Frequency-shifting analysis of electrostatically tunable micro-mechanical actuator

Abstract

A numerical approach for eigenvalue analysis of electrostatically tunable micro-mechanical actuators is proposed. An effi- cient algorithm for calculating the natural frequwncy shifting due to applied DC turning voltage is proposed. In the calculation of the coupled field problem, 3-D FEM and BEM are employed. The numerical examples are presented and the numerical analysis results are compared with experimental data. Previous works on the analysis of tunable electrostatic actuators used very simplified models in which micro- mechanical actuators are assumed as single mass-spring systems (3-5). However, actuator systems must be modeled to continuum structures because the actuators assumed as rigid body in previous works are generally deformable. Moreover, in real situations, the electrostatic field and the elastic deformation field of the system are 3-dimensionally coupled. An advanced numerical approach, which can treat continuous fields, is used only in the calculation of the electrostatic force applied on the device (6). In order to accurately predict the behavior of the real system, the analysis of the coupled fields in 3-D continuous system and an algorithm for evaluating the effects of electrostatic field on natural frequency are necessary. In this work, a numerical scheme for eigenvalue analysis of electrostatically tunable micro-mechanical actuators is proposed. We have used the boundary element method (BEM) and the finite element method (FEM) for the 3-D coupled analysis of electrostatic field and deformation of a device. In order to consider the effects of electrostatic field on natural frequency, an equivalent stiffness matrix for electrostatic tuning voltage is introduced. We can perturb the equilibrium structure using a concerned eigenvector of operating mode and then linearize the corresponding electrostatic force variation in order to determine the equivalent stiffness. For examples, a simple beam-shaped structure and a tunable micromirror are analyzed and the numerical analysis results are compared with experimental data.