1994 Hashimoto Contact Stiffness

8/4/2019 1994 Hashimoto Contact Stiffness

1/9

E s t i m a t i o n o f c o n t a c t s t i f fn e s sa t i n t e r f a c e s in m a c h i n es t r u c t u r e s b y a b e a m m o d e l o na n e l a s t i c f o u n d a t i o nM a s a tos h i Ha s h i m oto* , E ts uo M a ru i + a nd Sh i nobu Ka to *T h e d y n a m i c p r o p e r t i e s o f m a c h i n e s t r u c t u r e s a r e s i g n i f i c a n t l yi n f lu e n c e d b y a n i n t e r a c ti o n a t t h e m a t i n g s u r f a c e s o f m a c h i n ee l e m e n t s . T h i s i n t e r a c t i o n i s c a l l e d c o n t a c t s t if f n e s s , a n d th ed e v e l o p m e n t o f a s i m p l e m e t h o d f o r c o n t a c t s t if fn e s s e s t i m a t i o n isa n i m p o r t a n t t r i b o l o g i c a l o b j e c t i v e . I n t h i s p a p e r t h e c o n t a c ts t if fn e s s is e s t i m a t e d b y a b e a m m o d e l v i b r a t in g o n a n e l a s ti cf o u n d a t i o n . T h e e f f e c ts o f c l a m p i n g c o n d i t i o n a n d m a t i n g s u r f a c et o p o g r a p h y o n t h e c o n t a c t s t i f f n e s s o b t a i n e d a r e q u a n t i t a t i v e l y a n da c c u r a t e ly r e p r e s e n t e d b y e x p e r i m e n t a l e q u a t i o n s .Ke y w ords : in te r face , su r face con tact , con tact s t i f fness, es t imat ion by f reev ib ra t io n , e xp e r ime n ta l e q u a t io n

N o t a t i o n IkA r e a l a r e a o f c o n t a c tA * c r o s s - s e c ti o n a l a r e a o f t h e c o l u m n k *a l , a 2 , b l , c o e f f i c ie n t s f o r th e e x p e r i m e n t a l c o n t a c tb 2 , c l , c 2 s t if f n e s s e q u a t i o n La * c o n t a c t l e n g t h b e t w e e n c o l u m n a n d mh o l d e r b l o c k s Nb d i s t r i bu t i on coef f i c i en t fo r ~bE Y o u n g ' s m o d u l u s N oF c l a m p i n g l o a df n a t u r a l f r e q u e n c y o f t h e c o l u m n s y s t e m Pf or n a t u r a l f r e q u e n c y o f t h e f u n d a m e n t a l P mm o d e o f n o r m a l v i b ra t io n o f P . . .c a n t i l e v e r b e a m RG s h e a r m o d u l u s s tn m ax d e p t h i n t e rv a l b e t w e e n t h e h i g h e s t a n d T

t h e l o w e s t a s p e r i t y p e a kh a r b i t r a r y d e p t h o f a s p e r i t y p e a k Pd ,

m o m e n t o f i n e r ti a o f a r e a o f th e c o l u m ncon t ac t s t i f fness per un i t l eng t h a t e l as t i cf o u n d a t i o nc o n t a c t s t i f f n e s s p e r u n i t a p p a r e n tc o n t a c t a r e al e n g th o f th e c a n t i l e v e r e d p a r ti n d e x n u m b e r f o r d ~su f f i x i nd i ca t i ng no rmal d i r ec t i on t o t hem a t i n g s u r f a c e st o t a l a s p e r i t y n u m b e r o n t h e n o m i n a lc o n t a c t a r e ap r o b a b i l i t y f o r a s p e r i t y c o n t a c ty i e l d p r e s s u r e o f a s p e r i t ym e a n c o n t a c t p r e s s u r ee q u i v a l e n t r a d i u s o f a s p e r i t y t i ps u r f a c e t o p o g r a p h y c o e f f i c i e n tsu f fi x i nd i ca t i ng t angen t i a l d i r ec t i on t ot h e m a t i n g s u r f a c e sd e n s i t y o f t h e c o l u m nc u m u l a t i v e p e a k h e i g h t d i s t r i b u t i o n

I n t r o d u c t i o nM o s t p r a c ti c a l d e s i g n s f o r m a c h i n e s t r u c t u r e s i n c o r p o r -a t e s o m e f o r m o f c o n n e c t io n b e t w e e n t h e b a s ice l e m e n t s . T h e t o t a l d y n a m i c p r o p e r t y o f m a c h i n e* Department of Mechanical Engineering, Toyota College of Tech-nology, 2-1 Eisei-cho, Toyota-shi 471, Japan.+Department o f Mechanical Engineering, Faculty o f Engineering,Gifu University, 1-1 Yanagido, Gifu-shi 501-11, Japan.* Emeritus Professor o f Nagoya University, 4-1-76 Tokugawayama-cho, Chikusa-ku, Nagoya 464 , Japan.Received 16 September 1993; revised 24 January 1994; accepted 14June 1994

s t ruc t u res i s c l o se l y r e l a t ed t o t he i n t e r f ace charac t e r -i st ic s o f th e m a c h i n e e l e m e n t s . T h e s e c h a r a c t e ri s ti c si n b o t h t h e d i r e c t i o n n o r m a l t o t h e m a t i n g s u r f a c e s( n o r m a l d i r e c t i o n ) a n d t h a t p a r a l l e l t o t h e m a t i n gs u r f a c e s ( t a n g e n t i a l d i r e c t i o n ) d e p e n d o n t h e m i c r o -scop i c su r f ace p rope r t y , t ha t i s , t he d i s t r i bu t i on o fa s p e r i t i e s o n m a t i n g s u r f a c e s , t h e a s p e r i t y s h a p e a n dt h e e l a s t i c o r p l a s t i c m a t e r i a l p r o p e r t y . T h e i n t e r f a c ec h a r a c t e r is t ic r e l a t e d t o t h e r i gi d it y b e t w e e n t h e m a t i n gsu r f aces i s t he so -ca l l ed con t ac t s t i f fness , and i s oneo f t h e i m p o r t a n t t r ib o l o g ic a l p r o p e r t i e s o f m a t i n gs u r f a c e s , w h i c h d e s c r i b e s t h e d y n a m i c b e h a v i o u r o fm a c h i n e s t r u c t u r e s .

