Electrostatic force nonlinearity is widely present in MEMS systems, which could impact the system sensitivity performance. The Frequency modulation (FM) method is proposed as an ideal solution to solve the problem of environmental fluctuation stability. The effect of electrostatic force nonlinearity on the sensitivity performance of a class of FM micro-gyroscope is investigated. The micro-gyroscope consists of a tapered cantilever beam with a tip mass attached to the end. Considering the case of unequal width and thickness, the motion equations of the system are derived by applying Hamilton's principle. The differential quadrature method (DQM) was used to analyze the micro-gyroscope's static deflection, pull-in voltage, and natural frequency characteristics. We observed that from the onset of rotation, the natural frequencies of the drive and sense modes gradually split into a pair of natural frequencies that were far from each other. The FM method directly measures the angular velocity by tracking the frequency of the drive and sense modes. Then, based on the linear system, the reduced-order model was used to analyze the influence of the shape factor and DC voltage on the sensitivity performance. Most importantly, the nonlinear frequency of system was obtained using the invariant manifold method (IMM). The influence of electrostatic force nonlinearity on the performance of the FM micro-gyroscope was investigated. The results show that the different shape factors of width and thickness, as well as the different DC voltages along the drive and sense directions, break the symmetry of the micro-gyroscope and reduce the sensitivity of the system. The sensitivity has a non-linear trend with the rotation speed. The DC voltage is proportional to the electrostatic force nonlinearity coefficient. As the DC voltage gradually increases, the nonlinearity is enhanced, resulting in a significant decrease in the sensitivity of the micro-gyroscope. It is found that the negative shape factor (width and thickness gradually increase along the beam) can effectively restrain the influence of electrostatic force nonlinearity, and a larger dynamic detection range can be obtained.
The parameter tuning of a multi-stable energy harvester is crucial to enhancing harvesting efficiency. In this paper, the bifurcation theory is applied to quantitatively reveal the effects of structural parameters on the statics and dynamics of a quad-stable energy harvester (QEH). Firstly, a novel QEH system utilizing the geometric nonlinearity of springs is proposed. Static bifurcation analysis is carried out to design quad-stable working conditions. To investigate the cross-well and high-energy vibration, the complex dynamic frequency (CDF) method, suitable for both weakly and strongly nonlinear dynamic problems, is then applied to deduce the primary response solution. By using the unfolding analysis in singularity theory, four steady-state properties and dozens of primary resonance modes are demonstrated. Based on the transition set, the effective bandwidth for energy harvesting can be customized to adapt well to various vibration environments by parametric adjustment. Finally, the experimental tests verify that the output power can reach up to 1 mW. The proposed QEH and its mechanics optimization can guide energy supply for next-generation wireless systems and low-power sensors under magnetic forbidding environments.
In this paper, we apply the leverage amplification principle to improve the gain of a three-degrees-of-freedom (3-DoF) micro-gyro. The gain of the micro-gyro can be improved by designing linear and nonlinear micro-gyros with an anchored lever mechanism (ALM). First, the sensor system of the micro-gyro is designed as a complete 2-DOF system with an ALM. The effect of the leverage rate (LR) on the mass ratio and frequency coupling parameter (FCP) of the complete 2-DOF sense system is studied. We analyze the variation rule of the gain of the lever's input and output as the LR increases. Afterwards, the bandwidth and gain performance of linear and nonlinear micro-gyros with an ALM is investigated by applying the arbitrarily tunable characteristics of peak spacing of the complete 2-DOF system. The influence of LR, FCP, nonlinear strength, damping, and peak spacing on bandwidth and gain of the 3-DOF micro-gyro is analyzed. The results indicate that both LR and FCP have a large effect on the gain and bandwidth of a micro-gyro with an ALM. The LR parameter mainly improves the gain of the micro-gyro, and the FCP parameter mainly adjusts the bandwidth performance. Adding levers can effectively improve the gain performance of the linear micro-gyro. The linear micro-gyro with an ALM can improve the gain by 4.5 dB compared to the one without an ALM. The nonlinear micro-gyro with an ALM combines two characteristics: the nonlinear micro-gyro can improve the bandwidth, while the lever structure can improve the gain. Compared with the linear micro-gyro without an ALM, the gain can be increased by 17.6 dB, and the bandwidth can be improved as well. In addition, the bandwidth of a micro-gyro with an ALM is related to the gain difference between the peaks of the lever output. The increase in the gain difference leads to a flattening of the left peak, which effectively broadens the bandwidth. For nonlinear micro-gyros with an ALM, the bandwidth can be further improved by increasing the nonlinear stiffness coefficient, and better gain and bandwidth can be obtained using a vacuum package.
