1. P04731
Advanced Chassis Engineering
Vehicle Dynamics Portfolio
Oier Bañuelos-Arteagoitia (15059319)
Oxford Brookes University

2. 1
Table of Contents
1 Straight line Acceleration:.................................................................................................. 2
1.1 Introduction...............................................................................................................................2
1.2 Tire Effects ................................................................................................................................2
1.2.1 Pacejka Tire Model..........................................................................................................2
1.2.2 Rolling Resistance ............................................................................................................3
1.2.3 Mu Value...........................................................................................................................3
1.3 Weight Transfer........................................................................................................................4
1.4 Aerodynamics............................................................................................................................4
1.5 Inertia.........................................................................................................................................5
1.6 Mechanical Components:.........................................................................................................6
1.6.1 Gear shift duration...........................................................................................................6
1.6.2 Drive train efficiency........................................................................................................6
1.7 Road Slope:................................................................................................................................6
1.8 Formula Student Question .......................................................................................................7
2 Suspension Optimization.................................................................................................... 9
2.1 Introduction...............................................................................................................................9
2.2 Adams Setup..............................................................................................................................9
2.3 Optimization Procedure .........................................................................................................10
2.4 Cosh Function and Results.....................................................................................................13
2.5 Suspension Travel ...................................................................................................................16
2.6 Conclusion ...............................................................................................................................16
3 Derivative Analysis............................................................................................................ 17
3.1 Introduction.............................................................................................................................17
3.2 Theoretical Background.........................................................................................................17
3.3 Results Spreadsheet and Bicycle Model................................................................................18
3.3.1 Spreadsheet.....................................................................................................................18
3.3.2 Adams Model..................................................................................................................20
3.3.3 Comparison.....................................................................................................................21
3.3.4 Pacejka Tire Model in Adams.......................................................................................22
3.4 Yaw Rate and Curvature Response.......................................................................................23
3.5 Tangential Speed.....................................................................................................................25
4 Tyre Model......................................................................................................................... 28
4.1 Introduction.............................................................................................................................28
4.2 The Fiala Model ......................................................................................................................28
4.3 The Brush Model.....................................................................................................................30
4.4 The Magic Formula ................................................................................................................32
4.5 The MF-SWIFT Model...........................................................................................................33
4.6 Comparative Table .................................................................................................................35
4.7 Formula Student .....................................................................................................................36

3. 2
1 Straight line Acceleration:
1.1 Introduction
The goal of this first portfolio is to develop further the simplified excel table, trying to
represent a more realistic straight line acceleration. In order to be able to reach this goal, different
parameters will have to be considered. The main aspects that will be treated in this report are:
tires effect on performance, weight transfer, aerodynamics, inertia and road slope among others.
Other variables could also be taken into account, such as effect of the suspension stiffness’s,
which have not been bear on mind due to the lack of space.
1.2 Tire Effects
1.2.1 Pacejka Tire Model
The most important part of a car is the tires. This is since they are the only part in contact
with the road and where most of the forces acting will take place. To introduce a tyre model in
the spreadsheet is very important to make a more realistic scenario. The biggest effect in the
performance is at the beginning of the acceleration, since the engine can produce a very high
amount of longitudinal force. The question is: can the tyres support such an amount? Here it is
where the Pacejka Model comes handy. At the start the performance of the car will be delimitated
by the amount of force the tires can deal with. However when the car reaches a specific velocity,
the engine will be the one that limits its performance. This change can be explained using the
Pacejka Model, which main formula is:
𝐹𝑥 = 𝐷 ∙ sin(𝐶 ∙ atan(𝐵𝑋 − 𝐸 ∙ (𝐵𝑋 − atan(𝐵𝑋)))) + 𝑉
From the formula it can be seen that the aim is to calculate the longitudinal force the tires
can produce. This force is dependent especially on the vertical load and the slip ratio.
Depending in the experience of the driver the slip ration will vary. This variable relates the
car velocity with the angular velocity of the wheels to calculate the performance losses due to the
spinning of the tires. With this ratio the real angular velocity of the wheels has been calculated at
the beginning of the spreadsheet. An experience driver is capable of getting the maximum
performance which is at 6%, while a less experience driver would make the tires spin losing some
performance (slip ration about 4%).In the table below the results for the needed variables and the
longitudinal force can be observed.
The procedure to implement the Model is an iterative process and therefore the software
excel has to run an iteration to find an equilibrium between the input and the output. Looking to
the flowchart will make the understanding easier of how the loop works:
6,606 1,500 10569,900 1,982 0,000 0,000 -1,600 0,000 0,000008 10591,819
6,800 1,500 10879,540 2,040 0,000 0,000 -1,600 0,000 0,000008 10902,101
7,380 1,500 11808,458 2,214 0,000 0,000 -1,600 0,000 0,000008 11832,945
8,348 1,500 13356,656 2,504 0,000 0,000 -1,600 0,000 0,000008 13384,353
9,703 1,500 15524,132 2,911 0,000 0,000 -1,600 0,000 0,000008 15556,325
11,444 1,500 18310,888 3,433 0,000 0,000 -1,600 0,000 0,000008 18348,859
13,447 1,500 21514,909 4,034 0,000 0,000 -1,600 0,000 0,000008 21559,525
15,508 1,500 24812,840 4,652 0,000 0,000 -1,600 0,000 0,000008 24864,294
17,740 1,500 28384,766 5,322 0,000 0,000 -1,600 0,000 0,000008 28443,628
20,225 1,500 32360,754 6,068 0,000 0,000 -1,600 0,000 0,000008 32427,861
23,196 1,500 37114,034 6,959 0,000 0,000 -1,600 0,000 0,000008 37190,998
26,570 1,500 42511,712 7,971 0,000 0,000 -1,600 0,000 0,000008 42599,870
30,155 1,500 48248,420 9,047 0,000 0,000 -1,600 0,000 0,000008 48348,473
33,999 1,500 54397,671 10,200 0,000 0,000 -1,600 0,000 0,000008 54510,476
38,361 1,500 61377,936 11,508 0,000 0,000 -1,600 0,000 0,000008 61505,216
43,049 1,500 68878,190 12,915 0,000 0,000 -1,600 0,000 0,000008 69021,024
48,037 1,500 76858,848 14,411 0,000 0,000 -1,600 0,000 0,000008 77018,231
53,374 1,500 85399,080 16,012 0,000 0,000 -1,600 0,000 0,000008 85576,173
59,000 1,500 94399,923 17,700 0,000 0,000 -1,600 0,000 0,000008 94595,682
64,947 1,500 103915,100 19,484 0,000 0,000 -1,600 0,000 0,000008 104130,591
71,245 1,500 113992,679 21,374 0,000 0,000 -1,600 0,000 0,000008 114229,067
New rear load
[kN]
Curvature
factor E
Horizontal
shift H [%]
Stiffness
factor B [-]
Stiffness
BCD [kN]
Peak factor D
[kN]
Shape factor C
[-]
Composite
Bx1 [-]
Vertical
shift V
Final longitudinal
force Fx [kN]
Table 1: Pacejka results to calculate the longitudinal force.

4. 3
1.2.2 Rolling Resistance
In the chart the rolling resistance (Fr) is introduced, which is another effect of the tyres. It
is not as important as the Pacejka Model, but still has an effect on the final acceleration
performance. The rolling resistance is used to represent the friction force needed to move the car
in any direction. This quantity is subtracted to the longitudinal force acting in the tire, but first the
rolling resistance coefficient has to be calculated following:
𝑐 𝑟𝑟 = 𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑜𝑓 𝑟𝑜𝑙𝑙𝑖𝑛𝑔 𝑓𝑟𝑖𝑐𝑡𝑖𝑜𝑛 +
1
𝑇𝑖𝑟𝑒 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒
∗ 0.01 ∗ 0.0095 ∗ 𝑣2
Now the rolling resistance can be obtained using:
𝐹𝑟𝑟 = 𝑐 𝑟𝑟 ∗ 𝑔 ∗ 𝑚
It is important to mention that in the rolling resistance the pressure of the tire is considered.
Depending on the pressure the contact patch will vary, having a larger rolling resistance for low
pressure tires. This can be observed in Plot 1. At the same time the rolling resistance will rise
exponentially with increasing velocity (Plot 2).
1.2.3 Mu Value
On the motorsport competitions, the handling of the vehicles are very dependent on the
type of tires they use, since they are in charge of the interaction between car and road transmitting
all the forces. Therefore depending on the weather conditions the tyres will change, having: wet
and dry tyres. Wet tyres are softer than dry ones because the forces acting on them cannot be as
large and the tyre runs cooler, since the water lowers their temperature. That is why wet tyres
Figure 1: Flowchart describing the implementation of Pacejka as an iteration.
Plot 1: Rolling resistance vs tire pressure. Plot 2: Rolling resistance ve speed.

