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

Huabiao Zhang; Xinye Li; Lijuan Zhang

doi:10.3390/mi12020107

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

Micromachines ◽

10.3390/mi12020107 ◽

2021 ◽

Vol 12 (2) ◽

pp. 107

Author(s):

Huabiao Zhang ◽

Xinye Li ◽

Lijuan Zhang

Keyword(s):

Primary Resonance ◽

Singularity Theory ◽

Driving Voltage ◽

Nonlinear Stiffness ◽

Resonance Transition ◽

Residue Theorem ◽

Electrostatic Forces ◽

Mechanical Sensitivity ◽

Periodic Response ◽

Stable Periodic

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.

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Primary Resonance of a Vehicle Suspension System With Nonlinear Stiffness and Nonlinear Damping

Volume 2: Automotive Systems, Bioengineering and Biomedical Technology, Fluids Engineering, Maintenance Engineering and Non-Destructive Evaluation, and Nanotechnology ◽

10.1115/esda2006-95027 ◽

2006 ◽

Cited By ~ 1

Author(s):

Shaohua Li ◽

Shaopu Yang

Keyword(s):

Primary Resonance ◽

Singularity Theory ◽

Nonlinear Damping ◽

Suspension System ◽

Vehicle Suspension ◽

Model Parameters ◽

Nonlinear Stiffness ◽

Single Degree Of Freedom ◽

Single Degree ◽

Vehicle Suspension System

In this work, primary resonance of a single-degree-of-freedom (SDOF) vehicle suspension system with nonlinear stiffness and nonlinear damping under multi-frequency excitations is investigated. The primary resonance equation is obtained by average method, and then the system’s bifurcation behaviors are studied by singularity theory. In addition, the effect of changing physical model parameters on the system’s primary resonance is studied.

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A Study on Vibration Characteristics and Stability of the Ambulance Nonlinear Damping System

Abstract and Applied Analysis ◽

10.1155/2013/501081 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 1

Author(s):

Meng Yang ◽

Xinxi Xu ◽

Chen Su

Keyword(s):

Approximate Solution ◽

Primary Resonance ◽

Nonlinear Behavior ◽

Singularity Theory ◽

Nonlinear Damping ◽

Nonlinear Stiffness ◽

First Order ◽

Damping Effect ◽

Vibration Model ◽

The Impact

Considering the impact of the nonlinear stiffness, a 2-DOF vibration model with cubic terms was established according to the structural feature and nonlinear behavior. Ignoring the impact of nonlinear terms, the system was linearly analyzed. In the case of primary resonance and 1 : 1 internal resonance, a multiscale method was used to obtain a first-order approximate solution. Taking the parameters of one tracked ambulance for instance, the approximate solution was corroborated and the influence of the parameters on damping effect was investigated. Finally, motion stability of the damping system was analyzed with singularity theory. The theoretical bases for improving efficiency of the damping system were provided.

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Boundary of the Basin of Attraction for Weakly Damped Primary Resonance

Journal of Applied Mechanics ◽

10.1115/1.2896026 ◽

1995 ◽

Vol 62 (4) ◽

pp. 941-946 ◽

Cited By ~ 9

Author(s):

R. Haberman ◽

E. K. Ho

Keyword(s):

Homoclinic Orbit ◽

Initial Conditions ◽

Basin Of Attraction ◽

Primary Resonance ◽

Analytic Formula ◽

Melnikov Integral ◽

Small Dissipation ◽

Stable Periodic ◽

The Basin Of Attraction ◽

Selection Of

The dissipatively perturbed Hamiltonian system corresponding to primary resonance is analyzed in the case in which two competing stable periodic responses exist. The method of averaging fails as the trajectory approaches the unperturbed homoclinic orbit (separatrix). By using the small dissipation of the Hamiltonian (the Melnikov integral) near the homoclinic orbit, the boundaries of the basin of attraction are determined analytically in an asymptotically accurate way. The selection of the two competing periodic responses is influenced by small changes in the initial conditions. The analytic formula is shown to agree well with numerical computations.

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On the Dynamics of the Micro-Ring Driven by Traveling Bias Voltages

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.311-313.1027 ◽

2011 ◽

Vol 311-313 ◽

pp. 1027-1031 ◽

Cited By ~ 3

Author(s):

Xiu Qian Ye ◽

Yi Bao Chen ◽

Bih Sheng Hsu ◽

Yuh Chung Hu

Keyword(s):

Analytical Approach ◽

Electromechanical Coupling ◽

Coupling Effect ◽

Ring Structure ◽

Distributed Model ◽

Driving Voltage ◽

Electrostatic Forces ◽

Small Deflection ◽

First Time ◽

Traveling Speed

There is no literature mentioned the modeling of the microstructures subjected to traveling electrostatic forces. This paper is the first time to present an analytical approach to investigate the dynamics of a micro-ring structure driven by the traveling bias voltage. The traveling electrostatic forces may come from the sequentially-actuated actuating electrodes arranged around the flexible ring. A linearized distributed model considering the electromechanical coupling effect is derived based on the small deflection assumption. According to the analytical results, the stiffness of the micro-ring will be softened periodically with the traveling speed of the driving voltage and the variation increases with the increasing of the voltage.

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Nonlinear Coupled Vibration of Electrically Actuated Arch with Flexible Supports

Micromachines ◽

10.3390/mi10110729 ◽

2019 ◽

Vol 10 (11) ◽

pp. 729 ◽

Cited By ~ 1

Author(s):

Ze Wang ◽

Jianting Ren

Keyword(s):

Multiple Scales ◽

Primary Resonance ◽

Single Mode ◽

Coupled Vibration ◽

Order Ordinary Differential Equation ◽

Electrically Actuated ◽

Second Mode ◽

Flexible Supports ◽

Stable Periodic ◽

Phase Angles

The nonlinear coupled vibration of an electrically actuated arch microbeam has attracted wide attention. In this paper, we studied the nonlinear dynamics of an electrically actuated arch microbeam with flexible supports. The two-to-one internal resonance between the first and second modes is considered. The multiple scales method is used to solve the governing equation. Four first-order ordinary differential equation describing the modulation of the amplitudes and phase angles were obtained. The equilibrium solution and its stability are determined. In the case of the primary resonance of the first mode, stable periodic motions and modulated motions are determined. The double-jumping phenomenon may occur. In the case of the primary resonance of the second mode, single-mode and two-mode solutions are possible. Moreover, double-jumping, hysteresis, and saturation phenomena were found. In addition, the approximate analytical results are supported by the numerical results.

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Time Series and the Dynamics of Demand Pacing

Methods of Information in Medicine ◽

10.1055/s-0038-1634284 ◽

2000 ◽

Vol 39 (02) ◽

pp. 114-117

Author(s):

D. T. Kaplan

Keyword(s):

Time Series ◽

Periodic Orbits ◽

Chaotic Systems ◽

Unstable Periodic Orbits ◽

Periodic Response ◽

Stable Periodic

Abstract:Motivated by a common practice in cardiology, we analyze the dynamics of a demand paced system where one seeks to create a stable periodic response. By using techniques originally developed for controlling chaotic systems, one can enhance the information contained in time series regarding hidden, unstable periodic orbits. This makes it possible, for example, to track drifts in a system‘s dynamics.

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