Application of Homotopy Perturbation Method with an Auxiliary Term for Nonlinear Dropping Equations Arisen in Polymer Packaging System

Abstract

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 [1–3]. 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 [4] 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) [5], the homotopy analysis method (HAM) [6], He’s max-min method, and the homotopy perturbation method (HPM) [7–11]. HPM is an analytical method providing an alternative approach to introducing an expanding parameter and applied to many areas of science and engineering [12–17]. Nawaz [18] 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 [19] modified the HPM with an auxiliary term which makes the HPM more flexible. In recent studies, He [20] 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 [21] can be expressed as 𝑚𝑥̈+𝛽₃tanh𝛽₁𝑥+𝛽₄tan𝛽₂𝑥+𝛽₅tan³𝛽₂𝑥(0)𝑥̇(0)==0,𝑥=0,2𝑔ℎ, (1) 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 𝛽_i 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.

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: 𝑋̈+tanh𝑋+𝜆₂tan𝜆₁𝑋+𝜆₃tan³𝜆₁𝑋𝑋(0)𝑋̇(0)𝑇=0,=0,=𝑉=⁰𝐿2𝑔ℎ=2𝛽₁𝑚𝑔ℎ𝛽₃. (2)

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

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

According to He’s recent study [20], the homotopy equation can be constructed with an auxiliary term as 𝐿𝑢+𝑝𝐿𝑢−𝐿𝑢+𝑁𝑢+𝛼𝑝1−𝑝𝑢=0, (5) 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: 𝑢=𝑢₀+𝑝𝑢₁+𝑝²𝑢₂+⋯. (6)

3. Nonlinear Polymer Packaging System

By the Taylor series, (2) can be expanded by Taylor series as follows to simplify the calculation: 𝑋̈+𝑋𝜆₁𝜆₂+1+𝑋³𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13+𝑋⁵2𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+215=0. (7)

According to [16], the homotopy equation with an auxiliary term can be constructed as 𝑋̈+Ω²𝑋+𝑝𝜆₁𝜆₂+1−Ω²𝑋+𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13𝑋³+2𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝑋15⁵+𝛼𝑝1−𝑝𝑋=0. (8)

According to the classical perturbation method, the iteration equations can be constructed as 𝑋̈₀+Ω²𝑋₀=0,𝑋₀(0)=0,𝑋̇₀(0)𝑋̈=𝑉,₁+Ω²𝑋₁+𝑋₀𝜆₁𝜆₂+1−Ω²+𝛼+𝑋³₀𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13+𝑋⁵₀2𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝑋̈15=0,₂+Ω²𝑋₂+𝑋₁𝜆₁𝜆₂+1−Ω²+𝛼+3𝑋²₀𝑋₁𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13+5𝑋⁴₀𝑋₁2𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+215=0. (9)(10)(11)

Solve (9) and then obtain the initial approximate solution 𝑋₀=𝐴sinΩ𝜏,𝐴Ω=𝑉, (12) 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 𝑋̈₁+Ω²𝑋₁+𝐾₁sinΩ𝑡+𝐾₂sin3Ω𝑡+𝐾₃sin5Ω𝑡=0, (13) where 𝐾₁=𝜆₁𝜆₂+1−Ω²𝐴+3+𝛼4𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13𝐴³+582𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝐴15⁵,𝐾₂1=−4𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13𝐴³−5162𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝐴15⁵,𝐾₃=1162𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝐴15⁵. (14)

In order to eliminate the secular term, it must be satisfied that 𝐾₁=0. Thus, Ω²=𝜆₁𝜆₂+3+1+𝛼4𝜆³₁𝜆₂3+𝜆³₁𝜆₃−13𝐴²+582𝜆⁵₁𝜆₂15+𝜆⁵₁𝜆₃+2𝐴15⁴. (15)

A special solution of the ordinary differential equation (13) can be easily obtained as 𝑋₁=𝐾₂8Ω²𝐾sin3Ω𝜏+³24Ω²sin5Ω𝜏. (16)

Thus, by substituting (16) into (11), the second-order iteration equation can be expressed as 𝑋̈₂+Ω²𝑋₂+𝑀₁sinΩ𝜏+𝑀₂sin3Ω𝜏+𝑀₃sin5Ω𝜏+𝑀₄sin7Ω𝜏+𝑀₅sin9Ω𝜏=0, (17) where 𝑀₁=192𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+2²69120Ω²𝐴⁸+𝜆³₁𝜆₂+3𝜆³₁𝜆₃−12𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+2576Ω²𝐴⁶+𝜆³₁𝜆₂+3𝜆³₁𝜆₃−1²384Ω²𝐴⁴𝑀−𝛼,₂=2𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+29𝐾₂−2𝐾₃576Ω²𝐴⁴+𝜆³₁𝜆₂+3𝜆³₁𝜆₃−16𝐾₂−𝐾₃96Ω²𝐴²−𝐾₂Ω²−𝛼−𝜆₁𝜆₂−18Ω²,𝑀₃=−2𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+22𝐾₂−𝐾₃192Ω²𝐴⁴−𝜆³₁𝜆₂+3𝜆³₁𝜆₃−13𝐾₂−2𝐾₃96Ω²𝐴²−𝐾₃Ω²−𝛼−𝜆₁𝜆₂−124Ω²,𝑀₄=2𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+23𝐾₂−4𝐾₃1152Ω²𝐴⁴−𝐾₃𝜆³₁𝜆₂+3𝜆³₁𝜆₃−196Ω²𝐴²,𝑀₅=2𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+23𝐾₂−4𝐾₃1152Ω²𝐴⁴. (18)

And no secular terms require 𝑀₁=0; thus 192𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+2²69120Ω²𝐴⁸+𝜆³₁𝜆₂+3𝜆³₁𝜆₃−12𝜆⁵₁𝜆₂+15𝜆⁵₁𝜆₃+2576Ω²𝐴⁶+𝜆³₁𝜆₂+3𝜆³₁𝜆₃−1²384Ω²𝐴⁴−𝛼=0. (19)

By the same way, the second-order iteration solution can be obtained by substituting (12) into (17): 𝑋₂=𝑀₂8Ω²𝑀sin3Ω𝜏+³24Ω²+𝑀sin5Ω𝜏₄48Ω²𝑀sin7Ω𝜏+⁵80Ω²sin9Ω𝜏. (20)

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.

Figure 2 

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.

Table 1 

Comparison between the HPM solution and the numerical solution.

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
𝛽1 and 𝛽2:	The displacement impendence, m−1
𝛽3, 𝛽4, and 𝛽5:	The elasticity, N
𝑔:	The acceleration of gravity, m/s−2
ℎ:	The dropping height, m
𝑇0, 𝐿, and 𝜆i:	Nondimensionalized parameters
𝑋:	Nondimensionalized displacement
𝜏:	Nondimensionalized time
𝑉:	Nondimensionalized velocity
𝐴:	Nondimensionalized displacement amplitude
Ω:	Nondimensionalized frequency
𝐾i 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).