At present, most commercial computer-aided manufacturing （CAM） systems are deficient in efficiency and performances on generating tool path during machining impellers. To solve the problem, this article develops a s...At present, most commercial computer-aided manufacturing （CAM） systems are deficient in efficiency and performances on generating tool path during machining impellers. To solve the problem, this article develops a special software to plan cutting path for ruled surface impellers. An approximation algorithm to generate cutting path for machining integral ruled surface impellers is proposed. By fitting sampling data points of an impeller blade into a curve, a model of ruled surface blade of an impeller is built up. Furthermore, by calculating the points where the cutter axis vector intersects the free-form hub surface of an impeller, problems about, for instance, the ambiguity in calculation and machining the wide blade surface with a short flute cutter are solved. Finally, an integral impeller cutting path is planned by way of an integrated cutter location control algorithm. Simulation and machining tests with an impeller are performed on a 5-axis computer numerically controlled （CNC） mill machine, which shows the feasibility of the proposed algorithm.展开更多
Predicting the cutting forces required for five-axis flank milling is a challenging task due to the difficulties involved in determining the Undeformed Chip Thickness(UCT) and CutterWorkpiece Engagement(CWE). To solve...Predicting the cutting forces required for five-axis flank milling is a challenging task due to the difficulties involved in determining the Undeformed Chip Thickness(UCT) and CutterWorkpiece Engagement(CWE). To solve these problems, this paper presents a new mechanistic cutting force model based on the geometrical analysis of a flank milling process. In the model,the part feature and corresponding cutting location data are taken as input information. The UCT considering cutter runout is calculated according to the instantaneous feed rate of the element cutting edges. A solid-discrete-based method is used to precisely and efficiently identify the CWE between the end mill and the surface being machined. Then, after calibrating the specific force coef-ficients, the mechanistic milling force can be obtained. During the validation process, two practical operations, three-axis flank milling of a vertical surface and five-axis flank milling of a nondevelopable ruled surface, are conducted. Comparisons between predicted and measured cutting forces demonstrate the reliability of the proposed cutting force model.展开更多
Motivated by the definition of the machining errors induced by tool path planning methods, a mapping curve of the tool axis of a cylindrical cutter is constructed on the tool surface. The mapping curve is a typical on...Motivated by the definition of the machining errors induced by tool path planning methods, a mapping curve of the tool axis of a cylindrical cutter is constructed on the tool surface. The mapping curve is a typical one that can be used to express the closeness between the tool surface and the surface to be machined. A novel tool path planning method is proposed for flank or plunge milling ruled surfaces based on the minimization of the one-sided Hausdorff distance （HD） from the mapping curve to the surface to be machined. It is a nonlinear optimization problem in best uniform approximation （BUA） or Chebyshev sense. A mathematical programming model for computing the minimum one-sided HD is proposed. The linearization method of the programming model is provided and the final optimal solutions are obtained by simplex method. The effectiveness of the proposed BUA method is verified by two numerical examples and compared with the least squares （LS） and double point offset （DPO） methods. The variation in tool orientation induced by the optimization of the tool positions is also evaluated.展开更多
In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the ...In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.展开更多
On the tasis of study in the mathematical model of 3-dimensional ruled surface (RS),this paper introduces a new concept of distance paramcter (DP) and also puts forward that themethod of modeling a RS depends on not o...On the tasis of study in the mathematical model of 3-dimensional ruled surface (RS),this paper introduces a new concept of distance paramcter (DP) and also puts forward that themethod of modeling a RS depends on not only two boundary curves but also DP. According toabove theory, the formulas to calculate corresponding point coordinates to any kind of top and bot-tom profile of a workpiece and formulas to calcuate the maximum inclination angle of ruling linehave been obtained. Then a different top and bottom RS mathining method including profile withline-are combination as well as parametric curves has been achieved by 4-axes simultancous con-trol programming proposed.展开更多
Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermit...Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O to be a minimal submanifold of M 6.展开更多
We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a speci...We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.展开更多
In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M2(c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.
