INTRODUCTION
With increasing growth of nanotechnology, micro and nano electromechanical
switches (MEMS and NEMS) have become the center of interest for researchers
(Babazadeh and Keshmiri, 2009; Miskam
et al., 2009). Many mechanical engineers have been focused on solving
the nonlinear governing equation and modeling the instability of electromechanical
switches. It is well-established that the governing equation of most engineering
and physical systems is nonlinear in its nature. Many efforts have been conducted
by scientists to solve the mathematical nonlinear equations of the systems.
Recently, various mathematical methods, such as Adomian decomposition (Adomian,
1994), variational iteration (Noorzad et al.,
2008; Shakeri et al., 2009; Barari
et al., 2008), homotopy perturbation (Sharma
and Methi, 2011; Fazeli et al., 2008) etc.,
have been proposed for solving nonlinear problems. Among these methods, the
Adomian decomposition has been widely used to investigate engineering problems
i.e. stochastic systems (Jaradat, 2008), oscillation
(Momani et al., 2008) and heat transfer (Biazar
and Amirtaimoori, 2005). After introducing the conventional Adomian method,
several investigators made attempt to improve the abilities and convergence
speed of the decomposition method. Rach (1984) proposed
a systematic formula for computing the Adomian's polynomials. Further modification
of the polynomials was also provided by Gabet (1994).
Furthermore, comparison between the decomposition method and the Taylor series
approximation shows that the decomposition method is much more efficient than
the Taylor series method (Wazwaz, 1998). A modified
Adomian decomposition method has been applied to simulate the static deflection
of electrostatic micro-actuators (Kuang and Chen, 2005).
Wazwaz and El-Sayed (2001) proposed a powerful modification
of the Adomian decomposition method. This modification highly accelerates the
convergence of the decomposition polynomials and has been applied for solving
higher order boundary value problems (Wazwaz, 2000,
2001).
The aim of this paper was to evaluate the limitations/abilities of conventional
and modified Adomian decomposition methods in solving constitutive equation
of NEMS. In this regards, numerical solution was obtained using MAPLE commercial
software and Adomian solutions were compared with the numerical results. The
precision and convergence speed of both methods were compared.
GOVERNING EQUATION OF NEMS
Figure 1 shows the typical cantilever and doubly-supported
beam-type NEMS constructed from a conductive electrode suspended over a conductive
substrate. Applying voltage difference between the electrode and ground causes
the electrode to deflect towards the ground. At a critical voltage/deflection,
which is known as pull-in instability voltage/deflection, the electrode becomes
unstable and pulls-in onto the substrate. The pull-in voltage and pull-in deflection
of a NEMS are named as the pull-in parameters of the switches. Determining the
electrode deflection and pull-in parameters of NEMS are crucial issues for engineers.
Considering the van der Waals force, the governing equation of beam-type NEMS
can be derived into (Ramezani et al., 2008):
where, W is the deflection of the electrode, Z is the distance from the clamped
end and I is the moment of inertia of the electrode cross section, Eeff
is the effective electrode material modulus, ε0=8.854x10-12
C2N-1m-2 is the permittivity of vacuum, V is
the applied voltage, g is the initial gap between the electrode and the substrate,
d is the width of cross section and A is the Hamaker constant. Using the substitutions
w=W/g and z=Z/L, Eq. 1 becomes:
|
| Fig. 1(a-b): |
(a) Schematic representation of (a) a cantilever NEMS and
(b) doubly-supported NEMS |
In above equations, the dimensionless parameters, α, β and γ
are defined according to:
Using numerical computations, the variation range of above parameters which
satisfies physical considerations (Ramezani et al.,
2008) approximately could be defined as:
| • |
0≤α≤1.21, 0≤β≤1.68, 0≤γ≤0.65
For cantilever NEMS |
| • |
0≤α≤50.09, 0≤β≤70.06, 0≤γ≤0.65 For doubly-supported
NEMS |
Note that at the onset of the instability, the maximum deflection of the electrode
increases without requiring any further increase in voltage. In mathematical
view, the slope of w-β curve reaches infinity when instability occurs,
i.e., dw/dβ(z = 1)→∞ and dw/dβ(z = 0.5)→∞ for cantilever
and doubly-supported NEMS, respectively. As a convenient approach, the pull-in
instability voltage, βPI and pull-in deflection, uPI,
of NEMS can be determined via plotting w(z = 1) vs. β for cantilever and
w (z = 0.5) vs. β for doubly-supported NEMS.
FUNDAMENTALS OF DECOMPOSITION METHODS
Consider a differential equation of a fourth-order boundary-value problem (Wazwaz,
2001):
With boundary conditions:
Equation 4 can be represented as:
Where, L(4) is a differential operator which is defined as:
The corresponding inverse operator L-(4) is defined as a 4-fold integral operator, that is:
Employing the decomposition method (Wazwaz, 2001),
the dependent variable in Eq. 4 can be written as:
where, constants C1 and C2 can be determined from the boundary condition at another boundary point. In above relations, function An approximates nonlinear function f (x,y) and is determined as a polynomial series:
According to Conventional Adomian Decomposition (CAD), series An is obtained using the following formula:
On the other hands, according to Modified Adomian Decomposition (MAD), the
following convenient equations can be utilized to obtain an appropriate solution
for An (Adomian, 1986; Rach,
1984):
where,
 ,n>0, |
0≤i≤n, l≤pi≤n,-v+l and ki is the number of
repetition of the fpi, the values of pi are selected from
the above range by combination without repetition.
