Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method

Gamal M. Ismail; Alwaleed Kamel; Abdulaziz Alsarrani

doi:10.1515/phys-2024-0011

Article Open Access

Studying nonlinear vibration analysis of nanoelectro-mechanical resonators via analytical computational method

Gamal M. Ismail , Alwaleed Kamel and Abdulaziz Alsarrani

Published/Copyright: April 19, 2024

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From the journal Open Physics Volume 22 Issue 1

Abstract

Periodic behaviour analysis of nano/microelectromechanical systems (N/MEMS) is an important area due to its numerous prospective applications in micro instruments. The intriguing and unique qualities of these systems, notably their tiny size, batch manufacturing, low power consumption, and great dependability have piqued the attention of academics and enterprises in using these structures to manufacture various microdevices. This article presents the parameter expansion method (PEM) to obtain the approximate solutions of N/MEMS. The present approach, as well as its speed and simplicity in providing analytical solutions that converge quickly to the exact numerical ones, distinguishes this study. The PEM has the benefit of immediately providing analytical solutions to nonlinear differential equations while avoiding costly calculations. Furthermore, in terms of establishing numerous terms of semi-analytic solutions, this approach is very faster and superior to other established analytical techniques in the literature.

Keywords: nanoelectro-mechanical resonators; nonlinear vibration; analytical method; cantilever beam; numerical solution

1 Introduction

The scientific phenomena frequently take the form of nonlinear types. Nonlinear differential equations (NDEs) are used in various fields to express numerous problems related to mathematics, biology, engineering, and physics. In particular, most of the branches can be modelled by using derivatives in NDEs [1,2,3,4]. The nano/microelectromechanical system (N/MEMS) has sparked global interest, and its applications have transformed technologies in a wide range of sophisticated industries, including wearable sensors, 5G communication technology, aircraft, and energy harvesting. When the applied force is sufficiently strong, however, the pull-in instability occurs, and dependable functioning is prohibited. MEMS are of significant attention and have been the topic of various studies.

MEMS are widely used in a variety of disciplines, including medicine, industry, robotics, aerospace, automotive, and many others [5,6,7,8]. MEMS are also widely employed in many engineering domains, including optical and bio-medical engineering, and are widely used in many applications, such as micro-pumps, accelerometers, micro-switches, and so on [8]. MEMS is a technology that integrates computers with small mechanical devices contained in semiconductor chips such as gears and actuators. These systems can detect, control, and trigger mechanical processes on the microscale, and they can work alone or in groups to produce effects on the macroscale. MEMS devices, which need minimal mechanical components and low-voltage levels for actuation, are rapidly expanding in current technology [9,10,11]. Numerous researches works in the literature deal with nonlinear differential equations in diverse sectors of science and engineering, therefore emphasizing the relevance of mathematical calculations was critical. Many nonlinear differential equations can be statistically studied, but only a handful can be directly solved. Numerous approximation strategies have been employed in the literature to determine the interaction between frequency and amplitude. The perturbation technique, which was commonly used to obtain approximate analytic solutions to nonlinear differential equations, was the most versatile tool in evaluating nonlinear engineering issues. The determination of periodic solutions of nonlinear oscillators is an important subject in physical nonlinear research. Traditional analytical approaches were utilized to address nonlinear differential equations, which are often employed to solve nonlinear oscillators with modest parameters. There are several powerful analytical and approximate methods have been studied to solve a large number of NDEs with various assumptions such as variational iteration method [12,13], modified multistage decomposition method [14], homotopy perturbation method [15,16,17,18,19], He’s frequency formulation [20,21], generalized residual power series method [22], harmonic balance method [23,24], Exp-function method [25], Laplace variational iteration method [26], reduced differential transform method [27], variational iteration method Padé technique [28], energy balance method (EBM) [29,30], global residue harmonic balance method (GRHBM) [31,32], Gamma function method [33,34], global error minimization method [35], Levenberg–Marquardt backpropagation neural networks [36], Hamiltonian approach [37], parameter expanding method (PEM) [38,39,40,41,42], and finite element method [43,44,45]. Recently, refs. [46,47] proposed a variational principle for a fractal N/MEMS, where the new variational theory opens a new direction in fractal in these systems, it provides an excellent physical knowledge of the iteration approach, and variational numerical techniques always have a conservation scheme with a fast rate of convergence. Many analytical methods compatible with nonlinear fractional differential equations can be constructed, and the results simulated using other existing numerical and analytical techniques in the literature. The current work provides considerable insight and critical analysis of the dynamic features of nano-scaled structures, as well as the essential guidelines for assessing the dynamic response of nano-components in advanced MEMS/NEMS. The advantage of the PEM is that it provides direct analytical solutions to nonlinear differential equations without requiring costly computations. Furthermore, this technique is faster and superior to other analytical approaches in the literature in developing a variety of analytic solutions.