T R I BO L O GY I N T E R N AT I O N AL 0 30 1 -6 79 X /9 4/ 06 /0 4 23 -0 9 1 99 4 Bu t te r w o rt h -H e i n e ma n n L t d 4 2 3


2/9

M . H a s h i m o t o et a l . - - E s t i m a t i o n o f c o n t a c t s t i ff n e s s a t i n t e rf a c e s

M a n y s t u d i e s h a v e b e e n c a r r i e d o u t a n d c o n s i d e r a b l ek n o w l e d g e h a s a c c u m u l a t e d o n c o n t a c t s t i f f n e s s . F o re x a m p l e , D o l b e y et al. j i nves t i ga t ed t he con t ac ts t i f fne ss o f me ta l - t o -me ta l and me ta l - t o -p l a s t i c j o in t si n t h e l o w - c o n t a c t p r e s s u r e r a n g e . T h e y s h o w e d t h a tt he power l aw re l a t i onsh ip can be used t o de sc r ibethe con t ac t p re ssu re -de f l ec t i on and t he con t ac t s t i f fne sscha rac t e r is t i c s o f many ma te r i a l s and su r face f i n ishes .T h o r n l e y et al. 2 m e a s u r e d t h e s t a t i c a n d d y n a m i cs t i f fne ss o f j o in t s sub j ec t ed t o l oad ing and exc i t a t i onin t he p l ane norm a l t o t he j o in t ' s su r face . T he i n f luenceof l ub r i ca t i on was a l so i nves t i ga t ed . From the i r r ev i ewof t he r e sea rch o n t he s t i f fne ss , damp ing , f r i c ti on andwea r o f t he f i xed and s l i d ing j o in t s , Back et al. 3demons t ra t ed t ha t t he t o t a l de f l ec t i on a t a j o in t edconnec t ion i s dep end en t upo n the e l a s t i c it y o f thec o m p o n e n t s s u r r o u n d i n g t h e m a t i n g s u r f a c e s , a n d t h e ya l so d i scussed fac to r s a f fec t i ng t he norma l s t i f fne ss .D e k o n i n c k 4 c a rr i e d o u t a n e x p e r i m e n t o n e l as t icn o r m a l d e f o r m a t i o n p r o p e r t i e s o f f i x e d m e t a l j o i n t sh a v in g t w o - d i m e n s i o n a l p r i s m a t ic r o u g h n e s s p e a k s . H eind i ca t ed t ha t t he p la s t i c de fo rma t ion o f roughn esspeaks i nc rea se s t he norma l e l a s t i c s t i f fne ss , due t op re load o f a j o in t .Yo sh imu ra -~'6 r ep re sen t ed me ta l j o in t s by a s im pl i fi edm o d e l o f a s p r i n g e l e m e n t a n d a v i s c o u s d a m p e re l e m e n t a n d o b t a i n e d j o i n t c h a r a c t er i s ti c s b y c o m p u t e rs imula t i on . I t i s u se fu l t o d i scuss t he i n t e rac t i onth rough the ma t ing su r face us ing a mode l i n t e r facec o m p o s e d o f d i st r i b u t e d s p ri n g a n d d a m p e r e l e m e n t s( see F ig 1 , fo l l owing Yo sh im ura ' s mo de l ) . T he s t i ffne sso f t h e s e s p r i n g e l e m e n t s c o r r e s p o n d s t o t h e c o n t a c ts t i f fne ss o f t he ma t ing su r faces .The con tac t s t i f fne ss depends on t he a spe r i t y d i s t r i -bu t ion pa t t e rn i n t he dep th d i r ec t i on , t he shape ande l a s t i c and /o r p l a s t i c ma te r i a l p rope r ty . Name ly , t hec o n t a c t s i t u a t i o n b e t w e e n t w o m a t i n g s u r f a c e s i sd e t e r m i n e d b y t h e a s p e r it y d i s t r ib u t i o n p a t t e r n a n dthe i r p l a s t i c ma te r i a l p rope r ty . In t h i s c a se , when ac o m p r e s s i v e f o r c e i s a p p l i e d b e t w e e n m a t i n g s u r f a c e sa con t ac t s t i f fne ss i n t he norma l d i r ec t i on r e su l t s f romthe e l a s t i c cha rac t e r i s t i c o f t he a spe r i t i e s i n t hec o m p r e s s i v e d i r e c t i o n ( n o r m a l t o t h e s u r f a c e ) . W h e na shea r fo rce i s app l i ed t o t he ma t ing su r faces acon tac t s t i f fne ss i n t he t angen t i a l d i r ec t i on i s due t othe e l a s t ic cha rac t e r i s t i c o f t he a sp e r i t i e s i n t he shea rd i rec t i on ( t angen t i a l t o t he su r face ) . From the above ,Y o u n g ' s m o d u l u s a n d t h e s h e a r m o d u l u s a r e t h epr inc ipa l f ac to r s i n f luenc ing t he co n t ac t s t i f fne ss i n t henorma l o r t angen t i a l d i r ec t i ons . In t h i s sense , i t i s

Upper Surface

~ ~ r-~ t i n ~ r J -n ~ . S prin g Elem ent+ + + + + + ~ + ~ - ~ V i s c o u s D am p erL 2 ~ ~ ~ t _ ~ E l e m en t

