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Int , J. M ach. Too ls Ma nufac t . Vol . 29, No. 4~ pp.589-600, 1989. 0890-6955/8953.00 + .00

Pr i n t e d i n Gre a t Br i ta i n Pe rga m on Pre ss p i e

C O N T R O L O F C O N T A C T S T IF F N ES S IN M A C H I N E P A R T S

B . N O W I C K I * a n d M . C I CH O W I C Z *

( R e c e i v e d 18 J a n u a r y 1989)

A b s t r ac t - -R ig id i t y i s one o f t he m a in f ac to r s dec i s i ve fo r t he qua l i t y o f m ach ine t oo l s , m easu r ing i n s t rum en t sand m any o the r dev i ce s . O ne o f t he p r i nc ipa l de fo rm a t ion com ponen t s o f a m ach ine e l em en t i s con t ac tde fo rm a t ion du e t o su r f ace roughnes s . Th e con t ac t r i g id it y can be con t ro l l ed by s e lec t i ng app rop r i a t e w ork ingmethods and parameters , inf luencing di rect ly the surface roughness character is t ics . I t i s for th is reason thatm any t heo re t i ca l and expe r im en ta l w orks have been devo t ed t o t he p rob l em m en t ioned . A s a r e su l t o f t hef ir st l oad be ing app l i ed t he su r f ace roughnes s u nde rgoes e l a s t i c -p l a s t i c de fo rm a t ion , w h ich has no t ye t bee ndesc r ibed by any cons i s t en t t heo ry . D ur ing un load ing and subsequen t l oad ing p roces se s t he de fo rm edaspe r i t ie s r em a in e l a s t ic ( i f t he r e a r e no t angen t i a l d i sp l acem en t s ) . H o w ev e r , a s a r e su lt o f no da t a be ing

ava i l ab l e a s rega rds t he s t e r eom e t r i c f ea tu r e s (3 -D ) o f r oughnes s and t he i r changes due t o p l a st i c de fo rm a t ion ,t he t heo re t i ca l r e su l t s a r e cha rged w i th cons ide rab l e e r ro r s . I n eng inee r ing p rob l em s t he va lue o f a con t ac tde fo rm a t ion m ay be com pu ted m ak ing use o f s t a ti s ti c a l r e l a t ions based on roughnes s m ode l s o f va r i ousdeg rees o f com pl i ca t i on . These r e l a t i ons a r e e s t ab l i shed on t he bas i s o f t he r e su l t s o f som e t e s t s o f t hesu r f aces o f e l em en t s m ach ined by t he s am e m e thods and m ade o f t he s am e m a te r i a l s a s t he e l em en t s t o beana lysed . These m e thods enab l e u s t o eva lua t e con t ac t de fo rm a t ions w i th an accu racy w h ich i s m uch be t t e rt han t ha t o f t he t heo re t i ca l m e thods . A n o the r m e th od fo r ensu r ing the r equ i r ed r i g id i ty o f m anufac tu redob j ec t s i s by d i r ec t m easu rem en t o f con t ac t de fo rm a t ion i n t he cou r se o f t he m anufac tu r i ng p roces s , f o rw h ich a spec i a l appa ra tus i s r equ i r ed . T he dev i ce s w h ich have been des igned fo r t he m easu rem en t o f con t ac tde fo rm a t ions o f a su r f ace enab l e such m easu rem en t by us ing a check su r f ace o f h igh ha rdnes s and l owroughnes s a s com p ared w i th those o f t he pa r t s cons ide red .

b ~ v

E

0

F , ~G

H B , H R B ,H R C

J

KK iqA qr

RT

R M A X p

R~, R mu~, R~,R~k, R, , R~ ,

S, , , t~ , HSC

N O M E N C L A T U R E

to t a l con t ac t de fo rm a t ionsubsc r ip t s - -p l , sp : p l a s t ic and e l a s t ic de fo rm a t ionssubscr ipts- - - l , 2 : deformat ions for the f i rs t and second load cyclespa ram e te r s o f bea r ing r a t i o cu rve : t , = b . e vY oung ' s e l a s t i c i t y m odu lussubscr ipts- - - l , 2 : for two contact ing mater ia lsPoisson ' s ra t iosubscr ipts- - - l , 2 : for two contact ing mater ia lsequ iva l en t Y oung ' s m odu lus : 1 /0 = (1 - p,~)/EI + (1 - ~.~)/E2r e l a t i ve app roach : ¢ = a/Rm~xFi sche r ' s num bers lope o f t he bea r ing r a t i o cu rve

subscr ipts- - -20, 50: for an approa ch o f 20% and 50% . G2~ = dt2o/da, G~o = dts~/da

ha rdnes s

contact r ig idi tysubscr ipts- - l , 2 : for the f i rs t and second load cyclessuperscr ipts- - - ' , " : for loading and unloadingprof i le sol id i ty fac torc o n s t a n t d e p e n d i n g o n v - - e q u a t i o n ( 8 )l oad pe r nom ina l a r ea o f con t ac tprof i le s lopep ro f i l e peak cu rva tu re r ad iuscoe f f i c ien t o f m u l t i p l e co r r e l a t i on

T - S t u d e n t ' s n u m b e rth r ee d im ens iona l r oughnes s he igh ts t anda rd roughnes s pa ram e te r s

*Pol i tech nika W arszawsk a, Warsza wa, AI . N iepodlegl'o~ci 222 p . 161, Poland.

