Generalized parametric resonance

We consider the dynamic response of systems subjected to principal parametric resonant excitation in which the full potential, including nonlinearities, is modulated in a time-periodic manner. This work was motivated by the model of Rhoads et al. [J. Sound Vibration, 296(2006), pp. 797-829], which was used to describe an interesting bifurcation structure that was experimentally observed in a microelectro-mechanical system. The goal of the present investigation is to more fully explore this class of systems, described by a generalized nonlinear Mathieu equation, and ascertain general features of their response. The method of averaging is us ed to derive equations governing the slowly varying amplitude and phase for parametrically excite d systems with weak nonlinearity, small damping, and near resonant excitation, allowing for a de tailed analysis of the system steady-state response. Results for a general class of models are presented first, followed by details for two examples: (i) generalized parametric resonance with cubic nonlinearities, for which the model of Rhoads et al. [J. Sound Vibration, 296(2006), pp. 797-829] is extended by including nonlinear damping and the system response is more fully described, and (ii) a vertically excited inverted pendulum with a stiff linear torsional spring, for which the time-periodic modulation acts on the full gravitational potential. The analysis, which is supported by numerical simulations, shows that nonlinear damping and higher-order nonlinearities lead to a bounded primary response, in contrast to the unbounded primary response found in some parameter regio ns in the Rhoads model. More interesting is the fact that the steady-state frequency response can exh ibit a sequence of isolas along the nonlinear response branches associated with the usual parametric re sonance, where the number of isolas depends on the level of damping, the form of the potential, and the level of excitation. These results indicate that a relatively simple generalization of the Mathieu equation is able to capture a variety of responses that are important in applications.