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(1)x¨+αx˙+x=−λ(1+x)2, which is a simplified model for an electrostatically actuated MEMS device. The static pull-in value is λ⁎=427, 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: touchdown occurs. The maximal displacement achieved by a stationary solution, known as the pull-in distance, is equal to −13 in this model. Assuming that the motion starts from rest, we establish the existence of a dynamic pull-in value λd⁎(α)∈(0,λ⁎), defined for α∈[0,∞), which is a threshold in the sense that x(t) approaches a stable stationary solution as t→∞ for 0<λ<λd⁎(α), while touchdown occurs for λ>λd⁎(α). This dynamic pull-in value is a continuous, strictly increasing function of α and limα→∞⁡λd⁎(α)=λ⁎. A similar result is obtained for initial conditions of the form x(0)∈(−13,1), x˙(0)=0.