Power and Machines

The mechanisms lesson ended with gears trading torque for speed, and a gear train can multiply torque by 15 without breaking a sweat. What it cannot do is create power, and a motor's spec sheet makes that plain. A small 12 V motor lists two headline numbers: stall torque 0.6 N m and no-load speed 3,000 rpm. The two best numbers on the sheet are the two ends of one curve, and the motor never gets to be at both ends at once.

Power is torque times speed

Power is the rate of doing work, measured in watts, and for anything spinning it equals torque times angular speed: P = tau times omega. The trap lives in omega. Angular speed must be in radians per second, and a spec sheet gives rpm. One rpm is 2 pi radians per 60 seconds, so multiply rpm by 2 pi / 60, about 0.105, to get rad/s.

Worked example: a motor delivering 0.5 N m while spinning at 4,000 rpm. Omega is 4,000 times 0.1047 = 418.9 rad/s, so P = 0.5 times 418.9 = 209 W. Skip the conversion and multiply 0.5 by 4,000 for 2,000 W, ten times too big, a mistake that has ended more than one student's drivetrain calculation. The units are the check: N m times rad/s gives joules per second, and joules per second are watts.

The formula survives a change of clothes. One revolution costs work equal to torque times 2 pi, so a motor turning at r revolutions per second delivers tau times 2 pi times r watts, which is the same statement in different units. Spec sheets usually list a rated point too, the torque and speed the motor can hold continuously without overheating. A motor can often exceed its rating for short bursts, and that burst ability is real design currency for machines that start, lift, and rest.

The torque-speed curve

A permanent-magnet DC motor under steady voltage runs on a nearly straight line between its two headline numbers. At zero speed the motor produces its full stall torque; at its no-load speed it produces no torque at all; everywhere between, torque falls linearly as speed rises. The reason is the motor working against itself: a spinning motor is also a generator, and the voltage it generates, called back-EMF, opposes the supply. High speed means high back-EMF means little current means little torque.

Power along that line is torque times speed, zero at both ends and maximal in the middle. Halfway between stall and no-load, this motor delivers 0.3 N m at 1,500 rpm: omega = 157.1 rad/s and P = 0.3 times 157.1, about 47 W. That peak sits at half the no-load speed for any linear DC motor, a fact worth memorizing.

Designers rarely park a motor exactly on the peak. The peak is what the machine can do for a short push; for continuous work the sensible region sits below it, at higher speed and lower torque, where the motor runs cooler and lives longer. A drivetrain that spends its whole life at peak power is a drivetrain with a short life, and the curve is read with that in mind.

The same matching logic reaches past brushed DC motors. Brushless motors, steppers, and servos all publish torque and speed limits and get matched to loads the same way. A stepper holds position at standstill but loses torque as steps per second climb, which is why a 3D printer axis skips steps when pushed too fast. The curve shapes differ; the conversation between motor and load does not.

The stall end is also the dangerous end. At stall there is no back-EMF at all, so the only thing limiting current is the tiny resistance of the copper windings, and the motor turns that current into heat at a rate it cannot survive for long.

Matching motor to load

Put the pieces together on a real job: a winch must lift 40 kg at a steady speed, winding rope onto a drum of radius 0.05 m. The drum needs torque equal to the weight times the radius: 40 times 9.8 times 0.05 = 19.6 N m. Direct drive is hopeless: the motor stalls at 0.6 N m, a factor of 33 short.

A 100:1 gearbox fixes the torque. With 70 percent of the motor's work surviving the gearbox and drum, the motor must supply 19.6 / (100 times 0.7) = 0.28 N m. That point sits well clear of the 0.6 N m stall limit. Now read the curve for speed: 3,000 times (1 - 0.28 / 0.6) = 1,600 rpm. The drum follows at 16 rpm, and the lift speed is the drum's circumference times its rotation rate: 2 pi times 0.05 times 16 / 60 = 0.084 m/s, a calm 8.4 cm per second. The motor happens to deliver 0.28 times 167.6 = 47 W, right near its peak, which is no accident: a well-matched motor works near the middle of its curve.

