From digest.v7.n1019 Mon Jan 26 10:22:31 1998
From: Pete Read <read_at_arl.bna.boeing.com>
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
-S + 1 -0.25 + 1
L = ------- = -------- = 0.375
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
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
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.
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