Differential Geometrical Conditions of Hypoid Gears with Conjugate Tooth Surfaces

1998 ◽  
Vol 122 (3) ◽  
pp. 323-330 ◽  
Author(s):  
Norio Ito ◽  
Koichi Takahashi
Keyword(s):  
Contact Line ◽  
Necessary Conditions ◽  
Contact Point ◽  
Theoretical Solution ◽  
Curved Surfaces ◽  
Principal Curvatures ◽  
Hypoid Gears ◽  
Tooth Surfaces ◽  
Principal Directions

Hypoid gears are one of the most general form of gearing, and a theoretical solution for them has been studied by many researchers. Many hypotheses and theorems about these gears have been proposed—some of them correct and many of them wrong. The tooth surfaces are parts of general curved surfaces and they must have principal directions and principal curvatures on every contact point. However, there has been no detailed research on the fundamental elements of the surface. This study develops necessary conditions for determining these curvatures and principal directions for conjugate gearing with a contact line by introducing the concept of geodesic torsions. [S1050-0472(00)00503-1]

Study on Induced Principal Curvature and Induced Principal Direction Based on Normal Vector of Instantaneous Contact Line

2015 ◽  
Author(s):  
Yaping Zhao ◽  
Chenru Xi ◽  
Yimin Zhang
Keyword(s):  
Contact Line ◽  
Principal Curvature ◽  
Principal Direction ◽  
Normal Curvature ◽  
Normal Vector ◽  
Principal Curvatures ◽  
Proof Techniques ◽  
Principal Directions ◽  
Transient Contact ◽  
Worm Drive

The computing formulae, in different forms, for the normal vector of the instantaneous contact line are summarized systematically. For some of them, the distinct and sententious proof techniques are put forward. Based on the normal vector of the transient contact line, the computing formulae for the induced normal curvature and the induced geodesic torsion are deduced laconically and strictly. Owing to making use of the normal vector of the transient contact line, the style of the obtained formulae is more elegant. Particularly, a novel developing approach for the computing formula of the induced geodesic torsion is proposed. On the basis of the induced geodesic torsion, the computing formulae for the induced principal directions are derived. From this, the calculating formulae for the induced principal curvatures are obtained rigorously and conveniently. All these work reveal the pivotal position of the normal vector of the momentary contact line in the meshing theory for the line conjugate gearing. By right of the meshing theory established, the meshing analysis for the modified TA worm drive is performed. A number of basic and important formulae are attained and the numerical outcome of the induced principal curvature is given out.

On the Choice of the Contact Frame in Railroad Vehicle Dynamics

2009 ◽  
Author(s):  
Ali Afshari ◽  
Ahmed A. Shabana
Keyword(s):  
Angular Velocity ◽  
Contact Point ◽  
Rolling Direction ◽  
Contact Forces ◽  
Normal Vector ◽  
Principal Curvatures ◽  
Hertz Theory ◽  
Principal Directions ◽  
Railroad Vehicle ◽  
Contact Ellipse

In some of the wheel/rail creep theories used in railroad vehicle simulations, the direction of the tangential creep forces is assumed to be the wheel rolling direction (RD). When Hertz theory is used, an assumption is made that the rolling direction is the direction of one of the axes of the contact ellipse. In principle, the rolling direction depends on the wheel motion, while the direction of the axes of the contact ellipse (CE) are determined using the principal directions which depend only on the geometry of the wheel and rail surfaces and do not depend on the motion of the wheel. The RD and CE directions can also be different from the direction of the rail longitudinal tangent (LT) at the contact point. In this investigation, the differences between the contact frames that are based on the RD, LT, and CE directions, that enter into the calculation of the wheel/rail creep forces and moments, are discussed. The choice of the frame, in which the contact forces are defined, can be determined using one longitudinal vector and the normal to the rail at the contact point. While the normal vector is uniquely defined, different choices can be made for the longitudinal vector including the RD, LT, and CE directions. In the case of pure rolling or when the slipping is small, the RD direction can be defined using the cross product of the angular velocity vector and the vector that defines the location of the contact point. Therefore, this direction does not depend explicitly on the geometry of the wheel and rail surfaces at the contact point. The LT direction is defined as the direction of the longitudinal tangent obtained by differentiation of the rail surface equation with respect to the rail longitudinal parameter (arc length). Such a tangent does not depend explicitly on the direction of the wheel angular velocity nor does it depend on the wheel geometry. The CE direction is defined using the direction of the axes of the contact ellipse used in Hertz theory. In the Hertzian contact theory, the contact ellipse axes are determined using the principal directions associated with the principal curvatures. Therefore, the CE direction differs from the RD and LT directions in the sense that it is function of the geometry of the wheel and rail surfaces. In order to better understand the role of geometry in the formulation of the creep forces, the relationship between the principal curvatures of the rail surface and the curvatures of the rail profile and the rail space curve is discussed in this investigation. Numerical examples are presented in order to examine the differences in the results obtained using the RD, LT and CE contact frames.

