Car Differential Simulator: How a Differential Works, in Interactive 2D and 3D Animation
A differential is the gear set in a driven axle that splits the drive between the left and right wheels while letting them turn at different speeds. This differential simulator puts it in a rear-wheel-drive car you can steer, drive onto ice or lift on a jack, and shows the crown wheel and pinion, the case, the spider gears and the side gears turning in a 2D section and a 3D cutaway. Every speed, torque and tyre force is computed, so you can see why each kind of differential behaves as it does.
How a differential works
The propeller shaft turns the pinion, which drives the crown wheel (ring gear) bolted to the differential case. The case carries a cross pin with two (sometimes four) spider gears, and they mesh with two side gears, one splined to each half-shaft. Driving straight, both wheels turn at the case speed: the spiders are carried round with the case and do not turn on their pin, so the whole set turns as one. In a corner the outer wheel must turn faster, so the spiders roll on the side gears: one side gear turns faster than the case by exactly as much as the other turns slower. That is the rule every bevel-gear differential obeys:
ωleft + ωright = 2 ωcase
Why a car needs a differential
On a 10 m turn (measured at the centre of the rear axle) with a 1.55 m track, the inner rear wheel runs on a 9.225 m arc and the outer on a 10.775 m arc, so at 20 km/h with 0.315 m tyres the wheels must turn at 155.4 and 181.5 rpm while the case turns at 168.4 rpm. A solid axle would force one speed on both, so one tyre would be dragged and the other pushed: the tyres scrub, the axle winds up and the car resists the turn. Run the Corner and Tight turn tests with the Open and Locker settings to see the difference.
Parts of a differential
- Pinion: the small bevel gear on the end of the propeller shaft.
- Crown wheel (ring gear): the large bevel gear driven by the pinion; with it, the final drive.
- Differential case (carrier): bolted to the crown wheel and running in two bearings in the axle housing.
- Cross pin and spider gears (planet gears): carried round by the case, free to turn on the pin.
- Side gears (sun gears): meshed with the spiders and splined to the half-shafts.
- Half-shafts (axle shafts): from the side gears to the wheel hubs.
In road cars the pinion is set below the crown wheel's centre line, a hypoid gear set: it lowers the propeller shaft and lets the pinion be larger and stronger for the same ratio. It changes nothing in how the differential divides speed and torque; this simulator draws the gears as straight bevel gears so their teeth can be followed.
Types of differential
An open differential always gives the two wheels equal torque. That is perfect in a corner and the reason for its weakness: the wheel with less grip sets the torque for both. A limited-slip differential resists a speed difference between the wheels, which lets it give more torque to the slower wheel. In a clutch-pack (Salisbury) LSD the resistance is friction from plates squeezed by springs (the preload) and by ramps that push harder as the drive torque rises. In a viscous LSD it is silicone fluid sheared between plates, so it grows with the speed difference. A Torsen uses worm gears whose friction holds a fixed torque bias ratio between the wheels. A locker joins the wheels so they always turn together.
| Differential | Torque split | Left wheel on ice: distance | Left wheel lifted: distance | Typical use |
|---|---|---|---|---|
| Open | Always 50 / 50 | 4.1 m | 0.2 m | Most road cars |
| Clutch-pack LSD | Up to preload + ramp friction | 10.2 m | 3.4 m | Performance and rally cars |
| Viscous LSD | Grows with speed difference | 23.6 m | 20.0 m | Centre and rear axles of road cars |
| Torsen | Up to the bias ratio (3:1 = 75 / 25) | 10.6 m | 1.9 m | All-wheel-drive centre and rear axles |
| Locker | Whatever keeps both wheels together | 30.4 m | 27.6 m | Off-road and heavy vehicles |
The Torsen's figure with a wheel in the air is the one that surprises people: three times nothing is still nothing. Braking the spinning wheel gives it a torque to multiply, which is how a brake-based traction control helps an open or Torsen differential; try it with the rear wheel brake sliders.
