ABSTRACT
In this work we review the phenomena associated with internal resonances in multi-degree-of-freedom nonlinear mechanical systems, and then describe their consequences for the system response. In particular, we first show how an internal resonance produces a strong coupling between the normal modes. This is shown to result in amplitude modulated motion in the free and undamped response of a system with quadratic nonlinearities. We then use the method of averaging to derive the averaged equations that govern, to a first-order approximation, the response of harmonically excited two-degree-of-freedom nonlinear mechanical systems with cubic nonlinearities. Restricted forms of these averaged equations for systems with 3:1 and 1:1 internal resonances characterize the nonlinear response of many physical systems, including pendulums, stretched strings, beams with various boundary conditions, and plates and shells with different geometries. We provide a brief review of the recent literature and present examples of the response of systems with 3:1 and 1:1 internal resonances. We show that these internal resonances, in the presence of an external resonance, give rise to a coupling between the modes involved in the resonance, leading to nonlinear periodic, almost periodic, and chaotic amplitude modulated motions.
