Parametric resonance is a dynamic phenomenon where one or more parameters of the system (i.e., inertia, stiffness or damping) are periodically modulated in time. Parametric resonators are governed by a 2nd order, linear non-autonomous differential equation. In microelectromechanical systems (MEMS), parametric resonators may be simply implemented using linearly tapered comb-drives. Therefore, it is interesting and relevant to consider MEMS parametric resonators for sensing applications.
There are two classic examples of parametric resonators, governed by the Mathieu equation and the Meissner equation. In the case of the Mathieu equation, stiffness is modulated by a single harmonic, whereas in the Meissner equation, stiffness is modulated by a square waveform. The response of a parametric resonator can be bounded (i.e., stable) or unbounded (i.e., unstable). On the boundary between the bounded and unbounded responses, the response may be periodic. The stability of the response is presented using an Ince-Strutt diagram (a.k.a. stability map), which relates the stability of the response to the parameters of the system. Both the Mathieu and Meissner equations are strongly unstable in regions of high relative over-modulation.
In this talk, the Whittaker-Hill (W-H) equation that is related to Quantum Mechanics is presented. It was recently identified that the stability map of the W-H equation includes crossover points, in regions of high relative over-modulation.
In our investigation we discovered that in the vicinity of these crossover points, tiny disjoined regions of stable response exist. When linear damping is added to the equation, the crossover points vanish and each two disjoined regions of stability merge into a single island of stability. We derived a new analytic expression for the vertical location of crossover points of the W-H equation.
We generalized the W-H equation by adding a new tuning parameter. This tuning parameter allows us to dictate the vertical location of islands of stability, and even enables to shift islands of stability to the left side of the stability map, where the mechanical stiffness is negative.
