Limited Slip Differential

This was originally posted by tagfat at RSC here.

Make your LSD work faster

Diffs for beginners The first time in recent times I thought about how the workings of a differential - the "diff" in the following - was when I read Steve Smith's GPL-tutorial "Four-Wheel Drift". As a happy newbie I had totally missed discovering that you could "turn" the setup menu-pages and get from "suspension" to "drivetrain". I guess I had enough to think about on the suspension-page already. And I was quite capable of running off the track without the clever use of advanced differential settings!

The few things that Steve Smith writes about the diff were interesting but never really detailed enough to enlighten my dim soul: He does a quick run through of the diff's function and purpose, mostly focusing on some user friendly rules of thumb and less on the details of the construction. Which is great as most people don't need anything but that. In fact most people reading this should probably just skip the long technical introduction and get down to the guide on how to adjust the diff for better performance on track:

http://geocities.com/n_heusink/setupguide/guide.htm

Most of the technical stuff will in any case be much harder to grasp if you haven't fooled around with the settings a bit beforehand. When you have had a bit of success with that - and even if you haven't - it's time to get your head down and take a look at what is really going on there.

The following is my attempt to make the diff and its adjustments more accessible to the average driver, giving him a starting point for developing his own setups. If you have no experience with mechanics at all there will be a steep learning curve to climb, but it should really be worth the effort. If you think of your GP-car as a female being, that will actually give you a clue to where the really important stuff can be found. Yes - the differential is perhaps the most significant piece of engineering in a Grand Prix car and even if you don't get to look at it very often it should always be on your mind.

Sadly there has been a common lack of understanding of the rampangles´ and clutchplates´ actual function in a differential, something which has meant that while they are some of the most significant variables of the vehicle, most people have refrained from fiddling with the diff settings.

The diff settings are however the natural starting-point for making your own setups, and with a basic understanding of the diff's construction you will know how to get started.

Purpose

The main engine shaft of your F1 - be it a Brabham or a BRM - has a longitudinal orientation, just like the main engine in a ship. But whereas a ship gets propelsion from an actual propeller pushing water backwards, the F1 car gains propulsion by means of the wheels "pushing" the tarmac backwards. To make this work you have to turn the driving wheels 90 degrees in relation to the propeller-shaft, so that the periferi movement at the bottom of the rotation - where the wheels touch the track - is opposite to the desired movement (forward in most cases). Motorbikes often don't have that problem as the engine-shaft often has a transverse orientation. One illustrative exception is the classic BMW which has a boxer engine with a longitudinal power shaft, which (through the gearbox) drives a ring gear mounted on the rear wheel. In principle you could construct the transmission in a F1-car in similar fashion, driving the two rear wheels with a ring gear mounted on a solid axle; this would be excellent if you were only going straight ahead all the time. In curves and corners there is a slight problem with this, something which we could call the "whisky-glass problem". Take a whisky-glass and roll it over you desk and you will see it roll round in a nice regular curve. If your desk is big enough it will actually go round in a circle. If the glass was wet it would paint a ring on the table. A long-drink glass with the same diameter at top and bottom would roll straight ahead and over the edge. The problem with a solid rear axle (using equal sized wheels) is that during a turn, we need the outer wheel to travel a longer distance than the inner rear wheel; it must therefore rotate faster to achieve this in the same time. So, with equally sized rear wheels it is necessary for these wheels to rotate at different speeds - the "diff" is what makes this possible.

Construction To obtain different rotational speeds for each rear wheel you need a slightly more complicated transmission: Instead of the propeller-shaft driving one ring gear on a solid rear wheel shaft you have a small ring gear for each rear wheel, and a larger single ring gear (usually called the 'Crown Wheel') driven by the engine shaft. The link between the latter and the two smaller ring gears connected to the rear wheels is what does the trick: Mounted inside the crown wheel are some "spider- gears" which are engaged with the ring-gears on each side, but which are free to rotate on their own axis. When the crown wheel gear is driven by the motor the whole assembly will rotate, and thereby drive the two rear wheels. But as they can rotate they will allow one of the wheels to rotate at a slower rate if the other wheel goes that much faster. The greater the difference between the speed of the wheels, the faster the spider gears will have to rotate.

With kind permision from Ian Frechette:

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The practical implementation is something like this: The outside ring gear or crown wheel is solidly mounted on the outside of a cylinder and driven by the engine. Inside the cylinder a number of spider gears are mounted between the two independent ring gears attached to the rear wheel. Bearings at each end of the cylinder holds the two rear axles. When the cylinder is rotated by the engine the spider gears distribute the force to each of the rear wheels according to the resistance they each offer. The force of the engine will accelerate the wheel on the "line of least resistance". This is called an 'open differential' and is what you have in most normal cars. It would be fine for racing if it wasn't for a small but significant loophole.

