Vehicle suspension research

8/11/2019 Vehicle suspension research

1/54

REPORTDOCUMENTATION

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EPORTDATE

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August

2002

Final

TechnicalReport

4.ITLEAN DSUBTITLE

Controllable WheeledVehicleSuspensionResearch

6.AUTHOR(S )

Prof.N.J.Theron,P.S.

Els

5.

UNDINGNUMBERS

C-N68171-01-M-5852

7. PERFORMING

ORGANIZATION

NAME(S)AN D ADDRESS(ES)

ResearchEnterprises,

University

ofPretoria

PO

Box

14679,

Hatfield

0028,

South

Africa

M0016

9.

SPONSORING/MONITORING

AGENCY

NAME(S)

AN D

ADDRESS(ES

US Naval

RegionalContracting

CenterDetachment

London,

GovernmentBuildings,

Block2,Wingl2

U.S.

Army Tank-Automotive

Command,

ATTN:

Dr.

F.Hoogterp,

Warren,

MI

48397-

50000

10.

SPONSORING/MONITORING

AGENCYREPORTNUMBER

R&D9086-AN-01S

11.

SUPPLEMENTARY NOTES

FinalTechnical

Report

fo rcontractno .N68171-00-M-5852,52pages.

12a.

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STATEMENT

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C O D E

ABSTRACT(Maximum

200

words)

Th e

classic

compromisebetweenwheeled

vehicle

ridecomfort

an d

handling

iswellknown.or off-roadvehicles(a susedby

the

military),itisverydifficultto

achieve

agoodcompromisedue

to

the

fact

that

thesevehiclesareused

on

highways

at

high

speeds.ontrollablesuspension systemsofferthepossibilityto

change

the spring

an d

dampercharacteristicswhilethe

vehicle

is

moving,

therebyadaptingtodifferentterrainsan dspeeds.

hi sresearchinvolvedthedesign,development,

manufacturing,

modelingan dtestingofatwo-stage,semi-active,hydro-pneumatic spring,combinedwithatw ostagesemi-activedamper.

This

system

promises

to

improveboththe

ride

comfort

an d

handling

(and

therefore

the

mobility)ofmilitary

wheeled

vehicles.

Testresults

indicate

thatth erequiredcharacteristics

ca n

beachieved,

an dadesignstudy provesthefeasibilityoffittingthe

system

toa

vehicle.

t

is

concludedthat

the

proposed

suspension

system

is

feasiblean d

that

further

development

ofthesystem

should

continue.

14.S U BJ E C TTERMS

USArmyResearch,

South

Africa,

Wheeledvehicle,Suspension,

Control,

Semi-active,

Vehicle

Dynamics,

Simulation,

Hydropneumatic,

Ride

comfort,Handling

17.

SECURITY

CLASSIFICATION

OF

REPORT

Unclassified

18. SECURITYCLASSIFICATION

OF THIS

PAGE

Unclassified

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SECURITY

CLASSIFICATION

OF ABSTRACT

Unclassified

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PAGES

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PRICE

CODE

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N S N7540-01-280-5500

Standardor m9 8Rev.

-89)

Prescribed

by

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Std.

239-18

298-102


2/54

A D

Controllable

Wheeled

Vehicle

SuspensionResearch

Final

Technical

Report

by

Prof.

N.J.

Theron,

Mr.

P.S.

Els

August2002

United

StatesArmy

EUROPEAN

RESEARCHOFFICE

OF

THEU.S.ARMY

London,England

CONTRACT

NUMBER:

N68171-01-M-5852

P . - ?

n

c

t

o

s

sk-weis

ResearchEnterprises

atUniversity

of

Pretoria

(PTY)

LTD

Approvedfor

Public

Release,

Distribution

Unlimited

2 0 0 2 1 2 0 2

2 5

Av

fo3-u[-^m


3/54

Abstract

Theclassiccompromise

betweenwheeled

vehicle

ridecomfortan d

handlingiswellknown.

Forff-roadehiclesas

se d

yhe

ilitary),

tseryifficult

ochieve ood

compromiseueoheac t

hat

hese

ehicles

re

lso

se d

nhighwayst

ig h

peeds.

Controllableuspensionystemsffer

he

ossibility

ohange

hepringnd

amper

characteristics

while

the

vehicle

is

moving,

thereby

adapting

to

different

terrains

an d

speeds.

This

esearchnvolved

he

esign,

evelopment,

manufacturing,

modeling

nd

estingof

a

two-stage,

emi-active,

ydro-pneumatic

pring,

ombined

ith

wo

tage

emi-active

damper.Thissystempromises

toimprove

bothth e

ride

comfort

an d

handling

(and

therefore

th e

obility)

f

militaryheeledehicles.es t

esults

ndicate

hat

he

equired

characteristicsca n

be

achieved,

an d

a

design

studyprovesthefeasibilityoffittingthe

system

toavehicle.

t

isconcludedthat

theproposed

suspensionsystemiseasiblean dthat

further

development

ofthe

system

should

continue.


4/54

List

of

Keywords

Wheeled

vehicle

Suspension

Control

Semi-active

Vehicle

Dynamics

Simulation

Hydropneumatic

Spring

Damper

Ride

comfort

Handling


5/54

TableofContents

1 .

tatement

ofproblem

2.

ackground

to

problem

3.

pproach

to

solving

problem

3.1.

asic

vehicledynamics

simulation

model

3.2.equired

suspension

characteristics 1

3.3.evelopment

of

suspension

hardware 3

3.4.

anufacture

of

prototype

suspension

system

7

3.5.

estingan dcharacterisationofsuspensionsystem8

3.5.1.

pring

characteristics

8

3.5.2.

ampingcharacteristics

3

3.5.3.

alve

response

times

4

3.6.athematicalmodelofsuspensionunit 5

4.onclusions

1

5.

ecommendations

2

6.

iterature

cited 4

List

of

Appendixes

AppendixA-

MATLABmodel l

Appendix

B

-

Basic

dimensions

of

prototypesuspension

unitl


6/54

1.tatement

of

problem

The

esignfwheeledehicle

uspension

ystems

lways

nvolved ompromise

between

ideomfort

nd

andling.

orood

ideomfort ompliantuspension

systemis

normallyrequiredwhile

goodhandlingdemands

a

stiffsuspensionsystemto

control

od yroll.Withnormalpassiveuspensionystem,hecharacteristicsof

th e

springs

an d

dampers

are

fixed

at

the

design

stage

an d

cannot

be

changed

afterwards.

By

usingcontrollablespringsan d

dampers,

hese

characteristicsanbechangedwhileth e

vehiclesmoving.thereforebecomesossibleoave

oft

ettingsorgoodide

comfortwhilsttraveling

in

astraightline

on

a

goodroad,while

th esuspension

ca nbe

changed

to

ahard

setting

moments

laterto

give

good

handling

when

th evehiclehas

to

change

direction

as

requiredfo r

lanechangingor

even

accident

avoidance.

Controllable

suspension

systems

ca n

thereforereduceoreveneliminate

th e

ridecomfort

vs .

handling

compromise.

