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micro-beams under the effect of van der Waals force, a semi-analytical approach
Accepted Manuscript
Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach Amir R. Askari , Masoud Tahani PII: DOI: Reference:
S1007-5704(15)00345-7 10.1016/j.cnsns.2015.10.014 CNSNS 3672
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
15 May 2014 21 September 2015 21 October 2015
Please cite this article as: Amir R. Askari , Masoud Tahani , Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach, Communications in Nonlinear Science and Numerical Simulation (2015), doi: 10.1016/j.cnsns.2015.10.014
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ACCEPTED MANUSCRIPT
Highlights A new procedure for pull-in analysis of beam-type N/MEMS under vdW force has been introduced. This procedure able us to represent analytical and semi-analytical solutions for pull-in problems. This method can extract all pull-in parameters simultaneously. It is found that this new method agrees better than previous ones with FE results.
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Some closed-form expressions are also presented for vdW and electrical pull-in
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parameters.
ACCEPTED MANUSCRIPT
Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach
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Amir R. Askari, Masoud Tahani* Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
ABSTRACT
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The objective of the present paper is to determine pull-in parameters (pull-in voltage and its corresponding displacement) of nano/micro-beams with clamped-clamped, clamped-free, clamped-hinged and hinged-hinged boundary conditions, when they are subjected to the electrostatics and van der Waals (vdW) attractions. The governing non-linear boundary value
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equation of equilibrium is derived, non-dimensionalized and reduced to an algebraic
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equation, which describes the position of the maximum deflection of the beam, utilizing the Galerkin decomposition method. The equation which governs on the stability condition of the
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system is also obtained by differentiating the reduced equilibrium equation with respect to the maximum deflection of the beam. These two equations are solved simultaneously to
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determine pull-in parameters. Closed-form solutions are provided for cases under electrical
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loading and vdW attraction alone. The combined effect of both electrostatic and vdW loadings are also investigated using the homotopy perturbation method (HPM). It is found that the present semi-analytical findings are in excellent agreement with those obtained numerically. In addition, it is observed that the present semi-analytical approach can provide results which agree better with available three-dimensional finite element simulations as well *
Corresponding author. Tel.: +98-511-8806055; fax: +98-511-8763304. E-mail address: [email protected] (Masoud Tahani).
ACCEPTED MANUSCRIPT as those obtained by nonlinear finite element method than other available analytical or semianalytical findings in the literature. Non-dimensional electrostatic and vdW parameters, which are defined in the text, are plotted versus each other at pull-in condition. It is found that there exists a linear relationship between these two parameters at pull-in condition. Using this fact, pull-in voltage, detachment length and minimum allowable gap of electrostatically
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actuated nano/micro-beams are determined explicitly through some closed-form expressions.
Keywords: Nano/micro-electro-mechanical beams, Pull-in instability, vdW attraction,
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Homotopy perturbation method, Detachment length, Closed-form expressions.
1. Introduction Nano/micro-systems
have
applications
in
many
engineering
fields
such
as
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communications, automotive and robotics [1]. Nano/micro-electro-mechanical systems (N/MEMS) can be considered as a largest collection of these systems, because of their fast
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response, low power consumption, reliability and their capability of batch fabrications [2].
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Electrically actuated nano/micro-beams represent a major structural component and play a crucial role in many N/MEMS devices [2]. The main components of a typical N/MEMS
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device can be considered as a fixed electrode and a movable one. The movable electrode can be modeled as an electrically actuated nano/micro-beam or plate which deflects toward the
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fixed one through the application of an external voltage on the system. The applied voltage has an upper limit in which the electrostatic attraction overcomes the elastic restoring force of the movable electrode. Therefore, the movable electrode suddenly collapses toward the fixed one in this manner. This unstable phenomenon is called pull-in instability. The maximum deflection occurred just before the collapsing of movable electrode is also called pull-in
ACCEPTED MANUSCRIPT displacement. Pull-in displacement and pull-in voltage, the so-called pull-in parameters, can be considered as the most important parameters in designing electrically actuated nano/micro-beams. By decreasing the dimensions of electrically actuated systems from micro-scales to nanoscales, the intermolecular surface forces significantly influence on the behaviors of
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nano/micro-beams. The most important forces at the scale of N/MEMS are the Casimir and vdW attractions. The vdW force arises from the correlated oscillation of the instantaneously induced dipole moments of the atoms placed at the close parallel conductive plates [3]. The vdW force is a short range force in nature, but it can lead to long range effects more than 0.1
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μm [4]. The Casimir force can be simply understood as the long range analog of the vdW force, resulting from the propagation of retarded electromagnetic waves [5]. The main purpose of the present paper is to investigate the effect of the vdW attraction on
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the characteristics of pull-in instability in beam-type N/MEMS. To date, lots of researchers have dealt with the mechanical behavior of electrically actuated nano/micro-beams and
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plates. A comprehensive review of different models of electrostatically actuated micro-
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systems are presented by Batra et al. [6]. Herein, some of the most pioneering works in the field of modelling pull-in instability in N/MEMS devices are reviewed.
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Nathanson et al. [7] and Taylor [8] simultaneously observed pull-in instability in microsystems. Pamidighantam et al. [9] studied electrically actuated micro-cantilevers as well as
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clamped-clamped micro-beams using equivalent spring-mass model. They present a closedform expression for pull-in voltage and validated their solutions with finite element (FE) simulations provided by Coventorware (CW) commercial software. Their model improved the estimation of previous model presented by Nathanson et al [7] for pull-in voltage; however, their model represent this value in terms of pull-in displacement which was
ACCEPTED MANUSCRIPT empirically chosen. Chowdhury et al. [10] also presented a closed-form solution for cantilever micro-beams using another equivalent spring-mass model. They used the third order Maclaurin's series expansion of the electrostatic forcing term to calculate the effective stiffness coefficient of the micro-beam. Kuang and Chen [11] solved the non-linear boundary value equation which governs on the equilibrium of both micro-cantilevers and doubly
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clamped micro-beams semi-analytically using Adomian decomposition method (ADM). Mojahedi et al. [12] investigated static pull-in instability using the combination of Galerkin’s projection method and the HPM. They converted the governing boundary value equation of equilibrium to an algebraic equation using the first linear un-damped mode-shape of an un-
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deformed micro-beam and solved the resulting non-linear algebraic equation through the HPM. It should be noted that they expanded the electrostatic forcing term about the static deflection of the movable electrode which was updated iteratively by increasing in the value
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of the input voltage. Therefore, their iterative approach cannot be considered as an analytical procedure. Ramezani et al. [13] proposed a distributed parameter model to study pull-in
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instability of electrically actuated nano-cantilevers subjected to the vdW and the Casimir forces. They transferred non-linear differential equation of the model into the integral form
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by using the Green’s function of the clamped-free beam and the integral equation was solved
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using the shape function of the beam deflection. It is to be noted that, their solutions could not satisfy all boundary conditions and suffered from the over-estimation of pull-in voltage.
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Soroush et al. [14] also investigated the effect of the vdW attraction on pull-in instability of nano/micro-beams using the modified Adomian decomposition method (MADM). Although they could present some expressions for the detachment length and minimum allowable gap of nano/micro-systems, there exists a gap between their findings and those obtained by FE simulations [9].
ACCEPTED MANUSCRIPT It is worth noting that the solution of the equation which governs on the static equilibrium has been investigated in previous studies to capture pull-in instability when the slope of deflection-voltage graph reaches infinity. However, in present paper the equilibrium equation and the governing equation on the stability conditions are considered simultaneously. The governing equation of equilibrium is derived using the Euler-Bernoulli beam theory and
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reduced to an algebraic equation which describes the position of the beam maximum deflection through the Galerkin decomposition method. The stability equation is also obtained by differentiating the reduced equilibrium equation with respect to the maximum deflection of the beam. This set of non-linear algebraic equations is solved analytically for
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cases under electrical loading alone and vdW attraction alone to present some closed-form expressions for pull-in voltage and displacement as well as the minimum allowable gap and maximum allowable length of the nano/micro-beam. A semi-analytical solution is also
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performed using the HPM to investigate the interaction between electrical and vdW loadings, where the solution of the system under electrical loading alone is selected as the initial guess
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to construct the homotopy series solution. It should be noted that unlike the traditional
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perturbation methods, the HPM does not depend on the assumption of small parameters and can solve strongly non-linear problems [15].
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It is found that the present semi-analytical results agree better with available FE predictions than those obtained semi-analytically through beam model in previous studies. In
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addition, one of the advantages of the present approach is that both pull-in voltage and pull-in displacement are obtained simultaneously which is not possible in previous solution procedures. In other words, the previous studies focused on the solution of the governing equilibrium equation which determined the static deflection in terms of input voltage. This procedure captured pull-in instability form deflection-voltage graph when its slope reaches
ACCEPTED MANUSCRIPT infinity. However, the present approach provides pull-in parameters straightly. It is also found that the dimensionless pull-in voltage varies linearly versus the non-dimensional vdW parameter. Based on this important finding, some closed-form expressions are introduced to predict pull-in voltage in terms of the system parameters. It is worth mentioning that the present solution procedure is developed in terms of the
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nano/micro-beam mode-shapes. Therefore, one just needs to update the mode-shape function to obtain the proper results for any desired boundary conditions. In this way, the present findings are provided for usual beam-type N/MEMS with clamped-free (C-F), clamped-
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clamped (C-C), clamped-hinged (C-H) and hinged-hinged (H-H) boundary conditions.
2. Problem formulation
Fig. 1 shows a typical structure of electrostatically actuated nano/micro-systems, where
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the main components are fixed and movable electrodes. The fixed electrode modeled as a ground plane and the movable one modeled as a nano/micro-beam. The length, width,
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thickness and density of the nano/micro-beam are L, b, h and , respectively. The initial gap
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between the non-actuated beam and the stationary electrode is d. Also, x, y and z are the coordinates along the length, width and thickness, respectively. W is the deflection of the
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beam, I is the second moment of cross-sectional area about the y axis, is Poisson’s ratio and E is the effective Young’s modulus of the nano/micro-beam which is replaced
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by E / (1 2 ) when b 5h [16]. It is to be noted that, due to the finite dimensions of the electrodes, the lines of
electrostatic field are not parallel over the edges. The influence of these unparalleled lines of electrostatic field over the edges is called the fringing field effect. Considering the fringing
ACCEPTED MANUSCRIPT field effect, the electrostatic excitation by polarized DC voltage V per unit length of the beam can be expressed as [17]
Fes
bV 2 2d W
2
d W 1 0.65 b
(1)
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where is the dielectric constant of the medium. It is worth noting that the fringing field effect can be neglected for the case of wide nano/micro-beams [18].
