wong,j & reece, a.r. 1967. prediction of rigid wheel performance based on the analysis of soil -...

7/28/2019 Wong,j & Reece, A.r. 1967. Prediction of Rigid Wheel Performance Based on the Analysis of Soil - Wheel Stresses ...

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Journal o/Terramechanics, 1967, Vol. 4, No. 1, pp. 81 to 98. Pergamon Press Ltd.Printed in Great Britain.

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E B A S E DO N T H E A N A L Y S I S O F S O I L - W H E E L S T R E S S E S

P A R T I . P E R F O R M A N C E O F D R I V E N R I G I D W H E E L SJO-YUNG WONG* and A. R. REECE*

INTRODUCTIONRECENT inves t iga tions b y the au th or s have show n tha t a soun d theore t ica l bas i sf o r th e p r e d i c t io n o f wh e e l p e r f o r m a n c e o n s o il c a n o n l y b e p r o v i d e d b y k n o w l e d g eof the ac tua l so i l f a i lu r e phen om ena bene a th w hee ls [1] . Fo r a ve ry wide whee l( e .g . a wide ro l le r o r poss ib ly a t e r r a - ty r e ) , so i l f a i lu r e on ly occur s in the d i r ec t iono f m o t i o n a n d i t is , t h e re f o re , a t wo d i m e n s i o n a l p r o b l e m . O n t h e o t h e r h a n dexis t ing theo ry for whee ls a s sum es so i l fa i lu r e and f low to be a lm os t en t i r e lys i d e wa y s . T h e i n v e s t i g a t i o n s r e f e r r e d t o a b o v e h a v e s h o wn h o we v e r , t h a t b e n e a t ha convent iona l r a the r na r row whee l so i l f a i l s in a compos i te way , pa r t ly s idewaysa n d p a r t l y l o n g i t u d i n a l . T h e l o n g i t u d i n a l f a i l u r e p h e n o m e n o n u s u a l l y c a n n o t b eignored even for a qu i te na r row whee l , and the f a i lu r e pa t te rn i s the r e fore muchm o r e c o m p l i c a t e d t h a n e x i s t in g t h e o r y a s s u m e s .A grea t dea l o f e ffor t wi l l be ne eded be fore a th r ee d im ens iona l f a i lu r e pa t te rnc a n b e e s t a b l i s h e d a n d o n l y wh e n t h i s h a s b e e n c o m p l e t e d c a n a s o u n d t h e o r e t i c a la n a l y s i s o f t h e p r o b l e m b e a t t e m p t e d . Ho we v e r , r e c e n t i n v e s t i g a t i o n s d o i n d i c a t etha t the s ideways so i l f a i lu r e s t i l l p lays an impor tan t ro le benea th convent iona lwhee ls [1] . There fore , a s an ex pedien t a t the p r esen t s tage of the deve lo pm ent o fso i l -veh ic le mechanics , a theory for p r ed ic t ing the pe r formance of convent iona lwh e e l s b a s e d o n t h e p l a t e p e n e t r a t i o n t e s t b u t w i t h c o n s i d e r a t i o n s o f t h e i m p o r t a n ta s p e c t s o f t h e a c t u a l i n t e r a c t i o n b e t we e n wh e e l s a n d s o i l s h o u l d b e d e v e l o p e d .

T h i s p a p e r d e s c r ib e s s u c h a n a p p r o a c h c o n s i d e ri n g t h e d i s t r ib u t i o n o f b o t h r a d i a land tangent ia l s t r e s ses on the so i l -whee l in te r f ace and tak ing in to account tha tt h e y n o t o n l y d e p e n d o n s o i l p r o p e r t i e s a n d wh e e l d i m e n s i o n s b u t a l s o o n t h e s l i p( o r sk i d) . A c o m p r e h e n s i v e p r o c e d u r e f o r p r e d ic t in g t h e p e r f o r m a n c e o f c o n v e n t i o n a lwh e e l s i s p r o d u c e d . I t i s s h o wn t h a t t h is m e t h o d c a n g i v e b e t t e r p r e d i c ti o n o f wh e e lpe r formance than ex is t ing theory .This w ork cons i s t s o f two pa r t s. Pa r t I conce nt r a tes on the d i scuss ion of theper formance of d r iven r ig id whee ls , pa r t i cu la r ly wi th r e f e r ence to sand . Towedr ig id wh ee ls wi l l be d i scussed in Pa r t I I .

1. THE DISTRIBU TION OF RADIAL AND TAN(~ENTIAL STRESSES ON THESOIL-WHEEL INTERFACEAs i s we l l k n o wn , t h e e q u i l i b r i u m o f a wh e e l , wo r k i n g o n h o r i z o n t a l g r o u n d

wi t h c o n s t a n t v e l o c i t y , c a n b e d e s c r i b e d b y t h e f o l l o wi n g e q u a t i o n s :*Department of Agricultural Engineering, University of Newcastle upon Tyne.

81


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8 2 J O - Y U N G W O N G a n d A . R . R E E C E

01 o 1

0 , 0 2( I )

1 o 1

0 '2 0 '2( 2 )

o j

l r (0)= r ~ b d O .0 2

( 3 )

W h e r e o r ( 0 ) a n d 7 ( 0 ) r e p r e s e n t t h e a v e r a g e r a d i a l a n d t a n g e n t i a l s t r es s a c r o s st h e w h e e l w i d t h . F i g u r e 1 i l l u s tr a t e s t h e f o r c e s , t o r q u e a n d s t r e ss e s a c t i n g o n aw h e e l .

T

o "F IG . 1 F o r c e s , t o r q u e a n d s t r e s se s a c t i n g o n a d r i v e n r ig i d w h e e l .