58 9

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590 B. NOWICKIand M. CICHOWICZ

INTRODUCTION

AMONG other assumptions which are made for the design of a machine, there is that

of a definite contact rigidity, which must be achieved by choice of an appropriate

material, form and dimensions of the machine elements to be designed and appropriate

surface finish, the deformation of its elements and the contact deformation of their

surfaces. In most cases the overall deformation can be computed with an accuracy

sufficient for the engineering practice. However, it constitutes a small proportion of

the total deformation. Contact deformation is the sum of the elastic deformation of

surface waviness, form errors and the elastic, plastic or combined elastic-plastic defor-

mation of asperities. It usually constitutes 50-95% of the total deformation of a machine

and is usually decisive for the overall rigidity. The values of the contact deformation

of the waviness and the form errors can be evaluated on the grounds of the Hertz theory.

However, essential difficulties arise due to the fact that their form and dimensions are

not known beforehand. Thus, every surface must be measured in order to determine

the type of the form error, its height, curvature radius, etc. as well as the height and

the length of the wave. It is only in rare cases that these errors have repeated forms

and dimensions typical, for instance, of a definite machine-object -tool system for whichthe values measured on a surface may be expected to be met on any other surface

machined under the same conditions. Under fixed machining conditions the type of

surface roughness is reproducible as are the values of the roughness parameters. The

geometrical microstructure of the surface, that is the lay, their height, spacing, profile

slope, etc. can, under certain conditions, be controlled in the course of the production

process by selecting suitable machining methods and parameters and, possibly, by

changing those parameters in the course of the production process if the required values

of the roughness parameters have not been attained. The possibility of controlling the

production process to ensure the required roughness and its repeatability for other

machined elements was one of the causes of intense research work being initiated in

order to devise methods for computing contact deformations of rough surfaces. Theroughness summits may undergo elastic, plastic or combined elastic-plastic deformation,

depending on their geometry and the load applied. By the term "elastic-plastic defor-

mation" we mean a state in which some roughness summits or their fragments undergo

plastic deformations, the deformation of the remaining summits being elastic.

METHODS FOR COMPUTING CONTACT DEFORMATION OF A ROUGH SURFACE

The computat ion methods of contact deformations are based on a number of simplify-

ing assumptions concerning the model of the surface and the properties of the material

of the superficial layer. If the plastic model of deformation is used, it is assumed that

the pressure is equal to the H B hardness of the material and the value of the real

contact area can be determined directly from the bearing ratio of the profile. In thiscase, the differences in height between particular roughness summits and profile peaks

are rejected. These differences are essential [7] and range from several score % to

300%.

It is also assumed that the properties of the superficial layer are the same as those

of the core and the material is not work hardened. Under such assumptions the value

of the real contact area per unit area is

q (1)t p = l i b "

Assuming that the bearing ratio is described, for small approach values by theequation

tp = b. C (2)

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C o n t r o l o f C on t a c t S t i f fne s s 591

w h e r e b a n d v a r e t h e p a r a m e t e r s o f t h e b e a r i n g r a t i o c u r v e a n d e is t h e r e l a t iv e

a p p r o a c h ( e = a/Rm~x), t h e v a l u e o f t h e c o n t a c t d e f o r m a t i o n is

a \b. HB] Rmax' (3)

For typ ica l su r f aces , equa t ion (3 ) i s u sed in i t s s im pl i f i ed fo rm [10]

a = 4 . 1 R a ~ . (4 )

I n m o d e l c o n t a c t s , f o r w h i ch i t is a s s u m e d t h a t t h e d e f o r m a t i o n o f th e s u m m i t s is

e l a s t ic , t h e s p h e r i c a l f o r m o f a s u m m i t ( w h i c h i s d e s c r i b e d b y i ts r a d i u s r ) i s m o s t

f r e q u e n t l y u s e d . T h e a s p e r it i e s a n d t h e p h y s ic a l p r o p e r t i e s o f b o t h m a t e r i a l s a r e a s c r i b e d

t o a s in g l e s u r fa c e , t e r m e d t h e e q u i v a l e n t s u r f a c e , th e p a r a m e t e r s o f w h i c h r e s u lt f r o m

t h e p r o p e r t i e s o f b o t h s u r f a c e s . T h a t i s

r l r 2r - (5)

r 1 + r 2

R a = R a , + R a 2 ( 6 )

1 - ~ I 1 - ~ 2

0 - E1 + E2 (7)

M a k i n g t h e a b o v e a s s u m p t i o n s , w i t h t h e a d d i t i o n a l a s s u m p t i o n t h a t t h e r o u g h n e s s

d i s t r ib u t i o n i s d e s c r i b e d b y t h e e q u a t i o n o f t h e b e a r i n g r a t i o o f t h e p r o f i le , D e m k i n

[ 1] , K o m b a t o w [ 13 ] a n d M i c h i n [5 ] d e r i v e d a n u m b e r o f re l a t io n s f o r t h e r e l a t i v e v a l u e so f c o n t a ct d e f o r m a t i o n ( e = a/Rmax), t h e m o s t c o n v e n i e n t f o r c o m p u t a t i o n h a v i n g t h e

fo rm

= / 2 ~ r q O { r 1½/2/(2v+ )

w h e r e g l is a c o n s t a n t d e p e n d i n g o n v . M a k i n g s o m e a d d i t i o n a l s i m p li f yi n g a ss u m p t i o n s

a n d i n t r o d u c i n g t h e m e a n v a l u e s o f b a n d v , t h is r e l a ti o n c a n b e r e d u c e d t o

= 4 .5 R~ 8 q O . 4 rO.2 00 .4. (9 )

S o m e t h e o r e t i c a l m e t h o d s b a s e d o n t h e G a u s s , W e i b u l l a n d e x p o n e n t i a l d i s t r i b u t i o n s

o f r o u g h n e s s s u m m i t s a s w e l l a s t h e d e n s i t i e s a n d t h e c u r v a t u r e r a d i i o f r o u g h n e s s

s u m m i t s h a v e b e e n d e v e l o p e d b y N a y k o , B u s h , G i b s o n , O d e n , M a r ti n s a n d o th e r s .