2.

ackground

to

problem

The

outh

African

ontrollableuspension

esearch

ffort

overhe

as t

welve

ears

concentrated

on

semi-active

dampers

an d

hydro-pneumatic

springs

fo r

wheeledoff-road

vehicles.

A n

overview

of

al l

th eresearch

ctivitiesuring

thisperiod

an

be

ound

n

references[1 ]to

[18].

The

ankAutomotive

Command

TACOM)

f

he

S

Army's

esearch

ctivities

focusednemi-actives

well

s

ully

ctive

uspension

ystems.his

ncluded

he

developmentf

n

lectric

ctive

uspensionctuator

nd

ullyctiveydraulic

suspensionfo r

ahigh

mobilityoff-road

wheeled

vehicle.

TACOM

experience

sbased

on

test

results

fo r

bothwheeled

an d

trackvehicles.

Experiencenouth

Africa

orrelates

er y

wellwith

US

xperiencendheessons

learned

and

conclusionsreachedareingood

agreement.

mprovementsareinth esame

order

of

magnitude

and

problemsidentified

with

current

semi-activesuspension

systems

are

also

very

similar.A

similarapproach

to

advanced

suspension

research

is

followed

in

that,lthough

mathematical

nalysis

nd

imulationserformed,he

alidation

f

resultssobtaineduringieldestswithuspension

hardwareittedoehiclesnd

testedunderreallifeconditions.

Duringdiscussions

between

M r.

P.S.

Els(Universityof

Pretoria,

outh

Africa)

an d

Dr.

F.B.

Hoogterp

TACOM,

Detroit)

n

September2000

19], efinitemutual

esearch

interest

in

th e

field

of

semi-active

suspension

systems

was

identified.

The

idea

of

adding

a

emi-active

ydro-pneumatic

pring

o

he

emi-active

amperechnology,

s

proposedyls2] ,snovel

ndwarrants

more

detailednvestigation.heesulting

research

project

isdefinedin[20].

3.

pproach

to

solving

problem

The

purpose

of

this

esearch

so

esign,

evelop,

manufacturend

test wo-stage,

semi-active,

hydropneumatic

pring,ombined

with

wo

tageemi-active

amper.

The

esultinguspension

ardwaresested

nd

haracterized

o

btain

ll

he

parameters

required

fo r

mathematicalmodeling.

In

order

to

investigate

th e

feasibility

of

theproposedsuspension

system,

th e

projectincludedsix

tasksnamely


7/54

i) Developing

a

basic

vehicle

dynamics

simulation

model

to

predict

ridecomfort

an d

handling,

ii)

Determiningtherequiredsuspensioncharacteristicsfo r

th e

"best"

ride

comfort

an d

"best"

handling

respectively,using

thevehicledynamics

model,

iii) Designing

rototype

uspension

ystem

apablef

roducing

he

equired

characteristics,

iv)

Manufacturing

th e

prototype

suspension

system

according

to

th e

design,

v)

Testing

ndharacterisation

fherototype

uspension

ystemo

etermine

feasibilityan dconformanceto

specification,

vi )

Developing

amathematical

model

of

the

prototype

suspension

system

that

ca n

be

incorporated

into

thevehicledynamics

model

at

a

laterstage.

These

six

tasks

will

nowbe

described

in

more

detail.

3 .1 . Basicvehicledynamics

simulation

model

Inrder

o

imulateheideomfortndhandling

of

a

ehicle,

imulationmodels,

based

on

parametersfo raLandrover

Defender10sports

utilityvehicle

(seefigure

),

were

eveloped

n

A DS

Dynamic

Analysis

nd

Design

ystem)

nd

MATLAB

respectively.

Figure

1

-Landrover

Defender

110

vehicle

TheD AD Smodel

has

81egreesof

freedom,butafter

addingjoints,

constraints

an da

driver

model,

4

unconstraineddegrees

of

freedom

remain.These

consist

of

th evehicle

body

displacements

(lateral,

longitudinal,vertical,roll,pitch

an d

yaw),

wheel

rotations,

frontxle

erticaldisplacement

nd

ol l

nd

ea r

xleertical

displacementnd

oll.

Non-linearspring,

amper,

um p

top

nd

tire

characteristics

re

used.The

vehicle

s

steeredve r

redetermined

ourse

y impleriver

model

whichstimates

he

lateral

ositional

rror

based

on

theya wangle

of

th evehicle

body

at

the

current

time

step

and

thedesiredlateralposition

at

a

specified

driverpreview

time.

Thedrivermodel

is

implemented

using

amplifiers,

summers

an d

input

elements.

The

basic

components

of

theD AD S

model

ar e

summarizedin

table

1 .

AimpleynamicmodelorimulatingheideesponseoftheLandroverDefender

110

sport

utilityvehiclewasalsodevelopedandcodedinMATLAB.t

th e

time

when

thiswasdone

th e

comprehensivemodelof

th evehicle,including

th e

suspensionsystem

geometry

and

kinematics,was

already

runningsuccessfullywithinthe

D AD S

(Dynamic

Analysis

an d

DesignSystem)environment.


8/54

Modelentities

Components

Quantity

Vehiclebody

2

Wheels

4

Front

axle

Rigid

bodies

Rear

axle

(13)

Ground

(fixedin

space)

Front

hubs

(left

&

right)

2

Anti-rollbars

2

Frontwheels

to

fronthubs

2

Revolute

joints

Front

hubs

to

front

axle

2

(9 )

Rear

wheels

to

rearaxle

2

Body

torsionalstiffness

Anti-rollbar

left

an d

right

2

Spherical-sphericaljoints

Axlelocating

and

push-pull

rods,

steering

links

5

(5 )

Revolute-revolute

oint

Radius

ro d

(1 )

Revolute-sphericaljoints

A-armrear

(2 )

Panhard

rod

front

Constraints

Steering

control

input

(2 )

Forward

speed

Force

elements

Non-lineardampers

4

(18)

Springs

(choice

of

hydropneumatic

and

coil

springs)

4

Bump

stops

4

Generic

tires

4

Body

torsional

stiffness

spring

Anti-rollbar

stiffness

Controlelements

Amplifiers

2

(9 )

Summers

2

Inputs

2

Steering

angle

limiter

Output

torquesleft

an d

right

2

Initial

conditions

Vehicle

forward

speed

(1 )

Table

1-

Components

of

th e

D AD S

model

Theurpose

of

theMATLAB

model

wasnot

oeplaceheD AD S

model.

t

was

assumedhatuch

ode,

based

n

imple

nd

airlyough

pproximations

o

he

suspension

kinematicsand

limited

to

only

smallangle

rotations

(i.e.,

excluding

yaw

and

thus

teering,

handling

an d

an ylateralynamics)

nd

withaverysimplepointcontact

tiremodelwouldexecutesignificantlyfaster

thanthe

full

D AD S

model.heplanwas

tosehi sodenheesign

f

th e

ontrol

ystem

f

theemi-activeuspension

system,

withspecial

emphasis

onth e

vehicle

ride.

n

addition

to

th eexpected

quicker

execution

peed,

econd

easonor

developing

he

MATLAB

ode

was

hat,

t

he


9/54

time,

only

one

license

to

D AD S

was

available(a s

opposed

to

anumber

of

licenses

tothe

A D A M S

program,acquired

recently).

aving

available

an

additionaldynamic

model

of

th eehiclewas

eemed

beneficialn

reeingth e

ingleD AD S

icenseor

otherwork

while

the

ride

characteristics

of

the

semi-active

suspension

systemwas

investigated.

