Abstract
Due to various sources of nonlinearities, micro/nano-electro-mechanical-system (MEMS/NEMS) resonators present highly nonlinear behaviors including softening- or hardening-type frequency responses, bistability, chaos, etc. The general Duffing equation with quadratic and cubic nonlinearities serves as a characterizing model for a wide class of MEMS/NEMS resonators as well as lots of other engineering and physical systems. In this paper, after brief reviewing of various sources of nonlinearities in micro/nano-resonators and discussing how they contribute to the Duffing-type nonlinearities, we propose a Homotopy Analysis Method (HAM) approach for derivation of analytical solutions for the frequency response of the resonators. Toward this aim, we first apply the HAM to the proposed Duffing equation, and through this procedure, we derive the first-order and second-order HAM-based analytical solutions for the frequency response of the resonator. As the main novelty, we show that the second-order solution benefits from a tunable parameter, known as the convergence-control parameter, which is a distinguishing aspect of the HAM and plays a key role in enhancing the accuracy of the obtained analytical expressions in strongly nonlinear problems. We use the obtained analytical solutions for the study of nonlinear dynamics in two types of electrostatically actuated MEMS resonators proposing hardening, softening or mixed behaviors near their primary resonance frequency. Numerical simulations are performed to validate the analytical results.
Similar content being viewed by others
References
Abbasbandy S (2006) The application of homotopy analysis method to nonlinear equations arising in heat transfer. Phys Lett A 360:109–113
Abbasbandy S, Shirzadi A (2011) A new application of the homotopy analysis method: solving the Sturm-Liouville problems. Commun Nonlinear Sci Numer Simul 16:112–126. doi:10.1016/j.cnsns.2010.04.004
Abbasbandy S, Shivanian E (2011) Predictor homotopy analysis method and its application to some nonlinear problems. Commun Nonlinear Sci Numer Simul 16:2456–2468. doi:10.1016/j.cnsns.2010.09.027
Abbasbandy S, Shivanian E, Vajravelu K (2011) Mathematical properties of h-curve in the frame work of the homotopy analysis method. Commun Nonlinear Sci Numer Simul 16:4268–4275. doi:10.1016/j.cnsns.2011.03.031
Akgöz B, Civalek Ö (2011) Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams. Int J Eng Sci 49:1268–1280. doi:10.1016/j.ijengsci.2010.12.009
Akgöz B, Civalek Ö (2015) A novel microstructure-dependent shear deformable beam model. Int J Mech Sci 99:10–20. doi:10.1016/j.ijmecsci.2015.05.003
Amorim TD, Dantas WG, Gusso A (2014) Analysis of the chaotic regime of MEMS/NEMS fixed–fixed beam resonators using an improved 1DOF model. Nonlinear Dyn. doi:10.1007/s11071-014-1715-4
Azizi S, Ghazavi MR, Rezazadeh G et al (2013) Tuning the primary resonances of a micro resonator, using piezoelectric actuation. Nonlinear Dyn 76:839–852. doi:10.1007/s11071-013-1173-4
Bakkyaraj T, Sahadevan R (2014) On solutions of two coupled fractional time derivative Hirota equations. Nonlinear Dyn 77:1309–1322. doi:10.1007/s11071-014-1380-7
Batra RC, Porfiri M, Spinello D (2006) Electromechanical model of electrically actuated narrow microbeams. J Microelectromech Syst 15:1175–1189
