tire stiffness and damping determined_nasa_technical paper 1671
TRANSCRIPT
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NASA Technical
Paper
1671
NASA
TP
1 6 7 1
c. 1
Robert
K.
Sleeper
and
Robert C. Dreher
JULY 1980
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TECH
LIBRARY KAFB, NM
NASA
Technical
Paper 1671
Tire Stiffness and Damping Determined
From Static and Free-Vibration Tests
Robert
K.
Sleeper
and
Robert
C.
Dreher
Langley ResearchCenter
Hatnpton Virginia
National Aeronautics
and Space Administration
Scientific and Technical
Information Office
1980
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SUMM RY
Stiffness anddampingof a nonro lling ti r e a re determined experimentally
from both s t a t i c force-displacement re la tio ns and thefree-vibration behavior
of a cable-suspended platenpressedagainst he t i r e periphery.Lateral and
fore-and-aftspringconstants and damping fac to rs of a 4 9
x
1 7 s i z e a i r c r a f t
t i re fo r dif fer en t t i r e pr es su res and v ert ical loads are measuredassuming
a rate-independent dampingorm.
I n
addition, a technique
i s
applied or
estim ating he magnitude of the t i r e masswhich participates i n the vibratory
motion of the dynamic te st s.Re sults show th at both the l a t e r a l and fore-and-
a f t spring constants generally ncrease
w i t h
t i r e pressure
b u t
only the lat ter
increased significantly
w i t h
v e rt ic a l ir e loading. The fore-and-aftspring
con stan ts were gre ate r than those
i n
the a te ra ldir ec tio n. The static -spr ing -
constan t variation s were s im ilar o t h e
dynamic
varia t ions
b u t
exhibited lower
magnitudes. Dampingwas small and in se nsi ti ve o ir e oad in g . Furthermore,
s t a t i c damping accounted for a sig ni fi ca nt po rt io n of th at found dynamically.
Ef fec tiv e tir e masseswere a ls o small.
INTRODUCTION
Ti re stif fn ess anddamping i n the la te r a l and fore-and-aft dire ctio ns are
important properties
i n
dynamic analyses of a i r c r a f t wheel
shimmy
and antiskid
braking systems. S ta ti c e s ts on nonrolling t i r e s havebeenused for a number
of years to measure ti r e s t i f fn ess (e.g. e f.
1 ) .
Tests on a ro ll in g ir ea r e
preferred b u t equipnent and f ac ili ty lim it at io ns make such te st s d if f ic u lt to
implement.
As
a resu l t , i reproper t i e sa regenerally measured
us ing
a platen
loaded vert ical ly
w i t h
a t i r e and supported on bearings (e.g. re fs . 2 and 3
where the propertiesar e deduced rom the response of the pl at en oapplied
fo rc es . Such a support system, however, ty p ic a ll y n je c ts indeterm inant motion
e f f e c t s
and
limits
t e s t s os ta t i cap pl ica t io ns . While uch st a t i c e st s remain
a primary source
of
stiffness anddamping information, measurements ob tained
from vi br at io n es ts appear to be more rep res en tat ive of the operati ng
environment.
The objective of
t h i s
report i s todiscuss the re su lts of
an
experimental
ef fo rt to measure s tif fn es s anddamping prop ert ies of a nonro lling t i r e
us ing
a cable-suspended platenpressedagainst the t i r e periphery. Both s t a t i c and
dynamic t e s t s were performed t o determine spr ing cons tan ts anddamping factors
of a large aircraft i re displace d
i n
ei ther the la te ra l orfore-and-aftdirec-
tion. Damping is t rea ted i n a rate-independent form.Three pl at en s were
employed
i n
the dynamic t e s t s to provide
an
in di ca tio n of t i r e mass involvement
i n
thevibratory motion. The
s t u d y
was conducted on
a
4 9 x 1 7
s i ze t i r e over
a range of v e r t i c a l loads and i nfl at io n pre ss ur es extending to th e i r maximum
ratedvalues.
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SYMBOLS
Values are
g i v e n n b o t h
S I
and U.S. CustomaryUnits.
C
damping forceo e f f i c i e n t ,
N-sec/m
( lb f - sec / in . )
c.
g. c e n t e r of g r a v i t y
F
complex appliedo r c e ,
N
( l b f )
Fmaxaximum appliedorcemagni tude , N ( l b f
1
FO i n i t i a lp p l i e do r c ea g n i t u d e ,
N
( l b f )
F V
t i r e v e r t i c a l load, N ( l b f )
Fx=0 appliedor ce when displac ement
i s
zero ,
N
( l b f )
f o s c i l l a t i o nr e q u e n c y ,
Hz
- = / z i -
k
kC
k t
Q.
m
mP
m t
N
t
X
X 0
XN
t o t a l
s p r i n gc o n s t a n t , N/m ( l b f / i n . )
c a b l e n t e r a c t i o ns t i f f n e s s ,
N/m
( l b f / i n . )
t i r e s p r i n gc o n s t a n t ,
N/m
( l b f / i n . )
cable
l e n g t h , m ( f t )
v i b r a t i n g
mass,
kg (lbm)
p l a t e n
mass,
kg l h )
e f f e c t i v e tire mass, kg lbm)
number ofc y c l e s
time, sec
complexdisplacement, m ( i n . )
or ig ina ld i sp lacementampl i tude ,
m
( i n .
