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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  t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements  f o r an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available f o r reference and study,  I further  agree that permission f o r extensive copying of t h i s thesis f o r scholarly purposes may  be granted by the Head of my Department or by his  representatives.  It i s understood that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of Mechanical Engineering, The University of B r i t i s h 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 o c c u r when two s o l i d b o d i e s rubbed together,  a r e a n a l y z e d m a t h e m a t i c a l l y and o b s e r v e d  In the mathematical a n a l y s i s ,  experimentally.  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 m o t i o n  d u r i n g t h e s l i p p e r i o d i s d e r i v e d making use o f t h e e x p e r i m e n t a l v e l o c i t y curve.  are  A q u a l i t a t i v e graphical s o l u t i o n of t h i s  friction-  differential  e q u a t i o n o f m o t i o n 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 f o r m and b e h a v i o r of the motion.  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 m o t i o n t o undergo s t a n d a r d a n a l y t i c a l solution.  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 surfaces  and s i x d i f f e r e n t s u p p o r t i n g s y s t e m s .  confined to s l i d i n g i n the negative the p a r t i c u l a r surfaces  used.  The experiments were  slope r e g i o n of the f r i c t i o n curve  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 t h e t r a n s l a t i n g s u r f a c e a r e c o n s i d e r e d and t h e r e s u l t s s u g g e s t t h a t decay o f t h e v i b r a t i o n s , as t h e s p e e d o f t h e moving s u r f a c e  is  used.  U s i n g t h e 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 t h e shape o f t h e  accordingly.  are  the  increased,  corresponds i n f o r m t o t h e f r i c t i o n - v e l o c i t y c u r v e f o r t h e s u r f a c e s  slope r e g i o n of the f r i c t i o n curve, the t h e o r e t i c a l r e s u l t s  for  negative  altered  Good c o r r e l a t i o n i s o b t a i n e d between t h e a n a l y t i c a l r e s u l t s  the experimental  observations.  and  - iv -  TABLE OF CONTENTS  PAGE  CHAPTER  1  1. INTRODUCTION  1  2. FRICTIONAL VIBRATIONS - A LITERATURE SYNOPSIS .......  2  3. SUMMARY  7  2  1. THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS .......  8  3  1. DESCRIPTION OF EXPERIMENTAL APPARATUS  ii  5  a) THE VIBRATION APPARATUS  18  b) THE APPARATUS FOR FRICTION MEASUREMENT.  22  2. INSTRUMENTATION  23  3. PREPARATION OF THE SLIDING SURFACES  21*  1. EXPERIMENTAL RESULTS a) FRICTION-VELOCITY CURVES FOR THE STEEL SURFACES ..  26  b) FRICTIONAL VIBRATION RESULTS FOR THE STEEL SURFACES  28  1. DISCUSSION OF RESULTS  $d  2. CONCLUSIONS OF THE INVESTIGATION  62  3. RECOMMENDATIONS  63  APPENDICES DERIVATION OF THE ANALYTICAL RELATIONSHIP DESCRIBING  1.  THE SHAPE OF THE FRICTION-VELOCITY CURVE  6U  CALIBRATION OF THE APPARATUS AND SUBSIDIARY TECHNIQUES  6$  2. 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 .  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  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  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-  VELOCITY OF SLIDING  lU  2 ............. 30  FICIENT OF FRICTION AND THE TABLE VELOCITY FOR THE 6 SYSTEMS  U5  - v iLIST OF FIGURES ( c o n t ' d )  FIGURE 2k  PAGE GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -  SYSTEM IL 25  Oa0o»00O0OOOO0OO0000000000000*00<»*0**********O6* • « «  OO0O000O9*0O0O0O0*00«O0000*00000OO**000«00O«*O0*00O  GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -  SYST EM 28  •OeO00O0OOa000«O0O00*000»000000«000**000«0«*000««00  GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY SYSTEM 5  29  0e«oo0000»oo00000O0O00000o*oa0000ttft*0000Ooeo000000O  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY  SYSTEM 1 31  50  GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -  SYST EM 6 30  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY 53  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY SYSTEM 3  33  •  55  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY SYSTEM 5  35  5U  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY SYSTEM k  3k  -  oo«0000000tt»O0000OO0OOOOO00000000OO0000000000O000OO  SYSTEM 2 32  ^4-7  GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -  SYSTEM 3 27  ^4-^  GRAPH OF AMPLITUDE OF VIBRATION VERSUS TABLE VELOCITY -  SYSTEM 2 26  o00O«0000«e0oeo«0e0O0e00eeoe0O0oo«a«»o0**oooe«0O0OO  56  GRAPH OF MAXIMUM SLIP VELOCITY VERSUS TABLE VELOCITY SYSTEM 6  57  - vii  -  LIST OF TABLES  TABLE  PAGE  1  RANGE OF THE CONTROLLED VARIABLES  2  NUMERICAL VALUES OF THE PARAMETERS FOR THE SIX EXPERIMENTAL SYSTEMS e « o * * o » « e * « a « 9 « o o * o o o o 9 9 « « « » 0 O o e « Q 0 o « » o o « e « « « a 6 o o 6 O o  23  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 a t 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 , force of s t a t i c f r i c t i o n , d i f f e r e n c e between F _ . stiffness  (lbs.). (lbs.).  and F ^  o f t h e e l a s t i c system,  mass o f t h e v i b r a t i n g p a r t s ,  ,  (lbs.). (lbs„/in.).  (lbs.  sec„2/in.).  s t r u c t u r a l damping c o e f f i c i e n t o f t h e e l a s t i c s y s t e m , 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 ,  (lbs.sec./in,)  (lbs.sec/in.).  s l o p e of the negative slope 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  (sees./in.).  slope of the p o s i t i v e slope 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, time,  (sees./in.), (sees.).  t i m e o f t h e 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 ,  (sees.).  p e r i o d of the motion, ( 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 ,  (in./sec).  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 , table velocity,  (in./sec).  (in./sec).  c r i t i c a l table velocity,  (in./sec).  t o t a l normal l o a d between t h e s u r f a c e s ,  (lbs.).  d i s p l a c e m e n t , v e l o c i t y and a c c e l e r a t i o n o f t h e s l i d e r r e s p e c t i v e l y , d i s p l a c e m e n t o f t h e s l i d e r due t o t h e breakaway and s t a t i c coefficients  of f r i c t i o n r e s p e c t i v e l y ,  (in.),  d i s p l a c e m e n t o f t h e s l i d e r due t o t h e minimum c o e f f i c i e n t friction,  (in.).  of  - ix LIST OF SYMBOLS ( c o n t ' d )  x  t h e minimum d i s p l a c e m e n t o f t h e s l i d e r ,  06  amplitude of v i b r a t i o n , ( i n . ) .  <\>  t h e d i m e n s i o n l e s s parameters  d>  t h e d i m e n s i o n l e s s damping p a r a m e t e r :  /*•  }  J*  S  c o e f f i c i e n t of f r i c t i o n ,  -  (in.).  . • ~^ -  and c o e f f i c i e n t  of s t a t i c  friction  respectively. -^B 1 F /^breakaway, K>  y^x  k i n e t i c and minimum c o e f f i c i e n t s  of f r i c t i o n  respectively.  i s t h e c o e f f i c i e n t o f f r i c t i o n a t s l i d i n g speeds g r e a t e r  l)  combined damping r a t i o .  t),  damping r a t i o f o r t h e e l a s t i c  l)  damping r a t i o f o r t h e  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,  6Jj '  damped n a t u r a l c i r c u l a r f r e q u e n c y o f t h e s u p p o r t i n g system, (rad,/sec.).  than^  system.  surfaces. (rad./sec.)  M  .  ACKNOWLEDGEMENT  The a u t h o r i s g r a t e f u l f o r t h e a d v i c e and encouragement 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 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 assistance  given  discussions  was r e c e i v e d f r o m  t h e N a t i o n a l R e s e a r c h C o u n c i l o f 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 t h e L u b r i c a t i o n L a b o r a t o r y , of Mechanical Engineering, U n i v e r s i t y of B r i t i s h Columbia.  