The effect of capillary and intermolecular forces on instability of the electrostatically actuated microbeam with T-shaped paddle in the presence of fringing field

Applied Mathematical Modelling 71 (2019) 243–268

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

The effect of capillary and intermolecular forces on instability of the electrostatically actuated microbeam with T-shaped paddle in the presence of fringing field Behnam Firouzi, Mehdi Zamanian∗ Department of mechanical engineering, Faculty of engineering, Kharazmi University, P.O. Box 15719-14911, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 4 April 2018 Revised 18 January 2019 Accepted 12 February 2019 Available online 16 February 2019 Keywords: Microbeam Pull in instability Capillary force Casimir force Van der waals force Primary resonance

a b s t r a c t The present study is aimed to investigate pull-in instability, static deformation, natural frequency, primary and subharmonic resonances in the clamped–clamped microbeam with middle T-shaped paddle in presence of electrostatic force, Casimir force, van der Waals force and fringing field effect; besides, it is focused on investigating the effect of capillary force on instability and adhesion considering the effects of middle layer elongation and axial force. The microbeam is modeled as an Euler Bernoulli beam. In order to solve the nonlinear equations, the Galerkin-based rank reduction method was used. The discretized equations are solved by the perturbation theory in the neighborhood of primary and subharmonic resonances. The obtained results showed that the van der Waals and Casimir forces had attractive nature and led to the increased static deformation as well as occurrence of the pull-in at lower voltages; besides, the effects of attraction in Casimir was more than that in van der Waals. Results showed that with increase in the dimensionless length, which is influenced by the capillary force, the maximum static deformation was increased as well, leading to instability in capillary coefficients. Furthermore, presence of the electrostatic force also accelerated the instability resulting from the capillary force. Effects of the axial force and elongation in the clamped state resulted in the reduced static deformation as well as delayed instability. It is shown that changing the Casimir and Van der Waals forces as well as the fringing field effect, in addition to causing changes in the voltage and range, considerably affected determination of the bifurcation points. To validate the analytical results, numerical simulation is performed. © 2019 Elsevier Inc. All rights reserved.

1. Introduction In recent years, MEMS systems have been widely used in various industries due to their special mechanical and electrical features. Since these systems are manufactured by micromachining techniques, a very high level of performance, reliability, and complexity can be made at a low cost on a small silicon chip. Elements of many of the MEMS systems, as cantilever and clamped microbeams, are under electrostatic actuation, which is widely used in various mechanical, electrical, optical, and biomedical fields; therefore, the electrostatic actuation is created by applying a voltage between the microbeam, which served as the movable electrode, and the fixed electrode, which is placed in front of the microbeam [1]. Investigation of the parameters related to pull-in, such as pull-in voltage and deflection, is one of the most important issues in studying the ∗

Corresponding author. E-mail addresses: be.fi[email protected] (B. Firouzi), [email protected] (M. Zamanian).

https://doi.org/10.1016/j.apm.2019.02.016 0307-904X/© 2019 Elsevier Inc. All rights reserved.

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behavior of these systems. The study of pull-in parameters as well as the factors affecting the pull-in has been regarded in many researches in order to make progress in design and production of the MEMS systems by preventing the pull-in and increasing the reliability. Abdelrahman et al. used the shooting method to investigate the static and dynamic behavior of the electrostatically actuated clamped microbeam. In this work, static deformation and natural frequency were evaluated by taking into account the effects of middle layer elongation for clamped microbeam [2]. The instability resulted from the static and dynamic pullin micro-resonators has been evaluated in the studies conducted using numerical, analytical, and laboratory methods [3–5]. Rasekh and Khadem considered the effects of non-linear curvature and non-linear inertia in study of the pull-in voltage in a nano-tube under electrostatic excitation. Using Galerkin method, they discretized the partial differential equations and the resulting equations were solved by direct numerical integration method [6]. During the microstructures drying process, the liquid entrapped by the microstructure’s bottom surface creates a strong capillary force, which inclines the micro-sensor towards the bottom electrode. Also, moisture can cause densification of the fluid, and in some micro-sensors, due to their large surfaces, when the capillary force is stronger than the restoring forces the microbeam is pulled toward the bottom surface. This feature is similar to the pull-in instability when the microbeam undergoes electrostatic actuation. If the elastic forces of microstructures, during collapse, cannot resist against both capillary and adhesion forces induced at the microbeam and bottom surface contact point, the microstructure will be adhered to the bottom surface permanently [1]. In a paper, Nahavandi and Korayem investigated the forces applied on the tip of an AFM cantilever microbeam. In this micro-system, the capillary force is created due to the contact of a water thin layer of water accumulated between the surface and sample. They showed that roughness and smoothness of the surface and geometry of the cantilever microbeam’s tip had significant effect on the modeling and the applied forces [7]. Oukad and Younis modeled and analyzed the static behavior and capillary instability of the cantilever and clamped microbeams. In this work, the model was assumed as a continuous microbeam, and the capillary force as a displacement-dependent nonlinear force and an extensive force along a certain length of the microbeam. Galerkin method was used to reduce the order of equations. In order for validation, the obtained results were compared with those of previous theoretical studies conducted using Ritz method for linear and nonlinear models. [8]. Young-Dupré model was employed in [9] to analyze the instability of clamped–clamped microbeam and microcantilever under the effect of capillary and electrostatic forces. They used Galerkin method to solve the nonlinear equations. In this work, the non-ideal boundaries are modeled by adding springs. Guo et al. studied the static and dynamic instability of the MEMS/NEMS actuators under capillary force. For this purpose, they calculated the dimensionless equations of motion when the system was affected by the capillary force with nonlinear vibrations. Furthermore, qualitative analysis of the nonlinear equations, phase curve as well as bifurcation phenomenon were investigated [10]. By reducing the system’s dimensions from micro to nano, the effect of some forces such as Casimir and van der Waals become more influential. The van der Waals force stands for the interaction between the magnetic poles in the atomic scale. The Casimir force is indeed the gravity force between two parallel flat plates, which is resulted from quantum fluctuations in the ground state of the electromagnetic field [11]. Several studies have been conducted on the effect of Casimir and van der Waals forces on behavior of the NEMS systems. Farrokhabadi et al. [12], showed that the Casimir force has significant effect on mechanical behavior of the actuators with small dimension, and also the actuator’s geometry directly affects effectiveness of the Casimir force. Furthermore, they studied the effect of Casimir force on stability and adhesion, and calculated the appropriate length and distance to prevent the adhesion caused by Casimir force. Liang et al. used variational method to investigate the effect of Casimir force on the electrostatically actuated NEMSs. In this work, the generalized differential quadrature technique was used to solve the equations numerically. They showed that the Casimir force, which is in fact a spontaneous force between two electrodes, can reduce the external applied voltage. Moreover, in nano dimensions, by considering the Casimir force, pull-in occurs without applying voltage [13]. In another study, Yu and Wu presented an approximate analytical method to predict static response of then clamped microbeam with axial stress. In this research, the microbeam underwent unilateral (one-sided) or bilateral (two-sided) symmetric electrostatic actuation, and also effects of the fringing field and Casimir force were taken into account. The proposed method was based on the combination of potential energy and selection of form function. In this paper, the solution stability analysis was performed. They showed that this method can be applied to investigate the effect of various physical parameters on mechanical behavior of the electrostatically actuated microbeams [14]. Jia et al. investigated the pull-in instability in the electrostatically actuated micro-switches considering the nonlinear geometric effects, axial force, electrostatic force, and Casimir force. In order to calculate the governing differential equations, the virtual work method was used, and the equations were solved by DQM method. The pull-in voltage and the static deformation were analyzed for four types of boundary conditions [15]. The effect of Casimir force on nano-actuators composed of nano-wire, CNTs, and nanotweezers, has been evaluated in several studies [16–19]; furthermore, the effect of Casimir force on static and dynamic pull-in on beam-type NEMS systems has been studied using various numerical and analytical methods., For instance, Sadighi et al. [20,21] used parameter expansion methods (PEM), Oukad [22] used Newton–Raphson method and finite element method, and Karimpour et al. [23] used DTM method. The effect of Casimir force on micromirrors, CNTs, and nanoactuatrs, has been evaluated in several studies [24–27]. In numerous articles, the cantilever microbeam with rigid paddle at the free end and clamped microbeam with middle rigid paddle have been used as paddle-type MEMS\NEMS for applications such as sensor and actuator [28–34], resonator [35–37], CNT [38,39], and AFM [40,41]; also, in some studies, the T-shaped paddle has been used for designing the sensors [42–46].