I t is t h e r e f o r e n e c e s s a r y t o d e t e r m i n e t h e r e l a t i o n s h i ps b e t w e e n r a d i a l a n dt a n g e n t i a l s t r e s s es a n d s o i l p r o p e r t i e s , w h e e l d i m e n s i o n s a n d t h e d e g r e e o f s li p , i no r d e r t o p r e d i c t t he w h e e l p e r f o r m a n c e .1 .1 R a d i a l s t r e s s d i s t r i b u t i o n

E x i s t i n g t h e o r y a s s u m e s t h a t t h e r a d i a l s o i l r e a c t i o n a t a n e l e m e n t o f t h e r i mi s t h e s a m e a s t h a t b e n e a t h a p l a t e p e n e t r a t e d t o t h e s a m e d e p t h [ 2 ] . T h i s i m p l i e st h a t t h e b a s i c c h a r a c t e r i s t i c s o f th e d i s t r i b u t i o n o f r a d i a l s t r e s s a n d m o t i o n r e s i s t a n c ea r e i n d e p e n d e n t o f s li p. E x p e r i m e n t s o n c l a y b y U f f e l m a n n s h o w t h a t t h e r a d i a ls t r es s d i s t r i b u t i o n s e e m s t o b e q u i t e u n i f o r m a n d q u i t e i n d e p e n d e n t o f sl i p [ 3 ].B u t a n u m b e r o f e x p e r i m e n t s o n s a n d h a v e s h o w n t h a t t h e r a d i a l s tr e ss d i s t r ib u t i o ni s a f u n c t i o n o f s li p a n d t h e m a x i m u m r a d i a l s t r es s d o e s n o t a c t a t t h e p o i n tb e n e a t h t h e a x l e ( " b o t t o m ~ l e a d - c e n t r e " ) a s w o u l d b e e x p e c t e d f r o m t h e p la t es i n ka g e a n a l o g y . I t a c t u a l l y o c c u r s in f r o n t o f b o t t o m - d e a d - c e n t r e a n d s h if tsf o r w a r d w i t h i n c r e a s i n g s l i p [ 4 - 7 ] . T h i s i n d i c a t e s t h a t s p e c i a l a t t e n t i o n s h o u l d b e


3/18

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E . P A R T I 83p a i d t o a n a l y s i n g t h e c h a r a c t e ri s t ic s o f r a d i a l s t re s s d i s t r i b u t i o n o n s a n d a n d t h ee x i s t i n g t h e o r y s h o u l d b e m o d i f i e d a c c o r d i n g l y .

T h e a u t h o r s h a v e p r e v i o u s l y s h o w n t h a t t h e m a x i m u m r a d ia l s t re s s o c cu r s a t t h ep o i n t w h e r e t h e t w o s o i l f a i l u r e z o n e s b e n e a t h t h e w h e e l j o i n e a c h o t h e r , a s a tp o i n t A s h o w n i n F i g . 2 [ 1] . B u t i t i s n o t p o s s i b l e t o l o c a t e t h i s p o i n t f o r a d r i v e n

FI~. 2.

. / j j / ~ / / ' / " J/Diagram showing the two fai lure zones l :eneath a dr iven r ig id wheel on thelongi tudinal p lane.

w h e e l b y a t h e o r e t i c a l a p p r o a c h a t t h e m o m e n t , o w i n g t o t h e d if f i c u lt y i n e s t a b -l is h i n g th e t h r e e d i m e n s i o n a l f a i lu r e p a t t e r n . H o w e v e r , s e v e r a l s e ts o f e x p e r i m e n t a lr e s u lt s a r e n o w a v a i l a b l e [ 4 - 7 ] a n d t h e se s h o w t h a t t h e lo c a t i o n o f t h e p o i n t o ft h e m a x i m u m r a d i a l s tr e ss c a n b e e x p r e s s e d b y t h e f o l l o w i n g e q u a t i o n , w h e n t h ew h e e l i s w o r k i n g o n s a n d :

0M0 " -~ = c l + c , ,_ i . (4 )F i g u r e 3 s h o w s t h e e x p e r i m e n t a l r e su l ts f o r th e r e l a ti v e p o s i t i o n o f t h e m a x i m u m

r a d i a l s t r es s f o r d i ff e re n t w h e e l s u n d e r d i f f e re n t w o r k i n g c o n d i t i o n s [ 4 - 7 ] .T h e c o e f f ic i e n ts c i a n d c ~ a r e t a b u l a t e d i n T a b l e 1 .

TABLEAn gle of Coeffic ients forin ternal Soi l determining the re la t iveSo i l shea r ing cohes ion Dens i ty pos i ti on o f ma x imu m Rem arksresistance c 2/ rad ial stress& (Ib/in . 2) ( lb/in . a) cl c~

C om pac t sand 33-3 0" 10 0-0575 0"43 0- 32 Ac cord ing toLoo se sand 31 .1 0 -12 0 .048 0"18 0 " 3 2 On afekoSand 36"0 0"10 0-0617 0-285 0" 32 Ac cordin g toH e g e d u sD r y s a n d 2 4 .0 - - - - 0 "3 8 0 - 4 1 A c c o r d i n g t oSelaI t is i n t e re s t i n g t o n o t e t h a t e v e n t h o u g h t h e e x p e r i m e n t a l c o n d i t i o n s a r e v a r i o u s ,

t h e c o e f f i c ie n t c~ h a s a l m o s t t h e s a m e v a l u e , o n l y c l i s d i f fe r e n t. T h i s i m p l i e s t h a tt h e re l a t i o n s h i p b e t w e e n t h e r e l a t iv e p o s i t i o n o f t h e m a x i m u m r a d i a l s t re s s a n ds l ip h a s t h e s a m e b a s i c c h a r a c t e r i s t i c , t h a t is t o s a y th e c h a r a c t e r i s t i c l i n e s [ f r o me q u a t i o n ( 4) ] h a v e a l m o s t t h e s a m e s l o p e. T h e o n l y d if fe r e n ce b e t w e e n t h e m i s


4/18

84 J O - Y U N G W a N G a n d A . R . R E E C E9 0

8 C7O

6 0

o~ 50

d;~ 403 0

(a ) LC~ poct sand 49.4x6 in . r i g id wheel , ax le load2 0 0 0 b2 . L o o s e sandx 4 9 . 4 x 6 i n . rigid wheel , ax le load1986 lb .

o 49. 4x1 2 in . r ig id wheel , ax le load2 0 8 5 I b , I

F I G . 3 .( a )(b )(c)

2 0 - ~.....,

,o I0 IO

2

2 0 30 4 0 5 0 6 0i , %7 0 ()6 O

50

~ 4 0 ~ J~ . . 2 ~ ~ -

20

( b ) I9 G8 C7 0

6 C

4 03 02 0I 0

0

f x ' /w

7 0 I0 2 0 3 0 8 0

D r y sondW h e e l dio . 20 in .

