Mathematical Problems in Engineering

Volume 2013 (2013), Article ID 135957, 3 pages

http://dx.doi.org/10.1155/2013/135957

Research Article

Inner-Resonance Conditions for Honeycomb Paperboard Cushioning Packaging System with Critical Component

Jun Wang,¹ Zhi-geng Fan,² Xiang Hong,¹ and Li-xin Lu¹

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

Received 25 March 2013; Accepted 3 August 2013

Academic Editor: Changfeng Ge

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.

A dynamic model was proposed for a honeycomb paperboard cushioning packaging system with critical component. Then the coupled equations of the system were solved by the variational iteration method, from which the conditions for inner-resonance were obtained, which should be avoided in the cushioning packaging design.

1. Introduction

Honeycomb paperboard is widely applicated in packaging industry due to its excellent performance in energy absorption and vibration attenuation. During the past decades, the dynamic behaviors of honeycomb paperboard under dynamical compression and impact loading are studied thoroughly. Experimental studies, theoretical modeling, and numerical simulations are all involved [1–3]. However, the oscillation in the honeycomb paperboard packaging system is of inherent nonlinearity [4, 5], and it should be treated as a double-degree-of-freedom system when the critical component should be considered [6]. It is desirable to obtain the inner-resonance conditions for a coupled packaging system since the packaged product will be damaged even at a very low dropping height for packaged product with critical component [7]. However, it remains a problem to obtain the resonance condition for nonlinear packaging system, especially for multidegree-of-freedom nonlinear cushioning packaging system [7]. The variational iteration method (VIM) has been extensively used in various nonlinear sciences, the nonsmooth problem [8], the -difference and the -fractional equations [9–11], the fractional calculus [12, 13], the fuzzy equation [14], and many other nonlinear problems [15, 16].

The governing equations of the honeycomb paperboard cushioning packaging system with critical component can be expressed as [17]

Here the coefficients and denote, respectively, the mass of the critical component, and the main part of product, defines the gravity, while represents the nonlinear elastic coefficient of honeycomb paperboard cushioning pad. is the coupling stiffness of the critical component, and is the dropping height.

By introducing these parameters , and letting , , , and , (1) can be equivalently written in the following forms:

where

2. Variational Iteration Method

The variational iteration method, VIM [18], first proposed by He, has been widely applicated in solving many different kinds of nonlinear equations [19, 20]. Applying the variational iteration method [8], the following iteration formulae can be constructed:

Beginning with the initial solutions

we have

Substituting (6) into (2) yields

where

3. Inner-Resonance

The inner-resonance can be expected when one of the following conditions is met:

These conditions should be avoided during the cushioning packaging design procedure.

4. Conclusion

The conditions for inner-resonance, which should be avoided in the cushioning packaging design procedure, can be easily obtained using the variational iteration method.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 51205167), Research Fund of Young Scholars for the Doctoral Program of Higher Education of China (Grant no. 20120093120014), and Fundamental Research Funds for the Central Universities (Grant no. JUSRP51302A).

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