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Mokhtari et al. [28] investigated a cantilever microbeam with end rigid paddle as an electrostatic micro-accelerometer. In this modeling, electrostatic actuation was applied on the whole cantilever microbeam, and then the effect of factors such as Casimir force, geometric parameters, size phenomenon, and acceleration on system’s instability was investigated. Nayfeh et al. [29] studied pull-in voltage, static deformation, and natural frequency in a gas sensor composed of a rigid micro-paddle attached to the end of micro-cantilever. They used Hamilton technique to obtain the equations of motion and Galerkin method to separate the equations. In this modeling, the electrostatic actuation was applied merely on the end micro-paddle. The obtained results showed that this type of design was efficient for modeling more accurate mass sensors. Oukad et al. studied mechanical shock in paddle-type micro-cantilever, and evaluated the effect of electrostatic and capillary forces on mechanical shock separately [30]. Alsaleem et al. [31], Sahai et al. [32], and Krylov et al. [33] conducted a laboratory study on mechanical behavior of the paddle-type configuration. Furthermore, recently, Zamanian et al. [44,45] and Firouzi et al. [46] investigated static and dynamic behavior of the cantilever and clamped–clamped microbeam with T-shaped paddle. They showed that these configurations could be used to design a microsensor with uniform deflection advantage, with its mechanical behavior being the same as that of classical microbeams with non-uniform deflection. According to the reviewed papers, it is clear that effects of Casimir and van der Waals as well as the adhesion resulting from capillary force are very important in designing the sensors. The history of studies in this regard indicates that the simultaneous effects of capillary and Casimir forces on mechanical behavior of the electrostatically actuated paddle-type microbeams have not been studied so far [28–41]; furthermore, investigations imply that the effect of Casimir and capillary forces on mechanical behavior of the clamped–clamped microbeams with T-shaped paddle has not been studied so far, and thus the present study is focused on these subjects. In this research, the configuration was considered as a clamped– clamped microbeam with middle T-shaped paddle. In this study, effects of the Casimir, van der Waals, and capillary forces on static behavior, stability, pull-in, natural frequency, primary and subharmonic resonances were studied. For this purpose, the nonlinear equations governing the studied models were solved using Galerkin method. The system linear mode shapes is used as comparison functions. The results were obtained using an analytical (Perturbation) and a numerical (Rung–Kutta) method; then, the two methods were compared in order to validate the obtained results. Effect of the Casimir, van der Waals, and capillary attractive forces on static deformation, natural frequency and frequency response were evaluated for clamped–clamped microbeam with T-shaped paddle model in the presence of electrostatic force and fringing fields. In previous studies [8,9,30], it has been shown that the Young-Dupré model can be used for evaluating the effects of capillary force on the mechanical behavior of clamped microbeams and microcantilevers [8,9] as well as paddle microcantilevers [30]. They have also demonstrated that results of this modeling are in agreement with experimental results. So, this model is used to analyze the capillary instability. In order for validation, results of the present study were compared with those of the previous works. The obtained results showed that the van der Waals and Casimir forces had attractive nature and lead to the increased static deformation as well as occurrence of the pull-in at lower voltages; besides, effects of attraction in Casimir were greater than that in van der Waals. It was shown that changing the Casimir and Van der Waals forces as well as the fringing field effect, in addition to causing changes in the voltage and range, considerably affected the determination of the bifurcation points. The results showed that with a longitudinal increase under the capillary force, the maximum static deformation was increased, which led to instability in the lower capillary coefficients. Moreover, presence of the electrostatic force accelerated the instability resulting from capillary force. 2. Modeling and formulation In the assumed configuration, the T-shaped paddle was connected to the middle of the clamped–clamped microbeam with length of l. As shown in Fig. (1), the left and right sides of the vertical part of the T-shaped paddle have, respectively, length of l1 and l2 relative to the end section of the left side; also, length of the left and right sections of the horizontal part of the T-shaped section is l3 and l4 , respectively. The DC voltage is applied between the horizontal part of the T-shaped object and the opposite electrode, which is at a distance of d to the microbeam. In this configuration, the thickness of the vertical part of the T-shaped paddle (l2 -l1 ) which is, in fact, where the T-shaped paddle is connected to the clamped microbeam, has been considered higher than the thickness of the microbeam by many folds. Moreover, the thickness of the horizontal part of the T-shaped paddle (h2 ) has been considered to be large. As a result, based on these assumptions, the behavior of the T-shaped paddle may be assumed to be rigid. Considering the electric fringing field, the electrostatic force per unit of length for microbeams is calculated as following [1]:

Fel =



ε bT V 2 2 ( d − wT )

2

d − wT 1 + 0.65 bT



(1)

where ε = 8.854 × 10−12 C2 N−1 m−2 is the permittivity of vacuum. V is the applied electrostatic voltage and bT is the width of the T-shaped paddle. A microbeam with a droplet of liquid trapped underneath it causes capillary force. In this configuration, local capillary force exists in each point of the contact surface of the horizontal part of the T-shaped paddle and the liquid as well as between the liquid and the lower surface, whose value equals γ cos (θ ) based on Young-Dupré [47], perpendicular to the width of the microbeam. The value of the equivalent F force is γ A/r in which r denotes the distance between the horizontal part of the T-shaped paddle and the substrate. Moreover, A represents the contact surface between the horizontal part of

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Fig. 1. Schematic representation of clamped–clamped microbeam with T-shaped paddle.

Fig. 2. Effect of capillary and electrostatic forces on microsystem.

the T-shaped paddle and the fluid. When the capillary force is created between the horizontal part of the T-shaped paddle and the lower surface, the microbeam undergoes deflection, and thus the value of distance r varies proportionate to the capillary force. By considering the changes in r based on previous research [48], the resulting capillary force per unit of length between the T-shaped paddle and substrate is expressed as follows [8]:

Fcap =

2γ bT v f cos(θ f ) ( d − wT )

(2)

Where Vf is the ratio between the fluid volume and the total volume underneath microbeam. γ is fluids surface tension and θ f is angel that fluids forms with both the microbeam and the substrate. It is noteworthy that, in this modeling, the material of both parallel plates has been considered equal, and the surfaces are highly hydrophilic. Therefore, the angles created between the fluid and both plates is equalAlso, it has been assumed that the perfect wetting condition holds. Thus, angle theta has been considered zero in the analysis of results. Another important point in this modeling is that due to the high thickness of the T-shaped part, this part has little bending and, therefore, its length is not increased. Consequently, it is assumed that the T-shaped paddle shows rigid behavior. Therefore, the length of the T-shaped part is not increased. In addition, it is assumed that, during the deflection of the microbeam, a leakage of fluids occurs along the width of the horizontal part of the T-shaped paddle. Thus, the volume of the fluid trapped under the horizontal part of the T-shaped paddle can be assumed constant. In the studied model, as can be seen in Fig. (2), the electrostatic and capillary forces have been considered as extensive forces, which are created between the horizontal part of the T-shaped paddle and the opposite electrode, and thus these forces are calculated through integration in the specified interval. Since the T-shaped part has rotational motion, the torque resulted from the electrostatic and capillary forces are created as well. The force and torque resulting from the electrostatic

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Fig. 3. (a) Forces applied on microsystem; (b) microsystem after separation.

actuation and capillary effect are calculated as following:

Fel =

1 ε bT (VDC + VAC cos(t ))2 2

Mel =

1 ε bT (VDC + VAC cos(t ))2 2

FCap = 2γ bT cos(θ f )ν f MCap = 2γ bT cos(θ f )ν f

 

l4 l3 l4 l3

 

l4 l3 l4 l3

1

( d − wT )

1

dx + 0.325ε (VDC + VAC cos(t )) 2

2



l4

1

dx

l3 (d − wT (x )) (d − wT (x ))  l4 ( x − lc ) ( x − lc ) 2 dx + 0.325ε (VDC + VAC cos(t )) dx l3 (d − wT (x )) (d − wT (x ))2

(3)

dx

( x − lc ) dx ( d − wT )

