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A study of friction induced vibration Potter, Allan Freer 1962

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A STUDY OF FRICTION INDUCED VIBRATION by ALLAN FREER POTTER B.Sc. University of Manitoba, 1958 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1962 - i i -In presenting this thesis i n pa r t i a l fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make i t freely available for reference and study, I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering, The University of British Columbia, Vancouver 8, Canada. Date: l a y , 1962. - i i i -ABSTRACT F r i c t i o n a l v i b r a t i o n s , w h i c h occur when two s o l i d b o d i e s are rubbed t o g e t h e r , are a n a l y z e d m a t h e m a t i c a l l y and observed e x p e r i m e n t a l l y . I n the mathemat ica l a n a l y s i s , the n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n o f mot ion d u r i n g the s l i p p e r i o d i s d e r i v e d making use o f the e x p e r i m e n t a l f r i c t i o n -v e l o c i t y c u r v e . A q u a l i t a t i v e g r a p h i c a l s o l u t i o n o f t h i s d i f f e r e n t i a l e q u a t i o n o f motion i s p r e s e n t e d t o i l l u s t r a t e t h e g e n e r a l form and b e h a v i o r o f the m o t i o n . The e x p e r i m e n t a l f r i c t i o n - v e l o c i t y curve i s t h e n l i n e a r i z e d a l l o w i n g t h e d i f f e r e n t i a l e q u a t i o n o f motion t o undergo s t a n d a r d a n a l y t i c a l s o l u t i o n . The e x p e r i m e n t a l i n v e s t i g a t i o n s were c a r r i e d out u s i n g u n l u b r i c a t e d s t e e l s u r f a c e s and s i x d i f f e r e n t s u p p o r t i n g systems. The experiments were c o n f i n e d t o s l i d i n g i n the n e g a t i v e s l o p e r e g i o n o f t h e f r i c t i o n curve f o r the p a r t i c u l a r s u r f a c e s u s e d . The e f f e c t s o f l o a d , s t i f f n e s s and v e l o c i t y o f the t r a n s l a t i n g s u r f a c e are c o n s i d e r e d and t h e r e s u l t s suggest t h a t t h e decay o f t h e v i b r a t i o n s , as the speed o f the moving s u r f a c e i s i n c r e a s e d , corresponds i n form t o the f r i c t i o n - v e l o c i t y curve f o r the s u r f a c e s u s e d . U s i n g the o r i g i n a l a n a l y t i c a l r e l a t i o n s h i p d e s c r i b i n g the shape o f the n e g a t i v e s l o p e r e g i o n o f the f r i c t i o n c u r v e , the t h e o r e t i c a l r e s u l t s are a l t e r e d a c c o r d i n g l y . Good c o r r e l a t i o n i s o b t a i n e d between the a n a l y t i c a l r e s u l t s and the e x p e r i m e n t a l o b s e r v a t i o n s . - iv -CHAPTER 1 1. 2. 3. 2 1. 3 1. 2. 3. ii 1. 5 1. 2. 3. APPENDICES 1. 2. TABLE OF CONTENTS PAGE INTRODUCTION 1 FRICTIONAL VIBRATIONS - A LITERATURE SYNOPSIS ....... 2 SUMMARY 7 THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS ....... 8 DESCRIPTION OF EXPERIMENTAL APPARATUS a) THE VIBRATION APPARATUS 18 b) THE APPARATUS FOR FRICTION MEASUREMENT. 22 INSTRUMENTATION 23 PREPARATION OF THE SLIDING SURFACES 21* EXPERIMENTAL RESULTS a) FRICTION-VELOCITY CURVES FOR THE STEEL SURFACES .. 26 b) FRICTIONAL VIBRATION RESULTS FOR THE STEEL SURFACES 28 DISCUSSION OF RESULTS $d CONCLUSIONS OF THE INVESTIGATION 62 RECOMMENDATIONS 63 DERIVATION OF THE ANALYTICAL RELATIONSHIP DESCRIBING THE SHAPE OF THE FRICTION-VELOCITY CURVE 6U CALIBRATION OF THE APPARATUS AND SUBSIDIARY TECHNIQUES 6$ BIBLIOGRAPHY 67 - V -LIST OF FIGURES FIGURE PAGE 1 DIAGRAMMATIC REPRESENTATION OF THE SLIDING SYSTEM .......... ,8 2 THE FORM OF THE STICK SLIP OSCILLATION ... 9 3 THE VARIATION IN COEFFICIENT OF FRICTION WITH RELATIVE VELOCITY OF SLIDING . 9 k GRAPHICAL SOLUTION FOR THE MOTION IN THE SUBCRITICAL ZONE .. 12 5 GRAPHICAL SOLUTION FOR THE MOTION IN THE SUPERCRITICAL ZONE 13 6 THE LINEARIZED FRICTION-VELOCITY CURVE lU 7 DIAGRAMMATIC SKETCH OF THE APPARATUS 20 8 THE SELF—ALIGNING JOINT 20 9 PHOTOGRAPHS OF THE VIBRATION APPARATUS 21 10 GRAPH OF COEFFICIENT OF FRICTION VERSUS RELATIVE VELOCITY .. 27 11 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 1 29 12 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 2 ............. 30 13 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 3 31 lU TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM k 32 15 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 5 33 16 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 6 ............. 3U 17 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM 1 39 18 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM 2 kO 19 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM 3 Ul 20 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM h k2 21 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM 5 U3 22 GRAPH OF SLIDER DISPLACEMENT VERSUS TABLE VELOCITY-SYSTEM 6 kk 23 GRAPH SHOWING THE RELATIONSHIP BETWEEN THE BREAKAWAY COEF-FICIENT OF FRICTION AND THE TABLE VELOCITY FOR THE 6 SYSTEMS U5 - v i -LIST OF FIGURES ( c o n t ' d ) FIGURE PAGE 2k GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYSTEM IL o00O«0000«e0oeo«0e0O0e00eeoe0O0oo«a«»o0 * *oooe«0O0OO ^4-^  25 GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYSTEM 2 Oa0o»00O0OOOO0OO0000000000000*00<»*0**********O6* •«« 4^-7 26 GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYSTEM 3 OO0O000O9*0O0O0O0*00«O0000*00000OO**000«00O«*O0*00O 27 GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYST EM •OeO00O0OOa000«O0O00*000»000000«000**000«0«*000««00 28 GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYSTEM 5 50 29 GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -SYST EM 6 0e«oo0000»oo00000O0O00000o*oa0000ttft*0000Ooeo000000O 30 GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM 1 oo«0000000tt»O0000OO0OOOOO00000000OO0000000000O000OO 31 GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM 2 53 32 GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM 3 • 5U 33 GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM k 55 3k GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM 5 56 35 GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY -SYSTEM 6 57 - v i i -LIST OF TABLES TABLE PAGE 1 RANGE OF THE CONTROLLED VARIABLES 23 2 NUMERICAL VALUES OF THE PARAMETERS FOR THE SIX EXPERIMENTAL SYSTEMS e«o**o»«e*«a«9«oo*oooo99«««»0Ooe«Q0o«»oo«e«««a6oo6Oo 35^  - v i i i •= LIST OF SYMBOLS f o r c e o f k i n e t i c f r i c t i o n at s l i d i n g speeds g r e a t e r t h a n t L , ( l b s . ) minimum f o r c e o f f r i c t i o n , ( l b s . ) . f o r c e o f s t a t i c f r i c t i o n , ( l b s . ) . d i f f e r e n c e between F _ . and F ^ , ( l b s . ) . s t i f f n e s s o f the e l a s t i c system, ( l b s „ / i n . ) . mass o f the v i b r a t i n g p a r t s , ( l b s . s e c „ 2 / i n . ) . s t r u c t u r a l damping c o e f f i c i e n t o f the e l a s t i c system, ( l b s . s e c . / i n , ) r e s u l t a n t e f f e c t i v e v i s c o u s damping c o e f f i c i e n t , ( l b s . s e c / i n . ) . s l o p e o f t h e n e g a t i v e s l o p e p o r t i o n of the f r i c t i o n - v e l o c i t y curve ( s e e s . / i n . ) . s l o p e of t h e p o s i t i v e s l o p e p o r t i o n o f the f r i c t i o n - v e l o c i t y c u r v e , ( s e e s . / i n . ) , t i m e , ( s e e s . ) . t i m e of the s l i p and s t i c k p e r i o d s r e s p e c t i v e l y , ( s e e s . ) . p e r i o d of t h e m o t i o n , ( s e e s . ) . r e l a t i v e v e l o c i t y of s l i d i n g , ( i n . / s e c ) . t h a t v a l u e o f r e l a t i v e v e l o c i t y o f s l i d i n g t h a t produces a minimum c o e f f i c i e n t o f f r i c t i o n , ( i n . / s e c ) . t a b l e v e l o c i t y , ( i n . / s e c ) . c r i t i c a l t a b l e v e l o c i t y , ( i n . / s e c ) . t o t a l normal l o a d between t h e s u r f a c e s , ( l b s . ) . d i sp lacement , v e l o c i t y and a c c e l e r a t i o n o f the s l i d e r r e s p e c t i v e l y , displacement of the s l i d e r due t o t h e breakaway and s t a t i c c o e f f i c i e n t s o f f r i c t i o n r e s p e c t i v e l y , ( i n . ) , d isplacement o f the s l i d e r due t o the minimum c o e f f i c i e n t o f f r i c t i o n , ( i n . ) . - i x -LIST OF SYMBOLS ( c o n t ' d ) x the minimum displacement o f the s l i d e r , ( i n . ) . 