Dynamical Solutions of Singular Wave Equations Modeling Electrostatic MEMS

作者：Guo Yujin^*

来源：SIAM Journal on Applied Dynamical Systems, 2010, 9(4): 1135-1163.

DOI：10.1137/09077117X

• electrostatic MEMS
• Euler-Bernoulli plate
• singular plate equations
• hinged boundary conditions
• global existence
• exponential convergence
• finite-time quenching

We study a fourth-order singular wave equation involving a singular nonlinear term lambda/(1-u)(2) in a bounded domain of R-N. This equation models a simple electrostatic microelectromechanical system (MEMS) device consisting of a thin elastic plate with boundary supported at 0 above a rigid ground plate located at 1. Here u is modeled to describe the dynamical deflection of the elastic plate. When a voltage-represented here by lambda-is applied, the elastic plate deflects towards the ground plate, and snap-through (quenching) may occur when it exceeds a certain critical value lambda* (the pull-in voltage), creating a so-called pull-in instability, which greatly affects the design of many devices. For 1 <= N <= 3, analytic results show that there exist 0 < lambda(1) <= lambda* < infinity such that for 0 <= lambda < lambda(1) the elastic plate globally exists and exponentially converges to a regular steady state, while for lambda > lambda* the elastic plate quenches at finite time.

• 出版日期2010