^{1, 2}

^{1}

^{2}

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.

Newton’s damage boundary concept [

For the complexity of nonlinear systems, the numerical method is mainly used for the analysis of dropping shock characteristic [

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.

For the cubic nonlinear cushion packaging system [

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 (

To check the correctness of the solution of first-order approximation, substituting (^{−1}, ^{−3}), the two key parameters ^{−1}. The extended period of the shock was obtained as

For dynamics question of the dropping shock, the numerical solution to (

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

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

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,

We set (^{−1}, and they can be denoted as

As

In a cushioning packaging system, any small vibration might lead to serious damage due to inner resonance. The inner resonance [

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

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.