Abstract
An analytical and experimental investigation into the response of a nonlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotropic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the response consisted of the first, third, and fourth modes. In these cases, 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 interactions 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 unconventional coordinate transformation. In the analytical model, we used experimentally determined damping coefficients. From this nonautonomous system, we obtained a first approximation of the response by using the method of averaging. The analytically predicted responses and bifurcation diagrams show good qualitative agreement with the experimental observations. The current study brings to light a new type of nonlinear motion not reported before in the literature and should be of relevance to many structural and mechanical systems. In this motion, a static response of a low-frequency mode interacts with the dynamic response of two high-frequency modes. This motion loses stability, resulting in oscillations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.
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Anderson, T.J., Nayfeh, A.H. & Balachandran, B. Coupling between high-frequency modes and a low-frequency mode: Theory and experiment. Nonlinear Dyn 11, 17–36 (1996). https://doi.org/10.1007/BF00045049
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DOI: https://doi.org/10.1007/BF00045049