Analytical study of the vibrating double-sided quintic nonlinear nano-torsional actuator using higher-order Hamiltonian approach

Abstract

In this work, we investigate and apply higher-order Hamiltonian approach (HA) as one of the novelty techniques to find out the approximate analytical solution for vibrating double-sided quintic nonlinear nano-torsional actuator. Periodic solutions are analytically verified, and consequently, the relationship between the initial amplitude and the natural frequency are obtained in a novel analytical way. The HA is then extended to the second-order to find more accurate results. To show the accuracy and applicability of the technique, the approximated results are compared with the homotopy perturbation method and numerical solution. According to the numerical results, it is highly remarkable that the second-order approximate solutions produce better than previously existing results and almost similar in comparing with the numerical solutions.

Keywords

Higher-order Hamiltonian approach numerical method double-sided nano-torsional switch periodic solutions

Introduction

Nonlinear differential equations are mathematical equations that are significant in modeling physical, chemical, biological phenomena that occur in nature. Applications can be found in physics, ocean acoustics, ocean engineering, mathematics, fluid dynamics, mechanics, and finance. Examples include modeling mechanical vibration, heat, linear and nonlinear water waves, coastal ocean, propagation of nonlinear waves, lakes pollution, sound vibration, elasticity, and fluid dynamics, just to name a few. Although nonlinear DEs have a wide range of applications to real-world problems in science and engineering, the majority of DEs do not have analytical solutions. It is, therefore, important to be able to obtain an accurate solution numerically. The advancements of high-speed computers have made it possible to find numerical solutions to complex DEs while minimizing the time it requires to perform the computations. Many computational methods have been developed and implemented to successfully approximate solutions. Traditionally, mesh methods such as the finite difference method, finite element method, and boundary element method have been used. These methods require a mesh to connect nodes inside the computational domain or on the boundary. Complications of these methods include a slow rate of convergence, spatial dependence, instability, low accuracy, and difficulty of implementation in complex geometries. However, meshless approximation techniques using radial basis functions have been developed over the last several decades. These techniques are easy to implement, highly accurate, and truly meshless, which avoids troublesome mesh generation for high-dimensional problems. While this method along with other methods has been proven to be effective, a significant disadvantage to these methods is that they are not able to handle the solution of problems on a large scale. In this case, the matrices can become dense and are often poorly conditioned. Not only is stability an issue but the computational cost and memory allocation become very high thereby rendering the method ineffective. Furthermore, the solution is very sensitive to the choice of the shape parameter in the RBFs. To circumvent these issues, a number of efforts have been made in the literature in which their results have a good accuracy with the numerical solutions. Among these methods, a domain decomposition introduced by Main-Duy and Tran Cong,¹ a multi-grid approach in Ref. 2, compact support radial basis functions by Chen et al.,³ variational iteration techniques,^4–6 the greedy algorithm,⁷ local RBF method,⁸ semi inverse method,^9,10 iteration perturbation method,¹¹ variational principle,^12–16 Taylor series method¹⁷ and local methods such as the local Kansa method and local method of approximate particular solutions,¹⁸ He’s frequency formulation,¹⁹ and many more.^20–29

Nonlinear oscillators have been widely utilized to represent numerous physical systems particularly in applied sciences and engineering including the vibrations of plates and beams, the vibrations induced on different structures by fluid flow, the large amplitude oscillations of centrifugal governor systems, the free vibration of a restrained uniform beam undergoing large amplitudes of oscillation and carrying intermediate lumped mass, the oscillations of magneto-elastic mechanical systems or pendulum-like systems, the oscillations of pendulum-like systems or propagation of a short electromagnetic pulse in a nonlinear medium, and so on (see, for example, Refs. 30–34).

