Final drive and differential

Final-drive gear types and ratios, how the bevel-gear differential lets the wheels turn at different speeds while splitting torque equally, and the speed and traction calculations that follow.

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Why it matters

The final drive gives the last, permanent torque multiplication between gearbox and wheels and, in a rear-wheel-drive car, turns the drive through 90°. The differential then shares that torque between the two driving wheels while letting them turn at different speeds in a corner. Their gear types, ratios and kinematics decide top speed, acceleration, tyre wear and why a car with one wheel on ice cannot move – all common viva and exam material.

Key ideas

Final drive

  • A permanent reduction (typically about 3–4.5 in cars, higher in trucks) between the propeller shaft or gearbox output and the differential. Every gear's overall ratio is the gearbox ratio times the final-drive ratio.
  • In front-engine rear-drive and 4×4 axles the final drive is a bevel pinion driving a crown wheel (ring gear), turning the drive through 90°. Types:
    • Straight bevel – simple but noisy; rarely used now.
    • Spiral bevel – curved teeth, more teeth in contact, quieter and stronger; pinion axis meets the crown-wheel axis.
    • Hypoid – spiral-type teeth with the pinion axis offset below the crown-wheel centre line. This lowers the propeller shaft (lower floor tunnel) and allows a larger, stronger pinion for a given ratio, and runs very quietly. The offset introduces sliding along the teeth, so hypoid gears are slightly less efficient and need extreme-pressure (EP) hypoid oil.
    • Worm and wheel – high ratio in one stage, quiet, used in some heavy vehicles; lower efficiency.
  • In transverse front-wheel-drive transaxles the final drive is usually a helical spur pair, since no 90° turn is needed.
  • Double-reduction final drives (bevel stage plus spur stage, or hub reduction gears at the wheels) are used in heavy trucks to get high ratios without a huge crown wheel.

Differential

  • Why it is needed. In a turn, the outer wheel travels a larger arc than the inner. If both were locked to one shaft, one tyre would have to slip, causing scrub, wear, steering resistance and wind-up. The differential allows different speeds while still driving both wheels.
  • Construction (bevel-gear type). The crown wheel is bolted to the differential case (cage). Inside, a cross-pin carries two (or four) differential pinions (planet/star pinions) that mesh with two side (sun) gears splined to the half-shafts.
  • Straight ahead: both wheels turn at the same speed; the pinions do not rotate on their pin; the whole assembly turns as one.
  • Cornering: the pinions rotate on their pin, so one side gear speeds up as much as the other slows down. The cage speed is always the average of the two wheel speeds.
  • One wheel held: the other wheel turns at twice cage speed.
  • Torque split. Because the pinions are free to rotate (and ignoring friction), they act as balance beams: an open differential always delivers equal torque to both wheels. So the total tractive effort is limited to twice what the wheel with less grip can transmit – the classic one-wheel-on-ice problem, solved by limited-slip and locking differentials (next topic).
  • Inter-axle (centre) differentials in 4×4s do the same between front and rear axles.
  • The final drive also produces a torque reaction on the axle casing, and the propeller-shaft torque tends to lift one side of the axle slightly, loading one rear wheel more than the other (most noticeable in powerful live-axle cars).

Formulas

i_f = Z_c / Z_p

  • i_f: final-drive ratio (–); Z_c: teeth on the crown wheel; Z_p: teeth on the pinion.

N_c = N_p / i_f, T_c = T_p · i_f · η

  • N_c, T_c: speed (rev/min) and torque (N·m) of the crown wheel / differential case; N_p, T_p: pinion (propeller-shaft) speed and torque.

N_c = (N_L + N_R) / 2

  • N_L, N_R: left and right wheel (side gear) speeds (rev/min). If one wheel is stationary, the other turns at 2 N_c.

T_L = T_R = T_c / 2 (open differential, friction neglected)

  • T_L, T_R: torques to the left and right half-shafts (N·m).

N_o / N_i = (R + t/2) / (R − t/2)

  • N_o, N_i: outer and inner wheel speeds; R: turning radius of the axle centre (m); t: track width (m).

T_max,total = 2 · μ_low · W_low · r

  • Maximum total wheel torque an open differential can deliver (N·m) when the lower-grip wheel has adhesion coefficient μ_low and normal load W_low (N); r: wheel radius (m).

