Analysis of the Orbits of Electrostatic MEMS Resonators

2008 ◽  
Author(s):  
F. Najar ◽  
E. M. Abdel-Rahman ◽  
A. H. Nayfeh ◽  
S. Choura
Keyword(s):  
Initial Conditions ◽  
Order Approximation ◽  
Differential Quadrature Method ◽  
Basin Of Attraction ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Electrostatic Mems ◽  
Differential Integral Equation ◽  
First Order Approximation ◽  
The Basin Of Attraction

We study the dynamic behavior of an electrostatic MEMS resonator using a model that accounts for the system nonlinearities due to mid-plane stretching and electrostatic forcing. The partial-differential-integral equation and associated boundary conditions representing the system dynamics are discretized using the Differential Quadrature Method (DQM) and the Finite Difference Method (FDM) for the space and time derivatives, respectively. The resulting model is analyzed to determine the periodic orbits of the resonator and their stability. Simultaneous resonances are identified for large orbits. Finally, we develop a first-order approximation of the microbeam dynamic response, which reveals an erosion of the basin of attraction of the stable orbits that depends heavily on the amplitude and frequency of the AC excitation. Simulations show that the smoothness of the boundary of the basin of attraction can be lost to be replaced by fractal tongues, which increase the sensitivity of the microbeam response to initial conditions. As a result, the locations of the stable and unstable fixed points are likely to be disturbed.

Stability Analysis of Electrostatically Actuated Resonators With Delayed Feedback Controller

2010 ◽  
Author(s):  
Fadi Alsaleem ◽  
Mohammad I. Younis
Keyword(s):  
Initial Conditions ◽  
Basin Of Attraction ◽  
Capacitive Sensor ◽  
Delayed Feedback ◽  
Feedback Controller ◽  
Electrostatically Actuated ◽  
Mems Resonators ◽  
Electrostatic Mems ◽  
Delayed Feedback Controller ◽  
The Stability

In this work we investigate the stability of parallel-plate electrostatic MEMS resonators using a delayed feedback controller. Two case studies are investigated: a capacitive sensor made of cantilever beams with a proof mass at their tip and a clamped-clamped microbeam. Dover-cliff integrity curves and basin-of-attraction analysis are used for the stability assessment of the frequency response of the resonators for several scenarios of positive and negative gain in the controller. It is found that, in the case of a positive gain, a velocity or a displacement feedback controller can be used to effectively enhance the stability of the resonators. This is confirmed by an increase in the area of the safe basin of attraction and in shifting the Dover-cliff curve upward. On the other hand, it is shown that a negative gain can significantly weaken the stability of the resonators. This can be of useful use in MEMS for actuation applications, such as in the case of capacitive switches, to lower the activation voltage of these devices and to ensure their trigger under all initial conditions.

Boundary of the Basin of Attraction for Weakly Damped Primary Resonance

1995 ◽  
Vol 62 (4) ◽  
pp. 941-946 ◽  
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.

Frequency analysis of nonlinear oscillations via the global error minimization

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
M Kalami Yazdi ◽  
P Hosseini Tehrani
Keyword(s):  
Variational Approach ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Global Error ◽  
Error Minimization ◽  
Free Oscillations ◽  
First Order ◽  
First Order Approximation

AbstractThe capacity and effectiveness of a modified variational approach, namely global error minimization (GEM) is illustrated in this study. For this purpose, the free oscillations of a rod rocking on a cylindrical surface and the Duffing-harmonic oscillator are treated. In order to validate and exhibit the merit of the method, the obtained result is compared with both of the exact frequency and the outcome of other well-known analytical methods. The corollary reveals that the first order approximation leads to an acceptable relative error, specially for large initial conditions. The procedure can be promisingly exerted to the conservative nonlinear problems.

