Computational Dynamics of modeling BMW M3 and
Chevrolet Cavalier’s U-turn and Lane Change
Haiwei Shi1, Kangrui Zhu2, Lyukangcheng Wang3, and Xiaowei Ma4.
1
Swanson School of Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania, US.
3
Suzhou High Scholl of Jiangsu Province, Suzhou, Jiangsu Province, China.
4
Shanxi Experimental Secondary School, Taiyuan, Shanxi Province, China.
Corresponding
author
e-mail:
HAS196@pitt.edu
or
jiawei.m@columbia.edu
or
cywang@suda.edu.cn
Abstract
The model of dynamics of cars, specifically the U-turn and lane change trajectory of BMW M3
and Chevrolet Cavalier is investigated. The models are shown via linear and nonlinear force
model of cars and motions of cars which are computed and plotted in Python. Then, we plotted
the trajectories of the two cars in two force models. Since the Chevrolet Cavalier has a larger
turning radius and slip angle, which causes it more difficult to finish turns, we concluded that
BMW M3 has a better turning performance.
Keyword
Motion of cars
Linear and nonlinear force model
U-turn and Lane change
BMW M3
Chevrolet Cavalier
1. Introduction
The paper will study the U-turn and lane-change turning problem based on models of BMW
M3 [1] and Chevrolet Cavalier [2]. Since 1769, when cars were invented, the process of
development has never stop. There are many features of cars experiencing changes during the
process, for example, engine, tire, differential, etc. People have done a lot of research on
improvement of engine efficiency, refining the shape of car shell, designing new brake
system. In 1769, the first steam-powered automobile which was capable for human
transportation was built by Nicolas-Joseph Cugnot [3]. At the 19th century, the first internal
combustion engine and an early electric motor were invented [4]. At the mid-19th century, the
first electric vehicle appeared. Nowadays, there are so many different types of cars, for
example, Tesla’s electric vehicles and Benz’s automobile.
During the development of cars, the compliments are also experiencing huge progress. The
first commercial differential system used on cars was introduced in 1832 by Richard Roberts
[5]. The differential system makes the car differ the speeds of wheels at left and right during
the turning process, reduce the possibility of turnover. Anti-lock braking system (ABS) is a
safety anti-skid braking system used for preventing wheels from locking during braking
therefore maintain driver’s control during emergency [6]. It was invented by Mario Palazzetti
and sold to Robert Bosch [7].
However, there is a lack of investment on car’s turning. The input speed and angle’s effect on
U-shape turn and lane-change turn still needs further investigation. In spite of the great
progress in the past decades, this paper will study the turning problem of two popular car
models, BMW M3 and Chevrolet Cavalier. We show that different speed and angle input will
