In this work, we present an analytical methodology to find mode shapes and the corresponding natural frequencies that can be applied to any system of coupled resonators. To the best of our knowledge, there have been few attempts to analytically generate the natural frequencies and mode shapes of multiresonator micromechanical structures characterized by distributed-parameter systems. Mode shapes and frequencies of these resonators are commonly approximated numerically using finite-element packages, such as ANSYS, COMSOL, and Coventor. Mechanically coupled microbeam resonators have attracted attention recently in the microscale realm, especially in RF MEMS. In addition, the effect of the coupling beam location on the natural frequencies, and hence the filter behavior, is investigated. We note that the weak coupling beam splits the frequency of the single resonator into two close frequencies. We found that the agreement is excellent. To validate our methodology, we compare our analytical results to these obtained numerically using ANSYS. This methodology can be simply extended to accommodate any configuration of mechanically coupled arrays.

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. We suggest a method to reduce the large system of equations into a smaller system.

This implies that the system of linear homogeneous algebraic equations becomes larger as the array of resonators becomes complicated. 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. To demonstrate the methodology, we analyze a mechanical filter composed of two beams coupled by a weak beam. We present a methodology to calculate analytically the mode shapes and corresponding frequencies of mechanically coupled microbeam resonators.

Filtering mode mass effect