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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


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


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


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


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


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



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


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. 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


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


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



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



2. Engine model

3. Transmission (Simplified Model)

4. Tire models

5. Brake

35


(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


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



• 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




• 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


(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


(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


(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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