1 Introduction Since the United States pioneered the concept of “advanced manufacturing technology†in the late 1980s, the research and development of advanced manufacturing technologies have received widespread attention worldwide, and China has also included advanced manufacturing technologies in the “Ninth Five-Year Planâ€. "One of the major areas of national science and technology development. Modern advanced manufacturing technology continuously absorbs the latest achievements of modern machinery, electronics, materials, information, computers, management, and other cutting-edge technologies through the innovative development of traditional manufacturing technologies, and applies them comprehensively to the development, design, manufacture, and inspection of products. , management, after-sales service and other manufacturing processes to achieve high quality, high efficiency, low consumption, clean, agile manufacturing. Modern advanced manufacturing technologies are advanced, versatile, systematic, and integrated, and are moving in the direction of precision, automation, flexibility, intelligence, and digitization. The research of the method of generating the digital surface is proposed to meet the needs of advanced manufacturing technology. It has important theoretical and practical significance for the precision processing of analytical surfaces with unknown or non-analytical expressions. 2 Research on the method of digital surface preparation The essence of the digital surface preparation method is to study the geometric relationship and relative motion relationship between the tool forming surface and the discretely expressed digital surface. This research mainly involves the following four aspects: 1 digitization of the surface; 2 digitized surfaces and their microscopic properties; 3 the conjugate theory of digitized surfaces; 4 the development and application of digitized surfaces. The digitization of surfaces can provide data for the formation of digital surfaces. The study of digital surfaces and their microscopic properties can provide a guarantee for the breakthrough of the theory of digital surface conjugation; the conjugate theory of digital surfaces can solve the problem of digital surface formation. Provides a theoretical basis; the application of digitized surface formation can verify the effectiveness of the method for generating a digital surface. The digitization of the digital surface of the surface is to determine the number and distribution of surface data points reasonably under the premise of control accuracy. The ultimate purpose is to provide reasonable and accurate three-dimensional data for the formation of digital surfaces. The digitalization of surfaces can be divided into digitization of real surfaces and digitization of virtual surfaces. The digitization of the real surface is achieved by three-dimensional measurement of the physical model or the processed workpiece entity; digitization of the virtual surface is based on modern design theory. The digitization of the digital surface of the real surface is the use of three-dimensional measurement technology to extract the three-dimensional data of the entity, thereby realizing the discretization of the real surface. In recent years, the development of modern three-dimensional measurement technologies (including sensing technology, control technology, laser technology, computer technology, and other related technologies) has provided the necessary technical means for the digitization of real surfaces. When realizing the digitization of real surfaces, the most commonly used three-dimensional non-contact measurement method is laser measurement. The digitization of the digital virtual surface of the virtual surface is a digitized surface represented by arsenic in the form of discrete data. The development and application of modern design theory and methods (including optimization design theory, modern finite element theory and boundary analysis methods, CAD/CAE technology, etc.) make it possible to digitize virtual surfaces. The design of new surfaces represented by discrete data has become an important development direction of surface design. Digital Surfaces and Their Microscopic Properties In recent decades, there have been many surfaces in production that cannot be described by mathematical expressions, such as sculpting surfaces with sharply varying surfaces in profile surfaces, free-form surfaces based on aircraft, body surfaces, and many Simple surface spliced ​​complex surfaces. Since the digitized surface is formed by digitizing the surface, the concrete mathematical expression is generally unknown. At present, the domestic research on digital surfaces is in its infancy. Although foreign countries began to study digital surfaces in the 1980s, there is no universally accepted definition. It is generally believed that a digitized surface represents a set of digitized points of a continuous surface. The digitized surface is not necessarily a real surface, it can be stored in a computer in a three-dimensional matrix. The digitized surface studied in this paper is divided into two forms: one is calculated by digital design; the other is measured after processing. In general, the microscopic properties of a surface include its principal (main) vector, principal direction, tangent vector, Gaussian curvature, and so on. The microscopic characteristics of the surface affect the surface's transmission efficiency, processing quality, stress conditions and so on. The differential geometry theory can be used to describe the micro-characteristics of analytically continuous general surfaces, but it is ineffective for digitized surfaces with only discrete point information. Because the digitized surface has discrete characteristics, the method for solving its microscopic characteristics is also different from the general surface. For example, when the normal vector is used, the normal vector formula can be discretized and solved by the numerical approximation method; the normal vector of a triangle surface around a certain point can also be used. Instead of the point normal vector, someone uses the local spline method to solve the midpoint tangent of three or five points; there is also a parametric quadratic curve to solve the main