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 + 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 $λ^{*}=\frac{4}{27}$, which corresponds to the largest value of $λ$ for which there exists at least one stationary solution. For $λ> λ^{*}$ 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 $λ_{d}^{*}(α) \in (0, λ^{*})$, defined for $α\in [0,\infty)$, which is a threshold in the sense that $x(t)$ approaches a stable stationary solution as $t \to \infty$ for $0 < λ< λ_{d}^{*}(α)$, while touchdown occurs for $λ> λ_{d}^{*}(α)$. This dynamic pull-in value is a continuous, strictly increasing function of $α$ and $\lim_{α\to\infty} λ_{d}^{*}(α)= λ^{*}$.

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