Axial Disk Clutch / Brake (Uniform Wear vs Uniform Pressure)

torque-power

An axial disk clutch or brake is a ring of friction lining squeezed between two plates: press with an axial force $F$, and friction across the annulus from $r_i$ to $r_o$ passes a torque. The formula for that torque looks like it should be settled — but it depends on something you cannot measure directly and that changes over the clutch’s life: how the contact pressure is distributed across the face. Two classical models bracket reality, and this page shows both, side by side, without picking a winner — the same honesty the combined bending-and-torsion shaft uses for its two yield criteria.

Two models over one piece of hardware

Slice the annulus into rings. A ring at radius $r$, width $dr$, has area $dA = 2\pi r\,dr$; under local pressure $p(r)$ and friction coefficient $\mu$ it passes torque $dT = \mu\,p(r)\,r\,dA$. Everything turns on what $p(r)$ is:

• Uniform pressure — a new, rigid clutch presses evenly, $p$ constant. Fixing $p$ from $F = \int p\,dA$ and integrating gives

$T_{up} = N\,\frac{2}{3}\,\mu F\,\frac{r_o^{3} - r_i^{3}}{r_o^{2} - r_i^{2}}$

• Uniform wear — a real lining wears fastest where it works hardest. Wear rate goes as pressure times sliding speed, $p\,v = p\,\omega r$, so a rigid plate wears until $p\,r$ is constant: pressure peaks at the inner edge and falls as $1/r$. That single change collapses the torque integral to a strikingly simple mean-radius form,

$T_{uw} = N\,\mu F\,\frac{r_o + r_i}{2}$

Both are multiplied by $N$, the number of friction faces — a single-plate clutch has two (one each side of the disk); a multi-disk stack has more. Getting $N$ right, and remembering these are per-face formulas, is the classic transcription trap.

Uniform wear is always the safe number

Subtract the two and something clean falls out — the difference is a perfect square over a positive quantity:

$T_{up} - T_{uw} = N\,\mu F\,\frac{(r_o - r_i)^{2}}{6\,(r_o + r_i)} \ge 0$

So the uniform-pressure model always predicts at least as much torque as uniform wear, with equality only in the thin-annulus limit $r_i \to r_o$ (where the two pressure profiles coincide). This bracket is machine-proven in the test pipeline. Its design lesson is why the uniform-wear result is the one engineers actually use: after a brief run-in the pressure redistributes to the $1/r$ shape, so the worn clutch delivers the smaller torque — and quoting the smaller number is the conservative choice. Uniform wear also predicts a higher peak pressure, so it governs lining life too. The widget shows both torques so you can watch the gap: it is widest for a wide annulus and closes as $r_i$ approaches $r_o$.

The best bore is $r_o/\sqrt{3}$

Given a lining that can survive a maximum pressure, where should the bore go? Hold $p_{max}$ fixed instead of $F$. Then the uniform-wear torque is $T = \pi\mu\,p_{max}\,r_i(r_o^{2} - r_i^{2})$, and setting $dT/dr_i = \pi\mu\,p_{max}(r_o^{2} - 3r_i^{2}) = 0$ gives

$r_i^{*} = \frac{r_o}{\sqrt{3}} \approx 0.577\,r_o$

A bore smaller than this buries lining area near the axis, where the moment arm is tiny; a larger bore throws away area. The optimum is marked on the friction face in the widget — slide $r_i$ toward it and watch the torque peak. The peak lining pressure itself is

$p_{max} = \frac{F}{2\pi\,r_i(r_o - r_i)}$

checked against a cited allowable value; exceed it and the page warns (the run-in and burnup story is in the failure note).

Slip power is what actually destroys clutches

While a clutch engages, the driving and driven sides turn at a relative slip speed $\omega_{slip}$, and the friction torque does no useful work — it dissipates power $P = T\,\omega_{slip}$ as heat right at the rubbing face. That heat, not the torque, sets how hard and how often a clutch can be cycled; it is why a slipping clutch smells and a hard-braked rotor glows. The page reports the slip power for each torque model, paired, so neither is silently preferred. The clutch’s job is usually to spin up an inertia — the flywheel disk is exactly what it accelerates, and the engagement that stores energy there is the heat dumped here.

