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Mathematical Problems in Engineering
Volume 2013 (2013), Article ID 382509, 4 pages
http://dx.doi.org/10.1155/2013/382509
Research Article

He’s Max-Min Approach for Coupled Cubic Nonlinear Equations Arising in Packaging System

1Department of Packaging Engineering, Jiangnan University, Wuxi 214122, China
2Key Laboratory of Food Packaging Techniques & Safety of China National Packaging Corporation, Jiangnan University, Wuxi 214122, China

Received 10 March 2013; Accepted 29 March 2013

Academic Editor: Mufid Abudiab

Copyright © 2013 Jun Wang. 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.

Abstract

He’s inequalities and the Max-Min approach are briefly introduced, and their application to a coupled cubic nonlinear packaging system is elucidated. The approximate solution is obtained and compared with the numerical solution solved by the Runge-Kutta algorithm yielded by computer simulation. The result shows a great high accuracy of this method. The research extends the application of He’s Max-Min approach for coupled nonlinear equations and provides a novel method to solve some essential problems in packaging engineering.

1. Introduction

Various kinds of nonlinear oscillation problems exist in the engineering field, which are usually difficult to be solved analytically. However, the analytical solution is significant for the further intensive study. Among the methods for analytical solution, the Perturbation method [1] is one of the most well-known approaches and is based on the existence of small or large parameters which is not commonly contained in many nonlinear problems. Besides, in order to avoid some restrictions of Perturbation Method, some other methods are developed, including the homotopy perturbation method (HPM), the variational interation method (VIM), many well-established asymptotic methods [2], a novel Max-Min method [3]. The Max-Min approach is developed from the idea of ancient Chinese math and owns the property of convenient application, less calculation and high accuracy, and so forth. Among current researches about He’s Max-Min approach and its applications [410], few involve coupled nonlinear problems such in packaging engineering, especially the higher-dimensional coupled nonlinear problems.

In this paper, He’s Max-Min approach is applied to the second order coupled cubic nonlinear packaging system to get its frequencies and periods under different situations. What’s more, the obtained analytical solution is compared with the solution of computer simulation by Matlab. Consequently, the comparison shows the efficiency of this method.

2. He’s Inequalities and the Max-Min Approach [3]

According to He’s Max-Min approach, in order to obtain the exact solution of certain variable x, its minimum of Max values and maximum of Min values should firstly gained as follows: where a, b, c, and d are real numbers, and then and is approximated by where and are weighing factors and .

The changing progress of from zero to infinite is just that of from to . Thus there must exist a certain value of   while the corresponding value of    locates at its exact solution.

However, the method to determine the value of is varied. In this paper, the method in [3] is used to determine the value of .

3. Modelling and Equations

Packaged products can be potentially damaged by dropping [11, 12], and it is very important to investigate the oscillation process of the packaging system. Most products, especially mechanical and electronic products, are composed of large numbers of elements, and the damage generally occurs at the so-called critical component [13]. In order to prevent any damage, a critical component and a cushioning packaging are always included in a packaging system [14], as shown in Figure 1. Here the coefficients and denote, respectively, the mass of the critical component and main part of product, while and are, respectively, the coupling stiffness of the critical component and that of cushioning pad.

382509.fig.001
Figure 1: The model of a packaging system with a critical component.

The oscillation in the packaging system is of inherent nonlinearity. The governing equations of cubic nonlinear cushioning packaging system with the critical component can be expressed as [14] where

Here is the incremental rate of linear elastic coefficient for cushioning pad, and h is the dropping height. Equation (4) can be equivalently written in the following forms: where

4. Application of He’s Max-Min Approach

From (6), we can easily obtain

Rewrite (16) in the following form:

According to the Max-Min method, we choose a trial function in the following form: which meets the initial conditions as described in (13) and (15).

By simple analysis, from (17)-(18) we know that

The maximal and minimal value of are, respectively, 1 and 0. So we can immediately obtain

According to He Chengtian’s interpolation [6], we obtain where and are weighting factors, , .

