A universal application of the general integral transform: New technique for the analysis of nonlinear oscillator systems

Abstract

The wide applicability of a nonlinear oscillator is the main reason why it has gained so much attention from mathematicians and scientists, while it is difficult to be solved both numerically and theoretically. Variational iteration method is one of the numerical approaches which are used to investigate these nonlinear systems. In this paper, we present a nonlinear oscillator arising in a micro-electromechanical system as an example and derive its analytical solution by He’s frequency formula method and variational iteration method coupled with a new general integral transformation. The Lagrange multiplier is formulated and the new iterative format for the correction functional of the variational iteration method is given by the general integral transformation. The approximate nonlinear frequencies are achieved by this new technique and compared with the numerical ones obtained by the Runge–Kutta method. The new general integral transformation has the same essential qualities as those for Laplace and Fourier transform, and it becomes a promising tool to nonlinear problems when coupled with the variational iteration method.

Introduction

Nonlinear systems are ubiquitous in nature, and the laws controlling their behaviors are fundamental in engineering machinery and transportation equipment.^1–4 These systems can exhibit bifurcations, chaos, counterintuitive, or unexpected dynamic behaviors that most researchers are interested in.^5–7 The process of vibration is the energy conversion process between the kinetic energy and potential energies of the system. The solutions of nonlinear vibration equations can provide deep insights into the essence of general relationships applicable not only to physical phenomena, chemical systems, and geophysical systems but also to biological systems, so it is very important to study the analytical or approximate solutions.

The behaviors of complex systems are usually modeled by nonlinear dynamical systems, while lots of nonlinear equations cannot be precisely solved. In engineering applications, a simple calculation is always required and how to solve a nonlinear equation quickly is important. Therefore, identifying approximate solutions that meet certain accuracy demands is a main task, and many efforts have been made toward the studying of numerical analysis methods. There exist many analytical techniques for discussing the characteristics of nonlinear systems, for example, He’s frequency formula method,^8,9 the homotopy analysis method,^10,11 the homotopy perturbation method,^12–17 Taylor series method,^18,19 the bisection method, ²⁰ and Newton–Raphson method.^21–24 The variational iteration method was used by many researchers to solve linear and nonlinear models and has been proved to be reliable and efficient for a wide variety of scientific applications.^25,26 It can be employed for numerical computations with a high degree of accuracy as its reliability and validity,^27,28 where the identification of Lagrange multiplier was difficult. The series solution obtained by the variational iteration method converges rapidly, but the calculation is complex in most issues.

Abbasbandy successfully applied He’s homotopy perturbation method to compute Laplace transform. ²⁹ Watugala employed the Sumudu transformation to investigate differential equations. ³⁰ Basit et al. proposed the formable transform to study a nonlinear oscillator.^31–34 Farooq et al. have reported progress in understanding the Cattaneo–Christov heat and mass fluxes through a nonlinear stretching cylinder. ³⁵ Rehman et al. studied modified Laplace transformation based on the variational iteration method for the mechanical vibrations. ³⁶ Burqan et al. proposed a novel approach for obtaining precise solutions of the time-fractional Navier–Stokes equation utilizing the Laplace transform and power series method. ³⁷ Inspired by the newly defined generalized integral transformation, ³⁸ we attempt to combine the variational iteration method with this integral transformation to investigate the numerical solutions of nonlinear oscillators. This iteration scheme integrates several existing iterative methods and has the potential to generate numerous new iteration schemes. We will take the micro-electromechanical system (MEMS) as an example for detailed explanation.