06 ampl i tude o f v i b r a t i o n , ( i n . ) . <\> the d imensionless parameters - . d> t h e d imensionless damping parameter: • ~^ -/*•} J*S c o e f f i c i e n t o f f r i c t i o n , and c o e f f i c i e n t o f s t a t i c f r i c t i o n r e s p e c t i v e l y . -^B 1 FK>/^breakaway, k i n e t i c and minimum c o e f f i c i e n t s o f f r i c t i o n r e s p e c t i v e l y . y ^ x i s the c o e f f i c i e n t o f f r i c t i o n at s l i d i n g speeds greater t h a n ^ M . l) combined damping r a t i o . t), damping r a t i o f o r the e l a s t i c system. l) damping r a t i o f o r the s u r f a c e s . CO n a t u r a l c i r c u l a r frequency of the s u p p o r t i n g system, ( r a d . / s e c . ) 6Jj ' damped n a t u r a l c i r c u l a r f requency o f t h e s u p p o r t i n g system, ( r a d , / s e c . ) . ACKNOWLEDGEMENT The author i s g r a t e f u l f o r the a d v i c e and encouragement g i v e n by h i s r e s e a r c h d i r e c t o r , D r . C. A . B r o c k l e y and f o r h e l p f u l d i s c u s s i o n s w i t h f e l l o w graduate s t u d e n t s . F i n a n c i a l a s s i s t a n c e was r e c e i v e d from the N a t i o n a l Research C o u n c i l of Canada under Grant Number A 1065 and t h e e x p e r i m e n t a l work was c a r r i e d out i n the L u b r i c a t i o n L a b o r a t o r y , Department o f M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Columbia. CHAPTER ONE 1. INTRODUCTION 2. FRICTIONAL VIBRATIONS - A LITERATURE SYNOPSIS 3. SUMMARY - 1 -1. INTRODUCTION I f one o f two s u r f a c e s i n f r i c t i o n a l c o n t a c t i s d r i v e n s l o w l y f o r w a r d w h i l e the o t h e r i s e l a s t i c a l l y suspended t o a f i x e d p o s i t i o n , i t i s f o u n d t h a t s l i d i n g i s not c o n t i n u o u s , but r a t h e r proceeds w i t h a s e r i e s o f " s t i c k s " and " s l i p s " . I n p o s i t i o n - c o n t r o l servomechanisms o p e r a t i n g a t creep speeds, the presence o f these s e l f - e x c i t e d f r i c t i o n a l v i b r a t i o n s , a p a r t f r o m c a u s i n g i n -c r e a s e d wear, des tro ys t h e accuracy and s e n s i t i v i t y o f any f i n a l p o s i t i o n i n g movement. The purpose, t h e r e f o r e , o f t h i s i n v e s t i g a t i o n i s t o s tudy not o n l y the form o f the v i b r a t i o n but the i n f l u e n c e o f v a r i o u s parameters d e f i n i n g the s u p p o r t i n g system, on t h e s t a b i l i t y o f the f r i c t i o n a l v i b r a t i o n s . I t i s hoped t h a t the r e s u l t s , p r e s e n t e d i n the form o f f u n c t i o n a l r e l a t i o n s h i p s , w i l l be o f g e n e r a l v a l u e i n d e s i g n . The i n v e s t i g a t i o n was c a r r i e d out u s i n g c l e a n , u n l u b r i c a t e d f l a t s t e e l s u r f a c e s , the p r e p a r a t i o n o f which i s c a r e f u l l y d e f i n e d . The d r i v e n s u r f a c e was moved w i t h pure t r a n s l a t i o n thus e l i m i n a t i n g any r o t a t i o n a l e f f e c t s . P r o v i s i o n was made i n the e x p e r i m e n t a l apparatus f o r v a r i a t i o n i n l o a d , s t i f f n e s s , and v e l o c i t y o f the d r i v e n s u r f a c e . - 2 -2. FRICTIONAL VIBRATIONS - A LITERATURE SYNOPSIS E a r l y i n v e s t i g a t i o n s on the s l i d i n g o f b o d i e s a t low v e l o c i t i e s under boundary f r i c t i o n r e v e a l e d t h a t the motion may not be a cont inuous p r o c e s s . L o r d R a y l e i g h ( l ) * d i s c u s s e s b r i e f l y the mot ion o f the v i o l i n s t r i n g under t h e a c t i o n o f the bow, but o f f e r s no d e t a i l e d development o f the motion s i n c e "some o f the d e t a i l s were s t i l l o b s c u r e " . I n 1929, W e l l s (2), i n an e f f o r t t o measure k i n e t i c boundary f r i c t i o n a t low speeds, d i s c o v e r e d an u n s t a b l e r e g i o n where " a l t e r n a t e s t i c k i n g and s l i p p i n g took p l a c e under c e r t a i n c i r c u m s t a n c e s " . No v e r y s e r i o u s attempt was made a t t h a t t i m e t o e x p l a i n the b e h a v i o r i n t h i s r e g i o n . Thomas (3) s t u d i e d f r i c t i o n a l v i b r a t i o n s employing a n a l y t i c a l and g r a p h i c a l t e c h n i q u e s . The d i f f e r e n t i a l e q u a t i o n o f motion o f t h e v i b r a t i n g s l i d e r was s o l v e d and v a r i o u s cases were s t u d i e d . On the phase p l a n e , Thomas shows the t r a j e c t o r i e s r e p r e s e n t i n g the s o l u t i o n o f the e q u a t i o n o f motion f o r t h e s i m p l e harmonic case w i t h no damping. Under l u b r i c a t e d c o n d i t i o n s a v i s c o u s damping term i s added t o the d i f f e r e n t i a l e q u a t i o n and t h e phase p l a n e r e p r e s e n t a t i o n i s t h e n a l t e r e d a c c o r d i n g l y . Kaidanovsky and H a i k i n (U) s t u d i e d f r i c t i o n a l v i b r a t i o n s i n s l i d i n g systems hav ing f r i c t i o n v a r y i n g w i t h v e l o c i t y , and t h e y observed t h a t a necessary c o n d i t i o n f o r such v i b r a t i o n s i s the e x i s t e n c e o f a r e g i o n i n which the f r i c t i o n decreases as the v e l o c i t y i n c r e a s e s . I n . s u c h c i rcumstances e q u i l i b r i u m i s u n s t a b l e . As a matter o f i n t e r e s t t h e y p o i n t out t h a t the r e l a t i o n between f r i c t i o n and speed f o r hydrodynamic l u b r i c a t i o n possesses a * Numbers i n b r a c k e t s des ignate r e f e r e n c e s w h i c h a r e l i s t e d i n t h e B i b l i o g r a -phy. A Supplementary B i b l i o g r a p h y i s i n c l u d e d f o r completeness. - 3 -narrow r e g i o n where f r i c t i o n decreases w i t h i n c r e a s i n g v e l o c i t y . I n the J narrow r e g i o n r e p r e s e n t i n g boundary l u b r i c a t i o n t h e s l o p e o f the curve i s n e g a t i v e w h i l e i n the hydrodynamic r e g i o n the s l o p e i s p o s i t i v e . They suggest t h a t e q u i l i b r i u m i s u n s t a b l e i n the boundary o r n e g a t i v e s l o p e r e g i o n w h i l e s l i d i n g i n t h e hydrodjmamic r e g i o n i s i n h e r e n t l y s t a b l e . B l o k (5) c o n f i r m e d t h a t f r i c t i o n a l r e l a x a t i o n o s c i l l a t i o n s depend upon the p a r t i c u l a r shape o f the f r i c t i o n - v e l o c i t y c u r v e . His a n a l y t i c a l t reatment i s based on the l i n e a r i z e d f r i c t i o n - v e l o c i t y curve and he extends h i s a n a l y s i s t o show t h a t t h e o s c i l l a t i o n s depend on the amount o f damping i n the s u p p o r t i n g system. He develops a d imens ionless parameter , f - F. and shows t h a t f r i c t i o n a l v i b r a t i o n s w i l l not occur i n a s l i d i n g system i f the damping r a t i o , ~i) , i s l a r g e r t h a n some c r i t i c a l v a l u e depending on t h e magnitude o f ^ . I f t h e damping i n t h e system i s e q u a l t o the c r i t i c a l v a l u e , then he shows t h a t the system hovers between v i b r a t i o n and smooth s l i d i n g . Bowden and Leben (6) c a r r i e d out a s e r i e s o f experiments on the f r i c t i o n between s l i d i n g metals i n the absence o f a l u b r i c a t i n g f i l m . They show t h a t t h e f r i c t i o n a l f o r c e does not remain' constant d u r i n g s l i d i n g and t h a t the process may not be c o n t i n u o u s . S l i d i n g may p r o c e e d w i t h a s e r i e s o f " s t i c k s " and " s l i p s " . They suggest t h a t f r i c t i o n between metals can be a t t r i b u t e d i n p a r t t o l o c a l i z e d c o l d w e l d i n g p r o c e s s e s . S i n c e i s o l a t e d a s p e r i t i e s c a r r y the l o a d , the r e s u l t i s t h a t e x c e s s i v e l o c a l p r e s s u r e s a r e e s t a b l i s h e d thus g i v i n g r i s e t o c o l d w e l d i n g o r adhes ion ( 2 0 ) . Dur ing t h e i r exper iments , s imultaneous measurements were t a k e n o f the s u r f a c e temperature . T h i s r e v e a l e d l a r g e f l u c t u a t i o n s i n temperature , and, a t the i n s t a n t o f s l i p - k -t h e r e was a sudden temperature " f l a s h " . Thus i n a d d i t i o n t o the c o l d w e l d i n g process t h e y suggest t h a t hot w e l d i n g processes a l s o e x i s t w h i c h c o n t r i b u t e t o the s u r f a c e damage. F u r t h e r q u a n t i t a t i v e i n v e s t i g a t i o n s as t o the peak temperatures a c h i e v e d d u r i n g s l i p were c a r r i e d out by Morgan, Muskat and Reed (7). Dudley and S w i f t (8) employed L i e n a r d ' s method o f g r a p h i c a l c o n -s t r u c t i o n o f i n t e g r a l curves i n t h e i r a n a l y s i s o f f r i c t i o n a l v i b r a t i o n s . The advantage i n t h i s g r a p h i c a l approach i s t h a t d i r e c t use i s made o f the e x p e r i m e n t a l f r i c t i o n - v e l o c i t y c u r v e . I t i s not necessary t o express t h i s curve a n a l y t i c a l l y or t o l i n e a r i z e i t . The g r a p h i c a l c o n s t r u c t i o n i s a p p l i e d t o v a r i o u s shapes o f e x p e r i m e n t a l f r i c t i o n curves and the r e s u l t s show t h a t under c e r t a i n c o n d i t i o n s o s c i l l a t i o n s b u i l d up o r p e r s i s t w h i l e i n o ther cases they decay. The method serves t o i l l u s t r a t e the s o l u t i o n o f t h e governing d i f f e r e n t i a l e q u a t i o n o f motion p r e v i o u s l y i n t r o d u c e d by B l o k ( £ ) . I n 19U8, B r i s t o w (9) conf i rmed t h a t the e x i s t e n c e o f t h e n e g a t i v e s l o p e r e g i o n was a necessary c o n d i t i o n f o r the e x c i t a t i o n o f f r i c t i o n a l v i b r a t i o n s . H i s o b s e r v a t i o n s i n d i c a t e d t h a t t h e mot ion i s dependent on the r e l a t i v e magnitude o f the damping c o e f f i c i e n t o f t h e s u p p o r t i n g system. D u r i n g the " s t i c k " p e r i o d , s l i g h t r e l a t i v e movement between the two s u r f a c e s was n o t e d w h i c h r e s u l t e d i n a decrease i n t h e . a m p l i t u d e o f v i b r a t i o n as the v e l o c i t y o f the moving s u r f a c e i n c r e a s e d . However, no e x p e r i m e n t a l r e l a -t i o n s h i p was p r e s e n t e d t o show the form o f t h i s a m p l i t u d e decrease . Rabinowitz (10) found t h a t two s u r f a c e s undergo s l i g h t d isplacement b e f o r e the s t a t i c f r i c t i o n c o e f f i c i e n t f a l l s t o some k i n e t i c v a l u e . From h i s exper iments , w h i c h were n o n - r e l a x a t i o n t e s t s , i t was n o t e d t h a t the s t a t i c f r i c t i o n p e r s i s t e d f o r a d isplacement o f the o r d e r o f 1+0 m i c r o i n c h e s . Wide v a r i a t i o n i n t h i s f i g u r e c o u l d e x i s t depending upon the g e n e r a l s u r -face c o n d i t i o n s . As p r e v i o u s l y suggested by Bowden, when m e t a l s u r f a c e s are i n c o n t a c t , p l a s t i c f l o w occurs and m e t a l l i c j u n c t i o n s are formed. R a b i n o w i t z suggested t h a t p r e l i m i n a r y displacement i s t h e y i e l d o f t h e j u n c t i o n s p r i o r t o t h e i r s h e a r i n g . A j u n c t i o n o f somewhat g r e a t e r s t r e n g t h i s formed a f t e r a l o n g e r p e r i o d o f r e s t , t h e process b e i n g analogous t o creep . T h i s r e s u l t s i n i n c r e a s e d c r o s s e e t i o n a l a r e a o f the j u n c t i o n s w h i c h t h e n r e q u i r e l a r g e r s h e a r i n g f o r c e s . Thus i t appears t h a t t h e c o e f f i c i e n t o f s t a t i c f r i c t i o n becomes a f u n c t i o n o f the " t ime o f r e s t " . A c c o r d i n g l y i n 1953, B u r w e l l and Rabinowitz ( l l ) s t u d i e d t h e i n -f l u e n c e o f t h i s " t i m e o f r e s t " on the s t r e n g t h o f j u n c t i o n s . I t was c o n -c l u d e d from the experiments t h a t a f i n i t e t ime i s r e q u i r e d f o r a j u n c t i o n t o r e a c h i t s f u l l s t r e n g t h . There are however, some p r i n c i p l e f e a t u r e s o f t h i s s t r e n g t h i n c r e a s e t h a t are s t i l l obscure . The e f f e c t s o f f r i c t i o n a l v i b r a t i o n s i n servomechanisms i s d i s c u s s e d by H. L a u e r . ( 1 2 ) . He d e s c r i b e s the d i f f i c u l t i e s i n the o p e r a t i o n o f a s i m p l e p o s i t i o n i n g servo a r i s i n g from n o n - l i n e a r f r i c t i o n e f f e c t s . The n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n d e s c r i b i n g these e f f e c t s i s s t u d i e d by means o f g r a p h i c a l phase p l a n e d i s p l a c e m e n t s . D e r j a g u i n , Push and T o l s t o i (13) have p r e s e n t e d a v e r y d e t a i l e d t h e o r e t i c a l a n a l y s i s based on B l o k ' s e a r l i e r work. L i n e a r i z a t i o n o f t h e e x p e r i m e n t a l f r i c t i o n - v e l o c i t y curve was necessary i n order t o s o l v e the d i f f e r e n t i a l e q u a t i o n o f mot ion d u r i n g s l i p . However, they f a i l t o adequate ly d e f i n e the l i n e a r i z e d f r i c t i o n - v e l o c i t y curve upon w h i c h t h e i r a n a l y s i s i s based. Boundary c o n d i t i o n s , r e p r e s e n t i n g the r e g i o n o f systems f o r w h i c h - 6 -f r i c t i o n a l v i b r a t i o n s are i m p o s s i b l e , are a p p l i e d t o t h e equat ions f o r v e l o c i t y and a c c e l e r a t i o n . T h i s r e s u l t s i n a parameter <j) , w h i c h i s s i m i l a r i n form t o t h a t developed by B l o k . i A F where A F r e p r e s e n t s the d i f f e r e n c e between the s t a t i c and k i n e t i c f o r c e s o f f r i c t i o n . I n a p p l y i n g the boundary c o n d i t i o n s t h e y do not d e f i n e whether o r not the t a b l e v e l o c i t y i s i n the p o s i t i v e o r n e g a t i v e s l o p e r e g i o n o f t h e f r i c t i o n - v e l o c i t y c u r v e . I n h i g h p r e c i s i o n machines o p e r a t i n g under s e r v o - c o n t r o l , i t i s n e c e s s a r y t o m i n i m i z e the v a l u e o f c r i t i c a l v e l o c i t y i n o r d e r t o p r o v i d e s e n s i t i v e r e s p o n s e . F o l l o w i n g the t h e o r e t i c a l a n a l y s i s o f D e r j a g i u n , Push and T o l s t o i (13), S i n g h ( l i t ) , (18) and (19) p r e s e n t s an e x p e r i m e n t a l s tudy o f c r i t i c a l v e l o c i t y . E f f o r t i s made t o p r e s e n t t h e necessary c o n d i t i o n s t h a t must e x i s t i n the s u p p o r t i n g system w h i c h w i l l produce a minimum c r i t i c a l v e l o c i t y . C o n c e n t r a t i n g on s t r u c t u r a l and s u r f a c e damping r a t i o s , r e s u l t s are p r e s e n t e d i n g r a p h i c a l form w h i c h s e r v e t o p r e d i c t the magnitude o f c r i t i c a l v e l o c i t i e s f o r a wide range o f damping. S i n g h concludes t h a t t o suppress o r e l i m i n a t e f r i c t i o n a l v i b r a t i o n s i t i s necessary t o : 1. Reduce t h e d i f f e r e n c e between t h e s t a t i c and k i n e t i c f o r c e s o f f r i c t i o n . 2. I n c r e a s e the s t i f f n e s s - i n e r t i a r a t i o . 3. I n c r e a s e the damping i n the" system. - 7 -3. SUMMARY To.-, make f u l l use o f t h e e x i s t i n g i n f o r m a t i o n on the mechanics o f the phenomenon and t o serve as a s t a r t i n g p o i n t f o r the p r e s e n t i n v e s t i g a t i o n , a c o n c i s e summary w i l l be i n c l u d e d a t t h i s p o i n t . 1. The n e g a t i v e s l o p e r e g i o n o f the f r i c t i o n - v e l o c i t y curve i s a n e c e s s a r y c o n d i t i o n f o r the e x c i t a t i o n o f f r i c t i o n a l v i b r a t i o n s i n s l i d i n g systems. 2. The v i b r a t i o n depends on t h e magnitude o f t h e damping i n the system. 3. S l i g h t r e l a t i v e a d j u s t i n g movement between the s u r f a c e s d u r i n g the " s t i c k " p e r i o d has been observed. U. A p r e l i m i n a r y displacement immediate ly p r i o r t o s l i p has been n o t i c e d by p r e v i o u s i n v e s t i g a t o r s and i t has been suggested t h a t t h i s movement r e p r e s e n t s the y i e l d of the j u n c t i o n s i n s h e a r . 5>. To decrease the c r i t i c a l v e l o c i t y f o r any g i v e n system i t i s n e c e s s a r y t o : a) decrease the d i f f e r e n c e between the s t a t i c and k i n e t i c c o e f f i -c i e n t s o f f r i c t i o n . b) r a i s e the s t i f f n e s s o f the s u p p o r t i n g system. c) i n c r e a s e t h e damping i n t h e system. CHAPTER TWO THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS - 8 -1. THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS Cons ider the g e n e r a l case o f a s l i d e r o f mass "m" r e s t r a i n e d by an e l a s t i c system o f e l a s t i c i t y "K" and i n c o n t a c t w i t h a p lane s u r f a c e w h i c h i s d r i v e n w i t h a constant v e l o c i t y " V " . Such a system i s shown d i a g r a m m a t i c a l l y i n F i g . 1, i n w h i c h " r " r e p r e s e n t s the c o e f f i c i e n t o f v i s c o u s s t r u c t u r a l damping o f the s u p p o r t i n g system. F i g . 1 Diagrammatic r e p r e s e n t a t i o n o f the s l i d i n g system . The displacement "x" o f the s l i d e r i s measured r e l a t i v e t o the p o s i t i o n o f the u n s t r a i n e d s p r i n g . I f no r e l a t i v e mot ion e x i s t s between the s l i d e r and the lower s u r f a c e , t h e n the e q u a t i o n d e s c r i b i n g the mot ion of t h e s l i d e r may be w r i t t e n ass A V + K<* = -AW D u r i n g t h e " s t i c k " p e r i o d e q u a t i o n 1 governs, and the displacement o f t h e s l i d e r i s r e p r e s e n t e d by the p o r t i o n o f the curve l a b e l l e d AB i n F i g . 2. Up t o p o i n t B, the f o r c e o f s t a t i c f r i c t i o n i s capable o f w i t h s t a n d i n g the sum o f the s p r i n g and damping f o r c e s . S i n c e the f o r c e o f f r i c t i o n cannot exceed the s t a t i c v a l u e , d isplacement o f the e l a s t i c system beyond p o i n t B r e s u l t s i n r e l a t i v e mot ion between t h e s l i d e r and the moving s u r f a c e . - 9 -^ 3 M Of, M I N B f APERIODIC A [ C V H E A V y DAMPING-X F i g . 2 The form o f the " s t i c k s l i p " o s c i l l a t i o n . The s l i p p e r i o d t h e n occurs d u r i n g w h i c h the s l i d e r moves r a p i d l y from B t o C. F o r many s u r f a c e s , the f r i c t i o n - v e l o c i t y r e l a t i o n s h i p takes the g e n e r a l form i l l u s t r a t e d i n F i g . 3 . I f r e l a t i v e mot ion e x i s t s between t h e s u r f a c e s t h e e q u a t i o n d e s c r i b i n g t h e mot ion o f the s l i d e r can be w r i t t e n as: where J*- i s the c o e f f i c i e n t o f f r i c t i o n between the s u r f a c e s , and v a r i e s as i l l u s t r a t e d i n F i g . 3 . ' Z 4 M S/.OPE = -^F <U-\l-rp F i g . 3 The v a r i a t i o n i n c o e f f i c i e n t o f f r i c t i o n w i t h r e l a t i v e v e l o c i t y o f s l i d i n g . F o r the case o f l i g h t damping, c o m p a r a t i v e l y l a r g e s l i p v e l o c i t i e s are encountered as suggested by the s l o p e o f the curve BC i n F i g . 