Department  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 contact  i s driven slowly  f o r w a r d w h i l e t h e 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 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 , b u t r a t h e r proceeds w i t h a s e r i e s  is of  " 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 c r e e p speeds,  the  presence of 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 , apart from causing i n c r e a s e d wear,  d e s t r o y s t h e a c c u r a c y a n d 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 p u r p o s e ,  therefore,  o f t h i s i n v e s t i g a t i o n i s t o s t u d y not o n l y  t h e f o r m o f t h e v i b r a t i o n b u t t h e i n f l u e n c e o f v a r i o u s parameters t h e s u p p o r t i n g system, hoped t h a t t h e r e s u l t s ,  defining  on t h e s t a b i l i t y o f t h e f r i c t i o n a l v i b r a t i o n s .  It  is  presented 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 surfaces,  the p r e p a r a t i o n of 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 effects. P r o v i s i o n was made i n t h e e x p e r i m e n t a l a p p a r a t u s f o r v a r i a t i o n i n load, stiffness,  and v e l o c i t y o f t h e d r i v e n  surface.  - 2 2.  FRICTIONAL VIBRATIONS - A LITERATURE SYNOPSIS Early investigations  on t h e s l i d i n g o f b o d i e s a t l o w 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 t h e m o t i o n may n o t be a c o n t i n u o u s process.  L o r d R a y l e i g h ( l ) * discusses b r i e f l y the motion of 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 t h e bow, b u t o f f e r s  no d e t a i l e d development o f t h e m o t i o n  s i n c e "some o f t h e d e t a i l s were s t i l l o b s c u r e " . effort  I n 1929, W e l l s ( 2 ) , i n an  t o measure k i n e t i c boundary f r i c t i o n a t low s p e e d s , 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 c e r t a i n circumstances".  s t i c k i n g and s l i p p i n g t o o k p l a c e under  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 behavior 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 e m p l o y i n g a n a l y t i c a l and graphical techniques.  The d i f f e r e n t i a l e q u a t i o n o f m o t i o n 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 . shows t h e t r a j e c t o r i e s  Thomas  r e p r e s e n t i n g the s o l u t i o n of the equation of motion  f o r t h e s i m p l e harmonic case w i t h no damping. viscous  On t h e phase p l a n e ,  Under l u b r i c a t e d c o n d i t i o n s a  damping t e r m i s added t o t h e d i f f e r e n t i a l e q u a t i o n and t h e phase  representation i s then a l t e r e d  plane  accordingly.  K a i d a n o v s k y 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 h a v i n g 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 , necessary  and t h e y o b s e r v e d t h a t a  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 existence  t h e f r i c t i o n d e c r e a s e s as t h e v e l o c i t y i n c r e a s e s . equilibrium i s unstable.  of a r e g i o n i n which  In.such  circumstances  As a m a t t e r 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 p o s s e s s e s a  * Numbers i n b r a c k e t s d e s i g n a t e 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 c o m p l e t e n e s s .  - 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 .  In 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 t h e c u r v e i s n e g a t i v e w h i l e i n t h e hydrodynamic r e g i o n t h e s l o p e i s p o s i t i v e .  They  s u g g e s t 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 t h e boundary o r n e g a t i v e  slope  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 t h e p a r t i c u l a r shape 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 .  His a n a l y t i c a l  t r e a t m e n t i s b a s e d on t h e 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 c u r v e a n d he extends his  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 t h e amount o f damping  i n the s u p p o r t i n g system.  He develops a d i m e n s i o n l e s s p a r a m e t e r ,  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 o c c u r i n a s l i d i n g s y s t e m i f t h e 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 t h e  critical  v a l u e , t h e n he shows t h a t t h e system hovers between v i b r a t i o n and smooth sliding. Bowden and Leben (6) c a r r i e d out a s e r i e s o f e x p e r i m e n t s on t h e f r i c t i o n between s l i d i n g m e t a l s i n t h e 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 n o t r e m a i n ' c o n s t a n t d u r i n g s l i d i n g a n d t h a t t h e p r o c e s s may n o t be c o n t i n u o u s . of  " s t i c k s " and " s l i p 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  They s u g g e s t t h a t f r i c t i o n between m e t a l s can be  a t t r i b u t e d i n part to l o c a l i z e d c o l d welding processes.  Since i s o l a t e d  a s p e r i t i e s carry the l o a d , the r e s u l t i s that excessive l o c a l pressures e s t a b l i s h e d thus g i v i n g r i s e to c o l d welding or adhesion ( 2 0 ) . e x p e r i m e n t s , s i m u l t a n e o u s measurements were t a k e n o f t h e s u r f a c e  are  During t h e i r 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 t e m p e r a t u r e , and, a t t h e i n s t a n t o f  slip  - k t h e r e was a sudden t e m p e r a t u r e  "flash".  Thus i n a d d i t i o n t o t h e  cold  w e l d i n g p r o c e s s t h e y s u g g e s t t h a t hot w e l d i n g p r o c e s s e s a l s o e x i s t w h i c h c o n t r i b u t e to the surface  damage.  Further quantitative investigations  as t o t h e 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 Dudley and S w i f t (8)  (7).  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 experimental f r i c t i o n - v e l o c i t y curve. 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 .  I t i s not necessary  the  t o express t h i s  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 a n d t h e  results  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 t h e r cases t h e y decay.  The method s e r v e s t o i l l u s t r a t e t h e s o l u t i o n o f  the  g o v e r n i n g d i f f e r e n t i a l e q u a t i o n o f m o t i o n 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) c o n f i r m e d t h a t t h e e x i s t e n c e s l o p e r e g i o n was a n e c e s s a r y vibrations.  His observations  of the  negative  c o n d i t i o n f o r the e x c i t a t i o n of f r i c t i o n a l i n d i c a t e d t h a t t h e m o t i o n i s dependent on t h e  r e l a t i v e magnitude o f t h e damping c o e f f i c i e n t  of the s u p p o r t i n g system.  D u r i n g t h e " 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 t h e two  surfaces  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 v e l o c i t y o f t h e moving s u r f a c e i n c r e a s e d . t i o n s h i p was p r e s e n t e d  the  However, no e x p e r i m e n t a l r e l a -  t o show t h e f o r m o f t h i s a m p l i t u d e d e c r e a s e .  R a b i n o w i t z (10)  f o u n d t h a t two s u r f a c e s  before 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  undergo s l i g h t  displacement  f a l l s t o some k i n e t i c v a l u e .  h i s e x p e r i m e n t s , 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  From 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 displacement 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 face conditions. are i n c o n t a c t ,  o f t h e o r d e r o f 1+0 m i c r o i n c h e s .  