Fractal theory was put forward in 1970s. Because it can solve a large number of realistic set problems in life, such fractal theory is developing continuously. With the improvement of fractal theory, fractal theory has been applied to various industries and achieved good results. Mechanical engineering is an important index to evaluate the development level of science and technology in China, and the application of fractal theory in mechanical engineering further promotes the faster development of mechanical engineering. Through the application of fractal theory, we can more intuitively reflect the state of mechanical objects in the actual movement process, more accurately check mechanical faults and mechanical fractures, and solve most problems in mechanical engineering. In the future development, fractal theory will be more widely used in mechanical engineering.
Concept and main characteristics of 1 fractal theory
For fractal theory, the earliest origin is to describe and study the set problem from the mathematical point of view. The fractal theory was put forward by Mandelbrot, who put forward the problem of describing the length of British coastline in Science magazine. The coastline is a very irregular curve. In order to distinguish the coastline from the actual structure, we can't just describe it from the shape. Judging from the shape of the coastline taken in the air, the two photos compared look very similar. This similarity refers to the similarity between the local shape and the overall shape, which is widespread in nature. Similarly, this similarity is also reflected in the current mechanical engineering.
For example, there are many irregular and complex objects in mechanical engineering, which cannot be described in detail, but after the fractal theory is put forward, we can evaluate and study it with the help of this theory. In view of the current definition of fractal theory, it is impossible to define it scientifically and clearly because it is constantly developing, but literally, we can describe it as follows: Fractal in fractal theory mainly refers to some complex and fragmented substances, which have some similar characteristics and can form a certain system, but from a mathematical point of view, fractal has a certain proportion, and it can be correct in the whole fractal system.
In addition, in the fractal system, these changes are similar in rotation, displacement and amplification. From the above description, we can draw the main characteristics of fractal theory, including proportionality and permutation invariance, which are the basic characteristics of fractal system. These two characteristics can ensure that the actual fractal system is not completely chaotic and untraceable, and from the definition of fractal theory, we can also draw another characteristic of fractal theory-it can describe and study things based on fractal dimension and mathematical methods. Based on the two characteristics described above, we can see that in the real material world, all objects can reflect the irregular shape of objects through their proportions, and then describe and study objects more clearly.
2 the development process of fractal theory
Fractal theory was first born in the 1970s, when it was only a simple description of objective things, but in the later development process, fractal theory gradually matured, and its development process can be roughly divided into the following stages.
2. 1 fractal theory put forward stage
This stage occurs from 1870 to 1930. At this stage, people first put forward the basic definition of fractal theory and studied it from various angles. On this basis, some objects to which fractal theory can be applied are developed. The representative event of this stage is that Wilstrass proved that there is no finite or infinite derivative at any point in a continuous function. After this theory was put forward, people developed various fractal objects, including peano curve, Brownian motion and Cantor's triple diversity, which showed the basic characteristics of fractal theory.
2.2 the mature stage of fractal theory
After more in-depth research and analysis of fractal theory, people find that fractal objects are embodied in all aspects of nature. On the basis of the increasing number of fractal objects, fractal theory is gradually maturing. At this stage, people not only defined the fractal theory more strictly and scientifically, but also formed the corresponding fractal theory system. On the basis of describing fractal theory from a mathematical point of view, the concept of dimension is put forward. Although fractal theory has been greatly supplemented, it also has some limitations. Fractal theory is in the theoretical research stage, which has little connection with other fields and is not practical.
2.3 the perfect stage of fractal theory
At this stage, fractal theory not only formed an independent specialty and discipline, but also scientifically expounded the content, significance and methods of molecular geometry, and had great connections with other disciplines. In the research, the practicability of fractal theory has been greatly improved. At present, fractal theory has been applied to mechanical engineering, mechanics, molecular chain and material control. In the process of development, we can see that fractal theory has strong application. As an important theory in modern society, mechanical engineering has also begun to use fractal theory to solve various mechanical problems in the actual development process.
3 The concrete practice of fractal theory in mechanical engineering
Fractal theory is an important part of nonlinear science. With people's in-depth study, fractal theory can objectively and accurately describe and evaluate irregular things in a more rigorous mathematical description way. At the same time, it has the characteristics of proportionality and displacement invariance, so that fractal theory can solve problems in mechanical engineering. Its concrete practice and application are embodied in the following aspects.
3. The application of1in the study of mechanical friction
Mechanical friction has many aspects. As far as the practical application of fractal theory in mechanical friction is concerned, it is mainly used to predict the contact wear of rough surfaces and study the friction temperature distribution. Because of the relative independence of fractal parameters, the shape characteristics of rough surfaces and mechanical structures can be described intuitively and objectively. In addition, according to the results of tribology research model established by fractal system, the influence of measuring instrument precision and sampling length can be ignored, which can further improve the accuracy and effectiveness of the model.