5. 4
have a lower friction coefficient Mu value. The Mu parameter is defined with the Pacejka tire
model, where the longitudinal force over the vertical load gives its value
1.3 Weight Transfer
Weight transfer occurs during both straight line acceleration and cornering. To understand
a vehicle during acceleration and cornering it is of main importance to understand how weight
transfer occurs. Accelerating in a corner could lead to reach an undesirable behaviour of the car
such as understeering due to the weight transfer, since the vertical load at the front will decrease
and loosing grip. Alongside during corner weight is transferred to the outer wheels shrinking the
available grip in the inner tires. In order to minimize such an effect anti-roll bars can be installed
or a wider track width. On the other side, depending on the car, it could be a good option to try to
increase the weight transfer. The weigh distribution is dependent on the height of the centre of
gravity, the weight force and the wheel base or the track width depending on the situation. From
the formula is know that the higher the centre of gravity is located, the more weight transfer will
occur.
𝑊𝑇 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒
=
ℎ∙𝑚∙𝑔
𝐿
; 𝑊𝑇 𝑐𝑜𝑟𝑛𝑒𝑟𝑖𝑛𝑔
=
ℎ∙𝑚∙𝑔
𝑇
1.4 Aerodynamics
Aerodynamics is a very important matter in car competition. The purpose is to create down
force to keep the car in contact with the road. It is very useful above all during cornering,
nevertheless it is also important for straight line acceleration, although its configurations will vary
depending on the track style. Nevertheless aerodynamics has a negative aspect that needs to be
considered: drag. Therefore both situations will be explained.
Down force is in charge of pushing the car against the road having an effect on the vertical
load. There are different ways to increase down force, such as: introducing wings, flaps or a
diffuser. In the spreadsheet it is given that the weight of the car is 2.2 larger than the static weight.
This way the lift coefficient (𝐶𝐿) can be calculated which is needed for the lift force formula.
𝐹𝐿 =
1
2
∗ 𝐴 ∗ 𝜌 𝑎𝑖𝑟 ∗ 𝐶𝐿 ∗ 𝑣2
This force is added to the vertical load after applying the weight transfer to calculate the
real vertical load in the rear. This new force is then used in the Pacejka Model as mentioned
before.
Drag is the force that an object has when moving along a fluid. The drag formula is
basically the lift force but using the drag coefficient instead.
𝐹𝐷 =
1
2
∗ 𝐴 𝑓𝑟𝑜𝑛𝑡 ∗ 𝜌 𝑎𝑖𝑟 ∗ 𝐶 𝐷 ∗ 𝑣2
The frontal area is used, since is it is the one hitting the air. That is why the drag force
harms the performance, having to subtract it to the forward longitudinal force. When introducing
a wing or a flap to create more down force, at the same time there will be an induced drag which
has to be added to the chassis drag. Both drag and down force have a different result when
changing the angles of attack of a flap, exposing a lower or higher area to the air.
As it can be seen from the lift and drag force, both are dependent on the velocity, expecting
the following behaviour from the plots below.

6. 5
Another variable it could be used to extend the spreadsheet is the density of the fluid, being
air in this particular case. The density is dependent on pressure and temperature. When pressure
increases the density increases too. Therefore depending on the racing track the competition will
take part, the height might vary, resultantly the atmospheric pressure and the density. This can be
simply demonstrated with the ideal gas formula:
𝜌 =
𝑝 𝑎𝑡𝑚 ∗ 𝑀
𝑅 ∗ 𝑇
Having: 𝑝 𝑎𝑡𝑚for the atmospheric pressure, M defines the molecular mass of air, R being
the ideal gas constant and T the temperature.
As it is well known the atmospheric pressure changes dependent on altitude and
temperature. The higher the altitude, the smaller the pressure will be since less amount of air is
sustained in that area. In order to define this, the following formula has been used in the excel
table:
𝑝 𝑎𝑡𝑚 = 𝑝0 ∗ (
0.0065 ∗ ℎ
𝑇 + 273.15 + 0.0065 ∗ ℎ
)
𝑔∗𝑀
𝑅∗𝐿 𝑏
Where:
𝑝 𝑎𝑡𝑚
: Atmospheric pressure
𝑝0: Pressure at sea level
T: Temperature
h: Altitude
1.5 Inertia
To calculate the inertia the moments around the CoG of the vehicle have to be considered.
It seems to be a small effect, but when looking into top teams, they reduce the inertia even with
small details. For example in some cases the position of the brake callipers are as close as it is
possible to the CoG to minimize its inertia. In the spreadsheet the inertia for the front and rear tire
has been calculated. To try to make a more realistic calculation tires and rim have been considered.
The inertia of the wheel is dependent on the mass and the radius, where the tire has been taken as
a ring and the rim as a cylinder. The total inertia for a single wheel would be the sum of these
two. Hence the total inertia of the complete car would be the sum of two front and two rear wheels.
The calculation method is hereby presented:
𝐼𝑓𝑟𝑜𝑛𝑡 = 0.5 ∗ 𝑚 𝑟𝑖𝑚 ∗ 𝑟𝑟𝑖𝑚
2
+ 0.5 ∗ 𝑚 𝑓𝑡𝑦𝑟𝑒 ∗ (𝑟 𝑤ℎ𝑒𝑒𝑙
2
+ 𝑟𝑟𝑖𝑚
2
)
𝐼𝑟𝑒𝑎𝑟 = 0.5 ∗ 𝑚 𝑟𝑖𝑚 ∗ 𝑟𝑟𝑖𝑚
2
+ 0.5 ∗ 𝑚 𝑟𝑡𝑦𝑟𝑒 ∗ (𝑟 𝑤ℎ𝑒𝑒𝑙
2
+ 𝑟𝑟𝑖𝑚
2
)
Where:
𝐼𝑡𝑜𝑡 = 2 ∗ 𝐼𝑓𝑟𝑜𝑛𝑡 + 2 ∗ 𝐼𝑟𝑒𝑎𝑟
Plot 3: Lift force vs velocity. Plot 4: Drag force vs velocity.

7. 6
1.6 Mechanical Components:
Some mechanical effects will briefly be discussed, having a small repercussion in the time
and the engine power output.
1.6.1 Gear shift duration
Nowadays it is possible to find a theoretical zero gear shift. However this is not true,
although the time needed during gear changes are very small. Nevertheless it has be regarded in
the spreadsheet with an approximation. In excel a value can be given for the gear shift duration,
which will be added to the time each time a new gear is selected. In real life during the time in
which the gears are changing the velocity of the car will decrease due to drag and mechanical
friction of parts. This implies that an extra time is needed to reach once again the speed where the
change first started. This can be seen in the plot for a shift duration of 0.2s.
1.6.2 Drive train efficiency
With drive train it is considered all the components in charge of transmitting the power of
the engine to the wheels. In an ideal case no losses would be found, but in reality the friction of
all components has to be taken into account. At a given rpm the engine offers a specific torque
value which is multiplied by the efficiency in order to account all losses.
1.7 Road Slope:
Despite the big effect the inclination of the road can have in the performance of the car, it
has been located in the last position. This is because the road is not an effect of the car but an
external one. Additionally the straight line acceleration test are done in a flat surface. Yet it has
been included and it will be discussed in this section.
The slope of the road where the acceleration test will be carried out is of main importance,
since the time needed to reach any velocity will strongly vary. This is due to the weight
distribution of the car when having an inclination. The weight force of the car will distribute
among the x and y axis using simple trigonometric rules. This leads to a reduction of the vertical
load but on the contrary to an increase in the longitudinal force. This distribution has a huge effect
on the total time needed to reach a velocity, depending on the sign of the angle. If a positive angle
is set then the car will have to drive up a hill. This contributes to a reduction of the final
longitudinal force, so a smaller force pushing the car forward is applied. On the other hand if the
road has a negative angle, the car will roll down a hill, so the longitudinal force will increase
ending with a higher force in x direction. In the free body diagram the situation for a positive
angle can be observed.
Plot 5: Time vs acceleration with and without gear shift duration.

8. 7
The below presented formulas describe the situation in x and y axis for both situations.
For a positive angle (driving up a hill)
In X-axis: Ʃ𝐹𝑥 = 𝐹𝑙𝑜𝑛𝑔 − 𝑀 ∗ 𝑔 ∗ sin(𝛼)
In Y-axis: 𝐹𝑦 = 𝑀 ∗ 𝑔 ∗ cos(𝛼)
Similarly for a negative angle (driving down the hill)
In the X-axis: Ʃ𝐹𝑥 = 𝐹𝑙𝑜𝑛𝑔 + 𝑀 ∗ 𝑔 ∗ sin(𝛼)
In the Y-axis: 𝐹𝑦 = 𝑀 ∗ 𝑔 ∗ cos(𝛼)
1.8 Formula Student Question
‘’ A formula student team is trying to improve its performance in the acceleration event where
the quickest car over 100m wins. Some team members have suggested sitting the driver on two
cushions to raise the centre of gravity. Other team members say the opposite and the driver should
be strapped even lower for this event than normal racing. What pressures act in favour and against
each proposal and which should they follow? ’’
The purpose is to change the height of the centre of gravity. This modification has a direct
effect in the weight transfer of the car that occurs while accelerating. From the equation for the
weight transfer it can be seen that the higher the centre of gravity is the more weight transfer will
occur. In order to be able to choose one of the suggestions, the type of car has to be first defined.
The results will not be the same for a rear wheel drive than a four wheel drive.
𝑊𝑡 =
ℎ ∙ 𝑚 ∙ 𝑔
𝐿
Rear wheel drive:
If the formula student team is competing with a rear wheel drive, then the first suggestion
would improve the car performance. Due to the higher weight distribution, more vertical load will
be resting on the rear wheels. As it can be seen in the section where the Pacejka Model is
explained, the higher the vertical load is the more longitudinal force the tires can support. As a
result the car will be able to accelerate faster due to the greater weight transfer.
Figure 2: Free body diagram during the upwards direction.

9. 8
All-wheel drive:
On the other hand if the formula team has an all-wheel drive car, then lowering the centre
of gravity would be the appropriate choice to minimize the weight transfer. The difference is that
in this case the engine power is provided to all four wheels. As explained for the rear wheel drive,
the vertical load at the wheels due to the weight transfer will change. The front wheels will lose
vertical load, while the rear ones will win. However the losses at the front are much greater than
the gain at the rear so at the end the car will lose longitudinal force available at the tires,
consequently the car will accelerate slower having a higher centre of gravity.

