Investigation of dynamic instability of three plates switch under step DC voltage actuation using modified couple stress theory

In this paper dynamical instability of three-layer micro-switch 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 pull-in 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 pull-in 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.

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 non-negligible 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][19][20][21][22][23][24][25][26].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] modeled static and dynamic pull-in instability of nanoactuator based on the strain gradient theory.Pull-in instability of cantilever and fixed-fixed beam-type nano structure and nano switches using strain gradient and couple stress theory have been studied in refs.[5,6,28,29].Keivani et al. [7] investigated the instability analysis of nanotweezers using couple stress theory.Rashvand et al. [30][31][32] studied size-dependent behavior of circular and rectangular micro-plates.
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] have studied pull-in instability in a torsional nano-actuator considering Casimir force.Vakili et al. [33], Koochi et al. [20] and Soroush et al. [22] have studied instability of nano-beams subjected to electrostatic force and considering van der Waals force.Koochi et al. investigated pull-in instability of carbon nanotube based actuator under the Coulomb force and considering van der Waals attraction [24].Dequesnes et al. [34,35] have studied the pull-in phenomena and pull-in voltage of a carbon-based nano-electromechanical switch.
Three layer micro switch is a type of micro structure, in which micro-beam/micro-plate 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].Azizi et al. [37] studied bifurcation Behavior of a Capacitive three layer microswitch.Mobki et al. [38] designed and analyzed a new micromachined tunable three layer capacitor with extended tenability.Azimloo et al. [39] Shah-Mohammadi-Azar [40] presented Angular Velocity Sensors based three layer switch and studied their pull-in 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 pull-in 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 micro-switch 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.

MODEL DESCRIPTION
In this section schematic view of studied micro-switch is illustrated in the coming section and the governing equations of the micro-switch 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 0 G .Applied voltages of 1 V and 2 V are induced from lower and upper elec- trode to micro beam.These voltages cause to attraction electrostatic force between micro beam and electrodes.If indicate thickness and width of micro beam.Micro beam is considered as isotropic media with elastic modulus of E , density of  , length of L , cross moment inertia of I , cross section of A .Parameter p denotes applied volt- ages ratio and equals with

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]: where V is the occupied volume of the material.σ, ε, m and  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  and G are Lame constants, l is length scale parameter of material, u and  are displacement and rotation vectors.Also correspondence of these vectors is as: Considering the x-z coordinate of Figure 1, which x axis coincide centroidal one; arrays of displacement vectors can be shown in the form of [42]: Where u, v and w are components of displacement vectors in direction of x, y and z axes respectively.Also rotation angle () can be obtained based on the classic beam theory: Investigation of dynamic instability of three plates switch under step DC voltage actuation using modified couple stress theory Latin American Journal of Solids and Structures, 2018, 15 (7), e54 4/17 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).
Where x M and xy Y indicate resultant and couple moment and are as: Kinetic energy also can be shown as Eq. ( 16): Latin American Journal of Solids and Structures, 2018, 15(7), e54 5/17 Virtual work of electrostatic force may be assumed as: Where elec q 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 is residual force, and r  and  are the biaxial effective residual force and Poisson's ratio [2].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] as: Where  (20).
The governing dynamic equation of three layer micro-switch subjected to electrostatic force and considering length scale parameter of micro-beam are presented in this section.Due to the nonlinearity of term of electrostatic force, Galerkin method are implemented for dynamic analysis of the micro-switch.This method is proper and reliable one for analysis of static and dynamic behavior of micro and nano-switches.The procedure of Galerkin implementation for solving the Eq. ( 19) will be extended in the next section.

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]: Latin American Journal of Solids and Structures, 2018, 15(7), e54 6/17 Where: Where indicate the effective mass, stiffness, damping, and actuating force matrices respectively.  q t can be obtained from above set of ordinary differential equations using an integration scheme.

