Sage Journals: Discover world-class research

Abstract

The microelectromechanical devices have triggered rocketing interest in various advanced technologies, that is, sensors, material science, and energy harvesting, and now the devices tend to a nanoscale size, making the technology much more attractive in both academic and industrial communities. This paper takes a graphene nano/microelectromechanical system as an example to study its dynamical property, which is the foundation for the optimal design and reliable operation. As the system is inextricably complex with singularity and zero conditions in a fractal space, the iteration perturbation method is used to obtain the periodic solution of the system. The results show clearly the low-frequency property of the system and pull-in instability; furthermore, the fractal dimension affects greatly the system’s dynamical property.

Keywords

Nano/microelectromechanical system fractal iteration perturbation method nonlinear oscillator pull-in instability criterion

Introduction

Recently, the nano/microelectromechanical system (N/MEMS) has attracted extensive attention due to its attractive characteristics such as the extremely small size, the remarkably high reliability, the easy batch manufacturing, and the low power consumption. Its numerous applications to the highly sensitive sensors, the excellently reliable actuators, bridges, and switches have been extensively studied.^1–4 During the operation of MEMS, the pull-in instability will occur,^5–9 which should be avoided in practical applications. There are various methods in the literature to deal with the pull-in instability: Anjum et al.¹⁰ gave a systematical review on various analytical methods for MEMS systems, Anjum et al.¹¹ showed that the variational iteration method is an efficient tool to the analysis of the periodicity of an N/MEMS, Ji-Huan He et al. established a variational theory for MEMS systems,¹² Anjum et al. applied Li-He’s modification of the homotopy perturbation method with great success,¹³ Rehman et al. suggested a modified variational iteration method coupled with Laplace transform,¹⁴ and Ji-Huan He et al. discovered the pull-in plateau.¹⁵

Carbon-based nanomaterials, including one-dimensional carbon nanotubes (CNTs) and two-dimensional graphene,^16,17 are more attractive in various N/MEMS applications (such as electronic components, mass/gas sensors, and supercapacitors) due to their unique mechanical, electrical, optical, and chemical properties and have become the first choice for sensing applications in MEMS (microelectromechanical systems) and NEMS (nanoelectromechanical systems) devices.^18–22 Graphene is widely used in mass and gas sensing applications because of its low mass and high surface to volume ratio and its good mechanical and electronic properties.

In 2021, Tian et al.²³ suggested a totally new concept of the fractal MEMS system, that is a microelectromechanical system working in a porous medium like plasma. In this paper, we will extend the fractal MEMS system to a graphene N/MEMS.^24,25 Considering the environmental pollution, these electronic devices must work in a fractal space filled with particles. Under such conditions, the operation of electronic devices will inevitably be affected, which will also affect the pull-in instability. In this paper, the two-scale fractal theory^26–28 is adopted to establish a fractal-fractional differential model for the graphene N/MEMS. The two-scale fractal is now widely used for fractal vibration systems,^29–33 the fractal diffusion,^34,35 porous concretes,^36–38 and dynamic economics.³⁹

Nonlinear problems are relatively complex, and it is difficult for us to obtain an exact solution of a nonlinear equation. Therefore, analytical approximation methods are mostly used in the study of nonlinear systems. In particular, for the N/MEMS model, many of these analytical methods are also applicable. For example, the variational iteration method,^40,41 the homotopy perturbation method,⁴² the energy balance method,^43,44 He’s frequency formula,^45,46 the parameter expansion method,^47–49 the residual power progression method,⁵⁰ and Hamiltonian method.^51,52 Recently, a powerful method, namely, the iteration perturbation method (IPM), has been proved to be an extremely simple, effective, and convenient method which is adequate for both weak and strong nonlinear oscillators proposed by Ji-Huan He.⁵³ The IPM is exceedingly simple in principle, easy to apply, and has good accuracy.⁵³ A comprehensive overview of various analysis methods for nonlinear oscillators has been provided by He.^54,55

In this paper, firstly, the fractal derivatives in the fractal space are used to describe the dynamic behavior of graphene N/MEMS, and the generalized fractal MEMS equation is constructed. The model is characterized by second-order nonlinear ordinary differential equations with zero initial conditions. Secondly, the fractal model is transformed into differential forms by using the two-scale transformation. Finally, considering the complex nonlinear problem in fractal space, the periodic solution of graphene N/MEMS model is obtained by using the He’s IPM. This simple and effective method gives the approximate nonlinear frequency and approximate analytical solution. We compare the results of IPM with those of the fourth-order Runge–Kutta method (RK4) to verify the solution procedure.

