Voltage Response for Parametrically Actuated MEMS Cantilever Beam Using Homotopy Analysis Method and Method of Multiple Scales

2018 ◽  
Author(s):  
Christopher Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Parametric Resonance ◽  
Homotopy Analysis Method ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Excitation Frequency ◽  
Voltage Response ◽  
Analysis Method ◽  
Softening Effect ◽  
Homotopy Analysis

The purpose of this paper is to investigate the nonlinear dynamics governing the behavior of electrostatically actuated micro electro mechanical systems (MEMS) cantilever undergoing parametric resonance. The MEMS consists of a cantilever parallel to a ground plate. The beam is actuated via an A/C voltage with excitation frequency near first natural frequency of the cantilever. The model includes damping, electrostatic, and Casimir (or van der Waals) forces. The electrostatic force is modeled to include the fringe effect. The amplitude-voltage response of the parametric resonance and the effects of varying the magnitudes of the fringe, Casimir (or Van der Waals), and damping forces along with varying the detuning parameter are reported. The response is obtained using two different methods, namely the method of multiple scales (MMS), and the homotopy analysis method (HAM). In this study approximations up to a 2nd order HAM are used. HAM is a deformation technique that begins with an initial guess and continuously deforms it to the exact answer. For the 1st Order HAM, a softening effect is reported. The 1st Order HAM matches the MMS results in low amplitude and begins to soften and deviate away from the MMS solution in higher amplitudes. For the 2nd Order HAM deformation the softening effect is slightly more pronounced with a slightly lower prediction of the maximum deflection of the cantilever tip. For the 2nd order deformation solution the stable branch of the amplitude-voltage response obtained by the HAM shifts leftward from the MMS solution with the unstable branches between the two methods continue to agree in low amplitudes and deviate in high amplitudes. As a remark, the higher order HAM solutions are obtained symbolically with the software Mathematica and numerically ran with the software Matlab.

Parametric Resonance of Electrostatically Actuated MEMS Cantilever Resonators: Homotopy Analysis Method Versus the Method of Multiple Scales

2018 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christopher Reyes
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Second Order ◽  
Analysis Method ◽  
Casimir Forces ◽  
Electrostatically Actuated ◽  
Softening Behavior ◽  
Order Deformation ◽  
Homotopy Analysis

This paper investigates the parametric resonance of electrostatically actuated MicroElectroMechanicalSystems (MEMS) cantilever resonators. The electrostatic force is modeled to include fringe effect. The MEMS consists of a cantilever over a parallel ground plate and an AC voltage between them. The actuation frequency is near first natural frequency of the cantilever beam. This leads to parametric resonance. It is of interest to investigate the amplitude frequency response of MEMS cantilever resonators. This paper uses the Homotopy Analysis Method (HAM), which is able to capture nonlinear behaviors for higher amplitudes, large parameters, and strong nonlinearities. The base method used for comparison in this work is the method of multiple scales (MMS). MMS is a perturbation method. It requires a relatively short computational time for simulations. Although the CPU time is advantageous, MMS is only accurate for weak nonlinearities and low amplitudes. It is in the interest to compare how well HAM captures the softening behavior of this system as opposed to MMS. In this paper the influences of Casimir forces and Van der Waals effects are included. Electrostatic, Van der Waals and Casimir forces are nonlinear. HAM is a deformation technique that continuously deforms the initial guess, provided to the procedure, to the exact solution. In this work the first and second order deformation equations are constructed for the equation of motion governing the behavior of the MEMS cantilever beam. In the first order deformation, HAM deviates from the solution obtained by MMS. This deviation demonstrates the power of the method to capture the softening behavior more accurately than MMS even at the 1st order deformation HAM. In the second order deformation construction, the HAM’s solution softens more than the previous, demonstrating that higher order deformation approximations result in higher accuracy. In the second order deformation, HAM contains the convergence control parameter. This parameter is chosen via the c0 curve approach. Up to 2nd order HAM deformations are evaluated for this paper. These higher order homotopy deformation solutions were developed and automated symbolically in the software Mathematica and tested numerically using Matlab software.

On Nonlinear Dynamics of Nanotweezers

2016 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Bin Liu
Keyword(s):  
Nonlinear Dynamics ◽  
Frequency Response ◽  
Parametric Resonance ◽  
Taylor Series ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Van Der Waals Forces ◽  
The Other ◽  
Amplitude Frequency Response

This paper deals with amplitude-frequency response of electrostatic nanotube nanotweezer device system. Soft alternating current (AC) of frequency near natural frequency actuates the nanotubes. This leads the system into parametric resonance. The Method of Multiple Scales (MMS) in which the nonlinear electrostatic and van der Waals forces are expanded in Taylor series is used to compare two expansions, one up to third power and the other up to fifth power. The frequency response of the system is reported and the effects of van der Waals forces, electrostatic forces, and damping forces on the frequency response are investigated.

