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MECA0492 : Vehicle dynamics
Pierre DuysinxResearch Center in Sustainable Automotive
Technologies of University of Liege
Academic Year 2017-2018
1
Bibliography
T. Gillespie. « Fundamentals of vehicle Dynamics », 1992, Society of Automotive Engineers (SAE)
W. Milliken & D. Milliken. « Race Car Vehicle Dynamics », 1995, Society of Automotive Engineers (SAE)
R. Bosch. « Automotive Handbook ». 5th edition. 2002. Society of Automotive Engineers (SAE)
J.Y. Wong. « Theory of Ground Vehicles ». John Wiley & sons. 1993 (2nd edition) 2001 (3rd edition).
M. Blundel & D. Harty. « The multibody Systems Approach to Vehicle Dynamics » 2004. Society of Automotive Engineers (SAE)
G. Genta. «Motor vehicle dynamics: Modelling and Simulation ». Series on Advances in Mathematics for Applied Sciences - Vol. 43. World Scientific. 1997.
2
INTRODUCTION TO HANDLING
3
Introduction to vehicle dynamics
Introduction to vehicle handling
Vehicle axes system
Tire mechanics & cornering properties of tires
Terminology and axis system
Lateral forces and sideslip angles
Aligning moment
Single track model
Low speed cornering
Ackerman theory
Ackerman-Jeantaud theory
4
Introduction to vehicle dynamics
High speed steady state cornering
Equilibrium equations of the vehicle
Gratzmüller equality
Compatibility equations
Steering angle as a function of the speed
Neutral, understeer and oversteer behaviour
Critical and characteristic speeds
Lateral acceleration gain and yaw speed gain
Drift angle of the vehicle
Static margin
Exercise
5
Introduction
In the past, but still nowadays, the understeer and oversteercharacter dominated the stability and controllability considerations
This is an important factor, but it is not the sole one…
In practice, one has to consider the whole system vehicle + driver
Driver = intelligence
Vehicle = plant system creating the manoeuvering forces
The behaviour of the closed-loop system is referred as the « handling », which can be roughly understood as the road holding
6
Introduction
Model of the system vehicle + driver
7
Introduction
However because of the difficulty to characterize the driver, it is usual to study the vehicle alone as an open loop system.
Open loop refers to the vehicle responses with respect to specific steering inputs. It is more precisely defined as the ‘directional response’ behaviour.
The most commonly used measure of open-loop response is the understeer gradient
The understeer gradient is a performance measure under steady-state conditions although it is also used to infer performance properties under non steady state conditions
8
AXES SYSTEM
9
Reference frames
Local reference frame oxyz attached to the vehicle body -SAE (Gillespie, fig. 1.4)
XY Z
O
Inertial coordinate system OXYZ
10
Reference frames
Inertial reference frame
X direction of initial displacement or reference direction
Y right side travel
Z towards downward vertical direction
Vehicle reference frame (SAE):
x along motion direction and vehicle symmetry plane
z pointing towards the center of the earth
y in the lateral direction on the right hand side of the driver towards the downward vertical direction
o, origin at the center of mass
11
Reference frames
z
x
y x
Système SAE
Système ISO
x z
yComparison of conventions of SAE and ISO/DIN reference frames
12
Local velocity vectors
Vehicle motion is often studied in car-body local systems
u : forward speed (+ if in front)
v : side speed (+ to the right)
w : vertical speed (+ downward)
p : rotation speed about a axis (roll speed)
q : rotation speed about y (pitch)
r : rotation speed about z (yaw)
13
Forces
Forces and moments are accounted positively when acting ontothe vehicle and in the positive direction with respect to the considered frame
Corollary
A positive Fx force propels the vehicle forward
The reaction force Rz of the ground onto the wheels is accounted negatively.
Because of the inconveniency of this definition, the SAEJ670e « Vehicle Dynamics Terminology » names as normal force a force acting downward while vertical forces are referring to upward forces
14
VEHICLE MODELING
15
The bicycle model
When the behaviours of the left and right hand wheels are not too different, one can model the vehicle as a single track vehicle known as the bicycle model or single track model.
