Thermo-elastic Damping in a Capacitive Micro-beam Resonator Considering Hyperbolic Heat Conduction Model and Modified Couple Stress Theory
الموضوعات :M Najafi 1 , G Rezazadeh 2 , R Shabani 3
1 - Mechanical Engineering Department, Urmia University
2 - Mechanical Engineering Department, Urmia University
3 - Mechanical Engineering Department, Urmia University
الکلمات المفتاحية: Length-scale parameter, Modified couple stress theory, Thermo-elastic damping, Electrostatic force,
ملخص المقالة :
In this paper, the quality factor of thermo-elastic damping in an electro-statically deflected micro-beam resonator has been investigated. The thermo-elastic coupled equations for the deflected micro-beam have been derived using variational and Hamilton principles based on modified couple stress theory and hyperbolic heat conduction model. The thermo-elastic damping has been obtained discretizing the governing equations over spatial domain and applying complex frequency approach. The effects of the applied bias DC voltage, playing simultaneously role of an external force and softening parameter, on the quality factor have been studied. The obtained results of the modified couple stress and classic theories are compared and the effects of the material internal length-scale parameter on the differences between results of two theories have been discussed. In addition, the effects of different parameters such as beam length and ambient temperature on the quality factor have been studied.
[1] Tilmans H.A., Legtenberg R., 1994, Electrostatically driven vacuum-encapsulated polysilicon resonators, Part II, Theory and performance, Sensors and Actuators A 45: 67-84.
[2] Rezazadeh G., Khatami F., Tahmasebi A., 2007, Investigation of the torsion and bending effects on static stability of electrostatic torsional micromirrors, Microsystem Technologies 13: 715-722.
[3] Ruhan M., Shen J., Wheeler C.B., 2001, latching micro electromagnetic relays, Sensors and Actuators A 91: 346-350.
[4] Chen J.Y., Hsu Y.C., Lee S.S., Mukherjee T., Fedder G.K., 2008, Modeling and simulation of a condenser microphone, Sensors and Actuators A 145–146: 224-230.
[5] Liu J., Martinn D.T., Kardirvel K., Nishida T., Cattafesta L., Sheplak M., Mann B., 2008, Nonlinear model and system identification of a capacitive dual-backplate MEMS microphone, Journal of Sound and Vibration 309: 276-292.
[6] Saif M.T.A., Alaca B.E., Sehitoglu H., 1999, Analytical modeling of electrostatic membrane actuator micro pumps, Journal of Microelectromechanical Systems 8: 335–345.
[7] Zener C., 1937, Internal friction in solids. I. Theory of internal friction in reeds, Physical Review 52 (3): 230-235.
[8] Zener C., 1938, Internal friction in solids. II. General theory of thermo-elastic internal friction, Physical Review 53: 90–99.
[9] Rozshart R.V., 1990, The effect of thermo-elastic internal friction on the Q of the micromachined silicon resonators, IEEE Solid State Sensor and Actuator Workshop, Hilton-Head Island, SC, 13–16.
[10] Landau L.D., Lifshitz E.M., 1959, Theory of Elasticity, Pergamon Press, Oxford.
[11] Evoy S., Oikhovets A., Sekaric L., Parpia J.M., Craighead H.G., Carr D.W., 2000, Temperature-dependent internal friction in silicon nano-electromechanical systems, Applied Physics 77(15): 2397-2399.
[12] Duwel A., Gorman J., Weinstein M., Borenstein J., Warp P., 2003, Experimental study of thermo-elastic damping in MEMS 350 gyros, Sensors and Actuators A 103: 70-75.
[13] Lifshitz R., Roukes M.L., 2000, Thermo-elastic damping in micro and nano mechanical systems, Physical Review B 61: 5600-5609.
[14] Guo F.L., Rogerson G.A., 2003, Thermo-elastic coupling effect on a micro-machined beam resonator, Mechanics Research Communications 30: 513-518.
[15] Sun Y., Fang D., Soh A.K., 2006, Thermo-elastic damping in micro-beam resonators, International Journal of Solids and Structures 43: 3213-3229.
