Third order curvature analysis of conjugate surfaces

The planar Euler-Savary equation and its spatial extensions reveal the curvature relations between two rigid contacting curves and surfaces under relative motion at contact points. But these equations usually do not concern curvature derivatives which are considered as the third order local properties of regular surfaces and are very important for engineering surfaces, e.g. gear tooth surfaces. Based on the previous work, the curvature derivative tensor defined by the sum of tensor products of a vector and a second order tensor is introduced for the third order local properties of surface first. Then the relations between the curvature derivatives of mating surfaces undergoing generalized relative motion are obtained directly, including both line contact and point contact cases. The derivation and formulation are clear and free of reference systems. The compact and invariant results can be used for the design and manufacturing of gears and other sculptured surfaces. For illustration and validation, the application for the third order contact analysis of spiral bevel gears is presented with a calculation example.