Regular and Interference Tangents of Tooth Surfaces in Point Contact Gearing

2000 ◽  
Author(s):  
Ningxin Chen
Keyword(s):  
Point Contact ◽  
Hypoid Gear ◽  
Principal Curvatures ◽  
Numerical Examples ◽  
Surface Normal ◽  
Tooth Surfaces ◽  
Principal Directions ◽  
Spiral Bevel

Abstract The presented paper studies regular and interference tangents of two tooth surfaces in point contact gearing. The relative surface of two tangent surfaces is introduced and algorithm of its principal curvatures and principal directions are given. By analysis of three types of the relative surfaces, their principal curvatures and surface normal directions, the conditions of the regular and the interference tangents are obtained. Two numerical examples, a spiral bevel and a hypoid gear sets, both face-hobbed, illustrate the above conditions and the TCA results.

Directions of the Tangential Creep Forces in Railroad Vehicle Dynamics

2010 ◽  
Vol 5 (2) ◽  
Author(s):  
Ali Afshari ◽  
Ahmed A. Shabana
Keyword(s):  
Angular Velocity ◽  
Contact Point ◽  
Rolling Direction ◽  
Contact Forces ◽  
Normal Vector ◽  
Principal Curvatures ◽  
Hertz Theory ◽  
Principal Directions ◽  
Railroad Vehicle ◽  
Contact Ellipse

In some of the wheel/rail creep theories used in railroad vehicle simulations, the direction of the tangential creep forces is assumed to be the wheel rolling direction (RD). When the Hertz theory is used, an assumption is made that the rolling direction is the direction of one of the axes of the contact ellipse. In principle, the rolling direction depends on the wheel motion while the direction of the axes of the contact ellipse (CE) are determined using the principal directions, which depend only on the geometry of the wheel and rail surfaces and do not depend on the motion of the wheel. The RD and CE directions can also be different from the direction of the rail longitudinal tangent (LT) at the contact point. In this investigation, the differences between the contact frames that are based on the RD, LT, and CE directions that enter into the calculation of the wheel/rail creep forces and moments are discussed. The choice of the frame in which the contact forces are defined can be determined using one longitudinal vector and the normal to the rail at the contact point. While the normal vector is uniquely defined, different choices can be made for the longitudinal vector including the RD, LT, and CE directions. In the case of pure rolling or when the slipping is small, the RD direction can be defined using the cross product of the angular velocity vector and the vector that defines the location of the contact point. Therefore, this direction does not depend explicitly on the geometry of the wheel and rail surfaces at the contact point. The LT direction is defined as the direction of the longitudinal tangent obtained by differentiation of the rail surface equation with respect to the rail longitudinal parameter (arc length). Such a tangent does not depend explicitly on the direction of the wheel angular velocity nor does it depend on the wheel geometry. The CE direction is defined using the direction of the axes of the contact ellipse used in Hertz theory. In the Hertzian contact theory, the contact ellipse axes are determined using the principal directions associated with the principal curvatures. Therefore, the CE direction differs from the RD and LT directions in the sense that it is function of the geometry of the wheel and rail surfaces. In order to better understand the role of geometry in the formulation of the creep forces, the relationship between the principal curvatures of the rail surface and the curvatures of the rail profile and the rail space curve is discussed in this investigation. Numerical examples are presented in order to examine the differences in the results obtained using the RD, LT and CE contact frames.

A Method of Local Synthesis of Gears Grounded on the Connections Between the Principal and Geodetic Curvatures of Surfaces

1981 ◽  
Vol 103 (1) ◽  
pp. 114-122 ◽  
Author(s):  
F. L. Litvin ◽  
Y. Gutman
Keyword(s):  
Contact Point ◽  
Optimal Conditions ◽  
Principal Curvatures ◽  
Local Synthesis ◽  
Double Curvature ◽  
Tooth Surfaces ◽  
Gear Mechanism ◽  
The Relationship

A spatial gear-mechanism with double curvature tooth surfaces is discussed. The following results are obtained: a) the relationship between the principal curvatures of the surfaces, b) the conditions under which the path of the contact point on the surface is a geodetic line in the local sense. A method of synthesis of mismatched gearing proposed here permits determination of optimal conditions of meshing in the vicinity of the main contact point.