(4) l +l

Where bT and wT indicate width and deflection of the T-shaped paddle. Also, according to Fig. (1), lc = 1 2 2 . The expressions of intermolecular forces are derived by using vacuum quantum fluctuation of the electromagnetic field modes of a cavity comprising two flat conductors. The van der Waals force per unit of length of the microbeam proportionate to the third power of separation is expressed as following:

Fvan =

Ah b 6π ( d − w )

3

(5)

Where Ah is Hamaker constant, the value of which is in the interval [0.4, 4] × 10−19 . The intermolecular force is another Casimir force, which is proportionate to the fourth power of separation. The Casimir force per unit of length is calculated as following [1]:

Fcas =

π 2 hcb 4 240(d − w )

(6)

Where h = 1.055 × 10−34 is the Plank’s constant divided by 2π and c = 2.998 × 108 ms is the speed of light. As shown in Fig. (3), in order for modeling the effects, the T-shaped paddle is considered as a force and a moment on a point at distance of lc from the left end of the clamped microbeam. Assuming rigid displacement for the T-shape section,

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the equations of equilibrium are written based on Newton’s law as following:

F = Fel + Fcap − mT ⇒F=

∂ 2 w ( lc , t ) ∂t2

 l4  l4 1 1 1 2 ε bT (VDC + VAC cos(t ))2 dx + 0 . 325 ε ( V + V cos ( t )) dx DC AC 2 2 ( d − w T (x )) l3 (d − wT (x )) l3  l4 1 ∂ 2 w ( lc , t ) + 2γ bT cos(θ f )ν f dx − mT ( d − w ) ∂t2 T l3

M = Mel + Mcap − IT

∂ 3 w ( lc , t ) ∂ x∂ t 2

 l4  l4 1 ( x − lc ) ( x − lc ) 2 ε bT (VDC + VAC cos(t ))2 dx + 0.325ε (VDC + VAC cos(t )) dx 2 2 l3 (d − wT (x )) l3 (d − wT (x ))  l4 ( x − lc ) ∂ 3 w ( lc , t ) + 2γ bT cos(θ f )ν f dx − IT ∂ x∂ t 2 l3 ( d − wT )

⇒M=

(7)

Where t is time, x is the axial coordinate, w is the transverse deflection of the microbeam. mT , IT are Mass and mass moment of inertia, these expression are expressed as:

m T = [ρ b ( l 2 − l 1 ) × h 1 ] + [ρ b T ( l 4 − l 3 ) × h 2 ]



IT =

  1 ρ bh1 × (l2 − l1 ) × (l2 − l1 )2 + h21 ) + ρ bh1 × (l2 − l1 ) × 12 

+ h¯ =



h1 h¯ − 2

  1 ρ bT h2 × (l4 − l3 ) × (l4 − l3 )2 + h22 ) + ρ bT h2 × (l4 − l3 ) × 12

h1 × (l2 − l1 ) × ( h21 ) × bT + h2 × (l4 − l3 ) × (h1 + h1 × ( l2 − l1 ) × bT + h2 × ( l4 − l3 ) × b

h2 2



2  2 

h2 + h1 − h¯ 2

)×b

(8)

Where h¯ is distance of the mass center of the T-shaped paddle to the microbeam’s upper surface and ρ is material density. The equation of motion and boundary conditions of the clamped microbeam with T-shaped paddle, considering the Casimir, van der Waals, and capillary forces in accordance with Eqs. (3)–(7) and regarding the effects of middle layer elongation and axial force, will be as following (see the appendix of reference [46]):

EI

 l 2 2 ∂w ∂ 4w ∂ 2w π 2 hcb Ah b EA ∂ w + ρ A = + + + N 4 3 2l 0 ∂ x ∂ x4 ∂t2 ∂ x2 240(d − w ) 6π ( d − w )



 ∂ 2 wT ∂ 3 w ∂ Dirac(x − lc ) − mT Dirac (x − lc ) + IT 2 ∂x ∂t ∂ x∂ t 2

  l4 1 1 + dx Dirac (x − lc ) ε bT (VDC + VAC cos(t ))2 2 2 l3 (d − wT (x ))

  l4 1 2 + 0.325 ε (VDC + VAC cos(t )) dx Dirac (x − lc ) l3 (d − wT (x ))

  l4 1 ( x − lc ) ∂ Dirac(x − lc ) 2 ε bT (VDC + VAC cos(t )) + dx 2 2 ∂x l3 (d − wT (x ))

  l4 ( x − lc ) ∂ Dirac(x − lc ) 2 + 0.325 ε (VDC + VAC cos(t )) dx ( d − w ( x )) ∂x T l3

  l4 1 + 2 γ bT cos(θ )ν f dx Dirac (x − l ) ( d − w T (x )) l3

  l4 ( x − lc ) ∂ Dirac(x − lc ) + 2 γ bT cos(θ )ν f dx ( d − w ) ∂x T l3  ∂w  w|x=0 = 0, =0 ∂ x x=0  ∂w  w|x=l = 0, =0 ∂ x x=l

(9)

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249

Where I is the area moment of inertia of the microbeam cross section, E is modulus of elasticity, A is cross-section area and N is the induced axial load on the beam. In the above formula, Dirac denotes the Dirac delta function defined as follows:

l

Dirac (x − lc ) dx = 1



0

(10) x = lc x = lc

0 ∞

Dirac (x − lc ) =

Displacement of the free (free at both ends) microbeam (T-shaped section) and the θ angle are obtained as following using Fig. 4:





h wT (x, t ) = w(lc , t ) + h¯ + (1 − cos θ ) + (lc − x ) tan θ , 2 ∂ w ( lc , t ) θ= ∂x

tan θ ≈ θ (11)

For convenience, the following set of non-dimensional parameters is introduced:

x xˆ = , l

αcas = αel =

w ˆi = i w d

π 2 hcbl 4 240EId5

εb pl 4 2EId3

mT ˆT = m , ρ bhl

,

,

t i = 1, 2, p, tˆ = T

αvdw = α f r−pal =

Ah bl 4 , 6EIπ d4

0.325ε l 4 , EId3

IT IˆT = , bhl 3

 T =

αcap =

ρ Al 4 EI

,

2γ a p cos θ¯ l 4 EId2

α1 = 6

d 2 h

Nl 2 Nˆ = EI

(12)

Now, in non-dimensional form, the equation of motion and associated boundary conditions can be stated as:

  2  lˆ  ∂ 4 wˆ ∂ 2 wˆ αvan αcas ∂ 2 wˆ ∂ wˆ + = + + α1 + Nˆ 3 4 4 ∂ xˆ ∂ xˆ ∂ xˆ2 ∂ tˆ2 0 (1 − wˆ ) (1 − wˆ )

2  

3   ∂ wˆ T  ∂ wˆ 1  ∂ Dirac(xˆ − lˆc ) ˆ ˆ ˆT −m xˆ=l Dirac(xˆ − lc ) + IT xˆ=l ∂ xˆ ∂ tˆ2 ∂ xˆ∂ tˆ2

  lˆ4 1 2 + αel (VDC + VAC cos(t )) dxˆ Dirac (xˆ − lˆc ) 2 ˆl3 (1 − w ˆ T (xˆ))

  lˆ4 1 2 + α f r (VDC + VAC cos(t )) dxˆ Dirac (xˆ − lˆc ) ˆ T (xˆ) lˆ3 1 − w

  l4 (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) 2 2 + αel (VDC + VAC cos(t )) VDC dxˆ 2 ∂ xˆ l3 ( 1 − w ˆ T (xˆ))

  l4 (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) 2 + α f r (VDC + VAC cos(t )) dxˆ ˆ ˆ 1 − w ( x ) ∂ xˆ l3 T

 lˆ 

l  4 4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) + αcap V f dxˆ Dirac (xˆ − lˆc ) + αcap V f dxˆ ˆ T (xˆ) ˆ T (xˆ) ∂ xˆ lˆ3 1 − w l3 1 − w  ∂ wˆ  ˆ |xˆ=0 = 0, w  =0 ∂ xˆ xˆ=0  ∂ wˆ  ˆ |xˆ=lˆ = 0, w  =0 ∂ xˆ xˆ=lˆ

 hˆ ˆ T (xˆ, tˆ) = w ˆ (lˆc , tˆ) + h¯ + w (1 − cos θˆ ) + (lˆc − xˆ)θˆ 2

∂ wˆ (lˆc , tˆ) θˆ = ∂ xˆ

(13)

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Fig. 4. Schematic diagram of T-shaped body displacement.

