Disc Clutch Torque (Uniform Wear)

An annular friction disc does not press evenly, and the question of how the pressure is distributed is what separates the two standard clutch formulas.

Wear at any point on the facing goes roughly as the pressure times the sliding velocity there, and sliding velocity is proportional to radius. So the outer edge of a new, flat facing wears fastest, taking material away until the pressure has redistributed. The equilibrium the disc converges on is $pr = \text{constant}$, which means pressure is HIGHEST AT THE INNER RADIUS and falls off outward. That is the uniform-wear assumption, and integrating $dT = 2\pi\mu p r^2\,dr$ with $pr = p_a r_i$ held constant gives $T = \pi\mu p_a r_i (r_o^2 - r_i^2)$ per rubbing surface.

The rival assumption — uniform PRESSURE — describes a facing that is brand new, perfectly flat and rigidly backed, and gives a torque roughly ten to fifteen per cent higher on typical geometry. Design on uniform wear. It is the condition the clutch spends nearly all of its life in, and it is the smaller of the two answers, so it is conservative in both senses at once.

Uniform wear also delivers a tidy design result. For a given outer radius and maximum pressure, differentiate $r_i(r_o^2 - r_i^2)$ with respect to $r_i$ and the torque is greatest at $r_i = r_o/\sqrt{3} \approx 0.577\,r_o$. This is why clutch facings are relatively narrow annuli rather than full discs: material inboard of that radius adds rubbing area that contributes almost no torque, while raising the pressure the inner edge must survive. Anyone whose instinct is to fill the available circle with friction material is designing a worse clutch than the one that leaves a hole in the middle.

Three practical notes. The peak pressure has to stay under the facing material's own allowable, a published property that sits in the hundreds of kilopascals to a low megapascal for organic materials and higher for sintered metal; exceeding it wears the facing out fast and cooks the inner edge, which is where the peak lives. When torque will not fit in the space available, add SURFACES rather than diameter — that is the whole idea behind the multi-plate pack in a motorcycle or an automatic transmission. And a clutch's real design case is often the ENERGY it absorbs while slipping into engagement rather than the torque it holds afterwards: every engagement dumps heat proportional to the slip, which is why a clutch that holds its rating comfortably can still be destroyed by being slipped.