Lower SurfaceF i g 1 M o d e l o f m e t a l i n t e r fa c e4 2 4

impor t an t i n t r i bo logy tha t t he con t ac t s t i f fne ss i se s t i m a t e d f r o m t h e s u r f a c e t o p o g r a p h y a n d t h e e l a s t i cand p l a s t i c ma te r i a l p rope r t i e s .In t h i s r epor t a s impl i f i ed me thod fo r e s t ima t ing t hecon tac t s t i f fne ss i s p roposed . Tha t i s , t he con t ac ts t i f fne ss a t i n t e r face s i n bo th norma l and t angen t i a ld i r ec t i ons i s e s t ima ted f rom the v ib ra t i on o f a can t i -l eve red s t ruc tu re on an e l a s t i c founda t ion hav ingd i s t r i bu t ed sp r ing e l em en t s . On the ' ba s i s o f t he sed i scuss ions t he e f fec t s o f t he c l amping cond i t i on andma t ing su r face t opography on t he con t ac t s t i f fne sso b t a i n e d a r e q u a n t i t a ti v e l y a n d a c c u r a t e l y r e p r e s e n t e db y e x p e r i m e n t a l e q u a t i o n s f o r s im p l e e s t i m a t io n o fcon tac t s t i f fne ss . A d e t a i l ed d i scuss ion o f t he m e tho dof t heore t i ca l c a l cu l a t i ons o f con t ac t s t i f fne ss by su r facet o p o g r a p h y a n d m a t e r i a l p r o p e r t i e s w i l l b e p r e s e n t e di n a n o t h e r p a p e r .E x p e r i m e n t a l e s t i m a t i o n o f s t a ti c c o n t a c ts t i f f n e s sIn t he p re s en t pap e r t he s t a t ic con t ac t s t i f fne ss o f t hein t e r face i s ob t a ined t h rough a f r ee v ib ra t i on t e s t o fa can t i l eve red s t ruc tu re .E x p e r i m e n t a l a p p a r a t u s a n d m e t h o dFigure 2 shows the expe r imen ta l appa ra tus . In t hef igu re a squa re co lumn B i s he ld be tween ho ld ingb locks Hj and H2 unde r a ca l i b ra t ed c l amping l oadb y a v i c e V . T h e c l a m p i n g l o a d i s m e a s u r e d b y t w oload ce l ls L~ and Lz . The m om ent o f i ne r ti a o f t hea rea o f b locks H~ and H2 i s 25 t imes l a rge r t han t ha to f co lumn B . Two sph e r i ca l ho l lows a re cu t i n to b lockH~ a t t he p re ssed pa r t s by l oad ce l l s . The r ad ius o fthe se sphe r i ca l ho l l ows i s equa l t o t he sphe r i ca li nden t e r r ad ius o f t he l oad ce l l s . Th i s p rocedureemphas i ze s t he r e l a t i ve r i g id i t y be tween H~ and l oadce l ls L t and L2 . H ence , t he e f fec t o f c l amping by t hev i ce t h rough load ce l ls on t he v ib ra t i on o f thecan t i l eve red co lumn B i s sma l l .C o l u m n B a n d b l o c k s H ~ a n d H 2 a r e m a d e o f m i lds t ee l o r a lumin ium a l l oy . Ma te r i a l cons t an t s o f ma t ingsur faces a re g iven i n Tab le 1 . The t ab l e a l so i nc ludesY o u n g ' s m o d u l u s , t h e s h e a r m o d u l u s a n d t h e y i e l dpre ssu re .The ma t ing su r faces o f B , H t and H2 a re fi n i shed bysand b l a s t i ng wi th va r ious g ra in s i ze s a f t e r su r facegr ind ing . The f l a tne ss o f t he g round su r faces i s l e ss

N

100

v

B

F i g 2 E x p e r i m e n t a l a p p a r a t u s f o r f r e e v i b r a t i o n1 9 9 4 V O L U M E 2 7 N U M B E R 6


3/9

M. Hashimoto et al.--Est imation of contact stiffness at interfaces

T a b le 1 M a t e r ia l c o n s t a n t o f m a t in g s u r f a c e sM ate r ia l Yo ung ' s m o du lus Shea r m od u lus Y ie ld p ressu reE (GPa) G (GPa) Pm (GPa)Mi ld s te e l 2 0 0 8 0 , 2Alum in ium 70 28 0 .23

2 5

y l c ~ n H M - O 2 2 . . R an kT ay lo rl -I d, m .~ = . - , '

Fig 3 Surface roughness profile o f sand blasted steel surface ($1)

than 1 p~m, and a uniform contac t pressure dis tributionis obtained over the whole nominal contact areabetween column B and blocks H1 and H2.An impulse is input at the top end of column B inthe normal (N) or tangential (T) directions to themating surfaces. Free damped vibration in the inputdirection of the column end is detected by an eddycurrent type displacement meter set opposite the input.The meter is at a point 10 mm inwards from the topend. Experiments were carried out for combinationsof the column and the holder blocks having the samelevel of surface topography.S u r f a c e t o p o g r a p h yAs mentioned in the Introduction, one importantfactor, which has a considerable influence on thecontact stiffness, is surface topography. A uniquetechnique for evaluating the surface topography isgiven in this section.The mating surfaces are sand blasted with abrasivesof various grain sizes. A surface S1 is sandblasted byabrasive #46 and surfaces $2, $3 and $4 by abrasives#70, #100 and #200. Therefore the surface roughnessdistribution and the shape of the asperity peak aresimilar regardless of the direction of surface profilemeasurement. The grain size of abrasive #46 isdistributed between 0.40 and 0.60 mm. Those of #70,#100, #220 are distributed between 0.30 and 0.35 mm,0.15 and 0.18 mm and 0.075 and 0.088 mm, respect-ively.Figure 3 shows the surface roughness profile of steelsurface S1 sand blasted by abrasive #46, which wasmeasured by a Talysurf profilometer. Many asperitiesare distributed on the surface at various depths. FigureTRIBOLOGY INTERNATIONAL

4 illustrates the cumulative peak height distribution inthe depth direction of these asperity peaks on thesteel surface S1 shown in Fig 3 on probability paper,according to the standard Greenwood & Williamsoncontact model7. Cumulative peak height distributionis the ratio of asperity number N between the depthh = 0 and h = h to the total asperity number No onthe whole nominal contact area. Cumulative peakheight distribution is approximately linear at the meanpeak height (50%) on probability paper as shown inFig 4. However, the cumulative height distribution inthe vicinity of 0 I~m depth on the abscissa deviated

9 9 . 9 99 9 . 9

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Fig 4 Cumulative pe ak height distribution on pro b-ability paper4 2 5


4/9

M . H a s h i m o t o e t a l . ~ E s t i m a t i o n o f c o n t a c t s t if fn e s s a t i n t e rf a c e sf r o m t h e l i n e a r d i s t r i b u t i o n . T h e a s p e r i t i e s i n t h ev i c in i ty o f 0 ~ m d e p t h a r e l o c a t e d i n th e h i g h e r r a n g ea n d h a v e a g r e a t e r p o s s i b il i t y o f c o n t a c t w i t h t h ea s p e r i t i e s o n t h e o p p o s i t e s u r f a c e . A s a r e s u l t , i t i sn o t a l w a y s c o n v e n i e n t t o r e p r e s e n t t h e p e a k h e i g h td i s t ri b u t io n b y G r e e n w o o d & W i l l i a m s o n ' s n o r m a ld i s t r i b u t i o n m o d e l .T h e n t h e c u m u l a t i v e p e a k h e i g h t d i s t r i b u t i o n w a sp l o t t e d o n t h e C a r t e s i a n c o o r d i n a t e s y s t e m s a s s h o w ni n F i g 5 . H e r e , c u m u l a t i v e p e a k h e i g h t d i s t r i b u t i o nh a s a q u a d r a t i c d i s t r i b u t i o n c h a r a c t e r i s t i c i n t h e v i c i n i t yo f h = 0 p .m d e p t h . I t w a s e a r l i e r a s c e r t a i n e d b y t h ea u t h o r s 9 t h a t t h e c u m u l a t i v e p e a k h e i g h t d i s t r i b u t io no f a s a n d b l a s t e d s u r f a c e c a n b e r e p r e s e n t e d b y th ef o l l o w i n g e q u a t i o n p r e c i s e l y i n t h e d i s t r i b u t i o n r a n g eb e t w e e n h/Hmax = 0 - 0 .7 :