Electrostatic torsional actuators are extensively applied in nano-electromechanical system devices such as tunable torsional capacitors, torsional mirrors, and torsional radio frequency switches. Recently, several experimental studies have been conducted on the pull-in instability and dynamic behavior of nano-electromechanical system devices.^35–39

In the last two decades, significant progress is made in analytical approximate solutions of nonlinear oscillator’s differential equations.^40–42 Nowadays, several methods have been used in the literature to investigate the vibrating double-sided quintic nonlinear nano-torsional actuator such as variational iteration method,^43–45 energy balance method,^46,47 homotopy perturbation method,^48,49 global error minimization method,⁵⁰ homotopy analysis method,⁵¹ global residue harmonic balance method,⁵² and Hamiltonian approach (HA)^53–56 have been applied to investigate the governing nonlinear differential equations.

The main objective of this article is to get the accurate analytical natural frequency for the nonlinear vibration of double-sided-actuated nano-torsional switches with different actuation voltages as a second-order nonlinear differential equation up to second-order Hamiltonian approaches. The results obtained show that the proposed technique is convenient and very effective for the highly nonlinear governing equations of nonlinear beam vibration. The planned analytical technique proves that to obtain a highly accurate solution of micro-beam vibration, the two terms in series expansions are sufficient. The relationship between frequency and amplitude is then gained in an approximate analytical form. Also, a comparison is carried out between the obtained frequency responses of the systems and the numerical solutions.

The basic description of the Hamiltonian approach

A HA to nonlinear oscillators was suggested by He in 2010,⁵⁵ and recently, it has been advanced for the transverse vibration of a reinforced concrete pillar.⁵⁷ In order to clarify the basic idea of the HA, we consider the following nonlinear differential equation of an oscillatory system as^55,58

\ddot{u} + f (u) = 0, u (0) = A, \dot{u} (0) = 0.

(1)

For equation (1), a variational principle can be established easily, which reads⁵⁹

J (u) = \int_{0}^{T / 4} (- \frac{1}{2} {\dot{u}}^{2} + F (u)) d t

(2)

where the period of nonlinear oscillator is denoted by

T

\partial F / \partial u = f

. In equation (2),

F (u)

is the potential energy and

(1 / 2) {\dot{u}}^{2}

is its kinetic energy, so this equation is the least Lagrangian action, from which its Hamiltonian can be obtained immediately, which reads

H (u) = \frac{1}{2} {\dot{u}}^{2} + F (u) = constant

(3)

From equation (3), we have

\frac{\partial H (u)}{\partial A} = 0.

(4)

Introducing $\tilde{H} (u)$ as a new function, defined as in Refs. 55 and 60

\tilde{H} (u) = \int_{0}^{T / 4} (\frac{1}{2} {\dot{u}}^{2} + F (u)) d t = \frac{1}{4} T H

(5)

It is clear that

\frac{\partial \tilde{H} (u)}{\partial T} = \frac{1}{4} H

(6)

Rearranging equation (6), we get the following

\frac{\partial}{\partial A} (\frac{\partial \tilde{H} (u)}{\partial T}) = 0 or \frac{\partial}{\partial A} (\frac{\partial \tilde{H} (u)}{\partial (1 / ω)}) = 0.

(7)

Mathematical modeling

The schematic of a nano-torsional switch³⁸ has been shown in Figure 1 under double-sided electrostatically actuated voltages. The nano-torsional system has a spring torque coefficient $κ_{u}$ and rotational inertia $J$ .

Figure 1.

The configuration of a double-sided nano-torsional switch.

The initial gap of air is $g$ and an electrostatic force of attraction which originates from voltage $V_{1}$ and voltage $V_{2}$ from lower and upper plates, respectively, causes the nano-actuator to tilt.

For the nano-actuator system, the governing equation of motion can be expressed as below³⁸

\ddot{u} + β_{1 n} u + β_{2 n} u^{2} + β_{3 n} u^{3} + β_{4 n} u^{4} + β_{5 n} u^{5} + β_{0 n}, u (0) = A, \dot{u} (0) = 0

(8)

where the parameters

β_{0 n}, \dots, β_{5 n}

for the Casimir and Van der Waals effects have been described in the Appendix.

