ACME-186; No. of Pages 10

Original Research Article

The influence of dispersion forces on the dynamicpull-in behavior of vibrating nano-cantilever basedNEMS including fringing field effect

H.M. Sedighi a,*, F. Daneshmand b,c,d, J. Zare a

aDepartment of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University, Ahvaz 61357-43337,IranbDepartment of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec,Canada H3A 2K6cDepartment of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Sainte-Anne-de-Bellevue, Quebec,Canada H9X 3V9dSchool of Mechanical Engineering, Shiraz University, Shiraz, Iran

a r c h i v e s o f c i v i l a n d m e c h a n i c a l e n g i n e e r i n g x x x ( 2 0 1 4 ) x x x – x x x

a r t i c l e i n f o

Article history:

Received 5 July 2013

Accepted 7 January 2014

Available online xxx

Keywords:

Dynamic pull-in instability

NEMS actuators

Dispersion forces

Parameter Expansion Method

Second-order frequency–amplitude

relation

a b s t r a c t

Dynamic pull-in instability of vibrating nano-actuators in the presence of actuation voltage

is studied in this paper through introducing the closed form expression for the fundamental

frequency of beam-type nano-structure. The fringing field effect and dispersion forces

(Casimir and van der Waals attractions) are taken into account in the dynamic governing

equation of motion. The influences of initial amplitude of vibration, applied voltage and

intermolecular forces on the dynamic pull-in behavior and fundamental frequency are

investigated by a modern asymptotic approach namely Parameter Expansion Method (PEM).

It is demonstrated that two terms in series expansions are sufficient to produce an accept-

able solution of the actuated nano-structure. The obtained results from numerical methods

by considering three mode assumptions verify the strength of the analytical procedure. The

qualitative analysis of system dynamic shows that the equilibrium points of the autono-

mous system include stable center points and unstable saddle nodes. The phase portraits of

the nano-beam actuator exhibit periodic and homoclinic orbits.

# 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All

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1. Introduction

Nanoelectromechanical systems (NEMS) are widely used innovel technologies and significantly impact many areas ofapplied sciences due to its great potential for applications suchas nano relays and switches, nanoaccelerometers and high

* Corresponding author. Tel.: +98 6113330010x5665; fax: +98 611333664E-mail addresses: h.msedighi@scu.ac.ir, hmsedighi@gmail.com (H.

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1644-9665/$ – see front matter # 2014 Politechnika Wrocławska. Publhttp://dx.doi.org/10.1016/j.acme.2014.01.004

frequency resonators and ultrasensitive sensors. In recenttimes, there has been considerable interest in developingnanomechanical and nanoelectromechanical systems, suchas capacitive sensor, switches and actuators at nano-scale[1]. Consider a nano-cantilever beam-type actuator whichconsists of two conducting electrodes; one is movable and theother is fixed. When the voltage difference between the two

2.M. Sedighi).

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

ished by Elsevier Urban & Partner Sp. z o.o. All rights reserved.

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ACME-186; No. of Pages 10

electrodes exceeds the critical voltage, known as the pull-involtage, the movable electrode becomes unstable and pulls-inonto the fixed one [2].

With the reduction in the dimensions of electro-staticallyactuated structures to the nanoscale, new phenomena such asvan der Waals [3] and Casimir forces [4] should be taken intoaccount. The van der Waals effect stands for the electrostaticinteraction among pair of magnetic poles at the atomicdimension. The Casimir force is related to the attractive forcebetween two flat parallel plates of solids that results fromquantum fluctuations in the ground state of the electromag-netic field [5]. At micro-separations (typically less than severaltens of micrometers), the attraction between two surfacescould be described by the Casimir interaction [6]. In thissituations, the interaction between the many fluctuatingdipoles present within the bodies leads to Casimir forces [7](see Fig. 1a). Considering the ideal case, the Casimir interactionis proportional to the inverse fourth power of the separation[8]. When separation is well below the plasma (for metals) orabsorption (for dielectrics) wavelength of the surface material(typically less than several tens of nanometers), the Casimirforce should be replaced by the van der Waals force. In thiscase, the retardation is not significant and the attractionbetween two surfaces varies with the inverse cube of theseparation [9]. Physically, this attraction arises as shown inFig. 1b; a fluctuating dipole p1 induces a fluctuating electro-magnetic dipole field, which in turn induces a fluctuatingdipole p2 on a nearby particle, leading to van der Waals forcesbetween the particles [7].