Then, the approximate solution of (16) can be written as

To determine the value of k, substituting (22) into (16) results in the following residual [4]: And by setting where , we obtain the value as

Substituting (25) into (21) yields

From (26), we can easily obtain the frequency value , which can be used to obtain the approximate solution of (16). Figure 2 shows the approximate solution, (22), agrees well with the numerical solution by the Runge-Kutta method for various different values of   and , where the initial velocity is assumed as , and , as illustrated in (14) and (15). The parameter for typical packaging system ranges from 3 to 5, and from 1 to 3. As shown in Figure 2, the deviation of the solution by the Max-Min approach from that by the Runge-Kutta method is very small, taking Figure 2(a), for example, the whole deviation , where represents the relating error of the solution by the Max-Min approach from that by the Runge-Kutta method.

fig2
Figure 2: Comparison of the approximate solution by the Max-Min approach with the numerical simulation solution solved by the Runge-Kutta algorithm. (Asterisk: solution by the Max-Min approach; continuous line: solution by the Runge-Kutta method).

5. Conclusion

The Max-Min method, which has been widely applied to many kinds of strong nonlinear equations such as pendulum and Duffing equations, is applied to study the nonlinear response of coupled cubic nonlinear packaging system in this study for the first time. The method is a well-established method for analyzing nonlinear systems and can be easily extended to many kinds of nonlinear equation. We demonstrated the accuracy and efficiency of the method in solving the coupled equations, showing that this method can be easily used in engineering application with high accuracy without cumbersome calculation.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant number: 51205167) and Research Fund of young scholars for the Doctoral Program of Higher Education of China (Grant number: 20120093120014), Fundamental research funds for the Central Universities (Grant number: JUSRP51302A).

References

  1. A. H. Nayfeh, Perturbation Methods, Wiley-VCH, Weinheim, Germany, 2007.
  2. J.-H. He, “Some asymptotic methods for strongly nonlinear equations,” International Journal of Modern Physics B, vol. 20, no. 10, pp. 1141–1199, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. J. H. He, “Max-min approach to nonlinear oscillators,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 9, no. 2, pp. 207–210, 2008. View at Scopus
  4. R. Azami, D. D. Ganji, H. Babazadeh, A. G. Dvavodi, and S. S. Ganji, “He's max-min method for the relativistic oscillator and high order duffing equation,” International Journal of Modern Physics B, vol. 23, no. 32, pp. 5915–5927, 2009. View at Publisher · View at Google Scholar · View at Scopus
  5. D. Q. Zeng and Y. Y. Lee, “Analysis of strongly nonlinear oscillator using the max-min approach,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 10, no. 10, pp. 1361–1368, 2009. View at Scopus
  6. D. Q. Zeng, “Nonlinear oscillator with discontinuity by the max-min approach,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 2885–2889, 2009. View at Publisher · View at Google Scholar · View at Scopus
  7. Y. Y. Shen and L. F. Mo, “The max-min approach to a relativistic equation,” Computers and Mathematics with Applications, vol. 58, no. 11-12, pp. 2131–2133, 2009. View at Publisher · View at Google Scholar · View at Scopus
  8. S. A. Demirbaǧ and M. O. Kaya, “Application of He's max-min approach to a generalized nonlinear discontinuity equation,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 4, pp. 269–272, 2010. View at Scopus
  9. S. S. Ganji, D. D. Ganji, A. G. Davodi, and S. Karimpour, “Analytical solution to nonlinear oscillation system of the motion of a rigid rod rocking back using max-min approach,” Applied Mathematical Modelling, vol. 34, no. 9, pp. 2676–2684, 2010. View at Publisher · View at Google Scholar · View at Scopus
  10. S. S. Ganji, A. Barari, and D. D. Ganji, “Approximate analysis of two-mass-spring systems and buckling of a column,” Computers & Mathematics with Applications, vol. 61, no. 4, pp. 1088–1095, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. R. E. Newton, Fragility Assessment Theory and Practice, Monterey Research Laboratory, Monterey, Calif, USA, 1968.
  12. G. J. Burgess, “Product fragility and damage boundary theory,” Packaging Technology and Science, vol. 15, no. 10, pp. 5–10, 1988.
  13. J. Wang, Y. Khan, R. H. Yang, et al., “Dynamical behaviors of a coupled cushioning packaging model with linear and nonlinear stiffness,” Arabian Journal for Science and Engineering, 2013. View at Publisher · View at Google Scholar
  14. J. Wang, Z. W. Wang, L. X. Lu, Y. Zhu, and Y. G. Wang, “Three-dimensional shock spectrum of critical component for nonlinear packaging system,” Shock and Vibration, vol. 18, no. 3, pp. 437–445, 2011. View at Publisher · View at Google Scholar · View at Scopus