The

M A T L A B

odes

asedn

he

quations

of

motionofth e

ehicle

ystem.he

derivation

of

these

equations

using

Lagrangean

dynamics

is

shown

in

Appendix

A.

The

imulationesults

of

th eMATLABod e

ompares

well

withimulationesults

predicted

by

DADS.

number

of

figures

comparing

outputsfromth e

tw oprograms

fo r

the

same

excitationconditionare

included

below.

Simulations

ofthe

Landrover

Defender110

being

driven

at

a

constantspeed

of

60

km/h

overastretchof

rough

road

identified

asBelgian

paving"

weredonewithbothmodels.

Figure

2

shows

th e

comparisonof

th everticaldisplacement

of

th e

vehicle

bodycenter

of

gravity,

as

predicted

by

thetw omodels.hecomparisonisgenerallyfairlygood.he

seconderivativeofthis

ata,

presentednigure

s

he

erticalcceleration

ofthe

vehicleod yenterfravitylearlyhowshathe

A D S

odel

redicted

significantly

more

high

frequency

activity

than

the

M A T L A B

model.

his

m ay

be

du e

toth efact

that

theD AD Smodelha s

a

larger

number

of

degrees

offreedom,giving

rise

to

ig h

requency

modes.

he

DADSodel

lso

airlyccuratelyccounts

or

he

suspensionkinematics

an d

it

is

expected

that

modelingthe

kinematicswill

also

giverise

to

ig hrequency

ehavior.

omparisons

of

the

esultspredictedbyth e

womodels

withrespecttothefrontaxle

vertical

displacementan d

thebody

rollangleareincluded

in

figures

4

an d5.

hese

figures

confirmthatth e

tw o

modelsgenerally

agree

well

bu t

thatthe

D AD S

modelpredicts

high

frequencyactivitythat

is

missedbyth e

M A T L A B

model.

To

nvestigate

or e

pecificallyod yol l

ynamics

f

he

ATLAB

odel,

simulations

wereperformedof

th eLandroverbeingdrivenwith

nly

its

ef t

hand

side

wheels

ver

a50m m

high

00

mm

wideplatform-likebstacle

with

straight

upan d

downamps

t1.3.

herighthand

ide

wheels

ollowed

la t

road

urface.

his

obstacle

is

locally

referredtoasth e

APG

obstacle.he

M A T L A B

model

indicatedthat

if

this

obstacle

is

crossedoverat

0

km/h

th e

bump

stopshit

through,

inwhich

case

the

simulationserminatedince

t

oe sot

llow

xtrapolation

n

raphs.heDADS

modelmay

also

indicate

this,

bu t

itwas

not

specifically

investigated.

he

D AD Smodel

doesno t

necessarily

terminate,

though,

at

such

an

occurrence.

Becauseof

th e

M A T L A B

ermination

roblems

he

imulation

was

on et

lo w

km/h.

he

vehiclebody

center

of

gravityverticaldisplacement

androllangle

results

ar e

comparedn

igures

nd.heomparison

between

he

M A T L A B

ndDADS

models

are

generally

good.

These

resultsshowthat

the

M A T L A Bcode

is

performing

inanacceptable

manner

an d

m ay

be

used

in

future

fo r

the

design

of

the

controlsystem,withrespectto

vehicle

ride,

under

conditions

ofsmallangles

ofrollan d

pitch.


10/54

-0.16

-0.17

-0.18

BelgianPaving ,6 0

m /h

o

03

Q

l -

-a

o

G O

-0.19

-0.2

0.21

-0.22

-0.23

I

i

i

i

i

IJ\

Jrfll

i

r

i

A

fr

jpf

V

'

r f \ n \ F s

i

...

|r

rax

1 1 /

'''

'

\j

MATLAB

AD S

i

i

i

7

Time

(s )

1 1

1 2

Figure-omparison

f

enter

f

ravity

ertical

isplacementredicted

y

M

ATLAB

and

D AD S

modelsfo ra

Belgianpavingroadtraversed

at60km/h.

BelgianPaving ,

6 0

km/h

1012

Figure

3

-

Comparison

of

center

ofgravityvertical

accelerationpredicted

by MATLAB

an d

D AD S

models

fo r

a

Belgianpavingroad

traversed

at

60

km/h.


11/54

Belgian

Paving ,60km/h

Figure

4

-Comparisonof

frontaxle

vertical

displacement

predicted

byMATLABan c

D AD S

modelsfo r

a

Belgian

paving

road

traversed

at

60

km/h.

Belgian

Paving ,

6 0

m /h

0.03

0.02

0.01

a

1

c

0

-0.01

IM

5

-0.02

TZ

-0.03

C D

-0.04

-0.05

-0.06

i

l

l

i__

ATLAB -

AD S

i

J

'

.....A-L.-|

L|.i

j\\ AI

Wi

ft

/

A )L

l/l

y

LJiu

'v

;-^\(r

1

l'

F r

r

i

- -

i

i

i

i

7

Times )

1 0

1 1 1 2

Figure

Comparisonof

vehiclebody

roll

ngle

predicted

by

MATLABnd

D AD S

modelsfo r

a

Belgianpaving

road

traversedat60km/h.


12/54

-0.12

-0.13

0.14

-0.15

APG

with

left

wheels

only ,5

km/h

0.16

Q

1^

C D

O

f

-0.17

C D

-0.18

-0.19

2

ATLAB

AD S

I

is A

1

t

X

-N

/l-TSZ ,

)

\

i

3

3 .5

4

4.5

5

5 .5

Time

(s )

6

6 .5

7 7. 5 8

Figure

6-

Comparison

of

vehicle

body

center

ofgravity

verticaldisplacementpredicted

byMATLAB

and

DADS

models

fo ran APG

obstacle

at

5km/h

with

left

wheels

only.

AP G

with

left

wheels

only,

5

km/h

0 .1

0.08

f.06

-a

c c

If

.04

en

c

03

e

.0 2

> -.

T3

O

m

0

-0.02

-0.04

ATLAB

:

\

AD S

i

\

\N \

An

.

_i^

W

\_._

...XJ--

i

2.5 3

3.5

4

4.5

5

5.5

6

6 .5

7

7.5

Times )

Figure7Comparisonofvehicleodyoll

ngle

predictedyheMATLAB

nd

D AD Smodels

fo r

an A PG

obstacle

traversed

at

5km/hwith

th e

left

wheels

only.

10


13/54

3.2. Required

suspension

characteristics

TheD AD S

model

was

se do

redict

ide

omfort

ndhandling

ofthe

ehiclewith

differentombinationsofspringndamperharacteristics.

imulation

esultswere

used

toetermineirst

order

indicationsofthe

best"oftan dhardcharacteristics

or

both

the

spring

and

damper.

