GPL Differential Information
Introduction:
Much has been written over the years at SRMZ about car differentials and how GPL models them. The vast majority of this information has been good, but there have also been a few inaccuracies and some misunderstandings. This article is an attempt to clear up some things.
1. The purpose of the differential is simple--it permits two wheels on the same end of the car to rotate at different speeds. This makes it easier for the two tires to negotiate different radius curves when the car is turning a corner.
2. Several types of differentials have been developed. Here are the four that have interest for GPL:
A. Open--this type uses an ingenious set of gears that allows the two wheels to rotate at different speeds at all times as they are never locked together. Consequently, it handles corners very nicely. Its main drawback is that torque to each tire/wheel is limited by the tire with the poorest grip. For example, if one tire is slipping on a slick surface such as ice, torque is reduced so that the slipping tire spins while the other doesn't spin at all and the car won't move. A similar situation occurs on race cars in a corner because the inside tire is less loaded and has less grip than the outside tire. The open differential by itself is rarely seen on race cars although it is almost universally used on passenger cars.
Surprisingly, the open differential was invented back in 1827 by French watchmaker Onésiphore Pecqueur.
B. Locked--this type is the opposite of the open differential. The two wheels are always locked together and rotate at the same speed. It is commonly used on trucks, dragsters, and karts. Locked differentials are seldom seen on race cars as they promote understeer. However, the 1970s Porsche 917 Turbo CanAm cars used a locked differential as other differential types of that era could not handle the car's massive torque.
What is really needed for a race car is a differential that is open for entering the corners and locked for exiting the corners and maximum acceleration on the straights. The limited slip differential (LSD) was developed to accomplish these goals. An LSD has a mechanism that locks the two wheels together under certain conditions. The two LSDs that apply to 1960s race cars and GPL are the Cam and Pawl and the Salisbury.
C. Cam and Pawl LSD--this type uses a set of plungers (pawls) that extend into slots to lock the two wheels together completely when power is applied to the differential. When power is removed, the plungers retract and the two wheels are free to rotate independently of each other. The Cam and Pawl is one of the simplest LSDs and is relatively cheap and reliable; however by its very nature, the plungers tend to wear quickly and have to be replaced often. Also, the locking action is relatively quick so essentially it is either completely locked or completely open.
David Wright researched the subject and determined that Cam and Pawl LSDs were used on all F1 cars of the 1960s. Hewland in England and ZF in Germany manufactured their own versions of the Cam and Pawl LSD. The first use of a LSD other than the Cam and Pawl didn't occur until Ferrari used a Salisbury LSD on their mid 1970s F1 cars.
This is all well and good, but strangely GPL does not model the Cam and Pawl LSD; rather, it models the Salisbury LSD.
D. Salisbury LSD--this type uses mechanical wedges (ramps) and a series of clutch plates to lock the two wheels together. The advantage of the Salisbury is that the amount of lock is individually adjustable when the power is applied and when removed. Also, the two wheels are locked together up to a certain point then are free to rotate somewhat independently. The locking action is more progressive than with the Cam and Pawl. By all accounts, the Salisbury is a vast improvement over the Cam and Pawl.
3. GPL Salisbury LSD Model:
A. GPL simulates the Salisbury LSD fairly well. The player setup menu contains settings for the ramp angles for "power" and "coast" and the number of clutches. These settings are applied to a formula to determine a "locking percent" limit. GPL's code continually keeps both wheels/tires turning at the same rate by apportioning torque from one wheel to the other; however, when the torque difference reaches the locking percent limit, the two wheels begin to rotate somewhat independently of each other.
B. The formulae for locking percent are:
Power Locking Percent = Cosine(Power Ramp Angle) * (Number of Clutches + 1) * 5%
Coast Locking Percent = Cosine(Coast Ramp Angle) * (Number of Clutches + 1) * 5%
This chart shows the locking percent for various combinations:
Note that ramp angle and number of clutches both contribute to the locking percent calculation. Also note that the number of clutches affects both power and coast locking percent.Therefore, it is possible to have the same locking percent by using different combinations of ramp angle and number of clutches.
GPL Setup Manager displays locking percent according to these formulae.
C. As best as I can determine from reviewing GPL's code, the car should handle the same when using the same locking percent regardless of the ramp angle and number of clutches used. The ramp angle and number of clutches values are not used anywhere else in the code.
D. The power locking percent is used when the player's throttle is on by any amount.
E. The coast locking percent is used only when the player's throttle is completely off! This has a major implication for left foot brakers as they will be using the coast locking percent only if they completely remove their right foot from the throttle pedal. It's all too easy to keep a slight amount of pressure on the throttle pedal which causes the power locking percent to be used inadvertently.
F. Disregard any reference to what real world cars used for ramp angles and number of clutches. They may or may not correlate to what GPL needs for a good handling car. Years of practical experience with GPL has determined that power ramp angles between 60 to 85 degrees, coast ramp angles between 30 to 45 degrees, and number of clutches between 1 to 3 can be used depending on the car characteristics and the player's skill level. These values correspond to locking percents of 1% to 10% for power and 7% to 17% for coast.