2. The r e a s o n f o r these h i g h v e l o c i t i e s i n the underdamped case i s apparent from - 10 Fig. 3 since the coefficient,of f r i c t i o n achieves a low kinetic value very rapidly. This allows the term "Kx" i n equation 2 to govern, returning the slider to point C i n Fig. 2. At point G, the velocity of the slider becomes equal to the velocity of the moving surface and a "stick" period commences. The cycle ABC i s repeated continuously, the motion being referred to as "st i c k - s l i p " sliding. otherwise the displaced slider w i l l d r i f t asymptotically from point A of Fig.2 back to a displacement corresponding to the velocity of the moving surface. Under these conditions further relaxation oscillations are im-possible unless the velocity of the moving surface drops to zero momentarily. This would result i n another transient followed by smooth sliding. s l i p (equation 2) is a second order non-linear equation, the motion can be represented graphically on a phase plane diagram which consists of a plot of velocity versus position. The general zero slope isocline method w i l l be used to establish the phase plane plot (1$). Scale restrictions do not permit the application of the method to accurate analysis of normal relaxa-tion-type oscillations but qualitative use of the phase plane diagram does serve to i l l u s t r a t e the general behavior. It should be mentioned that the damping is less than aperiodic, Since the differential equation of motion of the slider during Rewriting the differential equation 2 i n the form: and letting x » y we have: - 11 -F o r the zero s l o p e i s o c l i n e we se t g i v i n g ; ^ - nr" " nr 3 S e l e c t i n g v a r i o u s v a l u e s o f k = y and o b t a i n i n g the c o r r e s p o n d i n g v a l u e s o f from the e x p e r i m e n t a l f r i c t i o n curve s i m i l a r t o t h a t shown i n F i g . 3, and s u b s t i t u t i n g these v a l u e s i n t o e q u a t i o n 3 r e s u l t s i n the zero s l o p e i s o c l i n e curve p l o t t e d on the phase p lane i n F i g . i+. U s i n g M e n a r d ' s g r a p h i c a l c o n -s t r u c t i o n t h e s l o p e s o f the t r a j e c t o r i e s o f the d i f f e r e n t i a l e q u a t i o n can be drawn on the phase p l a n e . The complete s l i p t r a j e c t o r y f o r the i n i t i a l c o n d i t i o n r e p r e s e n t e d by p o i n t B can be drawn as shown by f o l l o w i n g , i n a c l o c k w i s e d i r e c t i o n , t h e s h o r t s l o p e l i n e s i n the i n t e g r a l f i e l d . F i g . \\ i s drawn f o r a t a b l e v e l o c i t y i n the n e g a t i v e s l o p e r e g i o n o f the f r i c t i o n curve and t h i s r e s u l t s i n a w e l l d e f i n e d " s t i c k - s l i p " o s c i l l a t i o n o f approximate ampl i tude AB. The e f f e c t s o f s t a t i c f r i c t i o n becoming o p e r a t i v e at p o i n t A v i r t u a l l y cut o f f t h e phase p l a n e diagram and a b r u p t l y s tops the s p i r a l l i n g t r a j e c t o r y . F i g . 5 shows a system o p e r a t i n g i n the p o s i t i v e s l o p e r e g i o n o f the f r i c t i o n curve where t h e p o s s i b i l i t y e x i s t s f o r t h e s p i r a l l i n g t r a j e c t o r y t o " m i s s " the v e r t i c a l k = V a x i s a t some p o i n t D. T r a j e c t o r i e s t h a t do not encounter the a x i s k = V are not i n f l u e n c e d by t h e e f f e c t s o f s t a t i c f r i c t i o n consequent ly f u r t h e r s t i c k i n g i s i m p o s s i b l e and these t r a j e c t o r i e s s p i r a l i n t o the s t a b l e p o i n t E. The v a l u e of k a t p o i n t E i s zero so t h a t smooth s l i d i n g r e s u l t s (13). The e f f e c t s o f s t r u c t u r a l damping i n the system i s d e s c r i b e d by t h e term -K - 12 -h-X=Y= V Fig.li Graphical solution for the motion in the subcritical zone. in equation 3 . Increasing the value of "r" produces a family of straight lines having increasing slope through the origin of the phase plane. Adding this component to the first term of equation 3 results in a zero slope isocline possessing a steeper positive slope on the phase plane. This in turn results in a more rapid spiralling in of the whole integral fi e l d to the stable point E. Returning to Fig. 3> the friction-velocity curve consists of two portionsj an exponentially-shaped portion at low*U values, and a linear F i g . 5 G r a p h i c a l s o l u t i o n f o r the motion i n t h e s u p e r c r i t i c a l zone p o r t i o n at higherX( v a l u e s . The f r i c t i o n - v e l o c i t y curve can be expressed q u i t e a c c u r a t e l y by the r e l a t i o n s h i p * ; S i n c e * Refer t o Appendix 1 f o r the d e r i v a t i o n o f t h i s e q u a t i o n . - lU -substitution yields: Substitution of equation 6 into equation 2 produces a non-linear differential equation by virtue of the exponential term. It i s now apparent that the slider i s under large negative damping for a short period immediately after s l i p . Thus each oscillation i s i n i t i a l l y self-excited by the negative slope of the friction-velocity curve. In order to solve the non-homogeneous second order differential equation 2 by analytical methods the coefficients i n equation 2 must be con-stant. This condition i s f u l f i l l e d only when is a single linear function of"!/. Correspondingly Fig. 6 illustrates the linearized friction-velocity curve which w i l l be used i n the following analytical solution. Essentially, relationship 5> becomes: The effect of static f r i c t i o n , , w i l l be added as a boundary condition in the solution. This approach is used by Derjaguin, Push and Tolstoi (13). A Fig. 6 The linearized friction-velocity curve. -15 -Re-writing equation 2 and substituting 7, gives: 8 Expressing the coefficient of x as R and the terra on the right as we have the general form: /vrv rp The solution to this differential equation can be written as: + e i n c o s CAJ^JO t- D S I N OJ$X f i o ip- FK.  * ' | / \ c o s + B  o j ^ * J Differentiation with respect to time yields: (B-4>A)cos out -(A+<I>B)SIN cojt i i and CP = -co, e where 12 /vw /I /VYV •J - A.W. 13 lit 15 16 / v w - 16 -CJ*---£-4> = 17 18 19 I n o r d e r t o e v a l u a t e t h e c o n s t a n t s o f i n t e g r a t i o n , A and B, the f o l l o w i n g boundary c o n d i t i o n s a p p l y ; When t « 0, the s l i d e r i s on t h e verge o f s l i p , r e p r e s e n t e d by p o i n t B i n F i g . 2, p o i n t B i n F i g . k and p o i n t F i n F i g . 5. T h e r e f o r e , from e q u a t i o n 1; a l s o K Equat ions 10, 11 and 12 become; — K ai rp = V e cp = - V o ^ e cos cjjtt - SIN V cos a y * + <f>(^- S'N WJI* 20 21 22 23 21* f = A F = W Q s - . A * ) 25 - 1 7 -S i n c e the f o l l o w i n g e x p e r i m e n t a l i n v e s t i g a t i o n s are l i m i t e d t o v e r y s m a l l v a l u e s o f R and V, powers o f ~jj and terms i n V i n equat ions 2 2 , 2 3 and 2k can be n e g l e c t e d , g i v i n g : K K cos + T^S/M oujtj; 2 6 - ~i)wiic r rp = - < ^ L ^ - p ^ e SIN u>A£ 2 7 cos out - ~T)SJN O J^^T 2 8 R e f e r r i n g t o F i g . 2 the minimum displacement can be w r i t t e n as: rP = W ^ M _ W ( > » - > * M V e Thus the ampl i tude o f v i b r a t i o n becomes: 2 9 K V 3 0 From e q u a t i o n 8 i t w i l l be n o t i c e d t h a t the p o s i t i v e s l o p e , s , o f t h e f r i c t i o n - v e l o c i t y curve c o n t r i b u t e s t o the v i s c o u s damping i n the system. The s t r u c t u r a l v i s c o u s damping c o e f f i c i e n t r i s e f f e c t i v e l y i n c r e a s e d by an amount SpW which can be r e f e r r e d t o as " s u r f a c e damping". The c o r r e -sponding v i s c o u s s u r f a c e damping r a t i o ~jj i s g i v e n by e q u a t i o n 1 6 . The magnitude o f the a m p l i t u d e o f v i b r a t i o n i s seen t o depend on the r a t i o W K" and t h e d i f f e r e n c e between the c o e f f i c i e n t s o f s t a t i c and k i n e t i c f r i c t i o n . CHAPTER THREE DESCRIPTION OF EXPERIMENTAL APPARATUS a) THE VIBRATION APPARATUS b) THE APPARATUS FOR FRICTION MEASUREMENT INSTRUMENTATION PREPARATION OF THE SLIDING SURFACES - 18 -l . a ) THE VIBRATION APPARATUS The work o f p r e v i o u s i n v e s t i g a t o r s shows t h a t f r i c t i o n a l v i b r a -t i o n s depend l a r g e l y on the c h a r a c t e r i s t i c s o f the s u p p o r t i n g system as w e l l as the shape o f the f r i c t i o n - v e l o c i t y r e l a t i o n s h i p f o r the s u r f a c e s i n v o l v e d . I t has been shown t h a t the f o l l o w i n g parameters of the s u p p o r t i n g system are i m p o r t a n t ; 1) S t i f f n e s s o f the e l a s t i c system 2) Normal l o a d on the s u r f a c e s 3) S t r u c t u r a l damping i n the s u p p o r t i n g system ii) V e l o c i t y o f the t r a v e r s i n g s u r f a c e A c c o r d i n g l y , the e x p e r i m e n t a l a p p a r a t u s , c o n s t r u c t e d by the a u t h o r , was des igned t o a l l o w convenient v a r i a t i o n i n some o f t h e above parameters . The a p p a r a t u s , i n i t s p r e s e n t form, however, has no p r o v i s i o n f o r v a r i a t i o n i n s t r u c t u r a l damping. W i t h r e s p e c t t o s t i f f n e s s , l o a d , and t r a v e r s i n g v e l o c i t y , the i n f l u e n c e o f these parameters on the r e s u l t i n g form o f the v i b r a t i o n w i l l be shown. F o r s t i l l g r e a t e r v e r s a t i l i t y , i t was d e s i r e d t o keep t h e mass o f the v i b r a t i n g member independent o f t h e normal l o a d on the s u r f a c e s . To a c c o m p l i s h t h i s , i t i s necessary t h a t t h e e l a s t i c system t r a n s m i t the normal l o a d , consequent ly the suspens ion was des igned i n the form o f a c a n t i l e v e r beam, shown i n F i g , 7, w h i c h possesses a h i g h n a t u r a l f requency i n the normal d i r e c t i o n . The s t i f f n e s s o f the c a n t i l e v e r beam i n the t r a n s -v e r s e d i r e c t i o n i s e a s i l y a l t e r e d by a s i m p l e change i n l e n g t h w h i c h can be a c c o m p l i s h e d by a d j u s t i n g the c lamping b l o c k shown i n F i g . 