Steering geometry

Tech Tip: Steering Geometry
Designing Steering Geometry
When you’re designing steering kinematics, the goal
is to orient the tire to the road in the optimal orientation. But, how do you know the optimal orientation?
Tire data, of course!
One of the basic decisions when designing a steering system is how much Ackermann you want. The
answer to this is determined directly by the tire characteristics, and you can answer this question by using OptimumT in a clever way.
So, let’s try it! We will design the steering geometry for a small autocross car (210kg) as an example.
The data that we’ll use was collected at Calspan by
the FSAE TTC (Tire Testing Consortium). Since
we’re using a race car for the example, our goal is to
generate the maximum lateral force from the tires.
We’ll start by taking raw tire data that was collected
on a tire testing machine and import it into OptimumT. Next we’ll fit a Pacejka 2002 tire model to
this data using the OptimumT Model Fitting Tool.
For more information about this process, please refer to the OptimumT documentation. What we end
up with is shown in Figure 1.
Once we have a tire model, we can use OptimumT’s visualization tools to look at some derived
quantities. We can also use the OptimumT Add-In
to directly incorporate the tire model into Excel or
Matlab. We’ll look at the slip angle at which the
peak lateral force occurs at different loads and camber angles. This will allow us to design the steering
geometry to suit the tire. Before we get to that, however, we need to find the vertical load and camber of
each of the front wheels. We’ll do this in two steps:
first using a simple method, then a more accurate
way.

Figure 1: Tire data and model
weight transfer), the CG height, the weight distribution, the front and rear roll stiffnesses and the track.
For this example, the vehicle configuration used
is shown in Table 1.
Mass (m)
Weight Distribution (ρ)
CG Height (h)
Front Track (tf )
Front Roll Stiffness (Kf )
Rear Roll Stiffness (Fr )

210kg
45% front
140mm
1300mm
220N m/o
200N m/o

Table 1: Basic vehicle parameters
To calculate the lateral weight transfer, we’ll use
the following formulas:

Mroll = Ay mh
Mroll
θroll =
Kf + Kr
θroll Kf
Fzr = 0.5ρmg +
tf
θroll Kf
Fzl = 0.5ρmg −
tf
(Fyr + Fyl )
Ay =
ρm

Simplified Steering Geometry
If, for a moment, we ignore the camber of the front
tires, we can find the desired Ackermann angle very
easily. We need to estimate the maximum lateral acceleration that the car can achieve. We can do this
by constructing a simple weight transfer spreadsheet
and by using OptimumT. We need to know the mass
of the car (for now we’ll ignore the non-suspended

• Ay is the lateral acceleration
• Mroll is the roll moment
• θroll is the roll angle
1

(1)
(2)
(3)
(4)
(5)


Figure 3: Ideal slip angle versus load

Figure 2: Maximum lateral force versus load
• Fz r and Fz l are the right and left tire vertical
loads
• Fy r and Fy l are the right and left tire lateral
forces
• g is the acceleration due to gravity
We’ll perform an iterative calculation to find the
maximum lateral acceleration using the equations
above. Using Figure 2, we will also use them to find
the vertical load on the two front tires. This vertical
load will be used to find the ideal slip angle. We find
that the right tire has a vertical load of 686N and
the left tire has a load of 241N . In these calculations, we implicitly assume that the front axle is the
limiting axle; thus the car has terminal understeer.
We can use OptimumT to plot Ideal Slip Angle
versus Normal Load (Figure 3). By picking out the
slip angle at the two vertical loads that we calculated
earlier, we find that the outside tire should operate
at 4.38o slip angle and the inside tire should operate
at 4.11o . This can be used to find the appropriate
steering geometry, and indicates that that the car
needs positive Ackermann.