The limitations of the open diff

Consider some basic physics of racing: When a F1-car is driven through a curve at speeds near the limit set by the available traction there will be a great difference between the load on the inner and the outer wheels. The inertia is working in the direction given by the tangent to the trajectory at the place where the car is located. Even if the wheels are following the curve the body will tend to follow the tangent.

In curves the car is turned a bit clockwise if the curve is clockwise, relative to the tangent of the trajectory. In other words the car is not pointing in the direction of the momentarily movement but rather in the desired direction. As the inertia will have its effect evenly spread out in all parts (relative to weight) you can view the force as acting through the center of gravity. The force of the inertia can, for practical purposes, be split into a component pointed in the same direction as the wheels and one component normal to this. The first one supports the desired movement, the second one is nothing but trouble.

The latter will be opposed by the traction offered by the tires, but the combined forces of inertia and resistance will work to roll the car because the center of gravity is the contact-patch for the four tires. Hence you get roll when turning the car, and hence you have much better traction on the "loaded" outside wheels compared to the inside "unloaded" wheels.

In racing you will tend to take this situation to its extreme, and using all available traction also means creating a lot of roll-force. Because of the roll the unloaded inner rear wheel will not have a lot of traction. Therefore the traction needed for resisting a) sliding and b) spinning at the same time is often just not there. In other words: the inside rear wheel very often looses its grip.

While the loss of traction from the spinning inner rear wheel is no big deal - as the inner wheel doesn't have a lot of available traction in the first place - the open differential makes this a bit more serious. The problem is that the open diff will permit the one wheel to be held still if the other can rotate freely.

To make a very simple analogy it's like having two buckets - one empty and one filled with water - which are connected with a string. If you have a second string attached to the middle of that string and you pull, the unloaded bucket will move, but the loaded bucket won't move as long as the unloaded one can move.

Quoting an expert: "When one wheel starts to spin, the torque being recieved by the slower wheel drops to twice that at the spinning wheel, which is very little." In other words, with no load on the inside rear, all the torque applied at the crown wheel acts to spin this wheel, preventing the loaded outside tyre from putting down any power.

This constitutes the loophole of the open diff.

The Limited Slip Differential The limitid slip differential - the "LSD" - is a compromise between a spool (solid axle) and the open differential: It allows for sufficient difference between the rotational speed of each rear wheel to not waste the grip potential, but not so much difference that all acceleration can be lost. There are different types of LSD´s but all cars in GPL has a salisbury LSD, which is especially well suited for racing purposes.

Construction On each of the driveshafts from the rear wheels and inside the diff-housing a number of clutch plates are mounted. They are facing another set of clutch plates which are rotating with the crown wheel/housing assembly. When pressure is applied to these clutch plates they will try to lock each driveshaft and the motor-driven housing together and tend to prevent each driveshaft moving relative to the diff housing assembly. With no pressure they permit each driveshaft to maintain its own rotational speed.

The way pressure is applied is where the well known 'ramp angles' come into play: Between the clutch plates two freely rotating pressure rings/ramp cages are mounted. The two pressure rings/ramp cages face each other in the center of the differential, but have a number of v-formed profiles which allows the shafts of the spider gears to sit between the pressure rings.

When the shafts of the spider gears are rotated by the diff housing, they will exert a force on the ramps of the ramp cages/pressure rings, which will tend to spread the cages apart, and thereby put pressure on the clutch plates, producing frictional locking force.

Strictly speaking nothing would happen if the pressure rings/ramp cages could rotate absolutely freely, so they do have a frictional preload on them to make it work. Any added pressure from the spidergear shafts will increase the friction from the clutch plates and make the pressure rings/ramp cages even more susceptible to pressure.

The angle of the v-profiles determines the magnitude of the forces exerted on the clutch plates (as a function of the torque applied at the crown wheel): with a lower angle (measured from the rotational plane) the forces will be greater. With an angle of 90 degrees all force would just rotate the pressure rings and no force would be applied to the clutchplates, leaving the diff 'open'.

When accelerating the car the forces are coming from the engine (applied at the crown wheel), while the wheels drive the motor under engine-braking. As you have different needs under acceleration and braking it is fortunate that you can choose different angles on the front and the backside of the spider-shafts: The frontside determines the action at acceleration, the backside while (engine)braking. Hence the terms "powerside angle" and "coastside angle".