From equation (8), amplitude and frequency relationship of nonlinear oscillators can be obtained. The Hamiltonian equation for the current special problem can be written as

\tilde{H} (u) = \int_{0}^{T / 4} (\frac{1}{2} u^{' 2} + \frac{1}{2} β_{1 n} u^{2} + \frac{1}{3} β_{2 n} u^{3} + \frac{1}{4} β_{3 n} u^{4} + \frac{1}{5} β_{4 n} u^{5} + \frac{1}{6} β_{5 n} u^{6} + β_{0 n} u) d t

(9)

According to Ref. 38, the electrostatic torques per unit length, applied to the nano-actuator from the lower and upper plates, can be computed using a standard capacitance model.

Following Refs. 35 and 38, the electrostatic torque from the lower plate can be expressed as

M_{1 - e s} (θ) = \frac{ε_{0} w V_{1}^{2}}{2 θ^{2}} [\ln (\frac{g - L θ}{g}) + \frac{L θ}{g - L θ}]

(10)

and then, the electrostatic torque per unit length applied to the nano-actuator from the upper plate can be written as

M_{2 - e s} (θ) = \frac{ε_{0} w V_{2}^{2}}{2 θ^{2}} [\ln (\frac{g + L θ}{g}) - \frac{L θ}{g + L θ}]

(11)

In the same way, the vdW and the Casimir torques from the upper and lower surfaces can be, respectively, described in the form

M_{v d W} (θ) = \frac{A w L^{2}}{12 π g} [\frac{1}{{(g - L θ)}^{2}} - \frac{1}{{(g + L θ)}^{2}}]

(12)

M_{C} (θ) = \frac{π^{2} ℏ c w L^{2}}{1440 g^{2}} [\frac{3 g - L θ}{{(g - L θ)}^{3}} - \frac{3 g + L θ}{{(g + L θ)}^{3}}]

(13)

The index $n$ is 2 for the van der Waals force and 3 for the Casimir effect. According to Ref 35, Lin and Zhao introducing the following non-dimensional variables as:

τ = \sqrt{\frac{J}{κ_{θ}}} t, u = \frac{θ}{θ_{\max}}, a_{u - 1} = \frac{ε_{0} w L^{3} V_{1}^{2}}{2 κ_{θ} g^{3}}, a_{u - 2} = \frac{ε_{0} w L^{3} V_{2}^{2}}{2 κ_{θ} g^{3}}, b_{2} = \frac{A w L^{3}}{12 π κ_{θ} g^{4}}, b_{3} = \frac{π^{2} ℏ c w L^{3}}{1440 κ_{θ} g^{5}}

(14)

Hamiltonian approach approximation of first order

For the first order approximation, we assume the first approximate solution of equation (8) as:55 $ω$

u (t) = A \cos (ω t)

(15)

Substituting equation (15) into (9), we get

\tilde{H} (u) = - \frac{1}{8} π A^{2} ω^{2} + \frac{1}{8} π A^{2} β_{1 n} + \frac{2}{9} A^{3} β_{2 n} + \frac{3}{64} π A^{4} β_{3 n} + \frac{8}{75} A^{5} β_{4 n} + \frac{5}{192} π A^{6} β_{5 n} + A β_{0 n}

(16)

Setting

\begin{array}{l} \frac{\partial}{\partial A} (\frac{\partial \tilde{H} (u)}{\partial (1 / ω)}) = 0 & \Rightarrow - \frac{1}{4} π A ω^{2} + \frac{1}{4} π A β_{1 n} + \frac{2}{3} A^{2} β_{2 n} + \frac{3}{16} π A^{3} β_{3 n} + \frac{8}{15} A^{4} β_{4 n} + \frac{5}{32} π A^{5} β_{5 n} + β_{0 n} = 0. \end{array}}

(17)

Solving equation (17), an approximate frequency as a function of amplitude can be obtained as