10. 9
2 Suspension Optimization
2.1 Introduction
In this section the aim is to optimize the suspension of a vehicle. In a suspension not only
the kinematics is of importance but the dynamics behind too. That is why an analysis of a moving
car has to be studied, so that the best setting can be found. The suspension can be tuned in different
ways, depending on the objective of the car. During this chapter the suspension of an Audi R8
will be optimezed.
The procedure requires:
1. to calculate the centre of gravity
2. Create the model in Adams
3. To optimize the suspension
4. Analyse the results
Calculating the position of the centre of gravity (CoG):
To know where the CoG is located, the lengths ‘a’ and ‘b’ have to be first calculated.
Knowing the weight distribution in each axle and the wheelbase length this is a simple task:
𝑎 = 2650𝑚𝑚 ∗ 0.56 = 1484𝑚𝑚 ; 𝑏 = 2650𝑚𝑚 ∗ 0.44 = 1166𝑚𝑚
2.2 Adams Setup
Total Mass 𝑚 𝑇𝑜𝑡 1650 kg
Sprung mass 1485 kg
Unsprung mass 165 kg
Weight distribution at the front axle 44 %
Weight at the front axle 𝑚 𝑓 726 kg
Weight distribution at the rear axle 56 %
Weight at the rear axle 𝑚 𝑟 924 kg
Table 1: Audi R8 properties.
Figure 1: Audi R8 specifications.
Figure 2: 4DoF Adams model.

11. 10
In order to create the above model in ADAMs the following steps have to be followed:
1. The uprights and the surface where the road is in contact with the tyres are first created
and the mass of each upright has to be defined by the user (75kg at the front and 90kg at
the rear).
2. Then the body of the car is introduced, setting its sprung mass and the inertia in its
corresponding axis.
3. All bodies are joined together with springs and the needed joints. For the springs
connecting the tyre with the upright no damper has been used. On the other hand the
suspension (spring between the body and the upright) has a spring and damper stiffness.
4. For the road input a spline function has been introduced in the impose motion at the road
box. Here a delay due to the distance between the front and rear axle has been used. For
this calculation it has been assumed that the car is moving at a constant speed of 100km/h,
which implies a delay of 0.09s.
𝑡 𝑑𝑒𝑙𝑎𝑦 =
𝑙
𝑣𝑐𝑎𝑟
=
2.650𝑚
100
𝑘𝑚
ℎ
∗
1000
3600
= 0.095𝑠
2.3 Optimization Procedure
When optimizing a suspension, diverse parameters can be considered depending on its goal.
Furthermore plenty of different methods can be used to reach the desired final aim. In this section
the applied method will be discussed with its corresponding results.
The optimization will start by calculating the optimized theoretical values for the spring
coefficients. The ideal case would be to have the smallest value as possible, but unfortunately
they are limited by the driving height at certain cases. For example when driving over a bump the
suspension will shrink considerable and the riding height would remain constant allowing a crush
between the car and the bump. The spring coefficient is dependent on the car frequency which is
defined depending on the type of car to optimize. A normal road car frequency is between 0.8 Hz
and 1.4Hz, if the road car has a higher performance and is considered a sport car then the
frequency will be a bit higher. Racing cars are normally between 2 and 2.5 Hz, while F1 cars can
even reach frequencies of about 4 Hz. In this case a smaller frequency at the front has been used,
due to the car properties: 𝑓𝑓 = 1.5 𝐻𝑧 𝑎𝑛𝑑 𝑓𝑟 = 1.7 𝐻𝑧.
𝑓 =
1
2𝜋
∗ √
𝑘
𝑚
→ 𝑘 = (𝑓 ∗ 2𝜋)2
∗ 𝑚
Optimized Theoretical Spring Coefficients:
Front Spring Stiffness: 𝑘 𝑓 = (𝑓𝑓 ∗ 2𝜋)2
∗
𝑚 𝑓
1000
= (1.5 ∗ 2𝜋)2
∗
726𝑘𝑔
1000
= 64.488
𝑁
𝑚𝑚
Rear Spring Stiffness: 𝑘 𝑟 = (𝑓𝑟 ∗ 2𝜋)2
∗
𝑚 𝑟
1000
= (1.7 ∗ 2𝜋)2
∗
924𝑘𝑔
1000
= 105.422
𝑁
𝑚𝑚
With the help of the spring coefficients the critical damping can be calculated, which will
then allow the estimation of the optimized theoretical damping coefficients. The critical damping
defines the value range for the underdamped, critically damped and over damped behaviours,
which will be discussed in the next topic into more detail, using: 𝐶𝑐 = 2 ∗ √𝑘 ∗ 𝑚
Front critical damping: 𝐶𝑐 𝑓
= 2 ∗ √𝑘 𝑓 ∗
𝑚 𝑓
1000
= 2 ∗ √64.488
𝑁
𝑚𝑚
∗
726𝑘𝑔
1000
= 13.685
𝑁
𝑚𝑠−1
Rear critical damping: 𝐶𝑐 𝑟
= 2 ∗ √𝑘 𝑟 ∗
𝑚 𝑟
1000
= 2 ∗ √105.422
𝑁
𝑚𝑚
∗
924𝑘𝑔
1000
= 19.739
𝑁
𝑚𝑠−1
Now a theoretical estimation of the optimized value can be determined. The value 0.4 has
been chosen since it is roughly the regime where the optimized value usually sits.

12. 11
Front theoretical damping value: 𝐶𝑓 𝑡ℎ
= 𝐶𝑐 𝑓
∗ 0.4 = 13.685
𝑁
𝑚𝑠−1 ∗ 0.4 = 5.474
𝑁
𝑚𝑠−1
Rear theoretical damping value: 𝐶𝑟 𝑡ℎ
= 𝐶𝑐 𝑟
∗ 0.4 = 19.739
𝑁
𝑚𝑠−1 ∗ 0.4 = 7.896
𝑁
𝑚𝑠−1
From this point there is enough values known to start the optimization using the software
Adams. During the entire simulations the spring stiffness’s will remain constant and the best
fitting damper coefficient will be found. During the optimization four different parameters will
be considered:
 Rear contact patch.
 Heave acceleration.
 Front contact patch.
 Pitch acceleration.
The main reasons behind this selection is the importance of grip between the tyre and the
asphalt. Therefore the minimization of the force variation at the front and rear contact patch is
essential. The remaining two options are more directed towards the drivers comfort, trying to
minimize the most uncomfortable movements.
Two different approaches with the same purpose have been used, an automatic optimization
and a manual one, being a more arduous method.
During the manual optimization the aim is to find the minimum RMS of the desired
parameter. The rear contact patch case will be discussed. The carried method is an iterative
process, due to the strong connection between the front and rear damping stiffness’s.
1. The procedure starts with the simulation using the calculated theoretical values and
getting from the post processor the RMS for the rear contact patch (RMS=583.3968)(See
Plot 1).
2. The rear damping stiffness is replaced by a created design variable which goes from 0.1
to 30, since a margin to the critical damping value has been used (see Table 2). From the
plot in the post processor the minimum RMS of the rear contact patch is read and the
damping stiffness where it is located (𝑅𝑀𝑆 𝑚𝑖𝑛 = 583.2705; 𝐶𝑟1
= 8.3483) (Plot 2).
3. The new rear damping coefficient is used, while a design variable at the front is inserted.
After running the simulation, a new plot will be created, where the RMS for the rear
contact force can be read (𝑅𝑀𝑆 𝑚𝑖𝑛 = 581.9456; 𝐶𝑓1
= 4.2895) (Plot 3).
4. Again the design variable for the rear has to be set and the new front damping stiffness is
typed in. From the new plot, the minimum RMS is estimated (𝑅𝑀𝑆 𝑚𝑖𝑛 = 581.9456;
𝐶𝑟2
= 8.3483) being the minimum value for the rear contact patch (Plot 4).
Table 2: Design variable range and critical damping coefficients.
Plot 1: Rear patch with theoretical values
(RMS=583.3968).
Plot 2: Cf theoretical and Cr as a design variable
(𝐑𝐌𝐒 𝐦𝐢𝐧 = 𝟓𝟖𝟑. 𝟐𝟕𝟎𝟓; 𝐂 𝐫 𝟏
= 𝟖. 𝟑𝟒𝟖𝟑).
Front Damper Design Variable 0.1-20 Ccf 13,685
Rear Damper Design Variable 0.1-30 Ccr 19,739
Design Variable Values

13. 12
As it can be seen the minimum RMS from step three is identical to the last, so the minimum
value has been reached. Beside the rear damping stiffness 𝐶𝑟1
𝑎𝑛𝑑 𝐶𝑟2
are equal, meaning that the
loop has been completed.
To simplify the work the optimization feature in Adams has been used, where both, front
and rear design variables for the damping coefficients can be done together. The software will
run the needed iterations until a minimum value has been obtained. With the help of three different
plots, the RMS and the front and rear damping coefficients can be read. Once more the rear contact
patch is the one of interest, since the obtained results can be validated with the manual method to
proof that both options get the same behaviours. In this case the minimum RMS for the rear
contact patch is at 3 iterations with and RMS of 581.9202, giving coefficients of 𝐶𝑓 =
4.1786 𝑎𝑛𝑑 𝐶𝑟 = 8.4696. The difference between both methods is very small and therefore the
remaining parameters will be analysed using the automatic option.
The minimum RMS and the best fitting damping coefficient for each of the parameters can be
observed in the table below:
Plot 3: Cf as a design variable and Cr=8.3483
(𝑹𝑴𝑺 𝒎𝒊𝒏 = 𝟓𝟖𝟏. 𝟗𝟒𝟓𝟔; 𝑪 𝒇 𝟏
= 𝟒. 𝟐𝟖𝟗𝟓).
Plot 4: Cf=4.2895 and Cr as a design variable
(𝑹𝑴𝑺 𝒎𝒊𝒏 = 𝟓𝟖𝟏. 𝟗𝟒𝟓𝟔; 𝑪 𝒓 𝟐
= 𝟖. 𝟑𝟒𝟖𝟑).
Plot 5: Iteration vs RMS for rear contact patch.
Plot 6: Iteration vs Cf for rear contact patch. Plot 7: Iteration vs Cr for rear contact patch.