S i n c e n o a p p a r a t u s f o r m e a s u r i n g t h e q u a n t i t i e s i n v o l v e d i s a v a i l a b l e , t h e u s e o f t h e s e

m e t h o d s f o r e n g i n e e r i n g p u r p o s e s i s a s y e t l i m i t e d .

A m o n g o t h e r n u m e r ic a l m e t h o d s f o r d e t er m i n in g t h e c o n t a c t d e fo r m a t i o n s o f r o u g h

s u r f a c e s , t h e m e t h o d o f T s u k a d a a n d A n n o [ 1 1, 1 2] y i e l d s r e s u lt s w h i c h b e s t a p p r o a c h

t h e m e a s u r e d d a t a . I t i s b a s e d o n s o m e s t a t i s t i c a l r e l a t i o n s b e t w e e n t h e p a r a m e t e r s

d e s c r i b i n g t h e r o u g h n e s s f e a t u r e s i n d i r e c t i o n s p a r a l l e l a n d n o r m a l t o t h e l a y a n d o n

t h e a s s u m p t i o n t h a t t h e p e a k d i s t ri b u t i o n o f t h e p r o f il e o b e y s , f o r a s m a l l a p p r o a c h ,

t h e W e i b u l l l a w . T h e f a c t t h a t t h e r o u g h n e s s s u m m i t s o f a s u r f a c e a r e h i g h e r t h a n t h ep ro f i l e peak s RMAX P = (2.5 --3.2 ) Ra i s a l so t aken in to con s idera t ion . T h e r esu l t s o f

c o m p u t a t i o n h a v e b e e n v e r i f i e d b y t h e a u t h o r s f o r s m a ll l o a d s o n l y ( q < 0 . 03 HB), th e

a g r e e m e n t w i t h t h e e x p e r i m e n t a l r e s u l t s b e i n g f o u n d t o b e g o o d . I t h a s b e e n s h o w n b y

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5 9 2 B . N o w l c ~ l a n d M . C l c n o w i c z

t e n t a t i v e e x p e r i m e n t a l v e r if i c a ti o n o f t h e a b o v e r e l a t io n s t h a t t h e d i f f e r e n c e s b e t w e e n

t h e m e a s u r e d a n d c o m p u t e d v a l u e s a m o u n t , f o r s t e e l s u r f a c e s m a c h i n e d b y t y p i c a l

m e t h o d s , t o a b o u t 5 0 - s e v e r a l h u n d r e d % ( T a b l e 1 ) . T h e v a l u e s o f t h e r e a l d e fo r -

m a t i o n s , b o t h e l a s t i c a n d p l a s t i c , a r e u s u a l l y g r e a t e r t h a n t h e t h e o r e t i c a l v a l u e s . I t

m a y b e s u p p o s e d t h a t o n e o f t h e m a i n c a u s e s o f t h is f a c t is th a t t h e p r o f i le c h a r a c t er i st i c s

such as the he igh t o f the aspe r i t i e s , the bear ing r a t io an d the cu rv a tu re r ad i i o f thes u m m i t s a r e , i n m o s t o f th e w o r k s s tu d i e d ( e x c e p t t ha t o f T s u k a d a a n d A n n o ) a s c r ib e d

d i r e c t l y t o t h e 3 - D s u r f a c e r o u g h n e s s , a l t h o u g h m a n y e x p e r i m e n t a l [ 7 ] a n d t h e o r e t i c a l

w o r k s [ 1 2 ] s h o w t h a t t h e r e a r e e s s e n t i a l d i f f e r e n c e s b e t w e e n t h e m .

A t t h e p r e s e n t s t a g e o f k n o w l e d g e o f t h e s u rf a c e r o u g h n e s s p r o b l e m a n d w i t h t h e

e x i s t i n g m e a s u r e m e n t m e t h o d s , t h e r e a r e n o a d e q u a t e d a t a f o r c o n s t r u c t i n g m o d e l s o f

s u r f a c e r o u g h n e s s e n a b l i n g u s t o o b t a i n a c c u r a t e t h e o r e t i c a l r e s u l t s . T h i s w a s o n e o f

t h e r e a s o n s f o r i ni ti a ti n g w o r k t o c o n s t r u c t d e v i c e s f o r r a p i d m e a s u r e m e n t o f c o n t a c t

d e f o r m a t i o n s o f s u r f a c e s o f m a c h i n e e l e m e n t s a n d t o e s t a b l i s h s t a t i s t i c a l r e l a t i o n s

b e t w e e n t h e r o u g h n e s s p a r a m e t e r s a n d t h e c o n t a c t d e f o r m a t i o n .

D E V I C E S F O R M E A S U R I N G C O N T A C T D E F O R M A T I O N S O F R O U G H S U R F A C E S

B y a n a l y s i n g t h e m e t h o d s o f c o n t r o l o f t h e c o n t a c t r i g i d i t y o f a r o u g h s u r f a c e i t w a s

f o u n d t h a t , i n s t e a d o f m e a s u r i n g t h e h a r d n e s s a n d t h e r o u g h n e s s o f a s u r fa c e , t h e

r e s u l t s b e i n g u s e d f o r c o m p u t i n g t h e c o n t a c t d e f o r m a t i o n , i t i s p r e f e r a b l e t o m e a s u r e

t h e m d i r e c t ly w i t h m u c h h i g h e r a c c u r a c y , i f a p p r o p r i a t e i n s t r u m e n t s a r e a v a i l ab l e .

T h e s e i n s t r u m e n t s s h o u l d e n a b l e u s t o m e a s u r e a n d o b t a i n t h e c o n t a c t r ig i d it y re q u i r e d

f o r th e p r o d u c t i o n , r e s e a r c h o r d e v e l o p m e n t o f p r o t o t y p e m a c h i n e s o r m a c h i n e e l e m e n t s

o r , i n c a s e o f a n y c h a n g e o f m a t e ri a l s , t h e i r s t a t e o r m a c h i n i n g p r o c e s s e s .