ω_{H A} = \frac{\sqrt{75 π A^{5} β_{5 n} + 256 A^{4} β_{4 n} + 90 π A^{3} β_{3 n} + 320 A^{2} β_{2 n} + 120 π A β_{1 n} + 480 β_{0 n}}}{2 \sqrt{30 π} \sqrt{A}}

(18)

Hamiltonian approach approximation of second order

To improve the precision and accurateness of the HA, as the response of the system, we consider the following equation

u (t) = a \cos (ω t) + b \cos (3 ω t)

(19)

where the initial condition is

A = a + b

(20)

Substituting equation (19) into (9), we get

\begin{array}{l} \tilde{H} (u) = - \frac{1}{8} π ω^{2} (a^{2} + 9 b^{2}) + \frac{1}{8} π β_{1 n} (a^{2} + b^{2}) + \frac{2}{945} β_{2 n} (105 a^{3} + 63 a^{2} b + 243 a b^{2} - 35 b^{3}) \\ + \frac{1}{64} π β_{3 n} (3 a^{4} + 4 a^{3} b + 12 a^{2} b^{2} + 3 b^{4}) + \frac{8}{225225} β_{4 n} (3003 a^{5} + 6435 a^{4} b + 18590 a^{3} b^{2} \\ + 5590 a^{2} b^{3} + 10935 a b^{4} - 1001 b^{5}) + \frac{5}{192} π β_{5 n} (a^{6} + 3 a^{5} b + 9 a^{4} b^{2} + 6 a^{3} b^{3} + 9 a^{2} b^{4} + b^{6}) \\ + β_{0 n} (a - \frac{b}{3}) \end{array}}

(21)

Setting

\begin{array}{l} \frac{\partial}{\partial a} (\frac{\partial \tilde{H} (u)}{\partial (1 / ω)}) = 0 \\ \Rightarrow - \frac{1}{4} π a ω^{2} + \frac{1}{4} π a β_{1 n} + \frac{2}{945} β_{2 n} (315 a^{2} + 126 a b + 243 b^{2}) + \frac{1}{64} π β_{3 n} (12 a^{3} + 12 a^{2} b + 24 a b^{2}) \\ + \frac{8}{225225} β_{4 n} (15015 a^{4} + 25740 a^{3} b + 55770 a^{2} b^{2} + 11180 a b^{3} + 10935 b^{4}) \\ + \frac{5}{192} π β_{5 n} (6 a^{5} + 15 a^{4} b + 36 a^{3} b^{2} + 18 a^{2} b^{3} + 18 a b^{4}) + β_{0 n} = 0, \end{array}}

(22)

\begin{array}{l} \frac{\partial}{\partial b} (\frac{\partial \tilde{H} (u)}{\partial (1 / ω)}) = 0 \\ \Rightarrow \frac{1}{4} π b β_{1 n} + \frac{9}{4} π b ω^{2} + \frac{2}{945} β_{2 n} (63 a^{2} + 486 a b - 105 b^{2}) + \frac{1}{64} π β_{3 n} (4 a^{3} + 24 a^{2} b + 12 b^{3}) \\ + \frac{8}{225225} β_{4 n} (6435 a^{4} + 37180 a^{3} b + 16770 a^{2} b^{2} + 43740 a b^{3} - 5005 b^{4}) \\ + \frac{5}{192} π β_{5 n} (3 a^{5} + 18 a^{4} b + 18 a^{3} b^{2} + 36 a^{2} b^{3} + 6 b^{5}) - \frac{β_{0 n}}{3} = 0. \end{array}}

(23)

Getting a relation between different parameters from equations (20), (22), and (23) is not easy manually. Using Mathematica software program, the second-order algebraic equation set will be ready to solve to get the natural frequency

ω

and values of a and b, so the second-order approximation solution is given by equation (19) after knowing the values of the variables, some of the results are listed in Table 1 for the Van der Waals force and Table 2 for the Casimir effect.