14. 13
Table 3: Minimum RMS obtained with the optimization feature for all four parameters.
If the optimization would be based on single parameters these would be enough, since the
minimum RMS with its Cf and Cr values have been already found. However the aim is to
minimize the values for all four variables at the same time, having the best behaviour as possible.
Hence 16 different configurations (each rear damping stiffness with each front one) will be
simulated and the average RMS for each parameter will be estimated. As an example the first
case will be presented in detail, while the remaining ones will simply show their results.
Case 1: 𝐶𝑓 = 4.1786 𝑎𝑛𝑑 𝐶𝑟 = 8.4696
After running the simulation with this setup, the four parameters need to be analysed in the
post processor, reading the average RMS for each one (see results below).
2.4 Cosh Function and Results
Additionally all 16 cases need to be compared to each other, so that the best configuration
contemplating the rear and front contact patch, heave acceleration and pitch acceleration can be
determined. For it a cosh function will be introduced. The aim of such a function is to classify the
Cf=4,1786
Cr=8,4696
Rear Patch 581,9202
Front Patch 336,7607
Heave Acc 511,2261
Pitch Acc 24,5895
RMS
Case 1
Plot 8: Case 1 rear contact patch. Plot 9: Case 1 front contact patch.
Plot 10: Case 1 heave acceleration. Plot 11: Case 1 pitch acceleration.
Table 4: Case 1 Adams results.
Number of Iterations Min RMS Cf Cr
Rear Contact Patch 3 581,9202 4,1786 8,4696
Front Contact Patch 5 331,1292 5,5676 9,8983
Heave Acceleration 1 476,9012 8,6679 11,259
Pitch Acceleration 4 20,188 0,8253 8,9002

15. 14
parameters in a ranking, giving each a percentage proportional to the weight of importance in the
final solution.
During this optimization, the most important parameter is the rear patch with 50%, since
we are dealing with an Audi R8 with more weight at the rear. This would reduce the force
variation at the rear tyre, having more grip and consequently lower performance losses. The front
patch is the second most important, reducing the contact force variation between the tyre and the
road too. The two following parameters are concerning the drivers comfort having the heave
acceleration at the third position. The main
difference with the pitch acceleration is that during
heave the internal organs of the human body are
moving up and down, causing the drive to feel
uncomfortable after a while. The human body
displacement during pitch is a more natural
gesture, since humans use their heads to node.
In order to be able to successfully complete the cosh function the obtained RMS will have
to be first normalized. This step is important when comparing different cases, since the aim is to
reach a value of the same magnitude to make them comparable to each other. To normalize the
average RMS the appropriate minimum RMS for each parameter will be used. Finally the cosh
function can be implemented, where the sum of all parameters multiplied by its percentage is
fulfilled (see Table below).
One finished applying this procedure for all 16 cases, the final value can be directly
compared. The aim is to minimize the RMS value, so the case with the lowest cosh function value
is the best configuration for the Audi R8 suspension setup. To make easier the localization of the
desires configurations, the values have been sorted out in a list from lowest to highest and in a
table with the damping stiffness configuration.
In the tables above the three best and worst configurations have
been highlighted. From the point of view of minimizing the rear and
front contact patch, the heave acceleration and the pitch acceleration,
case 14 has shown the best results. The setup of the car for this
particular case would be:
 𝐾𝑓 = 64.488
 𝐾𝑟 = 105.422
 𝐶𝑓 = 5.5676
 𝐶𝑟 = 8.9002
8,4696 9,8983 11,259 8,9002
4,1786 1,0268 1,028015 1,035033 1,026323
5,5676 1,023133 1,02512 1,032716 1,022911
8,6679 1,044288 1,047092 1,05557 1,044297
0,8253 1,199351 1,200411 1,209899 1,198429
Cf
CrDamping
Configuration
Cf Rear Patch 581,9202 581,9202 1 50,00%
4,1786 Front Patch 336,7607 331,1292 1,017006957 30,00%
Cr Heave Acc 511,2261 476,9012 1,071974866 15,00%
8,4696 Pitch Acc 24,5895 20,188 1,21802556 5,00%
RMS min
Damping
Configuration
Case 1
RMS
Normalized RMS
(RMS/RMSmin)
Porcentage
RMS Cosh
Function
1,026799595
Rear Contact Patch 50%
Front Contact Patch 30%
Heave Acceleration 15%
Pitch Acceleration 5%
Total 100%
Cosh Function
Table 5: Percentages used in the cosh function.
Table 6: Cosh function calculation and result for case1.
Table 7: Lowest to highest
cosh function results.
Table 8: Cosh function results for each configuration possible (16 cases).
Table 8: Case 14 results.
Cf RearPatch 584,1046 581,9202 1,003753779 50,00%
5,5676 Front Patch 331,5535 331,1292 1,001281373 30,00%
Cr Heave Acc 492,6065 476,9012 1,032931978 15,00%
8,9002 Pitch Acc 26,5311 20,188 1,314201506 5,00%
Case 14 1,022911174
1 1,022911174 Case 14
2 1,023133022 Case 2
3 1,025119572 Case 6
4 1,026322838 Case 13
5 1,026799595 Case 1
6 1,028014589 Case 5
7 1,032716017 Case 10
8 1,035033013 Case 9
9 1,044287972 Case 3
10 1,044297092 Case 15
11 1,047091525 Case 7
12 1,055569752 Case 11
13 1,198428789 Case 16
14 1,199351433 Case 4
15 1,200410974 Case 8
16 1,209899478 Case 12
List from lowest to higest Cosh
Function results (Case 1-16)

16. 15
Looking into the obtained results for case 14, having the lowest cosh function RMS at the
first combined analysis, the following can be stated:
The parameter with the biggest weight is the rear contact patch, which has a Cf=4.8 and
Cr=8.4. As in the cosh function three more parameters are considered, they will directly affect
these damping coefficients.
Concerning the rear damping stiffness all three remaining parameters have a higher rear
damping than rear contact patch, therefore in the cosh function it is expected an increase in Cr
coefficient. The two parameters with higher weight are: the rear contact patch with 50%, having
Cr=8.4, and the front contact patch with 30%, having Cr=9.8. The corresponding damping
coefficient at which the minimum RMS have been found at case 14 fits with the Cr for the pitch
acceleration ; Cr=8.9.
The front damping coefficient for case 14 is Cf=5.5 the value for the front contact patch
min RMS. when the rear contact patch min RMS has a Cf of 4.8 but the heave acceleration min
RMS require Cf=8.6. However the pitch acceleration has a much lower one (Cf=0.8). The analysis
shows that when giving even a small weight to the front contact patch, the front damping
coefficient needs to be increased to find the best configuration.
To improve the optimization further, it is convenient to consider that the main difference
in front damping analysis with respect to the rear damping analysis is that the distance between
Cf values (4.1 and 5.5) is very large, while in the rear damping value the differences between Cr
values are much more progressive.
In order to have smaller changes an intermediate value for
the front damping (Cf=4.8) has been introduced to find out, how
this will affect the final results. So four extra cases (17-20) have
been added to the previous 16 cases. This can be seen in the tables
below. With this value the best configuration sits between the two
most important parameters, being case 20. Thanks to this new
intermediate front damping an updated best configuration has
been found. In the plot below the difference in the cosh function
general optimized results can be seen.
1 1,022801569 Case20
2 1,022911174 Case14
3 1,023133022 Case2
4 1,023136111 Case17
5 1,024773001 Case18
6 1,025119572 Case6
7 1,026322838 Case13
8 1,026799595 Case1
9 1,028014589 Case5
10 1,032087891 Case19
11 1,032716017 Case10
12 1,035033013 Case9
13 1,044287972 Case3
14 1,044297092 Case15
15 1,047091525 Case7
16 1,055569752 Case11
17 1,198428789 Case16
18 1,199351433 Case4
19 1,200410974 Case8
20 1,209899478 Case12
List from lowest to higest Cosh
Function results
Table 9: Lowest to highest cosh
function results case (1-20).
Table 10: Cosh function results with each configuration possible (case 1-20).
Plot 12: Optimization results with and without intermediate Cf.
8,4696 9,8983 11,259 8,9002
4,1786 1,026799595 1,028014589 1,035033013 1,026322838
4,8731 1,023136111 1,024773001 1,032087891 1,022801569
5,5676 1,023133022 1,025119572 1,032716017 1,022911174
8,6679 1,044287972 1,047091525 1,055569752 1,044297092
0,8253 1,199351433 1,200410974 1,209899478 1,198428789
Damping Configuration
Cr
Cf

