Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 2, pp. 317-329, Warsaw 2015 DOI: 10.15632/jtam-pl.53.2.317 EFFECT OF HIGH ELECTROSTATIC ACTUATION ON THERMOELASTIC DAMPING IN THIN RECTANGULAR MICROPLATE RESONATORS Ardeshir Karami Mohammadi, Nassim Ale Ali Department of Mechanical Engineering, Shahrood University of Technology, Iran e-mail: akaramim@yahoo.com In this paper, a micro resonator is modeled as a thin rectangular microplate with ther- moelastic damping that is actuated electrostatically. Large static deformation due to high polarization voltage is considered, and vibration of microplate occurs around the static de- flection. Due to the effect of thermoelastic damping, the frequency of vibration is a complex value that is used to determine the quality factor of thermoelastic damping.Also, the pull-in voltage is considered because nonlinear properties aremore appearedwhen approaching the polarization voltage to the pull-in voltage. Keywords: microplate resonator, thermoelastic damping, high polarization voltage 1. Introduction Thermoelastic damping is the intrinsic damping inMicro ElectroMechanical Systems (MEMS). It arises from the entropy generation due to irreversible heat flux in vibrating device (Sun and Saka, 2009). Zener (1937, 1938) was the first who predicted thermo-elastic damping. He found an expression for quality factor of thermoelastic damping in beams. Conversion of mechanical energy into heat in vibrating elastic beamswas treated byAlblas (1981). He also found that this damping is negligible for macro structures. Lifshitz and Roukes (2000) investigated the quality factor of a thermo-elastic microbeam and found that thermoelastic damping has important effects in micro and nano scales. There are two methods for calculating the quality factor of thermoelastic damping: energy method, in which dissipated and maximum stored energy should be calculated (Sudipto et al., 2006; Prabhakar and Vengallatore, 2008; Serra and Bonaldi, 2009; Guo and Rogerson, 2003), and eigenfrequencymethod, in which real and imaginary parts of the eigenfrequency should be calculated (Sun and Saka, 2009; Nayfeh and Younis, 2004b; Yi and Matin, 2007; Choi et al., 2010). Each method can be done with numerical or analytical procedures or combination of them. Nayfeh andYounis (2004b)modeled the electrostatically actuatedmicroplate by considering thermoelastic damping. They obtained an expression for quality factor analytically by using perturbation theory. Sudipto and Aluru (2006) investigated thermoelastic damping in an elec- trostatically actuated microbeam by the thermal energy method. They studied the effect of applied voltage on thermoelastic damping. In MEMS, there are different actuation and sensing properties such as thermal, optical, electrostatic, electromagnetic, piezoresistive andpiezoelectric but electrostatics is oftenpreferred (FargasMarquès et al., 2005). In electrostatics actuation, an elastic conductor is located above a stationary conductor.The electrical load can be composed of two components, includingDCand ACvoltage. TheappliedDCvoltage deforms the elastic surface that causes to change the system capacitance and to stretch themid-plane of the elastic surface. The applications are transistors, switches,micro-mirrors, pressure sensors,micro-pumps,moving valves andmicro-gripperswhich 318 A. KaramiMohammadi, N.A. Ali have no harmonic motion in their systems. If AC voltage is added then resonators are obtained (Batra et al., 2007). There are many works and papers that investigated the electrical actuation in micro struc- tures. Batra et al. (2007) reviewed them in their work. Abdel-Rahman et al. (2002) presented a nonlinearmodel of electrically actuatedmicrobeams consideringmid-plane stretching.They sho- wed static deflection of the microbeam due to DC polarization and vibration of the microbeam around its statically deflected position. MEMS resonators are devices that vibrate with AC voltage around the static deflection due to DC polarization voltage. The thermoelastic damping as well as frequency of the structure of the resonator is affected by theDCvoltage because the thermoelastic damping is directly related to the imaginary part of the frequency. In addition, in MEMS resonators, high sensitivity and resolution are needed (Nayfeh and Younis, 2004a), so for achieving this purpose, the damping in such devices should be decreased. However, studying the behavior of thermoelastic damping in resonators is necessary for manufacturers of MEMS. In this paper, a resonator is modeled as a thin rectangular microplate with thermoelastic damping that is actuated electrostatically. Large static deformation due to high polarization voltage is considered, andvibrationof themicroplate occurs aroundthe static deflection.Because of thermoelastic damping, the frequencyof vibration is a complex value that is used todetermine the quality factor of thermoelastic damping. Also, the pull-in voltage is investigated because nonlinear properties are more appeared when approaching the polarization voltage to the pull- in voltage. 2. Model description and assumptions A resonator is modeled as a rectangular microplate subject to the effect of a high electrostatic polarization voltage Vp. The equations of motion of the isotropic thinmicroplate are derived by using the combination of the classical plate theory and von-Karman type nonlinearity (Nayfeh andPai, 2004).ACartesian coordinate system(x,y,z) is attached to themicroplate such that the xy plane corresponds to themid-plane of the rectangular microplate over the domain 0¬ x ¬ a and 0¬ y ¬ b ∂2 ∂x2 + 1 2 (1+ν) ∂2v ∂x∂y + 1 2 (1−ν) ∂2 ∂y2 + ∂w ∂x ∂2w ∂x2 + 1 2 (1+ν) ∂w ∂y ∂2w ∂x∂y + 1 2 (1−ν) ∂w ∂x ∂2w ∂y2 = 1−ν2 Eh ( ρh ∂2 ∂t2 + ∂NT ∂x ) ∂2v ∂y2 + 1 2 (1+ν) ∂2 ∂x∂y + 1 2 (1−ν) ∂2v ∂x2 + ∂w ∂y ∂2w ∂y2 + 1 2 (1+ν) ∂w ∂x ∂2w ∂x∂y + 1 2 (1−ν) ∂w ∂y ∂2w ∂x2 = 1−ν2 Eh ( ρh ∂2v ∂t2 + ∂NT ∂y ) D∇4w+ρh ∂2w ∂t2 +NT∇2w = 1 12 ρh3 ∂2 ∂t2 (∇2w)−∇2MT + Eh 1−ν2 {[∂u ∂x + 1 2 (∂w ∂x )2 +ν (∂v ∂y + 1 2 (∂w ∂y )2)]∂2w ∂x2 +(1−ν) (∂u ∂y + ∂v ∂x + ∂w ∂x ∂w ∂y ) ∂2w ∂x∂y + [∂v ∂y + 1 2 (∂w ∂y )2 +ν (∂u ∂x + 1 2 (∂w ∂x )2)]∂2w ∂y2 } + ε0V 2 p 2(d−w)2 (2.1) where Effect of high electrostatic actuation on thermoelastic damping... 319 MT = EαT 1−ν h/2∫ −h/2 zθ dz NT = EαT 1−ν h/2∫ −h/2 θ dz D = Eh3 12(1−ν2 ∇ 2 = ∂2 ∂x2 + ∂2 ∂y2 (2.2) which are the thermal moment, thermal axial force, plate flexural rigidity and two-dimensional Laplacian operator, respectively, and θ = T −T0, in which T(x,z,t) and T0 are defined as the temperature field of the beam, and stress-free temperature (in equilibrium), respectively. Also, t, αT , E, ν, h, d, ε0 and ρ are time, coefficient of thermal expansion, Young’s modulus, Pois- son’s ratio, thickness of the microplate, capacitor gap, dielectric constant andmaterial density, respectively. u, v and w are displacement components along with the x, y and z directions, respectively. The thermal conduction equation, containing the thermoelastic coupling term, canbewritten as (Sun and Saka, 2009; Nayfeh and Pai, 2004) κ∇2θ+κ ∂2θ ∂z2 = ρcv ∂θ ∂t +βT0z ∂ ∂t (∂u ∂x + ∂v ∂y −z (∂2w ∂x2 + ∂2w ∂y2 )) (2.3) where cv and κ are the specific heat at constant volume and the thermal conductivity, respecti- vely. β = EαT/(1−2ν) is the thermal modulus. So equations (2.1) and (2.3) represent the governing equations of nonlinear vibration of the micro-plate with thermoelastic damping (TED). In addition, thermal and elastic properties are assumed independent of temperature, and the temperature change due to TED is assumed to be small, thus the vibration of the micro-plate is investigated in a constant temperature T0. For convenience, the following nondimensional variables are introduced (denoted by hats) x̂ = x a ŷ = y b ẑ = z h ŵ = w d û = u u v̂ = v v t̂ = t t θ̂ = θ θ (2.4) where u = d2 a v = d2 b t = 2ab h √ 3ρ E θ = βT0h 2d2 tκb2 (2.5) Substituting equations (2.4) into equations (2.1), the following equations are obtained α1 ∂2û ∂x̂2 + 1 2 (1+ν) ∂2v̂ ∂x̂∂ŷ + 1 2 (1−ν) ∂2û ∂ŷ2 +α1 ∂ŵ ∂x̂ ∂2ŵ ∂x̂2 + 1 2 (1+ν) ∂ŵ ∂ŷ ∂2ŵ ∂x̂∂ŷ + 1 2 (1−ν) ∂ŵ ∂x̂ ∂2ŵ ∂ŷ2 =(1−ν2)α2 ∂2û ∂t̂2 +(1+ν)α3 ∂N̂T ∂x̂ α−11 ∂2v̂ ∂ŷ2 + 1 2 (1+ν) ∂2û ∂x̂∂ŷ + 1 2 (1−ν) ∂2v̂ ∂x̂2 +α−11 ∂ŵ ∂ŷ ∂2ŵ ∂ŷ2 + 1 2 (1+ν) ∂ŵ ∂x̂ ∂2ŵ ∂x̂∂ŷ + 1 2 (1−ν) ∂ŵ ∂ŷ ∂2ŵ ∂x̂2 =(1−ν2)α4 ∂2v̂ ∂t̂2 +(1+ν)α5 ∂N̂T ∂ŷ 320 A. KaramiMohammadi, N.A. Ali α1 ∂4ŵ ∂x̂4 +2 ∂4ŵ ∂x̂2∂ŷ2 +α−11 ∂4ŵ ∂ŷ4 +(1−ν2) ∂2ŵ ∂t̂2 +(1+ν)N̂T ( α6 ∂2ŵ ∂x̂2 +α7 ∂2ŵ ∂ŷ2 ) (2.6) = (1−ν2) ∂2 ∂t̂2 ( α2 ∂2ŵ ∂x̂2 +α4 ∂2ŵ ∂ŷ2 ) − (1+ν) ( α8 ∂2M̂T ∂x̂2 +α9 ∂2M̂T ∂ŷ2 ) +12α210 {[ α1 ∂û ∂x̂ + α1 2 (∂ŵ ∂x̂ )2 +ν (∂v̂ ∂ŷ + 1 2 (∂ŵ ∂ŷ )2)]∂2ŵ ∂x̂2 +(1−ν) (∂û ∂ŷ + ∂v̂ ∂x̂ + ∂ŵ ∂x̂ ∂ŵ ∂ŷ ) ∂2ŵ ∂x̂∂ŷ ] + [ α−11 ∂v̂ ∂ŷ + 1 2 α−11 (∂ŵ ∂ŷ )2 +ν (∂û ∂x̂ + 1 2 (∂ŵ ∂x̂ )2)]∂2ŵ ∂ŷ2 } +α11 V 2p (1− ŵ)2 In this model, Lifshitz and Roukes (2000) assumption is used. So the thermal gradient in the z direction is much larger than the gradients in the x and y directions. Therefore, κ∇2θ in equation (2.3) can be ignored. Thus the equation is simplified, and substituting equations (2.4) into equations (2.3), yields ∂2θ̂ ∂ẑ2 = α12 ∂θ̂ ∂t̂ + ∂ ∂t̂ ( α1 ∂û ∂x̂ + ∂v̂ ∂ŷ − ẑ α10 ( α1 ∂2ŵ ∂x̂2 + ∂2ŵ ∂ŷ2 )) (2.7) Equations (2.6) and (2.7) represent the nondimensional governing equations of the system, where α1 = b2 a2 α2 = h2 12a2 α3 = αT θ̂b 2 d2 α4 = h2 12b2 α5 = αT θ̂a 2 d2 α6 = 12αT θ̂b 2 h2 α7 = 12αT θ̂a 2 h2 α8 = 12αT θ̂b 2 dh α9 = 12αT θ̂a 2 dh α10 = d h α11 = ε0a 2b2 2d3D α12 = ρcph 3 2abκ √ E 3ρ (2.8) and N̂T = 1/2∫ −1/2 θ̂ dẑ M̂T = 1/2∫ −1/2 θ̂ẑ dẑ (2.9) It should be noted that the parameters of equations (2.8) and (2.9) are related as follows α2 α4 = α1 α8 α9 =α2 α8 = αα10α12 α = 12α2TET0 (1−2ν)ρcp (2.10) 3. Large deformations under electrostatic load The microplate undergoes deflection under electrostatic voltage Vp. For calculating this deflec- tion, the dynamic and thermoelastic terms should be eliminated form equations (2.6) Effect of high electrostatic actuation on thermoelastic damping... 321 α1 ∂2ûs ∂x̂2 + 1 2 (1+ν) ∂2v̂s ∂x̂∂ŷ + 1 2 (1−ν) ∂2ûs ∂ŷ2 +α1 ∂ŵs ∂x̂ ∂2ŵs ∂x̂2 + 1 2 (1+ν) ∂ŵs ∂ŷ ∂2ŵs ∂x̂∂ŷ + 1 2 (1−ν) ∂ŵs ∂x̂ ∂2ŵs ∂ŷ2 =0 α−11 ∂2v̂s ∂ŷ2 + 1 2 (1+ν) ∂2ûs ∂x̂∂ŷ + 1 2 (1−ν) ∂2v̂s ∂x̂2 +α−11 ∂ŵs ∂ŷ ∂2ŵs ∂ŷ2 + 1 2 (1+ν) ∂ŵs ∂x̂ ∂2ŵs ∂x̂∂ŷ + 1 2 (1−ν) ∂ŵs ∂ŷ ∂2ŵs ∂x̂2 =0 α1 ∂4ŵs ∂x̂4 +2 ∂4ŵs ∂x̂2∂ŷ2 +α−11 ∂4ŵs ∂ŷ4 =12α210 {[ α1 ∂ûs ∂x̂ + α1 2 (∂ŵs ∂x̂ )2 +ν (∂v̂s ∂ŷ + 1 2 (∂ŵs ∂ŷ )2)]∂2ŵs ∂x̂2 +(1−ν) (∂ûs ∂ŷ + ∂v̂s ∂x̂ + ∂ŵs ∂x̂ ∂ŵs ∂ŷ )∂2ŵs ∂x̂∂ŷ + [ α−11 ∂v̂s ∂ŷ + 1 2 α−11 (∂ŵs ∂ŷ )2 +ν (∂ûs ∂x̂ + 1 2 (∂ŵs ∂x̂ )2)]∂2ŵs ∂ŷ2 } +α11 V 2p (1− ŵs)2 (3.1) where ûs, v̂s and ŵs are static displacements. Nonlinear static equations (3.1) can be solved by Galerkin’s method, using the following approximations (Szilared, 2004) ŵs(x̂, ŷ)= ∑ m ∑ n Wsmnϕmn(x̂, ŷ) ûs(x̂, ŷ)= ∑ m ∑ n Usmnψmn(x̂, ŷ) v̂s(x̂, ŷ)= ∑ m ∑ n V smnψmn(x̂, ŷ) (3.2) To simplify the solution only the first term (m = n = 1) of equations (3.2) is considered. Substituting into equations (3.1) the functions ϕ11 and ψ11 as is listed in Table 1 with respect to the boundary conditions, equations (3.1) become ∫∫ [ α1U s 11 ∂2ψ11 ∂x̂2 + 1 2 (1+ν)V s11 ∂2ψ11 ∂x̂∂ŷ + 1 2 (1−ν)Us11 ∂2ψ11 ∂ŷ2 +α1 ( Ws11 ∂ϕ11 ∂x̂ )( Ws11 ∂2ϕ11 ∂x̂2 ) + 1 2 (1+ν) ( Ws11 ∂ϕ11 ∂ŷ )( Ws11 ∂2ϕ11 ∂x̂∂ŷ ) + 1 2 (1−ν) ( Ws11 ∂ϕ11 ∂x̂ )( Ws11 ∂2ϕ11 ∂ŷ2 )] ψ11 dx̂ dŷ =0 ∫∫ [ α−11 V s 11 ∂2ψ11 ∂ŷ2 + 1 2 (1+ν)Us11 ∂2ψ11 ∂x̂∂ŷ + 1 2 (1−ν)V s11 ∂2ψ11 ∂x̂2 +α−11 ( Ws11 ∂ϕ11 ∂ŷ )( Ws11 ∂2ϕ11 ∂ŷ2 ) + 1 2 (1+ν) ( Ws11 ∂ϕ11 ∂x̂ )( Ws11 ∂2ϕ11 ∂x̂∂ŷ ) + 1 2 (1−ν) ( Ws11 ∂ϕ11 ∂x̂ )( Ws11 ∂2ϕ11 ∂ŷ2 )] ψ11 dx̂ dŷ =0 ∫∫ [ α1W s 11 ∂4ϕ11 ∂x̂4 +2Ws11 ∂4ϕ11 ∂x̂2∂ŷ2 +α−11 W s 11 ∂4ϕ11 ∂ŷ4 −12α210 {[ α1U s 11 ∂ψ11 ∂x̂ + α1 2 ( Ws11 ∂ϕ11 ∂x̂ )2 +ν ( V s11 ∂ψ11 ∂ŷ + 1 2 ( Ws11 ∂ϕ11 ∂ŷ )2)] Ws11 ∂2ϕ11 ∂x̂2 +(1−ν) ( Us11 ∂ψ11 ∂ŷ +V s11 ∂ψ11 ∂x̂ +Ws11 ∂ϕ11 ∂x̂ Ws11 ∂ϕ11 ∂ŷ ) Ws11 ∂2ϕ11 ∂x̂∂ŷ + [ α−11 V s 11 ∂ψ11 ∂ŷ + 1 2 α−11 ( Ws11 ∂ϕ11 ∂ŷ )2 +ν ( Us11 ∂ψ11 ∂x̂ + 1 2 ( Ws11 ∂ϕ11 ∂x̂ )2)] Ws11 ∂2ϕ11 ∂ŷ2 } − α11V 2 p (1−Ws11ϕ11) 2 ] ϕ11 dx̂ dŷ =0 (3.3) 322 A. KaramiMohammadi, N.A. Ali Table 1. List of functions ψ11 and ϕ11 with respect to the boundary condition (Leissa, 1969) CCCC φ11 = [cos(2πx̂)−1][cos(2πŷ)−1] ψ11 =sin(πx̂)sin(πŷ) CFCF φ11 =cos(2πx̂)−1 ψ11 =sin(πx̂cos(πŷ) Introducing new parameters A1 = ∫∫ [ α1 ∂2ψ11 ∂x̂2 + 1 2 (1−ν) ∂2ψ11 ∂ŷ2 ] ψ11 dx̂ dŷ A2 = 1 2 (1+ν) ∫∫ ∂2ψ11 ∂x̂∂ŷ ψ11 dx̂ dŷ A3 = ∫∫ [ α1 ∂ϕ11 ∂x̂ ∂2ϕ11 ∂x̂2 + 1 2 (1+ν) ∂ϕ11 ∂ŷ ∂2ϕ11 ∂x̂∂ŷ + 1 2 (1−ν) ∂ϕ11 ∂x̂ ∂2ϕ11 ∂ŷ2 ] ψ11 dx̂ dŷ A4 = ∫∫ [ α−11 ∂2ψ11 ∂ŷ2 + 1 2 (1−ν) ∂2ψ11 ∂x̂2 ] ψ11 dx̂ dŷ A5 = ∫∫ [ α−11 ∂ϕ11 ∂ŷ ∂2ϕ11 ∂ŷ2 + 1 2 (1+ν) ∂ϕ11 ∂x̂ ∂2ϕ11 ∂x̂∂ŷ + 1 2 (1−ν) ∂ϕ11 ∂x̂ ∂2ϕ11 ∂ŷ2 ] ψ11 dx̂ dŷ A6 = ∫∫ ( α1 ∂4ϕ11 ∂x̂4 +2 ∂4ϕ11 ∂x̂2∂ŷ2 +α−11 ∂4ϕ11 ∂ŷ4 ] ϕ11 dx̂ dŷ A7 =−12α 2 10 ∫∫ [ α1 ∂ψ11 ∂x̂ ∂2ϕ11 ∂x̂2 +(1−ν) ∂ψ11 ∂ŷ ∂2ϕ11 ∂x̂∂ŷ +ν ∂ψ11 ∂x̂ ∂2ϕ11 ∂ŷ2 ] ϕ11 dx̂ dŷ A8 =−12α 2 10 ∫∫ [ ν ∂ψ11 ∂ŷ ∂2ϕ11 ∂x̂2 +(1−ν) ∂ψ11 ∂x̂ ∂2ϕ11 ∂x̂∂ŷ +α−11 ∂ψ11 ∂ŷ Ws11 ∂2ϕ11 ∂ŷ2 ] ϕ11 dx̂ dŷ A9 =−12α 2 10 ∫∫ {[α1 2 (∂ϕ11 ∂x̂ )2 + ν 2 (∂ϕ11 ∂ŷ )2]∂2ϕ11 ∂x̂2 +(1−ν) ∂ϕ11 ∂x̂ ∂ϕ11 ∂ŷ ∂2ϕ11 ∂x̂∂ŷ + [1 2 α−11 (∂ϕ11 ∂ŷ )2 + ν 2 (∂ϕ11 ∂x̂ )2]∂2ϕ11 ∂ŷ2 } ϕ11 dx̂ dŷ FE = α11V 2 p ∫∫ ϕ11 (1−Ws11ϕ11) 2 dx̂ dŷ (3.4) Equations (3.3) can be rewritten as A1U s 11+A2V s 11+A3W s 11 =0 A2U s 11+A4V s 11+A5W s 11 =0 A6W s 11+A7U s 11W s 11+A8V s 11W s 11+A9W s 11 =0 (3.5) Now, by solving these equations, Us11, V s 11 and W s 11 can be calculated, and then substituting them into equations (3.2), the static deflection is obtained. Effect of high electrostatic actuation on thermoelastic damping... 323 4. Transverse vibration around the static deflection The micro plate deflections have two components. The static deflections, as discussed in the previous section, due to the polarization voltage Vp and the dynamic vibration deflections, occur around the static state. Consider only lateral vibration, thus the deflections û(x̂, ŷ), v̂(x̂, ŷ) and ŵ(x̂, ŷ, t̂) can be written as ŵ(x̂, ŷ, t̂)= ŵs(x̂, ŷ)+ ŵd(x̂, ŷ, t̂) v̂(x̂, ŷ, t̂)= v̂s(x̂, ŷ) û(x̂, ŷ, t̂)= ûs(x̂, ŷ) (4.1) Theequationsdescribingvibrationof themicroplate aroundthe staticdeflectionsareobtainedby substituting equation (4.1) into equations (2.6)3 and (2.7) and dropping the terms representing the equilibrium position that is the static deflection, equation (3.1)3, and high order terms of the dynamic deflections. The result can be written as α1 ∂4ŵd ∂x̂4 +2 ∂4ŵd ∂x̂2∂ŷ2 +α−11 ∂4ŵd ∂ŷ4 +(1−ν2) ∂2ŵd ∂t̂2 +(1+ν)N̂T ( α6 ∂2ŵs ∂x̂2 +α7 ∂2ŵs ∂ŷ2 ) =(1−ν2) ∂2 ∂t̂2 ( α2 ∂2ŵd ∂x̂2 +α4 ∂2ŵd ∂ŷ2 ) − (1+ν) ( α8 ∂2M̂T ∂x̂2 +α9 ∂2M̂T ∂ŷ2 ) +12α210 {[ α1 ∂ûs ∂x̂ + α1 2 (∂ŵs ∂x̂ )2 +ν (∂v̂s ∂ŷ + 1 2 (∂ŵs ∂ŷ )2)]∂2ŵd ∂x̂2 + [ α1 ∂ŵs ∂x̂ ∂ŵd ∂x̂ +ν (∂ŵs ∂x̂ ∂ŵd ∂x̂ )]∂2ŵs ∂x̂2 +(1−ν) (∂ûs ∂ŷ + ∂v̂s ∂x̂ + ∂ŵs ∂x̂ ∂ŵs ∂ŷ )∂2ŵd ∂x̂∂ŷ +(1−ν) (∂ŵs ∂x̂ ∂ŵd ∂ŷ + ∂ŵd ∂x̂ ∂ŵs ∂ŷ )∂2ŵs ∂x̂∂ŷ + [ α−11 ∂v̂s ∂ŷ + 1 2 α−11 (∂ŵs ∂ŷ )2 +ν (∂ûs ∂x̂ + 1 2 (∂ŵs ∂x̂ )2)]∂2ŵd ∂ŷ2 + [ α−11 ∂ŵs ∂ŷ ∂ŵd ∂ŷ +ν (∂ŵs ∂x̂ ∂ŵd ∂x̂ )]∂2ŵs ∂ŷ2 } + 2α11V 2 p (1− ŵs) 3 ŵd+ 2α11Vp (1− ŵs) 2 v(t̂) ∂2θ̂ ∂ẑ2 = α12 ∂θ̂ ∂t̂ + ∂ ∂t̂ ( − ẑ α10 ( α1 ∂2ŵd ∂x̂2 + ∂2ŵd ∂ŷ2 )) (4.2) For calculating thequality factor ofTED, coupled thermoelastic equations (4.2) shouldbe solved in the case of harmonic vibrations. So, ŵd and θ̂ are set as in the following ŵd(x̂, ŷ, ẑ, t̂)= ∞∑ n=0 ∞∑ m=0 Wmn(x̂, ŷ, ẑ)e iΩmnt̂ θ̂(x̂, ŷ, ẑ, t̂)= ∞∑ n=0 ∞∑ m=0 Θmn(x̂, ŷ, ẑ)e iΩmnt̂ (4.3) where Wmn(x̂, ŷ) and Θmn(x̂, ŷ, ẑ) are the (m,n)-th transverse mode shapes of the plate, and the associated temperature variation, respectively, and Ωmn is the complex frequency that has the real part ωmn, and the imaginary part λmn which is related to the damping. Therefore, substituting equations (4.3) into the equation of transverse vibration around static deflection (4.4)1 and thermal conduction equation (4.4)2 yields 324 A. KaramiMohammadi, N.A. Ali α1 ∂4Wmn ∂x̂4 +2 ∂4Wmn ∂x̂2∂ŷ2 +α−11 ∂4Wmn ∂ŷ4 −Ω2mn(1−ν 2)Wmn +(1+ν)N̂Tmn ( α6 ∂2ŵs ∂x̂2 +α7 ∂2ŵs ∂ŷ2 ) =−Ω2mn(1−ν 2) ( α2 ∂2Wmn ∂x̂2 +α4 ∂2Wmn ∂ŷ2 ) − (1+ν) ( α8 ∂2M̂Tmn ∂x̂2 +α9 ∂2M̂Tmn ∂ŷ2 ) +12α210 {[ α1 ∂ûs ∂x̂ +α1 2 (∂ŵs ∂x̂ )2 +ν (∂v̂s ∂ŷ + 1 2 (∂ŵs ∂ŷ )2 ) ]∂2Wmn ∂x̂2 + [ α1 ∂ŵs ∂x̂ ∂Wmn ∂x̂ +ν (∂ŵs ∂x̂ ∂Wmn ∂x̂ )]∂2ŵs ∂x̂2 +(1−ν) (∂ûs ∂ŷ + ∂v̂s ∂x̂ + ∂ŵs ∂x̂ ∂ŵs ∂ŷ )∂2Wmn ∂x̂∂ŷ +(1−ν) (∂ŵs ∂x̂ ∂Wmn ∂ŷ + ∂Wmn ∂x̂ ∂ŵs ∂ŷ )∂2ŵs ∂x̂∂ŷ + [ α−11 ∂v̂s ∂ŷ + 1 2 α−11 (∂ŵs ∂ŷ )2 +ν (∂ûs ∂x̂ + 1 2 (∂ŵs ∂x̂ )2)]∂2Wmn ∂ŷ2 + [ α−11 ∂ŵs ∂ŷ ∂Wmn ∂ŷ +ν (∂ŵs ∂x̂ ∂Wmn ∂x̂ )]∂2ŵs ∂ŷ2 } + 2α11V 2 p (1− ŵs)3 Wmn ∂2Θmn ∂ẑ2 = α12 ∂Θmn ∂t̂ + ∂ ∂t̂ ( − ẑ α10 ( α1 ∂2Wmn ∂x̂2 + ∂2Wmn ∂ŷ2 )) (4.4) where N̂Tmn = 1/2∫ −1/2 Θmn dẑ M̂ T mn = 1/2∫ −1/2 Θmnẑ dẑ (4.5) Assuming that there is no heat flow across the upper and lower surface of the beam, the boundary conditions for solving equation (4.4)2 are ∂Θmn/∂ẑ = 0 at ẑ = ±1/2. Then solving equation (4.4)2, and substituting the results into equation (4.5), the following equations are obtained M̂Tmn = 1 α10α12 CTmn ( α1 ∂2Wmn ∂x̂2 + ∂2Wmn ∂ŷ2 ) N̂Tmn =0 (4.6) where CTmn = 1 12 − 2 N3mn [ tan (Nmn 2 ) − Nmn 2 ] Nmn =(1− i) √ Ωmnα12 2 (4.7) Therefore, equations (4.6) should be substituted into equation (4.4)1. Since in the case of free vibration the amplitude of higher harmonic terms are dramatically small relative to the primary amplitude, their effects on thermoelastic damping are negligible. Thus by considering m = n = 1 and setting Wmn = ϕmn, Galerkin’s method is used for calculating Ωmn, see Hagedorn and Gupta (2007). The function ϕ11 should satisfy the boundary conditions and is selected with respect to the boundary conditions that are listed in Table 1. Therefore, equation (4.4)1 should be rewritten as Effect of high electrostatic actuation on thermoelastic damping... 