Dear friends, I hope the following information from my clockmaking experience adds something for those interested in making a hypocycloidal rack.
Every gear tooth has two parts. The dedendum is below the pitch circle, - simply put - the point of contact. We can ignore the dedendum as it is just an area shaped to provide clearance for the meshing pinion, or smaller gear. The addendum is the part above, or outside, the pitch circle. It is the part that contacts the pinion and its form is based on one of three basic curves that can be easily visualized by using a coin, a ruler and a jar top. *Cycloidal* teeth are used when a straight rack meshes with a pinion. The cycloidal curve can be visualized by placing your coin against the edge of the ruler. Note the point where the coin touches the ruler, then roll the coin along the ruler. The path generated by the point is a cycloidal curve. *Epicycloidal* teeth are used when a round gear meshes with a pinion. The epicycloidal curve can be visualized by similarly rolling the coin around the outside of the round jar top. This is the gear form used in clocks. *Hypocycloidal* teeth are used when an internal gear meshes with a pinion. This is the form that would be used on the internal rack we are discussing. The hypocycloidal curve can be visualized by rolling the coin around the *inside* of the jar top. In every case the meshing pinion should have a semi-circular addendum and a radial dedendum. It should be noted that when properly designed and executed, gears have no friction. The surfaces roll over each other. Hypocycloidal gears can be cut on a milling machine with a cutter shaped specifically for the gear (or rack) to be made. The problem, especially if cutting a complete gear of small diameter, is providing clearance so the cutter can get inside the material. Another consideration is that cycloidal gearing is best applied to situations where the gear is driving the pinion - as in clocks. In our example the pinion will be driving the hypocycloidal rack. I suggest this can be ignored as there will be no continuous or heavy torque involved. In cases where pinions are driving gears under considerable torque, the involute curve is used to generate the tooth shape. An involute curve is "a curve traced by a point on an inextensible string which is unwound from a circular disk." For what it's worth... Bob Terwilliger Certified Master Clockmaker http://www.shadow.net/~bobt
