Study of the Nonlinear Dropping Shock Response of Expanded Foam Packaging System

Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers [1–3]. In most cases, the constitutive model of cushioning package materials is strong nonlinear. The variational iteration method (VIM), first proposed by Professor He [4], can be used to solve some strong nonlinear engineering problems.VIMcan avoid somedefects of Adomian method and some other kinds of perturbation methods. And by a few steps of iteration, the convergence solution can be easily obtained. After investigated in some VIM researching, He and Wu [5] developed this method into a general basic framework. Khan et al. [6] researched the application of VIM in fractional nonlinear differential equations with initial boundary problem. Rezazadeh et al. [7] studied the parametric oscillation of an electrostatically actuated microbeam using variational iteration method. Bildik et al. [8, 9] compared the VIM, differential transform method, and the Adomian decompositionmethod for partial nonlinear differential equations, and the results showed that VIM was more reliable. And in the packaging dynamics area, Wang et al. [10] obtained the inner-resonance conditions of tangent cushioning packaging system by applying VIM with good agreement. Jafari and Khalique [11] applied the variational iteration methods for solving fuzzy differential equations. Most recently, Wu soluted the fractional heat equations by variational iteration method [12]. According to [4], if a differential equation can be written as


Introduction
Dropping is an unavoidable situation for a packaged product while delivered, which is investigated by many researchers [1][2][3].In most cases, the constitutive model of cushioning package materials is strong nonlinear.
The variational iteration method (VIM), first proposed by Professor He [4], can be used to solve some strong nonlinear engineering problems.VIM can avoid some defects of Adomian method and some other kinds of perturbation methods.And by a few steps of iteration, the convergence solution can be easily obtained.After investigated in some VIM researching, He and Wu [5] developed this method into a general basic framework.Khan et al. [6] researched the application of VIM in fractional nonlinear differential equations with initial boundary problem.Rezazadeh et al. [7] studied the parametric oscillation of an electrostatically actuated microbeam using variational iteration method.Bildik et al. [8,9] compared the VIM, differential transform method, and the Adomian decomposition method for partial nonlinear differential equations, and the results showed that VIM was more reliable.And in the packaging dynamics area, Wang et al. [10] obtained the inner-resonance conditions of tangent cushioning packaging system by applying VIM with good agreement.Jafari and Khalique [11] applied the variational iteration methods for solving fuzzy differential equations.Most recently, Wu soluted the fractional heat equations by variational iteration method [12].
According to [4], if a differential equation can be written as the corresponding iteration equation can be identified as This presented paper investigated for the first time the applicability and the validity of this VIM-2 for EPS foam 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 VIM-2 and Runge-Kutta method were compared.

EPS Foam Nonlinear Packaging System
While dropping, the nondimensional motive equation of EPS foam packaging system can be described as [13]: with initial boundary conditions: where  is the nondimensional displacement while dropping,  is the nondimensional initial velocity, and  1 ,  2 , and  3 are the nondimensional system parameters.
By the fifth-order Taylor series, (5) can be expanded as the following to simplify the calculation: In order to simplify the calculation, we set Thus, with the initial solution  0 = sin Ω, (2) can be rewritten and solved as In order to eliminate the secular term, the coefficient of  must be zero.Thus, which can be solved to obtain the frequency Ω.

Results
In order to verify the previous method, the approximate solution by the new VIM was compared with the numerical solution solved by the Runge-Kutta method, as illustrated in Table 1, and the results show that for different parameters, the VIM solutions are all in good agreement with the numerical solutions which can be almost equal to the exact solution.

Conclusions
The dropping shock equation of polymer-based packaging system was soluted by the VIM-2.The first-order frequency

Table 1 :
Comparison of the VIM method with the numerical method by Runge-Kutta method. 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 VIM-2 with convenient calculation. solution