A x l e load 3 0 0 1 bx 2 5 0 1 b0 2 0 0 1 b IO 0 1 b

I I40 50 60 70i, %

fI

Sandx 2O x3in . r i g id wheel, ax le load 150 Ib10 * 20x Sin . r i g id wheel , ax le load 150 Ib

I I J J i I0 I 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 I 0 0i, %

Variation of the relative position of the m axim um rad ial stress point wil~h slip.4 9 - 4 X 6 i n . a n d 4 9 " 4 x 1 2 i n. ri g id w h e e l s o n c o m p a c t a n d l o o s e s a n d( a c c o rd i n g t o O n a f e k o [ 4] ).2 0 i n . d ia . r ig i d w h e e l o n d r y sa n d ( a c c o r d i n g t o S e l a [ 6 ] ).2 0 5 i n . a n d 2 0 x 3 in . r i g id w h e e l s o n s a n d ( a c c o r d i n g t o H e g e d u s [ 7 ]) .

in t h e i n i t i a l v a l u e .I t t h e r e f o r e s e e m s t h a t t h e r e l a t i v e p o s i t i o n o f t h e m a x i m u m r a d i a l s t r e s s d e p e n d s

a n t h e c o m p r e s s i b i l i ty o f th e s a n d , a n d i t i s q u i t e i n d e p e n d e n t o f w h e e l d i m e n s i o n st n d s i n k a g e . O f c o u rs e , m o r e t h e o r e t i ca l a n d e x p e r i m e n t a l i n v e s ti g a t io n s a r e n e e d e d) e f o re s u c h a c o m p r e h e n s i v e c o n c l u s i o n c a n b e c o m p l e t e l y j u st if ie d .

E x p e r i m e n t a l e v i d e n c e [ 4 ], '.[5 ] h a s a l s o s h o w n t h a t o n s a n d f r o m t h e b e g i n n i n g) f c o n t a c t 01 t o t h e m a x i m u m r a d i a l s t r e ss p o i n t 0M ( F i g . 4 ), t h e r a d i a l s t r e s s a r o u n d:h e r i m c a n b e q u i t e w e l l d e s c r i b e d b y t h e p r e s s u r e - s i n k a g e r e l a ti o n s h i p p r o p o s e d) y R e e c e : [8 ]. I n o t h e r w o r d s , i n t h e r e g i o n b e t w e e n 01 a n d 0M ( c a l le d t h e " f r o n t: e g i o n " a n d i l l u s t r a t e d i n F i g . 4 ) , t h e r a d i a l s t r e s s c a n b e c a l c u l a t e d b y t h e f o l l o w i n g; q u a t i o n :

o -~ = ( k ~ + k 2 b ) (5 )


5/18

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E . P A R T I 85I f t h e r e la t i o n s h i p z = ( c o s 0 - c o s 0 1 )r is u s e d , th e n e q u a t i o n ( 5) c a n b e r e - w r it t e n

a s f o l l o w s :

o'~ (0 ) = (k~ + k2b) - - f f (cos 0 - cos G)" . (6 )F r o m t h e m a x i m u m r a d i a l p r e s s u r e p o i n t 0 ~ t o t he e n d o f c o n t a c t 02 ( c a ll e d th e

" r e a r r e g i o n " a n d i l l u s t r a t e d i n F i g . 4 ) , t h e r a d i a l s t r e s s a r o u n d t h e r i m d e c r e a s e s .B u t i t i s i n t e r e s t i n g t o n o t e t h a t t h e r a d i a l s t r e s s d i s t r i b u t i o n s t i l l h a s s i m i l a rc h a r a c t e r i s t i c s . T h i s s u g g e s t s th a t i f t h e r e la t i v e p o s i t i o n t o t h e c o r r e s p o n d i n g f r e es u r f a c e o f a p o i n t i n t h e r e a r r e g i o n i s t h e s a m e a s t h a t o f o n e i n t h e f r o n t r e g i o n ,t h e n b o t h w i l l h a v e t h e s a m e v a l u e o f r a d i a l s t r e s s . T h i s c a n b e d e s c r i b e d b y t h e

FIo. 4.

o ', Rear Frontr e g i o n [ _ regionO'max

j ' ,Oz o OR OM Ov O, ~ *

D iag ram illustrating the characteristic distribution of radial stress on thesoil-wheel interface of a driven rigid wheelf o l l o w i n g e q u a t i o n s a n d i s i l l u s t r a t e d i n F i g . 4 , i .e .

OR 0 2 0 ~ - OF0 ~ - 0 , , 0 1 - 0 ~ ' (7 )

th en tron = o-oF. (8)A c c o r d i n g t o t h e a b o v e p r o p o s a l , t h e r a d i a l s t r e s s a c t i n g i n t h e r e a r r e g i o n c a n

b e c a l c u l a te d b y t h e f o l l o w i n g e q u a t i o n :

0 - 2 I n

02 r e p r e s e n t s t h e d e g r e e o f r u t r e c o v e r y . I t i s q u i t e c l e a r t h a t r u t r e c o v e r y i s af u n c t i o n o f s l i p a n d w h e e l w i d t h [ 1 ] , b u t t h e q u a n t i t a t i v e r e l a t i o n s h i p b e t w e e nt h e m h a s n o t y e t b e e n e s t a b l i s h ed . I f r u t r e c o v e r y is s m a l l a n d n e g l e c t ed ( e .g . th ew h e e l i s n o t w i d e a n d t h e d e g r e e o f s l i p i s n o t v e r y h i g h ) , t h e a b o v e e q u a t i o n c a n


6/18

8 6 J O - Y U N G W O N G a nd A . R . R E E C Eb e s i m p l i f i e d a s f o l l o w s :

o - 2 ( O ) = ( k ~ + k 2 b )- ~ - ) " [ c o s ( 0 ~ - 0 - ~ - ( 0 ~ - 0 ~ ) ) - c o s G ] " ( 1 0)A c c o r d i n g t o e q u a t i o n ( 4) , 0M c a n b e r e p l a c e d b y t h e f o l l o w i n g e q u a t i o n "

0~ = ( c l + c 2 0 01 .E q u a t i o n ( 10 ) c a n t h e n b e r e w r i t te n a s fo l l o w s :

c r 2 ( O ) = ( k t+ k 2 b ) ( b ) " c o s ( 0 1 _ 0 ( 1 - ( c , + c d ) "c ,+ c j ) ) ) -c o s O ~ ( 1 1 )F r o m t h e f o r e g o i n g a n a l y s i s , it i s c l e a r th a t i f t h e p r e s s u r e - s i n k a g e c o n s t a n t s

G , k 2 a n d n , t h e c o e f fi c ie n t s c i a n d c2, t h e w h e e l d i m e n s i o n s r a n d b , s i n k a g e za n d s l i p i a r e k n o w n , t h e n u s i n g e q u a t i o n s ( 4) , ( 6) a n d ( 9 ) o r ( 1 1) , t h e r a d i a l s t r e s sd i s t r i b u t io n o n t h e s o i l - w h e e l i n t e r f a c e c a n b e d e t e r m i n e d q u a n t i t a t i v e ly .

