Extending the validity of squeezed-film damper models with elongations of surface dimensions

Compact models for micromechanical squeezed-film dampers with gap sizes comparable to the surface dimensions are presented. Two different models considering both the border flow and non-uniform pressure distribution effects are first derived for small squeeze numbers. In the first 'surface extension' model the border effects are considered simply by calculating the damping with extended surface dimensions, and in the second 'border flow channel' model an additional short fictitious flow channel is placed at the damper borders. Utilizing a large amount of two-dimensional (2D) FEM simulation results by varying the damper dimensions, mainly the ratio a/h between the surface length and the air gap height, surface elongations are extracted using both elongation models. Both linear and torsional modes of motion are considered at the continuum flow regime. These results show that the `surface extension' model is superior, since the extracted elongation Delta a is almost constant (Delta a = 1.3h), leading to a very simple model. Next, the rare gas effects are included in the `surface extension' model in the slip flow regime (Knudsen number 0 < Kn < 0.13). Considering slip boundary conditions, 2D FEM simulations are repeated and the elongations, now functions of the Knudsen number, are extracted from the results. A simple fitted equation for calculating the surface extension Delta a finally results. The maximum relative errors of the models are smaller than 10% for a/h > 4 in the linear motion and for a/h > 10 in the torsional motion. The model assumes incompressible flow and thus the maximum frequency where the models are valid is limited. In typical MEMS topologies where the elongations must be considered, this means that the models are valid below frequencies of 500 kHz. To also model rectangular 2D squeezed-film dampers, these elongations are applied directly in the surface length. and width used in the compact models. Comparison with three-dimensional (3D) FEM simulations shows that the new model gives excellent results, and it extends the validity range of existing compact models. The maximum relative error of the models is smaller than 10% for a/h > 16 in the linear motion and for a/h > 16 in the torsional motion. The new surface extension model is useful in simulating both the circuit level and the system level behavior of gas-damped microelectromechanical devices with aspect ratios greater than 2 in the time and frequency domains.