Effect of flexural modes on squeeze film damping in MEMS cantilever resonators

We present an analytical model that gives the values of squeeze film damping and spring coefficients for MEMS cantilever resonators taking into account the effect of flexural modes of the resonator. We use the exact mode shapes of a 2D cantilever plate to solve for pressure in the squeeze film and then derive the equivalent damping and spring coefficient relations from the back force calculations. The relations thus obtained can be used for any flexural mode of vibration of the resonators. We validate the analytical formulae by comparing the results with numerical simulations carried out using coupled finite element analysis in ANSYS, as well as experimentally measured values from MEMS cantilever resonators of various sizes vibrating in different modes. The analytically predicted values of damping are, in the worst case, within less than 10% of the values obtained experimentally or numerically. We also compare the results with previously reported analytical formulae based on approximate flexural mode shapes and show that the current results give much better estimates of the squeeze film damping. From the analytical model presented here, we find that the squeeze film damping drops by 84% from the first mode to the second mode in a cantilever resonator, thus improving the quality factor by a factor of 6 to 7. This result has significant implications in using cantilever resonators for mass detection where a significant increase in the quality factor is obtained by using a vacuum.

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