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Construction of conical axoids on the basis of congruent spherical ellipses

Kresan T., Pylypaka S., Ruzhylo Z., Rogovskii I., Trokhaniak O.

Archives of Materials Science and Engineering

2022

Vol. 113, nr 1

13--18

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.