A study of branching phenomena in nonautonomous piecewise‐linear systems with asymmetrical restoring force in the case of the external force with a sum of several harmonics

Abstract

AbstractIn the study of nonlinear oscillations, it is well known that, in general, the fundamental harmonic solutions are either in phase with the external force or in opposite phase with it in a lossless system represented by a second‐order differential equation for a single external force. This paper considers a system with asymmetrical restoring force for several external forces. In the following steps, it is shown that there exist solutions in this system which are neither in‐phase nor in opposite phase with the external forces. First, we consider a second‐order system with an asymmetrical restoring force composed of two line segments as an asymmetrical system, and obtain a periodic solution in the same way as in the case of a single external force. The stability of the obtained periodic solution is studied by using Hill's equation and the boundary conditions of the stable and unstable regions are shown clearly. Moreover, the branching phenomena in the boundary are analyzed in four cases. In three of these four cases, there are branching or jump phenomena of the harmonic solutions of fractional orders (2n + 1) /2 (n = 0, 1, 2) due to asymmetry as in the case of a single external force. However, the fourth case shows the branching phenomena of periodic solutions which are neither in‐phase nor in opposite phase with the external forces. The results of numerical analysis and the appropriateness of the analysis are shown.