As it is mentioned in the previous section, the vdW and Casimir attractions significantly affect the nano/micro-beam deflection at sub-micron separations. The vdW and Casimir
f Cas
Ab 6 d W
3
2 cb 240 d W
4
(2a)
(2b)
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f vdW
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forces per unit length of the beam, respectively, take the following forms [5, 19]:
1.055 1034 J is the Plank constant divided by 2
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where A is the Hamaker constant,
media
(Ag,
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and c 2.998 108 m/sec is the light speed. The Hamaker constant for two identical metal Au,
Cu)
interacting
across
vacuum
(air)
has
values
in
range
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30 50 1020 J [19]. It is to be mentioned here that the Casimir force can be considered as
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the long range analog of the vdW attraction [5]. Hence, the effects of these two attractions should not be considered simultaneously. The differential equation which governs on the equilibrium of the nano/micro-beam
subjected to the combined effects of the electrostatic and dispersion forces (i.e. vdW or Casimir attraction) can be expressed as follows [20]:
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EI
d 4W Fes Fdisp dx 4
(3)
where Fdisp refers to the vdW attraction or the Casimir force. In present paper, it is assumed that the nano/micro-beam deflection is subjected to C-C, C-
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F, C-H and H-H boundary conditions. For convenience, the following non-dimensional variables are introduced:
W x Wˆ , xˆ d L
(4)
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Upon substitution of the non-dimensional quantities given in Eq. (4) into Eq. (3) and dropping the hats, the following result will be obtained: W
(1 W )
(1 W )
2
m (1 W )m
(5)
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where the prime sign denotes derivative with respect to x. Also, m is the dispersion force
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index which specifies the type of the applied force at sub-micron separations. That is, m 3 refers to systems under the vdW force and m 4 is related to those subjected to the Casimir
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attraction. The non-dimensional parameters , , 3 and 4 , which are, respectively, called
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as the fringing field effect, electrostatic, vdW and Casimir parameters, take the form: d b
6 V 2 L4 2 AL4 2 cL4 , , 3 4 Eh3d 3 Eh3d 4 20 Eh3d 5
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0.65 ,
(6)
It is to be mentioned here that, in the present paper, we focus on the effect of the vdW
force on the pull-in characteristics of wide nano/micro-beams which represent major applications in N/MEMS devices [18, 21, 22]. 3. Solution procedure
ACCEPTED MANUSCRIPT Due to the high non-linearity involved in Eq. (5), an exact solution for this equation has not been found to date. Hence, an approximate solution will be developed here through the Galerkin weighted residual method. Based on this procedure, the nano/micro-beam deflection can be expressed as a linear combination of a complete set of linearly independent basis functions [23]. It is noted that these functions must satisfy all kinematic boundary conditions
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[23]. Therefore, linear un-damped mode-shapes of the un-deformed nano/micro-beam can be used as these basis functions. It is proved that using only the first mode for pull-in analysis of electrically actuated nano/micro-beams maybe very accurate [12, 24, 25]. Hence, the deflection of the nano/micro-beam based on one-mode solution can be expressed as
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W x w x
(7)
where w is an un-known generalized coordinate, which should be determined through the Galerkin procedure, and (x ) is the first linear un-damped mode-shape of the un-deformed
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nano/micro-beam. Herein, ( x) for C-C, C-F and C-H boundary conditions is determined as
ED
[26]
(8)
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x cosh x cos x sinh x sin x
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where the parameters , and for aforementioned boundary conditions are given in Table 1 [27]. It is noteworthy to mention that the normalization parameter is determined
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such that the unknown generalized coordinate w describes the maximum deflection of the nano/micro-beam whose position is also presented in Table 1 [25]. Furthermore, the normalized mode-shape of the H-H nano/micro-beam is selected as follows:
x sin x
(9)
ACCEPTED MANUSCRIPT As it is mentioned at the end of the previous section, the present paper focuses on the effect of the vdW force on wide nano/micro-beams pull-in characteristics. For this object, we set the values of and m in Eq. (5) to 0 and 3, respectively, and multiply this equation by
x (1 W )3 , substitute Eq. (7) into the resulting equation and integrate the outcome from
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x 0 to 1 and obtain
a0 a1 w a2 w2 a3 w3 a4 w4 0
(10)
where the coefficients ai i 0,1,..., 4 are given in the Appendix.
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Eq. (10) governs on the equilibrium of electrostatically actuated wide nano/micro-beams subjected to the combined effects of electrostatic and vdW forces. In other words, in this condition all the acting force upon the nano/micro-beam including the elastic restoring force, vdW attraction and electrostatic actuation are in balance. Hence, one can write
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f total w, , 3 a0 a1 w a2 w2 a3 w3 a4 w4 0
(11)
(12)
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f total 0 w
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The stable equilibrium provided [28]
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Therefore the threshold of stability is f total / w 0 . For this condition one can write (13)
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f total / w a1 2a2 wPI 3a3 wPI2 4a4 wPI3 0
It should be noted that Eq. (13) refers to pull-in condition. Therefore, the dimensionless
maximum deflection (w ) has been replaced by non-dimensional pull-in displacement (w PI ) in this equation. Non-dimensional pull-in voltage ( PI ) and displacement (w PI ) can be determined thorough the simultaneous solution of Eqs. (10) and (13). For the sake of brevity,
ACCEPTED MANUSCRIPT the non-dimensional pull-in displacement is called pull-in displacement hereinafter. Using the equation of the coefficient a1 in (A1), the non-dimensional pull-in voltage can be written as
PI
a dx dx 1
1
2
1
1
(14)
0
0
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The coefficient a1 can also be determined using Eq. (13) as
a1 2a2 wPI 3a3 wPI2 4a4 wPI3
(15)
Substituting Eqs. (14) and (15) into Eq. (10), may lead to the following equation in terms of
b0 b1 wPI b2 wPI2 b3 wPI3 b4 wPI4 0
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pull-in displacement (w PI ):
(16)
and given in the Appendix.
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where the coefficients bi ( i 0,1,..., 4 ) are known in terms of the input values of the problem
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Pull-in displacement (w PI ) can be found by solving Eq. (16). This equation is a non-linear algebraic equation which can be solved using the HPM. The coefficient a1 can also be
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determined by substituting w PI , which is calculated from Eq. (16), into Eq. (15). Non-
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dimensional pull-in voltage is also determined by substituting coefficient a1 into Eq. (14).
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3.1 The basic idea of the HPM The HPM is one of the most effective methods for finding semi-analytical solutions of
strongly non-linear problems. This method converts a general non-linear problem to infinite linear problems by embedding an auxiliary parameter p [15]. To illustrate the basic idea of the HPM briefly, let us consider a general non-linear problem as follows
ACCEPTED MANUSCRIPT u
u
u 0
(17)
where u is a general non-linear equation and u is an un-known parameter.
u is the
u is its non-linear part. Using
p [0,1] as
linear and homogeneous part of Eq. (17) and
an embedding parameter, the homotopy function for Eq. (17) can be constructed as [29]
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u, p 1 p u u p u 1 p u u p u u u u p u u
(18)
where u is an initial guess for the non-linear problem in (17). It is obvious that as p increases from 0 to 1, the solution of Eq. (18) varies from the initial guess to the solution of Eq. (17).
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Unlike the traditional perturbation techniques, the HPM, which does not depend on the assumption of a small parameter, applies these techniques on the homotopy function of a non-linear equation instead of the non-linear equation itself [29]. To do so, the un-known
u u 0 pu1 p 2u 2 p 3u 3 ...
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parameter u is expanded as a power series in terms of p as
(19)
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The solution of Eq. (17) can be found by substituting Eq. (19) into Eq. (18), collecting the
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terms with the identical powers of p i ( i 0,1, 2,... ) and setting the coefficients of p i to zero. By following this procedure, the non-linear problem is converted to a set of linear equations
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with respect to u i ( i 0,1, 2,... ) which can be solved easily. After finding sufficient
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approximations, by setting p 1 , the solution of the non-linear Eq. (17) can be found as
u u 0 u1 u 2 u 3 ... u n
where n is the order of the approximated solution.
3.2 Application of the HPM to the nano/micro-beam problem
(20)
ACCEPTED MANUSCRIPT Now, let us apply the HPM to the nano/micro-beam problem to find the solution of Eq. (16). The linear and non-linear parts of this equation can be considered as
wPI b1 wPI ,
wPI b0 b2 wPI2 b3 wPI3 b4 wPI4
(21)
Next, we expand the un-known parameter (w PI ) as a power series in terms of p, substitute it
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in the homotopy function of Eq. (16) and equate the terms with the identical powers of p and obtain
p 0 : w PI , 0 w PI
1 b0 b2 wPI2 , 0 b3 wPI3 , 0 b4 wPI4 , 0 b1
p 2 : wPI , 2
wPI ,1
p 3 : wPI , 3
wPI , 0
b1
b1
2b w 2
PI , 0
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p1 : wPI ,1
3b3 wPI2 , 0 4b4 wPI3 , 0
2b2 wPI , 2 3b3 wPI , 0 wPI , 2 wPI2 ,1
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2b4 wPI , 0 2wPI , 0 wPI , 2 3w
(22b)
(22c)
b2 wPI2 ,1 b 1
(22d)
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2 PI ,1
(22a)
The higher-order approximations for w PI can be established by considering more terms in Eq.
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(19) and following the same procedure.
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The initial guess (w PI ) has a sizable effect on adjusting convergence criteria for homotopy series solution. Therefore, it is very important to choose an appropriate value for the initial
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guess. Herein, the pull-in displacement for a system without the effect of the vdW force is considered as the initial guess. Using the Galerkin procedure discussed above, the governing equation of such system can be reduced as
c 0 c1 w c 2 w 2 c 3 w 3 0 where the coefficients ci i 0,1, 2,3 are presented in the Appendix.
(23)
ACCEPTED MANUSCRIPT As discussed already, pull-in condition can be obtained as
f total / w c1 2c 2 w 3c3 w 2 0
(24)
Eq. (24) is a quadratic equation, which can be solved analytically as
w PI
c 2 c 22 3c1c 3
(25)
3c 3
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It should be noted that only the positive root of Eq. (24) is acceptable for the pull-in displacement and if two positive roots are obtained, the minimum one will be selected. The corresponding pull-in parameter for this case can also be determined as
1 1
dx
c w 1
PI
c 2w PI2 c 3w PI3
(26)
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PI
0
3.3 Detachment parameters
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The maximum allowable length of a nano/micro-beam which prevents it from sticking to the substrate without the application of external voltage is called detachment length. It should
ED
be noted that a nano/micro-beam can be pulled-in under the effect of vdW attraction alone
PT
when the vdW parameter reaches its critical value. The detachment length and also the minimum allowable gap between non-actuated nano/micro-beam and its substrate can be
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determined through finding the critical value of the vdW parameter. To find this parameter, consider Eq. (5) without the electrostatic forcing terms. This equation can be reduced to the
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following algebraic equation using Galerkin’s procedure:
d0 d1 w d2 w2 d3 w3 d4 w4 0
(27)
where the coefficients di i 0,1,..., 4 are given in the Appendix. As discussed above, the threshold of stability is f total / w 0 . Therefore, one can write
ACCEPTED MANUSCRIPT f total / w d1 2d2 w 3d3 w2 4d4 w3 0
(28)
Eq. (28) is a cubic equation whose real root can be determined analytically as [30]
wPI , vdW
1 3 d3 C 0 12 d 4 C
(29)
where
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C
1 12 4 30
3
2
0 9 d32 24 d2 d4
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1 54 d33 216 d2 d3 d4 432 d1 d42
(30a)
(30b) (30c)
The critical vdW parameter which corresponds to the pull-in displacement (w PI , vdW ) is also determined as
1 1
dx
d
1
wPI , vdW d 2 wPI2 , vdW d3 wPI3 , vdW d 4 wPI4 , vdW
(31)
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3Cr
0
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After finding the critical value of the vdW parameter ( 3Cr ), the minimum allowable gap
respectively, as
4
L max
4
2AL4 Eh 33Cr
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d min
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and the maximum allowable length of the beam can also be found using Eqs. (32a) and (32b),
AC
Eh 3d 43Cr 2A
(32a)
(32b)
It is worth noting that these two parameters are also very important in designing N/MEMS devices.
4. Results and discussions
ACCEPTED MANUSCRIPT To validate the present approach, a micro-cantilever without the effect of the vdW force is considered. The properties of this micro-beam are presented in Table 2. Table 3 represents a comparison between the present results for pull-in voltage provided through Eq. (26) and those obtained in some previous studies [9, 10, 13, 14]. Based on the results of this table, the present procedure can predict pull-in voltage more accurately than previous continuous
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models obtained by Ramezani et al. [13] and Soroush et al. [14] in comparison to published FE simulations obtained by Pamidighantam et al. [9]. It is noted that the lumped models introduced by Pamidighantam et al. [9] and Chowdhury et al. [10], could represent even better results than the continuous models presented by Ramezani et al. [13] and Soroush et al.
