Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor

Abstract

Parametric resonance has been well established in many areas of science, including the stability of ships, the forced motion of a swing and Faraday surface wave patterns on water. We have previously investigated a linear parametrically driven torsional oscillator and along with other groups have mentioned applications including mass sensing, parametric amplification, and others. Here, we thoroughly investigate the design of a highly sensitive mass sensor. The device we use to carry out this study is an in-plane parametrically resonant oscillator. We show that in this configuration, the nonlinearities (electrostatic and mechanical) have a large impact on the dynamic response of the structure. This result is not unique to this oscillator—many MEMS oscillators display nonlinearities of equal importance (including the very common parallel plate actuator). We report the effects of nonlinearity on the behavior of parametric resonance of a micro-machined oscillator. A nonlinear Mathieu equation is used to model this problem. Analytical results show that nonlinearity significantly changes the stability characteristics of parametric resonance. Experimental frequency response around the first parametric resonance is well validated by theoretical analysis. Unlike parametric resonance in the linear case, the jumps (very critical for mass sensor application) from large response to zero happen at additional frequencies other than at the boundary of instability area. The instability area of the first parametric resonance is experimentally mapped. Some important parameters, such as damping co-efficient, cubic stiffness and linear electrostatic stiffness are extracted from the nonlinear response of parametric resonance and agree very well with normal methods.