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Major Course ContentsMajor Course Contents
Part 1: Lateral Vehicle Dynamics
Part 2: Longitudinal Vehicle DynamicsPart 2: Longitudinal Vehicle Dynamics
Part 3: Vehicle Control Systems
Part 4: Suspensions
Part 5: Three-dimensional rigid body model of a vehicle
Major Course Contents
Part 2: Longitudinal Vehicle Dynamics
j
Part 2: Longitudinal Vehicle Dynamics2.1 Longitudinal Dynamic Model2.2 Engine model 2.3 Transmission2 4 Ti d l2.4 Tire models2.5 Brake
P t 3 V hi l C t l S tPart 3: Vehicle Control Systems3.1 Driver Model
driver-in-the-loop systems3 2 Lateral Stability Control (sec 1 9)3.2 Lateral Stability Control (sec. 1.9)3.3 Lane Keeping Systems 3.4 Adaptive Cruise Control3.5. Tire-road friction estimation 3.6 Tire force estimation 3.7 Vehicle state estimation:
- side slip angle, vehicle speed (3 5 Autonomous Driving Systems)(3.5 Autonomous Driving Systems)(3.8 Fault detection and Failsafe Systems)
Part.2L it di l V hi l D iLongitudinal Vehicle Dynamics
1 Longitudinal Dynamic Model1. Longitudinal Dynamic Model
2. Engine model
3. Transmission
4. Tire models
5. Brake
3
Vehicle Model
3D Vehicle Model Powertrain Model (2-Wheel Drive)Yaw
Pitch
Roll
Z
Y
X
Y
4-QuarterCar Model
• Engine Model
Roll bar
• Engine Model• Torque Converter• Transmission• Axle Shaft6 DOF V hi l B d • Axle Shaft• Differential Gear
• 6-DOF Vehicle Body• 4-Quarter Car Model• Tire Model
4 Wheel Dynamic Model
4
• 4-Wheel Dynamic Model• 14 DOF Vehicle Model
1. Longitudinal Dynamic Model
▪ Free Body Diagram of Vehicle Body
aF
aMv
rfM
M T
FbfT
rrMsT
gMv tF
trfFbrT
gv trrF
5
1. Longitudinal Dynamic Model
Equations of Motion for Vehicle Body
Vehicle Body where,Vehicle Body
Ltrrtrfv FFFdtdvM
where,
: vehicle mass
: tractive forces of front/rear wheels
vM
trrtrf FF ,
Driving Load
singMFF vaL
/
: aerodynamic force
: gravitational constant
trrtrf
aFg
Rear wheel (driving)
gvaL
: moment of inertia of front/rear wheel
: driving shaft torquewrwf JJ ,
sT
brrrtrrrsr
wr TMFrTdt
dwJ
F h l (d i )
: radius of front/rear wheel
: rolling resistance of front/rear wheel
rf rr ,
rrrf MM ,Front wheel (driven)
bfrftrfff
wf TMFrdt
dwJ
: brake torque of front/rear wheelbrbf TT ,
6
1. Longitudinal Dynamic Model
Aero Dynamic (Drag Force)Aero Dynamic (Drag Force)
2wind
1 ( )2a d F xF C A v v 2
where,
: the mass density of air y
: the aero dynamic drag coefficient
: the front area of the vehicle, which is the projected area of the
dC
FAvehicle in the direction of travel
: the longitudinal vehicle velocity
F
xv: the wind velocity, positive for a head wind and negative for a tail windwindv
1.6 0.00056 ( 765)FA m
Example for 800~2000 kg vehicle
7Wong, 2001 in Rajamani, page.97~99
1. Longitudinal Dynamic Model
Rolling Resistance
▪ Tire rotates, both the tire and the road are subjected to deformation in the contact patch.
▪ The loss of energy in tire deformation results in viscous dissipation.
Contact Patch
tziFtziAsymmetric Normal Force Distribution
8
1. Longitudinal Dynamic Model
Rolling ResistanceRolling Resistance
▪ The moment due to the offset normal load, , is balanced by the moment due to the riM
rolling resistance force, .static xir R
ibiT
siTz
si
x
xiRix
staticr
F
riM
tziF tziF
9
1. Longitudinal Dynamic Model
Rolling ResistanceRolling Resistance
▪ Typically, the rolling resistance is modeled as being roughly proportional to the normal
force on each tires, i.e.
