A machine drive train

gear drive designshaft sizingtorque through a gearboxkey and bearing selection

Motor to bearing: torque from power, the gear reduction, tooth load, the shaft diameter torsion demands, key shear and bearing life.

Shaft Torque from Power and Angular Speed

T = \frac{P}{\omega}

The torque in a rotating shaft: transmitted power divided by angular speed, with ω in radians per second. The relation that decides how thick every shaft in a machine has to be, because a shaft feels the torque and never the power.

Compound Gear Train Value

e = \frac{N_2 \, N_4}{N_1 \, N_3}

The overall ratio of a two-stage compound gear train: the product of the driven tooth counts over the product of the driver tooth counts. The rule that lets a 40:1 reduction be built from two ordinary meshes instead of one enormous wheel.

Gear Pitch Diameter

d = m \, N

The pitch diameter of a spur gear from its module and tooth count — the diameter of the imaginary rolling cylinder the teeth are cut around. The metric module form, d = m N, with the North American diametral pitch convention P = N/d explained and bridged rather than silently substituted.

Gear Tooth Tangential Force

W_t = \frac{2T}{d}

The tangential load a gear tooth carries: torque divided by the pitch radius, which is twice the torque over the pitch diameter. The starting number for every bending and contact stress calculation, and for sizing the shaft and bearings behind the gear.

Shaft Diameter from Allowable Torsional Shear

d = \sqrt[3]{\frac{16 T}{\pi \tau}}

The diameter a solid round shaft needs so that the torsional shear stress at its surface stays within an allowable value. The torsion formula rearranged, and the reason a shaft carrying eight times the torque only has to be twice as thick.

Shear Stress in a Parallel Key

\tau = \frac{2T}{d \, w \, L}

The average shear stress across a parallel key in a shaft keyway: the tangential force at the shaft surface spread over the key's width times its length. The check that decides whether the key or the shaft gives way first — and it is meant to be the key.

Bearing Basic Rating Life (L₁₀)

L_{10} = \left( \frac{C}{P} \right)^{p}

How long a rolling bearing lasts before one in ten of a large group of identical bearings has failed by fatigue, in millions of revolutions. The load ratio raised to a power — cubed for ball bearings, 10/3 for roller bearings — which is why halving the load multiplies life eightfold.

How they fit together

Follow one load path from the motor nameplate to the bearing catalogue and every formula in this set gets used exactly once. Shaft torque from power starts it, and the relationship it encodes is the one that decides the size of everything downstream: for a given power, torque is inversely proportional to speed. A 15 kW motor at 3000 rpm makes 48 N·m and one at 300 rpm makes 477 N·m, which is why a slow-speed drive is a physically large drive and why gearboxes get expensive at the output end rather than the input end.

Gear train value is the reduction, and it is also the torque multiplier — that same 10:1 that dropped your speed raised your torque tenfold, minus efficiency, and the shaft on the far side of it is carrying that. Size the output shaft for the output torque, not the motor's. Pitch diameter from module and tooth count gives the geometry, and tangential tooth force is 2T/d, the load the tooth actually feels. Note the direction that goes: for a given torque, a small pinion carries a higher tooth load than a large gear, which is why the pinion is the part that wears out and why it is usually the harder material of the pair.

The last three check that the parts survive it. Shaft diameter from allowable torsional shear sizes the shaft, and diameter enters as the cube root of torque, so a doubled torque needs only a 26% larger shaft — reassuring, and also the reason a marginal shaft is rarely fixed by a small increase. Key shear stress is the step most often skipped, and it should not be: a key is deliberately the weakest element in many drives, a sacrificial fuse protecting the gearbox, so checking it is checking whether it will do that job at the right torque rather than at start-up. Bearing rating life closes the design, and its exponent is the thing to remember — cubed for ball bearings, 10/3 for rollers. That means a 25% overload cuts life roughly in half, and it means a bearing quoted at L₁₀ has a 10% chance of failing before the number you just calculated. L₁₀ is not a service life, it is the point at which one in ten are gone.