THE INFLUENCE OF SMALL SCALE ON THE PULL-IN BEHAVIOR OF NONLOCAL NANOBRIDGES CONSIDERING SURFACE...

5/26/2018 THE INFLUENCE OF SMALL SCALE ON THE PULL-IN BEHAVIOR OF NONLOCAL NANOBRIDGES CONSIDERING SURFACE EFF

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International Journal of Applied MechanicsVol. 6, No. 3 (2014) 1450030 (22pages)c Imperial College Press

DOI: 10.1142/S1758825114500306

THE INFLUENCE OF SMALL SCALE ON THE PULL-IN

BEHAVIOR OF NONLOCAL NANOBRIDGES CONSIDERING

SURFACE EFFECT, CASIMIR AND

VAN DER WAALS ATTRACTIONS

HAMID M. SEDIGHI

Department of Mechanical Engineering

Shahid Chamran UniversityAhvaz 61357-43337, Iran

[email protected]

[email protected]

Received 8 October 2013Revised 24 January 2014

Accepted 28 January 2014Published 10 April 2014

This paper is proposed to study the dynamic pull-in instability of nonlocal nanobridges

incorporating the surface effect and intermolecular forces. The second-order frequency-amplitude relation is introduced via an asymptotic approach namely homotopy pertur-bation method (HPM). The effects of applied voltage and intermolecular parameters onpull-in instability as well as the natural frequency are investigated. Furthermore, theinfluence of nonlocal parameter and surface energy on the dynamic pull-in voltage isconsidered. It is shown that two terms in series expansions are sufficient to produce anacceptable solution of the mentioned nanostructure. The obtained results from numeri-cal methods verify the strength of the analytical procedure. The qualitative analysis ofthe system dynamic shows that the equilibrium points of the autonomous system includecenter points with periodic trajectories and unstable saddle points with homoclinic orbits.

Keywords: Nonlocal nanoactuator; surface effect; Casimir and van der Waals attractions;

dynamic pull-in instability; homotopy perturbation method.

1. Introduction

Over the last few years, the applications of nanoelectromechanical systems (NEMS)

due to their excellent mechanical and electrical properties have been developed and

accordingly interest in the nonlinear analysis of nanoscale structures has been grown

[Guo and Zhao, 2006; Tadi Beniet al., 2011; Kacem, 2011; Chan et al., 2011; Koochi

et al., 2012; Hasheminejad et al., 2012; Sedighiet al., 2014]. Nanotechnological inves-

tigation on vibration properties of nanobeams under certain support conditions isimportant because such components can be used in components such as nanosen-

sors and nanoactuators. As the dimensions of a structure approach the nanoscale,

the properties and elastic field can be size-dependent and new phenomena such

as van der Waals [Soroush et al., 2010] and Casimir forces [Noghrehabadi et al.,

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H. M. Sedighi

2012] should be taken into consideration. The effect of vacuum fluctuations can

be modeled through the dispersion forces, i.e., Casimir and van der Waals attrac-

tions. The van der Waals force represents the electrostatic interaction between pair

of magnetic poles at the atomic scale. The Casimir effect stands for the attrac-tive force between two flat parallel plates of solids which originates from quantum

fluctuations in the ground state of the electromagnetic field [Moghimi Zand and

Ahmadian, 2010]. Several investigations have studied the pull-in instability and

nonlinear analysis of nanoscale structures by employing different assumptions and

theories [Rasekh and Khadem, 2011; Mahmoud et al., 2012; Tadi Beni et al., 2013].