Batra RC, Porfiri M, Spinello D (2008) Vibrations of narrow microbeams predeformed by an electric field. J Sound Vib 309:600–612. doi:10.1016/j.jsv.2007.07.030
Belardinelli P, Brocchini M, Demeio L, Lenci S (2013) Dynamical characteristics of an electrically actuated microbeam under the effects of squeeze-film and thermoelastic damping. Int J Eng Sci 69:16–32. doi:10.1016/j.ijengsci.2013.03.011
Bazaei A, Maroufi M, Reza Moheimani SO (2016) On the modeling of tilted fixed-guided flexible beams under tension. Acta Mech 227(2):333–352. doi:10.1007/s00707-015-1448-6
Cacan MR, Leadenham S, Leamy MJ (2014) An enriched multiple scales method for harmonically forced nonlinear systems. Nonlinear Dyn 78:1205–1220. doi:10.1007/s11071-014-1508-9
Caruntu DI, Luo L (2014) Frequency response of primary resonance of electrostatically actuated CNT cantilevers. Nonlinear Dyn 78:1827–1837. doi:10.1007/s11071-014-1537-4
Civalek Ö, Akgöz B (2013) Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix. Comput Mater Sci 77:295–303. doi:10.1016/j.commatsci.2013.04.055
Daqaq MF (2010) Response of uni-modal duffing-type harvesters to random forced excitations. J Sound Vib 329:3621–3631. doi:10.1016/j.jsv.2010.04.002
Das K, Batra RC (2009) Symmetry breaking, snap-through and pull-in instabilities under dynamic loading of microelectromechanical shallow arches. Smart Mater Struct 18:115008. doi:10.1088/0964-1726/18/11/115008
Fahsi A, Belhaq M (2009) Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator. Commun Nonlinear Sci Numer Simul 14:244–253. doi:10.1016/j.cnsns.2007.07.010
Farokhi H, Ghayesh MH, Amabili M (2013) Nonlinear dynamics of a geometrically imperfect microbeam based on the modified couple stress theory. Int J Eng Sci 68:11–23. doi:10.1016/j.ijengsci.2013.03.001
Fu Y, Zhang J, Jiang Y (2010) Influences of the surface energies on the nonlinear static and dynamic behaviors of nanobeams. Phys E Low-dimensional Syst Nanostructures 42:2268–2273. doi:10.1016/j.physe.2010.05.001
Ghayesh MH, Farokhi H, Amabili M (2013a) Nonlinear behaviour of electrically actuated MEMS resonators. Int J Eng Sci 71:137–155
Ghayesh MH, Farokhi H, Amabili M (2013b) Nonlinear dynamics of a microscale beam based on the modified couple stress theory. Compos Part B 50:318–324. doi:10.1016/j.compositesb.2013.02.021
Haghighi HS, Markazi AHD (2010) Chaos prediction and control in MEMS resonators. Commun Nonlinear Sci Numer Simul 15:3091–3099. doi:10.1016/j.cnsns.2009.10.002
Hammad BK, Abdel-Rahman EM, Nayfeh AH (2009) Modeling and analysis of electrostatic MEMS filters. Nonlinear Dyn 60:385–401. doi:10.1007/s11071-009-9603-z
Hassanpour PA, Esmailzadeh E, Cleghorn WL, Mills JK (2010) Nonlinear vibration of micromachined asymmetric resonators. J Sound Vib 329:2547–2564. doi:10.1016/j.jsv.2009.10.033
He X, Wu Q, Wang Y et al (2009) Numerical simulation and analysis of electrically actuated microbeam-based MEMS capacitive switch. Microsyst Technol 15:301–307. doi:10.1007/s00542-008-0702-4
Ibrahim MI, Younis MI, Alsaleem F (2010) An investigation into the effects of electrostatic and squeeze-film non-linearities on the shock spectrum of microstructures. Int J Non Linear Mech 45:756–765. doi:10.1016/j.ijnonlinmec.2010.05.005
Jia XL, Yang J, Kitipornchai S, Lim CW (2012) Resonance frequency response of geometrically nonlinear micro-switches under electrical actuation. J Sound Vib 331:3397–3411. doi:10.1016/j.jsv.2012.02.026
