Asymmetric Roll Centers
Author(s): Wm. C. Mitchell
Source: SAE Transactions , 1998, Vol. 107, SECTION 6: JOURNAL OF PASSENGER CARS
(1998), pp. 2632-2639
Published by: SAE International
Stable URL: https://www.jstor.org/stable/44741226
JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide
range of content in a trusted digital archive. We use information technology and tools to increase productivity and
facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.
Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at
https://about.jstor.org/terms
SAE International is collaborating with JSTOR to digitize, preserve and extend access to SAE
Transactions
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
983085
Asymmetric Roll Centers
Wm. C. Mitchell
Wm. C. Mitchell Software
Copyright © 1998 Society of Automotive Engineers, Inc.
many applications they produce the same result. But
ABSTRACT
when we apply the definitions to asymmetric suspensions
the differences can be critical. Indeed, not all authors
The roll center is an important analysis tool for vehicle
dynamics. But most analysis of the roll center is agree
basedon the definition. We must understand the
assumptions
that underlie the definitions before we
on production cars, which usually have symmetric
susthem.
pensions and a center of gravity near the centerlineextend
of the
vehicle. Racing cars, particularly oval track stock cars,
The best discussion of roll centers I have found is a paper
often have asymmetric suspensions and usually have a
by J.C. Dixon (1) titled "The roll-centre concept in vehicle
weight bias. For a car that is only going to turn left there
handling dynamics."" This appeared in the Procedings of
is no reason for the left side front suspension to be anythe Institute of Mechanical Engineers, volume 20, numthing like the right side. Oval track cars usually have
ber 1. as
much weight on the inside as the rules allow.
THE
KINEMATIC ROLL CENTER
Analytical tools adapted from the standard industry
texts
or production car use do not properly address asymmetThis is the most common definition of a roll center, but
ric suspensions. This paper will analyze the asymmetric
suspension and discuss the role of the roll center. relies
It will
upon some assumptions that are dubious for production
begin with a theoretical analysis of the roll center and
the cars and impossible for oval track racing cars.
underlying assumptions. It will show that the roll center is
Ellis (2) has the following definition:
a clever device for calculating forces where you do not
know how lateral force is distributed between the inside
"Traditionally the vehicle has been assumed to roll
and outside tires.
about a 'roll-axis' which has been defined as an
The paper will include an analysis of the relationship
centres' ...
between lateral movement of the roll center as a result of
roll and vertical movement of the roll center as a result of
axis joining two imaginary points, the 'roll
The roll centres themselves have been taken as
bump movement. The paper will prove that a symmetric
suspension which exhibits no lateral movement of the roll
kinematic centres of rotation of the suspension
assuming that the wheels are rigid and do no
center as a result of chassis roll will also have a roll cen-
move sideways on the road surface.
ter which moves vertically with the center of gravity when
the chassis exhibits vertical movement. This desirable
Dixon (1) does a good job of discussing the underlyi
assumptions of the kinematic roll center. These assum
property preserves the moment arm and contributes to
tions are severely violated in the motorsports arena.
stability. It also explains why lateral movement of the roll
Assumption #1 : The contact point between tire and ro
center, which is calculated by popular software programs,
remains fixed even as the tire changes camber.
can be used as a design criteria.
This analysis will then be extended to asymmetric sus- This also means there is no lateral deflection of the
pensions and will derive stability criteria for suspensionswheel or tire. Photographs and under-car TV camthat do not have the roll center on the centerline of the
eras show this is not true. This definition depends
upon a pin-joint between tire and road. With a wide
car. Techniques will be developed which allow the
pneumatic tire there is no simple description of the
designer to place the roll center wherever he wants and
contact between tire and road.
keep it near there as the vehicle moves.
Assumption #2: There is no track change.
WHAT IS A ROLL CENTER?
