Simultaneous combination resonances in an autoparametrically resonant system

Abstract

There have been many reports to date of the effects of seemingly small, non-linear, autoparametric interactions in certain structures undergoing forced vibration, and it is known that large intermodal exchanges of energy can occur under several prescribed internal resonance conditions. This paper purports to illustrate the highly complex responses that can be generated within a system of coupled cantilever beams when two such internal resonances, each in the form of a combination resonance, exist in very close proximity to each other. The resulting four mode interaction can exhibit non-synchronous large amplitude responses of the indirectly excited modes, in addition to the well known phenomenon of saturation of the directly excited mode, and the stationary system responses are also shown to be highly susceptible to very small shifts in system tuning. Thus it is proposed that the non-linear response of the system can be dependent on the dominance, or otherwise, of one of the internally generated combination resonances over its near neighbour. A four degree of freedom model is proposed and a multiple scales treatment of the governing equations is presented. A selection of experimental results are presented which highlight the above phenomena.