significantly change the results of turning, and also different linearity will provide different
results.
2. Mechanisms
Fig1. Schematics of car
2.1. Translational Motion
Translational motion refers to the linear motions of the cars along the road which is a basic and
significant situation in cars’ operation. The cars can be divided into different parts and have
external forces ⃗⃗𝐹𝑖 and internal forces ⃗𝑓𝑖 exerting on them. From Newton’s Second Law, the
motion of any part i is given by:
⃗⃗𝐹𝑖 + ⃗𝑓𝑖 = 𝑚𝑖 ⃗⃗⃗
𝑎𝑖
(1)
In order to show motion of the whole car, sum up the equation (2) over i and introduce the
center of mass. And the sum of internal forces is zero because any two body parts occur in equal
but opposite collinear pairs. The equation is given by:
⃗⃗𝑖 = 𝑀𝐺 𝑎
∑𝐹
⃗⃗⃗⃗𝐺
(2)
𝑖
As shown in Fig.1, the external forces mainly exert on the front and rear parts of cars. So, the
net forces can be divided into two parts of the front force ⃗⃗⃗
𝐹𝑓 and the rear force ⃗⃗⃗
𝐹𝑟 . And the
equation of translational motion is determined:
𝑀𝐺 ⃗⃗⃗⃗
𝑎𝐺 = ⃗⃗⃗
𝐹𝑓 + ⃗⃗⃗
𝐹𝑟
2.2 Rotational Motion
(3)
The rotational motion is a motion form for cars to determine how cars operate during different
types of changes. In the analyze of car’s rotational operation, car-fixed form is helpful to solve
the problems. As shown in Fig.1, components of car fixed frame, ⃗⃗⃗⃗
𝑢1 and ⃗⃗⃗⃗
𝑢2 , can be expressed
in the form of χ:
cos 𝜒
𝑢1 = [ sin 𝜒 ]
⃗⃗⃗⃗
𝑢2 = [
⃗⃗⃗⃗
− sin 𝜒
]
cos 𝜒
(4)
(5)
As shown in the Fig.1, the front force ⃗⃗⃗
𝐹𝑓 is perpendicular to the front tire and makes an angle
π
2
+ δ with respect ⃗⃗⃗⃗
𝑢1 . Also, the rear force is perpendicular to the rear tire and makes an angle
with respect to ⃗⃗⃗⃗
𝑢2 . The front force and rear force are defined as:
𝜋
𝜋
⃗⃗⃗
⃗⃗⃗𝑓 ‖ cos ( + 𝛿) ⃗⃗⃗⃗
⃗⃗⃗𝑓 ‖ sin ( + 𝛿) ⃗⃗⃗⃗
⃗⃗⃗𝑓 ‖ sin 𝛿 ⃗⃗⃗⃗
⃗⃗⃗𝑓 ‖ cos 𝛿 ⃗⃗⃗⃗
𝐹𝑓 = ‖𝐹
𝑢1 + ‖𝐹
𝑢2 = −‖𝐹
𝑢1 + ‖𝐹
𝑢2
2
2
⃗⃗⃗
⃗⃗⃗𝑟 ‖𝑢
𝐹𝑟 = ±‖𝐹
⃗⃗⃗⃗2
(6)
(7)
With the fact that the body of the car is rigid, the position of each part of the car can be computed
with center of mass, shown in Fig.2, in order to calculate the kinetic equation:
⃗𝑟𝑖 = ⃗⃗⃗
𝑟𝐺 + ⃗⃗⃗
𝑟𝑃
(8)
Where ⃗⃗⃗
𝑟𝐺 is the position of the center of mass, and ⃗⃗⃗
𝑟𝑃 is the position of part i relative to the
center of mass.
Fig2. Polar coordinates of cars
On the base of equation (2), the kinetic equation can be determined as:
⃗⃗𝑖 + ⃗𝑓𝑖 ) = ∑(𝑟⃗𝑖 − ⃗⃗⃗
∑(𝑟⃗𝑖 − ⃗⃗⃗
𝑟𝐺 ) × (𝐹
𝑟𝐺 ) × 𝑚𝑖 ⃗⃗⃗
𝑎𝑖
𝑖
(9)
𝑖
With the polar coordinates in Fig.2, velocity can be differentiated by the position of parts of the
cars and equation (6), and the acceleration can be differentiated by velocity from equation (11)
and Fig.2:
𝑟𝑃 = 𝑙𝑖 [
⃗⃗⃗
− sin 𝜒
]
cos 𝜒
𝑣𝑃 = 𝑙𝑖 𝜒 ′ [
⃗⃗⃗⃗
(10)
− sin 𝜒
]
cos 𝜒
(11)
𝑎
⃗⃗⃗⃗𝑝 = 𝑙𝑖 𝜒 ′′ ⃗⃗⃗
𝑢𝑡 + 𝑙𝑖 (𝜒 ′ )2 ⃗⃗⃗⃗
𝑢𝑛
(12)
Based on the equation (11) and Fig.1, the front and back velocity can be defined:
𝑣𝑓 = ⃗⃗⃗⃗
⃗⃗⃗⃗
𝑣𝐺 + 𝑎𝜒 ′ ⃗⃗⃗⃗
𝑢2
(13)
𝑣𝑟 = ⃗⃗⃗⃗
⃗⃗⃗
𝑣𝐺 − 𝑏𝜒 ′ ⃗⃗⃗⃗
𝑢2
(14)
Plug equation (8) and (9) into the kinetic equation (12) to get the final expression for angular
