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

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 RungeKutta 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.


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 wellknown 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 [4][5][6][7][8][9][10], 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.

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  = /.

Critical component Product
Packaging The model of a packaging system with a critical component.
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 .

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  1 and  2 denote, respectively, the mass of the critical component and main part of product, while  1 and  2 are, respectively, the coupling stiffness of the critical component and that of cushioning pad.
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  2 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

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 sin 2 Ω are, respectively, 1 and 0. So we can immediately obtain According to He Chengtian's interpolation [6], we obtain where  and  are weighting factors,  = /( + ),  = (3Ω 2 −  2 1 ) 2 .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  = 2/Ω, we obtain the  value as Substituting (25) into (21) yields From (26), we can easily obtain the frequency value Ω, 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  1 and  2 , where the initial velocity is assumed as (0) = (0) = 0, and Ẋ(0) = Ẏ (0) = Ω = 1, as illustrated in ( 14) and (15).The parameter  1 for typical packaging system ranges from 3 to 5, and  2 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 ∑ 120 =1 (|Δ  /  |) = 1.72%,where Δ  /  represents the relating error of the solution by the Max-Min approach from that by the Runge-Kutta method.

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.

3 Figure 2 :
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).