^{1}Department of Marine Engineering, Khorramshahr University of Marine Science &amp; Technology

^{2}Department of Mechanical Engineering, Shahrood University of Technology, Shahrood

Abstract

This paper presents a nonlinear model of a clamped-clamped microbeam actuated by an electrostatic load with stretching and thermoelastic effects. The frequency of free vibration is calculated by discretization based on the Differential Quadrature (DQ) Method. The frequency is a complex value due to the thermoelastic effect that dissipates energy. By separating the real and imaginary parts of frequency, the quality factor of thermoelastic damping is calculated. Both the stretching and thermoelastic effects are validated by the referenced papers. This paper shows that the main nonlinearity of this model is voltage, which makes the difference between linear and nonlinear models. The variation of thermoelastic damping (TED) versus geometrical parameters, such as thickness, gap distance and length, is investigated and these results are compared by linear and nonlinear models in high voltages. This paper also shows that in high voltages the linear model has a large margin of error for calculating thermoelastic damping (TED) and thus the nonlinear model should be used.

[1] Ali H. Nayfeh, Mohammad I. Younis, "Modeling and simulations of thermoelastic damping in microplates", Journal of Micromechanics and Microengineering, 14 pp 1711–1717, 2004.

[2] Nayfeh A H and Younis M I., "A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping", Journal of Micromechanics and Microengineering, 14, pp 170–181, 2004.

[3] C. Zener, "Internal friction in solids I. Theory of internal friction in reeds", Physical Review, Volume 32, pp 230-235, 1937.

[4] C. Zener, "Internal friction in solids II. General theory of thermoelastic internal friction", Physical Review, Volume 53, pp 90-99, 1937.

[5] J. B. Alblas, "A note on the theory of thermoelastic damping", Journal of Thermal Stresses, Volume 4, Issue 3-4, pp 333-355, 1981.

[6] R. Lifshitz, M. L. Roukes, "Thermoelastic damping in micro- and nanomechanical systems", Physical Review B, Volume 61, Number 8, pp 5600-5609, 2000.

[7] Sudipto K. De, N. R. Aluru, "Theory of thermoelastic damping in electrostatically actuated microstructures", Physical Review B, 74, 144305, pp 1-13, 2006.

[8] S. Prabhakar, S. Vengallatore, "Theory of thermoelastic damping in micromechanical resonators with two-dimensional heat conduction", Journal of Microelectromechanical Systems, Vol. 17, No. 2, pp 494-502, 2008.

[9] Enrico Serra, and Michele Bonaldi, "A finite element formulation for thermoelastic damping analysis", International Journal for Numerical Methods in Engineering, 78, pp 671–691, 2009.

[10] F.L. Guo, G.A. Rogerson, "Thermoelastic coupling effect on a micro-machined beam resonator", Mechanics Research Communications, 30, pp 513–518, 2003.

[11] Yuxin Sun and Masumi Saka, "Thermoelastic damping in micro-scale circular plate resonators", Journal of Sound and Vibration 329, pp 328–337, 2009.

[12] Jinbok Choi, Maenghyo Cho, Jaewook Rhim, "Efficient prediction of the quality factors of micromechanical resonators", Journal of Sound and Vibration, 329, pp 84–95, 2010.

[13] Yun-Bo Yi, Mohammad A. Matin, "Eigenvalue Solution of Thermoelastic Damping in Beam Resonators Using a Finite Element Analysis", Journal of Vibration and Acoustics, Vol. 129, pp 478-483, 2007.

[14] Fargas Marqu`es A, Costa Castell´o R and Shkel A M, "Modelling the electrostatic actuation of MEMS: state of the art" Technical Report, pp 1-33, 2005.

[15] R. C. Batra, M. Porfiri, and D. Spinello, "Review of modeling electrostatically actuated microelectromechanical systems", Smart Materials and Structures, 16, pp 23–31, 2007.

[16] Abdel-Rahman E. M., Younis M. I. and Nayfeh A. H., "Characterization of the mechanical behavior of an electrically actuated microbeam", Journal of Micromechanics and Microengineering, 12, pp 759–66, 2002.

[17] Nayfeh A. H. and Younis M. I., "Dynamics of MEMS resonators under superharmonic and subharmonic excitations", Journal of Micromechanics and Microengineering, 15, pp 1840–7, 2005.