acceleration of the cars:
∑(𝑟⃗𝑖 − ⃗⃗⃗
𝑟𝐺 ) × 𝑚𝑖 ⃗⃗⃗
𝑎𝑖 = ∑(𝑟⃗𝑖 − ⃗⃗⃗
𝑟𝐺 ) × ⃗⃗𝐹𝑖 = (∑ ⃗⃗⃗⃗
𝑚𝑖 𝑙𝑖2 ) 𝜒 ′′
𝑖
𝑖
(15)
𝑖
From the equation (15), there is also the explanation of net torque. And from the net torque, it
can be divided into the front and rear torque with different orientations according to Fig.1, Fig.3
and Fig.4:
⃗⃗⃗𝑓 ‖ cos 𝛿 (𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛)
+𝑎‖𝐹
𝑀𝑓 = (𝑟⃗⃗⃗𝑓 − ⃗⃗⃗
𝑟𝐺 ) × ⃗⃗⃗
𝐹𝑓 = {
⃗⃗⃗𝑓 ‖ cos 𝛿 (𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑜𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛)
−𝑎‖𝐹
𝑀𝑟 = (𝑟⃗⃗⃗𝑓 − ⃗⃗⃗
𝑟𝐺 ) × ⃗⃗⃗
𝐹𝑓 = {
⃗⃗⃗𝑓 ‖ (𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑜𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛)
+𝑏‖𝐹
⃗⃗⃗𝑓 ‖ (𝑛𝑒𝑔𝑎𝑡𝑖𝑣𝑒 𝑜𝑟𝑖𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛)
−𝑏‖𝐹
(16)
(17)
Fig.3 Orientation of the front part
Fig.4 Orientation of the rear part
Where in [8], there is more details about cars’ orientations.
Then introduce yaw moment of inertia 𝐼𝑧 , which is measured in [9], to complete the whole
equation of rotational motion with equation (15), (16), and (17):
⃗⃗⃗𝑓 ‖ ± 𝑏‖𝐹
⃗⃗⃗𝑓 ‖
𝐼𝑧 𝜒 ′′ = ±𝑎‖𝐹
(18)
2.3 Mathematical model of tire-road interaction
Due to the assumption that the left side is almost equal to the right side, two-wheel model and
four-wheel model have a relationship:
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗
⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐹𝑐𝑒𝑛𝑡𝑒𝑟 = 𝐹
𝑙𝑒𝑓𝑡 + 𝐹𝑟𝑖𝑔ℎ𝑡
(19)
From the equation (5), (13), and (14), and Fig.5, 6, calculate the slip angle, which is carefully
investigated in [10], that between velocity and force exerted on the tires:
𝛼𝑓 = δ − sin−1
𝛼𝑟 = − sin−1
𝑣𝑓 · ⃗⃗⃗⃗
⃗⃗⃗⃗
𝑢2
(20)
‖𝑣
⃗⃗⃗⃗𝑓 ‖
⃗⃗⃗𝑟 · ⃗⃗⃗⃗
𝑣
𝑢2
‖𝑣
⃗⃗⃗𝑟 ‖
Fig.5 Velocity of front tire
(21)
Fig.6 Velocity of rear tire
The basic formula in the linear model is:
F = Cα
(22)
From the equation (20), (21), (22), and Fig.3, Fig.4, torque of each tire can be derived as:
𝑀𝑓 = aC𝛼𝑓 cos 𝛿
(23)
𝑀𝑟 = −bC𝛼𝑟
(24)
2.4 Model of force
Because linear model is only accurate for small angles, when α becomes large this model
overestimates the force, non-linear model is needed which is slightly different from linear
model shown in Fig.7:
F = C tan 𝛼 𝑓(𝜆)
𝑓(𝜆) = {
(2 − 𝜆)𝜆 𝑖𝑓 𝜆 < 1
1 𝑖𝑓 𝜆 ≥ 1
𝜇𝐹𝑁
λ=|
|
2𝐶 tan 𝛼
(25)
(26)
(27)
Fig.7 Force model
2.5 Turning
The experiment will be set up with two types of turns: U-turn and lane change that needs
functions of δ to indicate how the wheels are being turned by the driver:
Mathematical formula for the U-turn:
0
𝑖𝑓 𝑡 < 𝑡𝑠𝑡𝑎𝑟𝑡
𝛿0
𝜋(𝑡 − 𝑡𝑠𝑡𝑎𝑟𝑡 )
[1 − cos (
)] 𝑖𝑓 𝑡𝑠𝑡𝑎𝑟𝑡 ≤ 𝑡 < 𝑡𝑠𝑡𝑎𝑟𝑡 + 𝑡0
2
𝑡0
𝛿0 𝑖𝑓 𝑡𝑠𝑡𝑎𝑟𝑡 + 𝑡0 ≤ 𝑡 < 𝑡𝑒𝑛𝑑
δ=
𝛿0
𝜋(𝑡𝑒𝑛𝑑 + 𝑡0 − 𝑡)
[1 − cos (
)] 𝑖𝑓 𝑡𝑒𝑛𝑑 ≤ 𝑡 < 𝑡𝑒𝑛𝑑 + 𝑡0
2
𝑡0
0 𝑖𝑓 𝑡𝑒𝑛𝑑 + 𝑡0 ≤ 𝑡
{
(28)
Mathematical formula for the lane change:
0 𝑖𝑓 𝑡 < 𝑡𝑠𝑡𝑎𝑟𝑡
(𝑡 − 𝑡𝑠𝑡𝑎𝑟𝑡 )
δ = 𝛿0 sin (
) 𝑖𝑓 𝑡𝑠𝑡𝑎𝑟𝑡 ≤ 𝑡 < 𝑡𝑠𝑡𝑎𝑟𝑡 + 2𝜋𝑡0
𝑡0
0 𝑖𝑓 𝑡𝑠𝑡𝑎𝑟𝑡 + 2𝜋𝑡 ≤ 𝑡
{
(29)
2.6 Basic data
