Parametric Characteristics and Bifurcation Analysis of Multi-Degree-of-Freedom Micro Gyroscope with Drive Stiffness Nonlinearity.

The dynamic equations of a four-degree-of-freedom micro gyroscope system were developed considering the nonlinearity of driving stiffness, the primary resonance, and the 1:1 internal resonance. Then, the perturbation analysis was carried out using the method of multiple scales. The influence of stiffness nonlinearity and system parameters on micro-gyro dynamic characteristics, output sensitivity, detection bandwidth, and working stability were discussed based on the analytic and numerical solutions of the dynamic equations. Through the singularity theory, the influence of system parameters on bifurcation behavior was analyzed. The results show that the amplitude jump and multi-stable solutions caused by the nonlinear hardening characteristics of the high robust two-degree-of-freedom drive-mode occur outside the detection bandwidth. In addition, the influence on the bandwidth was weak and the sensitivity of the bandwidth area was slightly reduced. Moreover, saturation existed in the response amplitude of the second drive-mode in spite of the primary resonance being completely tuned or detuned. As a result, although the electrostatic force amplitude was out of the unstable region and even took a larger value, the micro gyroscope obtained a larger stable output. Besides, nonlinearity will lead to energy transfer between various modes of multi-degree-of-freedom micro gyroscopes. That means the response amplitudes could change greatly due to the variation of the external environment even the system is under a constant excitation frequency. Therefore, increasing the stiffness coefficient of the micro beam and the electrostatic force amplitude can maintain the robustness of the system to environmental changes and avoid the occurrence of bifurcation.