This assumption would restrict the vehicle to one ride
The formal definitions of roll center distinguish between aheight for each roll angle. But the vehicle goes
through ride height changes (heave) as it moves
kinematic roll center and a force-based roll center. For
2632
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
THE SPRING CENTER
around the race track. This is particularly true of
banked tracks where the lateral force is applied with
a normal force resulting from the banking or aerody-
If the chassis truly did rotate about the 'roll center' the
namic downforce.
we could use the lateral location of the roll center to
determine how much one spring compresses and ho
Assumption #3: The roll center is really an instantaneous
center because it can migrate as the car moves.
much the other extends. But this kinematic definition
ignores the springs entirely. The springs define a 'spring
This is why many industry texts restrict the definition
center' which relates to the relative stiffness of the
to low lateral forces. Race cars are designed for
springs. Oval track cars often have different springs
the inside and outside of the car. If you consider a c
large lateral forces.
with a massive spring on the outside and a light spring
The kinematic roll center concept also depends upon
the inside, that car is going to roll about the massi
spring. This example contradicts the kinematic
approach.
rigid links without bushings. In this case race cars fit the
assumption better than street cars.
m ^ l' _[ III J m r >
'ns'an'
^
ļ
l'
_[
Center ^
Fi
i
I
/
Roll
''
Center
'
Contaci Patch Contact Patch
Figure 2. The Roll Center is located at the intersection o
with the tire Contact Patch. For a symmetric vehicle i
2633
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
i
i
r~r~]
Figure
3.
In
Figure
FORCE-BASED ROLL CENTER
vertical plane through any pair of wheel centres at
which lateral forces may be applied to the sprung
The force-based roll center relies upon the same projec-mass without producing suspension roll."
tion of A-arms to an instant center. But some definitions
specify a roll center where these lines intersect the centerline of the car. Applying the forces at a point beneath
the CG makes it easier to understand jacking forces.
Gillespie (3) states:
"A lateral force in the contact patch of the left wheel
reacts along the line from the contact point to the
Dixon goes on to say:
"This does not call for the roll-centre to be in the
centre-plane and is therefore ambiguous, although
it is usually taken to be there."
If you assume the forces from the tire contact patches are
applied at the intersection of the lines from tire contact
patch to IC, then you can combine the forces from the left
and right tires into one force without knowing how the latcenter plane of the vehicle establishes the roll cen- eral force is distributed between the tires. The force vecter R."
tor from the outside tire adds to the force vector from the
pivot point ... Its elevation where it crosses the
Others use the intersection of the two lines. For a sym- inside tire giving a combined force with magnitude equal
to the total lateral force. This allows you to define a
metric car the lines intersect on the centerline of the car.
height of the roll center.
Dixon (2) quotes the SAE definition:
TotalForce = p * G * Force + (1 - p) * G * Force = G *
"The roll centre is defined in SAE J670c Vehicle
Dynamics Terminology in the following way: 'TheForce irregardless of proportion p of force from outside
tire
[S.A.E.] roll-centre is the point in the transverse
2634
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
THE LATERAL ROLL CENTER LOCATION
For determining the jacking effect, it makes more sense
to apply these forces at a point beneath the CG of the
vehicle. Then the lateral force can be translated into a
None of these definitions really deal with the lateral loca
jacking force (or de-jacking force from the inside tire) tion of the roll center. However a belief has develope
without creating a moment. But this construction that if the roll center does not move laterally when th
requires a knowledge of how the lateral force is distribvehicle rolls then the car will be "stable". There is no par
uted. In this case the two forces do not add; they subticular reason stated for this, but it seems to have prove
tract. Designers often assume that most of the lateral
out in many practical examples. In the next portion of th
force comes from the outside tire and thus ignore the depaper we will describe the relationship between later
of the roll center and the moment arm
jacking effect from the inside tire. This assumption movement
is
questionable on high-banked tracks where the inside tire
between RC height and CG. It turns out that if the ro
may contribute a significant amount to the lateral force.
center does not move laterally in roll then the momen
arm remains constant. That is, if the vehicle moves dow
Milliken and Milliken (4) have a good explanation of this in
one mm (bump) then the RC height also moves dow
their chapter on Suspension Geometry, which was writone mm and preserves the length of the moment arm.
ten by Terry Satchell.
