Abstract
In this paper dynamical instability of threelayer microswitch under DC voltage actuation has been studied. Recent studies have used the classical beam theory while leaving out the length scale parameter. In this paper dynamic behavior of the switch has been investigated based on couple stress theory and considering the length scale parameter. To this end, governing dynamic equation of the micro switch has been extracted and presented. Considering the nonlinearity of governing equation due to the existence of electrostatic force, Galerkin method has been implemented to overcome this nonlinearity and solve the mentioned equation and obtain the dynamic response. Dynamic response of micro switch has been investigated with and without considering the damping effects. Variation of dynamic pullin voltage versus micro beam length and primary gap between micro beam and stationary electrodes have been studied using couple stress and classic beam theory and obtained results have been compared to each other. Also dependency of dynamic pullin voltage to damping factor has been studied with considering two theories. Furthermore switching time of micro switch have been determined and compared using couple stress and classic beam theories.
Keywords:
micro switch; dynamic instability; electrostatic; couple stress theory; length scale parameter
1 INTRODUCTION
Nowadays micro electro mechanical devices have shown remarkable significance in various industries such as automation, military, medical, telecom. This issue is because of significant advantages of these structures such as small size, low energy consumption, high performance, low manufacturing costs and feasibility for batch fabrication. Simulating the mechanical behavior of these devices, particularly their static and dynamic behavior, is required for design and manufacturing purposes. Therefore, electrostatically actuated MEM/NEM devices such as, micro switches [^{1}[1] W. Tian, Z. Chen, (2015), “Analysis of bistable inductive microswitch based on surface micro size effect”, Applied Surface Science, 334(15), 3239., ^{2}[2] G. Rezazadeh, M. Fathalilou, M. Sadeghi, (2011), “Pullin Voltage of ElectrostaticallyActuated Microbeams in Terms of Lumped Model Pullin Voltage Using Novel Design Corrective Coefficients”, Sens Imaging, 12, 117131.] micromirrors [^{3}[3] M. Taghizadeh, H. Mobki, (2014), “Bifurcation analysis of torsional micromirror actuated by electrostatic forces”, Archives of Mechanics, 66(2), 95111., ^{4}[4] Y. Wang, K. Kumar, L. Wang, X. Zhang, (2012), “Monolithic integration of binaryphase fresnel zone plate objectives on 2axis scanning micromirrors for compact microscopes”, Optics express, 20(6), 66576668.], nanoswitches [^{5}[5] A. Koochi, H. HosseiniToudeshky, (2015), “Coupled effect of surface energy and size effect on the static and dynamic pullin instability of narrow nanoswitches”, International Journal of Applied Mechanics, 7(04), p.1550064., ^{6}[6] A. Kanani, A. Koochi, M. Farahani, E. Rouhic, M. Abadyan, (2016), “Modeling the size dependent pullin instability of cantilever nanoswitch immersed in ionic liquid electrolytes using strain gradient theory”, Scientia Iranica. Transaction B, Mechanical Engineering, 23(3), 976989.], nanotweezers [^{7}[7] M. Keivani, A. Koochi, M. Abadyan, (2017), “A New Bilayer Continuum Model Based on GurtinMurdoch and Consistent CoupleStress Theories for Stability Analysis of BeamType Nanotweezers”, Journal of Mechanics, 33(2), 137146., ^{8}[8] J. Berthelot, S.S. Aćimović, M.L. Juan, M.P. Kreuzer, J. Renger, R. Quidant, (2014), “Threedimensional manipulation with scanning nearfield optical nanotweezers”, Nature nanotechnology, 9(4), 295299.], micro capacitors [^{9}[9] H. Mobki, M. H. Sadeghia, G. Rezazadeh, (2015), “State Estimation of MEMs Capacitor Using Taylor Expansion”, International Journal of EngineeringTransactions B: Applications, 28(5), 764770.
