December
2002
ACKERMANN RECOMMENDATION
Question:
I am modifying a road racing Formula Ford
for SCCA Solo 2 [American autocross]. I am considering adding
more Ackermann effect to make the car work better in tight
turns. Is this a sound idea, and if so, what do you suggest
for geometry?
Answer:
Without writing a really long piece on Ackermann,
yes you will probably help the car. I don't know what geometry
you have now, but as a general rule a car needs more Ackermann
for events with tight turns, e.g. autocross or hillclimbs.
There isn't a universally agreed way to express
how much Ackermann (toe-out increase with steer) a car has.
The closest thing we have is to take the plan-view (top-view)
distance from from the front axle line to the convergence
point of the steering arm lines, divide the wheelbase by that
number, and express the quotient as a percentage. If the steering
arms converge to a point on the rear axle line, that's said
to be 100% Ackermann. If they converge to a point twice the
wheelbase back, that's said to be 50%. If they converge to
a point 2/3 of the wheelbase back, that's said to be 150%.
If they are parallel, that's zero Ackermann. If they converge
to a point twice the wheelbase ahead of the front axle, that's
said to be -50%.
Supposedly, with 100% Ackermann, the front wheels
will track without scuffing in a low-speed turn, where the
turn center (center of curvature of the car's motion path)
lies on the rear axle line in plan view. This is actually
not strictly true, even for the simplest steering linkage,
which would be a beam axle system with a single, one-piece
tie rod. With either a rack-and-pinion steering system or
a pitman arm, idler arm, and relay rod or center link, we
can't fully predict what the Ackermann properties will be
at all, merely by looking at the plan view geometry of the
steering arms. The whole mechanism affects toe change with
steer.
Even knowing what instantaneous toe we want
in a specified dynamic situation is not simple. We don't necessarily
want equal slip angles on both front tires. For any given
steer angle, the turn center might be anywhere, depending
on the situation. All the infinitely numerous possible situations
will have different optimum toe conditions. Therefore, there
is no relationship between steer and toe that is right for
all situations.
The toe we have at any particular instant results
not only from Ackermann effect, but also from static toe setting
and toe change with suspension movement (roll and ride Ackermann).
Because of these complexities, there is no single
obvious way to define what constitutes theoretically correct
Ackermann. It is possible to come up with a rationally defensible
definition for your own purposes, but there is no standard
rule, and it is unlikely that there ever will be.
TORQUE, RPM, AND POWER DISTRIBUTION IN DIFFERENTIALS
Question:
I would like some clarification on the issue
of torque distribution between the front and rear axles on
4wd vehicles. I find the matter fairly easy to understand
when you have wheels spinning, and a limited-slip differential,
but I find it more confusing when I read statements that a
vehicle has a permanent torque distribution of, say, 32% front
and 68% rear.
To me, torque and revolutions go hand in
hand: reduce rpm and you increase torque, as in a ring and
pinion. Doesn't that mean that if you want different torque
at the front and rear axles, they have to turn at different
speeds?
I know that in vehicles with viscous coupling
drive to one axle, one can have a different overall drive
ratio at each end, and this is often deliberately employed
just to load the system in normal driving, and make it respond
quicker to traction loss. But how does a rigid system, with
a planetary differential for example, split torque unequally?
When we are dealing with one input torque,
from one gear or shaft, and one output torque on a single
shaft or other member, the relationship you describe between
torque and speed does hold. Neglecting friction, power in
equals power out. If rpm is changed, torque must change too,
in inverse proportion, for the product of the two (power)
to remain constant.
However, when the output power is divided between
two shafts by a differential, things change a bit. Total power
in still equals total power out (again neglecting friction),
but power at each of the two output shafts is not necessarily
equal to power at the other shaft. Any non-locking differential
maintains a fixed distribution of torque between the two output
shafts, while letting their relative speeds vary freely. In
a conventional differential, the torque split is 50/50. In
a planetary differential with one planetary gearset, the torque
split is unequal but still fixed, while the shafts can turn
at different speeds.
