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Advances in Mathematical Physics
Volume 2017 (2017), Article ID 5214616, 8 pages
Research Article

The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

1College of Mathematics, Physics and Information Engineering, Zhejiang Normal University, Jinhua, Zhejiang 321004, China
2College of Mathematics, Xiamen University of Technology, Xiamen 361024, China
3College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, China

Correspondence should be addressed to Y. H. Qian; nc.unjz@4002hyq

Received 2 October 2016; Revised 21 February 2017; Accepted 20 March 2017; Published 10 April 2017

Academic Editor: Zhi-Yuan Sun

Copyright © 2017 Y. H. Qian et al. 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.


The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.