F i g u r e s 5 a n d 6 s h o w t h a t t h is c a n g i v e a s a t i s f a c t o r y d e s c r i p t i o n o f th e r a d i a ls t re s s d i s t r i b u t io n a r o u n d a d r i v e n r i g i d w h e e l w o r k i n g o n d i ff e r en t s an d s . T h ep r e d i c t e d d i s t r i b u t i o n i s o b t a i n e d f r o m t h e m e a s u r e d s i n k a g e u s i n g e q u a t i o n s ( 4) ,

4 0

3 O

2 Od

0 - 1 0

, Y,;?/J 0 I 0 20

8o

- - M e o s u r e d- - - - P r e d i c ted

30 4 0

F I 6 . 5. C o m p a r i s o n o f t h e m e a s u r e d a n d p r e d i c t e d d i st r ib u t i o n o f r a d i a l s t re s s o f a49"4 6 i n. r i g i d w he e l a t 22" 1 pe r c e n t s l ip w i t h t he s a m e s i nk a ge on c om pa c t s a nd" (m e a su re d v a l u e s f r o m O n a f e k o ' s e x p e r i m e n t s , s o i l v a lu e s a s i n T a b l e 2 ).TABLE 2

SoilS o i l p r o p e r t i e s P r e s s u r e - s i n k a g e S h e a r C o e f f ic i en t s f o rA n g l e o f c o n s t a n t s d e f o r - d e t e r m i n i n g t h ei n t e rna l S o il De ns i t y m a t i on r e l a t i ve pos i t i ons h e a r in g c o h e s i o n 3 ' k l k 2 n m o d u l u s o f m a x i m u mres i s tan ce c ( lb / i n ? ) K r a d i a l s t r e s s~b (l b /in . ~) (in.) o c2

C o m p a c tsan d 33 '3 0"10 0 '057 5 20 2"5 0"4706 1 '5 0"43 0 '32L o o s esan d 31"1 0 " 1 2 0 .048 0 2 1"1504 1"5 0-18 0-32


7/18

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E . P A R T I 87( 6 ) a n d ( 1 1 ) , n e g l e c t i n g t h e e f f e c t o f r u t r e c o v e r y . S o i l p r o p e r t i e s a n d o t h e r c o n s t a n t su s e d i n t h e c o m p u t a t i o n a r e t a b u l a t e d i n T a b l e 2 .

25

20

J5 /

M e o s u r e di P r e d i c t e d

5

-co -to o io 2o 30 40 50 6o0*1~o. 6. Co m pariso n of the m easured and predicted distribution of radial stress of a49" 4 6 in. r igid w heel at 22"1 per cen t slip with the sa m e sinkage on loose sand(measured values fr om O nafe ko's experiments, soil values as in T able 2).

1.2 Tangential stress distributionI t h a s b e e n s h o w n t h a t s h e a r s t re s s d e v e l o p e d i n so i l d e p e n d s o n s h e a r d e f o r m a t i o n

( d i s p l a c e m e n t ) , a n d t h e r e l a t i o n s h i p b e t w e e n t h e m c a n b e q u i t e w e l l d e s c r i b e d b yt h e f o l l o w i n g s i m p l i fi e d B e k k e r e q u a t i o n :

-r = ( c + o - t a n 6 ) ( 1 - e - i / x ) .I n o r d e r t o e v a l u a t e t h e s h e a r ( t a n g e n t i a l ) s t r e s s d i s t r i b u t i o n a r o u n d t h e r i m , i t

i s , t h e r e f o r e , n e c e s s a r y t o f i n d o u t t h e s h e a r d e f o r m a t i o n j a l o n g t h e s o i l - w h e e li n t e r f a c e .

T h e s h e a r d e f o r m a t i o n b e n e a t h a t r a c k h a s b e e n i n v e s t i g a t e d b y B e k k e r b a s e do n t h e a n a l y s i s o f t h e s l i p v e l o c i t y v j o f t h e t r a c k a n d i s i l l u s t r a t e d i n F i g . 7 ( a) .

F o l l o w i n g a s i m i l a r a p p r o a c h , t h e s h e a r d e f o r m a t i o n o n t h e s o i l -w h e e l i n te r f a c ec a n b e f o u n d . B e n e a t h a d r i v e n r i g i d w h e e l , t h e s l i p v e l o c i ty vj o f t h e r i m r e l a ti v et o t h e s o i l is th e t a n g e n t i a l c o m p o n e n t o f th e a b s o l u t e v e l o c i t y v , a t t h e s a m ep o i n t , a s i l lu s t r a t e d i n F i g . 7 ( b ). T h e s l i p v e l o c i t y v j c a n , t h e r e f o r e , b e e x p r e s s e db y t h e e q u a t i o n :

v j = r t o [ 1 - (1 - i ) cos 0 ] .T h e s h e a r d e f o r m a t i o n j a l o n g t h e s o i l - w h e e l i n t e r f a c e c a n t h e n b e c a l c u l a t e d

i n th e f o l l o w i n g w a y :

j=01t

f v j d t = / r w [ t - ( 1 - i ) c s ] d-toa o

= r [(01 - O) - (1 - 0 ( sin 01 -- s in 0)] . (12)


8/18


z z / / / f J f J ~ f j J . ~ J / J / J . ~ J ~ z .(O ) v =r toij=[~vjd l = ix

F I o . 7 .

-------~-v

( b ) Vov = rw [I- (I-i) cos 0]J = . f o ' v j d t= r ~ O i - O - ( l - i ) ( s i n O f - s i n 0 ) ]

D i a g r a m i l lu s t r a t i n g t h e d e v e l o p m e n t o f s h e a r d e f o r m a t i o n b e n e a t h ( a) a t r a cka n d ( b ) a r ig i d w h e e l .

