basics of automotive engineering part 3: basics of vehicle...

University of Novi Sad

FACULTY OF TECHNICAL SCIENCES

Basics of Automotive Engineering

Part 3:

Basics of Vehicle Dynamics

Dr Boris Stojić, Assistant Professor

Department for Mechanization and Design Engineering

Chair for Engines and Vehicles

Part 3:

IntroductionIntroduction

Basics of Vehicle DynamicsBasics of Vehicle Dynamics

IntroductionIntroduction

Introduction

• Tasks and contents of vehicle dynamics

Basic overview

Some basic topics: to study...

...interaction between vehicle and its surroundings, w/ or w/o driver

...how forces influence vehicle motion and vice versa...how forces influence vehicle motion and vice versa

...what the vehicle response will be in certain driving situation

...how design changes will affect vehicle behavior

etc.

Introduction

• Approaches and assumptions

Basic overview

Full vehicle model, general driving situation:

Many degrees of freedom

Many inputs and outputs, complex relationships

carthrottle.com

Many inputs and outputs, complex relationships

System of coupled non-linear differential equations

Not appropriate for analytical study

Experimental approach, CAE modeling and simulations

popularmechanics.com

carsim.com

newslincolncounty.com

Introduction


Basic overview

Simplified models, restricted driving maneuvers:

Less DOF

Restricted number of I/O’sRestricted number of I/O’s

Possibility of ODE linearization

Manageable math, appropriate for analytical study

Gaining insight into main physical relationships

Some aspects of basic engineering analysis carried out easily

Introduction


Basic overview

Longitudinal vehicle dynamics

Forces and motions in longitudinal direction, smooth road surface

Predicting top speed, acceleration and braking performances, gradeability, fuel Predicting top speed, acceleration and braking performances, gradeability, fuel consumption...

ni.com

Introduction


Basic overview

Lateral vehicle dynamics

Forces and motions mainly in lateral direction

Predicting cornering performances, handling, stability...Predicting cornering performances, handling, stability...

tut.fi

Introduction


Basic overview

Vertical vehicle dynamics

Forces and motions mainly in vertical direction

Ride, vibration behavior, tyre/road contact...Ride, vibration behavior, tyre/road contact...

scielo.br

Introduction

• Examples of usage in engineering and everyday life

Basic overview

Let’s name a few...

What is the maximum velocity of the vehicle?

How many horsepower does the vehicle need?How many horsepower does the vehicle need?

What will be the fuel consumption of the vehicle?

How long does it take for the vehicle to come to stop?

What happens if shock-absorbers don’t work?

How to re-gain lost adhesion of the tyre?

What happens when the brake is suddenly applied during cornering?

Understanding active vehicle safety!

And so on and on and on...

Introduction

• 3rd law

• Reaction forces, Free-body diagram

• Important application: no action without reaction!

Overview of Newtonian laws of motion

• 1st law: body equilibrium

• Net force

• 2nd law: force, mass, acceleration

• Rotational motion: torque, moment of inertia, angular acc.

Introduction

• Some important reaction forces:

Overview of Newtonian laws of motion

Ground force

Friction (adhesion) forceF

FFT

T

FT

Introduction

Engine-to-wheel torque transmission

output

inputg

n

ni Definition of gear ratioDefinition of gear ratio

Poutput = Pinputg

Transmission power loss:Transmission power loss:

Transmitted torque:Transmitted torque: Toutput = Tinputigg

Transmission

itr, tr

Vehicles:Vehicles:

Input element:

ENGINE

Output element:

WHEEL

ne, Te nw, Tw

Tw = Teitrtrtr

ew

i

nn

Introduction


Transmission system of gear pairs connected in seriesPassenger car: transmission = gearbox + final drive

otomoto.com.au

k-m-p.nl

k-m-p.nl

car-mri.com

otomoto.com.au k-m-p.nl

Tw = Teitrtr

tr

ew

i

nn

iitrtr = i= igg iifftrtr = = gg ff

Introduction


ig = iI, iII, iIII, iIV, ... – FOR EVERY GEARGEAR APPROPRIATE GEAR RATIOGEAR RATIO

LOWER GEAR LARGER GEAR RATIO

E.g. iI = 4.05 || iII = 2.82 || iIII = 1.75 || iIV = 1.04 || iIV = 0.80

Forces acting on the vehicleForces acting on the vehicle


Forces acting on the vehicleForces acting on the vehicle

Forces acting on the vehicle

• Gravity effects

• Aerodynamic forces

• Tyre-road interaction

Overview of forces

• Tyre peculiarities

• Side slip – very special property of pneumatic tyre

• Load dependence – importance of CG position


• Causes axle loads

• Motion resistance on the graded road (vector decomposition)

Gravity force – vehicle weight

a

hCG

WfW

decomposition) a

b

Wr

W

l AMA = 0 Wf·l = W·cos· b – W·sin·hCG

Zi = 0 Wf + Wr = W·cos

= 0:

sinαWh

cosαWb

W CGf

ll

sinαWh

cosαWa

W CGr

ll

Wb

Wf l

Wa

Wr l


• Motion resistance

• Lift force

• Lateral force

Aerodynamic forces

2

vAcF

2

WW

Rill

racingcardynamics.com


• Contact pressure distribution of non-moving tyre

Tyre behaviour: rolling resistance


• Rolling tyre: hysteresis effect


WTF

e

WT

RZ RX

rD

FX

Internal elastic force

Internal friction force


• Rolling tyre: hysteresis effect


WTF

RX = frWT – rolling resistance of a single free-rolling tyre

fr – rolling resistance coefficient

e

WT

RZ RX

rD

FX

fr – rolling resistance coefficient

From: Genta/Morello

Dependency of fr on velocity (example)


• Definition of slip

Tyre behaviour: longitudinal slip

Theoretical wheel speed: vt = rD

Real speed: v

v = v : NO SLIP v

v = vt: NO SLIP

v < vt: DRIVE WHEEL

v > vt: BRAKE WHEELrD

v

v

ωr1

v

vvs TDt

TDt

t

ωr

v1

v

vvs

BRAKE WHEEL

DRIVE WHEEL

s=1: car moving, wheel locked

s=1: car standing, wheel sliding

s=0: wheel rotating freely


• Definition of slip


Rigid wheel: just a geometric interpretation!

FREE WHEEL BRAKING DRIVING

vs=0 vs vs

Real tyre: pronounced elasticity far more complex slip mechanism!


• Slip mechanism and longitudinal force generation


• Looking at the single particle of tyre contact patch

• Undeformed at the beginning

• Longitudinal deformation increases as the particle “travels” through the contact patch“travels” through the contact patch

• Particle tip “glued” to the ground due to adhesion

• (i) Deformation propagates with the velocity vS

• (ii) Local longitudinal force increases with the growth of the deformation




• (i) Deformation u(x) propagates with the velocity vS

u(x)vS

x

vS

rD - vS

No sliding of the particle at the ground but slip exists (vvt)!

rD




• (ii) Local longitudinal force increases with the growth of the deformation

u(x)

x

Ftan(x)

Ftan(x)

x

Local force distribution

Net longitudinal force – sum of elementary (local) tangential forces

Proportional to the area under the line




Low torqueLow deformationLow slipLow longitudinal force

High torqueHigh deformationHigh slipHigh longitudinal force




RX

Where does this nonWhere does this non--linearity come from??linearity come from??

Net longitudinal

s

RX linearity come from??linearity come from??longitudinal force

Wheel slip

We said: particle tip remains “glued” to the road!

This can not be true all the time –there is not enough adhesion at the end of the patch!




Local vertical force distribution

Maximum AVAILABLElocal longitudinal force =

= Local vertical force Friction coeff.




xA0 B

x

Zone of local particle slidingRequired long. force(zone of stick)

1 2Available long. force




Area increases force increases

Not linearly with slip anymore!Not linearly with slip anymore!

s

RX

Torque increasesDeformation increasesSlip increases

Further slip increase Further slip increase ––what will happen now?what will happen now?