[10] S. Afrang, H. Mobki, M.H. Sadeghi, G. Rezazadeh, (2015), “A new MEMS based variable capacitor with wide tunability, high linearity and low actuation voltage”, Microelectronics Journal, 46(2), 191197.^{11}[11] H. Mobki, M. H. Sadeghi, G. Rezazadeh, (2015), “Design of Direct Exponential Observers for Fault Detection of Nonlinear MEMS Tunable Capacitor”, International Journal of EngineeringTransactions A: Basics, 28(4), 634641.], sensors [^{12}[12] A. G.P. Kottapalli, M. Asadnia, J. Miao, M. Triantafyllou, (2015), “Soft polymer membrane microsensor arrays inspired by the mechanosensory lateral line on the blind cavefish”, Journal of Intelligent material systems and structures, 26(1), 3846., ^{13}[13] N. Sugita, K. Ishii, T. Furusho, K. Harada, M. Mitsuishi, (2015), “Cutting temperature measurement by a microsensor array integrated on the rake face of a cutting tool”, CIRP AnnalsManufacturing Technology, 64(1), 7780.], resonators [^{14}[14] F. Tajaddodianfar, H.N. Pishkenari, M.R.H. Yazdi, E.M. Miandoab, (2015), “On the dynamics of bistable micro/nano resonators: analytical solution and nonlinear behaviour”, Communications in Nonlinear Science and Numerical Simulation, 20(3), 10781089., ^{15}[15] J. F. Rhoads, S.W. Shaw, K.L. Turner, (2010), “Nonlinear dynamics and its applications in microand nanoresonators”, Journal of Dynamic Systems, Measurement, and Control, 132(3), 034001.] and oscillators [^{16}[16] V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. Schmitz, S.O. Demokritov, (2012), “Magnetic nanooscillator driven by pure spin current”, Nature materials, 11(12), 10281031., ^{17}[17] R.H. Liu, W.L. Lim, S. Urazhdin, (2013), “Spectral characteristics of the microwave emission by the spin Hall nanooscillator”. Physical review letters, 110(14), 147601.] are widely designed, fabricated, used and analyzed.
The majority of the micro structures perform using electrostatic actuation. The nature of this actuation is nonlinear which results in the nonlinearity of the governing equations. Presence of such nonlinearity along with length scale parameter cause difficulty in analysis with simulation soft wares and may result in deviation of obtained results. Therefore theoretical analysis may be considered as firm technique to examination of mechanical behavior. Precise modeling of micro switch is necessary to obtain accurate behavior of the structures. Length scale parameter is an important parameter which must be considered for analysis of mechanical behavior of micro structures. This parameter is an inherent characteristic of movable part of micro structure (micro beam or micro plate), and if it is less than specific limit, its effects is exposed. Application of classic beam theory (classic elasticity theory) for micro structures with nonnegligible length scale parameter may result in inaccurate results. So considering the length scale parameter and application of couple stress theory instead of classic beam theory is an essential issue in the case of mechanical analysis of micro structures. Numerous researches have been done in the case of static and dynamic stability analysis of micro switches without considering this effect [^{18}[18] H. Mobki, M.H. Sadeghi, G. Rezazadeh, (2015), “Application of Thau observer for fault detection of micro parallel plate capacitor subjected to nonlinear electrostatic force” International Journal of Engineering, 28(2), 270276.
[19] Y.T. Beni, M. Abadyan, A. Koochi, (2011), “Effect of the Casimir attraction on the torsion/bending coupled instability of electrostatic nanoactuators”, Physica scripta, 84(6), 065801.
[20] A. Koochi, A. Noghrehabadi, M. Abadyan, (2011), “Approximating the effect of van der Waals force on the instability of electrostatic nanocantilevers”, International Journal of Modern Physics B, 25(29), 39653976.
[21] H. Mobki, M.H. Sadeghi, S. Afrang, G. Rezazadeh, (2011), “On the tunability of a MEMS based variable capacitor with a novel structure”, Microsystem technologies, 17(9), 1447.
[22] R. Soroush, A. L. I. Koochi, A.S. Kazemi, M. Abadyan, (2012), “Modeling the effect of Van Der Waals attraction on the instability of electrostatic cantilever and doublysupported nanobeams using modified adomian method”, International Journal of Structural Stability and Dynamics, 12(05), 1250036.
[23] A. Azizi, N.M. Fard, H. Mobki, A. Arbi, (2018), “Bifurcation Behaviour and Stability Analysis of a NanoBeam Subjected to Electrostatic Pressure”, Applied and Computational Mathematics, 7(12), 111.
[24] A. Koochi, N. Fazli, R. Rach, (2014), “Modeling the pullin instability of the CNTbased probe/actuator under the Coulomb force and the van der Waals attraction” Latin American Journal of solids and structures, 11(8), 13151328.
