Rolling of a developable helicoid along its bending

Abstract

The topic of rolling surfaces one by one in the scientific literature has not received wide coverage. The rolling surfaces of rotation, which are used to transmit rotational motion between skew and crossed axes, are studied in detail. When designing gears for torque transmission, both developable and nondevelopable surface surfaces are considered as axoids. Well-known surfaces for transmitting torque between crossed axes are cones with combined vertices. If the vertices of the cones are removed to infinity, then the surface generators become parallel, that is, the cones turn into cylinders with parallel axes. Both in the cones and in the cylinders, the common line of contact is the rectilinear generatrix of both surfaces. When transmitting torque between the crossed axes, the axoids are single-cavity hyperboloids of rotation with a common rectilinear tangent generatrix. They are nondevelopable ruled surfaces. There is a significant difference between the rolling of developable and nondevelopable surfaces one after the other. For pairs of cones and cylinders, rolling occurs without sliding, and for nondevelopable surfaces of hyperboloids, sliding occurs along a common line of contact. However, this does not mean that nondevelopable ruled surfaces cannot roll one over another without sliding.
It is known from surface theory that a surface can roll without sliding along its bend. If the bending of the initial ruled surface occurs with the preservation of rectilinear generators, then the rolling of the initial and bended surfaces occurs without sliding with the contact line along the common generatrix surface, regardless of whether they a developable or nondevelopable. You can control the bending of a developable surface by deforming its edge of regression. Such a deformation occurs by changing its torsion while maintaining curvature as a function of the length of the arc. For a developable helicoid, the edge of regression is a screw line. By changing its torsion, you can get another screw line with a different angle of its rise. These two screw lines correspond to two developable helicoids, each of which can be obtained by bending the other. Both helicoids have a common involute. To roll the helicoids one by one, they must be combined so that the contact line is the common rectilinear generatrix of both surfaces. This means that at the corresponding points of the edges of regression with equal values of the lengths of the arcs, the accompanying trihedral of both curves must coincide. To ensure contact of both surfaces along a common rectilinear generatrix, one of them was rotated using Euler angles and parallel transfer. Surfaces with a common contact line are constructed using the obtained equations.