The goal of this study is to find an analytical approximation solution to the free vibration equation that arises in MEMS. To show the efficiency and accuracy of the current technique, the acquired findings are compared with the EBM [30], GRHBM [32], and numerical solutions. The current technique produces more dependable findings for the current situation. Finally, it is demonstrated that the PEM has significant promise and that it might be applied to other substantially nonlinear situations.

2 Basic concept of the PEM

Consider the following differential equation:

(1) u ̈ + α u + β f ( u , t ) = 0 ,

where α and β are the coefficients of u and the nonlinear function f ( u , t ) and will be defined in Eqs. (3) and (4).

According to PEM [48,49], the solution and coefficients of Eq. (1) are expanded in a similar way:

(2) u ( t ) = u 0 ( t ) + p u 1 ( t ) + p 2 u 2 ( t ) + … ,

(3) α = ω 2 + p ω 1 + p 2 ω 2 + … ,

(4) β = p a 1 + p 2 a 2 + … .

Inserting Eqs. (2)–(4) into Eq. (1) and then equating the terms in powers of p , we obtain

(5) p 0 : u ̈ 0 + ω 2 u 0 = 0 ,

(6) p 1 : u ̈ 1 + ω 2 u 1 + ω 1 u 0 + a 1 f ( u 0 , t ) = 0 .

The solutions of Eq. (5) is

(7) u 0 = A cos ( ω t ) .

By substituting Eq. (7) into Eq. (6), we obtain

(8) p 1 : u ̈ 1 + ω 2 u 1 + ω 1 A cos ( ω t ) + a 1 f ( A cos ( ω t ) , t ) = 0 .

We utilize the Fourier expansion series to achieve the secular term as follows:

(9) ( A cos ( ω t ) , t ) = ∑ k = 0 ∞ b 2 k + 1 cos ( ( 2 k + 1 ) ω t ) .

Inserting (9) into (8) yields:

(10) p 1 : u ̈ 1 + ω 2 u 1 + ( ω 1 A + a 1 b 1 ) cos ( ω t ) = 0 .

We examine the following to avoid using secular terms:

(11) ω 1 A + a 1 b 1 = 0 .

Putting p = 1 in Eqs. (3) and (4), we have:

(12) ω 1 = α − ω 2 ,

(13) a 1 = β .

By inserting Eqs. (12) and (13) into Eq. (11), we shall attain the oscillator’s first-order approximation frequency (1). It is worth noting that, according to Eqs. (4) and (13), we can discover that a i ≠ 0 for all i = 1 , 2 , 3 , 4 , …

3 Mathematical model

In this section, we show a severely nonlinear issue with enormous motion amplitudes and physical parameters by considering a nano resonator made up of a moving electrode and two stationary substrates. The mobile electrode might be thought of as a clamped-clamped or cantilever beam with a rectangular cross section that is implanted between two fixed substrates [30,32]. Figure 1 depicts a schematic for a clamped-clamped resonator.