Several researchers have been studied the pull in behaviorof nano structures [10–14]. Nanotechnological investigation onvibration properties of nano-beams under certain supportconditions and suddenly DC actuation is important becausethese components can be used in structures such as nano-sensors and nano-actuators.

Rasekh and Khadem [15] studied the nonlinear behavior ofelectrostatically actuated carbon nanotubes including nonlin-earity in curvature, inertia and electrostatic force. Fu andZhang [16] presented a modified continuum model forelectrically actuated nanobeams by incorporating surfaceelasticity. They investigated the effects of the surface energieson the static and dynamic responses, pull-in voltage and pull-intime. Ramezani et al. [17] determined the detachment lengthand the minimum initial gap of freestanding nano-switches

Fig. 1 – Physical description of (a) Casi

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subjected to intermolecular and electrostatic forces using adistributed parameter model.

Duan and Wan [18] investigated the ‘‘Pull-in’’ of pre-stressed thin film by electrostatic potential for 1-D rectangularbridges and 2-D circular diaphragms. Boyd and Lee [19]investigated the deflection and pull-in instability of nano-scale beams in liquid electrolytes. They revealed that the pull-in voltage of a double-wall carbon nanotube suspended over agraphite substrate in liquid can be less than or greater than thepull-in voltage in air, depending on the bulk ion concentration.Size-dependent bending elastic properties of nanobeams withthe influence of the surface relaxation and the surface tensionwere studied by Guo and Zhao [20]. They considered a three-dimensional (3D) crystal model for a nanofilm with n layers ofrelaxed atoms. The influence of Casimir force on the nonlinearbehavior of nanoscale electrostatic actuators was studied byLin and Zhao [21]. They utilized one degree of freedom mass-spring model to investigate the bifurcation properties of thenano-actuators. The dynamic behavior of nanoscale electro-static actuators by considering mass-spring model wasdeveloped by Lin and Zhao [22] in order to study the Hopfbifurcation properties of nano-structure. Bansal and Clark [23]presented a lumped model of a single walled carbon nanotube(CNT) using structural matrix mechanics. They presented thedynamic response of a zigzag or armchair chirality, with thedesired diameter, length, and distributed loading parameters.Nonlinear dynamic response of beam and its application innanomechanical resonator was developed by Zhang et al. [24].They showed that for the nanomechanical resonator of tension-dominant nonlinearity, its dynamic nonlinearity decreasesmonotonically with increasing axial loading and increasesmonotonically with the increasing aspect ratio of length tothickness. The influences of the van der Waals and the Casimirforces on the stability of the electrostatic torsional nano-electromechanical system (NEMS) actuators have been investi-gated by Guo and Zhao [25]. They also derived the critical gapsunder the actions of the vdW and the Casimir torques whenthere is no electrostatic torque. Casimir effects on the criticalpull-in gap and pull-in voltage of nano-electromechanicalswitches have been studied by Lin and Zhao [26]. Theypresented an approximate analytical expression of the criticalpull-in gap with the Casimir force using perturbation theory.

Recently, many approaches for approximating the solu-tions to nonlinear oscillatory systems were developed. The

mir and (b) van der Waals forces.

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

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ACME-186; No. of Pages 10

most widely studied approximation methods are perturbationmethods [27]. Perturbation methods provide the most versatiletools available in nonlinear analysis of engineering problemsand they are constantly being modified and applied to manyproblems. But these methods have a main shortcoming; thereis no small parameter in the equation, no approximation couldbe obtained. Regarding the above restriction, new analyticalmethods without depending on presence of small parameterin the equation were developed for solving complicatednonlinear systems. These techniques include HomotopyAnalysis Method (HAM) [28], Parameter Expansion Method(PEM) [29–32], Homotopy Perturbation Method (HPM) [33–35],Iteration Perturbation Method (IPM) [36], Variational IterationMethod (VIM) [36,37], Max–Min Approach [38] and HamiltonianApproach (HA) [39]. Among these methods, Parameter Expan-sion Method (PEM) proposed by He [40] is considered to be onepowerful method that capable to handle strongly nonlinearbehaviors. PEM is straightforward approach to solve thenonlinear governing equations, analytically. The ParameterExpansion Method entails the bookkeeping parameter method[41] and the modified Lindstedt–Poincare method and has beenapplied to various nonlinear oscillators with nonlinearboundary conditions [29–32].

