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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 879564, 11 pages
http://dx.doi.org/10.1155/2014/879564
Research Article

Normal Form for High-Dimensional Nonlinear System and Its Application to a Viscoelastic Moving Belt

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

Received 10 May 2014; Revised 24 July 2014; Accepted 11 August 2014; Published 27 August 2014

Academic Editor: Yongli Song

Copyright © 2014 S. P. Chen and Y. H. Qian. 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.

Abstract

This paper is concerned with the computation of the normal form and its application to a viscoelastic moving belt. First, a new computation method is proposed for significantly refining the normal forms for high-dimensional nonlinear systems. The improved method is described in detail by analyzing the four-dimensional nonlinear dynamical systems whose Jacobian matrices evaluated at an equilibrium point contain three different cases, that are, (i) two pairs of pure imaginary eigenvalues, (ii) one nonsemisimple double zero and a pair of pure imaginary eigenvalues, and (iii) two nonsemisimple double zero eigenvalues. Then, three explicit formulae are derived, herein, which can be used to compute the coefficients of the normal form and the associated nonlinear transformation. Finally, employing the present method, we study the nonlinear oscillation of the viscoelastic moving belt under parametric excitations. The stability and bifurcation of the nonlinear vibration system are studied. Through the illustrative example, the feasibility and merit of this novel method are also demonstrated and discussed.