In order to analyze the different actions of BMW M3 and Chevrolet Cavalier in different
situations, the necessary data for them are shown in the chart:
Table.1 Data of BMW M3 and Chevrolet Cavalier
Property
BMW M3
Chevrolet Cavalier
Front wheel a (m)
a=1.36
a=0.98
Rear wheel b (m)
b=1.37
b=1.66
Mass （kg）
mass=1549
mass=1187
Yaw of inertia (kg𝑚2 )
yaw = 2886
yaw = 1928
Frictional coefficient μ
μ=0.9
μ=0.9
Front tire cornering stiffness
(N/rad/axle)
𝐶𝑓 =194,000
𝐶𝑓 = 58,000
Rear tire cornering stiffness
(N/rad/axle)
𝐶𝑟 = 240,000
𝐶𝑟 = 58,000
2.7 Flow Diagram
Start
Import packages of python
Define the class of cars’
Define functions of calculations of
Output the picture of each car element
Find interaction between
Find interaction between force
Plot the simulation of car in progress
Input data of Chevrolet
Input data of BMW M3
Linear
Nonlinear
Linear
Nonlinear
U-turn
Lane change
U-turn
Lane
Output the plots
Output the plots
Compare the plots of two cars
Stop
3. RESULTS:
3.1
On carrying out our simulation, obtained a graph of the delta function against time In which
I observed that the graph was steady at delta 0 for a time of one second, on which it gradually
started rising steadily up to a mafunctioninglue of 0.1 radians. I observed that it took
approximately 2.5 seconds for the graph to obtain this value. The graph then stabilized at this
maximum point up to approximately 8 seconds, then started dropping steadily towards the 0
radians and after 10 seconds, I observed that the graph returned to the 0 radians point and
maintained this position to the end of the simulation.
1:Delta against time
On changing the turn parameters a change in the graph was observed in that on plotting the
turn after a change in the lane change turn type, the graph was observed to maintain a zero
radian motion for about 1 second, on which it rose to a maximum value of 0.1 radians in
about 1.5 seconds. On which it dropped again almost immediately, to a negative radian of -0.1
after 2 seconds of the simulation time. It rose again to radian zero
and it maintained this location for the rest of the simulation.
A graph of force against the slip angle in degrees for the BMW M3 vehicle was plotted and
the results obtained Ire analyzed as stated below: On starting the simulation, various
characteristics under test such as the Linear Front, Linear Rear, the Nonlinear rear as Ill as the
nonlinear front Ire initialized. On starting the simulation, it was observed that the linear rare
characteristic was directly proportional to the force, as Ill as the slip angle that had been
applied to the vehicle. It was observed to rise from a minimum value of 0 force at slip angle 0
degrees to a maximum value of approximately 20KN, with a slip angle of 4.5 degrees. The
other variables such as the linear front, the nonlinear front, as Ill as nonlinear rear, Ire also
observed to be directly proportional to the amount of force applied to the BMW M3 as Ill as
the slip angle.
