Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Autonomous quasi-harmonic and forced vibration of frictional systems Ko, Pak Lim 1969

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1970_A1 K62.pdf [ 11.06MB ]
Metadata
JSON: 831-1.0102151.json
JSON-LD: 831-1.0102151-ld.json
RDF/XML (Pretty): 831-1.0102151-rdf.xml
RDF/JSON: 831-1.0102151-rdf.json
Turtle: 831-1.0102151-turtle.txt
N-Triples: 831-1.0102151-rdf-ntriples.txt
Original Record: 831-1.0102151-source.json
Full Text
831-1.0102151-fulltext.txt
Citation
831-1.0102151.ris

Full Text

AUTONOMOUS QUASI-HARMONIC AND FORCED VIBRATION OF FRICTIONAL SYSTEMS by PAK LIM KO B . S c , U n i v e r s i t y o f S t r a t h c l y d e , Glasgow, S c o t l a n d , 1963 M.A.Sc., U n i v e r s i t y o f B r i t i s h C o l u m b i a , Vancouver, B r i t i s h C o l u m b i a , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department o f M e c h a n i c a l E n g i n e e r i n g > We a c c e p t t h i s t h e s i s as co n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA Oc t o b e r , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e -quire m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of.my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t p u b l i c a t i o n , i n p a r t o r i n whole, o r the c o p y i n g o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . PAK LIM KO Department o f M e c h a n i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The b e h a v i o u r o f a system s u b j e c t t o q u a s i - h a r m o n i c type f r i c t i o n a l o s c i l l a t i o n was i n v e s t i g a t e d . The same f r i c t i o n a l system w i t h e x t e r n a l e x c i t a t i o n was a l s o i n v e s -t i g a t e d b o t h e x p e r i m e n t a l l y and t h e o r e t i c a l l y . V a r i o u s f r i c t i o n a l m a t e r i a l c o m b i n a t i o n s i n c l u d i n g s t e e l , polymer, r u b b e r and f i b r e m a t e r i a l s and l u b r i c a n t s were used t o p r o v i d e d i f f e r e n t forms o f f r i c t i o n c h a r a c t e r -i s t i c s . The dynamic f r i c t i o n - v e l o c i t y c u r v e s were o b t a i n e d by r e c o r d i n g s i m u l t a n e o u s l y the a c c e l e r a t i o n f o r c e , damp-i n g f o r c e , s p r i n g f o r c e and f r i c t i o n f o r c e d u r i n g one c y c l e o f the q u a s i - h a r m o n i c o s c i l l a t i o n . The c u r v e s were e x p r e s s e d as a f u n c t i o n o f s l i d i n g v e l o c i t y and were i r e p r e s e n t e d by n t h o r d e r p o l y n o m i a l s as w e l l as by exponen-t i a l e x p r e s s i o n s . The f i r s t a p p r o x i m a t i o n methods by K r y l o v and B o g o l i u b o f f were used t o s o l v e 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 s o f motion i n b o t h t h e autonomous and non-autonomous c a s e s . I n a d d i t i o n , the method o f , harmonic b a l a n c e was a l s o used i n the non-autonomous c a s e . In b o t h c a s e s , the Runge-Kutta n u m e r i c a l method was used t o i n v e s t i g a t e the t r a n s i e n t s t a t e o f t h e o s c i l l a t i o n s . T h e o r e t i c a l r e s u l t s f o r the autonomous system i n d i c a t e d t h a t the humped form f r i c t i o n - v e l o c i t y c u r v e was 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 i s t e n c e o f q u a s i - h a r m o n i c o s c i l l a t i o n . Subharmonic e n t r a i n m e n t a t the f r e q u e n c y o f t h e a u t o p e r i o d i c o s c i l l a t i o n o r harmonic e n t r a i n m e n t a t t h e e x t e r n a l ex-c i t a t i o n f r e q u e n c i e s , depending on the magnitude o f the e x t e r n a l e x c i t a t i o n , were a l s o p r e d i c t e d from the a n a l y s i s . E x p e r i m e n t a l r e s u l t s were o b t a i n e d from a p i n on d i s c t y pe f r i c t i o n a l system h a v i n g a t r a c k v e l o c i t y range of 0.04 i n / s e