Differentials
by Brian Brown
With the recent discussion on differential swaps, I tried to write a description of how they work.

I wish I could include pictures for the following descriptions. I'll try my best to describe things with words. It might help to find some pictures to look at, or even better, some units to actually hold in your hands and play around with.

CONVENTIONAL DIFFERENTIAL:

A conventional differential consists of a housing that is directly attached to the ring gear. On the same axis as the differential housing are two output shafts. These pass through to the inside of the housing, but are not directly connect to the housing. Inside the housing attached to each output shaft is a 45 degree angle bevel gear (referred to as output gears). Placed between the two output gears are some more 45 degree bevel gears called spider gears. The spider gears pivot on rods that are attached to the housing (these rods are oriented radially to the housing axis).

When the housing is stationary, rotation of one output gear causes the other output gear to turn in the opposite direction.

When a car is driving in a straight line, and the wheels aren't slipping, both output gears are turning at the same speed, because they are coupled together through the ground. The entire differential assembly is spinning together as an assembly, but internally, there is no movement of the gears with respect to each other.

When the car goes around a corner, the outer wheel has to spin faster. The differential will be spinning at a speed that is halfway between the speed of the inner wheel and the outer wheel. Internally, the gears are moving a speed that is the difference of the outer wheel speed minus the inner wheel speed. With respect to the housing, the inner wheel output gear is actually spinning backwards, and the outer wheel output gear is spinning forwards.

All of this works great as long as neither of the driven wheels exceeds it's traction ability. Force can be distributed to both wheels while any difference in cornering speed is compensated for. The main thing is that this design is dependent on the two outputs being coupled together through the ground.

When one wheel exceeds its traction, there is no reaction force to enable power to be sent to the other wheel, so the system loses its ability to function.

Note that if one wheel is stationary and the other spinning, the spinning wheel will actually be turning at *twice* the speed that is indicated by the speedometer.

LIMITED SLIP DIFFERENTIALS:

Most limited slip differential designs work by attempting to detect slippage and reacting (I'll refer to these as Reactive LSD's). Some other designs rely on directional force transmission characteristics, so that power to the wheel with good traction isn't dependent on a reaction force from the wheel with bad traction (I'll refer to these as Directional LSD's).

REACTIVE LSD's:

All reactive mechanical designs face a fundamental issue: they can't tell the difference between one wheel going faster due to a corner, or due to slippage. This means that some type of compromise must be made in how it is adjusted. Setting the adjustment for very little slip (lots of lock-up) helps maximize the amount of power that is transferred to the ground, but will cause problems going around corners because it won't allow enough of a difference between the inner and outer wheel. Adjusting to allow more slip (less lock-up) helps cornering, but reduces the LSD's effectiveness at increasing traction.

Typically, LSD's in production street cars are adjusted to have somewhere between 25% to 40% lock-up. 70% lock-up is right on the edge of where cornering issues start to become a problem.

Other considerations with Reactive LSD's are response time, and smoothness of the reaction.

PLATE CLUTCH LSD:

This is the type of LSD that BMW uses. It falls under the reactive category. The general construction is the same as a conventional differential with some added features.

The output gears are connected to the output shafts by splines so they can slide. Each output gear has a spring (coil type or spring washer) that pushes it into engagement with the spider gears. A characteristic of bevel gears is that they try to push each other out of engagement. This characteristic results in an axial thrust force that pushes the output gear against its spring. The amount of this thrust force is proportional to the difference in the force being transmitted to the two wheels. When both wheels are spinning at the same speed, this thrust force is zero.

Between the output gear and the differential housing are one or more plate clutch disks. The thrust force of the output gear squeezes the gear against the clutch disks and the housing. This causes a coupling between the output gear and the housing which overrides the spider gears.

Because this design uses sliding friction as part of its operation, it will wear with use. It requires the use of a friction modifier in the differential lubricant. It can also generate a significant amount of heat under heavy use (note that BMW adds a differential oil cooler to the M roadster).

This design can be adjusted by changing the spring strength, number of clutch disks, size of clutch disks, and to a small extent, the amount of friction modifier in the gear oil.

CONE CLUTCH LSD:

Similar in concept to the plate clutch design. The main difference is that cones are used instead of disks.

FERGUSON LSD:

Also called a Viscous LSD. BMW used this design in the 325iX. It also can be classified as a reactive design based around a conventional differential.

This design adds a viscous coupling between the two output gears.

