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From digest.v7.n1019 Mon Jan 26 10:22:31 1998
From: Pete Read <>
Date: Mon, 26 Jan 1998 06:53:24 -0500
Subject: <MISC> Limited Slip Diff Percentage, Open Diff in Snow

A couple of people have asked about the percentage rating of limited slip differentials. I'm repeating my post from a while back on the subject. If the theory isn't interesting, maybe the handbrake trick for getting unstuck will help.

(Trivia question for long-time digesters -- what does this have to do with Airplanes? Don't answer, just think about it and smile about how these threads go round and round.)

...old post on limited slip percentage...

Harris Yong asks:
>Exactly what does BMW mean when they say that a car is equipped
>with a 25% LSD...
>So is an open differential a 0% differential or a 100% differential...
>How about an axle with no differential?

Limited Slip Differential (LSD) Percentage (S)

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.

Open Differentials and Snow

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.

Open Diff Traction = 2 x Least Traction Wheel

With one wheel on ice and one on dry pavement, two times zero equals zero torque applied to the road. The differential keeps trying to balance the torque by applying more power to the spinning, lesser traction, wheel.

You can see why it is not good to spin tires in the snow. Once a tire starts spinning, more torque is applied which makes it spin more, further lessening traction. Any traction regained on the spinning wheel side is doubled because it is also applied to the other, non-spinning wheel.

So common sense says to increase traction any way possible to keep the tires from spinning. Snow tires are an obvious solution as is 150 lbs of sand in the trunk between the rear drive wheels. If you get stuck, sand can also be thrown under the spinning wheel (floor mats also work in a pinch).

Trick for Getting Unstuck When One Wheel Spins

Spreading sand and throwing floor mats under wheels can be a little messy. Here is a trick that is worth a try before you get dirty. It is perfect when one wheel is spinning on ice while the other sits on dry pavement.

Try partially applying the handbrake. The spinning wheel will slow down and the extra torque needed to overcome the handbrake will be transferred to other wheel with more traction. Traction control works the same way, except it only applies the brake to the spinning wheel, not to both rear wheels.

Pete Read
'88 M5
Arlington, VA

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