1
displacement amplitude of Nthcycle, m ( i n . )
5
viscous damping fac tor
T
f requency period, sec
w c i r c u l a ro r c i n grequency, sec-1
2
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APPROACH
Tire
sp r in g c o n s t a n t s anddamping f a c t o r s i n b o t h t h e
l a t e r a l
and ore-
a n d - a f t d i r e c t i o n s were determined rom
s t a t i c
anddynamic tests using a cable-
s u s p e n d e dp l a t e np r e s s e da g a i n s t h ep e r ip h e ry of t h e t i re . S t a t i cc h a r a c t e r -
is t ics were derived rommeasurementsof platen d i s p l a c e m e n t r e s u l t i n g from
s l o w l yapp l ie d o rc es . The s t a t i c s p r i n gc o n s t a n t was de te rmined rom hes lope
of
t h e a x i s of t h e h y s t e r e s i s loop des cr i bed by the o rce -d isp lacement re la t ion-
ship , and
a
damping factor was der ived rom
i t s
width . Dynamic ch ar ac te r i s t ic s
were obtained rom
simple,
s ing ledegreeof reedom ree-v ibra t ion tests of the
t e s t p la t e n . Thus, fo r h e l a t t e r
tests
t h es p r i n gc o n s t a n t was derived rom
t h ev i b r a t i o n a lf r e q u e n c ya n dp l a t e n mass sp ec i f ic a t io ns , and th e damping
fac tor
was de te rmined run hed isp lacementampl i tudedecay
rate. Estimates
o f th e
e f f e c t i v e
t i r e
masses p a r t i c i p a t i n g i n t h e o s c i l l a t o r y m o t i o n s of thedynamic
tests
were determined romc ha ng es i n h e f r e q u e n c y r e s u l t i n g f r o m
similar
t es t s
w i t h d i f f e r e n t mass platens.
-PARATUS
AND
TEST
PROCEDURE
Figure 1 i s a photographof he
t e s t
apparatusand t e s t t i r e . The appara-
t u s i s
shown preparedf o r
a
l a t e r a l dynamic
t es t .
Test F i x t u r e
Themain s t r u c t u r e of t h e t e s t f i x t u r e is conf igured as two three-bay
por-
t a l frames oinedoverhead by fo ur beams a n dalo ng he loo r by a t h i c k p l a t e .
The frames,cons t ruc tedsfwelded 10-in. s t e e l H-beams, are nominally 3.0 m
(10 f t ) deep,2.2
m
(7.1 f t ) highand a r e spaced a d i s t a n c eof 2.1
m
( 7 f t )
a p a r t . The p l a t e l o n g h e l o o r
i s
2.5
c m
(1
in . ) h i ck . The
t i r e rim
i s
s u p p o r t e don he ef t by a taperedwelded box s t ru c tu re , c o n s t r u c t e d from
2.5-cm (1-in. ) t h i c k p l a t e
s tee l
which i s su sp e n d e d ru n h eu p p e rp a r t of t h e
f i x t u r e and s t a b i l i z e d by 0.2-cm (4-in . )
diameter
pipe. A v e r t i c a l
beam a l so
suspended rom heupper par t o f h e f i x t u r es u p p o r t s h e r i g h t s i d e of t h e
rim
and
clamps it to
t h e f i x t u r e
t o
preven t t i r e r o t a t i o n .
T h e sp e c i a l f ea tu re o f h ea p p a ra tu s
i s
t h e s u p p o r t i n g of t h e t e s t p l a t e n
by fourcables . Each ca bl e i s 1/2-in.
s t ee l wire rope
and i s suspended rom
a
force-measuring oad
ce l l
connected t o
a
h y d r a u l i c c y l i n d e r as shown i n f i g -
u r e
1. The cab le ree -sw ing e n g th 8 i s approximately .83 m ( 6 t ) . Ti re
load ing is accomplished by ene rg iz in g he h y d r a u l i cc y l i n d e r s
to
l i f t t h e p l a t e n
v e r t i c a l l ya g a i n s t h e
t i r e ;
i n d i v i d u a lc y l i n d e rc o n t r o l
i s
a v a i l a b l e
t o
equal-
i z e h e c a b l e o a d i n g or l e v e l h e p l a t e n .
A l l
t e s t
platens
were 66
cm
(26 in . )s q u a r ew i t hd i f f e r e n t h i c k n e s s e s and
mater ia l compositions. The t w o l i g h t e rp l a t e n s
were
made of aluminum p la t e .
They
were
7.6
cm
(3 n.)and13.2 cm (5.19 n.) hickandweighed102.1 kg
(225
lh
and
1
73.3 kg (382
lh ,
e sp e c t iv e ly . The h e a v ie s tp l a t e n was a
15.4-cm (6.06-in. ) t h i c k
s t e e l
p l a t e andweighed536.1 kg (1 182 lh). The
p l a t e n t e s t weigh ts nc luded 4.5 kg 1 0 l h ) or c a b le s a n dattachments. The
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upper su rfa ce of each pl at en was painte d i n the center w i t h a g r i t - f i l l e d
enamel toprevent
t i r e
slippage.
A separ ate hyd raulic cylind er was used to displa ce he platen durin g he
s t a t i c t e s t s . A mechanical ratcheting device and a quick-release mechanism
wereemployed t o provide the i n i t i a l displacement
and
rel ea se fo r he dynamic
te st s. The d ir ec ti on of t e s t motion was varied
by
changing theorientation of
the hydra ulic cy linder or the di sp lac ing mechanism depending on the type of
tes t .
Test Tire
The te s t s were conducted w i t h a na tur al rubber, recapped, si ze 4 9
x
1 7 ,
type V I I , 26-ply ra te d a ir cr a ft t i re of bias-ply construction having a rated
inf latio n pre ssu re of
1220
kPa
17 7
ps i ) and a ra te d maximum v e r t i c a l load of
1 7 8 kN 4 0 000 l b f ) . The nominal t i r e masswas 7 9 .4 kg 7 75 I h . The t i r e
was the same t i r e used
i n
reference 2.