17. 16
2.5 Suspension Travel
Now that the optimization has finalized, the suspension travel for the best configuration,
being case 20. The aim is to verify if the chassis of the car can get in contact with the wheels at
any stage of the road. An Audi R8 has approximately a 20mm gap between the body and the
wheel. To measure the maximum compression of the rear and front suspension a measure function
has been created in Adams to calculate its deformation, which cannot exceed 20mm to avoid
contact.
As it can be seen in the Table showing
the results, the maximum suspension travel
is 14.27 at the front. Satisfying this clearance
necessity, the suspension optimization has
concluded
2.6 Conclusion
It has implemented an optimization process that includes the manual and automatic
methods to verify the results and a combined analysis of the 4 parameters to find the case with
the lowest RMS using a cosh function. This results have been further optimized with a final
adjustment to avoid too wide ranges (intermediate Cf value).
As a conclusion it is important to point out that the theoretical approximation for the
damping coefficients are quite accurate, where the difference between the Adams and the
theoretical values are of 0.0936 at the front damping and 1.0042 at the rear.
The additional cases considering a more progressive intervals with an intermediate Cf value
confirms that the new value is the best option (case 20), improving the optimization process,
providing results that agrees with expected, always keeping Cr higher than Cf, as it should be due
to the weight distribution
This means that the results obtained considering the 4 parameters tend to intermediate
values, depending on the weight of each parameter, to provide Cf and Cr values for a better overall
performance.
Cf Rear Patch 582,8248 581,9202 1,001554509 50,00%
4,8731 Front Patch 332,5625 331,1292 1,004328522 30,00%
Cr Heave Acc 500,152 476,9012 1,048753914 15,00%
8,9002 Pitch Acc 25,6035 20,188 1,268253418 5,00%
Case 20 1,022801569
Plot 13: Front suspension travel. Plot 14: Rear suspension travel.
Table 9: Front and rear suspension travel results.
14,27
13,26
Max
Cr=8,9002
Cf=4,8731
Case 20
Suspension Travel For The Best Case (1-20)
Front Suspension Travel
Rear Suspension Travel
Table 11: Results best case (case 20).

18. 17
3 Derivative Analysis
3.1 Introduction
In this chapter the derivative analysis will be discussed. It is the part of vehicle dynamics
where the behaviour of a car during cornering is investigated. This is a very important aspect to
consider during the construction of a car to get the maximum performance possible during
cornering.
A spreadsheet with the derivative analysis and a bicycle model in Adams will be used to
validate the obtained results. The effects of the front and rear tire, the position of the centre of
gravity and the steering angle which define the vehicles path and behaviour will be analysed. This
chapter will cover the following topics
 Spreadsheet calculations
 Bicycle model in Adams
 Introducing Pacejka tire model in Adams
 Curvature and yaw rate responses
 Velocity estimation
While a car is driving around a corner, different variables are interacting in the system. The
car is moving with a velocity which is tangential to the line from the centre of gravity of the
curvature. This can be split into the x and y components. At the same time each wheel will have
its own velocity which can be divided into x and y component too. Besides the car will be rotating
around its Centre of Gravity, having a yaw angle and a yaw rate. During a corner the vehicle will
suffer a centrifugal force pointing outward, inducing a reaction force at the tires in the opposite
direction. This reaction force is dependent on the tire slip angles. The side force will increase with
increasing slip angle.
The derivative analysis is based on three stages:
 Transient turn entry
 Steady state cornering
 Transient turn exit
On one hand during both transient stages, the yaw rate and the lateral velocity are changing
with time. During the entry it will build up from zero, while at the exit everything will return to
zero for the straight line. On the other hand at the steady state phase the slip angles, the yaw rate
and the lateral velocity are constant, keeping the car along an unvarying curvature radius R. Here
the Force and Moment equilibrium are considered.
Assumptions:
During the derivative analysis and the bicycle model in Adams the following assumptions
have been made:
 No lateral or longitudinal weight transfer
 No rolling or pitch motions
 Linear tires behaviour
 Constant forward velocity (In Adams with a speed controller)
 No suspension or aerodynamics effects included
3.2 Theoretical Background
The entire study has been fulfilled for three different cases: underdamped, critically
damped and over damped, depending on the damper coefficient used. These three cases represent

19. 18
the different behaviours a car can show being understeering, neutral and over-steering
respectively. To get a more thorough understanding of the presented cases, a brief comparison at
the steady state stage for all cases will be presented.
 Neutral steering car:
In a neutral steering car the centre of gravity is located at the middle of its wheelbase. The
tire size and pressure and the cornering stiffnesses at the front and rear will be assumed to be
equal, consequently the side forces too. The required force and moment at the front tire are
directly proportional to the yaw angle of the body during the establishment of the rear tire. This
will results into a steering angle which is not dependent on velocity nor lateral acceleration, since
it follows the Ackermann steering angle (see formula).
𝛿 𝑛𝑒𝑢𝑡𝑟𝑎𝑙 = 𝛿𝐴𝑐𝑘𝑒𝑟𝑚𝑎𝑛𝑛 =
𝑙
𝑅
In a path with constant radius R, as lateral force increases the slip angle at both tires will
increase too. Due to all the assumptions used this would lead to a same slip angle at the front and
at the rear: 𝛼 𝑓 = 𝛼 𝑟.
 Understeer car:
A car with its centre of gravity closer to the front axle is considered an understeering
vehicle. This implies that the distance ‘a’ from the centre of gravity is shorter than the distance to
the rear axle (b). Therefore more weight will be resting in the front tires, having a larger slip angle
at the front tire (𝛼 𝑓 > 𝛼 𝑟).
In order to be able to stay at the same curvature angle than in a neutral steering car, the
steering angle should be increase as an effect of having a larger slip angle at the front tire.
𝛿 𝑢𝑛𝑑𝑒𝑟𝑠𝑡𝑒𝑒𝑟 = 𝛿𝐴𝑐𝑘𝑒𝑟𝑚𝑎𝑛𝑛 + (−𝛼 𝑓 + 𝛼 𝑟) =
𝑙
𝑅
+ +(−𝛼 𝑓 + 𝛼 𝑟)
 Over-steering car:
On the contrary an over steering vehicle has more weight in the rear axle, so the distance
from the centre of gravity to rear axle is now shorter. Hence the rear slip angle will predominate
where 𝛼 𝑓 < 𝛼 𝑟. As a result of this difference the car will suffer a tail swing behaviour which leads
to an increase in the front slip angle. To compensate the extra slip angle the steer input will need
to be reduced. The over steer car is defined using the same formula as in the understeering car.
3.3 Results Spreadsheet and Bicycle Model
3.3.1 Spreadsheet
Now that the basic concepts have been introduced, the procedure used in the spreadsheet
will be discussed for the underdamped case. The remaining two case follow the same procedure
but their formulas need to be adjusted for its steering behaviour. This has been done for two
velocities: 40 and 80 mph.
The aim of the spreadsheet is to calculate the yaw rate of a vehicle at a constant velocity
with a step steer input of 𝛿 = 0.043633. The exponential term will get smaller with time and
the sinusoidal term has an offset from the input motion by a phase angle 𝜙.The formula below
has been used to calculate the understeering yaw rate.
𝑟 = 𝑋 ∗ 𝑒−𝜍∗𝑤 𝑛∗𝑡
∗ sin(𝑤 𝑑 ∗ 𝑡 + 𝜙) +
𝑐2
𝐾
∗ 𝛿0

20. 19
Where: 𝑤 𝑛 = √
𝐾
𝐼
; 𝐶 = 𝑁𝑟 +
𝐼∗𝑌 𝛽
𝑚∗𝑣
𝜍 =
𝐶
𝐶 𝑐
; 𝑤 𝑑 = √1 − 𝜍2 ∗ 𝑤 𝑛
The initial condition at t=0 no yaw rate will appear (r=0) and the initial constant X can be
estimated:
𝑋 =
−𝐶2 ∗ 𝛿
𝐾 ∗ sin(𝜙)
By differentiating the yaw rate equation and eliminating the steering input delta, the phase
shift can be calculated:
𝜙 = atan(
𝐶2 ∗ 𝑤 𝑑 ∗ 𝐼
𝑐𝑓 ∗ 𝑎 ∗ 𝐾 + 𝜍 ∗ 𝑤 𝑛 ∗ 𝐶2 ∗ 𝐼
)
Finally the yaw rate will be normalized, so that all three cases can be directly compared to
each other. To do so, the yaw rate needs to be divided by the yaw rate at t equals infinity
(𝑟(𝑡 = ∞)).
𝑟(𝑡 = ∞) =
𝐶2 ∗ 𝛿
𝑣
The distribution of the spreadsheet can be seen in the Figure below:
After having fulfilled all the calculation steps for the underdamped, the critically damped
and the over damped case, the normalized solutions can be compared in a single plot.
Figure 1: Spreadsheet layout and results.

21. 20
From the plot it can be seen that all three behaviours tend to the value one, which
corresponds to the normalized yaw rate. On the x-axis the time in seconds is plotted. The three
behaviours will now be explained. The underdamped case, which corresponds to the blue line,
gets to the desired value almost at the same time than the critically damped case. However the
behaviour of the critically damped is much smoother than the underdamped. Additionally the
critically case does not have an oscillating behaviour. Looking into the green line (Over damped)
it can be seen that the car needs a longer period of time to reach the same value, emphasizing that
the critically damped case is the best choice. From the plot the following conclusion can be
withdrawn: the higher the damping coefficient are the longer it takes to reach the demanded yaw
rate. Nevertheless, if the damping ratio of the underdamped case is reduced the required yaw rate
will be reached earlier, but a stronger non desirable oscillating motion will be seen, which
overshoots the value.
3.3.2 Adams Model
With the help of Adams a bicycle model will be created in order to validate the above
obtained yaw rates. The location of the centre of gravity will vary for each case considered above.
The steps followed have been:
1. Generate all the bodies and required joints to attach the body and the wheels.
2. Create multiple design variable for the velocity of the bicycle, the x and y component of
the tire velocities, the front and rear slip angles.
3. Create the required forces: a speed controller force to fulfil the constant velocity
assumption and the corresponding lateral forces at each wheel.
4. Introduce a step input to the joint at the front wheel with the same steering angle than in
the spreadsheet.
From the Adams results the yaw velocity needs to be imported to make the comparison
with the spreadsheet viable. In the plots below both Adams and the spreadsheet results for all
three cases can be seen for 40 and 80 mph.
Plot 1: Spreadsheet results for under-, critically and over damped.

