He Chengtian’s Inequalities for a Coupled Tangent Nonlinear System Arisen in Packaging System

Mathematical Problems in Engineering, Sep 2013

He Chengtian’s inequalities from ancient Chinese algorithm are applied to strong tangent nonlinear packaging system. The approximate solution is obtained and compared with the solution yielded by computer simulation, showing a great high accuracy of this method. The suggested approach provides a novel method to solve some essential problems in packaging engineering.

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He Chengtian’s Inequalities for a Coupled Tangent Nonlinear System Arisen in Packaging System

He Chengtian’s Inequalities for a Coupled Tangent Nonlinear System Arisen in Packaging System Jun Wang,1 Zhi-geng Fan,2 Li-xin Lu,1 An-jun Chen,1 and Zhi-wei Wang3 1Jiangsu Province Key Laboratory of Advanced Food Manufacturing Equipment and Technology, Department of Packaging Engineering, Jiangnan University, Wuxi 214122, China 2School of Light Industry, Zhejiang University of Science and Technology, Hangzhou 310023, China 3Key Laboratory of Product Packaging and Logistics of Guangdong Higher Education Institutes, Jinan University, Zhuhai 519070, China Received 19 July 2013; Accepted 27 August 2013 Academic Editor: Guo-Cheng Wu Copyright © 2013 Jun Wang 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. Abstract He Chengtian’s inequalities from ancient Chinese algorithm are applied to strong tangent nonlinear packaging system. The approximate solution is obtained and compared with the solution yielded by computer simulation, showing a great high accuracy of this method. The suggested approach provides a novel method to solve some essential problems in packaging engineering. 1. Introduction In order to avoid some restrictions of perturbation method [1], some other methods are developed, such as the homotopy perturbation method (HPM) [2, 3], the variational iteration method (VIM) [4–6], the homotopy analysis method (HAM) [7], and He Chengtian’s inequalities which cannot be found in the literature but recently reported in [8]. The max-min approach which is developed from the idea of ancient Chinese mathematics is demonstrated to be of convenient application, less calculation, and high accuracy. Among current researches about He Chengtian’s inequalities and their applications [9–11], few involved coupled nonlinear problems. In our previous research [12], He Chengtian’s inequalities were introduced to study the nonlinear dropping shock response for coupled cubic nonlinear packaging system, showing the effectiveness of the method. In packaging system, many cushioning materials behave as the tangent nonlinear characteristics [13, 14], and the dropping shock response of tangent packaging system with critical component is also studied [15]. In this paper, He Chengtian’s inequalities are applied to the coupled nonlinear tangent packaging system with critical component, and the obtained analytical solution is compared with the solution of computer simulation. The aim of this research is to suggest a new and simple mathematical method for solving the nonlinear dropping shock equations arisen in packaging system. 2. Modelling and Equations The governing equations of tangent nonlinear cushioning packaging system with the critical component can be expressed as [15] where Here the coefficients and denote the mass of the critical component and the main part of the product, respectively, while and are the coupling stiffness of the critical component and that of cushioning pad, respectively, is the compression limit of the cushioning pad, and is the dropping height. Equation (1) can be equivalently written in the following forms: where 3. Application of He Chengtian’s Inequalities From (3), we can easily obtain Rewrite (13) in the form According to He Chengtian’s inequalities, we choose a trialfunction in the form which meets the initial conditions as described in (10) and (12). By simple analysis, from (14)-(15), 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 [8, 12], we obtain where and are weighting factors, , and . Then, the approximate solution of (13) can be written as To determine the value of , substituting (19) into (13) results in the following residual [8]: And by setting where , we obtain the value as Substituting (22) into (18) yields From (23), we can easily obtain the frequency value . Table 1 gives the values of with different and , and Figure 1 shows that the approximate solution, (19), agrees well with the exact solution for various different values of and , where the initial velocity is assumed as . Table 1: Values of Ω from (23) with different values of and . Figure 1: Comparison of the approximate solution with the exact solution (asterisk: the approximate solution; continuous line: the exact solution). 4. Conclusion He Chengtian’s inequalities are for the first time applied to study the nonlinear response of coupled tangent packaging system. The results show that this method can be easily used in engineering application with high accuracy without cumbersome calculation. Acknowledgments The authors would like to appreciate the financial support of National Natural Science Foundation of China (Grant no. 51205167), Research Fund for the Doctoral Program of Higher Education of Chi (...truncated)


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Jun Wang, Zhi-geng Fan, Li-xin Lu, An-jun Chen, Zhi-wei Wang. He Chengtian’s Inequalities for a Coupled Tangent Nonlinear System Arisen in Packaging System, Mathematical Problems in Engineering, 2013, 2013, DOI: 10.1155/2013/604850