The significant aim of the present article is to asymptoti-cally predict the dynamic pull-in behavior of nano-structuresin the presence of applied voltage and intermolecular forces.The effect of vibrational amplitude on pull-in instability,natural frequency and dynamic pull-in voltage is investigated.In this direction, analytical expressions for vibrational responseand second-order frequency of nano actuated beams incorpo-rating Casimir and van der Waals effects are presented. Theproposed asymptotic solution demonstrates that the two termsin series expansions is sufficient to obtain a highly accuratesolution of beam-type nano-cantilever vibrations. Finally, theinfluences of amplitude and significant parameters on the pull-in behavior and natural frequency are studied.

2. Mathematical formulation

A nano-cantilever beam considered here has length l,thickness h, width b, density r and modulus of elasticity Eas illustrated in Fig. 2. The air initial gap is g and an attractiveelectrostatic force which originates from voltage V causes thenano-beam to deflect.

The electrostatic force per unit length of the beam can beexpressed as [4]:

Fes ¼ beV2

2ðg � yÞ2 1 þ besg � yb

� �(1)

Fig. 2 – Schematic of an actuated nano-cantilever beam.

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where the parameter bes represents the first order fringing-field correction which is equal to 0.65. The van der Waalseffect per unit length of the beam which is proportional tothe inverse cube of the separation can be written as [3]:

F3 ¼ Ahb

6pðg � yÞ3 (2)

where Ah is the Hamaker constant with values in therange [0.4,4] � 10�19. The second effect is the intermolecularCasimir force per unit length of the beam, which is defined asfollows:

F4 ¼ p2�hcb

240ðg � yÞ4 (3)

where �h ¼ 1:055 � 10�34 is the Planck's constant divided by 2pand c = 2.998 � 108 m/s is the speed of light. For cantilevernano-beams by incorporating electrostatic actuation and in-termolecular forces, the nonlinear governing equation of mo-tion can be expressed as:

rbhytt þ EIyxxxx � Fes � Fn ¼ 0 (4)

where the index n is 3 for the van der Waals force and 4 for theCasimir effect. The beam deflection is subjected to four kine-matic boundary conditions as:

yð0; tÞ ¼ 0; y0ð0; tÞ ¼ 0; y00ðl; tÞ ¼ 0; y000ðl; tÞ ¼ 0; (5-a)

and the following initial conditions:

yðx; 0Þ ¼ A fðxÞ; _yðx; 0Þ ¼ 0; (5-b)

where f(x) is the first eigenmode of the cantilever beam. Byintroducing the following nondimensional variables

t ¼ffiffiffiffiffiffiffiffiffiffiffiffiEI

rbhl4

st; Y ¼ y

g; j ¼ x

l; a ¼ g

b; l2 ¼ 24eV2l4

Eh3g3;

l3 ¼ Ahbl4

6EIpg4; l4 ¼ p2hcbl4

240EIg5(6)

the nondimensional nonlinear equation of motion can bewritten as

@2Y@t2

þ @4Y

@j4� l2

4ð1 � YÞ2 ð1 þ abesð1 � YÞÞ � ln

ð1 � YÞn ¼ 0 (7)

The nondimensional forces Fes and Fn in Eq. (4) can beapproximated by Taylor's series as

Fes ¼ l2

4½ð1 þ abesÞ þ ð2 þ abesÞY þ ð3 þ abesÞY2

þ ð4 þ abesÞY3 þ ð5 þ abesÞY4 þ � � � �(8)

F3 ¼ l3ð1 þ 3Y þ 6Y2 þ 10Y3 þ 15Y4 þ � � � Þ (9)

F4 ¼ l4ð1 þ 4Y þ 10Y2 þ 20Y3 þ 35Y4 þ � � � Þ (10)

substituting Fes and Fn from Eqs. (8)–(10) into Eq. (7), the non-dimensional nonlinear equation of motion can be written as