RXNet longitudinal force

•• Further slip increase Further slip increase –– whole contact patch slideswhole contact patch slides

•• Tyre friction decreases with sliding velocity increaseTyre friction decreases with sliding velocity increase

•• Net force decreasesNet force decreases

PEAK POINTPEAK POINT

s

force

Wheel slip

•• Net force decreasesNet force decreases

Chassis Handbook


• Introduction of adhesion coefficient


T

X

W

R

longitudinal force

tyre/axle load (vertical force)

FSometimes we use approximation: traction force FT instead of real longitudinal force RX:

T

T

W

F

What’s the difference?

TW

WT

WT

RX

FXT

DD

WX W

r

e

r

TR

FTfr

frWT = Frol

RX = FT – Frol When FT >> Frol RX FT

rD

e


• Introduction of adhesion coefficient


MAX

s<MAX

s

s<MAX

s=100%s10-15%

MAX will decrease with load increase!

RXMAX

WT


• Some example values of adhesion coefficient


s (%)

From: Wallentowitz


• Introduction of tyre side slip angle

Tyre behaviour: side slip

When subjected to side force, tyre rolls at an angle with respect to longitudinal axis

TYRE TRAVELLING DIRECTION

LONGITUDINAL DIRECTION

LATERAL FORCE


• Lateral tyre deformation due to side force


Vehicle pushes tyre with FY

Tyre particles deform sideways

Net ground lateral force arises – RY

v Net ground lateral force arises – RY

RY acts behind wheel centre

tP – “pneumatic trail”

- wheel side slip angleFY

RY

v

tP

x

z

y




Tyre structure elasticity additionally affects deformation path

v

FY

RY




v

Lateral deformation distributionSide slip angle

RY

v

FY

tP

Force from vehicle

Pneumatic trail

Ground reaction

RYtP = ALIGNING MOMENT


• Pneumatic trail / aligning moment behaviour


stanford.edu


• Lateral force and aligning moment vs. slip angle


MS

fromThe Automotive Chassis Vol. 1

from Chassis Handbook Notice vertical load (FZ) impact!

δ

Pneumatic trail decreases when increases!


• Non-linear tyre behavior


c - tyre lateral stiffness (depends on vertical load!)

Linear approximation: FY = cApplies for small

Zone of pronounced non-linearity


• Factors affecting lateral force / side slip dependency:


Vertical load

Pressure

Camber angleCamber angle

Longitudinal force

etc.wikipedia


• Impact of tyre load WT


Vertical load increase

Source: Wallentowitz

Sid

e fo

rce

F y

Side slip angle

Vertical load increase

Contact length increases larger Fy

for the same

For same Fy - decreases when WT

increases c increases with WT

Relation between WT and c is degressive


• Simultaneous presence of longitudinal and side force

Tyre behaviour: combined slip

FXRealizedRealized longitudinal

Wheel lock or full Wheel lock or full slide: no side slide: no side force available!force available!

YXR FFF

FR2 = FX

2 + FY2

FR

FY

FX

FRMAX = WT· MAX

FX2 + FY

2 = (WT· MAX)2 = const

Available Available side force

RealizedRealized longitudinal force

Free rolling tyre: Free rolling tyre: maximum side maximum side force available!force available!


• ...

Overall tyre behaviour and modelling issues

Performances: acceleration, top Performances: acceleration, top speed, gradeability speed, gradeability


speed, gradeability speed, gradeability

Vehicle performances

• Summary of motion resistances, impact of velocity

• Engine torque curve

• Powertrain parameters

Introduction


Summary of motion resistances

Frol = frW

GRADE RESISTANCE

ROLLING RESISTANCE

F = Wsin

WIND RESISTANCE2

vAcF

2

WW


• Velocity impact: comparison rolling vs. aerodynamic resistance (force, power)

• Impact of grade resistance


Infinitesimal work =

Definition of power:Definition of power:dt

dsFP

Infinitesimal work =Force F infinitesimal displacement ds

Velocity v

P = FP = FvvRate at which energy is utilizedRate at which energy is utilized

i.e.i.e.Velocity at which we can overcome resistance forceVelocity at which we can overcome resistance force

P = TP = T -- for rotary motionfor rotary motion = 2n/60

[radians/sec]n [RPM]