2.1. Static deformation By assuming the time intervals equal to zero, the equations governing the static deformation are obtained:

  2  lˆ  ∂ 4 wˆ s αvan αcas ∂ 2 wˆ s ∂ wˆ s = + + α1 + Nˆ 3 4 4 ∂ xˆ ∂ xˆ ∂ xˆ2 0 (1 − wˆ s ) (1 − wˆ s )



  lˆ4  lˆ4 1 1 2 2 ˆ + αel VDC dxˆ Dirac (xˆ − lc ) + α f r VDC dxˆ Dirac (xˆ − lˆc ) 2 ˆ T (xˆ) lˆ3 (1 − w lˆ3 1 − w ˆ Ts (xˆ))

 lˆ 

l  4 4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) ˆ + αcap V f dxˆ Dirac (xˆ − lc ) + αcap V f dxˆ ˆ Ts (xˆ) ˆ ˆ ˆl3 1 − w 1 − w ( x ) ∂ xˆ l3 Ts



  l4  l4 (xˆ − lc ) (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) ∂ Dirac(xˆ − lˆc ) αel VDC 2 dxˆ + α f r VDC 2 dxˆ 2 ˆ T (xˆ) ∂ xˆ ∂ xˆ l3 ( 1 − w l3 1 − w ˆ Ts (xˆ))  ∂ wˆ s  ˆ s |xˆ=0 = 0, w  =0 ∂ xˆ xˆ=0  ∂ wˆ s  ˆ s |xˆ=lˆ = 0, w  =0 ∂ xˆ xˆ=lˆ

 hˆ ˆ Ts (xˆ, tˆ) = w ˆ s (lˆc , tˆ) + hˆ¯ + w (1 − cos θˆs ) + (lˆc − xˆ)θˆs 2

∂ wˆ s (lˆc , tˆ) θˆs = ∂ xˆ

(14)

Eq. (14) is solved by Galerkin and using the shape of clamped microbeam modes as comparative functions. Therefore,  ˆ s (x ) = 3i=1 ai ψi (x ) (where ai indicates constant coefficients and ψ i (x) is shape of the symmetrical mode by substituting w of the simple clamped microbeam) in Eq. (14), multiplying ψ j (x), j = 1..3 in the obtained equation, and integrating from xˆ = 0 to xˆ = 1, three coupling algebraic equations are obtained in accordance with Eq. (14), so that by solving the equation and obtaining the coefficients, the static deformation is obtained (Appendix 1). By numerically calculating the coefficients, the values of mode shape can be obtained. Fig. 5 illustrates the first three mode shapes. Fig. 6 illustrates that, for w values less than 0.5, a very good convergence exists between the real value and the value of Taylor’s expansion. It is noteworthy that, due to the non-linear effect of the electrostatic force in MEMS systems, when the dimensionless deflection is increased above 0.5, the pull-in phenomenon occurs. As a result, these systems will not have a deflection of more than 0.5, indicating that the use of Taylor’s expansion will yield more acceptable results. 2.2. Natural frequency Deformation of the system can be considered as the sum of static deformation and a dynamic deformation (vibration) ui (x,t) around this position. Therefore:

ˆ (x, t ) = w ˆ S (x ) + u(x, t ) w ˆ T (x, t ) = w ˆ Ts (x ) + uT (x, t ) w

(15)

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251

Fig. 5. Mode shapes of the clamped–clamped microbeam.

Fig. 6. A comparison between the exact and approximate (Taylor expansion) values of (a)

1 1−wT (x )

(b)

1

(1−wT (x ))2

.

By substituting the Eq. (15) in (13), expanding the electrostatic, capillary, Casimir, and van der Waals forces around the static position, and omitting the terms of Eq. (14) from the obtained equation, equation of the model’s free vibration around the static position is calculated as following:

∂ 4 uˆ ∂ 2 uˆ αvan αcas + = (3uˆ ) + (4uˆ ) 4 ∂ xˆ4 ∂ tˆ2 (1 − wˆ s ) (1 − wˆ s )5   2  lˆ   2  lˆ  ∂ 2 uˆ ∂ wˆ s ∂ wˆ s ∂ uˆ ∂ uˆ ˆ ˆ + α1 +N + α1 × +N ∂ xˆ ∂ xˆ2 ∂ xˆ ∂ xˆ ∂ xˆ2 0 0   2  lˆ

 lˆ  ∂ wˆ s ∂ uˆ  ˆ ∂ 2 wˆ s ∂ 2 uˆ ∂ uˆ ˆ + α1 +N + × + N α 1 ∂ xˆ ∂ xˆ2 ∂ xˆ ∂ xˆ ∂ xˆ2 0 0

2  

3   ∂ uˆT  ∂ uˆ  ∂ Dirac(xˆ − lˆc ) ˆ ˆ ˆT ˆ −m Dirac ( x − l ) + I  xˆ=l c T xˆ=l 2 2 ∂ xˆ ∂ tˆ ∂ xˆ∂ tˆ



  lˆ4  lˆ4 2uˆ uˆ 2 2 ˆ ˆ ˆ ˆ + αel VDC dx Dirac (x − lc ) + α f r VDC dx Dirac (xˆ − lˆc ) 3 2 lˆ3 (1 − w lˆ3 (1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ))

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  l4 ∂ Dirac(xˆ − lˆc ) ∂ Dirac(xˆ − lˆc ) (xˆ − lc ) ˆ + α f r VDC 2 d x 2 ∂ xˆ ∂ xˆ l3 ( 1 − w l3 ( 1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ))

 lˆ 

l  4 4 ˆ uˆ (xˆ − lc ) ∂ Dirac(xˆ − lc ) αcap V f dxˆ Dirac (xˆ − lˆc ) + αcap V f dxˆ 2 2 ∂ xˆ lˆ3 (1 − w l3 ( 1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ))  ∂ uˆ  uˆ|xˆ=0 = 0,  =0 ∂ xˆ xˆ=0  ∂ uˆ  uˆ|xˆ=lˆ = 0,  =0 ∂ xˆ xˆ=lˆ   hˆ uˆT (xˆ, tˆ) = uˆ (lˆc , tˆ) + h¯ + (1 − cos θˆd ) + (lˆc − xˆ)θˆd + αel VDC 2



l4

2uˆ (xˆ − lc )



dxˆ 3

2

∂ wˆ (lˆc , tˆ) θˆd = ∂ xˆ

(16)

By solving the equation using Galerkin method, the natural frequency and shape of the system modes will be obtained. 3 Therefore, ui (x,t) in this equation is assumed as i=0 qi (t )ψi (x ) where qi (t) is time response and ψ i (x) is shape of the dimensionless modes of the simple clamped beam. Then, by multiplying the resulted equation by the shape of mode and integrating from 0 to 1, the three differential equations are obtained as described in Appendix 2. By assuming Pi (t) = Ai eIωt (where ω is the natural frequency and Ai indicates the coefficients) and substituting it in the following equation, forming the matrices of mass [M] and rigidity [K] in the resulted equations, and putting the determinant of the coefficients [K − Mω2 ] equal to zero, the system’s natural frequency is obtained. The Eq. (16) can be restated as follows:

S1 q(t ) + S2 q2 (t ) + S3 q3 (t ) + S4 q˙ (t ) + S5 q¨ (t ) + S6Vac cos(t ) + S7Vac 2 cos2 (t ) + S8Vac q(t ) cos(t ) = 0 ⇒ N1 q(t ) + N2 q2 (t ) + N3 q3 (t ) + N4 q˙ (t ) + q¨ (t ) + F1Vac cos(t ) + F2Vac 2 cos2 (t ) + F3Vac q(t ) cos(t ) = 0 S1 S2 S3 S4 S6 S7 S8 N1 = , N2 = , N3 = , N4 = , F1 = , F2 = , F3 = , S5 S5 S5 S5 S5 S5 S5

(17)

where the multipliers Si; i = 1..8 are constant, which has been obtained in previous section. These constants are described in Appendix 3. 3. Primary resonance Now it is assumed that:

q(t ) = ε q1 (T0 , T1 , T2 ) + ε 2 q2 (T0 , T1 , T2 ) + ε 3 q3 (T0 , T1 , T2 )