Homotopy Perturbation Method for Nonlinear Vibration Analysis of Functionally Graded Plate

2013 ◽  
Vol 135 (2) ◽  
Author(s):  
Ali A. Yazdi
Keyword(s):  
Perturbation Method ◽  
Material Properties ◽  
Homotopy Perturbation Method ◽  
Initial Conditions ◽  
Order Approximation ◽  
Oscillation Amplitude ◽  
Functionally Graded ◽  
Homotopy Perturbation ◽  
Graded Material ◽  
First Order Approximation

In this paper, the Homotopy perturbation method (HPM) is used to analysis the geometrically nonlinear vibrations of thin rectangular laminated functionally graded material (FGM) plates. The Von Karman's strain-displacement relations have been employed to model structural nonlinearity of the system. The material properties of the plate are assumed to be graded continuously in direction of thickness. The effects of initial deflection, aspect ratio and material properties are investigated. Based on the results of this study, the first order approximation of the HPM leads to highly accurate solutions for geometrically nonlinearity vibration of FGM plates. Moreover, HPM in comparison with other traditional analytical methods (e.g., perturbation methods) has excellent accuracy for the whole range of oscillation amplitude and initial conditions.

scholarly journals Chaotic Dynamics of a Mixed Rayleigh–Liénard Oscillator Driven by Parametric Periodic Damping and External Excitations

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-18
Author(s):  
Yélomè Judicaël Fernando Kpomahou ◽  
Laurent Amoussou Hinvi ◽  
Joseph Adébiyi Adéchinan ◽  
Clément Hodévèwan Miwadinou
Keyword(s):  
Chaotic Dynamics ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Melnikov Method ◽  
Constant Term ◽  
Geometrical Shape ◽  
Analytical Prediction ◽  
Coexistence Of Attractors ◽  
The Basin Of Attraction

In this paper, chaotic dynamics of a mixed Rayleigh–Liénard oscillator driven by parametric periodic damping and external excitations is investigated analytically and numerically. The equilibrium points and their stability evolutions are analytically analyzed, and the transitions of dynamical behaviors are explored in detail. Furthermore, from the Melnikov method, the analytical criterion for the appearance of the homoclinic chaos is derived. Analytical prediction is tested against numerical simulations based on the basin of attraction of initial conditions. As a result, it is found that for ω = ν , the chaotic region decreases and disappears when the amplitude of the parametric periodic damping excitation increases. Moreover, increasing of F 1 and F 0 provokes an erosion of the basin of attraction and a modification of the geometrical shape of the chaotic attractors. For ω ≠ ν and η = 0.8 , the fractality of the basin of attraction increases as the amplitude of the external periodic excitation and constant term increase. Bifurcation structures of our system are performed through the fourth-order Runge–Kutta ode 45 algorithm. It is found that the system displays a remarkable route to chaos. It is also found that the system exhibits monostable and bistable oscillations as well as the phenomenon of coexistence of attractors.

A Chaotic Quadratic Bistable Hyperjerk System with Hidden Attractors and a Wide Range of Sample Entropy: Impulsive Stabilization

2021 ◽  
Vol 31 (16) ◽  
Author(s):  
M. D. Vijayakumar ◽  
Alireza Bahramian ◽  
Hayder Natiq ◽  
Karthikeyan Rajagopal ◽  
Iqtadar Hussain
Keyword(s):  
Impulsive Control ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basin Of Attraction ◽  
Sample Entropy ◽  
Hidden Attractors ◽  
System Equilibrium ◽  
Wide Range ◽  
Hyperjerk System ◽  
The Basin Of Attraction

Hidden attractors generated by the interactions of dynamical variables may have no equilibrium point in their basin of attraction. They have grabbed the attention of mathematicians who investigate strange attractors. Besides, quadratic hyperjerk systems are under the magnifying glass of these mathematicians because of their elegant structures. In this paper, a quadratic hyperjerk system is introduced that can generate chaotic attractors. The dynamical behaviors of the oscillator are investigated by plotting their Lyapunov exponents and bifurcation diagrams. The multistability of the hyperjerk system is investigated using the basin of attraction. It is revealed that the system is bistable when one of its attractors is hidden. Besides, the complexity of the systems’ attractors is investigated using sample entropy as the complexity feature. It is revealed how changing the parameters can affect the complexity of the systems’ time series. In addition, one of the hyperjerk system equilibrium points is stabilized using impulsive control. All real initial conditions become the equilibrium points of the basin of attraction using the stabilizing method.

scholarly journals Family of Bistable Attractors Contained in an Unstable Dissipative Switching System Associated to a SNLF

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
J. L. Echenausía-Monroy ◽  
J. H. García-López ◽  
R. Jaimes-Reátegui ◽  
D. López-Mancilla ◽  
G. Huerta-Cuellar
Keyword(s):  
Global Attractor ◽  
Control Parameter ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Nonlinear Function ◽  
Bifurcation Parameter ◽  
Switching System ◽  
Generator System ◽  
The Basin Of Attraction

This work presents a multiscroll generator system, which addresses the issue by the implementation of 9-level saturated nonlinear function, SNLF, being modified with a new control parameter that acts as a bifurcation parameter. By means of the modification of the newly introduced parameter, it is possible to control the number of scrolls to generate. The proposed system has richer dynamics than the original, not only presenting the generation of a global attractor; it is capable of generating monostable and bistable multiscrolls. The study of the basin of attraction for the natural attractor generation (9-scroll SNLF) shows the restrictions in the initial conditions space where the system is capable of presenting dynamical responses, limiting its possible electronic implementations.

