Pull-In Effect Analysis in a MEMS Actuator

Problem Statement

In microelectromechanical systems (MEMS), electrostatic actuators that use electrostatic force to control movable parts are widely used. Consider the physical instability (pull-in effect) inherent in such a system.

A fixed lower plate of area $S$S is placed horizontally. Directly above it, an upper plate (movable plate) of area $S$S and mass $m$m is suspended by a light insulating spring with spring constant $k$k. The two plates form a parallel plate capacitor facing each other. The space between the plates is vacuum with permittivity $\varepsilon_0$ε0​. Edge effects at the plate boundaries can be neglected.

Initially, no voltage is applied between the plates, and the movable plate is at rest in equilibrium between gravity and the spring's elastic force. At this point, the plate separation is $d$d and the capacitance of the capacitor is $C_0$C0​.

From this state, a DC power supply is connected between the plates, and the applied voltage $V$V is slowly increased from $0$0 (slowly enough that the plates do not vibrate, always maintaining force equilibrium). As the voltage increases, the plate separation decreases. When the voltage $V$V exceeds a critical value $V_0$V0​, the spring's restoring force can no longer support the electrostatic force, and the movable plate is suddenly attracted to and collides with the lower plate. This phenomenon is called the "pull-in effect," and the critical voltage $V_0$V0​ is called the "pull-in voltage."

Find the value of $V_0^2$V02​, the square of the pull-in voltage.

Constraints

• Spring constant: $k = 1.35 \text{ N/m}$k=1.35 N/m
• Initial plate separation: $d = 6.4 \times 10^{-6} \text{ m}$d=6.4×10−6 m
• Initial capacitance: $C_0 = 2.4 \times 10^{-12} \text{ F}$C0​=2.4×10−12 F
• Mass: $m = 2.0 \times 10^{-6} \text{ kg}$m=2.0×10−6 kg
• Gravitational acceleration: $g = 9.8 \text{ m/s}^2$g=9.8 m/s2
• All physical quantities are in SI units under ideal vacuum conditions.

Input Format

The square of the pull-in voltage $V_0^2$V02​ (in units of $\text{V}^2$V2) is expressed as an irreducible fraction $\frac{A}{B}$BA​. Give the value of $A + B$A+B as a positive integer.