The influence of dispersion forces on the dynamic pull-in behavior of vibrating nano-cantilever based NEMS including fringing field effect

ACME-186; No. of Pages 10 archives of civil and mechanical engineering xxx (2014) xxx–xxx

Available online at www.sciencedirect.com

ScienceDirect journal homepage: http://www.elsevier.com/locate/acme

Original Research Article

The influence of dispersion forces on the dynamic pull-in behavior of vibrating nano-cantilever based NEMS including fringing field effect H.M. Sedighi a,*, F. Daneshmand b,c,d, J. Zare a a

Department of Mechanical Engineering, Faculty of Engineering, Shahid Chamran University, Ahvaz 61357-43337, Iran b Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6 c Department of Bioresource Engineering, McGill University, 21111 Lakeshore Road, Sainte-Anne-de-Bellevue, Quebec, Canada H9X 3V9 d School of Mechanical Engineering, Shiraz University, Shiraz, Iran

article info

abstract

Article history:

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

Received 5 July 2013

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

Accepted 7 January 2014

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

Available online xxx

(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

Keywords:

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

Dynamic pull-in instability

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

NEMS actuators

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

Dispersion forces

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

Parameter Expansion Method

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

Second-order frequency–amplitude

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

relation

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 rights reserved.

1.

Introduction

Nanoelectromechanical systems (NEMS) are widely used in novel technologies and significantly impact many areas of applied sciences due to its great potential for applications such as nano relays and switches, nanoaccelerometers and high

frequency resonators and ultrasensitive sensors. In recent times, there has been considerable interest in developing nanomechanical and nanoelectromechanical systems, such as capacitive sensor, switches and actuators at nano-scale [1]. Consider a nano-cantilever beam-type actuator which consists of two conducting electrodes; one is movable and the other is fixed. When the voltage difference between the two

* Corresponding author. Tel.: +98 6113330010x5665; fax: +98 6113336642. E-mail addresses: [email protected], [email protected] (H.M. Sedighi). 1644-9665/$ – see front matter # 2014 Politechnika Wrocławska. Published by Elsevier Urban & Partner Sp. z o.o. All rights reserved. http://dx.doi.org/10.1016/j.acme.2014.01.004 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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

2

archives of civil and mechanical engineering xxx (2014) xxx–xxx

electrodes exceeds the critical voltage, known as the pull-in voltage, the movable electrode becomes unstable and pulls-in onto the fixed one [2]. With the reduction in the dimensions of electro-statically actuated structures to the nanoscale, new phenomena such as van der Waals [3] and Casimir forces [4] should be taken into account. The van der Waals effect stands for the electrostatic interaction among pair of magnetic poles at the atomic dimension. The Casimir force is related to the attractive force between two flat parallel plates of solids that results from quantum fluctuations in the ground state of the electromagnetic field [5]. At micro-separations (typically less than several tens of micrometers), the attraction between two surfaces could be described by the Casimir interaction [6]. In this situations, the interaction between the many fluctuating dipoles present within the bodies leads to Casimir forces [7] (see Fig. 1a). Considering the ideal case, the Casimir interaction is proportional to the inverse fourth power of the separation [8]. When separation is well below the plasma (for metals) or absorption (for dielectrics) wavelength of the surface material (typically less than several tens of nanometers), the Casimir force should be replaced by the van der Waals force. In this case, the retardation is not significant and the attraction between two surfaces varies with the inverse cube of the separation [9]. Physically, this attraction arises as shown in Fig. 1b; a fluctuating dipole p1 induces a fluctuating electromagnetic dipole field, which in turn induces a fluctuating dipole p2 on a nearby particle, leading to van der Waals forces between the particles [7]. Several researchers have been studied the pull in behavior of nano structures [10–14]. Nanotechnological investigation on vibration properties of nano-beams under certain support conditions and suddenly DC actuation is important because these components can be used in structures such as nanosensors and nano-actuators. Rasekh and Khadem [15] studied the nonlinear behavior of electrostatically actuated carbon nanotubes including nonlinearity in curvature, inertia and electrostatic force. Fu and Zhang [16] presented a modified continuum model for electrically actuated nanobeams by incorporating surface elasticity. They investigated the effects of the surface energies on the static and dynamic responses, pull-in voltage and pull-in time. Ramezani et al. [17] determined the detachment length and the minimum initial gap of freestanding nano-switches