Fractal mathematical modeling

We consider a model of a dual parallel plate driven by electrostatic power. A graphene strip of cross-sectional area $S^{'}$ is movable and connected to a fixed base unit. We assume that its initial distance is set to be $h$ . The schematic diagram of the model is shown in Figure 1. Based on the existing literature, graphene material satisfies the following relationship

σ = E ε - D ε | ε |,

(1)

in which

σ

denotes the axial stress,

ε

indicates the axial strain,

E

is Young’s modulus, and

D

is the elastic stiffness constant. Since the displacement of graphene strip, the movable flat plate is impacted by the restoring force

F_{r}

and is subject to the electrostatic force

F_{e}

due to the imposing the voltage

V

. According to equation (1), the formulae

F_{r}

and

F_{e}

are given as follows

F_{r} = - S^{'} E \frac{\hat{x}}{l} + S^{'} D \frac{\hat{x}}{l} | \frac{\hat{x}}{l} |,

(2)

F_{e} = \frac{ε_{0} S V^{2}}{2 {(h - \hat{x})}^{2}},

(3)

in which

l

is the length of the dual parallel plate,

ε_{0}

denotes the electric emissivity,

S

is the area of the movable flat plate, and

h

is the initial distance of the dual parallel plate. By applying Newton’s second law of motion, the motion behavior of the movable flat plate can be written in the following form

m \frac{d^{2} \hat{x}}{d {\hat{t}}^{2}} + S^{'} E \frac{\hat{x}}{l} - S^{'} D \frac{\hat{x}}{l} | \frac{\hat{x}}{l} | = \frac{ε_{0} S V^{2}}{2 {(h - \hat{x})}^{2}},

(4)

in which

\hat{x}

is the displacement between two parallel plates at time

\hat{t}

and

m

is the mass of area of the movable flat plate.

Figure 1.

Graphene N/MEMS model.

In order to simplify equation (4), we introduce the following dimensionless variables

x = \frac{\hat{x}}{h}, t = t^{'} \sqrt{\frac{S^{'} E}{m l}}, α = \frac{h D}{l E}, k = \frac{ε_{0} S l V^{2}}{2 S^{'} E h^{3}} .

(5)

Then equation (4) can be converted as follows

\frac{d^{2} x}{d t^{2}} + x - α x | x | = \frac{k}{{(1 - x)}^{2}} .

(6)

We suppose that the following zero initial conditions:

x (0) = 0, {\frac{d x}{d t} |}_{t = 0} = 0 .

(7)

If the microelectromechanical system in Figure 1 is carried out in the porous medium, then the model needs to consider in the fractal space. Therefore, the generalized fractal microelectromechanical system is given as follows

\frac{d^{2} x}{d t^{2 γ}} + x - α x | x | = \frac{k}{{(1 - x)}^{2}},

(8)

where

d^{2} x / d t^{2 γ}

means He’s fractal derivative and is defined as^26,56

\frac{d x}{d t^{γ}} = Γ (1 + γ) \lim_{t \to t_{0}} \frac{x (t) - x (t_{0})}{{(t - t_{0})}^{γ}} .

(9)

The fractal derivative can be used to model various discontinuous problems, for example, shallow water waves along an unsmooth boundary,⁵⁷ the fractal modification of Chen–Lee–Liu equation,⁵⁸ the fractal boundary layer theory,⁵⁹ fractal Duffing equation,⁶⁰ heat prevention through porous materials,^61,62 and a cement mortar’s fluidity.⁶³

Equation (8) is a microelectromechanical system in the fractal space investigated by Tian et al.^7,23 The air can be viewed as a porous medium, and the fractal dimension $γ$ indicates the distribution of air. If the vibrational behavior of the system occurs in a vacuum, then we have $γ = 1$ . For the equation (8) with $γ = 1$ , when $α = k = 0$ , the model becomes the classical harmonic oscillator. Skrzypacz, et al.⁴ studied the dynamic pull-in analysis when $α > 0$ and $k \geq 0$ . Very recently, Anjum et al.¹⁰ discussed the pull-in analysis and obtained analytical solution by using the combination of variational iteration method and the Laplace transform.