( n h ~ m f : ) m- - b ( 1 )I n th i s e q u a t i o n t h e i n d e x n u m b e r m i s a l m o s t c o n s t a n t( m ~ 2 ) f o r a ll g r a i n s i z e s a n d m a t e r i a l s t e s t e d , a n dH m ax i s t h e d e p t h i n t e r v a l b e t w e e n t h e h i g h e s t a n dt h e l o w e s t a s p e r i t y p e a k s . I n t h i s p a p e r e v a l u a t i o n o fs u r f a c e t o p o g r a p h y b y e q u a t i o n ( 1 ) is a d o p t e d .T h e r e a l c o n t a c t a r e a i s t h e m o s t i m p o r t a n t t r ib o l o g i c alf e a t u r e , a n d h a s a s i g n if i c a n t e f f e c t o n c o n t a c t s t i f fn e s sa t t h e i n t e r f a c e . B y u s i n g K r a g e l s k ii ' s r o d m o d e P o fr o u g h s u r f a c e s f o r c u m u l a t i v e p e a k h e i g h t d i s t r ib u t i o nc h a r a c t e r is t i c o f e q u a t i o n ( 1 ) ( m ~ 2 ) t h e r e a l c o n t a c ta r e a A c a n b e o b t a i n e d .T h e n u m b e r o f c o n t a c t i n g a s p e r it i e s is e x a m i n e d b yK r a g e l s k ii ' s r o d m o d e l i n F i g 6 , w h e n t h e t w o s u r f a c e sS ( i = 1 ) a n d G ( i = 2 ) c o m e i n t o c o n t a c t . I n t h ef i g u r e t h e c h a i n l i n e i n d i c a t e s t h e r e l a t i v e p o s i t i o n o ft h e t w o s u r f a c e s w h e n t h e h i g h e s t a s p e r i t i e s o n b o t hs u r f a c e s b e g i n t o c o n t a c t e a c h o t h e r .F r o m t h i s p o s i t i o n , s u r f a c e S ( i = 1) i s d i s p l a c e dd o w n w a r d s b y a n a m o u n t a , a n d a s p e r i t i e s A ~ w h i c h

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D e p t h t i mFig 5 Cum ulative pe ak height distribution on Cartesiancoordinate system4 2 6

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A / J i~ / / / / ~ / / / / S / 4 / 5 ; . (l 2)

Fig 6 Kragelskii's rod m ode l fo r rough surfaces

a r e l o c a t e d i n t h e n a r r o w dhi l a y e r a t a d e p t h h ~ a r ed i s p l a c e d t o a n e w p o s i t i o n A ' . A t t h i s t i m e , t h ea s p e r i t i e s i n t h e dhl l a y e r c o n t a c t t h o s e i n t h e n a r r o wdh 2 l a y e r a t a d e p t h h 2 o f t h e s e c o n d s u r f a c e G ( i = 2 ) .T h e p r o b a b i l i t y o f c o n t a c t b e t w e e n t h e t i p s o f a sp e r i ti e si n b o t h dhl a n d dh 2 l a y e r s i s d e t e r m i n e d b y t h ea s p e r i t y d i st r i b u ti o n r u l e . T h e r a t e o f c o n t a c t P(a) o rt h e p r o b a b i l i t y i s g i v e n a s a f u n c t i o n o f t h e r e l a t i v ea p p r o a c h a o f t w o s u r f a c e s a s f o ll o w s :

NP (a ) - N of " (" h i dcb i dO2= dhi 21~i- dh 2 dh2) d O ( 2 )w h e r e N o = Nol = No2 w h e n t h e s u r f a c e t o p o g r a p h i e so f th e t w o m a t i n g s u r f a c e s a r e i d e n t i c a l a n d N o i se q u a l t o t h e s m a l l e r o n e b e t w e e n N o~ a n d N o2 w h e nt h e s u r f a c e t o p o g r a p h i e s o f th e t w o m a t i n g s u r f a c e sa r e d i f f e r e n t .T h e r e a l c o n t a c t a r e a ~ A o f a p a i r o f c o n t a c t i n ga s p e r i t i e s a t d e p t h s h ~ a n d h 2 i s o b t a i n e d a s f o l l o w s :

R1R28 A = ~ R a ' , R . . . . . . ( 3 )R1 + Rzw h e r e t h e r e l a t i v e a p p r o a c h a ' o f tw o a s p e r i t ie s i sg iv en b y a ' = a - ( h i + h 2 ) . S e m i - s p h e r i c a l a s p e r i t -i e s 9 , w h o s e r a d i u s i s R i, c o v e r t h e s a n d b l a s t e d m a t i n gs u r f a c e s S ( i = 1 ) a n d G ( i = 2 ) . T h e q u a n t i t y R i ne q u a t i o n ( 3 ) i s a n e q u i v a l e n t r a d i u s o f a s p e r i t i e s int h e c o m b i n a t i o n o f t h e s u r fa c e s S a n d G .T h e t o t a l r e a l a r e a o f c o n t a c t A i s t h e n w r i t te n a sfo l l o ws :

f~ I a-h , drbl drY2A = No dhl dh, ~ 8Adh2 ( 4 )) )A p p l y i n g e q u a t i o n ( 1 ) t o t h e s a n d b l a s t e d s u r f a c e t h er e a l c o n t a c t a r e a o f t h i s s u r f a c e i s o b t a i n e d b y i n s e r t i n gm = 2 i n t o e q u a t i o n ( 1 ) :

wRblb2No a5A = 3 0 n 2 m a x i n2max2 =St as ( 5 )w h e r e

1 99 4 V O L U M E 2 7 N U M B E R 6


5/9

M . H a s h i m o t o e t a l . - - E s t i m a t i o n o f c o n t a c t s t if f n e s s a t i nt e rf a c e s