This means the vehicle remains at a constant roll angle
rather than trying to change roll angle. This gives a fee
WHY A ROLL CENTER?
ing of stability to the driver.
this roll center remains fixed in the vehicle frame of refThe roll center is used in two types of analysis.IfThe
height of the roll center relates to the amount of lateral
erence then the forces acting through the suspension
load transfer through the suspension links. It also reflects
links to not attempt to change the roll angle of the vehicle.
allows the car to "take a set" and gives a feeling of
the amount of jacking force exerted on the chassisThis
from
lateral force. The higher the roll center the more jacking
stability.
force. The jacking force raises the chassis, which has the
ROLL CENTER CONTROL
detrimental effect of raising the center of gravity ACHIEVING
of the
vehicle and also decreasing the aerodynamic forces.
This is an argument for having a roll center near, or
even
Designing
a suspension with the desired roll center
below, the ground.
migration properties can be difficult. The roll center is the
intersection of two lines which connect the contact pat
Swing-axle cars are the classic example of this situawith the instant center, which is also an intersection o
tion. It explains why swing-axles are not used on modern
two geometric lines. This type of problem does not easi
race cars.
lead to a closed-form equation that can be solved. It
The distance between the height of the roll center
possibleand
to develop analytical techniques that addre
the height of the CG defines a moment arm.this
Thequestion,
longer but the easiest way to explain them is t
the moment arm the more the car will try to look
roll at
inthe
reacproblem from a different vantage point.
tion to lateral forces. A tendency to roll may force the use
Instead of using the road as a frame of reference a
having the car move, consider the problem from t
of heavier springs or anti-roll bars to restrict the roll. This
is an argument for a roll center height near the CG.
frame of reference of the vehicle. The body will rema
These two conflicting consequences of roll center
height
stationary
while the road moves under it. Or the body
make it impossible to specify the best roll center.
But stationary
roll
remains
while the suspension moves up a
center height does remain a useful tuning tool.
down. Now the problem of finding a roll center that do
not move laterally in roll and preserves the moment arm
/K
Figure 5. In the road frame of reference the read remains fixed while the chassis moves in ride and roll.
2635
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
under ride changes becomes a problem of having a fixed
area by designing a suspension that projects through a
roll center. What we want is a suspension design that
desired point in this frame of reference.
keeps the line from instant center to contact point moving
We are not guaranteed a solution, or a practical solution.
through a specific point on the chassis. This point may
be on the centerline of the vehicle or not. If the suspension from either side keeps the force vector line going
through the desired point then we will have a stationary
The desired suspension lengths may place the pickup
points inside the engine block. In this case the location of
the Instant Center may have to be moved to achieve a
stable roll center.
point.
And the solution is really an instantaneous solution. It
will be very accurate for small movements but the roll
center will begin to migrate as the movements increase.
But the RC envelope will still be smaller than with an
uncontrolled suspension.
This formulation has several nice properties. First of all,
it produces a fixed target rather than a moving target.
This is always easier to hit. Secondly, it splits the problem into two parts. We can design each side of the suspension to achieve the desired characteristics rather than
designing two suspensions that interact in the desired
way. It is always easier to solve two distinct linear equations with one unknown rather than one pair of equations
This effect can work against designers converting production cars to race cars. A production car with a sixinch ride height may be designed with excellent roll cen-
with two unknowns.
ter control. If the car is to be raced it will often be lowered
an inch or more to lower the center of gravity. It may be
impossible to achieve roll center control at this ride height
without changing pickup points, which may be against the
The way to achieve roll center stability while maintaining
the instant center (IC) location, and thus the first-order
kinematic effects, is to change the lengths of the suspension arms. We can shorten or lengthen the arms without
changing the IC location. This preserves our first-order
ride-camber, roll-camber, and scrub-change effects. But
the changes will effect roll center stability. You can even
rules.