[16] Nayfeh H., Younis M.I, 2004, Modeling and simulations of thermo-elastic damping in microplates, Journal of Micromechanics Microengineering 14: 1711-1717.
[17] Sun Y., Saka M., 2008, Vibrations of microscale circular plates induced by ultra-fast lasers, International Journal of Mechanical Sciences 50: 1365-1371.
[18] Sun Y., Tohmyoh H., 2009, Thermo-elastic damping of the axisymmetric vibration of circular plate resonators. Journal of Sound and Vibration 319: 392-405.
[19] Sun Y., Saka M., 2010, Thermo-elastic damping in micro-scale circular plate resonators, Journal of Sound and Vibration 329: 328-337.
[20] Sharma J.N., Sharma R., 2011, Damping in micro-scale generalized thermo-elastic circular plate resonators, Ultrasonics 51(3): 352-358.
[21] Rezazadeh G., Vahdat A.S, Pesteii S.-M., Farzi B., 2009, Study of thermo-elastic damping in capacitive micro-beam resonators using hyperbolic heat conduction model, Sensors and Transducers Journal 108(9): 54-72.
[22] Vahdat A.S., Rezazadeh G., 2011, Effects of axial and residual stresses on thermo-elastic damping in capacitive micro-beam resonator, Journal of the Franklin Institute 348: 622-639.
[23] Shengli K., Shenjie Zh., Zhifeng N., Kai W., 2009, Static and dynamic analysis of micro-beams based on strain gradient elasticity theory, International Journal of Engineering Science 47:487-498.
[24] Wang B., Zhao J., Zhou S., 2010, A microscale timoshenko beam model based on strain gradient elasticity theory, European Journal of Mechanics-A/Solids 29: 591-599.
[25] Mindlin R.D., Tiersten H.F., 1962, Effects of couple-stresses in linear elasticity, Archive for Rational Mechanics and Analysis 11: 415–448.
[26] Mindlin R.D., 1963, Influence of couple-stresses on stress-concentrations, Experimental Mechanics3: 1-7.
[27] Toupin R.A., 1962, Elastic materials with couple-stresses, Archive for Rational Mechanics and Analysis 11(1): 385-414.
[28] Yang F., Chong A.C.M., Lam D.C.C., Tong P., 2002, Couple stress based strain gradient theory of elasticity, International Journal of Solidsand Structures39: 2731-2743.
[29] Rezazadeh G., Vahdat A.S., Tayefeh-rezaei S., CetinkayaCe., 2012, Thermo-elastic damping in a micro-beam resonator using modified couple stress theory, Acta Mechanica 223(6): 1137-1152.
[30] Cao Y., Nankivil D.D., Allameh S., Soboyejo W., 2007, Mechanical properties of Au films on silicon substrates, Materials and Manufacturing processes 22: 187-194.
[31] Shrotriya P., Allameh S.M., Lou J., Buchheit T., Soboyejo W.O., 2003, On the measurement of the plasticity length-scale parameter in LIGA nickel foils, Mechanics of Materials 35: 233-243.
[32] Rezazadeh G., Tahmasebi A., Zubstov M., 2006, Application of piezoelectric layers in electrostatic MEM actuators: controlling of pull-in voltage, Microsystem Technologies 12: 1163-1170.
[33] Sad M.H, 2009, Elasticity Theory Application and Numerics, Elsevier Inc.
[34] Kong S., Zhou S., Nie Z., Wang K., 2008, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science 46: 427-37.
[35] Park S.K., Gao X.-L., 2006, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics Microengineering 16: 2355-2359.
[36] Khisaeva Z.F., Ostoja-starzewski M., 2006, Thermo-elastic damping in nano mechanical resonators with finite wave speeds, Journal of Thermal stresses 29(3): 201-216.
[37] Osterberg P.M., Senturia S.D., 1997, A test chip for MEMS material property measurement using electrostatically actuated tests tructures, Journal of Microelectromechanical Systems 6: 107-188.
