Dynamic Characteristics And Delay Feedback Analyses Of Double Drive And Double Sense Micro-Gyroscope Under Stiffness Nonlinearity

MEMS micro-gyroscope is a kind of inertial device used to measure angular velocity and displacement,which is developed by micro-electromechanical system processing technology and measurement and control technology.It is widely used in military and civil fields because of its small size,low power consumption,reliability and stability.Traditional single-degree-offreedom micromachined gyroscopes improve their mechanical sensitivity by matching the natural frequencies of drive and sense modes,but results in a significant reduction in bandwidth and robustness.In order to overcome the inherent design defects of single-degree-of-freedom micro-gyroscopes,the methods of increasing the degree of freedom of drive and sense modes have been proposed successively.On the other hand,the micro-gyroscope which is a highprecision dynamic sensor has a high geometric nonlinearity and exhibits nonlinear dynamic characteristics during operation.Therefore,the study of the dynamic behavior and control problems of multi-degree-of-freedom micro-gyroscopes under stiffness nonlinearity is of great significance for improving the research and development level of multi-degree-of-freedom micro-gyroscopes and broadening the application field of micro-gyroscopes.In this paper,the combination of theoretical solution and numerical simulation is used to study a 4-DOF microgyroscope with two drive and two sense modes as the research object.The effect of stiffness nonlinearity on the system response amplitude and natural frequency drift under the condition of primary resonance and 1:1 internal resonance is studied.And the control effect of time-delay velocity and displacement feedback control on the dynamic performance of a 4-DOF microgyroscope with stiffness nonlinearity.The research results can provide theoretical basis for controlling or eliminating the influence of stiffness nonlinearity on the dynamic performance and stability of multi-DOF micro-gyroscopes.In order to compare the influence of cubic non-linear stiffness and linear stiffness on the output signal of the micro-gyroscope with double drive and double sense modes,the dynamic equation of micro-gyroscope is solved by complex exponential method at first.Then the highdimensional nonlinear dynamic equation of micro-gyroscope is analyzed by the method of multiple scales,and the effects of stiffness nonlinearity and system parameters on resonance amplitude-frequency curve,resonance frequency,sensitivity and frequency are explored.Finally,the influence of environment and structural parameter disturbance on the dynamic performance of the system is analyzed by the local bifurcation theory.It is found that the instability phenomena such as amplitude jump and multi-stable solution caused by nonlinearity occur outside the output bandwidth and have little influence on the sensitivity within the bandwidth.Under different primary resonance parameters,the energy transfer of the sense modes is similar to that of the drive mode two,and the change of the internal resonance parameter will have a greater impact on the system amplitude response.The local bifurcation analysis of the micro-gyroscope nonlinear system is studied.It is found that the stiffness coefficient and the electrostatic driving force amplitude of the micro-beams connected with the decoupled frame play an important role in the parameter perturbation.The dynamic behavior of the 4-DOF micro-gyroscope system with stiffness nonlinearity under time-delay velocity feedback control is studied.The method of multiple scales is used to solve the dynamic equations of the controlled system,and the effects of velocity feedback gain and time-delay parameters on the output response are analyzed.It is found that the velocity feedback gain has only frequency modulation effect on the output response amplitude when the time-delay is zero.When the time delay is semi-period,the feedback gain has a modulating effect on the response amplitude,and is opposite to the case without time-delay.When the time delay is between semi-period and a period,the feedback gain has frequency modulation and amplitude modulation effect on the response amplitude at the same time.A method of controlling the response amplitude by using the amplitude modulation effect of velocity feedback gain is proposed.This method improves the sensitivity stability and weaken the nonlinear effect by coordinating the electrostatic force amplitude and the velocity negative feedback gain while stabilizing the sensitivity of the gyroscope.The dynamic behavior of the 4-DOF micro-gyroscope system with stiffness nonlinearity under time-delay displacement feedback control is studied.The influence of displacement feedback gain and time-delay parameters on the output response amplitude is analyzed from two different perspectives of frequency domain and time domain.It is found that the effect of displacement feedback gain on the amplitude-frequency response is mainly reflected in the adjustment of resonance frequency when there is no time delay.When the time delay is half of the excitation period,the effect of feedback gain on the response amplitude is opposite to that without time delay.When the delay is one quarter of a period or three quarters of a period,the effect of feedback gain on the amplitude-frequency response is mainly reflected in the magnitude.The influence of time-delay parameters on the output response in time domain is intuitively expressed in response time.According to the value of the gain,the time required for the response to reach the steady-state amplitude is controlled.The effect of displacement feedback gain is intuitively expressed in response amplitude.The occurrence of feedback control parameters may cause the response of the system to have a monostable response at the multi-stable amplitude,but it may also lead to complex periodic motions similar to chaos and quasi-periodic motions.