We investigate modeling the dynamics of an electrostatically actuated resonator using the perturbation method of multiple time scales (MTS). First, we discuss two approaches to treat the nonlinear parallel-plate electrostatic force in the equation of motion and their impact on the application of MTS: expanding the force in Taylor series and multiplying both sides of the equation with the denominator of the forcing term. Considering a spring–mass–damper system excited electrostatically near primary resonance, it is concluded that, with consistent truncation of higher-order terms, both techniques yield same modulation equations. Then, we consider the problem of an electrostatically actuated resonator under simultaneous superharmonic and primary resonance excitation and derive a comprehensive analytical solution using MTS. The results of the analytical solution are compared against the numerical results obtained by long-time integration of the equation of motion. It is demonstrated that along with the direct excitation components at the excitation frequency and twice of that, higher-order parametric terms should also be included. Finally, the contributions of primary and superharmonic resonance toward the overall response of the resonator are examined.
References
1.
Nathanson
, H. C.
, Newell
, W. E.
, Wickstrom
, R. A.
, and Davis
, J. R.
, 1967
, “The Resonant Gate Transistor
,” IEEE Trans. Electron. Devices
, 14
(3
), pp. 117
–133
.2.
Zhang
, W.
, and Turner
, K. L.
, 2005
, “Application of Parametric Resonance Amplification in a Single-Crystal Silicon Micro-Oscillator Based Mass Sensor
,” Sens. Actuators A
, 122
(1
), pp. 23
–30
.3.
Hafiz
, M. A. A.
, Kosuru
, L.
, and Younis
, M. I.
, 2016
, “Microelectromechanical Reprogrammable Logic Device
,” Nat. Commun.
, 7
, p. 11137
.4.
Ilyas
, S.
, Jaber
, N.
, and Younis
, M. I.
, 2017
, “MEMS Logic Using Mixed-Frequency Excitation
,” J. Microelectromech. Syst.
, 26
(5
), pp. 1140
–1146
.5.
Wong
, A. C.
, and Nguyen
, C. C.
, 2004
, “Micromechanical Mixer-Filters (“Mixlers”)
,” J. Microelectromech. Syst.
, 13
(1
), pp. 100
–112
.6.
Ilyas
, S.
, Chappanda
, K. N.
, and Younis
, M. I.
, 2017, “Exploiting Nonlinearities of Micro-Machined Resonators for Filtering Applications
,” Appl. Phys. Lett.
, 110
(25
), p. 253508
.7.
Pourkamali
, S.
, and Ayazi
, F.
, 2005
, “Electrically Coupled MEMS Bandpass Filters—Part I: With Coupling Element
,” Sens. Actuators A
, 122
(2
), pp. 307
–316
.8.
Kacem
, N.
, Hentz
, S.
, Pinto
, D.
, Reig
, B.
, and Nguyen
, V.
, 2009
, “Nonlinear Dynamics of Nanomechanical Beam Resonators: Improving the Performance of NEMS-Based Sensors
,” Nanotechnology.
, 20
(27
), p. 275501
.9.
Guerra
, D. N.
, Bulsara
, A. R.
, Ditto
, W. L.
, Sinha
, S.
, Murali
, K.
, and Mohanty
, P.
, 2010
, “A Noise-Assisted Reprogrammable Nanomechanical Logic Gate
,” Nano Lett.
, 10
(4
), pp. 1168
–1171
.10.
Jalil
, J.
, Zhu
, Y.
, Ekanayake
, C.
, and Ruan
, Y.
, 2017
, “Sensing of Single Electrons Using Micro and Nano Technologies: A Review
,” Nanotechnology
, 28
(14
), p. 142002
.11.
Nayfeh
, A. H.
, 2011
, Introduction to Perturbation Techniques
, Wiley
, Weinheim, Germany
.12.
Nayfeh
, A. H.
, and Mook
, D. T.
, 2008
, Nonlinear Oscillations
, Wiley
, Weinheim, Germany
.13.
Younis
, M. I.
, and Nayfeh
, A. H.
, “A Study of the Nonlinear Response of a Resonant Microbeam to an Electric Actuation
,” Nonlinear Dyn.
, 31
(1
), pp. 91
–117
.14.
Abdel-Rahman
, E. M.
, and Nayfeh
, A. H.
, 2003
, “Secondary Resonances of Electrically Actuated Resonant Microsensors
,” J. Micromech. Microeng.
, 13
(3
), p. 491
.15.
Nayfeh
, A. H.
, 1984
, “Quenching of Primary Resonance by a Superharmonic Resonance
,” J. Sound Vib.
, 92
(3
), pp. 363
–377
.16.
Nayfeh
, A. H.
, 1983
, “The Response of Single Degree-of-Freedom Systems With Quadratic and Cubic Non-Linearities to a Subharmonic Excitation
,” J. Sound Vib.
, 89
(4
), pp. 457
–470
.17.
Kacem
, N.
, Baguet
, S.
, Duraffourg
, L.
, Jourdan
, G.
, Dufour
, R.
, and Hentz
, S.
, 2015
, “Overcoming Limitations of Nanomechanical Resonators With Simultaneous Resonances
,” Appl. Phys. Lett.
, 107
(7
), p. 073105
.18.