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[14].
To validate the accuracy of the present approach for other types of boundary conditions, we develop the present procedure for narrow micro-beams with fringing field effect. Doing
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so, one can easily reach to a cubic equation which can be solved analytically. The results of dimensionless pull-in voltages and displacements for C-C, C-H and H-H micro-beams with
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d / b 1 are compared with both semi-analytical and numerical findings of Tadi Beni et al.
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[31] in Table 4. It is to be mentioned here that the numerical and MADM results of Tadi Beni et al. [31] are extracted from Figs. 9(a) - (c) of their paper. As it is seen from Table 4, the
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present predictions for both pull-in voltages and displacements are in good agreement with numerical findings of Tadi Beni et al. [31]; even better than their semi-analytical results
AC
provided by the MADM. Herein a convergence study is also conducted to find the number of terms which must be
included in the approximated solution of the HPM when the effect of the vdW force has been taken into account. To do so, the vdW parameter of the system is assumed to be 3 0.13Cr , where the values of the critical vdW parameter for nano/micro-beams with C-C, C-F, C-H
ACCEPTED MANUSCRIPT and H-H boundary conditions are calculated using Eq. (31) and presented in Table 5. Table 6 shows a comparison between different orders of the present HPM results (i.e. both nondimensional pull-in voltage and displacement) and the present numerical findings as well as the MADM predictions presented by Soroush et al. [14]. It should be noted that the present numerical results for Eq. (16) are prepared by two different methods: I. the fsolve command
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of the MATLAB commercial software with the zero initial guess, and II. the secant method n n 1 2 [32] with the initial guesses w1PI 0 and wPI 0.05 and the tolerance of wPI wPI 106 .
It is worth noting that, unless stated otherwise, the numerical results of the present paper are
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calculated using the aforementioned values for the initial guesses and the tolerance. Also the results of this table are rounded up to four digits. Based on the results of Table 6, the present HPM findings are converged when the order of the approximation is set to n 41 . Hence, unless stated otherwise, the results of the present paper are calculated using this order of
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approximated solution. As it is seen from Table 6, although the converged results of the HPM
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for both pull-in displacement and voltage agree excellently with present numerical findings which match exactly with each other, there exists a gap between them and those obtained by
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the MADM [14]. Additionally, it is worth noting that the present approach can provide pullin voltage and displacement straightly, while Soroush et al. [14] extracted pull-in parameters
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using the deflection-voltage graph when its slope reaches infinity.
AC
Verification is also performed here for systems in which the effect of the vdW force has been taken into account. Fig. 2 shows a comparison between the present HPM and numerical results, available semi-analytical findings in the literature [13, 14] as well as those obtained through non-linear finite element method (NFEM) [22]. Based on the results of this figure, the present numerical and semi-analytical results match each other excellently and agree very well with NFEM predictions. This fact can successfully justify our faith in the accuracy of the
ACCEPTED MANUSCRIPT present semi-analytical and numerical approaches. However, there exists a gap between the previous semi-analytical findings published by Ramezani et al. [13] and Soroush et al. [14]. The results of the present HPM series solution and numerical findings, which are obtained by both the MATLAB command fsolve and the secant method, are also compared with each other as well as those obtained by Soroush et al. [14] for both C-C and C-F boundary
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conditions in Fig. 3. As it is seen from this figure, although the present semi-analytical and numerical findings agree excellently with each other, there exists a gap between them and the results reported by Soroush et al. [14]. So, as it can be observed from the aforementioned comparisons and validations presented in Tables 3, 4 and 6 as well as Figs. 2 and 3, one can
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concluded that the present approach provides more accurate results than other available semianalytical procedures in the literature [13, 14]. In addition, since the present semi-analytical approach can provide pull-in characteristics straightly, there is no need to find the solution of
M
the equilibrium equation for some different values of the input voltages in the range of zero to pull-in voltage of the system. Hence, not only it does not suffer from long run-time, but also
ED
it is so rapid and can represent both pull-in voltage and displacement simultaneously.
PT
Therefore, due to the aforementioned desired features of the present procedure, it can be considered as an alternative approach for extracting pull-in characteristics of beam-type
CE
N/MEMS.
Fig. 3 shows that increasing the vdW parameter decreases both non-dimensional pull-in
AC
voltage and displacement. In addition, this figure shows a linear relationship between the non-dimensional pull-in voltage and the vdW parameter. This linear descending behavior will be utilized here to present a closed-form expression for pull-in voltage. To investigate this linear relationship, variations of the pull-in parameter for nano/micro-beams with C-C, C-F, C-H and H-H boundary conditions versus the vdW parameter are depicted in Fig. 4 using the
ACCEPTED MANUSCRIPT HPM as well as the present numerical methods. Furthermore, some lines are fitted to the results of these graphs. It is to be mentioned here that the results of the secant method and those obtained by the MATLAB command fsolve match each other exactly. Hence, for the sake of improving the quality of presentation, the findings of these two methods are depicted using the same marker in Fig. 4. As it is seen from Fig. 4, the accuracy of the present
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approach for all investigated boundary conditions is also successfully justified. According to the equations of the fitted lines, which are given for each type of the investigated boundary
V PI
A 1 d
2
Eh3d 3 L4
where the values of the coefficients
1
conditions are given in Table 7.
(33)
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conditions on Fig. 4, one can write
and
2
for each type of the investigated boundary
M
Eq. (33) can be utilized to predict pull-in voltage for electro-mechanical beams with submicron separations. It is noted that for micro-beams with a negligible vdW effect, a closed-
3
Eh3d 3 L4
(34)
PT
V PI
ED
form expression for pull-in voltage can also be determined analytically through Eq. (26) as
3
for all investigated boundary conditions are also provided in Table 7.
CE
where the values of
It is to be noted that the maximum relative error between pull-in voltages calculated from
AC
Eqs. (33) and (34) for systems without the vdW effect, which corresponds to nano/microbeams with C-F boundary conditions, is 1.75%. This small relative error can verify the approximated results of Eq. (33) successfully. The critical values of the vdW parameter can be approximated by equating the linear pullin formulas, which are shown on Fig. 4 for all investigated boundary conditions, to zero.
ACCEPTED MANUSCRIPT These values can also be determined analytically using Eq. (31) as those presented in Table 5. The negligible difference between the analytical results of Eq. (31) and the approximated findings of linear pull-in formulas can also verify the results of Eq. (33) again. Using the analytical results of Eq. (31), the corresponding expressions for minimum allowable gap and maximum allowable length, the so-called detachment length, can be reduced to
d min
4
L max
1 4
d
4
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L 4 A / Eh3
Eh3 / A
4
(35b)
for all investigated boundary conditions are also presented in Table
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where the coefficient
(35a)
7.
It is noteworthy to mention that the closed-form Eqs. (33) - (35) can be employed to determine the pull-in voltage, minimum allowable gap and the detachment length of beam-
M
type N/MEMS explicitly. Furthermore, according to these equations, it is obvious that the effect of the vdW attraction on pull-in instability decreases with an increase of the structural
ED
stiffness of the system. Therefore, one can conclude that investigating the effect of the vdW
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5. Conclusions
PT
force on pull-in instability of cantilever-type N/MEMS is more essential than the others.
The present paper has been focused on the pull-in instability of electrostatically actuated
AC
nano/micro-beams with C-C, C-F, C-H and H-H boundary conditions subjected to the vdW force through a semi-analytical approach. The Euler-Bernoulli beam theory has been employed to derive the governing equation of equilibrium. This equation has been nondimensionalized and reduced to a non-linear algebraic equation which describes the position of the nano/micro-beam maximum deflection through the Galerkin decomposition method.
ACCEPTED MANUSCRIPT The resulting reduced equilibrium equation and the one which governs on the stability conditions have been considered simultaneously to extract both pull-in voltage and displacement. Closed-form expressions for pull-in voltage and displacement under electrical loading alone as well as the maximum allowable length and the minimum allowable gap of beam-type N/MEMS have been presented. The effect of the vdW attraction on pull-in voltage
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and displacement has also been investigated semi-analytically through the HPM. The present results have been compared and validated by numerical findings, which were obtained by two different methods: I. the fsolve command of the MATLAB commercial software, and II. the secant method, as well as those obtained numerically, analytically and semi-analytically in
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previous studies. It was found that the present results agree better with available threedimensional FE simulations as well as NFEM findings than other available analytical or semi-analytical approaches in the literature. It was also found that non-dimensional pull-in
M
voltage varies linearly versus the vdW parameter. Furthermore, based on this important finding some closed-form expressions for calculating pull-in voltage in beam-type N/MEMS
ED
with C-C, C-F, C-H and H-H boundary conditions which work at sub-micron separations
PT
have also been introduced.
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Appendix
AC
The coefficients ai i 0,1,..., 4 , bi i 0,1,..., 4 , ci i 0,1,...,3 and di i 0,1,..., 4 appearing in Eqs. (10), (16), (23) and (27) are, respectively, defined as:
a0 3 dx, a1 dx 2 dx, a2 3 2 dx 1
0
1
1
0
0
1
1
1
0
0
0
a3 3 3 dx, a4 4 dx
(A1)
ACCEPTED MANUSCRIPT
1 1 1 dx 1 2 dx 3 dx dx, b 0 b0 3 0 1 a2 , b2 1 0 a3 a2 1 1 0 2 2 2 dx dx dx 0 0 0 1
b3
4 dx 0
1
2 dx
(A2)
a4 2a3 , b4 3a4
0
1
1
1
0
0
0
1
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c0 dx, c1 dx, c2 2 2 dx, c3 3 dx 0
1
1
1
0
0
0
1
d0 3 dx, d1 dx, d 2 3 2 dx, d3 3 3 dx 1
d 4 4 dx
0
(A4)
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0
(A3)
References
Senturia, SD. Microsystem Design. Dordrecht, Kluwer Academic Publishers; 2001.
[2]
Younis, MI. MEMS Linear and Nonlinear Statics and Dynamics. New York, Springer;
ED
M
[1]
2011.
Lifshitz, EM. The theory of molecular attractive forces between solids, Sov Phys JETP.
PT
[3]
[4]
CE
2 (1956) 73-83.
Liu, H, Gao, S, Niu, S, Jin, L. Analysis on the adhesion of microcomb structure in
AC
MEMS, Int J Appl Electrom. 33 (2010) 979–84.
[5]
Lamoreaux, SK. The Casimir force: background, experiments, and applications, Rep Prog Phys. 68 (2005) 201-36.
[6]
Batra, RC, Porfiri, M, Spinello, D. Review of Modeling Electrostatically Actuated Microelectromechanical Systems, Smart Mater Struct. 16 (2007) R23-31.
ACCEPTED MANUSCRIPT [7]
Nathanson, HC, Newell, WE, Wickstrom, RA, Davis, JR. The resonant gate transistor, IEEE T Electron Dev. 14 (1967) 117-33.
[8]
Taylor, GI. The coalescence of closely spaced drops when they are at different electric potentials, Proc Roy Soc A. 306 (1968) 423-34. Pamidighantam, S, Puers, R, Baert, K, Tilmans, HAC. Pull-in voltage analysis of
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[9]
electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions, J Micromech Microeng. 12 (2002) 458-64.
[10] Chowdhury, S, Ahmadi, M, Miller, WC. A closed-form model for the pull-in voltage of
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electrostatically actuated cantilever beams, J Micromech Microeng. 15 (2005) 756-63. [11] Kuang, JH, Chen, CJ. Adomian decomposition method used for solving nonlinear pullin behavior inelectrostatic micro-actuators, Math Comput Model 41 (2005) 1479-91.
