COUPLING BETWEEN HIGH-FREQUENCY MODES AND A LOW-FREQUENCY MODE - THEORY AND EXPERIMENT

An analytical and experimental investigation into the response of a no nlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotro pic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the r esponse consisted of the first, third, and fourth modes. In these case s, the modulation frequency of the amplitudes and phases of the third and fourth modes was equal to the response frequency of the first mode . Subsequently, we developed an analytical model to explain the intera ctions between the widely separated modes observed in the experiments. We used a three-mode Galerkin projection of the partial-differential equation governing a thin, isotropic, inextensional beam and obtained a sixth-order nonautonomous system of equations by using an unconventi onal coordinate transformation. In the analytical model, we used exper imentally determined damping coefficients. From this nonautonomous sys tem, we obtained a first approximation of the response by using the me thod of averaging. The analytically predicted responses and bifurcatio n diagrams show good qualitative agreement with the experimental obser vations. The current study brings to light a new type of nonlinear mot ion not reported before in the literature and should be of relevance t o many structural and mechanical systems. In this motion, a static res ponse of a low-frequency mode interacts with the dynamic response of t wo high-frequency modes. This motion loses stability, resulting in osc illations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.