In recent decades, the MEMS has numerous applications in engineering research as a tiny mechanical module driven by electricity. It possesses several attractive advantages that can make up for the shortcomings of traditional electronic components, including high intelligence, low energy consumption, small size, and high integration. ³⁹ So far, there are a large number of literature studies on the dynamic characteristic analysis of MEMS models. Tian et al. established a fractal MEMS system.^40,41 He proposed a variational principle for both analytical and numerical analyses of the MEMS system. ⁴²

Consider the following nonlinear oscillator arising in a MEMS system

$y^{″}+y+\frac{\theta }{y-1}=0,y\left(0\right)=0,y^{\prime }\left(0\right)=0,\theta >0$

(1)

As shown in Figure 1, it can be described the movement of the wire as a point mass. Here y is the dimensionless distance, $y=\frac{u}{b}$ , $y<1$ , $\theta$ represents a voltage-related parameter, $\theta =\frac{2\times {10}^{-7}{i}_1{i}_{2}h}{kb}$ , k is the stiffness of the spring, i₁ and i₂ are the direct currents through the wires, and h is the length of a current-carrying wire. When $\theta$ < 0.203632188, the system exhibits periodic behavior. ⁴³

Figure 1.

MEMS system with a current-carrying wire.

Seeking out a high-precision solution for an MEMS model is an extremely important and meaningful research project. In this work, high-precision solutions are obtained by He’s frequency formula method and a new general integral transformation method, both of which are efficient and easy to operate.

A frequency formula method

He’s frequency formula has been considered as a simple and accurate frequency–amplitude expression based on an ancient Chinese mathematical algorithm in which the frequency–amplitude of various nonlinear oscillators was found.

A general nonlinear oscillator has the following form:

$x^{″}+f\left(x\right)=0,x\left(0\right)=A,x^{\prime }\left(0\right)=0$

(2)

He’s frequency formula is as follows:

${\omega }_{He}={\sqrt{\frac{df\left(x\right)}{dx}}|}_{x=\frac{A}{2}}$

(3)

The frequency formula method is mainly used to find the frequency form of differential equations with non-zero initial conditions. We attempt to convert the zero initial conditions into non-zero initial conditions by assuming $x=A-y$ and substituting it into equation (1), resulting in the following form:

$x^{″}+x-A+\frac{\theta }{1-A+x}=0,x\left(0\right)=A,x^{\prime }\left(0\right)=0$

(4)

the nonlinear term $\frac{\theta }{1-A+x}$ can be changed to $\frac{\theta }{1-A}·\frac{1}{1-\frac{x}{A-1}}$ . Using the truncated Taylor series $\frac{1}{1-x}=1+x+x^2+\cdots$ , the nonlinear term could be expanded into the following form:

$\frac{\theta }{1-A+x}=\frac{\theta }{1-A}\left(1+\frac{x}{A-1}+\frac{x^2}{{\left(A-1\right)}^2}+\cdots \right)$

(5)

and its convergence radius is $\left|x\right|<|1-A|$ .

Equation (4) becomes as follows:

$x^{″}+x+\frac{\theta }{1-A}\left(\frac{x}{A-1}+\frac{x^2}{{\left(A-1\right)}^2}+\cdots \right)-A+\frac{\theta }{1-A}=0$

(6)

for eliminating the constant term, this equation $-A+\frac{\theta }{1-A}=0$ needs to be assumed, and it gets $\theta =A\left(1-A\right)$ .

He’s frequency formula ${\omega }_{He}$ of equation (4) is approximated as follows:^8,9

${\omega }_{He}={\sqrt{\frac{d\left(x-A+\frac{\theta }{1-A+x}\right)}{dx}}|}_{x=\frac{A}{2},\theta =A\left(1-A\right)}=\sqrt{\frac{5A^2-8A+4}{{\left(A-2\right)}^2}}$

(7)

The frequency formula method is a simple and intuitive calculation way to discuss the effect of the parameter A on the frequency properties. Table 1 displays the values of the frequency obtained by the frequency formula method and the numerical method at various values of the parameters. The relative error reveals a high correctness of the frequency obtained by the frequency formula method when $\theta$ is small. The frequency formula method has vast potential for studying nonlinear problems.

Table 1.

Comparison of frequencies.