22. 21
3.3.3 Comparison
From Both plots it can be clearly seen that all the values obtained for the normalized yaw
rate in the spreadsheet calculations and the Adams models are identically the same. It is even hard
to distinguish between both, just the over damped case at 80 mph is slightly different.
Comparing the results for 40 and 80 mph a big difference in behaviour can be observed.
The underdamped case at 80 is much more aggressive and reaches a maximum at approximately
1.3, while at a lower speed it just goes slightly above 1. This induces a bigger oscillating motion
needing more time to settle down at the desired yaw rate. Contemplating the critically damped
case, it can be seen that at higher speed the yaw rate needs more time to reach its final value.
However its behaviour remains the same (smooth behaviour). The over damped case is the one
which most varies between 40 and 80 mph. At 80 mph the yaw rate was not able to reach the
normalized behaviour after 2 seconds, which clearly shows that is the case which most time needs
to get to the value, stating clear that the higher the damping coefficient, the more time it needs to
get to the value. Nonetheless the conclusion still remain untouched, having the critically damped
case as the best option for racing cars.
Plot 2: Spreadsheet and Adams comparison plot at 40 mph.
Plot 3: Spreadsheet and Adams comparison graph at 80mph.

23. 22
The figure above corresponds to the underdamped case, where the front slip angle and the
rear slip angle have been plotted in a single graph. It clearly shows that the front slip angle
predominates over the rear slip angle. This represents the above theoretical background section
where it has been stated that 𝛼 𝑓 > 𝛼 𝑟. This is due to the specific weight distribution, since more
weight is applied at the front axle.
3.3.4 Pacejka Tire Model in Adams
As mentioned before, until now the tire where considered linear. To make the model more
realistic and upgrade the model in Adams, the Pacejka tire model has been introduced in the
bicycle model.
The aim is to see the difference in behaviour using the exposed assumption or defining a
tire model. For this purpose, the Pacejka tire model for cornering has been used in the front and
rear tire as a function of the slip angle design variable in Adams. The procedure used can be seen
below:
1. Calculate the vertical load: It is important to know the weight distribution at the front and
rear axles, since the magic formula will be introduced for both wheels.
2. Estimate the needed parameters to input in the magic formula for cornering.
3. Replace the front and rear tire forces used in the first analysis for the magic formula using
the equation:
Underdamped case:
At the front tire:
(6355.899*SIN(1.4*ATAN((0.10398*(VARVAL(.MODEL_1.FRONT_SLIP_ANGLE)*(180/P
I)))+2*(0.10398*(VARVAL(.MODEL_1.FRONT_SLIP_ANGLE)*(180/PI))-
ATAN(0.10398*(VARVAL(.MODEL_1.FRONT_SLIP_ANGLE)*(180/PI)))))))
Rear tire force:
(6355.899*SIN(1.4*ATAN((0.105392*(VARVAL(.MODEL_1.REAR_SLIP_ANGLE)*(180/P
I)))+2*(0.105392*(VARVAL(.MODEL_1.REAR_SLIP_ANGLE)*(180/PI))-
ATAN(0.105392*(VARVAL(.MODEL_1.REAR_SLIP_ANGLE)*(180/PI)))))))
The difference between using a tire model or the assumption can be seen in the plot below.
Plot 4: Front slip angle vs rear slip angle for underdamped case.

24. 23
As expected, the Pacejka yaw rate is lower. This is mainly due to the smaller force
following the magic formula, which leads to a larger curvature radius R. This can be clearly seen
by looking into the formula 𝑟 =
𝑣
𝑅
. If the radius of the curvature is greater than the yaw rate is
going to be smaller, since the velocity remains constant with the installed speed controller.
3.4 Yaw Rate and Curvature Response
In a vehicle at cornering stage different linear steady state control responses can be analysed:
 Curvature response
 Yaw velocity response
 Lateral acceleration response
 Sideslip angle response
In this case only the curvature and the yaw rate responses will be considered. Once again the
spreadsheet results will be used to validate the obtained results from the Adams model. Since the
under-, neutral- and over-steer cases have been presented before, it is known that each will have
a different curvature and yaw rate response.
The curvature response defines the needed steer wheel angle to enter a specific corner
radius. The equations used in the spreadsheet is:
1
𝑅⁄
𝛿
=
1
𝑙⁄
1 + 𝐾 ∗ 𝑣2
The yaw rate response dictates the amount of yawing moment it occurs at a concrete
steering angle input. In order to obtain the yaw rate response, the curvature response needs to be
multiplied by the velocity.
𝑟
𝛿
=
𝑣
𝑙⁄
1 + 𝐾 ∗ 𝑣2
From both equation it can be seen that the responses will vary with velocity and where K
is the stability factor which can be calculated using:
Plot 5: Adams results with and without Pacejka model.

25. 24
𝐾 =
𝑚 ∗ 𝑁𝛽
𝑌𝛿 ∗ 𝑁𝛽 − 𝑌𝛽 ∗ 𝑁𝛿
∗
1
𝑙
Two different values can be assigned to the stability factor, depending on the steer case.
This parameter is dependent on the derivative terms, which makes it independent of speed.
 K=0: If the curvature and yaw responses formulas are looked, it can be stated that the
responses will become those of a neutral steering car, since the K term will be equal to
zero, ending with:
1
𝑅⁄
𝛿
=
1
𝑙
;
𝑟
𝛿
=
𝑣
𝑙
If the curvature response were to be solved towards the steering wheel angle, it can be seen
that the Ackermann steering is obtained having 𝛿 =
𝑙
𝑅
, which perfectly agrees with the
explanation in the theoretical background section for neutral steer cars.
 If K≠0: In this case the curvature and yaw velocity response will depend on the assigned
sign of K, which is determined by the denominator of the stability factor equation. Since
𝑌𝛽 is always negative, the directional stiffness 𝑁𝛽 will be the variable which determines
the sign of K.
 If K is positive the denominator of the yaw rate response and the curvature response
will get smaller. As the velocity of the car increases the yaw rate and curvature
response will tend to decrease. This case defines the behaviour of an understeer car.
 If K is negative the denominator will get smaller and smaller when increasing
velocity. The effect on the curvature and the yaw rate response is that they will both
get infinitely large, describing the behaviour of an oversteer car.
Relaying on the sign of the stability factor (understeer for positive K and over-steer for negative
K) a velocity defining its behaviour can be estimated.
For an understeer car the characteristic speed is of interest. This velocity dictates the point
where the maximum response is achieved. The characteristic speed is a function of the derivative
terms or can be simplified by using the stability factor. This makes the characteristic speed
independent of velocity.
𝑣 𝑐ℎ𝑎𝑟 = √
−𝑁𝛽 ∗ 𝑉 ∗ 𝑌𝑅 + 𝑌𝛽 ∗ 𝑉 ∗ 𝑁𝑟
𝑁𝛽 ∗ 𝑚
= √
1
𝐾
= 28.0306
On the other hand in the over damped case the response value will tend to infinity. At this
point the critical speed can be found. The critical speed can be calculated with the help of the
stability factor too, being speed independent.
𝑣 𝑐𝑟𝑖𝑡 = √
−1
𝐾
= 43.3964
The critical speed in the Adams model can also be estimated, since at the point where the
critical speed is achieved the car will start spinning, as an effect of the responses tending to
infinity.

26. 25
Since the curvature and the yaw rate responses are a function of velocity, the speed
controller has to be replaced by a simple forward force which gives the car a constant acceleration.
The curvature and yaw rate responses cannot be directly exported. Therefore the yaw rate with
respect to the car velocity will be used and then transformed to the needed parameter. To calculate
the curvature response with respect to the yaw rate, the following has been used:
𝐶𝑢𝑟𝑣𝑎𝑡𝑢𝑟𝑒 𝑟𝑒𝑠𝑝𝑜𝑛𝑠𝑒 =
1
𝑅 ∗ 𝛿
=
𝑟
𝑣 ∗ 𝛿
Since 𝑟 =
𝑣
𝑅
and solving towards R we get 𝑅 =
𝑣
𝑟
.
The yaw rate response can be directly calculated using
𝑟
𝛿
.
The following plots have been obtained for the different responses, where the critical speed and
the characteristic speed can be found:
Once again, looking into the under and critically damped cases, the results in the
spreadsheet and in the Adams model match perfectly, while the over damped is slightly different.
However the difference in values at which the over damped case tends is very small (between the
theoretical and the Adams model). The strong influence of the stability factor in the yaw rate and
curvature response can be clearly seen. When having a negative K, the values will tend to infinity
as it can be seen in the green and orange lines for the over steer behaviour. The critically damped
case shows a constant curvature response and a linear yaw rate response, as it can be expected
from the Ackermann steering. Finally in the underdamped case the maxima, where the
characteristic speed is located and how the yaw rate and curvature responses decreases with
increasing velocity can be distinguished. This is mainly caused by the positive stability factor.
3.5 Tangential Speed
During the steady state the slip angle at the front and rear tyres will change with respect to
velocity. If the car is travelling at low speed the front wheel will suffer a small slip angle while
the rear one will have zero slip angle. Now if the car increases its velocity the front slip angle will
increase more and a slip angle at the rear tyre will start to appear. These would result in very small
lateral forces at the front and rear tyres, since they are pointing approximately perpendicular to
the tyre direction. In case the car should increase its velocity even further both slip angles will
increase and their lateral forces respectively. As a result of the increase in speed the yaw angle of
the body at its CoG will decrease, having at a specific point no horizontal velocity. At this point
the vehicle is pointing in the same direction as it is travelling and a tangential direction to the
Plot 6: Curvature response vs car velocity. Plot 7: Yae rate response vs car velocity.

