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REPORTDOCUMENTATION
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blank)
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EPORTDATE
.
EPORT
TYPE
AN DDATES
COVERED
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.
DISTRIBUTION/AVAILABILITY
STATEMENT
Approved
fo r
Public
Release.
12b.
DISTRIBUTION
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
19 ,
SECURITY
CLASSIFICATION
OF ABSTRACT
Unclassified
15. NUMBEROF
PAGES
16 .
PRICE
CODE
20 .LIMITATIONOF ABSTRACT
Unlimited
N S N7540-01-280-5500
Standardor m9 8Rev.
-89)
Prescribed
by
ANSI
Std.
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298-102
8/11/2019 Vehicle suspension research
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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
8/11/2019 Vehicle suspension research
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.
8/11/2019 Vehicle suspension research
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List
of
Keywords
Wheeled
vehicle
Suspension
Control
Semi-active
Vehicle
Dynamics
Simulation
Hydropneumatic
Spring
Damper
Ride
comfort
Handling
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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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
8/11/2019 Vehicle suspension research
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.
8/11/2019 Vehicle suspension research
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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
8/11/2019 Vehicle suspension research
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.
8/11/2019 Vehicle suspension research
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-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.
8/11/2019 Vehicle suspension research
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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.
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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*
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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
8/11/2019 Vehicle suspension research
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.
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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
8/11/2019 Vehicle suspension research
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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
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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< i >
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
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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,
< p ,
6,4>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
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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
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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
,
< p ,
9,
( j > 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 ,< i > 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
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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,
< p
r
)
+
h
g/d
jDi{z,Zf e\if)+
h
g/d
jD2{z,Zf ,e\if)
+ h
g/d
jDz{z,z
r
, e
r
)
+ h
g/d
jD4(Z,Zr, ' i>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 )
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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
< i > , 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
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.
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 , < P r )
-fs4(Z,Z
r
, < j > , 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 ,
< p r )
+
fs*(Z,
Z
r
,
< t >
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
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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,
< p ,
0 ,
4 >
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
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AppendixB
Basic
Dimensionsof
Prototype
Suspension
Unit
(All
dimensions
in
m m
-
do
not
scale)
Bl
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400
:
1
w
0
1
15
k~
h
105
75
o
u- >
20
H
0115
20
tvjl
B2
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20
LO
to
to;
CO
LO
to
o
ID
Recommended