Use Of Transformations With The Higher Order Method Of Multiple Scales To Determine The Steady State Periodic Response Of Harmonically Excited Non-linear Oscillators, Part II: Transformation Of Detuning

Abstract

Use of the higher order method of multiple scales with reconstitution to determine the periodic steady state response of harmonically excited non-linear oscillators is considered. In this method, frequency detuning can be introduced in several different ways. The main objective of this work is to compare the second order results determined by introducing detuning in the square of the excitation frequency against the second order results determined by introducing detuning in the excitation frequency itself. A modified reconstitution procedure is used to uncover the relative approximations used in these second order solutions. The well known primary resonance in the Duffing oscillator is used as a prototype example; and it is shown that the combined effects of using transformed time and introducing a detuning parameter in the square of the excitation frequency can lead to a non-uniform expansion within the excitation frequency band in which large amplitude primary response is excited.