M
[12] Mojahedi, M, Moghimi Zand, M, Ahmadian, MT. Static pull-In analysis of electrostatically actuated microbeams using homotopy perturbation method, Appl Math
ED
Model. 34 (2010) 1032-41.
PT
[13] Ramezani, A, Alasty, A, Akbari, J. Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces, Int J Solids
CE
Struct. 44 (2007) 4925-41.
AC
[14] Soroush, R, Kooch, A, Kazemi, AS, Abadyan, M. Modeling the effect of van der Waals attraction on the instability of electrostatic cantilever and doubly-supported nano-beams using modified Adomian method, Int J Struct Stab Dy. 12 (2012) 1250036 (18 pages).
[15] He, JH. Homotopy perturbation technique, Comput Method Appl M. 178 (1999) 25762.
ACCEPTED MANUSCRIPT [16] Younis, MI, Abdel-Rahman, EM, Nayfeh, AH. A reduced-order model for electrically actuated microbeam-based MEMS, J Microelectromech S. 12 (2003) 672-80. [17] Gupta, RK. Electrostatic Pull-In Test Structure Design for In-Situ Mechanical Property Measurements of Microelectromechanical Systems [Ph.D. Dissertation]. Cambridge,
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MA: Massachusetts Institute of Technology (MIT); 1997. [18] Chao, PCP, Chiu, CW, Liu, TH. DC dynamic pull-In predictions for a generalized clamped–clamped microbeam based on a continuous model and bifurcation analysis, J Micromech Microeng. 18 (2008).
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[19] Israelachvili, JN. Intermolecular and Surface Forces, 3rd ed. University of California, Santa Barbara, California, Elsevier; 2011.
[20] Meirovitch, L. Fundamentals of Vibrations. New York, McGraw-Hill; 2001.
M
[21] Tahani, M, Askari, AR. Accurate electrostatic and van der Waals pull-in prediction for fully clamped nano/micro-beams using linear universal graphs of pull-in instability,
ED
Physica E. 63 (2014) 151-9.
PT
[22] Moghimi Zand, M, Ahmadian, MT. Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces, J Mech Eng Sci. 224
CE
(2010) 2037-47.
AC
[23] Reddy, JN. Energy Principles and Variational Methods in Applied Mechanics. New York, John Wiley & Sons; 2002.
[24] Batra, RC, Porfiri, M, Spinello, D. Vibrations of narrow microbeams predeformed by an electric field, j Sound Vib. 309 (2008) 600-12.
ACCEPTED MANUSCRIPT [25] Askari, AR, Tahani, M. An alternative reduced order model for electrically actuated micro-beams under mechanical shock, Mech Res Commun. 57 (2014) 34-9. [26] Rao, SS. Vibraion of Continuous Systems. New Jersey, John Wiley & Sons; 2007. [27] Balachandran, B, Magrab, E. Vibrations, 2nd ed. Toronto, Cengage Learning; 2009.
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[28] Seydel, R. Practical Bifurcation and Stability Analysis, 3rd ed. New York, Springer; 2009.
[29] He, JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int J Nonlinear Mech. 35 (2000) 37-43.
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[30] Uspensky, JV. Theory of equations. New York, McGraw-Hill; 1948.
[31] Tadi Beni, Y, Koochi, A, Abadyan, M. Theoretical study of the effect of Casimir force, elastic boundary conditions and size dependency on the pull-in instability of beam-type
M
NEMS, Physica E. 43 (2011) 979-88.
AC
CE
PT
ED
[32] Faires, JD, Burden, RL. Numerical methods, 4th ed., Cengage Learning; 2012.
ACCEPTED MANUSCRIPT Table captions Table 1. Characteristics of nano/micro-beams mode-shapes. Table 2. Geometric and material properties of the micro-cantilever studied in Table 3. Table 3. A comparison between pull-in voltages (V) calculated by different methods for a
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micro-cantilever with the geometric and material properties presented in Table 2.
Table 4. Comparisons between normalized pull-in voltages and displacements for narrow
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micro-beams ( d / b 1) with C-C, C-H and H-H boundary conditions.
Table 5. Critical values of the vdW parameter for nano/micro-beams with different boundary conditions.
Table 6. Convergence of the homotopy series solution for nano/micro-beams with different
ED
M
boundary conditions and the vdW parameter 3 0.13Cr .
Table 7. Values of the coefficients
1
and
2
in Eq. (33),
3
in Eq. (34) and
4
in Eq. (35)
CE
Figure captions
PT
for all investigated boundary conditions.
AC
Fig. 1. Schematic of an electrically actuated nano/micro-beam. Fig. 2. Non-dimensional pull-in voltage of an electrically actuated nano/micro-cantilever versus the vdW parameter calculated by different methods. Fig. 3. Comparisons between present numerical and semi-analytical results with those obtained by the MADM [14] for nano/micro-beams with C-C and C-F boundary conditions.
ACCEPTED MANUSCRIPT Fig. 4. Linear relationship between the non-dimensional pull-in voltage and vdW parameter for nano/micro-beams with (a) C-C, (b) C-F, (c) C-H and (d) H-H boundary conditions.
Table 1. Characteristics of nano/micro-beams mode-shapes.
Position of the maximum deflection
C-C
x 0.5000
0.6297
4.7300
0.9825
C-F
x 1.0000
0.5000
1.8751
0.7341
C-H
x 0.5808
0.6626
3.9266
1.0008
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CR IP T
Boundary conditions (B.C.’s)
Table 2. Geometric and material properties of the micro-cantilever studied in Table 3. b (μm)
300
50
h (μm)
d (μm)
E (GPa)
2.5
77
0.33
M
L (μm)
CE
PT
ED
1
Table 3. A comparison between pull-in voltages (V) calculated by different methods for a micro-
AC
cantilever with the geometric and material properties presented in Table 2. Case
Pull-in voltage (V)
Present
FE simulations [9]
[9]
[10]
[13]
[14]
2.25
2.25
2.33
2.25
2.37
2.16
ACCEPTED MANUSCRIPT Table 4. Comparisons between normalized pull-in voltages and displacements for narrow microbeams ( d / b 1 ) with C-C, C-H and H-H boundary conditions. Tadi Beni et al.
Tadi Beni et al.
[31] (Numerical)
[31] (MADM)
Present B.C.’s wPI
PI
wPI
PI
wPI
C-C
47.2456
0.4505
48.9883
0.4272
50.7865
0.4538
C-H
22.6722
0.4471
23.4831
0.4263
H-H
9.4052
0.4431
9.6992
0.4229
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PI
25.2809
0.5188
10.2857
0.5066
conditions.
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Table 5. Critical values of the vdW parameter for nano/micro-beams with different boundary
C-C
C-F
3Cr
48.6244
1.1581
C-H
H-H
23.2963
9.6456
M
B.C.’s
ED
Table 6. Convergence of the homotopy series solution for nano/micro-beams with different boundary
B.C.’s
C-F
C-H
Secant
MADM command
method
13
19
25
31
41
[14] fsolve
wPI
0.4069
0.4016
0.3993
0.3985
0.3981
0.3980
0.3979
0.3979
0.3979
0.3961
PI
64.0295
61.4117
60.2903
59.8353
59.6597
59.5954
59.5678
59.5642
59.5642
64.2373
AC
C-C
MATLAB
HPM (n)
7
CE
1
PT
conditions and the vdW parameter 3 0.13Cr .
wPI
0.4692
0.4661
0.4646
0.4640
0.4637
0.4636
0.4635
0.4635
0.4635
0.4684
PI
1.4618
1.4321
1.4183
1.4121
1.4093
1.4081
1.4074
1.4072
1.4072
1.5765
wPI
0.4036
0.3984
0.3962
0.3953
0.3950
0.3949
0.3948
0.3948
0.3948
-
ACCEPTED MANUSCRIPT
30.7307
29.4874
28.9591
28.7464
28.6650
28.6354
28.6228
28.6211
28.6211
-
wPI
0.3998
0.3947
0.3926
0.3918
0.3915
0.3914
0.3913
0.3913
0.3913
-
PI
12.7446
12.2395
12.0268
11.9419
11.9096
11.8979
11.8930
11.8923
11.8923
-
Table 7. Values of the coefficients
1
and
2
in Eq. (33),
investigated boundary conditions.
in Eq. (34) and
4
in Eq. (35) for all
C-F
C-H
H-H
1
0.1451
0.1439
0.1459
0.1460
2
10.9909
0.2597
5.2809
2.1941
3
3.3658
4
0.3383
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C-C
0.5187
2.3301
1.4996
0.8611
0.4066
0.5069
AC
CE
PT
ED
M
Coefficient
3
CR IP T
H-H
PI
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ACCEPTED MANUSCRIPT
AC
CE
PT
ED
M
Fig. 1. Schematic of an electrically actuated nano/micro-beam.
ACCEPTED MANUSCRIPT 2 HPM (Present) M ATLAB command fsolve (Present) Secant method (Present) NFEM (M oghimi Zand and Ahmadian [22]) Semi-analytical method (Ramezani et al. [13]) M ADM (Soroush et al. [14])
1
0.5
0
0
0.31
0.62
3
CR IP T
PI
1.5
0.93
1.24
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Fig. 2. Non-dimensional pull-in voltage of an electrically actuated nano/micro-cantilever versus the
AC
CE
PT
ED
M
vdW parameter calculated by different methods.
ACCEPTED MANUSCRIPT 0.42
72 C-C nano/micro-beam
C-C nano/micro-beam 54
PI
0.36
0.33
0.3
18 (a) 0
13
0.5
26
39
3
0
52
(b) 0
13
1.8
C-C nano/micro-beam
1.35
PI
w PI
36
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0.45
54
26
3
39
52
C-F nano/micro-beam
C-F nano/micro-beam
72
PI
36
CR IP T
w PI
0.39
0.4
0.9
0.45
18
0
13
1.8
0.31
26 39 3 (Present) HPM
C-F nano/micro-beam
0.62
3
0.93
1.24
M
(b)
(c) 0
52 MATLAB command fsolve (Present)
ED
0
0.35
0
(d)
0
0.31
0.62
3
Secant method (Present)
0.93
1.24
MADM [14]
Fig. 3. Comparisons between present numerical and semi-analytical results with those obtained by the 1.35
PT
MADM [14] for nano/micro-beams with C-C and C-F boundary conditions.
CE
0.9
0
(d) 0
AC
0.45
0.31
0.62
3
0.93
1.24
HPM (Pres MATLAB Secant met MADM [1
ACCEPTED MANUSCRIPT 70
1.68
PI = -1.3671 3 + 65.9451
PI = -1.3559 3 + 1.5581
52.5
1.26
PI
35
17.5
0
0.84
0.42 (a) 0
12.5
25
3
34
37.5
0
50
(b) 0
14
PI = -1.3712 3 + 31.6853
0.6
3
0.9
1.2
10.5
PI
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PI
0.3
PI = -1.3762 3 + 13.1646
25.5
17
7
3.5
(c) 6
12
3
18
ED
PI = -1.3559 3 +01.5581
M
8.5
0
CR IP T
PI
T
24
The fitted line
0
(d) 0
HPM
2.5
5
3
7.5
10
Numerical
PT
Fig. 4. Linear relationship between the non-dimensional pull-in voltage and vdW parameter for
0.6
3
0.9
1.2
AC
0.3
CE
nano/micro-beams with (a) C-C, (b) C-F, (c) C-H and (d) H-H boundary conditions.
