Email Alert    RSS

Applied Mathematics and Mechanics >

2020 , Vol. 41 >Issue 10: 1517 - 1538

DOI: https://doi.org/10.1007/s10483-020-2658-6

Articles

Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance

Expand
  • 1. Mechanical Engineering Department, Tarbiat Modares University, Tehran 14115-111, Iran;
    2. Mechanical Engineering Department, Urmia University of Technology, Urmia 57166-93187, Iran

Received date: 2020-04-09

  Revised date: 2020-07-01

  Online published: 2020-10-09

Fold

Abstract

The nonlinear interactions of a microarch resonator with 3:1 internal resonance are studied. The microarch is subjected to a combination of direct current (DC) and alternating current (AC) electric voltages. Thin piezoelectric layers are thoroughly bonded on the top and bottom surfaces of the microarch. The piezoelectric actuation is not only used to modulate the stiffness and resonance frequency of the resonator but also to provide the suitable linear frequency ratio for the activation of the internal resonance. The size effect is incorporated by using the so-called modified strain gradient theory. The system is highly nonlinear due to the co-existence of the initial curvature, the mid-plane stretching resulting from clamped anchors, and the electrostatic excitation. The eigenvalue problem is solved to conduct a frequency analysis and identify the possible regions for activating the internal resonance. The effects of the piezoelectric actuation, the electric excitation, and the small-scale effect are investigated on the internal resonance. Exclusive nonlinear phenomena such as Hopf bifurcation and hysteresis are identified in the microarch response. It is shown that by applying appropriate piezoelectric actuation, one is able to activate microarch internal resonance regardless of the initial rise level of the microarch. It is also disclosed that among all the parameters, AC electric voltage has the greatest effect on the energy exchange between the interacting modes. The results can be used to design resonators and internal resonance based micro-electro-mechanical system (MEMS) energy harvesters.

Key words： microarch resonator; internal resonance; multiple time scales method; micro-electro-mechanical system (MEMS); piezoelectric actuation; electrostatic excitation

Cite this article

A. NIKPOURIAN, M. R. GHAZAVI, S. AZIZI . Size-dependent modal interactions of a piezoelectrically laminated microarch resonator with 3:1 internal resonance[J]. Applied Mathematics and Mechanics, 2020 , 41(10) : 1517 -1538 . DOI: 10.1007/s10483-020-2658-6

