# Nonlinear Response of Strong Nonlinear System Arisen in Polymer Cushion

Authors: Jun Wang; Li-xin Lu; Huan-xin Jiang; Yong Zhu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A dynamic model is proposed for a polymer foam-based nonlinear cushioning system. An accurate analytical solution for the nonlinear free vibration of the system is derived by applying He's variational iteration method, and conditions for resonance are obtained, which should be avoided in the cushioning design.


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## Body

## 1. Introduction

Packaged products can be potentially damaged by dropping. In order to prevent any damage, a product and a cushioning packaging are always included in a packaging system [1, 2], and it is very important to investigate the condition for resonance. However, the oscillation in the packaging system is of inherent nonlinearity [3–5], and it remains a problem to obtain the resonance condition for nonlinear packaging system. Polymer foams, especially EPS (expanded polystyrene), are widely used for cushion or protective packaging, and the governing equations can be expressed as
(1)mx¨+β3th(β1x)+β4tan(β2x)+β5tan3(β2x)=0,x(0)=0,x˙(0)=2gh.Here, the coefficientm denotes the mass of the packaged product, while βi denote, respectively, the characteristic constants of polymer foams which could be obtained by compression test, and h is the dropping height.By introducing these parameters:T0=m/β1β3, L=1/β1 and let X=x/L, T=t/T0, λ1=β2/β1, λ2=β4/β3, and λ3=β5/β3, (1) can be written in the following forms
(2)X¨+thX+λ2tan(λ1X)+λ3tan3(λ1X)=0,X(0)=0,X˙(0)=V=T0L2gh=2β1mghβ3.By using Taylor series for sinX and tanX, (2) can be equivalently written as
(3)X¨+ω012X+(-13+λ3λ13+13λ2λ13)X3+(215+215λ2λ15+λ3λ15)X5+1115λ3λ17X7=0,X(0)=0,X˙(0)=V=T0L2gh=2β1mghβ3,
where
(4)ω01=1+λ1λ2.


## 2. Variational Iteration Method

The variational iteration method [6–13] has been widely applied in solving many different kinds of nonlinear equations [6–16], and is especially effective in solving nonlinear vibration problems with approximations [17–20]. Applying the variational iteration method [6–13], the following iteration formulae can be constructed:
(5)X1=X0+1ω01×∫0tsinω01(s-t){13X¨0+ω012X0+(-13+λ3λ13+13λ2λ13)X03+(215+215λ2λ15+λ3λ15)X05+1115λ3λ17X07}ds.Beginning with the initial solutions,(6)X0=Asin(Ωt).We have(7)X1=Asin(Ωt)-1ω01(Ω2-ω012)×(aA+34bA3+5256cA5-6371024dA7)(Ωsin(ω01t)+ω01sin(Ωt))-14ω01(9Ω2-ω012)(bA3+564cA5-189256dA7)(3Ωsin(ω01t)+ω01sin(3Ωt))+116ω01(25Ω2-ω012)(cA5+74dA7)(5Ωsin(ω01t)+ω01sin(5Ωt))-dA764ω01(49Ω2-ω012)×(7Ωsin(ω01t)+ω01sin(7Ωt)),
where
(8)a=1+λ1λ2,b=-13+13λ2λ13+λ3λ13,c=215+215λ2λ15+λ3λ15,d=1115λ3λ17.


## 3. Resonance

The resonance can be expected when one of the following conditions is met:(9)Ω=ω01,Ω=13ω01,Ω=15ω01,Ω=17ω01.These conditions should be avoided during the cushioning packaging design procedure.


## 4. Conclusion

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



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*Source: 891914-2013-02-05.xml*