subjected to intermolecular and electrostatic forces using a distributed parameter model. Duan and Wan [18] investigated the ‘‘Pull-in’’ of prestressed thin film by electrostatic potential for 1-D rectangular bridges and 2-D circular diaphragms. Boyd and Lee [19] investigated the deflection and pull-in instability of nanoscale beams in liquid electrolytes. They revealed that the pullin voltage of a double-wall carbon nanotube suspended over a graphite substrate in liquid can be less than or greater than the pull-in voltage in air, depending on the bulk ion concentration. Size-dependent bending elastic properties of nanobeams with the influence of the surface relaxation and the surface tension were studied by Guo and Zhao [20]. They considered a threedimensional (3D) crystal model for a nanofilm with n layers of relaxed atoms. The influence of Casimir force on the nonlinear behavior of nanoscale electrostatic actuators was studied by Lin and Zhao [21]. They utilized one degree of freedom massspring model to investigate the bifurcation properties of the nano-actuators. The dynamic behavior of nanoscale electrostatic actuators by considering mass-spring model was developed by Lin and Zhao [22] in order to study the Hopf bifurcation 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 the dynamic response of a zigzag or armchair chirality, with the desired diameter, length, and distributed loading parameters. Nonlinear dynamic response of beam and its application in nanomechanical resonator was developed by Zhang et al. [24]. They showed that for the nanomechanical resonator of tensiondominant nonlinearity, its dynamic nonlinearity decreases monotonically with increasing axial loading and increases monotonically with the increasing aspect ratio of length to thickness. The influences of the van der Waals and the Casimir forces on the stability of the electrostatic torsional nanoelectromechanical system (NEMS) actuators have been investigated by Guo and Zhao [25]. They also derived the critical gaps under the actions of the vdW and the Casimir torques when there is no electrostatic torque. Casimir effects on the critical pull-in gap and pull-in voltage of nano-electromechanical switches have been studied by Lin and Zhao [26]. They presented an approximate analytical expression of the critical pull-in gap with the Casimir force using perturbation theory. Recently, many approaches for approximating the solutions to nonlinear oscillatory systems were developed. The

Fig. 1 – Physical description of (a) Casimir and (b) van der Waals forces. 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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

3

archives of civil and mechanical engineering xxx (2014) xxx–xxx

most widely studied approximation methods are perturbation methods [27]. Perturbation methods provide the most versatile tools available in nonlinear analysis of engineering problems and they are constantly being modified and applied to many problems. But these methods have a main shortcoming; there is no small parameter in the equation, no approximation could be obtained. Regarding the above restriction, new analytical methods without depending on presence of small parameter in the equation were developed for solving complicated nonlinear systems. These techniques include Homotopy Analysis Method (HAM) [28], Parameter Expansion Method (PEM) [29–32], Homotopy Perturbation Method (HPM) [33–35], Iteration Perturbation Method (IPM) [36], Variational Iteration Method (VIM) [36,37], Max–Min Approach [38] and Hamiltonian Approach (HA) [39]. Among these methods, Parameter Expansion Method (PEM) proposed by He [40] is considered to be one powerful method that capable to handle strongly nonlinear behaviors. PEM is straightforward approach to solve the nonlinear governing equations, analytically. The Parameter Expansion Method entails the bookkeeping parameter method [41] and the modified Lindstedt–Poincare method and has been applied to various nonlinear oscillators with nonlinear boundary conditions [29–32]. The significant aim of the present article is to asymptotically predict the dynamic pull-in behavior of nano-structures in 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 response and second-order frequency of nano actuated beams incorporating Casimir and van der Waals effects are presented. The proposed asymptotic solution demonstrates that the two terms in series expansions is sufficient to obtain a highly accurate solution of beam-type nano-cantilever vibrations. Finally, the influences of amplitude and significant parameters on the pullin behavior and natural frequency are studied.

2.


g  y 1 þ bes b 2ðg  yÞ beV 2

2

Ah b

F3 ¼

(2)

6pðg  yÞ3

where Ah is the Hamaker constant with values in the range [0.4,4]  1019. The second effect is the intermolecular Casimir force per unit length of the beam, which is defined as follows: p2 h
cb

F4 ¼

(3)

240ðg  yÞ4

where h
¼ 1:055  1034 is the Planck's constant divided by 2p and c = 2.998  108 m/s is the speed of light. For cantilever nano-beams by incorporating electrostatic actuation and intermolecular forces, the nonlinear governing equation of motion 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 the Casimir effect. The beam deflection is subjected to four kinematic boundary conditions as: yð0; tÞ ¼ 0;

y0 ð0; tÞ ¼ 0;

y00 ðl; tÞ ¼ 0;

y000 ðl; tÞ ¼ 0;

yðx; 0Þ ¼ A fðxÞ;

