Periodic Oscillations in MEMS under Squeeze Film Damping Force

Abstract

We provide sufficient conditions for the existence of periodic solutions for an idealized electrostatic actuator modeled by the Liénard-type equation $\ddot{x}+F_D\left(x,\dot{x}\right)+x=\beta {\mathcal{V}}^2\left(t\right)/{\left(1-x\right)}^2,x\in \left]-\infty ,1\right[$ with $\beta \in {ℝ}^{+}$ , $\mathcal{V}\in C\left(ℝ/T\mathbb{Z}\right)$ , and $F_D\left(x,\dot{x}\right)=\kappa \dot{x}/{\left(1-x\right)}^3$ , $\kappa \in {ℝ}^{+}$ (called squeeze film damping force), or $F_D\left(x,\dot{x}\right)=c\dot{x}$ , $c\in {ℝ}^{+}$ (called linear damping force). If $F_D$ is of squeeze film type, we have proven that there exists at least two positive periodic solutions, one of them locally asymptotically stable. Meanwhile, if $F_D$ is a linear damping force, we have proven that there are only two positive periodic solutions. One is unstable, and the other is locally exponentially asymptotically stable with rate of decay of $c/2$ . Our technique can be applied to a class of Liénard equations that model several microelectromechanical system devices, including the comb-drive finger model and torsional actuators.