T h e m e a s u r i n g d e v i c e s w h i c h h a v e b e e n d e s i g n e d a t t h e I n s t i tu t e o f M e c h a n i c a l

P r o d u c t i o n M e t h o d s , W a r s a w T e c h n i c a l U n i v e r s i t y c a n b e u s e d i n a l a b o r a t o r y f o r

m e a s u r i n g c o n t a c t d e f o r m a t i o n s o f t h e s u r f a c e s o f m a c h i n e p a r t s , t h e d i m e n s i o n s o f

w h i c h m a k e i t p o s s i b l e f o r t h e m t o b e m o u n t e d o n a B r i n e l l p r e s s o r d i r e c t l y o n t h e

s u r f a c e o f a l a rg e m a c h i n e p a r t d u r i n g p r o d u c t i o n . S i n c e d i r e c t m e a s u r e m e n t o f c o n t a c t

d e f o r m a t i o n s b e t w e e n l a r g e e l e m e n t s i s d i f f i c u l t , t h e u s u a l p r o c e d u r e i s t o m e a s u r e t h e

d e f o r m a t i o n o f th e r o u g h s u r f ra c e t e s t e d i n c o n t a c t w i t h t h e c o n t r o l s u r fa c e o f a ch e c k i n g

e l e m e n t m a d e o f si n t e r e d c a r b i d e a n d c h a r a c t e r i z e d b y h i gh v a l u e s o f E a n d HRC a n d

v e r y l o w ro u g h n e s s . P r e l i m i n a r y t e s t s o f t h e p r o t o t y p e i n s t r u m e n t s s h o w e d t h a t t h e

e s s e n t ia l p r o b l e m s d e c i s iv e f o r t h e c o r r e c t n e s s o f m e a s u r e m e n t a r e t h o s e o f e n s u ri n g

a c c u r a t e p o s i t i o n i n g o f t h e w o r k i n g h e a d ( a x ia l , w i th r e f e r e n c e t o t h e l o a d i n g s y s t e m )

and very good con tac t w i th the su r f ace t es ted . T he pos s ib i l i ty o f sm al l l a te r a l d i sp lace-

m e n t s o f t h e u p p e r p a r t o f t h e w o r k i n g h e a d i n t h e c o u r s e o f m o u n t i n g o n t h e s u r f a c e

t e s t e d w a s e n s u r e d t o f a c i li t a te i t s c o n t a c t w i t h t h e r o u g h n e s s s u m m i t s l o c a t e d a t v a r i o u s

he igh t s .

T h e t w o v a r i e t i e s o f m e a s u r i n g d e v i c e s w h i c h h a v e s o f a r b e e n b u i l t a r e t h e M O S -0 2 l a b o r a t o r y d e v i c e m o u n t e d o n a B r i n e l l p r e s s a n d u s e d f o r m e a s u r i n g c o n t a c t

d e f o r m a t i o n [ 3] a n d t h e M O S - 0 2 p n e u m a t i c i n s t r u m e n t f o r si m i la r m e a s u r i n g o p e r a t i o n s

p e r f o r m e d o n t h e s u r f a c e s o f l a r g e m a c h i n e p a r t s . T h e r e a r e f o u r o p e r a t io n a l s y s t e m s

which can be d i s t ingu i shed in the M OS -02 dev ice (F ig . 1 ): ( a ) a load ing sys tem inc lud ing

a p i s ton (3 ) and a ba l l coup l ing (4 ) [4] ; (b ) a load m easur ing sys tem cons i s t ing o f a

d y n a m o m e t e r ( 5 ) a n d a l o a d t r a n s d u c e r o f i n d u c t a n c e o r st r ai n g a u g e t y p e ( 8 ) s u s p e n d e d

o n a l e v e r s y s t e m ( 6 , 7 ) ; ( c ) a d e f o r m a t i o n m e a s u r i n g s y s t e m , i . e . a m e a s u r i n g h e a d

(10) w i th an induc t ive t r ansd uce r o f d i sp lacem en t (9 ) and a con t ro l in se r t (11 ) ; and (d )

a pos i t ion ing sys tem wi th a s e t o f cam s (13 , 15 ) and a gu id ing s leev e (12 ) fo r the

m e a s u r i n g h e a d ( 1 0 ). T h e i n s t r u m e n t i s m o u n t e d o n t h e p i s t o n o f a B r i n el l p re s s .

D u r i n g o p e r a t i o n o f t h e p r e s s , t h e p i s t o n ( 3 ) m o v e s d o w n w a r d s u n t il i t t o u c h e s t h eu p p e r e l e m e n t o f t h e s p h e r i c a l j o i n t . F r o m t h a t t i m e f o r t h t h e l o a d i n c r e as e s a n d i s

t r a n s m i t t e d f r o m t h e p i s t o n ( 3 ) th r o u g h t h e b a l l c o u p l i n g ( 4) , t h e d y n a m o m e t e r ( 5 ) a n d

the s lee ve o f the m easur ing head (10 ) to the in se r t (11 ) , wh ich i s in con tac t w i th the

s u r f a c e t e s t e d .

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5 94 B . N o w l c r t a n d M . C t c n o w l c z

FI G . 1 . T he M O S- 02 l a bo r a t o r y de v i c e f o r m e a s u r i ng c on t a c t de f o r m a t i ons .

