Construction of conical axoids on the basis of congruent spherical ellipses

Purpose: To carry out the transition from a cylindrical gear in which the centroids are congruent ellipses with centres of rotation in the foci, to a bevel gear on the basic of congruent spherical ellipses. Design/methodology/approach: Congruent ellipses with centres of rotation in the foci serve as centroids for the design of cylindrical gears with non-circular wheels. The article analytically shows that the analogues of ellipses on the plane - congruent spherical ellipses are the basis for the construction of the axoids of the corresponding bevel gears. An analogue of the centre-to-centre distance for ellipses in the plane is the angle between the axes of rotation of conical axoids. Findings: Based on the equality of the arcs of ellipses, the dependence of the angle of rotation of one axoid on the angle of rotation of another is found. Graphs of this dependence for separate cases are given. It is shown under what conditions the axes of axoids intersect at right angle. The parametric equations of spherical ellipses and corresponding axoids are given. They were used to construct spherical ellipses and corresponding conical axoids for different cases. For gears with right angle between the axes, separate positions of the axoids with different angles of their rotation around their axes are constructed. Practical implications: Spherical ellipses are directing curves for the construction of the corresponding conical axoids. Originality/value: The paper shows that congruent spherical ellipses act as centroids for the design of axoids of bevel gears. They roll one by one without sliding, rotating around axes that intersect in the centre of the sphere. To design such gears, it is important to know the interdependence between the geometric parameters, especially for common gears with a right angle between the axes.

OPTIMAL DESIGN OF CYLINDRICAL GEARS WITH CONVEX-CONCAVE CONTACT: OBJECTIVE FUNCTION AND VARIABLES PLANNING

Reducing the mass and dimensions of gears is an actual task of modern mechanical engineering. One of the perspective ways to solve it is the use of gearing with a convex-concave contact of the teeth. Therefore, the study is devoted to the development of methods for the optimal design of cylindrical gears with convex-concave contact of the working surfaces. Optimality criteria: minimum contact stresses and (or) minimum relative sliding velocities, taking into account design, geometrical and technological constraints. C-C gearing was chosen as the object of research. It was proposed by the Slovak scientists M. Boshanski and M. Veresh. An objective function is constructed for the case of minimizing contact stresses. The optimality criterion is formulated as follows: contact stresses σH in the mesh must take the minimum possible value when all constraints are met. An objective function is also constructed for the case of minimizing the relative sliding velocities of profiles. The optimality criterion is formulated as follows: the relative sliding s velocities of profiles λ at the extreme points of mesh must take the minimum possible value when all the constraints are met. Variables planning are defined. These are pressure angle at the pole αС, the curvature radius at the upper part of contact path rkh, and the curvature radius at the lower part of contact path rkd. A method for solving the problem of optimal design is chosen. The method of probing the space of design parameters was chosen from all the variety. The points of the LPτ-sequence are used as test points. The method allows you to operate with a significant number of parameters – up to 51, provides a sufficiently large number of evenly distributed test points – up to 220. In further studies, it is planned to form a system of constraints on variables planning, to develop methods and algorithms for solving the problem. Also carry out test and verification calculations to confirm and evaluate the theoretical results. Keywords: gear, convex-concave contact, optimal design, objective function, variables planning

PROCESSING OF HARDENED CYLINDRICAL GEAR WHEELS OF THE CUTTING GEARBOX OF THE COMBINE UKD 200-500

The article discusses the latest developments of unique technological methods of gear milling of cylindrical gears for preliminary blade gear processing of hardened cylindrical gears of the cutting reducer of the UKD200-500 coal mining harvester for the final gear grinding of gear teeth with modulus m = 16 mm, with hardness HRC 56 ... 62. The peculiarity of the design of special hob cutters is that a circle passing through the lower boundary points of the involute is used as the palloid of the machine gearing of the tool and part. Pre-cutting the teeth of hardened wheels with carbide milling cutters allows you to remove the main allowance for the final gear grinding.