(18)

Where T0 = t, T1 = ε t, T2 = ε 2 t, are time scales and ε is a dimensionless book keeping parameter. In order to balance the damping and the excitation forces with the nonlinear term, it is assumed that S4 and S6 has order (ε 2 ) and (ε 3 ). By this assumption and substituting Eq. (18) into (17), it is found:Order ε 1 :

∂ 2 q1 + N1 q1 = 0; ∂ T0 2

N1 =

S1 S5

(19)

Order ε 2 :

∂ 2 q2 ∂ 2 q1 + N q = −2 − N2 q1 2 ; 1 2 ∂ T0 ∂ T1 ∂ T0 2

N2 =

S2 , S5

(20)

Order ε 3 :

∂ 2 q3 ∂ 2 q1 ∂ 2 q2 ∂ 2 q1 ∂ q1 + N1 q3 = −2 −2 q2 − − 2N2 q1 q2 − N3 − N4 q1 3 − F cos(T0 ) − F  cos(2T0 ) 2 ∂ T0 ∂ T2 ∂ T0 ∂ T1 ∂ T0 ∂ T0 ∂ T1 2 N3 = Since ω =



S3 S4 S6 S7 , N4 = , F= , F = S5 S5 S5 S5

(21)

N1 , the homogeneous solution of (19) can be expressed in the following term:

q1 = A(T1 , T2 )eiωT0 + A¯ (T1 , T2 )e−iωT0

(22)

where A(T1 , T2 ) is a complex constant. By substituting (22) into Eq. (20):

  ∂ 2 q2 ∂ A iωT0 + N q = −2 i ω e + (N2 A2 )e2iωT0 + N2 AA¯ + Cc 1 2 ∂ T1 ∂ T0 2

(23)

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where CC stands for the complex conjugate of the preceding terms. Eliminating the secular terms from q2 demands that:

A = B(T2 )

(24)

The particular solution of (23) is:

q2 =

N2 2 2iωT0 N2 ¯ 2 −2iωT0 N2 A e A e + − 2 2 AA¯ 3ω 2 3ω 2 ω

(25)

By substituting q1 and q2 from (22) and (25), into (21), considering resonance frequency as  = ω + ε 2 σ , and keeping the terms that contains secular terms, it gets:

 ∂ 2 q3 F + N1 q3 = −iωN4 A − 2iωA + SA2 A¯ + eiσ T2 eiωT0 + Cc + NST ; 2 2 ∂ T0

S=

10N2 2 − 3N3 3ω 2

(26)

Where σ is detuning parameter and NST stands for the rest of the terms, which do not produce secular terms. By expressing A in the polar form (A = 12 a(T2 )eiβ ) and substituting it into Eq. (26), elimination of secular terms demands:

1 3 1 F F Sa − N4 iωa + cos(σ T2 − β ) + i sin(σ T2 − β ) = 0 8 2 2 2

−ia + aβ  +

(27)

Assuming γ = σ T2 − β , and separating real and imaginary parts in (27) yields

1 F a = − N4 a + sin γ 2 2ω

(28)

1 F S a3 + cos γ 8ω 2ω

aγ  = σ a +

(29)

For non-transient solution a = 0 , γ  = 0 so, for equilibrium solution one must set:

1 F N4 ωa = sin γ 2 2



(30)

 F 1 σ ω + Sa2 a = − cos γ 8

(31)

2

By squaring (30) and (31) and adding the results, we obtain the frequency response equation as:

1 2 2 2 N4 ω a + 4



2 1 F2 σ ω + S a2 a2 = 8

(32)

4

From this equation, it is obvious that the maximum amplitude occurs when terms inside the parenthesis are equal to zero. So, maximum amplitude and resonant shift are as follows: 1 N 2 4 4

ω 2 a2 =

−ω =−

F2 4



F2

(33)

Samax 2 S F2 =− 8ω 8 ω3 N4 2

(35)

⇒ amax =

ω2 N4 2

The characteristic equation of Jacobian matrix of Eq. (32) is

λ + N4 λ + 2



N4 2 S a0 2 3S a0 2 + (σ + )(σ + ) =0 4 8ω 8ω

(35)

Considering Eq. (35), the eigenvalue of the Jacobian matrix will be as follows



λ1,2 =

−N4 ±

−4(σ + S8aω0 )(σ + 2 2

3S a0 2 8ω

)

(36)

Instability condition occurs when one of the eigenvalues of the system gets a positive value, and the bifurcation point occurs when one of the eigenvalues is equal to zero. It is obtained from Eq. (35) that

N4 2 + 4



σ

S a0 2 + 8ω



 3S a0 2 σ+ =0 8ω

(37)

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Fig. 7. Cantilever beam with a rigid paddle attached to end section when l3 = l, h1 = h.

Fig. 8. Maximum deflection of microbeam with respect to the variation of Vf ; where θ f = 0,

γ f = 75 mN/m, l1 = 200 μm, l2 = 300 μm, d = 4 μm.

Fig. 9. Variation of the first natural frequency with the variation of DC voltage, where l1 = 250 μm, l2 = 300 μm, d = 4 μm, h = 1.5 μm, h1 = 3 μm f or x = 0..l1 , b = 25 μm f or x = l1 ..l2 ..

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Fig. 10. Variation of response amplitude a with respect to /ω.

Fig. 11. Maximum static deflection with respect to the electrostatic voltage by considering fringing field effect based on Palmers’ and Mejis’ methods [49,50].

3.1. Subharmonic resonance If one assumes the frequency of AC voltage as  = 2ω + ε 2 σ , then secular terms will be appeared in the equation obtained by equating the terms with same power of ε 3 . It means that system has a subharmonic resonance frequency at ∼ =2ω. Similar to work done in previous section we can obtain the frequency response equation of subharmonic resonance as:

1 2 2 N4 a + 4



aσ S a3 + 2 8

2 =

K 2 a2

ω2

,

K=

N F  F3 2 V + V ac ac 4 6ω 2

The stability of the stationary solution may be studied by constructing the Jacobian matrix.

(38)

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Fig. 12. (a) Schematic view (b) Maximum static deformation based on applied voltage when the T-shaped paddle is placed in different position.

3.2. Numeric analysis The numerical solution was obtained using Maple 2017 software based on the Rung–Kutta Fehlberg algorithm. The algorithm called RKF45 finds numerical solutions using both fifth- and fourth-order Runge–Kutta. For solving the equation, first, the initial condition was considered equal to zero, and the excitation frequency was considered less than the obtained frequency from the perturbation solution for maximum amplitude. Then, the amplitude of the steady state solution was called from the time history of the numerical solution. Subsequently, the excitation frequency slightly increased and the steady state of the previous step was considered as the initial condition of the next step. In all the steps, the velocity of the initial condition was considered equal to zero. It must be noted that here the numerical solution was obtained from a forward change of the value of the excitation frequency and thus not all the points of the resonance frequency were covered due to the jump phenomenon. If one wants to cover all the points, a backward change of the value of the excitation frequency must be performed. Therefore, the unstable solution may not be obtained using this numerical method and instead, the numerical shooting method may be used [1]. 4. Discussion and results Since considering the clamped-free boundary conditions and the values of l3 = l and h1 = h are considered in the codes, then, the microbeam will become in the form of a cantilever beam with a rigid paddle attached to end section

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Fig. 13. Maximum deflection variations of the clamped–clamped microbeam with T shped paddle with respect to the variation of electrostatic voltage for different values of van der waals coefficient.

Fig. 14. Maximum deflection variations of the clamped–clamped microbeam with T shped paddle with respect to the variation of electrostatic voltage for different values of casimir coefficient.