Worked examples

Example 1 (standard). A rear-wheel-drive car with track 1.5 m and wheel radius 0.30 m turns at 36 km/h on a radius of 20 m measured to the centre of the rear axle. The final-drive ratio is 4.0. Find the speeds of the two rear wheels, the differential case and the propeller shaft.

  1. v = 36/3.6 = 10 m/s at the axle centre.
  2. Outer wheel: v_o = 10 × (20 + 0.75)/20 = 10.375 m/s; N_o = 60 × 10.375 / (2π × 0.30) = 330.2 rev/min.
  3. Inner wheel: v_i = 10 × (20 − 0.75)/20 = 9.625 m/s; N_i = 60 × 9.625 / (2π × 0.30) = 306.4 rev/min.
  4. Case: N_c = (330.2 + 306.4)/2 = 318.3 rev/min (the same as both wheels straight ahead at 10 m/s).
  5. Propeller shaft: N_p = 4.0 × 318.3 = 1273 rev/min.

Answer: outer 330.2, inner 306.4, case 318.3 and propeller shaft 1273 rev/min. The outer wheel turns 7.8% faster than the inner.

Example 2 (GATE level). The same car stands with its right rear wheel on ice (μ = 0.10) and the left on dry road (μ = 0.8). Each rear wheel carries 4000 N. The engine runs at 2000 rev/min in first gear (3.5). Neglecting differential friction, find the maximum total tractive effort with the open differential, and the speed of the spinning wheel if the car does not move.

  1. Maximum torque at the icy wheel: μ · W · r = 0.10 × 4000 × 0.30 = 120 N·m.
  2. Open differential, equal torques: left wheel also receives only 120 N·m, though it could transmit 0.8 × 4000 × 0.30 = 960 N·m.
  3. Total wheel torque = 240 N·m; tractive effort = 240 / 0.30 = 800 N.
  4. Case speed: N_c = 2000 / (3.5 × 4.0) = 142.9 rev/min.
  5. Left wheel stationary, so right wheel: N_R = 2 × 142.9 = 285.7 rev/min.

Answer: only 800 N of tractive effort, against 3600 N if each wheel could use its own grip ((120 + 960)/0.30); the spinning wheel turns at about 286 rev/min. If 800 N is less than the resistance to start, the car cannot move – the reason for limited-slip and locking differentials.

Common mistakes

  • Saying an open differential sends more torque to the wheel with more grip; it always splits torque equally (ignoring friction). It is speed, not torque, that it divides unequally.
  • Forgetting that the cage speed is the average of the wheel speeds, so a spinning wheel turns at twice cage speed when the other is stopped.
  • Using the vehicle's turning radius at the outer wheel instead of the axle centre (or the wrong half-track).
  • Saying hypoid gears are more efficient than spiral bevels; the offset adds sliding and needs EP oil.
  • Using ring teeth ÷ pinion teeth correctly but then multiplying speed instead of dividing.

For GATE ME

Differential kinematics is the usual numerical: wheel speeds in a turn, cage speed as the mean of wheel speeds, one wheel stationary, and overall speed and torque through gearbox and final drive. The open-differential torque split and the resulting traction limit also appear, as does the bevel-gear differential treated as an epicyclic train (case = carrier). Know the reasons for hypoid gearing and double reduction for short-answer questions.

Quick check

  1. Crown wheel 41 teeth, pinion 10 teeth: what is the ratio?
  2. Left wheel 300 rev/min, right wheel 340 rev/min: what is the cage speed?
  3. With one wheel jacked up and the other on the ground, the engine turns the cage at 100 rev/min. How fast does the free wheel spin?
  4. Why does a hypoid final drive need special oil?
  5. How does an open differential divide torque between the wheels?

Answers: 1. 4.1. 2. 320 rev/min. 3. 200 rev/min. 4. The pinion offset causes sliding between the teeth under high pressure, which needs extreme-pressure additives. 5. Equally, ignoring internal friction.

Interview questions

All Automotive Transmission and Driveline interview questions

Try answering each one aloud before you open it.