depending upon t h e g e n e r a l  As p r e v i o u s l y s u g g e s t e d by Bowden, when m e t a l s u r f a c e s p l a s t i c f l o w o c c u r s and m e t a l l i c j u n c t i o n s a r e  Rabinowitz suggested that p r e l i m i n a r y displacement junctions p r i o r to t h e i r shearing. i s formed a f t e r This r e s u l t s  i n increased crosseetional  of t h i s  "time of r e s t "  c l u d e d from t h e experiments to reach i t s f u l l strength.  i s the y i e l d of  strength to  creep.  area of the junctions which then  Thus i t appears t h a t t h e c o e f f i c i e n t  of  rest".  B u r w e l l and R a b i n o w i t z ( l l ) s t u d i e d t h e i n on t h e 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 -  that a f i n i t e time i s r e q u i r e d f o r a j u n c t i o n There a r e however,  t h i s s t r e n g t h increase that are s t i l l  some p r i n c i p l e f e a t u r e s o f  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 by H. L a u e r . ( 1 2 ) .  the  t h e p r o c e s s b e i n g analogous  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 t h e " t i m e o f A c c o r d i n g l y i n 1953,  formed.  A j u n c t i o n o f somewhat g r e a t e r  a longer p e r i o d of r e s t ,  require larger shearing forces.  fluence  sur-  He d e s c r i b e s  the d i f f i c u l t i e s  is  discussed  i n the operation of a simple  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 t h e s e 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  Derjaguin,  displacements.  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  t h e o r e t i c a l a n a l y s i s b a s e d on B l o k ' s e a r l i e r work.  detailed  L i n e a r i z a t i o n of  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 was n e c e s s a r y i n o r d e r t o s o l v e  the  d i f f e r e n t i a l equation of motion during s l i p .  However, t h e y f a i l t o  adequately  d e f i n e t h e 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 based.  is  Boundary c o n d i t i o n s , r e p r e s e n t i n g t h e 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 the equations 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)  for  , which i s  s i m i l a r i n f o r m t o t h a t d e v e l o p e d by B l o k .  AF  i  where A F  represents  of f r i c t i o n .  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  forces  I n a p p l y i n g t h e boundary c o n d i t i o n s t h e y do not d e f i n e  whether  or 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 or negative slope r e g i o n of the friction-velocity  curve.  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  is  necessary to minimize the value of c r i t i c a l v e l o c i t y i n order to provide s e n s i t i v e response. and T o l s t o i (13),  F o l l o w i n g t h e 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  Singh ( l i t ) ,  of c r i t i c a l v e l o c i t y .  (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 t u d y  E f f o r t i s made t o p r e s e n t t h e n e c e s s a r y  conditions  t h a t must e x i s t i n t h e s u p p o r t i n g s y s t e m w h i c h w i l l produce a minimum critical velocity.  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 a r e p r e s e n t e d i n g r a p h i c a l f o r m w h i c h s e r v e t o p r e d i c t t h e 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.  Singh concludes  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 i t i s necessary 1.  to:  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 friction.  2. I n c r e a s e t h e 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 t h e damping i n the" s y s t e m .  that  of  - 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 t h e mechanics o f  t h e phenomenon and t o s e r v e as a s t a r t i n g p o i n t f o r t h e 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 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 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 of 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 t h e s y s t e m . 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 t h e s u r f a c e s  during the  " s t i c k " p e r i o d has been o b s e r v e d . U. A p r e l i m i n a r y d i s p l a c e m e n t i m m e d i a t e l y 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 s u g g e s t e d t h a t t h i s movement represents the y i e l d of the j u n c t i o n s i n shear. 5>. To d e c r e a s e t h e 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 s y s t e m i t i s  necessary  to: a) decrease 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 c o e f f i cients of  friction.  b) r a i s e t h e 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.  c) i n c r e a s e t h e damping i n t h e s y s t e m .  CHAPTER  TWO  THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS  - 8 -  1.  THEORETICAL ANALYSIS OF FRICTIONAL VIBRATIONS C o n s i d e r t h e 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 s y s t e m 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 l a n e which i s driven w i t h a constant v e l o c i t y " V " . d i a g r a m m a t i c a l l y i n F i g . 1,  surface  Such a s y s t e m i s shown  i n which " r " represents  the c o e f f i c i e n t  of  v i s c o u s s t r u c t u r a l damping o f t h e s u p p o r t i n g system.  Fig.  1  Diagrammatic r e p r e s e n t a t i o n o f t h e s l i d i n g s y s t e m .  The d i s p l a c e m e n t " x " o f t h e s l i d e r i s measured r e l a t i v e t o t h e 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 . the lower surface,  I f no r e l a t i v e m o t i o n e x i s t s between t h e s l i d e r and  then the equation d e s c r i b i n g the motion of the 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 t h e d i s p l a c e m e n t 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 t h e p o r t i o n o f t h e c u r v e l a b e l l e d AB i n F i g . 2. Up t o p o i n t B, t h e f o r c e o f s t a t i c f r i c t i o n i s c a p a b l e o f w i t h s t a n d i n g t h e sum o f t h e s p r i n g and damping f o r c e s . exceed the s t a t i c v a l u e ,  S i n c e t h e f o r c e o f f r i c t i o n cannot  d i s p l a c e m e n t o f t h e 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 m o t i o n between t h e s l i d e r and t h e moving s u r f a c e .  - 9f  B  ^3  APERIODIC  M  Of,M I N  A  [  C  V  HEAVy  DAMPING-  X Fig.  2  The f o r m o f t h e " s t i c k  The s l i p p e r i o d t h e n o c c u r s t o C.  slip" oscillation.  d u r i n g w h i c h t h e s l i d e r moves r a p i d l y from B  F o r many s u r f a c e s , t h e f r i c t i o n - v e l o c i t y  general form i l l u s t r a t e d i n F i g . 3.  r e l a t i o n s h i p takes the  I f r e l a t i v e m o t i o n 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 m o t i o n o f t h e s l i d e r can be w r i t t e n  where J*-  i s the c o e f f i c i e n t  as:  o f f r i c t i o n between t h e 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  S/.OPE = - ^ F  M  <U-\l-rp Fig. 3  The v a r i a t i o n i n c o e f f i c i e n t  of f r i c t i o n w i t h  relative  v e l o c i t y of s l i d i n g . F o r t h e case o f l i g h t damping, c o m p a r a t i v e l y  large s l i p  velocities  are e n c o u n t e r e d as s u g g e s t e d by t h e s l o p e o f the curve BC i n F i g . 2. reason for these high v e l o c i t i e s  i n t h e underdamped case i s apparent  The from  - 10  F i g . 3 since the c o e f f i c i e n t , o f f r i c t i o n achieves a low k i n e t i c value very rapidly.  This allows the term "Kx" i n equation 2 to govern, returning the  s l i d e r to point C i n F i g . 2.  