Research on Linear Simplification Technology of Nonlinear Cushion Packaging System

[Abstract] Most common buffer packaging systems are non-linear systems, which make it difficult to design buffer packaging. Because the analysis method of dynamics of linear system is very mature at present, when doing buffered package design, the nonlinear buffer packaging system should be simplified as a linear system to analyze and calculate. This paper analyzes and studies the method of simplifying the nonlinear buffer packaging system into a linear system and the problems that should be paid attention to in the linearization process.

Keywords: buffer packaging; nonlinear system; linearization; simplified method

In the buffer packaging, the key step is to establish a dynamic model of a reasonable buffer packaging system. The actual buffer packaging system is often a complex dynamic system. How to abstract a simple dynamic model that can meet the needs of actual packaging engineering design is still a problem worthy of attention.

L1] analyzed how to reduce the two-degree-of-freedom buffer packaging system into two independent single-degree-of-freedom systems to analyze the computational problems. The literature [2] further analyzes the influence of the design goal on the simplification of the dynamic model, and points out that it is suitable to calculate and analyze a simplified model of the maximum displacement amplitude of a moderate to medium packaging system. It is often inappropriate to analyze and calculate the velocity of the buffer packaging system. Amplitude, because the simple two-degree-of-freedom vibration system is forced to use two independent single-degree-of-freedom systems to calculate the acceleration of the system, the result will be far from the actual acceleration, and it can not meet the needs of engineering design. .

Both the literature [1] and the literature [2] use the linear system as the research object to study the simplification of the easing and middle packaging system. The use of a linear dynamics system to describe the buffer packaging system is a common and simplified method. The biggest advantage of using a linear system is that it can use the superposition principle. Under the superposition principle, the response of the system to different excitations can be linearly added, but for the nonlinear system, the superposition principle is not established. In addition, the analysis methods for linear systems have also been developed.

Dealing with nonlinear dynamic systems often requires a completely different approach than dealing with linear systems. The development of nonlinear differential equations in mathematics is far inferior to linear differential equations, which makes the study of nonlinear dynamics much more complex than that of linear systems. Even though it is now possible to perform numerical simulations using computers, the analysis of nonlinear dynamics is still a very difficult task. In many cases, only approximate solutions can be obtained. Therefore, when analyzing and studying the buffer packaging system, linear dynamics should be used as far as possible. Learning system model. In fact, in many cases, the use of equivalent linearization of buffer packaging systems for simplified calculations can meet the needs of packaging engineering. This paper analyzes and studies the method of nonlinear easing: the simplified packaging system into a linear system and the issues that should be paid attention to when performing linearization.

1 Linearization technology

There are two basic methods for linearizing the cushioning packaging system: the actual buffer packaging system is directly described as a linear dynamics system. For example, in buffer packaging, a large number of buffer packaging materials are used as non-linear cushioning materials, most of which have the technology reflected by the tangent elastomer. Such cushioning packaging materials include high-foaming polyethylene plastics, foam rubbers, rubberized fibers, air cushions for cushioning packaging, and the like. The load deformation relationship of the tangent elastomer under static load is:

In the formula, f (x) is the load of the cushioning packaging material; x is the initial elastic modulus of the artificial cushioning packaging material for the deformation amount of the cushioning packaging material; d1 is the ultimate deformation amount of the cushioning packaging material. When the deformation of the cushioning material reaches or exceeds, the cushioning properties are lost.

At present, the buffer packaging design using tangent-type elastomer material is basically analyzed and calculated from the perspective of statics, mainly considering the response of the packaged object to the mid-blast when it is dropped from a high place. The basic method is to assume that when the buffer packaging system is subjected to the maximum impact, all the mechanical energy of the packaged material is converted into the deformation energy of the buffer material, and then the buffer packaging design is performed according to the law of energy conservation. This method is not suitable for dynamic analysis of buffered packaging systems subjected to continuous vibration excitation.

Non-linear cushioning systems are used to construct nonlinear dynamic systems. Without considering the damping of the system, the dynamic equation of the tangential-type elastomer material in the cushioning-middle packaging system can be expressed by the following formula:

In the formula, z is the relative displacement of the packaged object and the buffer packaging material; m is the mass of the packaged object; f(t) is the equivalent excitation force received by the packaged object.

Equation (2) does not mathematically solve the general feasible method. If it is a deterministic incentive, it can be analyzed numerically by means of a sub-computer; if it is a random excitation, although it can also be analyzed by numerical simulation, it is much more difficult. but if Very small, then:

Substituting Equation (3) into Equation (1), the original nonlinear system is transformed into a linear system:

In fact, in order to ensure the reliability of the buffer packaging system, most buffer packaging systems should be designed to work in this equivalent linear range. Therefore, in this case, using the linear system model, the calculation accuracy can fully meet the needs of packaging engineering design, there is no need to adopt a complex nonlinear dynamic model.

The second method to achieve linearization is to describe the actual slow and medium packaging system as a nonlinear dynamic system, and establish a corresponding nonlinear dynamics mathematical model, and then perform the equivalent linearization of the mathematical model. A number of different equivalent linear methods have been developed. The choice of parameters for the equivalent linear differential equation must be such that the difference between the equivalent linear system and the nonlinear system before the equivalent is a minimum (usually the minimum mean square error).

(to be continued)