Mathematical Modeling and Generating Method of Hourglass Worm Gear Hob’s Rake Face Based on a Rotating Paraboloid Surface

Abstract

The rake angles on both sides of the cutting edges of the hourglass worm gear hob significantly influence its cutting performance, which, in turn, plays a decisive role in the surface quality of the machined worm wheel. To balance the rake angles along the tooth height direction of each hob tooth and enhance the overall cutting performance of the hob, this paper proposes a method that utilizes a rotating paraboloid surface to generate the helical rake face of the hourglass worm gear hob. First, the conjugate condition equations for the rake face generated by the rotating paraboloid surface are derived. A mathematical model for the helical rake face of planar double-enveloping hourglass worm gear hob is established. This study explores the influence of two machining parameters on the rake angle, specifically the milling drive ratio coefficient k and the geometric parameter of a parabolic milling cutter p. Through a systematic analysis of the variations in rake angle at the dividing toroidal surface and along the tooth height direction, the optimal parameter values were identified as k = 0.9115 and p = 0.6834. The results show that, after optimization, the hob rake angle range is around ±4.7°, with a maximum rake angle difference of 6.3072° along the tooth height direction, and the rake angles on both sides of the teeth are more balanced. The structure of the rake face is more reasonable, reflecting the feasibility of rotating paraboloid tools for forming tools in the machining of complex surfaces.

Keywords

• hourglass worm
• worm gear hob
• rake face
• rake angle
• rotating paraboloid surface