On the internal resonance of multi-degree-of-freedom systems with cubic non-linearity

Abstract

A new method of multiple scales procedure is used to analyze the forced responses of multiple degrees of freedom systems with cubic non-linearity, in which the internal resonance takes place simultaneously together with the forced resonances. Attention is focused on the superharmonic and subharmonic resonance caused by internal resonance as Ω is near Ω1 or Ω2, where Ω is the forcing frequency, and Ω1 and Ω2 are the first and the second natural frequencies. It is found that the responses are usually excitation-level dependent and there is a critical excitation amplitude f{hook}c. In the case of Ω near Ω1, when f{hook}>f{hook}c the "out-of-phase" superharmonic resonance disappears. In the case of Ω near Ω2, when 0<f{hook}<f{hook}c, there are two branches of response; when f{hook}>f{hook}c, one branch of responses vanishes. The critical excitation amplitudes in different forced vibration are determined analytically in this paper. © 1989.