Now, according to decomposition methods, the recursive relations of Eq. 9 can be provided as follows:
In order to apply decomposition methods for simulating deflection and pull-in
behavior of NEMS, the substitution y=1-w is used to rewrite Eq.
2 into the following form:
The solution of Eq. 2 can be represented as:
where, the constants C1 and C2 can be determined by solving
the resulted algebraic equations from B.C. at z = 1, i.e., using Eq.
14c and 14d for cantilever and doubly-supported NEMS,
respectively.
Conventional Adomian method (CAD): In order to solve Eq. 15 using CAD, formula (11) is expanded to obtain:
Substituting Eq. 16 in recursive Eq. 13, we obtain:
Therefore the solution of Eq. 2 is obtained as:
Modified Adomian method (MAD): In the case of modified domain methods
(Eq. 12), it is obtained:
Substituting Eq. 19 in Eq. 13, we obtain:
Therefore, the solution of Eq. 2 can be summarized to:
Case studies and comparing of the methods: In order to compare decomposition methods, typical cantilever and a doubly-supported NEMS are simulated and the results are compared with numerical data.
Figure 2a-c shows the variation of tip
deflection as a function of series terms for three typical cantilever NEMS (β
= γ = 0.5) with different van der Waals coefficients (Fig.
2a-c) correspond to a = 0, 0.25 and 0.5, respectively.
This figure reveals that the value of α coefficient has a great influence
on the convergence of the conventional series. As seen, the CAD series might
not converge for large α values (Fig. 2a). However, this
shortcoming is not observed in the case of the MAD series (Fig.
2d-f) where the series solution rapidly converges to the
numerical solution. Figure 3 shows the convergence of pull-in
value for typical cantilever NEMS obtained by various series terms. This figure
reveals that CAD converges to a pull-in value which is different from numerical
values. However, the pull-in value obtained by modified method converges to
that of the numerical value. Figure 4 shows the variation
of pull-in voltage for cantilever NEMS as a function of van der Waals force
parameter (α). This figure shows that the difference between Adomian and
numerical solutions increases by increasing the a value. As seen, no
solution exist, when α exceeds its critical value.
Figure 5 shows the variation of tip deflection for typical
doubly-supported NEMS (a =β = 5, γ = 0.5) as a function of series
terms. This figure reveals that conventional decomposition can not be applied
for modeling pull-in performance of doubly-supported NEMS.
|
| Fig. 2 (a-f): |
Convergence check for the NEMS tip deflection (b = g = 0.5)
vs, number of seies terms for three typical cantilever cases: (a) and (d)
a = 0, (b) and (e) a = 0.25 (c) and (f) a = 0.5 |
|
| Fig. 3: |
Variation of βPI for typical cantilever NEMS
(α = 0.5 and γ = 0.65) |
|
| Fig. 4: |
Variation of pull-in voltage (βPI) of cantilever
NEMS as a function of van der Waals force (a) (γ = 0.65) |
As seen, while the MAD method rapidly converges to the numerical solution,
CAD series converges to an unacceptable value. Furthermore, Table
1 shows the convergence of pull-in voltage of typical doubly-supported NEMS
obtained by Adomian method using various series terms. As seen βPI
values obtained by MAD series converge to that of numerical value, i.e., βPI
= 43.575. In Table 1, only the βPI values
obtained by MAD have been presented since the CAD method is not reliable for
simulating double-supported NEMS. Note that the MAD series which are not able
to capture the instability of the switch are physically meaningless and cannot
be used for investigating the pull-in performance of the NEMS.
|
| Fig. 5: |
Convergence check for the tip deflection of a typical doubly-supported
NEMS (α = β = 5, γ= 0.5) vs. number of series terms |
| Table 1: |
Convergence check of pull-in voltage for typical NEMS (α
= 5 and γ = 0.65). As seen, βPI values obtained by
Adomian series converge to that of numerical value (i.e., βPI
= 43.575) |
 |
CONCLUSION
Modified and conventional Adomian decomposition methods were applied to solve nonlinear governing equation of beam-type NEMS. The deflection and pull-in parameters of cantilever and doubly-supported NEMS were computed and the result was compared with the numerical solution. It was observed that the modified Adomian method provides accurate results and converges rapidly to numerical solution. However, the convergence of the conventional series highly depends on the values of constant coefficients in the NEMS governing equation. Interestingly, we showed that there are some cases that using CAD method could lead to physically incorrect results. Specially, for doubly-supported NEMS, the deflection value computed by conventional decomposition series might be very different from that of numerical method. Interestingly, none of these shortcomings was observed for modified Adomian decomposition series.
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