On a Graphical Solution for the Forced Vibration of a System with Non-linear Restoring Force

Abstract

This paper deals with the forced vibration of a single-degree-of-freedom system under simple harmonic force. The author discussed this problem in two cases separately; namely, the one of symmetric restoring force and the other of unsymmetric one. As the result of studying, the author expounded a graphical solution which is an approximate one but will be very conveniently used to get the steady vibration of this system. In this paper, the author proves by adequate calculations that the graphical solution is practically available for obtaining resonance curves and phase difference curves in the system with any non-linear restoring force. Generally, the equation of motion in this case is expressed as mx⁻⁻+cx⁻+f(x)=P₀sinωt, (1) where m is the vibrating mass, c the viscous damping coefficient, f(x) the restoring force with any non-linear characteristic and P₀sinωt the simple harmonic force acting on the vibrating mass, and x means displacement of the mass, t time, x⁻ and x⁻⁻ velocity and accelaration respectively. Dealing with this problem, it is conventional to discuss the solutions in the two cases ; that is, the case when f(x) is symmetric as to x=O and the other case f(x) is non-symmetric.