Y€ þ Yð4Þ þ b1;nY þ b2;nY2 þ b3;nY

3 þ b4;nY4 þ b0;n ¼ 0 (11)

where the parameters b0;n . . . b4;n have been described in theAppendix. Assuming Yðj; tÞ ¼ qðtÞ fðjÞ, where fðjÞ is the firsteigenmode of the cantilever beam and can be expressed as:

fðjÞ ¼ coshðlcjÞ � cosðlcjÞ

� coshðlcÞ þ cosðlcÞsinhðl Þ þ sinðl Þ ðsinhðlcjÞ � sinðlcjÞÞ (12)

c c

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

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ACME-186; No. of Pages 10

where lc ¼ 1:875 is the root of characteristic equation for firsteigenmode. Applying the Bubnov–Galerkin method, nonlineargoverning equation can be extracted as follows:

d2qðtÞdt2

þ b01;nqðtÞ þ ½b02;nðqðtÞÞ2 þ b03;nðqðtÞÞ3 þ b04;nðqðtÞÞ4 þ b00;n� ¼ 0

(13-a)

where

b01;n ¼ b1;n þ l4c

b0i;n ¼Z 1

0bi;nf

iþ1dj; ði 6¼ 1Þ(13-b)

subjected to the following initial conditions:

qð0Þ ¼ A; _qð0Þ ¼ 0; (13-c)

In the current study, in order to obtain the closed-formexpression for the second order approximate frequency ofactuated nano-beams, Eq. (13-a) is solved analytically, usingPEM. On the other hand, in order to validate the integrity ofasymptotic approach, Matlab built-in function ‘‘ode45’’ isemployed for numerical simulations by considering threemode assumptions.

3. Asymptotic analysis by PEM

The basic character of the PEM is to expand the solutionand some parameters in the nonlinear governing equation.This asymptotic approach involves the bookkeeping parame-ter method and the modified Lindstedt–Poincare method.Consider the nonlinear equation of motion described inEq. (13-a) subject to the following initial conditions:

qð0Þ ¼ A; _qð0Þ ¼ 0 (14)

According to the PEM, the solution q(t) and the factors 1 andb01;n may be expanded in series of p:

qðtÞ ¼ q0ðtÞ þ pq1ðtÞ þ p2q2ðtÞ þ � � � (15)

b01;n ¼ v2 � pb1 � p2b2 þ � � � (16-a)

1 ¼ pc1 þ p2c2 þ � � � (16-b)

Substituting Eqs. (15) and (16) into Eq. (13-a) and equatingthe terms with the same powers of p results in:

p0 : €q0ðtÞ þ v2q0ðtÞ ¼ 0; q0ð0Þ ¼ A; _q0ð0Þ ¼ 0 (17)

p1 : €q1ðtÞ þ v2q1ðtÞ ¼ b1q0ðtÞ � c1½b02;nðq0ðtÞÞ2 þ b03;nðq0ðtÞÞ3

þ b04;nðq0ðtÞÞ4 þ b00;n�; q1ð0Þ ¼ 0; _q1ð0Þ ¼ 0 (18)

p2 : €q2ðtÞ þ v2q2ðtÞ ¼ b1q1ðtÞ þ b2q0ðtÞ � c1½2b02;nq0ðtÞq1ðtÞþ 3b03;nðq0ðtÞÞ2q1ðtÞ þ 4b04;nðq0ðtÞÞ3q1ðtÞ� � c2½b02;nðq0ðtÞÞ2

þ b03;nðq0ðtÞÞ3 þ b04;nðq0ðtÞÞ4 þ b00;n�; q2ð0Þ ¼ 0;

_q2ð0Þ ¼ 0 (19)

When p = 0, Eq. (13-a) becomes a linear differentialequation and the approximated solution can be obtainedfor p = 1. The solution of the first equation is q0ðtÞ ¼ A cosðvtÞ,substitution of this result into the right-hand side of Eq. (18)gives

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q::1ðtÞ þ v2q1ðtÞ ¼ b1A � 3

4c1b03;nA

3� �

cosðvtÞ

þ �12c1b04;nA

4 � 12c1b02;nA

2� �

cosð2vtÞ

� 12c1b02;nA

2 � 38c1b04;nA

4

� 14c1b03;nA

3 cosð3vtÞ � c1b00;n

� 18c1b04;nA

4 cosð4vtÞ; (20)