7. The normal l o a d W, i s a p p l i e d by means o f the l o a d i n g system w h i c h p i v o t s the c a n t i l e v e r beam assembly about the a x i s A - A . This a x i s i s s u p p o r t e d on an a d j u s t a b l e - 1 9 -base (not shown) by means o f two b a l l b e a r i n g b l o c k s . To e l i m i n a t e any c u r v a t u r e e f f e c t s caused by r o t a t i n g t u r n t a b l e s i t was d e c i d e d t o employ pure t r a n s l a t i o n o f t h e lower s u r f a c e . This was a c c o m p l i s h e d by the use o f a 2 - | i n c h diameter power screw on which a l a r g e t h r e a d e d nut i s gu ided t r a n s v e r s e l y by s l i d i n g ways. The power screw i s d r i v e n by a t h y r a t r o n t u b e - c o n t r o l l e d v a r i a b l e speed motor c o u p l e d t o the power screw by two worm d r i v e r e d u c t i o n gear boxes g i v i n g a combined r a t i o o f 7 0 0 ; 1 . The r e s u l t i n g maximum t r a n s l a t i o n a l speed o f the lower s u r f a c e i s . 0 3 0 i n c h per second. I n f r i c t i o n measurements i t i s e s s e n t i a l t h a t t h e s l i d i n g s u r f a c e s under e x a m i n a t i o n are m a i n t a i n e d i n constant u n i f o r m c o n t a c t at a l l t i m e s throughout the s l i d i n g p r o c e s s . To a c c o m p l i s h t h i s c o n d i t i o n , w i t h a m i n i -mum amount o f p r e c i s i o n machine work, the s e l f - a l i g n i n g j o i n t i l l u s t r a t e d i n F i g . 8 was des igned. I t w i l l be observed f rom F i g . 8 t h a t the upper specimen o r s l i d e r i s f r e e t o r o t a t e i n two p l a n e s about axes t h a t are p e r -p e n d i c u l a r t o each o t h e r . Consequently t h i s s e l f - a l i g n i n g j o i n t a l l o w s f o r s l i g h t a d j u s t i n g movement o f t h e upper specimen t o occur d u r i n g the s l i d i n g p r o c e s s . T h i s i s an important f e a t u r e o f the a p p a r a t u s . The mass, m, o f the v i b r a t i n g p a r t s i s made up o f the t o t a l weight o f t h i s j o i n t p l u s a p r o p o r t i o n o f the beam weight t h a t p a r t i c i p a t e s i n the m o t i o n . The s t r u c t u r a l damping c o e f f i c i e n t , r , o f the c a n t i l e v e r beam, determined by f r e e v i b r a t i o n t e s t s , was f o u n d t o be v i s c o u s i h form but q u i t e s m a l l i n magnitude. A b r i e f d e s c r i p t i o n o f t h e f r e e v i b r a t i o n t e s t s , as w e l l as the g e n e r a l c a l i b r a t i o n procedure f o r the a p p a r a t u s , w i l l be f o u n d i n Appendix 2 . - 20-FIG.7 DIAGRAMMATIC S K E T C H O F T H E A P P A R A T U S FIG. 8 T H E S E L F - A L I G N I N G JOINT - 22 -1. b) THE APPARATUS FOR FRICTION MEASUREMENT As previously mentioned, the investigations were carried out using unlubricated steel surfaces. To obtain the experimental friction-velocity relationship for the steel surfaces, i l l u s t r a t e d qualitatively i n Fig. 3, a subsidiary f r i c t i o n apparatus was used. This was necessary since s l i p velocities i n the neighborhood of 3 inches per second were recorded on several t r i a l tests. This speed i s far beyond the range of the vibration apparatus. To produce such comparatively large speeds, the subsidiary f r i c t i o n apparatus was designed using a horizontally mounted 16 inch diameter revolving steel plate made of the same material as the steel surfaces under study. The rotational speed of the steel plate can be controlled by means of a variable speed electric motor. Curvature effects, which were considered to have negligible effect i n the friction-velocity measurements, were however minimized by conducting the experiments at the largest possible radius on the-steel plate. Since the cantilever beam suspension system i s easily removed from the vibration apparatus, this unit was used to support the upper f r i c t i o n surface. - 23 -2. INSTRUMENTATION D e f l e c t i o n o f the s l i d e r i s measured by means o f a Brush "Type One" l i n e a r displacement e l e c t r o m e c h a n i c a l t r a n s d u c e r c o u p l e d t o a B r u s h Model BL 202 d i r e c t w r i t i n g magnetic o s c i l l o g r a p h so t h a t a permanent r e -c o r d i n g o f s l i d e r movement i s o b t a i n e d . The moving core o f the t r a n s d u c e r i s connected t o the s l i d e r by means o f a s m a l l t e n s i o n s p r i n g . The o s c i l l o -graph i s e l e c t r i c a l l y c o u p l e d t o a B r u s h Model BL 320 a m p l i f i e r w h i c h produces 0 a minimum s e n s i t i v i t y o f 1.2 s c a l e d i v i s i o n s on the o s c i l l o g r a p h c h a r t per . 0 0 1 i n c h s l i d e r d i s p l a c e m e n t . Consequently t h e maximum ampl i tude t h a t can be r e c o r d e d i s ,0U0 i n c h c o r r e s p o n d i n g t o f u l l c h a r t w i d t h . To produce a convenient means o f c e n t e r i n g t h e t r a c e o f the v i b r a -t i o n on the o s c i l l o g r a p h c h a r t , the complete t r a n s d u c e r i s mounted on an a d j u s t a b l e base or c r o s s f e e d . By means o f a handwheel, graduated t o r e a d .0005 i n c h d isp lacement , the t r a n s d u c e r can be p o s i t i o n e d t o " c e n t e r " the t r a c e o f the v i b r a t i o n . From the o s c i l l o g r a p h c h a r t s o f t h e v i b r a t i o n , u s i n g t h e s e n s i t i v i t y as l i s t e d above, s l i d e r d e f l e c t i o n s o f the o r d e r o f .0005 i n c h can be a c c u r a t e l y determined. Thus the accuracy o f the c r o s s f e e d and t h e o s c i l l o g r a p h are matched. Table 1 i n d i c a t e s the range o f the v a r i a b l e s t h a t can be c o n t r o l l e d by adjustments i n the a p p a r a t u s . V a r i a b l e Symbol Range Convenient Increment Load W 0 t o 30 pounds 2 pounds S t i f f n e s s K 12 t o 180 pounds per i n c h 13 pounds per i n c h Table V e l o c i t y V 0 t o . 0 3 i n c h per second .0001 i n c h per second TABLE 1. Range o f t h e c o n t r o l l e d v a r i a b l e s . - 2h -3 . PREPARATION OF THE SLIDING SURFACES C o n s i d e r a b l e care and a t t e n t i o n was devoted t o the p r e p a r a t i o n o f the s l i d i n g s u r f a c e s b e f o r e any f r i c t i o n .or v i b r a t i o n t e s t s were made. The bes t c r i t e r i o n , o f c o u r s e , i s r e p r o d u c i b i l i t y o f r e s u l t s . P r e v i o u s ex-p e r i m e n t e r s have u s e d many "recommended" f i n i s h i n g and c l e a n i n g t e c h n i q u e s but t h e r e seems t o be g e n e r a l agreement i n the l i t e r a t u r e t h a t the bes t procedure i s t o expose a f r e s h s u r f a c e by c u t t i n g o r a b r a s i o n . I n a d d i t i o n , the experiments must be performed as soon as p o s s i b l e a f t e r the f i n i s h i n g and c l e a n i n g p r o c e s s . I n t h i s i n v e s t i g a t i o n no p r o v i s i o n was made i n e i t h e r apparatus f o r removing f r e s h l y accumulated contaminant m a t e r i a l f rom d i r e c t l y i n f r o n t o f the s l i d e r , however the experiments were performed d i r e c t l y a f t e r the s u r f a c e treatment process w i t h a t ime i n t e r v a l o f o n l y a few m i n u t e s . The s u r f a c e s were f i r s t p r e p a r e d from G 1020 ( c o l d r o l l e d ) s t e e l , t h e n de-greased u s i n g t r i - c h l o r o e t h y l e n e . U s i n g grade number 80-G, B e h r -Manning "Tufbak" d u r i t e w a t e r p r o o f emery paper , t h e s u r f a c e s were p o l i s h e d i n the d i r e c t i o n o f s l i d i n g . The emery paper was u s e d d r y , and a f t e r p o l i s h i n g , the s u r f a c e s were t h o r o u g h l y washed a g a i n w i t h t r i - c h l o r o e t h y l e n e . A f t e r one e x p e r i m e n t a l t r a v e r s e o f t h e lower s u r f a c e , b o t h s u r f a c e s were r e f i n i s h e d and c l e a n e d . The s u r f a c e roughness o f the specimens produced by t h i s f i n i s h i n g process was measured by means o f a B r u s h " S u r f i n d i c a t o r " Model BL-110, equipped w i t h a power t r a v e r s e . The range o f the S u r f i n d i c a t o r i s 1 t o 1000 m i c r o i n c h e s and the i n s t r u m e n t i s c a l i b r a t e d t o measure the RMS ( r o o t mean square) roughness h e i g h t i n m i c r o i n c h e s . Measurements were made at c l o s e i n t e r v a l s a l o n g each f r i c t i o n s u r f a c e i n d i r e c t i o n s b o t h p a r a l l e l and - 25 -perpendicular to the direction of sliding. RMS roughness readings are listed in Table 2 using the following symbols: Ijj = average RMS surface roughness value in microinches parallel to the direction of sliding. YjL => average RMS surface roughness value in microinches perpendi-cular to the direction of sliding. CHAPTER FOUR EXPERIMENTAL RESULTS a) FRICTION-VELOCITY CURVES b) FRICTIONAL VIBRATION RESULTS - 26 -1. a) FRICTION-VELOCITY CURVES The f r i c t i o n - v e l o c i t y curves f o r t h e s t e e l specimens a r e shown i n F i g . 