c t o 13.5 i n / s e c . E x t e r n a l e x c i t a t i o n f o r c e s were o b t a i n e d by a p p l y i n g the p r i n c i p l e o f o u t - o f - b a l a n c e mass. The f r e q u e n c y range o f the e x t e r n a l e x c i t a t i o n i s 0-90 c p s . The growth and decay o f the q u a s i - h a r m o n i c o s c i l l a t i o n was o b s e r v e d . I n t h e non-autonomous c a s e , 'quenching' o f t h e a u t o p e r i o d i c o s c i l l a t i o n by t h e e x t e r n a l e x c i t a t i o n was r e c o r d e d . I n g e n e r a l , the e x p e r i m e n t a l r e s u l t s s u b s t a n t i a t e t h e p r e d i c t i o n s o f the t h e o r e t i c a l a n a l y s e s . The e x p e r i m e n t a l r e s u l t s a l s o showed t h a t t h e v e r t i c a l e x t e r n a l e x c i t a t i o n has the e f f e c t o f r e d u c i n g t h e maximum s t a t i c f r i c t i o n and s u b s e q u e n t l y e x t i n g u i s h i n g s t i c k - s l i p o s c i l l a t i o n . TABLE OF CONTENTS Chapter Page I I n t r o d u c t i o n 1 I I H i s t o r i c a l Background 6 I I I T h e o r e t i c a l . . . . 3.1 I n t r o d u c t i o n 18 3.2 Form o f F r i c t i o n - V e l o c i t y F u n c t i o n R e q u i r e d f o r t h e E x i s t e n c e o f Q u a s i -Harmonic F r i c t i o n - I n d u c e d V i b r a t i o n . . . 23 3.3 Autonomous System 28 3.4 Non-Autonomous System 40 3.5 Summary 61 IV E x p e r i m e n t a l 4.1 I n t r o d u c t i o n 64 4.2 A p p a r a t u s 66 .4.3 I n s t r u m e n t a t i o n 72 4.4 Specimen 80 4.5 E x p e r i m e n t a l Methods 83 V R e s u l t s and D i s c u s s i o n s 5.1 Autonomous Case 88 5.2 Non-Autonomous Case 108 VI C o n c l u s i o n 132 R e f e r e n c e s 136 Appendix I D e r i v a t i o n o f Steady S t a t e A m p l i t u d e and Phase E q u a t i o n s from the E x p o n e n t i a l E x p r e s s i o n 4^2 V Chapter Page Appendix I I Harmonic B a l a n c e Method 144 Appendix I I I (1) System Parameters 147 (2) D e t e r m i n a t i o n o f System Damping C o e f f i c i e n t 148 Appendix IV C a l i b r a t i o n and S c a l i n g o f 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 S i g n a l s . . . 151 Appendix V Specimen C o m p o s i t i o n 154 LIST OF FIGURES F i g u r e Page 1.1.1 Displacement-Time Waveforms o f F r i c t i o n -Induced V i b r a t i o n 156 1.1.2 Schematic Diagrams o f Three F r i c t i o n a l Systems 157 3.1.1 T o p o l o g i c a l Diagrams I l l u s t r a t i n g S o f t and Hard S e l f - E x c i t a t i o n s 3.2.1 P o s s i b l e Forms o f F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curves 3.2.2 Humped Form o f a F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve 160 3.2.3 x-x Phase P l a n e Diagrams 161 3.3.1 Humped Form o f a F r i c t i o n - V e l o c i t y C h a r a c t e r -i s t i c Curve R e p r e s e n t e d by a S i m p l i f i e d E x p o n e n t i a l E x p r e s s i o n 162 3.3.2 T h e o r e t i c a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y Curve From the Humped F r i c t i o n -V e l o c i t y Curve o f F i g . 3.3.1 16 3 3.3.3 y,, y. vs V e l o c i t y P l o t Showing Regions o f s t a b i l i t y and i n s t a b i l i t y 164 "i 4.2.1 G e n e r a l Arrangement o f A p p a r a t u s and I n s t r u m e n t a t i o n 165 4.2.2 C l o s e - u p View o f A p p a r a t u s 166 4.2.3 I s o m e t r i c Diagram o f Apparatus 167 4.2.4 Diagram o f E x t e r n a l E x c i t a t i o n G e n e r a t o r . . 16 8 4.3.1 B l o c k Diagram o f I n s t r u m e n t a t i o n C i r c u i t r y 169 4.3.2 V e l o c i t y T r a n s d u c e r and d.c. A m p l i f i e r . . . 170 v i i F i g u r e Page 4.3.3 One C y c l e Sequence T r i g g e r C i r c u i t 171 5.1.1 One C y c l e O s c i l l o s c o p e Trace o f a F r i c t i o n -V e l o c i t y Curve and a x-x Phase P l a n e Diagram 172 5.1.2 Graph o f E x p e r i m e n t a l and T h e o r e t i c a l Curves - Type A l . . . 