The

coil

springs

on

th e

baseline

suspension

were

replaced

with

hydro-pneumaticsprings

where

th espring

stiffnessis

determined

by

thega svolume

inthe

static

position.

tatic

ga s

olumeswere

aried

between

.01iteran d3.0

iter.

This

ives

range

of

spring

stiffness

from

about

0to

0.1imes

that

ofthebaseline

coil

spring

stiffness.

To

simplify

thedampercharacteristics,thebaselinedamperforcewas

scaled

ithaconstant

factor

that

varied

between

0.8

(i.e.,

softer

than

baseline)

up

to

3

(3

timeshigherthan

baseline).

Simulations

wereperformed

or7dampercharacteristicsnd

0

pring

characteristics

within

hese

anges,

iving otal

of

70

imulation

uns.

Although

hi srocesswas

performedmanually

or

th eproject,

tudy

not

part

of

this

roject)snprogressto

investigate

the

applicability

of mathematical

ptimisationto

the

problem

in

an

attempt

toecrease

henumber

of

requiredimulation

uns.

Verypositivepreliminary

esults

have

been

obtained

as

discussed

in

[27].

Ride

omfortas

imulatedve r ypicalff-roaderrain

Belgianavinglock

course)

t

ehicle

peed

of60m/h.Ridecomfortwas

valuated

using

th eertical

accelerationatthedriver

position

(right

front)

as

well

as

th e

left

rear

passenger

position.

Theertical

ccelerationwasweighted

using

the

Britishtandard

BS84 1

weighting

filter

an dcalculatingaweighted

root

mean

square

(RMS)

value([21],[22]

and

[23]).

A

three-dimensionalplot

of

weighted

RM Scceleration

vs.

pringstaticas

volume

an d

damper

scale

factorisindicated

infigure

8.

The

owest

cceleration

evels

best

id e

omfort)

re

btained

with

owamping

(dampercale

actor

of

0.8)nd

of t

pringsstatic

asolume

.5

iters).Motion

sickness

values

do

however

increase

with

very

soft

springs.

Handlingwas

simulated

by

performing

a

severe

double

lane

change

manoeuvre

[24]

at

a

speedof60km/hfo r

the

same

valuesof

spring

an ddamper

characteristicsusedfo r

ride

comfortnalysis.Maximumodyol l

ngleasse ds

valuationarameteror

handling.igure ndicates

the

results

ofthe

handlingsimulations.Thesmallest

body

roll

ngleisachievedwithth e

stiffest

spring

(static

ga s

volume

of

0.01iter)

whilethe

roll

angle

is

insensitive

tothedamperscale

factor

asca n

be

expected.

Th e

"best"

handlingsuspensionisthereforeatashigha

spring

stiffness

as

possible.

The

areaswherethere

re

ap s

n

thegraph,

re

where

the

vehicle

ould

notcomplete

the

lane

change

without

rolling

over.

It

s

oncluded

hat

or

es t

ide

omfort,

of tuspension

sneeded

nd

or

es t

handling

a

stiff

suspension

is

needed.Thisisin

linewith

generalesignrules

an d

was

themotivation

orinitializingthisresearch

project.

The

imulationresults

ohowever

indicate

that

fo r

th e

hardsuspension

setting,a

staticga s

volume

of0.1

iter

an d

damping

scale

actor

of

between

nd

s

uitable

nd

or

he

of t

uspension

etting,

as

volumeofgreater

than

0.5

iteran d

a

damping

scalefactor

of

0.8

will

be

uitable

irst

orderalues

or

heesign.

he

ig h

damper

haracteristicse dn

he

esign

ofthe

1 1


14/54

Suspensionsystem

will

therefor

be

between

2an d3

times

th e

baseline

values,

while

th e

lowdampingshouldbeless

than

0.8

times

the

baseline

value.

More

simulation

will

be

performed

at

different

speeds

and

overdifferent

terrain

profilesat

a

laterstage,

butth e

current

results

ar everyusefulfo r

developingth e

suspensionhardware.

'>...

4

_ _

--'

- . . . . _

-L.._

.1

3

'

5

-

o

1--..

Mfc^jl--..

1.5 -

^^^^^^MMH^Mrf|

" S

4

1.5- :

' " N L A -

\

Y

2

L

\

^ '>

" - ' . v . '

" "

-^.^iite|i

3

2 .5

\ _

...V-----V.

2 \^

....-''--Z^

IN*


23/54

endstops).Th ecompressibilityis

thereforesignificant

forthestiff

spring

characteristics

an d

needs

tobetakeninto

accountduring

spring

calculations.

Thefigure

also

indicates

the

verygood

correlation

achieved

when

the

spring

characteristicis

correctedusing

the

bulk

modulus.Figure22indicatesmeasuredan dcalculatedcharacteristicsfor

both

the

softan d

stiff

springs.

l

t

4

oft

Spr ing

Measured

1 oftSpr ingCalcula ted(0.5iterstat ic

ga s

volume)

jW

sl

A*Y

-

i* K*fr

:

i,/r*

H

h ; -:

,.-/

*

>

K

.. .

I1A

^wf l*

i

i

i

-40200

Spr ing

Displacement

[mm]

Figure19-Soft

spring

characteristic

ti ff

Spr ing

Measured

Sti ff

Spr ingCalcula ted

r

* S/

Spr ing

Displacement

[mm]

Figure

20

-Stiff

springcharacteristic

21


24/54

40

35

30

25

z

=

0

Ol

'c

a.

15

10

5

0

-1C

I

ii

TJ t

If:

1

/Measuredbulkmodu lus f

if

1

T

t

u

./....:

n S /

'

77

Calcula ted

(Cor rected)

.

//

;

//

B V

/

II

*

/. . / .

easured

Character ist ic

_ _

Measured

BulkModu lus

4BulkModu lusof1.5e09N/ m

2

M

m

Calculated

Stiff

Spr ing

Character ist ic

Corrected

wi th

BulkModulus

a j

tf Measured //J

;

;

/

.jrLtvs

i

0806040

-20 0 20

40

BO

Spr ing

Displacement

[mm)

Figure21-

Stiff

springcharacteristic,

correctedwith

bulk

modulus

j

Sti ff jSpr ing

Measured .

IY

N

Stiif

Spr ing

(Calculated

an d

corrected

:

:

oftSpr ingMeasured

So f tSpr ing

Calculated

tiffSpr ing

Measured

-e~

Stiff

Spr ing

Calculated

an dCorrected

I

J

B/

afl.Spjing-Measured..-,..4

^

/ Soft

Sj i r ing

Calculatfed- - JV^

yVSy

rrT&ll&^Z..

Wj^

.

^ i i5|p5*j|L^M i?'

MM***

tS00&^ '

i

-8 0

-2 0

0

Spr ing

Displacement

[mm]

Figure22

-

Soft

an d

stiff

spring

characteristics

2 2


25/54

3 .5 .2 .

Damping

ch r cteristics

The

damper

packsinth e

strutwere

taken

from

standardLandrover

reardampers.

