1980 Sachdeva Interface Friction

8/4/2019 1980 Sachdeva Interface Friction

1/11

Int..L Mech. Sci. Vol. 23, pp. 149-1_';9, 1981 0020--7403181/030149-1150 2.00/0P r i n t e d i n G r e a t B r i t ai n . P e r g a m o n P r e s s L t d .

TH EON

EFFECT OF INTERFACE FRICTIONTHE INPLANE FLEXIBILITIES OFMACHINE TOOL JOINTS

T. D. SACHDEVADelhi College of Engineering, Delhi-! 10006, India

a n dC. V. RAMAKRISHNAN

Department of Applied Mechanics, Indian Institute of Technology, New Delhi-110029, India(Received 30 January 1980; in revised form 6 November 1980)

Summary--The effect of interface friction between contact surfaces on the flexibilities ofmachine-tool joints is studied. The joint flexibility is determined by applying unit oad successivelyat one end of the joint while keeping the other end fixed. Only plane problems have been aken up forconsideration. A finite element based numerical procedure is used for solving the problem. Resultsare presented for the care of a hypothetical machine tool joint and dovetail joint for various valuesof friction coefficient. It is observed that the decrease in flexibility is only 15% for values of frict ioncoefficient/z -- 0.2.

c , ~ , c~

#j,#;pj, p;IT,]0,~,0,

NOTATIONelements of the compliance matrix for bodies A and B respectively. Corresponds to displacement atpoint "i" due to a unit load at "j"external force at node "1"rigid body displacement vector for body B ~ r = (avO)vectors of contact loads at point j of bodies A and B respectivelynormal contact loads at joint "j" for a frictionless contact problem for bodies A and B respectivelytransformation m atrix corresponding to n ode " i "vector of displacements at n ode "i" for b odies A and B respectivelyinitial clearance at n ode "i "coefficient of friction

1. INTRODUCTIONJ o i n t s a r e p r o v i d e d i n t h e m a c h i n e t o o l s f o r f u n c t i o n a l p u r p o s e s , f o r t h e e a s e o fm a n u f a c t u r e a n d e a s e o f tr a n s p o r t a t i o n . T h e j o i n t s c a n b e c l a s s if i e d a s f ix e d j o i n t s a n d t h es l id i n g j o i n t s . T h e p e r f o r m a n c e o f t h e m a c h i n e t o o l s is i n f l u e n c e d t o a g r e a t e x t e n t b y t h es t i f f n e s s a n d t h e d a m p i n g p r o p e r t i e s o f t h e s e j o i n t s .

M o s t o f t li e w o r k c a r r ie d o u t s o f a r o n t h e s t if f n e ss o f t h e jo i n t s h a s a s s u m e d t h e c o m -p o n e n t s s u r r o u n d i n g t h e j o i n t s u r f a c e s a s r ig id a n d h e n c e o n l y t h e s u r f a c e c o m p l i a n c eh a s b e e n c o n s i d e r e d f o r d e te r m i n i n g t h e s t i ff n e ss . R e c e n t w o r k b y B a c k e t a l . [1 -3 ] hass h o w n t h a t d e f o r m a t i o n o f t h e j o i n t s i s d u e t o s u r f a c e c o m p l i a n c e a s w e l l a s d u e t od e f o r m a t i o n o f t h e b o d y o f t h e jo i n t. T h e i r w o r k h a s s h o w n t h a t in s o m e j o i n ts , t h e e f f e c t o fs u r f a c e c o m p l i a n c e o n t h e t o t a l d e f o r m a t i o n is o f th e o r d e r o f" 1 0% o n l y . H e n c e f o rc o m p u t i n g t h e s t i ff n e s s o f th e j o i n ts , t h e d e f o r m a t i o n o f t h e b o d y o f t h e j o i n t m u s t b e t a k e ni n t o a c c o u n t . T h i s p r o b l e m c a n b e t r e a t e d a s a n e l a s t i c c o n t a c t p r o b l e m . A s t h i s i s an o n - l i n e a r p r o b l e m , i t e r a t i v e p r o c e d u r e i s t h e o n l y p o s s i b l e m e t h o d f o r s o l u t i o n . A s t h es h a p e o f th e m a c h i n e t o o l j o i n t s is v e r y c o m p l e x , t h e c o n v e n t i o n a l s o l u t i o n o f t w o e la s t i cb o d i e s i n c o n t a c t i s n o t a p p l i c a b l e t o t h i s c a s e . H e n c e r e c o u r s e h a s t o b e m a d e t o t h ep o w e r f u l f i ni te e l e m e n t m e t h o d .

