MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 982859 10.1155/2013/982859 982859 Research Article Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System 0000-0002-3899-3410 Hong Xiang 1 Wang Jun 1,2 Lu Li-xin 1,2 Lee Dong Sun 1 Department of Packaging Engineering Jiangnan University Wuxi 214122 China jiangnan.edu.cn 2 Key Laboratory of Food Packaging Techniques and Safety of China National Packaging Corporation Wuxi 214122 China 2013 14 5 2013 2013 06 02 2013 28 03 2013 23 04 2013 2013 Copyright © 2013 Xiang Hong et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The homotopy perturbation method (HPM) with an auxiliary term was applied to obtain approximate analytical solutions of polymer cushioning packaging system. The second-order solution of the equation of motion was obtained and compared with the numerical simulation solution solved by the Runge-Kutta algorithm. The results showed the high accuracy of this modified HPM with convenient calculation.

1. Introduction

Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers . In most cases, the constitutive models of cushioning package materials are strong nonlinear.

Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically. However, the analytical solution is more significant for the further intensive study. Among the methods for analytical solution, the perturbation method  is one of the most well-known approaches, and it is based on the existence of small parameters which are not commonly existed in many nonlinear problems. Besides, in order to avoid some restrictions of perturbation method, some other methods are developed, including the variational iteration method (VIM) , the homotopy analysis method (HAM) , He’s max-min method, and the homotopy perturbation method (HPM) . HPM is an analytical method providing an alternative approach to introducing an expanding parameter and applied to many areas of science and engineering . Nawaz  studied nonlinear boundary value problems for fourth-order fractional integrodifferential equations using both VIM and HPM, and the comparison results showed that both methods were very effective and convenient to solve this problem. Noor  modified the HPM with an auxiliary term which makes the HPM more flexible. In recent studies, He  summarized the modification of the HPM by introducing an auxiliary term in the homotopy equation, and Duffing equation was used as an example to illustrate the solution procedure.

Polymer foams, especially expanded polystyrene (EPS), are widely used for cushion or protective inner packaging, and the governing equations  can be expressed as (1)mx¨+β3tanh(β1x)+β4tan(β2x)+β5tan3(β2x)=0,  x(0)=0,x˙(0)=2gh, where x denotes the displacement of the product while dropping, m; the coefficient m denotes the mass of the packaged product, kg; β1 and β2 are the displacement impendence, m−1; β3, β4, and β5 are the elasticity, N, and βi denote, respectively, the characteristic constants of polymer foams which could be obtained by compression test; g is the acceleration of gravity, m/s−2; and h is the dropping height, m. The polymer packaging system can be shown in Figure 1.

Polymer packaging system.

By introducing these parameters: T0=m/β1β3,L=1/β1,λ1=β2/β1,λ2=β4/β3,andλ3=β5/β3, and letting X=x/L,τ=t/T0, (1) can be written in the following nondimensional forms: (2)X¨+tanhX+λ2tan(λ1X)+λ3tan3(λ1X)=0,X¨+tanhX+λ2tan(λ1X)+λ3tan3X(0)=0,X˙(0)=V=T0L2gh=2β1mghβ3.

This paper investigated for the first time the applicability and the validity of the modified HPM for EPS polymer cushioning packaging system. Besides, in order to show the accuracy of this method, some specific parameters were used in the constitutive equation based on real situation, and solutions of the modified HPM and Runge-Kutta method were compared.

2. Homotopy Perturbation Method with an Auxiliary Term

Considering a general nonlinear equation (3)Lu+Nu=0, where L and N are the linear operator and nonlinear operator, respectively.

According to the classic HPM, the homotopy equation can be constructed as (4)L~u+p(Lu-L~u+Nu)=0, where L~, constructed based on the model property, is a linear operator and L~u=0 can depict the solution property. With the increase of the embedding parameter p from 0 to 1, (4) will transform from L~u=0 to the original one.