_ 0Þ ¼ 0; yðx;

(5-b)

where f(x) is the first eigenmode of the cantilever beam. By introducing the following nondimensional variables sffiffiffiffiffiffiffiffiffiffiffiffi EI y x g 24eV2 l4 t¼ ; t; Y ¼ ; j ¼ ; a ¼ ; l2 ¼ 4 g l b rbhl Eh3 g3 Ah bl4 ; 6EIpg4

l3 ¼

l4 ¼

p2 hcbl4 240EIg5

(6)

the nondimensional nonlinear equation of motion can be written as (7)

The nondimensional forces Fes and Fn in Eq. (4) can be approximated by Taylor's series as l2 ½ð1 þ abes Þ þ ð2 þ abes ÞY þ ð3 þ abes ÞY 2 4 þ ð4 þ abes ÞY 3 þ ð5 þ abes ÞY4 þ    

Fes ¼

F3 ¼ l3 ð1 þ 3Y þ 6Y 2 þ 10Y 3 þ 15Y 4 þ    Þ 2

3

4

F4 ¼ l4 ð1 þ 4Y þ 10Y þ 20Y þ 35Y þ    Þ (1)

(5-a)

and the following initial conditions:

@2 Y @4 Y l2 ln þ 4  ð1 þ abes ð1  YÞÞ  ¼0 @t2 ð1  YÞn @j 4ð1  YÞ2

Mathematical formulation

A nano-cantilever beam considered here has length l, thickness h, width b, density r and modulus of elasticity E as illustrated in Fig. 2. The air initial gap is g and an attractive electrostatic force which originates from voltage V causes the nano-beam to deflect. The electrostatic force per unit length of the beam can be expressed as [4]: Fes ¼

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

(8) (9) (10)

substituting Fes and Fn from Eqs. (8)–(10) into Eq. (7), the nondimensional nonlinear equation of motion can be written as € þ Y ð4Þ þ b Y þ b Y2 þ b Y 3 þ b Y 4 þ b ¼ 0 Y 1;n 2;n 3;n 4;n 0;n

(11)

where the parameters b0;n . . . b4;n have been described in the Appendix. Assuming Yðj; tÞ ¼ qðtÞ fðjÞ, where fðjÞ is the first eigenmode of the cantilever beam and can be expressed as: fðjÞ ¼ coshðlc jÞ  cosðlc jÞ Fig. 2 – Schematic of an actuated nano-cantilever beam.



coshðlc Þ þ cosðlc Þ ðsinhðlc jÞ  sinðlc jÞÞ sinhðlc Þ þ sinðlc Þ

(12)

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

4

archives of civil and mechanical engineering xxx (2014) xxx–xxx

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

:: q 1 ðtÞ þ v2 q1 ðtÞ ¼

d2 qðtÞ þ b01;n qðtÞ þ ½b02;n ðqðtÞÞ2 þ b03;n ðqðtÞÞ3 þ b04;n ðqðtÞÞ4 þ b00;n  ¼ 0 dt2 (13-a) where b01;n ¼ b1;n þ l4c Z 1 b0i;n ¼ bi;n fiþ1 dj;

(13-b)

ði 6¼ 1Þ

0

subjected to the following initial conditions: _ qð0Þ ¼ 0;

qð0Þ ¼ A;

(13-c)

In the current study, in order to obtain the closed-form expression for the second order approximate frequency of actuated nano-beams, Eq. (13-a) is solved analytically, using PEM. On the other hand, in order to validate the integrity of asymptotic approach, Matlab built-in function ‘‘ode45’’ is employed for numerical simulations by considering three mode assumptions.

3.