The two-scale transform is a powerful tool to convert a discontinuous space into its continuous one. It can approximately transform the fractional calculus into the traditional calculus,⁶⁴ which is easy to solve. For example, consider a fractal differential equation:

\frac{D u}{D t^{β}} + \frac{D u}{D x^{η}} + F (u) = 0 .

(10)

where

\frac{D u}{D t^{β}}

and

\frac{D u}{D x^{η}}

are He’s fractal derivatives. We use the two-scale transform

T = t^{β},

(11)

X = x^{η},

(12)

Therefore, equation (10) can be rewritten as follows:

\frac{D u}{D T} + \frac{D u}{D X} + F (u) = 0 .

(13)

With help of the two-scale transform

τ = t^{γ}

, equation (8) can be converted into the following form

x^{″} + x - α x | x | = \frac{k}{{(1 - x)}^{2}},

(14)

in which dot denotes the derivative with respect to

τ

In the case of zero initial values, the use of existing asymptotic approaches can make the approximation process really complicated. In order to overcome the influence of the calculation difficulty caused by the zero initial conditions, we consider the following transformation

x = A - u,

(15)

in which

A < 1

indicates the amplitude of the solution

u (τ)

. Then equation (14) can be converted to

\frac{d^{2} u}{d τ^{2}} - A + u + α A^{2} - 2 α A u + α u^{2} + \frac{k}{{(1 - A + u)}^{2}} = 0 .

(16)

The nonlinear term can be expanded as follows:

\frac{k}{{(1 - A + u)}^{2}} = \frac{k}{{(- 1 + A)}^{2}} + \frac{2 k}{{(- 1 + A)}^{3}} u + \frac{3 k}{{(- 1 + A)}^{4}} u^{2} + \frac{4 k}{{(- 1 + A)}^{5}} u^{3} + O (u^{4}) .

(17)

Therefore, equation (16) can be written in the following form:

\frac{d^{2} u}{d τ^{2}} - A + u + α A^{2} - 2 α A u + α u^{2} + \frac{k}{{(- 1 + A)}^{2}} + \frac{2 k u}{{(- 1 + A)}^{3}} + \frac{3 k u^{2}}{{(- 1 + A)}^{4}} + \frac{4 k u^{3}}{{(- 1 + A)}^{5}} = 0 .

(18)

Equation (18) can also be rearranged as follows:

\frac{d^{2} u}{d τ^{2}} + (- 2 α A + 1 + \frac{2 k}{{(- 1 + A)}^{3}}) u + (α + \frac{3 k}{{(- 1 + A)}^{4}}) u^{2} + \frac{4 k u^{3}}{{(- 1 + A)}^{5}} - A + α A^{2} + \frac{k}{{(- 1 + A)}^{2}} = 0 .

(19)

We re-write equation (19) in the form

\frac{d^{2} u}{d τ^{2}} + β_{1} u + β_{2} u^{2} + β_{3} u^{3} + β_{0} = 0,

(20)

where

β_{1} = 1 - 2 α A + \frac{2 k}{{(- 1 + A)}^{3}}, β_{2} = (α + \frac{3 k}{{(- 1 + A)}^{4}}), β_{3} = \frac{4 k u^{3}}{{(- 1 + A)}^{5}}, β_{0} = - A + α A^{2} + \frac{k}{{(- 1 + A)}^{2}} .

(21)

Basic ideas of the IPM

The IPM was developed by He⁵³ which is based on the combination of the perturbation technique and the iterative method. It can be used for weak nonlinear problems and strong nonlinear differential equations. Sedighi et al.⁶⁵ had obtained high accurate approximate solution to transversely vibrating quintic nonlinear beams in order to demonstrate the efficiency of IPM. In this section, it offers a concise introduction about IPM to solve nonlinear differential equations. Consider the following nonlinear oscillation:

x^{″} + x + p f (x, x^{'}) = 0, x (0) = A, x^{'} (0) = 0 .

(22)

The above nonlinear equation is rewritten as follows:

x^{″} + x + p x \cdot g (x, x^{'}) = 0 .