Answer:
Usually the differential carrier or planet carrier
is driven by a gear, which receives power from another gear
driven by the input shaft. At the carrier, the simple inverse
relationship between speed and torque applies. Torque at the
carrier is input torque times rpm reduction factor. The sum
of the output torques equals the carrier torque. The average
of the output speeds equals the carrier speed. Power at each
individual output shaft can be any value at all. It is even
possible to have negative power (retardation) at one output
shaft if that shaft is being forced to turn backward (opposite
to torque). But the sum of the two power outputs must equal
the power input. (That's the sum of their signed values, not
their absolute values.)
It is helpful to think of each spider or planet
gear as being similar to a beam, with a load applied at its
midpoint, and reaction or support forces at two points equidistant
from the load. The load is the drive force applied at the
spider or planet gear's shaft. The reaction forces are the
output shaft resistances to vehicle motion, acting at the
points of mesh between spider and side gears, or between planet
and sun and planet and annulus. Since the spider or planet
shaft is always at the gear's center, the forces at the mesh
points are always equal. This is true regardless of the rotational
speeds of the various elements.
In a conventional differential, the side gears
are equal diameter, so the equal forces at the mesh points
act on equal moment arms, and produce equal torques. In a
planetary, the annulus is larger than the sun, so the output
torque at the annulus is greater than the output torque at
the sun. The ratio of the output torques is the ratio of the
pitch diameters of the annulus and sun. So the bigger the
planet gears are in comparison to the sun, the more unequal
the torque split becomes. Usually, the annulus drives the
rear axle and the sun drives the front axle.
We can, in fact, regard the conventional differential
as a unique version of the planetary, cleverly reconfigured
by the use of bevel gears to allow the sun and annulus to
be the same size.
All of this determines the torques at the front
and rear drive shafts. Usually, the main rpm reduction and
torque multiplication (after the transmission) happens at
the axle, not at the transfer case. It is possible to use
different ring and pinion ratios at the front and rear axles,
and/or different tire sizes front and rear, and further alter
the drive force distribution at the tire contact patches.
At the axles, the usual rpm/torque inverse proportionality
applies. To get more front torque and less rear by using dissimilar
axle ratios, the front drive shaft must turn faster than the
rear. That will increase wear at the center diff, rather like
traveling a long distance with unequal size tires on an axle.
Actually, the least wear at the center diff comes with slightly
less torque multiplication at the front axle than at the rear
- say a 4.10:1 ring and pinion at the front and a 4.11 at
the rear. This is because even on a straight road, the car
doesn't quite go perfectly straight, and in most turns the
front wheels will track outside the rears. Consequently, the
front wheels travel a few more revolutions per mile more than
the rears, even if the effective radii of the tires are equal.
A spool or completely locked differential drives
both output shafts at the same rpm, and does not split the
torque in any fixed proportion. This is opposite to an open
differential, which controls relative torque at the output
shafts but not relative speed. With a spool, torque distribution
depends on relative resistance at the two output shafts. It
is quite possible for one output shaft to have negative resistance
(wheel dragging and trying to drive the axle), while the other
output shaft has a torque greater than the sum of the two
(wheel driving the car plus overcoming drag from the other
wheel). The former condition exists on the outside wheel,
and the latter on the inside wheel, when making a turn with
a spool and no tire stagger.
A partially locking or limited-slip differential
is midway between. It allows some difference in speed, but
adds torque to the slower output shaft and takes that torque
from the faster output shaft.
A viscous coupling transmits torque according
to the amount of slippage at the coupling. The faster the
input shaft turns relative to the output, the greater the
torque at the output shaft. Unlike a gear set, however, the
relationship is usually not a simple linear function of the
rpm ratio.
Note that none of these alternatives split power
equally. No known passive mechanical device does that.
