Curvature interference characteristic of conical surface enveloping conical worm

Abstract

The theory to distinguish whether the curvature interference happens or not is well established for the enveloping conical worm. Some important results, such as the meshing function, the equation of the enveloping conical worm helicoid, and the curvature interference limit function, are all obtained. The foundation of computing the curvature interference limit line is to solve the system of nonlinear equations. A technique based on the elimination method and the geometric construction is proposed, which can be employed to study the existence of the solution of a system of nonlinear equations and to solve all the solutions of this system within the given solving domain. By means of the theory and technique brought forward, it is discovered that there usually are two curvature interference limit lines separately on each side of a tooth of an enveloping conical worm. When the number of the enveloping conical worm threads is smaller, the curvature interference in general does not happen on the both sides of a tooth. The avoiding mechanism of the curvature interference is that the limit line does not exist during the real cutting mesh because the limit line is inside the entity of the enveloping conical worm and its conjugate line is outside the entity of the grinding wheel. The numerical outcome shows that, the toe on the $$e$$ flank has the greatest potential risk to be subjected to the undercutting.