Instrumentation
Cable loads determined from load c e l l s were monitored p ri or to te st in g and
a lin ea r potentiometer was i n st a ll e d to measure l a t e r a l or fore-and-aft d i s -
placements during testing. A l inea rs t ra in gage accelerometer was mployed i n
the dynamic t e s t s o measure pl ate naccele ration . For s ta ti c e s ti n g an addi-
t ional load ce l l
w a s
ut i l i ze d to measure external forces that displaced the
platen.
Tape record ings of the pla ten ac ce le ra tion and displacement weremade dur-
ing
the dynamic tests
and
a time-code generator was incorporated t o provide a
millisecond time reference.
Test Procedure
After i nf lat in g the unloaded t i r e to the test pressu re thepl aten was pre-
pared fo r either the s t a t i c or dynamic te st s by centering heplaten beneath
the t i re and uniformly raising it ag ain st he ire periphery. ndividual
hydrauliccylinde r adjustm ents weremade to equa lize he cable oad ing and le ve l
theplaten. I n general,ve rtica l oad ing s were w i t h i n
3
percent of spe ci fie d
nominal loadings.Plate n displacements were kept small to minimizeboth t i re
slippage
and
nonlinear effects.
.- The s t a t i c te s ts wereperformed
by
slowly forcing he platen
from
i t s
ne utra l po sitio n a distan ce of approximately 0 . 6 4 cm (0.25 i n . ) both
la ter a l l y and fore and a f t through two complete cycles. Corresponding fo rc es
and displacements were recorded du ring he t e s t s whichwere repeate d for each
combination of t i r e pres sur e,vertical load, and motion dir ec tio n. For the se
tes ts , hree i re pressu res ranging f rm 6 8 9
(1
00) t o 1 2 4 1 kPa 1 8 0 psi) and
the ollowing our v e r ti c a l loads were xamined:
2 2 . 2
5 0 0 0 ) , 4 . 51 0
000),
8 9 . 0
20 0 0 0 ) , and
1 7 7 . 9 kN 4 0 0 0 0 l b f ) .
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Dynam c t est s. - The dynamc t est i ng was perf or med by di spl aci ng t he pl at e
appr oxi mat el y 0 . 6 4 cm 0.25 i n.), r el easi ng i t , and r ecor di ng t he r esul t i ng
damped f r ee- vi br at i on di spl acement and accel erat i on t i me hi st ori es. Test s wer e
conduct ed f or several combi nat i ons of pl at en masses, t i r e pr essures, and ver -
t i cal l oads wi t h bot h l at er al and f or e- and- af t mot i on. W t hi n t he dynam c test
t he t i r e was i nf l at ed t o one of t hr ee t i r e pr essur es r angi ng f r om8 9 1 00)
t o 1 2 4 1 kPa 1 8 0 psi ) and was subj ect ed t o ei ght ver t i cal l oads r angi ng f r om
22.2 5000)
t o 1 7 7 . 9 kN 4 0
000
l bf ) .
DATA REDUCTI ONAND ANALYSES
The t echni ques f or comput i ng t he spr i ng const ant and dampi ng f act or f r o
t he f or ce- di spl acement r el at i onshi ps of t he st at i c t est s and t he mot i on of t
dynam c t est s ar e gi ven i n t hi s sect i on. Al so descr i bed
s
t he met hod devel oped
f or r emovi ng t he ef f ect of cabl e i nt er act i ons w t h t he comput ed spr i ng con-
st ant s. I n addi t i on, a t echni que f or comput i ng t he ef f ect i ve t i r e mass f r om
dynam c t est s wi t h di f f er ent mass pl at ens i s gi ven.
Spr i ng Const ant
Cabl e i nt er act i on. - The f ol l owi ng sket ch shows t he f or ces act i ngn the
di spl aced pl at en and i ndi cat es t hat t hey are der i ved f r om a combi nat i on
f
t he
t i r e st i f f nes s
kt
and a component of t he cabl e f or ces whi ch maye t r eat ed as
a cabl e i nt er act i on st i f f ness kc def i ned by
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wher e Fv i s t he ver t i cal l oad and & i s t he f r ee- swi ng cabl e l engt h. Thus,
t he t ot al spr i ng const ant
k
act i ng on t he pl aten may
e
r esol ved i nt o
k = kt + kc
or
where t he t i r e spr i ng const ant s kt deri ved f r om t he syst em must be r educed b
t he cabl e i nt er act i on st i f f ness kc. I n t hi s paper i t i s assumed t hat cabl e
i nt er act i on does not af f ect t he dampi ngr t he ef f ecti ve t i r e mass.
St at i c t est s. - Typi cal f or ce- di spl acement cur ves f or bot h l at er al and f
and- af t t est s ar e pr esent ed i n f i gur e These hyst er esi s l oops or i gi nat e at
t he or i gi n and af t er t wo l oadi ng cycl es t er m nat e at zer o l oad. The l oad di s
cont i nui t y at t he ext r eme posi t i ons
s
at t r i but ed t o t i r e creep t hat occur ss
t he l oadi ng di r ect i ons ar e manual l y swi t ched.
For t hese t est s t he sl ope of t he f or ce- di spl acement hyst er esi s- l oop ax
( t he dashed l i ne connect i ng t he l oop extr emes) def i nes t he t ot al st i f f ness
appl i ed t o t he pl at en. The t i r e spr i ng const ant kt
i s
f ound by subt r act i ng t he
cabl e i nt er act i on st i f f ness kc f r om t he t ot al spr i ng const ant k.
