Application of Variational Iteration Method with Energy Method for Nonlinear Equation Arisen from Packaging System

Abstract

The variational iteration method (VIM) is widely applied for solving various kinds of nonlinear equations. Despite its simplicity and effectiveness, the accuracy of the method may depend on the iteration steps. However, the more iteration steps one makes, the more complex the results may become. To overcome this shortcoming, a new method combining the VIM with energy method (EM) is proposed and applied to study the nonlinear response of cubic packaging system. The analytical expressions of the important parameters such as the maximum displacement response, the maximum acceleration response of the system, the extended period of the shock, and the conditions for inner resonance of system were obtained. The results show that the maximum of the acceleration and the displacement and the extended period of the shock got by this method are very similar to the ones got by Runge-Kutta method. The result provides the new method for the dropping shock problem of nonlinear packaging system.

1. Introduction

Newton’s damage boundary concept [1] is the foundation for the present packaging design. In this theory, the product packaging system was considered to be undamped single degree of freedom linear system. These may not be valid because of the complexity of products configuration and the diversity of cushioning material. In procedure of transportation and storage, the dropping shock might lead to serious damage of product. Wang [2, 3] and Wang et al. [4] developed the concept of the dropping damage boundary curve for linear and nonlinear packaging system to evaluate the damage of product as a result of drop impact because most packaged products are damaged as a result of shocks in transportation. In the dropping shock dynamic evaluation of the nonlinear packaging system including cubic and tangent nonlinear system and other systems, it is very important to obtain the dynamic response, the maximum displacement response, the maximum acceleration response of the system, and the extended period of the shock.

For the complexity of nonlinear systems, the numerical method is mainly used for the analysis of dropping shock characteristic [2–5]. Most recently, various analytical approaches for solving nonlinear differential equations were widely applied in the analysis of engineering practical problem [6–21], such as the homotopy perturbation method [6], parameter expansion method [7], energy balance method [8], and variational iteration method (VIM) [9–12]. The VIM has been widely applied in solving kinds of nonlinear equations. For given an initial approximate solution (may contain unknown parameters), a first-order approximate analytic solution is obtained, which is not limited by small parameter.

In this paper, the VIM is used to solve the dropping shock dynamic equation of the cubic nonlinear packaging system, and the first-order approximate solution was obtained. In order to improve the accuracy of the solutions, the new method combining the first-order approximate solution of the VIM with the energy method (EM) of packaging dynamics is developed, and the analytical expressions of the important parameters such as the maximum displacement response, the maximum acceleration response of the system, the extended period of the shock, and the conditions for inner resonance of system were obtained. The results show that the maximum of the acceleration and the displacement and the extended period of the shock got by this method are very similar to the ones got by Runge-Kutta (R-K) numerical method of order4. The result provides a new method for the dropping shock problem of the cubic nonlinear packaging system.

2. The First-Order Approximation Solution of the VIM

For the cubic nonlinear cushion packaging system [2], the dynamic model of the system is depicted in Figure 1; in the condition of no system damping, the dropping shock dynamic equation and the dropping shock initial conditions of system can be written as 𝑥̈+𝜔²₀𝑥+𝑘𝑥³𝑥(0)𝑥̇(0)==0,=0,2𝑔ℎ, (1)(2) where 𝑘=𝜀𝜔²₀,  𝜀=𝑟/𝑘₀,  𝜔₀=𝑘₀/𝑚 is the frequency parameter, 𝑚 is the mass of product, ℎ is the dropping height of the packaging system, 𝑘₀ and 𝑟 denote the linear elastic constant coefficient of the cushioning material and its incremental rate, respectively, and 𝑔 is the gravity acceleration.

Figure 1 

The dynamic model of cubic packaging system.

