scholarly journals Influence of System and Actuator Nonlinearities on the Dynamics of Ring-Type MEMS Gyroscopes

Vibration ◽  
2021 ◽  
Vol 4 (4) ◽  
pp. 805-821
Author(s):  
Ibrahim F. Gebrel ◽  
Samuel F. Asokanthan
Keyword(s):  
Differential Equations ◽  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Electrostatic Force ◽  
Essential Element ◽  
Ring Structure ◽  
Dynamic Responses ◽  
Ongoing Research ◽  
Actuator Dynamics ◽  
External Forces

This study investigates the nonlinear dynamic response behavior of a rotating ring that forms an essential element of MEMS (Micro Electro Mechanical Systems) ring-based vibratory gyroscopes that utilize oscillatory nonlinear electrostatic forces. For this purpose, the dynamic behavior due to nonlinear system characteristics and nonlinear external forces was studied in detail. The partial differential equations that represent the ring dynamics are reduced to coupled nonlinear ordinary differential equations by suitable addition of nonlinear mode functions and application of Galerkin’s procedure. Understanding the effects of nonlinear actuator dynamics is essential for characterizing the dynamic behavior of such devices. For this purpose, a suitable theoretical model to generate a nonlinear electrostatic force acting on the MEMS ring structure is formulated. Nonlinear dynamic responses in the driving and sensing directions are examined via time response, phase diagram, and Poincare’s map when the input angular motion and nonlinear electrostatic force are considered simultaneously. The analysis is envisaged to aid ongoing research associated with the fabrication of this type of device and provide design improvements in MEMS ring-based gyroscopes.

Dynamics of MEMS-Based Angular Rate Sensors Excited via External Electrostatic Forces

2020 ◽  
Author(s):  
Ibrahim F. Gebrel ◽  
Ligang Wang ◽  
Samuel F. Asokanthan
Keyword(s):  
Dynamic Behavior ◽  
Electrostatic Force ◽  
Angular Rate ◽  
Equations Of Motion ◽  
Ring Structure ◽  
Dynamic Responses ◽  
Actuator Dynamics ◽  
Vibratory Gyroscopes ◽  
Thin Ring ◽  
The Mathematical Model

Abstract This paper investigates the dynamic behavior of rotating MEMS-based vibratory gyroscopes which employs a thin ring as the vibrating flexible element. The mathematical model for the MEMS ring structure as well as a model for the nonlinear electrostatic excitation forces are formulated. Galerkin’s procedure is employed to reduce the equations of motion to a set of ordinary differential equations. Understanding the effects of nonlinear actuator dynamics is considered important for characterizing the dynamic behavior of such devices. A suitable theoretical model to generate nonlinear electrostatic force that acts on the MEMS ring structure is formulated. Dynamic responses in the driving and the sensing directions are examined via time responses, phase diagram, and Poincare’ map plots when the input angular motion and the nonlinear electrostatic force are considered simultaneously. The analysis is envisaged to aid fabrication of this class of devices as well as for providing design improvements in MEMS Ring-based Gyroscopes.

scholarly journals Nonlinear Dynamic Modeling and Analysis of an L-Shaped Multi-Beam Jointed Structure with Tip Mass

Materials ◽  
2021 ◽  
Vol 14 (23) ◽  
pp. 7279
Author(s):  
Jin Wei ◽  
Tao Yu ◽  
Dongping Jin ◽  
Mei Liu ◽  
Dengqing Cao ◽  
...  
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Natural Frequencies ◽  
Great Influence ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Nonlinear Odes ◽  
Global Mode ◽  
Orthogonality Relations ◽  
Tip Mass