Table 1.

Values of $a,$ $b$ , and $ω$ in the Van der Waals effect for some known values of amplitude $A$ .

$A$	$a$	$b$	$ω$
$0.45$	$0.452377$	$- 0.00237662$	$0.787592$
$0.65$	$0.660866$	$- 0.010866$	$0.715118$
$0.75$	$0.772796$	$- 0.022796$	$0.650417$
$0.85$	$0.904103$	$0.054103$	$0.540949$

Table 2.

Values of $a,$ $b$ , and $ω$ in the Casimir effect for some known values of amplitude $A$ .

$A$	$a$	$b$	$ω$
$0.35$	$0.352754$	$- 0.00275385$	$0.697767$
$0.45$	$0.457771$	$- 0.00777132$	$0.644485$
$0.50$	$0.512938$	$- 0.0129381$	$0.605165$
$0.60$	$0.642411$	$- 0.0424115$	$0.469677$

Numerical results and discussion

The Hamiltonian approach as a suggested technique is used to find the approximate analytical expressions for the displacement and frequency of vibrating double-sided quintic nonlinear nano-torsional actuator and compare the obtained results with those obtained from the HPM.³⁸ Also, the results from numerical calculations (RK4) are considered the reference for such comparisons.

According to Figures 2 and 3, it is observed that when the order of the suggested technique increases, more accurate results and higher agreement are gained. Hence, the second-order solution was the best in terms of computational efficiency and accuracy. The time history obtained for the initial condition is illustrated in Figures 2 and 3. It is seen that in the time domain, a very excellent correlation is still preserved.

Figure 2.

Comparison of the results obtained from the analytical solution (black line), HPM (blue line), and the numerical solution (green line).

Figure 3.

Comparison of the results obtained from the analytical solution (black line), HPM (blue line), and the numerical solution (green line).

In Figures 2 and 3, we compare the results of analytical and numerical solutions. From the previous, we can conclude the following:

Figure 2 shows the comparison between the analytical HA of second-order and numerical solutions gained from the Runge–Kutta method of fourth-order by considering the Van der Waals effect for $A = 0.35$ , $0.45$ , $0.75$ , and $0.85$ . It can be perceived that the results of our suggested technique agree exceptionally well with the numerical results and gives more accurate results than first-order homotopy perturbation method.³⁸

Figure 3 shows the comparison between analytical and numerical solutions by considering the Casimir effect for $A = 0.35$ , $0.45$ , $0.5$ , and $0.6$ ; it can be seen that there is an agreement between our results and the numerical results.

The obtained results in this work confirm that the HA of higher order is an effective and powerful technique for the solution of nonlinear differential equations in various fields of applied sciences and engineering. Hence, this method may be used to study physical problems with strong nonlinearity efficiently.

Conclusion

The HA of higher order as a modern powerful analytical method was employed successfully to obtain the frequency–amplitude relationship of the nonlinear nano-electromechanical system. The HA of second-order provides a direct and easy process for determining approximations of the periodic solutions. The idea of this research has important outcomes for biological, physical models, and their related complications. Thus, in describing physical, chemical, medical, social, and engineering processes, the conclusion is that the HA of higher order is helpful.

The results obtained along with the comparisons between analytical solutions and numerical ones show excellent agreement between analytical and numerical ones and do provide strong evidence to the utility of the approach and support the belief that the proposed method has a substantial future in solving a wide range of difficult problems that occur in the natural sciences and engineering. A simple solution procedure with high accuracy in the obtained results from the benchmark problem reveals the novelty, reliability, and wider applicability to the proposed analytical technique.