PI = -1.3762 3 + 13.1646
2.5
5
3
7.5
10
Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach Amir R. Askari , Masoud Tahani PII: DOI: Reference:
S1007-5704(15)00345-7 10.1016/j.cnsns.2015.10.014 CNSNS 3672
To appear in:
Communications in Nonlinear Science and Numerical Simulation
Received date: Revised date: Accepted date:
15 May 2014 21 September 2015 21 October 2015
Please cite this article as: Amir R. Askari , Masoud Tahani , Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach, Communications in Nonlinear Science and Numerical Simulation (2015), doi: 10.1016/j.cnsns.2015.10.014
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Highlights A new procedure for pull-in analysis of beam-type N/MEMS under vdW force has been introduced. This procedure able us to represent analytical and semi-analytical solutions for pull-in problems. This method can extract all pull-in parameters simultaneously. It is found that this new method agrees better than previous ones with FE results.
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Some closed-form expressions are also presented for vdW and electrical pull-in
AC
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PT
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M
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parameters.
ACCEPTED MANUSCRIPT
Stability analysis of electrostatically actuated nano/micro-beams under the effect of van der Waals force, a semi-analytical approach
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Amir R. Askari, Masoud Tahani* Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran
ABSTRACT
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The objective of the present paper is to determine pull-in parameters (pull-in voltage and its corresponding displacement) of nano/micro-beams with clamped-clamped, clamped-free, clamped-hinged and hinged-hinged boundary conditions, when they are subjected to the electrostatics and van der Waals (vdW) attractions. The governing non-linear boundary value
M
equation of equilibrium is derived, non-dimensionalized and reduced to an algebraic
ED
equation, which describes the position of the maximum deflection of the beam, utilizing the Galerkin decomposition method. The equation which governs on the stability condition of the
PT
system is also obtained by differentiating the reduced equilibrium equation with respect to the maximum deflection of the beam. These two equations are solved simultaneously to
CE
determine pull-in parameters. Closed-form solutions are provided for cases under electrical
AC
loading and vdW attraction alone. The combined effect of both electrostatic and vdW loadings are also investigated using the homotopy perturbation method (HPM). It is found that the present semi-analytical findings are in excellent agreement with those obtained numerically. In addition, it is observed that the present semi-analytical approach can provide results which agree better with available three-dimensional finite element simulations as well *
Corresponding author. Tel.: +98-511-8806055; fax: +98-511-8763304. E-mail address: [email protected] (Masoud Tahani).
ACCEPTED MANUSCRIPT as those obtained by nonlinear finite element method than other available analytical or semianalytical findings in the literature. Non-dimensional electrostatic and vdW parameters, which are defined in the text, are plotted versus each other at pull-in condition. It is found that there exists a linear relationship between these two parameters at pull-in condition. Using this fact, pull-in voltage, detachment length and minimum allowable gap of electrostatically
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actuated nano/micro-beams are determined explicitly through some closed-form expressions.
Keywords: Nano/micro-electro-mechanical beams, Pull-in instability, vdW attraction,
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Homotopy perturbation method, Detachment length, Closed-form expressions.
1. Introduction Nano/micro-systems
have
applications
in
many
engineering
fields
such
as
M
communications, automotive and robotics [1]. Nano/micro-electro-mechanical systems (N/MEMS) can be considered as a largest collection of these systems, because of their fast
ED
response, low power consumption, reliability and their capability of batch fabrications [2].
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Electrically actuated nano/micro-beams represent a major structural component and play a crucial role in many N/MEMS devices [2]. The main components of a typical N/MEMS
CE
device can be considered as a fixed electrode and a movable one. The movable electrode can be modeled as an electrically actuated nano/micro-beam or plate which deflects toward the
AC
fixed one through the application of an external voltage on the system. The applied voltage has an upper limit in which the electrostatic attraction overcomes the elastic restoring force of the movable electrode. Therefore, the movable electrode suddenly collapses toward the fixed one in this manner. This unstable phenomenon is called pull-in instability. The maximum deflection occurred just before the collapsing of movable electrode is also called pull-in
ACCEPTED MANUSCRIPT displacement. Pull-in displacement and pull-in voltage, the so-called pull-in parameters, can be considered as the most important parameters in designing electrically actuated nano/micro-beams. By decreasing the dimensions of electrically actuated systems from micro-scales to nanoscales, the intermolecular surface forces significantly influence on the behaviors of
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nano/micro-beams. The most important forces at the scale of N/MEMS are the Casimir and vdW attractions. The vdW force arises from the correlated oscillation of the instantaneously induced dipole moments of the atoms placed at the close parallel conductive plates [3]. The vdW force is a short range force in nature, but it can lead to long range effects more than 0.1
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μm [4]. The Casimir force can be simply understood as the long range analog of the vdW force, resulting from the propagation of retarded electromagnetic waves [5]. The main purpose of the present paper is to investigate the effect of the vdW attraction on
M
the characteristics of pull-in instability in beam-type N/MEMS. To date, lots of researchers have dealt with the mechanical behavior of electrically actuated nano/micro-beams and
ED
plates. A comprehensive review of different models of electrostatically actuated micro-
PT
systems are presented by Batra et al. [6]. Herein, some of the most pioneering works in the field of modelling pull-in instability in N/MEMS devices are reviewed.
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Nathanson et al. [7] and Taylor [8] simultaneously observed pull-in instability in microsystems. Pamidighantam et al. [9] studied electrically actuated micro-cantilevers as well as
AC
clamped-clamped micro-beams using equivalent spring-mass model. They present a closedform expression for pull-in voltage and validated their solutions with finite element (FE) simulations provided by Coventorware (CW) commercial software. Their model improved the estimation of previous model presented by Nathanson et al [7] for pull-in voltage; however, their model represent this value in terms of pull-in displacement which was
ACCEPTED MANUSCRIPT empirically chosen. Chowdhury et al. [10] also presented a closed-form solution for cantilever micro-beams using another equivalent spring-mass model. They used the third order Maclaurin's series expansion of the electrostatic forcing term to calculate the effective stiffness coefficient of the micro-beam. Kuang and Chen [11] solved the non-linear boundary value equation which governs on the equilibrium of both micro-cantilevers and doubly
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clamped micro-beams semi-analytically using Adomian decomposition method (ADM). Mojahedi et al. [12] investigated static pull-in instability using the combination of Galerkin’s projection method and the HPM. They converted the governing boundary value equation of equilibrium to an algebraic equation using the first linear un-damped mode-shape of an un-
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deformed micro-beam and solved the resulting non-linear algebraic equation through the HPM. It should be noted that they expanded the electrostatic forcing term about the static deflection of the movable electrode which was updated iteratively by increasing in the value
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of the input voltage. Therefore, their iterative approach cannot be considered as an analytical procedure. Ramezani et al. [13] proposed a distributed parameter model to study pull-in
ED
instability of electrically actuated nano-cantilevers subjected to the vdW and the Casimir forces. They transferred non-linear differential equation of the model into the integral form
PT
by using the Green’s function of the clamped-free beam and the integral equation was solved
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using the shape function of the beam deflection. It is to be noted that, their solutions could not satisfy all boundary conditions and suffered from the over-estimation of pull-in voltage.
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Soroush et al. [14] also investigated the effect of the vdW attraction on pull-in instability of nano/micro-beams using the modified Adomian decomposition method (MADM). Although they could present some expressions for the detachment length and minimum allowable gap of nano/micro-systems, there exists a gap between their findings and those obtained by FE simulations [9].
ACCEPTED MANUSCRIPT It is worth noting that the solution of the equation which governs on the static equilibrium has been investigated in previous studies to capture pull-in instability when the slope of deflection-voltage graph reaches infinity. However, in present paper the equilibrium equation and the governing equation on the stability conditions are considered simultaneously. The governing equation of equilibrium is derived using the Euler-Bernoulli beam theory and
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reduced to an algebraic equation which describes the position of the beam maximum deflection through the Galerkin decomposition method. The stability equation is also obtained by differentiating the reduced equilibrium equation with respect to the maximum deflection of the beam. This set of non-linear algebraic equations is solved analytically for
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cases under electrical loading alone and vdW attraction alone to present some closed-form expressions for pull-in voltage and displacement as well as the minimum allowable gap and maximum allowable length of the nano/micro-beam. A semi-analytical solution is also
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performed using the HPM to investigate the interaction between electrical and vdW loadings, where the solution of the system under electrical loading alone is selected as the initial guess
ED
to construct the homotopy series solution. It should be noted that unlike the traditional
PT
perturbation methods, the HPM does not depend on the assumption of small parameters and can solve strongly non-linear problems [15].
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It is found that the present semi-analytical results agree better with available FE predictions than those obtained semi-analytically through beam model in previous studies. In
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addition, one of the advantages of the present approach is that both pull-in voltage and pull-in displacement are obtained simultaneously which is not possible in previous solution procedures. In other words, the previous studies focused on the solution of the governing equilibrium equation which determined the static deflection in terms of input voltage. This procedure captured pull-in instability form deflection-voltage graph when its slope reaches
ACCEPTED MANUSCRIPT infinity. However, the present approach provides pull-in parameters straightly. It is also found that the dimensionless pull-in voltage varies linearly versus the non-dimensional vdW parameter. Based on this important finding, some closed-form expressions are introduced to predict pull-in voltage in terms of the system parameters. It is worth mentioning that the present solution procedure is developed in terms of the
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nano/micro-beam mode-shapes. Therefore, one just needs to update the mode-shape function to obtain the proper results for any desired boundary conditions. In this way, the present findings are provided for usual beam-type N/MEMS with clamped-free (C-F), clamped-
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clamped (C-C), clamped-hinged (C-H) and hinged-hinged (H-H) boundary conditions.
2. Problem formulation
Fig. 1 shows a typical structure of electrostatically actuated nano/micro-systems, where
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the main components are fixed and movable electrodes. The fixed electrode modeled as a ground plane and the movable one modeled as a nano/micro-beam. The length, width,
ED
thickness and density of the nano/micro-beam are L, b, h and , respectively. The initial gap
PT
between the non-actuated beam and the stationary electrode is d. Also, x, y and z are the coordinates along the length, width and thickness, respectively. W is the deflection of the
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beam, I is the second moment of cross-sectional area about the y axis, is Poisson’s ratio and E is the effective Young’s modulus of the nano/micro-beam which is replaced
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by E / (1 2 ) when b 5h [16]. It is to be noted that, due to the finite dimensions of the electrodes, the lines of
electrostatic field are not parallel over the edges. The influence of these unparalleled lines of electrostatic field over the edges is called the fringing field effect. Considering the fringing
ACCEPTED MANUSCRIPT field effect, the electrostatic excitation by polarized DC voltage V per unit length of the beam can be expressed as [17]
Fes
bV 2 2d W
2
d W 1 0.65 b
(1)
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where is the dielectric constant of the medium. It is worth noting that the fringing field effect can be neglected for the case of wide nano/micro-beams [18].
