He’s multiple scales method for nonlinear vibrations

Abstract

He’s multiple scales method is a couple of the homotopy perturbation method and the multiple scales technology in the classic perturbation method. This method has been proved to be a powerful mathematical tool to various nonlinear equations, and it is extremely effective for forced nonlinear oscillators. This paper shows that the method can be further improved by incorporating some known technologies, e.g., the parameter-expanding technology, the enhanced perturbation method and the homotopy perturbation method with an auxiliary term. Due to the wide application of the homotopy perturbation method, He’s multiple scales method cleans solutions of nonlinear equations while the classic perturbation method fails.

Introduction

Recently El-Dib made a great progress in the homotopy perturbation method,^1–5 and suggested a powerful modification by incorporating the multiple scales technology in the classic perturbation method ⁶ into the solution process of the homotopy perturbation method,^7,8 it can solve various nonlinear problems, especially nonlinear vibration systems with forced terms and damping terms. In this paper, we will show that El-Dib’s modification, which is called as He’s multiple scales method, has obvious advantages over its original one when it further couples with the dynamical frontier of modern mathematics.

He’s multiple scales method

Let’s begin with a brief introduction to He’s multiple scales method^7,8 by considering a forced Duffing equation in the form $$u^{″}+u+\eta u^{\prime }+\epsilon u^3=B\hspace{0.17em}\cos \hspace{0.17em}\beta t$$ (1)with initial conditions $$u(0)=A,\hspace{1em}{u}^{\prime }(0)=0$$ (2)

He’s multiple scales method^7,8 is to construct a homotopy in the form $$u^{″}+u+\eta u^{\prime }+p\epsilon u^3=B\hspace{0.17em}\cos \hspace{0.17em}\beta t,\hspace{1em}p\in [0,1]$$ (3)and to expand the solution in the form $$u=u_0(T_0,T_1,T_2,\dots )+p{u}_1(T_0,T_1,T_2,\dots )+p^2{u}_2(T_0,T_1,T_2,\dots )+\dots \dots$$ (4)and to expand the operators $d/\mathit{dt}$ and $d^2/d{t}^2$ in the forms, respectively $$\frac{d}{\mathit{dt}}=D_0+p{D}_1+p^2{D}_2+\dots$$ (5) $$\frac{d^2}{d{t}^2}={(D_0+p{D}_1+p^2{D}_2+\dots )}^2=D_0^2+2p{D}_0{D}_1+p^2(D_1^2+2D_0{D}_2)+\dots$$ (6)where $D_n=\partial /\partial T_n$ .

The solution process is extremely simple as that for the multiple scales technology in the classic perturbation, ⁶ and it is not necessary to repeat it hereby.

Some effective modifications of He’s multiple scales method

Hereby we show the last achievements on the homotopy perturbation method^9–16 can be used to further improve the He’s multiple scales method,^7,8 making it accessible to wide classes of nonlinear problems. We write equation (1) in the form $$u^{″}+1\times u+\eta u^{\prime }+p\epsilon u^3=B\hspace{0.17em}\cos \hspace{0.17em}\beta t$$ (7)

We can expand the coefficient of the linear term into the following form^16,17 $$1={\Omega }^2+a_{1}p+a_2{p}^2+\dots \dots$$ (8)where $\Omega$ is the unknown frequency to be further determined late, $a_i(i=1,2,\dots )$ are constants.

This parameter expanding technology^16,17 has been widely applied in the homotopy perturbation method,^18–22 and we also omit the solution process due to too much literature, the solution process is available in He. ¹⁶

In order to control convergence of the iteration process in He’s multiple scales method, we can add an auxiliary term in the homotopy equation^23,24 $$u^{″}+({\Omega }^2+a_{1}p+a_2{p}^2+\dots \dots )u+\eta u^{\prime }+p\epsilon u^3+\mathit{ap}(1-p)=B\hspace{0.17em}\cos \hspace{0.17em}\beta t$$ (9)where a is a parameter used for controlling the convergence rate, its optimal identification is referred to He. ²³

We can also take the advantages of the enhanced perturbation method,^25,26 which is the newest achievement in the perturbation method. Incorporating this last achievement into He’s multiple scales method will certainly reveal this already astonishing well-resourced advancement.

