Dynamic Analysis of the Electrostatically Controlled Membrane Mirror using Multiple Scales

The vibrational dynamics of an electrostatically controlled membrane mirror are investigated. The motion is described by a nonlinear partial differential equation with nonconstant coefficients. The system equation is first linearized and then parameterized. The resulting equation is a singularly perturbed linear partial differential equation with variable coefficients. Asymptotic approximations are then developed systematically by the generalized multiple scales method. Thus, the rapid and slow aspects of the membrane dynamics in both the spatial and temporal sense are separated systematically. The results of our approach are compared with direct numerical integration. An excellent agreement between the two solutions is seen in the comparisons. Further reduction of the errors by including higher order approximations is discussed.