173 5.1.3 One C y c l e O s c i l l o s c o p e T r a c e s a t V a r i o u s D i s c V e l o c i t i e s - Type A l 174 5.1.4 Graph o f E x p e r i m e n t a l and T h e o r e t i c a l Curves - Type A2 175 5.1.5 Graph o f E x p e r i m e n t a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y Curves a t V a r i o u s Normal Loads - Type A2 176 5.1.6 Graph o f E x p e r i m e n t a l and T h e o r e t i c a l Curves - Type B l . . 177 5.1.7 Graph o f E x p e r i m e n t a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y Curves a t V a r i o u s Normal Loads - Type B2 178 5.1.8 F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curves o f Type C 179 5.1.9 x-x Phase P l a n e Diagrams o f t h e S t i c k -S l i p Type 180 5.1.10 Graph o f E x p e r i m e n t a l Curves Showing E f f e c t o f E x t e r n a l Damping - S t e e l on S t e e l . . . . 181 5.1.11 M i c r o p h o t o g r a p h s o f a Carbon F i b r e - R e s i n S l i d e r 182 5.1.12 F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curves - Carbon F i b r e - R e s i n on S t e e l and R e s i n on S t e e l 183 5.1.13 F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve Reproduced From a |One C y c l e O s c i l l o s c o p e Trace - Rubber on S t e e l 184 5.2.1 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - L i n e a r System 185 v i i i F i g u r e Page 5.2.2 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type C 186 5.2.3 x-x Phase P l a n e Diagram d u r i n g F o r c e d V i b r a t i o n - Type B l 187 5.2.4 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type B l 188 5.2.5 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency ? R a t i o - Type A2 189 5.2.6 Graphs Showing E f f e c t o f A m p l i t u d e o f V i b r a t i o n on the Average D i s p l a c e m e n t o f the S l i d e r D u r i n g F o r c e d V i b r a t i o n o f a F r i c t i o n a l System 190 5.2.7 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Parameters a t V a r i o u s D i s c V e l o c i t i e s - Non-Resonance 191 5.2.8 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Magnitude - Non-Resonance 192 5.2.9 Displacement-Time and V e l o c i t y - T i m e T r a c e s - Non-Resonance . 19 3 5.2.10 Graph o f A m p l i t u d e o f V i b r a t i o n vs Frequency R a t i o - Fundamental Resonance 194 5.2.11 Displacement-Time and V e l o c i t y - T i m e T r a c e s a t V a r i o u s E x t e r n a l E x c i t a t i o n F r e q u e n c i e s . 195 5.2.12 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s a t V a r i o u s E x t e r n a l E x c i t a t i o n F r e q u e n c i e s 196 5.2.13 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type A l 197 5.2.14 Disp l a c e m e n t - T i m e , and V e l o c i t y - T i m e and D i s p l a c e m e n t - V e l o c i t y T r a c e s - ^ Harmonic. . 19 8 2 5.2.15 Disp l a c e m e n t - T i m e , V e l o c i t y - T i m e and ; D i s p l a c e m e n t - V e l o c i t y T r a c e s - i. Harmonic. . 199 2 5.2.16 Displacement-Time, V e l o c i t y - T i m e ^ and D i s p l a c e m e n t - V e l o c i t y T r a c e s - — Harmonic. . 200 i i x F i g u r e Page 5.2.17 Displacement-Time, V e l o c i t y - T i m e and D i s p l a c e m e n t - V e l o c i t y T r a c e s - j Harmonic. . 201 5.2.18 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Magnitude 202 5.2.19 Graph o f E x t e r n a l E x c i t a t i o n Magnitude f o r the E x t i n c t i o n o f A u t o p e r i o d i c O s c i l l a t i o n vs E x t e r n a l E x c i t a t i o n Frequency 20 3 5.2.20 Displacement-Time and V e l o c i t y - T i m e T r a c e s a t V a r i o u s E x t e r n a l E x c i t a t i o n Magnitudes - 1_ Harmonic 204 4 5.2.21 D i s p l a c e m e n t - V e l o c i t y Phase P l a n e Diagrams 2 05 5.2.22 Displacement-Time and V e l o c i t y - T i m e T r a c e s - L i n e a r i s e d F r i c t i o n - V e l o c i t y Curve . . . 