The viscous coupling utilizes a characteristic of silicone fluid. When silicone fluid is between two closely spaced moving surfaces, it acts as a lubricant at slow speeds. When the speed of the two surfaces increases to a certain point, the fluid develops a shear force that couples the two surfaces together. The crossover between these two conditions happens very rapidly.

The coupler has one side connected to a can and the other side connected to a shaft that is inside the can. There are a series of disk mounted on the shaft. Between each of these disks are disks that are attached to the can at their outer edge. All of the disks have a bunch of holes drilled in them. The entire assembly is immersed in silicone fluid.

Under normal conditions, the can and the shaft are free to rotate independently. The silicone fluid allows the closely spaced disks to slide next to each other. When the speed difference between the can and the shaft reaches a certain point, they are coupled together when the shear action of the interleaved plates causes a transition in the fluid.

The Ferguson design is adjusted by the total surface area of the plates, the spacing of the plates, and the formulation of the silicone fluid.

An important distinction between the way a Ferguson and clutch plate unit behave is that the Ferguson reacts to differences in *speed*, while the clutch plate unit reacts to differences in *torque* transmitted.

An advantage to the Ferguson is that it can be adjusted for up to 90% lock-up without losing the ability to go around corners.

There are several disadvantages to the Ferguson: A slipping wheel has to spin a bit before a reaction occurs. When the reaction does occur, it's fairly abrupt. Because it's speed reactive, it can cause problems with braking because a locked up wheel's force will be transferred to the wheel that didn't lock. This is a non-issue with a torque reactive differential because there is no torque being transmitted through the differential due to the brakes. For this reason, the Ferguson design is more popular to use for the center differential in four wheel drive vehicles.

DIRECTIONAL LSD's:

Unlike the Reactive LSD's, Directional LSD's aren't based on the conventional differential. They use various means to directly couple the force applied to the differential to each output. These designs distinguish between the different directions that forces are applied.

When power is being transferred from the engine to the wheels, the direction of the power transfer is from the differential housing to each output shaft. When going around a corner, the difference in wheel speeds applies a force from one output shaft to the other output shaft.

LOCKER LSD:

No sane person would put one of these on a ti. But it's useful to briefly discuss it to illustrate a principle.

Loosely described, the differential housing is connected to each output shaft by a ratchet.

When the vehicle is traveling in a straight line, applied power is transferred from the differential housing to each output by locking both ratchets. If one wheel looses traction the other one doesn't care, because it's still directly connected. 100% lock-up.

When the vehicle goes around a corner, the outer wheel needs to spin faster. The *direction* of the force that causes the outer wheel to spin faster is from the ground, through the wheel, and *into* the output shaft of the differential. This causes the output shaft for the outer wheel to spin faster than the housing, it's ratchet just starts clicking away (these things are pretty noisy). The inner wheel is then left with the burden of handling all of the applied power (this makes for some strange handling traits).

The actual design is a little more complicated. (Simple ratchets like I described wouldn't allow any power to be applied in reverse). The main use for this design is for drag cars that want to have maximum traction in a straight line, with the ability to drive around corners just to transport the car, or for off-road vehicles that often lift one wheel off of the ground.

GLEASON (TORSEN) LSD:

TORSEN is a concatenation of 'Torque Sensing'.

This design uses a characteristic of worm and ring gears. A worm gear is usually a long thin helical gear that mates with the edge of a large diameter ring gear. When the worm gear is turned, it causes the ring gear to rotate. Tried the other way: attempting to turn the ring gear directly will just cause everything to lock solid. Motion is allowed in one direction only. This is partly due to the typical high gear ratio involved with a conventional worm and ring gear pair, but a major factor is the interaction of the shape of the gear teeth.

The actual gears inside the torsen don't have the appearance of conventional worm and ring gears, but they still are classified as worms and rings. In this case, the worm gears are significantly larger in diameter than the ring gears.

Each output shaft passes into the differential housing, where it is driven by an output gear. This output gear is a worm.

The differential housing has several pairs (3-5) of ring gears (they don't actually look like rings) that are mounted in cutouts in the housing. Each pair of ring gears are connected to each other by conventional gears that act as synchronizers. One ring gear of the pair meshes with the right output worm gear. The other ring gear of the pair meshes with the left output worm gear. The reason several pairs (3-5) of ring gears are used is to increase load capacity.

When the differential housing rotates, the ring gears are rotated around the output worm gears. Thus, force is being applied from a ring gear to a worm gear. As described above, no relative motion can occur between these two gears because they lock up solid. This means that full force is being applied from the differential housing to the output shaft. This occurs regardless of whether or not the other output shaft has any load (traction).

