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Journal for Geometry and Graphics 7 (2003), No. 2, 201--210
Copyright Heldermann Verlag 2003

Explicit Calculation Methods for Conjugate Profiles

Gert Bär
Institute for Geometry, Dresden University of Technology, Zellescher Weg 12-14, Willersbau B 119, 01062 Dresden, Germany,

The tooth profiles of two meshing spur gears are considered. A given profile may envelop another gear profile. The paper presents an explicit parametrization of the gear profile. It is also shown that the corresponding conjugate rack profile is explicitly determined. Analogously, the line of action is described. Afterwards, the inverse problem is studied. If the gear centers, the gear ratio, and a line of action are given then conjugate gear profiles are determined by the solution of an ordinary differential equation of first order. An example of application is shown. Finally, this differential equation is geometrically interpreted by introducing the pole distance. At any instant the rate of change of the pole distance relates to the inclination of the rack profile. It is shown that for a specific linear relation, corresponding conjugate profiles are trochoids of 2nd or 3rd order.

Keywords: plane kinematics, conjugate tooth profiles, line of action, rack profile.

MSC: 53A17.

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