[25] N. Kacem, S. Baguet, S. Hentz, R. Dufour, (2011), “Computional and quasianalytical models for nonlinear vibrations of resonant MEMS and NEMS sensors”, International Journal of NonLinear Mechanics, 46(3), 532542.^{26}[26] J.G. Guo, L.J. Zhou, Y.P. Zhao, (2009), “Instability analysis of torsional MEMS/NEMS actuators under capillary force”, Journal of Colloid and Interface Science, 331(2), 458462.]. Some works have devoted to analysis of static and dynamic behavior of micro and nano structures considering the length scale parameter. Sedighi et al. [^{27}[27] H.M. Sedighi, A. Koochi, M. Abadyan, (2014), “Modeling the size dependent static and dynamic pullin instability of cantilever nanoactuator based on strain gradient theory”, International Journal of Applied Mechanics, 6(05), 1450055.] modeled static and dynamic pullin instability of nanoactuator based on the strain gradient theory. Pullin instability of cantilever and fixedfixed beamtype nano structure and nano switches using strain gradient and couple stress theory have been studied in refs. [^{5}[5] A. Koochi, H. HosseiniToudeshky, (2015), “Coupled effect of surface energy and size effect on the static and dynamic pullin instability of narrow nanoswitches”, International Journal of Applied Mechanics, 7(04), p.1550064., ^{6}[6] A. Kanani, A. Koochi, M. Farahani, E. Rouhic, M. Abadyan, (2016), “Modeling the size dependent pullin instability of cantilever nanoswitch immersed in ionic liquid electrolytes using strain gradient theory”, Scientia Iranica. Transaction B, Mechanical Engineering, 23(3), 976989., ^{28}[28] H. Mobki, M.H. Sadeghi, G. Rezazadeh, M. Fathalilou, (2014), “Nonlinear behavior of a nanoscale beam considering length scaleparameter” Applied Mathematical Modelling, 38(5), 18811895., ^{29}[29] A. Koochi, H.M. Sedighi, M. Abadyan, (2014), “Modeling the size dependent pullin instability of beamtype NEMS using strain gradient theory”. Latin American Journal of Solids and Structures, 11(10), 18061829.]. Keivani et al. [^{7}[7] M. Keivani, A. Koochi, M. Abadyan, (2017), “A New Bilayer Continuum Model Based on GurtinMurdoch and Consistent CoupleStress Theories for Stability Analysis of BeamType Nanotweezers”, Journal of Mechanics, 33(2), 137146.] investigated the instability analysis of nanotweezers using couple stress theory. Rashvand et al. [^{30}[30] K. Rashvand, G. Rezazadeh, H. Mobki, M.H. Ghayesh, (2013), “On the sizedependent behavior of a capacitive circular microplate considering the variable lengthscale parameter”, International Journal of Mechanical Sciences, 77, 333342.
[31] K. Rashvand, G. Rezazadeh, H. Madinei, (2014), “Effect of lengthscale parameter on pullin voltage and natural frequency of a microplate” International Journal of Engineering, 27(3), 375384.^{32}[32] K. Rashvand, G. Rezazadeh, R. Shabani, M. Sheikhlou, (2012), “On the sizedependent nonlinear behavior of a capacitive rectangular microplate considering modified couple stress theory”, In Proceedings of the 1st International Conference of Mechanical Engineering and Advanced Technology. ICMEAT Isfahan, Iran (pp. 1012).] studied sizedependent behavior of circular and rectangular microplates.