1rRblb2N (6 )S t = 3 0 n 2 m a x 1 n 2 m a x 2

The coefficient st is called the surface topographycoefficient and R, b, H . . . . No are the equivalentradius of curvature at the asperity tip, the distributioncoefficient in equation (1), the depth interval and thetotal asperity number on the nominal contact surface,respectively. For a surface having a large value of st,surface roughness is small and load-carrying capacityis large. These parameters for the surface topographyaffect the contact situation of a rough metal surface 11and are given in Table 2, including the surfacetopography coefficient st.The contact situation for the two mating surfaces isdetermined by equations (5) and (6) and the yieldpressure of the asperities pm 8. The deforming mechan-ism of the asperities at this stage is the plasticdeformation phase and is a function of the surfacetopography coefficient st. Contact stiffness is genera tedas the integration of the elastic characteristic atcontacting asperities (junctions). Contact stiffness inthe normal direction is a function of the elasticcompressive deformation property of junctions. Also,the contact stiffness in the tangential direction is afunction of the elastic shearing deformation propertyof junctions.

E x p e r i m e n t a l r e s u l t o f f r e e v i b r a t io nExamples of the frequency of free damped vibrationare given in Figs 7 and 8. These frequencies can beapproximately regarded as the fundamental naturalfrequencies of the cantilevered structures because thedamping ratio is small. Figure 7 is the result of a steelcolumn structu re and Fig 8 of an aluminium alloycolumn structure. The frequency increases slightly withan increase in the clamping load F or mean clampingpressure p . . . . The frequency of normal vibration isslightly higher than that of tangential vibration. Boththe steel and the aluminium alloy column structureshave the same order of frequency. The frequency offree vibration (natu ral frequency) rises with an increasein the surface topogra phy coefficient st, for bothdirections and both materials.

1 5 0 0

1 0 0 0

z5 0 o

i

0a

S t e e l

1 0F k N

I

o S 1 $ 2$ 3 $ 4

I

N o r m a l D i r e c ti o n, I L1 5 2 0

1 0Pme. , , MPa

1 5

1 5 0 0

1 0 0 0

5 0 0

Ste e l

oSa S S S

T a n g e n t i a l D i r c c l k m!

i

5 10 15 20F k N

i i i i i i i

0 5 10 15b P,.c~,,, M PaF i g 7 R e l a t i o n b e t w e e n n a t u r a l f r e q u e n c y a n d c l a m p i n gl o a d o r m e a n c l a m p i n g p r e s s u r e . ( a ) S t ee l , n o r m a ldirec t ion , (b) s tee l , tangent ia l d irec t ionN a t u r a l f r e q u e n c y c a l c u l a t io n b y a b e a m m o d e lo n a n e l a s ti c f o u n d a t i o n a n d e s t i m a t i o n o fc o n t a c t s t i f f n e s sTo calculate the natural frequency of the experimentalstructure in Fig 2, when considering the interfacial

T a b l e 2 S u r f a c e t o p o g r a p h y o f s a n d b l a s t e d s u r fa c e sSur fa ce Hrna ( t xm) b m R ( l im ) No "106 s t ( I xm -3 )$1 (M i ld s tee l ) 15 2 .5 2 20 2 .4 6 .21 "102( A l u m i n i u m ) 3 0 2 .2 2 1 7 2 .0 2 . 1 3 " 1 0$ 2 ( M i l d s t e e l ) 8 2 . 4 2 1 5 2 .3 5 .08 "103( A l u m i n i u m ) 2 3 2 .3 2 1 5 2 . 4 7 . 1 3 " 1 0S 3 ( M i l d s t e e l ) 6 2 . 8 2 1 3 2 . 5 2 . 0 6 "1 0 4(A lu m in iu m ) 17 2.6 2 11 2.8 2 .61 * 102$ 4 ( M i l d s t e e l ) 4 3 . 7 2 1 2 2 . 6 1 . 7 5 "1 0 s(A lum in u m ) 12 3 .5 2 9 3 .3 1 .84 " 103

T R I B O L O G Y I N T E R N A T I O N A L 4 2 7


6/9

M . H a s h i m o t o et a l . ~ E s t i m a t i o n o f c o n t a c t s ti ff n e ss a t i n t e rf a c es

1500[ A l u m i n i u m I

~z

1000

50 0

0 L0L0a

5i

10F kN

i

o S1o $ 2$3 $ 4- - 2

Normal Direction15 20

5 10 15Pmcan MPa

1500

1 0 0 05 0 }0

t i0b

Alumin ium

, I5 10F kNi i

o S1o $ 2e $ 3 $ 4

rFangential Direction '] 5 20

_L_ I10 15Pmcan MPa

F i g 8 R e l a t i o n b e t w e e n n a t u r a l f r e q u e n c y a n d c l a m p i n gl o a d o r m e a n c l a m p i n g p r es s u re . ( a ) A l u m i n i u m a l lo y ,n o r m a l d i r e c t i o n , ( b ) a l u m i n i u m a l l o y , t a n g e n t i a l d i r e c -t ion

e f f e c t b e t w e e n c o l u m n B a n d b l o c k s H 1 a n d H 2 , th ec o l u m n i s t r e a t e d a s a b e a m o n a n e l a s t i c f o u n d a t i o n( F i g 9 ) w i t h k a s t h e s p r i n g c o n s t a n t p e r u n i t l e n g t ha l o n g t h e c o l u m n 12. T h e f a c t o r k r e p r e s e n t s t h e c o n t a c ts i t u a t i o n ( c o n t a c t s t i f f n e s s ) a t t h e i n t e r f a c e . I n F i g 9a * i s t h e c o n t a c t l e n g t h o f c o l u m n a n d b l o c k s a n d Li s t h e l e n g t h o f th e c a n t i l e v e r e d p a r t . Y o u n g ' s m o d u l u sa n d t h e m o m e n t o f i n e r t i a o f t h e a r e a o f th e c o l u m na r e E a n d I , r e s p e c t i v e l y .