A l t h o u g h a n u m b e r o f r e s e a r c h e r s h a v e a p p l i e d t h e fi ni te e l e m e n t m e t h o d t o t h e e la s t icc o n t a c t p r o b l e m s , t h e t e c h n i q u e s u s e d b y t h e m a r e h i g hl y it e r a ti v e a n d n e e d a l a rg e n u m b e ro f i t e r a t i o n s a n d , t h e r e f o r e , t h e s e a r e e x p e n s i v e t o u s e . A l l t h e s e m e t h o d s n e g l e c t t h es u r f a c e c o m p l i a n c e . T h e o n l y m e t h o d s t o d a t e w h i c h t a k e i n t o a c c o u n t t h e s u r f a c ec o m p l i a nc e a r e t h e m e t h o d s u s e d b y B a c k e t a l . [2 ]. H o w e v e r , B a c k e t a l . h a v e d i s c o v e r e dt h a t i n m o s t o f t h e m a c h i n e t o o l j o in t s , t h e s u r f a c e c o m p l i a n c e i s sm a l l a s c o m p a r e d t o t h ec o m p l i a n c e o f t h e b o d y o f t h e j o in t . T h e s e m e t h o d s a l s o n e e d a l ar g e n u m b e r o f i te r a ti o n s .

149


2/11

150 T.D. SACHDEVAand C. V. RAMAKRISHANAN2. METHOD OF ANALYSIS

The simple technique used by Francavilla and Zienkiewicz[4] for frictionless elastic contact withdisplacement boundary condit ions has been extended by the authors [5] to handle frictionless contact with forceboundary conditions and further generalized to handle situations where the effect of interface friction is to beincluded [6]. A brief description of these two methods is presented here. Further details of the methods can befound in the references cited above. It is to be noted that proportionate loading has been assumed. For arbitraryloading histories incremental loading procedure has to be used.Two elastic bodies A and B are shown in Fig. 1, before they come in contact. The body A is acted upon by aforce vector [ while the body B has fixed boundary conditions. The necessary rigid body movement of thesupports of body B to bring about the contact and result in the force vector [ at the boundary nodes can beexpressed by the vector G such that

= . ( l )

The displacements of the points i B in the direction normal to the boundary in contact are given bymu , ~ = ~ . c , ~ p ; + [ : r , l d ( 2 )

where "m" is the number of nodes in contact, plis the contact force normal to the boundaryof body B, C~ are theflexibility coefficients of body B, obtained after static condensation and inversion and [T,] is a kinematictransformation matrix.For the nodes in contact, the compatibility of displacements of the two bodies in the direction normal to thecontact surface gives the equation.m k( c ~ , c , ~ ) p , - [ r , ] d = - ~ c ~ . . . , f , + t ~ ; ~3 )

where "m" is the number of nodes in contact, Pi is the contact force normal to the boundary of body A, k is thenumber of nodes with external forces, [t is the Ith element of the vector of external forces, C~ are the flexibilitycoefficients of the body A and U/ is the vector of clearances in the normal direction between the correspondingnodes of the two bodies. In equation (3) it is obvious that the node numbers 1,2 .. . m are contact nodes while(m + 1), (m + 2 ). . . (m + k) correspond to the nodes with external loads. In the analysisprocedure, however, thenode numberingcan be arbitrary but a one to one correspondence exists between these (m + k) nodes and thecorresponding nodal nicknames. The compliance matrices [C~] and [C~] are obtained through an appropriatestatic condensationprocedure described in Ref. [5]. The transformationmatrix [Tl] is given n Appendix 1. As theflexibility coefficient of the body A are obtaiued after assuming wo imaginary support points, he additional ~hreeequations necessary for solving equation (3) and releasing the imaginary fixity of body A are the three equililJriumequations of body A.The above technique for the frictionless contact has been generalized to handle contact problems withfriction. In this case the compatibility of displacements is applied for both normal and tangential directions forthose nodes which adhere to the corresponding nodes. For the nodes which slip, compatibility of displacements isapplied for the normal direction and slip condition s applied in the tangential direction. The following equation isobtained for the nodes which adhere to each other.

m - _ _, E ., ( c ~ + c ~ ) : , - [ r , ] a = - ~ . , C ~ . + d k + u ( 4 )

T

FIG. 1. Bodies A and B before contact.


3/11

T h e e f f e c t o f in t e r f a c e f r i c t io n 15 1f o r i = 1 , 2 , 3 . . . . m w h e r e " m " i s th e n u mb e r o f n o d e s in c o n ta c t a n d e i s th e n u mb e r o f n o d e s w i th e x te r n a lf o rc e s l C ~ ] a n d [ C ~ ] a r e 2 x 2 s u b m a t r i c e so f f l e x i b il i t y c o e ff i c i e n t s, p ~ i s t h e v e c to r o f c o n t a c tf o r c e s a tt h e n o d e j ,

' - " " -- -- o in the tangent ia l and norm al d~rec t~on ,. fk is the ve c to r o f ex te r na l fo rc es a t the k th loaded node an d U~ ~s the ve c to ro f c l e a r a n c e s a t t h e n o d e i .F o r th e n o d e s w h ic h s l ip in th e t a n g e n t ia l d i r e c t io n , th e e q u a t io n s c o r r e s p o n d in g to t a n g e n t i a l d i r e c t io n ine q u a t io n ( 4 ) a r e r e p la c e d b y