(23)

The iteration formula of the above equation can be constructed as follows:

x_{n + 1}^{″} + x_{n + 1} + p x_{n + 1} g (x_{n}, x_{n}^{'}) = 0,

(24)

in which

x_{n}

represents the

n_{t h}

approximate solution. In the asymptotic approach, the initial approximate solution is presumed as

x (0) = A \cos (ω t)

, in which

ω

is the angular frequency of the determined oscillation. Substituting the first approximation into equation (24) yields the following:

x^{″} + x + p x \cdot g (x_{0}, x_{0}^{'}) = 0 .

(25)

Then, the solution

u

and its coefficients are expanded as follows:

x = x_{0} + p x_{1} + p^{2} x_{2} + \dots,

(26)

c o e f f (x) = ω^{2} + p c_{1} + p^{2} c_{2} + \dots .

(27)

By substituting equations (26) and (27) into equation (25) and equalizing the coefficients with the same power

p

, we can obtain a series of linear differential equations. After removing the coefficients of the secular terms, the uncertain parameters

c_{1}, c_{2},

… of equation (27) can be decided. And solving the remaining differential equations for

x_{1}, x_{2},

… gives the solution of the problem.

Application of IPM in a fractal space

Consider the nonlinear equation, equation (20), for the purpose of gaining the iteratively perturbed solution of the governing equation. An artificial parameter $p$ is presented as follows:

\frac{d^{2} u}{d τ^{2}} + p u (β_{3} u^{2} + β_{2} u + β_{1}) + p β_{0} = 0 .

(28)

Equation (28) can be approximated by⁵³

\frac{d^{2} u}{d τ^{2}} + p u (β_{3} u_{0}^{2} + β_{2} u_{0} + β_{1}) + p β_{0} = 0,

(29)

in which

u_{0}

is the initial approximate solution. The initial solution can be represented by

u_{0} = A \cos (ω τ)

, where

ω

is the unknown fundamental frequency. Substituting

u_{0}

into equation (29) yields:

\frac{d^{2} u}{d τ^{2}} + p u {(β_{3} A^{2} (\cos (ω τ))}^{2} + β_{2} A \cos (ω τ) + β_{1}) + p β_{0} = 0 .

(30)

Equation (30) can be converted in the following form:

\frac{d^{2} u}{d τ^{2}} + p u β_{1} + p u β_{2} Acos (ω τ) + \frac{1}{2} p u β_{3} A^{2} \cos (2 ω τ) + \frac{1}{2} p u β_{3} A^{2} + p β_{0} = 0 .

(31)

According to the standard perturbation method, solution and the coefficient

p β_{1}

are expanded in a series of

p

as follows:

u = u_{0} + p u_{1} + p^{2} u_{2} + \dots,

(32)

p β_{1} = ω^{2} + p c_{1} + p^{2} c_{2} + \dots .

(33)

This expansion method is proposed by He⁶⁶ in the homotopy perturbation method with two expansion parameters. This method is very suitable for nonlinear equations with two nonlinear terms, and there may be diverse influence on the solution. By substituting equations (32) and (33) into equation (31) and equalizing the coefficients with the same power, we can obtain differential equation for $u_{1}$ as follows:

\frac{d^{2} u_{1}}{d τ^{2}} + ω^{2} u_{1} + (c_{1} A + \frac{3 β_{3} A^{3}}{4}) \cos (ω τ) + \frac{1}{2} β_{2} A^{2} \cos (2 ω τ) + \frac{1}{4} β_{3} A^{3} \cos (3 ω τ) + \frac{1}{2} β_{2} A^{2} + β_{0} = 0 .

(34)

Assuming that there exists the term $\cos (ω τ)$ in the right-hand side of equation (34), the secular term $τ \cos (ω τ)$ will be appeared in the final solution. Consequently, the coefficient of the term $\cos (ω τ)$ of equation (34) needs to be equal to zero. The following equation can be obtained:

c_{1} = - \frac{3}{4} β_{3} A^{2} .