The bicycle model proved to be able to account for numerous properties of the dynamic and stability behaviour of vehicle under various conditions.
x,u,p
y,v,qz,w,r
f
Velocity
r
Fyf
Tr
Tf
Fyr
Fxr
Fxf
L
b
c
t
16
The bicycle model
Geometrical data:
Wheel base: L
Distance from front axle to CG: b
Distance from rear axle to CG: c
Track: t
Tire variables
Sideslip angles of the front and rear tires: f and r
Steering angle (of front wheels)
Lateral forces developed under front and rear wheels respectively: Fyf and Fyr.
Longitudinal forces developed under front and rear wheel respectively: Fxf and Fxr.
x,u,p
y,v,qz,w,r
f
Velocity
r
Fyf
Tr
Tf
Fyr
Fxr
Fxf
L
b
c
t
17
The bicycle model
Assumptions of the bicycle model
Negligible lateral load transfer
Negligible longitudinal load transfer
Negligible significant roll and pitch motion
The tires remain in linear regime
Constant forward velocity V
Aerodynamics effects are negligible
Control in position (no matter about the control forces that are required)
No compliance effect of the suspensions and of the body
x,u,p
y,v,qz,w,r
f
Velocity
r
Fyf
Tr
Tf
Fyr
Fxr
Fxf
L
b
c
t
18
The bicycle model
Remarks on the meaning of the assumptions
Linear regime is valid if lateral acceleration<0.4 g
Linear behaviour of the tire
Roll behaviour is negligible
Lateral load transfer is negligible
Small steering and slip angles, etc.
Smooth floor to neglect the suspension motion
Position control of the command : one can exert a given value of the input variable (e.g. steering system) independently of the control forces
The sole input considered here is the steering, but one could also add the braking and the acceleration pedal.
19
The bicycle model
Assumptions :
Fixed: u = V = constant
No vertical motion: w=0
No roll p=0
No pitch q = 0
Bicycle model = 2 dof model :
r=wz, yaw speed
v, lateral velocity or b, side slip of the vehicle
Vehicle parameters:
m, mass,
Jzz inertia bout z axis
L, b, c wheel base and position of the CG
20
The bicycle model
x,u,p
y,v,qz,w,r
f
Velocity
f
Velocity
rr
Fyf
Fyr
Fyf
Tr
Tf
rv
u Vb
Fyr
Fxr Fxr
FxfFxf
L
b
c
h
m, J
21
LOW SPEED TURNING
22
Low speed turning
At low speed (parking manoeuvre for instance), the centrifugal accelerations are negligible and the tire have not to develop lateral forces
The turning is ruled by the rolling without (lateral) friction and without slip conditions
If the wheels experience no slippage, the instantaneous centres of rotation of the four wheels are coincident.
The CIR is located on the perpendicular lines to the tire plan from the contact point
In order that no tire experiences some scrub, the four perpendicular lines pass through the same point, that is the centre of the turn.
23
Ackerman-Jeantaud theory
24
Ackerman-Jeantaud condition
One can see that
This gives the Ackerman Jeantaud condition
Corollary e · i
25
Ackerman-Jeantaud condition
The Jeantaud condition is not always verified by the steering mechanisms in practice, as the four bar linkage mechanism
26
Jeantaud condition
The Jeantaud condition can be determined graphically, but the former drawing is very badly conditioned for a good precision
In practice one resorts to an alternative approach based on the following property
Point Q belongs to the line MF when the Jeantaud condition is fulfilled
The distance from Q to the line MF is a measure of the error from Jeantaud condition
27
Ackerman theory
rv
u Vb
L
R
centre du virage
vf
vr
RCG
28
Ackerman theory
The steering angle of the front wheels
The relation between the Ackerman steering angle and the Jeantaud steering angles 1 and 2
R=10 m, L= 2500 mm, t=1300 mm
1 = 15.090° 2= 13.305°
= 14.142°
(1+2)/2=14.197°
rv
u Vb
L
R
centre du virage
vf
vr
RCG
29
Ackerman theory
Curvature radius at the centre of mass
Relation between the curvature and the steering angle
Side slip b at the centre of mass
30
Ackerman theory
The off-tracking of the rear wheel set
rv
u Vb
L
R
centre du virage
vf
vr
RCG
31
HIGH SPEED STEADY STATE CORNERING
32
High speed steady state cornering
At high speed, it’s required that tires develop lateral forces to sustain the lateral accelerations.