T h e s h e a r s t r e s s a r o u n d t h e r i m c a n , t h e r e f o r e , b e e x p r e s s e d b y t h e f o l l o w i n ge q u a t i o n :

r (0 )= (c + t r (0 ) t an th) (1 - e - j/K)

= ( c + o " ( 0 ) t a n th) (1 - e-r/K l l l - ) - ~ ' - ' > I ' i " l - s i n ) J ) ( 1 3 )I n t h e f r o n t a n d r e a r r e g i o n s , o - ( 0 ) c a n b e d e t e r m i n e d b y e q u a t i o n s ( 6 ) a n d ( 9 )

o r ( 1 1 ) r e s p e c t i v e l y .I n c i d e n t a l l y , t h e s h e a r d e f o r m a t i o n a l o n g t h e s o i l - w h e e l i n t e r f a c e h a s a l s o b e e n

i n v e s t i g a te d b y O n a f e k o u s i n g a d i ff e re n t a p p r o a c h a n d t h e s a m e r e s u l t h a s b e e no b t a i n e d [ 4 , 5 ] .

I n o r d e r t o c a r r y o u t c a l c u l a t io n o f t a n g e n t ia l s t re s s f r o m e q u a t i o n ( 13 ), t h es h e a r d e f o r m a t i o n m o d u l u s K s h o u l d b e d e t e r m i n e d b e f o r e h an d . E x p e r i m e n t s b yW i l l s [ 9 ] a n d t h e a u t h o r s h a v e s h o w n t h a t t h e s h e a r d e f o r m a t i o n m o d u l u s K s e e m st o d e p e n d o n n o r m a l s t r e s s . B u t t h e q u a n t i t a t i v e r e l a t i o n s h i p b e t w e e n t h e m s t i l ln e e d s t o b e d e t e r m i n e d b y f u r t h e r i n v e s t i g a ti o n s . I n t h is a n a l y s is , a n a v e r a g e v a l u eo f K i s t a k e n i n t h e c o m p u t a t i o n f o r a p a r t i c u l a r s o il .

I t h a s b e e n s h o w n t h a t e q u a t i o n ( 1 3 ) c a n g i v e f a i r l y g o o d p r e d i c t i o n o f t a n g e n t i a ls t re s s e s a r o u n d t h e ri m o f a d r i v e n w h e e l p a r t i c u l a r l y a t h i g h s li p a n d o n c o m p a c ts a n d [ 4 , 5 ] . B u t i t n e e d s s o m e m o d i f i c a t i o n b e f o r e i t c a n b e a p p l i e d t o p r e d i c t t h et a n g e n t i a l s t r e s s e s o n t h e s o i l - w h e e l i n t e r f a c e o f a t o w e d w h e e l , a n d t h is w i ll b ed i s c u s s ed i n P a r t I I o f th i s p a p e r .


9/18

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E . P A R T I 892. P R O C E D U R E F O R P R E D I C T I N G T H E P E R F O R M A N C E O F D R I V E NR I G I D W H E E LSI f s o i l v a l u e s c , th , K , k t , k 2, n , w h e e l d i m e n s i o n s r , b , t h e c o e f fi c ie n t s f o r d e t e r -

m i n i n g t h e r e l a t iv e p o s i t i o n o f t h e m a x i m u m r a d i a l s t re s s Cl, c2 a n d v e r t ic a l a x l el o a d o f th e w h e e l W a r e k n o w n , t h e w h e e l p e r f o r m a n c e c a n b e p r e d i c t e d b y t h ef o l lo w i n g p r o c e d u r e .2 .1 P r e d i c t i o n o [ s i n k a g e

I f r u t r e c o v e r y i s n e g l e c te d , b y u s i n g e q u a t i o n s ( 4) , ( 6) , ( 11 ) a n d ( 13 ), t h ee q u i l i b r i u m e q u a t i o n ( 1 ) c a n b e r e w r i t t e n a s f o l l o w s :

W : rb01 (c1+c2 i) 01

( c 1 + c 2 i } 0 1 0

0 1 ( c 1 + c 2 i ) 0 1

(c1+c21101 o0 1

( C l - } - 2 i ) 0 1(cos 0 - cos Oa)" co s 0 dO

(c1+c2 t) 01+ ; , ~ + ~ ( ~ ) . (~ o s o 1 _ o (,-,~ ,+ ~ ,~ 1+ ~)

0-- COS 01 ) n cos 0 dO

0 1

(c i+c2 i) 01x ( I - e - r ~ K t% -o)-II-~l I,i.ol-.i.olj ) s in 0 dO

( e l - t - c 2 i ) 0 1+ f [ ~ + ( ~ l + ~ , ~ ) ( ~ ) ( c o s ( 0 1 0 C l + C ~ , , , 0 ,

0

x (1 - e-r/K t%-o)-(1-i~ (.i.o:-,i,o)]) sin 0 dO . (14)W h e n W is g i v e n a n d o t h e r c o n s t a n t s a r e k n o w n , t h e c o n t a c t a n g l e 0 1 c a n b e

f o u n d f r o m t h e a b o v e e q u a t i o n ( 14 ), a n d t h e w h e e l s i n k a g e z0 c a n t h e n b ed e t e r m i n e d f r o m

z0 = (1 - co s 01)r. (15)


10/18

PREDICTION OF RIGID WHEEL PERFORMANCE. PART 1n f o r t u n a t e l y , e q u a t i o n ( 14 ) is t o o c o m p l e x t o b e i n t e g r a t e d b y t h e n o r m a le s s . T h e r e f o r e , i n p r a c t i c e , S i m p s o n ' s r u l e i s u s e d t o e v a l u a t e t h e i n t e g r a l s a n dm c c e s s i v e a p p r o x i m a t i o n m e t h o d i s t h e n a p p l i e d t o f i n d o u t t h e c o n t a c t a n g l e7 ais p r o c e s s c a n e a s i l y b e c a r r i e d o u t o n a h i g h s p e e d d i g i t a l c o m p u t e r .i s i n t e r e s t i n g t o p o i n t o u t t h a t b y u s i n g e q u a t i o n ( 1 4 ) t o p r e d i c t t h e w h e e la ge , t h e e f fe c t o f s u p p o r t i n g p a r t o f t h e a x l e l o a d b y t h e v e r t i c a l c o m p o n e n t s

a n g e n t i a l s t r e s s e s i s c o n s i d e r e d . T h i s i s n e g l e c t e d i n e x i s t i n g t h e o r y w h e nic t i ng s inka ge (2) .