Asymptotic analysis by PEM

The basic character of the PEM is to expand the solution and some parameters in the nonlinear governing equation. This asymptotic approach involves the bookkeeping parameter method and the modified Lindstedt–Poincare method. Consider the nonlinear equation of motion described in Eq. (13-a) subject to the following initial conditions: _ qð0Þ ¼0

qð0Þ ¼ A;

(14)

According to the PEM, the solution q(t) and the factors 1 and b01;n may be expanded in series of p: qðtÞ ¼ q0 ðtÞ þ pq1 ðtÞ þ p2 q2 ðtÞ þ   

(15)

b01;n ¼ v2  pb1  p2 b2 þ   

(16-a)

1 ¼ pc1 þ p2 c2 þ   

(16-b)

1

p :

€0 ðtÞ þ v2 q0 ðtÞ ¼ 0; q

€1 ðtÞ þ v q1 ðtÞ ¼ b1 q0 ðtÞ  q 2

þ p2 :

q0 ð0Þ ¼ A;

b04;n ðq0 ðtÞÞ4

þ

b00;n ;

q_ 0 ð0Þ ¼ 0

c1 ½b02;n ðq0 ðtÞÞ2

q1 ð0Þ ¼ 0;

þ

3 b1 ¼ c1 b03;n A2 4

thereby, solving Eq. (18) for q1(t) yields the following secondorder approximation for the solution q(t)as: cosðvtÞð48b04;n A4 þ 160b02;n A2  15b03;n A3 þ 480b00;n Þ q1 ðtÞ ¼ 480v2 cosð2vtÞð80b04;n A4 þ 80b02;n A2 Þ b03;n A3 cosð3vtÞ þ þ 480v2 32v2 4 0 0 b4;n A cosð4vtÞ 480b0;n  180b04;n A4  240b02;n A2 þ þ 120v2 480v2 (22) Eqs. (16-a) and (16-b) for two terms approximation of series respect 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) and eliminating the secular terms gives: 5 2 7 1 3 SðvÞ ¼  b02;n A3  b04;n b02;n A5 þ b03;n b02;n A4 þ b03;n b00;n A2 6 4 2 2 63 0 2 7 3 0 0 6 0 2  b4;nA þ b3;n b4;n A þ b1;n Av  2b02;n b00;n A 80 10 3 0 3 02 5 3 2 b A  3b04;n b00;n A3  Av4 ¼ 0 (24) þ b3;n A v  4 128 3;n solving Eq. (24) for the frequency of oscillation v, the following second-order frequency–amplitude relation for vibrating actuated nano-cantilever cab be obtained: " 4 b01;n 3 0 b02 3b01;n b03;n A2 15b02 1;n 3;n A þ b3;n A2 þ þ þ vðAÞ ¼ 8 2 4 8 128 

2 7b02;n b04;n A4 b02;n b03;n A3 3b00;n b03;n A 5b02 2;n A þ þ  4 2 6 2

6 3b03;n b04;n A5 63b02 4;n A  3b04;n b00;n A2   2b02;n b00;n þ 10 80

!1=2 #1=2

(25)

b03;n ðq0 ðtÞÞ3

q_ 1 ð0Þ ¼ 0

(18)

þ 3b03;n ðq0 ðtÞÞ2 q1 ðtÞ þ 4b04;n ðq0 ðtÞÞ3 q1 ðtÞ  c2 ½b02;n ðq0 ðtÞÞ2 q_ 2 ð0Þ ¼ 0

(21)

(17)

€2 ðtÞ þ v2 q2 ðtÞ ¼ b1 q1 ðtÞ þ b2 q0 ðtÞ  c1 ½2b02;n q0 ðtÞq1 ðtÞ q þ b03;n ðq0 ðtÞÞ3 þ b04;n ðq0 ðtÞÞ4 þ b00;n ;

(20)

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

Substituting Eqs. (15) and (16) into Eq. (13-a) and equating the terms with the same powers of p results in: p0 :

  3 b1 A  c1 b03;n A3 cosðvtÞ 4   1 1 þ  c1 b04;n A4  c1 b02;n A2 cosð2vtÞ 2 2 1 3 2 0  c1 b2;n A  c1 b04;n A4 2 8 1 3 0  c1 b3;n A cosð3vtÞ  c1 b00;n 4 1  c1 b04;n A4 cosð4vtÞ; 8

q2 ð0Þ ¼ 0; (19)

When p = 0, Eq. (13-a) becomes a linear differential equation and the approximated solution can be obtained for p = 1. The solution of the first equation is q0 ðtÞ ¼ A cosðvtÞ, substitution of this result into the right-hand side of Eq. (18) gives

4.