As it is mentioned in the previous section, the vdW and Casimir attractions significantly affect the nano/micro-beam deflection at sub-micron separations. The vdW and Casimir
f Cas
Ab 6 d W
3
2 cb 240 d W
4
(2a)
(2b)
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f vdW
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forces per unit length of the beam, respectively, take the following forms [5, 19]:
1.055 1034 J is the Plank constant divided by 2
ED
where A is the Hamaker constant,
media
(Ag,
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and c 2.998 108 m/sec is the light speed. The Hamaker constant for two identical metal Au,
Cu)
interacting
across
vacuum
(air)
has
values
in
range
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30 50 1020 J [19]. It is to be mentioned here that the Casimir force can be considered as
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the long range analog of the vdW attraction [5]. Hence, the effects of these two attractions should not be considered simultaneously. The differential equation which governs on the equilibrium of the nano/micro-beam
subjected to the combined effects of the electrostatic and dispersion forces (i.e. vdW or Casimir attraction) can be expressed as follows [20]:
ACCEPTED MANUSCRIPT
EI
d 4W Fes Fdisp dx 4
(3)
where Fdisp refers to the vdW attraction or the Casimir force. In present paper, it is assumed that the nano/micro-beam deflection is subjected to C-C, C-
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F, C-H and H-H boundary conditions. For convenience, the following non-dimensional variables are introduced:
W x Wˆ , xˆ d L
(4)
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Upon substitution of the non-dimensional quantities given in Eq. (4) into Eq. (3) and dropping the hats, the following result will be obtained: W
(1 W )
(1 W )
2
m (1 W )m
(5)
M
where the prime sign denotes derivative with respect to x. Also, m is the dispersion force
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index which specifies the type of the applied force at sub-micron separations. That is, m 3 refers to systems under the vdW force and m 4 is related to those subjected to the Casimir
PT
attraction. The non-dimensional parameters , , 3 and 4 , which are, respectively, called
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as the fringing field effect, electrostatic, vdW and Casimir parameters, take the form: d b
6 V 2 L4 2 AL4 2 cL4 , , 3 4 Eh3d 3 Eh3d 4 20 Eh3d 5
AC
0.65 ,
(6)
It is to be mentioned here that, in the present paper, we focus on the effect of the vdW
force on the pull-in characteristics of wide nano/micro-beams which represent major applications in N/MEMS devices [18, 21, 22]. 3. Solution procedure
ACCEPTED MANUSCRIPT Due to the high non-linearity involved in Eq. (5), an exact solution for this equation has not been found to date. Hence, an approximate solution will be developed here through the Galerkin weighted residual method. Based on this procedure, the nano/micro-beam deflection can be expressed as a linear combination of a complete set of linearly independent basis functions [23]. It is noted that these functions must satisfy all kinematic boundary conditions
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[23]. Therefore, linear un-damped mode-shapes of the un-deformed nano/micro-beam can be used as these basis functions. It is proved that using only the first mode for pull-in analysis of electrically actuated nano/micro-beams maybe very accurate [12, 24, 25]. Hence, the deflection of the nano/micro-beam based on one-mode solution can be expressed as
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W x w x
(7)
where w is an un-known generalized coordinate, which should be determined through the Galerkin procedure, and (x ) is the first linear un-damped mode-shape of the un-deformed
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nano/micro-beam. Herein, ( x) for C-C, C-F and C-H boundary conditions is determined as
ED
[26]
(8)
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x cosh x cos x sinh x sin x
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where the parameters , and for aforementioned boundary conditions are given in Table 1 [27]. It is noteworthy to mention that the normalization parameter is determined
AC
such that the unknown generalized coordinate w describes the maximum deflection of the nano/micro-beam whose position is also presented in Table 1 [25]. Furthermore, the normalized mode-shape of the H-H nano/micro-beam is selected as follows:
x sin x
(9)
ACCEPTED MANUSCRIPT As it is mentioned at the end of the previous section, the present paper focuses on the effect of the vdW force on wide nano/micro-beams pull-in characteristics. For this object, we set the values of and m in Eq. (5) to 0 and 3, respectively, and multiply this equation by
x (1 W )3 , substitute Eq. (7) into the resulting equation and integrate the outcome from
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x 0 to 1 and obtain
a0 a1 w a2 w2 a3 w3 a4 w4 0
(10)
where the coefficients ai i 0,1,..., 4 are given in the Appendix.
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Eq. (10) governs on the equilibrium of electrostatically actuated wide nano/micro-beams subjected to the combined effects of electrostatic and vdW forces. In other words, in this condition all the acting force upon the nano/micro-beam including the elastic restoring force, vdW attraction and electrostatic actuation are in balance. Hence, one can write
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f total w, , 3 a0 a1 w a2 w2 a3 w3 a4 w4 0
(11)
(12)
PT
f total 0 w
ED
The stable equilibrium provided [28]
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Therefore the threshold of stability is f total / w 0 . For this condition one can write (13)
AC
f total / w a1 2a2 wPI 3a3 wPI2 4a4 wPI3 0
It should be noted that Eq. (13) refers to pull-in condition. Therefore, the dimensionless
maximum deflection (w ) has been replaced by non-dimensional pull-in displacement (w PI ) in this equation. Non-dimensional pull-in voltage ( PI ) and displacement (w PI ) can be determined thorough the simultaneous solution of Eqs. (10) and (13). For the sake of brevity,
ACCEPTED MANUSCRIPT the non-dimensional pull-in displacement is called pull-in displacement hereinafter. Using the equation of the coefficient a1 in (A1), the non-dimensional pull-in voltage can be written as
PI
a dx dx 1
1
2
1
1
(14)
0
0
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The coefficient a1 can also be determined using Eq. (13) as
a1 2a2 wPI 3a3 wPI2 4a4 wPI3
(15)
Substituting Eqs. (14) and (15) into Eq. (10), may lead to the following equation in terms of
b0 b1 wPI b2 wPI2 b3 wPI3 b4 wPI4 0
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pull-in displacement (w PI ):
(16)
and given in the Appendix.
M
where the coefficients bi ( i 0,1,..., 4 ) are known in terms of the input values of the problem
ED
Pull-in displacement (w PI ) can be found by solving Eq. (16). This equation is a non-linear algebraic equation which can be solved using the HPM. The coefficient a1 can also be
PT
determined by substituting w PI , which is calculated from Eq. (16), into Eq. (15). Non-
CE
dimensional pull-in voltage is also determined by substituting coefficient a1 into Eq. (14).
AC
3.1 The basic idea of the HPM The HPM is one of the most effective methods for finding semi-analytical solutions of
strongly non-linear problems. This method converts a general non-linear problem to infinite linear problems by embedding an auxiliary parameter p [15]. To illustrate the basic idea of the HPM briefly, let us consider a general non-linear problem as follows
ACCEPTED MANUSCRIPT u
u
u 0
(17)
where u is a general non-linear equation and u is an un-known parameter.
u is the
u is its non-linear part. Using
p [0,1] as
linear and homogeneous part of Eq. (17) and
an embedding parameter, the homotopy function for Eq. (17) can be constructed as [29]
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u, p 1 p u u p u 1 p u u p u u u u p u u
(18)
where u is an initial guess for the non-linear problem in (17). It is obvious that as p increases from 0 to 1, the solution of Eq. (18) varies from the initial guess to the solution of Eq. (17).
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Unlike the traditional perturbation techniques, the HPM, which does not depend on the assumption of a small parameter, applies these techniques on the homotopy function of a non-linear equation instead of the non-linear equation itself [29]. To do so, the un-known
u u 0 pu1 p 2u 2 p 3u 3 ...
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parameter u is expanded as a power series in terms of p as
(19)
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The solution of Eq. (17) can be found by substituting Eq. (19) into Eq. (18), collecting the
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terms with the identical powers of p i ( i 0,1, 2,... ) and setting the coefficients of p i to zero. By following this procedure, the non-linear problem is converted to a set of linear equations
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with respect to u i ( i 0,1, 2,... ) which can be solved easily. After finding sufficient
AC
approximations, by setting p 1 , the solution of the non-linear Eq. (17) can be found as
u u 0 u1 u 2 u 3 ... u n
where n is the order of the approximated solution.
3.2 Application of the HPM to the nano/micro-beam problem
(20)
ACCEPTED MANUSCRIPT Now, let us apply the HPM to the nano/micro-beam problem to find the solution of Eq. (16). The linear and non-linear parts of this equation can be considered as
wPI b1 wPI ,
wPI b0 b2 wPI2 b3 wPI3 b4 wPI4
(21)
Next, we expand the un-known parameter (w PI ) as a power series in terms of p, substitute it
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in the homotopy function of Eq. (16) and equate the terms with the identical powers of p and obtain
p 0 : w PI , 0 w PI
1 b0 b2 wPI2 , 0 b3 wPI3 , 0 b4 wPI4 , 0 b1
p 2 : wPI , 2
wPI ,1
p 3 : wPI , 3
wPI , 0
b1
b1
2b w 2
PI , 0
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p1 : wPI ,1
3b3 wPI2 , 0 4b4 wPI3 , 0
2b2 wPI , 2 3b3 wPI , 0 wPI , 2 wPI2 ,1
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2b4 wPI , 0 2wPI , 0 wPI , 2 3w
(22b)
(22c)
b2 wPI2 ,1 b 1
(22d)
ED
2 PI ,1
(22a)
The higher-order approximations for w PI can be established by considering more terms in Eq.
PT
(19) and following the same procedure.
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The initial guess (w PI ) has a sizable effect on adjusting convergence criteria for homotopy series solution. Therefore, it is very important to choose an appropriate value for the initial
AC
guess. Herein, the pull-in displacement for a system without the effect of the vdW force is considered as the initial guess. Using the Galerkin procedure discussed above, the governing equation of such system can be reduced as
c 0 c1 w c 2 w 2 c 3 w 3 0 where the coefficients ci i 0,1, 2,3 are presented in the Appendix.
(23)
ACCEPTED MANUSCRIPT As discussed already, pull-in condition can be obtained as
f total / w c1 2c 2 w 3c3 w 2 0
(24)
Eq. (24) is a quadratic equation, which can be solved analytically as
w PI
c 2 c 22 3c1c 3
(25)
3c 3
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It should be noted that only the positive root of Eq. (24) is acceptable for the pull-in displacement and if two positive roots are obtained, the minimum one will be selected. The corresponding pull-in parameter for this case can also be determined as
1 1
dx
c w 1
PI
c 2w PI2 c 3w PI3
(26)
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PI
0
3.3 Detachment parameters
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The maximum allowable length of a nano/micro-beam which prevents it from sticking to the substrate without the application of external voltage is called detachment length. It should
ED
be noted that a nano/micro-beam can be pulled-in under the effect of vdW attraction alone
PT
when the vdW parameter reaches its critical value. The detachment length and also the minimum allowable gap between non-actuated nano/micro-beam and its substrate can be
CE
determined through finding the critical value of the vdW parameter. To find this parameter, consider Eq. (5) without the electrostatic forcing terms. This equation can be reduced to the
AC
following algebraic equation using Galerkin’s procedure:
d0 d1 w d2 w2 d3 w3 d4 w4 0
(27)
where the coefficients di i 0,1,..., 4 are given in the Appendix. As discussed above, the threshold of stability is f total / w 0 . Therefore, one can write
ACCEPTED MANUSCRIPT f total / w d1 2d2 w 3d3 w2 4d4 w3 0
(28)
Eq. (28) is a cubic equation whose real root can be determined analytically as [30]
wPI , vdW
1 3 d3 C 0 12 d 4 C
(29)
where
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C
1 12 4 30
3
2
0 9 d32 24 d2 d4
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1 54 d33 216 d2 d3 d4 432 d1 d42
(30a)
(30b) (30c)
The critical vdW parameter which corresponds to the pull-in displacement (w PI , vdW ) is also determined as
1 1
dx
d
1
wPI , vdW d 2 wPI2 , vdW d3 wPI3 , vdW d 4 wPI4 , vdW
(31)
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3Cr
0
ED
After finding the critical value of the vdW parameter ( 3Cr ), the minimum allowable gap
respectively, as
4
L max
4
2AL4 Eh 33Cr
CE
d min
PT
and the maximum allowable length of the beam can also be found using Eqs. (32a) and (32b),
AC
Eh 3d 43Cr 2A
(32a)
(32b)
It is worth noting that these two parameters are also very important in designing N/MEMS devices.
