對於很多微機電系統諧振器而言能量耗散機制的控制與對其物理效應的研究有著關鍵的作用，微機電系統諧振器元件可以被應用在很多方面例如加速度計、微形陀螺儀、微扭轉鏡、麥克風等。微機電諧振器的重要性能指標為其共振頻率以及品質因子，高品質因子的諧振器具有高靈敏度、高解析度以及高度穩定等優點，故追求諧振器的高品質因子是一項重要的研究課題，微機電系統的擠壓氣膜阻尼是其主要的阻尼來源，在常壓環境下氣體被困在微型樑諧振器基材間的超薄間隙中，當微型樑諧振器振動時會推擠氣體進而造成振動能量損失產生阻尼，此阻尼是影響諧振器品質因子的主要原因之一，為了減輕此一阻尼必須使環境氣壓下降使間隙中的氣體成為稀薄氣體，且將代表間隙中固體表面與稀薄氣體分子的相互交互作用行為的表面適應係數納入考量，另一方面由橫向等向性的材料構成懸臂樑結構在其振動時產生的的熱彈性阻尼也必須納入考慮，因為橫向等向性不像等向性材料高度對稱，所以在震盪運動時的熱彈性阻尼行為會不同，故需另外調查，在微機電諧振器的工作環境中，溫度的角色十分重要，在擠壓氣膜阻尼中環境溫度會直接影響稀薄氣體的行為，在熱彈性阻尼中環境溫度更直接與品質因子掛勾所以溫度的變化也是另一探討的重點。
從結果中可以觀察出隨溫度升高，稀薄氣體的擠壓氣膜阻尼會逐漸喪失主導地位，其在總阻尼中的貢獻會被熱彈性阻尼所取代，橫向等向性的材料參數會對熱彈性阻尼產生明顯的影響，另外稀薄氣體的表面調節係數會對擠壓氣膜阻尼產生重大的影響。

In this research we construct a model via numerical method to analyze the dynamic properties of MEMS resonator structure, high quality factor is a highly demanded performance in many high resolution high sensitivity and high stability MEMS components. At normal pressure the gas trapped in thin gap between cantilever resonator and substrate, trapped gas in the gap will be squeezed when resonator vibrate lead to the squeeze film damping, this source of damping is one major source of reduce quality factor, the gas pressure have to decrease in order to decrease this source of damping the gas pressure. Mean while in low pressure rarefied gas, the surface accommodation factor must take in to account. On the other hand the thermal-elastic damping of transversely isotropic material is another source of damping need to consider because the stress strain behavior of transversely isotropic material is different form normal isotropy material so further intestate is needed. Temperature is a important parameter to study because temperature will heavily influence the gas behavior and the material of resonator itself.

[1] C. Zener (1937), “Internal Friction in Solids. I. Theory of Internal Friction in Reeds, ” Phys.Rev. 52 pp.230-235.
[2] R.Lifshitz, M.L.Roukes (2000), “Thermoelastic damping in micro- and nanomechanical systems, ” Phys. Rev. B 61(8), pp.5600-5609.
[3] H.W.Lord, Y.Shulman (1967), “ A generalized dynamical theory of thermoelasticity, ” Journal of the Mechanics and Physics of Solids Volume 15, (Issue 5), pp.299-309.
