MECA0525 : Vehicle dynamics Pierre Duysinx Research Center in Sustainable Automotive Technologies of University of Liege Academic Year 2020-2021 1
MECA0525 : Vehicle dynamics
Pierre DuysinxResearch Center in Sustainable Automotive
Technologies of University of Liege
Academic Year 2020-2021
1
Lesson 1: Steady State Cornering
2
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.
3
INTRODUCTION TO HANDLING
4
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
5
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
6
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 closed loop system vehicle + driver
◼ Driver = intelligence
◼ Vehicle = plant system creating the manoeuvring forces
◼ The behaviour of the closed-loop system is referred as the « handling », which can be roughly understood as the road holding
7
Introduction
Model of the system vehicle + driver
8
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
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AXES SYSTEM
10
Reference frames
Local reference frame oxyz attached to the vehicle body -SAE (Gillespie, fig. 1.4)
XY Z
O
Inertial coordinate system OXYZ
11
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
12
Reference frames
z
x
y x
Système SAE
Système ISO
x z
yComparison of conventions of SAE and ISO/DIN reference frames
13
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 x axis (roll speed)
◼ q : rotation speed about y (pitch)
◼ r : rotation speed about z (yaw)
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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
15
VEHICLE MODELING
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The bicycle model
◼ When the behaviours of the left and right hand wheels are not that much 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
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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
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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
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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 ground 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.
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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 about z axis
◼ L, b, c wheel base and position of the CG
21
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
22
LOW SPEED TURNING
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Low speed turning
◼ At low speed (parking manoeuvre for instance), the centrifugal accelerations are negligible and the tire have not to develop any lateral forces
◼ The turning is ruled by the rolling conditions 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 have to pass through the same point, that is the centre of the turn.
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Ackerman-Jeantaud theory
25
Ackerman-Jeantaud condition
◼ One can see that
◼ This gives the Ackerman Jeantaud condition
◼ Corollary
26
Ackerman-Jeantaud condition
◼ The Jeantaud condition is not always verified by the steering mechanisms in practice, as the four bar linkage mechanism
27
Genta Fig. 5.2
Jeantaud condition
◼ The Jeantaud condition can be determined graphically, but the former drawing is very badly conditioned for a good precision
◼ Actually, 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
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Wong Fig. 5.2
Wong Fig. 5.4
Ackerman theory
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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°
30
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
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Ackerman theory
◼ The off-tracking of the rear wheel set
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HIGH SPEED STEADY STATE CORNERING
33
High speed steady state cornering
◼ At high speed, the tires have todevelop 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 kinematics of the motion, the IC is located at the intersection of the normal lines to the local velocity vectors under the tires.
◼ The IC, which was located at the rear axel for low-speed turn, is now moving to a point in front.
34
High speed steady state cornering
− f
f r
v f
v r
35
Sideslip angles have been assumed to be on left side of the wheel. We consider the modulus of .
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)
e J x y = 0
e t J y z = 0
36
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
◼ Inertia tensor
37
e J x y = 0
e t J y z = 0
Dynamics equations of the vehicle motion
◼ It comes
◼ And finally
◼ The only nontrivial equations are
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Dynamics equations of the vehicle motion
◼ Circular motion
◼ wz: rotation speed about vertical axis
◼ V tangent velocity
◼ R radius of the turn
◼ Steady state
◼ Equation of motion
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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
40
Equilibrium equations of the vehicle
◼ Solving
◼ Can be made by using Horner rule
41
Behaviour equations of the tires
◼ Cornering force for small slip angles
Gillespie, Fig. 6.242
Gratzmüller equality
◼ Using the equilibrium and the behaviour condition, one gets
◼ One yields the Gratzmüller equality
43
Compatibility equations
◼ Compatibility equation consists in evaluating the side slip angles in terms of the velocities
44
Because of assumption r<0!
Because of steering action !
Compatibility equations
◼ Evaluation of velocities under front and rear axles thanks to the Poisson transport equation
45
Compatibility equations
◼ The velocity under the rear wheels are given by
◼ The compatibility of the velocities yields the slip angle under the rear wheels
46
Compatibility equations
◼ The velocity under the front wheels are given by
◼ The compatibility of the velocities yields the slip angle under the front wheels
47
Steering angle
◼ Steering angle as a function of the slip angles under front and rear wheels
◼ This gives relation between the steering angles and the
48
Correction due to side slipAckerman angle
Steering angle
◼ Steering angle as a function of the slip angles under front and rear wheels
◼ Let’s insert the expression of the side slip angles in terms of lateral forces and cornering stiffness
49
Steering angle
◼ The expression of the steering angle as a function of the slip angles under front and rear wheels
◼ Inserting the values of the side slip angles as a function of the velocity and the cornering stiffness of the wheels sets yields
◼ Or
◼ with
50
Understeer gradient
◼ The steering angle is expressed in terms of the centrifugal acceleration
◼ So
◼ With the understeer gradient K of the vehicle
51
Steering angle as a function of V
Gillespie. Fig. 6.5 Modification of the steering angle as a function of the speed
52
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
53
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
54
Lateral acceleration and yaw speed gains
◼ Lateral acceleration gain
◼ Yaw speed gain
55
Lateral acceleration gain
◼ Purpose of the steering system is to produce lateral acceleration
◼ For neutral steer, K=0 and the lateral acceleration gain is increasing constantly with the square of the speed : V²/L
◼ 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 perturbation produces an infinite lateral acceleration
56
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 and increases with the speed (slope 1/L)
◼ 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.
57
Yaw velocity gain
Gillespie. Fig. 6.6 Yaw rate as a function of the steering angle
58
Sideslip angle at centre of mass
◼ Definition (reminder)
◼ Value
◼ Value as a function of the speed V
◼ Becomes zero for the speed
independent of R !
59
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
60
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
61
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 perturbation lateral forces F do not produce any steady-state yaw velocity
◼ That is:
62
Static margin
◼ It comes
◼ So the static margin writes
◼ A vehicle is
◼ Neutral steer if e = 0
◼ Under steer (K>0) if e<0 (behind the CG)
◼ Over steer (K<0) if e>0 (in front of the CG)
◼ Remember that
63
Static margin
Gillespie. Fig. 6.10 Maurice Olley’s definition of understeer and over steer
64
Exercise
◼ Let a vehicle A with the following characteristics:
◼ Wheelbase 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 :
◼ Wheelbase 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
65
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
66
Exercise
◼ Compute:
◼ The 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 (%)
67
Exercise 1
◼ Data
mb 562,0= 2,522 0,562 1,960c L b m= − = − =
2228,0=L
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=
68
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 = =69
Exercise 1
◼ Side slip angles under the front tires
²4992,91yf
c VF m N
L R= =
²/4893,4²
smR
Va
y== Nma
y1883,6424=
yfffFC =
3110 / degfC N =
radC
F
f
yf
f0281,06106,1 ===
70
Exercise 1
◼ Side slip angles under the rear tires
◼ Side slip angle at CG
²1431,3092yr
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 = − = − = −
71
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= =
72
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= = = =
−73
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
= = = =
74
Exercise 1
◼ Neutral maneuver point
◼ Static margin
75