3. 1.65438+ Moreover, on the other hand, due to the different resolutions of measuring instruments, the results will be different. In this case, the traditional parameters can only describe the morphology of rough parts in a certain scale. However, fractal theory can show the unique characteristics of rough parts' surfaces, which is mainly manifested by using the definition of fractal dimension and the corresponding calculation method to determine its fractal dimension and fractal parameters, thus describing the actual morphology of rough parts' surfaces.
3. 1.2 describes the temperature distribution of sliding friction. In the traditional friction temperature research model, only the highest temperature and average temperature in the contact area of parts can be predicted, and the results are not clear. In order to accurately describe the temperature distribution of friction performance of sliding contact surface of mechanical parts, fractal theory can be used to describe the traditional temperature research theory. The corresponding temperature distribution density function and temperature cumulative distribution function are established, which can infer the temperature distribution in the contact surface to the greatest extent. Under such research, it can provide conditions for studying the use of oil, the oxidation of the surface of mechanical parts and the high temperature wear of mechanical parts.
3. 1.3 Study on fractal model of wear prediction Wear prediction has always been an important part of mechanical engineering. Mechanical parts will be worn by friction during movement. In order to study the wear law of materials, it is necessary to establish the corresponding wear numerical model, but the friction coefficient is uncertain. At present, the established wear numerical model can only predict the friction law in a wide range and cannot be applied to practice. The establishment of fractal theory can provide a good foundation for the establishment of wear model to a great extent. This fractal theory can study the performance parameters and fractal parameters of materials, and quantitatively analyze the wear on this basis, so as to get the lowest fractal dimension value of wear rate, which is basically similar to the experimental value, thus reflecting the effectiveness and accuracy of the application of fractal theory in mechanical engineering.
3.2 Application in Fatigue Fracture Analysis
The fatigue fracture of mechanical parts often has irregular characteristics, which is not only manifested in the fracture morphology, but also in the propagation path of fracture cracks. Such irregularity also makes the fracture phenomenon of mechanical body more uncertain. In order to accurately analyze the fatigue fracture phenomenon of mechanical parts, the traditional methods mainly include standard variance method and peak distribution method. These methods analyze fatigue fracture by calculating and determining fracture area and variance, but their analysis results are often inaccurate. In this case, fractal theory can be used to solve the problem of linear extension of fatigue fracture. At present, people have established the corresponding fatigue crack analysis system and analyzed the influence of fatigue crack. On the basis of this analysis, the propagation speed and range of fatigue cracks can be controlled to a certain extent, which has high applicability.
3.3 Application in Mechanical Fault Research
At present, the fractal theory is used to study and analyze mechanical faults, mainly in the following two aspects: (1) extracting the fractal characteristics of mechanical parts wear debris, and calculating the wear rate of mechanical equipment on the basis of studying and calculating the fractal dimension of the model, thus providing basis for on-line fault diagnosis of mechanical equipment; ② Measure the operation characteristic signal of mechanical equipment, and extract the fractal characteristics of the signal according to the actual measurement results, so as to analyze the fault state and actual operation of mechanical equipment. According to the different operating characteristics of machinery, different fractal theory fault diagnosis methods can be adopted. The above research methods of mechanical faults can generally be applied to the fault diagnosis of mechanical friction and rotating machinery.
3.4 Applied to the design of complex products
In current mechanical engineering, product design is an important part. Good product design can greatly improve the internal quality, quality and design cost of products. Different from other mechanical engineering activities, design is a creative activity. In the current application of modern science and technology, intelligent design and automatic design will be the most important development direction of mechanical product design. However, there are many problems in traditional product design, mainly in the fact that only the symbolic reasoning technology in a single knowledge field is used in the actual design process, but this technology can only solve some local problems to a small extent. In the current product design, in addition to the computer integrated system occupying the main position in product design, people are also an important part in the design field. In order to reflect the main mode of innovative design, some designers put forward the viewpoint of product complexity design on the basis of fractal theory, that is, in order to reflect the complexity of design objects, the design method of fractal theory can be adopted, which can ensure the innovation and regularity of product design and promote the rapid development of mechanical product design.
4 conclusion
Fractal theory is an important branch of natural science. With the continuous development of fractal theory, fractal theory has gradually penetrated into various fields. In the current field of mechanical engineering, fractal theory can play an important role in product design, fault diagnosis and mechanical parts analysis, and then deduce the hidden laws in mechanical engineering to promote the continuous development of mechanical engineering.
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