[18] Younis M. I. and Nayfeh A. H., "A study of the nonlinear response of a resonant microbeam to an electric actuation", Nonlinear Dynamics, 31, pp 91–117, 2003.

[19] Younis M. I., Abdel-Rahman E. M. and Nayfeh A. H., "A reduced-order model for electrically actuated microbeam-based MEMS", Journal of Microelectromech. System, 12, pp 672–80, 2003.

[20] Abdel-Rahman E. M. and Nayfeh A. H. "Secondary resonances of electrically actuated resonant microsensors", Journal of Micromechanics and Microengineering. 13, pp 491–501, 2003.

[21] Nayfeh A. H. and Younis M. I., "Dynamics of MEMS resonators under superharmonic and subharmonic excitations", Journal of Micromechanics and Microengineering, 15, pp 1840–7, 2005.

[22] Najar F., Choura S., Abdel-Rahman E. M., El-Borgi S. and Nayfeh A. H., "Dynamic analysis of variable-geometry electrostatic microactuators", Journal of Micromechanics and Microengineering, 14, pp 900–6, 2006.

[23] Zhao X., Abdel-Rahman E. M. and Nayfeh A. H., "A reduced-order model for electrically actuated microplates", Journal of Micromechanics and Microengineering, 14, pp 900–906, 2004.

[24] Vogl G. W. and Nayfeh A. H., "A reduced-order model for electrically actuated clamped circular plates", Journal of Micromechanics and Microengineering, 15, pp 684–90, 2005.

[25] R.E. Bellman, J. Casti, "Differential quadrature and long term integration", Journal of Mathematical Analysis and Applications, Volume 34, 235–238, 1971.

[26] Feng Y., Bert CW., “Application of the quadrature method to flexural vibration analysis of a geometrically nonlinear beam”, Nonlinear Dynamics, Volume 156, pp 3-18, 1993.

[27] Guo Q. Zhong H., “Nonlinear vibration analysis of beams by a spline-based differential quadrature method”, Journal of Shock and Vibration, Volume 269, pp 413-420, 2004.

[28] Zhong H., Guo Q., “Nonlinear vibration analysis of Timoshenko beams using the differential quadrature method ”, Nonlinear Dynamics, Volume 32, pp 223-234.

[29] Han K. M., Xiao J. B., Du Z. M., “Differential quadrature method for Mindlin plates on Winkler foundations”, International Journal of Mechanical Sciences, Volume 38, pp 405-421, 1996.

[30] Liew K. M., Han J. B., Xiao Z. M., “Differential quadrature method for thick symmetric cross-ply laminates with first-order shear flexibility”, International Journal of Solids and Structures, Volume 33, pp 2647-2658, 1996.

[31] Liew K. M., Han J. B., “A four-node differential quadrature method for straight-sided quadrilateral Reissner/Mindlin plates”, Communications in Numerical Methods in Engineering, Volume 13, pp 73-81, 1997.

[32] Han J. B., Liew K. M., “An eight-node curvilinear differential quadrature formulation for Reissner/Mindlin plates”, Computer Methods in Applied Mechanics and Engineering, Volume 141, pp 265-280, 1997.

[36] P. Malekzadeh, A. R. Vosoughi, “DQM large amplitude vibration of composite beams on nonlinear elastic foundations with restrained edges”, Communications in Nonlinear Science and Numerical Simulation, Volume 14, pp. 906–915, 2009.

[37] Nayfeh A. H., Frank P. P., linear and nonlinear structural mechanics, New Jersey, John Wiley & Sons, pp 215-225, 2004.

[38] B. S. Sarma and T. K. Varadan, “Lagrange-Type Formulation for Finite Element Analysis of Non-Linear Beam Vibrations”, Journal of sound and vibration, Volume 86, pp. 61-70, 1983.

[39] A. Koochi, Hamid M. Sedighi, M. Abadyan, "Modeling the size dependent pull-in instability of beam-type NEMS using strain gradient theory" Latin American Journal of Solids and Structures, Volume 11, pp. 1806-1829, 2014.

[40] A. Koochi, H. Hosseini-Toudeshky, H. R. Ovesy, "modeling the influence of surface effect on instability of nano-cantilever in presence of van der waals force" Volume 13, No. 4, 1250072, 2013.