p / = ~ p~ " (5 )w h e re p / a n d pj" a r e th e t a n g e n t i a l a n d th e n o r ma l f o r c e s r e s p e c t iv e ly a t t h e j th c o n ta c t n o d e a n d ~ t i s t h ec o e f f i c i e n t o f f r i c t io n a t t h e in t e r f a c e .I n p r a c t i c e , t h e c o n ta c t r e g io n i s n o t k n o w n a p r i o r i a n d s o m e n o d e s m a y b e s l ip p in g wh i l e th e o th e r s m a y n o t .He n c e f o r s u c h p r o b le ms i t e r a t iv e p r o c e d u r e i s n e c e s s a r y a n d c h e c k s a r e n e c e s s a r y to id e n t i f y th e n o d e s lo s in gc o n ta c t a n d th o s e wh e r e in s l ip o c c u r s .A b r i e f d e s c r ip t io n o f th e a lg o r i th m i s a s f o l lo ws f u r th e r d e ta i l s w i th i l l u s tr a t iv e e x a mp le s c a n b e f o u n d in Re f .[6]. 1 . A s e t o f n o d a l p o in t s a r e a s s u me d to b e in c o n ta c t .2 . E q u a t io n ( 4) a lo n g wi th th e e q u i l ib r iu m e q u a t io n s a r e s o lv e d f o r th e n o r ma l a n d t a n g e n t i a l c o m p o n e n t s o fn o d a l lo a d s .3 . T h e r a t io o f n o d a l f o r c e in th e t a n g e n t i a l d i r e c t io n to th a t in th e n o r ma l d i r e c t io n i s c a lc u la te d a t a l l t h en o d e s . I f t h i s r a t io i s g r e a te r th a n th e c o e f f i c i e n t o f f r i c t io n ( ~ ) b e tw e e n th e two s u r f a c e s a t a n y n o d e , s l ip p in g o fn o d e s i s imp l i e d .4 . T h e p o r t io n o f e q u a t io n ( 4 ) c o r r e s p o n d in g to th e t a n g e n t i a l d i r e c t io n is r e p la c e d b y e q u a t io n ( 5 ) a n d a n e ws o lu t io n o b ta in e d .5 . S tep s -4are repea tedu nt i la nrm af rces in thecn tac tz necom eut tbeps i t ive nyand thera t i o f t a n g e n t i a l t o n o r ma l f o r c e f o r a l l t h e n o d e s in c o n ta c t i s e i th e r e q u a l to o r l e s s th a n /~ .T o s tu d y th e e f f e c t o f f u n c t io n a t t h e in t e r f a c e o n th e in p la n e f l e x ih i li t i e s o f ma c h in e to o l jo in t s , t h e t e c h n iq u e ss t a t e d a b o v e h a v e b e e n u s e d . F o l lo w in g a s s u mp t io n s h a v e b e e n ma d e in th e s tu d y :( i) T h e e f f e c t o f s u r f a c e c o m p l ia n c e h a s b e e n n e g le c te d .( i i ) Me ta l to me ta l c o n ta c t i s a s s u me d .( i i i ) Contac t sur faces a re assumed to be f la t .( iv ) T h e e f f e c t o f th e p r e s e n c e o f c l e a r a n c e s i s n e g le c te d .( v ) T h e e f f e c t o f t i g h te n in g o f th e jo in t o n th e jo in t f l e x ih i li t i e s i s n e g le c te d .T h r o u g h o u t th e p r e s e n t s tu d y , e ig h t n o d e d q u a d r a t i c i s o p a r a m e t r i c e l e me n t s h a v e b e e n u s e d a n d a s t a t e o fp la n e s t r a in h a s b e e n a s s u m e d . A 2 x 2 Ga u s s ia n in t e g r a t io n i s u s e d . T h e s o lu t io n o f e q u a t io n s i s a c h ie v e d u s in gG a u s s i a n e l i m i n a t i on p r o c e d u r e w i t h F r o n t a l H o u s e k e e p i n g a l g o r it h m .T h e f o l l o w i n g j o i n t s h a v e b e e n t a k e n u p f o r t h e s t u d y :( i ) Hy p o th e t i c a l jo in t w i th c o n ta c t s u r f a c e s n o r ma l to th e a x i s o f th e jo in t .( i i ) Do v e - ta i l j o in t .I n e a c h o f th e tw o jo in t s , t h e e f f e c t o f f r i c t io n o n th e a x ia l a s we l l a s r o ta t io n a l f l e x ib i l it i e s a n d th e e f f e c t o f lo a df o r a p a r t i c u la r v a lu e o f f ri c t io n a l c o e f f i c i e n t h a s b e e n s tu d ie d . T h e mo d u lu s o f e l a s t i c i ty o f th e ma te r i a l i s ta k e n a s0 .9 3 1 108 k P a . I n e a c h c a s e th e lo a d h a s b e e n a p p l i e d th r o u g h a l a y e r o f s t if f e l e m e n t s o u t s id e th e p o r t io n o f th ejo in t u n d e r c o n s id e r a t io n . T h e s e e l e me n t s a r e a s s u me d to h a v e mo d u lu s o f e l a s ti c i ty 1 0 0 t ime s th o s e o f th en o r m a l e l e m e n t s .3. H Y P O T H E T I C A L J O I N T W I T H C O N T A C T S U R F A C E S N O R M A L T O T H E A X IS O F T H E J O I N T3.1 A x i a l f l e x i b i l i tyT h e h y p o th e t i c a l s l id in g jo in t s h o wn in F ig . 