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.

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.

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]

Ligand Hyperfine Interaction in the Complex [Mn(H2O)6] ++ in La2(Mg,Mn)3(NO3)12·24H2O

1969 ◽  
Vol 24 (11) ◽  
pp. 1746-1751 ◽  
Author(s):  
D. van Ormondt ◽  
R. de Beer ◽  
M. Brouha ◽  
F. de Groot
Keyword(s):  
Simple Model ◽  
Water Molecule ◽  
Hyperfine Interaction ◽  
Principal Direction ◽  
The Other ◽  
Water Molecules ◽  
Manganese Ion ◽  
Interaction Tensor ◽  
Principal Directions

Abstract The elements of the hyperfine interaction (h.f.i.) between the manganese ion and the protons in the complex [Mn(H2O)6]++ in one of the two possible sites in La2(Mg, Mn)3(NO3)12 · 24 H2O have been measured with ENDOR at 15 to 20 K. The six water molecules in the complex at the chosen site are equivalent for reasons of symmetry.One principal direction of the h.f.i. tensor of each proton is found to be perpendicular to the Mn, O line. With the assumption that each proton is located in the plane of the other two principal directions of its interaction tensor the positions of the protons are evaluated from the anisotropic parts of the h.f.i. tensors. In this calculation the effect of covalency on the anisotropic h.f.i. is ac-counted for with the aid of a simple model.The isotropic h.f.i.'s with the two protons of a water molecule appear to be very nearly equal (+ 0.890 MHz for both). This latter result is remarkable in view of the fact that one proton is distinctly nearer to the manganese ion than the other.

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.

Determination of the principal directions of azimuthal anisotropy from P-wave seismic data

Geophysics ◽  
1998 ◽  
Vol 63 (2) ◽  
pp. 692-706 ◽  
Author(s):  
Subhashis Mallick ◽  
Kenneth L. Craft ◽  
Laurent J. Meister ◽  
Ronald E. Chambers
Keyword(s):  
Anisotropic Medium ◽  
Seismic Data ◽  
Principal Direction ◽  
Azimuthal Anisotropy ◽  
P Wave ◽  
Amplitude Variation ◽  
S Waves ◽  
Wave Data ◽  
Principal Directions ◽  
Qualitative Measure

In an azimuthally anisotropic medium, the principal directions of azimuthal anisotropy are the directions along which the quasi-P- and the quasi-S-waves propagate as pure P and S modes. When azimuthal anisotropy is induced by oriented vertical fractures imposed on an azimuthally isotropic background, two of these principal directions correspond to the directions parallel and perpendicular to the fractures. S-waves propagating through an azimuthally anisotropic medium are sensitive to the direction of their propagation with respect to the principal directions. As a result, primary or mode‐converted multicomponent S-wave data are used to obtain the principal directions. Apart from high acquisition cost, processing and interpretation of multicomponent data require a technology that the seismic industry has not fully developed. Anisotropy detection from conventional P-wave data, on the other hand, has been limited to a few qualitative studies of the amplitude variation with offset (AVO) for different azimuthal directions. To quantify the azimuthal AVO, we studied the amplitude variation with azimuth for P-wave data at fixed offsets. Our results show that such amplitude variation with azimuth is periodic in 2θ, θ being the orientation of the shooting direction with respect to one of the principal directions. For fracture‐induced anisotropy, this principal direction corresponds to the direction parallel or perpendicular to the fractures. We use this periodic azimuthal dependence of P-wave reflection amplitudes to identify two distinct cases of anisotropy detection. The first case is an exactly determined one, where we have observations from three azimuthal lines for every common‐midpoint (CMP) location. We derive equations to compute the orientation of the principal directions for such a case. The second case is an overdetermined one where we have observations from more than three azimuthal lines. Orientation of the principal direction from such an overdetermined case can be obtained from a least‐squares fit to the reflection amplitudes over all the azimuthal directions or by solving many exactly determined problems. In addition to the orientation angle, a qualitative measure of the degree of azimuthal anisotropy can also be obtained from either of the above two cases. When azimuthal anisotropy is induced by oriented vertical fractures, this qualitative measure of anisotropy is proportional to fracture density. Using synthetic seismograms, we demonstrate the robustness of our method in evaluating the principal directions from conventional P-wave seismic data. We also apply our technique to real P-wave data, collected over a wide source‐to‐receiver azimuth distribution. Computations using our method gave an orientation of the principal direction consistent with the general fracture orientation in the area as inferred from other geological and geophysical evidence.

Optimal Design of Modified Cylindrical Gear Based on Minimum Flash Temperature of Tooth Surfaces

2013 ◽  
Vol 655-657 ◽  
pp. 573-577
Author(s):  
Jin Ke Jiang ◽  
Zong De Fang ◽  
Xian Long Peng
Keyword(s):  
Friction Coefficient ◽  
Film Thickness ◽  
Contact Line ◽  
Gear Tooth ◽  
Tooth Surface ◽  
Uniform Grid ◽  
Normal Vector ◽  
Flash Temperature ◽  
Oil Film ◽  
Tooth Surfaces

Considering the gap of the contact line of modified involute cylindrical gears influencing on loads, oil film thickness, the friction coefficient was determined on the basis theory of TCA、 LTCA and EHL. so oil film thickness and friction coefficient corresponded with loads on contact line were dispersed, which was used to computed discrete temperature according to the Blok flash temperature formula. and an approach of modified tooth surface optimum design based on the minimum flash temperature was proposed: the modified tooth surfaces was defined as a sum of theoretical tooth and cubic B-spline fit surface based on the uniform grid points created by double parabolas and a straight line and whose normal vector was deduced, besides, used genetic algorithm to optimize the parameter of curve, and get the best modified gear tooth surfaces. the results shows that oil film is thicker in engaging-out, coefficient of friction is contrary, which is responsible for lower flash temperature in engaging-in, besides the flash temperature has little changes in the single tooth meshing zone, and helical gear has a lower flash temperature than spur gear due to higher overlap ratio.

scholarly journals Palm Vein Recognition Algorithm using Multilobe Differential Filters

2019 ◽  
Author(s):  
Екатерина Сафронова ◽  
Ekaterina Safronova ◽  
Елена Павельева ◽  
Elena Pavelyeva
Keyword(s):  
Principal Curvature ◽  
Spatial Configuration ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Gaussian Kernel ◽  
Branch Points ◽  
Principal Curvatures ◽  
Feature Maps ◽  
Vein Recognition ◽  
Palm Vein

In this article the new algorithm for palm vein recognition using multilobe differential filters is proposed. After palm vein image preprocessing vein structure is detected based on principal curvatures. The image is considered as a surface in a three-dimensional space. Some vein points are selected using the maximum principal curvature values, and the other vein points are found from starting points by moving along the direction of minimum principal curvature. Multilobe differential filters are used to extract feature maps for vein images. These filters are flexible in terms of basic lobe choice and spatial configuration of lobes. The multilobe differential filters used in the article simulate vein branch points, and Gaussian kernel is used as the basic lobe. The normalized root-mean-square error is applied for image matching. Experimental results using CASIA multi-spectral palmprint image database demonstrate the effectiveness of the proposed method. The value of EER=0.01693 is obtained.

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