206 5.2.23 Graph o f A m p l i t u d e o f S t i c k - S l i p V i b r a t i o n and Maximum S t a t i c F r i c t i o n F o r c e vs Load R a t i o 207 5.2.24 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s I l l u s t r a t -i n g t h e E x t i n c t i o n o f S t i c k - S l i p V i b r a t i o n due t o Normal E x c i t a t i o n 208 5.2.25 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s a t V a r i o u s Normal E x c i t a t i o n - Type A l 209 5.2.26 x - t , x - t and x-x T r a c e s a t i - Harmonic -Normal E x c i t a t i o n 210 A . l C a l i b r a t i o n Curves o f E l a s t i c Beam . . . . 211 A.2 L o g a r i t h m i c Decrement O s c i l l o g r a p h T r a c e and One C y c l e O s c i l l o s c o p e Trace F o r The D e t e r m i n a t i o n o f System Damping C o e f f i c i e n t 212 LIST OF SYMBOLS Un F u n c t i o n s o f V i b r a t i o n A m p l i t u d e a F u n c t i o n s o f V i b r a t i o n A m p l i t u d e a C o e f f i c i e n t s o f e x p r e s s i o n s f o r the F r i c t i o n - v e l o c i t y C h a r a c t e r i s t i c Curves C o e f f i c i e n t s o f t h e N o n l i n e a r F u n c t i o n ( E x p o n e n t i a l ) C o e f f i c i e n t s o f the N o n l i n e a r F u n c t i o n ( P o l y n o m i a l ) 2, mo) h E x t e r n a l E x c i t a t i o n F o r c e Magnitude F r i c t i o n F o r c e F u n c t i o n F u n c t i o n o f D i m e n s i o n l e s s A b s o l u t e V e l o c i t y X F u n c t i o n o f V i b r a t i o n A m p l i t u d e , a F u n c t i o n s o f V i b r a t i o n A m p l i t u d e s , a and a e B e s s e l F u n c t i o n s B i n o m i a l C o e f f i c i e n t s F u n c t i o n o f V i b r a t i o n A m p l i t u d e and F r e -quency , a and a F u n c t i o n s o f V i b r a t i o n A m p l i t u d e and Phase, a and cf> E x t e r n a l E x c i t a t i o n Parameter, F Q a / ( l - a 2 ) F u n c t i o n s o f V i b r a t i o n A m p l i t u d e s , b^ and XI C o e f f i c i e n t s D e r i v e d from Cg, C^, ... e t c o f the P o l y n o m i a l E x p r e s s i o n C o e f f i c i e n t s D e r i v e d from E^, E 2 , ... e t c . C o e f f i c i e n t s D e r i v e d from Q^, Q 2, ... e t c . Damping F a c t o r , r/mto. F u n c t i o n s o f a, ty and a x D i m e n s i o n l e s s V e l o c i t y D i m e n s i o n l e s s 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 X - 6 X - [F / ( 1 - a 2 ) ] s i n a x o X 2 F u n c t i o n o f a Steady S t a t e A m p l i t u d e o f A u t o p e r i o d i c O s c i l l a t i o n Steady S t a t e A m p l i t u d e o f O s c i l l a t i o n (Fundamental Resonance) Steady S t a t e A m p l i t u d e o f H e t e r o p e r i o d i c O s c i l l a t i o n C o n s t a n t s E c c e n t r i c i t y o f Out-Of-Balance Weight i n E x t e r n a l E x c i t a t i o n F o r c e l b Dynamic F r i c t i o n F o r c e l b F u n c t i o n s o f X L i n e a r Parameter i n S p r i n g S t i f f n e s s , a l s o used as l b / i n V a r i a b l e I n t e g e r s Mass o f V i b r a t o r y System l b / i n / s e c x i i n Order o f P o l y n o m i a l s r Damping C o e f f i c i e n t l b / i n / s e c t Time sec v D i s c V e l o c i t y i n / s e c w Normal Load l b x D i s p l a c e m e n t i n x A b s o l u t e V e l o c i t y i n / s e c x A c c e l e r a t i o n i n / s e c y^, Y2 F u n c t i o n s o f V a Frequency R a t i o , v/co 3 Load R a t i o , nf/w Y V a l u e I n d i c a t e s S m a l l n e s s cb, \b Phase p Mass o f O u t - o f - B a l a n c e Weight l b / i n / s e c u, , u C o e f f i c i e n t o f F r i c t i o n JC s a) Damped N a t u r a l Frequency r a d / s e c u>n N a t u r a l Frequency r a d / s e c v E x t e r n a l E x c i t a t i o n Frequency r a d / s e c T D i m e n s i o n l e s s Time c;, 5 , e , a , n , 8 C o n s t a n t s fi $ A A F u n c t i o n s o f V i b r a t i o n A m p l i t u d e , a ACKNOWLEDGEMENT The a u t h o r i s g r a t e f u l f o r t h e many h e l p f u l s u g g e s t -i o n s