Consistent Vibration Analysis of Electrostatically Coupled Structures: Application to Microsystems

Analysing the vibration of microsystems is a fundamental issue in the design of sensors and actuators. The state of the art currently consists in using staggered procedures based on the iteration between a structural model loaded by electrostatic forces and an electrostatic model defined on a domain following the deformation of the structure. Staggered iteration then leads to a static equilibrium position. Performing a perturbation analysis around the static equilibrium to evaluate the tangent stiffness for every degree of freedom leads to very high computing costs. Therefore only the tan-gent stiffness associated with assumed modes (typically the purely structural modes) is computed. Such a procedure can lead to important inaccuracies in the estimation of eigenfrequencies for designs where the electrostatic coupling is not quasi-uniform.
In our work, we have developed a fully coupled electro-mechanical formulation that allows to find static equilibrium positions in a non-staggered way and which provides fully consistent tangent stiffness matrices for vibration analysis. The efficiency of the approach will be illustrated on modes of micro-electromechanical devices. The coupled electromechanical modes obtained in the vicinity of equilibrium positions can be significantly different from the approximations obtained using a structural reduction at forehand. Numerical results are checked against analytical results. The application examples highlight the fact that the strongly coupled formulation proposed here allows consistent vibration analysis of the system and yields more accurate eigenmodes and frequencies than the classical perturbed assumed mode approach.