2( a) i s c o n s id e r e d f o r s tu d y in g th e in p la n e f l e x ib i li ty in th e a x ia ld i r e c t io n . T h e f in i t e e l e m e n t me s h i s a l s o s h o wn in F ig . 2( a) . T h e p a r t " A " h a s 4 2 e l e m e n t s w i th to t a l 3 22 d e g r e e s o ff r e e d o m ( D F ) wh i l e p a r t " B " h a s 4 6 e l e me n t s w i th 3 9 2 DF . E q u a l lo a d i s a s s u me d to b e a p p l i e d a t 4 n o d e s a ss h o wn . D is p la c e m e n t o f th e f a c e C D i s c o mp u te d f o r a to t a l l o ad o f 9 8 0 N w h i l e th e c o e f f i c i e n t o f f r i c t io n a t t h ein te r f a c e o f p a r t " A " a n d p a r t " B " i s v a r i e d f r o m 0 . 0 to 0 .4 . Co m p u ta t io n s a r e a l s o c a r r i e d o u t b y k e e p in g th ec o e f f i c i en t o f f r i c t io n # = 0 .2 a n d v a r y in g th e ma g n i tu d e o f th e lo a d . T h e d e f o r m e d s h a p e o f th e jo in t a f t e ra p p l i c a t io n o f lo a d i s a ls o s h o w n in F ig . 2( a) to a n e n la r g e d s c a le f o r # = 0 . 2 a n d to ta l l o a d o f 98 0 N . B o th f o r th ef r i c t io n le s s c o n ta c t a n d c o n ta c t w i th f r i c t io n , a f t e r a p p l i c a t io n o f th e lo a d , o n ly 1 2 n o d e s r e ma in in c o n ta c ta l th o u g h 1 4 n o d e s a r e a s s u m e d in c o n ta c t to s t a r t i t e r a t io n . T o s o lv e f o r f r ic t io n le s s c o n ta c t , o n ly 2 i te r a t io n s a r en e e d e d . Ho we v e r , wh e n f r i c t io n a t t h e in t e r f a c e i s c o n s id e r e d , 3 i t e r a t io n s a r e n e c e s s a r y in a l l t h e c a s e s toc o n v e r g e to th e f in a l s o lu t io n . A l l t h e n o d e s in th e c o n ta c t r e g io n a r e f o u n d to b e s l ip p in g in th e t a n g e n t i a l d i r e c t io nr e la t iv e to th e c o r r e s p o n d in g n o d e s o f th e o th e r p a r t .T h e v a r i a t io n o f a x ia l d i s p la c e me n t w i th lo a d f o r # = 0 . 2 is s h o wn in F ig . 2 (b ) wh i l e th e p e r c e n ta g e c h a n g e inf lex ib i l i ty comp ared to the f r ic t ion l ess c ont ac t fo r d if fe ren t v a lue s of p~ is show n in F ig . 2 (c) . In F ig . 2 (c ) 80 is thea x ia l d i s p la c e m e n t f o r th e f r i c t io n le s s c o n ta c t wh i l e 8 , i s t h e d i s p la c e m e n t f o r th e c o n ta c t w i th f r i c t io n . As th eF ig . 2( b ) s h o ws , d i s p la c e me n t i s d i r e c t ly p r o p o r t io n a l to th e lo a d , i . e . t h e a x ia l f l e x ib i li ty s c o n s ta n t f o r a p a r t i c u la rv a lu e o f in t e r f a c e f r i c t io n . F r o m F ig . 2 (c ), i t i s e v id e n t th a t th e d e v ia t io n in f l e x ib il i ty g o e s o n in c r e a s in g w i thin c r e a s e in f r i c t io n b u t i t i s n o t a l in e a r v a r i a t io n . T h e c o mp u te r t im e n e e d e d f o r th e s o lu t io n o f th i s p r o b le m v a r i e df rom l l to 17 ra in .3. 2 R o t a t i o n a l f l e x i b i l i t yT h e h y p o th e t i c a l jo in t s h o w n in F ig . 3( a) i s c o n s id e r e d f o r c o m p u t in g th e in p la n e r o ta t io n a l f l e x ib i l ity . T h ef in it e e l e m e n t m e s h i s a ls o s h o w n in F ig . 3 (a ). T h e n u mb e r o f e l e me n t s a n d th e d e g r e e s o f f r e e d o m f o r th i s c a s e isth e s a m e a s f o r th e a x ia l l o a d in g . E q u a l a n d o p p o s i t e lo a d s o f 4 9 0 N a r e a p p l i e d a t two n o d e s a s s h o w n in F ig . 3 (a )a n d th e v a lu e o f / z i s v a r i e d f ro m 0 . 0 to 0 .4 . T h e e f f e c t o f v a r y in g th e lo a d f o r a c o n s ta n t v a lu e o f ~ = 0 ' 1 i s a l s os tu d ie d . T h e d e f o r m e d s h a p e f o r c o n ta c t w i th f r i c t io n f o r # = 0 .1 i s a l s o s h o wn in F ig . 3( a) . F o r f r i c t io n le s sMS V o l . 2 3 . N o . 3 - - C