To show the basic of the method, we write equation (1) in the form $$(D^2+1+\eta D+\epsilon u^2)u=B\hspace{0.17em}\cos \hspace{0.17em}\beta t$$ (10)where $D=d/\mathit{dt}$ is differential operator.

The enhanced perturbation method^25,26 is to apply the annihilator operator $D^2+{\beta }^2$ to equation (10) $$(D^2+{\beta }^2)(D^2+1+\eta D+\epsilon u^2)u=(D^2+{\beta }^2)(B\hspace{0.17em}\cos \hspace{0.17em}\beta t)=0$$ (11)

The forced term is eliminated completely in equation (11), which makes the solution process much simpler by the homotopy perturbation or its various modifications. Hereby we can construct a homotopy equation in the form $$(D^2+{\beta }^2)(D^2+1+\eta D+p\epsilon u^2)u=0$$ (12)for He’s multiple scales method.

Li and He found the enhanced perturbation method can be successfully incorporated into the homotopy perturbation method, ²⁶ and the most important and promising application of He’s multiple scales method is to solve nonlinear oscillators with fractal derivatives or fractional derivatives.^27-31 Consider the fractal derivative partner for equation (1) $$\frac{d^{2}u}{d{t}^{2\alpha }}+u+\eta \frac{\mathit{du}}{d{t}^{\alpha }}+\epsilon u^3=\cos \hspace{0.17em}\beta t$$ (13)where $\frac{\mathit{du}}{d{t}^{\alpha }}$ is a fractal derivative defined as^27–31 $$\frac{\mathit{dV}}{d{t}^{\alpha }}=\Gamma (1+\alpha )\underset{\Delta t=t_B-t_A=\Delta t_0}{\lim }\frac{u(t_B)-u(t_A)}{{(t_B-t_A)}^{\alpha }}$$ (14)where $\alpha$ is the fractal order, $\Delta t_0$ is the lowest hierarchical time, beyond which all functions are assumed to be continuous with respect to time. The fractal derivative partner of the nonlinear oscillator occurs in microphysics ³² and nanotechnology.^33–39

Using a transform ²⁷ $$s=t^{\alpha }$$ (15)

Equation (13) becomes $$\frac{d^{2}u}{d{s}^2}+u+\eta \frac{\mathit{du}}{\mathit{ds}}+\epsilon u^3=\cos \beta t$$ (16)

The transform of equation (15) is to convert a fractal time in a micro/nano scale to a continuous time in a macro scale. ²⁷

Now equation (16) can be easily solved by one of above methods.

An example

We use an attachment oscillator ⁴¹ as an example, which reads $$u^{″}+\frac{\epsilon }{u^3}=0,u(0)=A,u^{\prime }(0)=0$$ (17)

We assume the solution can be expressed as $$u(t)=a+b\hspace{0.17em}\cos \hspace{0.17em}\omega t$$ (18)where $\omega$ is the frequency, a and b are constants satisfying the relation $$a+b=A$$ (19)

The frequency can be obtained as $$\omega =\sqrt{\frac{\epsilon }{{(a+\frac{1}{2}b)}^4}}=\frac{\sqrt{\epsilon }}{{(A-\frac{1}{2}b)}^2}$$ (20)where b is solved from the following transcendental equation $$b{A}^3={(A-\frac{1}{2}b)}^4$$ (21)

Discussion and conclusions

This short paper sheds a light on the moving frontiers of applications of He’s multiple scales method, its advantages are obvious, and audience, who want to catch the frontier, should be familiar with a basic knowledge of the perturbation method, the homotopy perturbation method, the parameter expanding method, the enhanced perturbation, the fractal calculus, the fractional calculus, and He’s frequency formula can be used to give an initial guess for this purpose.^14,40