Free-Form Optimal Conjugation Design

The paper presents the first optimal conjugation design methodology based on free-form conjugation modeling theory. The methodology is implemented for planar circular gearing and general for any planar gearing. According to the previous research of free-form conjugation, conjugate profiles are modeled by contact path geometries or cutter geometries which are represented by NURBS (non-uniform rational B-splines). The interchangeability between control points and interpolation points of NURBS are introduced in general to offer reasonable constraints of special interpolation conditions in conjugation design. To adapt to the flexibility brought by free-form techniques, the determination of an important conjugate property—contact ratio is carried out through geometric relationship. To make use of NURBS for optimal design, conjugate properties and their differentiations are represented by important parameters and then by control points and interpolation points of NURBS. The interested properties are relative curvature, specific sliding ratio and nominal Hertz contact stress, which are the main factors of gear efficiency and wear. The properties are well-known for their difficulties in optimization. The paper shows that with appropriate manipulations in mathematics and programming, it is feasible for gradient based optimization methods, which are accurate and fast in convergence. The methodology is consistent with the regular optimization of geometry design, and can be integrated into geometry design systems. The examples show the effectiveness of the proposed optimization framework.