Figure

3

ndicates

he

ampingharacteristics

measured

n

he

uspension

nit.

Dampingcharacteristicswere

eterminedby

xciting

th e

trut

with inusoidal

nput

displacement

with

a

displacement

of

25

m m

(total

stroke

of

50

mm).

The

frequency

of

th e

sine

wavewasvaried

to

give

different

velocities.

Force

values

were

calculated

from

th epressure

readings.

Threeifferentharacteristics

weremeasured

n

he

trut

namely

he

of tamping

characteristicboth

ampers

ypassed),

ig h

ampingcharacteristicnoypass)with

soft

spring

and

highdampingcharacteristic(n o

bypass)

withstiff

spring.

For

reference,

the

required(baselineLandrover)characteristic,

as

measured

on

a

Landrover

damper,is

alsondicatedn

he

raph.hetrutampingharacteristicswerexpectedoe

higher

than

th e

baselinedampers

du e

to

th e

increased

pistonarea

as

well

as

increased

flow.

hiswasound

oto

e

heasend

he

ifferenceane

ttributedo

manufacturingrroryhe

ubcontractor.

hismanufacturingrrormeanshathe

damper

packs

don't

properly

ea l

nside

he

alve

block

cavities,

nd

luid

s

eaking

pastheamper.hisroblemsnherocessofbeingectified.heesultso

however

indicate

that

therear e

three

discretedampinglevelsassociatedwithth estrut,

i.e.he

ampers

an

be

witchedbetween

ig h

ndow

amping

haracteristics.he

damping

evel

willbencreased

toth e

requiredlevels

nce

th e

ealing

problems

have

been

rectified.

Damper

character ist ic

calculated

fromP2

3

2

1

0)

-1

-2

-3

m

.\

\,

about

the

x

axis.

The

Landrover

vehicle

considered

in

this

study

hasso-calledrigidaxlesboth

front

ndear.

ts

s-

sumed

that

both

these

axles

are

kinematicallyconstrained

in

such

away

that

their

respective

centers

of

mass,whichare

bothassumed

to

lie

in

the

x

zplane,isplace

with

the

ectors

_

x

f

T

and

L

x

r

T

expressedin

the

Bbase,

while

the

sequenceof

Euler

angle

rotationsthatapplies

tothe

bodyalsoapplies

tothetw oaxles,exceptthat

the

frontand

rear

axlesrollthe

third

Euler

rotation)

through

angles

0/

andj>

r

,respectively,

insteadof through

the

bodyroll

angle

of

j> .

Thewheels

and

corresponding

suspension

members

arenumbered

as

follows:

fo r

left

front,

fo r

right

front,

3

fo r

left

rear

and

4

fo r

right

rear.

The

followingpointsandlengths

are

defined:

Pointc:hecenter

ofmassof the

vehicle

body(unsprungmass).

A2


41/54

Pointd{ :he

axle

center

of

masscorresponding

to

the

z-th

wheel.

This

means

that

\

and

d are

the

samepoint,

namely

the

center

of

massof

the

front

axle.

Pointe, :point

attached

to

thevehiclebodyon

an

axis

that

isparallelto

thezaxis

and

inthe

x-z

plane

such

that

the

axiscontains

pointd{ .

This

means

thate\

and

e re

the

samepoint,

namely

thepointdetermined

bythe

front

axle

as

described

above.

ength

lof.

he

distance

that

ej

s

above

dj

i.e.,

the

difference

in

z

coordinates

between

points

ej

and

pointi,positiveif

ej

s

abovedj)under

static

loading

when

the

ehicle

is

stationaryon

a

horizontal

surface

with

its

weighton

its

wheels).fpoint

ej

is

chosen

tobe

coincident

with

point

diunder

staticloading(i.e.,as

on

a

virtual

extension

of

thevehiclebody),l

0

i

=0.

Pointf.he

oint

f

attachment

o

he

orresponding

xl ef

the

pring-damper-combination

corresponding

to

the

i-th

wheel.

Pointvt :

he

pointof

attachment

to

the

bodyof the

spring-damper-combination

corresponding to

the

i-th

wheel.

Point

wi:

he

center

of

the

tire-groundcontact

areaof wheel

i.

Initially

at

t

=

0

(i.e.,

beforeany

rotationha staken

place)the

basevectors

of theI

base

and

the

Bbase

correspondinglycoincide.

Thefollowing

position

vectorsare

denned

in

the

B

baseat

t=

0:

That

of

point

e, ,

relativetopoint

c :

j

c/cl

0 Z j

c/c3

T

components

constant

with

time)

That

of

point

V i,relative

to

point

e, : Z j

u/e2

k

v/s3

T

components

constantwithtime)

hatof

point

gi ,

relative

topointdj:

_

h

g/d2

0

T

componentsvary

with

time)

hat

of

pointt u , - ,

relative

to

point

dj,withthe

vehicleunderstatic

loading

conditions:

L

h/i2

h^/is

T

componentsvary

with

time).

Expressed

n

as e

B,

om e

fthese

ectorsthose

otatingtogetherwith

he

-y-z

xi s

ystem)

will

alwaysremain

the

same,

but

inthose

cases

where

thevectordoes

not

rotatewiththe

x-y-zaxis

system,

it s

componentswill

vary

withtime,

as

indicated

above.

Also

note

thatthe

above

definitions

also

imply

assumptions

on

the

positions

where

the

spring-damper-combinations

ar e

attached

to

the

axle

units

and

the

vehicle

body.

The

kineticenergy

expressionof especially

thesprung

mass

issignificantly

simplified

if

thedisplacement

componentsexpressed

in

theX

base

are

chosen

as generalizedcoordinates,but

then

the

expressionfor

the

potential

energy

is

much

morecomplicated.

The

analysis

is

simplified

by

assuming

thatboth

the

vehicle

body

rollandpitch

angles

j>

and6

remain

small

so

that

the

sines

of these

angles

maybe

approximated

by

the

anglesthemselves,

while

their

cosines

may

be

approximated

byone.

urthermore,since

only

a

ride

simulationwill

be

erformed,

olateral

excitation

is

allowed

and

theya wangle

ip

and

ya wrate

ip

are

both

assumed

to

beconstrainedto

zero.

nderthese

assumptions,

onsideringthe

motion

of theaxles

relativetothevehiclebody,thedistinctionbetweenthe

actualmotion

whichhappensinthey-z-plane

and

an

approximation

to

thismotion

which

is

assumed totakeplace

in

the

F-Z-plane becomes

negligible.

So,

while

earlierit

wa s

said

that

the

axles

are

assumed

todisplace

withthe

vectors

x

f

T

and

Ix

r

T

xpressed

n

he

B

base,

t

sowassumedthatheseisplacementsrequallywell

described

s

the

ectors

X

f

T

nd

X

r

T

directly

expressed

in

the

I

base,

.e.,

withouttransformation

since

the

transformation

matrix

by

approximationis

an

identitymatrix.