( )xi zi ziR f F f F xi zi zi
where, 0.01 ~ 0.05 0.015f
is typically for passenger cars.yp y p g
▪ The resulting rolling resistance due to tire deformation and asymmetric normal force
distribution
i i i i iM r R F x ri static xi zi iM r R F x
10
Part.2L it di l V hi l D iLongitudinal Vehicle Dynamics
1 Longitudinal Dynamic Model1. Longitudinal Dynamic Model
2. Engine model
3. Transmission
4. Tire models
5. Brake
11
Powertrain and Brake Systemy2. Engine model
12
2. Engine model
2 1 2-state Engine model2.1 2 state Engine model
aimVmm
aom
1) Define of States of 2-state engine model
: manifold pressure
: engine velocityemP
RT2) State equations
air
mairmm M
RTmVP ( )m
m airair m
RTP mM V
.constTm derivative
13
e e i f pJ T T T air ai Vm aom m m m Where,
2 1 2-state Engine model
2. Engine model
Where
: ni ersal gas constant ( )R lKN /3148
2.1 2 state Engine model
: universal gas constant ( )
: the molecular weight of air ( )
: indicated torque( )
R molKmN /314.8
airMT
kmolkg /964.28mN: indicated torque( )
iT mN
( ) 5.48, ( )
ao iti T it
m t tT c tt t
: mass of air in the intake manifold
( )e it et t
airm : mass of air in the intake manifold
: the intake manifold volume(0.0027 m3)
: the torque constant(498636 )TcmVair
mN q ( )
: engine speed( )
: friction torque : pump torque
e
fTsrad /
pT
T sec/kg
14
q p p q
: mass flow rate of air from the vacuum boosterVmmf p
2.1 2-state Engine model
2. Engine model
1ao e vol airm c m
2.1 2 state Engine model
Fuel mass flow rate into the cylinders
m
e
VVc41
m
mmairair RT
VPMm where
)352.01010.8()222167.0()1010.35.24( 424 eaireairevol mm
eV : the engine displacement ( 0.0038 )3m
m MAX TC PRI Fuel mass flow rate into the manifold
aim MAX TC PRI
MAX :the maximum flow rate ( 0.1843 )sec/kgwhere
TC )0660.114459.1cos(1 1
46.79 46.79
15
19exp1
atm
m
PPPRI
2.1 2-state Engine model
2. Engine model
2.1 2 state Engine model
Normalized throttle angle Normalized pressure ratio influence
1.0
c
1.0
e
0.6
0.8
arac
teris
tic
0.6
0.8
tio In
fluen
ce
0.4
Thro
ttle
Ch
0.4
ress
ure
Rat
0 15 30 45 60 75 900.0
0.2
T
0 0 0 2 0 4 0 6 0 8 1 00.0
0.2
PThrottle Angle [deg]
0.0 0.2 0.4 0.6 0.8 1.0
Pressure Ratio (Pm/Patm)
16
2.1 2-state Engine model (Summary)
2. Engine model
2.1 2 state Engine model (Summary)
State equations
( )mRTP f P
( , , )
1 ( , )
mm ai Vm ao p e m
air m
e i f p e m
P m m m f PM V
T T T f PJ
aim MAX TC PRI where
eJ
Vmm : mass flow rate of air from the vacuum booster
( ) 5 48m t t
( ) 5.48, ( )
ao iti T it
e it e
m t tT c tt t
1ao e vol airm c m
eVc1 V : the engine displacement ( 0.0038 m3 )mmairair
VPMm
17
mVc
41 eV : the engine displacement ( 0.0038 m )m
air RTm
2.1 2-state Engine model (Summary)
2. Engine model
2.1 2 state Engine model (Summary)
State equations
( )mRTP f P
( , , )
1 ( , )
mm ai Vm ao p e m
air m
e i f p e m
P m m m f PM V
T T T f PJ
)352.01010.8()222167.0()1010.35.24( 424 eaireairevol mm where
eJ
aim MAX TC PRI
MAX :the maximum flow rate(0 1843 )sec/kgMAX :the maximum flow rate(0.1843 )sec/kg
TC )0660.114459.1cos(1 1
46.79 4679 1 46.79
19e p1 mPPRI
18
19exp1atm
m
PPRI
2. Engine model
2.2 1-state Engine model
Engine model using Engine Map
Engine
Throttleangle
NetEngine Map
Engine model using Engine Map
gmapEngine
speed
torque
200
240
280Throttle Angle 90.0
63.0
36.0
m]
80
120
160 25.224.423.622.6
19.916.4
15.7rque
[Nm
-40
0
4015.0
12.310.0
5.60.0Sh
aft t
or
1000 2000 3000 4000 5000 6000
-80
Time [sec]
19
[ ]
2. Engine model
2.2 1-state Engine model
1) Definition of States of 1-state engine model
e : Engine Velocity
1d2) State equations
)),(),((1tepenet
e
e wwTwTJdt
d
where, : Engine angular speed : Throttle angle
Je
: Effective engine inertia : Turbine angular speed
: Net torque
eJ
netTtw
: Pump torque
20
pT
Powertrain and Brake Systemy2. Engine model (Torque Converter)
21
2. Engine model
Torque Converter
2.3 Torque Converter
turbine
pumpstator