No secular terms in q1(t) requires eliminating the coefficientof term cos(vt) on the right-hand side of Eq. (20), we obtain

b1 ¼ 34c1b03;nA

2 (21)

thereby, solving Eq. (18) for q1(t) yields the following second-order approximation for the solution q(t)as:

q1ðtÞ ¼ cosðvtÞð48b04;nA4 þ 160b02;nA2 � 15b03;nA

3 þ 480b00;nÞ480v2

þ cosð2vtÞð80b04;nA4 þ 80b02;nA2Þ

480v2 þ b03;nA3 cosð3vtÞ32v2

þ b04;nA4cosð4vtÞ120v2 þ�480b00;n � 180b04;nA

4 � 240b02;nA2

480v2

(22)

Eqs. (16-a) and (16-b) for two terms approximation of seriesrespect to p and for p = 1 yields:

c1 ¼ 1; c2 ¼ 0; b2 ¼ v2 � pb1 � b01;n (23)

substituting of Eq. (23) into the right-hand side of Eq. (19) andeliminating the secular terms gives:

SðvÞ ¼ � 56b02;n

2 A3 � 74b04;nb

02;nA

5 þ 12b03;nb

02;nA

4 þ 32b03;nb

00;nA

2

� 6380

b04;n2A7 þ 3

10b03;nb

04;nA

6 þ b01;nAv2 � 2b02;nb

00;nA

þ 34b03;nA

3v2 � 3128

b023;nA5 � 3b04;nb

00;nA

3 � Av4 ¼ 0 (24)

solving Eq. (24) for the frequency of oscillation v, the followingsecond-order frequency–amplitude relation for vibratingactuated nano-cantilever cab be obtained:

vðAÞ ¼"b01;n2

þ 38b03;nA

2 þ b021;n4

þ 3b01;nb03;nA

2

8þ 15b023;nA

4

128

� 7b02;nb04;nA

4

4þ b02;nb

03;nA

3

2þ 3b00;nb

03;nA

2� 5b022;nA

2

6

þ 3b03;nb04;nA

5

10� 3b04;nb

00;nA

2� 63b024;nA6

80� 2b02;nb

00;n

!1=2#1=2

(25)

4. Results and discussion

In order to verify the results of numerical simulations, thetime histories of vibrating nano-cantilever in the absence ofintermolecular forces by adding the modified couple stressparameter are compared with the reported results byRahaeifard et al. [42]. As can be seen in Fig. 3, the resultsof present work exhibits an excellent agreement with theresults obtained by Rahaeifard et al. [42] using a hybrid finitedifference method.

To investigate the accuracy of the asymptotic approach topredict the pull-in phenomenon, another comparison is

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

Fig. 3 – Comparison between the numerical results with thereported results in Ref. [42].

Fig. 5 – Comparison between the results of analytical andnumerical solutions, (a) van der Waals and (b) Casimirforces.

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performed considering the nano-cantilever studied byMoghimi Zand and Ahmadian [5] by employing the finite-element method (FEM). Fig. 4(a) and (b) depicts the effects ofvan der Waals parameter l3 and Casimir parameter l4 on thedynamic pull-in values of nano-actuators. According to thesefigures, for two intermolecular models, PEM predicts slightlyhigher dynamic pull-in value in comparison with the obtainedresults using FEM. In addition, these figures indicate that anincrease in the dispersion force parameters, results in adecrease in dynamic pull-in values.

Furthermore, to demonstrate the soundness of proposedsolution by PEM, the analytical solutions at the side ofcorresponding numerical results have been plotted. As canbe seen in Fig. 5, the second order approximation of q(t) using

Fig. 4 – Comparison between the numerical results ofdynamic pull-in voltage with the reported results in Ref. [5]:(a) van der Waals and (b) Casimir effect.

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analytical method exhibits a good agreement with numericalresults by considering three mode assumptions for both vander Waals and Casimir effects assumption.

In order to show the convergence of the asymptoticanalysis by PEM, the absolute error for different approximateorders at t = 1, is tabulated in Table 1. As can be observed, fordifferent values of initial amplitudes, the absolute errordecreases as the order of approximation employed in seriessolution increases.

Eq. (25) reveals that the initial amplitude of vibration has asignificant effect on the dynamic behavior of NEMS/MEMSstructures. To show the dependency of fundamental frequen-cy and pull-in instability of such systems on the initial

Table 1 – Comparison between the errors of differentestimations for approximate solution by assuming vander Waals attraction.