• Numerical example: generic car


E.g.m = 1400kg W = 14000NcW = 0.3


technical-illustration.com

cW = 0.3A = 2.8 m2



6000 6000 6000Aerodynamic resistance

12% Grade

Net resistance 12% grade



mo

tio

n r

esis

tan

ce f

orc

es (

N)


0

1000

2000

3000

4000

5000

0 100 200 300

0

1000

2000

3000

4000

5000

0 100 200 300

0

1000

2000

3000

4000

5000

0 100 200 300

resistance

Rolling resistance

5% Grade resistance

12% Grade resistance

velocity (km/h)

mo

tio

n r

esis

tan

ce f

orc

es (

N)





mo

tio

n r

esis

tan

ce p

ow

er (

kW)

400

450

500

500 kW car



velocity (km/h)

mo

tio

n r

esis

tan

ce p

ow

er (

kW)

0

50

100

150

200

250

300

350

0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300

100 kW car

20 kW car




mo

tio

n r

esis

tan

ce p

ow

er (

kW)

70

80

90




velocity (km/h)

mo

tio

n r

esis

tan

ce p

ow

er (

kW)

0

10

20

30

40

50

60

0 10 20 30 40 50 60 70 80 90 100 110 120


• Applying Newton’s Second Law:

Longitudinal dynamics: equation of motion

MASS ACCELERATION = NET FORCE

researchgate

Actual mass of the vehicle

Caution: rotational elements also have to be accelerated!

We adopt approach of “effective mass” and “equivalent mass”

meff = m + meq

Effective mass that has to be accelerated on account of engine torque

Equivalent mass of the rotational inertia lumped at vehicle’s CG

Equivalent mass i.e. effective mass can be calculated empirically or analytically.




meff a = Fi

Forces acting in longitudinal direction:direction:

FT – traction force

FW – aerodynamic resistance

Wsin - grade resistance

Frol,f/r – front and rear rolling resistance




meff a = FT – FW – Frol – Wsin

Frol = frWcos

1

Frol = Frol,f + Frol,r – net rolling resistance

D

WT

r

TF

TW - net drive torque at all drive wheels




Finally we obtain:

F = F + F + F + FD

WT

r

TF TRACTION FORCE

FT = FW + Frol + Fin + F

Frol = frW

GRADE RESISTANCE

INERTIA RESISTANCE

ROLLING RESISTANCE

WIND RESISTANCE

Fin = meff a

F = Wsin

2

vAcF

2

WW


• Starting from engine torque curve

• Engine torque transforms to traction force

• Engine RPM transforms to velocity

Traction force diagram

D

trfge

D

wT

r

ηiiT

r

TF

fg

ew

ii

nn

EngineGearbox

Final driveif,f

ig,g

WheelTe,ne

Tw,nw

Sum for all drive wheels

iitrtr = i= igg iifftrtr = = gg ff v = rDw = rDnw/30

= 2n/60


Traction force diagram

0

20

40

60

80

100

120

140

160

180

0 2000 4000 6000 8000

Engi

ne

torq

ue

F T

Engine RPM ne

5000

6000

7000

8000

Trac

tio

n fo

rce

F T

1st gear: ig=iI=3.3

2nd gear: ig=iII=1.99

3rd gear: i =i =1.36

Final drive: if=3.85

0 2000 4000 6000 8000

0

1000

2000

3000

4000

5000

0 20 40 60 80 100 120 140 160 180 200 220

Vehicle velocity km/h

D

trfgeT

r

ηiiTF

fg

eD

ii

nrv

30

3rd gear: ig=iIII=1.36

4th gear: ig=iIV=1.00

5th gear: ig=iV=0.79


Performance determination: graphical approach

5000

6000

7000

8000Tr

acti

on

forc

e

1st

FT = FW + Frol + Fin + FFFTT

0

1000

2000

3000

4000

0 20 40 60 80 100 120 140 160 180 200 220

v km/h

2nd

3rd

4th

5th



FW + Frol

FF

FFinin (12% grade)