Footnotes

Acknowledgements

The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University, Saudi Arabia, for funding this work through research groups program under grant RGP.1/86/42.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hijaz Ahmad

Appendix

The defined parameters for Van der Waals force in equation (8) are as follows

\begin{matrix} β_{12} = 1 - \frac{2 a_{Θ - 2}}{3} - \frac{2 a_{Θ - 1}}{3} - 4 b_{2}, β_{22} = \frac{3 a_{Θ - 2}}{4} - \frac{3 a_{Θ - 1}}{4}, \\ β_{32} = - \frac{4 a_{Θ - 2}}{5} - \frac{4 a_{Θ - 1}}{5} - 8 b_{2}, β_{42} = \frac{5 a_{Θ - 2}}{6} - \frac{5 a_{Θ - 1}}{6}, \\ β_{52} = - \frac{6 a_{Θ - 2}}{7} - \frac{6 a_{Θ - 1}}{7} - 12 b_{2}, β_{02} = \frac{a_{Θ - 2}}{2} - \frac{a_{Θ - 1}}{2} \end{matrix}

The defined parameters for the Casimir effect in equation (8) are as follows

\begin{matrix} β_{13} = 1 - \frac{2 a_{Θ - 2}}{3} - \frac{2 a_{Θ - 1}}{3} - 16 b_{3}, β_{23} = \frac{3 a_{Θ - 2}}{4} - \frac{3 a_{Θ - 1}}{4}, \\ β_{33} = - \frac{4 a_{Θ - 2}}{5} - \frac{4 a_{Θ - 1}}{5} - 48 b_{3}, β_{43} = \frac{5 a_{Θ - 2}}{6} - \frac{5 a_{Θ - 1}}{6}, \\ β_{53} = - \frac{6 a_{Θ - 2}}{7} - \frac{6 a_{Θ u - 1}}{7} - 96 b_{3}, β_{03} = \frac{a_{Θ - 2}}{2} - \frac{a_{Θ - 1}}{2} \end{matrix}

where

a_{Θ - 1} = a_{Θ - 2} = 0.2

and

b_{2} = b_{3} = 0.01

References

Mai-Duy

Tran-Cong

. Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson’s equations. Eng Anal Boundary Elem 2002; 26(2): 133–156.

Shu

Ding

Yeo

. Local radial basis funcion-based differential quadrature method and its application to solve two-dimensional incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 2003; 192(7–8): 941–954.

Chen

Ganesh

Golberg

, et al. Multilevel compact radial functions based computational schemes for some elliptic problems. Comput Math Appl 2002; 43(3–5): 359–378.

J-H

Latifizadeh

. A general numerical algorithm for nonlinear differential equations by the variational iteration method. Int J Numer Methods Heat Fluid Flow 2020; 30(11): 4797–4810.

Ahmad

Khan

Durur

, et al. Analytical approximate solutions of diffusion equations arising in oil pollution. J. Ocean Eng. Sci 2021; 30(11): 4797–4810. doi:10.1016/j.joes.2020.05.002.

Nadeem

Ahmad

. Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients. Comput Math Appl 2019; 78(6): 2052–2062.

Hon

Schaback

Zhou

. An adaptive greedy algorithm for solving large RBF collocation problems. Numer Algorithms 2003; 32(1): 13–25.

Ahmad

Siraj-ul-Islam

Khaliq

AQM

. Local RBF method for multi-dimensional partial differential equations. ComputMath Appli 2017; 74(2): 292–324.

J-H

. Variational multi-scale finite element method for the two-phase flow of polymer melt filling process. Int J Numer Methods Heat Fluid Flow 2019; 30(3): 1407–1426.

10.

J-H

. Variational principle and periodic solution of the Kundu-Mukherjee-Naskar equation. Results Phys 2020; 17: 103031.

11.

El-Naggar

Ismail

. Analytical solution of strongly nonlinear Duffing oscillators. Alexandria Eng J 2016; 55(2): 1581–1585.

12.

J-H

. Generalized variational principles for buckling analysis of circular cylinders. Acta Mechanica 2019; 231(3): 899–906.

13.

J-H

Sun

. A variational principle for a thin film equation. J Math Chem 2019; 57(9): 2075–2081.