[4] F.L.Guo (2013), “Thermo-Elastic Dissipation of Microbeam Resonators in the Framework of Generalized Thermo-Elasticity Theory,” Journal of Thermal Stresses, Volume 36 (Issue 11) pp.1156-1168
[5] A.E.Green, K.A.Lindsay (1967), “Thermoelasticity, ” Journal of Elasticity Volume 2, (Issue 1), pp.1-7
[6] I.A.Abbas (2016), “The Effect of Relaxation Times on Thermoelastic Damping in a Nanobeam Resonator,” Journal of Molecular and Engineering Materials,Volume. 04, (No. 02)
[7] F. L. Guo, W. J. Jiao, G. Q. Wang & Z. Q. Chen (2016), “Distinctive features of thermoelastic dissipation in microbeam resonators at nanoscale,” Journal of Thermal Stresses,Volume39,(Issue 3)pp.360-369
[8] J.E.Bishop,V.K.Kinra (1997), “Elastothermodynamic damping in laminated composites”International Journal of Solids and Structures Volume 34, (Issue 9), pp.1075-1092
[9] S.Vengallatore (2005), “Analysis of thermoelastic damping in laminated composite micromechanical beam resonators ,”Journal of Micromechanics and Microengineering, Volume 15
[10] S.Prabhakar and S.Vengallatore (2007), “Thermoelastic damping in bilayered micromechanical beam resonators," Journal of Micromechanics and Microengineering, Volume 17,(Number 3)
[11] J.N.Sharma (2011), “Thermoelastic Damping and Frequency Shift in Micro/Nanoscale Anisotropic Beams,” Journal of Thermal Stresses Volume 34(Issue 7)
[12] D. Grover, J.N. Sharma (2012), “Transverse vibrations in piezothermoelastic beam resonators,”Journal of Intelligent Material Systems and Structures Vol 23, (Issue 1)
[13] A.Bokaian (1990), “Natural frequencies of beams under tensile axial loads,” Journal of Sound and Vibration Volume142(Issue 3)pp.481-498
[14] S.Kumar, M.A.Haque (2008), “Reduction of thermo-elastic damping with a secondary elastic field,” Journal of Sound and Vibration Volume 318, (Issue 3) pp.423-427.
[15] Sandeep Kumar, M. Aman Haque (2010), “Stress-dependent thermal relaxation effects in micro-mechanical resonators,”Acta Mechanica, Volume 212, (Issue 1–2) pp.83–91.
[16] S.B.Kim, J.H.Kim (2011), “Quality factors for the nano-mechanical tubes with thermoelastic damping and initial stress,” Journal of Sound and Vibration Volume 330, (Issue 7, 28) pp.1393-1402.
[17] S.Kumar, T.Alam and A.Haque (2013), “Thermo-mechanical coupling and size effects in micro and nano resonators,” Micro and Nano Systems Letters 1:2
[18] C. Catteneo (1958), “A form of heat conduction equation which eliminates the paradox of instantaneous propagation,” Compte Rendus, Volume. 247 pp.431–433.
[19] Vernotte, P. (1958), “Les paradoxes de la théorie continue de l'équation de la chaleur,” C. R. Acad. Sci., 246, pp.3154.
[20] M. Chester (1963),“Second Sound in Solids,”  Physical Review, Volume 131(5)
[21] H.Y. Zhou, P. Li (2017), “Thermoelastic Damping in Micro- and Nanobeam Resonators With Non-Fourier Heat Conduction,” IEEE Sensors Journal, Volume 17 (Issue: 21) pp.6966 – 6977.
[22] Z.Y. Zhong, J.P. Zhou,and H.L. Zhang(2017), “Thermoelastic Damping in Functionally Graded Microbeam Resonators,” IEEE Sensors Journal ,Volume: 17 (Issue: 11) pp.3381 – 3390.
[23] T.V. Roszhart (1990), “The effect of thermoelastic internal friction on the Q of micromachined silicon resonators” IEEE 4th Technical Digest on Solid-State Sensor and Actuator Workshop
[24] H.Najar1, A.Heidari, M.L.Chan, H.A.Yang, L.W.Lin, D.G.Cahill, and D. A.Horsley (2013), “Microcrystalline diamond micromechanical resonators with quality factor limited by thermoelastic damping”Appl. Phys. Lett. 102, 071901
[25] H.Najar1, M.L.Chan, H.A.Y, L.W.Lin, D.G.Cahill, and D.A.Horsley (2014), “High quality factor nanocrystalline diamond micromechanical resonators limited by thermoelastic damping” Appl. Phys. Lett. 104, 151903
[26] Y.X.Sun, D.I.Fang, A.K.Soh (2006), “Thermoelastic damping in micro-beam resonators” International Journal of Solids and Structures Volume 43, Issue 10, 3213-3229
[27] Z. F. Khisaeva, M.Ostoja-Starzewski (2006) , “Thermoelastic Damping in Nanomechanical Resonators with Finite Wave Speeds” Journal of Thermal Stresses Volume 29, Issue 3
[28] Shaker, F.J.(1975), “Efect of Axial load on Mode Shapes and Frequencies of Beams”, vol. NASA TN D-8109, NASA, Ed.