(Fig. 7). In [30], static deformation has been studied in absence of the electrostatic force and under the effect of capillary force. Comparison of the results is shown in Fig. 8, which indicates a very good consistency between the results; furthermore, comparison of the results with the similar work [29], that takes into account the effect of electrostatic force but ignores the effects of Casimir and capillary forces is shown in Fig. 9, which represents the consistency of this study with previous works. The equation of motion is solved using two methods, multiple scale perturbation and Rung–Kutta numerical method. Comparison of the results of the perturbation and numerical solutions is shown in Fig. 10, indicating a very good consistency between the results. The method used in this article to calculate the fringing field force was based on Palmer’s model. Another common and acceptable method for evaluating the fringing field effect is based on Mejis–Fokkema model [49,50]. Fig. 11 depicts the fringing field effect on maximum static deformation based on Palmers’ and Mejis’ methods. It is evident that the electrostatic force is similar with little difference across the two methods by considering the fringing field effect. Therefore, the use of either method will lead to relatively similar results. It must be noted that, in other figures, Palmer’s model has been applied to considering fringing field effects. The physical and geometric parameters are presented in Table (1).

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Fig. 15. Maximum deflection variations of the clamped–clamped microbeam with T shped paddle with respect to the variation of electrostatic voltage for different values of fringing field.

Fig. 16. Interaction effect of the parameters fringing field (α fr ) and casimir force (α cas ) on (α el VDC 2 )pull in.

Fig. 17. Maximum deflection variations with respect to the variations of electrostatic voltage for different values of (L4 − L3 )/L for the clamped–clamped microbeam with T shaped paddle.

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Fig. 18. Maximum deflection variations with respect to the variation of the ratio Vf for different values of (L4 − L3 )/L for the clamped–clamped microbeam with T shaped paddle.

Fig. 19. Maximum deflection variations of the clamped–clamped microbeam with T shaped paddle with respect to the variation of the ratio Vf for different 2 in presence and absence of fringing field. values of αel VDC

Table 1 Geometric and physical parameters of the clamped–clamped model. l1

l2

l3

l4

l

h1

229.5 μm h2 4.5 μm

280.5 μm b 100 μm

382.5 μm h 1.5 μm

510 μm

3.5 μm E 169 Gpa

0

75 mN/m

127.5 μm bT 120 μm d 1.18 μm

θf

γf

ρ

2230 kg/m3

Fig. 12 depicts the maximum static deformation based on applied voltage. It is obvious that pull-in instability differs in the case where the T-shaped paddle is placed in the middle of the beam compared to the case where it is not. When the T-shaped paddle is a long distance from the middle, the angle of rotation contributes to the early occurrence of pull-in. However, when it approaches the middle, the pull-in phenomenon occurs with delay. Therefore, when the T-shaped paddle is placed far from the middle of the microbeam, pull-in occurs earlier than other cases due to the rotation of the paddle. It must be mentioned that, because we aimed for uniform deflection in this study and attempted to examine the system

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Fig. 20. Maximum deflection variations of the clamped–clamped microbeam with T shped paddle with respect to the variation of the ratio Vf for different values of axial force.

Fig. 21. Maximum deflection variations of the clamped–clamped microbeam with T shped paddle with respect to the variation of the ratio Vf for different values of middle elongation coefficient.

Fig. 22. First natural frequency variations of the clamped–clamped microbeam with T shaped paddle with respect to the variations of electrostatic voltage for different values of van der waals coefficient.

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Fig. 23. First natural frequency variations of the clamped–clamped microbeam with T shaped paddle with respect to the variations of electrostatic voltage for different values of casimir coefficient.

Fig. 24. First natural frequency variations of the clamped–clamped microbeam with T shaped paddle with respect to the variations of electrostatic voltage for different values of fringing field.

in this state, other results of the article were extracted by considering the system in the symmetric form and placing the T-shaped paddle in the middle of the microbeam. In this section, effects of attraction forces of van der Waals and Casimir on the static deformation were evaluated for clamped–clamped microbeam with T-shaped paddle. As shown in Figs. 13 and 14, the van der Waals and Casimir forces have attractive nature, and cause increase in the static deformation of the clamped microbeam, meanwhile, reduce the pullin voltage. It should be noted that the pull-in voltage in the shapes is a voltage at which the slope of the curve of static deformation versus voltage inclines towards infinity; besides, as can be seen, with increase in values of the coefficients of the van der Waals and Casimir forces the pull-in voltage is reduced, and the static deformation is increased. Fig. 15 indicates Wmax versus α el VDC 2 for different values of the fringing field. As is obvious in this figure, the fringing field had a great effect on the pull-in phenomenon. It can be observed that with an increase in the value of the fringing field, the pull-in voltage was reduced whereas the static deformation increased. The interaction effect of the parameters α fr and α cas on the pull-in voltage is demonstrated in Fig. 16. This figure shows that the presence of the fringing field and Casimir force decreased the pull-in voltage of the microsystem. Fig. 17 evaluates the effects of the geometry of the T-shaped part on system instability in the presence of electrostatic, fringing field, Casimir, Van der Waal, and capillary forces simultaneously. Based on this figure, the longer the length of

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Fig. 25. Variation of response amplitude a with respect to /ω in presence and in absence of the intermolecular forces.

Fig. 26. Variation of response amplitude a with respect to /ω in presence and in absence of the fringing field effect.

the horizontal part of the T-shaped beam, the earlier the pull-in instability. This is because the longer the length of the horizontal part of the T-shaped paddle, the higher the level of electrostatic force. Moreover, since the surface between the horizontal part and the substrate is increased, the level of capillary force resulting from the presence of the fluid between the two plates is increased. In fact, by increasing the surface of the horizontal part of the T-shaped beam, electrostatic and capillary forces and torques are increased, and therefore, pull-in instability occurs earlier. This section is focused on the effect of capillary force on the instability and adhesion of clamped–clamped microbeam with T-shaped paddle. L4 –L3 mentioned in the above figures is length of the section, on which the capillary and electrostatic forces were applied. Fig. 18 shows the maximum static deformation for coefficient (Vf ) for different values of dimensionless length. Where Vf is the ratio between the fluid volume and the total volume underneath microbeam. This figure indicates that with increase in length and, as a result, increase in capillary force, the instability occurs at lower Vf and the static deformation is also increased. Fig. 19 indicates the maximum deformation relative to the coefficient (Vf ) both in the presence and absence of the electrostatic force as well as the fringing field effect. As can be seen, in presence of the electrostatic force, the capillary force-caused instability and adhesion occur sooner, and the static deformation is also increased. The reason is that, firstly, the electrostatic and capillary forces are in the same direction, and secondly, they have nonlinear nature; as a result, the capillary force significantly affects the value of pull-in voltage and system’s deformation. Furthermore, as shown in the figure, the fringing field effect enhanced the electrostatic force by adding more amplitude to attractive actuation, and, as a result, increased the deformation of the system. Fig. 20 represents the effect of axial force change on maximum deformation of the clamped–clamped microbeam with middle rigid paddle for middle layer elongation coefficient of 0.1. Values of the axial force are equal to −10, 0, 10, and 20 N. It is observed that increasing the axial force leads to reduction in the curve’s slope; consequently, the capillary force-caused

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Fig. 27. Variation of response amplitude a with respect to σ for different values of VDC .

instability occurs for larger values of Vf . This means that the lower the axial force applied on the microbeam, the lower the capillary force with which the maximum deformation is maximized. Fig. 21 shows the effect of change in the middle elongation coefficient for the maximum static deformation in a case that the axial force is equal to zero. Values of the middle layer elasticity coefficients are equal to 0.1, 5, and 10. As can be inferred from the figure, for small values of the Vf coefficient and, in fact, small capillary force, the middle layer elasticity coefficient has no effect on the maximum static deformation; however, if the Vf coefficient is further increased, the lower is the middle layer elasticity coefficient, the lower will be the Vf coefficient at which the capillary-caused instability occurs, and also the maximum deformation will be increased as well. In this model, the higher value of Vf = 1.5 × 10−5 of the middle layer elasticity coefficient affects the maximum deformation. Figs. 22 and 23 show the effect of Casimir and van der Waals forces on the main natural frequency. As shown in the above figures, for low values of the applied voltage, the Casimir and van der Waals coefficients have no effect on the natural frequency, while by increasing the applied voltage, the increased Casimir and van der Waals coefficients result in the reduced natural frequency value; furthermore, due to attractive nature of the Casimir and van der Waals forces, the pull-in occurs at lower voltages by increasing these forces. Fig. 24 shows the effect of the fringing field on the first natural frequency around the static position for different applied voltage values. As shown in the figure, increasing the fringing field decreased the first natural frequency and the pull-in occurred at lower voltages by increasing the fringing field. Eq. (32) shows that the amplitude of the frequency response of the system at the primary resonance depends on the values of ω, N4 , S, F where ω is the natural frequency of the linear system, F depends on excitation that contains the term VAC cos (t), N4 is resulted from the viscous damping, and S is resulted from non-linear electrostatic terms. By considering 10N 2