325 ∫∫ [ α1 ∂4ϕ11 ∂x̂4 +2 ∂4ϕ11 ∂x̂2∂ŷ2 +α−11 ∂4ϕ11 ∂ŷ4 −Ω2mn(1−ν 2)ϕ11+Ω 2 mn(1−ν 2) ( α2 ∂2ϕ11 ∂x̂2 +α4 ∂2ϕ11 ∂ŷ2 ) +(1+ν)αCTmn ( α1 ∂4ϕ11 ∂x̂4 +2 ∂4ϕ11 ∂x̂2∂ŷ2 +α−11 ∂4ϕ11 ∂ŷ4 ) −12α210 {[ α1 ∂ûs ∂x̂ + α1 2 (∂ŵs ∂x̂ )2 +ν (∂v̂s ∂ŷ + 1 2 (∂ŵs ∂ŷ )2)]∂2ϕ11 ∂x̂2 + [ α1 ∂ψ11 ∂x̂ +α1 ∂ŵs ∂x̂ ∂ϕ11 ∂x̂ +ν (∂ψ11 ∂ŷ + ∂ŵs ∂x̂ ∂ϕ11 ∂x̂ )]∂2ŵs ∂x̂2 +(1−ν) (∂ûs ∂ŷ + ∂v̂s ∂x̂ + ∂ŵs ∂x̂ ∂ŵs ∂ŷ )∂2ϕ11 ∂x̂∂ŷ +(1−ν) (∂ψ11 ∂ŷ + ∂ψ11 ∂x̂ + ∂ŵs ∂x̂ ∂ϕ11 ∂ŷ + ∂ϕ11 ∂x̂ ∂ŵs ∂ŷ )∂2ŵs ∂x̂∂ŷ + [ α−11 ∂v̂s ∂ŷ + 1 2 α−11 (∂ŵs ∂ŷ )2 +ν (∂ûs ∂x̂ + 1 2 (∂ŵs ∂x̂ )2)]∂2ϕ11 ∂ŷ2 + [ α−11 ∂ψ11 ∂ŷ +α−11 ∂ŵs ∂ŷ ∂ϕ11 ∂ŷ +ν (∂ψ11 ∂x̂ + ∂ŵs ∂x̂ ∂ϕ11 ∂x̂ )]∂2ŵs ∂ŷ2 } − 2α11V 2 p (1− ŵs)3 ϕ11 ] ϕ11 dx̂ dŷ (4.8) Introducing new parameters L1 = α1[1+(1+ν)αC T 11] L2 =2[1+(1+ν)αC T 11] L3 = α −1 1 [1+(1+ν)αC T 11] P1 = ∫∫ ∂4ϕ11 ∂x̂4 ϕ11 dx̂ dŷ P2 = ∫∫ ∂4ϕ11 ∂x̂2∂ŷ2 ϕ11 dx̂ dŷ P3 = ∫∫ ∂4ϕ11 ∂ŷ4 ϕ11 dx̂ dŷ P4 = ∫∫ ϕ211 dx̂ dŷ P5 =(1−ν 2) ∫∫ [( α2 ∂2ϕ11 ∂x̂2 + α2 α1 ∂2ϕ11 ∂ŷ2 ) −ϕ11 ] ϕ11 dx̂ dŷ P6 = ∫∫ { −12α210 {[ α1 ∂ûs ∂x̂ + α1 2 (∂ŵs ∂x̂ )2 +ν (∂v̂s ∂ŷ + 1 2 (∂ŵs ∂ŷ )2)]∂2ϕ11 ∂x̂2 + [ α1 ∂ŵs ∂x̂ ∂ϕ11 ∂x̂ +ν (∂ŵs ∂x̂ ∂ϕ11 ∂x̂ )]∂2ŵs ∂x̂2 +(1−ν) (∂ûs ∂ŷ + ∂v̂s ∂x̂ + ∂ŵs ∂x̂ ∂ŵs ∂ŷ )∂2ϕ11 ∂x̂∂ŷ +(1−ν) (∂ŵs ∂x̂ ∂ϕ11 ∂ŷ + ∂ϕ11 ∂x̂ ∂ŵs ∂ŷ )∂2ŵs ∂x̂∂ŷ + [ α−11 ∂v̂s ∂ŷ + 1 2 α−11 (∂ŵs ∂ŷ )2 +ν (∂ûs ∂x̂ + 1 2 (∂ŵs ∂x̂ )2)]∂2ϕ11 ∂ŷ2 + [ α−11 ∂ŵs ∂ŷ ∂ϕ11 ∂ŷ +ν (∂ŵs ∂x̂ ∂ϕ11 ∂x̂ )]∂2ŵs ∂ŷ2 } − 2α11V 2 p (1− ŵs) 3 ϕ11 } ϕ11 dx̂ dŷ (4.9) Equation (4.8) can be written as follows L1P1+L2P2+L3P3+Ω 2 11P5+P6 =0 (4.10) Finally, by calculating Ω11 fromequation (4.10) and separating the real and imaginary parts, the quality factor of TED for large deformation of the microplate is obtained Q−1 =2 ∣∣∣ λ11 ω11 ∣∣∣ (4.11) 326 A. KaramiMohammadi, N.A. Ali 5. Pull-in voltage Beyond themaximumvalue ofDCvoltage, themicroplate of the resonator snaps and touches the rigid plate. This maximum value, denoted by VM, is called pull-in voltage (Batra et al., 2007). For calculating VM, the minimum voltage at which themicroplate becomes unstable, should be found. For example, for amicroplate whose properties are listed inTable 2, the pull-in voltage is shown in Fig. 1. As can be seen in this figure, beyond this voltage, that themicroplate becomes unstable and the deflection grows suddenly. In this case, the pull-in voltage is VM = 130.73V and the related deflection is ŵmax =0.45. Table 2. List of geometrical andmaterial properties of the microplate h [µm] d [µm] a [µm] b [µm] T0 [K] κ [Wm −1K−1] 1.5 1.2 200 100 300 148 cp [Jkg −1K−1] αT [10 −6K−1] E [Gpa] υ [–] ρ [kgm−3] ε0 [C 2m−2N−1] 712 2.6 170 0.25 2330 8.85 ·10−12 Fig. 1. Pull-in voltage in the microplate with