On The Elliptic Balance Method

Abstract

In many practical situations a pair of coupled, damped, homogeneous nonlinear ordinary differential equations model the dynamical behavior of mechanical systems. For example, these equations arise during the process of studying the mechanical response of systems such as strings, beams, absorbers, plates, and so on. In general, the exact solution of this sort of equation is unknown and hence, numerical integration, perturbation techniques or geometrical methods have been applied to obtain their approximate solution. A large number of studies of the nonlinear behavior of such systems have been made using perturbation techniques; however, the vast majority have dealt with weakly nonlinear systems, i.e. with small values of the nonlinear parameter ε or the damping parameter ν. The objective of this work is to apply a novel perturbation technique developed by Elías-Zúñiga, called the Elliptic Balance Method (EBM), to obtain the approximate solution of nonlinear two-degree-of-freedom systems. Two examples are presented to compare the EBM solution with numerical integration. The first is related to a damped, nonlinear system with two degrees of freedom that describes the dynamical behavior of a viscohyperelastic simple shear suspension system with an undamped linear absorber, and the second is related to a proposed damped, nonlinear mechanical model. It will be shown that its amplitude-time response can be accurately described by the EBM solution even for moderate values of the damping coefficients ν_i and the nonlinear parameter ε.