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

Bashar K Hammad

doi:10.1155/2014/939467

Abstract

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.

Highlights

Coupled microbeam resonators have attracted attention recently in the microscale realm, especially in RF MEMS [1,2,3,4]
To discuss the proposed methodology in this paper and without loss of generality, we present closed-form expressions for the natural frequencies and mode shapes of micromechanical filters made of two clamped-clamped beam resonators connected via a coupling beam
The boundary-value problem (BVP) governing the natural frequencies and mode shapes is composed of five equations and twenty boundary conditions

Summary

Introduction

Coupled microbeam resonators have attracted attention recently in the microscale realm, especially in RF MEMS [1,2,3,4]. We present an analytical methodology to find mode shapes and the corresponding natural frequencies that can be applied to any system of coupled resonators This methodology provides closed-form expressions for mode shapes that are easier to handle, more robust, and accurate in further analysis of coupled-resonator systems, especially in developing reduced-order models that describe the nonlinear static and dynamic characteristics of microstructures. These expressions allow designers to obtain a deeper insight into the relationship among performance metrics and the underlying microstructure dimensions, boundary conditions, and material properties.

Problem Formulation

Eigenvalue Problem

Normalization of Mode Shapes

Results and Discussion

Conclusions