Note that some "alien" drivers may use locking percentages that are different than these. If you are an alien, you probably don't need to be reading this article in the first place. :)
Robert Fluerke in an SRMZ post stated that some left foot braking aliens use extremely high coast ramp angles of 60 to 85 degrees, but normally keep a little bit of throttle applied with the right foot so that the power ramp angle is used during corner entry. However if they miss the racing line, carry too much speed, or have understeer during corner entry, they completely lift off the throttle to get extra rotation from the coast ramp angle...pretty clever.
Possible Bug In GPL's Differential Code:
While researching differentials for this article, I found a very technical and detailed source for how differentials work. It contained the locking torque formula for a Salisbury LSD. Among other factors such as the number of clutch disks, the cotangent of the ramp angle is used. This make sense if you look at how the cage half moves sideways when the planet axle moves fore and aft.
However as explained before, GPL uses the cosine of the ramp angle in calculating its locking percent. I now believe this is a minor bug in the code and Papyrus should have used the cotangent of the ramp angle instead.
This figure displays GPL's locking percent if the cotangent were used instead of the cosine in the calculation. It shows that the same locking percent can be achieved with lower ramp angles. This may explain why real world power on ramp angles are typically lower than what we need for GPL.
I could easily make a patch that fixes this bug, but it's not worth doing as essentially it's a cosmetic fix. As stated before, locking percent is what is important for handling regardless of the ramp angle, number of clutches, or trigonometric function used to determine it.
Additional Information On The Cam And Pawl Limited Slip Differential:
Even though GPL doesn't model the Cam and Pawl LSD, it's still interesting to see how it works. Figure 10 is a diagram of the Hewland style Cam and Pawl LSD.
When power is applied to the differential, the rotating set of cams (item 9) forces the pawls (item 10) to extend into the set of slots (item 11). This locks the two wheels together. When power is removed, the cams relax their force allowing the plungers to retract from the slots and the two wheels rotate independently. Pretty simple actually.
Additional Information On The Salisbury Limited Slip Differential:
To finish up this discussion, here is more detailed information about how the Salisbury LSD works.
The Salisbury LSD is much more complicated and expensive than a Cam and Pawl LSD, but is vastly superior for reasons mentioned before. It is based on a standard open differential, but modifies it to lock the two wheels together under specific conditions. A standard open differential contains a set of two or four planet (sometimes called spider) gears whose spin axles are rigidly fixed inside a cage. However, the Salisbury differential splits the cage vertically into two halves which are then free to slide in and out relative to the two wheels. When a cage half moves outward, it compresses a series of clutch disks. Half of the clutch disks are locked to the differential's output gear while the other disks are locked to the wheel's half shaft. As the clutch disks are pressed together, their increasing friction begins to lock the differential's output gear and wheel's half shaft together. As a result, both wheels progressively lock and rotate together too.
Usually, a small amount of clutch disk friction is always desired so a small spring is added to compress the disks a bit. The spring's force is called preload. A Belleville type spring washer is normally used for this purpose. GPL does not model preload.
Figure 11 is a diagram of a typical Salisbury differential.
The Belleville spring washer is labeled Item 2 "Belleville Clutch Plate". The planet gears are labeled Item 4 "Planet Pinions". The planet gear axles are labeled Item 5 "Cross Pins". The output gear is labeled Item 6 "Sun Pinion". The cage half is labeled Item 7 "Sun Pinion Ring". The clutch disks are labeled Item 8 "Splined Clutch Plates". The wheels and their half shafts aren't shown.
The inside edge of each cage half has angled slots (ramps) in which the planet gear axles can slide fore and aft. Figure 12 shows a real world cage half on its side with the ramps on the top edge.
Figure 13 shows a real world cage looking at the end of one of the planet gear axles. The planet gear itself is underneath and can't be seen. The ramps on the inside edges of the two cage halves are clearly shown. Note that the power and coast ramps have different angles.
Figure 14 depicts how the two cage halves are forced apart by planet gear axle movement.
When the throttle is on and engine torque applied, the planet gear axle rotates which forces the cage halves apart along the "power" ramp. When the throttle is off, engine braking torque rotates the planet gear axle in the opposite direction and the cage halves are forced apart along the "coast" ramp.
By using different combinations of ramp angles and number of clutch disks, the mechanic can control how much the two wheels lock together under power on and off conditions. It is possible to achieve complete lock which effectively is a locked differential, no lock which effectively is an open differential, or any amount of lock in between. In the last case, the two wheels will still rotate at different rates if the reactive torque difference from the two tires exceeds the locking friction.
There are three classifications or "ways" of operation for the Salisbury differential. The classification depends on the power and coast ramp angles:
1.0 Way--the coast ramp angle is 90 degrees. This classification operates as an open differential when the throttle is off.
1.5 Way--the power and ramp angles are not the same. This classification provides different locking for throttle on and off. GPL models this classification.
2.0 Way--the power and ramp angle are the same. This classification provides the same locking for throttle on and off.