A qnumeric � q0�� �� qnumeric � q1

�� �� qnumeric � q2�� ��

0 0.04973997 0.00014507 0.000145070.05 0.04975019 0.00052312 0.000170830.1 0.05066131 0.00095012 0.000226170.15 0.05238962 0.00141606 0.000246830.2 0.05526881 0.00206676 0.000274510.25 0.05934554 0.00309974 0.000349450.3 0.06509971 0.00478752 0.000913330.35 0.07285962 0.00758339 0.002374590.4 0.08403840 0.01234765 0.00549251

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

Fig. 6 – Fundamental frequency vs. nondimensional amplitude of nano-beams: (a) effect of van der Waals parameter l3 and(b) effect of Casimir parameter l4.

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conditions, the impact of nondimensional initial amplitude onthe dynamics of actuated cantilever nano-beams have beeninvestigated. Fig. 6 shows the characteristic curves of naturalfrequency for a nano-beam under some assigned values of vander Waals and Casimir parameters l3, l4. It is obvious thatthese nondimensional parameters have notable effect onpredicting the pull-in instability and natural frequency. Atlower values of initial amplitude, the beam performs a periodicmotion and the frequency of vibration decreases by increasingthe initial amplitude. In the vicinity of the dynamic pull-inpoint, a small increase in the amplitude causes the nano-beamto be dynamically unstable; the pull-in instability occurs andthe nano-beam collapses onto the substrate. It is obvious fromFig. 6 that under the effect of the Casimir force, the dynamicpull-in instability occurs in the long regions of nondimension-al amplitude A in comparison with van der Waals effects. Onthe other hand, the fundamental frequency decreases byincreasing the intermolecular parameters. When the nondi-mensional parameters l3, l4 increase, periodic motions takeplace in the short regions of vibrational amplitude. If theintermolecular parameters are equal to 0.3, the systembecomes dynamically unstable at A � 0.42 when the van derWaals force is taken into consideration; instead as the Casimireffect is included, the nano-actuator diverges to the rigidplate at A � 0.29.

Fig. 7 – Fundamental frequency vs. nondimensional amplitude oWaals force and (b) Casimir force.

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The influence of actuation voltage parameter l on thenatural frequency and pull-in instability of elector-actuatednano-beam as a function of nondimensional amplitude hasbeen illustrated in Fig. 7. It is clear from this figure that thefundamental frequency decreases as the parameter l

increases and pull-in instability occurs in the long domainsof initial amplitude when the natural frequency of nano-beamdrops to zero.

The influence of nondimensional parameter a is shown inFig. 8, where the nonlinear v–A curves of clamped-freeactuated nano-beams are compared. As can be observed,the fundamental frequency decreases as the parameter a

increases. The frequency vanishes and pull-in instabilityoccurs in the long regions of vibrational amplitude byincreasing the nondimensional parameter a, for both vander Waals and Casimir force models.

The phase portraits of nano-beam actuator employingCasimir and van der Waals attractions are plotted in Figs. 9–11.It is concluded from Fig. 9a and b that for voltage parameterl = 1, there exist the equilibrium point as a center point, andthere are periodic orbits around it and homoclinic orbitsstarting from unstable saddle node and going back to it. It canbe concluded that in this situation, the nano actuator exhibitsperiodic oscillation near the equilibrium center point anddrops to the substrate beyond the unstable saddle node. In

f nano-beams, the impact of voltage parameter l. (a) van der

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

Fig. 8 – Fundamental frequency vs. nondimensional amplitude of nano-beams, the impact of parameter a. (a) van der Waalsforce and (b) Casimir force.

Fig. 9 – The phase portrait of nano-beam actuator with actuation parameter l = 1: (a) effect of van der Waals force and (b) effectof Casimir force.

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particular, under the effect of Casimir attraction force, thestable periodic orbits convert to unstable orbits at lower valuesof initial conditions.

When the actuation voltage increases to l = 1.6, as can beseen in Fig. 10, the unstable saddle point becomes closer tostable center point. This means that, by increasing theactuation voltage parameter, the pull-in instability happensat lower values of vibrational amplitude. At particular value ofvoltage parameter (here l = 1.88 for van der Waals force and

Fig. 10 – The phase portrait of nano-beam actuator with actuatio(b) effect of Casimir force.

Please cite this article in press as: H.M. Sedighi et al., The influence of dcantilever based NEMS including fringing field effect, Archives of Cacme.2014.01.004

l = 1.81 for Casimir effect), the nano-beam becomes dynami-cally unstable for all values of initial conditions. The phaseplane shown in Fig. 11 indicates that in this situation, thestable center point and unstable saddle node merge togetherand there are no periodic orbits in the phase portrait of thesystem.