P r e d i c t io n o [ d r a w b a r p u l l a n d t o r q u e i n p u th e n t h e c o n t a c t a n g l e 01 h a s b e e n d e t e r m i n e d b y e q u a t i o n (1 4), t h e d r a w b a rD c a n b e p r e d i c t e d b y t h e f o l l o w i n g e q u a t i o n :

0 1 ( C l + C 2 i ) 1D . = r b { f ' r l ( O ) c o s O d O + f r 2 ( O ) c o s O d O }

( c 1 + c 2 t )0 1 0Ol ( C l + C 2 i )O 1- r b { f ~ l ( O ) s i n O d O+ f ~ 2 ( O ) s i n O d O }

( c 1 + c 2 i ) 0 1 0

= r b0 1

( C l + C 2 i ) 1(COS 0 - COS 0a)" t a n 4) ]

x (1 - e -','K I(l-'-(1-i) (8i'~5-~n0)1)COS 0 dO( C l + C 2 i ) 1

0 -- COS 01 ) n tan 4) ]

x (1 - e - r/K l ( e l - 0> - ( l - i ) ( s i n01 - s in0 ) ] ) COS 0 dO }0 1

( c 1 + c 2 ) 0 1(cos 0 - co s 01)" sin 0 dO

( C l + C 2 QO+ f ( k l0 ( c o s e \ c---~ ~ --CO S O~ ) " sin 0 dO .(16)


11/18

PREDICTION OF RIG ID WH EEL PERFORMANC E. PART I 91T h e f i r s t t e r m o n t h e r i g h t h a n d s i d e o f e q u a t i o n ( 1 6 ) i s u s u a l l y d e f i n e d a s t h r u s t

H a n d t h e la s t t e rm a s m o t i o n r e s is t an c e R . T h i s i n d i c a te s t h a t n o t o n l y t h r u s t b u ta l s o m o t i o n r e s i s ta n c e i s a f u n c t i o n o f s li p .

T h e t o r q u e T i s s i m i l a rl y o b t a in e d f r o m :01 (Cl-t-c2i)01

(c l+c2i) 01 0

= r ~ b01

{Cl+C2i)01t cos 0 - cos 01)" tan f f ]

x (1 - e -r/ K 1 ( 1 - ) - ( 1 - i ) ( s i n O l - s i n O ) l ) dO(c1~c2i) 01

0c o s( o l O (X , , c o s 0 , t a n ]

x (1 - e - r /K tll-)-Ii-~l (~na-~i~)l)dO }. (17)

A g a i n , S i m p s o n ' s r u l e i s u s e d t o c a l c u l a t e t h e i n t e g r a l s i n b o t h e q u a t i o n s ( 1 6 )a n d (1 7) o n a d i g i t a l c o m p u t e r .A b l o c k s c h e m e i l l u s t r a t i n g t h e c o m p u t e r p r o g r a m f o r p r e d i c t i n g t h e p e r f o r m -

a n c e o f d r iv e n r i g i d w h e e ls u s i n g t h e a b o v e p r o c e d u r e i s g i v e n in t h e a p p e n d i x .T h e c o m p a r i s o n o f t h e m e a s u r e d a n d p r e d i c t e d p e r f o r m a n c e o f t w o r i g i d w h e e l s

o n d i f f e r e n t s a n d s u s i n g th i s c o m p u t e r p r o g r a m i s s h o w n i n F i g s . 8, 9 a n d 1 0. S o i lp r o p e r t i e s a n d o t h e r c o n s t a n t s u s e d i n t h e c o m p u t a t i o n a r e s h o w n i n T a b l e 2 .

A s e x a m p l e s , F i g s . 1 1 , 1 2 a n d 1 3 s h o w t h e c o m p a r i s o n o f t h e m e a s u r e d a n dp r e d i c t e d r a d i a l a n d t a n g e n t i a l s t r e s s e s d i s t r i b u t i o n o n t h e s o i l - w h e e l i n t e r f a c e o fd r i v e n w h e e l s u n d e r d i f f e r e n t c o n d i t io n s .

I t i s s h o w n t h a t w i t h i n t h e e x p e r i m e n t a l r a n g e t h e r e i s a q u i t e c l o s e a g r e e m e n tb e t w e e n t h e m e a s u r e d a n d p r e d i c te d p e r f o r m a n c e , p a r t i c u l a r ly o n c o m p a c t s a n d .

I t i s h o p e d t h a t t h e p r o c e d u r e d e s c r i b e d c a n a l s o b e a p p l i e d t o p r e d i c t t h ep e r f o r m a n c e o f r i g i d w h e e l s o n c l a y , i f t h e c o e f f ic i e n ts i n t h e e q u a t i o n s a r e s u b -s t i t u t e d w i t h t h e a p p r o p r i a t e o n e s f o r c la y . H o w e v e r , t h i s n e e d s f u r t h e r i n v e s t ig a t i o n .

CONCLUSIONS1. A c o m p r e h e n s i v e fr a m e w o r k f o r p r e d ic t i n g t h e p e r f o r m a n c e o f d r i v e n ri g id

w h e e l s h a s b e e n e s t ab l is h e d . A l l k n o w n m a j o r f a c to r s a f f ec t in g d r i v e n w h e e lp e r f o r m a n c e , e s p e c i a l l y t h e e f f e c t o f s l i p o n s i n k a g e , m o t i o n r e s i s t a n c e a n d t h r u s t ,h a v e b e e n t a k e n i n t o a c c o u n t . I t i s s h o w n t h a t t h e m e t h o d d e s c r i b e d c a n g i v eb e t t e r p r e d i c t i o n t h a n e x i s t i n g t h e o r y .


12/18


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0 2 0 4 0 6 0 8 0 I 0 0 Z O 4 0 6 0 8 0i , % i , % 1 0 0

F IG . 8 . C o m p a r i s o n o f t h e m e a s u r e d a n d p r e d ic t e d p e r f o r m a n c e o f a 4 9 " 4 X 6 in .d r i v e n r i g id w h e e l w i t h a x l e l o a d 2 0 0 0 l b o n c o m p a c t s a n d ( m e a s u r e d v a l u e s f r o mO n a f e k o ' s e x p e r i m e n t s ) .( a ) t h r u s t a n d m o t i o n r e s i s t a n c e v s . s li p ;( b ) d r a w b a r p u ll v s . s li p ;( c ) t o r q u e i n p u t v s . s l ip :( d ) s i n k a g e v s . s l ip .


