Analytical solution to contact characteristics for a variable hyperbolic circular-arc-tooth-trace cylindrical gear

The variable hyperbolic circular-arc-tooth-trace (VH-CATT) cylindrical gear is a new type of gear. In order to research the contact characteristics of the VH-CATT cylindrical gear, tooth surface mathematical models of this kind of gear pair are derived based on the forming principle of the rotating double-edged cutting method with great cutter head in this regard. Then, according to the differential geometry theory and Hertz theory, the formula of the induced normal curvature and equation of the contact ellipse are derived based on the condition of continuous tangency of two meshing surfaces, which proves that the contact form of VH-CATT cylindrical gear is point contact. The present work establishes analytical solutions to research the effect of different parameters for the contact characteristic of the VH-CATT cylindrical gear by incorporating elastic deformation on the tooth surface, which have shown that the module, tooth number and cutter radius have a crucial effect on the induced normal curvature and contact ellipse of the VH-CATT cylindrical gear in the direction of tooth trace and tooth profile. Moreover, a theoretical analysis solution, a finite element analysis and the gear tooth contact pattern are carried out to verify the correctness of the computerized model and to investigate the contact type of the gear; it is verified that the contact form on the concave surface of the driving VH-CATT cylindrical gear rotates from dedendum at the heel to the addendum at toe and is an instantaneous oblique ellipse due to elastic deformation of the contact tooth profile, and the connecting line of the ellipse center is the contact trace path. It is indicated that the research results are beneficial for research on tooth break reduction, pitting, wear resistance and fatigue life improvement of the VH-CATT cylindrical gear. The results also have a certain reference value for development of the VH-CATT cylindrical gear.

Wear depth calculation and influence factor analysis for groove ball bearing

Based on Hertz contact theory and load distribution, the formulas for contact stress cycle times, slip distance and wear depth measurements are derived, and the influences of load, curvature coefficient, roll body diameter and friction coefficient on the contact region wear depth and distributions are thoroughly analyzed. The results show that the wear depth is zero at the pure rolling point and the long half-axle terminals of contact ellipse, and reaches maximum value near by the long half-axle terminals of the contact ellipse, and further shows that the wear depth increases with increase of the load and friction coefficient, however decreases with increase of the curvature coefficient and roll body diameter.

Determination of Wheel-Rail Contact Characteristics by Creating a Special Program for Calculation

The authors in this paper describe the steps of creating a special program in GUI tool in Matlab. The program is designed to calculate the main properties of wheel-rail contact zone, such as: contact ellipse dimensions, normal stress and friction coefficients. All the relevant equations, which were introduced by different researchers, are firstly presented and modified to be applicable to the programming environment, and then the program was built. In the end, the program working quality is discussed and some expected future developments on this program are suggested. The proposed program can make the comparison between theoretical and experimental results, when they are available, easier and faster.

Four-Point Contact Ball Bearing Model With Deformable Rings

In many applications, such as four-point contact slewing bearings or main shaft angular contact ball bearings, the rings and housings are so thin that the assumption of rigid rings does not hold anymore. In this paper, several methods are proposed to account for the flexibility of rings in a quasi-static ball bearing numerical model. The modeling approach consists of coupling a semianalytical approach and a finite element (FE) model to describe the deformation of the rings and housings. The manner in which this weak coupling is made differs depending on how the structural deformation of the ring and housing assemblies is injected into the set of nonlinear geometrical and equilibrium equations in order to solve them. These methods enable us to account for ring ovalization, ring twist, and raceway opening (including change of conformity) since a tulip deformation mode of the ring groove is observed for high contact angles. Either the torus fitting technique or mean displacement computation are used to determine these geometrical parameters. A comparison between the different approaches allows us to study, in particular, the impact of raceway conformity change. The loads used in this investigation are chosen in order that the maximum contact pressure (the Hertz pressure) at the ball-raceway interface remains below 2000 MPa, without any contact ellipse truncation. For the ball bearing example considered here, relative differences of up to 30% on the axial displacement, 10% on the maximum contact pressure, and 10% on the contact angle are observed by comparing rigid and deformable rings for a typical loading representative of the one encountered in operation. Despite the local change of conformity, which becomes significant at high contact angles and for thin ball bearing flanges, it is shown that this hardly affects the internal load distribution. The paper ends with a discussion on how the ring and housing flexibility may affect the loading envelope when the truncation of the contact ellipse is an issue.

Solution of the Contact Zone Orientation for Normal Elliptical Hertzian Contact

Many mechanical designs have parts that come into, or lose, contact with each other. When elastic bodies with second order surface geometries come into contact, the contact patch is expected to be approximately flat and to have an elliptical boundary. Classic Hertzian contact mechanics can be used to model such contacts, but since there is no closed-form analytical solution to predict the major and minor axes of the contact zone ellipse, approximate numerical methods have been developed, some of which are very accurate. Predictions of the mutual approach of the bodies and the contact pressure distribution can then be made. Although the shape of the contact ellipse has been modeled and solved for, to date there has been no solution for the orientation of the contact ellipse with respect to either of the contacting bodies. The contact ellipse orientation is needed in order to model the shear stress distributions that occur when sticking friction forces are developed and separate contact zones of sticking and slipping are expected. Using the results of a numerical solution for the conventional contact parameters, this paper presents an analytical solution of the orientation of the contact ellipse, which is shown to depend only on the curvatures and the relative orientation of the contacting bodies. In order to validate the analytical solution, the results are compared with those from ABAQUS™ finite element simulations for cases of identical bodies and bodies with dissimilar curvatures. The predictions of the contact ellipse orientation angles and the major and minor semi-axes agree very well for all cases considered.

Assessment of Elliptical Conformal Hertz Analysis Applied to Constant Velocity Joints

To analyze the contact between two spherical bodies with different radii of curvature, the three-dimensional (3D) Hertz theory for elliptical contact is typically used. When the two contacting bodies have high conformity, such as the case for ball-in-groove, the Hertz theory may break down. In this research, finite element analysis (FEA) was used to assess the validity of 3D Hertz theory as found in the roller-housing contact of constant velocity joints. The contact area, normal approach, and contact pressure results show that Hertz agrees with FEA predictions for low compressive loads, where the contact ellipse is within the geometrical contact dimensions. At higher loads the contact ellipse extends beyond the contacting geometrical dimensions and the simplified analytical Hertz results diverge from the FEA results.

Directions of the Tangential Creep Forces in Railroad Vehicle Dynamics

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.