A distributed parameter model was employed by Ramezani et al.[2007] to deter-

mine the minimum initial gap and detachment length of nanoprobes in the pres-

ence of dispersion effects and electrostatic actuation. Vibration characteristics of

non-uniform single-walled carbon nanotubes (SWCNTs) conveying fluid embeddedin viscoelastic medium has been investigated by Rafiei et al. [2012] using nonlocal

EulerBernoulli beam theory. Sahmani and Ansari [2011] investigated the buckling

analysis of nanobeams using nonlocal continuum beam models of the different clas-

sical beam theories. They presented their results for different geometric parameters,

boundary conditions, and values of nonlocal parameter. Daneshmand et al. [2013]

introduced a gradient-enriched shell formulation based on the first-order shear defor-

mation shell model to analyze dynamic behavior of SWCNTs. Their model includes

two length scale size parameters related to the strain gradients and inertia gradients.They investigate the effects of the aspect ratio, transverse shear, circumferential and

half-axial wave numbers and length scale parameters on different vibration modes

of the SWCNTs.

In classical continuum mechanics, the effect of surface energy is ignored. How-

ever, the experimental results have shown that for nanoscale structures, the surface

effects become significant due to the high surface/volume ratio [Hasheminejad et al.,

2011]. Classical elasticity cannot model the size effect and the significant surface con-

tribution in nanotype structures [Eltaher et al., 2013]. Gurtin and Murdoch [1978]

developed a surface elasticity theory for isotropic materials and modeled the sur-face layer of a solid as a membrane with negligible thickness. Mahmoudet al. [2012]

developed the nonlocal finite element model to study the static bending behavior of

nanobeams, taking into consideration the surface effects. Li et al. [2011] presented

the analytical solution for the transverse vibration of simply supported nanobeams

subjected to an initial axial force based on nonlocal elasticity theory. A modified

continuum model of electrically actuated nanobeams by incorporating surface elas-

ticity has been presented by Fu and Zhang [2011]. They solved the complex math-

ematical problem by the analog equation method (AEM) and discussed the effects

of the surface energies on the static and dynamic responses, pull-in voltage and

pull-in time.

The influence of surface effects on the pull-in instability of a cantilever nanoac-

tuators has been investigated by Koochi et al. [2012] incorporating the influence of

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Pull-in Behavior of Nonlocal Nanobridges Considering Surface Effect

Casimir attraction. A nonlocal finite element model was developed by Eltaher et al.

[2013] to study the vibration characteristics of nanobeams, taking into account the

surface effects.

There have been several approaches employed to solve the governing nonlineardifferential equations to study the nonlinear oscillations such as maxmin approach

[He, 2008], energy balance method [Sedighi et al., 2012a], variational iteration

method [Ghadimi et al., 2012], homotopy analysis method [Sedighi et al., 2012b],

ADM-Pade technique [Noghrehabadi et al., 2012], optimal homotopy asymptotic

method [Kaliji et al., 2012], parameter expansion method [Shou and He, 2007;

Sedighi et al., 2012c, 2012d], hamiltonian approach (HA) [He, 2010] and iteration

perturbation method [Sedighi et al., 2013]. Recently an asymptotic approach namely

homotopy perturbation method (HPM) proposed by He [2003] has proven to be a

very effective and convenient method for solving nonlinear governing equations.The present article intends to provide the second-order frequency amplitude

relation in order to study the dynamic pull-in behavior of vibrating nonlocal

nanobridges in the presence of intermolecular forces and surface effects. To this

end, analytical expressions for vibrational response of nanoactuated beam based on

nonlocal elastic theory with incorporating Casimir and van der Waals effects are

presented. The obtained approximate solution demonstrates that two terms in series

expansions is sufficient to obtain a highly accurate solution of nanobeam vibration.

Finally, the influences of vibration amplitude, actuation voltage, surface effect andnonlocal parameters on the pull-in instability and natural frequency are studied.

2. Mathematical Modeling

A doubly-clamped beam-type nanostructure illustrated in Fig. 1 has lengthl, cross-

section areaAwith thicknesshand widthb, densityand bulk modulus of elasticity

E. The air initial gap is g and an attractive electrostatic force which originates from

voltageVcauses the nanobridge to deform. The nanoscale beam, based on Gurtin

Murdoch model is adopted to have an elastic surface with zero thickness with specificmaterial characteristics which accounts for the surface energy effects and assumed

to be perfectly bonded to its bulk material.

