10.6 Design of Brakes & Clutches: Energy, Lining Pressure & Heat

The Design Question

The friction analysis developed in the Engineering Mechanics chapter gives the braking torque a device can generate. Design asks two further questions:

1. How much energy must be absorbed, and how fast?
2. Can the friction surface survive the resulting temperature and pressure?

For mining equipment the second usually governs. A haul truck descending a long pit ramp dissipates enormous energy continuously, and it is thermal capacity, not torque capacity, that limits the design.

Energy to Be Absorbed

The last row is a power, not an energy, and it never ends while the descent continues — the reason retarders exist. For a 100-tonne loaded hauler descending a 1-in-10 grade at 5 m/s:

$$P = 100000\times9.81\times5\times0.1 = 490 \text{ kW}$$

Dissipating half a megawatt continuously through friction linings is not feasible, which is why large mine trucks use dynamic or hydraulic retarders and reserve friction brakes for final stopping and parking.

Heat Generation and the pv Value

Heat generated per unit area of lining per unit time is

$$\frac{\text{Heat}}{\text{area}\cdot\text{time}} = \mu\,p\,v$$

Since $\mu$ varies little for a given lining, the product $pv$ — lining pressure times rubbing velocity — is the standard design proxy. Manufacturers quote a permissible $pv$ for each material, and the design must satisfy

$$p\,v \leq (pv)_{\text{allowable}}$$

as well as separate limits on $p$ alone (to avoid crushing the lining) and $v$ alone (to avoid disintegration).

Friction materials

Running wet in oil roughly halves the coefficient of friction but greatly improves heat removal and lining life, which is why heavy earth moving machinery favours oil-immersed multi-plate brakes and clutches.

Temperature Rise

For a single stop, assuming all heat is absorbed by the drum or disc of mass $m$ and specific heat $c$:

$$\Delta T = \frac{E}{m\,c}$$

For repeated or continuous braking, heat must be rejected as fast as it is generated:

$$E_{\text{rate}} = C\,A\,\Delta T$$

where $C$ is the heat dissipation coefficient in watts per square metre per degree and $A$ the exposed surface area. Typical values of $C$ range from about 30 W/m$^2$K in still air to 100 or more with forced ventilation — which is the entire reason disc brakes are ventilated and drums are finned.

Internal Expanding Shoe Brake

This is the classic drum brake of vehicles and hoists. Two shoes pivot inside the drum and are forced outward by a cam or hydraulic cylinder.

Leading and trailing shoes

The key design feature is that the two shoes behave differently under the same actuating force.

For a shoe with actuating force $F$ at distance $a$ from the pivot, normal force moment arm $b$, and friction moment arm $c$:

$$N_{\text{leading}} = \frac{F a}{b - \mu c}, \qquad N_{\text{trailing}} = \frac{F a}{b + \mu c}$$

The leading shoe therefore produces substantially more torque. If $\mu c \geq b$, the denominator vanishes or reverses and the brake becomes self-locking — it grips without any applied force. That is catastrophic in a service brake and is deliberately avoided by keeping $b > \mu c$ with margin, since $\mu$ rises when linings are cold or wet.

A brake with both shoes leading (two separate actuators) gives high torque in one direction of rotation but very little in the other. A leading-trailing arrangement gives equal performance in both directions, which is why it is used on rear axles and on hoists that must hold in either direction.

Pressure distribution

For a shoe with a lining subtending angle $2\theta$, the normal pressure varies as

$$p = p_{\max}\frac{\sin\theta}{\sin\theta_{\max}}$$

so the maximum pressure occurs where the shoe is furthest from the pivot, not at the middle of the lining. Integrating this distribution gives the braking torque, and the peak pressure must be checked against the lining limit.

Disc Brakes

A disc or caliper brake clamps annular pads against both faces of a rotating disc. Using the uniform-wear assumption, the braking torque for $n$ friction surfaces is

$$T = n\,\mu\,F\,R_m, \qquad R_m = \frac{R_o + R_i}{2}$$

Advantages over drum brakes are decisive for heavy duty: both faces are exposed for cooling, thermal expansion moves the disc towards the pads rather than away (so there is no brake fade from expansion), water is thrown off by rotation, and the design is not self-energising, which makes the response linear and predictable. The price is a higher required actuating force, met with hydraulic assistance.

Clutch Design

Sizing

Starting from the torque relations derived earlier — uniform wear giving $T = n\mu W R_m$ — the design procedure runs:

1. Compute the torque to be transmitted, including a service factor of 1.5 to 3 for shock loads.
2. Choose a lining material and its permissible pressure $p_a$.
3. Choose a radius ratio. For maximum torque from a given outer radius, the optimum is

$$R_i = \frac{R_o}{\sqrt3} \approx 0.577\,R_o$$

In practice $R_i/R_o$ between 0.5 and 0.7 is used. 4. Compute the axial force from $W = 2\pi p_a R_i (R_o - R_i)$ under uniform wear, noting that peak pressure occurs at the inner radius. 5. Check the number of plates required, then check heat.

Engagement heat

During engagement the two sides slip, and the energy dissipated is

$$E = \frac{I_1 I_2}{2(I_1 + I_2)}\left(\omega_1 - \omega_2\right)^2$$

This is independent of how long the engagement takes, and it is why frequently cycled clutches — in a shovel swing drive, for example — need far more thermal capacity than their steady torque rating alone would suggest.

Selection guide

The fluid coupling is worth noting for a coal context: long overland conveyors start under enormous inertia, and a fluid coupling allows the motor to reach speed before load is progressively applied, avoiding both belt damage and motor stalling.