Dynam c t est s. -
A
t ypi cal t i me hi st or y of a dynam c t est
s
di spl ayed i n
f i gure
3.
The recor d shows t he accel erat i on and di spl acement r esponse of t he
pl at en t o a f r ee- vi br at i on t est . Fi nal r ef er ence di spl acement and accel er at i on
l evel s are i ndi cat ed al ong wi t h t he di spl acement envel opes. The anal og out put
of t he t i me- code gener at or i s al so shown.
The di spl acement r esponse exhi bi t ed a shi f t i n equi l i br i um l evel , at t r
ut ed t o t i r e creep. Even af t er account i ng f or t he shi f t , vi br at or y per i ods of
t he accel er at i on were more uni f orm t han t hose of t he di spl acement . Hence, t h
accel er at i on t i me hi st or i es, speci f i cal l y t he aver agef
3 or 4
Cycl es, were
used t o comput e t he vi br at i on f r equenci es.
For a l i ght l y- damped si mpl e spr i ng- mass syst em t he f r equency of vi br at
i s r el at ed t o t he pr oper t i es of t he system by t he equat i on
1
27
f = I/G
or
k
m
2
-
=
(27rf)2
=
(
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where f is t h e s c i l l a t i o n r e q u e n c y , T
i s
therequency e r iod , ndhe
r a t i o
k/m
is term ed i n h i s t u d y
a
frequencyparameter. The assumption of
small
damping is s u b s e q u e n t l y u s t i f i e d by experiment.
To compute the t i r e s p r i n gc o n s t a n t , h e r e q u e n c yp a r a me t e r i s f i r s t
de te rmined rom heper iod of Vibra t ionand hen he t o t a l s p r i n g c o n s t a n t
i s
computed rom th ep r o d u c t
of
t h ep l a t e n
mass
and the reque ncy
parameter.
The
s p r i n g c o n s t a n t i s fou nd by s u b t r a c t i n g t h e c a b l e i n t e r a c t i o n s t i f f n e s s from
t h e
t o t a l
s p r i n g c o n s t a n t .
Damping Factor
E n e r g y d i s s i p a t i o n
i s
m a n i f e s t e d n h e s e
tests
by t h e h y s t e r e t i c c h a r a c -
te r of the t i r e s ta t ic - force-displacementcurvesand by th edecay ingampl i tudes
of t h e r e e - v i b r a t i o n response. To account for t h i s damping i n s t a t i c applica-
t i o n s a rate-independent orm i s r e q u i re d . One s u c h e p r e s e n t a t i o nc a l l e d
s t r u c t u r a l
damping e.g. r e f .
4 ) is
u sed i n
s t r u c t u r a l
v i b r a t i o na n a l y s e s
( r e f .
5).
This damping
i s
e s p e c i a l l yu s e f u l o r h i ss t u d y n h a ts i n c e
damp-
i n g
i s
smal l
it
c a nr e a d i l y be r e l a t e d
t o
t h e
more
conventionalviscous ormof
damping t y p i ca l ly assumed inv i b r a t i o na n a l y s e s .S i n c e n r e e - v i b r a t i o n time
h i s t o r i e s s t r u c t u r a l damping
is
ind i s t ingu i shab le romviscous damping, a l l
damping i s t r e a t e d as s t r u c t u r a l damping in h is p a p e rb u te x p r e s s e d n terms
of heviscous damping fac tor .
S t a t i c
tests . -
Ligh t s t r u c t u r a l damping may be mathematicallyformula ted
i n terms of heviscous damping f ac t o r 5 by the o l low ingc o mp l e x t i f f n e s s
express ion
where F
i s
the omplex ppl iedorce ,
C
i s the i scous damping fa c t o r ,
k is the onven t iona l
( t o t a l )
spr ing ons tan t , and
x
i s theomplex
displacement .
I n s i g h t n t o h i s f o r c e - d i s p l a c e m e n t r e l a t i o n s h i p may
be
ga ined by so lv ing
f o r h ed i s p l a c e m e n t r e s u l t i n g from the complex s inuso ida l force
F = Foeiwt
where Fo is t h e n i t i a l p p l i e d o r c ema g n i t u d e a n d
W
i s t h e ci rcu lar f o r c -
ing reque ncy. When th e o r c e
is
i n t r o d u c e d n t o h ee q u a t i o n h ed i s p l a c e me n t
response
becomes
Fo k (wt-2r)
x =
1 + 4 5 2
which when pl ot te d w it h
respect
t o t h e a p p l i e d f o r c e y i e l d s a t i l t e d e l l i p se
whose wi dth ncr eas esw i t h
C
and or
s m a l l
damping the major a x i s
slope
approx imates hespr ingcons tan t .
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T h e r e l a t i o n s h i p of t h e e l l i pse width to the damping fac tor may be d e r i v e
u s i n g h e r ea l par t of the complex applied f o r c e a n dcomplexdisplace-ment,
i.e.
and
F d k
1 +
452
x = cos ( U t - 25)
For x = 0
l 31T
2
u t
= -
+
25, - + 25, ..
and
for
small
damping
a t
corresponding
times
theapp l ied o rcemagnitude may
be approximated by
or
Thus, th e damping factor f o r small va lues
i s
one-half
t h e
r a t i o of the o rce
a t
zerodisplacement
t o
t h e maximum a p p l ie d
force.
