Compact damping models for laterally moving microstructures with gas-rarefaction effects

Compact models for the viscous damping coefficient in narrow air gaps between laterally moving structures are reported. In the first part of the paper, a simple frequency-independent first-order slip-flow approximation for the damping coefficient is derived and compared with a more accurate expression obtained from the linearized Boltzmann equation. The simple approximation is slightly modified and fitted to match the accurate model, The resulting simple approximation has a maximum relative error of less than +/-6% at arbitrary Knudsen numbers in viscous, transitional and free molecular regions, In the second part of the paper, dynamic models for the damping force are derived, considering again gas rarefaction, by applying various boundary conditions. The damping admittance of the first-order slip-flow model is implemented also as an electrical equivalent admittance, constructed of RC sections, to allow both frequency and time domain simulations with a circuit simulator. The dependence of the damping admittance on pressure and gap displacement is demonstrated with model simulations. The accuracy and validity range of the model are verified with comparative numerical simulations of the Navier-Stokes equation. Finally, the damping coefficient in a lateral resonator is calculated using the compact model and compared with measured data with good agreement.