On the dynamic pull-in instability in a mass-spring model of electrostatically actuated MEMS devices

Abstract

In this work we study the mass-spring system \begin{equation} \ddot x + α\dot x + x = - \fracλ {(1+x)^{2}}, \label{e:inertia} \end{equation} which is a simplified model for an electrostatically actuated MEMS device. The static pull-in value is ${\lambda }^{\ast }=\frac{4}{27}$, which corresponds to the largest value of $\lambda$ for which there exists at least one stationary solution. For $\lambda >{\lambda }^{\ast }$ there are no stationary solutions and $x(t)$ achieves the value $-1$ in finite time: {\it touchdown} occurs. We establish the existence of a dynamic pull-in value ${\lambda }_d^{\ast }(\alpha )\in (0,{\lambda }^{\ast })$, defined for $\alpha \in [0,\infty )$, which is a threshold in the sense that $x(t)$ approaches a stable stationary solution as $t\to \infty$ for $0<\lambda <{\lambda }_d^{\ast }(\alpha )$, while touchdown occurs for $\lambda >{\lambda }_d^{\ast }(\alpha )$. This dynamic pull-in value is a continuous, strictly increasing function of $\alpha$ and ${\lim }_{\alpha \to \infty }{\lambda }_d^{\ast }(\alpha )={\lambda }^{\ast }$.