Results and discussion

In order to verify the results of numerical simulations, the time histories of vibrating nano-cantilever in the absence of intermolecular forces by adding the modified couple stress parameter are compared with the reported results by Rahaeifard et al. [42]. As can be seen in Fig. 3, the results of present work exhibits an excellent agreement with the results obtained by Rahaeifard et al. [42] using a hybrid finite difference method. To investigate the accuracy of the asymptotic approach to predict the pull-in phenomenon, another comparison is

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

5

archives of civil and mechanical engineering xxx (2014) xxx–xxx

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

performed considering the nano-cantilever studied by Moghimi Zand and Ahmadian [5] by employing the finiteelement method (FEM). Fig. 4(a) and (b) depicts the effects of van der Waals parameter l3 and Casimir parameter l4 on the dynamic pull-in values of nano-actuators. According to these figures, for two intermolecular models, PEM predicts slightly higher dynamic pull-in value in comparison with the obtained results using FEM. In addition, these figures indicate that an increase in the dispersion force parameters, results in a decrease in dynamic pull-in values. Furthermore, to demonstrate the soundness of proposed solution by PEM, the analytical solutions at the side of corresponding numerical results have been plotted. As can be seen in Fig. 5, the second order approximation of q(t) using

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

analytical method exhibits a good agreement with numerical results by considering three mode assumptions for both van der Waals and Casimir effects assumption. In order to show the convergence of the asymptotic analysis by PEM, the absolute error for different approximate orders at t = 1, is tabulated in Table 1. As can be observed, for different values of initial amplitudes, the absolute error decreases as the order of approximation employed in series solution increases. Eq. (25) reveals that the initial amplitude of vibration has a significant effect on the dynamic behavior of NEMS/MEMS structures. To show the dependency of fundamental frequency and pull-in instability of such systems on the initial

Table 1 – Comparison between the errors of different estimations for approximate solution by assuming van der Waals attraction.       q  q  q  A numeric  q0 numeric  q1 numeric  q2

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

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

0.04973997 0.04975019 0.05066131 0.05238962 0.05526881 0.05934554 0.06509971 0.07285962 0.08403840

0.00014507 0.00052312 0.00095012 0.00141606 0.00206676 0.00309974 0.00478752 0.00758339 0.01234765

0.00014507 0.00017083 0.00022617 0.00024683 0.00027451 0.00034945 0.00091333 0.00237459 0.00549251

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

6

archives of civil and mechanical engineering xxx (2014) xxx–xxx

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.

conditions, the impact of nondimensional initial amplitude on the dynamics of actuated cantilever nano-beams have been investigated. Fig. 6 shows the characteristic curves of natural frequency for a nano-beam under some assigned values of van der Waals and Casimir parameters l3, l4. It is obvious that these nondimensional parameters have notable effect on predicting the pull-in instability and natural frequency. At lower values of initial amplitude, the beam performs a periodic motion and the frequency of vibration decreases by increasing the initial amplitude. In the vicinity of the dynamic pull-in point, a small increase in the amplitude causes the nano-beam to be dynamically unstable; the pull-in instability occurs and the nano-beam collapses onto the substrate. It is obvious from Fig. 6 that under the effect of the Casimir force, the dynamic pull-in instability occurs in the long regions of nondimensional amplitude A in comparison with van der Waals effects. On the other hand, the fundamental frequency decreases by increasing the intermolecular parameters. When the nondimensional parameters l3, l4 increase, periodic motions take place in the short regions of vibrational amplitude. If the intermolecular parameters are equal to 0.3, the system becomes dynamically unstable at A  0.42 when the van der Waals force is taken into consideration; instead as the Casimir effect is included, the nano-actuator diverges to the rigid plate at A  0.29.

The influence of actuation voltage parameter l on the natural frequency and pull-in instability of elector-actuated nano-beam as a function of nondimensional amplitude has been illustrated in Fig. 7. It is clear from this figure that the fundamental frequency decreases as the parameter l increases and pull-in instability occurs in the long domains of initial amplitude when the natural frequency of nano-beam drops to zero. The influence of nondimensional parameter a is shown in Fig. 8, where the nonlinear v–A curves of clamped-free actuated nano-beams are compared. As can be observed, the fundamental frequency decreases as the parameter a increases. The frequency vanishes and pull-in instability occurs in the long regions of vibrational amplitude by increasing the nondimensional parameter a, for both van der Waals and Casimir force models. The phase portraits of nano-beam actuator employing Casimir and van der Waals attractions are plotted in Figs. 9–11. It is concluded from Fig. 9a and b that for voltage parameter l = 1, there exist the equilibrium point as a center point, and there are periodic orbits around it and homoclinic orbits starting from unstable saddle node and going back to it. It can be concluded that in this situation, the nano actuator exhibits periodic oscillation near the equilibrium center point and drops to the substrate beyond the unstable saddle node. In

