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.

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.

Frequency Response of Parametric Resonance of Electrostatically Actuated Circular Plates Under Hard Excitations

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

Abstract In this paper, the Method of Multiple Scales, and the Reduced Order Model method of two modes of vibration are used to investigate the amplitude-frequency response of parametric resonance of electrostatically actuated circular plates under hard excitations. Results show that the Method of Multiple Scales is accurate for low voltages. However, it starts to separate from the Reduced Order Model results as the voltage values are larger. The Method of Multiple Scales is good for low amplitudes and weak non-linearities. Furthermore the Reduced Order Model running with AUTO 07p is validated and calibrated using the 2 Term ROM time responses.

Parametric Resonance of Electrostatically Actuated MEMS Cantilever Resonators Using Method of Multiple Scales and Homotopy Perturbation Method

2017 ◽  
Author(s):  
Christopher Reyes ◽  
Dumitru I. Caruntu
Keyword(s):  
Perturbation Method ◽  
Parametric Resonance ◽  
Homotopy Perturbation Method ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Plate Electrode ◽  
Electrostatically Actuated ◽  
Homotopy Perturbation ◽  
Ground Plate

This paper investigates the dynamics governing the behavior of electrostatically actuated MEMS cantilever resonators. The cantilever is held parallel to a ground plate (electrode) with an AC voltage between the plate and the electrode causing the electrostatic actuation (excitation). For the purposes of this paper this is soft excitation. The frequency of the excitation is near the natural frequency of the cantilever leading to what is known as parametric resonance. The electrostatic force in the problem investigated throughout the paper is nonlinear in nature and includes the fringe effect. Two methods are used in investigating this problem: the method of multiple scales (MMS) and the homotopy perturbation method (HPM). The two methods work well for small non-linearities and small amplitudes. The influence of voltage, fringe, damping, Casimir, and Van der Waals parameters will be investigated in this paper using MMS and HPM as a means of verifying the results obtained.

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.

scholarly journals Voltage–Amplitude Response of Superharmonic Resonance of Second Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Vol 14 (3) ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Martin A. Botello ◽  
Christian A. Reyes ◽  
Julio S. Beatriz
Keyword(s):  
Numerical Integration ◽  
Multiple Scales ◽  
Low Voltage ◽  
Second Order ◽  
Voltage Amplitude ◽  
Amplitude Response ◽  
Superharmonic Resonance ◽  
Bifurcation Points ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper investigates the voltage–amplitude response of superharmonic resonance of second order (order two) of alternating current (AC) electrostatically actuated microelectromechanical system (MEMS) cantilever resonators. The resonators consist of a cantilever parallel to a ground plate and under voltage that produces hard excitations. AC frequency is near one-fourth of the natural frequency of the cantilever. The electrostatic force includes fringe effect. Two kinds of models, namely reduced-order models (ROMs), and boundary value problem (BVP) model, are developed. Methods used to solve these models are (1) method of multiple scales (MMS) for ROM using one mode of vibration, (2) continuation and bifurcation analysis for ROMs with several modes of vibration, (3) numerical integration for ROM with several modes of vibration, and (4) numerical integration for BVP model. The voltage–amplitude response shows a softening effect and three saddle-node bifurcation points. The first two bifurcation points occur at low voltage and amplitudes of 0.2 and 0.56 of the gap. The third bifurcation point occurs at higher voltage, called pull-in voltage, and amplitude of 0.44 of the gap. Pull-in occurs, (1) for voltage larger than the pull-in voltage regardless of the initial amplitude and (2) for voltage values lower than the pull-in voltage and large initial amplitudes. Pull-in does not occur at relatively small voltages and small initial amplitudes. First two bifurcation points vanish as damping increases. All bifurcation points are shifted to lower voltages as fringe increases. Pull-in voltage is not affected by the damping or detuning frequency.

