Compact model for a MEM perforation cell with viscous, spring, and inertial forces

Abstract

A compact model for calculating damping, inertial, and spring forces in a perforated squeeze-film damper is reported. The repetitive pressure patterns around each perforation are utilized by analyzing the visco-acoustic wave transmission around the hole in a cylindrical volume, called perforation cell. The model is needed in applications where the acoustic wavelength of the oscillation is comparable with the dimensions of the perforation cell. The model is constructed of acoustic impedance two-ports. A novel model is derived for the air gap region, and a published two-port model is used for the hole. The impedances for these two-ports are derived from the low reduced frequency model that is equivalent with linearized, harmonic Navier–Stokes equations for acoustic wave propagation in thin channels. This model considers also the transition from the isothermal conditions at low frequencies to the adiabatic ones at high frequencies. The dimensions of MEMS structures are considered using slip conditions for velocities and temperatures. Also, an easy-to-use simplified model for frequencies where the squeeze number and the Reynolds numbers are below unity is derived. The analytical compact model is verified with FEM simulations using a harmonic solver for linearized Navier–Stokes equations with slip boundary conditions in a wide range of perforation ratios. The maximum relative error in the damping coefficient in the simulated cases was 20% upto the first resonant frequency.