Ilyas
, S.
, Alfosail
, F.
, and Younis
, M. I.
, 2018
, “On the Response of MEMS Resonators Under Generic Electrostatic Loadings: Experiments and Applications
,” Nonlinear Dynamics
(accepted).19.
Zhang
, W. M.
, and Guang
, M.
, 2007
, “Nonlinear Dynamic Analysis of Electrostatically Actuated Resonant MEMS Sensors Under Parametric Excitation
,” IEEE Sensors.
, 7
(3
), pp. 370
–380
.20.
Caruntu
, D. I.
, and Martinez
, I.
, 2014
, “Reduced Order Model of Parametric Resonance of Electrostatically Actuated MEMS Cantilever Resonators
,” Int. J. Non-Linear Mech.
, 66
, pp. 28
–32
.21.
Kacem
, N.
, Baguet
, S.
, Dufour
, R.
, and Hentz
, S.
, 2011
, “Stability Control of Nonlinear Micromechanical Resonators Under Simultaneous Primary and Superharmonic Resonances
,” Appl. Phys. Lett.
, 98
(19
), p. 193507
.22.
Nayfeh
, A. H.
, 1985
, “The Response of Non-Linear Single-Degree-of-Freedom Systems to Multifrequency Excitations
,” J. Sound Vib.
, 102
(3
), pp. 403
–414
.23.
Chen
, C.
, Zanette
, D. H.
, Guest
, J. R.
, Czaplewski
, D. A.
, and López
, D.
, 2016
, “Self-Sustained Micromechanical Oscillator With Linear Feedback
,” Phys. Rev. Lett.
, 117
, p. 017203
.24.
Elshurafa
, A. M.
, Khirallah
, K.
, Tawfik
, H. H.
, Emira
, A.
, Aziz
, A. K. A.
, and Sedky
, S. M.
, 2011
, “Nonlinear Dynamics of Spring Softening and Hardening in Folded-MEMS Comb Drive Resonators
,” J. Microelectromech. Syst.
, 20
(4
), pp. 943
–958
.25.
Ouakad
, H. M.
, and Younis
, M. I.
, 2010
, “The Dynamic Behavior of MEMS Arch Resonators Actuated Electrically
,” Int. J. Non-Linear Mech.
, 45
(7
), pp. 704
–713
.26.
Ouakad
, H. M.
, and Younis
, M. I.
, 2010
, “Nonlinear Dynamics of Electrically Actuated Carbon Nanotube Resonators
,” ASME J. Comput. Nonlinear Dyn.
, 5
(1
), p. 011009
.27.
Nayfeh
, A. H.
, and Younis
, M. I.
, 2005
, “Dynamics of MEMS Resonators Under Superharmonic and Subharmonic Excitations
,” J. Micromech. Microeng.
, 15
(10
), p. 1840
.28.
Alsaleem
, F. M.
, Younis
, M. I.
, and Ouakad
, H. M.
, 2009
, “On the Nonlinear Resonances and Dynamic Pull-In of Electrostatically Actuated Resonators
,” J. Micromech. Microeng.
, 19
(4
), p. 045013
.29.
Jin
, Z.
, and Wang
, Y.
, 1998
, “Electrostatic Resonator With Second Superharmonic Resonance
,” Sens. Actuators. A
, 64
(3
), pp. 273
–279
.30.
Younis
, M. I.
, and Alsaleem
, F.
, 2009
, “Exploration of New Concepts for Mass Detection in Electrostatically-Actuated Structures Based on Nonlinear Phenomena
,” ASME J. Comput. Nonlinear Dyn.
, 4
(2
), p. 021010
.31.
Ouakad
, H. M.
, Alofi
, A. M.
, and Nayfeh
, A. H.
, 2017
, “Dynamic Analysis of Multilayers Based MEMS Resonators
,” Math. Probl. Eng.
, 2017
, p. 1262650
.32.
Azizi
, S.
, Chorsi
, M. T.
, and Nejad
, F. B.
, 2016
, “On the Secondary Resonance of a MEMS Resonator: A Conceptual Study Based on Shooting and Perturbation Methods
,” Int. J. Non-Linear Mech.
, 82
, pp. 59
–68
.33.
Rhoads
, J. F.
, Shaw
, S. W.
, and Turner
, K. L.
, 2006
, “The Nonlinear Response of Resonant Microbeam Systems With Purely-Parametric Electrostatic Actuation
,” J. Micromech. Microeng.
, 16
(5
), p. 890
.34.
Elnaggar
, A. M.
, Khalil
, K. M.
, and Rahby
, A. S.
, 2016
, “Harmonic Solution of a Weakly Nonlinear Second Order Differential Equation Governed the Motion of a TM-AFM Cantilever
,” Br. J. Math. Comput. Sci.
, 15
(4
), pp. 1
–11
.35.
Younis
, M. I.
, Abdel-Rahman
, E. M.
, and Nayfeh
, A. H.
, 2003
, “A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS
,” J. Microelectromech. Syst.
, 12
(5
), pp. 672
–680
.36.
Younis
, M. I.
, 2011
, MEMS Linear and Nonlinear Statics and Dynamics
, Springer Science and Business Media
, New York
.Copyright © 2019 by ASME
You do not currently have access to this content.