4/11

1 52 T . D . S A C H D E V A a n d C . V . R A M A K R I S H A N A N

c o n t a c t a f t e r a p p l i c a t i o n o f t h e l o a d , 9 n o d e s a r e f o u n d t o b e i n c o n t a c t w h i l e f o r c o n t a c t w i t h f r i c t io n , 7 n o d e sr e m a i n i n c o n t a c t . A t t h e b e g i n n i n g , 1 2 n o d e s a r e a s s u m e d t o b e i n c o n t a c t i n b o t h t h e c a s e s . S o l u t i o n f o rf r i c t i o n l e s s c o n t a c t n e e d s 2 i t e r a t i o n s w h i l e 4 i t e r a t i o n s a r e n e c e s s a r y f o r s o l u t i o n w i t h f r i c t i o n .T h e v a r i a t i o n o f r o t a t i o n o f f a c e C D w i t h l o a d f o r ~ = O . 1 i s s h o w n i n F i g . 3 (b ) w h i le p e r c e n t a g e d e v i a t i o n i nr o t a t i o n f r o m f r i f io n l e s s c o n t a c t f o r d i f f e r e n t v a l u e s o f ~ i s s h o w n i n F i g . 3( c ). I n F i g . 3 ( c) , 0o s t h e r o t a t i o n f o r t h ef r i c t i o n l e s s c o n t a c t w h i l e # , i s t h e r o t a t i o n w h e n t h e e f f e c t o f f r i c t io n i s t a k e n i n t o a c c o u n t . T h e s e c u r v e s a l s os h o w t h e s a m e c h a r a c t e r i s t i c s a s f o r t h e a x i a l c a se . T h e c o m p u t e r t i m e v a r y i n g f r o m 1 5 t o 1 7 r a i n i s r e q u i r e df o r t h i s c a s e .

o F - - ~ 1

2 & $ N 2 4 5 N 2 & S N 2 4 S N

R I R R R

I I - - I Zm m e m m mS 0 m m

R R

,i

8

r r . ~ l r l r ~ . F i i / i J l 4 1

1 & 0 m m E

jJ , 0 . 2 ~ i i i . , *100 m m

D E F O R M A T IO N C A L E "~i i0 20 &O QO I 1 0 1 0 0 p r o

l " ) h m m h

2 0 0 r a mFIG. 2(a) .

E

o

.8

.7

O~4E~Jv

O

00 i I ]1 2 3L O A D ( N x l O 3 )FIG. 2(b).


5/11

Th e e ffec t o f in te rfac e f r ic t ion 153

3

~ [ o 2 A X I A L L O A D " 9 8 0 N" " j

o l , I I l Io . o o . 1 0 . 2 o . 3 o . ~ .

FIG. 2 . F in i t e e l e m e n t m e sh a n d d e fo rm a t io n o f t h e h y p o th e t i c a l j o in t u n d e r a x i a l l o a d in g(# = 0 .2). (b) . Var ia t ion of d isp lac em ent wi th ax ia l load for the hyp othe t ica l jo in t (/~ = 0 .2). (c ) .E f fe c t o f f r i c t i o n o n a x i a l d e fo rm a t io n fo r t h e h y p o th e t i c a l j o in t (Ax ia l l o ad = 9 8 0 N) .

4 . D O V E - T A I L J O I N T4.1 A x i a l f l e x i b i l i tyTo s tu d y th e e f f e c t o f f r i c t i o n o n a t y p i c a l m a c h in e t o o l j o in t , t h e k n e e -c o lu m n jo in t o f a n a c tu a l m i l l i n gm a c h in e i s c o n s id e re d . Th e j o in t , t h e f i n it e e l e m e n t m e sh a n d t h e l o a d in g fo r s t u d y in g t h e i n p l a n e a x i a l f l e x ib il i tya re sh o w n in F ig . 4 (a ). Th e p a r t " A " h a s 5 0 e l e m e n t s wi th 3 62 D F wh i l e p a r t " B " h a s a l so 5 0 e l e m e n t s wi th 3 9 8 DF.Th e d i sp l a c e m e n t o f t h e f a c e C D i s c o m p u te d fo r a t o t a l l o a d o f 4 90 0 N wh e n th e c o e f f i c ie n t o f f r i c t i o n a t t h ein t e r f a c e v a r i e s f ro m 0 -0 to 0 .4 . Di sp l a c e m e n t s a re a l so c o m p u te d fo r/ ,i . = 0 .2 wh e n th e m a g n i tu d e o f t h e l o a d isv a r i e d f ro m 2 4 50 N to 9 8 0 0 N.Th e d e fo rm e d sh a p e o f t h e r i g h t h a l f o f t h e j o in t is sh o wn in F ig . 4 (a ) fo r / z = 0 .2 a n d ~ = 0 .0 . Af t e r a p p l i c a t i o no f t h e l o a d , o n ly 1 0 n o d e s r e m a in i n c o n t a c t b o th fo r t h e f r i c t i o n l e ss c o n t a c t a n d c o n ta c t w i th f r i c ti o n a lt h o u g h 1 4n o d e s a re a s su m e d in c o n t a c t a t t h e o u t se t . So lu t i o n fo r f r i c t i o n l e ss c o n t a c t n e e d s 3 i te r a t i o n s . Fo u r i t e r a t i o n s a ren e c e ssa ry t o so lv e i f t h e e f f e c t o f f r i c t i o n is i n c lu d e d . Al l t h e n o d e s i n c o n t a c t a r e fo u n d to b e s l i p p in g .Th e v a r i a t i o n o f a x i a l d i sp l a c e m e n t wi th lo a d fo r # = 0 .2 is sh o wn in F ig . ~b ) . T h e p e rc e n t a g e c h a n g e i n a x ia lf l e x ib i l i ty c o m p a re d t o t h e f r i c t i o n l e ss c o n t a c t s fo r d i f f e re n t v a lu e s o f ~ i s sh o wn in F ig. 4(c) . Th e c o m p u te r t im ere q u i re d fo r so lv in g th i s c a se v a r i e d f ro m 1 7 to 2 0 r a in .