A unified approach to kinematic synthesis of mechanism is presented in this paper. Firstly a new approach to adaptive curve fitting is presented, which leads the normal fitting error to be minimum for a series of give...A unified approach to kinematic synthesis of mechanism is presented in this paper. Firstly a new approach to adaptive curve fitting is presented, which leads the normal fitting error to be minimum for a series of given discrete points, including a plane curve fitting, a spherical curve fitting and a ruled surface fitting in terms of invariants of ruled surface. Approximate characteristic points and lines are defined, such as an approximate circle point, an approximate slide point, an approximate spherical cone point and an approximate constant axis with an approximate spherical image cone point and an approximate striction curve. Then,the ruled surface fitting will be converted into a space curve fitting and a spherical curve fitting by differential geometry. Based on these definitions and the adaptive curve fitting approaches, the unified mathematical model is set up for the kinematic synthesis of mechanism from planar, spherical to spatial motion. Finally, a planar mechanism or a spatial mechanism can be synthesized by means of searching for two approximate characteristic points, or a characteristic point and a characteristic line, even two characteristic lines. This puper lays a theoretic base for the existence of the best solution and the convergence of the optimum algorithm.展开更多
Fundamental matrix,drawing geometric relationship between two images,plays an important role in 3- dimensional computer vision.Degenerate configurations of the space points and the two camera optical centers affect st...Fundamental matrix,drawing geometric relationship between two images,plays an important role in 3- dimensional computer vision.Degenerate configurations of the space points and the two camera optical centers affect stability of computation for fundamental matrix.In order to robustly estimate fundamental matrix,it is necessary to study these degenerate configurations.We analyze all the possible degenerate configurations caused by twisted cubic and give the corresponding degenerate rank for each case.Relationships with general degeneracies,the previous ruled quadric degeneracy and the homography degeneracy,are also reported.展开更多
A Blaschke hypersurface admits S symmetry if and only if S（X, Y） = S（Y, X）. We prove that the shape operator has only one eigenvalue. And such Blaschke surfaces are classified as affine spheres or ruled surfaces.
In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directr...In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.展开更多
We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space.A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-posi...We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space.A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given.展开更多
The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relations...The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.展开更多
A new concept of design and manufacturing of ruled surface based on line geometry is proposed. Some practical algorithm for CAD system is derived. Some problems in design and manufacturing of ruled surface can be solv...A new concept of design and manufacturing of ruled surface based on line geometry is proposed. Some practical algorithm for CAD system is derived. Some problems in design and manufacturing of ruled surface can be solved by using the algorithm.展开更多
Ruled surfaces found in engineering parts are often blended with a constraint surface,like the blade surface and hub surface of a centrifugal impeller.It is significant to accurately machine these ruled surfaces in fl...Ruled surfaces found in engineering parts are often blended with a constraint surface,like the blade surface and hub surface of a centrifugal impeller.It is significant to accurately machine these ruled surfaces in flank milling with interference-free and fairing tool path,while current models in fulfilling these goals are complex and rare.In this paper,a tool path planning method with optimal cutter locations(CLs)is proposed for 5-axis flank milling of ruled surfaces under multiple geometric constraints.To be specific,a concise three-point contact tool positioning model is firstly developed for a cylindrical cutter.Different tool orientations arise when varying the three contact positions and a tool orientation pool with acceptable cutter-surface deviation is constructed using a meta-heuristic algorithm.Fairing angular curves are derived from candidates in this pool,and then curve registration for cutter tip point on each determined tool axis is performed in respect of interference avoidance and geometric smoothness.On this basis,an adaptive interval determination model is developed for deviation control of interpolated cutter locations.This model is designed to be independent of the CL optimization process so that multiple CLs can be planned simultaneously with parallel computing technique.Finally,tests are performed on representative surfaces and the results show the method has advantages over previous meta-heuristic tool path planning approaches in both machining accuracy and computation time,and receives the best comprehensive performance compared to other multi-constrained methods when machining an impeller.展开更多
基金Key Development Program of Science and Technology of Heilongjiang Province, China (GB05A501)
文摘At present, most commercial computer-aided manufacturing （CAM） systems are deficient in efficiency and performances on generating tool path during machining impellers. To solve the problem, this article develops a special software to plan cutting path for ruled surface impellers. An approximation algorithm to generate cutting path for machining integral ruled surface impellers is proposed. By fitting sampling data points of an impeller blade into a curve, a model of ruled surface blade of an impeller is built up. Furthermore, by calculating the points where the cutter axis vector intersects the free-form hub surface of an impeller, problems about, for instance, the ambiguity in calculation and machining the wide blade surface with a short flute cutter are solved. Finally, an integral impeller cutting path is planned by way of an integrated cutter location control algorithm. Simulation and machining tests with an impeller are performed on a 5-axis computer numerically controlled （CNC） mill machine, which shows the feasibility of the proposed algorithm.
