Mathematical Principle for Calculating Contacting Curve Length of Involute Helicon Gearing

The mathematical principle for calculating the contacting curve length of the involute Helicon gearing is put forward. The transient contacting curves within the conjugate zone are attained. The approximate analytical formula of the contacting curve length is derived. Based on that, the lengths of the contacting curves are computed by three methods, which are the approximate analytical formula, the numerical integration method, and calculating the distance between the beginning and the end of the contacting curve on the worm gear tooth surface. Besides, to demonstrate the rationality of the third method, two novel formulae for calculating the principal curvatures and directions of the surface are derived from the curvature parameters of two perpendicular directions to each other. These two novel formulae are used to calculate the principal curvatures and directions of the worm gear tooth surface, and evaluate the flatness of the tooth surface quantitatively. The results show that the contacting curve lengths calculated in this paper are generally between 2.7087 mm and 4.4858 mm; most of the contacting curve lengths do not vary much. The contacting curve length calculation principle proposed in this paper has high precision, and the maximum relative error between three methods is not more than −3.8838%. The worm gear tooth surfaces are relatively flat, the minimum of the principal curvature radii is 43.5494 mm, and the maximum is 3.3152 × 10⁴ mm; most of the principal curvature radii are much larger than the contacting curve lengths.