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].The micro-switch specification are  shown in this table, with increase of N the obtained pull-in voltage converges to 10.11V and presented results are in good agreement with results of ref [37].
Latin American Journal of Solids and Structures, 2018, 15(7), e54 7/17    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 pull-in 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 pull-in voltage.This remarkable difference between two magnitudes of pull-in 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 pull-in voltage is increased too.Based on the reported results of ref [2]   Based on the couple stress theory and considering

EI GAL 
as bending rigidity, the above equation may be presented as: So following equation can be presented as ratio of pull-in voltage obtained using couple stress and classic beam theory: Based on the above relation mentioned ratio is about 1.381 for the studied micro-switch.As shown in Figure 6 and Figure 10 the ratio of pull-in voltages is 10.45 1. 38
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 full-symmetry.But with increasing applied voltage and in the proximity of pull-in voltage, phase diagram transmutes to asymmetric oval-shaped one.Phase portraits for applied voltages upper than pull-in 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.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 0.02 c  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       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 pull-in voltage of the micro switch were investigated with considering the damping effects and using two mentioned theories.The obtained results showed that the calculated pull-in voltage by couple stress theory is higher than the one calculated by classic beam theory.Also it was shown that the ratio of pull-in voltages based on the couple stress theory to classic beam theory is a specific value.Furthermore, the dependency of pull-in voltage to initial gap and micro-beam length were studied and it was shown that with increase of initial gap and decrease of micro-beam 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 pull-in voltage based on the couple stress theory is higher than the obtained one using classic beam theory.Moreover, the variation of pull-in voltage and pull-in 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 pull-in voltage as well as pull-in time.
electrostatic force from upper electrode prevails over imposed force from lower electrode and micro beam bends toward upper electrode and vice versa.If 1 2 V V  , 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 instability of three plates switch under step DC voltage actuation using modified couple stress theory Latin American Journal of Solids and Structures, 2018, 15(7), e54 3/17

Figure 1 :
Figure 1: schematic view of three layer micro switch of vacuum.Governing equation of micro beam based on classic beam theory can be extracted by setting 0 l  in Eq.
pull-in time for various values of applied voltages are shown in Figure2.As can be seen in this figure, results of this paper are in good agreement with theoretical and experimental results.

Figure 2 :.
Figure 2: Comparison of the pull-in time for various values of applied voltage 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 Figure3.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 Figure4.As shown in this figure, phase portrait of micro beam is a restricted oval-shaped environment, where asymmetry of this shape with respect to horizon axis is caused by nonlinear electrostatic force.

Figure 3 :Figure 4 :
Figure 3: End gap versus time under step DC voltage of 1 10.44 V V  actuation, using couple stress theory

Figure 5 :Figure 6 :
Figure 5: End gap versus time under step DC voltage of 1 10.45 V V  actuation, using couple stress theory

Figure 7 :Figure 8 :Figure 9 :
Figure 7: End gap versus time under step DC voltage of 1 7.56 V V  actuation, using classic beam theory

Figure 10 :
Figure 10: Velocity of end tip versus end gap under step DC voltage of 1 7.57 V V  actuation, using classic beam theory Investigation of dynamic instability of three plates switch under step DC voltage actuation using modified couple stress theory Latin American Journal of Solids and Structures, 2018, 15(7), e5411/17

Figure 11 :
Figure 11: Phase portrait of golden micro beam subjected various applied voltages

Figure 12 :Figure 13 :
Figure 12: Dynamic instable voltage versus length for golden micro beam

Figure 15 :
Figure14: End gap versus time, for damping coefficient 0.02 c  , using couple stress theory

Figure 16 :
Figure 16: End gap versus time, for damping coefficient 0.02 c  , using classic beam theory

Figure 17 :
Figure 17: Velocity of end tip versus end gap for damping coefficient 0.02 c  , using classic beam theory

Figure 18 :
Figure 18: variation of dynamic instable voltage versus damping coefficient using couple stress theory

Figure 19 :
Figure 19: variation of dynamic instable voltage versus damping coefficient using classic beam theory

Figure 20 :Figure 21 :
Figure 20: Instability diagrams of micro beam for various magnitude of damping coefficient, using couple stress theory

Table 1 :
Comparison of the pull-in voltageIn 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.

Table 2 :
physical and geometrical properties of studied three layer golden micro switch