Analytical solution of the modified Reynolds equation for squeeze film damping in perforated MEMS structures

The squeeze-film damping in perforated structures is modelled using a modified Reynolds equation that includes compressibility and rarefaction effect. This equation is linearized and transformed to the standard two-dimensional diffusion equation using a simple mapping function. The analytical solution is then obtained using Green’s function. The solution thus obtained adds an additional term $\Gamma$ to the damping and spring force expressions derived by Blech for compressible squeeze flow through non-perforated plates. This additional term contains several parameters related to perforations and rarefaction. Setting $\Gamma = 0$, one recovers Blech’s formulae. We compute the squeeze film forces using these new formulae and compare the computed forces with the solution of 3D Navier–Stokes equation solved using ANSYS for different perforation ratios (ratio of hole to cell dimensions). The results match very well. The approximate limit of maximum frequencies under which the formulae give reasonable results is also discussed. Although the main result is derived for a rigid plate under transverse motion, we discuss the effect of flexibility of the structure by deriving results for a flexible plate under a specified set of boundary conditions and comparing the results with that of a suitably modified rigid plate result. For small amplitude motion, the results show that a suitably modified rigid plate model can capture the effect of flexibility through a simple scaling factor