2: Force against slip angle
A graph of force as a function of the delta was also plotted for the BMW M3 model. In which,
a directly proportional relationship was established betIen the delta and the force applied on
the BMW M3 vehicle.
A graph of force as a function of chiprime was also plotted for the BMW M3 model, in which
it was observed that the force applied on the vehicle was indirectly proportional to the
chiprime. A drop in the force was observed to cause an increase in the chiprime.
3.2
A graph of force against v_x for the BMW M3 model was also plotted and the results
obtained shoId that there was a directly proportional relationship betIen the force and v_x in
which it was noted that an increase in force resulted in a similar increase in the v_x. On
plotting a graph of the force against the chiprime on the rare tire of the BMW M3 model, it
was observed that the force gradually increased as the chiprime increased up to a maximum
value of about 250KN. On plotting a graph of the nonlinear force as a function of the delta, it
was observed that the force increased gradually and finally maintained a steady-state at
6000N. An analysis to determine the motion of the vehicle was carried out. This was done to
determine the characteristic of the vehicle at different types of motion that that can be
experienced by the vehicle. The inputs for the simulation included the time range over which
the simulation of the ODE was to be done, the time step interval, the initial car motion, the car
parameters, the force type experienced on the car as Ill as the turn parameters. The expected
output of the simulation was the time points for the solution, the value of the solution at the
time points with its particular shape. With this output and animation for the motion of the
vehicle was done, in which it was possible to determine the distance covered by the vehicle at
a particular time. A plot for the trajectory of the vehicle was also plotted in which the
characteristics of the trajectory of the vehicle Ire observed from the results obtained from the
simulation. The trajectory plot for the BMW M3 included the U-turn as Ill as the turn angle
plot.
In the case of the U-turn, it was observed that the linear situation had a loIr value of x
compared to that of the nonlinear but it was also noted that, the nonlinear model had a larger
value of y compared to that of the linear of 45 as a result of them having different solutions to
operate the delta.
For the BMW M3 linear sharp corner, it was observed that the time in the second graph was
shorter but with the same change of turn angle. For the case of the BMW M3 linear lane
change sharp turn, it was observed that the graph had a lane change sharp turn that had a
larger change in y with a small change in x. In the nonlinear sharp u-turn, it was observed that
the car was not able to finish the change. On comparing the two cars for the whole change, it
was observed that it took the BMW M3 a shorter time to complete while on the other hand,
the Chevy took approximately 4.61 seconds and as a result of this, the BMW M3 is more
suitable for completing the turn.
3: Force against S
4: Force against chiprime
5: Time against chiprime
4. Conclusions
In this paper, we have studied the problem of modeling the dynamics of the car, specifically
the U-turn and Lane Change trajectory. We took BMW M3 and Chevrolet Cavalier as
examples. The four-wheel model was simplified to a two-wheel model for analyzing. With the
formulas, we calculated the tire force and slip angle in linear and non-linear force models.
Then, we plotted the trajectories of the two cars in two force models, respectively. And the
condition of sharp turns was included, too. The known data was used to analyze the reasons
for the results. Chevrolet Cavalier has a larger turning curvature so that it has a larger turning
radius. And it has a larger slip angle. So it is difficult for Chevrolet Cavalier to finish sharp
turn. Finally, the results and discoveries of our research were demonstrated. We concluded
that the BMW M3 has better cornering performance.
The results and the methods in this paper can be used to analyze the situations of other cars
during Lane Change and U-Turn. The further researches can utilize our method to study the
performance and other fields of different cars. With some initial required parameters, the
trajectory can be simulated. The research can also be used to analyze the influence of car
parameters such as mass and cornering stiffness on the car motions. There are still limitations
in our model. We didn’t consider the process of braking and accelerating during U-turns and
lane changes. So there was no net force in the direction of the tire in our model. The only
force exerted by road was perpendicular to the tire. To solve this problem, we think that more
complex physical models of tire force are needed to analyze the force in the direction the tires
are pointing to. Besides, the two-wheel model led to certain errors. There is a differential
mechanism between the left and right driving wheels. We assumed that the left wheels were
the same as right wheels. This is not exactly true since in a turn the velocity of the wheels on
the inside was different from the wheels on the outside. In addition, the weight on the left and
right wheels was slightly different in a turn. In the further study, we think that the force and
torque due to the left and right wheels should be calculated separately instead of center wheel.
As a result, the errors can be decreased.
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