Fig. 7 – Fundamental frequency vs. nondimensional amplitude of nano-beams, the impact of voltage parameter l. (a) van der Waals force and (b) Casimir force. 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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10 archives of civil and mechanical engineering xxx (2014) xxx–xxx

7

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

particular, under the effect of Casimir attraction force, the stable periodic orbits convert to unstable orbits at lower values of initial conditions. When the actuation voltage increases to l = 1.6, as can be seen in Fig. 10, the unstable saddle point becomes closer to stable center point. This means that, by increasing the actuation voltage parameter, the pull-in instability happens at lower values of vibrational amplitude. At particular value of voltage parameter (here l = 1.88 for van der Waals force and

l = 1.81 for Casimir effect), the nano-beam becomes dynamically unstable for all values of initial conditions. The phase plane shown in Fig. 11 indicates that in this situation, the stable center point and unstable saddle node merge together and there are no periodic orbits in the phase portrait of the system. As mentioned before, the amplitude of vibration has significant influence on the nonlinear dynamic behavior of nano-structures. When the initial condition increases, the

Fig. 10 – The phase portrait of nano-beam actuator with actuation parameter l = 1.6: (a) effect of van der Waals force and (b) effect of Casimir force. 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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

8

archives of civil and mechanical engineering xxx (2014) xxx–xxx

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 force for 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 of Casimir force.

system becomes unstable and the actuated nano-beam collapse onto the rigid substrate. According to Fig. 12, it is obvious that by increasing the initial amplitude A, the time period of oscillation increases until the specific point namely ‘‘pull-in’’ (here A = 0.767 for van der Waals force and A = 0.638 for Casimir attraction) where the beam touches the substrate. Furthermore, as can be observed, when the Casimir attraction force is taken into account, the instability occurs at lower value of initial condition. Dynamic pull-in voltage of actuated nano-structure as a function of initial amplitude and for some assigned values of intermolecular parameters l3, l4 is plotted in Fig. 13. It appears form the figure that by increasing the amplitude of vibration, the dynamic pull-in voltage decreases for both intermolecular force models. In addition, the dynamic pull-in voltage decreases when the parameters l3, l4 increase until the dynamic pull-in voltage vanishes. In this situation, in the absence of actuation voltage, the nano-beam is unstable for some values of initial conditions. For example, when the parameters l3 and l4 are equal to 0.4, for van der Waals assumption, the dynamic pull-in voltage disappear as the initial amplitude approaches to A = 0.51, however, when the Casimir force effect is considered, the dynamic pull-in voltage tends to zero as the initial amplitude approaches to A = 0.37. This means that, in the absence of applied voltage, the freestanding nano-beam collapse onto the substrate at the aforementioned values of initial amplitudes.

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

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10 archives of civil and mechanical engineering xxx (2014) xxx–xxx

5.

Conclusion

In the current work, dynamic pull-in instability of nanocantilever based NEMS was investigated. The second-order frequency–amplitude relationship of vibrating actuated nanobeams including dispersion forces was proposed in order to study the amplitude dependence of pull-in instability of actuated nano-structure. The influences of actuation voltage and intermolecular forces on the system dynamic were presented. It was illustrated that the pull-in voltage and fundamental frequency of the system decrease by increasing the initial amplitude of vibration. At specific value of initial condition, pull-in phenomenon occurs and the nano-beam collapse onto the substrate. Furthermore, the qualitative analysis of the system motion exhibits that the equilibrium points of the nano-actuators include stable center points and unstable saddle nodes.

Appendix A b1;3 ¼  b2;3 ¼  b3;3 ¼  b4;3 ¼  b0;3









l2 4

ð2 þ abes Þ þ 3l3

l2 4

ð3 þ abes Þ þ 6l3

l2 4

ð4 þ abes Þ þ 10l3

l2 4



ð5 þ abes Þ þ 15l3
2  l ¼  4 ð1 þ abes Þ þ l3

 

b1;4 ¼  b2;4 ¼  b3;4 ¼  b4;4 ¼  b0;4









l2 4

ð2 þ abes Þ þ 4l4

l2 4

ð3 þ abes Þ þ 10l4

l2 4

ð4 þ abes Þ þ 20l4

l2 4

ð5 þ abes Þ þ 35l4
2  l ¼  4 ð1 þ abes Þ þ l4

  

