Novel Methods of Curvature Analysis in the Meshing Theory for Gear Drives

Abstract

Two sorts of techniques for the curvature analysis for a surface are proposed. One is based on the principal curvature and the principal direction and the other is based on the simplified generalized Euler and Bertrand formulae, which is especially valuable for the case that the computation of the principal direction is extraordinarily complicated. The principal curvature and direction problem is handled from the angle of the generalized eigenvalue problem. Their characteristics can thus be proved concisely. Besides the formula for the principal direction is constructed. The simplified generalized Euler and Bertrand formulae are established by means of the mean curvature. Additionally, the result equivalency of these two methods is proved and verified laconically. The curvature analysis for the helicoidal surface of a modified TA worm is accomplished and a number of basic and important formulae are obtained. Both the theoretical analysis and the numerical consequence manifest that, the points on a modified TA worm surface are the hyperbolic points and the modification has no influence on the type of the points. The spiral surface of a TA worm is an undevelopable ruled surface and the straight cutting edge forms one of its two asymptotic directions.