References

[1] YOUNIS, M. I. MEMS Linear and Nonlinear Statics and Dynamics, Springer, New York (2011)
[2] YOUNIS, M. I., OUAKAD, H. M., ALSALEEM, F., MILES, M. R., and WEILI, C. Nonlinear dynamics of MEMS arches under harmonic electrostatic actuation. Journal of Microelectromechanical Systems, 19, 647-656(2010)
[3] ZHANG, Y., WANG, Y., LI, Z., HUANG, Y., and LI, D. Snap-through and pull-in instabilities of an arch-shaped beam under an electrostatic loading. Journal of Microelectromechanical Systems, 16, 684-693(2007)
[4] KRYLOV, S., BOJAN, R. I., DAVID, S., SHIMON, S., and HAROLD, C. The pull-in behavior of electrostatically actuated bistable microstructures. Journal of Micromechanics and Microengineering, 18, 055026(2008)
[5] DAS, K. and BATRA, R. C. Pull-in and snap-through instabilities in transient deformations of microelectromechanical systems. Journal of Micromechanics and Microengineering, 19, 035008(2009)
[6] DAS, K. and BATRA, R. C. Instabilites in Arch Shaped MEMS, Springer, Heidelberg, 147-155(2010)
[7] CHEN, X. and MEGUID, S. A. On the parameters which govern the symmetric snap-through buckling behavior of an initially curved microbeam. International Journal of Solids and Structures, 66, 77-87(2015)
[8] MEDINA, L., GILAT, R., and KRYLOV, S. Symmetry breaking in an initially curved pre-stressed micro beam loaded by a distributed electrostatic force. International Journal of Solids and Structures, 51, 2047-2061(2014)
[9] FAROKHI, H., GHAYESH, M. H., and HUSSAIN, S. Pull-in characteristics of electrically actuated MEMS arches. Mechanism and Machine Theory, 98, 133-150(2016)
[10] OUAKAD, H. M. and YOUNIS, M. I. The dynamic behavior of MEMS arch resonators actuated electrically. International Journal of Non-Linear Mechanics, 45, 704-713(2010)
[11] RUZZICONI, L., BATAINEH, A. M., YOUNIS, M. I., CUI, W., and LENCI, S. Nonlinear dynamics of an electrically actuated imperfect microbeam resonator:experimental investigation and reduced-order modeling. Journal of Micromechanics and Microengineering, 23, 075012(2013)
[12] KRYLOV, S. and DICK, N. Dynamic stability of electrostatically actuated initially curved shallow micro beams. Continuum Mechanics and Thermodynamics, 22, 445-468(2010)
[13] ABDALLAH, R., MOHAMMED, L. F. B., MD ABDULLAH AL, H., and MOHAMMAD, I. Y. Experimental investigation of snap-through motion of in-plane MEMS shallow arches under electrostatic excitation. Journal of Micromechanics and Microengineering, 26, 015012(2016)
[14] TAJADDODIANFAR, F., NEJAT PISHKENARI, H., HAIRI YAZDI, M. R., and MAANI MIANDOAB, E. On the dynamics of bistable micro/nano resonators:analytical solution and nonlinear behavior. Communications in Nonlinear Science and Numerical Simulation, 20, 1078-1089(2015)
[15] FLECK, N. A., MULLER, G. M., ASHBY, M. F., and HUTCHINSON, J. W. Strain gradient plasticity:theory and experiment. Acta Metallurgica et Materialia, 42, 475-487(1994)
[16] STÖLKEN, J. S. and EVANS, A. G. A microbend test method for measuring the plasticity length scale. Acta Materialia, 46, 5109-5115(1998)
[17] ANDREW, W. M. and JONATHAN, S. C. Role of material microstructure in plate stiffness with relevance to microcantileverg sensors. Journal of Micromechanics and Microengineering, 15, 1060-1067(2005)
[18] RAHAEIFARD, M., KAHROBAIYAN, M. H., AHMADIAN, M. T., and FIROOZBAKHSH, K. Size-dependent pull-in phenomena in nonlinear microbridges. International Journal of Mechanical Sciences, 54, 306-310(2012)
[19] RAHAEIFARD, M., AHMADIAN, M. T., and FIROOZBAKHSH, K. Vibration analysis of electrostatically actuated nonlinear microbridges based on the modified couple stress theory. Applied Mathematical Modelling, 39, 6694-6704(2015)
[20] NIKPOURIAN, A., GHAZAVI, M. R., and AZIZI, S. On the nonlinear dynamics of a piezoelectrically tuned micro-resonator based on non-classical elasticity theories. International Journal of Mechanics and Materials in Design, 14, 1-19(2016)
[21] HUU-TAI, T., THUC, P. V., TRUNG-KIEN, N., and SEUNG-EOCK, K. A review of continuum mechanics models for size-dependent analysis of beams and plates. Composite Structures, 177, 196-219(2017)