Thegeneralizedcoordinatesg *

n

which

theynamicsof theroblemisthereforedescribedar eZ,Zf,

Z

r

,

6 ,

j > > f

and

j>

T

.

he

vehicleis

assumed

tobeconstrainedin

its

motion

to

ensure

that

the

velocity

componentintheX

direction

remains

constant

and

that

Y

and

ipalways

remain

zero.

A3

42/54

Potential

Energy

and

Dissipation

Functions

It

should

be

noted

that

thisappendixdeviates

from

the

convention

adopted

in

the

rest

of

the

report

in

that

in

thisappendix

the

springdisplacementsin

compression

and

damper

compression

ratesare

taken

as

negativewhilespringdisplacements

in

extension

and

damper

extension

rates

are

taken

aspositive.

n

the

sameway,

inthis,

the

spring/damper

forces

are

taken

as

positivewhen

in

tension

and

negative

when

in

compression.

Theelastic

potential

energystored

in

the

springs

is

given

by:

U s=U si(z ,z

f

< t > e,< t > f )

+

u

S2

(z,z

f

,< j > e, f

f

+

u

S3

(z,z

r

,< j >

e,

t

r

)

+

u

Si

{z,z

r

, < / >

e,

t

r

)

wherefo rthe

springsof the

front

axle,i=

1,2:

Usi(z,z

f

,

c

i > , e , < l > f )=Usi(&i)

i

=

z-zj+ i

ig/d

M-h)

Ajbeing

theextension

of

thei-th

spring,

whilefo r

thespringsof therear

axle,

i=3,4:

Usi(Z,Z

r

, (/ > , e , < t > r )=Usi(A

i

)Ai=

Z-Z

e/cl

6

+

l

ig/d2

(

< l > - < t >

r

)

Hereitshouldberemembered

thattheexpressions

forAj

ivenabovehavebeen

derived

bylinearizing

the

kinematics.

Also,

dUs

i

=

dUs

i

dA

i=

rA

x

Ai

dq

t

0A;d

qi

JSt(

l}

d

qi

where

rr

fsi(Ai)

is

the

spring

force

(positive

in

tension;

a

non-linearfunction

of

Aj)

in

the

i-th

spring.

Th eelasticpotentialenergystored

in

theanti-rollbarsis

Uarb

=

2

k

r b / ( < f >

~

f f+

k

< * r K ( < t >

~

< t > r f

Thepotential

energy

due

to

gravityis:

U G=

m

s

gZ

+

nifgZf

+

m

r

gZ

r

The

totalpotential

energy

is

given

by:

U

=U s

+

U

G

The

Rayleighdissipation

function

is

givenby :

TD

= Tm

(Z,

Z

f

,

i \ > ,

9 , 4 >

f

+

T

D2

(

Z,Z

f

,j > , 9 , 4 >

f

+

T

D3

{Z,

Z

r

, 0,

9

4 >

r

)

+

F

D

i{Z,

Z

r

, j> ,

0,

t

y )

where

fo rthe

dampersof

the

front

axle(i=

1,2):

Toi{Z,

Zf,

4 > ,

9,4>f )=

JFoi(Ai)

where

fo r

the

dampers

of

the

rear

axle

(i

=

3,4):

T

Di

{Z,

Z

r

, < ,

9 , 4 >

r

)

=

?Di(i)

where

j

is

the

extension

rate

of

thei-th

damper,

and

when

linearized,

indeed,

{

=

Z

Z

f

ie/Cl

9

+

l

ig/i2

{ 4>-4> f )

A4


43/54

inthe

case

ofthefront

dampersand

i=Z-Zli .

/cl

6+h

g/d2

{ j > - > r

inthe

caseof the

rear.

Also,

dq iAi dq iqi

where

f

Di

(i

isthe

damper

force(positivein tension;

a

non-linearfunction

of

A;)

in

the

t-th

damper.

Th elinearizing

assumptions

thatwereapplied

to

derive

thelinearexpressions

fo r

bothAjandA *

re :

hepitchnd

three

rollngles

,

j>

/ > /nd>

r

emain

small

such

that

he

in e

f

an y

f

these

anglesis

by

approximation

the

angleitself

while

the

cosineisapproximated

by

1.

he

motion

remains

small(essentially

a

pertubational

analysis)sothatanyterms

containing

prod-

ucts

f

generalized

isplacements

rtheirtime

erivatives,

fthenaturetqj,

tqj

or

qiqj ,

were

ignored,compared

totermslinear

in

these

variables.

^/e2

hg/dt

Kinetic

Energy

Using

the

same

small

angle

assumptionas

above,

Genta

[28,

p.

61 ,

362]

derives

thefollowingexpression

for

the

angularvelocity

of

the

vehiclebody,

expressedinthe

B

base,

interms

of the

roll,

pitch

and

ya w

anglesandrates:

6+j > i p

Substitutingtheadditional

assumptionthattheyawrate

is

also

zerointheaboveleadsto

Inderivingthe

kineticenergy,thesameapproachisfollowedasin

Genta

[28,

p.

63].

Here,

however,

a

further

simplifying

assumptionis

made

that

the

vehicle

bodyproduct

of

inertia

J

xz

in

thevehicle

body

axis

slso

ero,

eadingto

diagonalinertia

tensorin

theB

base.

hi sssumptionsperhapsot

fullyjustified,

ince

neither

the

x

nor

the

z

axesof

the

body

areaxes

of

symmetry.

n

the

ase

of

the

Landrover,however,

this

assumption

is

not

too

bad

either.

Withrespect

toboth

the

frontand

rear

axles

it

s

assumed

that

he

moment

of

inertia

of

the

axle

about

the

xle

axis

i.e.,

he

y-axis)

s

egligibly

small,omparedtothat

f

the

ehicle

body.

urthermore,

t

isassumed

that

the

moments

f

inertia

abouttheothertw o

axesx

and

z

ar e

approximatelythesame,Jfnthefront

and

J

r

nthe

rear,

and

that

these

are

the

principle

moments

of inertiaof

the

axle

units.

The

kinetic

energy

of thewhole

vehicle

A5

44/54

is

thengivenby

approximation

by:

T

= \m

s

{

X

2

+

Z

2

)

+

{

+

1

-m

f

X

2

+Z

2

)

+

\\

8

+

l

r

{X

2

+

Zl)

+

\{

1 \

6

- < t > r 0

,

J

x

0

0

0

Jy

0

\

0

0

J

*.

T

\Jf

0

0

0

0

0

0

0

Jf

T

Jr

0

0 0

0

0 0

J

r

=

m

a

{x

+

z

2

)+\jj

+

\j

e

2

+

\j

2

e

2

+\m

f

X

2

+

Zj)

+

\jffi

+

\jf &

+\rn

r

(X

2

+Z

2

)

+

JJ

r

fi+

^< 0

'

it

'

< 0

( 9

l r )

Virtualworkdoneby

external

forces:

4

6W

=

J2

S

w

w i

Equations

of

Motion

Lagrange's

equationsarenow

derivedusing

&T_dT dU d?