A x i a l a p p l i c a t i o n o f t h e f o r c e i s e n s u r e d b y a s y s t e m o f p r e l i m i n a r y p o s i ti o n i n g w h i c h

e n a b l e s t h e m e a s u r i n g h e a d t o b e s e t o n t h e s u r f a c e t e s t e d s o t h a t i t s a x i s c o i n c i d e se x a c t l y w it h t h e l i n e o f a c t i o n o f t h e p i s t o n . T h e c a m s m o u n t e d o n t h e s h a f t ( 1 4 )

d i s p l a c e t h e m e a s u r i n g h e a d ( 1 0 ) a n d t h e d y n a m o m e t e r ( 5 ) i n th e v e r t ic a l d i re c t i o n a n d

c o n t r o l t h e p r e l i m i n a r y p o s i t i o n i n g s y s t e m o f t h e m e a s u r i n g h e a d . T h e f l oa t in g b a l l

c o u p l i n g ( 4 ) e n s u r e s t h a t t h e m o m e n t s o f t h e f o r c e s a r e e l i m i n a t e d a n d t h e l o a d o n t h e

m e a s u r i n g h e a d ( 1 0 ) a c ts in t h e a x i a l d ir e c t io n . T h e v a l u e o f th e l o a d i s m e a s u r e d

i n d i re c t l y b y m e a s u r i n g t h e d e f l e c ti o n o f t h e d y n a m o m e t e r ( 5 ) b y m e a n s o f t h e i n d u c t i v e

d i s p l a c e m e n t t r a n s d u c e r ( 8 ) a t ta c h e d t o t h e s e t o f le v e r s (6 , 7 ) , o r a se t o f s t ra i n g a u g e s

g l u e d t o t h e d y n a m o m e t e r ( 5) . T h e d e f o r m a t i o n i s m e a s u r e d b y m e a n s o f th e m e a s u r i n g

hea d (10 ) by p res s ing the f ron ta l su r f ace o f the co n t ro l in se r t (11 ) aga ins t the su r f ace

t o b e t e s t e d . T h i s d e f o r m a t i o n i s e q u a l t o t h e d i s p l a c e m e n t o f t h e f ro n t a l s u r f a c e o f

t h e c o n t r o l i n s e r t w i th r e f e r e n c e t o t h e u n d e f o r m e d p a r t o f t h e s u r f a c e , w h i c h is t o u c h e db y t h e t i p o f t h e i n d u c t i v e t r a n s d u c e r ( 9 ).

T h e f o r m o f t h e m e a s u r i n g h e a d ( 1 0 ) e n a b l e s t h e c o n t r o l i n s e r t ( 11 ) to b e e x c h a n g e d ,

a l lo w i n g e l e m e n t s o f v a r i o u s m a t e r i a l s a n d v a r i o u s h e i g h ts o f r o u g h n e s s t o b e u s e d .

T h i s a l so e n a b l e s t h e m e a s u r e m e n t o f t h e c o n t a c t d e f o r m a t i o n s o f t w o r o u g h s u r f r a c es .

T h e m a x i m u m l o a d w h i c h c a n b e a p p l i e d i s 1 5 , 0 0 0 N o r 2 0 0 0 M P a , w h i c h e n a b l e s t h e

m e a s u r e m e n t o f t h e c o n t a c t d e f o r m a t i o n s o f s u r f a c e s o f r o u g h n e s s R a / > 0 . 2 Ix m a n d a

h a r d n e s s u p t o 6 0 HRC. T h e a c c u ra c y o f th e m e a s u r e m e n t o f fo r c e s is a b o u t 0 . 1 % a n d

tha t o f con tac t de fo rm at ion s i s - - -0 . 05 I~m . T he overa l l d im e ns ions o f the dev ice a re

H x b x a = 2 7 0 x 1 60 x 8 5 m m .

Severa l yea r s o f t e s ts and use o f the dev ices ju s t desc r ibed have conf i rm ed fu l ly the i r

use fu lnes s . Owing to a l l the m easur ing sys tem s be ing un i ted in a s ing le as sem bly and

the pos s ib i l i ty o f in s ta l l a t ion on a Br ine l l p res s , i t i s pos s ib le to ca r ry ou t m easu rem en ts

o f c o n t a c t d e f o r m a t i o n s i n p r a c ti c a ll y a n y l a b o r a t o r y . I n v e s t ig a t i o n s c o n d u c t e d u s i n g

t h e s e d e v i c e s s h o w e d t h a t t h e d i s p e r s i o n o f t h e c o n t a c t d e f o r m a t i o n s o f a r o u g h s u r f a c e

i s m u c h l o w e r t h a n t h a t o f th e r o u g h n e s s o r h a r d n e s s ( T a b l e 2 ) , n u m e r o u s r e p e t i t io n s

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Contro l of Contac t Stiffness 595

TABLE 2. DISPERSION OF MEASURED VALUES OF THE ROUGHNESS PARAMETERS AND THE CONTACT DEFORMATION

AND RIGIDITYOF A GROUND SURFACE ON THE BASIS OF THE RESULTS OF NINE MEASUREMENTS

R , , R c . . x Sm

X (o-m) (0,m) (l/ ram ) Aqa a~p ap] j ' f '

HRB (o-m) (o-m) (o-m) (Pa/o-m) (Pa/o-m)

Xm,,,-Xm,, 0.11 2.41 33.15 0.!1 4.0 0.7 0.4 0.5 4.64 10.645( 0.78 6.99 58.91 0.19 76.0 7.8 6.5 1.3 30.47 39.80

o'x 0.013 0.239 3.906 0.001 0.412 0.071 0.048 0.053 0.440 1.300

O-x/X. 10 ~ 16.1 34.3 66.3 6.9 5.4 9.1 7.3 42.2 14.4 32.7

o f t h e m e a s u r e m e n t t h u s b e i n g u n n e c e s s a r y . A s in g le m e a s u r e m e n t is s u f fi c ie n t f o r

r o u t i n e i n s p e c t i o n a n d t h r e e m e a s u r e m e n t s f o r r e s e a r c h p u r p o s e s .