13/18

P R E D I C T I O N O F R I G I D W H E E L P E R F O R M A N C E . P A R T I 9 3

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x T h r u s t o b t a i n e d fr o m i n t e g r a t i n gt h e m e a s u r e dt a n g e n t i a l s t r e s s e s

M o t i o n r e s i s t a n c e o b t a i n e d f r o mi n t e g r a t i n g th e m e a s u r e d r a d i a lstressesP r e d i c t e d

0 2 0 4 0 6 0 8 0 I 0 0 0i , %

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x

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x T o r q u e i n p u t o b t a i n e d f r o mi n t e g r a t i n g t h e m e a s u r e dt a n g e n t i a l s t r e s s e s T o r q u e i n p u t o b t a i n e d f r o mt o r q u e m e t e r

-- - - P r e d i c t e d

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x, e

X S i n k a g e c a l c u l a t e d f ro m t h em e a s u r e d e n t r y a n g l e S i n k a g e m e a s u r ed f r o m t h e0 r i g i n a ~ s o i l s u r f a c e

- -- P r e d i c t e d

0 2 0 4 0 6 0 8 0 I 0 0 0 2 0 4 0 GO 8 0 I 0 0i , % i , %

F IG . 9 . C o m p a r i s o n o f t h e m e a s u r e d a n d p r e d i c te d p e r f o r m a n c e o f a 4 9 " 4 6 i n .d r i v e n r i g i d w h e e l w i t h a x l e l o a d 1 9 8 6 l b o n l o o s e s a n d ( m e a s u r e d v a l u e s f r o mO n a f e k o ' s e x p e r i m e n t s ) .( a ) t h r u s t a n d m o t i o n r e s i s t a n c e v s . s l ip ;( b ) d r a w b a r p u l l v s . s li p ;( c ) t o r q u e i n p u t v s . s l ip ;( d ) s i n k a g e v s . s l ip .


14/18

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D r a w b a r p u l l o b t a i n e d f r o md r a w b a r d y n a m o m e t e rP r e d i c t e d

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X S i n k a g e c a l c u l a t e d f r o m t h em e a s u r e d e n l r y a n g l e

S i n k a g e m e a s u r e d f ro m t h eo r i g i n a l s o i l s u r f a c e

- - - - P r e d i c t e dII

2 0 4 0 6 0 8 0i , %

FIG. 10. Comp ari son of the measu red and predicted per for manc e of a 49"41 2 in.driven rigid wheel with axle load 2085 lb on loose sand (measured values fromOnafeko's experiments).(a) thrust and moti on resistance vs. slip;(b) dra wbar pull vs. slip;( c ) torque input vs. slip;( d ) s i n k a g e v s . s l i p .


15/18

PREDICTION OF RIGID WHEEL PERFOR MANCE. PART I 95

4 0 l M e o s u r e d] - - - - P r e d i c t e d3 C c r

" 3 7 ~ T .

\t ( / / \

O-to o io 20 30 " 4oOoFIG. 1 l. Compar ison of the measured and predicted distribution of the radial andtangential stresses of a 49"46 in. driven rigid wheel with axle load 2000 lb at22-1 per cent slip on compact sand (measured values from Onafeko's experiments).

2 5

2 0

2 , 15

b "

5

0 - 2 0

(7

/ / r

/ ,

M e o s u r e d- - - - P r e d i c t e d

\ \

,N.x- - Io o . i o 2 o 3 0 40 50 6 08"I~G. 12. Comp ari son of the measu red and predi cted distribution of the radial andtangential stresses of a 49-46 in. driven rigid wheel with axle load 1986 lb at22"1 per cent slip on loose sand (measured values from Onafeko's experiments).

d .

2 OI

1 5 . . . . .

I 0

5

0 . 2 0

/ /S S

M e o s u r e dP r e d i c l e d

\ \-io 0 io 20 30 40 508"FIG. 13. Compar ison o f the measured and predicted distribution of the radial andtangential stresses of a 49-4 12 in. driven rigid whee l wit h axle load 2085 lb at22"1 per cent slip on loose s~nd (measured values from Onafeko's experiments).


16/18

96 JO- YUN G WONG and A. R. REEC E

2 . T h e r e l a t i o n s h i p b e t w e e n t h e ra d i a l s tr e ss d i s t r ib u t i o n o n t h e s o i l - w h e e li n t e r f a c e a n d s l i p i s d e s c r i b e d , t a k i n g i n t o a c c o u n t t h e f a c t t h a t t h e m a x i m u mr a d i a l s t r e s s s h i f t s f o r w a r d w i t h i n c r e a s i n g s l i p . T h i s l e a d s t o t h e r e s u l t t h a t m o t i o nr e s i s t a n c e a n d s i n k a g e a r e f u n c t i o n s o f s l i p , w h i c h i s c o n s i s t e n t w i t h t h e u s u a l l yo b s e r v e d " s l ip - s in k a g e " p h e n o m e n o n .3 . B a s e d o n th e a n a l y s i s o f t h e r e l a t i v e m o v e m e n t b e t w e e n t h e r im a n d t h e s o il ,a t h e o r e t i c a l e x p r e s s i o n t o d e s c r i b e t h e s h e a r d e f o r m a t i o n a l o n g t h e s o i l - w h e e li n t e r f a c e h a s b e e n d e r i v e d . I t g i v e s a s a t i s f a c t o r y p r e d i c t i o n o f s h e a r s t r e s s d i s t r i -b u t i o n a t h i g h s l i ps .

4 . T h e e f f ec t o f s u p p o r t i n g p a r t o f t h e a x l e l o a d b y t h e v e r t i c a l c o m p o n e n t s o ft a n g e n t i a l s t r e s s e s i s c o n s i d e r e d i n t h e p r e d i c t i o n o f w h e e l s i n k a g e . T h i s g i v e s m o r ea c c u r a t e r e s u l t s .

5 . W h e n u s i n g t h e m e t h o d d e s c r i b e d t o p r e d i c t t h e p e r f o r m a n c e o f d r i v e nr i g i d w h e e l s , t h e c o e f f ic i e n t s f o r l o c a t i n g t h e r e l a t i v e p o s i t i o n o f th e m a x i m u mr a d i a l s t re s s o n t h e s o i l - w h e e l i n t e r f a c e h a v e t o b e d e t e r m i n e d b y e x p e r i m e n t .A l t h o u g h , i t is b e l i e v e d t h a t t h e e m p i r i c a l v a l u e s o f t h e c o e f f i c i e n ts g i v e n i n t h ep a p e r c o u l d p r o b a b l y c o v e r q u i te a w i d e r a n g e o f w o r k i n g c o n d i t io n s o n s a n d ,a g e n e r a l t h e o r e t i c a l a p p r o a c h f o r lo c a t i n g th e m a x i m u m r a d i a l s t re s s p o i n t s ti lln e e ds t o b e d e v e l o p e d in o r d e r t o m a k e a m o r e c o m p r e h e n s i v e pi c tu r e .