Eq. (26), the amount of S is equal to S = 3ω22 − 3N3 where N2 and N3 are due to non-linear quadratic and cubic electrostatic terms, respectively. Figs. 25 and 26 illustrate the changes of the system’s equilibrium amplitude a versus the change of σ in the presence and absence of the intermolecular force and the fringing field, respectively. These figures show that the system had unstable solutions for negative values of σ . It was demonstrated that if σ was positive, the gradual decrease of σ would cause the value of a to move toward the upper branch until reaching the saddle-node bifurcation point A. Then, it would jump to the lower branch. Moreover, it was indicated that if the stationary solution was on the lower branch, then, increasing σ would slowly cause the value of a to move toward the lower branch until reaching the saddle-node bifurcation point B. Afterwards, it would jump to the upper branch. Fig. 25 demonstrates the influence of Casimir force (van der Waals force) on the resonance frequency. As shown in the figure, the presence of Casimir force (van der Waals force) increased the nonlinear shift of the resonance frequency as well as the maximum amplitude. As can be observed, the Casimir force had higher impact relative to the van der Waals force Fig. 26 indicates that by considering the fringing field effect, the nonlinear shift of the resonance frequency and also the maximum amplitude increased. This was due to the fact that the amount of S increased while the amount of ω decreased. It must be noted that the amount of F increased due to an increase of the value of the static deflection and thus according to Eq. (35),  − ω increased. It must be mentioned that the amount of N4 obtained due to damping was almost constant. Therefore, by considering Eq. (33), an increase of the value of F and a decrease of the value of ω caused the amplitude to increase. The variation of the equilibrium solution a as a function of the detuning parameter σ at the subharmonic condition is studied in Figs. 27–29. Eq. (38) shows that the amplitude of the frequency response of the system at subharmonic resonance depends on the values of ω, N4 , N5 , K, where K is caused by AC electrostatic terms; the other parameters are described in the last section. These figures show that the system had two nontrivial branches which were approximately parallel. As is obvious in Fig. 27, the branches inclined to the right side for the small value of DC electrostatic voltage and by increasing the value

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Fig. 28. Variation of response amplitude a with respect to σ in presence and in absence of the intermolecular forces when VDC = 3.

Fig. 29. Variation of response amplitude a with respect to σ in presence and in absence of the fringing field effect when VDC = 3.

of DC voltage, the branches inclined to the left. This figure demonstrates that by increasing the value of DC electrostatic voltage, the hardening behavior changed to softening behavior. In fact, for small values of DC voltage, hardening stretching effect overcame softening electrostatic effect. Here, the effect of intermolecular forces (Casimir and van der Waals) and the fringing field on the variations of the bifurcation diagram was considered according to the variation of the control parameter σ where σ =  − 2ω. It is observed from Figs. 28 to 29 that by an increase of the value of intermolecular forces (the fringing field), the interval of the unstable trivial solution increased. Comparing these figures revealed that the variations resulted from an increase of the value of Casimir force were larger than the variations resulted from an increase of the value of van der Waals force. Moreover, Figs. 28 and 29 show that depending on the value of σ , an increase of the value of intermolecular forces (the fringing field) may decrease or increase the amount of the nontrivial solution. 5. Conclusion In this study, the pull-in instability, static deformation, and natural frequency were investigated in clamped–clamped microbeams with T-shaped paddle in the simultaneous presence of electrostatic, capillary, Casimir, and van der Waals forces. In analyses of the clamped–clamped microbeam with T-shaped paddle, effects of the middle layer elongation as well as the axial force were taken into consideration. The system was considered as a cantilever microbeam with a T-shaped rigid paddle attached to the end of the cantilever microbeam and middle of the clamped microbeam. It was assumed that the beams follow Euler Bernoulli theory. In order to solve the nonlinear equations, the Galerkin-based rank reduction method was used. The obtained results showed that the van der Waals and Casimir forces have attractive nature and cause increase in the static deformation as well as occurrence of pull-in at lower voltages; besides, the effects of attraction in Casimir is more than that in van der Waals. The results showed that increasing the dimensionless length, which undergoes the capillary and electrostatic forces (horizontal part of the T-shaped paddle) leads to increase in the maximum static deformation as well as instability in the lower capillary coefficients. Furthermore, the presence of the electrostatic force also accelerates the instability caused by the capillary force. It is shown that changing the Casimir and Van der Waals forces as well as the

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fringing field effect, besides causing changes in the voltage and range, considerably affected determination of the bifurcation points. Acknowledgment The study was performed in Department of Mechanical Engineering, Kharazmi University, under a grant presented by Vice Chancellor in Research, which should be acknowledge. Appendix 1 3 

ψ j (x )

3  i=1

j=1

ai

3 3  ∂ 4 ψi ( x )  αvan αcas = ψ j (x )  + ψ j (x )   4  3 3 4 3 ∂ xˆ 1 − i=1 ai ψi (x ) 1 − i=1 ai ψi (x ) j=1 j=1 ⎧ ⎫  2  ⎨  lˆ  ⎬  3 3 3 2  ∂ ψi ( x ) ∂ ψ ( x) i + ψ j (x ) α1 ai + Nˆ ai ∂ xˆ ∂ xˆ2 ⎩ ⎭ i=1 0 j=1 i=1

+

3 

 ψ j (x )αel VDC 2

j=1

+

3 

+

3 

ψ j (x )α f r VDC 2





ψ j (x )αcap V f

j=1

1

lˆ4 lˆ3

lˆ4 lˆ3



dxˆ Dirac (xˆ − lˆc ) 2

(1 − wˆ Ts (xˆ))

lˆ3

j=1

lˆ4



1 dxˆ Dirac (xˆ − lˆc ) ˆ Ts (xˆ) 1−w



1 dxˆ Dirac (xˆ − lˆc ) ˆ Ts (xˆ) 1−w

 (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) + ψ j (x )αel VDC dxˆ 2 ∂ xˆ l3 ( 1 − w ˆ Ts (xˆ)) j=1

  l4 3  (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) + ψ j (x )α f r VDC 2 dxˆ ˆ ˆ 1 − w ( x ) ∂ xˆ l3 Ts j=1

l  3 4  (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) + ψ j (x )αcap V f dxˆ ˆ Ts (xˆ) ∂ xˆ l3 1 − w j=1 3 

ˆ Ts (xˆ, tˆ) = w

2

3  i=1

θˆs =

3  i=1



l4

 hˆ ai ψi (lˆc ) + hˆ¯ + (1 − cos θˆs ) + (lˆc − xˆ)θˆs 2

∂ ψi (lˆc ) ai ∂ xˆ

Appendix 2

   3 3   ∂ 4 ψi ( x ) ∂ 2 qi (t ) ψ j (x ) qi (t ) + ψ j (x ) ψi ( x ) ∂ xˆ4 ∂ tˆ2 j=1 i=0 j=1 i=0     3 3 3 3     αvan αcas = ψ j (x ) 3 qi (t )ψi (x ) + ψ j (x ) 4 qi (t )ψi (x ) (1 − wˆ s )4 (1 − wˆ s )5 j=1 i=0 j=1 i=0  lˆ   3 3   ∂ ψi ( x ) ∂ 2 ψi ( x ) ∂ wˆ s + ψ j (x ) α1 × qi (t ) + Nˆ qi (t ) ∂ xˆ ∂ xˆ ∂ xˆ2 0 j=1 i=0 ⎧ ⎫  2 ⎨  lˆ  ⎬ 3 3 3  ∂ ψi ( x ) ∂ 2 ψi ( x ) + ψ j (x ) α1 qi (t ) + Nˆ qi (t ) ∂ xˆ ∂ xˆ2 ⎩ ⎭ i=0 0 j=1 i=0

