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Race Car Vehicle Dynamics
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Race Car Vehicle Dynamics
Muhammad Abdul Rehman
Collge of Aeronautical Engineering
National University of Sciences and Technology
Risalpur, Pakistan
m.abdul.rehman004@gmail.com
Abstract— This research paper explores the dynamics of
race car vehicles, with a particular focus on the phenomena of
understeer, oversteer, stability in yawing, and the natural and
forced response of the car. These are fundamental aspects of
vehicle dynamics that are critical to the successful design and
operation of high-performance race cars. Vehicle dynamics
relates tire and aerodynamic forces to overall vehicle
accelerations, velocities and motions, using Newton’s Laws of
Motion. The paper reviews the current state of the art in this
area, drawing on a range of literature sources and case studies
to illustrate key concepts and techniques. In particular, the
paper examines the underlying physical principles that govern
these phenomena and their effects on vehicle performance, as
well as the various engineering approaches and methods that
can be used to analyze a car's dynamic behavior.
and the inertial properties relative to it are taken constant. The
Vehicle Axis System is illustrated in Fig. 1.
Keywords— race car vehicles, vehicle dynamics, understeer,
oversteer, yawing, natural and forced response
I. INTRODUCTION
Race car vehicle dynamics is a complex and exciting field
of study that is essential for achieving peak performance in
high-speed vehicles. This field of study covers a wide range
of dynamic phenomena that can significantly affect a vehicle's
performance, including understeer, oversteer, lateral stability,
and the natural and forced response of the car. Understanding
and optimizing these dynamics is critical to designing and
operating a competitive race car, and can mean the difference
between winning and losing a race.
Fig. 1. Vehicle Axis System [3]
The angles in the vehicle axis system are illustrated in Fig. 2
The vehicle dynamics of cornering is of particular interest.
Understeer and oversteer are two of the most fundamental
aspects of vehicle dynamics, which describe how a car's front
or rear wheels respond when it turns. Stability in yawing
describes the car's ability to maintain a straight line while
cornering, while natural and forced responses of the car refer
to how the car responds to without and with external inputs or
disturbances respectively.
The paper will explore the underlying physical principles
that govern these phenomena, as well as the methods and
models that can be used to analyze a car’s dynamic behavior.
The natural and forced response of the car are modelled in
MATLAB Simulink and the relevant stability graphs are
plotted.
II. METHODOLOGY
A. Vehicle Axis System
It is important to define an axis system to which the
accelerations, velocities, and forces/moments causing them
can be referred. The common axis systems used in vehicle
dynamics work in the United States have been defined by the
Society of Automotive Engineers (SAE). The two basic axis
systems are the Earth-Fixed Axis System and the Vehicle Axis
System. The axis system used in this work is the Vehicle Axis
System which has its origin in aircraft usage. This system is
called the “Moving Axis System” because it moves with the
vehicle. The point to remember is that it is fixed in the vehicle
Fig. 2. Angles in vehicle axis system [3]
B. Terminologies
The terminologies for the vehicle dynamics used in this
work are listed below:













v – Lateral velocity
u – Forward velocity
V – Total velocity
r – Yawing velocity
N – Yawing moment
β – Sideslip angle
δ – Steer angle
m – Vehicle mass
a – Distance of front wheel axle from CG
b – Distance of rear wheel axle from CG
l – Distance from front wheel axle to rear wheel axle
CF – Cornering stiffness of the combined front tires
CR – Cornering stiffness of the combined rear tires
C. Elementary Vehicle Defined
The basic model used to represent a vehicle system is the
“bicycle” model. The bicycle model is a widely used and
well-established model for analyzing the vehicle dynamics of
four-wheeled vehicles, particularly for cars. The model
simplifies the complex dynamics of a vehicle into a twodegree-of-freedom system, and with this we can investigate
the effects of front and rear tire cornering stiffnesses, center
of gravity (CG) location along the wheel- base, and geometric
steer angle on the yawing and sideslipping motions. The
bicycle model for two degrees of freedom is illustrated in Fig.
3.
The vehicle is understeering if K>0, neutrally steering if K=0,
and oversteering if K<0.
E. Lateral Stability
The equations of motion are developed using the
elementary bicycle model. This is a linear model with two
degrees of freedom (2DF) which enables the calculation of
the motion variables as a function of the forces and moments
acting on the vehicle. The complete equations of motions in
derivative notation are given below:

𝑚𝑉 𝑟 + 𝛽̇ = 𝑌 𝛽 + 𝑌 𝑟 + 𝑌 𝛿


𝐼 𝑟̇ = 𝑁 𝛽 + 𝑁 𝑟 + 𝑁 𝛿
(3)
There are six derivatives in this vehicle system and their name
and nature are shown in Table. 1.
Table I: The derivatives for simple 2DF automobile
Derivative
𝑁
𝑌
𝑁
𝑌
𝑁
𝑌
No rolling or pitching moments
Constant forward velocity
No aerodynamic effects
Damping
Coupling
𝛽
+
𝑟
(4)
𝛿
(5)
The state-space is further represented in geometrical
parameters of the vehicle for simplification because the data
in this form was easily available.
D. Neutral Steer, Understeer, Oversteer
The steady-state behavior of a vehicle is the steering
characteristics of the vehicle. The characteristic of a vehicle's
slip angles where both front and rear are the same is called
neutral steer. In neutral steering, a vehicle turns at a rate
exactly proportional to the rate at which the steering wheel is
turned.
Understeer and oversteer are vehicle dynamics terms used
to describe the sensitivity of a vehicle to steering. Oversteer
is what occurs when a car turns (steers) by more than the
amount commanded by the driver. Conversely, understeer is
what occurs when a car steers less than the amount
commanded by the driver. In understeering, the slip angle of
the front wheel is greater than that of rear wheel. In
oversteering, the slip angle of the rear wheel is greater than
that of front wheel.
The steering characteristic of a vehicle is governed by a
factor called understeer gradient K given by:

−1
𝛽̇ =
𝑟̇
1) Assumptions
𝐾=
Control
𝑋̇ = 𝐴𝑥 + 𝐵𝑢
For this representation of the vehicle, the two "degreesof-freedom" are the motion variables, v (the lateral velocity)
and r (the yawing velocity). The input variable is the front
wheel steer angle, δ, which is under driver control.

Nature
The state-space representation of this system of equations is
given by:
Fig. 3. The bicycle model (two degrees of freedom) [3]



Name
Control Moment
Control Force
Yaw Damping
Sideslip Damping
Static Directional Stability
Lateral Force/Yaw Coupling

−1 𝑣
+
𝑟
𝑣̇
=
𝑟̇
𝛿
(6)
The eigen values for the system matrix A are calculated as:

det(𝑠𝐼 − 𝐴) = 0
The characteristic equation comes out to be of the form:
𝑠 + 𝑏𝑠 + 𝑐 = 0

The eigen values are calculated as:
𝑠
,
±
=
(8)
Where,
𝑏=
(
)
(9)
𝑐=
(
)
(10)
The stability of the system is governed by the two eigen
values 𝑠 , of the characteristic equation. The conditions for
stability are as follows:

If𝑐 > , the eigen values 𝑠 , both have a negative
real part and therefore the system is stable and
oscillatory

If ≥ 𝑐 > 0, the eigen values 𝑠 , are both real and
negative, so the system is stable and non-oscillatory
If 𝑐 ≤ 0, the eigen values 𝑠 , are both real, with one
of them negative and the other one non-negative. So,
the system is unstable

Hence, the necessary and sufficient condition for yaw
stability is 𝑐 > 0 in (8). The eigen values of the system
matrix A can also be directly calculated using MATLAB.
F. Stability vs Understeer/Oversteer
The numerator of the understeer gradient K reveals a
close relationship between the understeer gradient and the
stability as:


Table II: Vehicle system parameters for a race car
Symbol
m
V
Iz
a
b
CF
CR
Value
1700 kg
80 m/s
2000 kgm3
1.5 m
1.7 m
-40000 N/rad
-40000 N/rad
III. RESULTS AND DISCUSSIONS
A. Stability
The eigen values in (8) are calculated through MATLAB
and analyzed for the stability behavior. Both the eigen values
came out to be complex conjugates of each other with negative
real parts. This confirms that the vehicle is stable and
oscillatory. The root locus of this system is shown in Fig. 4.
The eigen values are as follows:
𝑠 = −0.937 + 1.969𝑖
𝑠 = −0.937 − 1.969𝑖