The dispersion forces play an important role in the instability behavior of nano structures. Although the effects of these forces can be neglected in the modeling of static and dynamic behavior of micro structures. In nanoscale, Tadi Beni et al. [^{19}[19] Y.T. Beni, M. Abadyan, A. Koochi, (2011), “Effect of the Casimir attraction on the torsion/bending coupled instability of electrostatic nanoactuators”, Physica scripta, 84(6), 065801.] have studied pullin instability in a torsional nanoactuator considering Casimir force. Vakili et al. [^{33}[33] F.V. Tahami, H. Mobki, A.A.K. Janbahan, G. Rezazadeh, (2009), “Pullin phenomena and dynamic response of a capacitive nanobeam switch”, Sensors & Transducers Journal, 110 (11), 2637.], Koochi et al. [^{20}[20] A. Koochi, A. Noghrehabadi, M. Abadyan, (2011), “Approximating the effect of van der Waals force on the instability of electrostatic nanocantilevers”, International Journal of Modern Physics B, 25(29), 39653976.] and Soroush et al. [^{22}[22] R. Soroush, A. L. I. Koochi, A.S. Kazemi, M. Abadyan, (2012), “Modeling the effect of Van Der Waals attraction on the instability of electrostatic cantilever and doublysupported nanobeams using modified adomian method”, International Journal of Structural Stability and Dynamics, 12(05), 1250036.] have studied instability of nanobeams subjected to electrostatic force and considering van der Waals force. Koochi et al. investigated pullin instability of carbon nanotube based actuator under the Coulomb force and considering van der Waals attraction [^{24}[24] A. Koochi, N. Fazli, R. Rach, (2014), “Modeling the pullin instability of the CNTbased probe/actuator under the Coulomb force and the van der Waals attraction” Latin American Journal of solids and structures, 11(8), 13151328.]. Dequesnes et al. [^{34}[34] M. Dequesnes, S. V. Rotkin, N. R. Aluru, (2002), “Calculation of pullin voltages for carbon nanotubebased nanoelectromechanical switches, Nanotechnology, 13, 120131., ^{35} [35] M. Dequesnes, Z. Tang, N.R. Aluru, (2004) “Static and Dynamic Analysis of Carbon Nanotubebased Switches”, journal of Engineering Materials and Technology, 126, 230237] have studied the pullin phenomena and pullin voltage of a carbonbased nanoelectromechanical switch.
Three layer micro switch is a type of micro structure, in which microbeam/microplate is suspended between two stationary plates. Recently some investigations have been accomplished in the case of static and dynamic stability analysis of these switches. Comprehensive stability analysis of three plate micro switch subjected to electrostatic force has been presented in ref. [^{36}[36] H. Mobki, G. Rezazadeh, M. Sadeghi, F. VakiliTahami, MM. SeyyedFakhrabadi, (2013) “A comprehensive study of stability in an electrostatically actuated microbeam”, International Journal of NonLinear Mechanics, 48, 7885.]. Azizi et al. [^{37}[37] A. Azizi, H. Mobki, G. Rezazadeh, (2016), “Bifurcation Behavior of a Capacitive MicroBeam Suspended between Two Conductive Plates”, Int J Sens Netw Data Commun, 5(4), 110.] studied bifurcation Behavior of a Capacitive three layer microswitch. Mobki et al. [^{38}[38] H. Mobki, K. Rashvand, S. Afrang, M. H. Sadegh, G. Rezazadeh, (2014) “Design, simulation and bifurcation analysis of a novel micromachined tunable capacitor with extended tunability” Transactions of the Canadian Society for Mechanical Engineering, 38 (1), 1529.] designed and analyzed a new micromachined tunable three layer capacitor with extended tenability. Azimloo et al. [^{39}[39] H. Azimloo, G. Rezazadeh, and R. Shabani, (2015) “Development of a capacitive angular velocity sensor for the alarm and trip applications.” Measurement, 63, 282286.] ShahMohammadiAzar [^{40}[40] A. ShahMohammadiAzar, H. Azimloo, G. Rezazadeh, R. Shabani, & B. Tousi, (2013). “On the modeling of a capacitive angular speed measurement sensor” Measurement, 46(10), 39763981.] presented Angular Velocity Sensors based three layer switch and studied their pullin behavior. However most of the related researches are based on overlooking of length scale parameter.
With regard to the importance of dynamic analysis of these switches and considering this effect, this paper is devoted to study the dynamic behavior, instability and pullin voltage of three layer micro switch with considering length scale parameter and using couple stress theory instead of classic beam theory. Furthermore, the importance of considering or neglecting this effect has been discussed. For this aim, the governing dynamic equation of the microswitch has been presented. Moreover, due to the nonlinearity of this equation, which is a result of the existence of electro static force, Galerkin method has been implemented for solving the governing equation. In the rest of the paper, the model description, mathematical model of micro switch, obtained results and supplementary discussion for results are presented.
