Disk Clutch Torque⚠ unverified

Understanding the Parameters

• Axial force $F$ — the clamping load from the spring, hydraulic piston, or diaphragm. Torque is directly proportional to it, so a stronger clamp gives more capacity — until the facing pressure exceeds the lining limit (Clutch Facing Pressure) or generates too much heat.

• Friction coefficient $\mu$ — the facing material property ($\approx 0.3$–$0.4$ for organic linings, $\approx 0.1$ wet, higher for sintered/ceramic). It fades with temperature (see Brake Fade Factor), so design values allow margin.

• Mean friction radius $R_m$ — the effective radius at which the friction force acts. It is not the simple average of the radii in general: the uniform-wear model gives $R_m = (R_o + R_i)/2$ (see Clutch Torque Capacity Uniform Wear), while the uniform-pressure model gives $R_m = \tfrac{2}{3}(R_o^3 - R_i^3)/(R_o^2 - R_i^2)$. Uniform wear is the standard (conservative) design assumption.

• Number of surfaces $n$ — a single-plate clutch has $n = 1$ (or 2 if both faces work); multi-plate clutches stack many disks to multiply capacity in a compact package (see Multiple Disk Clutch).

Derivation (Approaching a Proof)

Consider one annular friction face between inner radius $R_i$ and outer radius $R_o$, with contact pressure $p(r)$ and friction coefficient $\mu$. A thin ring at radius $r$ of width $\mathrm{d}r$ has area $2\pi r\,\mathrm{d}r$, carries normal force $p\,2\pi r\,\mathrm{d}r$, and contributes a friction torque (force $\times$ radius):

$$\mathrm{d}T = \mu\, p\, (2\pi r\,\mathrm{d}r)\, r = 2\pi\mu\, p\, r^2\,\mathrm{d}r.$$

The clamping force is $F = \int_{R_i}^{R_o} p\,2\pi r\,\mathrm{d}r$. The pressure distribution depends on the wear model:

• Uniform wear (worn-in clutch): wear $\propto p\,r$ is constant, so $p\,r = C$ (pressure highest at the inner radius). Then $F = 2\pi C(R_o - R_i)$ and $T = 2\pi\mu C\int_{R_i}^{R_o} r\,\mathrm{d}r = \pi\mu C(R_o^2 - R_i^2)$. Eliminating $C$: $$T = \mu F\,\frac{R_o + R_i}{2} = \mu F R_m.$$

• Uniform pressure (new clutch, $p$ constant) gives instead $T = \tfrac{2}{3}\mu F\dfrac{R_o^3 - R_i^3}{R_o^2 - R_i^2}$.

Either way the result has the form $T = \mu F R_m$ with an appropriate mean radius; multiplying by the number of friction surfaces $n$ gives $T = F\mu R_m n$. The uniform-wear form is preferred for design because a clutch quickly wears to that state and it predicts the lower capacity.

Dimensional check. $[T] = \text{N}\cdot\text{m}$ ($\mu$, $n$ dimensionless). ✓

History and Development

Friction-clutch torque analysis with the uniform-wear and uniform-pressure models is classical machine design (Shigley), developed in the early automotive and industrial era. The uniform-wear assumption — that a running clutch wears until $pr$ is constant — is the standard conservative basis for automotive and industrial clutch/brake rating.

Related Concepts: Clutch Torque Capacity Uniform Wear, Multiple Disk Clutch, Cone Clutch Torque, Clutch Facing Pressure, Friction, Collar Torque

Notes: Uses the mean friction radius $R_m$ — uniform-wear $(R_o+R_i)/2$ (standard) or uniform-pressure $\tfrac23(R_o^3-R_i^3)/(R_o^2-R_i^2)$. Check facing pressure and heat separately. The same $\mu F R$ physics governs disk brakes and thrust collars.