As mentioned before, the amplitude of vibration hassignificant influence on the nonlinear dynamic behavior ofnano-structures. When the initial condition increases, the

n parameter l = 1.6: (a) effect of van der Waals force and

ispersion forces on the dynamic pull-in behavior of vibrating nano-ivil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.

Fig. 11 – The phase portrait of nano-beam actuator: (a) effect of van der Waals force for l = 1.88 and (b) effect of Casimir forcefor l = 1.81.

Fig. 12 – Time history of nano-cantilever beams for different initial conditions: (a) effect of van der Waals force and (b) effect ofCasimir force.

Fig. 13 – Dynamic pull-in voltage for actuated nano-beamsas a function of initial amplitude: (a) effect of van der Waalsforce and (b) effect of Casimir force.

a r c h i v e s o f c i v i l a n d m e c h a n i c a l e n g i n e e r i n g x x x ( 2 0 1 4 ) x x x – x x x8

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system becomes unstable and the actuated nano-beamcollapse onto the rigid substrate. According to Fig. 12, it isobvious that by increasing the initial amplitude A, the timeperiod of oscillation increases until the specific point namely‘‘pull-in’’ (here A = 0.767 for van der Waals force and A = 0.638for Casimir attraction) where the beam touches the substrate.Furthermore, as can be observed, when the Casimir attractionforce is taken into account, the instability occurs at lower valueof initial condition.

Dynamic pull-in voltage of actuated nano-structure as afunction of initial amplitude and for some assigned values ofintermolecular parameters l3, l4 is plotted in Fig. 13. It appearsform the figure that by increasing the amplitude of vibration,the dynamic pull-in voltage decreases for both intermolecularforce models. In addition, the dynamic pull-in voltagedecreases when the parameters l3, l4 increase until thedynamic pull-in voltage vanishes. In this situation, in theabsence of actuation voltage, the nano-beam is unstable forsome values of initial conditions. For example, when theparameters l3 and l4 are equal to 0.4, for van der Waalsassumption, the dynamic pull-in voltage disappear as theinitial amplitude approaches to A = 0.51, however, when theCasimir force effect is considered, the dynamic pull-in voltagetends to zero as the initial amplitude approaches to A = 0.37.This means that, in the absence of applied voltage, thefreestanding nano-beam collapse onto the substrate at theaforementioned values of initial amplitudes.

Please cite this article in press as: H.M. Sedighi et al., The influence of dispersion forces on the dynamic pull-in behavior of vibrating nano-cantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j.acme.2014.01.004

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ACME-186; No. of Pages 10

5. Conclusion

In the current work, dynamic pull-in instability of nano-cantilever based NEMS was investigated. The second-orderfrequency–amplitude relationship of vibrating actuated nano-beams including dispersion forces was proposed in order tostudy the amplitude dependence of pull-in instability ofactuated nano-structure. The influences of actuation voltageand intermolecular forces on the system dynamic werepresented. It was illustrated that the pull-in voltage andfundamental frequency of the system decrease by increasingthe initial amplitude of vibration. At specific value of initialcondition, pull-in phenomenon occurs and the nano-beamcollapse onto the substrate. Furthermore, the qualitativeanalysis of the system motion exhibits that the equilibriumpoints of the nano-actuators include stable center points andunstable saddle nodes.

Appendix A

b1;3 ¼ � l2

4 ð2 þ abesÞ þ 3l3

� �b1;4 ¼ � l2

4 ð2 þ abesÞ þ 4l4

� �b2;3 ¼ � l2

4 ð3 þ abesÞ þ 6l3

� �b2;4 ¼ � l2

4 ð3 þ abesÞ þ 10l4

� �b3;3 ¼ � l2

4 ð4 þ abesÞ þ 10l3

� �b3;4 ¼ � l2

4 ð4 þ abesÞ þ 20l4

� �b4;3 ¼ � l2

4 ð5 þ abesÞ þ 15l3

� �b4;4 ¼ � l2

4 ð5 þ abesÞ þ 35l4

� �b0;3 ¼ � l2

4 ð1 þ abesÞ þ l3

� �b0;4 ¼ � l2

4 ð1 þ abesÞ þ l4

� �

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