Visco-Elastic Modelling of MEMS Inertial Sensors

The mechanical modelling of MEMS requires the determination of the inertia, of the damping and of the stiffness of the various elements that constitute the device. Although some parameters seem easy to be determined (e.g., the inertia! parameters), at working frequencies typical of MEMS inertial sensors some elements, such as supporting beams, not only contribute to the elasticity of the system but also to its inertia. For what concerns damping, two main pressure levels have to be considered: atmospheric pressure level (from now on called "high pressure," i.e., 105 Pa) and vacuum (from now on called "low pressure," i.e., 26 Pa). At high pressure the mean free path of an air molecule is much smaller than typical MEMS dimensions. Thus, air can be considered as a viscous fluid and two phenomena occur: flow damping and squeeze film damping. These two terms can be evaluated through a simplified Navier-Stokes equation. In vacuum the air cannot be considered as a viscous fluid any more since the mean free path of an air molecule is of the same order of magnitude of typical MEMS dimensions. Thus, the molecular fluid theory must be used to estimate the damping. The present paper shows an approach to pass from a complex FEA model to a lumped parameter model of the considered MEMS inertial sensor at both ambient and low pressure levels that can easily be used during the design or optimisation phases. Although developed and validated for a specific MEMS inertial sensor, the proposed approach is fully general and could be used for any other MEMS device.