On the Dynamics Behavior and a Control Design to a Nonlinear 2-DOF Vibrating Gyroscopic-MEMS Model

2011 ◽  
Author(s):  
Nelson Jose´ Peruzzi ◽  
Fa´bio Roberto Chavarette ◽  
Jose´ Manoel Balthazar
Keyword(s):  
Mechanical System ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Time History ◽  
Electrical Energy ◽  
Micro Electro Mechanical System ◽  
Mems Gyroscope ◽  
Time Histories ◽  
System Equations ◽  
And Control

In this paper, we deal with the nonlinear dynamics, the transfer of energy and control of the vibrations of a Micro Electro-mechanical System (MEMS) gyroscope. The MEMS are micro-transducers whose operation is based on elastic and electrostatic forces that convert electrical energy into mechanical energy and vice-versa. These systems can be modeled by 2-DOF spring-mass-damper system and the coupling of the system equations is performed by Coriolis force. This coupling is responsible for the energy transfers of the two vibration modes (drive-mode and sense-mode) and for the resonance in MEMS gyroscope. The governing equations of motion have periodic coefficients and as the dimensions of the quantities involved in the system may be inconsistent it is not advisable the use of perturbation methods for the solution of the MEMS gyroscope. For this reason, in the dynamic analysis and control of the vibrations of the MEMS gyroscope, we used a technique based on Chebyshev polynomial expansion, the iterative Picard and transformation of Lyapunov-Floquet (L–F). For the analysis of the dynamic of the micro electro-mechanical system gyroscope, we did the diagram of stability, phase planes and time history of transfer of energy. Finally, we did the control of the unstable orbit to a desired periodic one and compared the phase planes of orbits desired and controlled and time histories of energy transfer of the controlled and non-controlled system.

scholarly journals Application of harmonic balance method for solving non-linear equations of motion

2018 ◽  
Vol 182 ◽  
pp. 02024
Author(s):  
Robert Kostek
Keyword(s):  
Harmonic Balance ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Harmonic Balance Method ◽  
Time History ◽  
Balance Method ◽  
Non Linear ◽  
Time Histories ◽  
History Of ◽  
Non Linear Equations

This article presents the advantages and limitations of a harmonic balance method applied for solving non-linear equations of monition. This method provides an opportunity to find stable and unstable periodic solutions, which was demonstrated for a few equations. An error of solution decreases rapidly with increase of number of harmonics for smooth time history of acceleration, which shows convergence; whereas, for discontinuous time histories, this method is not effective.

scholarly journals Analysis of the brachistochronic motion of a variable mass nonholonomic mechanical system

2016 ◽  
Vol 43 (1) ◽  
pp. 19-32 ◽  
Author(s):  
Bojan Jeremic ◽  
Radoslav Radulovic ◽  
Aleksandar Obradovic
Keyword(s):  
Differential Equations ◽  
Mechanical System ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Nonholonomic Constraints ◽  
Initial Position ◽  
Mass Particle ◽  
Control Forces ◽  
And Control

The paper considers the brachistochronic motion of a variable mass nonholonomic mechanical system [3] in a horizontal plane, between two specified positions. Variable mass particles are interconnected by a lightweight mechanism of the ?pitchfork? type. The law of the time-rate of mass variation of the particles, as well as relative velocities of the expelled particles, as a function of time, are known. Differential equations of motion, where the reactions of nonholonomic constraints and control forces figure, are created based on the general theorems of dynamics of a variable mass mechanical system [5]. The formulated brachistochrone problem, with adequately chosen quantities of state, is solved, in this case, as the simplest task of optimal control by applying Pontryagin?s maximum principle [1]. A corresponding two-point boundary value problem (TPBVP) of the system of ordinary nonlinear differential equations is obtained, which, in a general case, has to be numerically solved [2]. On the basis of thus obtained brachistochronic motion, the active control forces, along with the reactions of nonholonomic constraints, are determined. The analysis of the brachistochronic motion for different values of the initial position of a variable mass particle B is presented. Also, the interval of values of the initial position of a variable mass particle B, for which there are the TPBVP solutions, is determined.