In this paper, we use the nonlinear hardening stiffness of drive mode deal with the contradiction between gain and bandwidth of the linear micro-gyroscope, to improve the bandwidth and gain in sense direction. Firstly, in order to adjust the distance between two resonant peaks, we changed an incomplete two-degree-of-freedom(2-DoF) sense mode system of the micro-gyroscope into a complete 2-DoF system. Afterward, according to the given nonlinear coefficient of stiffness of drive mode, the structure size of driving micro-beams was designed to obtain a nonlinear micro-gyroscope with controllable stiffness. Finally, we investigated the effects of peaks spacing, damping, and driving nonlinearity on gain and bandwidth, and the nonlinear micro-gyroscope was optimized by orthogonal experiment method and response surface method. The results reveal that the peaks spacing has a great influence on the gain and bandwidth of both linear and nonlinear micro-gyroscopes. The larger the peaks spacing, the lower the gain, but higher gain can be achieved when the resonant frequency of the drive mode is close to the lower-order resonant frequency of the sense mode. Driving nonlinearity leads to the response peak of the Coriolis force to have a hardening characteristic, thus forming a wide platform in the sense direction. Hardening of the response peak of the Coriolis force allows the micro-gyroscope to obtain a higher gain while the bandwidth of the sense mode is also greatly improved. In addition, parameter optimization can make the gain and bandwidth of the micro-gyroscope optimal. When the peaks spacing is small and the nonlinear stiffness coefficient is about 1012.2, under the premise that the gain is basically constant, the bandwidth of the sense mode increases about 1.76 times compared with the linear gyroscope. Damping can suppress the influence of nonlinearity in a micro-gyroscope system. Within a certain range, the frequency response of the nonlinear micro-gyroscope tends to be a linear system with the increase in damping, resulting in narrower bandwidth and lower gain.
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.
A class of bipolar electrostatically actuated micro-resonators is presented in this paper. Two parametric equations are proposed for changing the microbeam shape of the upper and lower sections. The mechanical properties of a micro-resonator can be enhanced by optimizing the two section parameters. The electrostatic force nonlinearity, neutral surface tension, and neutral surface bending are considered in the model. First, the theoretical results are verified with finite element results from COMSOL Multiphysics simulations. The influence of section variation on the electrostatic force, pull-in behaviors and safe working area of the micro-resonator are studied. Moreover, the impact of residual stress on pull-in voltage is discussed. The multi-scale method (MMS) is used to further study the vibration of the microbeam near equilibrium, and the relationship between the two section parameters of the microbeam under linear vibration was determined. The vibration amplitude and resonance frequency are investigated when the two section parameters satisfy the linear vibration. In order to research dynamic analysis under the case of large amplitude. The Simulink dynamics simulation was used to study the influence of section variation on the response frequency. It is found that electrostatic softening increases as the vibration amplitude increases. If the nonlinearity initially shows hardening behavior, the frequency response will shift from hardening to softening as the amplitude increases. The position of softening-hardening transition point decreases with the increase of residual stress. The relationship between DC voltage, section parameters, and softening-hardening transition points is presented. The accuracy of the results is verified using theoretical, numerical, and finite element methods.
The micro-electro-mechanical system (MEMS) resonator developed based on surface processing technology usually changes the section shape either due to excessive etching or insufficient etching. In this paper, a section parameter is proposed to describe the microbeam changes in the upper and lower sections. The effect of section change on the mechanical properties is studied analytically and verified through numerical and finite element solutions. A doubly-clamped microbeam-based resonator, which is actuated by an electrode on one side, is investigated. The higher-order model is derived without neglecting the effects of neutral plane stretching and electrostatic nonlinearity. Further, the Galerkin method and Newtonâ»Cotes method are used to reduce the complexity and order of the derived model. First of all, the influence of microbeam shape and gap variation on the static pull-in are studied. Then, the dynamic analysis of the system is investigated. The method of multiple scales (MMS) is applied to determine the response of the system for small amplitude vibrations. The relationship between the microbeam shape and the frequency response is discussed. Results show that the change of section and gap distance can make the vibration soften, harden, and so on. Furthermore, when the amplitude of vibration is large, the frequency response softening effect is weakened by the MMS. If the nonlinearity shows hardening-type behavior at the beginning, with the increase of the amplitude, the frequency response will shift from hardening to softening behavior. The large amplitude in-well motions are studied to investigate the transitions between hardening and softening behaviors. Finally, the finite element analysis using COMSOL software (COMSOL Inc., Stockholm, Sweden) is carried out to verify the theoretical results, and the two results are very close to each other in the stable region.