(14) ( a 2 − 4 a 4 u 2 + 6 a 6 u 4 − 4 a 8 u 6 + a 10 u 8 ) u ̈ + K 1 u + K 2 u 3 + K 3 u 5 + K 4 u 7 + K 5 u 9 + K 6 u 11 = 0 ,

(15) u ( 0 ) = A , u ̇ ( 0 ) = 0 .

Figure 1

Electrostatically actuated microbeam geometry.

The coefficients of K 1 − K 6 , a 0 − a 10 , b 1 − b 5 , and c 1 − c 5 are listed in the Appendix.

To normalize Eq. (14), the following nondimensional parameters are used

(16) w = w ˆ g , x = X l , τ = t E ˆ I ρ p h l 4 , α = 6 g 0 h , α vdW = a b l 4 6 π g 0 4 E ˆ I , α Ca = π 2 h c b l 4 240 g 0 5 E ˆ I δ = μ b h λ 2 E ˆ I , γ = 0.65 g b , β = ε 0 b V 2 L 4 2 g 0 3 E ˆ I , N = N ˆ l 2 E ˆ I .

According to Fu et al. [50], assuming w ( ξ , τ ) = ϕ ( ξ ) u ( τ ) where ϕ ( ξ ) is the first eigen mode of the clamped-clamped beam, which can be written as ϕ ( ξ ) = 16 ξ 2 ( 1 − ξ 2 ) , and u ( τ ) constricts the time-dependent part of the solution.

Rewrite Eq. (14) in the following form:

(17) u ̈ + K 1 a 2 u + 1 − 4 a 4 a 2 u 2 u ̈ + 6 a 6 a 2 u 4 u ̈ − 4 a 8 a 2 u 6 u ̈ + a 10 a 2 u 8 u ̈ + K 1 a 2 u + K 2 a 2 u 3 + K 3 a 2 u 5 + K 4 a 2 u 7 + K 5 a 2 u 9 + K 6 a 2 u 11 = 0 .

Assume that the solution is a power series in p :

(18) u = u 0 + p u 1 + p 2 u 2 + … ,

let

(19) K 1 / a 2 = ω 2 + p ω 1 + p 2 ω 2 + … ,

(20) 1 = p b 1 + p 2 b 2 + … .

Substituting Eqs. (18)–(20) into Eq. (17) and the coefficient of p should be zero, then a set of linear differential equations was created.

(21) u ̈ 0 + ω 2 u 0 = 0 ,

(22) u ̈ 1 + ω 2 u 1 + b 1 K 2 a 2 u 0 3 + b 1 K 3 a 2 u 0 5 + b 1 K 4 a 2 u 0 7 + b 1 K 5 a 2 u 0 9 + b 1 K 6 a 2 u 0 11 − 4 a 4 b 1 a 2 u 0 2 u ̈ 0 + 6 a 6 b 1 a 2 u 0 4 u ̈ 0 − 4 a 8 b 1 a 2 u 0 6 u ̈ 0 + a 10 b 1 a 2 u 0 8 u ̈ 0 = 0 .

We achieved the following conclusions by solving Eqs. (21) and (22) using the initial conditions stated in Eq. (15), and we obtain

