A modification of homotopy perturbation method for a hyperbolic tangent oscillator arising in nonlinear packaging system

Abstract

This paper studies a hyperbolic tangent oscillator arising in a nonlinear packaging system. The high-order nonlinearity makes it difficult to study the system analytically. A modified homotopy perturbation method is proposed, where the initial guess is an approximate solution of the well-known Duffing oscillator; the solution accuracy can be improved by either improvement of the initial guess or continuous iteration. This paper gives a general approach to high-order nonlinear oscillators.

Introduction

A packaging system is extremely complex, and there is not a universal way to solve the nonlinear vibration system.^1,2 Recently the homotopy perturbation method, which was proposed by Ji-Huan He in later 1990s,^2,3 has been caught much attention in the packing community. In this paper we consider the following hyperbolic tangent oscillator^1,2

$$m{x}^{″}+F_0\tanh \left(\frac{x_0}{F_0}x\right)=0,x\left(0\right)=0,x^{\prime }\left(0\right)=\sqrt{2gh}$$

(1)

where m denotes the mass of the packaged product; x denotes the displacement of product while dropping; k₀ is the coupling stiffness of the cushioning pad; $F_0$ is the impact force; and h is the dropping height.

Equation (1) arises in a cushioning packaging system,^1,2 and an accurate solution is much needed to optimize a packaging system. Expanding equation (1) in Taylor series and introducing parameters $k=\epsilon {\omega }_0{\hspace{0.17em}}^2,\epsilon =r/k_0,b=\sigma {\omega }_0{\hspace{0.17em}}^2,\sigma =a/k_0,{\omega }_0=\sqrt{k_0/m}$ , we obtain

$$x^{″}+{\omega }_0^{2}x-k{x}^3+b{x}^5=0,x\left(0\right)=0,x^{\prime }\left(0\right)=\sqrt{2gh}$$

(2)

In this paper we will suggest an effective modification of the homotopy perturbation method^3,4 to solve equation (2).

Homotopy perturbation method

The traditional homotopy perturbation method is to construct a homotopy equation in the form^3–13

$$x^{″}+\left({\omega }^2+a_{1}p+a_2{p}^2+\cdots \right)x+p\left(-k{x}^3+b{x}^5\right)=0$$

(3)

where p is a homotopy parameter, when p = 0; we have a linear oscillator, when p = 1, equation (3) should be the original one, equation (2), this requires

$${\hspace{0.17em}}_{\omega }^2+a_1+a_2+\cdots ={\omega }_0^2$$

(4)

where $\omega$ is the frequency and $a_i$ is a constant to be further determined. Equation (4) is also used in the parameter expansion method (parameter-expanding method).^14,15

The homotopy perturbation method has become a universal mathematical tool to nonlinear problems,^3–13 but there is still space for further improvement. In this paper we construct a homotopy equation in the form

$$x^{″}+{\omega }_0^{2}x-\left(K_0+K_{1}p+K_2{p}^2+\cdots \right)x^3+pb{x}^5=0$$

(5)

where $K_i$ is a constant to be further identified. When p = 0, we have the well-known Duffing equation, and when p = 1, equation (5) should be turned out to be equation (2), that means

$$K_0+K_1+K_2+\cdots =k$$

(6)

Proceeding with the same method as requested by the homotopy perturbation method, we have

$$x_0^{″}+{\omega }_0^{2}x-K_0{x}_0^3=0,x_0\left(0\right)=0,x_0^{\prime }\left(0\right)=\sqrt{2gh}$$

(7)

$$x_1^{″}+{\omega }_0^2{x}_1-3K_0{x}_0^2{x}_1-K_1{x}_0^3+b{x}_0^5=0,x_1\left(0\right)=0,x_1^{\prime }\left(0\right)=0$$

(8)

Equation (7) is the Duffing equation, the frequency of equation (7) reads ¹⁰

$$\omega =\sqrt{{\omega }_0^2-\frac{3}{4}{K}_0^2{A}^2}$$

(9)

We assume that the approximate solution is

$$x_0=A\sin \omega t$$

(10)

where the amplitude can be determined using the initial conditions, which is

$$A=\frac{\sqrt{2gh}}{\omega }$$

(11)

We assume that $x_1$ is a small term, which means

$$x_1\hspace{0.17em}\approx \hspace{0.17em}0$$

(12)

From equation (8), we can approximately identify $K_1$ in the form

$$K_1=b{A}^2$$

(13)

If the first-order approximate solution is enough, then from equation (6) we have

$$K_0+K_1=k$$

(14)

Finally we obtain the frequency as follows

$$\omega =\sqrt{{\omega }_0^2-\frac{3}{4}{\left(k-b{A}^2\right)}^2{A}^2}$$

(15)

It is not difficult to continue the solution process to have a higher order approximate frequency.

Discussion and conclusion

This paper suggests an effective modification of the homotopy perturbation method, the accuracy of the frequency can be improved by the following ways:

A higher accurate solution for the Duffing equation, which has been discussed in many publications. In this paper we adopt the simplest solution by the homotopy perturbation method.

Higher order approximate solution is to be solved. In this paper we only solve $x_1$ . We can also assume that $x_i\hspace{0.17em}\approx \hspace{0.17em}0(i=2,3,4,\dots )$ for fast identification of $K_i(i=2,3,4,\dots )$ in equation (6).

We can assume the approximate solution is

$$x_0=A_1\sin \omega t+A_2\sin 3\omega t$$

(16)

Using the initial conditions, we have

$$A_1+A_2=0$$

(17)

$$A_1\omega t+3A_2\omega =\sqrt{2gh}$$

(18)

We can give a more general form for the initial guess

$$x_0={\displaystyle \sum _{i=1}^N{A}_i\sin \left(\left(2i+1\right)\omega t\right)}$$

(19)

This paper gives a simple but effective way to deal with high-order nonlinear oscillators, and the method is also valid for other nonlinear problems.