10. The c o e f f i c i e n t of s t a t i c f r i c t i o n was f o u n d t o be q u i t e c o n s i s t e n t w i t h the v a l u e The s u r f a c e s were a l l o w e d t o remain a t r e s t f o r a p e r i o d o f about 10 minutes b e f o r e a s t a t i c t e s t was t a k e n . The c o e f f i c i e n t o f k i n e t i c f r i c t i o n was found t o be l o a d dependent as i n d i c a t e d by t h e curves o f F i g . 10. The v a l u e o f the s l o p e , f o r the p o s i t i v e s l o p e r e g i o n o f the f r i c t i o n c u r v e s , i s s m a l l , w i t h the f r i c t i o n r i s i n g more r a p i d l y w i t h the l i g h t e r l o a d . The e x p e r i m e n t a l f r i c t i o n curves c o n s i s t o f an average o f f o u r t e s t s . The mean d e v i a t i o n i n c o e f f i c i e n t o f f r i c t i o n f o r the f o u r t e s t s was - ,0k g i v i n g a r e l a t i v e e r r o r w i t h r e s p e c t t o o f - 20$. From t h e average curves o f F i g . 10, the mean d e v i a t i o n i s - .00? g i v i n g a r e l a t i v e e r r o r w i t h r e s p e c t t o o f - y%. Corresponding loads o f s i x pounds and t e n pounds were u s e d i n the v i b r a t i o n e x p e r i m e n t s . C o n s i d e r a b l e care was t a k e n t o a c c u r a t e l y determine t h e v a l u e o f ' U M , and i n d i c a t i o n s from a r e a s o n a b l e number o f t e s t s p r e d i c t e d t h i s v a l u e t o be o f t h e order o f .06 i n c h e s per second. T h i s i s an average v a l u e and t h e mean d e v i a t i o n from t h i s average was found t o be as much as i $0%. Thus the v a l u e o f w i s q u i t e v a r i a b l e and i n g e n e r a l i t can be s t a t e d t h a t t h e c o e f f i c i e n t o f f r i c t i o n achieves a minimum v a l u e v e r y r a p i d l y , r e s u l t i n g i n a s t e e p n e g a t i v e s l o p e . The l i n e a r equat ions shown i n F i g . 10 s e r v e t o d e s c r i b e the ex-p e r i m e n t a l curves w i t h r e a s o n a b l e a c c u r a c y . The equat ions d e s c r i b i n g t h e p o s i t i v e s l o p e r e g i o n compare i n form w i t h e q u a t i o n 7. SURFACES •* C I 0 2 O COLD ROLLED S T E E L ROUGHNESS: Y// = 8 S-\H. | SO-PAPER ••yf^-lsO-C E M E R Y 2' O h O E k. u. o h 2 LU O u. ll. Ui o o R E L A T I V E VELOCITY *"U " ( INS./SEC.) FIG. 1 0 - GRAPH OF COEFFICIENT OF FRICTION VERSUS RELATIVE VELOCITY. - 28 -1. b) 1 FRICTIONAL VIBRATION RESULTS Table 2 g ives the v a l u e s o f the parameters d e f i n i n g t h e s i x systems w h i c h were u s e d i n t h e e x p e r i m e n t a l i n v e s t i g a t i o n . S u b s i d i a r y measurements u s e d i n the d e t e r m i n a t i o n o f K, W, OJ and r are d e s c r i b e d i n Appendix 2. Corresponding t o these s i x systems, t h e o s c i l l o g r a p h r e c o r d s o f t h e v i b r a t i o n s are shown i n F i g s . 11 t o 16. The v i b r a t i o n s p r o c e e d i n a d i r e c t i o n from l e f t t o r i g h t across each f i g u r e thus making the s l i p process p r o c e e d i n a d i r e c t i o n f rom the bottom t o t h e t o p o f each f i g u r e . Four t e s t s were conducted f o r each system a t the t a b l e v e l o c i t i e s n o t e d . From the r e s u l t i n g 6U o s c i l l o g r a p h t r a c e s , r e a s o n a b l e c o n s i s t e n c y i n the f e a t u r e s o f t h e v i b r a t i o n s was observed. F o r example, the mean d e v i a t i o n i n maximum displacement f o r system 1 a t V « .0015 i n c h per second i s Q% w i t h r e s p e c t t o e q u i l i b r i u m p o s i t i o n w h i l e f o r system 6 a t V = .030 i n c h per second the mean d e v i a t i o n i n maximum displacement i s 13$ w i t h r e s p e c t t o e q u i l i b r i u m p o s i t i o n . D e v i a t i o n s f o r the i n t e r m e d i a t e systems l i e w i t h i n t h i s - r a n g e . To o b t a i n an e s t i m a t e o f the maximum s l i p v e l o c i t y , the maximum o s c i l l o g r a p h c h a r t speed o f 5 i n c h e s per second was u s e d . From these h i g h speed t r a c e s w h i c h a r e mounted on the r i g h t hand s i d e o f F i g s . 11 t o 16, l a r g e r s c a l e p l o t s were made o f the s l i p p e r i o d thus o b t a i n i n g t h e v e l o c i t y v a r i a t i o n . The maximum s l i p v e l o c i t y a c h i e v e d d u r i n g each s l i p p e r i o d i s l i s t e d i n Table 2 . The low c h a r t speed on the o s c i l l o g r a p h was u s e d t o o b t a i n the g e n e r a l form o f the v i b r a t i o n and these low speed r e s u l t s are mounted 1 on the l e f t hand s i d e o f F i g s . 11 t o 16. Each displacement s c a l e Of) TABLE VELOCITY =.0015 IN. /SEC. CHART VELOCITY —.2 IN . /SEC. CHART VELOCITY=5 IN. / SEC. C ) TABLE VELOCITY=.03 IN . /SEC. CHART VEL0CITY= .2 IN. /SEC. w (O F I G . II T Y P I C A L T R A C E S O F T H E O S C I L L A T I O N F O R S Y S T E M I . S C A L E : I D I V I S I O N = .0012 I N C H E S S L I D E R D I S P L A C E M E N T . C) TABLE VELOCITY= .03 IN./SEC. CHART VEL0CITY=.2 IN./SEC. CHART VELOCITY= 5 IN./SEC. FIG. 12 T Y P I C A L T R A C E S OF T H E O S C I L L A T I O N F O R S Y S T E M 2. S C A L E : I DIVISION =.0012 I N C H E S S L I D E R D I S P L A C E M E N T . C)TABLE VEL. = .03 IN./SEC. CHART VEL.= .2 IN./SEC. CHART VEL0CITY=5 IN./SEC. FIG.I3 TYPICAL TRACES OF THE OSCILLATION FOR SYSTEM 3 . SCALE: I DIVISION =.0012 INCHES SLIDER DISPLACEMENT. d ) T A B L E VELOCITY = . 0 0 3 I N . / S E C . C H A R T VELOCITY = . 2 I N . / S E C . C H A R T VELOCITY = 5 I N . / S E C . C) T A B L E V E L . = . 03 I N . / S E C . C H A R T V E L . = .2 I N . / S E C . CHART V E L 0 C I T Y = 5 I N . / S E C . FIG. 14 T Y P I C A L T R A C E S OF T H E OSCILLAT ION FOR S Y S T E M 4 . S C A L E : I DIVISION = .0012 INCHES SLIDER D I S P L A C E M E N T . b) T A B L E VELOCITY = .03 I N . / S E C . CHART VELOCITY = 5 I N . / S E C FIG. 15 TYP ICAL T R A C E S OF T H E OSCILLATION FOR S Y S T E M 5 . ' S C A L E : I DIVISION = .0012 INCHES SLIDER D I S P L A C E M E N T . u Cl) T A B L E VELOCITY = .015 I N . / S E C . C H A R T V E L O C I T Y = . 2 I N . / S E C . CHART VELOCITY = 5 I N . / S E C . b) T A B L E V E L O C I T Y = . 0 3 I N . / S E C . CHART VEL0CITY = . 2 I N . / S E C . CHART V E L O C I T Y = 5 I N . / S E C . FIG. 16 T Y P I C A L T R A C E S OF THE OSCILLATION FOR SYSTEM 6 . S C A L E : I DIVISION = .0012 INCHES SL IDER D I S P L A C E M E N T . System Number K w OJ r m V • 1 M i c r o -Inches 12.8 M i c r o -Inches l b s / i n . l b s . Radians / s e c . l b s - s e c . / i n . l b s - s e c . c J i n . i n . / s e c . i n . / s e c . 7.U 155 10 300 .0U6 .0017 .0U5 .0015 -2.2 .015 -1.0 .030 - .21 2 15.2 8.1 119 10 21*0 .036 .0021 .036 .0015 -2.7 .015 -1.1 .030 - .kO 3 18.6 9.6 85 10 205 .026 .0020 .032 .0015 -3.1 .015 -1.5 .030 - .65 h 12.1 7.2 62.6 6 180 .027 .0019 .039 .0030 -3.8 .015 -1.6 .030 - .81 5 12.9 7.5 U1.U 6 150 .026 .0018 .01*8 .015 -2.01 .030 -1.3 6 18.3 8.3 38.U 6 135 .025 .0021 .014; .015 -2 .5 .030 * i . U 5 TABLE 2 N u m e r i c a l Values o f the Parameters f o r the S i x E x p e r i m e n t a l Systems. - 36 -division on the oscillograph traces corresponds to a slider displacement of 1.2 thousandths of an inch. This sensitivity is noted on Figs. 11 to 16. It wil l be observed that the amplitude of vibration depends on the velocity of the lower surface for each system. Increasing this table velocity for system 1 say, greatly reduces the amplitude of vibration and in general, for the remaining systems, the amplitude decreases with an in-crease in table velocity. In addition, the amplitude decreases as the ratio W. decreases which is readily predicted by the theory (equation 30). From the data provided by the oscillograph charts, Figs.17 to 22 illustrate graphically the variation in amplitude with table velocity for each system. Figs. 2h to 29 illustrate the rate of decay of amplitude of vibration with increasing table velocity for the six systems. Under frictional vibration no true static condition exists during the stick period since a process of continual adjustment between surface asperities takes place, consequently the experimental points giving the maximum displacement, in Figs. 17 to 22, are not proportional to the static coefficient of friction. Hence this apparent "static" coefficient of friction under vibration con-ditions wi l l be referred to henceforth as the "breakaway coefficient of friction", . This term seems appropriate since the slider actually breaks away from the lower traversing surface in such a manner as to produce a relationship with table velocity which is approximately exponential in shape. With this in mind, the experimental points were extrapolated beyond the point of maximum table velocity in an effort to determine where the amplitude of vibration seems to die out. With the aid of Figs. 21; to 29, i t appears that the amplitude dies out at a table velocity corresponding to the value of *U M from the friction-velocity curve. Thus, from these results, the maximum slider displacement seems to correspond to the value - 37 -o f k i n e t i c f r i c t i o n a t the p a r t i c u l a r t a b l e v e l o c i t y . A c c o r d i n g l y , u s i n g t h e a p p r o p r i a t e r a t i o o f F., the upper p o i n t s i n F i g s . 