Mating Involute Helicoids

Abstract

It has long been a matter for conjecture that a rational theory might be written for the constancy of angular velocity ratio among offset skew gear teeth. Speaking in terms of rigidity (in the absence of elasticity, vibration, wear and so on), the attainment of exact constant angular velocity ratio in spur and helical gears by means of involute action has been an exploitable and exploited possibility for 150 years [12] [13]. Matters have been at least explicable, also, among the so-called crossed helicals [33]. Within the wider range of general spatial gears, however, the geometry of an acceptable, involute transmission with exact angular velocity ratio has been seen, as yet, as being perhaps discoverable [59] [61], but mysterious [48]. During the recent forties, two important papers appeared on the subject of mating involute helicoids. They came from Giovanozzi [66], and Poritsky and Dudley [67]. Each of them showed that any two involute helicoids, axially mounted upon two shafts generally disposed, and in contact a single point, would under all circumstances (a) transmit rotary motion with uniform angular velocity, and (b) do so with the point of contact moving in a straight line. Various editions of a regular text by Giovanozzi [68] have since consolidated the basic material. This means that some of the aspects of this chapter 3 have been known before by others, but the fresh treatment here exposes new material that can be used effectively in mission-oriented design. I mention again Beam and the Beveloids [59] [61]. Since Beam, the Beveloids have been known, generically, as taper (or conical) gears. They were doubtlessly discovered during the testing of previously untested ‘taper’ methods of machining, but they are, on the other hand, best explained, as we shall see, by means of the ‘cylindrical’ principles outlined in this book (§8.02).