4 9 0 N

c2 '4 9 0 N

i-I DlI

~ , , 0 . 1

D E F O R M A T IO N S C A L E0 1 0 2 0 $ 0 4 O p m

,. ,

FIG. 3(a).


6/11

154 T.D. SACHDEVAand C. V. RAMAKP-JSHANAN

3 ,5

3.(]

2.5A

i 1.5

1.0

.5

0

14=0.1

J&0 80 120 100M O M E N T ( N m )

(b).

101-

x~ o:~ M O M E N T - 3

0 1 I I i J0 . 0 0 . 1 0 . 2 0 . 3 0 . 4p.

(c).F IG. 3 (a ) . F in i te e lem ent mesh and de fo rmat ion pa t te rn o f the hypo the t ica l jo in t under mom entload ing (p , = 0 . I ) . (b ) . Va r ia t ion o f roa t ion w i th app l ied mom ent fo r the hypo the t ica l jo in t (/~ = 0 -1 ) .( c ) E f fe c t o f f r i c t i o n o n ro ta t i o n fo r th e h y p o the t ic a l j o in t (M o m e n t = 3 9 .2 Nm ) .


7/11

The effect of interfa ce friction 155

L_ 100 r am _ I/ I / / / / / / / J

, , 100ram :/ i

. ,_ .~ .~ - -=~ . .~ -~ " -7 , / / /~= / / / t ' , , , ' / I _ - - - - - -A / / /~ " " ' ' ' "

,IR R R R R--~ 1 4 5 m m

J: 0 .2 )80N 98 0Np =0p = 0 /_.~ . . . . p = O 2L - 8 0 m m r I

D E F O R M A T I O N S C A LE0 100 200 300 1 . 00 50 0p r o

FIG. 4(a).

EEN2_

- - I r -EEt -EEeN

EE._L

.$

1 2 .2KE

UJII1,1

O

L p - 0 . 2

! J2 & 6 8 10

L O A D ( N x 1 0 3 )FIG. 4.(b).


8/11

1 5 6 T . D . SACHDEVA a n d C . V . RAMAKRISHANAN

30

2~

2C AXIAL LOAOm4000 N

' L

10~

S

0 I I I J0.0 (11 (12 (Z3 0.4(c )

F IG. 4 (a ). F in i t e e l e me n t me s h a n d d e f o r m a t io n p a t t e r n o f k n e e - c o lu m n jo in t u n d e r a x ial l o a d in g( ~ = 0 . 2) . ( b) . Va r i a t io n o f d i s p la c e m e n t w i th a x ia l l o a d f o r th e k n e e - c o lu m n jo in t ( t t = 0 . 2) . ( c ).E f f e c t o f f r i c t io n o n a x ia l d e f o r ma t io n f o r th e k n e e - c o lu mn jo in t ( Ax ia l lo a d = 4 9 0 0 N) .

4. 2 R o t a t i o n a l f l e x ib i l it yT h e f in i t e e l e m e n t m e s h a n d th e lo a d in g f o r s tu d y in g th e in p la n e r o ta t io n a l f l e x ib i l i ty f o r th e s a m e jo in t , i ss h o w n in F ig . 5 (a ). A s e a r li e r , f o r th i s c a s e a l s o c o mp u ta t io n s a r e c a r r i e d o u t f o r o n e v a lu e o f lo a d in g a n d d i f f e r e n tv a lu e s o f ~ a s we l l a s f o r o n e v a lu e o f tz a n d d i f f e r e n t v a lu e s o f lo a d in g . T h e d e f o r m e d s h a p e o f th e jo in t f o rc o n ta c t w i th f r i c t io n i s s h o w n in F ig , 5( a) f o r t z - - 0 . 2 . T h e v a r i a t io n o f r o ta t io n o f f a c e CD wi th lo a d f o r ~ = 0 . 2 iss h o wn in F ig . 5 (b ). T h e e f f e c t o f f r i c t io n o n th e r o ta t io n o f f a c e CD i s s h o wn in F ig . 5 (c ). T h i s c a s e n e e d e d ac o m p u te r t ime v a r y in g f r o m 2 3 to 2 8 r ain.