f r o m t h e f a c u l t y and g r a d u a t e s t u d e n t s i n t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g and f o r t h e a s s i s t a n c e o f t h e t e c h n i c a l s t a f f who c o n s t r u c t e d t h e e x p e r i m e n t a l a p p a r a t u s . S i n c e r e a p p r e c i a t i o n i s a l s o e x p r e s s e d t o Mr. J . E . J o n e s , a s t a f f member o f t h e T r i b o l o g y L a b o r a t o r y , whose e n t h u s i a s t i c a s s i s t a n c e g r e a t l y a c c e l e r a t e d t h e r e -s e a r c h programme. T h a n k s must a l s o be e x p r e s s e d t o t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g f o r t h e u s e o f t h e i r f a c i l i t i e s . P a r t o f t h e e x p e r i m e n t a l a p p a r a t u s u s e d i n t h i s i n v e s t i g a t i o n was p r e v i o u s l y c o n s t r u c t e d b y a f e l l o w g r a d u a t e s t u d e n t , Mr. H.R. D a v i s , and i s g r a t e f u l l y a c k n o w l e d g e d . S p e c i a l t h a n k s a r e due t o D r . C A . B r o c k l e y f o r h i s g u i d a n c e and c o n s t a n t e n c o u r a g e m e n t t h r o u g h o u t t h e r e s e a r c h programme. The e x p e r i m e n t a l w o r k was c a r r i e d o u t i n t h e T r i b o l o g y L a b o r a t o r y o f t h e D e p a r t m e n t 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 C o l u m b i a . 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 f r o m t h e D e f e n c e R e s e a r c h B o a r d o f C a n a d a u n d e r G r a n t No. 7 5 1 0 - 3 1 . I INTRODUCTION The f r i c t i o n c h a r a c t e r i s t i c s r e s u l t i n g from the motion o f one s u r f a c e over a n o t h e r form an i m p o r t a n t f a c e t o f the b e h a v i o u r o f many p h y s i c a l systems. When two s o l i d b o d i e s a r e rubbed t o g e t h e r , v i b r a t i o n o f some type f r e -q u e n t l y o c c u r s , w h i c h may, i n g e n e r a l , be c a l l e d ' f r i c t i o n -i n d u c e d v i b r a t i o n ' . F r i c t i o n - i n d u c e d v i b r a t i o n has been obser v e d i n a wide v a r i e t y o f systems. I n a u t o m a t i c t r a n s -m i s s i o n s , the engagement o f t h e c l u t c h depends on the f r i c t i o n c h a r a c t e r i s t i c o f the f l u i d and the f a c e m a t e r i a l s o f the c l u t c h . I n p o s i t i o n i n g systems t h e a c c u r a c y and the s e n s i t i v i t y o f response are g r e a t l y i m p a i r e d by v i b r a -t i o n i n d u c e d by t h e f r i c t i o n o f the s l i d i n g s u r f a c e s . The Froude pendulum and t h e mo t i o n o f v i o l i n s t r i n g under the a c t i o n o f a bow are f r e q u e n t l y c i t e d as examples o f f r i c t i o n - i n d u c e d v i b r a t i o n . Two forms o f autonomous f r i c t i o n - i n d u c e d v i b r a t i o n may be c l a s s i f i e d , namely s t i c k - s l i p v i b r a t i o n and q u a s i -harmonic v i b r a t i o n , F i g . 1.1.1. The s t i c k - s l i p o r r e l a x -a t i o n o s c i l l a t i o n i s c h a r a c t e r i s e d by the saw t o o t h d i s -p l a c e m e n t - t i m e waveform, whereas the q u a s i - h a r m o n i c v i b r a t i o n has a waveform which i s a p p r o x i m a t e l y s i n u s o i d a l . In t h e case o f s t i c k - s l i p o s c i l l a t i o n the regimes o f s t i c k and s l i p c o n s t i t u t e the complete v i b r a t i o n c y c l e . The s t i c k phase i s dependent on s t a t i c f r i c t i o n f o r c e s . A t 2 the end o f s t i c k , sudden r e l a x a t i o n o c c u r s and d u r i n g t h i s movement the system i s governed by dynamic f r i c t i o n f o r c e s . However, i n the case o f q u a s i - h a r m o n i c o s c i l l a t i o n t h e r e i s always r e l a t i v e movement between the two s l i d i n g s u r -f a c e s t h e r e f o r e the motion i s governed by dynamic f r i c t i o n f o r c e s