# Resonance Analysis for Tilted Support Spring Coupled Nonlinear Packaging System Applying Variational Iteration Method

Authors: An-Jun Chen
Journal: Mathematical Problems in Engineering
 (2013)

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## Abstract

The coupled nonlinear dynamical equations were developed for a tilted spring packaging system with critical components. The approximate solution and resonance conditions of system were obtained applying a variational iteration method. The resonance conditions, which should be avoided in the packaging design, can be easily obtained by VIM.


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## Body

## 1. Introduction

Damage of products frequently occurs at the circulating process, the vibration and shock are main dynamical reasons, it is very important to investigate the dynamical characteristics of packaging system for packaging design. Newton’s damage boundary concept [1] and succeeding modified damage evaluation approaches [2–4] were widely utilized in packaging design. The cushioning pad was treated as a simple linear or nonlinear spring, and packaging system was considered to be a single degree of freedom system [5]. However, the real packaging system, like the tilted spring system which was demonstrated to be a better candidate for products protection than linear system [6–8], includes product and many critical components, which is multi-degree-of-freedom system and nonlinear by nature. And it remains a problem to obtain the resonance condition for nonlinear packaging system, especially for multi-degree-of-freedom nonlinear cushioning packaging system. Most recently, various analytical approaches for solving nonlinear differential equations were widely applied in the analysis of engineering practical problems, such as EBM [9], PEM [10], and VIM [11, 12]. The variational iteration method does not need the assumption of linearization or weak nonlinearity and depends totally on Lagrange multiplier theory, which means that this method will only fail if the Lagrange multiplier for any partial differential equation/ordinary differential equation (PDE/ODE) or coupled PDE/ODE does not exist. Fortunately, the Lagrange multiplier for nonlinear equations can be simply obtained in most cases. In this paper, the variational iteration method was suggested to obtain the inner-resonance condition for a strong nonlinear packaging system.


## 2. Modeling and Equations

The model of tilted spring system with critical components can be modeled as shown in Figure1. The packaged product is supported by four springs which own the same stiffness k2 and length l0, φ0 is the angle of primary support position, m1 and m2 denote the mass of the critical component and the main part of the product, and k1 is the coupling stiffness of the critical component.Figure 1
The model of tilted spring system.The dropping shock approximate dynamic equations of system can be expressed as(1)m1d2x1dt2-k1(x2-x1)=0,m2d2x2dt2+4k2(a0x2+b0l0x22+c0l22x23)+k1(x2-x1)=0,
with the initial conditions
(2)x1(0)=0,dx1(0)dt=2gh,x2(0)=0,dx2(0)dt=2gh,
where a0=sin2φ0, b0=-(3/2)sinφ0cos2φ0, c0=(1/2)(1-6sin2φ0+5sin4φ0), h is the dropping height, and g is the gravity acceleration. Equation (1) can be rewritten in the following forms:
(3)d2y1dτ2+ωy12y1-ωy12y2=0,(4)d2y2dτ2+ωy22y2+b0y22+c0y23+(a0-ωy22)y1=0,
where
(5)y1=x1l0,y2=x2l0,ωy1=λ1,ωy2=a0+λ12λ2.
Here y1 and y2 are the dimensionless displacement parameters, ω1=k1/m1 and ω2=4k2/m2 are the frequency parameters, T=1/ω2 is the periodic parameter, τ=t/T is the dimensionless time parameter, λ1=ω1/ω2 is the frequency ratio, and λ2=m1/m2 is the mass ratio of system. The nondimensional form of the initial conditions can be written as
(6)y1(0)=0,dy1(0)dτ=Tl02gh,y2(0)=0,dy2(0)dτ=Tl02gh.