3 



3 

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     lˆ 3 3 2   ˆ ∂ ψ ( x ) ∂ w ∂ ψ ( x ) ∂ 2 wˆ s s i i ˆ + ψ j (x ) α1 + Nˆ qi (t ) + ψ ( x ) α × q ( t ) + N 1 j i ∂ xˆ2 ∂ xˆ ∂ xˆ ∂ xˆ2 0 0 j=1 i=0 j=1 i=0      2   3 3 3    ∂ 2 qi (t ) ∂ ψi (x ) ∂ Dirac(xˆ − lˆc ) ∂ uˆT ˆ ˆ − ψ j (x ) mˆ T Dirac ( x − l ) + ψ j (x ) IˆT | |xˆ=l c xˆ=l 2 2 ∂ xˆ ∂ xˆ ∂ tˆ ∂ tˆ j=1 j=1 i=0



  lˆ4  lˆ4 3 3   2uˆ T uˆT 2 2 ˆ ˆ ˆ ˆ + αel ψ j (x ) VDC dx Dirac (x − lc ) + α f r ψ j (x ) VDC dx Dirac (xˆ − lˆc ) 3 2 lˆ3 (1 − w lˆ3 (1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ)) j=1 j=1

 lˆ 

  l4 3 3 4   uˆT 2uˆT (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) 2 ˆ + αcap ψ j (x ) V f dxˆ Dirac (xˆ − lc ) + αel ψ j (x ) VDC dxˆ 2 3 ˆl3 (1 − w ∂ xˆ l3 ( 1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ)) j=1 j=1



  l4  l4 3 3   uˆT (xˆ − lc ) uˆT (xˆ − lc ) ∂ Dirac(xˆ − lˆc ) ∂ Dirac(xˆ − lˆc ) ˆ + αfr ψ j (x ) VDC 2 d x α ψ ( x ) V dxˆ cap j f 2 2 ˆ ∂ x ∂ xˆ l3 ( 1 − w l3 ( 1 − w ˆ Ts (xˆ)) ˆ Ts (xˆ)) j=1 j=1  ∂ uˆ  uˆ|xˆ=0 = 0,  =0 ∂ xˆ xˆ=0  ∂ uˆ  uˆ|xˆ=lˆ = 0,  =0 ∂ xˆ xˆ=lˆ   3  hˆ ˆ ¯ ˆ ˆ ˆ u T ( x, t ) = qi (t )ψi (lc ) + h + (1 − cos θˆd ) + (lˆc − xˆ)θˆd 3 



lˆ



∂ wˆ s ∂ xˆ

3 

qi (t )

i=0

3 

2

i=0

θˆd =



2

∂ ψi (lˆc ) ∂ xˆ

Appendix 3

  

 lˆ   lˆ4 4 ∂ 4 ϕ1 ϕ2 (xˆ, t ) ϕ2 (xˆ, t ) 2 2 ˆ ˆ S1 = − 2αel Vp dxˆ Dirac (xˆ − l ) ϕ1 dxˆ   3 dxˆ Dirac(xˆ − l ) − α f rVp ˆ 2 ∂ xˆ4 0 lˆ3 l3 ( 1 − ws2 ( x ˆ)) 1 − ws2 xˆ

 l 

 l    lˆ  4 4 (xˆ − lc )ϕ2 (xˆ, lˆ) (xˆ − lc )ϕ2 (xˆ, lˆ) ∂ Dirac(xˆ − lˆ) ∂ Dirac(xˆ − lˆ) 2 ˆ ˆ ϕ1 dxˆ + 2αal Vp2 d x + α V d x fr p ∂ xˆ ∂ xˆ 0 l3 l3 (1 − ws2 (xˆ))3 (1 − ws2 (xˆ))2 

lˆ



4αcas





10αcas



lˆ

20αcas



lˆ

lˆ lˆ 3αvdW ϕ12 dxˆ + ϕ12 dxˆ 5 0 (1 − ws1 (xˆ))5 0 1 − ws1 (xˆ)    lˆ  lˆ4  lˆ  l4 ϕ2 2 (xˆ − lˆc )ϕ2 2 ∂ Dirac(xˆ − lˆ) 2 2 ˆ S2 = −3αel Vp dxˆ Dirac (xˆ − l )ϕ1 dxˆ + 3αel Vp dxˆ ϕ1 dxˆ 4 4 ˆ ∂ xˆ 0 l3 ( 1 − ws2 ( x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))

 lˆ  lˆ    lˆ  l4 4 ϕ 2 2 (xˆ, t ) (xˆ − lˆc )ϕ2 2 ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + α f rVp 2 ˆ ˆ ˆ − α f r Vp 2 d x Dirac ( x − l d x ϕ1 dxˆ 3 4 ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))

+



+  S4 =

(1 − ws1 (xˆ))5 lˆ

0

cˆ1 ϕ12 dxˆ +

 0

lˆ



0



ϕ1 3 dxˆ +

6αvdW



lˆ

10αvdW



lˆ

ϕ1 3 dxˆ (1 − ws1 (xˆ))5 0 (1 − ws1 (xˆ))5 0    lˆ  lˆ4  lˆ  l4 ϕ2 3 (xˆ − lˆc )ϕ2 3 ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + 4αel Vp 2 ˆ ˆ ˆ S3 = −4αel Vp 2 d x Dirac ( x − l d x ϕ1 dxˆ 5 5 ˆ ∂ xˆ 0 l3 ( 1 − ws2 ( x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))

 lˆ  lˆ    lˆ  l4 4 ϕ 3 2 (xˆ, t ) (xˆ − lˆc )ϕ2 3 ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + α f rVp 2 ˆ ˆ ˆ − α f r Vp 2 d x Dirac ( x − l d x ϕ1 dxˆ 4 4 ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ)) +

ϕ1 4 dxˆ + lˆ4

lˆ3

(1 − ws1 (xˆ))5 

0

ϕ1 4 dxˆ

cˆ2 ϕ2 dxˆ Dirac (xˆ − lˆ)ϕ1 dxˆ −

 0

lˆ





lˆ4 lˆ3

 cˆ2 (xˆ − lˆc )ϕ2 dxˆ

∂ Dirac(xˆ − lˆ) ϕ1 dxˆ ∂ xˆ

B. Firouzi and M. Zamanian / Applied Mathematical Modelling 71 (2019) 243–268

267

  ∂ ϕ1  ∂ Dirac(xˆ − lˆ) ϕ1 dxˆ xˆ=l ∂ xˆ ∂ xˆ 0 0 0    lˆ  lˆ4  lˆ  lˆ4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆ) ˆ S6 = −2αell VDC dxˆ Dirac (xˆ − l )ϕ1 dxˆ + 2αel VDC dxˆ ϕ1 dxˆ 3 3 ˆ ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 l ˆ)) (1 − ws2 (xˆ)) 3



  lˆ  lˆ4  lˆ  lˆ4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + 2α f rVDC ˆ ˆ − 2α f rVDC d x Dirac ( x − l dxˆ ϕ1 dxˆ 2 2 ˆ ˆ ∂ xˆ 0 l3 ( 1 − ws2 ( x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))    lˆ  lˆ4  lˆ  lˆ4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + αel ˆ ˆ ˆ S7 = −αel d x Dirac ( x − l d x ϕ1 dxˆ 3 3 ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 lˆ3 (1 − ws2 (x ˆ)) ˆ))    lˆ  lˆ4  lˆ  lˆ4 1 (xˆ − lc ) ∂ Dirac(xˆ − lˆ) ˆ − αfr dxˆ Dirac (xˆ − l )ϕ1 dxˆ + α f r dxˆ ϕ1 dxˆ 2 2 ˆ ˆ ∂ xˆ 0 l3 ( 1 − ws2 ( x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))    lˆ  lˆ4  lˆ  l4 ϕ2 (xˆ − lc )ϕ2 ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + 2αel VDC ˆ ˆ ˆ S8 = −2αel VDC d x Dirac ( x − l d x ϕ1 dxˆ 3 3 ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ))    lˆ  lˆ4  lˆ  l4 ϕ2 (xˆ − lc )ϕ2 ∂ Dirac(xˆ − lˆ) ˆ)ϕ1 dxˆ + 2α f rVDC ˆ ˆ ˆ − 2α f rVDC d x Dirac ( x − l d x ϕ1 dxˆ 2 2 ∂ xˆ 0 lˆ3 (1 − ws2 (x 0 l3 ( 1 − ws2 ( x ˆ)) ˆ)) 

S5 =

lˆ

ϕ12 dxˆ + α3



lˆ

ϕ12 Dirac(xˆ − lˆ)dx − α4



lˆ

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