o n l y . The b a s i c mechanism o f f r i c t i o n between u n l u b r i c a t e d s u r f a c e s can be c o n s i d e r e d as a r i s i n g from two main f a c t o r s , namely a d h e s i o n and d e f o r m a t i o n . When two m e t a l s u r f a c e s are p l a c e d t o g e t h e r o n l y s m a l l a r e a s a re a c t u a l l y i n c o n t a c t [ 1 ] . Under c o n d i t i o n s o f s m a l l c o n t a c t i n g a r e a s p l a s t i c : f l o w a t t h e c o n t a c t a r e a s u s u a l l y o c c u r s under r e l a t i v e l y l i g h t l o a d s , and a j u n c t i o n i s formed between the t i p s o f the a s p e r i t i e s . The s t a t i c f r i c t i o n i s r e l a t e d t o the j u n c t i o n growth t h e o r y . I t i s g e n e r a l l y b e l i e v e d t h a t the s t a t i c f r i c t i o n i s time-dependent and t h a t t h e k i n e t i c f r i c t i o n i s v e l o c i t y dependent. More r e c e n t l y , t h e s t r a i n r a t e w h i c h i s a f u n c t i o n o f the d r i v i n g v e l o c i t y and o f the s p r i n g s t i f f n e s s o f the system has been c o n s i d e r e d t o be an i m p o r t a n t a s p e c t o f s t a t i c f r i c t i o n b e h a v i o u r [ 2 ] . The s t i c k - s l i p t y p e o f f r i c t i o n - i n d u c e d v i b r a t i o n i s u s u a l l y a t t r i b u t e d t o the k i n e t i c f r i c t i o n f o r c e b e i n g lower t h a n the s t a t i c f r i c t i o n f o r c e . Under t h e s e c o n d i t i o n s the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c has t h e e f f e c t o f i n t r o d u c i n g n e g a t i v e damping w h i c h a s s i s t s any s m a l l d i s -t u r b a n c e t o grow i n t o f u l l v i b r a t i o n c y c l e s . The phenom-enon of t h e s t i c k - s l i p t y pe o f f r i c t i o n - i n d u c e d v i b r a t i o n has r e c e i v e d e x t e n s i v e c o n s i d e r a t i o n , a l t h o u g h s a t i s f a c t o r y e x p l a n a t i o n s are l a c k i n g i n many a r e a s , such as the i n -t e r e s t i n g q u e s t i o n as t o j u s t how the i n i t i a l drop from s t a t i c f r i c t i o n o c c u r s , whether i n s t a n t a n e o u s l y o r o v e r a f i n i t e i n t e r v a l o f t i m e , o r o v e r a f i n i t e d i s t a n c e . U n t i l t h e p r e s e n t work v e r y l i t t l e was known about the q u a s i - h a r m o n i c type o f f r i c t i o n - i n d u c e d v i b r a t i o n o t h e r t h a n i t was somehow r e l a t e d t o the v a r i a t i o n o f f r i c t i o n f o r c e w i t h s l i d i n g v e l o c i t y . A c t u a l f r i c t i o n c o u p l e s d i s p l a y a v a r i e t y o f forms o f dynamic f r i c t i o n c u r v e . I t w i l l be shown t h a t t h e e x i s t e n c e o f t h e q u a s i -harmonic o s c i l l a t i o n i s c r i t i c a l l y dependent on the p a r t i c -u l a r shape o f t h e dynamic f r i c t i o n c u r v e . The s u b j e c t o f c o n t r o l l i n g o r r e d u c i n g t h e f r i c t i o n by a p p l y i n g v i b r a t i o n t o the f r i c t i o n p a r t s has been used i n p r a c t i c e f o r some t i m e . V i b r a t o r s are a t t a c h e d t o i n s t r u m e n t p a n e l s t o keep n e e d l e s f r e e and i n m o t i o n . ' D i t h e r i n g ' d e v i c e s were used on the g u i d i n g f i n s o f some e a r l y a i r t o a i r m i s s i l e s t o o b t a i n f a s t response from th e servomechanism. I t has been shown t h a t s o n i c v i b r a t i o n r educes th e s t a t i c f r i c t i o n d u r i n g u n l u b r i c a t e d s l i d i n g . However, t h e r e have been v e r y few attempts t o i n v e s t i g a t e t h i s s u b j e c t s y s t e m a t i c a l l y . The p r e s e n t work i s concerned m a i n l y w i t h the q u a s i -harmonic type o f f r i c t i o n - i n d u c e d v i b r a t i o n . Owing t o the l a r g e amount of work a l r e a d y i n v o l v