2 MODEL DESCRIPTION
In this section schematic view of studied microswitch is illustrated in the coming section and the governing equations of the microswitch will be extracted and presented. Figure 1(a) shows the schematic view of three layer micro switch, in which a micro beam is situated between two stationary electrodes. Initial distance of micro beam to upper and lower electrode is ${G}_{0}$. Applied voltages of ${V}_{1}$ and ${V}_{2}$ are induced from lower and upper electrode to micro beam. These voltages cause to attraction electrostatic force between micro beam and electrodes. If ${V}_{2}>{V}_{1}$, imposed electrostatic force from upper electrode prevails over imposed force from lower electrode and micro beam bends toward upper electrode and vice versa. If ${V}_{1}={V}_{2}$, micro beam is settled in its initial place and does not move toward any electrodes. Figure 1(b) shows the cross section view of the micro switch. $h$ and $b$ indicate thickness and width of micro beam. Micro beam is considered as isotropic media with elastic modulus of $E$, density of $\rho $, length of $L$, cross moment inertia of $I$, cross section of $A$. Parameter $p$ denotes applied voltages ratio and equals with $p=\raisebox{1ex}{${V}_{2}$}\!\left/ \!\raisebox{1ex}{${V}_{1}$}\right.$.
3 MATHEMATICAL MODELING
The governing dynamic equation of the micro switch is obtained and presented in this section.
Based on the couple stress theory strain energy $U$ of an isotropic material with linear elasticity behavior can be presented as [^{41}[41] F. Yang, A.C.M. Chong, D.C.C. Lam, P. Tong, (2002), “Couple stress based strain gradient theory for elasticity”, International Journal of Solids and Structures, 39(10), 27312743.]:
where $\overline{V}$ is the occupied volume of the material. σ, ε, m and 𝜒 are stress, strain, derivative part of couple stress and symmetric part of curvature tensors. Based on the linear elasticity theory, mentioned tensors can be presented as:
In this equation 𝜆 and G are Lame constants, l is length scale parameter of material, u and 𝜃 are displacement and rotation vectors. Also correspondence of these vectors is as:
Considering the xz coordinate of Figure 1, which x axis coincide centroidal one; arrays of displacement vectors can be shown in the form of [^{42}[42] S.K. Park, X.L. Gao, (2006), “BernoulliEuler beam model based on a modified couple stress theory”, J. Micromech. Microeng., 16(11), 23552359.]:
Where u, v and w are components of displacement vectors in direction of x, y and z axes respectively. Also rotation angle (𝜓) can be obtained based on the classic beam theory:
For plane stress and infinitesimal displacement, the components of strain tensor can be achieved using of Eqs. (3), (7), and (8) as:
Based on Eqs. (6)(8) following relations are accurate.
With substituting Eq. (10) into equation 5, arrays of symmetric curvature tensor can be presented as:
Similarly the form of stress tensor components can be extracted by substituting of Eq. (9) into Eq. (2).
Arrays of derivative part of couple stress tensor can be obtained by placing of Eq. (11) into Eq. (4):
Strain energy U can be presented as equation 14, with substituting of Eqs. (9), (11), (12) and (13) into equation 1.
Where ${M}_{x}$ and ${Y}_{xy}$ indicate resultant and couple moment and are as:
Kinetic energy also can be shown as Eq. (16):
Virtual work of electrostatic force may be assumed as:
Where ${q}_{elec}$ indicates induced electrostatic force to micro beam. Based on the Hamilton principal following relation is valid.
Governing dynamic equation of micro beam based on the couple stress theory and taking into account the residual stress and damping effects can be elicited and shown as:
Where ${N}_{r}={\sigma}_{r}\left(1\vartheta \right)bh$ is residual force, and ${\sigma}_{r}$ and $\vartheta $ are the biaxial effective residual force and Poisson's ratio [^{2}[2] G. Rezazadeh, M. Fathalilou, M. Sadeghi, (2011), “Pullin Voltage of ElectrostaticallyActuated Microbeams in Terms of Lumped Model Pullin Voltage Using Novel Design Corrective Coefficients”, Sens Imaging, 12, 117131.]. Considering the above equation; it is obvious that bending rigidity of micro beam is composed from two terms of EI (bending rigidity based classic beam theory) and GAl2 (bending rigidity caused by considering of length scale parameter effects).
Electrostatic force per length of micro beam can be shown based on the reports of ref [^{36}[36] H. Mobki, G. Rezazadeh, M. Sadeghi, F. VakiliTahami, MM. SeyyedFakhrabadi, (2013) “A comprehensive study of stability in an electrostatically actuated microbeam”, International Journal of NonLinear Mechanics, 48, 7885.] as:
Where ${\epsilon}_{0}=8.854\times {10}^{12}{C}^{2}{N}^{1}{m}^{2}$ is permittivity coefficient of vacuum. Governing equation of micro beam based on classic beam theory can be extracted by setting $l=0$ in Eq. (20).