According to He’s recent study , the homotopy equation can be constructed with an auxiliary term as (5)L~u+p(Lu-L~u+Nu)+αp(1-p)u=0, where α is the auxiliary parameter. It is obvious that when α=0, (5) is equivalent to the classical equation (4). And α can be determined in the iteration procedure.

The solutions of both (4) and (5) can be expanded into a series in p: (6)u=u0+pu1+p2u2+.

3. Nonlinear Polymer Packaging System

By the Taylor series, (2) can be expanded by Taylor series as follows to simplify the calculation: (7)X¨+X(λ1λ2+1)+X3(λ13λ23+λ13λ3-13)+X5(2λ15λ215+λ15λ3+215)=0.

According to , the homotopy equation with an auxiliary term can be constructed as (8)X¨+Ω2X+p[2λ15λ215(λ1λ2+1-Ω2)X+(λ13λ23+λ13λ3-13)X3+(2λ15λ215+λ15λ3+215)X5]+αp(1-p)X=0.

According to the classical perturbation method, the iteration equations can be constructed as (9)X¨0+Ω2X0=0,X0(0)=0,X˙0(0)=V,(10)X¨1+Ω2X1+X0(λ1λ2+1-Ω2+α)+X03(λ13λ23+λ13λ3-13)+X05(2λ15λ215+λ15λ3+215)=0,(11)X¨2+Ω2X2+X1(λ1λ2+1-Ω2+α)+3X02X1(λ13λ23+λ13λ3-13)+5X04X1(2λ15λ215+λ15λ3+215)=0.

Solve (9) and then obtain the initial approximate solution (12)X0=AsinΩτ,AΩ=V, where A and Ω are, respectively, the displacement amplitude and the frequency, and can be further determined.

Substitute (12) into (10) and rewrite it as (13)X¨1+Ω2X1+K1sinΩt+K2sin3Ωt+K3sin5Ωt=0, where (14)K1=(λ1λ2+1-Ω2+α)A+34(λ13λ23+λ13λ3-13)A3+58(2λ15λ215+λ15λ3+215)A5,K2=-14(λ13λ23+λ13λ3-13)A3-516(2λ15λ215+λ15λ3+215)A5,K3=116(2λ15λ215+λ15λ3+215)A5.

In order to eliminate the secular term, it must be satisfied that K1=0. Thus, (15)Ω2=(λ1λ2+1+α)+34(λ13λ23+λ13λ3-13)A2+58(2λ15λ215+λ15λ3+215)A4.

A special solution of the ordinary differential equation (13) can be easily obtained as (16)X1=K28Ω2sin3Ωτ+K324Ω2sin5Ωτ.

Thus, by substituting (16) into (11), the second-order iteration equation can be expressed as (17)X¨2+Ω2X2+M1sinΩτ+M2sin3Ωτ+M3sin5Ωτ+M4sin7Ωτ+M5sin9Ωτ=0, where (18)M1=19(2λ15λ2+15λ15λ3+2)269120Ω2A8+(λ13λ2+3λ13λ3-1)(2λ15λ2+15λ15λ3+2)576Ω2A6+(λ13λ2+3λ13λ3-1)2384Ω2A4-α,M2=(2λ15λ2+15λ15λ3+2)(9K2-2K3)576Ω2A4+(λ13λ2+3λ13λ3-1)(6K2-K3)96Ω2A2-K2(Ω2-α-λ1λ2-1)8Ω2,M3=-(2λ15λ2+15λ15λ3+2)(2K2-K3)192Ω2A4-(λ13λ2+3λ13λ3-1)(3K2-2K3)96Ω2A2-K3(Ω2-α-λ1λ2-1)24Ω2,M4=(2λ15λ2+15λ15λ3+2)(3K2-4K3)1152Ω2A4-K3(λ13λ2+3λ13λ3-1)96Ω2A2,M5=(2λ15λ2+15λ15λ3+2)(3K2-4K3)1152Ω2A4.

