Exact Mode Shapes of T-shaped and Overhang-shaped Microcantilevers

Abstract

Resonance frequencies and mode shapes of microcantilevers are of important interest in micro-mechanical systems for enhancing the functionality and applicable range of the cantilevers in vibration transducing, energy harvesting, and highly sensitive measurement. In this study, using the Euler-Bernoulli theory for beam, we figured out the exact mode shapes of cantilevers of varying widths such as the overhang- or T-shaped cantilevers. The obtained mode shapes have been shown to significantly deviate from the approximate forms of a rectangular cantilever that are commonly used in mechanics and physics. They were then used to figure out the resonance frequencies of the cantilever. The analytical solutions have been confirmed by using the finite element method simulations with very low deviation. This study suggested a method for correctly obtaining the resonance frequency of microcantilevers with complicated dimensions, such as the doubly clamped cantilever with the undercut, with the overhangs at the clamped positions, or with an attached mass in the middle.

Tuning the flexural frequency of overhang-/T-shaped microcantilevers for high harmonics

“…These analytical results could then be confirmed by using the finite element method (FEM) simulation. We based our method on the analytical method used in recent studies 31,32) where Ref. 31 examined the mode shape and frequency of a cantilever without assuming an interaction with the sample (therefore, matrix elements K 45 -K 48 had simple form and did not involve the interaction terms) and Ref.…”