T h e a = f ( q ) c h a r a c t e ri s t ic m a k e s i t p o s s i b l e to e s t i m a t e t h e e l a st ic p r o p e r t i e s o f t h e

su r faces in con tac t , wh ich i s es s en t ia l fo r m any app l ica t ions .

T h e M O S - 0 3 p o r t a b l e m e a s u r i n g i n s t r u m e n t h a s b e e n d e s i g n e d f o r p r o d u c t i o n c h e c k -

i n g o f t h e c o n t a c t d e f o r m a t i o n o f th e s u r f a c e s o f l a r g e e l e m e n t s s u c h a s m a c h i n e b o d i e s ,e t c . T h i s p n e u m a t i c d e v i c e ( F ig . 2 ) is c o m p o s e d o f a l o a d in g s y s t e m w i t h a p n e u m a t i c

c h a m b e r ( 6 ) , a l o a d m e a s u r i n g s y s t e m w i t h a p o t e n t i o m e t e r p r e s s u r e g a u g e , a d e f o r -

m a t i o n m e a s u r i n g s y s t e m , i .e . a m e a s u r i n g h e a d ( 1 0 ) i d e n ti c a l w i th t h a t o f t h e M O S -

02 and a pos i t io ner (7 , 8 ) the a im o f wh ich is to a l ign the m e asur ing head , i . e . to s e t

t h e m e a s u r i n g h e a d ( 1 0 ) f r e e l y o n t h e s u r f a c e t e s t e d e x a c t l y a l o n g t h e a x i s o f t h e

i n s t ru m e n t . T h e m e a s u r i n g h e a d i s d i s p l a c e d v er t ic a l ly b y m e a n s o f th e s c r e w ( 1 ). W h e n

t h e m e a s u r i n g h e a d ( 1 0 ) h a s b e e n l o w e r e d a n d s e t o n t h e s u r f a c e t o b e t e s t e d , f u r t h e r

r o t a t i o n o f t h e s c r e w ( 1 ) s e p a r a t e s t h e t w o c o n i c a l sl e e v e s (7 , 8 ) , th u s e n s u r i n g f r e e

s e t ti n g o f t h e m e a s u r i n g h e a d o n t h e s u r f a c e t o b e t e s t e d a n d p e r f e c t c o n t a c t w i th t h e

c o n t r o l s u r f a c e. B y t a n g e n ti a l m i c r o d i s p l a c e m e n t o f th e m e a s u r i n g h e a d , t h e c o v e r ( 4)

o f th e c h a m b e r ( 6 ) c a n b e s u s p e n d e d o n t h e e l a s ti c d i a p h r a g m ( 5 ). T h e i n s t r u m e n t i s

m o u n t e d b y m e a n s o f m a g n e t i c s u p p o r t s ( 1 2 ) o n t h e s u r f a c e t o b e t e s t e d . B y l e t t i n g

c o m p r e s s e d a i r t h r o u g h a c a l i b r a t e d n o z z l e i n t o t h e p n e u m a t i c c h a m b e r ( 6 ) , a l o a d i s

app l ied to the m easur ing head (10 ) and the in se r t (11 ) i s s e t in con tac t w i th the

I 2 3 4

FIG. 2. The MOS-03 portable pne umati c device for measuring contact deform ations .

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596 B. Nowlcgl and M. CICHOWICZ

surface to be tested. The load is measured directly by measuring the pressure with a

potentiometer pressure gauge. The procedure of measurement of the deformation is

the same as with the measuring instrument MOS-02, the unified measuring head (10)

being used. Similarly to the MOS-02 measuring instrument, signals from the inductive

displacement transducer can be directed to an X Y recorder, thus obtaining a diagram

of contact de format ion as a function of the load. This instrument is fed with compressedair at 0.5 MPa. The use of magnetic supports to mount the instrument means that the

load which can be applied to the measuring head is limited to 1000 N. I t is possible,

however, to use stronger magnetic supports or reduce the working area of the control

element. This would enable us to raise the pressure to q = 140 MPa. The ins truments

described above have been used with success for detailed studies of the relation between

the roughness parameters and the contact rigidity, thus confirming their usefulness for

research purposes. In the case of the MOS-02 instrument which can be installed on a

Brinell press, the dimensions of the part to be tested are limited by the small dimensions

of the Brinell press. The pneumatic instrument MOS-03 can be used for measurement

on the surface of a machine part of any size. It is necessary, however, that the surface

should be plane and horizontal and the material should be ferromagnetic. Thus, the

range of application of the measurement instruments just described is seen to be large

and includes devices such as machine tools, measuring instruments, heavy engines, etc.

MOS measurement instruments can also be used for production inspection during the

manufacturing process and tests of prototype machines and for material tests and

production methods of machine elements. The reliable structure and rapidity of

measurement enable us to perform statistical analyses, for which a large number of

experiments is necessary.

Some experiments were performed for specimens made of 45 and 1H13 steel surfaces

machined by grinding, planing, turning and milling. After machining, the specimens

were subjected to annealing in a vacuum in order to make uniform the mechanical

properties of the core and the superficial layer. The roughness of the surface was

measured by means of a Talysurf 5 profilometer. The parameters determined were Ra,Rp, R . . . A q , S m H S C , Rsk, t2o and tso. In the second stage of the investigation the

slope of the bearing ratio curve was computed additionally (for an approach of 20%

and 50%): G2o and Gso (G2o = dtEo/da for a = 0.2 Rmax); as well as the coefficient K

of the profile, K = Rv/Rmax . This coefficient determines the relative position of the

mean line with reference to the extreme points of the profile and constitutes a generalized

measure of its load carrying capacity. The parameters of the bearing ratio curve (b, v)

are not included in the set of roughness parameters. They are characterized by high

dispersion and are not used, in general, in countries other than the Soviet Union. This

results from the fact that the coordinate system in which they are determined is attached

to the highest summit of the profile, the location of which has a random character. The

radii of rounding of the summits are measured in an indirect manner. The computationmethods of these radii are charged with considerable errors and enable us to determine

the radius of the profile peak only, not the highest summits of the surface in the

stereometric sense, i.e. those which are in contact with the opposite surface. The contact

deformations were measured with reference to a control surface, i.e. the surface, very

carefully machined, of a ring made of sintered carbide. For each surface the character-

istic of deformation was determined as a function of the approach (Fig. 3) and, knowing

this, the total (a), plastic ( a p l ) and elastic ( asp ) deformations for the first and second

load cycles and the six coefficients of contact rigidity (Fig. 3) were determined. In the

first stage of computation the coefficients of correlation of all the 12 parameters

describing the deformation (a~, apll, asp1, a2, apl2, a s p 2 ) and the contact rigidities (j~, j~,

