Hulet self-unloader image courtesy of ASME.
Interesting things happen when you start using “non-standard” gear geometry. As early as the 1880s, engineers understood that tooth shape and root fillet geometry could have a big effect on gear performance. The “Lewis factor” we still include in bending strength calculations was the work of Professor Lewis and he, like others of that period, wrote of his investigations and spoke about them at conferences. A speech that Lewis gave at the Engineers’ Club in Philadelphia is available on-line and is still relevant.
Back then there was no “standard” geometry. The involute curve was just one of several competing tooth systems. Pressure angles had not settled into the 14.5, 20, and 25 degree patterns we employ today. Tooth depths varied widely as well; the famous Hulet self-unloader that revolutionized the Great Lakes ore boat trade had about an 8 degree pressure angle and a whole depth/NDP value of around 4!
Many solutions “worked.” The Hulet had center distances that could vary as much as an inch during its operation; no “standard” tooth could do that. The debate over which form was “best” continued into the mid 1920’s and even then multiple options were endorsed. The “20 degree full depth” model was most popular but even Professor Lewis expressed regret that he did not hold out for a 22.5 normal pressure angle.
This is the context for why rack offset is such an important technique that fledgling gear designers must understand and embrace. Even if we move to a high contact ratio “norm” the X factor remains important because of its ability to take an existing cutting tool and produce a tooth shape that is customized to the application.
How does it do this magic? Through the wonder of the involute curve! I cringe when people inflict high level math on beginners and never cite the “string unrolling from the base circle” thing until the student asks about it. One does not have to know about petrochemicals to appreciate the internal combustion engine or to drive a car. The same in true with involute curves.
If you cannot wait to understand that “string thing” feel free to peruse the Gear Technology archives. Keywords are your friends!