The free-body diagram of an infinitely small nanobeam element with length dxis

shown in Fig. 2. The contact tractionsTx andTz stands for the interaction between

the surface layer and bulk material. The bending moment and shear force act on the

cross-section are denoted byMandQ, respectively. Using Newtons second law, the

governing equations for the bending moment and transverse force can be expressed

as [Fu and Zhang, 2011]:

Q

x +

s

Tzds+q(x, t)A2w

t2 = 0, (1)

M

x

s

Txzds+QNw

x = 0, (2)

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H. M. Sedighi

Fig. 1. Schematic representation of a nanobridge.

Fig. 2. Free-body diagram of nanobeam element.

where S is the perimeter of the cross-section and q(x, t) is the transverse load

per unit length of the nanobeam. The transverse force includes electrostatic actu-

ation qes(x, t) and dispersion force qn(x, t), where the index n is 3 for the van der

Waals force and 4 for the Casimir effect. The distributed electrostatic force can be

expressed as [Tadi beni et al., 2013]:

qes(x, t) = bV2

2(gw)2

1 +es

(gw)

b

, (3)

where = 8.8541012 C2N1m2 is the permittivity of vacuum, w represents

the transverse deflection and the parameter es = 0.65 represents the fringing-field

effect. The van der Waals attraction per unit length of the beam can be written as

[Soroush et al., 2010]:

q3(x, t) = Ahb

6(gw)3

, (4)

where Ah is the Hamaker constant with values in the range [0.4, 4] 1019. The

Casimir effect which originates from quantum fluctuations is defined as follows:

q4(x, t) = 2bc

240(gw)4, (5)

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where = 1.0551034 is the Plancks constant divided by 2 and c = 2.998

108 m/s is the speed of light. Substituting Eq. (2) into Eq. (1) results in:

2M

x2

xs

Txzds s

Tzdsq(x, t)

x

Nw

x

+A2w

t2 = 0. (6)

The equilibrium equation for the stresses of the surface layer is governed by

[GurtinMurdoch, 1978]:

ixx

Ti =02usit2

, (7)

wherei = x, z,0represents the mass density of surface layer and usi is the deflection

of the surface layer in the i direction. Substitution of Eq. (7) into Eq. (6) yields:

2M

x2

x

s

xxx

zds

s

zxx

dsq(x, t)

x

N

w

x

+ A2w

t2 +

s

02uszt2

ds= 0. (8)

The governing equations for the axial force, bending moment and constitutive rela-

tions of the surface layer are expressed as follows [Fu and Zhang, 2011]:

N(w) = EA2l

l0

wx

2dx+N0, (9)

M= EI2w

x2

2vI

h

0

2w

x2 0

2w

t2

, (10)

xx =0+E0

u

xz

2w

x2

, (11)

zx =0w

x

. (12)

Substituting of Eqs. (9)(12) into Eq. (8) gives the following vibrational equation

for a local nanobeam incorporating the surface energy effect and dispersion forces

as: EI+E0I0

2vI 0h

4w

x4 + (A+0S0)

2w

t2 +

2vI 0h

4w

x2t2 =q(x, t),

0S0+N0+ EA

2l l

0

w

x2

dx2w

x2,

(13)

where 0 is the initial residual surface stress under, E0 is the elasticity of surface

layer, v is the Poissons ratio of the bulk material and I0 =Sz

2dSrepresents the

perimeter moment of inertia. For the rectangular nanobeam we have S0 = 2b. By

assuming the identical nonlocal parameter for both bulk material and surface layer,

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H. M. Sedighi

the nonlocal constitutive relations for the nanoscale beams can be written as follows

[Mahmoud et al., 2012]:

1e20a2

2

x2

Mnl =Ml, (14)1e20a

2 2

x2

nlxx =

lxx, (15)

