A novel differential frequency micro-gyroscope

We present a frequency-domain method to measure angular speeds using electrostatic micro-electro-mechanical system actuators. Towards this end, we study a single-axis gyroscope made of a micro-cantilever and a proof-mass coupled to two fixed electrodes. The gyroscope possesses two orthogonal axes of symmetry and identical flexural mode shapes along these axes. We develop the equations of motion describing the coupled bending modes in the presence of electrostatic and Coriolis forces. Furthermore, we derive a consistent closed-form higher-order expression for the natural frequencies of the coupled flexural modes. The closed-form expression is verified by comparing its results to those obtained from numerical integration of the equations of motion. We find that rotations around the beam axis couple each pair of identical bending modes to produce a pair of global modes. They also split their common natural frequency into a pair of closely spaced natural frequencies. We propose the use of the difference between this pair of frequencies, which is linearly proportional to the speed of rotation around the beam axis, as a detector for the angular speed.