The followingske tchgraph i -
c a l l y d e p i c t s these q u a n t i t i e s :
Dynamic tests.- Damping frcan thedynamic
t e s t s
was soughtf rom he oga-
rithmic decrement
of
thedecayingdisplacementampl i tude of t h e f r e e - v i b r a t i o n
time
hi s t or y . However, the ogar i th micdec rementcanno t be d e t e r m i n e dd i r e c t l y
from he
displacement time
h i s t o r y because of
i t s
d r i f t i n ge q u i l i b r i u m e v e l .
This nonsymmetry is removed fran hed i s p l a c e m e n t data by computing
a double
amplitude der ived from t h e d i f f e re nce be tween sp l i ne c u r v e - f i t t e dd i s p l a c e m e n t
e n v e l o p e s h a t
pass
through hedisplacementextremes. Fran thedouble-
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ampl i tudeva lues ,damping ac to r s o reach t e s t
were
computedover
a
few
rep-
r e s e n t a t i v ec y c l e s u s i n g h e e q u a t i o n
where 2xN is thedoub le mpl i tudeof heNt h yc le and 2x0 is t h eo r i g i n a l
double mpl i tude .Shouldh e damping f o r c e o e f f i c i e n t C
be
d e s i r e d ,
it
may
becomputed rom th e f o l l o wi n ge q u a t i o n :
Because
of ensormeasurement imi ta t ions ,def lec t ionsbelow 0 . 2 5 cm
(0.1
i n . )
were d is rega rded .
E f f e c t i v e Tire Mass
The so lu t ion for t h ee f f e c t i v e t i r e mass assumes t h a t h e mass m of he
vi br a t in g body of quat ion
( 1 ) is
composed
of
t h ep l a t e n
mass
mp and the
e f f e c t i v e t i r e mass m t , t h a t i s
m = mp + m t
By re pl ac in g he v i b r a t i n g mass with heproductof he t o t a l s p r i n gc o n s t a n t
and the ec ip roca lo f h e r e q u e n c ypar ame ter , he o l l owi ng e la t ion may be
der ived :
mP = k g ) -
m t
Th e e f f e c t i v e
t i r e mass is
then ound from
a
c o e f f i c i e n to b t a i n e df r o m a l i n e a r
r e g r e s s i o na n a l y s i so fe q u a t i o n 8).
RESULTS AND DISCUSSION
S t a t i c anddynamic
t es t s were
c o n d u c t e d n h e l a t e r a l and ore-and-aft
d i r e c t i o n s
t o
determine t i r e s p r i n gcons tan t s anddamping f a c t o r s . Dynamic
t es t s
w i t h d i f f e r e n t
mass
p l a t e n s p r o v i d e d n s i g h t n t o h e amountof t i r e mass
p a r t i c i p a t i n g n h e d yn am icmotion. n he o l lowing ec t ionsdynamic esul ts
are d isc uss ed and s t a t i c r e s u l t s are presen ted o rcompar i son . To c o n f i r m h a t
thecable-suspended s y s t e m e x h i b i t e d no s i g n i f i c a n tc o u p l i n gb e t we e n h ep i t c h -
ing and t r an s la t ing m o t i o n so f i t s p l a t e n , a two-degree-of-freedomanalysis of
t h e s ep l a t e nmo t i o n s i s p r e s e n t e d n h e a p p e n d i x .
Summaries of he t e s t cond i t ions and r e s u l t s f o r h e
l a t e r a l
and ore-and-
a f t r e e - v i b r a t io n tes ts are g i v e n n a b l e s I and
11.
Test
c o n d i t i o n sa n d
r e s u l t s
f o r h e s t a t i c tests are g iven
i n
t a b l e
111.
As shown i n he ab le s ,
l a t e r a l
and ore-and-aftdynamic
tests were
c o n d u c t e du s i n g h r e e p l a t e n s
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r a n g i n g n
mass
from
1 0 2 2 2 5 ) t o 5 3 6
kg
1 82
lbm . The
t i r e
was i n f l a t e d
to
one of t h r e ep r e s s u r e sr a n g i n gf r o m
6 8 9 1
00) to
1 2 4 1 kPa
(1
80 ps i )
where the
rated
i n f l a t i o n
pressure was 1 2 2 0 kPa 1 7 7 p s i ) .
The
t i r e
was a lso
loaded w i t h
one of e i g h t n o m i n a l )v e r t i c a l loads ranging from
2 2 . 2 kN
5000 l b f )
to
t h e
rated maximum
load
of
1 7 7 . 9 kN 4 0 000
l b f )
.
One
of
the easons oremploying
small amplitudes
i n t h e
tes ts is
t o mini-
m ize n o n l i n e a r i t i e s h a t c a n occur f o rs y s t e msu n d e r g o i n g a r g ed e f l e c t i o n s .
Some i n s i g h t i n t o t h e e x t e n t o f t h i s type o f n o n l i n e a r i t y c a n be gained from
t h e data. The dynamic
tests
revea led a s l i g h t r e q u e n c y n c r e a s ew i t h
ampli-
tude decay .Th i snon l inea reffect, however, was deemed i n s i g n i f i c a n t i n c en o
c u r v a t u r eo f h es p i n e
of
t h e
s t a t i c
h y s t e r e s i s
loop
was apparent e .g . ig .
2 ) .
Thus, when frequencyv a r i a t i o n s
occurred
dur ing
a t e s t
t h e y
were
averaged.
The determinat ion of s p r i n gconstants , damping
fac tors ,
a n d e f f e c t i v e
t i r e
masses
is
discussed i n h e s e c t i o n s h a t
follow.
Spr ing Cons tan t s
Latera l
andfore -and-a f t f requencyparamete r s de r ivedf r o m h eosc i l l a t ion
periods o f h ea c c e l e r a t i o n
time
h i s t o r i e sf o re a c hp l a t e n mass, t i r e pressure ,
andn o mi n a l v e r t i c a l load are t abu la t ed i n a b l e s
I
and
11,
r e s p e c t i v e l y .