A	$\theta$	$\omega$ of equation (7)	${\omega }_{exact}$	Relative error (%)
0.05	0.0475	0.9747	0.9740	0.0719
0.10	0.0900	0.9488	0.9362	1.3459
0.15	0.1275	0.9225	0.9018	2.2954
0.20	0.1600	0.8958	0.8597	4.1991
0.25	0.1875	0.8690	0.8215	5.7821
0.27	0.1971	0.8582	0.8012	7.1143

The approximate solution of the MEMS system is as follows:

$y=A-\mathit{Acos}\left(\sqrt{\frac{5A^2-8A+4}{{\left(A-2\right)}^2}}t\right)$

(8)

As $\theta =A\left(1-A\right)$ , it is easy to find that $\theta \le 0.25.$ For equation (1), there exists a periodic solution when $\theta <$ 0.203632188. The phase trajectories will be open and the pull-in phenomenon will occur when $\theta >0.203632188$ . We know that it is impossible to discuss the frequency of equation (1) when $0.203632188<\theta <0.25.$ Thus we use another new method, that is, the integral transform method.

A new general integral transform

Recently, the integral transform has been widely used in numerical simulation of its fast convergence and simple operation. It has great practical implications in different real-life challenges of engineering, such as electrical engineering, industrial engineering, mechanical engineering, and civil engineering. There exist various integral transforms used in mathematical theory and for solving differential equations, including the Laplace transform and Fourier transform. In some cases, choosing an appropriate integration transformation can make the analysis simple. The choice of transform becomes much importance when we study different problems. He et al. proposed a general integral transformation with others being its special cases to overcome the problem of how to choose a suitable one. ³⁸ It can be used for various nonlinear problems and opens a new window for studying nonlinear science.

He’s integral transformation of an input function $f\left(t\right)$ ( $t\ge 0$ ) is defined as follows:

$\mathcal{H}\left[f\left(t\right)\right]={\int }_0^{+\infty }{e}^{-s^{n}t}\gamma \left(s\right)f\left(t\right)dt\triangleq \mathcal{H}\left(s\right)$

(9)

here $\mathcal{H}\left(s\right)$ is the image of $f\left(t\right)$ , $\mathcal{H}$ is the integral transformation operator, and $s$ represents the transformation variable that is from the complex domain. The kernel of He’s integral transformation is $e^{-s^{n}t}\gamma \left(s\right)$ ( $\gamma \left(s\right)\ne 0)$ . The variable $n$ is from the integer range.

The Laplace transformation and Fourier transformation are defined, respectively, as follows:

$\mathcal{L}\left[f\left(t\right)\right]={\int }_0^{+\infty }{e}^{-st}f\left(t\right)dt$

(10)

$\mathcal{F}\left[f\left(t\right)\right]={\int }_{-\infty }^{+\infty }{e}^{-ivt}f\left(t\right)dt$

(11)

It is easy to find a relationship between the generalized transformation and other various transformations. It not only encompasses different transforms that belong to the Laplace integral transformation category but also displays characteristics of the Fourier integral transformation. Especially, equation (9) becomes the Fareeha transform that contains the characteristics of Fourier and Laplace transform when $\gamma \left(s\right)=1,n=1$ and $s=u+iv$ . ³⁸ The effective applicability to solving a nonlinear model is shown by these relations.

He’s integral transformation summarizes various defined integral transformations together and has the general properties of transformation, such as existence and linearity. It represents integral transformations in a universal form to facilitate solving different nonlinear equations.

A general nonlinear oscillator that could be modeled as a second-order nonlinear differential equation is in the following form:

$y^{″}+h\left(t,y,y^{\prime }\right)$

(12)

Equation (12) can be re-written as follows:

$L\left(y\right)+NL\left(y\right)+H\left(t\right)=0$

(13)

where

$L\left(y\right)=y^{″}+{\omega }^{2}y,NL\left(y\right)=g\left(y,y^{\prime }\right)-{\omega }^{2}y,H\left(t\right)=h\left(t,y,y^{\prime }\right)-g\left(y,y^{\prime }\right)$

here, L, NL, and H represent a linear operator, a nonlinear operator, and an inhomogeneous term, respectively. The parameter $\omega$ is the frequency that needs to be found.

By the variational iteration method, the correction iteration for equation (13) can be presented as follows^25–28:

$y_{m+1}\left(t\right)=y_m\left(t\right)+{\int }_0^t\phi \left(t,\rho \right)\left[L\left(y_m\right)+NL\left(\stackrel{\sim }{y_m}\right)+H\left(\rho \right)\right]d\rho ,m=0,1,2,\cdots$

(14)

where $\stackrel{\sim }{y_m}$ is a restricted variation, which means $\delta \stackrel{\sim }{y_m}=0$ .