4. Results and discussions
ACCEPTED MANUSCRIPT To validate the present approach, a micro-cantilever without the effect of the vdW force is considered. The properties of this micro-beam are presented in Table 2. Table 3 represents a comparison between the present results for pull-in voltage provided through Eq. (26) and those obtained in some previous studies [9, 10, 13, 14]. Based on the results of this table, the present procedure can predict pull-in voltage more accurately than previous continuous
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models obtained by Ramezani et al. [13] and Soroush et al. [14] in comparison to published FE simulations obtained by Pamidighantam et al. [9]. It is noted that the lumped models introduced by Pamidighantam et al. [9] and Chowdhury et al. [10], could represent even better results than the continuous models presented by Ramezani et al. [13] and Soroush et al.
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[14].
To validate the accuracy of the present approach for other types of boundary conditions, we develop the present procedure for narrow micro-beams with fringing field effect. Doing
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so, one can easily reach to a cubic equation which can be solved analytically. The results of dimensionless pull-in voltages and displacements for C-C, C-H and H-H micro-beams with
ED
d / b 1 are compared with both semi-analytical and numerical findings of Tadi Beni et al.
PT
[31] in Table 4. It is to be mentioned here that the numerical and MADM results of Tadi Beni et al. [31] are extracted from Figs. 9(a) - (c) of their paper. As it is seen from Table 4, the
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present predictions for both pull-in voltages and displacements are in good agreement with numerical findings of Tadi Beni et al. [31]; even better than their semi-analytical results
AC
provided by the MADM. Herein a convergence study is also conducted to find the number of terms which must be
included in the approximated solution of the HPM when the effect of the vdW force has been taken into account. To do so, the vdW parameter of the system is assumed to be 3 0.13Cr , where the values of the critical vdW parameter for nano/micro-beams with C-C, C-F, C-H
ACCEPTED MANUSCRIPT and H-H boundary conditions are calculated using Eq. (31) and presented in Table 5. Table 6 shows a comparison between different orders of the present HPM results (i.e. both nondimensional pull-in voltage and displacement) and the present numerical findings as well as the MADM predictions presented by Soroush et al. [14]. It should be noted that the present numerical results for Eq. (16) are prepared by two different methods: I. the fsolve command
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of the MATLAB commercial software with the zero initial guess, and II. the secant method n n 1 2 [32] with the initial guesses w1PI 0 and wPI 0.05 and the tolerance of wPI wPI 106 .
It is worth noting that, unless stated otherwise, the numerical results of the present paper are
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calculated using the aforementioned values for the initial guesses and the tolerance. Also the results of this table are rounded up to four digits. Based on the results of Table 6, the present HPM findings are converged when the order of the approximation is set to n 41 . Hence, unless stated otherwise, the results of the present paper are calculated using this order of
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approximated solution. As it is seen from Table 6, although the converged results of the HPM
ED
for both pull-in displacement and voltage agree excellently with present numerical findings which match exactly with each other, there exists a gap between them and those obtained by
PT
the MADM [14]. Additionally, it is worth noting that the present approach can provide pullin voltage and displacement straightly, while Soroush et al. [14] extracted pull-in parameters
CE
using the deflection-voltage graph when its slope reaches infinity.
AC
Verification is also performed here for systems in which the effect of the vdW force has been taken into account. Fig. 2 shows a comparison between the present HPM and numerical results, available semi-analytical findings in the literature [13, 14] as well as those obtained through non-linear finite element method (NFEM) [22]. Based on the results of this figure, the present numerical and semi-analytical results match each other excellently and agree very well with NFEM predictions. This fact can successfully justify our faith in the accuracy of the
ACCEPTED MANUSCRIPT present semi-analytical and numerical approaches. However, there exists a gap between the previous semi-analytical findings published by Ramezani et al. [13] and Soroush et al. [14]. The results of the present HPM series solution and numerical findings, which are obtained by both the MATLAB command fsolve and the secant method, are also compared with each other as well as those obtained by Soroush et al. [14] for both C-C and C-F boundary
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conditions in Fig. 3. As it is seen from this figure, although the present semi-analytical and numerical findings agree excellently with each other, there exists a gap between them and the results reported by Soroush et al. [14]. So, as it can be observed from the aforementioned comparisons and validations presented in Tables 3, 4 and 6 as well as Figs. 2 and 3, one can
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concluded that the present approach provides more accurate results than other available semianalytical procedures in the literature [13, 14]. In addition, since the present semi-analytical approach can provide pull-in characteristics straightly, there is no need to find the solution of
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the equilibrium equation for some different values of the input voltages in the range of zero to pull-in voltage of the system. Hence, not only it does not suffer from long run-time, but also
ED
it is so rapid and can represent both pull-in voltage and displacement simultaneously.
PT
Therefore, due to the aforementioned desired features of the present procedure, it can be considered as an alternative approach for extracting pull-in characteristics of beam-type
CE
N/MEMS.
Fig. 3 shows that increasing the vdW parameter decreases both non-dimensional pull-in
AC
voltage and displacement. In addition, this figure shows a linear relationship between the non-dimensional pull-in voltage and the vdW parameter. This linear descending behavior will be utilized here to present a closed-form expression for pull-in voltage. To investigate this linear relationship, variations of the pull-in parameter for nano/micro-beams with C-C, C-F, C-H and H-H boundary conditions versus the vdW parameter are depicted in Fig. 4 using the
ACCEPTED MANUSCRIPT HPM as well as the present numerical methods. Furthermore, some lines are fitted to the results of these graphs. It is to be mentioned here that the results of the secant method and those obtained by the MATLAB command fsolve match each other exactly. Hence, for the sake of improving the quality of presentation, the findings of these two methods are depicted using the same marker in Fig. 4. As it is seen from Fig. 4, the accuracy of the present
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approach for all investigated boundary conditions is also successfully justified. According to the equations of the fitted lines, which are given for each type of the investigated boundary
V PI
A 1 d
2
Eh3d 3 L4
where the values of the coefficients
1
conditions are given in Table 7.
(33)
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conditions on Fig. 4, one can write
and
2
for each type of the investigated boundary
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Eq. (33) can be utilized to predict pull-in voltage for electro-mechanical beams with submicron separations. It is noted that for micro-beams with a negligible vdW effect, a closed-
3
Eh3d 3 L4
(34)
PT
V PI
ED
form expression for pull-in voltage can also be determined analytically through Eq. (26) as
3
for all investigated boundary conditions are also provided in Table 7.
CE
where the values of
It is to be noted that the maximum relative error between pull-in voltages calculated from
AC
Eqs. (33) and (34) for systems without the vdW effect, which corresponds to nano/microbeams with C-F boundary conditions, is 1.75%. This small relative error can verify the approximated results of Eq. (33) successfully. The critical values of the vdW parameter can be approximated by equating the linear pullin formulas, which are shown on Fig. 4 for all investigated boundary conditions, to zero.
ACCEPTED MANUSCRIPT These values can also be determined analytically using Eq. (31) as those presented in Table 5. The negligible difference between the analytical results of Eq. (31) and the approximated findings of linear pull-in formulas can also verify the results of Eq. (33) again. Using the analytical results of Eq. (31), the corresponding expressions for minimum allowable gap and maximum allowable length, the so-called detachment length, can be reduced to
d min
4
L max
1 4
d
4
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L 4 A / Eh3
Eh3 / A
4
(35b)
for all investigated boundary conditions are also presented in Table
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where the coefficient
(35a)
7.
It is noteworthy to mention that the closed-form Eqs. (33) - (35) can be employed to determine the pull-in voltage, minimum allowable gap and the detachment length of beam-
M
type N/MEMS explicitly. Furthermore, according to these equations, it is obvious that the effect of the vdW attraction on pull-in instability decreases with an increase of the structural
ED
stiffness of the system. Therefore, one can conclude that investigating the effect of the vdW
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5. Conclusions
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force on pull-in instability of cantilever-type N/MEMS is more essential than the others.
The present paper has been focused on the pull-in instability of electrostatically actuated
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nano/micro-beams with C-C, C-F, C-H and H-H boundary conditions subjected to the vdW force through a semi-analytical approach. The Euler-Bernoulli beam theory has been employed to derive the governing equation of equilibrium. This equation has been nondimensionalized and reduced to a non-linear algebraic equation which describes the position of the nano/micro-beam maximum deflection through the Galerkin decomposition method.
ACCEPTED MANUSCRIPT The resulting reduced equilibrium equation and the one which governs on the stability conditions have been considered simultaneously to extract both pull-in voltage and displacement. Closed-form expressions for pull-in voltage and displacement under electrical loading alone as well as the maximum allowable length and the minimum allowable gap of beam-type N/MEMS have been presented. The effect of the vdW attraction on pull-in voltage
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and displacement has also been investigated semi-analytically through the HPM. The present results have been compared and validated by numerical findings, which were obtained by two different methods: I. the fsolve command of the MATLAB commercial software, and II. the secant method, as well as those obtained numerically, analytically and semi-analytically in
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previous studies. It was found that the present results agree better with available threedimensional FE simulations as well as NFEM findings than other available analytical or semi-analytical approaches in the literature. It was also found that non-dimensional pull-in
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voltage varies linearly versus the vdW parameter. Furthermore, based on this important finding some closed-form expressions for calculating pull-in voltage in beam-type N/MEMS
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with C-C, C-F, C-H and H-H boundary conditions which work at sub-micron separations
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have also been introduced.
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Appendix
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The coefficients ai i 0,1,..., 4 , bi i 0,1,..., 4 , ci i 0,1,...,3 and di i 0,1,..., 4 appearing in Eqs. (10), (16), (23) and (27) are, respectively, defined as:
a0 3 dx, a1 dx 2 dx, a2 3 2 dx 1
0
1
1
0
0
1
1
1
0
0
0
a3 3 3 dx, a4 4 dx
(A1)
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1 1 1 dx 1 2 dx 3 dx dx, b 0 b0 3 0 1 a2 , b2 1 0 a3 a2 1 1 0 2 2 2 dx dx dx 0 0 0 1
b3
4 dx 0
1
2 dx
(A2)
a4 2a3 , b4 3a4
0
1
1
1
0
0
0
1
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c0 dx, c1 dx, c2 2 2 dx, c3 3 dx 0
1
1
1
0
0
0
1
d0 3 dx, d1 dx, d 2 3 2 dx, d3 3 3 dx 1
d 4 4 dx
0
(A4)
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0
(A3)
References
Senturia, SD. Microsystem Design. Dordrecht, Kluwer Academic Publishers; 2001.
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Younis, MI. MEMS Linear and Nonlinear Statics and Dynamics. New York, Springer;
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M
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Lifshitz, EM. The theory of molecular attractive forces between solids, Sov Phys JETP.
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[3]
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MEMS, Int J Appl Electrom. 33 (2010) 979–84.
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Lamoreaux, SK. The Casimir force: background, experiments, and applications, Rep Prog Phys. 68 (2005) 201-36.
[6]
Batra, RC, Porfiri, M, Spinello, D. Review of Modeling Electrostatically Actuated Microelectromechanical Systems, Smart Mater Struct. 16 (2007) R23-31.
ACCEPTED MANUSCRIPT [7]
Nathanson, HC, Newell, WE, Wickstrom, RA, Davis, JR. The resonant gate transistor, IEEE T Electron Dev. 14 (1967) 117-33.
[8]
Taylor, GI. The coalescence of closely spaced drops when they are at different electric potentials, Proc Roy Soc A. 306 (1968) 423-34. Pamidighantam, S, Puers, R, Baert, K, Tilmans, HAC. Pull-in voltage analysis of
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[9]
electrostatically actuated beam structures with fixed–fixed and fixed–free end conditions, J Micromech Microeng. 12 (2002) 458-64.