17 t o 22 were c o n -K. v e r t e d t o produce F i g . 23, a p l o t o f v e r s u s t a b l e v e l o c i t y . S i n c e the e x p e r i m e n t a l r e s u l t s suggest t h a t the breakaway c o e f f i c i e n t o f f r i c t i o n corresponds t o the p a r t i c u l a r t a b l e v e l o c i t y , e q u a t i o n k A , w i t h *U r e p l a c e d by V, takes t h e form: v. o (^-^») 31 U s i n g t h e f o l l o w i n g e x p e r i m e n t a l v a l u e s : = ,k9, y^°M a »22 and " L / M = .06 i n c h per second, e q u a t i o n 31 becomes: - 8 o V / B = .a7e + 32 T h i s e q u a t i o n i s shown by the s o l i d curve i n F i g . 23. Thus t h e breakaway c o e f f i c i e n t o f f r i c t i o n v a r i e s i n the same manner as the f r i c t i o n - v e l o c i t y curve f o r t h e s u r f a c e s under e x a m i n a t i o n . U s i n g t h i s e x p e r i m e n t a l l y observed f a c t , equat ions 26, 27, 28, 29 , and 30 can be e a s i l y a l t e r e d i n accordance w i t h t h e r e l e v a n t f r i c t i o n - v e l o c i t y c u r v e , t o p r e d i c t w i t h r e a s o n a b l e accuracy the dynamic performance o f the s l i d e r f o r t a b l e v e l o c i t i e s i n the n e g a t i v e s l o p e r e g i o n . R e p l a c i n g t h e c o n s t a n t w i t h t h e breakaway c o e f f i c i e n t , the above mentioned equat ions become: K M I N . = . A W MAX. |^  -ill? -i 33 3U 35 - 38 -C O S - l) SI N CJ^Jt J 36 37 38 39 The upper s o l i d curves i n F i g s . 17 t o 22 r e p r e s e n t e q u a t i o n 3k w h i l e t h e lower curves r e p r e s e n t e q u a t i o n 35. The s o l i d curves i n F i g s . 2it t o 29 r e p r e s e n t e q u a t i o n 39 and the curves i n F i g s . 30 t o 35 r e p r e s e n t e q u a t i o n 37. - 6C -- on -- m -(D - ZH -z o 5 a L, U. O h z u o u. LL Ul o u > < < LU a: CD .01 .oa. . 0 3 .04- . 0 5 . 0 6 . 0 7 T A B L E V E L O C I T Y ' V " ( | N C H E S / S E C O N D ) . FI&. 2 3 GRAPH SHOWING THE RELATIONSHIP BETWEEN THE BREAKAWAY COEFFICIENT OF FR ICTION AND THE TABLE VELOCITY FOR THE S l X SYSTEMS. •ET-U I - 9n -- en -- 6n -- os --IS -- 25 -- -CHAPTER F I V E 1. DISCUSSION OF RESULTS 2:. CONCLUSIONS 3 . RECOMMENDATIONS - 58 -1. DISCUSSION The e x p e r i m e n t a l r e s u l t s show the r e l a t i o n s h i p between the ampl i tude o f v i b r a t i o n , 06 , and t h e v e l o c i t y o f the lower t r a v e r s i n g s u r f a c e , V. F r i c t i o n a l v i b r a t i o n s were produced u s i n g s i x s u p p o r t i n g systems h a v i n g v a r i o u s r a t i o s o f ^ . The s u p p o r t i n g systems were c h a r -a c t e r i z e d by r e l a t i v e l y constant v i s c o u s or s t r u c t u r a l damping. The magnitude o f the damping c o e f f i c i e n t s was s m a l l consequent ly the i n f l u e n c e o f s t r u c t u r a l damping on the r e s u l t i n g v i b r a t i o n was c o n s i d e r e d n e g l i g i b l e . From the r e s u l t s o f the f r i c t i o n - v e l o c i t y measurements, i t i s apparent t h a t t h e i n v e s t i g a t i o n s of f r i c t i o n a l v i b r a t i o n s were performed f o r t a b l e v e l o c i t i e s i n the n e g a t i v e s l o p e r e g i o n o f the f r i c t i o n - v e l o c i t y . c u r v e . The shape o f t h i s n e g a t i v e s l o p e r e g i o n was approximated by an e x p o n e n t i a l r e l a t i o n s h i p hav ing c e r t a i n d e f i n e d end c o n d i t i o n s . The v i b r a t i o n e x p e r i -ments show t h a t the decay o f maximum s l i d e r displacement i s a l s o e x p o n e n t i a l i n shape w i t h s i m i l a r c o n d i t i o n s o f v e l o c i t y and c o e f f i c i e n t o f f r i c t i o n . I n a d d i t i o n , the v i b r a t i o n seemed t o v a n i s h at a p o i n t c o r r e s p o n d i n g t o the "knee" o f the f r i c t i o n - v e l o c i t y c u r v e . The g r a p h i c a l approach suggested t h a t f o r t a b l e v e l o c i t i e s i n the p o s i t i v e s l o p e p o r t i o n of the f r i c t i o n -v e l o c i t y c u r v e , t r a j e c t o r i e s c o u l d " m i s s " the v e r t i c a l a x i s k = V and p r o c e e d t o c o m p l e t i o n . This r e p r e s e n t s the case o f smooth s l i d i n g . Table v e l o c i t i e s i n the n e g a t i v e s l o p e r e g i o n were shown t o produce s t a b l e l i m i t c y c l e s . I t was t h e r e f o r e conc luded t h a t the maximum s l i d e r displacement seemed t o correspond t o the v a l u e o f the k i n e t i c f r i c t i o n a t a p a r t i c u l a r t a b l e v e l o c i t y . The a n a l y t i c a l s o l u t i o n was t h e n a l t e r e d , i n v iew o f t h e e x p e r i m e n t a l i n d i c a t i o n s , making use o f the g e n e r a l e x p o n e n t i a l e x p r e s s i o n f o r the shape o f the n e g a t i v e s l o p e r e g i o n o f t h e f r i c t i o n - v e l o c i t y c u r v e . T h e o r e t i c a l and e x p e r i m e n t a l r e s u l t s were t h e n compared and r e a s o n a b l e c o r r e l a t i o n was found. F i g s . 17 t o 22 are e s s e n t i a l l y the envelope o f the v i b r a t i o n as a f u n c t i o n o f t a b l e v e l o c i t y and i n g e n e r a l t h e r e i s s t r o n g agreement b e -tween t h e o r y and experiment i n the v e l o c i t y range 0 < V ^ .002 i n c h per second. F o r t h i s v e l o c i t y i n t e r v a l , F i g s . 2k t o 29 p r e d i c t a mean d e v i a t i o n between t h e o r y and experiment o f about -.006 i n c h f o r system 1 and +.002 i n c h f o r system 6, g i v i n g r e l a t i v e e r r o r s , w i t h r e s p e c t t o e q u i l i b r i u m p o s i t i o n , o f -10$ and +2% r e s p e c t i v e l y . I n t e r m e d i a t e systems are seen t o l i e w i t h i n t h i s range. W i t h t h i s accuracy e q u a t i o n 39 may be o f v a l u e i n p r e d i c t i n g t h e p o s i t i o n i n g e r r o r t h a t can be expected i n f i n a l p o s i t i o n i n g movement under servo c o n t r o l . The accuracy o f e q u a t i o n 39 i s poor at v e l o c i t i e s i n t h e range .02 < V < .03 i n c h per second s i n c e c o r r e s p o n d i n g r e l a t i v e e r r o r s are -k% and 120$ a p p r o x i m a t e l y . F i g s . 30 t o 35 compare t h e e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s f o r the maximum s l i p v e l o c i t y f o r t h e s i x systems. W h i l e t h e e x p e r i m e n t a l r e s u l t s show c o n s i d e r a b l e s c a t t e r , i t w i l l be observed t h a t the t h e o r y serves t o p r e d i c t g e n e r a l t r e n d s i n the v a r i a t i o n o f maximum s l i p v e l o c i t y w i t h t a b l e v e l o c i t y . E n l a r g i n g the e x p e r i m e n t a l r e c o r d s t o o b t a i n the v e l o c i t y v a r i a t i o n can i n t r o d u c e e r r o r which c o u l d account f o r some o f the s c a t t e r i n the v a l u e s o f maximum s l i p v e l o c i t y . I t was shown t h a t f r i c t i o n a l v i b r a t i o n s are s e l f e x c i t e d w i t h each o s c i l l a t i o n b e i n g d r i v e n f o r a v e r y s h o r t p e r i o d by heavy n e g a t i v e damping. T h i s i s apparent when t h e r e l a t i o n s h i p d e s c r i b i n g the n e g a t i v e s l o p e r e g i o n s - 60 -is substituted into equation 2. Using the experimental value of 3^ from Fig. 10 we find TJA = -kO for system 1. The existence of the negative slope region serves to excite the vibrations. For the positive slope region we f i n d that V=^ "Z/{ since the numerical value of Sp from Fig. 10 is very small. The term "effective surface damping" was introduced to describe the component that is added to the value of r in equation 8 by virtue of the slope of the f r i c t i o n curve. The oscillograph records of the vibrations shown in Figs. 11 to 16, i l l u s t r a t e several interesting features of the phenomenon. It w i l l be observed that at low table velocities, breakaway of the slider i s very sudden while at increasing table velocities breakaway becomes less sudden accompanied by decreased accelerations and s l i p velocities. The junction forming process at the end of each s l i p period i s marked by comparatively low accelerations as i l l u s t r a t e d by the high speed s l i p traces. In some cases i t w i l l be seen (Fig. 12 (a) ) that the junction forming process consists of slight irregularities suggesting the presence of minute f r i c t i o n -a l vibrations. At low table velocities, the "stick" period is quite uniform and with the sensitivity used shows no v i s i b l e relative motion. At higher table velocities the "stick" period shows slight "sub-relaxation oscillations" of which Fig. 13 (c) is an example. Possibly due to increased activity and adjustment between the surfaces at higher table velocities the "stick" period becomes less stable. No doubt some degree of adjustment does occur between the surfaces at low table velocities and this slight movement becomes noticeable at higher table velocities. - 61 -Fig. 13 (a.) is a good example of "dwell" that was sometimes noticed prior to the slip period. The relative motion between the surfaces usually produced no movement of the slider, the slider remaining motionless for periods up to about two seconds or sliding distances of the order of .003 inch. A slip period usually followed the dwell. - 62 -2. CONCLUSIONS F r i c t i o n a l v i b r a t i o n s between u n l u b r i c a t e d s l i d i n g s t e e l s u r f a c e s have been s t u d i e d i n some d e t a i l . T r a v e r s i n g v e l o c i t i e s were c o n f i n e d t o the n e g a t i v e s l o p e p o r t i o n o f the f r i c t i o n - v e l o c i t y c u r v e . The c o n c l u s i o n s t o be d e r i v e d f rom the work a r e : 1. The v i b r a t i o n s are b a s i c a l l y a n o n - l i n e a r phenomenon. 