Stabilizing the Dynamics of Electrostatic MEMS Resonators Using Delayed Feedback Controller

2009 ◽  
Author(s):  
Fadi M. Alsaleem ◽  
Mohammad I. Younis
Keyword(s):  
Delayed Feedback ◽  
Feedback Controller ◽  
Superior Performance ◽  
Mems Resonator ◽  
Mems Resonators ◽  
Electrostatic Mems ◽  
Delayed Feedback Controller ◽  
Simulation And Experiment ◽  
The Stability ◽  
Selection Of

We study the effect of delayed feedback controller on the dynamic stability of a MEMS resonator actuated with DC and AC voltages. We show that the delayed feedback controller, with a careful selection of its parameters, can be used to stabilize an originally unstable resonator operating in the escape (dynamic pull-in) frequency band. Also, the controller is shown to enhance the stability of the resonator near pull-in, where it experiences a strong fractal behavior. In both cases, the controller shows superior performance in rejecting disturbances. Experimental and theoretical results are presented to demonstrate the capability of the feedback controller to stabilize the performance of the capacitive resonator. A good agreement between simulation and experiment was achieved.

scholarly journals A New Circumscribed Self-Excited Spherical Strange Attractor

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ramesh Ramamoorthy ◽  
Sajjad Shaukat Jamal ◽  
Iqtadar Hussain ◽  
Mahtab Mehrabbeik ◽  
Sajad Jafari ◽  
...  
Keyword(s):  
Strange Attractor ◽  
Unique Feature ◽  
Initial Conditions ◽  
Basin Of Attraction ◽  
Basins Of Attraction ◽  
Spherical Coordinates ◽  
Chaotic Flows ◽  
Pass Through ◽  
Third Dimension ◽  
The Basin Of Attraction

Studying new chaotic flows with specific characteristics has been an open-ended field of exploring nonlinear dynamics. Investigation of chaotic flows is an area of research that has been taken into consideration for many years; thus, it helps in a better understanding of the chaotic systems. In this paper, an original chaotic 3D system, which has not been investigated yet, is presented in spherical coordinates. A unique feature of the proposed system is that its velocity becomes zero for a specific value of the radius variable. Hence, the system’s attractor is expected to be stuck on one side of a plane in spherical coordinates and inside or outside a sphere in the corresponding Cartesian coordinates. It means that the attractor cannot pass through the sphere or even touch it. The introduced system owns two unstable equilibria and a self-excited strange attractor. The 1D and 2D system’s bifurcation diagrams concerning the alteration of two bifurcation parameters are plotted to investigate the system’s dynamical properties. Moreover, the system’s Lyapunov exponents in the corresponding period of bifurcation parameters are calculated. Then, two 2D basins of attraction for two different third dimension values are explored. Based on the basin of attraction, it can be found that the sphere has attraction itself, partially, and some initial conditions are led to the sphere, not to the strange attractor. Ultimately, the connecting curves of the proposed system are explored to find an informative 1D set in addition to the system’s equilibria.

scholarly journals Vibration Control in MEMS Resonator Using Positive Position Feedback (PPF) Controller

2016 ◽  
Vol 12 (11) ◽  
pp. 6821-6834
Author(s):  
Y A Amer ◽  
A.T EL Sayed ◽  
A.M. Salem
Keyword(s):  
Order Approximation ◽  
Perturbation Technique ◽  
Multiple Scale ◽  
Steady State Solution ◽  
Positive Position Feedback ◽  
Mems Resonator ◽  
Position Feedback ◽  
Resonator System ◽  
The Stability ◽  
First Order Approximation

In this paper, the vibration of a micro-electromechanical resonator with positive position feedback controller is studied. The analytical results are obtained to the first order approximation by using the multiple scale perturbation technique. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. The effects of the controller and some system parameters on the vibrating system are studied numerically. The main result of this paper indicates that it is possible to reduce the vibration for the resonator system.

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