Stable Harmonic and Subharmonic Oscillations of a System With Piecewise Linear Restoring Forces

Abstract

In the study of steady-state solutions of a nonlinear, or piecewise linear dynamic system under periodic forcing, it is often necessary to examine the stability of the solutions. The reason is that not all the possible solutions can persist for an arbitrarily long time in the presence of small random disturbances of all kinds. In general, some of the solutions are stable, and others are unstable. Only stable solutions can be expected to occur in physical systems. This paper analyzes the response of a certain class of piece-wise linear systems. The piecewise linearities arise solely from restoring forces which are found frequently in many engineering problems. Harmonic and subharmonic solutions of the piecewise linear systems are constructed and their stability studied. System parameters which affect the various types of solutions, and interrelations between these parameters for the various types of solutions to exist, are determined. Numerical results for practical design applications are presented. In particular, jump phenomena involving a transition from an unstable harmonic oscillation to a stable subharmonic oscillation, and vice versa, are shown to exist for the present systems.