5 . C O N C L U S I O N SF r o m t h e p r e s e n t s t u d y , t h e f o l lo w i n g c o n c lu s i o n s c a n b e d r a w n :( l ) Fo r a pa r t icu lar s i ze a nd sh a pe o f t he j o in t a nd a par t icu lar t y p e o f l o a d ing , t hec o n t a c t r e g i o n i s n o t m u c h a f f e c t e d b y t h e m a g n it u d e o f t h e l oa d . T h i s i s s o b e c a u s e t h ed e f o r m a t i o n d u e t o b e n d i n g i s m o r e d o m i n a n t a s c o m p a r e d t o t h e lo c a l d e f o r m a t i o n s i n t h ec o n t a c t z o n e .I~, 0.2

/

I \ \ ~ N\ " " [ I i ~ / /I ,

~ 9 0 N / . 9 0 N9 8 0 N

.4 J- - . . . . . . / / / ~ . . .ZL-

0 I [FORMATION SCA LE; i0 50 100 150 200 250 [am

9 e 0 N

FIG. 5(a).


9/11

T h e e f f e c t o f i n t e r f a c e f r i c t i o n 157

15

100xZ9[C~q[CCZo sCC

00 I I I I I I I I1 2 3 ~ $ 6 ? eMOMNT ( N m x 1 0 2 )

5 ( b ) .4 c

3 (

sx 2(:~o

1

L I I I0.0 0.1 0 .2 0.3 0.4/ z

5 ( c ) .F I G . 5 ( a ) . Fini te e lement m esh and de form at ion pa t te rn o f the knee-column jo int under m o m e n tloading ( p = 0 . 2 ). ( b ) . V a r i a t i o n o f rota t ion wi th appl ied moment for the knee column jo int( /~ = 0 - 2 ) . ( c ) . E f f e c t o f f r i c t i o n o n rota tion for the knee-column jo in t ( A p p l i e d m o m e n t = 2 0 0 N m ) .

( i i ) T h e d e f o r m a t i o n o f t h e j o i n t i s d i r e c t l y p r o p o r t i o n a l t o t h e l o a d a p p l i e d , i . e .f l e x i b i l i t y o f t h e j o i n t i s c o n s t a n t .( ii i) T h e f l e x i b i l it y o f t h e j o i n t g o e s o n d e c r e a s i n g a s t h e f r i c t io n a t t h e i n t e r f a c e

i n c r e a s e s b u t t h e v a r i a t i o n i s n o t l i n e a r . T h e d e c r e a s e i n f l e x i b i l i t y u p t o / z = 0 . 2 i s s m a l l , i .e .i t i s o f t h e o r d e r o f 1 5 % .

( iv ) S o l u t i o n f o r f r i c ti o n l e s s c o n t a c t n e e d s l e s s e r n u m b e r o f i te r a t io n s a s c o m p a r e d t os o l u t i o n w i t h f r i c t i o n .


10/11

158 T .D . SACHDEVA and C. V. R A M A K R I S H A N A NA c k n o w l e d g e m e n t s - - T h e a u t h o r s a r e g r a t e f u l t o t h e s t a f f o f t h e D e l h i U n i v e r s i t y C o m p u t e r C e n t r e f o r t h e i r h e lp .T h e f i rs t a u t h o r i s t h a n k f u l t o t h e D e l h i A d m i n i s t r a t i o n f o r s p o n s o r i n g h i m t o I . I .T . D e l h i fo r h i g h e r s t u d i e s . B o t ht h e a u t h o r s a c k n o w l e d g e t h e i r u s e f u l d i s c u s s i o n s w i t h D r . R . N a t a r a j a n .

R E F E R E N C E S1. N . BACK, M. BURCLEKIN and A. C O W L E Y , R e v i e w o f t h e r e s e a r c h o n f i x ed a n d s l id i n g jo i n t s . Proc. 13thIn t . Machine Tool Design and Research Con[. 1972.2. N. B A C K . M. BURDEKIN and A. C O W L E Y , P r e s s u r e d i s t r i b u t i o n a n d d e f o r m a t i o n s o f m a c h i n e d c o m -p o n e n t s i n c o n t a c t . Int. J . Mech. Sci. 15(12), 993-1010 (1973),3 . N . B A C K , M . B U R D EK IN a n d A . C O W LE Y , A n a l y s i s o f m a c h i n e t o o l j o i n t s b y f i n i t e e l e m e n t m e t h o d . Proc.14th In t . Machine Tool Design and Research Con[. 529-537 (1973) .4. A. FRANCAVILLA an d O . C . Z I E N K I E W I C Z , A n o t e o n n u m e r i c a l c o m p u t a t i o n o f e l a s t i c c o n t a c t p r o b l e m s .In t . J . Num. Meth . in Engng 9, 913--924 (1975).5. T. O. S A C H D E V A , C . V. RAMAKRISHNAN and R. NATARAJAN, A f in i te e l em en t m eth od for the e las t i cc o n t a c t p r o b l e m s . P a p e r a c c e p t e d f o r p u b l i c a t i o n , J . En g n g Pro d u c t io n , T ra ns . A S M E6 . Z . D . SACHDEVA an d C. V. R A M A K R I S H N A N , A f i n i t e e l e m e n t s o l u t i o n f o r t h e e l a s t i c c o n t a c t p r o b l e m sw i t h f r i c t i o n . P a p e r c o m m u n i c a t e d t o In t . J . Num. Meth . Engng.7 . T . D . S A C rl DE V A , F i n i t e e l e m e n t a n a l y s i s o f e l a s t i c c o n t a c t p r o b l e m a n d i t s a p p l i c a t i o n f o r t h e s t u d y o f

t h e e f f e c t o f j o i n t f le x i b i l it y o n f r e e v i b r a t i o n c h a r a c t e r i s t i c s o f m a c h i n e t o o l s tr u c t u r e s . P h . D . T h e s i s ,I . I .T . De lh i (1979) .