[29] Harris, C.M., Piersol, A.G. (2002), Harris’ Shock and Vibration Handbook, 5th edn. New York McGraw-Hill,
[30] L. LI and C. T. HAO (2011), “Constraints on Anisotropic Parameters in Transversely Isotropic Media and the Extensions to Orthorhombic Media, ” Chinese Journal of Geophysics, vol. 54, pp.798-809
[31] Verbridge, S.S., Shapiro, D.F., Craighead, H.G., Parpia, J.M. (2007):, “Macroscopic tuning of nanomechanics: substrate bending for reversible control of frequency and quality factor of nanostring resonators” Nano Lett. 7, pp.1728–1735
[32] Scott, S.V., Jeevak, M.P., Robert, B.R., Leon, M.B., Craighead, H.G. (2006), “High quality factor resonance at room temperature with nanostrings under high tensile stress”. J. Appl. Phys. 99, 124304
[33] G. Sosale, K.Das, Luc Frechette and S.Vengallatore (2011), “Controlling damping and quality factors of silicon microcantilevers by selective metallization” J. Micromech. Microeng. 21 105010 (7pp)
[34] R Sandberg, K Mølhave, A Boisen and W Svendsen (2005), “Effect of gold coating on the Q-factor of a resonant cantilever” J. Micromech. Microeng. 15 pp.2249–2253
[35] G. Sosale, S. Prabhakar, L.G. Frechette,S. Vengallatore (2011), “A Microcantilever Platform for Measuring Internal Friction in Thin Films Using Thermoelastic Damping for Calibration” Journal of Microelectromechanical Systems Volume: 20, Issue: 3,
[36] M.J. Seitner, K.Gajo, and E.M. Weig (2014), “Damping of metallized bilayer nanomechanical resonators at roomtemperature” Applied Physics Letters 105(21)
[37] H.H.Wu, L.W.Sang, Y.M.Li, T.Teraji, T.F.Li, M.Imura, J.Q.You, Y. Koide, M.Toda, and M.Y Liao (2018), “Reducing intrinsic energy dissipation in diamond-on-diamond mechanical resonators toward one million quality factor ” Physical review materials 2, 090601(R)
[38] A.Duwel, J.Gorman, M.Weinstein, J.Borenstein, and P.Ward (2003), “Experimental study of thermoelastic damping in MEMS gyros” Sensors and Actuators A Physical 103(1): pp.70-75 ·
[39] S. Evoy, A.Olkhovets, L.Sekaric, J. M.Parpia, H.G.Craighead, and D. W.Carr (2000), “Temperature-dependent internal friction in silicon nanoelectromechanical systems” Applied Physics Letters 77(15) pp.2397 - 2399
[40] V. T. Srikar and S.D.Senturia (2002), “Thermoelastic Damping in Fine-Grained Polysilicon Flexural Beam Resonators” Journal of Microelectromechanical Systems 11(5) pp.499 - 504 ·
[41] Zhili Hao (2008), “Thermoelastic damping in the contour-mode vibrations of micro- and nano-electromechanical circular thin-plate resonators” Journal of Sound and Vibration Volume 313, Issues 1–2, 3, pp.77-96
[42] F.Ashida, T.R.Tauchert (2003), “Thermally-induced wave propagation in a piezoelectric plate” Acta Mechanica, Volume 161, Issue 1–2, 1–16
[43] H.G. Craighead (2000), “Nanoelectromechanical systems” ,Science 2901532
[44] B. Ilic, Y. Yang H.G. Craighead (2004), “Virus detection using nanoelectromechanical devices” Apply Physic Letters ,85 2604-2606
[45] H. Campanella (2010), “Acoustic Wave and Electromechanical Resonators”, Norwood,MA ,Artech House.
