Geometric characteristics of involute helicoid

Abstract

The involute helicoid is one of the most commonly-used surfaces in mechanical engineering. Its typical application is to be used as the tooth flank of a cylindrical gear and the helicoid of a kind of cylindrical worm, which is named as the type I worm in both ISO and DIN standards. In history, the earlier research regarding the geometry of an involute helicoid can be dated back to the study of grinding approach for the above type I worm though the related study is qualitative. Nowadays the geometry of an involute helicoid is briefly and superficially discussed in the courses of an undergraduate student. In the current work, the scientific fact is proved that an involute helicoid is the unique developable cylindrical helicoid with constant helix parameter. The geometries of the profiles in different sections are systematically and deeply studied for an involute helicoid. The analytical solution of the geodesic line on an involute helicoid is attained through solving nonlinear ordinary differential equation. The numerical examples are provided to verify and illustrate the theory and method set forward. The whole work aims at supplying more in-depth knowledge in regard to the geometry of the involute helical surface for the readers, mainly including the teacher, trainer and engineers in mechanical engineering. Facing to an involute helicoid, instead of via the screw rack in the classical methodology, the mathematical relationship between the normal and transverse pressure angles is obtained. As to an involute helicoid, the tooth number condition that its root circle is larger than its base circle is given out. The numerical examples are provided to verify and illustrate the theory and method set forward.