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

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 𝑞-difference and the 𝑞-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] 𝑚₁𝑑²𝑥𝑑𝑡²+𝑘₁𝑥−𝑦=0,𝑥(0)=0,𝑥̇(0)=𝑚2𝑔ℎ,₂𝑑²𝑦𝑑𝑡²+⁹_𝑖=1𝑎_𝑖𝑦^𝑖−𝑘₁𝑦(0)𝑦̇(0)=𝑥−𝑦=0,=0,2𝑔ℎ. (1)

Here the coefficients 𝑚₁ and 𝑚₂ denote, respectively, the mass of the critical component, and the main part of product, 𝑔 defines the gravity, while 𝛽_𝑖 represents the nonlinear elastic coefficient of honeycomb paperboard cushioning pad. 𝑘₁ is the coupling stiffness of the critical component, and ℎ is the dropping height.

By introducing these parameters 𝑇₀=𝑚₂/𝑎₁, 𝐿=𝑎₁/𝑎₂ and letting 𝑋=𝑥/𝐿, 𝑌=𝑦/𝐿, 𝑇=𝑡/𝑇₀, and 𝛾_𝑖=𝑎_𝑖𝑎₁^𝑖−2/𝑎₂^𝑖−1(𝑖=3,…,9), (1) can be equivalently written in the following forms: 𝑋̈+𝜔²₀₁𝑋−𝜔²₀₁𝑌=0,𝑋(0)=0,𝑋^′(0)=𝑇₀𝐿̈2𝑔ℎ,𝑌+𝜔²₀₂𝑌+𝑌²+⁹_𝑖=3𝛾_𝑖𝑌^𝑖+1−𝜔²₀₂𝑌(0)𝑋=0,=0,𝑌^′(0)=𝑇₀𝐿2𝑔ℎ, (2) where 𝜔₀₁=𝜆₁,𝜔₀₂=1+𝜆²₁𝜆₂,𝜆₁=𝜔₁𝜔₂,𝜆₂=𝑚₁𝑚₂,𝜔₁=𝑘₁𝑚₁,𝜔₂=𝑘₂𝑚₂. (3)

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: 𝑋₁=𝑋₀+1𝜔₀₁^𝑡₀sin𝜔₀₁(𝑠−𝑡)𝑋̈₀+𝜔²₀₁𝑋₀−𝜔²₀₁𝑌₁𝑌𝑑𝑠,₁=𝑌₀+1𝜔₀₂×^𝑡₀sin𝜔₀₂(𝑠−𝑡)𝑌̈₀+𝜔²₀₂𝑌₀+𝑌²₀+⁹_𝑖=3𝛾_𝑖𝑌^𝑖₀𝑑𝑠. (4) Beginning with the initial solutions 𝑋₀=𝐴₁sinΩ₁𝑡,𝑌₀=𝐴₂sinΩ₂𝑡, (5) we have 𝑌₁=⁹_𝑖=1𝑝𝐾_𝑖𝜔₀₂𝑖²Ω²₂−𝜔²₀₂𝜔₀₂sin𝑖Ω₂𝜏−𝑖Ω₂sin𝜔₀₂𝜏𝐾+𝑞^𝑖𝑖²Ω²₂−𝜔²₀₂cos𝑖Ω₂𝜏−cos𝜔₀₂𝜏+𝐴₂sinΩ₂𝜏. (6) Substituting (6) into (2) yields 𝑋₁=𝜔²₀₁−𝐴₁Ω²₁𝜔₀₁sinΩ₁𝜏−Ω₁sin𝜔₀₁𝜏𝜔₀₁Ω²₁−𝜔²₀₁+𝜁₁𝜆₂⁻²𝐴₁Ω₁cosΩ₁𝜏−cos𝜔₀₁𝜏Ω²₁−𝜔²₀₁−𝜔²₀₁𝑝⁹_𝑖=1𝐾_𝑖𝜔₀₂𝑖²Ω²₂−𝜔²₀₂⋅𝑖Ω₂𝜔₀₁⋅𝜔₀₁sin𝜔₀₂𝜏−𝜔₀₂sin𝜔₀₁𝜏𝜔²₀₁−𝜔²₀₂−𝜔₀₂𝜔₀₁sin𝑖Ω₂𝜏−𝑖Ω₂sin𝜔₀₁𝜏𝜔₀₁𝜔²₀₁−𝑖²Ω²₂−𝜔²₀₁𝑞⁹_𝑖=1𝐾_𝑖𝑖²Ω²₀₂−𝜔²₀₂⋅cos𝜔₀₁𝜏−cos𝑖Ω₂𝜏𝜔²₀₁−𝑖²Ω²₂−cos𝜔₀₁𝜏−cos𝜔₀₂𝜏𝜔²₀₁−𝜔²₀₂−𝜔²₀₁𝐴₂𝜔₀₁sinΩ₂𝜏−Ω₂sin𝜔₀₁𝜏𝜔₀₁Ω²₂−𝜔²₀₁, (7) where 𝐾𝑝=1,𝑖=1,3,5,7,9,0,𝑖=2,4,6,8,𝑞=0,𝑖=1,3,5,7,9,1,𝑖=2,4,6,8,₁=−Ω²₂𝐴₂+𝜔²₀₂𝐴₂+63𝜅128⁹𝐴⁹₂+35𝜅64⁷𝐴⁷₂+58𝜅₅𝐴⁵₂+34𝜅₃𝐴³₂,𝐾₂7=−𝜅16⁸𝐴⁸₂−15𝜅32⁶𝐴⁶₂−12𝜅₄𝐴⁴₂−12𝐴²₂,𝐾₃=−21𝜅64⁹𝐴⁹₂−21𝜅64⁷𝐴⁷₂−5𝜅16⁵𝐴⁵₂−14𝜅₃𝐴³₂,𝐾₄=7𝜅32⁸𝐴⁸₂+3𝜅16⁶𝐴⁶₂+18𝜅₄𝐴⁴₂,𝐾₅=9𝜅64⁹𝐴⁹₂+7𝜅64⁷𝐴⁷₂+1𝜅16⁵𝐴⁵₂,𝐾₆1=−𝜅16⁸𝐴⁸₂−1𝜅32⁶𝐴⁶₂,𝐾₇9=−𝜅256⁹𝐴⁹₂−1𝜅64⁷𝐴⁷₂,𝐾₈=1𝜅128⁸𝐴⁸₂,𝐾₉=1𝜅256⁹𝐴⁹₂. (8)

3. Inner-Resonance

The inner-resonance can be expected when one of the following conditions is met: Ω₁=𝜔₀₁,Ω₁=𝜔₀₂,Ω₂=1𝑖𝜔₀₁Ω,𝑖=1,2,…,9,₂=1𝑖𝜔₀₂𝜔,𝑖=1,2,…,9,₀₁=𝜔₀₂. (9)(10)(11)(12)(13) 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).