The variational iteration method-2 (VIM-2) is applied to obtain approximate analytical solutions of EPS foam cushioning packaging system. The first-order frequency 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 VIM 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 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
(1)u¨+f(u,u˙,u¨)=0,
the corresponding iteration equation can be identified as
(2)un+1(t)=un(t)+∫0t(s-t){u¨n(s)+f(un,u˙n,u¨n)}ds.
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.
2. EPS Foam Nonlinear Packaging System
While dropping, the nondimensional motive equation of EPS foam packaging system can be described as [13]:
(3)X¨+tanhX+λ2tan(λ1X)+λ3tan3(λ1X)=0,
with initial boundary conditions:
(4)X(0)=0,X˙(0)=V,
where X is the nondimensional displacement while dropping, V is the nondimensional initial velocity, and λ1, λ2, and λ3 are the nondimensional system parameters.
By the fifth-order Taylor series, (5) can be expanded as the following to simplify the calculation:
(5)X¨+κ1X+κ2X3+κ3X5=0.
In order to simplify the calculation, we set
(6)κ1=(λ1λ2+1),κ2=(λ13λ23+λ13λ3-13),κ3=(2λ15λ215+λ15λ3+215).
Thus, with the initial solution X0=AsinΩt, (2) can be rewritten and solved as
(7)X1(t)=X0(t)+∫0t(s-t){X¨0+κ1X0+κ2X03+κ3X05}=X0(t)+∫0t(s-t)×{[5κ38A5+3κ24A3+(κ1-Ω2)A]sinΩs+(-5κ316A5-κ24A3)sin3Ωs+κ316A5sin5Ωs}ds.=(5κ38A5+3κ24A3+κ1A)sinΩt+(-5κ3144A5-κ236A3)sin3Ωt+κ3400A5sin5Ωt+15Ω2-8κ3A4-10κ2A2-15κ115ΩAt.
In order to eliminate the secular term, the coefficient of t must be zero. Thus,
(8)15Ω2-8κ3A4-10κ2A2-15κ1=0,
which can be solved to obtain the frequency Ω.
3. 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.
Comparison of the VIM method with the numerical method by Runge-Kutta method.
Parameters
ΩVIM
Ωnum
Error, %
λ1=0.5
λ2=2, λ3=5
1.4434
1.4742
2.089268756
λ2=2, λ3=10
1.4995
1.5458
2.995212835
λ2=5, λ3=5
1.8921
1.9037
0.609339707
λ2=5, λ3=20
1.9699
2.0103
2.009650301
λ1=1
λ2=2, λ3=5
1.9350
2.0761
6.796397091
λ2=2, λ3=10
2.0741
2.2712
8.678231772
λ2=5, λ3=5
2.5634
2.6110
1.8230563
λ2=5, λ3=20
2.7688
2.8965
4.408769204
λ1=2
λ2=2, λ3=5
2.7222
3.2654
16.63502174
λ2=2, λ3=10
3.0180
3.6363
17.00354756
λ2=5, λ3=5
3.6235
3.8534
5.966159755
λ2=5, λ3=20
4.0809
4.5069
9.452173334
4. Conclusions
The dropping shock equation of polymer-based packaging system was soluted by the VIM-2. The first-order frequency 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 VIM-2 with convenient calculation.
MindlinR. D.Dynamics of package cushioningWangZ.-W.On evaluation of product dropping damageWangJ.DuanF.JiangJ. H.LuL. X.Dropping damage evaluation for a hyperbolic tangent nonlinear system with a critical componentHeJ.-H.Variational iteration method—a kind of non-linear analytical technique: some examplesHeJ.-H.WuX.-H.Variational iteration method: new development and applicationsKhanY.FarazN.YildirimA.WuQ.Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear scienceRezazadehG.MadineiH.ShabaniR.Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration methodBildikN.KonuralpA.The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equationsBildikN.KonuralpA.YalçınbaşS.Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equationsWangJ.KhanY.YangR.-H.LuL.-X.WangZ.-W.FarazN.A mathematical modelling of inner-resonance of tangent nonlinear cushioning packaging system with critical componentsJafariH.KhaliqueC.Homotopy perturbation and variational iteration methods for solving fuzzy differential equationsWuG. C.Laplace transform overcoming principal drawbacks in application of the variational iteration method to fractional heat equationsWangJ.LuL.JiangH. ZhuY.Nonlinear response of strong nonlinear system arisen in polymer cushion