The VIM has been widely applied in solving many different kinds of nonlinear equations and is especially effective in solving nonlinear vibration problems with approximation. Assuming the initial solutions for (1) can be written as below 𝑥₀=𝐴sin(𝛼𝑡) (3) where 𝐴 and 𝛼 are unknown parameters. By using the view of the VIM, we can construct the following iteration formula [9–12]: 𝑥_𝑛+1=𝑥_𝑛+1𝜔₀^𝑡₀sin𝜔₀(𝑠−𝑡)𝑥̈_𝑛(𝑠)+𝜔²₀𝑥_𝑛(𝑠)+𝑘𝑥³_𝑛(𝑠)d𝑠. (4) The first-order iteration approximate solution was obtained as 𝑥₁=−3𝑘𝐴³4𝜔²₀−𝛼²sin(𝛼𝑡)+𝑘𝐴³4𝜔²₀−9𝛼²()+𝛼𝐴𝜔sin3𝛼𝑡₂₀−𝛼²+(3/4)𝑘𝐴³𝜔₀𝜔²₀−𝛼²−3𝛼𝑘𝐴³4𝜔₀𝜔²₀−9𝛼²×sin𝜔₀𝑡. (5) In order to ensure that no secular terms appear in the next iteration, the coefficient ofsin(𝜔₀𝑡) is equal to zero, from which the 𝛼 can be determined. The frequency parameter can be obtained as 𝛼=𝜔₀10+6𝜀𝐴²+81+(15/8)𝜀𝐴²+(9/16)𝜀²𝐴⁴.18 (6) When 𝜀 is a small parameter, the approximation solution of the frequency can be expressed as 𝛼=𝜔₀1+3𝜀𝐴²4. (7) The extended period of the shock can be expressed as follows: 𝜋𝜏=𝛼. (8) The first-order approximation solutions of the shock dynamic equation (1) can be written as follows: 𝑥₁=−3𝑘𝐴³4𝜔²₀−𝛼²sin(𝛼𝑡)+𝑘𝐴³4𝜔²₀−9𝛼²sin(3𝛼𝑡) (9) with 𝛼 defined as (6). The velocity response and the acceleration response of the system can be expressed as 𝑥̇₁=−3𝛼𝑘𝐴³4𝜔²₀−𝛼²cos(𝛼𝑡)+3𝛼𝑘𝐴³4𝜔²₀−9𝛼²(),𝑥̈cos3𝛼𝑡₁=3𝛼²𝑘𝐴³4𝜔²₀−𝛼²sin(𝛼𝑡)−9𝛼²𝑘𝐴³4𝜔²₀−9𝛼²sin(3𝛼𝑡). (10)(11) When 𝛼𝑡=𝜋/2, substituting it into (9) and (11), respectively, the maximum displacement and maximum acceleration of the system can be written in the form 𝑥_𝑚=3𝑘𝐴³4𝜔²₀−𝛼²+𝑘𝐴³4𝜔²₀−9𝛼²,𝑥̈_𝑚=3𝛼²𝑘𝐴³4𝜔²₀−𝛼²+9𝛼²𝑘𝐴³4𝜔²₀−9𝛼². (12)(13)

To check the correctness of the solution of first-order approximation, substituting (2) into (10) and combining with (6) (here, 𝑚=10 kg, 𝑘₀=600 Ncm⁻¹, 𝑟=72 Ncm⁻³), the two key parameters 𝐴 and 𝛼 can be obtained as: 𝐴=3.2252 cm, 𝛼=106.3257 s⁻¹. The extended period of the shock was obtained as 𝜋𝜏=𝛼=0.02955s. (14) When 𝛼𝑡=𝜋/2, according to (9) and (11), the maximum displacement of the system is𝑥_𝑚=3.4772 cm and the maximum acceleration of the system is𝑥̈_𝑚=45.9323 g, respectively.

For dynamics question of the dropping shock, the numerical solution to (1) can be obtained by applying the R-K method of order4. Then the extended period of the shock is 𝜏=0.02851 s, the maximum displacement of shock response is𝑥_𝑚=3.4026 cm, and the maximum acceleration of shock response is 𝑥̈_𝑚=49.7949 g, respectively. Comparing of the first-order approximate solution with the numerical solution, the relative error of the extended period of the shock is 3.65% and the relative errors of the maximum displacement and the maximum acceleration of the first-order approximate solution are 2.2%and 7.75%, respectively. As shown in Figures 2 and 3, the dropping shock displacement response and acceleration response of the first-order approximation solution by VIM are compared with the R-K method of order4. These results show that the first-order approximate solution needs further discussion as regards how to satisfy the demand of engineering.

Figure 2 

Comparison of the displacement response by the VIM and CVIM with the numerical simulation solved by the R-K method.

Figure 3 

Comparison of the acceleration response by the VIM and CVIM with the numerical simulation solved by the R-K method.

