Limited-slip and locking differentials

Why it matters

An open differential always gives both wheels the same torque, so a single wheel on ice, mud or in the air limits the whole axle to almost nothing. Limited-slip and locking differentials break that rule by letting the wheel with grip receive more torque. They are used on performance cars for traction out of corners, on 4×4s and trucks for off-road mobility, and their torque-bias calculations follow directly from the open-differential analysis.

Key ideas

• The open-differential limit. Ignoring friction, T_L = T_R. If one wheel can only transmit a small torque before it spins, the other wheel receives the same small torque.
• Torque bias ratio (TBR). For any limited-slip device, the ratio of the torque on the high-traction wheel to that on the low-traction wheel at the point where slip begins. An open differential has TBR ≈ 1; a fully locked one has an unlimited TBR. Internal friction makes the difference: T_high − T_low = T_lock, the locking (friction) torque.
• Limited-slip differential (LSD) – still allows speed difference for cornering but resists it, biasing torque toward the slower (gripping) wheel. Main types:
  • Clutch-type (plate) LSD. Multi-plate clutch packs between each side gear and the case. A preload spring (Belleville or coil) gives a constant locking torque, so some torque reaches the gripping wheel even if the other is in the air. In many designs the differential pinion shafts sit in ramps in the case; drive torque forces the ramps apart and clamps the clutches harder, so locking torque rises with input torque (a roughly constant TBR). Clutch wear, chatter in tight turns and need for special friction-modified oil are drawbacks.
  • Viscous coupling. Interleaved perforated plates, alternately splined to the two members, run in silicone fluid. The coupling transmits torque roughly in proportion to the speed difference, so it is speed-sensing: it does nothing until a wheel actually slips, and it can heat and stiffen ("hump") under prolonged slip. Widely used in centre differentials of all-wheel-drive cars.
  • Torque-sensing gear type (e.g. worm-gear Torsen, helical Quaife-type). Gears with high internal friction (worm or helical gears thrusting against the case) resist relative motion in proportion to the torque transmitted. They react instantly, do not wear like clutches and give a fixed TBR (typically about 2–5; take the maker's figure). But since bias is a multiple of the low-side torque, if one wheel is completely unloaded (zero torque) the gripping wheel also gets almost none.
  • Electronically controlled LSD / torque vectoring. A clutch pack controlled by a hydraulic or electric actuator sets the locking torque to any value, commanded by the stability-control computer; torque-vectoring units can even overdrive the outer wheel.
  • Brake-based "electronic differential lock". The ABS/traction-control system brakes the spinning wheel. Its brake torque adds to the low-side torque, so the differential can send that much more to the gripping wheel. Cheap, but it wastes energy as heat.
• Locking differential (diff lock). Locks the two side gears together (or a side gear to the case) by a dog clutch, so both wheels turn at the same speed, like a solid axle. Each wheel then takes whatever torque its own grip allows – total traction is the sum of both wheels' grip. Operated by lever, cable, air or electric actuator (selectable lockers), or automatically by a ratchet mechanism that unlocks the overrunning wheel in turns (automatic lockers). A locked differential causes scrub, tyre wear, heavy steering and driveline wind-up on high-grip surfaces, so it is meant for low-speed, low-grip use.
• Handling effects. On the drive axle, an LSD reduces inside-wheel spin when accelerating out of a corner but tends to resist turning (understeer) when locked hard; settings are a compromise. Front-axle LSDs can cause torque steer.

Formulas

TBR = T_high / T_low

• TBR: torque bias ratio (–); T_high, T_low: torques on the high- and low-traction half-shafts (N·m).

T_high − T_low = T_lock, T_high + T_low = T_c

• T_lock: total locking (friction) torque of the differential (N·m); T_c: torque delivered to the differential case (N·m).

T_total,max = T_low,max · (1 + TBR)

• Maximum total axle torque (N·m) when the low-traction wheel can transmit at most T_low,max = μ_low · W_low · r (N·m).