When the car goes around a corner and one wheel needs to go faster, the force from the faster outer wheel goes *into* the differential through the output gear. Now we have a situation where a force is being applied from a worm gear to a ring gear. Relative motion between these two gears is allowed when the force is in this direction.

To summarize the two main characteristics in a different way: Forces between the housing and an output shaft (engine power to a wheel) are directly coupled. Forces between two output shafts (differences in speed between the two wheels) allow the internal gears to rotate.

The real beauty of this design is that these two characteristics are autonomous. Both things can be happening at the same time. Full power can be applied while going around a corner. The wheels are allowed to turn at independent speeds. Full torque can be applied to a wheel even if the other has lost traction. (Up to the equivalent of about 80% lock up). Changes in the situation are automatically adjusted for instantly by the inherent nature of the design. Everything operates in a precise balance.

There is no need to choose a trade off between maximum traction, and the ability to go around corners.

It's also important to note that while this design relies on the friction characteristics of the gear teeth to control its behavior, it *doesn't* use friction to transfer power (like a Clutch Plate LSD). This design doesn't have any more wear than a conventional differential.

The Torsen is probably one of the most elegant mechanical designs in automotive history.

Unfortunately, I'm not aware of any source of Torsen's to fit BMW's.

Fortunately there is the...

QUAIFE LSD:

The basic principle of the Quaife is similar to the Torsen. The actual implementation looks quite different.

There are pairs of helical gears riding in cutouts in the housing. One gear of the pair drives the right output gear, the other gear of the pair drives the left output gear. The output gears are also a type of helical design. While not technically classified as worm and ring gears, the tooth faces are cut at angles that exhibit similar behaviors. One other difference from the Torsen design is that there are springs that add supplimentary forces to adjust the frictional loading of the gear teeth.

The description of the behavior of the Quaife is the same as the Torsen.

There is debate as to whether the Torsen or the Quaife is a better design. Supposedly the Torsen can achieve a higher ratio of torque bias, while the Quaife has even smoother operation. In my opinion, both designs are ABSOLUTELY OUTSTANDING. I haven't been able to compare the two directly in the same type of car.

This is a moot issue anyway since it appears that the Quaife design is the only one available for BMW's.

MODIFYING A TI WITH A DIFFERENT DIFFERENTIAL:

Up to this point, I've been discussing just the actual differential. In the ti, there is an entire differential assembly that contains not only the differential, but also the ring and pinion gears, the input and output flanges, and the assembly housing.

The ring and pinion gears perform a gearing reduction from the driveshaft to the wheels. Sometimes it is desirable to change the gear ratio. The general topic of gearing needs to be treated separately, but I'll offer a brief summary of my opinions:

The stock ti with a five speed has a final drive ratio of 3.45:1. A car modified with forced induction should keep this ratio. A car modified with a six cylinder should keep this ratio (maybe, depending on other factors). A stock or slightly modified engine would have an acceleration benefit from a ratio around 3.90:1. A highly modified, higher RPM four cylinder would benefit from a ratio around 4.10:1.

The point is, potential engine mods should be considered before deciding to change the rear end ratio.

I'm still hopeful that the ASC + T can be reprogrammed to offer improved performance. It's possible that this could be good enough to make me skip doing a differential transplant (maybe).

The factory type Clutch Plate LSD differential can be a real good choice particularly because there's a bunch of good, relatively economical used ones available. For a little bit of money, the lock-up can be increased (I like somewhere between 50% and 60% for spirited street driving). This unit works well alone, or in conjunction with ASC + T.

The Quaife LSD is the most desirable choice (I really, really want one for my ti). I've had one before in a Scirocco, and it was simply fantastic :). There were many benefits, and no downsides at all. Because it's reaction is so smooth and balanced, it should be complimentary with the ASC +T. Ok, there is one downside, it's quite expensive :( . I haven't priced it yet, but I'm sure it's well over a thousand dollars, which doesn't include the cost of installing it into the differential assembly. (I still really, really want one).

One last thing to note, is that an LSD will increase throttle induced oversteer. Something else to be considered when thinking about suspension tuning.

I hope this is helpful to understanding these things. I realize that the descriptions got really wordy at times. I would again suggest finding some units to look at while re-reading the descriptions to help understand things.

Regards, Brian Brown. BMWCCA #130878 '96 318tiS


http://www.318ti.org/notebook/diffs/
September 17, 1999