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A modified Reynolds equation where the gas inertia effect is included is derived from the Navier-Stokes equation. Continuum and slip-flow regions are modelled. Small flow velocity is assumed, and border effects are not considered. By introducing an effective flow rate coefficient including the inertial and rare gas effects, existing linearized analytic squeezed-film damping models can be reused. Equivalent-circuit mechanical impedance and admittance implementations for a rectangular parallel-plate damper are given. The model response is compared against 2D and 3D FEM time-domain and frequency-domain simulations with an excellent agreement. The validity and limitations of the models are discussed extensively.

The concept of the perforation cell is specified for compact modelling of perforated gas dampers with micromechanical dimensions. Both, analytic expressions and FEM simulations, are used to derive its flow resistance. An extensive set of FEM simulations is performed to characterize the flow resistance of the cell, and to derive approximations for different flow regions by fitting simple functions to them. Sinusoidal small-amplitude velocities are assumed, and micromechanical dimensions are considered with rare gas effects in the slip flow regime (Knudsen number <0.1). The model is capable of modelling all practical combinations of the perforation cell dimensions in a wide range of perforation ratios (1,...,90%). Its validity is verified with a Navier-Stokes solver, and it is shown to be accurate (relative error <4.5%) in the continuum and slip flow regimes. Estimates for cut-off frequencies due to inertial and compressibility effects are specified in a way that the maximum operation frequency of the model can be easily tested. Using a harmonic FEM solver, these estimates are verified. The perforation cell model is also applied to estimate the damping in a perforated rectangular damper (4,...,64 square holes). The damping predicted by the simple model is in moderate agreement with that obtained with 3D FEM simulations.

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 a is almost constant ( 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 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.

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