5 PARAMETRIC RESONANCES IN A MICROELECTROMECHANICAL SYSTEM

The Mathieu equation(1) governs the forced motion of a swing(2), the s tability 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 supercon ducting devices(9). Theory predicts that parametric resonances occur n ear 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 typical ly be observed, because of damping and the exponential narrowing(10) o f the regions with increasing n. Here we report parametrically excited torsional oscillations in a single-crystal silicon microelectromechan ical system. Five instability regions can be measured, due to the low damping, stability and precise frequency control achievable in this sy stem. The centre frequencies of the instability regions agree with the oretical predictions. We propose an application that uses parametric e xcitation to reduce the parasitic signal in capacitive sensing with mi croelectromechanical systems. Our results suggest that microelectromec hanical systems can provide a unique testing ground for dynamical phen omena that are difficult to detect in macroscopic systems.