Then the solution of equation (12) is structured as $y=\underset{m\to \infty }{\lim }{y}_m$ , and $y_m$ denotes (m)th approximate solution. According to the basic idea of the variational iteration method, the function $\phi$ is a Lagrange multiplier that can be adopted as $\phi =\phi \left(t-\rho \right)$ .

Applying He’s integral transformation on both sides of equation (14), the correction functional could be written as follows:

$\mathcal{H}\left[y_{m+1}\left(t\right)\right]=\mathcal{H}\left[y_m\left(t\right)\right]+\mathcal{H}\left[{\int }_0^t\phi \left(t-\rho \right)\left[L\left(y_m\right)+NL\left(\stackrel{\sim }{y_m}\right)+H\left(\rho \right)\right]d\rho \right]=\mathcal{H}\left[y_m\left(t\right)\right]+\mathcal{H}\left[\phi \left(t\right)\right]\left\{\mathcal{H}\left[L\left(y_m\right)\right]+\mathcal{H}\left[NL\left(\stackrel{\sim }{y_m}\right)\right]+\mathcal{H}\left[H\left(t\right)\right]\right\}$

(15)

Based on the variational theory and taking the variation on both sides of equation (15), it yields the following equation:

${\delta \mathcal{H}[y}_{m+1}\left(t\right)]=\delta \mathcal{H}\left[y_m\left(\mathrm{t}\right)\right]+\frac{s^{2m}+{\mathrm{\omega }}^2}{\gamma \left(s\right)}\mathcal{H}\left[\phi \left(t\right)\right]{\delta \mathcal{H}[y}_m\left(t\right)]$

(16)

Thus we get $\mathcal{H}\left[\phi \right]=-\frac{\gamma \left(s\right)}{s^{2m}+{\mathrm{\omega }}^2}$ , and the Lagrange multiplier is as follows:

$\phi =-\frac{\mathit{sin}\hspace{0.17em}\mathrm{\omega }t}{\mathrm{\omega }}$

(17)

It can be found that by using the integral transformation it is easy to obtain the Lagrange multiplier. Then, we employ this method to solve nonlinear systems with zero initial conditions.

By employing equation (17), we can rewrite equation (15) as follows:

$\mathcal{H}\left[y_{m+1}\left(t\right)\right]=\mathcal{H}\left[y_m\left(t\right)\right]-\frac{1}{\omega }\mathcal{H}[{\int }_0^t\hspace{0.17em}\mathit{sin}\hspace{0.17em}\omega \left(t-\rho \right)[y_m^{″}\left(\rho \right)+{\omega }^2{y}_m\left(\rho \right)+h\left(\rho ,y_m\left(\rho \right),y_m^{\prime }\left(\rho \right)\right)-{\omega }^2{y}_m\left(\rho \right)]d\rho ]$

(18)

Consider the following integral:

${\int }_0^t\hspace{0.17em}\mathit{sin}\hspace{0.17em}\omega \left(t-\rho \right)\left[y_m^{″}\left(\rho \right)+{\omega }^2{y}_m\left(\rho \right)\right]d\rho =\omega y_m\left(t\right)-{y_m^{\prime }\left(t\right)|}_{t=0}\hspace{0.17em}\mathit{sin}\hspace{0.17em}\omega t-\omega y_m\left(0\right)\mathit{cos}\hspace{0.17em}\omega t$

(19)

After using initial conditions and employing equation (19), equation (18) has following the form:

$\mathcal{H}\left[y_{m+1}\left(t\right)\right]=-\frac{1}{\omega }\mathcal{H}\left[{\int }_0^t\hspace{0.17em}\mathit{sin}\hspace{0.17em}\omega \left(t-\rho \right)\left[h\left(\rho ,y_m\left(\rho \right),y_m^{\prime }\left(\rho \right)\right)-{\omega }^2{y}_m\left(\rho \right)\right]d\rho \right]$