CURVATURE ANALYSIS OF VARIABLE PITCH LEAD SCREWS WITH CYLINDRICAL MESHING ELEMENTS

1996 ◽  
Vol 20 (2) ◽  
pp. 139-157 ◽  
Author(s):  
Yaw-Hong Kang ◽  
Hong-Sen Yan
Keyword(s):  
Contact Point ◽  
Tangential Direction ◽  
Radius Of Curvature ◽  
Principal Curvatures ◽  
Surface Geometry ◽  
Variable Pitch ◽  
Transformation Matrices ◽  
Mathematical Expressions ◽  
Screw Surface ◽  
Principal Directions

Based on coordinate transformation matrices and theory of gearing, we derive the mathematical expressions of surface geometry and the location of contact point of variable pitch lead screws. According to curvature theory, we obtain the principal curvatures, the principal directions, and the orientation of the contacting line at any contact point. Furthermore, the condition of avoiding undercutting of the screw surface, the reduced radius of curvature along any tangential direction, and the angle between the normal of contact line and the relative velocity are also derived. The result of this work is necessary for the tasks of contact stress analysis and wear/lubrication analysis for variable pitch lead screws with cylindrical meshing elements.

STUDYING THE CONTACT POINT AND INTERFACE MOVING IN A SINUSOIDAL TUBE WITH LATTICE BOLTZMANN METHOD

2001 ◽  
Vol 15 (09) ◽  
pp. 1287-1303 ◽  
Author(s):  
HAI-PING FANG ◽  
LE-WEN FAN ◽  
ZUO-WEI WANG ◽  
ZHI-FANG LIN ◽  
YUE-HONG QIAN
Keyword(s):  
Boundary Conditions ◽  
Contact Line ◽  
Lattice Boltzmann ◽  
Complex Geometry ◽  
Contact Point ◽  
Point Contact ◽  
Immiscible Fluids ◽  
Lattice Boltzmann Model ◽  
Stick Slip ◽  
Bounce Back

The multicomponent nonideal gas lattice Boltzmann model by Shan and Chen (S-C) is used to study the immiscible displacement in a sinusoidal tube. The movement of interface and the contact point (contact line in three-dimension) is studied. Due to the roughness of the boundary, the contact point shows "stick-slip" mechanics. The "stick-slip" effect decreases as the speed of the interface increases. For fluids that are non-wetting, the interface is almost perpendicular to the boundaries at most time, although its shapes at different position of the tube are rather different. When the tube becomes narrow, the interface turns a complex curves rather than remains simple menisci. The velocity is found to vary considerably between the neighbor nodes close to the contact point, consistent with the experimental observation that the velocity is multi-values on the contact line. Finally, the effect of three boundary conditions is discussed. The average speed is found different for different boundary conditions. The simple bounce-back rule makes the contact point move fastest. Both the simple bounce-back and the no-slip bounce-back rules are more sensitive to the roughness of the boundary in comparison with the half-way bounce-back rule. The simulation results suggest that the S-C model may be a promising tool in simulating the displacement behaviour of two immiscible fluids in complex geometry.

Characterization of 3D Surface Texture Directionality Using Multi-Scale Curvature Tensor Analysis

2017 ◽  
Author(s):  
Tomasz Bartkowiak
Keyword(s):  
Curvature Tensor ◽  
Surface Texture ◽  
Sampling Interval ◽  
Tensor Analysis ◽  
Principal Curvatures ◽  
Multi Scale ◽  
Principal Directions ◽  
2D And 3D ◽  
3D Surface Texture

Anisotropy of surface texture can in many practical cases significantly affect the interaction between the surface and phenomena that influence or are influenced by the topography. Tribological contacts in sheet forming, wetting behavior or dental wear are good examples. This article introduces and exemplifies a method for quantification and visualization of anisotropy using the newly developed 3D multi-scale curvature tensor analysis. Examples of a milled steel surface, which exhibited an evident anisotropy, and a ruby contact probe surface, which was the example of isotropic surface, were measured by the confocal microscope. They were presented in the paper to support the proposed approach. In the method, the curvature tensor T is calculated using three proximate unit vectors normal to the surface. The multi-scale effect is achieved by changing the size of the sampling interval for the estimation of the normals. Normals are estimated from regular meshes by applying a covariance matrix method. Estimation of curvature tensor allows determination of two directions around which surface bends the most and the least (principal directions) and the bending radii (principal curvatures). The direction of the normal plane, where the curvature took its maximum, could be plotted for each analyzed region and scale. In addition, 2D and 3D distribution graphs could be provided to visualize anisotropic or isotropic characteristics. This helps to determine the dominant texture direction or directions for each scale. In contrast to commonly used surface isotropy/anisotropy determination techniques such as Fourier transform or autocorrelation, the presented method provides the analysis in 3D and for every region at each scale. Thus, different aspects of the studied surfaces could clearly be seen at different scales.

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