Report on Planar Vehicle Dynamics. Model included observing dynamic states of vehicle using linear and non linear tire models with 3 degrees of freedom.

1. Project Report
on
Modelling Planar Vehicle Dynamics using Bicycle model
Vehicle Dynamics - I
Winter – 2017
Detroit, MI
Under guidance of: Made by:
Dr. Valery Pylypchuk Akshay Mistri
Access ID: GF6919

2. Page 2 of 16
CONTENTS
Page No.
1 Introduction 3 - 6
1.1 Equations of Motion 3 - 5
1.1.1 Inputs/Outputs of Project 3
1.1.2 Equations for Linear case 4
1.1.3 Equations for Non-Linear case 5
1.2 Understeer Coefficient 5
1.3 Lateral Acceleration Gain 6
2 Result Validation of the Baseline Model 6 - 7
2.1 Inputs Given 6
2.2 Outputs Obtained 7
3 Results Obtained 8 - 12
3.1 Variation of Lateral Tire Forces with Side Slip Angles 8
3.2 Variation of Lateral Acceleration Gain with Time 8 – 10
3.2.1 Outputs 9
3.2.2 Inputs Taken 9 – 10
3.3 Understeer Coefficient 10
3.4 Dynamic States, Linear vs Non-Linear 11
3.4.1 Velocities in x-coordinate axis 11
3.4.2 Velocities in y-coordinate axis 11
3.4.3 Yaw rates in x-y Plane 12
3.4.4 Yaw Angle in x-y Plane 12
4 Simulink Model 13 – 16
4.1 Linear Vehicle Model – 1 13
4.2 Linear Vehicle Model – 2 & Non-Linear Vehicle Model 13
4.3 Dynamic States 14
4.4 Lateral Acceleration Gain Block 14 - 15
4.5 Understeer Coefficient Block 15 - 16

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1. INTRODUCTION
1.1 Equations of Motion
Bicycle model is the simplest model to interpret dynamics of a vehicle. Due to
symmetricity, the two wheels on the front and rear are combined, resulting into a
better model for understand the side slip angles and tire forces.
Linear Model consists of a linear relationship between the side slip angle and lateral
tire force. Whereas, the Non-linear model has a non-linear relationship for the two.
Fig.1 Bicycle Model
1.1.1 Inputs/Outputs of Project
Inputs:
1. Front wheel torque, 𝑄 𝑓 [N-m]
2. Rear wheel torque, 𝑄 𝑟 [N-m]
3. Front wheel steer angle, 𝛿𝑓 [rad]
4. Rear wheel steer angle, 𝛿 𝑟 [rad] (taken 0 in our case)
Outputs: (or the Dynamics States)
1. Velocity in x-coordinate axis, 𝑉𝑥 [m/sec]
2. Velocity in y-coordinate axis, 𝑉𝑦 [m/sec]

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3. Yaw rate in x-y plane, r [rad/sec]
4. Yaw angle, ψ [rad]
Symbols Used:
𝐹𝑥𝑓: Reaction force from ground to front tire in x-direction [N]
𝐹𝑥𝑟: Reaction force from ground to rear tire in x-direction [N]
𝑅𝑓𝑒: Effective radii of the front tire (𝑅𝑓𝑒=0.32 m)
𝑅 𝑟𝑒: Effective radii of the rear tire (𝑅 𝑟𝑒=0.32 m)
𝐹𝑦𝑓: Lateral force (y-direction) on front tire [N]
𝐹𝑦𝑟: Lateral force (y-direction) on rear tire [N]
𝐶 𝛼𝑓: Cornering stiffness of front tire [76666 N/rad]
𝐶 𝛼𝑟: Cornering stiffness of rear tire [186300 N/rad]
a: Is the distance between Centre of Gravity (CG) of vehicle and the centre of
front tire. (a = 1.65 m).
b: Is the distance between Centre of Gravity (CG) of vehicle and the centre of
rear tire. (b = 1.37 m).
L: Wheelbase [a + b = 3.02 m]
m: Mass of the vehicle [3090 𝐾𝑔]
𝐼 𝑍𝑍: Moment of inertia of vehicle about z-coordinate axis. [5980 𝐾𝑔. 𝑚2
]
1.1.2 Equations for linear case:
1. 𝐹𝑥𝑓 =
𝑄 𝑓
𝑅𝑓𝑒
⁄
2. 𝐹𝑥𝑟 =
𝑄 𝑟
𝑅 𝑟𝑒
⁄
3. 𝐹𝑦𝑓 = 𝐶 𝛼𝑓 ( 𝛿 𝑓 −
𝑉 𝑦 + 𝑟.𝑎
𝑉 𝑥
)
4. 𝐹𝑦𝑟 = − 𝐶 𝛼𝑟 (
𝑉 𝑦 − 𝑟.𝑏
𝑉 𝑥
)
5. 𝑚(𝑉𝑥
̇ − 𝑟𝑉𝑦 ) = 𝐹𝑥𝑓 cos 𝛿 𝑓 − 𝐹𝑦𝑓 sin 𝛿 𝑓 + 𝐹𝑥𝑟
6. 𝑚(𝑉𝑦
̇ + 𝑟𝑉𝑥 ) = 𝐹𝑥𝑓 sin 𝛿 𝑓 − 𝐹𝑦𝑓 cos 𝛿 𝑓 + 𝐹𝑦𝑟

