On the gear mechanics before the cycloid and involute profiling

After the extraordinary development of the machinery in the Hellenistic antiquity, the gear technique was transmitted to the Middle Ages through the Byzantine and Islamic culture and then to the Modern Era. The tooth profile was very crude, often trapezoidal or even rectangular and the gear behavior differed substantially from the modern involute profile. The kinematics of trapezoid profiles is here analyzed in detail, focusing on the temporal variation of the speed ratio due to the back and forth shifting of the relative instant center. Considering an isolated tooth pair, an approach phase is firstly observable, where the tip of the driven profile is pushed by the driver flank and then, after passing through the matching configuration, a recess phase follows, where the tip of the driver profile pushes the flank of the driven one. The acceptability of each configuration of this theoretical evolution is then checked according to the interference prevention requirement for the following and preceding tooth pairs and for the back inactive profiles. This yields some limitation to the tooth thickness, which may justify the frequent tooth slenderness in the design of the gear systems of the Renaissance. The periodic tooth collisions due to the jerky variability of the speed ratio are properly analyzed, together with the energy losses arising from the sliding friction. Overall, the results show that only one tooth pair is active at each time instant and the acceptable contacts may belong to the only approach region, or to the only recess region, or may be split into two separate sub-phases, in approach and in recess, or may even straddle both regions. The occurrence of each of these situations depends on the average speed ratio and on the assigned clearance between the two wheels.