A dynamic model of an L-shaped multi-beam joint structure is presented to investigate the nonlinear dynamic behavior of the system. Firstly, the nonlinear partial differential equations (PDEs) of motion for the beams, the governing equations of the tip mass, and their matching conditions and boundary conditions are obtained. The natural frequencies and the global mode shapes of the linearized model of the system are determined, and the orthogonality relations of the global mode shapes are established. Then, the global mode shapes and their orthogonality relations are used to derive a set of nonlinear ordinary differential equations (ODEs) that govern the motion of the L-shaped multi-beam jointed structure. The accuracy of the model is verified by the comparison of the natural frequencies solved by the frequency equation and the ANSYS. Based on the nonlinear ODEs obtained in this model, the dynamic responses are worked out to investigate the effect of the tip mass and the joint on the nonlinear dynamic characteristic of the system. The results show that the inertia of the tip mass and the nonlinear stiffness of the joints have a great influence on the nonlinear response of the system.

Chaos and Nonlinear Dynamic Analysis of Porous Air Bearing System

2015 ◽  
Vol 764-765 ◽  
pp. 204-207
Author(s):  
Cheng Chi Wang ◽  
Jui Pin Hung
Keyword(s):  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Effective Means ◽  
Nonlinear Dynamic Analysis ◽  
Transformation Method ◽  
Dynamic Responses ◽  
Air Bearing ◽  
Dynamic Behaviors ◽  
Rotor Mass ◽  
Bearing System

The chaos and nonlinear dynamic behaviors of porous air bearing system are studied by a hybrid numerical method combining the finite difference method (FDM) and differential transformation method (DTM). The numerical results are verified by two different schemes including hybrid method and FDM and the current analytical results are found to be in good agreement. Furthermore, the results reveal the changes which take place in the dynamic behavior of the bearing system as the rotor mass is increased. From the dynamic responses of the rotor center, they reveal complex dynamic behaviors including periodic, sub-harmonic motion and chaos. The results of this study provide an understanding of the nonlinear dynamic behavior of PAB systems characterized by different rotor masses. Specifically, the results have shown that system exists chaotic motion over the ranges of rotor mass 10.66≤Mr<13.7kg. The proposed method and results provide an effective means of gaining insights into the porous air bearing systems.

A Hybrid Finite Element-Analytical Approach for Dynamic Analysis of a Micro-Resonator Driven by Electrostatic Combs

2012 ◽  
Vol 226-228 ◽  
pp. 708-713
Author(s):  
Mi Tao Song ◽  
Deng Qing Cao
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Analytical Approach ◽  
Dynamic Responses ◽  
The Finite Element Method ◽  
Global Mode ◽  
Hybrid Finite Element ◽  
Micro Resonator

Combining the finite element method and the analytical method, a hybrid finite element-analytical approach is established to calculate the nonlinear dynamic responses of a micro-resonator driven by electrostatic combs accurately for the purpose of programmed dynamical simulations and great shortening of workloads. The spatially discretized equations obtained by using the analytical undamped global mode functions to the nonlinear integro-partial differential equations and the ordinary differential equations of the micro-resonator, in which the coefficients are estimated by the discrete global mode shapes from the finite element method, are used to calculate the nonlinear dynamic responses of the micro-resonator. The results are compared with those merely based on the analytical mode functions of the micro-resonator, which shows that they can reach high accuracy when the elements in the micro-resonator are sufficiently small.

Dynamics of a Ring-Type Macro Gyroscope Under Electromagnetic External Actuation Forces

2018 ◽  
Author(s):  
Ibrahim F. Gebrel ◽  
Ligang Wang ◽  
Samuel F. Asokanthan
Keyword(s):  
Electromagnetic Force ◽  
Dynamic Behaviour ◽  
Essential Element ◽  
Ring Structure ◽  
Ring System ◽  
Response Behavior ◽  
External Excitation ◽  
Actuator Dynamics ◽  
Vibratory Gyroscopes ◽  
Galerkin's Procedure

This paper investigates the dynamic behaviour of a rotating ring that forms an essential element in ring-based vibratory gyroscopes that utilize oscillatory electromagnetic forces. Understanding the effects of nonlinear actuator dynamics is considered important for characterizing the dynamic behavior of such devices. A suitable theoretical model to generate nonlinear electromagnetic force that acts on the ring structure is formulated. In order to predict the dynamic behaviour of a ring system subjected to external excitation and body rotation, discretized equations obtained via Galerkin’s procedure is employed to investigate the time as well as frequency response behavior. Dynamic response in the driving and the sensing directions are examined via time responses, phase diagram, Poincare’ map and bifurcation plots when the input angular motion and the nonlinear electromagnetic force are considered simultaneously. The analysis is envisaged to aid ongoing experimental research as well as for providing design improvements in Ring-based Gyroscopes.