The tire can develop forces if and only if they are subject to a side slip angle.
Because of the motion kinematics, the CIR is located at the intersection to the normal to the local velocity vectors under the tires.
The CIR, which was located at the rear axel for low speed turn, is now moved to a point in front.
33
High speed steady state cornering
f
f r
v f
v r
34
Dynamics equations of the vehicle motion
Newton-Euler equilibrium equation in the non inertial reference frame of the vehicle body
Model with 2 dof b & r
Equilibrium equations in Fy and Mz :
Operating forces
Tyre forces
Aerodynamic forces (can be neglected here)
f
Velocity
r
Fyf
Fyr
rv
u Vb
Fxr
Fxf
L
a
b M, J
e J x y = 0
e t J y z = 0
nt :
Fy = M ( _v + r u)
N = Jzz _r
35
Equilibrium equations of the vehicle
Equilibrium equations in lateral direction and rotation about z axis
Solutions
The lateral forces are in the same ratio as the vertical forces under the wheel sets
36
Behaviour equations of the tires
Cornering force for small slip angles
C ® =
@ F y
@ ®
® = 0
< 0
Gillespie, Fig. 6.237
Gratzmüller equality
Using the equilibrium and the behaviour condition, one gets
One yields the Gratzmüller equality
38
Compatibility equations
The velocity under the rear wheels are given by
The compatibility of the velocities yields the slip angle under the rear wheels
f
f r
v f
v r
39
Compatibility equations
The velocity under the front wheels are given by
The compatibility of the velocities yields the slip angle under the front wheels
f
f r
v f
v r
40
Steering angle
Steering angle as a function of the slip angles under front and rear wheels
The steering angle can also be expressed as a function of the velocity and the cornering stiffness of the wheels sets
41
Understeer gradient
The steering angle is expressed in terms of the centrifugal acceleration
So
With the understeer gradient K of the vehicle
42
Steering angle as a function of V
Gillespie. Fig. 6.5 Modification of the steering angle as a function of the speed
43
Neutralsteer, understeer and oversteer vehicles
If K=0, the vehicle is said to be neutralsteer:
The front and rear wheels sets have the same directional ability
If K>0, the vehicle is understeer :
Larger directional factor of the rear wheels
If K<0, the vehicle is oversteer:
Larger directional factor of the front wheels
44
Characteristic and critical speeds
For an understeer vehicle, the understeer level may be quantified by a parameter known as the characteristic speed. It is the speed that requires a steering angle that is twice the Ackerman angle (turn at V=0)
For an oversteer vehicle, there is a critical speed above which the vehicle will be unstable
45
Lateral acceleration and yaw speed gains
Lateral acceleration gain
Yaw speed gain
46
Lateral acceleration gain
Purpose of the steering system is to produce lateral acceleration
For neutral steer, lateral acceleration gain is constant
For understeer vehicle, K>0, the denominator >1 and the lateral acceleration is reduced with growing speed
For oversteer vehicle, K<0, the denominator is < 1 and becomes zero for the critical speed, which means that any pertubation produces an infinite lateral acceleration
47
Yaw velocity gain
The second raison for steering is to change the heading angle by developing a yaw velocity
For neutral vehicles, the yaw velocity is proportional to the steering angle
For understeer vehicles, the yaw gain angle is lower than proportional. It is maximum for the characteristic speed.
For oversteer vehicles, the yaw rate becomes infinite for the critical speed and the vehicles becomes uncontrollable at critical speed.
48
Yaw velocity gain
Gillespie. Fig. 6.6 Yaw rate as a function of the steering angle
49
Sideslip angle
Definition (reminder)
Value
Value as a function of the speed V
Becomes zero for the speed
independent of R !
