Study On Dynamic Response Of Nonlinear Autonomous Packaging System

In logistics, the bad environment such as vibration and impact will result in product damage. While optimizing the structure of the product, cushioning packaging is also an important part of product protection, and the design of reliable packaging structure can effectively reduce product damage from vibration and impact. However, the research of product protection dynamics in the bad environment is weak. The packaging design of product is lack of significant and effective theoretical support and guidance. The cushioning packaging system is an abstraction of a variety of actual cushioning packages. The dynamic response characteristics of various types of packaging systems under various typical excitations are the theoretical basis of cushioning packaging design. The response of the cushioning system depends on the external force, and also depends on the structural characteristics of the system itself, like stiffness, damping and mass. The cushioning packaging systems are often nonlinear systems, and the classical type are cubic, tangent and hyperbolic tangent nonlinear system. The main purpose of this paper is to analyze the dynamic response of a single degree of freedom nonlinear packaging system under dropping impact. The research consists of dynamic response analysis for three types: free vibration of general nonlinear conservative system, free vibration of general nonlinear nonconservative system and dropping impact of general nonlinear nonconservative system.The max-min approach (MMA) and the homotopy analysis method (HAM) are introduced and the advantages and disadvantages of the two method are illustrated by some examples. For the classical tangent and hyperbolic tangent nonlinear system, governing equations have to be simplified by approximate Taylor series when the above two methods are used. The approximate governing equation greatly increases the analytical error. A novel method is proposed to simply and effectively solve this kind of more general second order nonlinear differential equation in this paper. Considering the influence of linear damping,solution expressed by the so called Ateb function of pure nonlinear nonconservative system is introduced. And the solution in the form of trigonometric function is developed. Due to the diversity of nonlinear, different forms of nonlinear systems may fallow the same motion and have the same motion characteristics.For the free vibration problem of a nonconservative system with a more general restoring force, the equivalent pure nonlinear equation of a general nonlinear differential equation is proposed. On this basis, the general solution of the free vibration of nonlinear nonconservative system is obtained. Comparisons with the numerical solution show that the method is simple, accurate and valid.The cushioning packaging system has the same dynamic equation in the dropping impact and free vibration. But the motion characteristics are different for the different initial conditions. The dynamic response of cubic and tangent nonlinear system in conservative and nonconservative forms are analyzed respectively and the results are very close to the numerical solution.In the end of this paper, the theoretical calculation is verified by experiment. Air cushion samples are impacted in different height and static stress. The experimental results show that the air cushion has uniform mechanical behavior under different conditions. Thus, the dynamic stress-strain relationship is established and the dynamic control equation is obtained.By using the method proposed in the theoretical calculation, the relationships are between maximum impact acceleration and static stress are obtained. And the dynamic cushioning curves of air cushion in different height are drawn. The good agreement between the theoretical values and the measured valued shows the validity and the correctness of the theoretical results.