(35)

No secular term in the second equation for $u_{2} (τ)$ yields:

c (c_{2}, ω) = \frac{β_{3}^{2} A^{5}}{128 ω^{2}} - \frac{5 β_{2}^{2} A^{3}}{12 ω^{2}} + c_{2} A - \frac{β_{2} β_{0} A}{ω^{2}} = 0,

(36)

thereby, solving equations (36) and (33) with

p = 1

for the fundamental frequency gives the following frequency–amplitude relationship for the actuated vibrating MWCNT as follows:

ω = \sqrt{\frac{3 β_{3} A^{2}}{8} + \sqrt{\frac{19 A^{4} β_{3}^{2}}{128} + \frac{3 A^{2} β_{1} β_{3}}{8} - \frac{5 A^{2} β_{2}^{2}}{12} - β_{0} β_{2} + \frac{β_{1}^{2}}{4}} + \frac{β_{1}}{2}}

(37)

Solving $u_{1} (τ)$ in equation (34) gives the following second-order approximation of $u (τ)$ :

u (τ) = A \cos (ω τ) - \frac{\cos (ω τ) (3 A^{3} β_{3} - 8 A^{2} β_{2} - 24 β_{0})}{24 ω^{2}} + \frac{3 β_{3} A^{3} \cos (ω τ) + 8 β_{2} A^{2} \cos^{2} (ω τ) - 16 β_{2} A^{2} - 24 β_{0}}{24 ω^{2}} .

(38)

u (t) = A \cos (ω t^{γ}) - \frac{\cos (ω t^{γ}) (3 A^{3} β_{3} - 8 A^{2} β_{2} - 24 β_{0})}{24 ω^{2}} + \frac{3 β_{3} A^{3} \cos (ω t^{γ}) + 8 β_{2} A^{2} \cos^{2} (ω t^{γ}) - 16 β_{2} A^{2} - 24 β_{0}}{24 ω^{2}} .

(39)

Numerical experiments

This subsection considers the effects of different values of the parameters $α$ and $k$ on the nonlinear frequencies and approximate analytical solutions. One of the listed parameters changes, while the others remain the same. With this approach, the time–displacement plots are shown in Figures 2 and 3. According to equation (20) and the corresponding equation, we selected different values of $A$ and corresponding values of $k$ , respectively, to give the pull-in curve.

Figure 2.

When A = 0.03, k = 0.045, the horizontal axis is dimensionless time, and the vertical axis is the dimensionless displacement. (a)–(c): compared the results of first-order approximation with second-order approximation at fractal dimension $α = 0.3,0.4,0.7$ ; (d): compared the first-order approximation and the second-order approximation with its corresponding RK4 solution at fractal dimension $α = 1.0$ .

Figure 3.

When A = 0.065, k = 0.08, the horizontal axis is dimensionless time, and the vertical axis is the dimensionless displacement. (a)–(c): compared the results of first-order approximation with second-order approximation at fractal dimension $α = 0.3,0.4,0.7$ ; (d): compared the first-order approximation and the second-order approximation with its corresponding RK4 solution at fractal dimension $α = 1.0$ .

In order to show the completeness of the solution proposed by IPM, the asymptotic solution and the corresponding numerical results are plotted. As shown in Figures 2 and 3, the second-order approximation of $u (t)$ using the analytical method shows a good agreement with the numerical results of the Runge–Kutta method of order four (RK4).

Equation (39) shows the low-frequency property at the initial stage, equation (39) can be written in the following form

u (t) = A \cos (Ω t) - \frac{\cos (Ω t) (3 A^{3} β_{3} - 8 A^{2} β_{2} - 24 β_{0})}{24 ω^{2}} + \frac{3 β_{3} A^{3} \cos (Ω t) + 8 β_{2} A^{2} \cos^{2} (Ω t) - 16 β_{2} A^{2} - 24 β_{0}}{24 ω^{2}} .

(40)

where

Ω = ω t^{γ - 1}

For $γ > 1$ , we have $\lim_{t \to 0} Ω \to 0$ . The low-frequency property has many potential applications in the effective energy harvesting, the highly sensitive sensors, and actuators.

When

ω^{2} = \frac{3 β_{3} A^{2}}{8} + \sqrt{\frac{19 A^{4} β_{3}^{2}}{128} + \frac{3 A^{2} β_{1} β_{3}}{8} - \frac{5 A^{2} β_{2}^{2}}{12} - β_{0} β_{2} + \frac{β_{1}^{2}}{4}} + \frac{β_{1}}{2} < 0

(41)

The pull-in instability will occur. So $ω = 0$ is the pull-in instability criterion.

