Dynamical behaviour of electro-mechanical coupled problem

MEMS are very small devices in which electric as well as mechanical dynamics phenomena appear. Because of the microscopic scale, some strong coupling effects between the different physical fields appear, and some forces, which are negligible at macroscopic scales, have to be taken into account. In order to make a good design of these micro-systems, it is important to analyse the coupling between the electrical and mechanical fields. This paper concerns the modelling of the strong electromechanical coupling appearing in micro-electro-mechanical systems (MEMS). The finite element method (FEM) is used to perform dynamical analysis taking into account large mesh displacements. Analysing the vibration of microsystems is a fundamental issues in the design of a broad range of sensors and actuators.

The state of the art currently consists in using staggered procedures to compute quasi-static configurations 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 yields the electromechanical linearized stiffness needed for computing the eigenfrequencies. Obviously, performing the perturbation analysis to evaluate the tangent stiffness for every degree of freedom leads to very high computing costs and therefore onlythe tangent stiffness associated with presumed modes (typically some purely structural modes) are computed. Such a procedure can lead to important inaccuracies 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 results obtained indicate that using this approach pull-in can be computed very accurately and at low computational cost. Also 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 where appropriate. The formulation developed for the strongly coupled electromechanical problem allows consistent vibration analysis of the system and yields more accurate vibration eigenmodes and frequencies than the classical staggered approaches.
A second interest of the coupled tangent matrix is the computation of the dynamical behaviour of the coupled problem when we apply suddenly a voltage to the electrodes. Even under the pull-in voltage, the overshoot of the dynamical response may reduce the distance between the plate so that the electric force becomes dominant and the plates stick together. This phenomenon is called the dynamical pull-in. We will also treat it in this research.