e d i n the p r e s e n t 4 s t u d i e s , no attempt was made t o i n v e s t i g a t e the fundamental a s p e c t of the c o n t a c t i n g s u r f a c e s , r a t h e r , the i n v e s t i -g a t i o n d e a l s m a i n l y w i t h the dynamic b e h a v i o u r o f a system s u b j e c t t o q u a s i - h a r m o n i c f r i c t i o n - i n d u c e d v i b r a t i o n . Three systems s u b j e c t e d t o f r i c t i o n f o r c e s which v a r y as a f u n c t i o n o f r e l a t i v e s l i d i n g v e l o c i t y have been i n v e s t i g a t e d . The f i r s t system, which c o n s i s t e d o f a mass, a s p r i n g and a damper but w i t h no e x t e r n a l e x c i t a t i o n , was used f o r the i n v e s t i g a t i o n s o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e and o f f r i c t i o n - i n d u c e d v i b r a t i o n . An e x t e r n a l harmonic e x c i t a t i o n i n the d i r e c t i o n o f the f r i c t i o n f o r c e was a p p l i e d t o the b a s i c elements i n the second system, whereas i n the t h i r d a h a r m o n i c a l l y f l u c t u a t i n g normal f o r c e was a p p l i e d t o the l o a d i n g end o f the b a s i c elements ( F i g . 1.1.2). Many p a s t e x p e r i m e n t s on f r i c t i o n and f r i c t i o n -i n d u c e d v i b r a t i o n have shown t h a t p r e c i s e f r i c t i o n measure-ment ds not an easy t a s k and t h a t f r i c t i o n i s s e n s i t i v e t o a number o f f a c t o r s which are d i f f i c u l t t o c o n t r o l i n d i v i d u a l l y . G e n e r a l l y i t s h o u l d be o b s e r v e d t h a t r e -p r o d u c i b l e r e s u l t s are d i f f i c u l t t o o b t a i n and i n the p a s t th e c o r r e l a t i o n between e x p e r i m e n t a l r e s u l t s and t h e o r -e t i c a l a n a l y s i s has been u n s a t i s f a c t o r y . Owing t o the i mportance o f the f r i c t i o n measurements i n v o l v e d i n the p r e s e n t i n v e s t i g a t i o n , the d e s i g n and development o f a r e l i a b l e f r i c t i o n a p p a r a t u s and i t s accompanying i n s t r u m e n -t a t i o n p l a y e d an i m p o r t a n t r o l e . 5 The p r e s e n t i n v e s t i g a t i o n c o n s t i t u t e s the f i r s t complete and thorough s t u d y o f the q u a s i - h a r m o n i c f r i c t i o n -i n d u c e d v i b r a t i o n , p a r t i c u l a r l y i n t h e cases o f e x t e r n a l e x c i t a t i o n . The a p p l i c a t i o n o f n o n l i n e a r mechanics r e -v e a l e d the dynamic b e h a v i o u r o f the q u a s i - h a r m o n i c f r i c t i o n - i n d u c e d v i b r a t i o n b o t h w i t h and w i t h o u t e x t e r n a l e x c i t a t i o n . The r e l i a b l e f r i c t i o n a p p a r a t u s which was s u b s e q u e n t l y d e v e l o p e d t o g e t h e r w i t h the t e c h n i q u e f o r t h e a c c u r a t e measurement o f f r i c t i o n and f r i c t i o n - i n d u c e d v i b r a t i o n p r o v i d e d a s a t i s f a c t o r y c o r r e l a t i o n between t h e t h e o r e t i c a l a n a l y s i s and e x p e r i m e n t a l r e s u l t s . I I HISTORICAL BACKGROUND A l t h o u g h t h e phenomenon o f f r i c t i o n - i n d u c e d v i b r a t i o n has been o b s e r v e d f o r a l o n g t i m e , i t i s o n l y d u r i n g the p r e s e n t c e n t u r y t h a t r e a l advances have been made i n g a i n i n g some u n d e r s t a n d i n g o f the mechanics o f the p r o c e s s e s i n v o l v e d . The e f f e c t o f speed on the v a l u e o f c o e f f i c i e n t o f f r i c t i o n was f i r s t d i s c u s s e d by Coulomb [ 3 ] , he found t h a t when d i f f e r e n t m e t a l s were rubbed t o g e t h e r , the f r i c t i o n f o r c e i n c r e a s e d w i t h speed. I n 1929 W e l l s [4] experimented w i t h a machine f o r measuring k i n e t i c f r i c t i o n under c o n d i t i o n s o f boundary l