Modelling of Nonlinear Circular Plates Using Modal Analysis: Simulation and Model Validation

There are many micro-structures that can be modelled as simple geometrical elements, such as beams with rectangular cross-sections or circular or rectangular plates. Examples of such structures abound in the MEMS literature [1-2]. Because they are very easy to manipulate, models based upon plate or beam approximations are very interesting. However, they are very often limited to the so-called “small-displacements” regime of the deformable structure, i.e. when the displacements of the structure, say the deflection of the beam or plate, are much smaller than its thickness. In this case, the tensile stresses that are caused by the elongation of the structure can be neglected and this results in purely linear models that can be readily implemented and used. In the “large-displacement” regime, the tensile stresses must be taken into account; this results in nonlinear partial differential equations (PDEs), such as the Von Karman equation for plates, and, as a consequence, in nonlinear models. The most common way of handling these nonlinear PDEs consists in linearizing them close to a working point, as in [3-4]. This approach can lead to excellent results, provided the linearization hypothesis holds. For sensors/actuators operating over a wide range, as in microswitch applications, this is often not the case and a model valid over the whole range must be found [5-6]. The same goes, for example, for squeeze-film damping for which models that remain valid for large displacements with respect to the gap [7] must be used. In this paper, a method for modelling the large-displacement actuation of deformable microstructures is proposed and illustrated, in the case of a circular axisymmetric plate. First of all, the problem is formulated and its boundary conditions are expressed as functions of the two unknowns, displacements w and Airy stress function F. The problem is then split in two parts, each part corresponding to one equation of Von Karman, to which the techniques of modal analysis are applied in order to obtain a set of ordinary differential equations (ODEs). The implementation of the resulting model is then discussed and some simulation results are given, as a basis for comparison.