Friction is a cited knob, not a material axis

Every other stress page here binds a material from the database. This one deliberately does not — and, as on the bearing-life page, the reason is worth stating. The two design inputs a friction lining contributes, the coefficient $\mu$ and the allowable pressure $p_{allow}$, are cited free knobs here, with their default ranges taken from the standard dry-lining values of Shigley’s friction-materials table (Table 16-3): $\mu \approx 0.25$–$0.45$ and $p_{allow} \approx 1$–$2$ MPa for molded/organic linings. A proper friction-lining materials table — with per-value basis and errata discipline like the rest of the material database — is real future work belonging to the materials provenance regime; it is genuine material data, not the relation-table capability the gear and shoulder-fillet pages use. Until then, dialing $\mu$ and $p_{allow}$ as knobs is the honest interface. Friction here is the product (the torque you want), the mirror of the power screw and belt drive, where friction is a budget to be spent.

Try it

Governing relations

Derivation

Steps marked modeling step are where physics enters by citation — every other line is machine-proven to follow from them, and the cited models are independently re-derived in the test pipeline where possible. See what is and isn't machine-verified.

How it fails

A clutch or brake almost never fails by running out of torque. It fails by getting hot, by wearing out, or by the friction coefficient quietly changing on you — none of which the clean torque formulas on this page can see. The number this page computes is the capacity of a cool, intact lining; the failure modes are what happens when that assumption lapses.

Overpressure: the failure the page does warn about

The one failure mode inside the model is lining overpressure. Under uniform wear the contact pressure peaks at the inner edge, $p_{max} = F/(2\pi r_i(r_o - r_i))$, and every friction material has a maximum pressure it can carry before it crushes, extrudes, or bakes. The widget warns when $p_{max} > p_{allow}$. Notice how easy that is to trigger: shrink the annulus toward a thin ring ($r_i \to r_o$) and $p_{max}$ runs away even though the torque barely moves — all the force is crammed onto a sliver of lining. This is exactly why the torque-optimal bore $r_i^{*} = r_o/\sqrt{3}$ matters: it is also close to where the lining is used efficiently rather than punished at the bore.

The heat the model cannot see

The deeper failure is thermal, and it lives entirely outside these equations. Every engagement dumps the slip energy $\int T\,\omega_{slip}\,dt$ into the friction face as heat. This page reports the instantaneous slip power $P = T\,\omega_{slip}$, but it deliberately does not integrate the engagement transient or the temperature rise — that is time-domain, thermal-mass territory, out of scope for a static page. Yet the consequences dominate real design:

• Fade. For most linings $\mu$ falls as temperature climbs. A brake that grips cold can go slick when hot — so the torque this page computes with a fixed $\mu$ is optimistic exactly when you need the brake most. Repeated hard stops are the classic fade scenario.
• Burnup and glazing. Enough heat chars organic linings, glazes the surface, boils the oil out of a wet clutch, or heat-checks and warps the mating steel. A clutch “slipped” too long is destroyed by the power it dissipated, not the torque it transmitted.
• Thermal capacity, not torque, sets the duty rating. How often a clutch can be cycled is a heat-rejection limit. The torque tells you whether it engages; the slip energy tells you whether it survives engaging.

The model changes under you

There is a subtler trap here than a missing term: the two models are not alternatives you pick once — they are the same clutch at two ages. A new, rigid lining really does start near uniform pressure, and then it wears, fastest where $p\,r$ is largest, until it reaches the uniform-wear $1/r$ profile and stays there. So the honest design number is the uniform-wear torque (the smaller one, by the machine- proven bracket), because that is what the clutch becomes after its first hours of service. Designing to the uniform-pressure torque is designing for a condition the hardware leaves behind almost immediately.

Out of scope here

This page is the isolated, cool-lining torque capacity. Named but not modeled: the engagement transient and temperature rise (time integration), fade curves and wear-life prediction, cone and centrifugal clutches, and multi-disk stack subtleties beyond the $N$-face multiplier. The inertia a clutch spins up lives on the flywheel-disk page — the energy stored there is the heat released here — and the same friction that is the product on this page is the cost on the power-screw and belt-drive pages.

Sources

• Budynas, R. G., & Nisbett, J. K., Shigley's Mechanical Engineering Design, 10th ed., McGraw-Hill, 2015 — §16-5 (frictional-contact axial clutches and brakes): the uniform-pressure and uniform-wear torque derivations, the peak-pressure relation, and the torque-optimal inner radius r_i = r_o/√3; and Table 16-3 (friction materials — coefficient of friction and maximum contact pressure for dry linings).
• Juvinall, R. C., & Marshek, K. M., Fundamentals of Machine Component Design, 5th ed., Wiley, 2011 — ch. 18 (clutches and brakes): the same uniform-pressure and uniform-wear axial-clutch analyses and the run-in argument for preferring the uniform-wear result in design.