The governing dynamic equation of three layer microswitch subjected to electrostatic force and considering length scale parameter of microbeam are presented in this section. Due to the nonlinearity of term of electrostatic force, Galerkin method are implemented for dynamic analysis of the microswitch. This method is proper and reliable one for analysis of static and dynamic behavior of micro and nanoswitches. The procedure of Galerkin implementation for solving the Eq. (19) will be extended in the next section.
4 NUMERICAL METHOD
In this section applied numerical approach for solving the Eq. (19) is presented. For this case dynamic displacement can be assumed as:
Using Galerkin approximation method, the equation of dynamic response has been obtained as [^{43}[43] H. Mobki, M. H. Sadeghi, G. Rezazadeh, M. Fathalilou, Aa. Keyvanijanbahan, (2014), “Nonlinear behavior of a nanoscale beam considering length scaleparameter”, Applied Mathematical Modelling, 38(56), 18811895.]:
Where:
Where indicate the effective mass, stiffness, damping, and actuating force matrices respectively. $q\left(t\right)$ can be obtained from above set of ordinary differential equations using an integration scheme.
5 RESULTS AND DISCUSSION
5.1. Validation and Convergence of Numerical Method
This subsection deals with validation of numerical method. For this purpose obtained results of this paper have been compared with those obtained by Hung and Senturia [^{44}[44] E. S. Hung, S. D. Senturia, (1999), “Generating efficient dynamical models for microelectromechanical systems from a few finiteelement simulation runs”, Journal of Microelectromechanical Systems, 8, 280289.]. The microswitch specification are $E=149Gpa$, $\rho =2330\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{1ex}{${m}^{2}$}\right.$, $L=610\mu m$, $b=40\mu m$, $h=2.2\mu m$, ${G}_{0}=2.3\mu m$, ${\sigma}_{r}=3.7MPa$, and $p=0$. Calculated pullin time for various values of applied voltages are shown in Figure 2. As can be seen in this figure, results of this paper are in good agreement with theoretical and experimental results.
For more validation the obtained results of this paper are compared with those presented in ref [^{37}[37] A. Azizi, H. Mobki, G. Rezazadeh, (2016), “Bifurcation Behavior of a Capacitive MicroBeam Suspended between Two Conductive Plates”, Int J Sens Netw Data Commun, 5(4), 110.]. The microswitch is a fixedfixed one with $E=169Gpa$, $\rho =2330\raisebox{1ex}{$kg$}\!\left/ \!\raisebox{1ex}{${m}^{2}$}\right.$, $L=600\mu m$, $b=4\mu m$, $h=2\mu m$, ${G}_{0}=2\mu m$, ${\sigma}_{r}=3.7MPa$, and $p=0.5$. The results convergence and comparison are shown in table 1. As shown in this table, with increase of N the obtained pullin voltage converges to 10.11V and presented results are in good agreement with results of ref [^{37}[37] A. Azizi, H. Mobki, G. Rezazadeh, (2016), “Bifurcation Behavior of a Capacitive MicroBeam Suspended between Two Conductive Plates”, Int J Sens Netw Data Commun, 5(4), 110.].
5.2. Dynamic response of microswitch
In this section obtained results of the dynamic instability of golden micro switch are presented. Physical and spatial properties of the micro switch are shown in table 2. The micro beam is a cantilever gold one.
Variation of gap between end tip of micro beam and lower electrode versus to time (using couple stress theory) for applied voltage 10.44 V is depicted in Figure 3. As shown in this figure, with applying this voltage, micro beam oscillates and no instability occurs. Phase diagram relevant to this figure is shown in Figure 4. As shown in this figure, phase portrait of micro beam is a restricted ovalshaped environment, where asymmetry of this shape with respect to horizon axis is caused by nonlinear electrostatic force.
