Modeling and prediction of the dynamic behaviour of microbeams under electrostatic load

This paper investigates the features of different techniques to model the dynamic behaviour of mi- crobeams electrostatically actuated, mainly to identify the frequency shifting due to the electromechanical coupling. The aim is to define an engineering approach to be followed for a preliminary characterization of the microsystem. In a first part of the paper the main aspect addressed is the application of a systematic approach to write the equations of motion, applicable not only to this particular device, but in general to MEMs. The Lagrangian generalized method is outlined and applied to the current case of electromechanical coupling. A second issue concerns the application either of lumped or distributed parameters models, eventually discretized by FEM or equivalent techniques. A first model with a single degree of freedom of a double clamped microbeam is investigated: the analytic solution of the linearized equation of motion is compared to the numerical one of the nonlinear problem, solved by the Newmark’s method. FEM models are then introduced and compared to the Newmark numerical integration applied to the single d.o.f. Although an accurate prediction of the nonlinear dynamic behaviour of the system can be drawn by the FEM-Newmark approach, a new procedure for a fast and preliminary estimation of the frequency response has been developed: the relationship between frequency and voltage is approximated by means of a parabolic curve based on two interpolation points, corresponding to the steady-state frequency (pure mechanical actuation) and to an arbitrary value of voltage, for which the Newmark’s method is applied only once.