Brake Torque from Friction

A friction brake turns kinetic energy into heat, and the torque it makes while doing so is the friction force times the radius it acts at, once for each surface that rubs. The equation is trivial; the three inputs are each a small trap.

Count the surfaces. A caliper squeezing a disc presses on BOTH faces, so its clamping force generates friction twice and n = 2. A multi-plate clutch or brake counts the INTERFACES between plates rather than the plates themselves: five plates alternating with four mating discs gives eight rubbing surfaces, not five or nine. Getting this count wrong is the most common error on the page, and it always errs by an integer factor, which makes it large.

The coefficient of friction is a property of a PAIR, not of a material. There is no such thing as the friction coefficient of a brake lining; there is only the coefficient of that lining against that drum or rotor material, at that temperature, in that condition. Published values for organic and semi-metallic linings on cast iron sit near 0.3 to 0.45, sintered metal a little higher, wet running well below. All of them fall — sometimes sharply — once the lining is hot, wet or glazed. That fall is brake FADE, and it arrives precisely when the brake is working hardest, which is why an actuator should be sized on the low end of the range rather than the nominal value.

The effective radius is the mean radius of the rubbing band, not the outer radius of the disc. Using the outer radius flatters the answer by a few per cent every time. Radius is nonetheless the cheapest torque available: it multiplies directly, costs no extra clamping force, and brings more swept area and better cooling with it. That is why bicycle and motorcycle discs grew outward rather than thicker, and why a physically oversized brake on a small hub usually means somebody could not move the radius and had to buy the torque with force instead.

The last point is the one that actually sizes most brakes: their limit is THERMAL, not frictional. A brake must absorb the entire kinetic energy of the machine and then get rid of it. A vehicle brake dumps megajoules into a few kilograms of iron in seconds, and the disc's job between stops is to shed that heat to the air. This is why discs are vented, why repeated stops fade a brake that handles one stop easily, and why a hoist brake that holds a static load and a brake that stops a moving one are entirely different design problems even at identical torque.