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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 383947, 11 pages
Research Article

Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

1Centro de Innovación en Diseño y Tecnología, Tecnológico de Monterrey, Campus Monterrey, E. Garza Sada 2501 Sur, 64849 Monterrey, NL, Mexico
2Department of Mechanical Engineering and Industrial Construction, University of Girona, Maria Aurelia Capmany 61, 17071 Girona, Spain

Received 22 July 2013; Revised 17 September 2013; Accepted 17 September 2013

Academic Editor: Miguel A. F. Sanjuán

Copyright © 2013 Alex Elías-Zúñiga 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.


In this work we consider the nonlinear equivalent representation form of oscillators that exhibit nonlinearities in both the elastic and the damping terms. The nonlinear damping effects are considered to be described by fractional power velocity terms which provide better predictions of the dissipative effects observed in some physical systems. It is shown that their effects on the system dynamics response are equivalent to a shift in the coefficient of the linear damping term of a Duffing oscillator. Then, its numerical integration predictions, based on its equivalent representation form given by the well-known forced, damped Duffing equation, are compared to the numerical integration values of its original equations of motion. The applicability of the proposed procedure is evaluated by studying the dynamics response of four nonlinear oscillators that arise in some engineering applications such as nanoresonators, microresonators, human wrist movements, structural engineering design, and chain dynamics of polymeric materials at high extensibility, among others.