Concluding remarks

In this paper, the dynamic behavior of the graphene N/MEMS in the fractal space is studied, and a fractal mathematical model considering constitutive stress–strain law and electrostatic force interaction is constructed for the system. In the case of zero initial conditions and magnetostatic excitation, the IPM is used to solve the model. The frequency–amplitude relationship of the graphene N/MEMS is effectively provided. The IPM has much advantages over the variational iteration method⁶⁷ and the variational method,⁶⁸ and the obtained result can be used for optimization of the N/MEMS system like that in Ref. 69, and the fractal MEMS also shows advantages over fractional differential models.^70,71

The influence of vibration amplitude on natural frequency is obtained. Numerical results verify the accuracy of the analytical solutions obtained. Finally, two terms in the series expansion are sufficient to produce an acceptable approximate solution. The amplitude–frequency relationship is very important for a better understanding of the oscillatory behavior of the system. Compared with other methods, the method applied in this paper is very simple and effective and can be extended to other fractal-fractional nonlinear problems.^72,73

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by the Natural Science Foundation of Shaanxi Province (No. 2023-JC-QN-0016) and and the Natural Science Foundation of Shaanxi Provincial Department of Education in 2022 (22JK0437).

ORCID iD

Guojuan Lv

References

Sedighi

Koochi

Daneshmand

, et al. Non-linear dynamic instability of a double-sided nano-bridge considering centrifugal force and rarefied gas flow. Int J Non Linear Mech 2015; 77: 96–106.

Kumar

Yadav

Kumar

, et al. MEMS impedance flow cytometry designs for effective manipulation of micro entities in health care applications. Biosens Bioelectron 2019; 142: 111526.

Sedighi

Bozorgmehri

. Dynamic instability analysis of doubly clamped cylindrical nanowires in the presence of Casimir attraction and surface effects using modified couple stress theory. Acta Mech 2016; 227(6): 1575–1591.

Skrzypacz

Kadyrov

Nurakhmetov

, et al. Analysis of dynamic pull-in voltage of a graphene MEMS model. Nonlinear Anal Real World Appl 2019; 45: 581–589.

Anjum

. Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions. Alex Eng J 2020; 59: 4343–4352.

. A variational principle for a fractal nano/microelectromechanical (N/MEMS) system. Int J Numer Method H 2022; 33(1): 351–359(9). DOI: 10.1108/HFF-03-2022-0191.

Tian

Ain

Anjum

, et al. Fractal N/MEMS: from pull-in instability to pull-in stability. Fractals 2021; 29(2): 2150030.

Yang

, et al. A simple frequency formulation for the tangent oscillator. Axioms 2021; 10(4): 320.

Skrzypacz

Ellis

, et al. Dynamic pull-in and oscillations of current-carrying filaments in magnetic micro-electro-mechanical system. Commun Nonlinear Sci Numer Simul 2022; 109: 106350.

10.

Anjum

, et al. A brief review on the asymptotic methods for the periodic behaviour of microelectromechanical systems. J Appl Comput Mech 2022; 8(3): 1120–1140.

11.

Anjum

Rahman

, et al. An efficient analytical approach for the periodicity of nano/microelectromechanical systems’ oscillators. Math Probl Eng 2022; 12: 9712199.

12.

Anjum

Skrzypacz

. A variational principle for a nonlinear oscillator arising in the microelectromechanical system. J Appl Comput Mech 2021; 7(1): 78–83.

13.

Anjum

Ain

, et al. Li-He’s modified homotopy perturbation method for doubly-clamped electrically actuated microbeams-based microelectromechanical system. FU Mech Eng 2021; 19(4): 601–612.

14.

Rehman

Hussain

Rahman

, et al. Modified laplace based variational iteration method for the mechanical vibrations and its applications. Acta Mech et Autom 2022; 16(2): 98–102.

15.

Yang

, et al. Pull-in stability of a fractal MEMS system and its pull-in plateau. Fractals 2022; 30(9): 2250185. DOI: 10.1142/S0218348X22501857.

16.

Y Abd Elazem

. The carbon nanotube-embedded boundary layer theory for energy harvesting. FU Mech Eng 2022; 20(2): 211–235.

17.

Zuo

Liu

. Fractal approach to mechanical and electrical properties of graphene/sic composites. FU Mech Eng 2021; 19(2): 271–284.

18.

Ouakad

Sedighi

. Rippling effect on the structural response of electrostatically actuated single-walled carbon nanotube based NEMS actuators. Int J Non Linear Mech 2016; 87: 97–108.

19.