At point G, the v e l o c i t y of the s l i d e r becomes  equal to the v e l o c i t y of the moving surface and a " s t i c k " period commences. The cycle ABC i s repeated continuously, the motion being r e f e r r e d to as "stick-slip" sliding. It should be mentioned that the damping i s less than aperiodic, otherwise the displaced s l i d e r w i l l d r i f t asymptotically from point A of Fig.2 back to a displacement corresponding to the v e l o c i t y of the moving surface.  Under these conditions further r e l a x a t i o n o s c i l l a t i o n s are im-  possible unless the v e l o c i t y of the moving surface drops to zero momentarily. This would r e s u l t i n another transient followed by smooth s l i d i n g . Since the d i f f e r e n t i a l equation of motion of the s l i d e r during s l i p (equation 2) i s a second order non-linear equation, the motion can be represented graphically on a phase plane diagram which consists of a plot of v e l o c i t y versus p o s i t i o n .  The general zero slope i s o c l i n e method w i l l  be used to e s t a b l i s h the phase plane plot (1$).  Scale r e s t r i c t i o n s do not  permit the application of the method to accurate analysis of normal r e l a x a tion-type o s c i l l a t i o n s but q u a l i t a t i v e use of the phase plane diagram does serve to i l l u s t r a t e the general behavior.  Rewriting the d i f f e r e n t i a l equation 2 i n the form:  and l e t t i n g x » y we have:  - 11 F o r t h e z e r o s l o p e i s o c l i n e we s e t  giving;  ^ - 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 t h e c o r r e s p o n d i n g v a l u e s  of  from t h e e x p e r i m e n t a l f r i c t i o n c u r v e 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 values i n t o equation 3 r e s u l t s i n the zero slope i s o c l i n e c u r v e p l o t t e d on t h e phase p l a n e i n F i g . i+. s t r u c t i o n the slopes of the t r a j e c t o r i e s be drawn on t h e phase p l a n e .  Using M e n a r d ' s graphical con-  o f t h e d i f f e r e n t i a l e q u a t i o n can  The complete s l i p t r a j e c t o r y f o r t h e  initial  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 clockwise d i r e c t i o n , the short slope l i n e s i n the i n t e g r a l f i e l d . Fig.  \\ i s drawn f o r a t a b l e v e l o c i t y i n t h e 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 c u r v e 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 a m p l i t u d e A B .  The e f f e c t s o f s t a t i c  friction  becoming o p e r a t i v e a t p o i n t A v i r t u a l l y c u t o f f t h e phase p l a n e diagram and abruptly stops 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 s y s t e m o p e r a t i n g  i n t h e p o s i 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 c u r v e where t h e p o s s i b i l i t y e x i s t s f o r the s p i r a l l i n g t r a j e c t o r y to "miss" the v e r t i c a l k = V axis some p o i n t D.  T r a j e c t o r i e s t h a t do n o t e n c o u n t e r t h e a x i s  at  k = V a r e 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 c o n s e q u e n t l y 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 value of k  s p i r a l i n t o t h e s t a b l e p o i n t E.  a t p o i n t E i s z e r o so t h a t smooth s l i d i n g r e s u l t s (13).  e f f e c t s o f s t r u c t u r a l damping i n t h e s y s t e m i s d e s c r i b e d by t h e t e r m  The The -  K  - 12 -  h-X=Y= V  Fig.li  Graphical solution for the motion i n 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 f i r s t term of equation 3 results in a zero slope isocline possessing a steeper positive slope on the phase plane.  This  in turn results i n a more rapid spiralling in of the whole integral f i 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  Fig.  5  G r a p h i c a l s o l u t i o n f o r t h e m o t i o n 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 a t higherX( v a l u e s .  The f r i c t i o n - v e l o c i t y c u r v e can be  q u i t e a c c u r a t e l y by t h e r e l a t i o n s h i p * ;  Since  * R e f e r t o Appendix 1 f o r t h e d e r i v a t i o n o f t h i s  equation.  expressed  - lU substitution yields:  Substitution of equation 6 into equation 2 produces a non-linear d i f f e r e n t i a l equation by v i r t u e of the exponential term.  I t i s now apparent that the  s l i d e r i s under large negative damping f o r a short period immediately slip.  after  Thus each o s c i l l a t i o n i s i n i t i a l l y s e l f - e x c i t e d by the negative  slope of the f r i c t i o n - v e l o c i t y curve. In order to solve the non-homogeneous second order d i f f e r e n t i a l equation 2 by a n a l y t i c a l methods the c o e f f i c i e n t s i n equation 2 must be constant. of"!/.  This condition i s f u l f i l l e d only when  i s a s i n g l e l i n e a r function  Correspondingly F i g . 6 i l l u s t r a t e s 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 which w i l l be used i n the following a n a l y t i c a l solution.  Essentially,  r e l a t i o n s h i p 5> becomes:  The e f f e c t of s t a t i c f r i c t i o n , i n the solution.  , w i l l be added as a boundary condition  This approach i s used by Derjaguin, Push and T o l s t o i (13).  A  Fig. 6  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.  -15 Re-writing equation 2 and substituting  7, gives: 8  Expressing the c o e f f i c i e n t of x  as R and the terra on the r i g h t as  we  have the general form: /vrv  rp  The s o l u t i o n to t h i s d i f f e r e n t i a l equation can be written as:  ip-  FK. + + e e  *  '  |i/ \nccooss  CAJ^JO  + t- BSIN DSIN  f  OJ$X oj^*J  io  -(A+<I>B)SIN cojt  ii  D i f f e r e n t i a t i o n with respect to time y i e l d s :  (B-4>A)cos  out  and  CP  = -co, e  12  where  13  lit /vw  /I /VYV  •J  - A.W. /vw  15  16  - 16 -  CJ*---£-  17  18  4> =  19  In order to evaluate the constants of i n t e g r a t i o n , boundary c o n d i t i o n s  A and B, t h e  apply;  When t « 0, t h e s l i d e r i s on t h e v e r g e o f s l i p ,  represented  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. from e q u a t i o n  following  by  Therefore,  1;  20  K  also  21  E q u a t i o n s 10,  11 and 12 become;  —  rp  cp  22  ai  K  =Ve  = -Vo^  cos cjjtt  e  f =  -  SIN  V cos a y * + <f>(^-  AF  =  WQs-.A*)  23  S  'N  WJI*  21*  25  -  17  -  Since the f o l l o w i n g experimental i n v e s t i g a t i o n s are l i m i t e d t o very s m a l l v a l u e s o f R and V , powers o f ~jj can be n e g l e c t e d ,  and terms i n V i n e q u a t i o n s 2 2 , 2 3 a n d 2k  giving:  cos  K  +  T^S/M  oujtj;  26  K  - ~i)wiic r  rp = - < ^ L ^ - p ^ e  SIN  27  u> £ A  - ~T)SJN  cos out  OJ^^T  28  R e f e r r i n g t o F i g . 2 the minimum d i s p l a c e m e n t can be w r i t t e n as:  rP  =  W ^ M  _  W  (>»->*MV  e  29  Thus t h e a m p l i t u d e o f v i b r a t i o n becomes:  30  K  V  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 t h e 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 c u r v e c o n t r i b u t e s t o t h e v i s c o u s damping i n t h e s y s t e m . 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 an amount SpW w h i c h can be r e f e r r e d t o as " s u r f a c e damping". s p o n d i n g v i s c o u s s u r f a c e damping r a t i o  ~jj  i n c r e a s e d by The c o r r e -  i s g i v e n by e q u a t i o n 1 6 .  The magnitude o f t h e 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 kinetic  W and t h e d i f f e r e n c e between t h e c o e f f i c i e n t s K"  friction.  o f s t a t i c and  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 t h e c h a r a c t e r i s t i c s o f t h e s u p p o r t i n g s y s t e m as w e l l as 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 r e l a t i o n s h i p f o r t h e involved.  