14.

J-H

. Lagrange crisis and generalized variational principle for 3D unsteady flow. Int J Numer Methods Heat Fluid Flow 2019; 30(3): 1189–1196.

15.

. A modified Li-He’s variational principle for plasma. Int J Numer Methods Heat Fluid Flow 2021; 31(5): 1369–1372. DOI: 10.1108/HFF-06-2019-0523.

16.

. A fractal variational theory for one-dimensional compressible flow in a microgravity space. Fractals 2020; 28(2): 2050024.

17.

J-H

. Taylor series solution for a third order boundary value problem arising in Architectural Engineering. Ain Shams Eng J 2020; 11(4): 1411–1414.

18.

Lee

Liu

Fan

. Local multiquadric approximation for solving boundary value problems. Comput Mech 2003; 30(5–6): 396–409.

19.

Jin

. A short review on analytical methods for the capillary oscillator in a nanoscale deformable tube. Math Meth Appl Sci 2020; 1‐8. DOI: 10.1002/mma.6321

20.

Ismail

. An analytical coupled homotopy-variational approach for solving strongly nonlinear differential equation. J Egypt Math Soc 2017; 25: 434–437.

21.

Ahmad

Rafiq

Cesarano

, et al. Variational iteration algorithm-I with an auxiliary parameter for solving boundary value problems. Earthline J Math Sci 2020; 3(2): 229–247.

22.

Hosen

Chowdhury

MSH

Ismail

Yildirim

, A modified harmonic balance method to obtain higher-order approximations to strongly nonlinear oscillators. J Interdiscip Math 2020; 23(7): 1325–1345.

23.

Ahmad

. Numerical solution of Painlevé II differential equation. Commun Numer Anal; 2019(1): 32–38.

24.

Cvticanin

Ismail

. Higher order approximate periodic solutions for the oscillator with strong nonlinearity of polynomial type. Eur Phys J Plus 2019; 134(6): 266.

25.

Alal Hosen

Ismail

Yildirim

, et al. A modified energy balance method to obtain higher-order analytical approximations to the oscillators with cubic and harmonic restoring force. J App Comp Mech 2020; 6(2): 320–331.

26.

Ahmad

Seadawy

Khan

, et al. Analytic approximate solutions for some nonlinear Parabolic dynamical wave equations. J Taibah Univ Sci 2020; 14(1): 346–358.

27.

Ahmad

. Variational iteration algorithm-II with an auxiliary parameter and its optimal determination. Nonlinear Sci Lett A 2018; 9(1): 62–72.

28.

Ahmad

. Variational iteration method with an auxiliary parameter for solving telegraph equations. J Nonlinear Anal Appl 2018; 2018(2): 223–232.

29.

Yokus

Kuzu

Demiroǧlu

. Investigation of solitary wave solutions for the (3 + 1) -dimensional Zakharov-Kuznetsov equation. Int J Mod Phys B 2019; 33(29).

30.

Nayfeh

Mook

. Nonlinear Oscillations. New York: Wiley, 1973.

31.

Elías-Zúñiga

. Exact solution of the cubic-quintic Duffing oscillator. Appl Math Model 2013; 37(4): 2574–2579.

32.

Kovacic

. The Duffing Equation : Nonlinear Oscillators and Their Behaviour. Chichester West Sussex: John Wiley & Sons, 2011.

33.

J-H

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

34.

Ain

. New promises and future challenges of fractal calculus: from two-scale thermodynamics to fractal variational principle. Therm Sci 2020; 24: 659–681.

35.

Lin

W-H

Zhao

Y-P

. Influence of damping on the dynamical behavior of the electrostatic parallel-plate and torsional actuators with intermolecular forces. Sensors 2007; 7(12): 3012–3026.

36.

Guo

J-G

Zhao

Y-P

. Influence of van der Waals and Casimir forces on electrostatic torsional actuators. J Microelectromechanical Syst 2004; 13(6): 1027–1035.