Final drive ratio
The final drive ratio is the crown wheel's teeth divided by the pinion's teeth. With a 41-tooth crown wheel and an 11-tooth pinion it is 41 ÷ 11 = 3.73:1. The overall reduction in a gear is the gearbox ratio times the final drive, so in this car's 3.62 first gear it is 13.49:1. A numerically higher ratio (a “shorter” axle) gives more torque at the wheels and more engine rpm at a given road speed:
| Crown wheel : pinion | Ratio | Engine rpm in a 1.00 gear | Engine rpm in a 0.83 overdrive | Overall ratio in 1st (3.62) |
|---|---|---|---|---|
| 43 : 14 | 3.07 | 2,586 | 2,147 | 11.12 |
| 41 : 12 | 3.42 | 2,877 | 2,388 | 12.37 |
| 41 : 11 | 3.73 | 3,139 | 2,605 | 13.49 |
| 41 : 10 | 4.10 | 3,453 | 2,866 | 14.84 |
| 41 : 9 | 4.56 | 3,836 | 3,184 | 16.49 |
The pinion turns at the final drive ratio times the case speed, and the case at the average wheel speed. That is why an open differential with one wheel on a jack and the other held still turns the free wheel at twice the case speed; run the Jack test to see it.
Who uses this differential simulation?
Automotive technician students learning the parts and the faults of a final drive, engineering students working through bevel gear trains and the kinematics of a differential, driving instructors explaining why a car with an open differential gets stuck, and anyone who has wondered why one wheel spins on ice. Teachers can set a test, press Run and pause it at the moment that matters.
Frequently asked questions
Why is a differential needed in a car?
In a turn the outer wheel travels a longer arc than the inner wheel, so it must turn faster. A solid axle would force both wheels to the same speed and one tyre would be dragged across the road. The differential lets the two wheels turn at different speeds while it still drives both: the case turns at the average of the two wheel speeds, ωL + ωR = 2ωc.
Why does one wheel spin with an open differential?
An open differential always gives the two wheels the same torque. If one wheel is on ice or off the ground, it can only take a small torque before it spins, so the wheel with grip gets the same small torque. The spinning wheel then turns at up to twice the case speed while the car hardly moves.
How do you calculate the final drive ratio?
Divide the number of teeth on the crown wheel (ring gear) by the number of teeth on the pinion. A 41-tooth crown wheel and an 11-tooth pinion give 41 ÷ 11 = 3.73:1, so the propeller shaft turns 3.73 times for each turn of the differential case. The overall ratio in a gear is the gearbox ratio times the final drive ratio.
What is the difference between a limited-slip and a locking differential?
A limited-slip differential lets the wheels turn at different speeds but resists the difference with clutch plates, a viscous coupling or worm gears, so it can send more torque to the wheel with grip. A locking differential joins the two wheels so they always turn at the same speed: the best traction off road, but the tyres scrub in a tight turn.
What is a Torsen differential's torque bias ratio?
The torque bias ratio (TBR) is the largest ratio of the torque on the gripping wheel to the torque on the slipping wheel that the differential can hold. A Torsen with a TBR of 3:1 can give one wheel three times the torque of the other, a 75 / 25 split. Because it multiplies the slipping wheel's torque, a wheel in the air (zero torque) still leaves the other wheel with nothing.
Explore Related Simulators
- Gearbox Simulator: the manual gearbox in front of the final drive, with synchromesh and a 3D car.
- Automatic Transmission Simulator: torque converter, CVT, DCT and AMT driving the same kind of car.
- Planetary Gear Simulator: the Willis equation; a bevel differential is an epicyclic train with its case as the carrier.
- Gear Trains Simulator: simple, compound and bevel gear ratios.
- Clutch System Simulator: the friction plates a clutch-pack LSD is built from.
- Hydraulic Brake System Simulator: the brakes a traction control uses to stop a wheel spinning.
- Circular Motion Simulator: the lateral acceleration that loads the outer wheel in a corner.
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