Date: Fri, 09 Apr 1999 16:28:17 -0400
From: Jim El Nabli <nabli_at_ibm.net>
Subject: Re: [E36M3] LSD with 25% (stock) vs. 40%...pros/cons

Based on a little research here's your answer:

In a standard differential, if one wheel loses traction, it will get all the power and will spin, while the wheel with traction gets nothing. The idea of a limited-slip differential is to prevent all power from being applied to only one driving wheel when traction is lost. There are numerous types of limited-slip, positraction, locker, etc. units.

The percentage number denotes the percentage of torque applied to the slower turning wheel from
the faster turning wheel. In a straight line, both drive wheels turn at the same speed, so no limited slip action is occurring. In a turn, or when one tire is spinning more than the other (such as on snow or ice), with a limited slip differential, 25, 40, or 75 percent of the torque applied to the faster wheel is applied to the slower wheel, effectively 'limiting slip'. A higher lockup percentage will cause increased rear tire wear on the inside tire during cornering -- the tire itself will have to slip slightly to counteract the limited slip's desire to have both tires turning at the same speed. It will also increase oversteer in wet or slippery conditions, but it will also increase understeer in tight corners under dry conditions. This is simply due to the fact that with a limited slip, the drive wheels tend to want to turn at the same speed, making the car tend to want to go in a straight line. When it is slippery, however, both drive tires will tend to lose traction at the same time, increasing oversteer. The
advantages are less inside wheelspin when accelerating out of a tight corner. This also translates into more horsepower to the pavement and faster autocross times -- provided that the suspension is tuned for the limited slip. The ability to accelerate out of corners without excess wheel spin can be a great advantage.

Om a more technical note:

The limited slip percentage (S) is also called the locking factor. It describes the maximum applied torque
difference between rear wheels compared with total applied torque. Passenger car LSDs are usually in the
25-40% locking factor range. Most BMW LSDs are 25%.

Limited Slip Locking Factor or Percentage S: (note: drive torque is torque applied to road surface)

               Drive Torque Difference Between Rear Wheels
         S =   ------------------------------------------- x 100%
                  Total Drive Torque of Both Rear Wheels

Think of a situation where the two rear wheels are on different surfaces with different coefficients of friction:

           H = Higher traction, more torque can be applied to road surface
           L = Lower traction, less torque can be applied to road surface

                H - L
         S =   ------- x 100 %
                H + L

By rearranging the equation a little, you see that for a 25% LSD, the High torque side can be as much as
62.5% of the total while the Low torque side can be as little as 37.5% of the total.

25% LSD Example:

                    S + 1      0.25 + 1
             H =   -------  =  --------  = 0.625
                    2               2

                   -S + 1       -0.25 + 1
             L =   -------  =  --------  = 0.375
                     2             2

The H/L ratio, called the bias ratio, is easier for me to think about because it quickly shows how much more
torque can be sent to the high side. With a 25% limited slip, it is possible to have 1.67 times as much torque applied to the high side. A 40% LSD works out to a 2.33 bias ratio.

25% LSD Example:

         H       S + 1     0.25 + 1
        --- =   ------- =  --------  = 1.67 (Bias Ratio)
         L      -S + 1    -0.25 + 1

A locked differential has a 100% locking factor (infinite bias ratio) because all torque can be applied to one wheel (e.g. one wheel on ice or in the air). For a limited slip, the initial preload, or break-away torque, allows power application when one drive wheel is on ice or in the air. Open differentials are another story (see
snow/ice write-up below).

In theory, an open differential has 0% locking factor (1.00 bias ratio) because the torque to each wheel is
balanced (H = L). In actual practice, there is some bias because the differential is not friction free.

Differentials reduce tire wear and help a car turn more easily by allowing the rear wheels to travel at different
speeds while turning corners. The inside wheel must slow down (smaller radius turn) while the outside wheel
speeds up an equal amount (larger radius turn). To balance the drive torque at each wheel, more torque is
applied to the outside wheel, speeding it up, while less torque is applied to the inside wheel, allowing it to slow down.

Open differentials always work well turning. They also apply power very evenly when both rear wheels have
adequate traction. However, the big downside, is their torque balancing action when one wheel has much less traction, such as in ice and snow.

The torque applied to the wheel with the most traction can only equal the lesser traction wheel. Total applied
torque for both wheels is only twice the traction of the worst wheel.

Increasing Locking Precentage

How do they increase the locking percentage on a diff? Do they just pack more shims in there to make the clutch plates tighter? Or do they re engineer the ramp angles? I have also read that Metric Mechanic adds clutch plates. In this case is the carrier machined to accept more plates?

The shim alters pre-load, and too much pre-load provokes understeer. It'd be better to get the locking to
almost go away under braking & corner entry, and then come back for power-on exits. The pre-load should be just enough for a smooth transition between the action of the coast & power ramps, and to keep things together when unloaded. But the shim is what people can do. The shim in question is not the 'thrust washer'
that the shop manual refers to, but the 'spacer ring' which is shaped like a Belleville spring washer as big as
the discs themselves. These are made of stiff spring steel around 2 mm thick. These, as well as the 'dog- eared plates' are selected to shim up the static locking.

The torque path has the ramps driving the differential pinion's shaft, and the ramp angle leverages the clamping force applied. Less angle on the ramp offers a mechanical advantage for applying the clutches = more lock. I have not measured any, but it looks as though BMW has the same angle for both coast & power
ramps. BMW did use shallower ramp angles on euro high % LS. (Anyone want to sell an original 75% factory
unit?)

Due to the BMW's design, the ramp angles can't be modified. The ramps are located on the pressure covers'
that are also used to retain the spider gears. The back side of the gears fits into the rounded interior section
of the left & right pressure covers. This may be so the gears assist in their spreading action.

Adding clutches increases surface area for clamping = more lock available. You would have to machine the
housing to fit them, so it could weaken it.

Hope this is enough!

> Gruppe;
>
> I have been curious about this but have not seen information regarding
> this.
>
> Korman and others advertise differentials for our cars that have lockup
> of 40% instead of the usual 25%, which, I believe is what is stock on
> the 1995 and newer M3s.
>
> What does it mean to have a 40% or for that matter a 25% lockup diff?
>
> What are the advantages or disadvantages one would see on track,
> autocross course, or street of one versus the other?
>
> Though this question doesn't affect my car, does this change the way or
> the effectiveness of the ABS systems or the traction control systems at
> all?
>
> Thanks for any explanations,
>
> Marc
> 1995 M3 with stock LSD diff and 25% lockup.