Exact Solution of the Oscillatory Behavior Under Axial Force of a Beam with a Concentrated Mass Within its Interval

Abstract

This article addresses the exact solution of free vibration of a beam with a concentrated mass within its intervals when the beam is subjected to axial loadings. This problem is frequently encountered in the design and modeling of resonant micro-sensors, where an exact model is required to calibrate the frequency output with a physical measurand. The significance of this approach is its ability to determine the exact mode shapes of vibration, which are necessary in the study of the time-domain response of sensors and determination of stability regions. The effects of the concentrated mass ratio, its location on the beam, and the applied axial force on the natural frequencies and the mode shapes of the resonant beam were studied in detail. The results of this study reveal how the mass ratio and location can change the sensitivity of the sensor to the input axial force. As an application of the presented approach, the modeling of a fabricated micro-electro-mechanical system (MEMS) has been discussed.