S p r i n g constants computed from f requency parameters and t h e i r p l a t e n mass are
also
g i v e n n h e tables . S p r i n gc o n s t a n t sd e t e r m i n e ds t a t i c a l l y are g i v e n n
t a b l e 111.
Latera l d i r e c t i o n . - The lateral-frequency-parameter va luesde r ivedf r o m
v i b r a t i o n periods us ingequa t ion
(1) are
d i s p l a y e d n i g u r e
4. As
expected,
t h e l a t e r a l
f requency parameter
decreases
w i t h n c r e a s i n gp l a t e n
mass. For
each
p l a t e n
mass
the requency parameter i n c r e a s e s w i t h n f l a t i o n
pressure.
The t i r e
l a t e r a l
s p r i n gc o n s t a n t s computed f rom hese data, and l i s t e d i n
t a b l e
I
are noted t o be e s s e n t i a l l y n s e n s i t i v e t o p l a t e n mass. Thus, he
dynamic l a t e r a l s p r i n gc o n s t a n t sp r e s e n t e d nf i g u r e5 ( a )
as
a func t ionofve r -
t i c a l load were ob ta ined
for
e a c hp r e s s u r ea n d o a d i n gcon di t ion by averaging
t h e data o b t a i n e d o re a c hpl at en . The averageddynamic
l a t e r a l
sp r i ng con-
s t a n t s whi ch a ng e from
9 3 7 5 3 5 0 ) t o 1471
kN/m
8 4 0 0
l b f / i n . ) , o r h e
t e s t
c o n d i t i o n s
described
i n h i s p a p e r ,
are
shown
t o
i n c r e a s e w i t h n f l a t i o n
pres-
sure. When t h ep r e s s u r e i s h e l dc o n s t a n t h es p r i n gc o n s t a n t s e a c h a maximum
value a t
some
i n t e r m e d i a t e v e r t i c a l o a d i n g .
The spr ingc o n s t a n t s o b t a i n e d
from
s t a t i c
t es t s
are
p r e s e n t e d i n f i g -
u r e5(b) . The
s t a t i c
va lues
are
shown to e x h i b i t r e n d s similar
t o
thedynamic
va lues for e q u i v a l e n t t e s t cond i t ions ,bu t are 1 0 t o 2 0 p e r c e n t lower t h a n h o s e
found i n t h e dynamic
tests.
For
purposes ofcompar ing hese da ta wi t h h o s e from o t h e rs o u r c e s ,s p r i n g
c o n s t a n t s
are
d i s p l a y e d
as
f u n c t i o n so f
t i r e
v e r t i c a l d e f l e c t i o n n f i g u r e
6 .
T h e v e r t i c a l
t i r e
d e f l e c t i o n s are l i s t e d i n
t a b l e
IV Data t r e n d s n i g u r e
6
are
similar to
t h o s eo f e f e r e n c e 1 ; however , th e in e a r
empirical
equa t ion
of
t h e e f e r e n c e does no t
describe
t h e s e r e n d s n h e
low
d e f l e c t i o n a n g eo f h e
s tudy.
1 0
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Fore -and-a f tdir ecti on. - The dynamicfore-and-aftfrequency
parameters
are
d i s p l a y e d n f i g u r e 7 and, as expected , he requencyparameter is shown to
i n c r e a s ew i t hd e c r e a s i n gp l a t e n mass. Ingenera l , he o re -and-a f t r equency
parameter i s less s e n s i t i v e to v a r i a t i o n s n n f l a t i o n p r e s s u r e and more s e n s i -
t i v e
t o
v a r i a t i o n s n h e v e r t i c a l o a d h a n h e
l a t e r a l
frequencyparameters.
S i n c e h e
t i r e
f o r e - a n d - a f ts p r i n gcon sta nts computed rom th es e da ta were
a l s o
found
to
be
e s s e n t i a l l y n s e n s i t i v e
t o
p l a t e n
mass
(see
t a b l e
11 ,
thedynamic
s p r i n g c o n s t a n t s f o r h e h r e e p l a t e n s
were
averaged oreach pressure and oad-
i n g c o n d i t i o n ( f ig . 8 ( a ) ) .
The averageddynamic ore-and-af t i re -spr ing-constantvalues ange rom
201 4
1
1 500)
t o 3677 kN/m
(21 000 l b f / i n . )a n d are c o n s i d e r a b l y a r g e r h a n h e
l a t e r a l - s p r i n g - c o n s t a n tv a l u e s o rc o mp a r a b l e
test
c o n d i t i o n s . The d a t a
o
f i g -
u r e8 ( a ) show t h a t h e s e s p r i n g c o n s t a n t s n c r e a s e w i t h n f l a t i o n p r e s s u r e
a t
t h e h i g h e r v e r t i c a l o a d s and g e n e r a l l y n c r e a s e w i t h ve r t i ca l oad when th e
i n f l a t i o n p r e s s u r e i s h e l d c o n s t a n t .
The s t a t i c fore-and-af tspr ing-constantvalues ,which are presen ted as a
f u n c t i o no fv e r t i c a l o a d n i g u r e8( b) , show t r en ds
s imi lar to
thedynamic
da ta . However, the ta t i c -sp r ing -co nsta ntv a l u e s are 20 t o 35 p e r c e n t l ess
than hedynamicvalues.This eduction i s a t t r i b u t e d , n par t ,
t o
thev i sco-
e l a s t i c n a t u r eo f h e t i r e .