[46] J.H.Ginsberg (2001), “Mechanical and Structral Vibrations” New York: John Willy& Sons Inc,
[47] J.W. Lee (2011), “Analysis of fluid-structure interaction for predicting resonant frequencies and quality factors of a microcantilever on a squeeze-film” J. of Mech. Sci. and Technol., Volume 25, Issue 12, pp.3005–3013
[48] M.Li, H.Tang, M.L.Roukes (2007), “Ultra-sensitive NEMS-based cantilevers for sensing, scanned probe and very high-frequency applications” Nature Nanotechnology 2(2) pp.114-20 
[49] H.Hosaka, K.Itao, S.Kuroda (1995), “Damping characteristics of beam-shaped micro-oscillators” Sensors and Actuators A Physical 49(1) 87-95
[50] W. E. Newell (1968) , “Miniaturization of Tuning Forks” Science 27 Vol. 161, Issue 3848, pp.1320-1326
[51] R.G. Christian (1966), “The theory of oscillating-vane vacuum gauges” Vacuum 16(4):175-178
[52] J.D.Zook, D.W.Burns, H.Guckel, J.J.Sniegowski, R.L.Engelstad, Z.Feng (1992), “Characteristics of polysilicon resonant microbeams” Sensors and Actuators A: Physical Volume 35, Issue 1, pp.51-59
[53] R.Legtenberg, H.A.C.Tilmans (1994), “Electrostatically driven vacuum-encapsulated polysilicon resonators PartI.Designand fabrication” Sensors and Actuators A: Physical Vol 45, Issue 1, pp.57-66
[54] C.Q.Gui, R.Legtenberg, M. C. Elwenspoek, Jan H. Fluitman (1999), “Q-factor dependence of one-port encapsulated polysilicon resonator on reactive sealing pressure” Journal of Micromechanics and Microengineering 5(2):183 · 
[55] M.H.Bao, H.Yang(2002), “Energy transfer model for squeeze-film air damping in low vacuum Energy transfer model for squeeze-film air damping in low vacuum” Journal of Micromechanics and Microengineering 12(12):pp.341-341
[56] R.B.Darling, C.Hivick, J.Y.Xu(1998), “Compact analytical modeling of squeeze film damping with arbitrary venting conditions using a Green's function approach”, Sensors and Actuators A: Physical Volume 70, Issues 1–2, 1, pp.32-41
[57] E.S.Hung, S.D.Senturia(1999), “Generating efficient dynamical models for microelectromechanical systems from a few finite-element simulation runs”, Journal of Microelectromechanical Systems 8(3):pp.280 - 289 ·
[58] A.H.Nayfeh, M.I.Younis (2003), “A new approach to the modeling and simulation of flexible microstructures under the effect of squeeze-film damping” Journal of Micromechanics and Microengineering, Vol 14, No 2
[59] C.Zhang, G.Xu, Q.Jiang (2004), “Characterization of the squeeze film damping effect on the quality factor of a microbeam resonator” Journal of Micromechanics and Microengineering, Vol 14, No 10,
[60] S.K.De, N.R.Aluru (2006), “Coupling of hierarchical fluid models with electrostatic and mechanical models for the dynamic analysis of MEMS” Journal of Micromechanics and Microengineering 16(8):pp.1705
[61] A.Pandey, R.Pratap (2007), “Effect of flexural modes on squeeze film damping in MEMS cantilever resonators” Journal of Micromechanics and Microengineering 17(12):pp.2475
[62] C.C.Nguyen, W.L.Li (2016a), “Effect of gas rarefaction on the quality factors of micro-beam resonators”,  Microsystem Technologies 23(8)
[63] C.C.Nguyen, W.L.Li (2016b), “Effects of surface roughness and gas rarefaction on the quality factor of micro-beam resonators”, Microsystem Technologies 23(8)
[64] W.L.Li (1999), “Analytical modelling of ultra-thin gas squeeze film”, Nanotechnology 10(4):440
[65] B.S.Kim, M.A.Hopcroft, R.Candler, C.M.Jha, M.Agarwal, R.Melamud, S.Chandorkar, G.Yama, T.W.Kenny (2008), “Temperature Dependence of Quality Factor in MEMS Resonators” Journal of Microelectromech Systems 17(3):pp.755 - 766
[66] S.Ghaffari, E.J.Ng, C.H.Ahn, Y.Yang, S.Wang V.A. Hong, T.W.Kenny (2015), “Accurate Modeling of Quality Factor Behavior of Complex Silicon MEMS Resonators” Journal of Microelectromech Systems 24(2):pp.276-288  
[67] H.A.C.Tilmans and R.Legtenberg (1994b), “Electrostatically driven vacuum encapsulated polysilicon resonators, Part II: Theory and performance”, Sensors and Actuators A, vol. 45, pp.67-84,
[68] S.J.Kim, R.Dean, R.Jackson, G.T.Flowers (2011) “An Investigation of the Damping Effects of Various Gas Environments on a Vibratory MEMS Device” Tribology International 44(2):pp.125-133
[69] B.T.Porodnov, P.E.Suetin, S.F.Borisov, V.D.Akinshin (1974), “Experimental investigation of rarefied gas flow in different channels” Journal of Fluid Mechanics 64(03):pp.417 - 438
[70] E.B.Arkilic (1997), “Measurement of the Mass Flow and Tangential Momentum Accommodation Coeffcient in Silicon Micromachined Channels”M.I.T Cambridge U.K
[71] B.J.Hamrock (1994), “Fundamentals of Fluid Film Lubrication” New York ,McGraw-Hill.