Many designers of race cars are clearly using roll center
control as a design criteria. I have seen Indy cars with
roll centers that do not move even 0.1 inches with one
degree of roll.
maintain the length of one arm and change the other until
the desired stability is achieved. This is a single-valued
THE INCLINE RATIO
function of one variable. The variable is the length of one
arm. The result is the change in intersection height at the
desired point. We desire a length which gives a change
of zero. This formula can be solved many ways. It can
be done graphically or with numerical techniques which
The best technique for designing a suspension with roll
center control is the incline ratio. This is the ratio of ride
height change to change in the point where the line
provide clever guesses of the next approximation.
Once the proper lengths have been determined for each
side, we can put the suspension together and return to
the road as a frame of reference. You may quibble that a
fixed point in one frame of reference is not fixed in the
other frame. This is true, but we have devised a tech-
nique that produces a roll center which is constrained
within a limited area. We can even adapt the same technique to the road frame of reference and derive a suspension that will hold the roll center within a very limited
between tire contact point and Instant Center crosses a
reference line. This is based on the angle of the line from
the tire contact patch to the Instant Center. Both kinematic and force-based definitions agree on the location of
this Instant Center. This dimension could be expressed
as an angle, but I find it easier to express it as a distance
along a vertical line. The reference line may be the centerline of the vehicle, or a line through the CG, or a line
through the roll center when the car is measured, or any
other line. We seek an incline ratio of 1 .0, meaning each
unit of change in ride height moves the incline point one
unit.
'|/
Figure
moves
'|/
6.
In
the
Ve
up
and
down
2636
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
M -
The Incline Ratio can be calculated like the motion ratio.
sponds, to the same amount of movement in the intersecThe easiest way to calculate the motion ratio is to contion point as in ride height.
sider two ride heights perhaps 0.01 inch apart. Take the
difference in spring movement and divide by 0.01 . This EXAMPLE
gives a motion ratio calculated as a difference rather than
a true derivative. The amount of the difference can be
To demonstrate the method with an asymmetric suspe
reduced to better approximate the true differential but
sion, we take the previous suspension and raise the rig
round-off errors become significant if we use too small
a upper inner pickup point one inch. This creates an
side
difference.
asymmetric suspension with the roll center located
10.363 inches to the left of the centerline and 1.775
The Incline Ratio is calculated based upon the same 0.01
inches above ground. Since we did not change the
difference. We are seeking a ratio of 1 .0, which correside we can use 6.316 lower arm pickup point which
us a stable symmetric roll center.
'/
'/
Figure
center
Table
1
Lower Inner Incline - Roll Center Lateral - RC Height
Pickup lateral Ratio 0.1 deg | 1.0 deg | 2.0 deg
ãõÕ 1.3791 -0.274 -2.738 -5.431 1 .228
ÕÕÕ 1.3105 -0.224 -2.237 -4.435 1.296
8Í5Õ 1.2459 -0.177 -1.763 -3.494 1.361
ÉTÕÕ 1.1846 -0.132 -1.314 -2.603 1.422
7ÜÖ 1.1264 -0.089 -0.887 -1.757 1.480
7ÃÕ 1.0712 -0.048 -0.483 -0.954 1.535
Õ5Õ 1.0186 -0.010 -0.098 -0.189 1.587
&ÕÕ 0.9686 0.027 0.269 0.539 1.636
ÕlÕ 0.9784 0.020 0.197 0.396 1.627
Õ20 0.9883 0.013 0.125 0.252 1.617
Õ30 0.9885 0.005 0.051 0.106 1.607
ēāī 0.9994 0.004 0.044 0.091 1 .606
Õ32 1.0004 0.004 0.037 0.077 1.605
Õ33 1.0015 0.003 0.029 0.062 1.604
Õ34 1 .0025 0.002 0.022 0.047 1.603
Õ35 1.0035 0.002 0.014 0.033 1.602
036 1.0044 0.001 0.007 0.018 1.601
Õ37 1.0055 0.000 0.000 0.003 1 .600
Õ38 1.0064 -0.001 -0.008 -0.011 1.599
Õ39 1.0075 -0.001 -0.015 -0.026 1.598
Õ4Õ 1.0085 -0.002 -0.023 -0.041 1 .597
2637
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
Table 2. Verification of Roll Center movement with Lower Inner Pickup Lateral of 6.31 6
Roll Angle RC Lateral RC Height Change in CG-RC Moment Arm
ÕÕÕ
ÕÕÕÕ
2^587
0Ī0
0.004
2.588
ÕÕÕÕ
0.001
0.20
0.008
2.588
0.001
0.50
0.020
2.589
0.002
TÕÕ
ÕÕ4Õ
2J593
2.00
0.083
2.610
0.023
Ž689
0.102
3ÃÕ
0.133
0.006
Ride Height RC Height Change in CG-RC Moment Arm
^0
TĪ44
0.244
TÕ
066Ī
ÕÕ74
-1.0
1.605
0.018
0.0
2.587
0.000
1.0
3.607
0.020
2X>
4.663
0.076
Tab
Lower
Pickup
Inner
Incline
lateral
Ratio
ÕIÕ 1.3791
9JÕÕ 1.3105
ÍTÕÕ 1.2459
7XK)
1.1170
ãÕÕ 1.0049
ŠT0
0.9030
5^90 0.9943
5.95
0.9994
Table
4.
