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Mathematical Problems in Engineering
Volume 2013, Article ID 582865, 5 pages
Research Article

An Iteration Scheme Suitable for Solving Limit Cycles of Nonsmooth Dynamical Systems

Department of Mechanics, Sun Yat-sen University, Guangzhou 510275, China

Received 19 July 2013; Accepted 26 September 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Q. X. Liu 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 Mickens iteration method (MIM) is modified to solve self-excited systems containing nonsmooth nonlinearities and/or nonlinear damping terms. If the MIM is implemented routinely, the unknown frequency and amplitude of limit cycle (LC) would couple to each other in complicated nonlinear algebraic equations at each iteration. It is cumbersome to solve these algebraic equations, especially for nonsmooth systems. In the modified procedures, the unknown frequency is substituted by the determined value obtained at the previous iteration. By this means, the frequency is decoupled from the nonlinear terms. Numerical examples show that the LCs obtained by the modified MIM agree well with numerical results. The presented method is very suitable for solving self-excited systems, especially those with nonlinear damping and nonsmooth nonlinearities.