## 3. Variational Iteration Method

Being different from the other nonlinear analytical methods, variational iteration method [11, 12] does not depend on small parameters, such that it can find wide application in nonlinear problem without linearization or small perturbations. Using the variational iteration method, we can construct the following iteration formula for (4):
(7)y2n+1=y2n+1ωy2∫0τsin[ωy2(s-τ)]×(d2y2nds2+ωy22y2n+b0y2n2+c0y2n3+(a0-ωy22)y1nd2y2nds2)ds.
We choose the initial solution as
(8)y10=A1sin(Ω1τ),(9)y20=A2sin(Ω2τ),
where
(10)A1=Tl0Ω12gh,A2=Tl0Ω22gh.By the iteration formula (8), we have the following first-order approximation solution:
(11)y21=(A2Ω2ωy2+3c0A23Ω24ωy2(ωy22-Ω22)-3c0A23Ω24ωy2(ωy22-9Ω22)+A1Ω1(a0-ωy22)ωy2(ωy22-Ω12))sin(ωy2τ)+(b0A222ωy22-b0A22ωy22-4Ω22)cos(ωy2τ)-A1(a0-ωy22)ωy22-Ω12sin(Ω1τ)-3c0A234(ωy22-Ω22)sin(Ω2τ)+c0A234(ωy22-9Ω22)sin(3Ω2τ)+b0A22ωy22-4Ω22cos(2Ω2τ)-b0A222ωy22.Substituting the first-order approximation solution of (11) into (3), the stationary process solution of (3) can be written as
(12)y11=C1ωy12ωy12-ωy22sin(ωy2τ)+C2ωy12ωy12-ωy22cos(ωy2τ)+C3ωy12ωy12-Ω12sin(Ω1τ)+C4ωy12ωy12-Ω22sin(Ω2τ)+C5ωy12ωy12-9Ω22sin(3Ω2τ)+C6ωy12ωy12-4Ω22cos(2Ω2τ),
where
(13)C1=A2Ω2ωy2+3c0A23Ω24ωy2(ωy22-Ω22)-3c0A23Ω24ωy2(ωy22-9Ω22)+A1Ω1(a0-ωy22)ωy2(ωy22-Ω12),C2=b0A222ωy22-b0A22ωy22-4Ω22,C3=-A1(a0-ωy22)ωy22-Ω12,C4=-3c0A234(ωy22-Ω22),C5=c0A234(ωy22-9Ω22),C6=b0A22ωy22-4Ω22.Computation illustrates that for the tilted spring system with critical components, the first-order approximation solution is enough. As shown in Figure2, the nondimensional dropping shock response displacement (y11) of the critical components is calculated and compared with the numerical integration solutions by using a built-in ODE45 solver in MATLAB, showing good agreement. The results are obtained for the following amounts: λ1=10, λ2=0.01, and φ0=80°, and the initial conditions are y1(0)=y2(0)=0, dy1(0)/dτ=dy2(0)/dτ=0.3.Figure 2
The comparison between the VIM solutions with ordinary differential equation solver in MATLAB.


## 4. Resonance

Through (11) and (12), the resonance conditions can be simplified as
(14)Ω1=ωy2,Ω2=ωy2,Ω2=12ωy2,Ω2=13ωy2,Ω1=ωy1,Ω2=ωy1,Ω2=12ωy1,Ω2=13ωy1,ωy2=ωy1.These conditions should be avoided during the cushioning packaging design procedure.


## 5. Conclusion

Packaged products can be potentially dropped in the transportation, during which the packaged products may be seriously damaged by the inner-resonance, especially for those large and fragile products. Therefore, it is essential to obtain the inner resonance conditions for dropped packaging system. In this paper, the nonlinear dynamical equations were established for the tilted support spring system with critical components, and the resonance conditions of system are discussed by using the variational iteration method. The first-order approximation solution was obtained and compared with the numerical simulation results using ODE45 solver in MATLAB, showing good agreement. The resonance conditions, which should be avoided in the packaging design, can be easily obtained by VIM.



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*Source: 384251-2013-05-22.xml*