A simple excitation model of parametric resonators: Simulating and explaining the response at cross-over points

We present a new intuitive and rational model of parametric resonance. In parametric resonators one of the system parameters, usually stiffness, is modulated in time. Due to this time modulation the system may develop a periodic response. It is well known that when this modulation is sufficiently strong and at an appropriate frequency, the periodic response may be unbounded - even though the system is not driven directly by an external force. Our model assumes that stiffness is toggled between two distinct values, and that this toggling occurs either when motion is maximal or when velocity is maximal. We show that this model of parametric resonance converges to the classic Meissner parametric resonator, at discrete values of the amplitude and frequency of stiffness modulation. At these critical points the system response is periodic and on the verge of becoming unbounded. The relevance of these critical points is that their discrete nature makes them appealing for sensing and clocking applications in MEMS.