Five parametric resonances in a microelectromechanical system

The Mathieu equation(1) governs the forced motion of a swing(2), the stability of ships(3) and columns(4), Faraday surface wave patterns on water(5,6), the dynamics of the electrons in Penning traps(7), and the behaviour of parametric amplifiers based on electronic(8) or superconducting devices(9). Theory predicts that parametric resonances occur near drive frequencies of 2 omega(o)/n, where omega(o), is the system's natural frequency and n is an integer greater than or equal to 1. But in macroscopic systems, only the first instability region can typically be observed, because of damping and the exponential narrowing(10) of the regions with increasing n. Here we report parametrically excited torsional oscillations in a single-crystal silicon microelectromechanical system. Five instability regions can be measured, due to the low damping, stability and precise frequency control achievable in this system. The centre frequencies of the instability regions agree with theoretical predictions. We propose an application that uses parametric excitation to reduce the parasitic signal in capacitive sensing with microelectromechanical systems. Our results suggest that microelectromechanical systems can provide a unique testing ground for dynamical phenomena that are difficult to detect in macroscopic systems.