Research on linear simplification technology of th

2022-10-01
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Research on linear simplification technology of nonlinear cushioning packaging system

[Abstract] most common cushioning packaging systems are nonlinear systems, which makes it very difficult to design cushioning packaging. At present, the research on the dynamic analysis method of linear system has been very mature, so when designing cushioning packaging, we should try our best to simplify the nonlinear cushioning packaging system into a linear system for analysis and calculation. This paper analyzes and studies the method of simplifying the nonlinear cushioning packaging system into a linear system and the problems that should be paid attention to when linearizing

key words: buffer packaging; Nonlinear system; Linearization; Simplified method

when carrying out buffer packaging, the key step is to establish a reasonable dynamic model of the potentiometer designed by the buffer packaging system to achieve the purpose of measuring displacement. The actual cushioning 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

literature l1] analyzed and discussed how to simplify the two degree of freedom buffer packaging system into two independent single degree of freedom systems to analyze and calculate. Literature [2] further analyzes the influence of the design goal on the simplification of the dynamic model, and points out that the simplified model suitable for calculating and analyzing the maximum displacement amplitude of the medium packaging system is often not suitable for analyzing and calculating the force mouth velocity amplitude of the buffer packaging system, because when the original two degree of freedom vibration system is forcibly used as two independent single degree of freedom systems to calculate the acceleration of the system, The results obtained will be far from the actual acceleration, and can not meet the needs of engineering design at all

literature [1] and literature [2] both take linear system as the research object to study the simplification of packaging system. Using linear dynamic system to describe cushioning packaging system is a common simplified method. The biggest advantage of adopting linear system is that superposition principle can be adopted. Under the superposition principle, the responses of the system to different excitations can be added linearly, while for nonlinear systems, the superposition principle is not tenable. In addition, the analysis method of linear system has also been developed relatively well

dealing with nonlinear dynamic systems often requires completely different methods from dealing with linear systems. In mathematics, the development of nonlinear differential equation theory is far less than linear differential equation, which makes the study of nonlinear dynamics much more complex than linear system. Even now, it is still very difficult to analyze and study the nonlinear dynamics with the help of computer numerical simulation. In many cases, only approximate solutions can be obtained. Therefore, when analyzing and studying the cushioning packaging system, we should try to use the linear dynamics system model. In fact, in many cases, the accuracy of simplifying the calculation of the cushioning packaging system by equivalent linearization can meet the needs of packaging engineering design. This paper will analyze and study the method of simplifying the non-linear packaging system into a linear system and the problems that should be paid attention to when linearizing

1 linearization technology

there are two basic methods to linearize the buffer packaging system: the actual buffer packaging system is directly described as a linear dynamic system. For example, in cushioning packaging, a large number of cushioning packaging materials are nonlinear cushioning materials, most of which have the technology reflected by tangent elastomers. This kind of cushioning packaging materials include high foaming polyethylene plastic, foam rubber, gelatinized fiber, air cushion for cushioning packaging, etc. The load deformation relationship of tangent elastomer under static load is:

in the formula, f (x) is the buffer package, and Jinan gold assay spring tension and compression testing machine is mainly used to measure the load of spring loaded assembly materials; X is the deformation of cushioning packaging material, and the initial elastic coefficient of artificial cushioning packaging material; D1 is the limit deformation of cushioning packaging material. When the deformation of cushioning packaging materials reaches or exceeds, the third method is to use a centrifuge for "rapid mixing", which will lose the cushioning performance

at present, the cushion packaging design using tangent elastomer materials is basically analyzed and calculated from the perspective of statics, mainly considering the impact of the packaged object when it falls from a height; Response to a hit. The basic method is to assume that when the cushioning packaging system is under the maximum impact, all the mechanical energy of the packaged material is converted into the deformation energy of the cushioning material, and then the cushioning packaging design is carried out according to the law of energy conservation. This method is not suitable for the dynamic analysis of cushioning packaging system under continuous vibration excitation

a nonlinear dynamic system is composed of a nonlinear cushioning packaging system. Without considering the damping of the system, the dynamic equation of the moderate packaging system of tangent elastomer material can be expressed by the following formula:

where Z is the relative displacement between the packaged material and the cushioning packaging material; M is the mass of the package; F (T) is the equivalent excitation force received by the package

equation (2) lacks a general and feasible method to solve it mathematically. If it is deterministic excitation, the numerical calculation and analysis can be carried out with the help of a sub computer; If it is a random excitation, although the factors such as crystallinity and porosity can also affect its degradation rate to be analyzed by numerical simulation, it is much more difficult. But if it is very small, then:

replace equation (3) into equation (1), and the original nonlinear system will be 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, the calculation accuracy of linear system model can fully meet the needs of packaging engineering design, and it is not necessary to use complex nonlinear dynamic model

the second method to realize linearization is to reduce the actual delay; The medium packaging system is described as a nonlinear dynamic system, and the corresponding nonlinear dynamic mathematical model is established, and then the mathematical model is equivalent linearized. At present, many different equivalent linear methods have been developed. The parameter selection of equivalent linear differential equation must minimize the difference between the equivalent linear system and the non-linear system in a sense (usually the minimum mean square deviation)

(to be continued)

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