$=-\frac{1}{\omega \gamma \left(s\right)}\mathcal{H}\left[\mathit{sin}\hspace{0.17em}\omega t\right]\mathcal{H}\left[h\left(t,y_m,y_m^{\prime }\right)-{\omega }^2{y}_m\right]$

(20)

The form of the iteration formula of equation (1) is displayed as follows:

$\mathcal{H}\left[y_{m+1}\left(t\right)\right]=-\frac{1}{\mathrm{\omega }\gamma \left(s\right)}\mathcal{H}[\hspace{0.17em}\mathit{sin}\omega \mathrm{t}]\mathcal{H}[y_m+\frac{\theta }{y_m-1}-{\omega }^2{y}_m]$

(21)

Using the initial condition $y\left(0\right)=0$ , equation (21) becomes as follows:

$\mathcal{H}\left[y_1\left(t\right)\right]=\frac{1}{\mathrm{\omega }\gamma \left(s\right)}\mathcal{H}\left[\theta \right]\mathcal{H}\left[\mathit{sin}\hspace{0.17em}\omega \mathrm{t}\right]$

(22)

Applying inverse integral transformation, we have the following equation:

$y_1\left(t\right)=\stackrel{\sim }{A}-\stackrel{\sim }{A}\mathit{cos}\hspace{0.17em}\mathrm{\omega }t$

(23)

where $\stackrel{\sim }{A}=\frac{\theta }{{\omega }^2}$ .

For higher-order solution, use the truncated Taylor series ${\left(1-y\right)}^{-1}=1+y+y^2+\cdots$ , and equation (21) will achieve as follows:

$\mathcal{H}\left[y_{m+1}\left(t\right)\right]=-\frac{1}{\mathrm{\omega }\gamma \left(s\right)}\mathcal{H}[\hspace{0.17em}\mathit{sin}\omega \mathrm{t}]\mathcal{H}[y_m-\mathrm{\theta }\left(1+y_m+y_m^2\right)-{\omega }^2{y}_m]$

(24)

For getting an approximate nonlinear frequency, substitute equation (23) into equation (24)

$\mathcal{H}\left[y_2\right]=-\frac{\theta }{\mathrm{\omega }\gamma \left(s\right)}\{\left(1-\frac{1}{{\omega }^2}+\frac{\theta }{{\omega }^2}+\frac{2{\theta }^2}{{\omega }^4}\right)\mathcal{H}\left[\mathit{sin}\hspace{0.17em}\mathrm{\omega }t\right]\mathcal{H}\left[\mathit{cos}\hspace{0.17em}\mathrm{\omega }t\right]\hspace{0.17em}-\frac{{\theta }^2}{2{\omega }^4}\mathcal{H}\left[\mathit{sin}\hspace{0.17em}\mathrm{\omega }t\right]\mathcal{H}\left[\mathit{cos}\hspace{0.17em}2\mathrm{\omega }t\right]+\left(-2+\frac{1}{{\omega }^2}-\frac{\theta }{{\omega }^2}-\frac{3{\theta }^2}{2{\omega }^4}\right)\mathcal{H}\left[1\right]\mathcal{H}\left[\mathit{sin}\hspace{0.17em}\mathrm{\omega }t\right]\}$

(25)

Therefore, the second-order approximate solution of equation (1) could be achieved by the inverse He’s integral transformation on equation (25)

$y_2=\frac{2\theta }{{\omega }^2}+\frac{{\theta }^2-\theta }{{\omega }^4}+\frac{3{\theta }^3}{2{\omega }^6}-\left(\frac{2\theta }{{\omega }^2}+\frac{{\theta }^2-\theta }{{\omega }^4}+\frac{4{\theta }^3}{3{\omega }^6}\right)\mathit{cos}\hspace{0.17em}\mathrm{\omega }t-\frac{{\theta }^3}{6{\omega }^6}\mathit{cos}\hspace{0.17em}2\mathrm{\omega }t-\left(\frac{\theta }{2\omega }+\frac{{\theta }^2-\theta }{2{\omega }^3}+\frac{{\theta }^3}{{\omega }^5}\right)t\hspace{0.17em}\sin \hspace{0.17em}\omega t$