Maximum velocity at 0% gradeMaximum velocity at 0% grade


Acceleration performance

meff a = FT – FW – Frol – Wsineff

rolWT

m

sinWFFFa

2,50

3,00

a(m

/s^2

)

0,00

0,50

1,00

1,50

2,00

0 20 40 60 80 100 120 140 160 180

a(m

/s^2

)

v (km/h)


Acceleration performance

dt

dva

a

dvdt

v

0

dva

1t

1/a

• Calculating acceleration time

I

II

III

IV

V

v(km/h)

1/a

(s /m)2

I

II

III

IV

V

v(km/h)

1/a

(s /m)2

vv

v vv

vv

vv

vvv

vvv

1

1

2

2

3

3

4

4

10

10

11 12

13

14

14

1511 1312

7

7

6

6

5

5

9

9

8

8AA A A A A A A A A A

AA

A


Dynamic axle loads

ag

WhΔW CG

in l

Wf, dyn = Wf,stat Win - FLf

Wr, dyn = Wr,stat ∓ Win - FLr

Neglecting contribution of rotating mass and rolling resistance

lf+lr=l

sinαWh

cosαWW CGrstatf,

ll

l

sinαWh

cosαWW CGfstatr,

ll

l

Braking performancesBraking performances


Braking performancesBraking performances

Braking performances

• 3 main phases of braking

Basics of braking process

Time (s)

Dec

eler

atio

n (

m/s

2)

1

2

3

Delay

System activation

Full deceleration

Dec

eler

atio

n (

m/s

1 2 3

3 Full deceleration

• 3 main phases of braking



t1

t2 t3

APPROXIMATION: tL

tS – Stopping time

tB

tL – Lost time

tS

tB – Braking time

• Approximated by 2 phases



tL tBtL

Stopping time: tS = tL + tB

tB

Lost time: v=v0=const

tS

Braking time: a=aF=const- full deceleration


• Lost distance:

Stopping and braking distance

• Braking distance:

2v

F

20

Ba2

vs

sL=v0tLLost time: 0.51 sec

• Stopping distance:F

20

L0Sa2

vtvs


Stopping and braking distance

2vF

20

Ba2

vs

bbc.co.uk

F

20

L0Sa2

vtvs

Fa2

sL=v0tL


Theoretical maximum deceleration

Adhesion coefficient for braking case:

A

BA

W

F

=MAX FBA – net brake force on axle (front or rear)

WA – axle load

Maximum possible braking force (at axle): s

Maximum possible braking force (at axle): FBA,MAX=MAXWA

Vehicle equation of motion – 2nd Newton’s law for braking:

(On level road, neglecting rotational inertia and wind resistance; with maximum braking forces both front and rear; rolling resistance comprised by braking forces)

maMAX = MAXWf + MAXWr aMAX – theoretically possible maximum deceleration

Wf, Wr – front and rear axle weights

Maximum braking force rear

Maximum braking force front

juniordesigner.com

Wr Wf


Theoretical maximum deceleration

maMAX = MAXWf + MAXWr

maMAX = MAX(Wf +Wr) = MAXW= MAXmg

aa = = ggaaMAXMAX = = MAXMAXgg

Braking distance would then be:g2

vs

MAX

20

B


• What does full deceleration aF depend on?

Adhesion utilisation

Maximum braking force maximum deceleration minimum braking distance

Two axles braked

Optimal goal – maximum adhesion utilization on both axlesOptimal goal – maximum adhesion utilization on both axles

s

This is rarely possible!

s


• Brake force distribution should match that of ground forces


s

Ground forces distribution varies depending on deceleration

Sophisticated control system required to account for this – not always present and not always fully efficient

s


• Adhesion not fully exploited on both axles!


s

Net braking force less than physically possible braking distance increases

We introduce braking efficiency B:

s

1a

a

a

MAX

F

MAX

FB

g


• Braking distance is now:

Braking efficiency impact on braking distance

BMAX

20

B2

vs

g

• Defined by law regulation:

BMAX2

Braking efficiency has to provide certain minimum requirements concerning braking performances, in certain driving conditions (e.g. 75%).


• ...

Optimal distribution of braking forces

basics of automotive engineering part 3: basics of vehicle...

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