[22] SHOJAEIAN, M., TADI BENI, Y., and ATAEI, H. Size-dependent snap-through and pull-in instabilities of initially curved pre-stressed electrostatic nano-bridges. Journal of Physics D:Applied Physics, 49, 295303(2016)
[23] GHAYESH, M. H., FAROKHI, H., and ALICI, G. Size-dependent electro-elasto-mechanics of MEMS with initially curved deformable electrodes. International Journal of Mechanical Sciences, 103, 247-264(2015)
[24] GHAYESH, M. H. and FAROKHI, H. Bistable nonlinear response of MEMS resonators. Nonlinear Dynamics, 90, 1627-1645(2017)
[25] NIKPOURIAN, A., GHAZAVI, M. R., and AZIZI, S. Size-dependent secondary resonance of a piezoelectrically laminated bistable MEMS arch resonator. Composites Part B:Engineering, 173, 106850(2019)
[26] TAJADDODIANFAR, F., PISHKENARI, H. N., YAZDI, M. R. H., and MIANDOAB, E. M. Size-dependent bistability of an electrostatically actuated arch NEMS based on strain gradient theory. Journal of Physics D:Applied Physics, 48, 245503(2015)
[27] PRADIPTYA, I. and OUAKAD, H. M. Size-dependent behavior of slacked carbon nanotube actuator based on the higher-order strain gradient theory. International Journal of Mechanics and Materials in Design, 14, 393-415(2018)
[28] YOUNIS, M. I. and NAYFEH, A. H. A study of the nonlinear response of a resonant microbeam to an electric actuation. Nonlinear Dynamics, 31, 91-117(2003)
[29] POURKIAEE, S. M., KHADEM, S. E., SHAHGHOLI, M., and BAB, S. Nonlinear modal interactions and bifurcations of a piezoelectric nanoresonator with three-to-one internal resonances incorporating surface effects and van der Waals dissipation forces. Nonlinear Dynamics, 88, 1785-1816(2017)
[30] LI, L., ZHANG, Q., WANG, W., and HAN, J. Nonlinear coupled vibration of electrostatically actuated clamped-clamped microbeams under higher-order modes excitation. Nonlinear Dynamics, 90, 1593-1606(2017)
[31] FAROKHI, H. and GHAYESH, M. H. Nonlinear size-dependent dynamics of microarches with modal interactions. Journal of Vibration and Control, 22, 3679-3689(2015)
[32] GHAYESH, M. H. and FAROKHI, H. Internal energy transfer in dynamical behaviour of Timoshenko microarches. Mathematics and Computers in Simulation, 112, 28-39(2015)
[33] RAMINI, A. H., HAJJAJ, A. Z., and YOUNIS, M. I. Tunable resonators for nonlinear modal interactions. Scientific Reports, 6, 34717(2016)
[34] OUAKAD, H. M., SEDIGHI, H. M., and YOUNIS, M. I. One-to-one and three-to-one internal resonances in MEMS shallow arches. Journal of Computational and Nonlinear Dynamics, 12, 051025(2017)
[35] LAM, D. C. C., YANG, F., CHONG, A. C. M., WANG, J., and TONG, P. Experiments and theory in strain gradient elasticity. Journal of the Mechanics and Physics of Solids, 51, 1477-1508(2003)
[36] REZAZADEH, G., TAHMASEBI, A., and ZUBSTOV, M. Application of piezoelectric layers in electrostatic MEM actuators:controlling of pull-in voltage. Microsystem Technologies, 12, 1163-1170(2006)
[37] NIKPOURIAN, A., GHAZAVI, M. R., and AZIZI, S. Size-dependent nonlinear behavior of a piezoelectrically actuated capacitive bistable microstructure. International Journal of Non-Linear Mechanics, 114, 49-61(2019)
[38] NAYFEH, A. H. and EMAM, S. A. Exact solution and stability of postbuckling configurations of beams. Nonlinear Dynamics, 54, 395-408(2008)
[39] EL-BASSIOUNY, A. F. Structural modal interactions with internal resonances and external excitation. Physica Scripta, 72, 132-141(2005)
[40] NAYFEH, A. H. Introduction to Perturbation Techniques, John Wiley & Sons, New York (2011)
[41] AZIZI, S., GHAZAVI, M. R., REZAZADEH, G., AHMADIAN, I., and CETINKAYA, C. Tuning the primary resonances of a micro resonator, using piezoelectric actuation. Nonlinear Dynamics, 76, 839-852(2014)
Options
Outlines

/

〈 〉

APS Journals | CSTAM Journals | AMS Journals | EMS Journals | ASME Journals
Total visitors:
Copyright © Applied Mathematics and Mechanics (English Edition)
Address：P. O. Box 122, Shanghai University, 99 Shangda Road, Shanghai 200444, China
E-mail： amm@department.shu.edu.cn
Technical support: Beijing Magtech Co.ltd support@magtech.com.cn