Fundamentals of Torque Converter
22
2efp wCT fC : Capacity Factor
2.3 Torque Converter Model 1
efp
ptt TRT
f
tR : Torque Ratio
Torque Converter Capacity Factor Map Torque Converter Torque Ratio Map0.004
c2 ]
2.4
0 000
0.002
ctor
[ N
sec
1 6
2.0
e R
atio
-0.002
0.000
paci
ty F
ac
1.2
1.6
Torq
ue
0.0 0.2 0.4 0.6 0.8 1.0 1.2-0.004
Cap
S d R ti
0.0 0.2 0.4 0.6 0.8 1.0 1.20.8
w( )tw23Speed Ratio Speed Ratio ( )t
e
ww
( )t
ew
2 C
2.3 Torque Converter Model 1
2efp wCT
tt TRT
fC
tR
: Capacity Factor
: Torque Ratioptt TRT tR : Torque Ratio
ASME Paper June 20003 2 3 31.695 10 0.2946 10 1.874 10fC
1.412 2.2 0.9
1.412 2.2 0.91 0.9tR
i d titw is speed ratio.t
ew
3 2 3 3 21 874 10 0 2946 10 1 695 10T 3 2 3 3 21.874 10 0.2946 10 1.695 10p p p t tT w w w w
3 2 2 32.3933 4.1228 1.9979 4.1450 10 0.9p p t tw w w wT
24
0.9
p p t tt
p
TT
)90/( ww 22 wcwwcwcT
2.3 Torque Converter Model 2 (Static Model by Kotwichi – Simple)
)9.0/( pt ww 321 ttppp wcwwcwcT 2
652
4 ttppt wcwwcwcT Counter Mode
)9.0/( pt ww 298
27 ttppp wcwwcwcT
pt TT
w: turbine torqueT : turbine speedwhere
Fluid Coupling Mode
twpw
91 ~ cc
: turbine torque
: pump torque
: constantspT
tT : turbine speed: pump speed
where,
91 cc : constants
ASME Paper June 2000
1.8740X10-3 4.6501X10-3 1.8740X10-3
0.2946X10-3 -1.9979X10-3 0.2946X10-3
1c
2c4c
5c7c
8c
25
-1.6950X10-3 -54.1450X10-3 -1.6950X10-33c 6c 9c
Part.2L it di l V hi l D iLongitudinal Vehicle Dynamics
1 Longitudinal Dynamic Model1. Longitudinal Dynamic Model
2. Engine model
3. Transmission
4. Tire models
5. Brake
26
Powertrain ModelPowertrain Model
• Engine Model• Torque Converter• Transmission• Transmission• Axle Shaft• Differential Gear
27
3. Transmission
wrSchematics of Power Transmission
t crFinal
ReductionTransmission
sT/CeA/T dREngine
Differential gear
Speed ratio
tgicr wRw giR : ith gear ratio
pwhere,
crds wRw dR
sw : driving shaft velocity
: final drive speed reduction ratio
28
s g y
3. Transmission
1 Tdw
Shaft Model
)(1sd
gi
t
cri
cr TRRT
Idtdw
)( wrcrdss wwRKT
Where ,
I sK: carrier inertia at I-th gear ratio : stiffness of driving shaftcriI
giRcrw
T
dRs: carrier inertia at I th gear ratio
: i-th gear ratio
: shaft torque
: final drive speed ratio reduction
: carrier angular velocity
sTwrw: shaft torque
: rear wheel angular velocity
29
3. Transmission
Differential Gear– Allow different speed on left and right when turning– Distribute equal driving torque on both sides
G4G4G1
G4 G2G1 G2
G3
G1
G2G3G3
(a)(b)
30
G1, G2 : Planet (Differential Gear)G3, G4 : Sun Gear
3. Transmission
Differential Gear– Sun Gear
slsilsls TrFJ
srsirsrs TrFJ
)(
221
22 srsldssd
sd
s
appil TT
rJrJrJ
rT
F )(22
2
,
,
,srsl
stotr
totrr
stotr
sapp TT
JJJ
TJJ
JT
)(
221
22 srsldssd
sd
s
appir TT
rJrJrJ
rT
F 2, dgdrtotr rmJJ
31
3. Transmission
Differential Gear– Ring Gear
)()2( , srslrrstotr TTTJJ )()2( , srslrstotrr TTJJT
2 2 2{ ( 2 ) } ( )J R J J R R T R R T T ,{ ( 2 ) } ( )ti i r tot s i d r t i d sl srJ R J J R R T R R T T
)(2, tistotr TTJT
JJT
32
)(2)2( ,,
srsltistotr
tditistotr
r TTJJJ
TRRJJJ
T
3. Transmission
• Differential Gear– Comparison of differential gear models
Di id d d l Di t ib t 50% f d i i t b th id• Divided model : Distribute 50% of driving torque on both sides• Simple model : Individually calculate driving torque for each side
100kph 15deg step steer– 100kph, 15deg step steer
33
3. Transmission
• Differential Gear– Comparison of differential gear models– 100kph, 15deg step steer
34
Part.2L it di l V hi l D iLongitudinal Vehicle Dynamics
1 Longitudinal Dynamic Model1. Longitudinal Dynamic Model
2. Engine model
3. Transmission (Simplified Model)
4. Tire models
5. Brake
35
3. Transmission (Simplified Model)
(1) S t I t(1) System Input
: Turbine TorquetTT : Brake Torque
: Longitudinal Tire Force
bT
txiF
(2) System Output
: i-th gear ratiogiR
: Turbine Velocity
: Wheel Speed at each wheelit
ttT3.
