MEAM 535 Advanced Dynamics

Fall 1999

Homework #9

1. An electrostatically actuated micro-electro-mechanical structure (MEMS) is modeled as a parallel plate capacitor with bottom plate fixed and top plate restrained by a linear spring of spring constant, k. Assume that the plate has only one degree of freedom that lets it move vertically. The plate has a mass of m. Plate area is A. The initial gap between the plates is g0. A DC voltage vdc is applied between the plates. The potential energy due to electrostatic force is given by (-0.5*C*vdc^2). C is the capacitance and is given by eps0*A/(g0-x). x is the displacement of the plate measured positive downward.
   • When vdc reaches a critical value, the plate pulls in onto the fixed plate although for smaller values of vdc, it will have a stable equilibrium position well above the bottom plate. Using the potential energy function, show that this critical voltage, called "pull-in" voltage is given by sqrt(8*k*g0^3/(27*eps*A).
   • Use the following data and compute the pull-in voltage.
     m = 1.1615e-10 kg; k = 164 N/m; g0 = 1.6E-6 m; A = 25E=9 m^2; eps = 8.854E-12 SI units;
   • Using Lagrange's equations, write the equation of motion for this system symbolically.
   • Matlab script given to you (capdyn.m) numerically integrates the equation of motion of this system.
     Note then when vdc just a bit higher than 27.545 volts, the plate pulls in although the "pull-in" voltage is higher than 27.545. Try it with the Matlab script first to see for yourself. This is called dynamic pull-in.
     Explain why this happens and derive an expression for the "dynamic pull-in" voltage.
   
   
2. A plane double pendulum is constructed as follows. An inextensible rod of length L1 is attached to a fixed frame at one end and a mass m1 at the other end. Another inextensible rod of length L2 is attahed to the first mass m1 and another mass m2. Assume that the masses of the rods are negligible. This double pendulum oscillates in the vertical plane under the effect of gravity. Denote the angle made by the first string with the vertical line by theta1 and the angle made by the second string with the vertical line by theta2.
   
   • Obtain the static equilibrium configurations for this system using the potential energy function.
   • Investigate the stability of motion at each of the equilibrium configurations using the linearized equations of motion.
   • Investigate the stability using Lyapunov's indirect method without linearizing the equations at all the equilibrium points.