=

dtdqi dq i dq i dq i

where

Qi

=

d(SW)

dSqi

A6

45/54

Equationof

motionw.r.t.

generalizedcoordinateZ

dT

-

m

s

Z

dz

ddT

. -

m

s

Z

dtdZ

dZ

d U _

_

dUsi^

dUs2

ds

dUsi_

dU a

dz

az

z

z

z

dz

= fsi(Z ,

Z

f

,4 > ,M/)+

fs

2

{Z,

Z

f

,< f> 6, < ,)+f

3

(Z,

Z

r

,< f > 6 , < j>

r

)+f

i(Z,

Z

r

,

< j > 6 , 4 >

r

)+

m

s

g

9T _

Tp\

Fpi

Tpz

Tpn

~ d z

~

dz

dz dz dz

=

fpi

(Z ,

Zf,

< j >

0, < j > f )

+fp

2

(Z,

Zf,

f )+fp3{Z, Z

r

,

< t > 8 , 4 >

r

)+fpi{Z,

Z

r

,

< f >

,f>

r

)

Q-mW-o

m

s

z+

fsi(z,

Zf,

< j >

e,

t

f

+ fs

2

(z,

Zf,

< t >

e,

4 >

f

+ f

ss

(

z,z

r

,< t >

e,

t

r

)+ f

Si

(

z,z

r

, < j > e , j

r

)

+ m

s

g+ fD

1

(

z,z

f

Je,^f)+ fD

2

(

z,Zf,^,e,^f)

+ fp

3

(z,z

r

,^,e,^

r

)+ fp

i

(

z,z

r

J,eJ

r

)

=

o

Equationof

motionw.r.t.generalizedcoordinate

Zf

dT

az,

mfZf

d _

T_

=

-

dtdz,

m

8T

n

8Zf-

=

dU

_

dUsi

dU

S

2

dU

G

dZf ~ 8Zf

+

Zf

+

dZ

f

= -/si

(Z ,

Zf,4 > ,

6 ,< j> f ) -

fs2(Z,Zf,

< j >

6,cj>f)+

mfg

A7

46/54

dz

f

~

dz

f

dz

f

=

-f

D

i(Z,Zf,4>,6,4>

f

)

-

fm(Z,Z

f

,

< j > , e , < t >

f

Qz,

d6Zf

(SW)

=

f

wlz

+

f

w

2z

mfZf

-fsi(Z,Z

f

,

< f > , 0 ,

< j>

f

)

-f

2

{

Z,

Z

s

,

< > , 9 ,< j >

}

)

+m

f

g

-

fDl(Z,Zf,,d,f)

-fD2(Z,Zf,,9,(j>f) = fwlz+fw2z

Equationof motionw.r.t.

generalizedcoordinateZ

r

dT

dZ

r

m

r

Z

r

dT

=

m

A

dtdZr

dT

az

r

=

0

8U

U ss dUsi dU

G

dz

r

az

r

dz

T

dz

r

=

-fS3(Z,Z

r

, < / > 6 , 4 >

r

)-f

4(Z,

Z

r

, 4 > ,9 , < j > r )

+

m

r

9

dZ

r

dZ

r

dZ

T

=

-fD3{Z,Z

r

, < j > , 9 ,( j >

r

)

-

fDi{Z,Z

r

, < j > , 9 , ( j > r )

0*- >-

Jw3z

+

Jwiz

.

m

r

Z

T

-

fs3(Z,

Z

r

,

< f >

9 ,

j>

r

)

-

fsi(Z,

Z

r

< j >

0 ,

< j > r )

+

m

r

g

-

/03

(Z,

Z

r

, < } >

0 , < f > r )-

fDi(Z,Z

r

,

4 > ,

9 ,

4 > r )

=

fw3z i Jwiz

A8

47/54

Equationof

motion

w.r.t.generalizedcoordinate9

^L

=

j

e+

j

(

)

2

e +j f)e+j4 >

2

9

89

at

8 9

J

y

9+2J

Z

4> 4 > 9 +2 J f ( j > f 4 >

f

9+ 2J

r

< j)

r

4

r

9

where

the

terms

inf>

2

,

f i

and

.have

been

neglectedsince

roll

anglesj> f> f

and

j>

r

re

assumed

to

be

small.

8T

n

09=

8U

Usi

dU

S

2 dU

S3

dUsA

8 9

9999

-

-hc

/cl

fi(Z,Zf,,9,(j)f)-l2

c/c

Js2(Z,Zf ,

9

,

j>

r

)

-/

4

/cl

A(Z,

Z

r

,

< f > , 9 , 4 >

r

)

8T _

Tpi

Tp-j.

Tpz FD A

8 9

~

8 9

89

8 9

8 9

Ji

e/el

fm(Z,

Z

f

,

0,

9,

< j ) f )

-h

t/cl

f

2(Z,

Z

f

,

< j > 9,

f>

f

h.

/cl

fv3{Z,

Z

r

,

 r )

-* 4

/cl

fDA(Z ,

Z

r

,

$,

9,

4 >

r

)

. J

y

9+2J

z

( j > 4 > e

+

2J

f

f>f(ff9

+

2J

T

< j>

r

^

r

9-

l

U/c

Js\{Z,Zf,< >,9

< f > f ) -

-h

e/el

f2(Z,Zf ,,9

h)

ke

/cl

f

3{Z,

Z

r

, j> 9,>

r

)

-h

/< :1

f4{Z,<

Z

r

,

, 9 , (j >

r

)-h

,

/o

jDi{Z,Zf,(t>,9,f)

-k

c/c

i

/D2(Z,Zf,4 > ,9 , f )

-

h i JDz{Z-,Z

r,

< t >

9,4 >

r

)-U

c/

dfDA

(Z,Z

r

, 4 > , 9 , c l >

r

)

=0

Equation

of

motion

w.r.t.

generalized

coordinate

f>

8T

8* *+

ddT

-

it

A9

48/54

ddT dT

T

2

dtd < f > o < f >

^ L

=

dUsi

dUs2

dUss

dUsA

dUarb

d < j >

(j >

(j >

< f>

(f >

(f >

=

h

g/i

Jsx{Z,Z

f

,< t > , 6 , ct > f )+

h

g/d

Js2(Z,Z

f

,

0,0, t>

f

)+ k

arbf

( t> -4 > f )

+h

g/i

Jsz{Z,Z

r

,

< t >

6 , j>

r

)+ h

g/d

Js4(Z,Z

r

,0,6 ,< t>

T

)

+

k

arK

{

t>

-0r)

dT

_

Tr>\ TD2

?m

TDA

5 0 ~ 9 0 9 0 9 0 90

=

h,

/d

J

i(Z,

Z

f

0,

6,0/)

+

2

a/i

J

2(Z,

Zf, j>,e,

< / / )

+ h

g/

fDs(ZjrJ4>r)+h,

/da

fD4{Z,Zr,i4>r)

. .