Experimental and Analytical Study of Free Lift-Off Motion Induced Slip Behavior of Rectangular Rigid Bodies

2004 ◽  
Vol 126 (1) ◽  
pp. 53-58 ◽  
Author(s):  
Tomoyo Taniguchi
Keyword(s):  
Energy Balance ◽  
Rigid Body ◽  
Variational Approach ◽  
Analytical Study ◽  
Mechanical Energy ◽  
Equations Of Motion ◽  
Rigid Bodies ◽  
Time Histories ◽  
Analytical Accuracy ◽  
Lift Off

The mechanism of the slip of a rectangular rigid body during free lift-off motion is investigated analytically and experimentally. Equations of motion of slipping rectangular rigid body during free lift-off motion are derived by the variational approach. The time histories of both slip and lift-off motions are numerically computed and compared with corresponding experimental results to discuss the analytical accuracy; given an initial enforced lift-off angle to the rectangular rigid body and then gently released. The mechanical energy balance of simultaneous slip and lift-off motions is clarified to explain the considerable reduction of lift-off angle due to the slip.

scholarly journals Mechanics and Control of a Terrestrial Vehicle Exploiting a Nonholonomic Constraint for Fishlike Locomotion

2013 ◽  
Author(s):  
Tony Dear ◽  
Scott David Kelly ◽  
Matthew Travers ◽  
Howie Choset
Keyword(s):  
Mechanical System ◽  
Nonholonomic Constraint ◽  
Equations Of Motion ◽  
Geometric Mechanics ◽  
Velocity Control ◽  
Joint Velocity ◽  
And Control

We present a novel mechanical system, the “landfish,” which takes advantage of a combination of articulation and a nonholonomic constraint to exhibit fishlike locomotion. We apply geometric mechanics techniques to establish the equations of motion in terms of the system’s nonholonomic momentum and analyze the system’s equilibrium properties. Finally, we demonstrate its locomotion capabilities under several controllers, including heading and joint velocity control.

Reduction of structural vibrations with the piezoelectric stacks ring

2020 ◽  
Vol 64 (1-4) ◽  
pp. 729-736
Author(s):  
Jincheng He ◽  
Xing Tan ◽  
Wang Tao ◽  
Xinhai Wu ◽  
Huan He ◽  
...  
Keyword(s):  
Vibration Control ◽  
Mechanical Energy ◽  
Electrical Energy ◽  
Structural Vibrations ◽  
Rotor Systems ◽  
Fea Model ◽  
Piezoelectric Stacks ◽  
Shunt Damping ◽  
Reduction Effect ◽  
Euler Bernoulli Beam

It is known that piezoelectric material shunted with external circuits can convert mechanical energy to electrical energy, which is so called piezoelectric shunt damping technology. In this paper, a piezoelectric stacks ring (PSR) is designed for vibration control of beams and rotor systems. A relative simple electromechanical model of an Euler Bernoulli beam supported by two piezoelectric stacks shunted with resonant RL circuits is established. The equation of motion of such simplified system has been derived using Hamilton’s principle. A more realistic FEA model is developed. The numerical analysis is carried out using COMSOL® and the simulation results show a significant reduction of vibration amplitude at the specific natural frequencies. Using finite element method, the influence of circuit parameters on lateral vibration control is discussed. A preliminary experiment of a prototype PSR verifies the PSR’s vibration reduction effect.

Design of a slender turning cutting tool via a vibration absorber equipped with piezoelectric ceramic

2021 ◽  
pp. 107754632110144
Author(s):  
Yiqing Yang ◽  
Haoyang Gao ◽  
Qiang Liu
Keyword(s):  
Piezoelectric Ceramic ◽  
Cutting Tool ◽  
Vibration Suppression ◽  
Equations Of Motion ◽  
Electrical Energy ◽  
Diameter Ratio ◽  
Vibration Absorber ◽  
Machined Surface ◽  
Structural Part ◽  
Machined Surface Roughness

Turning cutting tool with large length–diameter ratio has been essential when machining structural part with deep cavity and in-depth hole features. However, chatter vibration is apt to occur with the increase of tool overhang. A slender turning cutting tool with a length–diameter ratio of 7 is developed by using a vibration absorber equipped with piezoelectric ceramic. The vibration absorber has dual functions of vibration transfer to the absorber mass and vibration conversion to the electrical energy via the piezoelectric effect. Equations of motion are established considering the dual damping from the piezoelectric ceramic and rubber gasket. The equivalent damping of piezoelectric ceramic is derived, and the geometries are optimized to achieve optimal vibration suppression. The modal analysis demonstrates that the cutting tool with the vibration absorber can reach 80.1% magnitude reduction. Machining tests are carried out in the end. The machining acceleration and machined surface roughness validate the vibration suppression of the VA, and the output voltage by the piezoelectric ceramic demonstrates the ability of vibration sensing.

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