Verification
of
Roll
Pickup
lateral
at
5.95
Roll Angle RC Lateral RC Height Change in CG-RC Moment Arm
ÕÕÕ
-10.363
1.775
0.000
0Ī0
-10.359
1.775
0.000
Õ2Õ
-10.355
1.775
0.000
Õ5Õ
-10.338
1.776
0.001
TÕÕ
-10.296
1.781
0.006
2M
-10.155
ãÕÕ
-9.935
1.800
0.025
1.834
0.059
Ride Height RC Lateral RC Height Change in moment arm
23.088
-0.436
^5
-444.580
-8.114
TÕ
-20.520
0.607
TŠ
-13.800
1.216
ÕÕ
-10.355
1.775
05
-8.259
2.318
Ū)
-6.850
2.859
Ž0
-5.074
3.947
3X)
-4.003
5.058
2638
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms
0.789
-9.889
0.168
0.059
0.000
0.043
0.084
0.172
0.283
Table 4 shows that the roll center is contained within a
fairly small area. For 3 degrees of roll the roll center
moves 0.4 inches laterally and less than a tenth of an
inch vertically. For ride height changes the roll center
moves laterally a bit but the moment arm is constrained
within 0.3 inches except for the area where one instant
center moves below ground and the roll center is outside
the track of the vehicle. This is a singularity of the way
the roll center is constructed. When one IC is above
ground and the other below ground the roll center must
be outside of the track of the vehicle. When both instant
centers move below ground then the roll center returns to
a position between the tires.
This suspension was chosen using a stability criteria that
constrained the inclination of the force vector on the centerline of the vehicle. Had we chosen an Incline Ratio
designed to optimize the asymmetric roll center location
we could have reduced the roll center movement farther.
CONCLUSIONS
The incline ratio is a technique that can be added to suspension geometry software to better understand a double
A-arm suspension. If you want to constrain the roll center
within a limited area then the incline ratio allows the user
to construct a suspension more quickly. It also lets you
quickly understand how you need to move pickup points.
It is far easier than the trial-and-error methods required
previously.
REFERENCES:
1. Dixon, J.C. 'The roll-centre concept in vehicle handling
dynamics", Procedings of the Institute of Mechanical Engineers, volume 201 , number D1 , 1 987.
2. Dixon, J.C. Tvres. Suspension and Handling. Cambridge
University Press, 1 991 .
3. Gillespie, Thomas D., Fundamentals of Vehicle Dynamics.
SAE Publications Group, Warrendale, PA, 1 992.
4. Milliken, William F., and Millikem, Douglas R., Race Car
Vehicle Dynamics. SAE Publications Group, Warrendale,
PA, 1995.
2639
This content downloaded from
13.245.201.252 on Mon, 21 Mar 2022 13:03:03 UTC
All use subject to https://about.jstor.org/terms