Non-linear vibratory interactions in systems of coupled beams

Abstract

Small non-linear interactions of the type referred to as autoparametric may have a considerable effect on the forced oscillatory behaviour of structures. Under conditions of internal resonance these effects may bring about complex forms of response where the normal linear resonance of a directly excited mode is absorbed, and indirectly excited modes show simultaneous large non-synchronous responses. In this paper the forced vibration of a system of coupled beams is examined and it is shown that violent non-synchronous torsion and bending vibration observed in such a system may be explained by the existence of quadratic non-linear coupling terms and internal resonance effects between three and four modes. A finite degree of freedom model was formulated and the method of multiple scales was applied. For the case of three-mode interactions stationary solutions were obtained which gave very satisfactory agreement with measured responses, and showed the effects of detuning of the internal resonance. The analysis also gave a clear indication of the cause of strong four-mode interactions but did not yield stationary solutions in this case. Experimental results are included to illustrate the four-mode interactive response.