1e20a

2 2

x2

nlzx =

lzx, (16)

wheree0anda represent the nonlocal effects dependent on material and an internal

characteristic length nanoscale. Therefore, according to Eq. (13), the governing

equation for the nonlocal nanobeams, can be written as:

2Mnl

x2

x

s

nlxxx

zds

s

nlzxx

dsq(x, t)

x

N

w

x

+ A2w

t2 +

s

02uszt2

ds= 0. (17)

In order to achieve the governing equation, Eq. (17) is multiplied by the nonlocal

operator (1e2

0a2

2

/x2

) and assuming nl

xx = l

xx, nl

zx = l

zx [Mahmoud et al.,2012], the governing equation (17) can be rearranged as:

2M

x2

x

s

xxx

zds

s

zxx

ds+

1e20a

2 2

x2

q(x, t)

x

N

w

x

+ (A+0S0)

2w

t2

= 0. (18)

Substituting Eq. (11) into Eq. (18) results in the following vibrational equation for

nonlocal nanobeams as:EI+E0I0

2vI 0h

4w

x4 +

2vI 0h

4w

x2t20S0

2w

x2

=

1e20a

2 2

x2

q(x, t) +

N0+

EA

2l

l0

w

x

2dx

2w

x2

(A+0S0)2w

t2 . (19)

The beam vibration is subjected to the following four kinematic boundary

conditions:

w(0, t) = 0, w(0, t) = 0, w(l, t) = 0, w(l, t) = 0, (20)

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By introducing the following nondimensional variables

= EI

bhl4t, W =

w

g, =

x

l, = 6

g

h2

, =g

b, =

E0I0EI

,

1 =2v0

Eh, 2 =

0S0l2

EI , V2 =

6V2l4

Eh3g3, 3 =

Ahbl4

6EIg4,

4 = 2hcbl4

240EIg5, fi =

N0l2

EI , 0=

e0a

l , 1=

v0h

6l2, 2=

0S0A

(21)

the nondimensional equation of motion for nonlocal nanobeam vibration incorpo-

rating surface effects and dispersion forces can be written as:

(1 +1)4W

4 +14W

22 22W

2

=

120

2

2

V2(1W)2

(1 +es(1W)) + n

(1W)n

+

fi+

10

W

2d

2W

2 (1 +2)

2W

2

. (22)

Assuming W(,) = q()(), where () is the first eigenmode of the doubly-

clamped beam and can be expressed as:

() = cosh()cos()cosh()cos()

sinh()sin()(sinh()sin()), (23)

where = 4.73 is the root of characteristic equation for first eigenmode. Using

Taylors series expansion for qes and qn and applying the BubnovGalerkin decom-

position method, the nondimensional nonlinear equation of motion can be written

as:

d2q

d2 +1nq() + [2n(q())

2 +3n(q())3 +4n(q())

4 +0n] = 0, (24)

where the parameters0n, . . . , 4n have been described in the appendix.

3. Application of Homotopy Perturbation Method

Consider the following nonlinear differential equation [He, 2003]:

A(u)f(r) = 0, r (25)

which is subjected to the following boundary condition:

B

u,

u

t

= 0 r , (26)

whereA andB represents the differential and boundary operators,f(r) is a known

function and is the boundary of domain . The operator Amay be separated into

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H. M. Sedighi

linear partL and nonlinear part N. Then, Eq. (25) can be rearranged as below:

L(u) +N(u)f(r) = 0. (27)

One may formulate the following homotopy equation for Eq. (25) as:

H(, p) = (1p)[L()L(u0)] +p[A()f(r)] = 0. (28)

In the above equation, p [0, 1] is an homotopy parameter and u0 is the trial

solution for the approximation which should satisfy the initial condition. The solu-

tion of Eq. (28) may be expressed as a power series in p as:

=0+p1+p22+ . (29)

The embedding parameter p is employed to expand the square of the unknown

oscillation fundamental frequency as follows:

0=2 p1p

22 , (30)

where0is the coefficient ofu(r) in Eq. (25) and should be substituted by the right

hand side of Eq. (20) and the coefficients i(i= 1, 2, . . .) are arbitrary parameters

that should be determined. Setting the embedding parameter p = 1, the approxi-

mations for the solution and the fundamental frequency are

u= limp1

=0+1+2+ , (31)

2 =0+1+2+ . (32)

Now the HPM can be applied to Eq. (24). In this direction, the homotopy equation

can be constructed in the following form:

H(q, p) = (1p)[q+1nq] +p[q+1nq+2nq2 +3nq

3 +4nq4 +0n] = 0.