Nonlinear Vibration and Control of Piezoelectric Laminates

2009 ◽  
Author(s):  
Li-hua Chen ◽  
Chang-Liang Liu ◽  
Wei Zhang ◽  
Jin-hong Fan
Keyword(s):  
Differential Equations ◽  
Rectangular Plate ◽  
Dynamic Behavior ◽  
Nonlinear Dynamic ◽  
Chaotic Motions ◽  
Nonlinear Dynamic Equations ◽  
Control Gain ◽  
Piezoelectric Layers ◽  
Nonlinear Relation ◽  
And Control

In this paper, the dynamic behavior of piezoelectric laminates is investigated. Thin piezoelectric layers are assumed to be embedded on the top and the bottom surfaces of the rectangular plate. The top and the bottom layers are taken as the actuator and sensor, respectively. Based on Von Karman theory, the geometrically nonlinear relation between strain and displacement is proposed and basic large deformation equations are established. Nonlinear dynamic equations of piezoelectric laminates are formulated using Hamilton’s principle. The Galerkin’s approach is applied to partial differential equations to obtain the ordinary differential equations. The numerical results show the existence of periodic, bifurcation and chaotic motions for the laminated piezoelectric rectangular plate with the changes of frequency and amplitude of forcing loads. Furthermore we can control the vibration of the piezoelectric laminates using a constant gain velocity minus control algorithm. Using the control gain, the free vibration of the plate is damped out more quickly, and the nonlinear dynamic behavior varies from the system without control. Finally, a numerical simulation example shows that the method suggested in this paper is effective and simply.

scholarly journals Free Vibrations and Nonlinear Responses for a Cantilever Honeycomb Sandwich Plate

2018 ◽  
Vol 2018 ◽  
pp. 1-12 ◽  
Author(s):  
Junhua Zhang ◽  
Xiaodong Yang ◽  
Wei Zhang
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Sandwich Plate ◽  
Natural Frequencies ◽  
Ritz Method ◽  
Shear Deformation Theory ◽  
Dynamic Responses ◽  
Chaotic Motions ◽  
Honeycomb Sandwich ◽  
Honeycomb Sandwich Plate

Dynamics of a cantilever honeycomb sandwich plate are studied in this paper. The governing equations of the composite plate subjected to both in-plane and transverse excitations are derived by using Hamilton’s principle and Reddy’s third-order shear deformation theory. Based on the Rayleigh–Ritz method, some modes of natural frequencies for the cantilever honeycomb sandwich plate are obtained. The relations between the natural frequencies and the parameters of the plate are investigated. Further, the Galerkin method is used to transform the nonlinear partial differential equations into a set of nonlinear ordinary differential equations. Nonlinear dynamic responses of the cantilever honeycomb sandwich plate to such external and parametric excitations are discussed by using the numerical method. The results show that in-plane and transverse excitations have an important influence on nonlinear dynamic characteristics. Rich dynamics, such as periodic, multiperiodic, quasiperiodic, and chaotic motions, are located and studied by the bifurcation diagram for some specific parameters.

