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We present a methodology to calculate analytically the mode shapes and corresponding frequencies of mechanically coupled microbeam resonators. To demonstrate the methodology, we analyze a mechanical filter composed of two beams coupled by a weak beam. The boundary-value problem (BVP) for the linear vibration problem of the coupled beams depends on the number of beams and the boundary conditions of the attachment points. This implies that the system of linear homogeneous algebraic equations becomes larger as the array of resonators becomes complicated. We suggest a method to reduce the large system of equations into a smaller system. We reduce the BVP composed of five equations and twenty boundary conditions to a set of three linear homogeneous algebraic equations for three constants and the frequencies. This methodology can be simply extended to accommodate any configuration of mechanically coupled arrays. To validate our methodology, we compare our analytical results to these obtained numerically using ANSYS. We found that the agreement is excellent. We note that the weak coupling beam splits the frequency of the single resonator into two close frequencies. In addition, the effect of the coupling beam location on the natural frequencies, and hence the filter behavior, is investigated.

Natural Frequencies and Mode Shapes of Mechanically Coupled Microbeam Resonators with an Application to Micromechanical Filters