Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms Alex Elías-Zúñiga,1 Daniel Olvera,1 Inés Ferrer Real,2 and Oscar Martínez-Romero1 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. Abstract 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. 1. Introduction The aim of this paper focuses on using a nonlinear approach to transform the forced nonlinear equation: with nonlinear damping terms into an equivalent forced, linearly damped Duffing’s equation. Here we assume that is the system restoring force which could have rational or irrational conservative force terms, represents the system nonlinear dissipative effects, and are damping constants, is the exponent of the velocity, is the driving force magnitude, is the system driving frequency, is the current time, and is the initial amplitude. The main motivation in studying the equivalent representation form of (1) with nonlinear damping terms for which comes from the fact that the addition of nonlinear damping to the system could remove undesirable effects over the nonresonant regions that can help to improve the overall performance of Duffing-type vibration isolators [1, 2]. Furthermore, during the study of the dynamics response of resonators made from carbon nanotubes and graphene, Lifshitz and Cross [3] and Eichler et al. [4] concluded that damping is strongly dependent on the amplitude of motion and that its effects are better described by nonlinear damping forces. They also concluded that the nonlinearities could be associated with a dissipation channel exterior to the resonator, such as the manner in which the resonator is clamped by its boundaries to the surrounding material, friction effects associated with the sliding between the nanotube/graphene and the metal electrode, and the phonon-phonon interactions, among others. To quantify the nonlinear dissipative effects observed during the performance of micromechanical oscillators, Zaitzev and coworkers designed a doubly clamped beam oscillator and performed several experimental studies to understand the phenomenon of nonlinear damping. They found that nonlinear damping plays an important role in the dynamics response of the micromechanical beam oscillator [5]. On the other hand, it is known that structural engineering design utilizes nonlinear damper devices to reduce the forces exerted in the dampers that could exceed the device force capacity during the structure dynamics response to earthquakes. In this case, the exponent of the velocity is selected on the interval of [6]. Martínez-Rodrigo and Romero [7] found that when the nonlinear dampers velocity exponent is slightly less that 1, the forces in the dampers can be reduced more than 35% during the retrofitting of a multistory that leads to a similar structural performance when compared to the usage of linear dampers. Similar results were reported in [8] in which the utilization of nonlinear viscous dampers reduces the displacement response of existing girder bridges and arch bridge structures. Since nonlinear dampers with fractional powers in the velocity terms are commonly used to model the rhythmic movement of the wrist, the inclusion of a nonlinear damping term for which is considered in (1). This dynamics model is known as the one-fifth power law model [9]. Of course, there are other models t (...truncated)