37.

Lin

W-H

Zhao

Y-P

. Casimir effect on the pull-in parameters of nanometer switches. Microsystem Tech 2005; 11: 80–85.

38.

Sedighi

Shirazi

. Dynamic pull-in instability of double-sided actuated nano-torsional switches. Acta Mechanica Solida Sinica 2015; 28(1): 91–101.

39.

Shabani

Tariverdilo

Rezazadeh

, et al.. Nonlinear vibrations and chaos in electrostatic torsional actuators. Nonlinear Anal Real World Appl 2011; 12(6): 3572–3584.

40.

Sedighi

Shirazi

Zare

. An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method. Int J Non-Linear Mech 2012; 47(7): 777–784.

41.

Sedighi

Shirazi

. A new approach to analytical solution of cantilever beam vibration with nonlinear boundary condition. J Comput Nonlinear Dyn 2012; 7(3): 034502.

42.

Sedighi

Shirazi

Zare

. Novel equivalent function for deadzone nonlinearity: applied to analytical solution of beam vibration using He’s parameter expanding method. Latin Am J Sol Structures 2012; 9(4): 443–452.

43.

Ahmad

Seadawy

Khan

. Numerical solution of Korteweg-de Vries-Burgers equation by the modified variational Iteration algorithm-II arising in shallow water waves. Physica Scripta 2020; 95(4): 045210.

44.

Wazwaz

. The variational iteration method for rational solutions for KdV, K (2, 2), Burgers, and cubic Boussinesq equations. J Comput Appl Math 2007; 207(1): 18–23.

45.

G-C

Austin

. The variational iteration method which should be followed. Nonlinear Sci Lett A 2010; 1(1): 1–30.

46.

Durmaz

Kaya

. High-order energy balance method to nonlinear oscillators. J Appl Math 2012; 2012: 1–7. DOI: 10.1155/2012/518684.

47.

Yildirim

Askari

Saadatnia

, et al.. Analysis of nonlinear oscillations of a punctual charge in the electric field of a charged ring via a Hamiltonian approach and the energy balance method. Comput Math Appl 2011; 62(1): 486–490.

48.

. Homotopy perturbation technique. Comput Methods Appl Mech Eng 1999; 178(3–4): 257–262.

49.

J-H

. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003; 135(1): 73–79.

50.

Ismail

Abu-Zinadah

. Analytic approximations to non-linear third order Jerk equations via modified global error minimization method. J King Saud Univ - Sci 2021; 33(1): 101219.

51.

Sedighi

Shirazi

. Using homotopy analysis method to determine profile for disk cam by means of optimization of dissipated energy. Int Rev Mech Eng 2011; 5: 941–946.

52.

Ismail

Abul-Ez

Farea

, et al.. Analytical approximations to nonlinear oscillation of nanoelectro-mechanical resonators. The Eur Phys J Plus 2019; 134(1): 47.

53.

Akbarzade

Kargar

. Application of the Hamiltonian approach to nonlinear vibrating equations. Math Comput Model 2011; 54: 2504–2514.

54.

Yildirim

Saadatnia

Askari

. Application of the Hamiltonian approach to nonlinear oscillators with rational and irrational elastic terms. Math Comput Model 2011; 54(1–2): 697–703.

55.

J-H

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

56.

Sadeghzadeh

Kabiri

. Application of higher order hamiltonian approach to the nonlinear vibration of micro electro mechanical systems. Latin Am J Sol Structures 2016; 13(3): 478–497.

57.

F-Y

C-H

Zhang

J-J

, et al.. A fractal Boussinesq equation for nonlinear transverse vibration of a nanofiber-reinforced concrete pillar. Appl Math Model 2020; 82: 437–448.

58.

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

59.

J-H

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

60.

J-H

. Determination of limit cycle by Hamiltonian approach for strongly nonlinear oscillators. Int J Nonlinear Sci Numer Simulation 2010; 11: 1097–1101.