^{1}

^{1,2}

^{1,2}

^{1}

^{2}

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.

Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers [

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 [

Polymer foams, especially expanded polystyrene (EPS), are widely used for cushion or protective inner packaging, and the governing equations [^{−1}; ^{−2}; and

Polymer packaging system.

By introducing these parameters:

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.

Considering a general nonlinear equation

According to the classic HPM, the homotopy equation can be constructed as

According to He’s recent study [

The solutions of both (

By the Taylor series, (

According to [

According to the classical perturbation method, the iteration equations can be constructed as

Solve (

Substitute (

In order to eliminate the secular term, it must be satisfied that

A special solution of the ordinary differential equation (

Thus, by substituting (

And no secular terms require

By the same way, the second-order iteration solution can be obtained by substituting (

By solving (

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

Comparison between the HPM solution and the numerical solution.

Different parameters may lead to the different accuracy of the HPM solution. Table

Comparison between the HPM solution and the numerical solution.

_{
1} |
_{
2} |
_{
3} |
Ω_{HPM} |
Ω_{num} |
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 |

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 (

The displacement of the product while dropping, m

The mass of the packaged product, kg

The displacement impendence,

The elasticity, N

The acceleration of gravity,

The dropping height, m

Nondimensionalized parameters

Nondimensionalized displacement

Nondimensionalized time

Nondimensionalized velocity

Nondimensionalized displacement amplitude

Nondimensionalized frequency

Parameters used to simplify the expression

The auxiliary term for HPM.

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).