  1. 1.What is the purpose of the final drive in an automotive transmission system?Concept

    The final drive is a component of the automotive transmission system that reduces the rotational speed from the transmission and increases torque before it is transferred to the wheels. It typically consists of a pair of gears, usually a ring and pinion gear set, that provide the necessary gear reduction. This allows the vehicle to move efficiently at various speeds while maintaining optimal engine performance.

  2. 2.Explain the function of a differential in a vehicle.Concept

    A differential lets the two driving wheels on an axle turn at different speeds while both are driven. In a turn the outer wheel travels a longer arc than the inner one, and without a differential one tyre would have to scrub, causing wear, heavy steering and driveline wind-up. In the usual bevel-gear differential the case speed is the average of the two wheel speeds, and the free-turning pinions divide the torque equally between the wheels. That equal split is also its weakness: if one wheel is on ice, the other gets no more torque than the slipping wheel can take.

  3. 3.Why is a limited-slip differential used in some vehicles?Application

    A limited-slip differential (LSD) is used to improve traction by limiting the difference in speed between the two wheels on an axle. In situations where one wheel loses traction, such as on slippery surfaces, an LSD can transfer more torque to the wheel with better grip, enhancing the vehicle's stability and performance. This is particularly beneficial in high-performance and off-road vehicles.

  4. 4.What happens if a vehicle's differential fails?Application

    If a vehicle's differential fails, it can lead to several issues. The most immediate effect is the loss of the ability to allow wheels to rotate at different speeds, which can cause tire wear and handling problems, especially during turns. In severe cases, a failed differential can lead to a complete loss of power transmission to the wheels, rendering the vehicle immobile.

  5. 5.How does a torque vectoring differential enhance vehicle performance?Application

    A torque vectoring differential enhances vehicle performance by actively distributing torque between the wheels on an axle. This allows for better handling and stability, especially during cornering. By directing more torque to the outer wheel, the vehicle can maintain better traction and reduce understeer, leading to improved agility and control.

  6. 6.Explain the difference between an open differential and a locking differential.Concept

    An open differential lets the wheels turn at different speeds but always gives them equal torque, so total traction is limited to twice what the wheel with less grip can transmit; with one wheel on ice the vehicle may not move. A locking differential, when engaged, locks the side gears to the case so both wheels turn at the same speed, as if on a solid axle. Each wheel can then take whatever torque its own grip allows, so torque can be very unequal and the wheel with grip does the work. Because it stops the wheels turning at different speeds, a locked differential is meant for low-traction, low-speed use and should be released on high-grip roads and for normal cornering.

  7. 7.Why is hypoid gear used in the final drive of many vehicles?Application

    In a hypoid final drive the pinion axis is offset below the crown-wheel centre line. This lowers the propeller shaft, so the floor tunnel can be lower, and for the same ratio it allows a larger-diameter, stronger pinion with more teeth in contact, which makes the gears quiet and durable. The offset adds sliding along the teeth, so hypoid gears are slightly less efficient than spiral bevel gears and need extreme-pressure hypoid gear oil to prevent scuffing.

  8. 8.Calculate the final-drive ratio if the crown wheel has 41 teeth and the pinion has 10 teeth.Numerical

    Final-drive ratio = crown-wheel teeth / pinion teeth = 41 / 10 = 4.1. The pinion, and so the propeller shaft, turns 4.1 times for every turn of the crown wheel and differential case, and the torque is multiplied by 4.1 less gear losses. Odd tooth counts such as 41:10 are chosen so the same teeth do not meet on every revolution, which spreads wear.

  9. 9.A vehicle's differential has a torque bias ratio of 3:1. If one wheel loses traction and can only handle 50 Nm of torque, how much torque can the other wheel receive?Numerical

    The torque bias ratio (TBR) indicates how much more torque can be sent to one wheel compared to the other. With a TBR of 3:1, if one wheel can only handle 50 Nm, the other wheel can receive 3 times that amount. Therefore, the other wheel can receive 3 * 50 Nm = 150 Nm of torque.

  10. 10.What are the potential consequences of using an incorrect gear oil in a differential?Application

    Using an incorrect gear oil in a differential can lead to inadequate lubrication, increased friction, and overheating. This can cause premature wear of the gears and bearings, leading to noise, reduced efficiency, and eventually mechanical failure. It is crucial to use the manufacturer-recommended gear oil to ensure proper performance and longevity of the differential.

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