End gap versus time under step DC voltage of ${V}_{1}=10.44V$actuation, using couple stress theory
Velocity of end tip versus end gap under step DC voltage of ${V}_{1}=10.44V$ actuation, using couple stress theory
End gap versus time for applied voltage 10.45 is shown in Figure 5. As it can be observed in this figure, applying of 10.45V results in instability and collision of micro beam to lower electrode. Minimal applied voltage (10.45V) which causes the instability of micro beam, is entitled as dynamic instability voltage or dynamic pullin voltage. Phase portrait of Figure 5 is shown in Figure 6. As shown in this figure increasing of end gap cause the tendency of velocity to higher values. These magnitude (pullin voltage) was obtained by trial and error. If the applied voltage causes to instability (stability) of microswitch the mentioned voltage decrease (increase) as long as obtaining dynamic pullin voltage.
End gap versus time under step DC voltage of ${V}_{1}=10.45V$ actuation, using couple stress theory
Velocity of end tip versus end gap under step DC voltage of ${V}_{1}=10.45V$ actuation, using couple stress theory
Dynamic responses of micro beam to applied voltages 7.56V and 7.57V are drawn (using classic beam theory) and shown in Figure 7 and Figure 9 Related phase portraits are shown in Figure 8 and Figure 10 respectively. With comparing Figure 7 and Figure 9 it can be concluded that pullin voltage of micro beam based on the classic beam theory is 7.57V. However, as mentioned before, using couple stress theory yields 10.45V for pullin voltage. This remarkable difference between two magnitudes of pullin voltage is because of considering and neglecting the effects of length scale parameter. Considering this effect, which causes the increase of bending rigidity of micro beam and consequently, the dynamic pullin voltage is increased too.
End gap versus time under step DC voltage of ${V}_{1}=7.56V$ actuation, using classic beam theory
Velocity of end tip versus end gap under step DC voltage of ${\text{V}}_{1}=7.56\text{V}$ actuation, using classic beam theory
End gap versus time under step DC voltage of ${V}_{1}=7.57V$ actuation, using classic beam theory
Velocity of end tip versus end gap under step DC voltage of ${V}_{1}=7.57V$ actuation, using classic beam theory
Based on the reported results of ref [^{2}[2] G. Rezazadeh, M. Fathalilou, M. Sadeghi, (2011), “Pullin Voltage of ElectrostaticallyActuated Microbeams in Terms of Lumped Model Pullin Voltage Using Novel Design Corrective Coefficients”, Sens Imaging, 12, 117131.]
Based on the couple stress theory and considering $EI+GA{L}^{2}$ as bending rigidity, the above equation may be presented as:
So following equation can be presented as ratio of pullin voltage obtained using couple stress and classic beam theory:
Based on the above relation mentioned ratio is about 1.381 for the studied microswitch. As shown in Figure 6 and Figure 10 the ratio of pullin voltages is $\frac{10.45}{7.57}=1.38$, where is conformable with Eq. (26).
Figure 11 shows phase portrait of micro beam subjected to various applied voltages (continuous and dashed curves related to couple stress and classic beam theories.). As can be observed in this figure for lower applied voltages, phase portraits are stable and fullsymmetry. But with increasing applied voltage and in the proximity of pullin voltage, phase diagram transmutes to asymmetric ovalshaped one. Phase portraits for applied voltages upper than pullin one, are instable and increase of voltage results in increase of micro beam velocity. Comparing of voltages of coinciding diagram shows considerable difference between obtained magnitudes of applied voltage of two theories.
Variation of pullin voltage versus micro beam length and initial gap are shown in Figure 12 and Figure 13 respectively. As can be seen in these figures pullin voltage decreases with increase of length and decrease of initial gap. Although there are considerable differences between diagrams of couple stress and classic beam theories, these differences are more evident for low length and high gap. As can be seen in these figures for identical length or gap $\frac{{\left({V}_{pullin}\right)}_{couple}}{{\left({V}_{pullin}\right)}_{classic}}=1.38$. So ${\left({V}_{pullin}\right)}_{couple}{\left({V}_{pullin}\right)}_{classic}\cong 0.38{\left({V}_{pullin}\right)}_{classic}$ and it is proved that difference of two obtained voltages is equal with $0.38{\left({V}_{pullin}\right)}_{classic}$ and with increase of ${\left({V}_{pullin}\right)}_{classic}$ this difference is increased too. As shown in Figure 12 and Figure 13 with increase of gap and decrease of length, ${\left({V}_{pullin}\right)}_{classic}$ is increased and by increasing of ${\left({V}_{pullin}\right)}_{classic}$, the difference of ${\left({V}_{pullin}\right)}_{couple}{\left({V}_{pullin}\right)}_{classic}$ is increased too.