(33)

According to HPM, we assume that the solution of Eq. (33) can be expressed in

a series of homotopy parameter p:

q() =q() +pq1() +p2q2() + (34)

the coefficient ofq is expanded into a series in p in a similar way [He, 2003]:

1n=2 p1p

22+ . (35)

Substituting Eqs. (34) and (35) into Eq. (33) and rearranging based on powers of

p-terms yields:

p0 : q0() +2q0() = 0, q0(0) =A, q0(0) = 0, (36a)

p1 : q1() +2q1() =1q0() 2n(q0())

2 +3n(q0())3

+ 4n(q0())4 +0n, q1(0) = 0, q1(0) = 0, (36b)

p2 : q2() +2q2() =1q1() +2q0()[22nq0()q1() + 33n(q0())

2q1()

+ 44n(q0())3q1()][2n(q0())

2 +3n(q0())3

+ 4n(q0())4 +0n], q2(0) = 0, q2(0) = 0. (36c)

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Since the solution of Eq. (36a) is q0 =A cos(), the solution of Eq. (36b) should

not contain the so-called secular term cos(). Substitution of this result into the

right-hand side of Eq. (36b) yields:

q1() +2q1()

=

1A

3

43nA

3

cos() +

1

24nA

4 1

22nA

2

cos(2)

1

22nA

2 3

84nA

4 1

43nA

3 cos(3)0n1

84nA

4 cos(4). (37)

No secular terms in q1() require eliminating contributions proportional to cos()

on the right-hand side of Eq. (37), we have:

1 =

3

4 3nA

2

. (38)Solving Eq. (37) for q1() gives the following second order approximation for

q1() as:

q1() =cos()(484nA

4 + 1602nA2 153nA

3 + 4800n)

4802

+cos(2)(804nA

4 + 802nA2)

4802 +

3nA3 cos(3)

322

+

4nA4 cos(4)

1202 +

4800n1804nA4 2402nA

2

4802 . (39)Equation (35) for two terms approximation of series respect to p and for p = 1

yields:

2=2 1n1 (40)

substitution of Eq. (40) into the right-hand side of Eq. (36c) for q2() and elimi-

nating the secular terms gives:

S() =5

622nA

3 7

44n2nA

5 +1

23n2nA

4 +3

23n0nA

2 63

8024nA

7

+ 3

103n4nA

6 +1nA2 22n0nA+

3

43nA

32 3

12823nA

5

34n0nA3 A4 = 0 (41)

solving Eq. (41) for the fundamental frequency results in the second-order frequency-

amplitude relation for vibrating nonlocal nanobridge actuators as follows:

(A) =

1n

2 +

3

83nA

2 +

21n

4 +

31n3nA2

8 +

1523nA4

128

72n4nA4

4

+2n3nA3

2 +3

0n3nA2

522nA2

6 +3

3n4nA5

10 34n0nA

2

6324nA

6

80 22n0n

1/21/2. (42)

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H. M. Sedighi

4. Results and Discussion

To elucidate the strength of approximate solution by HPM, the analytical solu-

tions together with the corresponding numerical results have been plotted in Fig. 3.

As can be observed, for both intermolecular forces, the second-order approximationfor q() displays excellent agreement with numerical solutions using RungeKutta

method.

The influence of initial amplitude on the natural frequency as well as dynamic

pull-in behavior of doubly-clamped nanoactuators has been illustrated in Figs. 47.