bT
txiF
Transmission
And
36giR iWheel Dynamic Model
3. Transmission (Simplified Model)
bfLT xfLF
/ 2T
fL(3) Shaft Model
giR DGdRtT
cr scrT sT
/ 2sT
t
1cri cr t d s
gi
I w T R TR
bfRT xfRF
cr s
/ 2sTWhere,
1 1w w w f
fRt cr s
gi gi d
w w wR R R
bfL bfRT T
TgiR Equivalent
WheeldRtT
xfL xfRF Fcr scrT sT
t
37
fL fR
(3) Simplified Wheel Dynamic Model
3. Transmission (Simplified Model)
(3) Simplified Wheel Dynamic Model
• Relation between Shaft Speed and Wheel Speed
1 1 1 1 2 2 2 2
fL fR fL fRs s s fL s fR s sT T T T
A i th t• Assuming that: fL fR f 2
2 2fL fR f
s f
bfLT xfLFfL
bfL bfRT T
2 2
sT/ 2sT
sT
s/ 2sT
xfL xfRF Fs
fL fR
38
bfRT xfRF
fR
(3) Simplified Wheel Dynamic Model
3. Transmission (Simplified Model)
(3) Simplified Wheel Dynamic Model
• Driving Wheel (Front Wheel)
d xfL xfR bfL bf
fwR RwL s F
dJ J T r
dtF T T
• Driven Wheel (Rear Wheel) Driven Wheel (Rear Wheel)
xrL xrR brLr
wL wR brRdJ F F TJ rdt
T
bfLT xfLFfL
bfL bfRT T
sT/ 2sT
sT
s / 2sTxfL xfRF Fs
fL fR
39
bfRT xfRF
fR
3. Transmission (Simplified Model)
(4) Transmission and Wheel Dynamic Model(4) Transmission and Wheel Dynamic Model
• Speed Relation
=f s d cr gi d tR w R R w
• Shaft Model
1fcr cri dwdw II T R T cri t d sd gi
I T R Tdt R dt R
• Simplified Wheel Dynamic Model (Front Wheel)
fdF F T TJ J T r
xfL xfR bfL bfR
xfL xfR bfL bf
wL wR s
fs wL RwR
F F T TJ J T rdt
dT J J r
dF
tF T T
40
dt
3. Transmission (Simplified Model)
(4) Transmission and Wheel Dynamic Model(4) Transmission and Wheel Dynamic Model
1fcrid s
dtT
dwI R TR dt R
1
d gi
ft d xfL xfR bwL wR
gifL bfRF F T
R dt R
dT R J J r
RT
dt
gi
• Simplified Transmission and Wheel Dynamic Model (Front Wheel)
1fcrid wL wR d d
d gt xfL xfR bfL b
ifRT
dwI R J J FR r RR d
F T Tt R
bfL bfRT T
giR dRtTF F
crT sT
41
xfL xfRF Fcr st
fL fR
3. Transmission (Simplified Model)
(5) Summary(5) Summary
1dwI
• Simplified Transmission and Wheel Dynamic Model (Front Wheel)
1fcrid wL wR d d
d gt xfL xfR bfL b
ifRT
dwI R J J FR r RR d
F T Tt R
• Driven Wheel (Rear Wheel) Driven Wheel (Rear Wheel)
xrL xrR brLr
wL wR brRdJ F F TJ rdt
T
=f s d cr gi d tR w R R w
• Speed Relation
ttT3.
bT
txiF
Transmission
And
42giR iWheel Dynamic Model
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