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7. 𝐼 𝑍𝑍 𝑟̇ = 𝑎 (𝐹𝑥𝑓 sin 𝛿 𝑓 − 𝐹𝑦𝑓 cos 𝛿 𝑓) − 𝑏. 𝐹𝑦𝑟
1.1.3 Equations for Non-Linear case: Equations 3 and 4 are replaced by equations
3a and 4a given below.
• Equation 3a: 𝐹𝑦𝑓 = 𝐴. sin [ 𝐵. tan−1
(
𝐶 𝛼𝑓. 𝛼 𝑓
𝐴𝐵
⁄ )]
• Equation 3b: 𝐹𝑦𝑟 = 𝐴. sin [ 𝐵. tan−1
(
𝐶 𝛼𝑟. 𝛼 𝑟
𝐴𝐵
⁄ )]
Where,
▪ 𝐴 = 𝜇. 𝐹𝑧𝑓 or 𝜇. 𝐹𝑧𝑟 respectively for 𝐹𝑦𝑓 and 𝐹𝑦𝑟.
& 𝐹𝑧𝑓 =
𝑏𝑚𝑔
𝐿
⁄ 𝐹𝑧𝑟 =
𝑎𝑚𝑔
𝐿⁄ .
▪ B = 2
▪ 𝛼 𝑓 = (𝛿𝑓 −
𝑉𝑦 + 𝑟.𝑎
𝑉𝑥
)
▪ 𝛼 𝑟 = − (
𝑉 𝑦 − 𝑟.𝑏
𝑉 𝑥
)
1.2 Understeer Coefficient: [𝐾 𝑈𝑆]
The understeer coefficient was calculated using two vehicle models running at front
steering angles which differ by a small value while the wheel torques were kept same
for both.
𝛿𝑓_1 : 0.09 radians for first linear vehicle model
𝛿𝑓_2 : 0.1 radians for second linear vehicle model
𝑄 𝑓_1 & 𝑄 𝑓_2: 100 N-m torque on front wheels for both models
𝑄 𝑟_1 & 𝑄 𝑟_2: 0 N-m torque on rear wheels for both models
For finding the understeer coefficient the following equation was used,
𝐾 𝑈𝑆 = 𝑔.
∆ 𝛿 𝑓
∆ 𝑎 𝑦
⁄ −
𝑔. 𝐿
𝑉2⁄
Where 𝑎 𝑦 = 𝑉𝑦
̇ + 𝑟. 𝑉𝑥 &
𝑉 = (𝑉1 + 𝑉2)/2, 𝑉1 & 𝑉2 being the absolute velocities of the two vehicles

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1.3 Lateral Acceleration Gain [Gacc]: It is found from both sides of the equation and
plotted versus time.
𝐺 𝑎𝑐𝑐 =
𝑎 𝑦
𝑔⁄
𝛿𝑓
= (
𝑉𝑥
2
𝑔. 𝐿 + 𝑉𝑥
2
. 𝐾 𝑈𝑆
)
2. RESULTS FOR THE VALIDATION OF THE BASELINE MODEL
2.1 Inputs:
Front steer
Angle, 𝛿𝑓 [rad]
Rear Steer
Angle, 𝛿 𝑟 [rad]
Front wheel
Torque, 𝑄 𝑓 [N-m]
Rear wheel
Torque, 𝑄 𝑟 [N-m]
Time [seconds]

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2.2 Outputs Obtained:
Hence, the outputs obtained are desirable and the baseline model is validated.