Casimir Effect on Voltage Response of Superharmonic Resonance of NEMS Resonators

2017 ◽  
Author(s):  
Martin Botello ◽  
Christian Reyes ◽  
Julio Beatriz ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Equation Of Motion ◽  
Second Order ◽  
Mode Shapes ◽  
Voltage Response ◽  
Casimir Forces ◽  
Superharmonic Resonance ◽  
Reference Length

This paper investigates the voltage response of superharmonic resonance of the second order of electrostatically actuated nano-electro-mechanical system (NEMS) resonator sensor. The structure of the NEMS device is a resonator cantilever over a ground plate under Alternating Current (AC) voltage. Superharmonic resonance of second order occurs when the AC voltage is operating in a frequency near-quarter the natural frequency of the resonator. The forces acting on the system are electrostatic, damping and Casimir. To induce a bifurcation phenomenon in superharmonic resonance, the AC voltage is in the category of hard excitation. The gap distance between the cantilever resonator and base plate is in the range of 20 nm to 1 μm for Casimir forces to be present. The differential equation of motion is converted to dimensionless by choosing the gap as reference length for deflections, the length of the resonator for the axial coordinate, and reference time based on the characteristics of the structure. The Method of Multiple Scales (MMS) and Reduced Order Model (ROM) are used to model the characteristic of the system. MMS transforms the nonlinear partial differential equation of motion into two simpler problems, namely zero-order and first-order. ROM, based on the Galerkin procedure, uses the undamped linear mode shapes of the undamped cantilever beam as the basis functions. The influences of parameters (i.e. Casimir, damping, fringe, and detuning parameter) were also investigated.

Amplitude-Frequency Response for Superharmonic Resonance of Fourth Order of Electrostatically Actuated MEMS Cantilever Resonators

2019 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Christopher Reyes
Keyword(s):  
Frequency Response ◽  
Fourth Order ◽  
Microelectromechanical Systems ◽  
Multiple Scales ◽  
Electrostatic Force ◽  
Damping Force ◽  
Superharmonic Resonance ◽  
Electrostatically Actuated ◽  
Homotopy Analysis ◽  
Ground Plate

Abstract This paper deals with the frequency response of superharmonic resonance of order four of electrostatically actuated MicroElectroMechanical Systems (MEMS) cantilever resonators. The MEMS structure in this work consists of a microcantilever parallel to an electrode ground plate. The MEMS resonator is elelctrostatically actuated through an AC voltage between the cantilever and the ground plate. The voltage is in the category of hard excitation. The AC frequency is near one eight of the natural frequency of the resonator. Since the electrostatic force acting on the resonator is proportional to the square of the voltage, it leads to superharmonic resonance of fourth order. Besides the electrostatic force, the system experiences damping. The damping force in this work is proportional to the velocity of the resonator, i.e. it is linear damping. Three methods are employed in this investigation. First, the Method of Multiple Scales (MMS), a perturbation method, is used predictions of the resonant regions for weak nonlinearities and small to moderate amplitudes. Second, the Homotopy Analysis Method (HAM), and third, the Reduced Order Model (ROM) method using two modes of vibration are also utilized to investigate the resonance. ROM is solved through numerical integration using Matlab in order to simulate time responses of the structure. All methods are in agreement for moderate nonlinearities and small to moderate amplitudes. This work shows that adequate MMS and HAM provide good predictions of the resonance.

Reduced Order Model for MEMS Cantilever Resonators Under Soft AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Israel Martinez
Keyword(s):  
Natural Frequency ◽  
Nonlinear Response ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Electrostatic Force ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
First Order ◽  
Electrostatically Actuated

The nonlinear response of an electrostatically actuated cantilever beam microresonator is investigated. The AC voltage is of frequency near resonator’s natural frequency. A first order fringe correction of the electrostatic force and viscous damping are included in the model. The dynamics of the resonator is investigated using the Reduced Order Model (ROM) method, based on Galerkin procedure. Steady-state motions are found. Numerical results for the uniform microresonator are compared with those obtained via the Method of Multiple Scales (MMS).

Second Order Perturbation Analysis of a Forced Nonlinear Mathieu Equation

2012 ◽  
Author(s):  
Venkatanarayanan Ramakrishnan ◽  
Brian F. Feeny
Keyword(s):  
Perturbation Analysis ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Mathieu Equation ◽  
Second Order ◽  
Order Perturbation ◽  
Cubic Nonlinearity ◽  
First Order ◽  
Order Expansion ◽  
Direct Forcing

The present study deals with the response of a forced nonlinear Mathieu equation. The equation considered has parametric excitation at the same frequency as direct forcing and also has cubic nonlinearity and damping. A second-order perturbation analysis using the method of multiple scales unfolds numerous resonance cases and system behavior that were not uncovered using first-order expansions. All resonance cases are analyzed. We numerically plot the frequency response of the system. The existence of a superharmonic resonance at one third the natural frequency was uncovered analytically for linear system. (This had been seen previously in numerical simulations but was not captured in the first-order expansion.) The effect of different parameters on the response of the system previously investigated are revisited.

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