The conjugate profile of the circular teeth of a spur gear. Part I: Problem statement

Abstract

The paper presents the manner of finding the conjugate profile of a circular tooth of a spur gear. A synthesis of the method of enveloping applied to the cam mechanisms is presented in order to relate it the profile of the spur gears. The main argument of employing cam mechanisms is the possibility of obtaining any follower law of motion using a minimum number of parts- the cam and the follower. In the case of the mechanisms with flat face follower, the cam is obtained as an envelope of successive positions of the follower. The gear mechanisms are a particular case of cam mechanisms. The major requirement imposed to this mechanism is to transmit the rotational motion between two shafts with a constant transmission ratio. From here it results that the profile a geared wheel can be completely identified when there are known the distance between the axes, the transmission ratio and the profile of one of the wheels. The most used curve as tooth flank is the involute of a circle, due to the fact that this curve has as conjugate curve an involute, too. Although the involute profiles are common in most of the technical appliances, there are cases when they cannot satisfy the functional constraints of certain devices. As example, in the mechanical watches technology, large transmission ratios are needed and the gears with small number of teeth are used as routine. But this necessity is better fulfilled by cycloidal profiles than the involute ones. The circular profiles for the spur gear are the oldest gears due to the simple profile. The exact conjugate profile of a circular tooth obtained by enveloping by means of dedicated software is presented.