direction, a B-spline surface to solve the curvature, and a three-point circle. The arc method solves normal vectors and other methods. Comprehensive analysis shows that the methods for solving the microscopic characteristics of digital surfaces can be mainly divided into numerical methods and computational methods. The numerical method is to discretize the continuous problem, for example, to solve the tangent vector at a certain point by using the Tail expansion method. The computational geometry method is mainly to convert the digitized surface to an analytical surface by a spline surface, and then replace the digitized surface by the microscopic characteristics of the analytical surface. Microscopic properties. However, when using the above method to solve the microscopic characteristics of the digitized surface, there is inevitably a certain degree of principle error (such as the normal vector obtained by fitting the digitized surface with different spline surfaces is not the same). The research object of the digital surface formation method is a pair of conjugate surfaces—the tool surface and the digitized surface. According to the principle of conjugation, it can be known that the microscopic characteristics of the conjugated digital surface can be obtained by solving the microscopic characteristics of the tool surface. The Conjugate Theory of Digital Surfaces The object of the conjugate surface theory is the geometry and its conjugate motion. Its task is to study the intrinsic relationship and mutual transformation between conjugate geometry and conjugation motion under mechanical processing and mechanical transmission conditions. In practical applications, it is the problem of solving five types of conjugate surfaces (curvatures). In recent decades, some challenging problems have been encountered in the production practice, such as the meshing of quasi-hyperbolic gears and the contact of elastic gears. These transmission methods have broken through the rigid body assumption specification and continuity assumption of traditional mechanical transmission theory. This promotes the study of conjugate surface theory. The typical contents of conjugate surface theory include the study of the principle of elastic conjugate surface of paired elastic surface geometry and its conjugate motion, and the principle of discrete analytical conjugate surface that converts manual derivation calculation into computer adaptive processing. The assumption that the surface must be continuously tangentially contacted, the conjugated surface digital simulation principle that converts the model to a benchmark function, etc. The theory of digital conjugate surface mainly studies the connection and motion between the digitized surface expressed by the discrete point set and the conjugate analytical surface. It breaks through the limitation that the two pairs of conjugate surfaces required by the traditional conjugate theory must be analytic surfaces. The essence is to digitally transform the traditional analytical conjugate surface principle, which is to discretize the continuous variables. For discretizable variables such as position vectors, the position vector can be discretely discretized to a certain point; for non-discreteable variables such as the normal vector of a certain point, curvature, etc., relying on the variables for deriving an analytical expression, it is necessary to adopt other mathematical methods. solve. Therefore, the theory of digitized conjugate surfaces needs to solve the problem of how to discretize non-discrete variables. The theory of digital conjugate surface is born out of the traditional conjugate analytical theory. It uses modern numerical methods to transform the macroscopic and continuous surface conjugate problem into a microscopic and discrete point-to-point by means of computer's discrete processing capability of the data. Conjugate problem. Digital conjugate surface theory will provide a theoretical basis for the development of digital surfaces. The development and application of digital surfaces The problem of digital surface formation is the fifth category of the theory of digital conjugate surfaces, namely the known conjugate analytically expressed tool surfaces and the discretely expressed digitized surfaces (meshing surfaces). , find the conjugate motion of two conjugate surfaces. For the conjugate problem with the introduction of discrete surface expressions, scholars at home and abroad have conducted extensive research. Some people tried to use the free-form surface processing method to process digitized surfaces, but it was found that the processing accuracy was often difficult to meet the requirements. It has been proposed to use a discretely expressed tool surface to solve the machined contour surface. The discrete points can be divided into two cases: 1 The coordinates of each discrete point and the slope of the tangent are known. The solution is to start from a discrete point on the tool surface. Find the coordinates of the point on the contour surface of the workpiece that meets the meshing requirements. 2 Only the coordinates of the discrete point are given. In this case, the tangent slope of each point can be obtained by using a cubic spline function, and this can be converted into the first type. Solve the situation. It has been proposed to solve the discrete surface of the contour of the machined workpiece based on a mathematical model (analytical formula) of the tool surface. The tool mathematical model is r1(1)=r0(1)+Ar2(2)AIn2(2)r0a+Aar2(2)=0 where r1, r2 are position vectors, and (i) is represented in the coordinate system i. The problem, r0 is the distance between the origins of different coordinate systems, A is the transformation matrix, In2 is the surface normal vector, a is the motion parameter, and the subscript a is the derivation of the function a. By transforming the above mathematical model into a computational model, a computer area search algorithm is used to find the discrete surfaces of the tool processing. Someone is studying the meshing tooth surface corresponding to the discrete tooth surface and put forward a mathematical model of the numerical meshing surface ¢[t, x1(u), y1(u), dx1(u)/du, dy1(u)/du ]=A[t, x1(u), y1(u)]-B[t, x1(u), y1(u)]dy1(u)/du, where