Abstract
Gear theory is re-examined
and we find optimal shapes for gears.  As optimality
criteria, we allow: (1) minimal frictional losses (highest efficiency)
assuming linear law of friction {\it or} (2) uniform maximum stress (it
will wear out slowly and last the longest) assuming Hertzian contacts
{\it 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
{\it 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 differential equations
(ODEs), and for each of these 3 criteria we find the ODE for both spur
(cylinder) and bevel (conical) gears, i.e. a total of 6 ODEs.


Keywords
minimum wear, uniform stress, minimum temperature,
maximal efficiency, minimal frictional loss,
optimal shape, form, structure, spur gears, bevel gears.