A MATHEMATICAL MODEL OF OPTIMIZATION FOR GEARS WITH CONVEX-CONCAVE CONTACT

Abstract

The article is devoted to the optimal design of gears with convex-concave tooth contact. A mathematical model of optimization is built according to the criteria of minimum contact stresses or relative sliding velocities. C-C gearing proposed by the Slovak scientists M. Boshanski and M. Veres was chosen as the object of research. A sufficient quantity of the initial contour's variants have been developed for them. A large volume of theoretical and experimental studies of contact strength and wear resistance has been carried out. An objective function is constructed: for the case of minimizing contact stresses F_σ=σ_H→min; for the case of minimizing the relative sliding velocities F_λ=λ→min. As variables planning for both objective functions, the following were chosen: pressure angle at the pole α_С; the curvature radius at the upper part of contact path r_kh; the curvature radius at the lower part of contact path r_kd. You can add two variables planning in the case of minimizing the relative sliding speeds: the addendum and dedendum coefficients of basic rack tooth h_a^* and h_f^*. Formed a system of constraints on the variables planning: on pressure angle at the pole α_С; on the curvature radiuses of contact path r_kh and r_kd; on the addendum and dedendum coefficients of basic rack tooth h_a^* and h_f^*; conditions of the absence of cutter interference for tooth dedendum and sharpening for the tooth addendum; condition of the absence of mesh interference; the minimum permissible values for the profile contact ratio ε_α and the radial clearance coefficient с^*. Probing of the parameter space by means of an LP-sequence was chosen as a method for solving the problem. The method allows you to operate with a significant number of parameters – up to 51, provides a sufficiently large number of evenly distributed test points – up to 2²⁰. In further studies, it is planned to develop algorithms for solving the problem, to carry out test and verification calculations to evaluate the theoretical results obtained, to propose variants of C-C gearing with increased loading capacity.

Key words: gear, convex-concave contact, optimization, objective function, variables planning.