[72] A,Burgdorfer (1958), “The Influence of the Molecular Mean Free Path on the Performance of Hydrodynamic Gas Lubricated Bearings” Journal of Basic Engineering 81
[73] Y.T.Hsia, G.A.Domoto (1983), “An Experimental Investigation Of Molecular Rarefaction Effects In Gas Lubricated Bearings At Ultralow Clearances” Journal of Lubrication Tech 105(1), pp.120-129
[74] Y.Mitsuya (1993), “Modified Reynolds Equation for Ultra-Thin Film Gas Lubrication Using 1.5Order Slip-Flow Model and Considering Surface Accommodation Coefficient” Journal. Tribology 115(2), pp.289-294
[75] S.Fukui, R. Kaneko (1988), “Analysis of Ultra-Thin Gas Film Lubrication Based on Linearized Boltzmann Equation: First Report—Derivation of a Generalized Lubrication Equation Including Thermal Creep Flow”, Journal of Tribology 110(2):253
[76] S.Fukui, R. Kaneko (1990), “A Database for Interpolation of Poiseuille Flow Rates for High Knudsen Number Lubrication Problems” Journal of Tribology 112(1)
[77] T.Veijola, H.Kuisma, J.Lahdenperä, T.Ryhänen(1995), “Equivalent-circuit model of the squeezed gas film in a silicon accelerometer” Sensors and Actuators A Physical 48(3):239-248
[78] G.X.Li, H.G.Hughes (2000), “Review of viscous damping in micromachined structures”  Proceedings of SPIE - The International Society for Optical Engineering 4176
[79] C.C.Hwang, et.al (1996), “A new modified Reynolds equation for ultrathin film gas lubrication” IEEE Transactions on Magnetics 32(2):pp.344 – 347
[80] S.C Kang (1997), “A kinetic theory description for molecular lubrication” Carnegie Mellon University,
[81] W.L.Li (2002), “A Database for Couette Flow Rate Considering the Effects of Non-Symmetric Molecular Interactions” Journal of Tribology 124(4)
[82] W.L.Li (2003), “A database for interpolation of poiseuille flow rate for arbitrary Knudsen number lubrication problems” Journal of the Chinese Institute of Engineers 26(4)  
[83] W.L.Li (2004), “Modeling of Head/Disk Interface—An Average Flow Model”  Tribology Letters 17(3):pp.669-676
[84] W.L.Li (2008), “Squeeze film effects on dynamic performance of MEMS μ-mirrors-consideration of gas rarefaction and surface roughness” Microsystem Technologies 14(3):pp.315-324
[85] O.Reynolds (1886), “On The Therory of Lubrication and Its Application to Mr. Beauchamp Tower’s Experiments, Inculding anExperimental Determination of the Viscosity of Olive Oil”, Philos.Trans.R.Soc. London,177 pp.157-234
[86] B.Tower (1883), “Fires Report on Friction Exoeriments (Friction of Lubricaed
Bearing), Proc.Inst.Mech. Eng.(London), pp.632-659”
[87] C.L.M.H.Navier (1823), “Memorie sur les lois du movement des fluidse”, Mem.Acad.Sci.Inst.Fr.6 pp.389-416
[88] G.G.Stokes (1845), “On the Theories of the Internel Friction in Motion, and of the Equilibrium and MOTION of Elastic Solids” ,Trans.Cambridge Philos.Soc.,8 pp.287-341