The Computational Fundamentals of Spatial Cycloidal Gearing

The tooth flanks of bevel gears with involute teeth are still cut using approximations such as Tredgold’s and octoid curves, while the geometry of the exact spherical involute is well known. The modeling of the tooth flanks of gears with skew axes, however, still represents a challenge to geometers. Hence, there is a need to develop algorithms for the geometric modeling of these gears. As a matter of fact, the modeling of gears with skew axes and involute teeth is still an open question, as it is not even known whether it makes sense to speak of such tooth geometries. This paper is a contribution along these lines, as pertaining to gears with skew axes and cycloid teeth. To this end, the authors follow and extend results reported by Martin Disteli at the turn of the 20th century concerning the general synthesis of gears with skew axes. The main goal is to shed light on the geometry of the tooth flanks of gears with skew axes. The dualization of the tooth profiles of spherical cycloidal gears leads to ruled surfaces as conjugate tooth flanks such that at any instant the contact points are located on a straight line. A main result reported herein is Theorem 5, which is original. All results are proven by means of a consistent use of dual vectors representing directed lines and rigid-body twists. Giorgio Figliolini Dipartimento di Meccanica, Stutture, Ambiente e Territorio, Università degli Studi di Cassino, Via Di Biasio 43, 03043 Cassino (Fr), Italy, e-mail: [email protected] Hellmuth Stachel Institute of Discrete Mathematics and Geometry, Vienna University of Technology, Wiedner Hauptstr. 8-10/104, A 1040 Wien, Austria, e-mail: [email protected] Jorge Angeles Department of Mechanical Engineering & CIM, McGill University, 817 Sherbrooke Street West, Montréal (Québec), H3A 2K6, Canada, e-mail: [email protected]