Bifurcation Analysis of a Micro-Machined Gyroscope with Nonlinear Stiffness and Electrostatic Forces.

Huabiao Zhang,Xinye Li,Lijuan Zhang

doi:10.3390/mi12020107

Abstract

The bifurcation of the periodic response of a micro-machined gyroscope with cubic supporting stiffness and fractional electrostatic forces is investigated. The pull-in phenomenon is analyzed to show that the system can have a stable periodic response when the detecting voltage is kept within a certain range. The method of averaging and the residue theorem are employed to give the averaging equations for the case of primary resonance and 1:1 internal resonance. Transition sets on the driving/detecting voltage plane that divide the parameter plane into 12 persistent regions and the corresponding bifurcation diagrams are obtained via the singularity theory. The results show that multiple solutions of the resonance curves appear with a large driving voltage and a small detecting voltage, which may lead to an uncertain output of the gyroscope. The effects of driving and detecting voltages on mechanical sensitivity and nonlinearity are analyzed for three persistent regions considering the operation requirements of the micro-machined gyroscope. The results indicate that in the region with a small driving voltage, the mechanical sensitivity is much smaller. In the other two regions, the variations in the mechanical sensitivity and nonlinearity are analogous. It is possible that the system has a maximum mechanical sensitivity and minimum nonlinearity for an appropriate range of detecting voltages.

Highlights

Micro-machined gyroscopes are widely used because of their small size, light weight, simple structure, low manufacturing cost, and high reliability [1]
We propose an investigation of the bifurcation of a micro-machined gyroscope with nonlinear stiffness and electrostatic forces
The nonlinear dynamics of a micro-machined gyroscope system are presented with an approximate and numerical method that focuses on the effects of the driving and detecting voltages on the periodic motions and their bifurcations

Summary

Introduction

Micro-machined gyroscopes are widely used because of their small size, light weight, simple structure, low manufacturing cost, and high reliability [1]. Considering the nonlinearity of electrostatic forces, Lestev et al [13] studied the steady-state response of a micro-machined gyroscope. Tsai et al [14] investigated a micro-machined gyroscope with nonlinear stiffness and electrostatic forces, and showed the unstable region on the driving/detecting frequency plane. Sharma et al [18] numerically and experimentally analyzed the pull-in phenomenon of a micro-machined gyroscope and discussed the effects of the dynamic pull-in voltage and the measured angular velocity. For micro-machined gyroscopes, the amplitude of the detecting direction tends to be close to the gap of the capacitance to improve its sensitivity In this case, the conclusions must be inaccurate with Taylor expansion. We propose an investigation of the bifurcation of a micro-machined gyroscope with nonlinear stiffness and electrostatic forces. Since ss is a certain value, Equation (12) has a constraint Y ≤ 1

Pull-In Analysis of the Detecting Equation

Approximate Analytical Solution of the Micro-Machined Gyroscope System

Bifurcation Analyses

Conclusions