Kacem N, Hentz S, Pinto D et al (2009) Nonlinear dynamics of nanomechanical beam resonators: improving the performance of NEMS-based sensors. Nanotechnology 20:275501. doi:10.1088/0957-4484/20/27/275501
Kacem N, Baguet S, Hentz S, Dufour R (2011) Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors. Int J Non Linear Mech 46:532–542. doi:10.1016/j.ijnonlinmec.2010.12.012
Kahrobaiyan MH, Asghari M, Rahaeifard M, Ahmadian MT (2011) A nonlinear strain gradient beam formulation. Int J Eng Sci 49:1256–1267. doi:10.1016/j.ijengsci.2011.01.006
Kirrou I, Belhaq M (2013) Contact stiffness modulation in contact-mode atomic force microscopy. Int J Non Linear Mech 55:102–109. doi:10.1016/j.ijnonlinmec.2013.04.013
Kong S, Zhou S, Nie Z, Wang K (2009) Static and dynamic analysis of micro beams based on strain gradient elasticity theory. Int J Eng Sci 47:487–498. doi:10.1016/j.ijengsci.2008.08.008
Kovacic I, Brennan MJ (2011) The Duffing equation, nonlinear oscillators and their behavior. Joun Wiley & Sons
Krylov S, Ilic BR, Schreiber D et al (2008) The pull-in behavior of electrostatically actuated bistable microstructures. J Micromech Microeng 18:055026. doi:10.1088/0960-1317/18/5/055026
Krylov S, Ilic B, Lulinsky S (2011) Bistability of curved microbeams actuated by fringing electrostatic fields. Nonlinear Dyn 66:403–426
Liang S, Jeffrey DJ (2010) Approximate solutions to a parameterized sixth order boundary value problem. Comput Math with Appl 59:247–253. doi:10.1016/j.camwa.2009.07.053
Liao S (1995) An approximate solution technique not depending on small parameters: a special example. Int J Non Linear Mech 30:371–380
Liao S (2003) Beyond perturbation introduction to homotopy Analysis method. CRC Press
Liao S (2004a) On the homotopy analysis method for nonlinear problems. Appl Math Comput 147:499–513. doi:10.1016/S0096-3003(02)00790-7
Liao S (2004b) An analytic approximate approach for free oscillations of self-excited systems. Int J Non Linear Mech 39:271–280
Liao S (2009) Notes on the homotopy analysis method: some definitions and theorems. Commun Nonlinear Sci Numer Simul 14:983–997. doi:10.1016/j.cnsns.2008.04.013
Liao S (2012) Homotopy Analysis Method in Nonlinear Differential Equations. Springer
Liao S (2013) Chance and challenge : a brief review of the homotopy analysis method. In: advances of the homotopy analysis method. World Scientific Press
Maani Miandoab E, Nejat Pishkenari H, Yousefi-Koma A, Tajaddodianfar F (2014) Chaos prediction in MEMS-NEMS resonators. Int J Eng Sci 82:74–83. doi:10.1016/j.ijengsci.2014.05.007
Maroufi M, Bazaei A, Mohammadi A, Reza Moheimani SO (2015a) Tilted beam piezoresistive displacement sensor: design, modeling, and characterization. J Microelectromech Syst 24(5):1594–1605. doi:10.1109/JMEMS.2015.2426180
Maroufi M, Fowler AG, Reza Moheimani SO (2015b) MEMS nanopositioner for on-chip atomic force microscopy: a serial kinematic design. J Microelectromech Syst 24(6):1730–1740. doi:10.1109/JMEMS.2015.2434390
Mestrom RMC, Fey RHB, Phan KL, Nijmeijer H (2009) Experimental validation of hardening and softening resonances in a clamped-clamped beam MEMS resonator. Procedia Chem 00:4–7
Miandoab EM, Yousefi-Koma A, Pishkenari HN, Fathi M (2014) Nano-resonator frequency response based on strain gradient theory. J Phys D Appl Phys 47:365303. doi:10.1088/0022-3727/47/36/365303