J~, ]'2, J~, J~) were determined as well as the roughness parameters Ra, Rp, R . . . . Aq,

S m , R s k , H S C . The computation was performed by means of programs of stepwise

multiple regression from the library CDC and some standard procedures. Table 3

contains the matrix of correlation factors between these parameters and the parameters

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C o n t r o l o f C o n t a c t S t i f fne s s 597

o (F ,m) a~ 2"

, :: : _ _. i .

D0 8 0 g O I 0 0 180 190 2 00

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q ( M P a )

FiG . 3. D e f o r m a t i o n a s a f unc t i on o f l oa d . D i a g r a m o f p r oc e s s i ng o f m e a s u r e m e n t r e s u l ts .

of the bear in g ra t io o f the p rof i le t20, t so , G20, Gso, and the coeff ic ient K an d HB, whi c h

a r e i n t r o d u c e d a d d i t i o n a l l y i n s u b s e q u e n t s t a g e s .

The pa ra me t e r s s e l e c t e d on t he ba s i s o f a n a na l ys i s o f t he c oe f f i c ie n t s o f c o r re l a t i on

b e t w e e n t h e p a r a m e t e r s d e s c r ib i n g t h e c o n t a c t d e f o r m a t i o n a n d t h e r i g id i ty , t o e v a l u a t e

t hose p rope r t i e s a re a 2 , a sp2 , a p l 2 , J 2 a n d j~ . I n s u b s e q u e n t s t a g e s th e o b j e c t s o f c o m p u -

t a t i on w e re t he e qua t i ons o f r e gre s s ion , mu l t ip l e c or re l a t i on c oe f f i c i e n ts R , F i sc he r ' s

a nd S t ude n t ' s t e s ts a nd t he s t a nda rd de v i a t i ons o f func t i ons .

By a na l ys i ng t he c omput a t i on re su l t s i t wa s found t ha t t he oc c ur re nc e , i n t he s e t o fi nde pe n de n t va r i a b l e s , o f suc h da t a a s Ra , Rp , Rma x , Sm a nd HSC w i t h m a r k e d

c o r r e l a ti o n s b e t w e e n t h e m , r e s u lt s in t h e e q u a t i o n s o f r eg r e s si o n o b t a i n e d b e i n g s u c h

t h a t o n l y s o m e o f t h e f e a t u r e s o f t h e p r o f i l e ( r e p r e s e n t e d b y a f e w p a r a m e t e r s ) a r e

i nvo l ve d a nd t he va l ue s o f t he c oe f f i c i e n t o f mul t i p l e c or re l a t i on a nd F i sc he r ' s a nd

S t ud e n t ' s n um be rs a re l ow. I t i s i n t e re s ti ng t o ob se rve t ha t t he va l ue o f t he c oe f f i c ie n t

o f c or re l a t i on be t we e n t he l e ve l l i ng de p t h o f p ro f i l e Rp a nd t he c on t a c t de forma t i on

w a s f o u n d t o b e h i g h e r t h a n t h o s e f o r t h e r em a i n i n g h e i g h t p a r a m e t e r s o f ro u g h n e s s .

I n t h e s e c o n d s t a g e o f c o m p u t a t i o n ( T a b l e 4 ) th e i n d e p e n d e n t v a r i ab l e s w e r e R a , Sm

a nd Aq, t he p a ra m e t e r s o f t he be a r i ng ra t i o we re t2o a nd t so , a nd t h e ha rdn e s s HB w as

i n t rod uc e d a dd i t i ona l l y ( t he re b e i ng d i f f e re nc e s i n ha rdne s s d e sp i t e t he a nne a l ing

proc e s s ) . O n t he ba s i s o f t he r e su l ts ob t a i ne d i t wa s found t ha t t he pa ra me t e r tso ha st h e w e a k e s t l i n k s w i t h t h e d e f o r m a t i o n s , t h e r e f o r e i t w a s r e m o v e d f r o m t h e a n a l y s i s .

I n t h e s u b s e q u e n t s t a g e o f c o m p u t a t i o n t h e p a r a m e t e r ts o w a s r e p l a c e d b y t h e q u a n t i t ie s

G2o a nd Gso whi c h de sc r i be t he s l ope s o f t he c urve o f t he be a r i ng ra t i o a nd t he

c oe f f i c i e n t o f t he p ro f i l e K . F rom t he a na l ys i s ma de , i t fo l l ows t ha t t he e qua t i on s

ob t a i ne d fo r suc h da t a a re s t il l va gue a nd t he va l ue s o f R , F a nd T a re l ow.

In t he l a st s t a ge o f t he c o mp ut a t i on t he va r i a b l e s we re Ra , Aq , S~ , G 2o , K a nd HB.

The c ha ra c t e r o f t he r e l a t i ons ob t a i n e d is s i mi la r t o tha t o f t he t he ore t i c a l r e l a t ions .

The va l ue s o f the de for ma t i on s a nd t he c on t a c t r ig i d i ty a re func t i ons o f t he fo l l owi ng

s t a n d a r d r o u g h p a r a m e t e r s .