Fore-and-aft t i r e s p r i n gc o n s t a n t s are presen ted as a f u n c t i o n of t i r e ver-
t i c a l d e f l e c t i o n n i g u r e 9.
Data
fromboth hedynamic tes ts ( f i g .9 ( a ) ) a n d
the s t a t i c tes ts ( f i g .9 ( b ) ) n d i c a t e h a t h e o r e - a n d - a f t t i r e s p r i n gc o n s t a n t
g e n e r a l l y n c r e a s e sw i t hv e r t i c a ld e f l e c t i o n s .
Reference 3 c o n t a i n s
l a t e r a l
s t a t i c sp r ingcons tan t smeasured rom he same
type of t i r e
used
i n h i s r e p o r t , and r e f e re n c e
2
c o n t a i n s o r e - a n d - a f t s t a t i c -
sp r ing-cons tan tda ta rom he
same
t i r e
used
i n h i s
report .
The scant
d a t a
f r o m h e e f e r e n c e s n d i c a t e s imi lar t r e n d sb u t h es t i f f n e s sv a l u e s r o mb o t h
s e t s of da ta
were
beow t h e s t a t i c va lues
of
t h i sstudy. One cause f o r h e s e
d i f f e r e n c e s may be t h a t t h e t e s t a mp li t ud e s of t h i s s t u d y were a p p r e c i a b l y lower
than hose of r e fe renc es 2 and 3 .
As
ment ioned in e fe re nc e 1 , s p r i n gc o n s t a n t s
inc reasewi th educed t e s t amplitude. Other causes may
be
due to t i r e age,
mater ia l , and c o n s t r u c t i o n n c o n s i s t e n c i e s h a t may occur i n h e
same t i r e
as
well
as
i n d i f f e r e n t
t i r e s
o f h e same s i z e .
Damping Factor
Latera l
and for e-a nd- aft damping fa cto rsde r ived rom hed i sp lacement
am pli tud es of the damped fr e e v i b r a t i o no f e a c h t e s t are t a b u l a t e d n a b l e s . 1
and 11, re sp ec t i ve ly . Damping fac torsde te rmined rom s t a t i c tests are g iven
i n t a b l e 111.
Latera l d i r e c t i o n . - Th edamping f a c t o r s d e r i v e d f r a n v i b r a t o r y m o t i o n i n
the l a t e r a l d i r e c t i o n ,p r e s e n t e d n f i g u r e 1 0 ( a ) , are
small
and ange rom 2
to 7 percen t o f c r i t i c a l damping. The dynamic
l a t e r a l
damping fac torsg e n e r a l l y
appear to
be
i n s e n s i t i v e
t o
v e r t i c a l o a d v a r i a t i o n s
and
n oc o n s i s t e n t r e n d s
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are
n o t e dw i t hv a r i a t i o n s n t i r e i n f l a t i o np r e s s u r e .
The
da ta do i n d i c a t e
a
t endency fo r he l a t e r a l damping fac tors
to
decrease w i t h n c r e a s i n g p l a t e n
mass.
The l a t e r a l damping fac to r s ob ta inedf rom he s t a t i c tests are p r e s e n t e d
i n f i g u r e
1 0 ( b )
and
are
approx ima te lyequal nmagn i tude
to
t h e dynamic-damping-
f a c t o rv a l u e s of theheavywe igh tp la ten .These e su l t s would in d i ca t e ha t he
increaseddynamicdamping fac tors
associated
w i t h h e two l i g h t e r p l a t e n s may be
t h e r e s u l to f some a d d i t i o n a lviscous damping.
Fore -and-a f td i rec t ion . - The damping fa c t or s de r i ve d
from
thefore-and-
a f t
tests
are
shown i n f i g u r e 11. The dynamic ore-and-a ftdamping actors
( f i g . l ( a ) ) r a n g e b et we enapproximately
4
and
9
p e r c e n t
of
c r i t i c a l
damping
andno c o n s i s t e n t r e n d s are o b s e r v e dw i t hv a r i a t i o n s n h e
t e s t
c o n d i t i o n s .
The fore -and -aft damping fa ct o rs o b t a i n e d
from
t h e s t a t i c tests
are
pre-
s e n t e d n f i g u r e l ( b ) and are noted
to
b e c o n s i s t e n t l y lower than hedynamic
damping f ac to r s , he re by nd ic a t i ng ha t
some
viscous damping
i s
p r e s e n td u r i n g
fore-and-af t
t i r e
v i b r a t i o n s . A comparison
of
t h e s t a t i c damping f a c t o r s f rom
t h e
l a t e r a l
tests and t he ore-a nd-a f t tests i n d i c a t e s l i g h t l y higher damping
i n h e f o r e - a n d - a f t d i r e c t i o n s .
The f indings
from
the damping
tes ts
i n
b o t h
d i r e c t i o n s n d i c a t e h a t damp-
i n g
was
s u f f i c i e n t l y
small t o
j u s t i f y h e d e l e t i o n ofdamping e f f e c t s i n t h e
s t i f f n e s sc o m p u t a t i o n s .
E f f e c t i v e
T i re
Mass
E f f e c t i v e
t i r e
masses
are
computed rom
t h e l a t e r a l
and ore-and-af t
dynamic tes ts for each t i re pressure a n d v e r t i c a l
load
combination.
Latera l
d i r e c t i o n . - The e f f e c t i v e t i r e
mass
i n t h e l a t e r a l d i r e c t i o n
was
computed
us ing a l l t h r e e d i f f e r e n t mass p la t ensa n d i s g i v e n n t a b l e
I
f o re a c h
t i r e
pressure and loadingcondi t ion .
