Optimal Shapes for Gears

| Gear theory is reexamined and we nd optimal shapes for gears. As optimality criteria, we allow: (1) minimal frictional losses (highest eeciency) assuming linear law of friction or (2) uniform maximum stress (it will wear out slowly and last the longest) assuming Hertzian contacts or (3) uniform maximal temperature, assuming we are in a high power limit in which all heat is removed by the lubricant. Some other criteria which have been used before are that (4) mis-spacing the gears still yields perfect gear action with the desired speed ratio or (5) minimal vibration in the low friction limit. Both (4) and (5) lead to \involute gears" which are the standard in engineering practice. Criteria 1, 2, and 3 lead to apparently new gear-tooth forms. We manage to describe these curves with ordinary diierential equations (ODEs), and for each of these 3 criteria we nd the ODE for both spur (cylinder) and bevel (conical) gears, i.e. a total of 6 ODEs. Contents 1 Introduction 2 2 The most general kind of gears with extended contact and \no" sliding. 2 3 Gear terminology 3 4 History of gears 4 5 Fundamental law of gear action 4 6 The two kinds of gears which have been commonly used 5 7 Known optimal properties of involute gears 6 8 The conjugate of a gear 6 9 Interchangeability, matched pairs, and Don Juan gears 8 10 Sliding friction cannot be avoided 9 11 Is friction linear? 10 12 Curvature and arclength: some useful formulas 11 13 Radii of curvature of a gear and its mate 11 14 Velocity of the point of contact 13 15 Gear that minimizes frictional losses 14