(1 ) The me a n de v i a t i on o f t he p ro f i l e f rom t he mi dd l e l i ne Ra , d i r e c t l y r e l a t e d t o

g m a x •

(2 ) The s l ope o f t he p ro f i l e Aq , whi c h is a me a sure o f t he i nc re a se i n e l e me nt a ry

a re a s o f c on t a c t a s a r e su l t o f a va r i a t i on i n t he l oa d .

(3 ) T he m e a n spa c i ng Sm, whi c h is a me a sure o f t he de ns i t y o f rough ne ss sum mi t s .

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6 0 0 B . N o w l c K I a n d M . C t c r ~ o w l c z

r e l a t i o n s m a y b e u s e d . T h e y a r e m u c h l e s s g e n e r a l t h a n t h e t h e o r e t i c a l r e l a t i o n s a n d

r e q u i r e m u c h t e d i o u s w o r k t o b e d o n e , b u t t h e y e n s u r e , f o r d e f in i te v a ri a b il it y r a n g e s

o f th e r o u g h n e s s p a r a m e t e r s a n d t h e m a t e r i a l p r o p e r t ie s , a n a c c u r a c y o f c o m p u t a t i o n

w h i c h is m u c h h i g h e r t h a n t h a t o f t h e t h e o r e t i c a l r e la t io n s . T h e a i m o f t h e r e s e a r c h

w a s t o d e t e r m i n e s t a ti s ti c al r e l a t i o n s f o r c o m p u t i n g t h e c o n t a c t d e f o r m a t i o n s o f s u r fa c e s

fo r typ ica l s t ee l s and to def ine the d i r ec t ions fo r se l ec t ing roughness paramete r s , wh ichs h o u l d b e t a k e n i n t o c o n s i d e r a t i o n i n a s t u d y o f r e l a t i o n s o f s u c h a t y p e .

I N F E R E N C E S

(1 ) Suf f i c ien t con tac t r ig id i ty , wh ich i s one o f the f ac to r s dec i s ive fo r the qua l i ty o f

a m a c h i n e , c a n b e e n s u r e d b y u s i n g a c c u r a t e c o m p u t a t i o n a n d m e a s u r e m e n t

m e t h o d s .

( 2 ) T h e r e s u l t s o f t h e o r e t i c a l d e t e r m i n a t i o n o f c o n t a c t d e f o r m a t i o n s o f a r o u g h

s u r f a c e a r e u s u a l ly c h a r g e d w i t h c o n s i d e r a b l e e r r o r s .

(3 ) S ta ti s t ica l re l a t ions m ake i t poss ib l e to com pute con tac t de fo rm at ion s wi th an

a c c u r a c y w h i c h is h i g h e r t h a n t h a t a t t a i n a b l e t h e o r e t i c a l ly . H o w e v e r , t h e i r d e t e r -

m i n a t i o n r e q u i r e s c o n s i d e r a b l e l a b o u r .( 4) T h e p a r a m e t e r s w h i c h h a v e t h e s t r o n g e s t i n f lu e n c e o n t h e c o n t a c t r ig i d it y a re

Ra , S in , K , G2o and the l ike .

( 5) T h e d e g r e e t o w h i c h t h e r i g id i ty r e q u i r e m e n t s f o r a m a c h i n e a r e s a t is f ie d c a n b e

i n c r e a s e d c o n s i d e r a b l y b y u s i ng s p e c i a l d e v ic e s f o r m e a s u r i n g t h e c o n t a c t r i g i d it y

o f r o u g h s u r f a c e s .

R E F E R E N C E S

[1] N . DEMI~IN, Kontaktirowanie szerochowatych pow ierchnostej . N a u k a , M o s k v a ( 1 9 7 0 ) .[2 ] K . JOHNSOn, Contact Mechanics. C a m b r i d g e U n i v e r s i t y P r e s s , L o n d o n ( 19 8 5) .[ 3 ] M . O C H O W l C Z a n d B . N O W l C K I , A d e v i c e f o r m e a s u r i n g c o n t a c t r i g i d i t y , P a t e n t P . 24 3 2 6 9.[ 4 ] B . N O W l C K I, A d e v i c e f o r a p p l y i n g a n o r m a l l o a d , P a t e n t P . 2 4 3 27 0 .[5 ] N . MICnIN , Wnesznee trenie twerdych t~ N a u k a , M o s k v a ( 1 9 7 7 ) .

[ 6 ] B . N o w l c r a , Zag ad. Eksploatac]i Maszyn z . 3 (5S) , 245 (1983) .[ 7] B . N o w I c I o , P r a c e n a u k o w e , P o l i t e c h n ik a W a r s z a w s k a , Mechanika z.70. (1980).[ 8 ] J . O D E N a n d J . M A R T IN S ,Computer Meth. appl. Mech. Engng 52, 527 (1985).[9] T. THOMAS,Rough Surfaces. L o n g m a n , L o n d o n ( 1 9 8 2 ) .

[ 1 0 ] Trenie iznasziwanie i smazka, Spraw. M a s z i n o s t r o j e n i e , M o s k v a ( 1 9 7 8 ) .[11] T. TSUKADAa n d Y . A N N O , 1st , 2nd , 3rd Report Bullet in of the ISM E 15, 982 (1972).[12] T. TSUKADAa n d Y . A N N O , 1st , 2nd , 3rd B ulletin of the ISM E 17, 376 (1974).[13] V. KOMBAI:OV,Ocenka tribotechniczeskich swojstw kontaktiryjuszczich powierchnostej. N a u k a , M o s k v a

(1983).[14] Z . LEW lN and D . RESZETOV,Kontaktnaja zestkost maszin. M a s z i n o s t r o j e n i j e , M o s k v a ( 1 9 7 1 ) .