properties listed in Table 2 6. Results Francais andDufour (1999)measured the centre deflection of a fully clamped squaremicroplate under various electrostatic actuations. They depicted the center deflection versus the following parameter CD = ε0V 2 p (ab) 2 32d3D (6.1) In Fig. 2, the static deflection at the centre of the plate ŵmax, which is calculated here using theGalerkin’s method, is compared with the experimental results of Francais andDufour (1999). There is a good agreement between Francais and Dufour’s measurments and the large deformation model of microplate. In Fig. 3, the large deformation model in two cases ν =0 and ν = 0.25 are compared with Lifshitz model (Lifshitz and Roukes, 2000) in which the TED is calculated for the clamped- clamped microbeam. The microplate and microbeam have the same properties that are listed in Table 2. The microplate is also considered with the CFCF boundary condition and without electrical load (Vp =0). As can be seen in Fig. 3, TED in the Lifshitzmodel and themicroplate with ν =0 are in good agreement. InFig. 4, theTEDof large deformationmodel is depicted versusαbased on the configuration of Table 2, for two cases of voltages: Vp =0.1VM and Vp =0.9VM. α is an important parameter Effect of high electrostatic actuation on thermoelastic damping... 327 Fig. 2. Comparison of ŵ max calculated using Galerkin’s method with the experimental results of Francais and Dufour (1999) Fig. 3. Comparison of TED in large deformationmodels of the microplate (ν =0 and ν =0.25) with the Lifshitz model because it represents the properties of thematerial. As can be seen in this figure, by increasing α the difference between two cases becomes larger. In small values of α, these two cases are coincided, so small α can change the nonlinear model to a linear model of the microplate. For example, α of silicon is α =0.005, thus silicon has linear properties. Fig. 4. TED of the large deformationmodel versus α In Fig. 5, TED is depicted versus α1 that is geometrical parameter for α = α10 = α11 = α12 =1, α2 =0.1, VM =10.0899 and ν =0.25. As can be seen in this figure, there is a critical value of α1 in which TED has the maximum value. Also, the maximum values of TED are increased by growing the electrostatic load. 328 A. KaramiMohammadi, N.A. Ali Fig. 5. TED of the large deformationmodel versus α 7. Conclusion In this paper, a resonator is modeled as a rectangular microplate. TED of themicroplate is cal- culated by linear and nonlinear assumptions. In the large deformationmodel, large deformation due to electrostatic load is considered by von-Karman assumptions. For calculating the thermo- elastic damping, the static and vibration equations are solved by usingGalerkin’s method. Thematerial propertiesmay exhibit nonlinear effects onTED, but silicon has linear proper- ties. 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