W x2 " I ~ k Xl ~

L a"(Cant i levered) l - (E las t ic Foundat ion)Y2"

F i g 9 B e a m m o d e l o n a n e la s ti c f o u n d a t i o n

t

Yl

428

T w o s e ts o f r e c ta n g u l a r c o o r d i n a t e s y s t e m s a r e d e t e r m -i n e d f o r t h e c o l u m n a s s h o w n i n F i g 9 . B y d i s r e g a r d i n gt h e v i s c o u s d a m p e r e l e m e n t i n F i g 2 , t h e d i f f e r e n ti a le q u a t i o n s f o r t h e d y n a m i c st a t e o f t h e b e a m c a n b ew r i t t e n a s f o l l o w s :I n t h e r e g i o n o f t h e c o l u m n o n t h e e l a s ti c f o u n d a t i o n(0 -< xl -< a* ):

o4yt o2ytE 1 a ~ x~ + p A * ~ t2 + k y t = 0I n t h e c a n t i l e v e r e d p a r t ( 0 - < x 2 -< L ) :

(7 )

aaY~ a2Y~ - 0 (8 )E 1 0 x ~ + p Z * 01,2w h e r e p i s t h e d e n s i t y o f t h e c o l u m n a n d A * t h e c r o s s -s e c t io n a l a r e a o f t h e c o l u m n .T o o b t a i n n o r m a l v i b r a t i o n o f a b e a m o n a n e l a st icf o u n d a t i o n , t h e f o l l o w i n g s o l u t io n s a r e a s s u m e d , c o r r e -s p o n d i n g t o t h e d i f f e r e n t i a l e q u a t i o n s ( 7 ) a n d ( 8 ) :

y~ - Y~' (A lco s t o t + A e s i n t o t ) (9 )y~ = Y~ (A lco s to t + A 2s in tot ) ( 1 0 )

Y ~ ', Y ~ a r e f u n c t i o n s o f X l o r x 2 o n l y . S u b s t i t u t i n ge q u a t i o n s ( 9 ) a n d ( 1 0 ) i n t o e q u a t i o n s ( 7 ) o r ( 8 ) , t h eo r d i n a r y d i f f e r e n t i a l e q u a t i o n s f o r Y ]' o r Y ~ a r eo b t a i n e d :In th e reg io n o f 0 -< x l -< a* :

d 4 y t- - + 4 13 " 4 y ~ ' = 0dx~

In the reg io n o f 0 -< x2 -< L :( 1 1 )

d 4 y ~d x 4 1 3 4 y ~ = - 0 ( 1 2 )

w h e r e t h e f o l l o w i n g e x p r e s s i o n s a r e u s e d :13,4 _ k - pA *to 2 134 _ PA - t o 2 ( 1 3 )4 E l ' E 1

T h e n o r m a l s o l u t i o n f o r t h e c o l u m n o n a n e l a s t i cf o u n d a t i o n i s c a l c u l a te d f r o m e q u a t i o n ( 1 1 ) a s f o ll o w s :YT = eB*x1(B11 cos 13"x, + B t2s in 13 'x l)

+ e-1 3* x '(B l3co s 13"xl + B14s in 13"Xl) (14 )T h e n o r m a l s o l u t i o n f o r t h e c a n t i l e v e r e d p a r t i so b t a i n e d f r o m e q u a t i o n ( 1 2 ) a s f o l l o w s :

Y~ = B 2 1 c o s 1 3 x2 + B22sin 13x2+ B 2 3 c o s h B x 2 + B2 4sin h 13x2 (15 )

T h e e i g h t q u a n t i t i e s B l l , B 1 2 , B 1 3 , B 1 4 , B 2 1 , B 2 2 , B 23a n d B 2 4 i n e q u a t i o n s ( 1 4 ) a n d ( 1 5 ) a r e t h e i n t e g r a lc o n s t a n t s a n d c a n b e d e t e r m i n e d b y e i g h t b o u n d a r yc o n d i t i o n s , w h i c h a r e s e t a s f o l l o w s : ( 1 ) b e n d i n gm o m e n t a n d s h e a r f o r ce a r e z e r o a t b o t h e n d s o f th ec o l u m n ; ( 2 ) d e f l e c t i o n , i n c l i n a t i o n , b e n d i n g m o m e n ta n d s h e a r f o r c e a r e c o n t i n u o u s a t t h e b o u n d a r y o f t h e1 9 94 V O L U M E 2 7 N U M B E R 6


7/9

M . H a s h i m o t o e t a l . - - E s t i m a t i o n o f c o n t a c t s t if f n e s s a t in t e rf a c e s

c o l u m n o n a n e l a s t i c f o u n d a t i o n a n d a c a n t i l e v e r e dp a r t . T h e n w e o b t a i n t h e f o l l o w i n g l i n e a r s i m u l t a n e o u se q u a t i o n w i t h e i g h t u n k n o w n s :0 1 0 -10 0 0 0

t - 1 1 1 10 0 0 0~ 1 ~ 2 ~ 3 ~ 4

R(, - @ 2 ) R ( 1 + 2 ) - R ( 3 + ~ 4 ) R ( 3 - 4 )-2R2@ 2 2R2@1 2R2@4 -2R2@ 3

- 2R 3 ( q , + 02 ) 2R3 ( 1 - C z ) 2R3 ( (I )3 - - d l )4 ) 2R 3(O3 + 4 )

0 0 0 0- c o s a - s i n a c o s hc t s i n h a

0sin

- 1010

t-c os a sinh c~ cosh a

0 - 1 0- 1 0 - 1

0 - 1 01 0 - 1

n i lB12B13B l4B2 1B22B23B2 4

= 0 (16)