Miandoab EM, Yousefi-Koma A, Pishkenari HN (2015) Nonlocal and strain gradient based model for electrostatically actuated silicon nano-beams. Microsyst Technol 21:457–464. doi:10.1007/s00542-014-2110-2
Moghimi Zand M, Ahmadian MT (2009) Application of homotopy analysis method in studying dynamic pull-in instability of microsystems. Mech Res Commun 36:851–858. doi:10.1016/j.mechrescom.2009.03.004
Mohammadi H, Mahzoon M, Mohammadi M, Mohammadi M (2014) Postbuckling instability of nonlinear nanobeam with geometric imperfection embedded in elastic foundation. Nonlinear Dyn 76:2005–2016. doi:10.1007/s11071-014-1264-x
Nayfeh AH (2004) Perturbation Methods. Wiley-VCH
Nayfeh AH, Mook DT (1995) Nonlinear oscillations. Wiley-VCH
Nayfeh AH, Younis MI, Abdel-Rahman EM (2005) Reduced-order models for MEMS applications. Nonlinear Dyn 41:211–236. doi:10.1007/s11071-005-2809-9
Nayfeh AH, Ouakad HM, Najar F et al (2010) Nonlinear dynamics of a resonant gas sensor. Nonlinear Dyn 59:607–618. doi:10.1007/s11071-009-9567-z
Nejat Pishkenari H, Afsharmanesh B, Tajaddodianfar F (2016) Continuum models calibrated with atomistic simulations for the transverse vibrations of silicon nanowires. Int J Eng Sci 100:8–24. doi:10.1016/j.ijengsci.2015.11.005
Nguyen V-N, Baguet S, Lamarque C-H, Dufour R (2014) Bifurcation-based micro-/nanoelectromechanical mass detection. Nonlinear Dyn 79:647–662. doi:10.1007/s11071-014-1692-7
Odibat ZM (2010) A study on the convergence of homotopy analysis method. Appl Math Comput 217:782–789. doi:10.1016/j.amc.2010.06.017
Ouakad HM, Younis MI (2010) The dynamic behavior of MEMS arch resonators actuated electrically. Int J Non Linear Mech 45:704–713
Ouakad HM, Younis MI (2014) On using the dynamic snap-through motion of MEMS initially curved microbeams for filtering applications. J Sound Vib 333:555–568
Pratiher B (2014) Stability and bifurcation analysis of an electrostatically controlled highly deformable microcantilever-based resonator. Nonlinear Dyn 78:1781–1800. doi:10.1007/s11071-014-1543-6
Rhoads JF, Shaw SW, Turner KL et al (2006) Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators. J Sound Vib 296:797–829. doi:10.1016/j.jsv.2006.03.009
Sahai T, Bhiladvala RB, Zehnder AT (2007) Thermomechanical transitions in doubly-clamped micro-oscillators. Int J Non Linear Mech 42:596–607. doi:10.1016/j.ijnonlinmec.2006.12.009
Seleim A, Towfighian S, Delande E et al (2011) Dynamics of a close-loop controlled MEMS resonator. Nonlinear Dyn 69:615–633. doi:10.1007/s11071-011-0292-z
Sharma M, Sarraf EH, Baskaran R, Cretu E (2012) Parametric resonance: amplification and damping in MEMS gyroscopes. Sensors Actuators A Phys 177:79–86. doi:10.1016/j.sna.2011.08.009
Tajaddodianfar F, Yazdi MH, Pishkenari HN (2014) Dynamics of bistable initially curved shallow microbeams: effects of the electrostatic fringing fields. In: 2014 IEEE/ASME international conference on advanced intelligent mechatronics (AIM2014), 8–11 July 2014, Besacon, France. doi:10.1109/AIM.2014.6878258
Tajaddodianfar F, Hairi Yazdi M, Nejat Pishkenari H (2015a) On the chaotic vibrations of electrostatically actuated arch micro/nano resonators: a parametric study. Int J Bifurc Chaos 25:1550106. doi:10.1142/S0218127415501060
Tajaddodianfar F, Hairi Yazdi MR, Nejat Pishkenari H et al (2015b) Classification of the nonlinear dynamics in an initially curved bistable micro/nano-electro-mechanical system resonator. Micro Nano Lett 10:583–588
