Dynamic analysis of piecewise linear oscillators with time periodic coefficients

Abstract

A new method is presented for locating periodic steady-state response of piecewise linear dynamical systems with time periodic coefficients. As an example mechanical model, a gear-pair system with backlash is examined, under the action of a constant torque. Originally, some useful insight is gained on the dynamics by investigating the response of a weakly non-linear Mathieu–Duffing oscillator, subjected to a constant external load. The information obtained is then used in seeking approximate periodic solutions of the piecewise linear system. These solutions are determined by developing a new analytical method, which combines elements from approaches applied to piecewise linear systems with constant coefficients as well as from classical perturbation techniques applied to systems with time varying coefficients. The existence analysis is complemented by appropriate stability analyses, for all the possible types of the located periodic motions. In the second part of the work, this analysis is employed and numerical results are obtained. Namely, a series of typical response diagrams is first presented, illustrating the effect of the variable stiffness, the damping and the constant load parameters on the gear-pair response. Moreover, results obtained by direct integration of the equation of motion are finally presented, showing that the system examined can also exhibit more complicated or irregular dynamics.