w h e r eot = 13L, 8 = 13*a*, R* = 13"/13

@1 = e~c os 8, @2 = e~sin g (17 )@ 3 = e - ~ c o s g , (I)4 = e - % i n

T h e d e t e r m i n a n t d e f i n e d b y t h e c o e f fi c i e n t m a t r i x o fe q u a t i o n ( 1 6 ) m u s t b e z e r o t o o b t a i n n o n - z e r o i n t e g r a lc o n s t a n t s B o ( i = 1 , 2; j = 1 , 2 , 3 , 4 ) . T h i s r e l a t i o n i st h e f r e q u e n c y e q u a t i o n t o o b t a i n t h e n a t u r a l f r e q u e n c yo f t h e c o l u m n o n a n e l a s ti c f o u n d a t i o n .T h e c o n t a c t s t i f f n e s s k i s e s t i m a t e d f r o m t h e a b o v en o r m a l v i b r a t i o n o f th e c o l u m n s y s t e m . F i g u r e 1 0i n d i c at e s t h e r e l a t i o n b e t w e e n t h e n a t u r a l f r e q u e n c y fa n d t h e c o n t a c t s t i ff n e s s k , c a l c u l a t e d f o r t h e c o l u m ns y s t e m u s e d i n t h e e x p e r i m e n t , w h i c h h a s a 1 5 m ms q u a r e c r o s s - s e c t io n a n d L = 1 00 m m a n da * = 8 5 m m , u s in g t h e a b o v e t h e o r y . T h e n a t u ra lf r e q u e n c y i n c r e a s e s g r a d u a l l y w i t h a n i n c r e a s e i n t h ec o n t a c t s ti f fn e s s k a n d a p p r o a c h e s t h e n a t u r a l f r e q u e n c yf c r o f t h e f u n d a m e n t a l m o d e o f n o r m a l v i b r a t io n o ft h e c a n t i l e v e r b e a m f o r b o t h t h e s t e e l a n d a l u m i n i u ma l l o y s t r u c t u re s . F o r t h e s a m e m a g n i t u d e s o f t h ec o n t a c t s t if f n es s t h e a l u m i n i u m a l l o y s t r u c t u r e h a s ah i g h e r n a t u r a l f r e q u e n c y t h a n t h e s t e e l o n e .A p p l y i n g t h e n a t u r a l f r e q u e n c y (F i g s 7 a n d 8) o b t a i n e di n th e e x p e r i m e n t t o t h e r e s u l t o f Fi g 1 0 , th e c o n t a c ts t if f n es s k c a n b e e s t i m a t e d . T o i m p r o v e t h e g e n e r a l i t y

200{)

1500

/~ " / : i t :

I)L .[ . . . . A . . . . . . . .I I 0 I ( l ( I 1 0 0 0k (;P.

10000

Fig 10 R e la t i on be tween na tura l f requ ency and con tac tst i f fness

o f th e r e s u l t o b t a i n e d , t h e c o n t a c t s t i f fn e s s k i sc o n v e r t e d i n t o t h e c o n t a c t s t i ff n e s s k * p e r u n i t a p p a r e n tc o n t a c t a r e a b y d i v i d i n g t h e c o n t a c t s t i f f n e s s k b y t h ec o l u m n w i d t h ( 1 5 m m ) . T h e c o n t a c t s t if f ne s s in t h en o r m a l a n d t a n g e n t i a l d i r e c t i o n s a r e r e p r e s e n t e d b yk ~ a n d k - }. T h e e s t i m a t e d r e s u l t s a r e g i v e n i n F i g 11( s t e e l s t r u c t u r e ) a n d F i g 1 2 ( a l u m i n i u m a l l oy s tr u c t u r e )a s f u n c t i o n s o f a c l a m p i n g l o a d F o r a m e a n c l a m p i n gp r e s s u r e p . . . . . I t i s c l e a r f r o m t h e f i g u r e s t h a t t h em o r e e x c e l l e n t th e s u r f a c e t o p o g r a p h y o f t h e m a t i n gs u r f a c e s o f t h e c o l u m n a n d t h e h o l d i n g b l o c k s , t h eh i g h e r t h e c o n t a c t s t i f f n e s s . T h e c o n t a c t s t i f f n e s s i sl a r ge f o r a l a r g e c la m p i n g p r e s s u r e . T h e m a g n i t u d e o fc o n t a c t r i g i d i t y i n t h e n o r m a l d i r e c t i o n i s l a r g e r t h a nt h a t i n t h e t a n g e n t i a l o n e .

E x p e r i m e n t a l e q u a t i o n o f c o n t a c t s t i f f n e s sI t i s c o n v e n i e n t t o m a k e a n e x p e r i m e n t a l e q u a t i o n o fc o n t a c t s t i f fn e s s a s a f u n c t i o n o f t h e c o n t a c t s u r f a c et o p o g r a p h y f o r t h e a c t u a l d e s ig n o f m a c h i n e s t r u c t u r ed y n a m i c s . T h e c o n t a c t s t i ff n e s s k r~ a n d k -} i n c r e a s ep a r a b o l i c a l l y w i t h a n i n c r e a s e i n th e m e a n c l a m p i n gp r e s s u r e p . . . . o n s e m i - l o g a r i t h m i c g r a p h p a p e r , a ss h o w n i n F i g s 1 1 a n d 1 2. H e r e , t o s i m p l i f y t h e

~ o o o ~ ~ ~

"- 10 ~ ~.a~~ k T'T"

1 5 10 15 20F kN

; ' ; ' 1'0 ' 1'5Pmean MPa

Fig 11 Re la t i on be tween con tac t s t i ff ne ss and c lam pingl o a d o r m e a n c l a m p i n g p r e s su r e ( s te e l)T R I B O L O G Y I N T E R N A T I O N A L 4 29


8/9

M . H a s h i m o t o et a l . - - E s t i m a t i o n o f c o n t a c t s t i ff n e s s a t i n t e r fa c e s

~ I0

1 0 0 0 pA l u m i n i u

! o S1$ 2' $ 3

$ 41 0 0 - -

J Y //~P~0 . 0 . - 0 " 0

ro . l l0F

t(i 5P m e a n

10 15 20k N

L ~ p - . J10 "15M P a

Fig 12 Relation between contact stiffness and clampingload or mean clamping pressure (aluminium alloy)

0 ri

g - o . s ]' S t e e l ~ ~N o r m a l

- 1 . 0 ] . . . . T a n g e n t i a lOA

- - O - . . . . . . . . . . O -~ __ .

1 0 1 0 0 1 0 0 0S l x l 0 3 / t i n - 3

Fig 13 Exam ple for numerical coef ficient o f experimen-tal equation as a func tion of surface topographycoefficient (steel)

e x p r e s s i o n , t h e s u f fi x N o r T o f t h e c o n t a c t s t i f f n e s si s d e l e t e d a n d t h e c o n t a c t s t i f f n e s s i s r e p r e s e n t e ds i m p l y a s k * . T h e c o n t a c t s t i f f n e s s k * c a n b e w r i t t e na s a f u n c t i o n o f m e a n c l a m p i n g p r e s s u r e p . . . . . a sfo l l o ws :

i ' , , i / / /

] 0 " 7 / / f ~-" .-" '\ J/ / / # / i I !

