Non-Iterative Rauscher Method for 1-DOF System: a New Approach to Studying Non-Autonomous System via Equivalent Autonomous One

Abstract

In the paper a new non-iterative variant of Rauscher method is considered. In its current statement the method can be used in analysis of forced harmonic oscillations in 1-DOF nonlinear system. It is shown that three different types o f equivalent authonomous dynamical systems can be built for a given 1-DOF non-autonomous one. Two of them (1st and 2nd type) have wider set of solutions than that of the initial system. These solutions correspond to various values of amplitude and phase of external excitation. Solutions of the equivalent system of 3rd type are exclusively periodic ones. Based on the equivalent system of 3rd type such a function W(x,x') can be constructed that its level curves correspond to periodic orbits of the initial non-autonomous system. This function can be built a priori via computation of the invariant manifold of the equivalent system of 1st type. Using the same approach the Rauscher expansions cos(Qt)=C(x,x'), sin(Qt)=S(x,x') can also be constructed. It is also shown that equivalent systems can be investigated by means of harmonic balance method which allows construction o f W(x,x'), C(x,x') andS(x,x') in semi-analytical manner.