surfaces  I t has been shown t h a t t h e f o l l o w i n g parameters o f t h e s u p p o r t i n g  system are important; 1) S t i f f n e s s  of the e l a s t i c  2) Normal l o a d on t h e  system  surfaces  3) S t r u c t u r a l damping i n t h e s u p p o r t i n g s y s t e m ii) V e l o c i t y o f t h e t r a v e r s i n g  surface  A c c o r d i n g l y , the experimental apparatus,  c o n s t r u c t e d by t h e a u t h o r ,  was  d e s i g n e d t o a l l o w c o n v e n i e n t v a r i a t i o n i n some o f t h e above p a r a m e t e r s . The a p p a r a t u s ,  i n i t s p r e s e n t f o r m , however,  i n s t r u c t u r a l damping. velocity,  has no p r o v i s i o n f o r v a r i a t i o n  With respect to s t i f f n e s s ,  l o a d , and t r a v e r s i n g  t h e i n f l u e n c e o f t h e s e parameters on t h e r e s u l t i n g f o r m o f t h e  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 t h e v i b r a t i n g member independent o f t h e normal l o a d on t h e  surfaces.  To a c c o m p l i s h t h i s , i t i s n e c e s s a r y t h a t t h e e l a s t i c system t r a n s m i t t h e n o r m a l l o a d , c o n s e q u e n t l y t h e s u s p e n s i o n was d e s i g n e d i n t h e f o r m 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 p o s s e s s e s a h i g h n a t u r a l f r e q u e n c y 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 t h e c a n t i l e v e r beam i n t h e 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 t h e c l a m p i n g b l o c k shown i n F i g . 7.  The n o r m a l  l o a d W, i s a p p l i e d by means o f t h e l o a d i n g system w h i c h p i v o t s t h e beam assembly about t h e a x i s A - A .  T h i s a x i s i s s u p p o r t e d on an  cantilever  adjustable  -  19  -  base ( n o t 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 c a u s e d 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 l o w e r s u r f a c e .  T h i s was  a c c o m p l i s h e d by t h e use o f a 2 - | i n c h diameter power screw on w h i c h a l a r g e t h r e a d e d n u t i s g u i d e d t r a n s v e r s e l y by s l i d i n g ways.  The power screw  is  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 t h e 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 of 7 0 0 ; 1 . is  The r e s u l t i n g maximum t r a n s l a t i o n a l s p e e d o f t h e l o w e r  surface  . 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 a r e m a i n t a i n e d i n c o n s t a n t u n i f o r m c o n t a c t a t a l l t i m e s throughout the s l i d i n g process.  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, t h e 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 d e s i g n e d .  I t w i l l be o b s e r v e d f r o m F i g . 8 t h a t t h e 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 a r e 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 allows  for  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 o c c u r d u r i n g t h e s l i d i n g process.  T h i s i s an i m p o r t a n t f e a t u r e o f t h e  apparatus.  The mass, m, o f t h e v i b r a t i n g p a r t s i s made up o f t h e 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 t h e beam weight t h a t p a r t i c i p a t e s  in  the motion. The s t r u c t u r a l damping c o e f f i c i e n t , d e t e r m i n e d by f r e e v i b r a t i o n t e s t s , q u i t e s m a l l i n magnitude.  r,  o f t h e c a n t i l e v e r beam,  was f o u n d t o be v i s c o u s i h f o r m b u t  A b r i e f d e s c r i p t i o n of the free v i b r a t i o n t e s t s ,  as w e l l as t h e g e n e r a l c a l i b r a t i o n p r o c e d u r e f o r t h e a p p a r a t u s , w i l l be found i n Appendix 2 .  - 20-  FIG.7  FIG. 8  DIAGRAMMATIC  THE  SKETCH  SELF-ALIGNING  OF  THE  JOINT  APPARATUS  - 22 -  1. b)  THE APPARATUS FOR FRICTION MEASUREMENT  As previously mentioned, the investigations were c a r r i e d out using unlubricated s t e e l surfaces.  To obtain the experimental 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 t e e l surfaces, i l l u s t r a t e d q u a l i t a t i v e l y i n F i g . 3, a subsidiary f r i c t i o n apparatus was used.  This was necessary since s l i p  v e l o c i t i e s i n the neighborhood of 3 inches per second were recorded on several t r i a l t e s t s . apparatus.  This speed i s f a r beyond the range of the v i b r a t i o n  To produce such comparatively large speeds, the subsidiary  f r i c t i o n apparatus was designed using a h o r i z o n t a l l y mounted 16 inch diameter revolving s t e e l plate made of the same material as the s t e e l surfaces under study.  The r o t a t i o n a l speed of the s t e e l plate can be c o n t r o l l e d by means  of a v a r i a b l e speed e l e c t r i c motor.  Curvature e f f e c t s , which were considered  to have n e g l i g i b l e effect i n the f r i c t i o n - v e l o c i t y measurements, were however minimized by conducting the experiments at the largest possible radius on t h e - s t e e l plate.  Since the cantilever beam suspension system i s  e a s i l y removed from the v i b r a t i o n apparatus, t h i s 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 t h e s l i d e r i s measured by means o f a B r u s h "Type  One" l i n e a r d i s p l a c e m e n t 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 . is  The moving c o r e o f t h e t r a n s d u c e r  c o n n e c t e d t o t h e 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 -  g r a p h 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 t h e o s c i l l o g r a p h c h a r t p e r  .001 i n c h s l i d e r displacement.  C o n s e q u e n t l y t h e maximum a m p l i t u d e 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 c o n v e n i e n t means o f c e n t e r i n g t h e t r a c e o f t h e v i b r a t i o n on t h e o s c i l l o g r a p h c h a r t , t h e 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 o r c r o s s f e e d .  By means o f a handwheel, g r a d u a t e d t o r e a d  .0005 i n c h d i s p l a c e m e n t , t h e 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 " trace of the v i b r a t i o n .  the  From t h e 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 t h e o r d e r o f .0005 i n c h can be a c c u r a t e l y d e t e r m i n e d .  Thus t h e a c c u r a c y o f t h e  crossfeed  and t h e o s c i l l o g r a p h a r e matched.  T a b l e 1 i n d i c a t e s t h e range o f t h e 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 t h e a p p a r a t u s . Variable  Symbol  Load  W  Stiffness  K  12 t o 180 pounds p e r i n c h  V  0 to .03 i n c h per second  Table V e l o c i t y  TABLE 1.  Range 0 t o 30 pounds  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 .  C o n v e n i e n t Increment 2 pounds 13 pounds p e r i n c h .0001 i n c h per second  - 2h 3.  PREPARATION OF THE SLIDING SURFACES C o n s i d e r a b l e c a r e a n d a t t e n t i o n was d e v o t e d t o t h e p r e p a r a t i o n o f  t h e 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. best c r i t e r i o n , of course,  i s r e p r o d u c i b i l i t y of r e s u l t s .  Previous  The  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 b u t 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 that the best  p r o c e d u r e 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 .  In addition,  t h e experiments must be p e r f o r m e d as soon as p o s s i b l e a f t e r t h e 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  a p p a r a t u s f o r r e m o v i n g f r e s h l y a c c u m u l a t e d contaminant m a t e r i a l f r o m d i r e c t l y i n f r o n t o f t h e s l i d e r , however t h e e x p e r i m e n t s were p e r f o r m e d d i r e c t l y a f t e r t h e s u r f a c e t r e a t m e n t p r o c e s s w i t h a t i m e i n t e r v a l o f o n l y a few minutes. The s u r f a c e s  were f i r s t p r e p a r e d f r o m G 1020  then de-greased u s i n g t r i - c h l o r o e t h y l e n e .  (cold rolled)  U s i n g grade number 80-G,  steel, Behr-  Manning "Tufbak" d u r i t e w a t e r p r o o f emery p a p e r , 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 of 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 , t h e 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 l o w e r s u r f a c e ,  both surfaces  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 t h e specimens p r o d u c e d by t h i s f i n i s h i n g p r o c e s s 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, e q u i p p e d w i t h a power t r a v e r s e .  The range o f t h e 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 t h e 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 t h e RMS ( r o o t mean s q u a r e ) roughness h e i g h t i n m i c r o i n c h e s .  Measurements were made a t  close  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  l i s t e d i n Table 2 using the following symbols: Ijj =  average RMS surface roughness value i n microinches parallel to the direction of sliding.  YjL => average RMS surface roughness value i n microinches perpendicular 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  w i t h the value  o f 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  consistent  The s u r f a c e s were a l l o w e d t o r e m a i n 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  coefficient  o f k i n e t i c f r i c t i o n was f o u n d t o be l o a d dependent as i n d i c a t e d by t h e c u r v e s o f F i g . 10.  The v a l u e o f t h e s l o p e , f o r t h e p o s i t i v e s l o p e r e g i o n  of the f r i c t i o n curves, the l i g h t e r l o a d . of four t e s t s .  i s s m a l l , w i t h t h e 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 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  The mean d e v i a t i o n i n c o e f f i c i e n t  of f r i c t i o n f o r the  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  of -  From t h e average c u r v e s o f F i g . 10, t h e mean d e v i a t i o n i s - .00? r e l a t i v e error w i t h respect to  o f - y%.  four  20$.  giving a  Corresponding loads of  six  pounds and t e n pounds were u s e d i n t h e 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 c a r e 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 of  'U  M  , a n d i n d i c a t i o n s f r o m 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 o r d e r o f .06 i n c h e s p e r s e c o n d .  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 f r o m t h i s average was f o u n d t o be as much as i $0%. Thus t h e v a l u e o f that the c o e f f i c i e n t  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 of f r i c t i o n achieves  r e s u l t i n g i n a steep negative  stated  a minimum v a l u e v e r y r a p i d l y ,  slope.  The l i n e a r e q u a t i o n s shown i n F i g . 10 s e r v e t o d e s c r i b e t h e p e r i m e n t a l curves w i t h reasonable accuracy.  ex-  The e q u a t i o n s 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 f o r m w i t h e q u a t i o n 7.  S U R F A C E S •* C  COLD ROLLED S T E E L  I02O  ROUGHNESS: Y// = 8  S-\H.  ••yf^-lsO-C  2'  O h O  E k. u. o h  2  LU  O  u. ll.  Ui  o o  RELATIVE FIG. 1 0 -  GRAPH  OF C O E F F I C I E N T OF  FRICTION  VERSUS  VELOCITY  RELATIVE  *"U  " ( INS./SEC.)  VELOCITY.  EMERY  | SOPAPER  - 28 -  1. b)  1  FRICTIONAL VIBRATION RESULTS  T a b l e 2 g i v e s t h e v a l u e s o f t h e parameters  defining the  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 . measurements u s e d i n t h e d e t e r m i n a t i o n o f K, W, OJ  six  Subsidiary  and r are  described  i n A p p e n d i x 2. Corresponding t o these s i x systems, t h e v i b r a t i o n s a r e shown i n F i g s . 11 t o 16. d i r e c t i o n from l e f t to r i g h t across  the o s c i l l o g r a p h records  of  The v i b r a t i o n s p r o c e e d i n a  each f i g u r e t h u s making t h e s l i p p r o c e s s  p r o c e e d i n a d i r e c t i o n f r o m t h e b o t t o m t o t h e t o p o f each f i g u r e . F o u r t e s t s were c o n d u c t e d f o r each s y s t e m a t t h e t a b l e noted.  From t h e 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 ,  i n the features  o f t h e v i b r a t i o n s was o b s e r v e d .  reasonable  velocities  consistency  F o r example, t h e mean  d e v i a t i o n i n maximum d i s p l a c e m e n t 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 p e r s e c o n d t h e mean d e v i a t i o n i n maximum d i s p l a c e m e n t i s 13$ w i t h respect to e q u i l i b r i u m p o s i t i o n .  Deviations f o r the intermediate  systems  l i e within this-range.  To o b t a i n an e s t i m a t e o f t h e maximum s l i p v e l o c i t y , t h e maximum o s c i l l o g r a p h c h a r t s p e e d o f 5 i n c h e s p e r s e c o n d was u s e d .  From t h e s e h i g h  speed t r a c e s w h i c h a r e mounted on t h e 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 t h e 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 variation.  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 e a c h 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 t h e 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 t h e g e n e r a l f o r m o f t h e v i b r a t i o n and t h e s e l o w s p e e d r e s u l t s mounted on t h e l e f t 1  hand s i d e o f F i g s . 11 t o 16.  Each d i s p l a c e m e n t  are scale  Of)  TABLE  C)  VELOCITY =.0015 I N . / S E C .  TABLE  VELOCITY=.03  CHART  IN./SEC.  VELOCITY —.2  IN./SEC.  CHART VEL0CITY= .2  CHART VELOCITY=5  IN./SEC.  IN. / SEC.  w  (O F I G . II  TYPICAL  TRACES  O F  SCALE:  I DIVISION =  T H E  .0012  OSCILLATION INCHES  SLIDER  FOR  SYSTEM  I .  DISPLACEMENT.  C) TABLE  F I G . 12  VELOCITY= .03 IN./SEC.  CHART  VEL0CITY=.2  IN./SEC.  CHART  VELOCITY= 5  TYPICAL TRACES OF T H E OSCILLATION FOR SYSTEM 2. S C A L E : I DIVISION =.0012 I N C H E S SLIDER DISPLACEMENT.  IN./SEC.  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) TABLE  C)  TABLE  FIG. 14  VELOCITY =  V E L . = .03  TYPICAL SCALE:  .003  IN./SEC.  IN./SEC.  CHART  CHART  VELOCITY = . 2  V E L . = .2  IN./SEC.  IN./SEC.  CHART  CHART  VELOCITY =  VEL0CITY=5  T R A C E S O F T H E O S C I L L A T I O N FOR S Y S T E M 4 . I DIVISION = .0012 I N C H E S SLIDER DISPLACEMENT.  5  IN./SEC.  IN./SEC.  b)  TABLE  F I G . 15  VELOCITY =  .03  IN./SEC.  CHART  VELOCITY =  TYPICAL T R A C E S OF T H E OSCILLATION FOR SYSTEM 5 . S C A L E : I DIVISION = .0012 I N C H E S SLIDER DISPLACEMENT.  5  IN./SEC  ' u  Cl)  b)  TABLE  VELOCITY =  TABLE  FIG. 16  .015  VELOCITY=.03  IN./SEC.  IN./SEC.  CHART  CHART  VELOCITY=.2  VEL0CITY = .2  IN./SEC.  IN./SEC.  TYPICAL T R A C E S OF THE OSCILLATION FOR SYSTEM 6. S C A L E : I DIVISION = .0012 I N C H E S SLIDER DISPLACEMENT.  CHART  VELOCITY = 5  IN./SEC.  CHART  VELOCITY=5  IN./SEC.  System Number MicroInches  1  2  3  h  5  6  12.8  15.2  18.6  12.1  12.9  18.3 TABLE 2  MicroInches  7.U  8.1  9.6  7.2  7.5  8.3  K  w  OJ  lbs/in.  lbs.  Radians /sec.  155  119  85  62.6  U1.U  38.U  10  10  10  6  6  6  300  21*0  205  180  150  135  r lbs-sec. /in.  .0U6  .036  .026  .027  .026  .025  V  m lbs-sec.  in./sec.  c  • in./sec.  Jin. .0017  .0021  .0020  .0019  .0018  .0021  .0U5  .036  .032  .039  .0015  -2.2  .015  -1.0  .030  - .21  .0015  -2.7  .015  -1.1  .030  - .kO  .0015  -3.1  .015  -1.5  .030  - .65  .0030  -3.8  .015  -1.6  .030  - .81  .015  -2.01  .030  -1.3  .015  -2.5  .030  *i.U5  .01*8  .014;  N u m e r i c a l V a l u e s o f t h e Parameters f o r t h e 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 w i l 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 i n 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, i n Figs. 17 to 22, are not proportional to the static coefficient of f r i c t i o n . Hence this apparent "static" coefficient of f r i c t i o n under vibration conditions w i 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 i n such a manner as to produce a relationship with table velocity which is approximately exponential in shape.  