references

[1] M.A. Eltaher, F.F. Mahmoud, A.E. Assie, E.I. Meletis, Coupling effects of nonlocal and surface energy on vibration analysis of nanobeams, Applied Mathematics and Computation 224 (2013) 760–774. [2] J. Abdi, A. Koochi, A.S. Kazemi, M. Abadyan, Modeling the effects of size dependence and dispersion forces on the pullin instability of electrostatic cantilever NEMS using modified couple stress theory, Smart Materials and Structures 20 (2011) 055011. [3] R. Soroush, A. Koochi, A.S. Kazemi, A. Noghrehabadi, H. Haddadpour, M. Abadyan, Investigating the effect of Casimir and van der Waals attractions on the electrostatic pull-in instability of nano-actuators, Physica Scripta 82 (2010) 045801. [4] A. Noghrehabadi, M. Ghalambaz, A. Ghanbarzadeh, A new approach to the electrostatic pull-in instability of nanocantilever actuators using the ADM-Padé technique, Computers and Mathematics with Applications 64 (9) (2012) 2806–2815. , http://dx.doi.org/10.1016/j.camwa.2012.04.013. [5] M. Moghimi Zand, M.T. Ahmadian, Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 224 (2010) 2037–2047, http:// dx.doi.org/10.1243/09544062JMES1716. [6] A. Gusso, G.J. Delben, Dispersion force for materials relevant for micro- and nanodevices fabrication, Journal of Physics D: Applied Physics 41 (2008) 175405, http://dx.doi.org/10.1088/ 0022-3727/41/17/175405.

9

[7] A.W. Rodriguez, F. Capasso, S.G. Johnson, The Casimir effect in microstructured geometries, Nature Photonics 5 (2011) 211–221. , http://dx.doi.org/10.1038/nphoton.2011.39. [8] M. Bordag, U. Mohideen, V.M. Mostepanenko, New developments in the Casimir effect, Physics Reports 353 (1–3) (2001) 1–205. [9] R.C. Batra, M. Porfiri, D. Spinello, Review of modeling electrostatically actuated microelectromechanical systems, Smart Materials and Structures 16 (2007) R23, http://dx.doi. org/10.1088/0964-1726/16/6/R01. [10] M. Janghorban, Static and free vibration analysis of carbon nano wires based on Timoshenko beam theory using differential quadrature method, Latin American Journal of Solids and Structures 8 (2011) 463–472. [11] J.G. Guo, Y.P. Zhao, Dynamic stability of electrostatic torsional actuators with van der Waals effect, International Journal of Solids and Structures 43 (2006) 675–685. [12] Y. Chan, N. Thamwattana, J.M. Hill, Axial buckling of multiwalled carbon nanotubes and nanopeapods, European Journal of Mechanics A: Solids 30 (6) (2011) 794–806. [13] A. Koochi, A. Noghrehabadi, M. Abadyan, Approximating the effect of van der Waals force on the instability of electrostatic nano-cantilevers, International Journal of Modern Physics B 25 (29) (2011) 3965–3976. [14] R. Soroush, A. Koochi, A.S. Kazemi, M. Abadyan, Modeling the effect of van der Waals attraction on the instability of electrostatic cantilever and doubly-supported nanobeams using modified adomian method, International Journal of Structural Stability and Dynamics 12 (05) (2012) 1250036. [15] M. Rasekh, S.E. Khadem, Pull-in analysis of an electrostatically actuated nano-cantilever beam with nonlinearity in curvature and inertia, International Journal of Mechanical Sciences 53 (2) (2011) 108–115. [16] Y. Fu, J. Zhang, Size-dependent pull-in phenomena in electrically actuated nanobeams incorporating surface energies, Applied Mathematical Modelling 35 (2) (2011) 941–951. [17] A. Ramezani, A. Alasty, J. Akbari, Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces, International Journal of Solids and Structures 44 (14–15) (2007) 4925–4941. [18] G. Duan, K.T. Wan, ‘‘Pull-in’’ of a pre-stressed thin film by an electrostatic potential: a 1-D rectangular bridge and a 2-D circular diaphragm, International Journal of Mechanical Sciences 52 (9) (2010) 1158–1166. [19] J.G. Boyd, J. Lee, Deflection and pull-in instability of nanoscale beams in liquid electrolytes, Journal of Colloid and Interface Science 356 (2) (2011) 387–394. [20] J.G. Guo, Y.P. Zhao, The size-dependent bending elastic properties of nanobeams with surface effects, Nanotechnology 18 (2007) 295701, http://dx.doi.org/10.1088/ 0957-4484/18/29/295701. [21] W.H. Lin, Y.P. Zhao, Nonlinear behavior for nanoscale electrostatic actuators with Casimir force, Chaos, Solitons and Fractals 23 (2005) 1777–1785. [22] W.H. Lin, Y.P. Zhao, Dynamic behavior of nanoscale electrostatic actuators, Chinese Physics Letters 20 (11) (2003) 2070–2073. [23] R. Bansal, J.V. Clark, Lumped modeling of carbon nanotubes for M/NEMS simulation, Microsystem Technologies 18 (12) (2012) 1963–1970. [24] Y. Zhang, Y. Liu, K.D. Murphy, Nonlinear dynamic response of beam and its application in nanomechanical resonator, Acta Mechanica Sinica 28 (1) (2012) 190–200. [25] J.G. Guo, Y.P. Zhao, Influence of van der Waals and Casimir forces on electrostatic torsional actuators, Journal of Microelectromechanical Systems 13 (6) (2004) 1027–1035.