基金co-supported by the Major National S&T Program(No.2014ZX04014-031)the National Natural Science Foundation of China(No.51225503)
文摘Predicting the cutting forces required for five-axis flank milling is a challenging task due to the difficulties involved in determining the Undeformed Chip Thickness(UCT) and CutterWorkpiece Engagement(CWE). To solve these problems, this paper presents a new mechanistic cutting force model based on the geometrical analysis of a flank milling process. In the model,the part feature and corresponding cutting location data are taken as input information. The UCT considering cutter runout is calculated according to the instantaneous feed rate of the element cutting edges. A solid-discrete-based method is used to precisely and efficiently identify the CWE between the end mill and the surface being machined. Then, after calibrating the specific force coef-ficients, the mechanistic milling force can be obtained. During the validation process, two practical operations, three-axis flank milling of a vertical surface and five-axis flank milling of a nondevelopable ruled surface, are conducted. Comparisons between predicted and measured cutting forces demonstrate the reliability of the proposed cutting force model.
基金supported by the National Natural Science Foundation of China (51175065)
文摘Motivated by the definition of the machining errors induced by tool path planning methods, a mapping curve of the tool axis of a cylindrical cutter is constructed on the tool surface. The mapping curve is a typical one that can be used to express the closeness between the tool surface and the surface to be machined. A novel tool path planning method is proposed for flank or plunge milling ruled surfaces based on the minimization of the one-sided Hausdorff distance （HD） from the mapping curve to the surface to be machined. It is a nonlinear optimization problem in best uniform approximation （BUA） or Chebyshev sense. A mathematical programming model for computing the minimum one-sided HD is proposed. The linearization method of the programming model is provided and the final optimal solutions are obtained by simplex method. The effectiveness of the proposed BUA method is verified by two numerical examples and compared with the least squares （LS） and double point offset （DPO） methods. The variation in tool orientation induced by the optimization of the tool positions is also evaluated.
基金This paper is partially supported by the National Fundamental Research 973 Program of China under Grant No.2004CB318000.
文摘In this paper, we present a proper reparametrization algorithm for rational ruled surfaces. That is, for an improper rational parametrization of a ruled surface, we construct a proper rational parametrization for the same surface. The algorithm consists of three steps. We first reparametrize the improper rational parametrization caused by improper supports. Then the improper rational parametrization is transformed to a new one which is proper in one of the parameters. Finally, the problem is reduced to the proper reparametrization of planar rational algebraic curves.
文摘On the tasis of study in the mathematical model of 3-dimensional ruled surface (RS),this paper introduces a new concept of distance paramcter (DP) and also puts forward that themethod of modeling a RS depends on not only two boundary curves but also DP. According toabove theory, the formulas to calculate corresponding point coordinates to any kind of top and bot-tom profile of a workpiece and formulas to calcuate the maximum inclination angle of ruling linehave been obtained. Then a different top and bottom RS mathining method including profile withline-are combination as well as parametric curves has been achieved by 4-axes simultancous con-trol programming proposed.
文摘Proves that cosymplectic hypersurfaces of six dimensional Hermitian submanifolds of the octave algebra are ruled manifolds and establishes a necessary and sufficient condition for a cosymplectic hypersurface of Hermitian M 6 O to be a minimal submanifold of M 6.
文摘We consider the Bonnet ruled surfaces which admit only one non-trivial isometry that preserves the principal curvatures. We determine the Bonnet ruled surfaces whose generators and orthogonal trajectories form a special net called an A-net.
文摘In this paper, we study a real hypersurface M in a non-at 2-dimensional complex space form M2(c) with η-parallel Ricci and shape operators. The characterizations of these real hypersurfaces are obtained.
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. 59675003 and 59975013).
