Mathematical Problems in Packaging EngineeringView this Special Issue
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Inner-Resonance Conditions for Honeycomb Paperboard Cushioning Packaging System with Critical Component
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.
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 . 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 . However, it remains a problem to obtain the resonance condition for nonlinear packaging system, especially for multidegree-of-freedom nonlinear cushioning packaging system . The variational iteration method (VIM) has been extensively used in various nonlinear sciences, the nonsmooth problem , the -difference and the -fractional equations [9–11], the fractional calculus [12, 13], the fuzzy equation , and many other nonlinear problems [15, 16].
The governing equations of the honeycomb paperboard cushioning packaging system with critical component can be expressed as 
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 , first proposed by He, has been widely applicated in solving many different kinds of nonlinear equations [19, 20]. Applying the variational iteration method , the following iteration formulae can be constructed: Beginning with the initial solutions we have Substituting (6) into (2) yields where
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.
The conditions for inner-resonance, which should be avoided in the cushioning packaging design procedure, can be easily obtained using the variational iteration method.
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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