Analytical Dependences of the Kinematic Forming Primary Surfaces of the Worm Gear

The run-in method for obtaining the screw surface of a worm is based on the use of the worm gearing principle. In this case, the shaping surface (cutting tool) and the workpiece constitute a gear pair [4; 7]. The use of geometric modeling methods [8; 9] to simulate the process of shaping the working surface is based on the relative movement of intersecting objects in the form of a “workpiece-tool” system. This allows to obtain the necessary geometrical model that accurately reproduces the geometric configuration of the surfaces of the teeth of spatial gears [14; 15], where the producing surface of the tool moves in the selected reference system and its position at an arbitrary time is determined by a certain parameter, the motion parameter. The position of the cutting tool at the beginning and at the end of each pass is calculated using parametric equations, which make it possible to calculate the tool path for accurate processing of spatially complex surfaces [16–19]. In the process of mechanical action of a tool on a solid (workpiece), shaping occurs, which consists in the movement of the tool relative to the workpiece [30; 31]. The use of modern methods of three-dimensional computer graphics allows us to improve and accelerate the process of designing technological operations of tooth profiling, providing the final forms of the surfaces of the teeth in the form of visual and accurate computer-based solid-state models [39; 40]. The method is based on a virtual representation of the process of shaping in the form of intersection of solid-state 3D models of two objects (tools and workpieces), which generally perform a screw relative motion. As a result, the working surfaces of the teeth are formed as the envelopes of the tool producing surface [32–34]. For the formation of fission surfaces, mathematical dependences were obtained, which allow one to describe the mutual motion of a worm, a worm gear and a disk cutter [35–37]. These analytical dependences make it possible to simulate the virtual process of forming the side surfaces of the worm gearing elements [1–3; 5; 6]