In the following part of this section, the instability of micro switch with considering damping effects are investigated. Dynamic response of micro switch for applied voltages of 10.68V and 10.69V and $c=0.02$ is shown in Figure 14. As shown in this figure applying of 10.68V causes the oscillation of micro beam and as a consequence of damping effects, oscillation amplitude decreases gradually. But for applied voltage 10.65V micro beam loses its stability and collapses on to lower electrode. As shown in Figure 5 pullin voltage for undamped case is 10.45V, but for the case with damping coefficient of $c=0.02$ is 10.69V, where it indicates increase of damping effect leads to the increase of pullin voltage. Relevant phase diagram of Figure 14 is depicted in Figure 15. As shown in this figure phase diagram for applied voltage of 10.68 is spiralshaped where with time elapsing, velocity tend to zero.
Velocity of end tip versus end gap for damping coefficient $c=0.02$, using couple stress theory
Dynamic response and phase portraits of the micro beam subjected to applied voltages 7.79V and 7.8V for $c=0.02$ are depicted in Figure 16 and Figure 17. From Figure 16, it can be concluded that pullin voltage based on the classic beam theory is 7.8V. As mentioned before, as a consequence of increase of damping effect, pullin voltage increase too.
Velocity of end tip versus end gap for damping coefficient $c=0.02$ , using classic beam theory
Variation of pullin voltage versus damping coefficient was extracted and depicted in Figure 18 and Figure 19. As shown in these figures for c <1.8 pull in voltage depends on the damping ratio and with increasing this ratio the pullin voltage increases too. But above this level (for c >1.8), pullin voltage remains constant and can be regarded as dampingindependent.
variation of dynamic instable voltage versus damping coefficient using couple stress theory
variation of dynamic instable voltage versus damping coefficient using classic beam theory
In the end of this section, dependency of pullin time to damping coefficient is demonstrated. Pullin diagram and pullin voltage of micro switch considering various damping coefficient are drawn and shown in Figure 20 and Figure 21. It can be concluded from two figures, increase of damping coefficient results in the increase of pullin time as well as pullin voltage. The results of these figures are in good agreement with the results of ref. [^{46}[46] G. Rezazadeh, F. VakiliTahami, H. Mobki, M. M. Seyyed Fakhrabadi, A. Alizadeh, (2010) “Static and Dynamic Pullin Phenomena of a Capacitive Carbon Nano Tube”, 18th Annual International Conference on Mechanical EngineeringISME 2010, Sharif University of Technology, Tehran, Iran, 1113 May. ].
Instability diagrams of micro beam for various magnitude of damping coefficient, using couple stress theory
Instability diagrams of micro beam for various magnitude of damping coefficient, using classic beam theory
6 CONCLUSION
Dynamic instability of three layer micro switch was studied using the classic beam and couple stress (with considering length scale parameter) theories. For this reason governing dynamic equation of micro switch was obtained with and without considering the length scale parameter. Considering the nonlinearity of governing equation, Galerkin method was implemented to solve the governing equation. The applied numerical method was validated with previous experimental and numerical methods. The validation results showed good ability of Galerkin method for solving the governing dynamic equation. Dynamic behavior and pullin voltage of the micro switch were investigated with considering the damping effects and using two mentioned theories. The obtained results showed that the calculated pullin voltage by couple stress theory is higher than the one calculated by classic beam theory. Also it was shown that the ratio of pullin voltages based on the couple stress theory to classic beam theory is a specific value. Furthermore, the dependency of pullin voltage to initial gap and microbeam length were studied and it was shown that with increase of initial gap and decrease of microbeam length, the effect of considering of length scale parameter gets more obvious. Obtained results indicate remarkable difference between results of classic beam and couple stress theories, where obtained pullin voltage based on the couple stress theory is higher than the obtained one using classic beam theory. Moreover, the variation of pullin voltage and pullin time with respect to damping coefficient were investigated. Results of dynamic case with considering damping effects confirmed that with increasing damping coefficient, the amplitude of oscillation decreases. Acquired results in this case showed that increase of damping coefficient results in increase of pullin voltage as well as pullin time.
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Publication Dates

Publication in this collection
2018
History

Received
31 Oct 2017 
Reviewed
03 Nov 2017 
Accepted
20 Mar 2018