Gearbox Output Shaft: Power, Torque and Heat
A small conveyor drive uses a 1.5 kW three-phase motor at 1440 rev/min geared down through a 12:1 reduction gearbox to drive a head shaft. The gearbox is 88% efficient. The head shaft runs in two ball bearings and drives a belt pulley of 250 mm pitch diameter. During commissioning, the gearbox oil temperature stabilises at 78 °C in a 25 °C workshop, above the 70 °C limit in the manufacturer's data sheet. The maintenance team proposes increasing the oil grade to a thicker oil to reduce wear, and the production team wants to raise the conveyor speed by 15% to meet a throughput target.
1. Calculate the output speed, output torque and the heat generated in the gearbox at the current duty, showing each step.
Read the worked answer
Output speed. n_out = 1440 / 12 = 120 rev/min.
Power at output. P_out = 1.5 kW × 0.88 = 1.32 kW = 1320 W.
Output torque. ω = 2πn/60 = 2π × 120/60 = 12.57 rad/s. T = P/ω = 1320 / 12.57 = 105 N·m.
Losses as heat. Loss = 1.5 − 1.32 = 0.18 kW = 180 W, dissipated through the casing and oil.
Belt pull. At the 250 mm pitch diameter, radius 0.125 m, the effective tangential force is F = T/r = 105 / 0.125 = 840 N, which is the figure the shaft and bearings must be checked against.
Judgement note. The 180 W loss is small in absolute terms, but a small gearbox has a small surface area, so 180 W can easily produce the observed 53 K temperature rise. The temperature is a thermal-balance problem, not a lubrication problem, and this matters for the next decision.
2. Assess the two proposals - thicker oil and a 15% speed increase - and recommend what should actually be done.
Read the worked answer
Thicker oil. A thicker oil raises the viscosity, which increases churning and shear losses in the mesh and bearings. At the same input power the losses rise, so the oil runs hotter, not cooler, and the 70 °C limit is breached further. It may reduce wear on a marginal film, but it treats a symptom while worsening the cause. It should not be adopted as the first response.
15% speed increase. Speed scales with input speed, so n_out rises to 138 rev/min and ω rises by 15%. If the conveyor torque stays roughly constant, power rises by 15% to about 1.52 kW output and losses rise in proportion to about 207 W. The temperature rise above ambient is roughly proportional to dissipated power for a given casing, so the existing 53 K rise would grow to about 61 K, giving an oil temperature near 86 °C - clearly outside the limit.
Recommendation. Neither proposal is acceptable as stated. The defensible actions are: verify the actual measured loss and check for over-filling or a blocked breather (a common cause of overheating); improve heat rejection by cleaning fins, adding a fan or increasing casing airflow; and if the speed increase is genuinely required, re-select a larger gearbox or add an oil cooler rather than accepting a duty outside the manufacturer's rating. The engineering judgement is that a thermal limit is a hard constraint, and production targets do not override it.