With this i n 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 of k i n e t i c f r i c t i o n at the p a r t i c u l a r t a b l e v e l o c i t y .  Accordingly, using  t h e a p p r o p r i a t e r a t i o o f F., t h e 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  versus table v e l o c i t y .  t h e e x p e r i m e n t a l r e s u l t s suggest t h a t t h e breakaway c o e f f i c i e n t 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 ,  Since  of f r i c t i o n  e q u a t i o n k A , w i t h *U  replaced  by V , t a k e s t h e f o r m : v.  o  (^-^»)  31  Using the f o l l o w i n g experimental values: = ,k9,  y^°M  a  »22  and  B  =.a7e  " L / = .06 i n c h p e r s e c o n d , e q u a t i o n 31 M  becomes:  -8oV  /  T h i s e q u a t i o n i s shown by t h e s o l i d c u r v e i n F i g . 23. coefficient  Thus t h e  breakaway  o f f r i c t i o n v a r i e s i n t h e same manner as t h e f r i c t i o n - v e l o c i t y  curve f o r the surfaces fact,  32  +  under e x a m i n a t i o n .  Using t h i s experimentally  observed  e q u a t i o n s 26, 27, 28, 2 9 , and 30 can be e a s i l y a l t e r e d i n accordance  w i t h the relevant f r i c t i o n - v e l o c i t y curve, to p r e d i c t w i t h reasonable a c c u r a c y t h e dynamic performance o f t h e s l i d e r f o r t a b l e v e l o c i t i e s negative slope r e g i o n . coefficient,  Replacing the constant  t h e above m e n t i o n e d e q u a t i o n s  breakaway  become:  33  K = MAX.  .AW  3U  |^  -ill? -i MIN.  w i t h the  i n the  35  - 38 -  36  37  COS  - l) SI N CJ^Jt J  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 l o w e r c u r v e s r e p r e s e n t e q u a t i o n 35. represent  The s o l i d c u r v e s i n F i g s . 2it t o 29  e q u a t i o n 39 and t h e c u r v e s 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.  .03  .04-  .05 T A B L E  FI&. 2 3  GRAPH  SHOWING  THE RELATIONSHIP  BETWEEN  F R ICTION AND THE T A B L E VELOCITY FOR THE  .06 V E L O C I T Y  . 07  'V"  (|NCHES/SECOND).  T H E BREAKAWAY COEFFICIENT OF S l X SYSTEMS.  •ET-  UI  - 9n -  - en -  - 6n -  - os -  -IS -  - 25 -  -  -  CHAPTER  FIVE  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 t h e r e l a t i o n s h i p between  a m p l i t u d e o f v i b r a t i o n , 06 surface,  V.  , and t h e v e l o c i t y o f t h e l o w e r  the  traversing  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 c o n s t a n t v i s c o u s magnitude o f t h e damping c o e f f i c i e n t s  o r s t r u c t u r a l damping.  The  was s m a l l c o n s e q u e n t l y t h e  influence  o f s t r u c t u r a l damping on t h e 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 t h e r e s u l t s o f t h e f r i c t i o n - v e l o c i t y measurements, the i n v e s t i g a t i o n s velocities  i t i s apparent  o f f r i c t i o n a l v i b r a t i o n s were p e r f o r m e d f o r  i n the negative  The shape o f t h i s n e g a t i v e  slope r e g i o n of the  that  table  friction-velocity.curve.  s l o p e r e g i o n was a p p r o x i m a t e d 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 h a v i n g 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 t h e decay o f maximum s l i d e r d i s p l a c e m e n t i s a l s o 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  of  exponential friction.  I n a d d i t i o n , t h e v i b r a t i o n seemed t o v a n i s h a t a p o i n t c o r r e s p o n d i n g t o t h e "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 . that for table velocities  The g r a p h i c a l a p p r o a c h s u g g e s t e d  i n the p o s i t i v e slope p o r t i o n of the  v e l o c i t y curve, t r a j e c t o r i e s  friction-  c o u l d " m i s s " t h e v e r t i c a l a x i s k = V and p r o c e e d  to completion.  This represents  i n the negative  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  I t was t h e r e f o r e  t h e case o f smooth s l i d i n g .  Table  c o n c l u d e d t h a t t h e maximum s l i d e r  velocities  cycles.  displacement  seemed t o c o r r e s p o n d t o t h e v a l u e o f t h e 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 table velocity.  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 view of  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 t h e g e n e r a l e x p o n e n t i a l f o r t h e shape o f t h e n e g a t i v e  the  expression  slope region of the f r i c t i o n - v e l o c i t y  curve.  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 f o u n d .  F i g s . 17 t o 22 a r e e s s e n t i a l l y t h e envelope o f t h e 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 t h e v e l o c i t y range second.  0 < V ^ .002  i n c h per  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 s y s t e m 1 and +.002  i n c h f o r s y s t e m 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 .  l i e w i t h i n t h i s range.  I n t e r m e d i a t e systems a r e seen t o  W i t h t h i s a c c u r a c y 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 e x p e c t e d i n f i n a l p o s i t i o n i n g movement under s e r v o c o n t r o l .  The a c c u r a c y o f e q u a t i o n 39 i s poor a t  v e l o c i t i e s i n t h e range .02 < V < .03 i n c h p e r s e c o n d 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 a r e -k% a n d 120$  approximately.  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 t h e maximum s l i p v e l o c i t y f o r t h e s i x s y s t e m s . 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 ,  While the  experimental  i t w i l l be o b s e r v e d t h a t t h e t h e o r y  s e r v e s t o p r e d i c t g e n e r a l t r e n d s i n t h e v a r i a t i o n o f maximum s l i p v e l o c i t y with table velocity.  E n l a r g i n g the experimental records to obtain 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 w h i c h c o u l d account f o r some o f  the  s c a t t e r i n t h e 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 a r e 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 damping. slope  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 t h e  regions  negative negative  - 60 i s substituted i n t o equation 2. F i g . 10 we f i n d  Using the experimental value of 3^ from  TJ = -kO f o r system 1. A  The existence of the  slope region serves to excite the v i b r a t i o n s . we f i n d that  negative  For the p o s i t i v e slope region  V=^"Z/{ since the numerical value of Sp from F i g . 10 i s very  small. The term " e f f e c t i v e surface damping" was  introduced to describe  the component that i s added to the value of r i n equation 8 by v i r t u e of the slope of the f r i c t i o n curve. The oscillograph records of the vibrations shown i n Figs. 11 to i l l u s t r a t e several i n t e r e s t i n g features of the phenomenon.  16,  I t w i l l be  observed that at low table v e l o c i t i e s , breakaway of the s l i d e r i s very sudden while at increasing table v e l o c i t i e s breakaway becomes less sudden accompanied by decreased accelerations and s l i p v e l o c i t i e s .  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 ( F i g . 12 (a) ) that the junction forming process consists of s l i g h t i r r e g u l a r i t i e s suggesting the presence of minute f r i c t i o n a l vibrations.  At low table v e l o c i t i e s , the " s t i c k " period i s quite uniform and with the s e n s i t i v i t y used shows no v i s i b l e r e l a t i v e motion. v e l o c i t i e s the " s t i c k " period shows s l i g h t "sub-relaxation of which F i g . 