Mathematical model of shaping toothed products using volumetric tool with one motion parameter

Introduction. The development of a mathematical model of one-parameter shaping of a toothed product is considered. As an example, shaping of the side surface of the teeth of the Novikov gearing is studied; the mode and magnitude of the change in the shaping error heightwise the wheel tooth are shown. The work objective was to develop a mathematical model of the surface of the product teeth as a generating surface envelope of the tool. A computational and experimental study is carried out. The mathematical models obtained can be used in devices with copiers when shaping the side surface of the teeth of the Novikov gear. As an example, we consider the deviation behavior of the teeth profile of the Novikov gear with the original profile of DLZ 0.7-0.15

Materials and Methods. When building the model and studying its characteristics, the mathematical tools of the gearing theory, calculation procedure for cylindrical gears (A.A. Silich’s author development) were used. The paper proposes new mathematical models of the equations of the lateral surface of the gear teeth formed with a tool whose axial profile coincides with the original one. In the model under consideration, the tool moves along the axis of the product while the latter rotates on its axis. In the course of the study, numerical modeling was carried out to determine the error value in shaping the product profile using the tool.

Results. New mathematical models and software have been developed for numerical simulation of the shaping of a toothed product using a tool with one independent motion parameter. An algorithm has also been developed to determine the deviation error of the real profile from the nominal one for the tooth of the Novikov gear. Solutions to accurately reproduce the tooth profile are provided.

Discussion and Conclusions. The parametric method of analytical description of the surface used in the work simplifies the calculation of the cutting tool displacements in the problems of numerical control. Solving the problem of synthesizing the technology of workpiece surface treatment on metal-cutting machines provides the development of a description of the entire shaping process and requires the representation of the workpiece surface in the form of a mathematical model. The results obtained can be used under creating finishing methods for processing teeth when improving the quality of gear wheels and gear drives, as well as production efficiency.

1. Силич, А. А. Передачи Новикова. Геометрический расчет цилиндрических передач Новикова / А. А. Силич. — Тюмень : Изд–во ТИУ, 2016. — 79 с.

2. Силич, А. А. Цилиндрические передачи Новикова. Методика геометрического расчета / А. А. Силич. — Дюссельдорф : Lambert Academic Publishing, 2013. — 90 с.

3. Ломова, О. С. Математическое моделирование структурных изменений в поверхностях заготовок при тепловых возмущениях в процессе шлифования / О. С. Ломова // Омский научный вестник. — 2013. — №2(120). — С. 95–98.

4. Силич, А. А. Технология обработки зубчатых колес цилиндрических передач Новикова / А. А. Силич. — Тюмень : Изд. ТюмГНГУ. — 2007. — 162 с.

5. Litvin, F. L. Computerized Generation and Simulation of Meshing and Contact of New Type of NovikovWildhaber Helical Gears / F. L. Litvin, P.-H. Feng, S. A. Lagutin // NASA / Contractor Report-2000-209415 / ARLCR-428, 2000. — 55 p.

6. Litvin, F. L. Development of Gear Technology and Theory of Gearing / NASA Reference Publication 1406, ARL-TR-1500, 1998.

7. Литвин, Ф. Л. Теория зубчатых зацеплений / Ф. Л. Литвин. — 2-е изд., перераб. и доп. — Москва : Наука, 1968. — С. 584.