We present a reduced-order model and closed-form expressions describing the response of a micromechanical filter made up of two clamped-clamped microbeam capacitive resonators coupled by a weak microbeam. The model accounts for geometrical and electrical nonlinearities as well as the coupling between them. It is obtained by discretizing the distributed-parameter system using the Galerkin procedure. The basis functions are the linear undamped global mode shapes of the unactuated filter. Closed-form expressions for these mode shapes and the coressponding natural frequencies are obtained by formulating a boundary-value problem (BVP) that is composed of five equations and twenty boundary conditions. This problem is transformed into solving a system of twenty linear homogeneous algebraic equations for twenty constants and the natural frequencies. We predict the deflection and the voltage at which the static pull-in occurs by solving another boundary-value problem (BVP). We also solve an eigenvalue problem (EVP) to determine the two natural frequencies delineating the bandwidth of the actuated filter. Using the method of multiple scales, we determine four first-order nonlinear ODEs describing the amplitudes and phases of the modes. We found a good agreement between the results obtained using our model and the published experimental results. We found that the filter can be tuned to operate linearly for a wide range of input signal strengths by choosing a DC voltage that makes the effective nonlinearities vanish.
A Discretization Approach to Modeling Capacitive MEMS Filters
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Hammad, BK, Nayfeh, AH, & Abdel-Rahman, E. "A Discretization Approach to Modeling Capacitive MEMS Filters." Proceedings of the ASME 2007 International Mechanical Engineering Congress and Exposition. Volume 9: Mechanical Systems and Control, Parts A, B, and C. Seattle, Washington, USA. November 11–15, 2007. pp. 289-301. ASME. https://doi.org/10.1115/IMECE2007-42186
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