[10] Chowdhury, S, Ahmadi, M, Miller, WC. A closed-form model for the pull-in voltage of
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electrostatically actuated cantilever beams, J Micromech Microeng. 15 (2005) 756-63. [11] Kuang, JH, Chen, CJ. Adomian decomposition method used for solving nonlinear pullin behavior inelectrostatic micro-actuators, Math Comput Model 41 (2005) 1479-91.
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[12] Mojahedi, M, Moghimi Zand, M, Ahmadian, MT. Static pull-In analysis of electrostatically actuated microbeams using homotopy perturbation method, Appl Math
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Model. 34 (2010) 1032-41.
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[13] Ramezani, A, Alasty, A, Akbari, J. Closed-form solutions of the pull-in instability in nano-cantilevers under electrostatic and intermolecular surface forces, Int J Solids
CE
Struct. 44 (2007) 4925-41.
AC
[14] Soroush, R, Kooch, A, Kazemi, AS, Abadyan, M. Modeling the effect of van der Waals attraction on the instability of electrostatic cantilever and doubly-supported nano-beams using modified Adomian method, Int J Struct Stab Dy. 12 (2012) 1250036 (18 pages).
[15] He, JH. Homotopy perturbation technique, Comput Method Appl M. 178 (1999) 25762.
ACCEPTED MANUSCRIPT [16] Younis, MI, Abdel-Rahman, EM, Nayfeh, AH. A reduced-order model for electrically actuated microbeam-based MEMS, J Microelectromech S. 12 (2003) 672-80. [17] Gupta, RK. Electrostatic Pull-In Test Structure Design for In-Situ Mechanical Property Measurements of Microelectromechanical Systems [Ph.D. Dissertation]. Cambridge,
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MA: Massachusetts Institute of Technology (MIT); 1997. [18] Chao, PCP, Chiu, CW, Liu, TH. DC dynamic pull-In predictions for a generalized clamped–clamped microbeam based on a continuous model and bifurcation analysis, J Micromech Microeng. 18 (2008).
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[19] Israelachvili, JN. Intermolecular and Surface Forces, 3rd ed. University of California, Santa Barbara, California, Elsevier; 2011.
[20] Meirovitch, L. Fundamentals of Vibrations. New York, McGraw-Hill; 2001.
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[21] Tahani, M, Askari, AR. Accurate electrostatic and van der Waals pull-in prediction for fully clamped nano/micro-beams using linear universal graphs of pull-in instability,
ED
Physica E. 63 (2014) 151-9.
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[22] Moghimi Zand, M, Ahmadian, MT. Dynamic pull-in instability of electrostatically actuated beams incorporating Casimir and van der Waals forces, J Mech Eng Sci. 224
CE
(2010) 2037-47.
AC
[23] Reddy, JN. Energy Principles and Variational Methods in Applied Mechanics. New York, John Wiley & Sons; 2002.
[24] Batra, RC, Porfiri, M, Spinello, D. Vibrations of narrow microbeams predeformed by an electric field, j Sound Vib. 309 (2008) 600-12.
ACCEPTED MANUSCRIPT [25] Askari, AR, Tahani, M. An alternative reduced order model for electrically actuated micro-beams under mechanical shock, Mech Res Commun. 57 (2014) 34-9. [26] Rao, SS. Vibraion of Continuous Systems. New Jersey, John Wiley & Sons; 2007. [27] Balachandran, B, Magrab, E. Vibrations, 2nd ed. Toronto, Cengage Learning; 2009.
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[28] Seydel, R. Practical Bifurcation and Stability Analysis, 3rd ed. New York, Springer; 2009.
[29] He, JH. A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int J Nonlinear Mech. 35 (2000) 37-43.
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[30] Uspensky, JV. Theory of equations. New York, McGraw-Hill; 1948.
[31] Tadi Beni, Y, Koochi, A, Abadyan, M. Theoretical study of the effect of Casimir force, elastic boundary conditions and size dependency on the pull-in instability of beam-type
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NEMS, Physica E. 43 (2011) 979-88.
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CE
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[32] Faires, JD, Burden, RL. Numerical methods, 4th ed., Cengage Learning; 2012.
ACCEPTED MANUSCRIPT Table captions Table 1. Characteristics of nano/micro-beams mode-shapes. Table 2. Geometric and material properties of the micro-cantilever studied in Table 3. Table 3. A comparison between pull-in voltages (V) calculated by different methods for a
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micro-cantilever with the geometric and material properties presented in Table 2.
Table 4. Comparisons between normalized pull-in voltages and displacements for narrow
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micro-beams ( d / b 1) with C-C, C-H and H-H boundary conditions.
Table 5. Critical values of the vdW parameter for nano/micro-beams with different boundary conditions.
Table 6. Convergence of the homotopy series solution for nano/micro-beams with different
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boundary conditions and the vdW parameter 3 0.13Cr .
Table 7. Values of the coefficients
1
and
2
in Eq. (33),
3
in Eq. (34) and
4
in Eq. (35)
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Figure captions
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for all investigated boundary conditions.
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Fig. 1. Schematic of an electrically actuated nano/micro-beam. Fig. 2. Non-dimensional pull-in voltage of an electrically actuated nano/micro-cantilever versus the vdW parameter calculated by different methods. Fig. 3. Comparisons between present numerical and semi-analytical results with those obtained by the MADM [14] for nano/micro-beams with C-C and C-F boundary conditions.
ACCEPTED MANUSCRIPT Fig. 4. Linear relationship between the non-dimensional pull-in voltage and vdW parameter for nano/micro-beams with (a) C-C, (b) C-F, (c) C-H and (d) H-H boundary conditions.
Table 1. Characteristics of nano/micro-beams mode-shapes.
Position of the maximum deflection
C-C
x 0.5000
0.6297
4.7300
0.9825
C-F
x 1.0000
0.5000
1.8751
0.7341
C-H
x 0.5808
0.6626
3.9266
1.0008
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Boundary conditions (B.C.’s)
Table 2. Geometric and material properties of the micro-cantilever studied in Table 3. b (μm)
300
50
h (μm)
d (μm)
E (GPa)
2.5
77
0.33
M
L (μm)
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PT
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1
Table 3. A comparison between pull-in voltages (V) calculated by different methods for a micro-
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cantilever with the geometric and material properties presented in Table 2. Case
Pull-in voltage (V)
Present
FE simulations [9]
[9]
[10]
[13]
[14]
2.25
2.25
2.33
2.25
2.37
2.16
ACCEPTED MANUSCRIPT Table 4. Comparisons between normalized pull-in voltages and displacements for narrow microbeams ( d / b 1 ) with C-C, C-H and H-H boundary conditions. Tadi Beni et al.
Tadi Beni et al.
[31] (Numerical)
[31] (MADM)
Present B.C.’s wPI
PI
wPI
PI
wPI
C-C
47.2456
0.4505
48.9883
0.4272
50.7865
0.4538
C-H
22.6722
0.4471
23.4831
0.4263
H-H
9.4052
0.4431
9.6992
0.4229
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PI
25.2809
0.5188
10.2857
0.5066
conditions.
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Table 5. Critical values of the vdW parameter for nano/micro-beams with different boundary
C-C
C-F
3Cr
48.6244
1.1581
C-H
H-H
23.2963
9.6456
M
B.C.’s
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Table 6. Convergence of the homotopy series solution for nano/micro-beams with different boundary
B.C.’s
C-F
C-H
Secant
MADM command
method
13
19
25
31
41
[14] fsolve
wPI
0.4069
0.4016
0.3993
0.3985
0.3981
0.3980
0.3979
0.3979
0.3979
0.3961
PI
64.0295
61.4117
60.2903
59.8353
59.6597
59.5954
59.5678
59.5642
59.5642
64.2373
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C-C
MATLAB
HPM (n)
7
CE
1
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conditions and the vdW parameter 3 0.13Cr .
wPI
0.4692
0.4661
0.4646
0.4640
0.4637
0.4636
0.4635
0.4635
0.4635
0.4684
PI
1.4618
1.4321
1.4183
1.4121
1.4093
1.4081
1.4074
1.4072
1.4072
1.5765
wPI
0.4036
0.3984
0.3962
0.3953
0.3950
0.3949
0.3948
0.3948
0.3948
-
ACCEPTED MANUSCRIPT
30.7307
29.4874
28.9591
28.7464
28.6650
28.6354
28.6228
28.6211
28.6211
-
wPI
0.3998
0.3947
0.3926
0.3918
0.3915
0.3914
0.3913
0.3913
0.3913
-
PI
12.7446
12.2395
12.0268
11.9419
11.9096
11.8979
11.8930
11.8923
11.8923
-
Table 7. Values of the coefficients
1
and
2
in Eq. (33),
investigated boundary conditions.
in Eq. (34) and
4
in Eq. (35) for all
C-F
C-H
H-H
1
0.1451
0.1439
0.1459
0.1460
2
10.9909
0.2597
5.2809
2.1941
3
3.3658
4
0.3383
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C-C
0.5187
2.3301
1.4996
0.8611
0.4066
0.5069
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Coefficient
3
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H-H
PI
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ACCEPTED MANUSCRIPT
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Fig. 1. Schematic of an electrically actuated nano/micro-beam.
ACCEPTED MANUSCRIPT 2 HPM (Present) M ATLAB command fsolve (Present) Secant method (Present) NFEM (M oghimi Zand and Ahmadian [22]) Semi-analytical method (Ramezani et al. [13]) M ADM (Soroush et al. [14])
1
0.5
0
0
0.31
0.62
3
CR IP T
PI
1.5
0.93
1.24
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Fig. 2. Non-dimensional pull-in voltage of an electrically actuated nano/micro-cantilever versus the
AC
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M
vdW parameter calculated by different methods.
ACCEPTED MANUSCRIPT 0.42
72 C-C nano/micro-beam
C-C nano/micro-beam 54
PI
0.36
0.33
0.3
18 (a) 0
13
0.5
26
39
3
0
52
(b) 0
13
1.8
C-C nano/micro-beam
1.35
PI
w PI
36
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0.45
54
26
3
39
52
C-F nano/micro-beam
C-F nano/micro-beam
72
PI
36
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w PI
0.39
0.4
0.9
0.45
18
0
13
1.8
0.31
26 39 3 (Present) HPM
C-F nano/micro-beam
0.62
3
0.93
1.24
M
(b)
(c) 0
52 MATLAB command fsolve (Present)
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0
0.35
0
(d)
0
0.31
0.62
3
Secant method (Present)
0.93
1.24
MADM [14]
Fig. 3. Comparisons between present numerical and semi-analytical results with those obtained by the 1.35
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MADM [14] for nano/micro-beams with C-C and C-F boundary conditions.
CE
0.9
0
(d) 0
AC
0.45
0.31
0.62
3
0.93
1.24
HPM (Pres MATLAB Secant met MADM [1
ACCEPTED MANUSCRIPT 70
1.68
PI = -1.3671 3 + 65.9451
PI = -1.3559 3 + 1.5581
52.5
1.26
PI
35
17.5
0
0.84
0.42 (a) 0
12.5
25
3
34
37.5
0
50
(b) 0
14
PI = -1.3712 3 + 31.6853
0.6
3
0.9
1.2
10.5
PI
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PI
0.3
PI = -1.3762 3 + 13.1646
25.5
17
7
3.5
(c) 6
12
3
18
ED
PI = -1.3559 3 +01.5581
M
8.5
0
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PI
T
24
The fitted line
0
(d) 0
HPM
2.5
5
3
7.5
10
Numerical
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Fig. 4. Linear relationship between the non-dimensional pull-in voltage and vdW parameter for
0.6
3
0.9
1.2
AC
0.3
CE
nano/micro-beams with (a) C-C, (b) C-F, (c) C-H and (d) H-H boundary conditions.
PI = -1.3762 3 + 13.1646
2.5
5
3
7.5
10