3. The Correction of the First-Order Approximate Solution

For the demand of engineering, it is necessary that the first-order approximate solution needs correction. The new method was suggested which integrates the VIM with the EM of packaging dynamic, and the new theoretical solution can be obtained for the nonlinear dropping shock. For that reason, 𝑊 denotes the weight of the product, if there is no system damping; in the idea of energy method, the gravitational potential energy of the system turned into completely elastic potential energy of the system while the deformation of the cushion material achieved maximum 𝑥_𝑚. Assume the dropping height of the packaging system be ℎ; the gravitational potential energy of the system can be written as 𝑈=𝑊ℎ. (15) For the cubic nonlinear packaging system, from (1), the corresponding restoring force is expressed as 𝑓(𝑥)=𝑘₀𝑥+𝑟𝑥³. (16) By using the EM, we have 𝑊ℎ=^𝑥𝑚₀𝑘₀𝑥+𝑟𝑥³d𝑥. (17) The maximum displacement is obtained: 𝑥_𝑚=𝑘²₀+4𝑊𝑟ℎ−𝑘₀𝑟. (18) In the condition of no system damping, the acceleration achieved the maximum while the replacement reached the maximum. From (1), the maximum of acceleration can be expressed as 𝑥̈_𝑚=𝜔²₀𝑥_𝑚+𝜀𝑥³_𝑚. (19)

We set (18) and (19) into (12) and (13), respectively, and the following relations were obtained: 𝑥_𝑚=3𝑘𝐴³4𝜔²₀−𝛼²+𝑘𝐴³4𝜔²₀−9𝛼²=𝑘²₀+4𝑊𝑟ℎ−𝑘₀𝑟,𝑥̈_𝑚=𝛼²3𝑘𝐴³4𝜔²₀−𝛼²+9𝛼²𝑘𝐴³4𝜔²₀−9𝛼²=𝜔²₀𝑥_𝑚+𝜀𝑥³_𝑚. (20)(21) Combining (20) with (21), the results are𝐴=3.4014 cm and 𝛼=111.2367 s⁻¹, and they can be denoted as 𝐴_𝑣 and 𝛼_𝑣. Substituting 𝐴_𝑣 and 𝛼_𝑣 into (9) and (11), then the correction solution of the first-order approximate by using VIM (denotes CVIM) is obtained as 𝑥₁=−3𝑘𝐴³_𝑣4𝜔²₀−𝛼²_𝑣sin𝛼_𝑣𝑡+𝑘𝐴³_𝑣4𝜔²₀−9𝛼²_𝑣sin3𝛼_𝑣𝑡,𝑥̈₁=3𝛼²_𝑣𝑘𝐴³_𝑣4𝜔²₀−𝛼²_𝑣sin𝛼_𝑣𝑡−9𝛼²_𝑣𝑘𝐴³_𝑣4𝜔²₀−9𝛼²_𝑣sin3𝛼_𝑣𝑡. (22)

As 𝛼_𝑣𝑡=𝜋/2, from (22), the maximum displacement and the maximum acceleration response of the system, respectively, are𝑥_𝑚=3.4015 cmand 𝑥̈_𝑚=49.7367 g. The extended period of the shock is 𝜏_𝑣=𝜋/𝛼_𝑣=0.02824 s. Compared with numerical solution, the relative errors of the extended period of the shock, the maximum displacement, and the maximum acceleration are less than 0.95%,0.04%, and 0.13%, respectively. Additionally, the dropping shock displacement response and acceleration response of the CVIM are compared with the R-K method; see Figures 2 and 3; the displacement and acceleration response are very close to the R-K method. These results show good agreement.

4. Resonance

In a cushioning packaging system, any small vibration might lead to serious damage due to inner resonance. The inner resonance [13–16] is the key problem to optimal design. By (5), the resonance can be expected when one of the following conditions is met: 𝛼=𝜔₀10+6𝜀𝐴²+81+(15/8)𝜀𝐴²+(9/16)𝜀²𝐴⁴,18𝛼=𝜔₀,𝜔𝛼=⁰3. (23)

These conditions should be avoided during the cushioning packaging design procedure.

5. Conclusions

In the dropping shock dynamic evaluation of the nonlinear packaging system, it is very important to obtain the maximum displacement response, the maximum acceleration response of the system, and the extended period of the shock.

The variational iteration method (VIM) is widely applied for solving various kinds of nonlinear equations. Despite its simplicity and effectiveness, the accuracy of the method may depend on the iteration steps. However, the more iteration steps one makes, the more complex the results may become. To overcome this shortcoming, a new method combining the VIM with energy method (EM) is proposed and applied to study the nonlinear response of cubic packaging system. The results show that the maximum of the acceleration and the displacement and the extended period of the time got by this method are very similar to the ones got by R-K numerical method of order4. The correction of the VIM has been shown to solve effectively, easily, and accurately the dropping shock problem of cubic nonlinear packaging system. The conditions for resonance, which should be avoided in the product packaging design procedure, can be obtained by the first-order iteration solution.

Although the example given in this paper is the cubic nonlinear packaging system, this new method can be applicable to other dropping shock problems of nonlinear packaging system.