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3. RESULTS OBTAINED
3.1 Variation of Lateral Tire Force with Side Slip Angle
The first thing to observe is the variation of lateral tire forces with side slip angles for
linear and non-linear case. It is quite evident that in the first two linear models the
variation is linear and for the non-linear it is linear at the first but as side slip angle
increases the variation is non-linear.
3. 2 Lateral Acceleration Gain [Gacc] vs Time: is found from both sides the equation
𝐺 𝑎𝑐𝑐 =
𝑎 𝑦
𝑔⁄
𝛿𝑓
= (
𝑉𝑥
2
𝑔. 𝐿 + 𝑉𝑥
2
. 𝐾 𝑈𝑆
)

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3.2.1 Output:
Thus, graph from the right-hand side of the equation, shown in blue, is approximately
a straight line due to its theoretical nature. While the graph from the left-hand side of
the equation shows the dynamics behaviour due to change of steering angle. Initially,
the gain is huge (negative) due to sudden change in steer angle. Gradually it coincides
with the other curve due to constant steering angle. This explanation can also be
verified through the inputs which were kept as follows:
3.2.2 Inputs taken: Input 1: 𝛿𝑓
(
𝑉𝑥
2
𝑔. 𝐿 + 𝑉𝑥
2
. 𝐾 𝑈𝑆
)
𝑎 𝑦
𝑔⁄
𝛿𝑓

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Input 2: 𝐹𝑥𝑓
Input 3: 𝐹𝑥𝑟 = 0 Input 4: 𝛿 𝑟 = 0
3.3 Understeer Coefficient: 𝑲 𝑼𝑺
To find the understeer coefficient two linear vehicle models were taken into
consideration with:
a. Same input torque 𝑄 𝑓 [32 N-m] for the front wheels &
b. Different front wheel steering angles 𝛿𝑓 [0.09 & 0.1 radians] for the two.
The understeer coefficient Kus was found to be varying in the small range of 0.03
which can show that it is almost constant.
𝑲 𝑼𝑺

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3.4 Dynamic States, Linear vs Non-Linear
3.4.1. Velocities in x-coordinate axis: 𝑉𝑥 Linear & Non-Linear.
3.4.2 Velocities in y-coordinate axis: 𝑉𝑦 Linear & Non-Linear
These velocities clearly show the differences due to non-linear side slips. Variation in
non-linear case is more realistic in the end as shown.

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3.4.3 Yaw rates: r Linear & Non-Linear
3.4.4 Yaw Angle: ψ Linear & Non-Linear
Note: These outputs were obtained keeping:
a) Front Steer Angle 𝛿 𝑟 = −0.2 𝑡𝑜 0.2 for both linear and non-linear model.
b) Front Wheel Torque 𝑄 𝑓 = 2200 𝑁 − 𝑚 for both models.

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4. SIMULINK MODEL
4.1 Linear Vehicle Model-1: ‘Linear Vehicle 1’ receives input from Block ‘Inputs 1’
and gives Lateral acceleration gain and Understeer Coefficient.
4.2 Linear Vehicle-2 & Non-Linear Vehicle: While ‘Vehicle 2’ & ‘Non-Linear
Vehicle’ gets inputs from block ‘Inputs 2’ as shown below.

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4.3 Dynamic States Comparison: For comparing variations in Linear vs Non-Linear
case outputs from ‘Linear Vehicle 2’ and ‘Non-Linear Vehicle’ are compared using
the block shown below.
Note: Dynamic State with suffix ‘_2’ are of linear vehicle – 2 model and the others
are from non-linear case.
4.4 Lateral Acceleration Gain Block:
It is calculated using the formula: 𝐺 𝑎𝑐𝑐 =
𝑎 𝑦
𝑔⁄
𝛿 𝑓
= (
𝑉𝑥
2
𝑔.𝐿+𝑉𝑥
2
. 𝐾 𝑈𝑆
)

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Note: The suffix ‘_1’ below shows that the outputs of Linear Vehicle – 1 are taken into
consideration.
LHS of the equation was formulated as follows:
While, RHS of the equation:
4.5 Understeer Coefficient:

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For calculating the Understeer coefficient two linear vehicle models were
considered. The following equation was used:
𝐾 𝑈𝑆 = 𝑔.
∆ 𝛿 𝑓
∆ 𝑎 𝑦
⁄ −
𝑔. 𝐿
𝑉2⁄
For calculating 𝑉 (as shown in formula), average of absolute velocities of the two
linear vehicles were taken & it was named as V_avg. (Also, shown below)
Note: suffix ‘_1’ & ‘_2’ represent outputs taken from linear models 1 & 2
respectively.
For 𝑲 𝑼𝑺 :