Nonlinear resonances in systems having many degrees of freedom

An analysis is presented of the main, superharmonic, subharmonic, combination and internal resonances in a weakly nonlinear system having many degrees of freedom. The system has cubic nonlinearities, modal linear viscous damping and is subject to harmonic excitations. The method of multiple scales, a perturbation technique, is used to develop a unified method for the study of the various resonances. The effects of an internal resonance are explored in depth.

The first approximation obtained by the method of multiple scales extracts the dominant features of the response and expresses them in terms of elementary functions. It is shown that in the absence of internal resonances, the steady-state response can contain only the modes which are resonantly excited. In the presence of an internal resonance, modes other than those that are resonantly excited can appear in the response.

The usefulness of the method developed in this work in providing clarity and insight into nonlinear phenomena is illustrated by applications to the nonlinear vibrations of beams and circular plates. Numerical examples show that the results obtained by the first approximation compare favorably with the results obtained by a numerical integration of the governing differential equations. The numerical examples indicate that a significant transfer of energy can take place from the highest mode involved in the internal resonance to the lower modes but not vice versa.