T_lock = 2 · n · μ · F · r_m (preloaded clutch-type, two packs)

• n: friction surfaces in each pack; μ: friction coefficient (–); F: axial preload (N); r_m: mean friction radius (m). With ramps, F increases with input torque.

T_v ≈ k · Δω

• Viscous coupling torque (N·m), roughly proportional to the slip speed Δω (rad/s); k (N·m·s/rad) depends on plate number, area, gap and fluid viscosity and is taken from the maker's data.

Locked differential: T_total,max = (μ_L · W_L + μ_R · W_R) · r

Worked examples

Example 1 (standard). A car's left rear wheel is on ice and can transmit at most 120 N·m; the right rear wheel on dry tarmac could take 960 N·m. Wheel radius 0.30 m. Find the maximum total tractive effort with (a) an open differential, (b) a Torsen-type LSD with TBR 3, (c) a locked differential.

1. (a) Open: T_total = 2 × 120 = 240 N·m; F = 240 / 0.30 = 800 N.
2. (b) LSD: right wheel = 3 × 120 = 360 N·m (below its 960 N·m limit); T_total = 120 + 360 = 480 N·m; F = 1600 N.
3. (c) Locked: T_total = 120 + 960 = 1080 N·m; F = 3600 N.

Answer: 800 N, 1600 N and 3600 N respectively. The TBR-3 differential doubles traction compared with the open one; only locking uses all the available grip.

Example 2 (GATE level). A clutch-type LSD has a clutch pack on each side gear, each with 4 friction surfaces, μ = 0.12, mean radius 45 mm, and a preload spring force of 4000 N (no ramps). One rear wheel of a stuck car is lifted clear of the ground. Find the locking torque and the maximum tractive effort from the other wheel (wheel radius 0.30 m). What would an open differential or a Torsen-type unit give?

1. Locking torque: T_lock = 2 · n · μ · F · r_m = 2 × 4 × 0.12 × 4000 × 0.045 = 172.8 N·m.
2. The lifted wheel transmits T_low = 0.
3. T_high = T_low + T_lock = 0 + 172.8 = 172.8 N·m.
4. Tractive effort = 172.8 / 0.30 = 576 N.
5. Open differential: T_high = T_low = 0, so no drive. Torque-sensing (Torsen-type) unit: T_high = TBR × 0 = 0, also no drive.

Answer: T_lock ≈ 173 N·m; about 576 N of tractive effort, which may be enough to rock the car free. Preload is what lets clutch-type LSDs work with a wheel in the air; pure torque-sensing designs need some load on the low side (drivers lightly apply the brakes to create it).

Common mistakes

• Averaging the two wheels' torques; nothing in a differential averages torque. The case speed is the average of wheel speeds.
• Saying a locked differential gives both wheels equal torque; it gives them equal speed, and each wheel's torque is set by its own grip.
• Treating viscous couplings as torque-sensing; they respond to speed difference.
• Applying TBR to a wheel with zero load: TBR × 0 = 0.
• Forgetting that an LSD with its own friction still lets the wheels differentiate in normal cornering, whereas a locked differential does not.
• Using normal gear oil in a clutch-type LSD; it usually needs a friction-modified oil to prevent chatter.

For GATE ME

Expect conceptual questions on why an open differential loses traction, and on clutch, viscous, gear-type and locking differentials. Numerical questions use the torque bias ratio or locking torque to find the torque on each wheel and the total tractive effort, compared with an open or locked differential. Practise the relations T_high = TBR · T_low and T_high − T_low = T_lock.

Quick check

1. What is the TBR of an ideal open differential?
2. TBR 2.5 and the low-grip wheel can take 150 N·m: what total torque can the axle deliver?
3. Which type of LSD responds to speed difference rather than torque?
4. Why does a preloaded clutch LSD still drive with one wheel in the air?
5. Why should a diff lock be released on dry roads?

Answers: 1. 1. 2. 150 × 3.5 = 525 N·m. 3. The viscous coupling. 4. The preload gives a locking torque even when the other side carries no torque. 5. It forces both wheels to the same speed, so in turns the tyres scrub and the driveline winds up.