(23) u 0 = A cos ( ω t ) ,

(24) u 1 = 1 8 ω 2 165 A 11 b 1 K 6 512 a 2 + 21 A 9 b 1 K 5 64 a 2 − 21 a 10 A 9 b 1 ω 2 64 a 2 + 21 A 7 b 1 K 4 64 a 2 + 21 a 8 A 7 b 1 ω 2 16 a 2 + 5 A 5 b 1 K 3 16 a 2 − 15 a 6 A 5 b 1 ω 2 8 a 2 + A 3 b 1 K 2 4 a 2 + a 4 A 3 b 1 ω 2 a 2 × cos ( 3 t ω ) + 1 24 ω 2 165 A 11 b 1 K 6 1024 a 2 + 9 A 9 b 1 K 5 64 a 2 − 9 a 10 A 9 b 1 ω 2 64 a 2 + 7 A 7 b 1 K 4 64 a 2 + 7 a 8 A 7 b 1 ω 2 16 a 2 + A 5 b 1 K 3 16 a 2 − 3 a 6 A 5 b 1 ω 2 8 a 2 × cos ( 5 t ω ) + 1 48 ω 2 55 A 11 b 1 K 6 1024 a 2 + 9 A 9 b 1 K 5 256 a 2 − 9 a 10 A 9 b 1 ω 2 256 a 2 + A 7 b 1 K 4 64 a 2 + a 8 A 7 b 1 ω 2 16 a 2 × cos ( 7 t ω ) + 1 80 ω 2 11 A 11 b 1 K 6 1024 a 2 + A 9 b 1 K 5 256 a 2 − a 10 A 9 b 1 ω 2 256 a 2 cos ( 9 t ω ) + 1 120 ω 2 A 11 b 1 K 6 1024 a 2 cos ( 11 t ω ) .

with the nonlinear frequency–amplitude relationship:

(25) ω 1 = ( Δ 1 + Δ 2 ) / Δ 3 ,

where

Δ 1 = − A 2 ( 231 a 2 A 8 K 6 − 252 a 10 A 6 K 1 + 252 a 2 A 6 K 5 + 1120 a 8 A 4 K 1 + 280 a 2 A 4 K 4 ) , Δ 2 = − A 2 ( − 1 , 920 a 6 A 2 K 1 + 320 a 2 A 2 K 3 + 1 , 536 a 4 K 1 + 384 a 2 K 2 ) , Δ 3 = 4 a 2 ( 63 a 10 A 8 − 280 a 8 A 6 + 480 a 6 A 4 − 384 a 4 A 2 + 128 a 2 ) .

Finally, we obtain the first-order approximation provided by putting Eqs. (23) and (24) into Eq. (18):

4 Results and discussion

To examine the accuracy of the parameter-expansion method, we plot the analytical approximate solutions with the numerical solutions and those in the literature, for example, the EBM [30] and the GRHBM [32]. The calculations are plotted for the values of Van der Waals parameter α vdW by choosing the given values of parameters δ = 0 and γ = 0 in the first four figures while δ = 0.65 and γ = 0.65 at the last six figures, respectively, and for different values of α vdW , α Ca , β , and A , where the values N = 10 and α = 24 in the all cases for a clamped-clamped beam. The comparison results are shown in Figure 2. Figure 2 shows that the numerical findings from the fourth-order Runge–Kutta technique correspond quite well with the analytical approximation, demonstrating the high precision of the solutions utilizing the suggested technique. The analytical solutions show that the second term in series expansions is enough to obtain a very accurate solution to the current model. Figure 2 shows a comparison of different beginning frequencies of micro-beams estimated using different approaches, revealing that the PEM results correspond quite well with the numerical and experimental results reported in the literature. Also, the PEM performance implies that it might be used for more sophisticated nonlinear differential equations. We may therefore infer that the current technique is a more powerful computational methodology for analysing nonlinear problems than other known methods currently in use.

Figure 2

Comparison of the present analytical solution, EBM [30], GRHBM [32], and the numerical solution. (a) A = 0.1, β = 25, α _vdW = 0, α _Ca = 0, (b) A = 0.3, β = 25, α _vdW = 0, α _Ca = 0, (c) A = 0.1, β = 100, α _vdW = 0, α _Ca = 0, (d) A = 0.3, β = 100, α _vdW = 0, α _Ca = 0, (e) A = 0.1, β = 25, α _vdW = 100, α _Ca = 0, (f) A = 0.1, β = 25, α _vdW = 25, α _Ca = 25, (g) A = 0.2, β = 25, α _vdW = 100, α _Ca = 0, (h) A = 0.4, β = 25, α _vdW = 0, α _Ca = 0, (i) A = 0.4, β = 25, α _vdW = 25, α _Ca = 25, (j) A = 0.5, β = 25, α _vdW = 25, α _Ca = 25.

