Vibration analysis of micro-machined beam-type resonators

This paper addresses the exact solution of the free vibrations of a beam subjected to an axial force and carrying a concentrated rotary mass along its length. The vibration problem is frequently encountered in the design and modeling of resonant double-ended tuning fork (DETF) micro-structures, where an exact model is needed to determine the natural frequencies of vibration as a function of design and operational parameters. The significance of the approach presented in this study is first to develop a model that includes all the contributing parameters and second that its solution has the ability to determine the exact mode shapes of vibration. These eigenvectors are necessary in the study of the time-domain response of resonators and also determine the stability regions for the operation of electrostatic comb-drive exciters/detectors. The effects of the axial force, location, mass ratio, and the radius of gyration on the natural frequencies and mode shapes of DETF are investigated. It has been shown that depending on the location of the concentrated mass, the inclusion of its rotary inertia may either decrease or increase the natural frequencies of the resonator compared with the case of no rotary inertia is included. In the case of designing a resonator to perform as a sensor, one can make use of the presented model to determine how the mass ratio, location and the radius of gyration can alter the sensitivity of the sensor in response to the input measurand.