The project was a portable rig: a 2.4 hp gasoline engine spinning at 3800 rpm driving a single-cylinder air compressor that wants about 1500 rpm. So a gearbox sits in between — about a 2.53:1 reduction — with shafts, a gear set, bearings, couplings, and the cap screws that hold the compressor head on. The thing I want to flag is the trap hiding in the word strength.
The compressor torque is not constant. Over a single crank revolution the cylinder pressure climbs from near-atmospheric to a peak and collapses again, so the torque the output shaft sees is a pulse train, not a steady load. A shaft sized to survive the peak torque treated as a static load can still crack after a few million of those pulses, because the failure mode isn’t yielding — it’s fatigue. So the output shaft gets designed against a fatigue criterion (a Goodman line that trades off the alternating stress against the mean stress) rather than a single static factor of safety. “It doesn’t yield” is the wrong question; “how many cycles before a crack grows” is the right one.
A small detail that catches people: the output shaft is the one to worry about, not the input. It turns slower, and for the same transmitted power, slower means more torque. The high-speed input shaft is comparatively lightly loaded.
The other half of the lesson is that you can’t design any one element in isolation. You can’t size the shaft until you know the gear diameters, because the gears set the bending loads on it — and you can’t finish the gears until you’ve picked bearings, and the bearing bore pushes back on the shaft diameter. It’s circular on purpose. You break the loop by assuming: take the gears at, say, 4″ and 10″ with a 20° pressure angle, size the shaft against the fatigue load, then come back and refine once the gears and bearings are real. The gear set itself is AGMA full-depth teeth, hardened steel, sized against that same cyclic load; the head bolts get an infinite-life design because the head force cycles from 0 to ~1000 lb every revolution, and even the gasket choice (confined vs. unconfined) is really a stiffness decision about how that fluctuating load splits between the bolts and the joint.
None of those pieces is hard on its own. What the project actually teaches is the bookkeeping: every machine element is a boundary condition for the next one, and “design” is mostly iterating the interactions until the numbers stop moving — then designing for the load that repeats, not the load that’s largest once.