Tire Stiffness and Damping Determined_NASA_Technical Paper 1671

8/10/2019 Tire Stiffness and Damping Determined_NASA_Technical Paper 1671

1/47

NASA Technical

Paper

1671

NASA

TP

1 6 7 1

c. 1

Robert

K.

Sleeper

and

Robert C. Dreher

JULY 1980


2/47

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


3/47

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.


4/47

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


5/47

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

3


6/47

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 ) .

4


7/47

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

5


8/47

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

=

(

6


9/47

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 .

7


10/47

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-

8


11/47

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

9


12/47

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


13/47

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

11


14/47

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

12


15/47

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.

42


45/47

.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.

43


46/47

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

Subject Category 03

19. Security Qassif. of this report) 20. Securitylassif. (of this p a g e )

I

21. o P

Unclassified

For

sale

by

the National Technical

In fo rmat ion

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Documents

Tire Stiffness and Damping Determined_NASA_Technical Paper 1671