13 (c) i s an example.  At higher table oscillations"  Possibly due to increased a c t i v i t y  and adjustment between the surfaces at higher table v e l o c i t i e s the period becomes less stable.  "stick"  No doubt some degree of adjustment does occur  between the surfaces at low table v e l o c i t i e s and t h i s s l i g h t movement becomes noticeable at higher table v e l o c i t i e s .  - 61 Fig. 13 (a.) is a good example of "dwell" that was sometimes noticed prior to the s l i p 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 s l i p 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 have been s t u d i e d i n some d e t a i l .  surfaces  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 negative slope 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.  The c o n c l u s i o n s  t o be d e r i v e d f r o m t h e work a r e : 1.  The v i b r a t i o n s a r e b a s i c a l l y a n o n - l i n e a r phenomenon.  2. E a c h o s c i l l a t i o n i s s e l f - e x c i t e d .  For a very short p e r i o d the  s y s t e m i s d r i v e n b y l a r g e n e g a t i v e damping by v i r t u e o f t h e s t e e p negative slope 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 f o r surfaces  the  i n question.  3 . The i n v e s t i g a t i o n s s u g g e s t e d t h a t t h e f o r m o f t h e decay o f v i b r a t i o n a m p l i t u d e 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 c u r v e i n t h e n e g a t i v e s l o p e r e g i o n were  similar.  l u The a m p l i t u d e 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 p o n d i n g t o t h e "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 order 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 theory p r e d i c t s t h a t the necessary conditions are: a) Reduce t h e r a t i o  -  by e i t h e r d e c r e a s i n g t h e l o a d between t h e  K surfaces  or i n c r e a s i n g the s t i f f n e s s  of the s u p p o r t i n g system.  b) Decrease 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 (>^SA) coefficients  of  friction.  c) Operate t h e system a t 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 <  f o r the surfaces  M  used.  I t i s c o n s i d e r e d t h a t t h e 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 i m p o r t a n t f e a t u r e s  o f t h e 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 c a p a b l e 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 t h e o r d e r o f .030 i n c h e s p e r second.  T a b l e v e l o c i t i e s i n excess o f t h i s v a l u e p r e v e n t adequate o b s e r v a -  t i o n of the v i b r a t i o n s .  If velocities  i n t h e range o f .1 t o .15  inches per  s e c o n d 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 w o u l d be  unnecessary.  The l i m i t a t i o n s on maximum a m p l i t u d e 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 s u g g e s t t h a t t h e s t i f f n e s s  of the  cantilever  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 p e r inch.  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 t h e  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  satisfactorily. '  To observe t h e e f f e c t o f damping, t h e p r e s e n t a p p a r a t u s 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 end o f t h e s l i d e r . damping s y s t e m .  damper on t h e  T h i s w q u l d p r o v i d e a c o n v e n i e n t and f l e x i b l e  variable  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 F i g . 3 can be closely approximated by an analytical relationship of the form;  y^=C,e where the terms "  ^plX+Z 4  P  " describe the positive slope region and the  6  exponential term describes the negative slope region. i t is clear that C^ = / ' -  .  A  y  IA  +^ V.  s  At the point ( ,  Referring to Fig. 3> , the numerical  value of C^ must be reduced to some negligible quantity to allow the remaining linear terms to describe the positive slope portion. the effective cutoff point for the exponential term. relative error with respect to  This point is  Introducing a 1%  , the boundary condition for evaluation  of the constant C 2 becomes: 11* V.*  ,  ^  2A  =^ M + . O I ^ M  giving;  for the negative slope region.  For the complete friction-velocity curve,  equation 3A becomes; -  i  w h i c h compares w i t h e q u a t i o n U .  n  )  - 65 APPENDIX 2.  CALIBRATION OF THE APPARATUS AND SUBSIDIARY TECHNIQUES  a) C a l i b r a t i o n o f t h e L o a d i n g System The n o r m a l l o a d W, between t h e 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 w e i g h t s t o t h e 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 w e i g h t s on t h e l o a d i n g p a n , t h e n o r m a l l o a d p r o d u c e d  a t t h e 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 t h e s l i d e r so t h a t known l o a d s 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 w e i g h t on t h e l o a d i n g p a n . b) C a l i b r a t i o n o f t h e T r a n s d u c e r 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 a n d t r a n s ducer d i s p l a c e m e n t 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 t h e a m p l i f i e r , t h e moving c o r e o f t h e 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 t h e o p p o s i t e end o f t h e moving c o r e a d i a l gage r e a d i n g  t o . 0 0 0 1 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 c o r e p r o d u c e d by s l i g h t movements i n t h e t a b l e .  c) C a l i b r a t i o n o f t h e 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 t h e 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 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 t h e c e n t e r - l i n e o f t h e s l i d e r a n d t h e 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 f r o m t h e end o f t h e s l i d e r t o t h e s t r a i n r i n g by means o f a small steel b a l l .  The d i a l gage was p l a c e d on t h e o p p o s i t e end o f t h e  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 . 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 t was not p o s s i b l e t o  i n t h i s instance since the ampli-  f i e r a n d 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 t h e C o e f f i c i e n t o f V i s c o u s  (Structural)  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 t h e 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  displacement.  The e q u a t i o n g o v e r n i n g t h e m o t i o n i s s i m i l a r t o e q u a t i o n 2 except y i e l d i n g a homogeneous free  d i f f e r e n t i a l equation.  W= 0  The o s c i l l o g r a p h t r a c e o f t h 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 paper.  The envelope was drawn over t h e peaks o f b o t h p o s i t i v e and  n e g a t i v e a m p l i t u d e s thus g i v i n g t h e v a l u e o f r . f r e q u e n c y o f t h e s y s t e m c o u l d a l s o be c a l c u l a t e d .  From t h i s , t h e n a t u r a l 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 T a b l e 2.  e) V e l o c i t y o f t h e D r i v e n S u r f a c e To determine t h e 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 t h e  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 t h e end o f t h e moving t a b l e .  against  Knowing t h e s p e e d o f t h e o s c i l l o g r a p h c h a r t ,  the slope of the r e s u l t i n g t r a c e produced the corresponding value of 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 .  table  A graph was t h e n drawn g i v i n g  t h e r e l a t i o n s h i p between v e l o c i t y V and c o n s o l e s e t t i n g N .  - 67 BIBLIOGRAPHY  1.  Lord Rayleigh, "Theory of Sound", Dover Publications 19U5, Second E d i t i o n I89U, p. 208.  2.  J . H. 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