Sedighi

Daneshmand

Abadyan

. Modeling the effects of material properties on the pull-in instability of nonlocal functionally graded nano-actuators. Z Angew Math Mech 2016; 96(3): 385–400.

20.

Sedighi

Daneshmand

. Static and dynamic pull-in instability of multi-walled carbon nanotube probes by He’s iteration perturbation method. J Mech Sci Technol 2014; 28(9): 3459–3469.

21.

Zang

Zhou

Chang

, et al. Graphene and carbon nanotube (CNT) in MEMS/NEMS applications. Microelectron Eng 2015; 132: 192–206.

22.

Sedighi

Malikan

. Stress-driven nonlocal elasticity for nonlinear vibration characteristics of carbon/boron-nitride hetero-nanotube subject to magneto-thermal environment. Phys Scr 2020; 95(5): 055218.

23.

Tian

. Fractal pull-in stability theory for microelectromechanical systems. Front Phys 2021; 9: 606011.

24.

Anjum

. Nonlinear dynamic analysis of vibratory behavior of a graphene nano/microelectromechanical system. Math Methods Appl Sci 2020: 1–16. DOI: 10.1002/mma.6699.

25.

Anjum

. Homotopy perturbation method for N/MEMS oscillators. Math Methods Appl Sci 2020: 1–15. DOI: 10.1002/mma.6583 .

26.

. Seeing with a single scale is always unbelieving: from magic to two-scale fractal. Therm Sci 2021; 25(2): 1217–1219.

27.

Ain

. On two-scale dimension and its applications. Therm Sci 2019; 23(3): 1707–1712.

28.

Qian

. Two-scale thermal science for modern life–making the impossible possible. Therm Sci 2022; 26(3B): 2409–2412.

29.

Moatimid

Zekry

. Forced nonlinear oscillator in a fractal space. FU Mech Eng 2022; 20(1): 001–020.

30.

Feng

Niu

. He’s frequency formulation for nonlinear vibration of a porous foundation with fractal derivative. Int J Geomath 2021; 12(1): 14.

31.

Elias-Zuniga

. on two-scale dimension and its application for deriving a new analytical solution for the fractal Duffing’s equation. Fractals 2022; 30(3) 2250061.

32.

El-Dib

. A tutorial introduction to the two-scale fractal calculus and its application to the fractal Zhiber-Shabat oscillator. Fractals 2021; 29(08) 2150268.

33.

Liu

. A modified frequency-amplitude formulation for fractal vibration systems. Fractals 2022; 30(3) 2250046.

34.

Qian

. A fractal approach to the diffusion process of red ink in a saline water. Therm Sci 2022; 26(3B): 2447–2451.

35.

Liu

Wang

. A fractal Langmuir kinetic equation and its solution structure. Therm Sci 2021; 25(2): 1351–1354.

36.

Liu

, et al. A novel bond stress-slip model for 3-D printed concretes. Discrete Cont Dyn-A 2022; 15(7): 1669–1683.

37.

Liu

, et al. A fractal model for the internal temperature response of a porous concrete. Appl Comput Math-Bak 2022; 21(1): 71–77.

38.

Liu

Bai

, et al. Influence of pore defects on the hardened properties of 3D printed concrete with coarse aggregate. Addit Manuf 2022; 55: 102843.

39.

Sedighi

. Evans model for dynamic economics revised. Aims Math 2021; 6(9): 9194–9206.

40.

. Variational iteration method-a kind of non-linear analytical technique: some examples. Int J Non Linear Mech 1999; 34(4): 699–708.

41.

Mohammadian

. Application of the variational iteration method to nonlinear vibrations of nanobeams induced by the van der Waals force under different boundary conditions. Eur Phys J Plus 2017; 132(4): 169–181.

42.

. Homotopy perturbation method for bifurcation of nonlinear problems. Int J Nonlinear Sci Numer 2005; 6(2): 207–208.

43.

. Preliminary report on the energy balance for nonlinear oscillations. Mech Res Commun 2002; 29(2-3): 107–111.

44.

Zhang

Wan

. Application of the energy balance method to a nonlinear oscillator arising in the microelectromechanical system (MEMS). Curr Appl Phys 2011; 11: 482–485.

45.

Qie

Houa

. The fastest insight into the large amplitude vibration of a string. Rep Mech Eng 2021; 2(1): 1–5.

46.

. The simplest approach to nonlinear oscillators. Results Phys 2019; 15: 102546.