2. Each o s c i l l a t i o n i s s e l f - e x c i t e d . F o r a v e r y s h o r t p e r i o d t h e system i s d r i v e n by l a r g e n e g a t i v e damping by v i r t u e o f the s teep n e g a t i v e s l o p e p o r t i o n o f t h e f r i c t i o n - v e l o c i t y curve f o r the s u r f a c e s i n q u e s t i o n . 3 . The i n v e s t i g a t i o n s suggested t h a t the form o f t h e decay o f v i b r a t i o n ampl i tude and t h e shape o f t h e f r i c t i o n - v e l o c i t y curve i n the n e g a t i v e s l o p e r e g i o n were s i m i l a r . l u The ampl i tude o f v i b r a t i o n seemed t o d i e out a t a p o i n t c o r r e s -ponding t o the "knee" o f t h e f r i c t i o n - v e l o c i t y c u r v e . 5 . I n o r d e r t o suppress or e l i m i n a t e f r i c t i o n a l v i b r a t i o n s , the l i n e a r -i z e d t h e o r y p r e d i c t s t h a t t h e necessary c o n d i t i o n s a r e : a) Reduce the r a t i o - by e i t h e r d e c r e a s i n g the l o a d between t h e K s u r f a c e s or i n c r e a s i n g the s t i f f n e s s o f t h e s u p p o r t i n g system. b) Decrease the d i f f e r e n c e between t h e s t a t i c and k i n e t i c (>^SA) c o e f f i c i e n t s o f f r i c t i o n . c) Operate the system at a t r a v e r s i n g v e l o c i t y i n excess o f <\i M f o r the s u r f a c e s u s e d . I t i s c o n s i d e r e d t h a t the i n v e s t i g a t i o n has s e r v e d t o p o i n t out some o f t h e important f e a t u r e s o f the phenomenon. - 63 -i 3. RECOMMENDATIONS F u t u r e e x p e r i m e n t a l i n v e s t i g a t i o n s s h o u l d be c a r r i e d out u s i n g a m o d i f i e d apparatus capable o f p r o d u c i n g h i g h e r t r a v e r s i n g v e l o c i t i e s . The p r e s e n t apparatus i s l i m i t e d t o speeds o f the order o f .030 i n c h e s per second. Table v e l o c i t i e s i n excess o f t h i s v a l u e prevent adequate o b s e r v a -t i o n o f the v i b r a t i o n s . I f v e l o c i t i e s i n the range o f .1 t o .15 inches p e r second c o u l d be a c h i e v e d t h e n e x t r a p o l a t i o n o f t h e r e s u l t s would be unnecessary . The l i m i t a t i o n s on maximum ampl i tude t h a t can be r e c o r d e d by t h e p r e s e n t i n s t r u m e n t a t i o n methods suggest t h a t the s t i f f n e s s o f t h e c a n t i l e v e r beam s h o u l d be i n c r e a s e d s l i g h t l y t o a range o f about 200 t o U00 pounds per i n c h . S i n c e t h e r e i s s u f f i c i e n t s e n s i t i v i t y adjustment r e m a i n i n g i n the equipment, d e f l e c t i o n s o f about 25 m i c r o i n c h e s c o u l d s t i l l be r e c o r d e d s a t i s f a c t o r i l y . ' To observe t h e e f f e c t o f damping, t h e p r e s e n t apparatus c o u l d be e a s i l y m o d i f i e d by i n s t a l l i n g an e l e c t r o m a g n e t i c v i s c o u s - t y p e damper on the end o f the s l i d e r . T h i s wquld p r o v i d e a convenient and f l e x i b l e v a r i a b l e damping system. APPENDICES DERIVATION OF THE ANALYTICAL RELATIONSHIP DESCRIBING THE SHAPE OF THE FRICTION-VELOCITY CURVE. CALIBRATION OF THE APPARATUS AND SUBSIDIARY TECHNIQUES. - 6h -APPENDIX 1. DERIVATION OF THE ANALYTICAL RELATIONSHIP DESCRIBING THE SHAPE OF THE FRICTION-VELOCITY CURVE. The friction-velocity curve shown qualitatively in Fig. 3 can be closely approximated by an analytical relationship of the form; y ^ = C , e +^PV. IA where the terms " ^plX+Z64 " describe the positive slope region and the exponential term describes the negative slope region. Referring to Fig. 3> i t is clear that C^ = y / A ' s - . At the point ( , , the numerical value of C^ must be reduced to some negligible quantity to allow the re-maining linear terms to describe the positive slope portion. This point is the effective cutoff point for the exponential term. Introducing a 1% relative error with respect to , the boundary condition for evaluation of the constant C2 becomes: 11* V.* , ^ = ^ M + . O I ^ M 2 A giving; for the negative slope region. For the complete friction-velocity curve, equation 3A becomes; - i n ) w h i c h compares w i t h e q u a t i o n U . - 65 -APPENDIX 2. CALIBRATION OF THE APPARATUS AND SUBSIDIARY TECHNIQUES a) C a l i b r a t i o n o f the Loading System The normal l o a d W, between the s l i d i n g s u r f a c e s i s a d j u s t e d by a p p l y i n g s u i t a b l e weights t o the l o a d i n g pan i l l u s t r a t e d i n F i g . 7 and F i g . 9. F o r v a r i o u s weights on the l o a d i n g pan, t h e normal l o a d produced at the s l i d e r was measured by means o f a s t r a i n r i n g . The s t r a i n r i n g was l o a d e d d i r e c t l y by the s l i d e r so t h a t known loads W, were t a b u l a t e d as f u n c t i o n s o f beam l e n g t h and t o t a l weight on the l o a d i n g pan. b) C a l i b r a t i o n o f t h e Transducer To determine t h e r e l a t i o n s h i p between pen d e f l e c t i o n and t r a n s -ducer displacement f o r a g i v e n s e n s i t i v i t y s e t t i n g on the a m p l i f i e r , the moving core o f the t r a n s d u c e r was. p l a c e d i n c o n t a c t w i t h t h e end o f t h e moving t a b l e . On the o p p o s i t e end o f the moving core a d i a l gage r e a d i n g t o .0001 i n c h was u s e d t o r e c o r d d e f l e c t i o n s o f t h e core produced by s l i g h t movements i n the t a b l e . c) C a l i b r a t i o n o f the S t i f f n e s s o f t h e C a n t i l e v e r Beam To determine the s t i f f n e s s o f the c a n t i l e v e r beam f o r v a r i o u s l e n g t h s , t h e s t r a i n r i n g was mounted h o r i z o n t a l l y on t h e d r i v e n s u r f a c e by means o f s u i t a b l e b r a c k e t s . The a x i s o f t h e s t r a i n r i n g was p l a c e d c o a x i a l w i t h the c e n t e r - l i n e o f the s l i d e r and the d e f l e c t i n g l o a d was t r a n s m i t t e d from t h e end o f t h e s l i d e r t o the s t r a i n r i n g by means o f a s m a l l s t e e l b a l l . The d i a l gage was p l a c e d on the o p p o s i t e end o f the s l i d e r so t h a t d e f l e c t i o n s were r e a d i l y o b t a i n e d . I t was not p o s s i b l e t o use t h e t r a n s d u c e r t o r e c o r d d e f l e c t i o n s i n t h i s i n s t a n c e s i n c e t h e a m p l i -f i e r and o s c i l l o g r a p h were u s e d t o r e c o r d l o a d s on t h e s t r a i n r i n g . - 66 -d) E x p e r i m e n t a l D e t e r m i n a t i o n o f the C o e f f i c i e n t o f V i s c o u s ( S t r u c t u r a l ) Damping " r " The e v a l u a t i o n o f t h i s c o e f f i c i e n t was c a r r i e d out u s i n g f r e e v i b r a t i o n t e s t s i n w h i c h the s l i d i n g s u r f a c e s were s e p a r a t e d a l l o w i n g t h e c a n t i l e v e r beam t o v i b r a t e f r e e l y a f t e r b e i n g g i v e n an i n i t i a l d i s p l a c e m e n t . The e q u a t i o n g o v e r n i n g the mot ion i s s i m i l a r t o e q u a t i o n 2 except W = 0 y i e l d i n g a homogeneous d i f f e r e n t i a l e q u a t i o n . The o s c i l l o g r a p h t r a c e o f the f r e e damped v i b r a t i o n was drawn t o a l a r g e r s c a l e and t h e n p l o t t e d on semi l o g p a p e r . The envelope was drawn over the peaks o f b o t h p o s i t i v e and n e g a t i v e ampl i tudes thus g i v i n g the v a l u e o f r . From t h i s , the n a t u r a l f requency o f the system c o u l d a l s o be c a l c u l a t e d . Four f r e e v i b r a t i o n t e s t s were c a r r i e d out f o r each o f t h e systems o f Table 2. e) V e l o c i t y o f the D r i v e n S u r f a c e To determine the t a b l e v e l o c i t y V, f o r v a r i o u s s e t t i n g s on the t h y r a t r o n speed c o n t r o l c o n s o l e , t h e t r a n s d u c e r was p l a c e d d i r e c t l y a g a i n s t the end o f the moving t a b l e . Knowing the speed o f the o s c i l l o g r a p h c h a r t , t h e s l o p e o f the r e s u l t i n g t r a c e produced the c o r r e s p o n d i n g v a l u e o f t a b l e v e l o c i t y f o r the p a r t i c u l a r console s e t t i n g . A graph was t h e n drawn g i v i n g the r e l a t i o n s h i p between v e l o c i t y V and console s e t t i n g N . - 67 -BIBLIOGRAPHY 1. Lord Rayleigh, "Theory of Sound", Dover Publications 19U5, Volume 1, Second Edition I89U, p. 208. 2. J. H. Wells, "Kinetic Boundary Friction", The Engineer, (London), Vol. U*7, 1929, p. hSh. 3. S. Thomas, "Vibrations Damped by Solid Friction", The Philosophical Magazine (London), Vol. 9, Series 7, March 1930, p. 329. U. N.L. Kaidanovsky and S.E. Haikin, Journal of Technical Physics, U.S.S.R., Vol. 3, 1933, p. 91. 5. H. Blok, "Fundamental Aspects of Boundary Lubrication", Journal of the Society of Automotive Engineers, Vol. I46, 19^0. 6. F.P. 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