(a)

(b)

Fig. 3. Comparison of the results of analytical solutions with numerical solution forA = 0.5, 3 =0.1, 4 = 0.1, = 1, fi = 1, V = 4.5. (a) van der Waals and (b) Casimir forces.

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(a)

(b)

Fig. 4. Fundamental frequency versus initial amplitude of nanobridge vibration: (a) the effect ofvan der Waals parameter 3, (b) the effect of Casimir parameter 4.

Figure 4 illustrates the fundamental frequency of nanobridge for some values of van

der Waals and Casimir parameters 3, 4. It is concluded that the fundamental

frequency of nanobridge actuator decreases as the dispersion force parameters

increase. In addition, as the initial amplitude increases, the fundamental frequency

decreases to pull-in point and the nanostructure diverges to the rigid plate. Near thedynamic pull-in point, a small increase in the initial amplitude causes the nanobeam

to be dynamically unstable. It is obvious from Fig. 4 that when the Casimir force

is taken into account, the dynamic pull-in instability happens in the long regions

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H. M. Sedighi

(a)

(b)

Fig. 5. Fundamental frequency versus initial amplitude of nanobridge vibration, the impact ofnonlocal parameter 0: (a) effect of van der Waals force, (b) effect of Casimir force.

of initial amplitude, in comparison with van der Waals attraction. Moreover, as

the nondimensional parameters3, 4 increase, periodic motions occur in the short

regions of vibrational amplitude.

The impact of nonlocal parameter 0 on the dynamic pull-in instability of

nanobeams is shown in Fig. 5. It appears from this figure that the fundamentalfrequency decreases as the nonlocal parameter 0 increases. In addition, for both

intermolecular forces, the pull-in instability occurs in the short regions of initial

amplitude by increasing the nondimensional nonlocal parameter 0.

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(a)

(b)

Fig. 6. Fundamental frequency versus initial amplitude of nanobridge vibration, the impact ofvoltage parameter V: (a) effect of van der Waals force, (b) effect of Casimir force.

The influence of actuation voltage Von the fundamental frequency as a function

of initial amplitude of vibration has been illustrated in Fig. 6. It is clear that the

fundamental frequency decreases as the parameterV increases and pull-in instability

happens in the long domains of normalized amplitude when the nanobeam natural

frequency drops to zero.The significant influence of axial force parameter fiis shown in Fig. 7, where the

characteristic curves of fundamental frequency are compared. As can be observed,

the fundamental frequency increases as the axial force increases. The frequency

vanishes and pull-in instability occurs in the long regions of vibrational amplitude

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H. M. Sedighi

(a)

(b)

Fig. 7. Fundamental frequency versus initial amplitude of nanobridges, the impact of axial loadparameterfi: (a) effect of van der Waals force, (b) effect of Casimir force.

by decreasing the nondimensional parameterfi, for both van der Waals and Casimir

force models.

The phase portraits of nonlocal nanobeam actuator employing Casimir and van

der Waals attractions in the absence of applied voltage are plotted in Fig. 8. Accord-

ing to Figs. 8(a) and 8(b), it is concluded that without the applied voltage, thereexist the equilibrium pointQ1as a center point, and there are periodic orbits around

it. Homoclinic orbits start from unstable saddle node Q2 and goes back to it. It can

be observed that in the absence of actuation voltage, the nanoactuator exhibits

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(a)

(b)

Fig. 8. The phase portrait of nonlocal nanoactuator without the actuation voltage (V = 0) for0 = 0.1: (a) effect of van der Waals force 3 = 30, (b) effect of Casimir force 4 = 30.

periodic oscillation near the equilibrium point Q1 and collapse onto the rigid plate

beyond the saddle point Q2. In particular, under the effect of Casimir attraction

force, the stable periodic orbits convert to unstable orbits at lower values of initial

conditions.