Scuffing of cylindrical gears with pitch line velocities up to 100 m/s

Increasing demands on the power density of gearboxes require a precisive gear design regarding common failure mechanism. Particularly in turbo gearboxes with low-viscosity lubricants, the damage mechanism scuffing is relevant. In this paper an innovative test rig for the experimental investigation of scuffing at pitch line velocities up to 100 m/s is presented. The scuffing load capacity depending on the pitch line velocity of two gear design variants running at constant temperatures and lubricant conditions was investigated. Furthermore, the morphology of scuffing was investigated with regard to the damage location and the surface condition. Based on the experimental results, a simulation approach with an accuracy superior to the existing standards for calculating the scuffing load capacity of highspeed gears has been derived.

A Simulation Method for Free-form Milling Cylindrical Gears With Disc Cutters

Free-form milling is a flexible gear machining method that allows using general disc cutters to machine various gear types on a 5-axis machine tool. This paper proposes a two-dimensional simulation method for free-form milling of cylindrical gears with disc cutters. A mathematical model for free-form milling of cylindrical gears with disc cutters is constructed. By analyzing the spatial positional relationships between the cutter and the workpiece, the instantaneous contact line is derived and projected onto the gear end face, and then the projected curves are intersected to obtain the final profile. This calculation method realizes the rapid simulation of the actual cutting condition on the gear end face, which is beneficial to the judgment of machining interference and the analysis of the tooth profile accuracy during the gear machining process.

Cylindrical Gears with Contact Areas Distributed Along the Arc Teeth Length

Проблема повышения надежности приводов высоконагруженных машин связана с обеспечением работоспособности цилиндрических передач при наличии углов перекоса зубьев в зацеплении, вызываемых ошибками изготовления и сборки передач, а также податливостью их элементов и корпусных деталей. Эффективным направлением решения проблемы является переход от цилиндрических передач с прямыми, косыми зубьями на передачи с арочными зубьями. По сравнению с прямыми или косыми зубьями изгибная прочность арочных зубьев на 20…30 % выше. Процессы формообразования поверхностей арочных зубьев позволяют в их зацеплении получать линейный, локально-линейный, точечный характер касания, обеспечивать оптимальный закон распределения нагрузки по площадке контакта. В работе рассмотрены особенности компенсации перекоса зубьев в зацеплении перемещением (самоустановкой) шестерни или колеса с арочными зубьями вдоль оси. Показано, что самоустановка колес происходит только для постоянного крутящего момента и неменяющегося угла перекоса. Для компенсации случайного угла перекоса в условиях изменяющегося крутящего момента предложена адаптивная цилиндрическая передача с арочными зубьями, в зацеплении которых имеются две разнесенные по длине зоны контакта. Колесо передачи состоит из двух полуколес, разделенных упругой прокладкой. Возникающие в зонах контакта осевые силы уравновешивают друг друга и обеспечивают самоустановку полуколес при изменении угла перекоса. Нарезание арочных зубьев на обоих полуколесах осуществляется одновременно, поэтому трудоемкость изготовления адаптивной передачи с арочными зубьями не изменяется. Для полуобкатного варианта формообразования поверхностей арочных зубьев адаптивной цилиндрической передачи представлены результаты расчета положения рабочих линий в зацеплении, размеры тормозных зон касания поверхностей арочных зубьев.

Providing of lateral clearance in tooth gears by additional displacement of the original contour

A method of multicriteria optimization of gearing with the use of the Kompas-3D program for calculating the degrees of correction of the lateral clearance in the tooth gear is proposed. The recommendations (GOST 16532—70, Appendix 2) on the selection of the degrees of correction of the gear wheels of cylindrical gears are analyzed, the necessary corrections are proposed. Keywords: tooth gear, lateral clearance, degree of correction, multicriteria optimization method, Kompas-3D program, blocking contournhanthong [email protected]