Some Substantial Effects of Nonlinear Coupling between Modes of Vibration

Abstract

Some remarkable and unexpected patterns of forced vibration response may be obtained in simple structural configurations in which for example the resonant motion of a directly excited mode of vibration is suppressed, while other modes respond in a vigorous and catastrophic manner at frequencies far removed from the excitation frequency. Such responses have been observed in a number of vibratory systems having as a common feature subsystems of connected beams with blade-like proportions such that forced vibration response of the primary system induces axial or transverse in-plane inertial loads in the plane of maximum stiffness of the connected beam. It is then possible for torsional and flexural modes of the connected beam to be excited and couple in a nonlinear manner with the primary system modes.

The interactive motions may be attributed to quadratic nonlinear coupling terms in the differential equations which are rendered significant by the realization of internal resonance conditions both singly or by the simultaneous achievement of more than one. In the case of interactive motion involving a low number of modes the method of Multiple Scales has been applied to obtain explicit steady-state response relationships which have given good correlation with experimental measurements and which have verified distinctive aspects of the vibratory responses. At high levels of excitation the first order perturbation solution ceases to give reasonable predictions of the responses. Experimental results indicate that interactive motions of great complexity can then occur. The paper surveys general aspects of the problem and illustrates the specific case of nonlinear interactive motion in systems of coupled beams.