J

x

0-

J

z

< f

+l

lg/d

Jsi(Z,Z

f

,4

> , 6 , < l > f )

+

h

g/d

Js2{z,

Zf,

0,

e,

< j>

}

)

+

h

d

Js3(z,

z

r

, < t > e,

0

r

)

+h

g/d

jsi(z,z

r

,

0,e,

J,J>r)+karb

/

(< l > - < t > f )

+

K r b A < t > ~

< t > r )

=

0

Equation

of motionw.r.t.

generalizedcoordinate0/

wr

Jsh

dt90/

iW f~Wf

=

Jf{h

~

4 > i

2 )

A10

49/54

au

dUsi

C/S2

U

aTb

d(f>f d(j>f d4>f

-h

/d

Jsi z,z,

,e,f)-h

9/d

Js2(z,z

f

,

,e,f)

-

Krbf(4>-

< t > f )

dT

=

TD\ TD2

d4>f

d(j)f

-h

/d

jDl(Z,Zf,j>j,,9,f)

. .

J

f

(

4 ,

f

>

f

6

2

)

-l

lg/d

jsi z,Zf^,e^

f

)

2g/d

j

2

z,z

f

 , e , 4 >

f

)

-ll

/d2

f

Dl{Z,Zf,(f>,

2

,

4>f)

-

h

g/d2

f

D2{Z,

Z

f

j >

6 ,

< / > / )

k

arbf

4 >

- r

=j

r

t >

r

v

OCpT

d

T

dT

i

2

dt

d ( j >

r

o ( j >

r

du

d ( j >

r

-

dU

S3

U

Si

U

arb

l__

d f )

T

d f>

r

d t

T

h

g/d2

f3(Z,Z

r

,ip,

0,

r

)-h

g/d2

fA{Z,Z

r

,

0,6

< j>

T

)

-

k

a

rb

r

(-> r)

dT

d 4 > r

=

TDZ TDA

d(p

r

d ( j >

r

-h

g/d2

f

D3(Z,

Z

r

,

4 > ,

6 ,< j > r )-h

g/d

jDi{Z,

Z

r

, 0,

6 ,

0

r

d

4

Q < p ,

=

d

,

sw

)

=

fwizh

v/d2

All

50/54

.

j

-t>

r

e

2

)

-

i

3g/d

js(z,z

,

( j > , e ,fa)

-

h

s/d

Js4(z,z

r

< f > , e , < t >

r

)

4

h

g/d

jDs{Z,

Z

T

, 4 > ,

9 ,

ir)

-h

g/i2

f

D

4 (Z,Z

r

,> ,

9,

j > r )

~k

arbr

( < A

~

< t > r ) = ^

f***K,i

i=3

Let

the

statevectorbe

defined

as

q=[z

z

r

j>t> f j > r z>a

T

Then

the

above

7

equationsof

motionm aybe

summarized

as

follows:

Mq

=

F(q,f)

(Al)

where

is

anonlinearcolumnvector

function

of the

statevectorandthe

wheel

forcevector

f _

w

,which

contains

the

verticalcomponentsof thewheel

forces.

The

matrix

M

is

given

by

M

=

h [0 ]

7

[0]

7

M[22]

where

h

and

[ 0 ]

7

are

a

7

x

7

identity

and

zero

matrix,respectively,

and

where

the

7

x

7

submatrixM

[22

isgivenby:

~m

s

0

f

0

r

M[22]

0y

0

x

0f

0

r

Thus

it

isseen

thatthe

matrix

M

isdiagonal.

If

is

divided

into

tw o

7x1

sub-vectors

F

and

2

],

such

that

T

=

_f ^

j

,

then

Z[I]

=[0]T

Ir]g

whilethe

column

vector

j

2

sgiven

asfollows:

1strow:

fi{Z,Z

}

,

, 9 , ( j > f ) fs2{Z,Z

f

,( > , 0 , < t >

f

-f

3

(

Z,Z

r

, , 0 , , 6 , (l >

r

)

-m

s

g

-fDi(z,

z

f

4 > ,

e,

j>

f

f

D

2(z,z

,

,e,

< t>

f

f

D3

(

z,

z,j > ,

e,

i>

r

)-f

D

i{z,

z,j> ,

e,

i >

r

)

2nd row:

3rdrow:

/si{z,

z,

4 > ,

M/)

+f

S

2

(z,

Zf,

< j > e ,< t> f ) -

m

fg

+

f

D1

(Z,

Zf,

j> ,9 ,< / )+

fm

(Z,

Zf,

< j >

9 , 4 >

f

+

f

w

iz

+

fw2z

f

3

(Z,

Z

T

,

< t >

9 ,



0 ,

< f>

r

)

-m

r

g

+

fD3(Z,Z

r

,

4 > ,

9 ,

fa.)

+

f

D

i(Z,

Z

T

, 4 > ,

9 , 4 > r )+fwZz+wAz

A12

51/54

4th

row:

-2J

z

< PW-2Jf

f

t

f

6J

r(

r

r

0+

li

c

/c

Jsi(Z,Z

f

,( > , 6 , ( t > f )+

l^

/c

Js2(Z,Z

f

,( > ,e

:

( > f )

+h.

/el

fss(Z,Z

r

, f>

0,

p

r

)+

U

m/el

fsi(Z,

Z

r

,c j > , 0 ,t>

r

)+

h

m/el

fm(Z,

Z

f

,

j > ,

0, d

2

h

g

/d

Jsi(z,Zf,

< f > , 0 , < j >

f

)

-

h

g/d

Js2(z,

Zf,



f

)

-

h

g/d

jsz{z,

z

T

,

< j >

0, t

r

)

-

h

g/i

jsi{z,

z

r

,

4 > ,

e,

fa )

-h

g/d

jDi(Z,Zf,i,\ f)-h

g/d

J

2

(ZjfJj,i

f

)

s/d

J

3

(Zj

,i,6,4>

r

)

-h

g/d2

fDi(Z,Z

r

,

4

> , d , 4 >

T

)

-k

a

rbf{4>

-4>f)

~

k

arb

r

(

t > ~

t > r )

6th

row:

jfh

2

+h

g/d

Jsi{z,Zf,,e^f)+i

2g/i

js2{z,Zf ,e,

> f ) + h

s/i

j

l

{z,Zf,ef)

i

+

h

g/d2

fD2{Z,Zf, >,6,4>f)+k

ar

b

{

{(t>-4>f)+'^2

fizk

v

/d2

i=l

7throw:

J r < t > r 0

2

+

h

g/d2

fS3(Z,Z

r

,

0,

0, < f>

r

)

+h

g/d

JsA(Z,

Z

r

,

< j >

0,< j>

r

)+ l

3g/d

J

3 (Z,

Z

r

, > ,

0 ,

fa)

4

+

h

g/d2

fDi{Z,Z

r

,( j > ,0 ,

fa)

+

k

arbr

< f >

-

j>

r

)

+

22fwizh

w/d2

i=3

The

equation

(Al)

srecognized

as

a

first

order

differential

equationina

form

that

MATLAB

ca nsolve

using

its

built

in

Runge-Kutta

solvers.

A1 3


52/54

AppendixB

Basic

Dimensionsof

Prototype

Suspension

Unit

(All

dimensions

in

m m

-

do

not

scale)

Bl


53/54

400

:

1

w

0

1

15

k~

h

105

75

o

u- >

20

H

0115

20

tvjl

B2


54/54

20

LO

to

to;

CO

LO

to

o

ID

Documents

Vehicle suspension research