Investigating the Influence of Effective Parameters on Tooth Geometry of Spherical Involute Bevel Gears Through Solving Napier’s Equations

Abstract

Bevel gears are a kind of gears that transmit power between two intersecting shafts. Bevel gears with Octidal and spherical involute profiles are the most common types of bevel gears. The profile of spherical involute teeth is a three-dimensional complex surface, requiring an accurate geometric definition calculation. Due to this complex geometry, modeling, stress-strain analysis, and mesh stiffness calculation are very difficult. In this study by extracting Napier’s Equations from the tooth profile geometry, geometric parameters of the spherical involute curve have been calculated. The results show that the pitch cone angle has an important effect on the shape of the spherical involute curve, so choosing the correct angle is particularly important. A significant increase in the azimuthal angle is observed by increasing the polar angle. In addition, increasing the angle of the pitch cone reduces the curvature of the side surfaces of the teeth and facilitates the manufacturing process. On the other hand, reducing the pressure angle will result in flat lateral surfaces and a tooth shape similar to the octoidal tooth. The sphere radius variations do not affect the shape of the tooth and only change the size of the tooth. Finally, a gear system comprising a pinion and gear was made from Polylactic Acid (PLA) utilizing the calculated angles to verify the accuracy of computed angles. Also, experimental and finite element methods determined the pinion and gear contact pattern. There was a good agreement between the finite element simulation and the experimental observation. ©2026 The author(s).