Construction of helical surfaces in special coordinates

A large number of machine-building forms, including metal cutting tools (taps, cutters, drills, countersinks, etc.) have helical surfaces. These surfaces are used for the formation of cutting edges, the placement of chips and lubricating-cooling liquids. In addition, the helical surfaces in various variants of gears, worms, shafts participate in the transmission of mechanical motion. The qualitative manufacturing of helical surfaces depends on a special theory, which considers various ways of forming helical surfaces. The helical surface is obtained as a result of the helical motion of the generator. The points forming, during the helical motion, describe the helical lines, each of which can serve as a guide for the helical surface. In the presented article two variants of obtaining helical parallels are shown depending on the choice of space coordination. One of the properties of helical surfaces is the possibility of a “shear”, i.e., displacement along the axis of the helical surface during the helical movement. Due to this, helical surfaces are widely used in mechanical engineering: augers of mining, snow and agricultural machinery, spiral staircases, drills, helical, springs, coils and many other applications of helical surfaces. The development of methods for shaping surfaces in special coordinates requires the application of such well-known design schemes and methods to such coordination. One of the most common, in engineering, is the helical surfaces, which are formed by the helical motion of the represented generator. To form the helical surfaces different curvilinear coordinates are used. The choice of these coordinates depends on the specific conditions for shaping the helical surfaces. In this article, we consider options for replacing the normal cylindrical coordination of space by introducing a new congruence of coordinate lines. This is done by replacing one of the two families of coordinate lines on the reference cylinder by a family of helical lines. One of the variants of using helical surfaces are cylindrical helical wheels in volute profile. Gear wheels are used to transfer movement and force from one shaft to another. Transmission of this effort is due to the adhesion of the working surfaces of the teeth. Therefore, to gear drives, various kinematic and strength requirements are applied, which ensure their qualitative functioning. The operation of the gears depends to a large extent on the geometrical parameters of the teeth in engagement, the dimensions of the wheels and the parameters of the gear cutting machines. In the present article, the possibility of modeling the theoretical lateral surface of a cylindrical helical gear of an in volute profile by creating an internal equation based on a cylindrical coordinate system is shown. The main objective of the article is to develop an analytical model of the congruence of the coordinate lines of a normal cylindrical system, which is formed by replacing one of the families of the coordinate grid of the reference cylinder by a family of permanent-pitch helical lines.