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]
(1)m1d2xdt2+k1(x-y)=0,x(0)=0,x˙(0)=2gh,m2d2ydt2+∑i=19aiyi-k1(x-y)=0,y(0)=0,y˙(0)=2gh.

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 βi 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 γi=aia1i-2/a2i-1(i=3,…,9), (1) can be equivalently written in the following forms:
(2)X¨+ω012X-ω012Y=0,X(0)=0,X′(0)=T0L2gh,Y¨+ω022Y+Y2+∑i=39γiYi+(1-ω022)X=0,Y(0)=0,Y′(0)=T0L2gh,
where
(3)ω01=λ1,ω02=1+λ12λ2,λ1=ω1ω2,λ2=m1m2,ω1=k1m1,ω2=k2m2.

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:
(4)X1=X0+1ω01∫0tsinω01(s-t){X¨0+ω012X0-ω012Y1}ds,Y1=Y0+1ω02×∫0tsinω02(s-t){Y¨0+ω022Y0+Y02+∑i=39γiY0i}ds.
Beginning with the initial solutions
(5)X0=A1sin(Ω1t),Y0=A2sin(Ω2t),
we have
(6)Y1=∑i=19{p[Kiω02(i2Ω22-ω022)(ω02siniΩ2τ-iΩ2sinω02τ)]+q[Kii2Ω22-ω022(cosiΩ2τ-cosω02τ)]}+A2sinΩ2τ.
Substituting (6) into (2) yields
(7)X1=(ω012-A1Ω12)ω01sinΩ1τ-Ω1sinω01τω01(Ω12-ω012)+ζ1λ2-2A1Ω1cosΩ1τ-cosω01τΩ12-ω012-ω012p∑i=19{Kiω02(i2Ω22-ω022)·[iΩ2ω01·ω01sinω02τ-ω02sinω01τω012-ω022-ω02(ω01siniΩ2τ-iΩ2sinω01τ)ω01(ω012-i2Ω22)]}-ω012q∑i=19[Kii2Ω022-ω022·(cosω01τ-cosiΩ2τω012-i2Ω22-cosω01τ-cosω02τω012-ω022)]-ω012A2ω01sinΩ2τ-Ω2sinω01τω01(Ω22-ω012),
where
(8)p={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,K1=-Ω22A2+ω022A2+63128κ9A29+3564κ7A27+58κ5A25+34κ3A23,K2=-716κ8A28-1532κ6A26-12κ4A24-12A22,K3=-2164κ9A29-2164κ7A27-516κ5A25-14κ3A23,K4=732κ8A28+316κ6A26+18κ4A24,K5=964κ9A29+764κ7A27+116κ5A25,K6=-116κ8A28-132κ6A26,K7=-9256κ9A29-164κ7A27,K8=1128κ8A28,K9=1256κ9A29.

3. Inner-Resonance

The inner-resonance can be expected when one of the following conditions is met:
(9)Ω1=ω01,(10)Ω1=ω02,(11)Ω2=1iω01,i=1,2,…,9,(12)Ω2=1iω02,i=1,2,…,9,(13)ω01=ω02.
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).

WangD.-M.WangZ.-W.LiaoQ.-H.Energy absorption diagrams of paper honeycomb sandwich structuresWangZ.-W.Yu PingE.Mathematical modelling of energy absorption property for paper honeycomb in various ambient humiditiesGuoY. F.ZhangJ. H.Shock absorbing characteristics and vibration transmissibility of honeycomb paperboardRouillardV.SekM. A.CrawfordS.The dynamic behaviour of stacked shipping units during transport. Part I: model validationGarcia-Romeu-MartinezM. A.SekM. A.Cloquell-BallesterV. A.Effect of initial pre-compression of corrugated paperboard cushions on shock attenuation characteristics in repetitive impactsWangJ.DuanF.JiangJ. H.LuL. X.Dropping damage evaluation for a hyperbolic tangent nonlinear system with a critical componentWangJ.KhanY.LuL. X.WangZ. W.Inner resonance of a coupled hyperbolic tangent nonlinear oscillator arising in a packaging systemWuG. C.New trends in the variational iteration methodWuG.-C.Variational iteration method for q-difference equations of second orderKongH.HuangL. L.Lagrange multipliers of q-difference equations of third orderWuG. C.BaleanuD.New applications of the variational iteration method-from differential equations to q-fractional difference equationsWuG.-C.BaleanuD.Variational iteration method for the Burgers' flow with fractional derivatives—new Lagrange multipliersHuC. H.ZengK. L.FengH. R.ZengY.ChenJ. P.Approximate solutions of the ENSO model with fractional derivativesJafariH.KhaliqueC. M.Homotopy perturbation and variational iteration methods for solving fuzzy differential equationsFouladiF.HosseinzadehE.BarariA.DomairryG.Highly nonlinear temperature-dependent fin analysis by variational iteration methodSadighiA.GanjiD. D.Exact solutions of nonlinear diffusion equations by variational iteration methodJiangJ. H.WangZ. W.The research on the vibration characteristics for honeycomb paperboard cushioing package systemHeJ.-H.Variational iteration method—a kind of non-linear analytical technique: some examplesHeJ.-H.Asymptotic methods for solitary solutions and compactonsXuL.Dynamics of two-strand yarn spinning in forced vibration