Why not skip the gearbox and buy a motor that makes 19.6 N m directly? Motors that make big torque make it slowly and heavily: the copper, magnets, and steel needed for 19.6 N m at low speed cost far more than a small fast motor plus a stack of gears. The gearbox is not a luxury, it is the trick that turns cheap fast rotation into slow strong shafts. Weight, current draw, and cost all vote for the gear train, which is why nearly every real actuator, from a cordless drill to a car window lift, is a fast motor wearing a gearbox.

A 100:1 ratio also has to be built, not bought in one piece. Spur gear stages top out near 5:1 each before the pinion gets silly, so 100:1 is three or four stages in series, each spending its own few percent of efficiency. Worm gearboxes reach huge ratios in one stage and can self-lock, holding position with power off, but they run hot at 40 to 70 percent efficiency. The 70 percent assumed above is what a few spur stages honestly deliver, and it was not pulled from the air.

import math

m, r = 40.0, 0.05        # 40 kg load, drum radius 0.05 m
gear, eff = 100.0, 0.7   # 100:1 gearbox, 70 percent survives it
stall = 0.6              # motor stall torque, N*m
no_load = 3000.0         # motor no-load speed, rpm

drum_torque = m * 9.8 * r
motor_torque = drum_torque / (gear * eff)
rpm = no_load * (1 - motor_torque / stall)
lift = 2 * math.pi * r * rpm / gear / 60

print(f"drum torque needed:  {drum_torque:.1f} N*m")
print(f"motor torque needed: {motor_torque:.2f} N*m of {stall} stall")
print(f"motor speed: {rpm:.0f} rpm, drum speed: {rpm/gear:.0f} rpm")
print(f"lift speed: {lift*100:.1f} cm/s")

The script reproduces every number above. Change the load to 60 kg and the motor torque rises to 0.42 N m, the speed falls to 900 rpm, and the lift crawls at 4.7 cm/s; that is the curve talking. A heavier load does not strain the motor dramatically, it just bargains with it for speed.

Efficiency is a chain

Every stage between motor and load keeps only a fraction of what enters. Multipliers compound by multiplication, never addition: a 90 percent gearbox feeding a 95 percent chain drive feeding drum bearings at 98 percent delivers 0.9 times 0.95 times 0.98 = 0.84, about 84 percent overall. String six stages together and even 95 percent each leaves 0.74 at the far end. The losses are not abstract; they leave as heat, which is why a gearbox that has been working hard is too hot to touch. The worked winch above loses 16 percent of 47 W, about 7.5 W, warming its grease, and a stalled one turns the entire electrical input into heat in the motor windings.

Friction deserves one honest paragraph, because it is both the enemy and the price of everything working at all. Dry metal on metal under load galls, tears microscopic chunks off both surfaces, and can seize entirely; a bearing run dry fails in minutes rather than years. Lubrication works by interposing a thin film that the surfaces ride on: oil is pumped or splashed where flow exists, grease is oil held in place by a thickener for spots that never see flow. Choosing the right one is not fussiness; it is the difference between a machine that ages and one that stops.

Friction even has numbers worth memorizing. Steel sliding on steel without lubrication has a coefficient of friction near 0.6: 60 N of drag for every 100 N pressing the surfaces together. Oiled, the same pair drops well below 0.2. Rolling replaces sliding, which is the entire idea of the ball bearing, and rolling resistance runs in the thousandths to low hundredths, an order of magnitude better. A cart on bearings outglides a slider by ten times, and that gap is bought with a few grams of grease and grade-five steel balls.

The drivetrain above now exists in full: numbers on paper that balance, hold, spin, and lift. The next lesson, From Design to Part, takes the final step and turns a design into a physical object that actually fits together: tolerances, fits, and how the part gets made.