The paper presents the first theory and practice of free-form conjugation modeling. It introduces the concept of master-slave that broadens the traditional computer aided single geometry modeling to dual geometry modeling. The concept is then applied to gear geometry to establish the first free-form conjugation modeling technique, in which a fictitious free-form rack-cutter or free-form contact path is proposed as the master geometry. Simple geometric relationship suitable for conjugation modeling and undercutting analysis is found between the master and the conjugate profiles. With a free-form master geometry, free-form conjugation modeling has desirable properties of guaranteed continuity, flexibility, and controllability. It offers unlimited representation capability crucial for conjugation modification and optimization. Undercutting is examined rigorously and extensively under the free-form technique via differential geometry. For general planar conjugation, including noncircular gearing, necessary and sufficient undercutting conditions in terms of the master geometry are obtained. Simple geometric explanation of undercutting is demonstrated in terms of the distance between a contact path and two centrode centers. The technique is demonstrated with B-spline as the master geometry.

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