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004

ACME-186; No. of Pages 10

10

archives of civil and mechanical engineering xxx (2014) xxx–xxx

[26] W.H. Lin, Y.P. Zhao, Casimir effect on the pull-in parameters of nanometer switches, Microsystem Technologies 11 (2005) 80–85. [27] A.H. Nayfeh, Perturbation Methods, Wiley-Interscience, New York, 1973. [28] H.M. Sedighi, K.H. Shirazi, J. Zare, An analytic solution of transversal oscillation of quintic nonlinear beam with homotopy analysis method, International Journal of NonLinear Mechanics 47 (2012) 777–784. [29] H.M. Sedighi, K.H. Shirazi, A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition, Journal of Computational and Nonlinear Dynamics 7 (3) (2012) 034502, http://dx.doi.org/ 10.1115/1.4005924. [30] H.M. Sedighi, K.H. Shirazi, J. Zare, Novel equivalent function for deadzone nonlinearity: applied to analytical solution of beam vibration using He's parameter expanding method, Latin American Journal of Solids and Structures 9 (2012) 443–451. [31] H.M. Sedighi, K.H. Shirazi, A.R. Noghrehabadi, A. Yildirim, Asymptotic investigation of buckled beam nonlinear vibration, Iranian Journal of Science and Technology, Transaction of Mechanical Engineering 36 (2012) 107–116. [32] H.M. Sedighi, K.H. Shirazi, A. Reza, J. Zare, Accurate modeling of preload discontinuity in the analytical approach of the nonlinear free vibration of beams, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 226 (10) (2012) 2474–2484, http://dx.doi.org/10.1177/0954406211435196. [33] M. Jalaal, D.D. Ganji, G. Ahmadi, Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media, Advanced Powder Technology 21 (3) (2010) 298–304.

[34] M. Jalaal, D.D. Ganji, On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids, Advanced Powder Technology 22 (1) (2011) 58–67. [35] M. Jalaal, D.D. Ganji, An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid, Powder Technology 198 (1) (2010) 82–92. [36] H.M. Sedighi, K.H. Shirazi, M.A. Attarzadeh, A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches, Acta Astronautica 91 (2013) 245– 250. [37] D.D. Ganji, A semi-analytical technique for non-linear settling particle equation of motion, Journal of HydroEnvironment Research 6 (4) (2012) 323–327. [38] J.H. He, Max–Min approach to nonlinear oscillators, International Journal of Nonlinear Sciences and Numerical Simulation 9 (2008) 207–210. [39] H.M. Sedighi, K.H. Shirazi, Asymptotic approach for nonlinear vibrating beams with saturation type boundary condition, Proceedings of the Institution of Mechanical Engineers. Part C: Journal of Mechanical Engineering Science 227 (11) (2013) 2479–2486. [40] J.H. He, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics B 20 (2006) 1141–1199. [41] J.H. He, Bookkeeping parameter in perturbation methods, International Journal of Nonlinear Sciences and Numerical Simulation 2 (2001) 257–264. [42] M. Rahaeifard, M.T. Ahmadian, K. Firoozbakhsh, Sizedependent dynamic behaviour of microcantilevers under suddenly applied DC voltage, ProcIMechE Part C: J Mechanical Engineering Science (2013), http://dx.doi.org/10.1177/ 0954406213490376.

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 nanocantilever based NEMS including fringing field effect, Archives of Civil and Mechanical Engineering (2014), http://dx.doi.org/10.1016/j. acme.2014.01.004