Closed Solution for the Elastic-Dynamic Behavior of an Industrial Press Feed Mechanism

1994 ◽  
Author(s):  
Tao Xu ◽  
Gerard G. Lowen
Keyword(s):  
Differential Equations ◽  
Dynamic Behavior ◽  
Shape Function ◽  
Degree Of Freedom ◽  
Linear Differential Equations ◽  
Dynamic Responses ◽  
Secondary Importance ◽  
Closed Solution ◽  
Linear Differential ◽  
Two Degree Of Freedom

Abstract A new linearized two degree of freedom model of an industrial press feed mechanism, containing an RSSR linkage, a bent coupler, an overrunning sprag clutch, a feed strip and a brake, is presented. By introducing a double cantilever model of the coupler with an assumed quarter sine shape function, simplifying certain terms of secondary importance and replacing the non-linear clutch spring by a linear torsional spring with a deflection dependent stiffness, it was possible to develop a set of two linear differential equations for the all important feed stroke, which could be fully solved in an analytic manner for the dynamic responses of the coupler strain and the clutch windup angle.

scholarly journals Nonlinear dynamic responses of sandwich functionally graded porous cylindrical shells embedded in elastic media under 1:1 internal resonance

2021 ◽  
Author(s):  
Yunfei Liu ◽  
Zhaoye Qin ◽  
Fulei Chu
Keyword(s):  
Differential Equations ◽  
Nonlinear Dynamic ◽  
Internal Resonance ◽  
Metal Foam ◽  
Multiple Scales ◽  
Porous Metal ◽  
Functionally Graded ◽  
Dynamic Responses ◽  
Elastic Media ◽  
The Galerkin Method

AbstractIn this article, the nonlinear dynamic responses of sandwich functionally graded (FG) porous cylindrical shell embedded in elastic media are investigated. The shell studied here consists of three layers, of which the outer and inner skins are made of solid metal, while the core is FG porous metal foam. Partial differential equations are derived by utilizing the improved Donnell’s nonlinear shell theory and Hamilton’s principle. Afterwards, the Galerkin method is used to transform the governing equations into nonlinear ordinary differential equations, and an approximate analytical solution is obtained by using the multiple scales method. The effects of various system parameters, specifically, the radial load, core thickness, foam type, foam coefficient, structure damping, and Winkler-Pasternak foundation parameters on nonlinear internal resonance of the sandwich FG porous thin shells are evaluated.

Dynamic analysis of a helical gear reduction by experimental and numerical methods

2020 ◽  
Vol 68 (1) ◽  
pp. 48-58
Author(s):  
Chao Liu ◽  
Zongde Fang ◽  
Fang Guo ◽  
Long Xiang ◽  
Yabin Guan ◽  
...  
Keyword(s):  
Numerical Methods ◽  
Dynamic Analysis ◽  
Dynamic Behavior ◽  
Fourth Order ◽  
Numerical Models ◽  
Helical Gear ◽  
Operating Conditions ◽  
Dynamic Responses ◽  
Lumped Mass ◽  
Gear Mesh

Presented in this study is investigation of dynamic behavior of a helical gear reduction by experimental and numerical methods. A closed-loop test rig is designed to measure vibrations of the example system, and the basic principle as well as relevant signal processing method is introduced. A hybrid user-defined element model is established to predict relative vibration acceleration at the gear mesh in a direction normal to contact surfaces. The other two numerical models are also constructed by lumped mass method and contact FEM to compare with the previous model in terms of dynamic responses of the system. First, the experiment data demonstrate that the loaded transmission error calculated by LTCA method is generally acceptable and that the assumption ignoring the tooth backlash is valid under the conditions of large loads. Second, under the common operating conditions, the system vibrations obtained by the experimental and numerical methods primarily occur at the first fourth-order meshing frequencies and that the maximum vibration amplitude, for each method, appears on the fourth-order meshing frequency. Moreover, root-mean-square (RMS) value of the acceleration increases with the increasing loads. Finally, according to the comparison of the simulation results, the variation tendencies of the RMS value along with input rotational speed agree well and that the frequencies where the resonances occur keep coincident generally. With summaries of merit and demerit, application of each numerical method is suggested for dynamic analysis of cylindrical gear system, which aids designers for desirable dynamic behavior of the system and better solutions to engineering problems.

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