Squeeze film damping effect on the dynamic response of a MEMS torsion mirror

Abstract

This paper presents analytical solutions for the effect of squeeze film damping on a MEMS torsion mirror. Both the Fourier series solution and the double sine series solution are derived for the linearized Reynold equation which is obtained under the assumption of small displacements. Analytical formulae for the squeeze film pressure variation and the squeeze film damping torque on the torsion mirror are derived. They are functions of the rotation angle and the angular velocity of the mirror. On the other hand, to verify the analytical modeling, the implicit finite difference method is applied to solve the nonlinear isothermal Reynold equation, and thus numerically determine the squeeze film damping torque on the mirror. The damping torques based on both the analytical modeling and the numerical modeling are then used in the equation of motion of the torsion mirror which is solved by the Runge-Kutta numerical method. We find that the dynamic angular response of the mirror based on the analytical damping model matches very well with that based on the numerical damping model. We also perform experimental measurements and obtain results which are consistent with those obtained from the analytical and numerical damping models. Although the analytical damping model is derived under the assumption of harmonic response of the torsion mirror, it is shown that with the air spring effect neglected, this damping model is still valid for the case of nonharmonic response. The dependence of the damping torque on the ambient pressure is also considered and found to be insignificant in a certain regime of the ambient pressure. Finally, the convergence of the series solutions is discussed, and an approximate one term formula is presented for the squeeze film damping torque on the torsion mirror.