Equivalent Representation Form of Oscillators with Elastic and Damping Nonlinear Terms

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.

Nonlinear free vibrations of quintic inextensional beams lying on Winkler elastic substrate based on three-mode assumptions

“…When p ¼ 0, the governing equations become the linear differential equations for which an exact solution can be calculated for p ¼ 1. Substituting equations (21), (22), and (23) into equations (8-a) to (8-c) and collecting the terms of the same power of p, we obtain two series of linear equations which the first equations are…”

“…It is very important to provide an accurate analysis towards the understanding of the nonlinear vibration characteristics of beam structures. Elias-Zuniga et al 21,22 proposed a nonlinear transformation method to develop equivalent equations of motion of nonlinear oscillatory systems with nonlinear odd damping terms including quintic nonlinear term. Most models dealing with nonlinear dynamics of flexible beams include first mode model terms in the equations of motion.…”