Figure 4: Example of OptimumT Add-In use in Excel

OptimumT, but an easier way is by using the OptimumT Add-In. This allows you to make calculations
with OptimumT inside Excel or Matlab. For example, if you want to find the lateral force calculated
by a tire model that you created in OptimumT, you
can easily do this with the OptimumT Add-In. This
is shown in Figure 4.
To find the desired steering geometry, we can create a new spreadsheet in Excel that calculates the
lateral weight transfer when both front tires are generating the peak lateral force. The finished spreadsheet is show in figure 5.
The same formulas used in the simplified calculations for the weight transfer are used here. In these
calculations, we can use the OptimumT Add-In to
More Advanced Steering Geometry
find the maximum lateral force that the tire can genWe can improve the calculations by including the erate. We use the OptimumT Add-In function:
camber. We could do this by using graphs created in
CalculateFyPeakNegative()
2


Figure 5: Ackermann design spreadsheet

3

By using this function in the Excel spreadsheet,
we can automate the process of finding the maximum
lateral acceleration. We create a "circular reference"
in the spreadsheet and Excel will automatically perform an iterative calculation.
To complete the rest of the calculation, we need
a little bit more information about the vehicle. For
this example, we use the following vehicle parameters. The parameters given earlier still apply.
Wheelbase (l)
Camber Coefficient (C)
Static Camber (γ0 )
Caster (ϑ)
KPI (λ)

l
R+
l
G
αl =
R−
G
αr =

t
2
t
2

+β

(8)

+β

(9)

G
G
• αr and αl are the right and left geometric slip
angles

• R is the turn radius

1650mm
−0.6o /o
−2o
6o
2o

• β is the body slip angle
We find the ideal steered angle of the two steered
wheels using the following formula:

Table 2: Vehicle parameters required for advanced
analysis

G
i
δr = αr − αr

−

i
αl

i
i
• αr and αl are the ideal right and left slip angles

• δr and δl are the required right and left steered
angles
Where the ideal slip angle is calculated with the
OptimumT Add-In.
Since these equations depend on the turn radius,
we repeat them for multiple steer angles. Since we’re
designing the steering geometry, we’re interested in
the difference in steered angle between the inside and
outside wheel. When we take the difference in left
and right steered angle, the body slip angle will cancel out.
We now have the difference in left and right steered
angles versus turn radius. We’re trying to design
steering geometry, though, so we would prefer to plot
this versus some steering angle. We’ll assume that
the car will be close to neutral steer, so the Ackermann steering angle relationship holds (δack = l/R).
The difference in left and right steered angle plotted against the Ackermann steering angle is shown
in Figure 6. This gives us a target curve when we
design the steering geometry.
When we read this graph, we find that these tires,
fitted on this particular car, need positive Ackermann. The graph also suggests adding about 0.15o

(6)

γl = −γr0 − λ (1 − cos δl ) − ϑ sin δl − θroll C

(11)

δl =

Once we have these values, we can expand the
calculations preformed earlier by taking into account
the camber of the two steered wheels. Including
camber in the calculations increases the accuracy because we include the camber thrust when finding the
peak lateral forces. We calculate the inclination angles as follows:
γr = γr0 + λ (1 − cos δr ) − ϑ sin δr + θroll C

(10)

G
αl

(7)

• γr and γl are the right and left inclination angles
• δr and δl are the right and left steered angles
(found later)
The slip angles at which the peak lateral force occurs can be calculated using the OptimumT Add-In.
It is dependent on vertical load and the inclination
angle. The function used is:
CalculateSAPeakNegative()
We use geometric relationships to find the slip
angles of the front wheels when they are not steered.
This is used to calculate the required steered angle
of the two front wheels. The geometric slip angle
of the two front wheels can be closely approximated
with the following expression:
4


Figure 6: Ideal steering geometry
of static toe in (indicated by the offset at zero steering angle). However, the toe will have a large effect
on the on-center directional stability of the car. It
is best to set the toe angle for handing and stability,
rather than for achieving the ideal slip angle during
mid-corner.
This tech-tip gives a brief outline of what can be
done with OptimumT to assist in designing a steering system. OptimumT and the OptimumT Add-In
can be useful tools when evaluating the compromises
involved in steering design.
You can easily expand this spreadsheet to give
a much better picture of reality. For example, you
can include the rear axle in the calculations, you can
include the weight transfer due to steering, the nonsuspended weight transfer and a multitude of other
factors (including the effect on longitudinal forces).
You can even create a simple vehicle simulation in
Excel or Matlab. This is all made possible through
the OptimumT Add-In.
If you have any questions about OptimumT or
any of OptimumG’s other products and services, please
email engineering@optimumg.com. Also, don’t forget to visit www. optimumg. com

5

Steering geometry

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