Computational Geometry in the Synthesis of Skew Gear Teeth

Abstract

In skew gears of whatever kind, in general (there being special cases), the normal to the tangent plane at the point contact between teeth, the contact normal, may reside instantaneously at any shortest distance d from the pitch line and be inclined at any angle ф to it. The parameters of d and ф may both vary as the point of contact moves from start to finish along its path (and that path need not be straight), but, for constancy of angular velocity transmission as we move from tooth to tooth across teeth, the value of d and ф must remain a constant p, namely the pitch p (mm/rad) of the screwing at the pitch line. That, briefly, is the first law of gearing [3]. After a short review of earlier material from [2,3], which we need here, further confirmation of the veracity of the law is provided. This is then used in further discussion of the iterative methods that might be employed to exploit the law. Paper thus begins to explore the kinds of computational geometry that might be used (a) to synthesise the shapes of teeth that are kinematically correct, these to be cut by NC machine tools if necessary, and (b) to analyse the shapes of teeth that are not correct, these having already been designed and cut by approximate methods.