𝑏𝐶 ≥ 𝑎𝐶 ⟷ 𝐾 ≥ 0
𝑏𝐶 < 𝑎𝐶 ⟷ 𝐾 < 0
In order to be stable, a vehicle must be understeering or
neutrally steering. An oversteering vehicle is only stable up
to a certain velocity and unstable above that velocity.
G. Natural and Forced Response
The natural and forced response of the system is analyzed
using Simulink MATLAB. The natural response is analyzed
using the eigen values of the system matrix A, and the results
are obtained on MATLAB. The control inputs for forced
response analysis are of four types: step, impulse, sinusoidal
and ramp. The Simulink block set diagram for the analysis of
input responses is shown in Fig. 4. The relevant stability
graphs are obtained, and the results are discussed in the next
section.
Fig. 5. Root locus for the vehicle system
B. Response to control inputs
The time response of a race car to different control inputs,
such as step, ramp, sinusoidal, and impulse signals, can
provide valuable insights into the vehicle's dynamic behavior.
To investigate this behavior, a Simulink model was
developed with transfer functions representing the vehicle's
dynamics. By simulating the response of the vehicle to
different inputs, it was possible to observe how the car
reacted to changes in steering. The use of a Simulink model
with transfer functions allowed for a detailed analysis of the
time response of a race car to different control inputs.
Fig. 4. Simulink blockset model for different types of inputs
H. Vehicle System Parameters
The vehicle system parameters used for this work are
taken from a conference paper[2]. These parameters are for a
race car and the values of these parameters are listed below:
The motion variables under observation are lateral
velocity (v) and yawing velocity (r). The time response of
these vehicle states under step input are shown in Fig. 6 and
Fig. 7. These responses show that after the step input has been
given by the driver, the vehicle being a stable one achieves
the equilibrium state at a certain value as can be observed
from the plots.
Fig. 6. Time response of lateral velocity with step input
Fig. 9. Time response of yawing velocity with impulse input
The time response of these parameters under sinusoidal input
is shown in Fig. 10 and Fig. 11. These plots show that the
vehicle after the control input follows the sine output and
becomes proportional to the input and keep on oscillating
about the equilibrium position.
Fig. 7. Time response of yawing velocity with step input
The time response of these parameters under impulse input is
shown in Fig. 8 and Fig. 9. These plots show that the vehicle
after the control input comes back to its original equilibrium
state.
Fig. 8. Time response of lateral velocity with impulse input
Fig. 10. Time response of lateral velocity with sinusoidal input
Fig. 11. Time response of yawing velocity with sinusoidal input
The time response of these parameters under ramp input is
shown in Fig. 12 and Fig. 13. These plots show that the
vehicle after the control input keeps on moving away from
the equilibrium position as long as the ramp input is there.
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IV. CONCLUSION
In conclusion, the use of the bicycle model for analyzing
the vehicle dynamics of high-speed race cars has proven to
be a valuable tool for understanding the behavior of complex
system of a race car. Through the use of Simulink model and
transfer functions, we have been able to investigate the time
response of a race car to different control inputs. The analysis
has shown that the response of the vehicle varies depending
on the input signal used, with different inputs resulting in
more gradual or sudden changes in the car's behavior.
Overall, the bicycle model has proven to be a valuable tool
for studying the vehicle dynamics of race cars, and the
insights gained from this research can inform future work in
the field.
Fig. 11. Time response of lateral velocity with ramp input
REFERENCES
[1]
[2]
[3]
Fig. 11. Time response of yawing velocity with ramp input
Nelson, R.C. (2010) Flight Stability and Automatic Control. Chennai:
McGraw-Hill Education (India) Private Limited.
Hammad, M. (2019) Safety and Lateral Dynamics Improvement of a
Race Car Using Active Rear Wing Control.
Milliken, W.F. and Milliken, D.L. (1995) Race Car Vehicle Dynamics.
SAE International.