5 Conclusion

In this article, the PEM was employed to obtain the approximate solution of a nanoelectro mechanical resonator system. The frequency–amplitude relationships are achieved in the closed forms. The high accuracy of the PEM is presented by comparing the current solutions with the EBM [30], GRHBM [32], and the numerical solutions. The results show that the PEM is very useful in analysing nonlinear oscillations. Finally, we can conclude that the PEM can produce the approximate analytic solution and the corresponding quite accurate frequency with the first-order approximation. The suggested approach in this study is superior to the traditional perturbation method since it does not rely on small parameter assumptions. The new solution’s correctness is validated by comparing the acquired findings to previously published results in the literature, as well as the numerical solution. Excellent agreement was found between the present and numerical solution, while better results have been obtained as compared to other techniques available in the literature.

Acknowledgments

The researchers wish to extend their sincere gratitude to the deanship of scientific research at the Islamic University of Madinah for the support provided to the post-publishing program.

Funding information: The deanship of scientific research at the Islamic University of Madinah.
Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
Conflict of interest: The authors state no conflict of interest.

Appendix

The coefficients of K 1 − K 6 , a 0 − a 10 , b 1 − b 5 and c 1 − c 5 in Eq. (14) are as follows:

K 1 = ( 1 + δ ) b 1 − χ typ Nc 1 − 8 α Ca a 2 − 6 α vdw a 2 − 4 β a 2 − 2 γ β a 2 , K 2 = − 4 ( 1 + δ ) b 2 + 4 χ typ Nc 2 − χ typ α c 1 a 0 − 8 α Ca a 4 + 4 α vdw a 4 + 8 β a 4 + 6 γ β a 4 , K 3 = 6 ( 1 + δ ) b 3 − 6 χ typ Nc 3 + 4 χ typ α c 2 a 0 + 2 α vdw a 6 − 4 β a 6 − 6 γ β a 6 , K 4 = − 4 ( 1 + δ ) b 4 + 4 χ typ Nc 4 − 6 χ typ α c 3 a 0 + 2 γ β a 8 , K 5 = ( 1 + δ ) b 5 − χ typ Nc 5 + 4 χ typ α c 4 a 0 , K 6 = − χ typ α c 5 a 0 .

a 0 = ∫ 0 1 ϕ ′ 2 d x , a 1 = ∫ 0 1 ϕ d x , a 2 = ∫ 0 1 ϕ 2 d x , a 3 = ∫ 0 1 ϕ 3 d x , a 4 = ∫ 0 1 ϕ 4 d x , a 5 = ∫ 0 1 ϕ 5 d x , a 6 = ∫ 0 1 ϕ 6 d x , a 7 = ∫ 0 1 ϕ 7 d x , a 8 = ∫ 0 1 ϕ 8 d x , a 9 = ∫ 0 1 ϕ 9 d x , a 10 = ∫ 0 1 ϕ 10 d x .

b 1 = ∫ 0 1 ϕ ϕ ( i v ) d x , b 2 = ∫ 0 1 ϕ 3 ϕ ( i v ) d x , b 3 = ∫ 0 1 ϕ 5 ϕ ( i v ) d x , b 4 = ∫ 0 1 ϕ 7 ϕ ( i v ) d x , b 5 = ∫ 0 1 ϕ 9 ϕ ( i v ) d x ,

c 1 = ∫ 0 1 ϕ ϕ ′ ′ d x , c 2 = ∫ 0 1 ϕ 3 ϕ ′ ′ d x , c 3 = ∫ 0 1 ϕ 5 ϕ ′ ′ d x , c 4 = ∫ 0 1 ϕ 7 ϕ ′ ′ d x , c 5 = ∫ 0 1 ϕ 9 ϕ ′ ′ d x .

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Received: 2023-12-09

Revised: 2024-02-29

Accepted: 2024-03-13

Published Online: 2024-04-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2024-0011

Keywords for this article

nanoelectro-mechanical resonators; nonlinear vibration; analytical method; cantilever beam; numerical solution

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