Primary Resonance Voltage Response of Electrostatically Actuated M/NEMS Circular Plate Resonators

2014 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Reynaldo Oyervides
Keyword(s):  
Casimir Force ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Voltage Response ◽  
Circular Plates ◽  
Amplitude Response ◽  
Weakly Nonlinear ◽  
Electrostatically Actuated

This paper investigates the voltage-amplitude response of soft AC electrostatically actuated M/NEMS clamped circular plates. AC frequency is near half natural frequency of the plate. This results in primary resonance. The system is analytically modeled using the Method of Multiple Scales (MMS). The system is assumed weakly nonlinear. The behavior of the system including pull-in instability as the AC voltage is swept up and down while the excitation frequency is constant is reported. The effects of detuning frequency, damping, Casimir force, and van der Waals force are reported as well.

Voltage Response of Parametric Resonance of Double Wall Carbon Nanotube Under Electrostatic Actuation

2017 ◽  
Author(s):  
Ezequiel Juarez ◽  
Dumitru I. Caruntu ◽  
Young-Gil Park
Keyword(s):  
Carbon Nanotube ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Van Der Waals ◽  
Voltage Response ◽  
First Order ◽  
Electrostatically Actuated ◽  
First Order Method ◽  
Euler Bernoulli Beam ◽  
Multiple Scales Solution

In this paper, the Method of Multiple Scales is used to investigate the influences of damping and detuning frequency parameters on the amplitude-voltage response of an electrostatically actuated double-walled carbon nanotube. The forces responsible for the nonlinearities in the vibrational behavior are intertube van der Waals and electrostatic forces. Herein, the coaxial case is investigated, which eliminates the influence of the cubic van der Waals in the first-order solution. The double-walled carbon nanotube structure is modelled as a cantilever beam with Euler-Bernoulli beam assumptions since the double-walled carbon nanotube is characterized with high length-diameter ratio. The results shown assume steady-state solutions in the first-order Method of Multiple Scales solution. The importance of the results in this paper are the effect of damping and detuning frequency on the Hopf bifurcations, as these define the intervals of voltage for nonzero amplitudes.

Voltage Response of Parametric Resonance of MEMS Circular Plates Under Hard Excitations

2019 ◽  
Author(s):  
Julio S. Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Parametric Resonance ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Voltage Response ◽  
Circular Plates ◽  
Order Model ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

Abstract This work deals with the voltage response of parametric resonance of electrostatically actuated microelectromechanical (MEMS) circular plates under hard excitations. Method of Multiple Scales (MMS) and Reduced Order Model (ROM) method using two modes of vibration are used to predict the voltage-amplitude response of the MEMS circular plates. ROM is solved using AUTO 07p, a software package for continuation and bifurcation. MMS used in this paper has one term in the electrostatic force being considered significant. This is the way MMS is used to model hard excitations. MMS shows results similar to those of ROM at lower amplitudes and lower voltages. The differences between the two methods, MMS and ROM, are significant in high amplitudes for all voltages, and the differences are significant in all amplitudes for larger voltages. Significant differences can be noted in the effect of different parameters such as the detuning frequency and damping on the voltage response. ROM AUTO 07p is calibrated using ROM time responses in which the ROM is solved using the solver ode15s in Matlab.

HOMOTOPY ANALYSIS METHOD TO SOLVE BOUSSINESQ EQUATIONS

2015 ◽  
Vol 10 (3) ◽  
pp. 2825-2833
Author(s):  
Achala Nargund ◽  
R Madhusudhan ◽  
S B Sathyanarayana
Keyword(s):  
Homotopy Analysis Method ◽  
Degrees Of Freedom ◽  
Initial Approximation ◽  
Boussinesq Equations ◽  
Convergent Series ◽  
Boussinesq System ◽  
Analysis Method ◽  
Analytical Technique ◽  
Homotopy Analysis ◽  
Region Of Convergence

In this paper, Homotopy analysis method is applied to the nonlinear coupleddifferential equations of classical Boussinesq system. We have applied Homotopy analysis method (HAM) for the application problems in [1, 2, 3, 4]. We have also plotted Domb-Sykes plot for the region of convergence. We have applied Pade for the HAM series to identify the singularity and reflect it in the graph. The HAM is a analytical technique which is used to solve non-linear problems to generate a convergent series. HAM gives complete freedom to choose the initial approximation of the solution, it is the auxiliary parameter h which gives us a convenient way to guarantee the convergence of homotopy series solution. It seems that moreartificial degrees of freedom implies larger possibility to gain better approximations by HAM.

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