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

In this work we study the mass-spring system \begin{equation} \ddot x + \alpha \dot x + x = - \frac{\lambda} {(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 $\lambda^{*}=\frac{4}{27}$, which corresponds to the largest value of $\lambda$ for which there exists at least one stationary solution. For $\lambda > \lambda^{*}$ 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 $\lambda_{d}^{*}(\alpha) \in (0, \lambda^{*})$, 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 $0 < \lambda < \lambda_{d}^{*}(\alpha)$, while touchdown occurs for $\lambda > \lambda_{d}^{*}(\alpha)$. This dynamic pull-in value is a continuous, strictly increasing function of $\alpha$ and $\lim_{\alpha\to\infty} \lambda_{d}^{*}(\alpha)= \lambda^{*}$.