x1, x2, y1, y2 are surfaces respectively The coordinates of the position on points 1 and 2, where % is the time parameter and ' is the geometric parameter. By translating the problem into the derivative dyi/dxi of each discrete point, it is then solved by the method of deriving the interpolation function. Some people have studied the problems of the meshing analysis of the real tooth surface (the non-theoretical analytical expression of the actual machined tooth surface) and the theoretical tooth surface, and proposed to use the interpolation method to fit the real tooth surface into an analytical surface for study. The solution. The study of the problem of digital surface formation has some similarities with the above problems. The key is to solve the problem of discretization. Its theoretical basis is the principle of digital conjugate surface. Two conjugate surfaces are known and numerical methods can be used to solve the corresponding conjugate motion. This problem can be regarded as the inverse proposition of the analysis of the gear mechanism, namely the known mechanism diagram and the tooth surfaces of the two moving members, and the numerical solution method is used to solve the motion law. The digitized surface is developed to find the corresponding point on the analytical surface of the tool corresponding to the discrete points on the digitized surface, and then the two points are obtained through their respective conjugate motions to reach the conjugate contact point. Curved surface conjugates must meet two basic conditions: 1 contact conditions: r1 = r2, e1 = e2, that is, the conjugate points of the two surfaces and the normals of the curved surface units respectively overlap. According to the contact conditions, the two surfaces can be found to reach the contact through their respective movements. Point; 2 continuous transmission conditions: e · v12 = 0 (e⊥v12). According to the continuous transmission conditions, the instantaneous transmission ratio that satisfies the continuous contact transmission can be obtained, and the required movement parameters of the machine tool in the actual machining can be obtained. The mathematical expressions of the two basic conditions are now analyzed as follows. The contact condition coordinate systems s1, s2 are fixedly connected with two conjugate surfaces, respectively, and s is a fixed stationary reference coordinate system. The equations for the two surfaces are given in the coordinate systems s1, s2. Among them, the analytical expression of the tool surface is {r1=e1(u1, v1, w1) f1(u1, v1, w1)=0 {e1=(u1, v1, w1) f1(u1, v1, w1)=0 In the formula, u1, v1, w1 are the coordinates of the surface position, and there are only two independent variables. E1 is the unit normal of the tool surface at points (u1, v1, w1). The analytical expression of the digital surface is a set of digital coordinate data points, ie, r2={(u, v, w)|(ui,vi,wi),i=0,...,n} e2 is represented by a digitized surface The unit normal vector determined by the point (u, v, w)|(ui, vi, wi) can be regarded as an expression of (u, v, w)|(ui, vi, wi). It is known from the contact conditions that r1=r1(u1,v1,w1,a1)=r2(ui,vi,wi,a2)
E1(u1,v1,w1,a1)=e2(ui,vi,wi,a2) where a1, a2 are the motion parameters (ie, rotation angles). For the sake of simplicity, it is assumed that the conjugate motion is a single-parameter motion, that is, both motions in the conjugate motion (rotation around the axis and translation along the axis) can be represented by the motion parameter a. The principle of multi-parameter motion is similar to this. The above surface equation can be converted into 7 independent equations, and there are 8 unknown parameters (ui, vi, wi, a2, u1, v1, w1, a1), so one of the parameters can be used to express the other 7 parameters. The data points (ui, vi, wi) of the digitized surface are known, so a point on the digitized surface can be used to determine the conjugation motion of the two conjugate surfaces. Fixed the value of this parameter, you can find other parameters, they are all ui expressions. Continuous transmission conditions Assuming that the tool speed is constant and the digitized surface revolutions to be machined are geared, the gear ratio i12=w1/w2 can be determined using the meshing equation. According to the continuous transmission condition, e1·v12=e1·[(w1−w2)r1−Rw2]=0, where w1 is the rotation speed of the tool, w2 is the rotation speed of the digitized surface, and v12 is the relative movement speed of the two conjugate surfaces. In this equation, the radial vector r1 and the unit vector e1 are known, the w2, w1 direction is known, and the size and direction of the radial vector R at any point on the w2 line are known. When a given w1 module is given, w2 can be found. Module, from which you can find the size of the transmission ratio i12. By using the above method, the conjugate motion of the corresponding point of the two conjugate surfaces determined at one point on the digitized surface can be obtained. Similarly find the conjugate motion determined by all points on the digitized surface. Finally, these conjugate motions are subjected to a certain sorting process (for example, the arbitrary corners qi are arranged in ascending order of the tool's incremental corners for processing). The gear surface is the most typical complex surface in the engineering surface, and gear machining plays a decisive role in the manufacturing industry. Therefore, the digitized tooth surface can be produced and processed by using the precision machining performance of the CNC grinding machine, thereby verifying the method of digitizing the surface. Effectiveness. 3 Conclusion The research on the method of generating digital surface surpasses the traditional theory of analytical generation and transformation. It has important theoretical and practical significance for high-precision and high-efficiency machining of gears. The research results can also be directly used for the profiling of unknown tooth surfaces. Processing. The research results of digital surface formation methods can provide a technical basis for CAD/CAM integrated manufacturing and digital manufacturing of gears, thus forming a complete gear digital processing technology.
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