(26)

only when the secular term is equal to zero can periodicity be ensured, so the coefficient of $tsin\mathrm{\omega }t$ is zero

$\frac{\theta }{2\omega }+\frac{{\theta }^2-\theta }{2{\omega }^3}+\frac{{\theta }^3}{{\omega }^5}=0$

(27)

Thus the approximate frequency is obtained as follows:

$\mathrm{\omega }=\sqrt{\frac{1-\theta +\sqrt{-7{\theta }^2-2\theta +1}}{2}}$

(28)

From equation (28), when the parameter $\theta$ increases, the frequency $\mathrm{\omega }$ will decrease, and the period T will increase. To show the correctness of the nonlinear frequency obtained by equation (28), we will compare it with the numerical solution for various values of the parameters, as displayed in Table 2. A high correctness of the frequency achieved by the integral transformation coupled with the variational iteration method is revealed by the relative error.

Table 2.

Comparison of frequencies.

$\theta$	${\mathrm{\omega }}_{exact}$	$\mathrm{\omega }$ of equation (28)	Relative error (%) of equation (28)	Relative error (%) of equation (7)
0.0475	0.9740	0.9735	0.0513	0.0719
0.0900	0.9362	0.9444	0.8759	1.3459
0.1275	0.9018	0.9130	1.2420	2.2954
0.1600	0.8597	0.8797	2.3264	4.1991
0.1875	0.8215	0.8450	2.8606	5.7821
0.1971	0.8012	0.8309	3.7069	7.1143

The approximate solution of equation (1) is as follows:

$y_2=\frac{2\theta }{{\omega }^2}+\frac{{\theta }^2-\theta }{{\omega }^4}+\frac{3{\theta }^3}{2{\omega }^6}-\left(\frac{2\theta }{{\omega }^2}+\frac{{\theta }^2-\theta }{{\omega }^4}+\frac{4{\theta }^3}{3{\omega }^6}\right)\cos \hspace{0.17em}\omega t-\frac{{\theta }^3}{6{\omega }^6}\cos \hspace{0.17em}2\mathrm{\omega }t$

(29)

We draw pictures to consider the results for various values by equation (29). Figure 2 illustrates that the approximate solution by He’s integral transformation coupled with the variational iteration method has a good agreement with the mathematical solution by the fourth-order Runge–Kutta method. The Runge–Kutta method is a widely used high-precision one-step algorithm in engineering for numerically solving differential equations. Due to its high accuracy and the measures taken to suppress errors, He’s integral transformation coupled with the variational iteration method is an important iterative method for solving nonlinear ordinary differential equations. We may also decrease the error by getting a higher-order approximate solution to repeat the above process of equation (24). The convergence of the solution depends on the continuity of the integrand function, that is to say, we can obtain analytical solutions with arbitrary accuracy.

Figure 2.

Analytical solution of equation (29) (red dot), and the numerical solution (blue curve) of equation (1). (a) $\theta =0.0475$ ; (b) $\theta =0.09$ ; (c) $\theta =0.1275$ ; (d) $\theta =0.16$ .a.

Conclusion

In practical engineering applications, there is an urgent need to develop a simple method for obtaining relatively high-accuracy analytical solutions. The variational iteration method is one of several numerical approaches used for studying nonlinear systems. This paper explores a nonlinear oscillator with zero initial conditions that arises in a micro-electromechanical system, as an illustration. The analytical solution is obtained using He’s frequency formula method and the variational iteration method coupled with a new general integral transformation. The approximate nonlinear frequencies and analytical solutions are compared with the exact ones obtained by the fourth-order Runge–Kutta method. These results are discussed in detail and shown graphically. The powerful and easy-to-use technique for handling nonlinear differential equations is displayed through the result comparison and theoretical analysis. Because the new integral transform has the same essential qualities as those for Laplace and Fourier transform, it becomes a promising tool for solving nonlinear problems when coupled with the variational iteration method. It could be easily extended and applied to solve other nonlinear situations.