Damping and stiffening forces of a squeeze-film between two plates

Silicon microstructures, especially sensors and actuators, are characterized by squeeze film between the moving elements and the fixed substrates. The effect of the squeeze film has been represented by the damping effect in previous studies. Our goal, however, is to quantify the stiffening effect, which affects the resonance frequency of the squeeze film and oscillator system. In this research, the finite difference method for rectangular geometries is applied to solve the nonlinear isothermal Reynold's equation for the squeeze film. The approach is validated by comparison to numerical and theoretical results for small squeeze numbers. Attention is drawn to the elasticity of the air and the stiffening effect actually appears as a negative mass. The damping effect obtained from numerical solutions matches perfectly with the equivalent damping formula derived from the analytical formula. The dynamic response of sinusoidally forced oscillator involving one squeeze film is investigated by solving dynamic equation numerically in order to demonstrate the stiffening effect of the squeeze film. Three main parameters are considered : the initial gap of the squeeze film, the excitation amplitude, and the excitation frequency. The relationship between the amplitude of the steady-state sinusoidal response and the excitation frequency is determined. We obtain the dependence of the resonance frequency on the initial gap and the forcing amplitude and identify conditions when the stiffening effect must be included in dynamic study of the structures with squeeze films