Analytical Methodology for Gear Tooth Number Synthesis in a Ravigneaux Planetary Gear with Seven Kinematic Links and Two Degrees of Freedom

Abstract

Existing methods for selecting the number of teeth in complex planetary gear systems are often methodologically demanding. They do not always ensure all conditions required for proper operation and assembly. This paper presents an analytical methodology for determining gear tooth numbers. The method is demonstrated on a Ravigneaux planetary gear set with seven kinematic links and two degrees of freedom. It ensures the simultaneous satisfaction of all meshing and assembly conditions. Starting from the known transmission ratios, the number of teeth of one central gear, and the selected angular displacement of the outer planet gear, analytical relationships are derived. These allow the determination of the tooth numbers of all remaining gear elements. The procedure is implemented in Python 3.13. This enables a systematic evaluation of predefined input ranges and an automatic verification of geometric constraints, including interference and undercutting conditions. The proposed method yields six feasible configurations. Compared with the Borg-Warner M85 automatic transmission, deviations in individual gear ratios reach up to 10%. Significantly lower tooth numbers are achieved for several gears. These results suggest that the proposed methodology can achieve comparable kinematic performance while offering more compact gear designs and a potential weight reduction. The developed model also provides a basis for extension to more complex configurations and integration with optimisation and dynamic criteria in planetary gear synthesis.