Tajaddodianfar F, Nejat Pishkenari H, Hairi Yazdi M, Maani Miandoab E (2015c) On the dynamics of bistable micro/nano resonators: analytical solution and nonlinear behavior. Commun Nonlinear Sci Numer Simul 20:1078–1089. doi:10.1016/j.cnsns.2014.06.048
Tajaddodianfar F, Pishkenari HN, Yazdi MRH, Miandoab EM (2015d) Size-dependent bistability of an electrostatically actuated arch NEMS based on stain gradient theory. J Phys D Appl Phys 48:245503. doi:10.1088/0022-3727/48/24/245503
Tajaddodianfar F, Nejat Pishkenari H, Hairi Yazdi MR (2016) Prediction of chaos in electrostatically actuated arch micro-nano resonators: analytical approach. Commun Nonlinear Sci Numer Simul 30:182–195. doi:10.1016/j.cnsns.2015.06.013
Tavakolian F, Farrokhabadi A, Mirzaei M (2015) Pull—in instability of double clamped microbeams under dispersion forces in the presence of thermal and residual stress effects using nonlocal elasticity theory. Microsyst Technol. doi:10.1007/s00542-015-2785-z
Timurdogan E, Alaca BE, Kavakli IH, Urey H (2011) MEMS biosensor for detection of hepatitis A and C viruses in serum. Biosens Bioelectron 28:189–194. doi:10.1016/j.bios.2011.07.014
Tocchio A, Caspani A, Langfelder G (2012) Mechanical and electronic amplitude-limiting techniques in a MEMS resonant accelerometer. Sens J IEEE 12:1719–1725
Van Gorder RA, Vajravelu K (2008) Analytic and numerical solutions to the Lane—Emden equation. Phys Lett A 372:6060–6065
Wan W, Wu HY, Chen L et al (2014) Demonstration of motion transduction in a single-ion nonlinear mechanical oscillator. Phys Rev A 89:063401. doi:10.1103/PhysRevA.89.063401
Wang S, Du P, Zhou N (2013) Power system transient stability analysis through a homotopy analysis method. Nonlinear Dyn 76:1079–1086. doi:10.1007/s11071-013-1191-2
Yang JSPL, Yang J (2015) Size effect on the dynamic analysis of electrostatically actuated micro—actuators. Microsyst Technol. doi:10.1007/s00542-015-2788-9
Younesian D, Sadri M, Esmailzadeh E (2014) Primary and secondary resonance analyses of clamped–clamped micro-beams. Nonlinear Dyn 76:1867–1884. doi:10.1007/s11071-014-1254-z
Younis MI, Ouakad HM, Alsaleem FM et al (2010) Nonlinear Dynamics of MEMS Arches Under Harmonic Electrostatic Actuation. J Microelectromechanical Syst 19:647–656
Zaitsev S, Shtempluck O, Buks E, Gottlieb O (2011) Nonlinear damping in a micromechanical oscillator. Nonlinear Dyn 67:859–883. doi:10.1007/s11071-011-0031-5
Zhang Y, Wang Y, Li Z et al (2007) Snap-through and pull-in instabilities of an arch-shaped beam under an electrostatic loading. J Microelectromech Syst 16:684–693
Zhang W-M, Yan H, Peng Z-K, Meng G (2014) Electrostatic pull-in instability in MEMS/NEMS: a review. Sens Actuators A Phys 214:187–218. doi:10.1016/j.sna.2014.04.025
Zhao Y, Sun C, Wang Z, Wang L (2014) Analytical solutions for resonant response of suspended cables subjected to external excitation. Nonlinear Dyn 78:1017–1032. doi:10.1007/s11071-014-1493-z
Acknowledgments
The authors would like to gratefully thank the Iran National Science Foundation (INSF) for their financial support.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Tajaddodianfar, F., Yazdi, M.R.H. & Pishkenari, H.N. Nonlinear dynamics of MEMS/NEMS resonators: analytical solution by the homotopy analysis method. Microsyst Technol 23, 1913–1926 (2017). https://doi.org/10.1007/s00542-016-2947-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00542-016-2947-7