Non-linear mode coupling in symmetrically kinked bars

Abstract

A theoretical and experimental study of energy transfer between the vibrational modes of a symmetrically kinked bar with clamped ends is described. Two non-linear mechanisms responsible for energy transfer from one mode to another at twice the frequency are identified. The first arises from the interaction between shear and tensional forces at the kink and the second from unbalanced moments across the kink. In the system studied, the first of these mechanisms is dominant. A further related mechanism is responsible for energy transfer to modes at three times the base frequency. When a kinked bar with mode frequencies of a few hundred hertz is shaped so that two modes have the desired 2 to 1 frequency relation and is excited by striking it at a node of the higher mode, then the amplitude of that mode rises from zero to a maximum in a time of order 0.1 s and then decays. Theory and experiment are in quite good agreement in relation to the time delay to the maximum amplitude and the magnitude of this maximum. The results contribute to an understanding of the vibrational behaviour of certain musical gongs and are also relevant to other systems.