50
Sideslip angle
Gillespie. Fig. 6.7 Sideslip angle for a low speed turn
Gillespie. Fig. 6.8 Sideslip angle for a high speed turn
This is true whatever the vehicle is understeer or oversteer
b > 0 b < 0
51
Static margin
The static margin provides another (equivalent) measure of the steady-state behaviour
Gillespie. Fig 6.9 Neutral steer linee>0 if it is located in front of the vehicle centre of gravity
52
Static margin
Suppose the vehicle is in straight line motion (=0)
Let a perturbation force F applied at a distance e from the CG (e>0 if in front of the CG)
Let’s write the equilibrium
The static margin is the point such that the lateral forces do not produce any steady-state yaw velocity
That is:
53
Static margin
It comes
So the static margin writes
A vehicle is
Neutral steer if e = 0
Under steer if e<0 (behind the CG)
Over steer if e>0 (in front of the CG)
54
Static margin
Gillespie. Fig. 6.10 Maurice Olley’s definition of understeer and over steer
55
Exercise
Let a vehicle A with the following characteristics:
Wheel base L=2,522m
Position of CG w.r.t. front axle b=0,562m
Mass=1431 kg
Tires: 205/55 R16 (see Figure)
Radius of the turn R=110 m at speed V=80 kph
Let a vehicle B with the following characteristics :
Wheel base L=2,605m
Position of CG w.r.t. front axle b=1,146m
Mass=1510 kg
Tires: 205/55 R16 (see Figure)
Radius of the turn R=110 m at speed V=80 kph
56
Exercise
Rigidité de dérive (dérive <=2°) :
0
200
400
600
800
1000
1200
1400
1600
1800
0 1000 2000 3000 4000 5000 6000
Charge normale (N)
Rig
idit
é (
N/°
)
175/70 R13
185/70 R13
195/60 R14
165 R13
205/55 R16
57
Exercise
Compute:
The d’Ackerman angle (in °)
The cornering stiffness (N/°) of front and rear wheels and axles
The sideslip angles under front and rear tires (in °)
The side slip of the vehicle at CG (in °)
The steering angle at front wheels (in °)
The understeer gradient (in °/g)
Depending on the case: the characteristic or the critical speed (in kph)
The lateral acceleration gain (in g/°)
The yaw speed velocity gain (in s-1)
The vehicle static margin (%)
58
Exercise 1
Data
mb 562,0 2,522 0,562 1,960c L b m
2228,0L
b0,7772
c
L
kgm 1431
NL
cmgW
f8714,10909 3127,6909r
bW mg N
L
smkphV /2222,2280
²/4893,4²
smR
Va
y mR 110
59
Exercise 1
Ackerman angle
Tire cornering stiffness of front wheels
Tire cornering stiffness of rear wheels
3134,10229,0)0229,0arctan()arctan( radR
L
NL
cmgW
f8714,10909
NFz
93,5454
deg/1550)1( NCf
3110 / deg 177616,98 /fC N N rad
3127,6909r
bW mg N
L
1563,84zF N
(1) 500 / degrC N
1000 / deg 57295,8 /rC N N rad 60
Exercise 1
Side slip angles under the front tires
NR
Vm
L
cFyf 879,54992
²
²/4893,4²
smR
Va
y Nma
y1883,6424
yfffFC
3110 / degfC N
radC
F
f
yf
f0281,06106,1
61
Exercise 1
Side slip angles under the rear tires
Side slip angle at CG
²1431,3092yf
b VF m N
L R
r r yrC F 1000 / degrC N
1,4313 0,0250yr
r
r
Frad
C
rr
R
c
V
crb
1,9600,0250 0,0072 0,4105
110radb
62
Exercise 1
Steering angle at front wheels
Understeer gradient
R
V
C
Lmb
C
Lmc
R
L
rf
²//
f r
L
R
1,3134 1,6106 1,4313 1,4927
/ / 1112,1732 318,8268
3100 1000f r
mc L mb LK
C C
2/0399,0 msK ' * 0,3918 deg/K K g g
63
Exercise 1
Understeer gradient: check!
Characteristic speed
r
r
f
f
gC
W
gC
WK
R
VK
R
L ²3,57
4925,14893,43134,1 K
R
L2
K
LV
carac
²//49639,6deg/0399,0 2 smradEmsK
2,52260,1793 / 216,64
6,9639 4carac
LV m s kph
K E
64
Exercise 1
Lateral acceleration gain
Yaw speed gain
deg/3066,04927,1
81,9/4893,4g
aG y
ay
/ 22,222 /1107,7543 deg/ / deg
1,4927r
r V RG s
65
Exercise 1
Neutral maneuver point
Static margin
10003100
1000.9600,13100.562,0
rf
rf
CC
cCbCe
me 0531,0
%11,2L
e
66