文摘A unified approach to kinematic synthesis of mechanism is presented in this paper. Firstly a new approach to adaptive curve fitting is presented, which leads the normal fitting error to be minimum for a series of given discrete points, including a plane curve fitting, a spherical curve fitting and a ruled surface fitting in terms of invariants of ruled surface. Approximate characteristic points and lines are defined, such as an approximate circle point, an approximate slide point, an approximate spherical cone point and an approximate constant axis with an approximate spherical image cone point and an approximate striction curve. Then,the ruled surface fitting will be converted into a space curve fitting and a spherical curve fitting by differential geometry. Based on these definitions and the adaptive curve fitting approaches, the unified mathematical model is set up for the kinematic synthesis of mechanism from planar, spherical to spatial motion. Finally, a planar mechanism or a spatial mechanism can be synthesized by means of searching for two approximate characteristic points, or a characteristic point and a characteristic line, even two characteristic lines. This puper lays a theoretic base for the existence of the best solution and the convergence of the optimum algorithm.
基金Supported by the National Natural Science Foundation of China under Grant Nos.60835003 and 60773039.
文摘Fundamental matrix,drawing geometric relationship between two images,plays an important role in 3- dimensional computer vision.Degenerate configurations of the space points and the two camera optical centers affect stability of computation for fundamental matrix.In order to robustly estimate fundamental matrix,it is necessary to study these degenerate configurations.We analyze all the possible degenerate configurations caused by twisted cubic and give the corresponding degenerate rank for each case.Relationships with general degeneracies,the previous ruled quadric degeneracy and the homography degeneracy,are also reported.
文摘A Blaschke hypersurface admits S symmetry if and only if S（X, Y） = S（Y, X）. We prove that the shape operator has only one eigenvalue. And such Blaschke surfaces are classified as affine spheres or ruled surfaces.
文摘In this paper, evolutions of ruled surfaces generated by the quasi normal and quasi binormal vector fields of space curve are presented. These evolutions of the ruled surfaces depend on the evolutions of their directrix using quasi frame along a space curve.
基金This work is partially supported by the National Science Foundation of China
文摘We study the volume growth of the geodesic balls of a minimal submanifold in a Euclidean space.A necessary condition for the isometric minimal immersion into a Euclidean space is obtained. A classification of non-positively curved minimal hypersurfaces in a Euclidean space is given.
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. 10261002 and 10671009)
文摘The ruled surfaces of normals and binormals of a space curve is locally classified under the left-right action according to the types of the curve. In order to do this some useful results are obtained on the relationship of the powers of terms in the Taylor series of an invertible function and its inverse.
文摘A new concept of design and manufacturing of ruled surface based on line geometry is proposed. Some practical algorithm for CAD system is derived. Some problems in design and manufacturing of ruled surface can be solved by using the algorithm.
基金supported by the National Natural Science Foundation of China(Nos.U22A20202 and 52205516)the China Postdoctoral Science Foundation(No.2022 M720641)。
文摘Ruled surfaces found in engineering parts are often blended with a constraint surface,like the blade surface and hub surface of a centrifugal impeller.It is significant to accurately machine these ruled surfaces in flank milling with interference-free and fairing tool path,while current models in fulfilling these goals are complex and rare.In this paper,a tool path planning method with optimal cutter locations(CLs)is proposed for 5-axis flank milling of ruled surfaces under multiple geometric constraints.To be specific,a concise three-point contact tool positioning model is firstly developed for a cylindrical cutter.Different tool orientations arise when varying the three contact positions and a tool orientation pool with acceptable cutter-surface deviation is constructed using a meta-heuristic algorithm.Fairing angular curves are derived from candidates in this pool,and then curve registration for cutter tip point on each determined tool axis is performed in respect of interference avoidance and geometric smoothness.On this basis,an adaptive interval determination model is developed for deviation control of interpolated cutter locations.This model is designed to be independent of the CL optimization process so that multiple CLs can be planned simultaneously with parallel computing technique.Finally,tests are performed on representative surfaces and the results show the method has advantages over previous meta-heuristic tool path planning approaches in both machining accuracy and computation time,and receives the best comprehensive performance compared to other multi-constrained methods when machining an impeller.
