Construction of spherical non-circular wheels formed by symmetrical arcs of loxodrome

Abstract

Bevel gears are used to transmit torque between intersecting axes. They demonstrate high reliability and durability of work, as well as a constant gear ratio. The disadvantage of such a transmission is the mutual sliding of the surfaces of the teeth of the gears, which leads to the emergence of friction forces and wear of their working surfaces. In this regard, there is a task to design such bevel gears that would have no slip. Non-circular wheels are understood as a pair of closed curves that rotate around fixed centers and at the same time roll over each other without sliding. They can serve as centroids for the design of cylindrical gears between parallel axes. If the axes of rotation of the wheels intersect, then the gears are called conical. An analog of gears between parallel axes, in which centroids are flat closed curves, for gears with intersecting axes are spherical closed curves. For a bevel gear with a constant gear ratio, such spherical curves are circles on the surface of the sphere, and with a variable gear ratio, spatial spherical curves. This paper considers the construction of closed spherical curves that roll around each other without sliding when they rotate around the axes intersecting in the center of the spheres. These curves are formed from symmetrical arcs of the loxodrome, a curve that crosses all the meridians of the ball at a constant angle. This angle should be 45°, which ensures the intersection of the loxodrome at right angles. Analytical dependences have been derived underlying the calculations of profiles of spherical non-circular wheels and their visualization by means of computer graphics. The results could be used to design non-circular wheels for textile machines, hydraulic machine pumps, pump dispensers, etc.