The re su l t s
are
shown
t o
vary rom 2.7 6.0) to 13.9 kg 30.7
lbm)
andhave
anaveragevalueof
7.5
kg
16.5
lbm). when
compared t o
t h e
t o t a l t i r e
mass of
79.4
kg
175 lbm)
t h ea v e r a g ee f f e c t i v e t i r e
mass i s small.
One re as on or he
v a r i a t i o n s n h e e f f e c t i v e - t i r e - m a s s
da t a
i s
a t t r i b u t e d to
a l ack of nstrumen-
t a t i o n p r e c i s i o n
as i l l u s t r a t e d
i n h e f o l l o w i n g
error
a n a l y s i s .
The mass
error
Am o c c u r r i n g
from
a period inaccuracy AT can
be
de r ived
fromequat ion
1 t o be
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Thus, for a
period
inaccuracyof 1 msec, t h ee q u a t i o n n d i c a t e s h a t Am
w i l l
bewi th in hefo l lowingrange :
3 . 0 kg 6 . 6 lbm)
Am
9.1 kg 2 0 . 1 lbm
For-e-and-aftdirect ion.- Upon ex am ina tio nof hefo re -and-a f tda ta , he
s p r i n g c o n s t a n t s f o r h e h e a v y p l a t e n
were
found
t o
be changingwith requency;
hence, no e f f e c t i v e t i r e mass was computed f o r t h a t p l a t e n
i n
the o re -and-a f t
d i r e c t i o n . The e f f e c t i v e t i r e masses a s s o c i a t e dw i t h h e t e s t d a t a r o m h e
remaining two p l a t e n s are g iven i n t a b l e 11. These masses were g e n e r a l l yh i g h e r
t h a n h o s ea s s o c i a t e dwi t h h e
l a t e r a l tes ts
and anged rom 7 . 8 17 . 2 )
t o
2 5 . 9 kg 5 7 . 2
lbm
with
an
averagevalueof 1 5 . 6 kg 3 4 . 4
lbm .
Equation 9 )
p r e d i c t s
mass er ro rs
i n h e a n g eo f
4 . 4 5 kg 9 . 8 lbm
1 2 4 118 0 ) w 1 2 4 1 (18 0 )
P l a t e n mass,
536 kg (1182 1bm)
T i r ep r e s s u r e ,
k P a ( p s i )
0
689
(100 )
17
965
(140 )
0
1241
(180 )
L I 1
0 30 60 90 1205080
V e r t i c a l o a d , kN
L I ,
I I
I
I I
I
0
10
20
30 40 x 103
V e r t i c a l o a d ' ,
1
b f
(a ) Dynamic te s t s .
Figure 1 1 .-Va ria tio n of fore-and-aft damping fa ct or w i t h platen mass,
t i re pressure ,
and
vertical loading.
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.08
L
c
0
2 .06
n
E
m
-o .04
4 2
0
Tire pressure,
kPa ( p s i )
1 2 4 11 8 0 )
0
30 60 90
120
150 180
Vertical load, kN
1 I I I
I I I I
0
10
20
30
40
x
l o 3
Vertical load,
l b f
b)
S t a t i c
tests
Figure
1 1
.- Concluded.
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1 . Reporto. 2. Governmentccession No.
3.
Recipients
a t a l o g
No.
NASA TP 1671
4. Title and Subtitle I 5. Report Date
TIRE STIFFNESS AND DAMPING
DETERMINED FROM
STATIC AND FREE-VIBRATION TESTS
7. Author(4 8. Performingrganizationeport No.
Robert
K.
Sleeper and
Robert
C.
Dreher
I
L-13500
10. Work Unit No.
9. Performing Organization Name and Address
505 44 33 01
NASA Langley
Research
Center
Hampton, VA 23665
11.Contract or Grant N o .
~~~~ ~
13.Type of R e p o r t and Period Covered
2. Sponsoringgencyame Address
Technical Paper
14. SponsoringAgency code
ational Aeronautics and Space Administration
Washington, DC 20546
I
5. Supplementary Notes
16.
Abstract
Stiffness anddamping ofa nonrolling t i r e are determinedexperimentally fran both
s t a t i c force-displacementrelati ons and the free-vibrationbehaviorof a cable-
suspended platenpressedagainst t h e t i r e perip hery. La tera l and ore-and-aft
spring constan ts anddamping
factors
of
a
49
x
17 s i z e
a i r c r a f t
t i r e
or
dif fe ren t
t i r e
pressures and ver ti ca l loads
are
measured as sm in g a rate-independe nt damping
form. Inaddi t ion ,
a
technique i s applied for esti mat ing he magnitudeof the
t i r e mass which pa rti cip ate s n he vi bra to ry motion of
t h e
dynamic tests. R e s u l t s
show th at bo th he
l a t e r a l and ore-and-aft
spr ing constants general ly ncrease
w i t h
t i r e pressure but only he
l a t t e r
inc reased s igni f icant ly wi th ve r t ica l t i r e
loading. The fore-and-aftspringconstants were greater han those i n h e l a t e r a l
direct ion. The sta t ic-spr ing-constantvar ia t ions were similar t o the dynamic
var ia t ionsbutexhibi ted
laver
magnitudes. Damping
was small
and insensitive t o
t i r e
loading.Furthermore,
s t a t i c
damping accounted
for
a
s igni f icantpor t ion
of
that found dynamically.Effective t i r e masses
were
also small.
17.
Key WordsSuggested byuthor(s) ) 18.istributiontatement
Tires
-
Unlimited
T i r e vibra t ion
Tire damping
Tire
spring constant
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