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From the standpoint of the development of possibilities of application in theory and practice, the works of Prof. B. A. Perepelitsa from Kharkov Polytechnic Institute and his disciples to develop an applied methodology of multiparameter mappings in relation to the profiling and functioning of complex curvilinear objects and transmission mechanisms in mechanical engineering, mainly with examples of gears, are presented. The work substantiates the relevance of the study of gears with a complex noninvolute profile of the side surfaces of the teeth, which in some applications have advantages over involute gears and are devoid of some of their drawbacks associated with quality indicators. A technique for obtaining mating surfaces of the teeth of non-invasive gears as envelopes of the specified surfaces of the teeth of tools is described. A scheme for forming pairs of non-involute gears, from which a gearing can be composed, is proposed. At the same time, diamond-abrasive tools are considered as shaping the working gear profile in its cutting according to the copying scheme and finishing according to the rolling honing scheme. In the first case, the profile of a special shaped tool on a high-strength metal bond is supported by a master electrode according to the scheme of the anodic connection of the tool into the electric circuit of dressing, similar to diamond spark grinding. In the second case, the use of gear wheels-hones on elastic ligaments is shown. It is shown that to obtain the mating surfaces of the teeth of two non-involute gears, two tool rails can be used with the profiles of the side surfaces of the teeth opposite to each other. As a nonlinear profile of the tooth lateral surface of the tool rail, some part of one of the simulated flat kinematic curves is considered. A description of the program developed in accordance with the described method is given, which allows you to calculate the geometric characteristics of the shaped profiles of the gear pair wheels, visualize the shaping process, and also determine the quality indicators of the gearing. Thus, the prerequisites were created for choosing from the resulting geometric modeling of the curve field of such tooth profiles of the tools, which would provide the most rational combination of the tooth profiles of the gears processed by them and the required quality parameters of the gear teeth. The results of the study of the pressure ratio between the teeth of a gear and the overlap ratio of gears when choosing the shape of the tooth profiles are presented. A series of numerical experiments for gearing, formed by pairs of tool rails with different profiles of the side surfaces of the teeth straight, convex and concave, as well as convexconcave were performed. It is shown that non-involute gearing can have large reduced radii of curvature (and consequently smaller pressure coefficients) at the points of tangency of the profiles compared to involute gearing with a slight increase or decrease in the gearing overlap ratio. The most preferable is the variant of the rails with convex and concave tooth profiles, which provides the best values of both quality indicators of the engagement.

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