And no secular terms require M1=0; thus (19)19(2λ15λ2+15λ15λ3+2)269120Ω2A8+(λ13λ2+3λ13λ3-1)(2λ15λ2+15λ15λ3+2)576Ω2A6+(λ13λ2+3λ13λ3-1)2384Ω2A4-α=0.

By the same way, the second-order iteration solution can be obtained by substituting (12) into (17): (20)X2=M28Ω2sin3Ωτ+M324Ω2sin5Ωτ+M448Ω2sin7Ωτ+M580Ω2sin9Ωτ.

By solving (15) and (19) simultaneously, we can obtain the frequency Ω and the auxiliary term α. And the final solution X can be uniformed by X=X0+X1+X2.

4. Results

In order to verify the above method, the approximate solution by the new HPM was compared with the numerical solution solved by the Runge-Kutta method, as illustrated in Figure 2 with parameters: V=1,λ1=2,λ2=10,andλ3=20. The results showed a good agreement.

Comparison between the HPM solution and the numerical solution.

Different parameters may lead to the different accuracy of the HPM solution. Table 1 showed the accuracy of HPM compared with the numerical solution in different λ1,λ2, and λ3, and the results showed that, for different parameters, the HPM solutions were all in good agreement with the numerical solutions which can be almost equal to the exact solution.

Comparison between the HPM solution and the numerical solution.

λ 1 λ 2 λ 3 ΩHPM Ωnum Error, %
2 5 5 3.78959836 3.83806242 1.26272204
2 5 10 4.02572209 4.04923784 0.58074509
2 10 5 4.83451694 4.84579542 0.23274775
2 10 20 5.18247480 5.22933694 0.89613923
5 5 5 6.75089158 7.10746548 5.01689246
5 5 10 7.36074401 7.8252235 5.93567059
5 10 5 8.11755921 8.28280247 1.99501631
5 10 20 9.15230023 9.42438670 2.88704696
10 5 5 10.95651190 12.22175860 10.3524112
10 5 10 12.09216730 13.21351496 8.48636917
10 10 5 12.56277414 13.17320032 4.63384876
10 10 20 14.66198678 15.63821607 6.24258730
5. Conclusion

It is desirable to obtain the solution of strong nonlinear equation arisen in polymer packaging system. In this paper, the homotopy perturbation method with an auxiliary term was applied, and the solution was obtained and compared with the Runge-Kutta method, showing good agreement. The results indicate that the HPM with an auxiliary term was suitable for solving the strong nonlinear vibration problems in packaging system. From the comparison results shown above, it can be concluded that, for the polymer packaging system with different parameters (λ1,λ2,andλ3), the HPM can be a vigorous method to obtain its frequency. What is more, different parameters will lead to different error value. And by roughly estimate, with the increase of λ1, the HPM solution’s relative error increases; with the increase of λ2 and λ3, the HPM solution’s relative error decreases. Thus, the future research can focus on the contribution of different λ1,λ2, and λ3 to the HPM’s accuracy and to what range of λ1,λ2,andλ3 values the HPM can be applied.

Parameter List x :

The displacement of the product while dropping, m

m :

The mass of the packaged product, kg

β 1 and β2:

The displacement impendence, m-1

β 3 , β4, and β5:

The elasticity, N

g :

The acceleration of gravity, m/s-2

h :

The dropping height, m

T 0 , L, and λi:

Nondimensionalized parameters

X :

Nondimensionalized displacement

τ :

Nondimensionalized time

V :

Nondimensionalized velocity

A :

Nondimensionalized displacement amplitude

Ω :

Nondimensionalized frequency

K i and Mi:

Parameters used to simplify the expression

α :

The auxiliary term for HPM.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 51205167), Research Fund of young scholars for the Doctoral Program of Higher Education of China (Grant no. 20120093120014), and Fundamental Research Funds for the Central Universities (Grant no. JUSRP51302A).

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