It appears from Fig. 9 that when the actuation voltage is applied, the unstable

saddle node Q2 becomes closer to stable point Q1. In other words, in the presence

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H. M. Sedighi

(a)

(b)

Fig. 9. The phase portrait of nonlocal nanoactuator with actuation parameterV= 7 for0 = 0.1:(a) effect of van der Waals force 3 = 30, (b) effect of Casimir force 4 = 30.

of actuation voltage, the pull-in instability occurs at lower values of initial ampli-

tudes. If the van der Waals attraction is taken into account, by increasing the voltage

parameter V, the nanoactuator becomes dynamically unstable for any initial condi-

tion at V = 10.18, as indicated in Fig. 10. On the other hand, as the Casimir effectis adopted, the nanobridge becomes unstable when the parameter V approaches

to 9.28.

The effect of nondimensional parameters2 and which account for the surface

energy on the dynamic pull-in behavior of nonlocal nanobridges are investigated

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(a)

(b)

Fig. 10. The phase portrait of nonlocal nanoactuator with actuation parameter for0 = 0.1: (a)effect of van der Waals force for V = 10.18, (b) effect of Casimir force for V = 9.28.

in Figs. 11 and 12. The obtained results elucidate that the dynamic pull-in voltage

decreases by increasing the nonlocal parameter. In addition, Fig. 11 reveals that the

positive surface stress parameter2 increases the dynamic pull-in voltage while the

negative one decreases the pull-in voltage. Furthermore, it is concluded from Fig. 12that the positive , which stands for the surface elasticity, can increase the pull-in

voltage of the nanostructure while the negative one will decrease it. It should be

noted that the surface and bulk material properties have been extracted according

to Fu and Zhang [2011].

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H. M. Sedighi

Fig. 11. Dynamic pull-in voltage versus nonlocal parameter0 and the parameter 2 forA = 0.4,1 = 0.12 105, 2 = 0.4.

Fig. 12. Dynamic pull-in voltage versus nonlocal parameter 0 and the parameter forA= 0.4, 1 = 0.12 105, 2 = 0.4.

5. Conclusion

In this paper, the influence of dispersion forces as well as surface energy param-

eters on the dynamic pull-in behavior of nonlocal nanobridges was investigated.

The obtained results revealed that the nanostructure diverges to the substrate atlower values of applied voltage, when the Casimir effect is taken into considera-

tion. It was concluded that the nonlocal parameter can decrease the dynamic pull-

in voltage of nanostructures. The positive values of surface parameters increase

the dynamic pull-in voltage while the negative values of these parameters will

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decrease it. Furthermore, the qualitative analysis of the system motion exhibits that

the equilibrium points of the nanobeam system include center points and unstable

saddle nodes.

Appendix A

The coefficients of the governing equation are:

i,n=i,n

110

d+ (1 +2)(12010

d), i= 0, 1, 2, 3, 4,

where for van der Waals attraction:

0,3 =(V2(1 +es) +3)

10

d,

1,3 = (1 +1)4 2

10

d

(V2(2 +es) + 33)

10

(2 20)dfi

10

( 20(4))d,

2,3 =(V2(3 +es) + 63)

10

(3 20(2))d,

3,3 =(V2(4 +es) + 103)

10

(4 20(3))d

10

10

2d

20

1

0

2d

d

,

4,3 =(V2(5 +es) + 153) 10

(5 20(4)

)d

and for Casimir attraction:

0,4 =(V2(1 +es) +4)

10

d,

1,4 = (1 +1)4 2

10

d

(V2(2 +es) + 44) 1

0

(2 20)dfi

1

0

( 20(4))d,

2,4 =(V2(3 +es) + 104)

10

(3 20(2))d,

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H. M. Sedighi

3,4 =(V2(4 +es) + 204)

10

(4 20(3))d

1

0

10

2d

20

1

0

2d

d

,

4,4 =(V2(5 +es) + 354)

10

(5 20(4))d.

References

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THE INFLUENCE OF SMALL SCALE ON THE PULL-IN BEHAVIOR OF NONLOCAL NANOBRIDGES CONSIDERING SURFACE...

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