Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2013 / Article
!A Erratum for this article has been published. To view the article details, please click the ‘Erratum’ tab above.
Special Issue

Mathematical Problems in Packaging Engineering

View this Special Issue

Research Article | Open Access

Volume 2013 |Article ID 645074 | https://doi.org/10.1155/2013/645074

Huan-xin Jiang, Yong Zhu, Li-xin Lu, "Study of the Nonlinear Dropping Shock Response of Expanded Foam Packaging System", Mathematical Problems in Engineering, vol. 2013, Article ID 645074, 3 pages, 2013. https://doi.org/10.1155/2013/645074

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

Academic Editor: Jun Wang
Received16 May 2013
Revised19 Jul 2013
Accepted21 Jul 2013
Published22 Aug 2013

Abstract

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 [13]. 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.

2. 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 , , and 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 , (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 .

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.


Parameters Error, %

1.44341.47422.089268756
1.49951.54582.995212835
1.89211.90370.609339707
1.96992.01032.009650301

1.93502.07616.796397091
2.07412.27128.678231772
2.56342.61101.8230563
2.76882.89654.408769204

2.72223.265416.63502174
3.01803.636317.00354756
3.62353.85345.966159755
4.08094.50699.452173334

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.

References

  1. R. D. Mindlin, “Dynamics of package cushioning,” Bell System Technical Journal, vol. 24, article 3, pp. 353–461, 1945. View at: Google Scholar
  2. Z.-W. Wang, “On evaluation of product dropping damage,” Packaging Technology and Science, vol. 15, no. 3, pp. 115–120, 2002. View at: Publisher Site | Google Scholar
  3. J. Wang, F. Duan, J. H. Jiang, and L. X. Lu, “Dropping damage evaluation for a hyperbolic tangent nonlinear system with a critical component,” Journal of Vibration and Control, vol. 18, pp. 1417–1421, 2012. View at: Google Scholar
  4. J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at: Google Scholar
  5. J.-H. He and X.-H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. Y. Khan, N. Faraz, A. Yildirim, and Q. Wu, “Fractional variational iteration method for fractional initial-boundary value problems arising in the application of nonlinear science,” Computers & Mathematics with Applications, vol. 62, no. 5, pp. 2273–2278, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. G. Rezazadeh, H. Madinei, and R. Shabani, “Study of parametric oscillation of an electrostatically actuated microbeam using variational iteration method,” Applied Mathematical Modelling, vol. 36, no. 1, pp. 430–443, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. N. Bildik and A. Konuralp, “The use of variational iteration method, differential transform method and adomian decomposition method for solving different types of nonlinear partial differential equations,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 7, no. 1, pp. 65–70, 2006. View at: Google Scholar
  9. N. Bildik, A. Konuralp, and S. Yalçınbaş, “Comparison of Legendre polynomial approximation and variational iteration method for the solutions of general linear Fredholm integro-differential equations,” Computers & Mathematics with Applications, vol. 59, no. 6, pp. 1909–1917, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  10. J. Wang, Y. Khan, R.-H. Yang, L.-X. Lu, Z.-W. Wang, and N. Faraz, “A mathematical modelling of inner-resonance of tangent nonlinear cushioning packaging system with critical components,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2573–2576, 2011. View at: Publisher Site | Google Scholar
  11. H. Jafari and C. Khalique, “Homotopy perturbation and variational iteration methods for solving fuzzy differential equations,” Communications in Fractional Calculus, vol. 3, no. 1, pp. 38–48, 2012. View at: Google Scholar
  12. G. C. Wu, “Laplace transform overcoming principal drawbacks in application of the variational iteration method to fractional heat equations,” Thermal Science, vol. 6, no. 4, pp. 1257–1261, 2012. View at: Google Scholar
  13. J. Wang, L. Lu, H. Jiang, and Y. Zhu, “Nonlinear response of strong nonlinear system arisen in polymer cushion,” Abstract and Applied Analysis, vol. 2013, Article ID 891914, 3 pages, 2013. View at: Publisher Site | Google Scholar

Copyright © 2013 Huan-xin Jiang 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views851
Downloads546
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.