/ S t e e l

! k r " - . . . . .

o l l i0 5 10 15 20F k Ni0 5 10 15

P m e a . M P a

Fig 14 Calculation o f contact stiffness by experimentalequation (steel)

l o g k * = 2 ( 1 8 )p . . . . . q - b e . . . . - [ - cA p p l y i n g e q u a t i o n ( 1 8 ) t o t h e e x p e r i m e n t a l r e s u l t s o fF ig s 1 1 an d 1 2 , t h e co e f f i c i en t s a , b an d c i n eq u a t i o n( 1 8 ) a r e d e t e r m i n e d . I n t h e c a l c u l a ti o n o f t h e s ec o e f f ic i e n ts th e m e t h o d o f l e a st s q u a r e s i s u s e d . A ne x a m p l e f o r t h e s t e e l s t r u c t u r e i s g i v e n i n F i g 1 3 . I nt h e f i g u r e , t h e o b t a i n e d c o e f f i c i e n t a i s p l o t t e d a s af u n c t i o n o f s u r f a c e t o p o g r a p h y c o e f f ic i e n t s t o n s e m i -l o g a r i t h m i c g r a p h p a p e r , o n r e f e r r i n g to T a b l e 2 . I t i sc l e a r t h a t t h e r e l a t i o n s h i p b e t w e e n c o e f f i c ie n t a a n dt h e s u r f a c e t o p o g r a p h y c o e f f i c i e n t s t i s l i n e a r o ns e m i - l o g a r i t h m i c g r a p h p a p e r . T h i s l i n e a r i t y i s a l s or e c o g n i z e d f o r b o t h s u r f a c e m a t e r i a l s o f s t e e l a n da l u m i n i u m a l l o y a n d f o r b o t h n o r m a l a n d t a n g e n t i a ld i r e c t i o n s . C o e f f i c i e n t s b a n d c h a v e t h e s a m e l i n e a r i t y .F r o m t h e a b o v e c o n s i d e r a t i o n s , t h e l in e a r i ty r e c o g n i z e din F ig 1 3 can b e wr i t t en as fo l l o ws :

T a b l e 3 N u m e r i c a l c o e f f ic i e n t s a s f u n c t i o n s o f s u r f a c e m a t e r i a l

Direction al a2 blSteelNormal -0.0 0164 -0.00016 0.0388Tangential -0.00049 -0.00322 0.0325AluminiumNormal -0.00353 0.00019 0.0652Tangential -0.00063 -0.00261 0.0297

b 2 c l

0.1114 0.0605 -0.3330.0630 0.1005 -0.681

0.1745 0.1083 -0.8580.0940 0.0638 -0.596

430 1994 VOLUME 27 NUMBER 6


9/9

M . H a s h i m o t o e t a l . - - E s t i m a t i o n o f c o n t a c t s t i ff n e s s a t in t e r f a c e s

a = a~log st + a2b = bll og st + b2 (19)c = cllo g st + c2

Applyi ng equat ion (19) to the results shown in Fig 13and using the method of least squares, all coefficientsai,bi,ci,(i = 1, 2) of equ ation (19) are determ ined.The result is given in Table 3 for each surface material.The yield pressure of asperity Pm determines the realcontact area and contact situation corresponding tothe clamping load F or mean clamping pressure p . . . . Young's modulus E and the shear modulus G determinethe elastic deformation of asperities on mating surfacesin normal and tangential directions. Therefore thecoefficients' values given in Table 3 are considered asa function of the yield pressure Pro, Young's modulusE and shear modulus G. It is impossible to obtainexperimental equations of coefficients ai, bi and cifrom experimental results for only two mating surfacematerials. Therefore only numerical values are givenin Table 3.The experimental equation of contact stiffness k* isobtained from equations (18) and (19) as follows:

a 2 ) P m e a n b 2 ) P m e a n c 2 )* = 1 0 ( a l l g s t + 2 + ( b l l O g s t + + ( C l l O g s t +( 20 )

The results of contact stiffness of a steel structureshown in equation (20) and Table 3 are given inFig 14. The experimental data (normal direction) forsteel S1 surface are also plotted in the same figure.The calculated value of contact stiffness in Fig 14 andthe experimental value in Fig 11 agree well with eachother. The contact stiffness in the normal and tangentialdirections can be estimated by the above experimentalequation (20). The same result and comments applyto the aluminium alloy structure.

Conc lus ionsThe method for obtaining contact stiffness is explainedin this paper, using cantilevered structures whichincorporate connections between the basic elements

of various surface topographies. The contact stiffnesscan be obtained by combining the experimental resultsof the natural frequency of the structure and thecalculation of the principal vibration of the cantileveredstructure on an elastic foundation. An experimentalequation is constructed as a function of the clampingload and surface topography. A forthcoming studywill deal with the theoretical prediction of the contactstiffness, based on the elastic compressive deformationmodel or the elastic shear deformation model of thejunctions.

References1. Dolbey M.P. and Bell R. The contact stiffness of joints at lowapplied interface pressures. Annals of the CIRP 1971, 19 , 67-792. Thornley R.H. and Koenigsberger F. Dynamic characteristicsof machined joint loaded and excited normal to the joint face.Annals of the CIRP 1971, 1 9 , 459-4693. Back N., Burdekin M. and Cowley A. Review of the researchon fixed and sliding joints. Proc. 13th MTDR Conf., Macmillan,London, 1973, 87-974. Dekoninck C. Experimental study of the normal static stiffnessof metallic contact surfaces of joints. Proc. 13th MTDR Conf.,Macmillan, London, 1973, 61-665. Yoshimura M. Measurement of dynamic rigidity and dampingproperty for simplified joint models and simulation by computer.Annals of the CIRP 1977, 25, 193-1986. Yoshimura M. Computer-aided design improvement of machinetool structure incorporating joint dynamics data. Annals of theCIRP 1979, 27, 241-2467. Greenwood J.N. a n d W i l l i a m s o n J.B.P. Contact of nominallyflat surfaces. Proc. Roy. Soc. London. A 1966, 295, 300-3198. Kato S., Sato N. and Matsubayashi T. Some considerations oncharacteristics of static friction of machine tool slideway.ASME

Journal of Lubrication Engineering 1972, 9 4 , 234-2479. HashimotoM., Marui E. and Kato S. Considerations on dampingcharacteristics of turning tools (2nd Report). Effects of clampingload and surface topography. Bulletin of the Japan Society ofPrecision Engineering 1990, 24, 263-26810. Kragelskii I.V. Friction and Wear, Butterworths, London, 1965,4411. Kato S., Marui E., Kobayashi A. a n d S e n d a S. The influenceof lubricants on static friction characteristics under boundarylubrication. ASME Journal of Tribology 1985, 107, 188-19412. Timoshenko S.P. Strength of Materials, Pt II, Advanced Theoryand Problems, 3rd edn, Van Nostrand, Princeton, NJ., 1956, 1

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1994 Hashimoto Contact Stiffness