47.

. Modified lindstedt-poincare methods for some strongly non-linear oscillations part I: expansion of a constant. Int J Non Linear Mech 2002; 37(2): 309–314.

. Using parameter expansion method and min-max approach for the analytical investigation of vibrating micro-beams pre-deformed by an electric field. Adv Struct Eng 2013; 16(4): 681–687.

49.

Sedighi

Shirazi

. Vibrations of micro-beams actuated by an electric field via parameter expansion method. Acta Astronaut 2013; 85: 19–24.

50.

Jena

Chakraverty

. Residual power series method for solving time-fractional model of vibration equation of large membranes. J Appl Comput Mech 2019; 5(4): 603–615.

51.

Sadeghzadeh

Kabiri

. Application of higher order Hamiltonian approach to the nonlinear vibration of micro electro mechanical systems. Lat Am J Solids Struct 2016; 13(3): 478–497.

52.

. Hamiltonian approach to nonlinear oscillators. Phys Lett A 2010; 374(23): 2312–2314.

53.

. Iteration perturbation method for strongly nonlinear oscillations. J Vib Control 2001; 7: 631–642.

54.

. Some asymptotic methods for strongly nonlinear equations. Int J Mod Phys B 2006; 20(10): 1141–1199.

55.

. Asymptotic methods for solitary solutions and compactons. Abstr Appl Anal 2012; 130: 916793.

56.

Anjum

. Two-scale fractal theory for the population dynamics. Fractals 2021; 29(7) 2150182.

57.

. Variational principle for the generalized KdV-burgers equation with fractal derivatives for shallow water waves. J Appl Comput Mech 2020; 6: 735–740.

58.

Saeed

. A fractal modification of Chen-Lee-Liu equation and its fractal variational principle. Int J Mod Phys B 2021; 35(21): 2150214.

59.

Kou

Men

, et al. Fractal boundary layer and its basic properties. Fractals 2022; 30(9): 2250172. DOI: 10.1142/S0218348X22501729.

60.

Jiao

. Homotopy perturbation method for fractal Duffing oscillator with arbitrary conditions. Fractals 2022; 30(9): 2250165. DOI: 10.1142/S0218348X22501651.

61.

Liu

Zhang

, et al. Thermal oscillation arising in a heat shock of a porous hierarchy and its application. FU Mech Eng 2022; 20(3): 633–645.

. A fractional model and its application to heat prevention coating with cocoon-like hierarchy. Therm Sci 2022; 26(3): 2493–2498.

63.

Liu

. Fractal approach to the fluidity of a cement mortar. Nonlinear Eng 2022; 11(1): 1–5.

64.

. Fractal calculus and its geometrical explanation. Results Phys 2018; 10: 272–276.

. A study on the quintic nonlinear beam vibrations using asymptotic approximate approaches. Acta Astronautica 2013; 91: 245–250.

66.

. Homotopy perturbation method with two expanding parameters. Indian J Phys 2014; 88(2): 193–196.

67.

Wang

. Variational iteration method for solving integro-differential equations. Phys Lett A 2007; 367(3): 188–191.

68.

Wang

. A variational approach to nonlinear two-point boundary value problems. Comput Math Appl 2009; 58(11): 2452–2455.

, et al. Subcarrier-pairing-based resource optimization for OFDM wireless powered relay transmissions with time switching scheme. IEEE Trans Signal Process 2017; 65(5): 1130–1145.

70.

Wang

. Solving a class of variable order nonlinear fractional integral differential equations by using reproducing kernel function. Aims Math 2022; 7(7): 12935–12951.

71.

Chen

Wang

, et al. Solving two-sided fractional super-diffusive partial differential equations with variable coefficients in a class of new reproducing kernel spaces. Fractal Fract 2022; 6(9): 492.

72.

Yang

. Solitary waves of the variant Boussinesq-Burgers equation in a fractal dimensional space. Fractals 2022; 30(3) 2250056.

73.

Shen

, et al. Taylor series solution for fractal Bratu-type equation arising in electrospinning process. Fractals 2020; 28: 2050011.

Dynamic behaviors for the graphene nano/microelectromechanical system in a fractal space

Abstract

Keywords

Introduction

Fractal mathematical modeling

Basic ideas of the IPM

Application of IPM in a fractal space

Numerical experiments

Concluding remarks

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References