27. 26
circle in which it is rotating will appear. The speed at which this phenomenon occurs is called
tangent velocity. Finally if the speed is increased further, the body yaw angle will become
negative. The lateral forces at the wheels are limited by the slip angle, suffering a decrease in grip
due to tyre saturation.
To calculate the speed at which no horizontal velocity can be found the formula below needs to
be used:
𝑣𝑡𝑎𝑛𝑔 = √
𝑁𝛿 ∗ 𝑌𝑅 − 𝑁𝑅 ∗ 𝑌𝛿
𝑚 ∗ 𝑁𝛿
Due to the different weight distributions, each steer case will have a different tangential velocity.
Two different methods in Adams have been used to verify the results:
 Using the animation feature
After running the simulation, Adams offers the user an animation feature which allows
seeing the physical motion of the bicycle. This method is not the most precise one since the user
has to stop the animation just when the front tracer coincides with the rear tracer. Therefore a
small error can be expected. From the animation feature the time needed to reach this point can
be obtained. Then going into the car velocity plot with respect to time, the velocity at that time
can be directly read from the plot giving the tangential speed.
 The other option is to plot the displacement in x with respect to displacement in y for the
front and rear wheel in the same plot. This has been used to validate the obtained result
in the previous method for the underdamped case, being a more accurate procedure.
However it is very complicated to get the result since a very high zoom is needed to really
appreciate the different trajectories. From this plot the x displacement is known, which
going into the body x displacement, the time needed to get to that position can be read.
Now that the time is known the tangential velocity can be estimated, using the speed vs
time plot.
Figure 1: Bicycle model at tangent speed.

28. 27
In the following table the tangential speeds for the different cases are presented.
It can be seen that the animation method has results very close to the plot method and
verifies that it is also a good way to estimate the tangential speed in a more comfortable way. Yet
the tangential speeds from Adams and the spreadsheet are located in the same range, once more
stating that the theoretical and the Adams results are very close to each other.
Lastly the natural steer point has been calculated in the spreadsheet following the formula below.
𝑁𝑆𝑃 =
𝑑
𝑙
=
𝑎
𝑙
−
𝑁𝛽
𝑌𝛽
∗
1
𝑙
This is the point in which applying a lateral force to the car, the car will displace entirely without
any yaw moment acting.
References:
William F. Milliken and Douglas L. Milliken, Race Car Vehicle Dynamics.
Underdamped 16,28 14,66 14,49
Critically damped 13,229 13,12
Over damped 12,15 12,37
Steer case Spreadsheet
Adams animation
method
Adams plot
method
Plot 8: Front vs rear wheel displacement in x vs y grapgh.
Plot 9: Time vs car velocity to get tangent speed.
Table 1: Tangent speed results in the spreadsheet and Adams.

29. 28
4 Tyre Model
4.1 Introduction
The last chapter of this portfolio covers the topic tires. The aim is to select four tire models
and explain briefly explain them, so that a self-made comparison table can be added at the end
with their advantages and disadvantages with respect to the other tire models.
Tire model are a very important part of vehicle dynamics, since their aim is to represent in
the most accurate way the real behaviour of a tire. For it the required inputs need to be known,
such as tire characteristics (dimension, stiffness, type of tire,…) and the conditions at which the
tire will have to work (slip angle, camber angle, vertical load, longitudinal slip,…). Different
types of model are offered, depending on the needed input data: empirical, semi-empirical and
analytical tire models.
The four tire models which have been selected are:
 The Fiala Model
 The Brush Model
 The Magic Formula
 The MF-SWIFT Model
4.2 The Fiala Model
Author:
Information source: Blundell, J. and Harty, D (2015).The Multibody Systems Systems Approach
to Vehicle Dynamics 1st
and 2nd
Edition.
Model Explanation:
The Fiala model was presented in the year 1954, being considered as an extension of the
original Brush model.It is a tyre model that has been adapted as a standard one supplied with the
MSC ADAMS program. Although it is limited in capability, the huge advantage is that it only
needs 10 input parameters which are directly related to the physical properties of the tyre.
The input parameters are the used to generate the output parameters being: longitudinal
forces, lateral forces and aligning moment.
Input parameter table
𝑅1 The unloaded tyre radius (units- length)
𝑅2 The tyre carcass radius (units- length)
𝑘 𝑧 The tyre radial stiffness (units - force/length)
𝐶𝑠 The longitudinal tyre stiffness. This is the slope at the origin of the braking force 𝐹𝑥 when
plotted against slip ratio (units- force)
𝐶 𝛼 Lateral tyre stiffness due to slip angle. This is the cornering stiffness or the slope at the
origin of the lateral force 𝐹𝑦 when plotted against slip angle a. (units - force I radians)
𝐶 𝛾 Lateral tyre stiffness due to camber angle. This is the cornering stiffness or the slope at the
origin of the lateral force 𝐹𝑦 when plotted against camber angle y (units force I radians)
𝐶𝑟 The rolling resistant moment coefficient which when multiplied by the vertical force 𝐹𝑧
produces the rolling resistance moment 𝑀 𝑦 (units - length)
𝜁 The radial damping ratio. A value of zero indicates no damping and a value of one indicates
critical damping (dimensionless)
𝜇0 The tyre to road coefficient of "static" friction. This is the y intercept on the friction
coefficient versus slip graph
𝜇1 The tyre to road coefficient of "sliding" friction occurring at 100% slip with pure sliding
Table 1: Fiala input parameters.

30. 29
The main function of the input values 𝑅1, 𝑅2, 𝑘 𝑧 and 𝜁 is to define the vertical load resting
in the tyre. It is also important to mention that the Fiala model does not work with camber angle
and the lateral stiffness due to camber 𝐶 𝛾 should be ignored.
Calculation for the lateral force and the aligning moment:
For the lateral force a critical slip angle α* is calculated using:
α∗
= 𝑡𝑎𝑛−1
|
3𝜇𝐹𝑧
𝐶 𝛼
|
Friction coefficient:
Thanks to a linear interpolation the road friction
coefficient can be directly calculated by following the
formula below.
𝜇 = 𝜇0 − 𝑆 𝐿𝛼 ∗ (𝜇0 − 𝜇1)
Application of the tire model
Despite its main advantage, which is its simplicity, the Fiala tyre model main limitations are:
 It is not suitable for combined slip since it cannot represent combined cornering and
breaking or cornering and driving.
 It does not take into account camber.
 The variation of cornering stiffness 𝑪 𝜶at zero slip angle𝜶 under tyre load is not
considered. Therefore, it has no use in ground vehicle modelling.
 Conicity and plysteer are not represented.
Usefulness:
Very simple and fast calculation for simple models. The outputs which can be obtained
with the Fiala model are the longitudinal and lateral forces and the aligning torque.
Figure 1: FSlip vs friction coefficient plot.

31. 30
4.3 The Brush Model
Author: Proposed by Fromm and Julien
Information source: Pacejka, H. B. (2006). Tyre and Vehicle Dynamics 2nd
Edition
Model explanation:
The Brush model represents the tire contact with the road with non-mass elements known
as treads. The treads are able to deform individually in both directions: longitudinal and lateral,
which will result in a force at the contact patch. The model focuses in the interaction between
road and tire, based on the slip and the developed force. The elasticity of these treads represent
the combination of carcass, belt and tread elements of a real tire. Looking into the figure it can be
seen that the model assumes to have a rectangular contact patch. The front part is known as the
leading edge where the tread is always located perpendicular to the road surface. Depending on
the slip acting in the tire the treads will behave differently. If slip is applied to the case, then the
treads will adopt an inclination due to the difference in velocity between the tip and the base.
However the last tread at the trailing
edge will also be always perpendicular
to the road surface. The rolling
resistance forces are neglected in the
model.
Assumptions used in the model:
 Carcass is considered infinitely stiff.
 Parabolic pressure distribution.
 Constant friction coefficient for the longitudinal and the side slip.
Pure longitudinal force: In the longitudinal slip only the acceleration and breaking torque are
considered. The important part to mention here is that the wheel has now a different velocity than
the car speed Vx due to the slip angle. The force that is applied can be calculated depending on
the difference in velocity. Looking into the figure it can be seen that the treads along the contact
patch have different directions. The section from the leading
edge up to point E and F is the adhesion regime, being in charge
of the generating the longitudinal force Fx. The pressure at this
region will increase and the difference in velocity between the
tip and the base is not too big. Therefore the tire has enough
time to build up the longitudinal force. On the contrary the next
section is called the sliding area, where the velocity difference
at both ends of the treads are very large. Hence the pressure at
this point will start to decrease and no longitudinal force will
be generated. Lastly the sliding and relaxing region (G-H
point), where the velocity in speed will increase further,
allowing the last tread at the trailing edge to catch up and sit
perpendicular to the road surface. The parabolic distribution can be clearly seen in the figure due
to its opposite behaviour for the adhesion and the sliding region.
Figure 2: Brush model layout.
Figure 3: Pure longitudinal slip using the
Brush model

