered in this paper. The purpose of this paper is to approximate the nonlinear response of system at super/sub harmonic resonance. For many situations, single resonance mode is often observed to be leading as system enters into super/sub harmonic resonance. In this case, the single modal natural resonance theory can be applied to reduce the system model and a simplified model with only a single DOF is always obtained. Thus, an approximate solution and the analytical expression of frequency response relation are then derived using classical perturbation analysis. While the system is controlled by multiple modes, modal analysis for linearized system is used to decide dominant modes. The reduced model governed by these relevant modes is found and results in an approximate numerical solutions. An illustrative example of the discrete mass-spring-damper nonlinear vibration system with ten DOFs is examined. The approximation results are validated by comparing them with the calculations from direct numerical integration of the equation of motion of the original nonlinear system. Comparably good agreements are obtained.