A P P E N D I X 1Kin e ma t i c t ra n s[o rma t io n ma t r i xL e t p o i n t R i n F i g . 6 b e a p o i n t o n t h e r i g i d s u p p o r t s o f b o d y B . T h e t r a n s f o r m a t i o n m a t r i x [ ~ ] f o r t h en o d e i s f o r d i s p l a c e m e n t s n o r m a l t o t h e b o u n d a r y a t i D i s g i v e n b y

[Ti] = [sin 4~ co s 4, - x cos ~ - y sin ,hi (9)w h e r e d , i s a n g l e m a d e b y t h e t a n g e n t t o t h e b o u n d a r y a t t h e p o i n t i s w i t h t h e g i o b a X a x i s as s h o w n i n F i g.6.

A l s o

Y j. B /TANGENTAT ~" ~ / /

/L

~ X

F IG . 6 . C o m p u t a t i o n o f t r a n s f o r m a t i o n m a t r i x .

X = Xl - - XY = Yi - - Yr

w h e r e x l a n d x , a r e t h e x c o o r d i n a t e s o f t h e p o i n t s i a a n d R r e s p e c t i v e l y a n d y l a n d y , a r e t h e y c o o r d i n a t e so f t h e p o i n t s i B a n d R r e s p e c t i v e l y .A P P E N D I X 2

The finite element techniq ue fo r f lexibili ty coe~fcientsI n t h e f i n it e e l e m e n t m e t h o d , t h e s t i f fn e s s m a t r i c e s o f t h e i n d i v i d u a l e l e m e n t s a r e a s s e m b l e d t o g i v e t h es t i ff n e s s m a t r i x o f t h e s t r u c t u r e . A s t h e s t i f f n e ss p r o p e r t i e s o f a n o d e a r e i n f l u e n c e d o n l y b y t h e e l e m e n t ss u r r o u n d i n g i t, t h e e l i m i n a t i o n is c a r r ie d o u t a t t h e s a m e t i m e a s a s s e m b l y . H e n c e i f t h e s t i f f n e ss m a t r i x o f af e w s p e c if i e d n o d e s i s n e e d e d , a l l t h a t i s n e c e s s a r y i s to a v o i d t h e e l i m i n a t i o n o f t h e s e n o d e s d u r i n g t h ep r o c e s s o f a s s e m b l y a n d s o l u t i o n . T h e f l e x i b il i ty m a t r i x a n d h e n c e t h e f l ex i b il it y c o e l ~ c i e n t s a r e o b t a i n e d b yi n v e r t i n g t h e c o n d e n s e d s t i ff n e s s m a t r i x o b t a i n e d a t t h e e n d o f e l i m i n a t i o n .1 . A d d i t i o n a l i m a g i n a r y e l e m e n t s f o r m e d b y t h e s e t o f n o d e s i n c o n t a c t a n d t h e n o d e s w i t h e x t e r n a lf o r c e s a r e a s s u m e d .2 . A f t e r a p p l y i n g t h e n e c e s s a r y b o u n d a r y c o n d i t i o n s a n d t r a n s f o r m a t i o n s , t h e a s s e m b l y a n d t h ee l i m i n a t i o n l o o p i s c a r r i e d t h r o u g h t h e a c t u a l n u m b e r o f e l e m e n t s o n l y .3 . A t t h e e n d o f a s s e m b l y a n d e l i m i n a t i o n , th e v e c t o r s t o r e s i n t h e w o r k i n g s p a c e i s c o n d e n s e d a n dr e o r d e r e d t o g i v e t h e s t i f f n e s s m a t r i x .


11/11

T h e e f f e c t o f in t e r f a c e f r i c t io n 1 594 . T h e s t i f fn e s s ma t r ix s o o b ta in e d i s in v e r t e d to g iv e th e f l e x ib i l ity ma t r ix .5 . A s f ri c t io n le s s c o n ta c t i s a s s u me d a n d th e d i s p la c e me n t s o f th e p o in t " i " in th e d i r e c t io n o n ly n o r ma lto th e c o n ta c t s u r f a c e a r e n e e d e d , a l t e r n a te r o w s a n d c o lu m n s o f th e fl e x ib i l ity ma t r ix o b ta in e d a b o v e g iv eth e c o e f f i c ie n t s C~ a n d C ~ . Ho we v e r a s th e e x te r n a l f o r c e c a n b e a lo n g th e g lo b a l X a s we l l a s Yd i r e c t io n s , a l t e r n a te r o w s a n d a l l t h e c o lu mn s f r o m th e r e l e v a n t p o r t io n o f th e f le x ib i l i ty ma t r ix g iv e th e

coef f ic ien ts C: .~+t .

Date post:	07-Apr-2018
Category:	Documents
Upload:	eholmes80
View:	233 times
Download:	0 times

1980 Sachdeva Interface Friction

Documents