Stress measurement in a cantilevered silicon beam undergoing coupled motion of torsion and bending

In this article we derive an analytical mathematical model for a micromachined silicon tuning-fork gyroscope which responds to an external angular rate by a rotation about its sensitive axis. The mechanical stress caused by the sense motion is detected with a piezoresistive read-out structure. Based on the mathematical model of a cantilevered beam with cubic crystal structure undergoing a coupled motion of bending and torsion, steady-state solutions for the mechanical stress components within the piezoresistive read-out structure are provided. The analytical results are compared with finite-element simulations and experimentally verified for a certain design of the read-out structure sensitive to shear stress. Providing fully analytical solutions of the mechanical stress distributions, this paper establishes a basis for optimizing the design of the read-out structure as well as its position and orientation of the angular rate sensor for maximum sensitivity.