32. 31
Pure side slip: The Brush model allows the analysis of a tire during cornering focusing in side
slip. The principal parameter is the slip angle, which is defined by the difference in direction
between the moving direction and the wheel plane direction. This discrepancy generates a lateral
force and an aligning moment. As the slip angle increases, the tread deflection and the side force
gets larger, moving the generated force forward. In the side slip case two region can be distinguish
in the figure. The adhesion region which is parallel to the velocity vector V and the sliding region
which follows the pressure parabola until the adhesion region finishes. In the sliding area the
friction force is lower, generating less lateral force. If the centre point of the parabola is reached
during cornering, sliding will occur and increasing the slip angle further will not help to stabilize
the car. The distance between the centre line and the lateral force is known as the pneumatic trial
(t). If the pneumatic trial is zero the generated torque will vanish. The behaviours for different
slip angles can be observed in the Figure.
Combined slip: The last scenario that the Brush model can manage
is the combination of the two previous discussed slips, being the
longitudinal and the side slip. In this case the tyres will be able to
generate a longitudinal force, a lateral force and a moment thanks to
the interaction between the slip angle and the driving or braking
torque. Observing the Figure, it can be appreciated that the treads no
longer deflect on a single direction but in both axis (longitudinally and
sideways). In order to simplify this case the assumption of an isotropic
model will be considered. This means that the lateral and the
longitudinal stiffness are equal (𝐶 𝑝 = 𝐶 𝑝𝑥 = 𝐶 𝑝𝑦) and that the value
for the friction remain constant and equal (𝜇 = 𝜇 𝑥 = 𝜇 𝑦).
Usefulness: The inputs needed for the Brush model are: the parabolic pressure distribution, the
tread stiffness, the wheel radius and the velocities describing the motion (Vx, Vy, w). The outputs
of the model are the self-aligning torque and longitudinal and side forces.
The Brush model is a very simple model due to the assumptions used which can be applied
for longitudinal, side and combined slip. Beside it needs just a few input parameters to allow its
calculation. The Brush model explains well how the forces and torques are generated in the
contact patch. It could be a good option for simple situations due to its simplicity. However
extended Brush models can be found allowing the analysis of more complex situations, such as
the Multi-Line Brush.
Figure 4: Lateral forcé at different
slip angles. Figure 5: Lateral forcé distribution at
different slip angles.
Figure 6: Combined slip using
the Brush model.

33. 32
4.4 The Magic Formula
Author: Hans B. Pacejka and Egbert Bakker
Information source: Pacejka, H. B. (2006). Tyre and Vehicle Dynamics 2nd
Edition
Model Explanation:
The Magic Formula is an empirical model, since it describes the tire behaviour based on
experimental tests. One of the advantages of this model is that it allows its use combinig
longitudinal and lateral slip. Although the formula used in this model is very straight forward, the
input parameters are normally required to be obtained from experimental tests. In some particular
cases it is possible to make an interpolation for the input data to get an approximation of the tire
behaviour.
𝑓(𝑥) = 𝐷 ∗ sin(𝐶 ∗ arctan(𝐵𝑥 − 𝐸 ∗ (𝐵𝑥 − arctan(𝐵𝑥))))
Where:
 𝑓(𝑥): Output variable Fx, Fy, Mz
 x: Input variables: longitudinal slip or slip angle
 Variables that need to be calculated with the inputs:
- D: Peak value
- C: Shape factor
- B: Stiffness factor
- E: Curvature facture
In order to include ply-steer, conicity, rolling resistance effect is the longitudinal and lateral
direction and camber the formula above has to be extended, following:
𝐹(𝑥) = 𝑓(𝑥) + 𝑆 𝑣
Where: 𝑥 = 𝑋 + 𝑆 𝐻
𝑆 𝐻 =Horizontal shift
𝑆 𝑣 =Vertical shicft
The corresponding plot for the Magic Formula normally goes through the origin, then
reaches a maximum point and subsequently tends to a horizontal value. The parameters in the
Magic Formula dictate the shape of the graph. This behaviour can be seen in the plot below.

34. 33
Usefulness:
The main disadvantages of the Magic Formula are:
 Limited to steady-state conditions.
 Just shows the final results (forces, torque) and not at various points of the contact patch.
 Does not describe temperature or pressure changes.
 Large list of inputs needed.
Nevertheless it is able to fit experimental data very accurately offering precise results for the
forces and moments acting on a tire. Therefore it is considered as the main model in vehicle
dynamics, used in most cases to validate extended or new methods.
4.5 The MF-SWIFT Model
Author: TNO and Delft University in cooperation with Prof. H. B. Pacejka
Information source: Delft-Tyre & TNO Automotive (2008). MF-TYRE & MF-SWIFT 6.1 user
manual.
John T. Tielking and Naveen K. Mital (January 1974). A Comparative Evaluation of
Five Tire Traction Models
Model Explanation:
The MF-SWIFT tyre model from its
acronyms in English (Short Wavelength
Intermediate Frequency) is a semi-empirical
model. Its aim is to reduce calculations time,
improve reliability, user-frendly and
compatibility with a larger number of
situations. The model is capable to extend to
higher frequencies levels (up to 60-100 Hz),
short wavelengths excitations and an uneven
distributed road surface. One of the main
Figure 7: Magic Formula behaviour plot.
Figure 8: MF-SWIFT tire model concept.

35. 34
features is that the pressure change in the tyre can be considered. For this purpose the model
consist of a rigid ring, contact patch model and envelope properties.
As mentioned above, in order to be able to analyse the effect of pressure changes, the model
will split in two parts. Thus it considers a single point in the contact patch. On one hand the
generated forces and moments and on the other hand the envelop characteristics.
The main elements regarding the forces will have to be studied:
The most clear changes are the longitudinal stiffness, lateral or cornering stiffness and
camber stiffness. These changes will have also an effect in the peak longitudinal and lateral
coefficient, while at the same time in the generated amount of aligning torque, since the pneumatic
trial length will vary.
Beside the forces the envelope characteristic will also be affected, which describe the
behaviour of the tyre around abrupt disturbances in the road such as bumps. These should be
considered when having shorter wavelengths than 1.5m.
Five main elements can be pointed out to summarize the above changes due to pressure:
1. Rigid ring with 6 degrees of freedom
2. Residual stiffness & damping
3. Contact patch model
4. Generic 3D obstacle enveloping model
5. The ‘Magic Formula steady-state slip model (Pacejka, H. B., 2006)
Usefulness:
Since the tyres are the main element which keeps the car in contact with the road constantly
it is very important to optimize their functionality. Since this model allows calculating the effect
of external excitations (bumps), durability analysis can be done. These contribute to improve
control, driving comfort and tyre wear through absorbing vibrations more efficiently.
Since this model has been developed from the Magic Formula tyre model, it contains all
the advantages of it. It is consider as good model to simulate the vehicle handling, due to the fact
that it is capable to calculate the steady-state forces and moments. This is basically because the
model gathers all the advantages of the Magic Formula tyre model since it is directly developed
from it. Lastly it is important to mention that thanks to its accuracy and reliability, some
simulation software’s such as MSC ADAMs have included their package in their library for the
steady-state simulations at high frequency ranges.
Typical application areas for MF-Swift are vehicle ride comfort analysis; suspension
vibration analysis; development of vehicle control systems, such as ABS or ESP; and handling
and stability analysis, such as braking and power-off in a turn on an uneven road.

36. 35
4.6 Comparative Table
Models Inputs Outputs Advantages Disadvantages
Brush -wheel
dimension r
-tread stiffness
-velocities
(vx,Vy;w)
forces 𝐹𝑥, 𝐹𝑦 and
aligning moment
𝑀𝑧
-combined slip
-quick and
simple
approximation
-few inputs
needed
-not very
accurate due to
assumptions
Fiala -10 inputs -5 outputs
(longitudinal
and lateral
forces, plus the
moments)
- 10 inputs
related to tyre
properties
-fast and simple
-no combined
slip
-no camber
-no conicity or
plysteer
- lateral
coefficient for
α=0
Magic Formula -around 13
inputs needed
forces 𝐹𝑥, 𝐹𝑦 and
aligning
moments 𝑀𝑧
-camber
- shows very
accurate
behaviours
-combined slip
-compensates
conicity. Ply-
steer, rolling
resistance
-no forces at
different point of
the cantact patch
-does not
consider
temperature or
pressure changes
-large amount of
inputs required
MF-SWIFT -forces 𝐹𝑥, 𝐹𝑦
and moments
(𝑀 𝑥, 𝑀 𝑦, 𝑀𝑧)
-accurate and
reliable
-external
excitations
-camber
-Pressure change
-tire wear
-has a cost
Table 2: Comparison table of all four presented tire models.

37. 36
4.7 Formula Student
After comparing all four chosen model characteristics it can be stated that the most accurate
tire model is the MF-SWIFT model from Delft University. It’s the only model listed which allows
the influence of external disturbances, the change in pressure and the wear calculation of the tires.
Beside since it is derived from the Magic Formula is also a very precise and reliable model. The
only issue is if the Formula Student Team would have enough resources to pay for their services.
In case the Formula Student Team cannot afford the MF-SWIFT then the best choice would
be the Magic Formula. The main reasons are that it is an accurate model, which allows an
appropriate calculation of the performance the tire will be capable to offer with their car setup.
Additionally the Team would not have any issues with the input parameters, since most of the
Formula Student Teams buy their tires to suppliers, who can give them the exact input data.
Knowing the input parameters the forces at the tire can be easily calculated due to the simplicity
of the formula.