Free and forced oscillations in a class of piecewise-linear dynamic systems

Abstract

A study is made of the free and forced oscillations in dynamic systems with hysteresis, on the basis of a piecewise -linear, nonlinear model proposed by Reid. The existence, uniqueness, boundedness and periodicity of the solutions for a single degree of freedom system are established under appropriate conditions using topological methods and Brouwer's fixed-point theorem. Exact periodic solutions of a specified symmetry class are obtained and their stability is also examined. Approximate solutions have been derived by the Krylov-Bogoliubov-Van der Pol method and comparison is made with the exact solutions. For dynamic systems with several degrees of freedom, consisting of "Reid oscillators", exact periodic solutions are derived under certain restricted forms of "modal excitation" and the stability of the periodic solutions has been studied. For a slightly more general form of sinusoidal excitation, a simple way of obtaining approximate solutions by "apparent superposition" has been indicated. Examples are presented on the exact and approximate periodic solutions in a dynamic system with two degrees of freedom.