6. B e f o r e t h is t h e o r y c a n b e a c c e p t e d a s g e n e r a l l y t r u e i t w i ll n e e d m o r ee x p e r i m e n t a l s u p p o r t .Acknowledgement--This paper is published by permission of H.M.S.O.reserved.

bcCl, c2

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Crown copyrightN O T A T I O N

w h e e l w i d t hs o i l c o h e s i o nc o e ff ic i en t s f o r d e t e r m i n i n g t h e r e l a t iv e p o s i t i o n o f t h e m a x i m u mr a d i a l s t r e ss o n t h e s o i l - w h e e l i n t e r f a c ed r a w b a r p u l lt h r u s ts l ips h e a r d e f o r m a t i o n a l o n g t h e s o il - w h e e l i n t e r f a c es h e a r d e f o r m a t i o n m o d u l u sp r e s s u r e s i n k a g e m o d u l ie x p o n e n t o f s in k a g e t o w i d t h r a t i or a d i u s o f t h e w h e e lm o t i o n ( r o l l i n g ) r e s i s t a n c et o r q u e i n p u tf o r w a r d v e l o c i t y o f th e w h e e l a x l ea b s o l u t e v e l o c i t y o f a p o i n t o n t h e r i ms l ip v e l o c i t y o f a p o i n t o n t h e r i m ( t he t a n g e n t i a l c o m p o n e n t o ft h e a b s o l u t e v e l o c i t y v a)v e r t i c a l a x l e l o a d o f t h e w h e e ls i n k a g ew h e e l s i n k a g er a d i a l s t r e s s o n t h e s o i l - w h e e l i n t e r f a c er a d i a l s t re s s i n t h e f r o n t r e g i o n o f t h e s o i l - w h e e l i n t e r f a c er a d i a l s t re s s i n t h e r e a r r e g i o n o f t h e s o i l - w h e e l i n t e r f a c et a n g e n t i a l ( s h e a r ) s t r e s s o n t h e s o i l - w h e e l i n t e r f a c et a n g e n t i a l ( s h e a r ) s t re s s i n t h e f r o n t r e g i o n o f t h e s o i l - w h e e li n t e r f a c et a n g e n t i a l ( s h e a r ) s t r e s s in t h e r e a r r e g i o n o f t h e s o i l - w h e e l i n t e r f a c e


17/18

PREDICTION OF RIGID WHEEL PERFORMANCE. PART I 970 angle

01 entr y angle (or contac t angle when rut recov ery is neglected)03 exit angl eO~ angu lar position of the ma xi mu m radial stressth angle of int ern al shear ing resistance of soilto ang ula r velocity of the wheel

REFERENCES[l] JO-VUNG WONG and A. R. REECE. Soil Fai lure Beneath Rigid Wheels. Proceedings ofthe Second International Conference o f the International Society for Terrain-VehicleSystems, Quebec, Canada (1966).[2] M.G. BEKKER.Theory of Land Locomotion, The University of Michigan Press 0956).[3] F. L. UFFELMANN. The Performance of Rigid Cylindrical Wheels on Clay Soil.Proceedings" of the First International Conference on the Mechanics of Soil -VehicleSystems, Turin (1961).[ 4 ] O. ONAFEKO. An Analysis of the Soil Stresses and Deformations Beneath a Rigid Wheel.

Ph.D. thesis, The University of Newcastle upon Tyne (1965).[ 5 ] O. ONAFEKO and A. R. RE~CE. Soil Stresses and Deformations Beneath Rigid Wheels.J. Terramechanics, 4, No. 1, 59 (1967).[ 6 ] A. D. SELA. The Shear to Normal Stress Relat ionship Between a Rigid Wheel and DrySand. Paper presented at the annual meeting of the A.S.A.E. (1964).[ 7 ] E. HEGEDUS. Pressure Distribution under Rigid Wheels. Trans. Am. Soc. Agric. Engrs.8, No. 3 (1965).[ 8 ] A. R. REECE. Principles of Soil-Vehicle Mechanics. Proc. Inst. Mech. Engrs. 180, No. 20965).[ 9 ] B. M. D. WILLS. The Measurement of Soil Shear Strength and Deformation Moduliand a Comparison of the Actual and Theoretical Performance of a Family of RigidTracks. J. Agric. Engng. Res. 8, No. 2 (1963).


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98 J()-Yt JN(I WON (i and A. R. REE( 'E

APPENDIXBLOCK SCHEME OF THE COMPUTER PROGRAMME FOR PREDICTING q-HE PERFORMANCE OF DRIVEN

RIGID WHEELS

.J.I Alter the valueof the contactangle 81

Input dataSoil properties: angle of internal shearing resistance q~,cohesion c,

shear deformation modulus K,pressure sinkage moclulikm, kz,exponent of s inkage to width ratio n.Wheel parameters radius r ,width b,vert ical axle load W .Coefficients for determining the relative posJhon of ma x. radial stress c h c2,

[Choose on init ial value of the contact angle O ltl For slip (i) at increments o f 5% , us ing Simp son's rulecalculate the vertical comp onents of the resultant radial andI tongential forces, W~ from the following equation:

a l ( C t C 2 , ) S tw'=~b~_J,~,.~,,~~ o ; ( 8 c os # d O + J "o , o - 2 ( 8 1 c o sO _I 8~ r [ 0 1 + C 2 : ) ~[ + J l c ,+ c 2 , 1 o , ' ( 8 ) s i n O d 8 + j o T 2 ( # ) s i n O d # ~

iCom pare the given vert ical axle load of the w he el W andthe calculated value W'The di fference between Wand W' ~ ti r 3s greater than o giv en error eps ,] I The difference between Wa nd W" ]I w - w ' l - e p s I I is s m alle r t h a n a given error eps,I

' L I W - W ' I - e p s jTUsing the correct contact a ng le e~ and Simp son's rule , calculate thelhrust H , mot ion resistance R, drawbar pul l L) and torque input Tfromthe followina equations:

8 ~ C1.C2~ 1H : , b ~ f r ,( 0 )c o s 8 d O + [_ r z (O ) c o s d 8 }L J ( C I C 2 z 8 1 u - -8 1 C I + C ) 8 1 .

D = H RT : , ~ [ o , r , ( O d O + o c t c 2 ' o , r z i O l d O }- LJ(C~+Cz~)8~Print ou tSlip i .Thrust H,Motion resistance /7,Drowbor pul l D,Torque input T,Contact angle ~l

wong,j & reece, a.r. 1967. prediction of rigid wheel performance based on the analysis of soil -...

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