u b r i c a t i o n . The f r i c t i o n f o r c e was measured by means o f a t r i f i l a r s u s p e n s i o n o f w e i g h t s a r r a n g e d i n such a way t h a t the r e s t o r i n g f o r c e i n c r e a s e d w i t h t h e d i s p l a c e m e n t . Thomas [5] was p o s s i b l y the f i r s t t o demonstrate the p o s s i b i l i t y o f s e l f - e x c i t e d type o s c i l l a t i o n s . H i s g r a p h i c a l a n a l y s i s showed t h a t i n the absence o f v i s c o u s damping, s t a b l e o s c i l l a t i o n s e i t h e r o f t h e s i m p l e harmonic type o r o f the s t i c k - s l i p t y p e c o u l d o c c u r depending on whether 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 e q u a l t o o r s m a l l e r t h a n the 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 , w i t h 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 b e i n g independent o f the s l i d i n g v e l o c i t y . H i s f i n d i n g s a l s o suggested t h a t the 7 s i m p l e harmonic t y p e o s c i l l a t i o n s can no l o n g e r be s u s t a i n e d i n t h e pr e s e n c e o f v i s c o u s damping b u t i f the damping i s not e x c e s s i v e the s t i c k - s l i p t y p e o s c i l l a t i o n s can be m a i n t a i n e d . However, he showed no e x p e r i m e n t a l r e s u l t s t o s u b s t a n t i a t e h i s a n a l y s i s . I t i s q u i t e o b v i o u s t h a t h i s ; c o n d i t i o n s f o r t h e s i m p l e harmonic t y p e o s c i l l a t i o n would be e q u i v a l e n t t o a m a s s - s p r i n g system w i t h f r e e o s c i l l a t i o n , w i t h t h e s t a t i c f r i c t i o n f o r c e b e i n g the i n i t i a l c o n d i t i o n . I n 19 33, K a i d a n o v s k y and Hay k i n [6] made a s t u d y o f r e l a x a t i o n o s c i l l a t i o n s as a p p l i e d t o m e c h a n i c a l 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 the v e l o c i t y . They a s s e r t e d t h a t a n e c e s s a r y 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 t h a t a r e g i o n must e x i s t f o r w h i c h the 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 . I t was Papenhuyzen [ 7 ] , i n h i s i n v e s t i -g a t i o n o f the mechanics o f t h e s k i d d i n g o f a u t o m o b i l e t i r e s , who c l a s s i f i e d f r i c t i o n - i n d u c e d v i b r a t i o n i n t o the two g e n e r a l t y p e s w i t h r e l a t i o n t o the d r i v e n s u r f a c e v e l o c i t y . He showed the o c c u r r a n c e o f the s t i c k - s l i p t y p e r e l a x a t i o n o s c i l l a t i o n and the s i m p l e harmonic t y p e o s c i l l a t i o n a t d i f f e r e n t s t a g e s o f the d r i v e n s u r f a c e v e l o c i t y v a r i a t i o n . He a l s o showed t h a t a t low d r i v e n s u r f a c e v e l o c i t i e s t he v i b r a t i n g member remained s t a t i o n a r y a t a c o n s t a n t d i s p l a c e d p o s i t i o n . Bowden and Lebon [8] c a r r i e d o u t a s e r i e s o f e x p e r -iments on the f r i c t i o n between s l i d i n g m e t a l s i n the absence o f a l u b r i c a t i n g f i l m . I n t h e i r e x p e r i m e n t s the-r e c o r d e d s l i d i n g v e l o c i t i e s were v e r y h i g h i n comparison w i t h t h e d r i v e n s u r f a c e v e l o c i t y . They s u g g e s t e d t h a t l o c a l w e l d i n g c o u l d r e s u l t from the h i g h temperature f l a s h e d u r i n g t h e s l i p s t a g e o f the s t i c k - s l i p t y p e o s c i l l a t i o n s . However, B l o k [9] i n d i c a t e d t h a t i t i s u n l i k e l y t h a t the w e l d i n g e f f e c t i s the main r e a s o n f o r the o c c u r r e n c e o f the f r i c t i o n a l o s c i l l a t i o n s . He showed t h a t such v i b r a -t i o n s may s i m p l y r e p r e s e n t a form o f r e l a x a t i o n o s c i l l a t i o n w h ich may, i n t u r n , depend upon the p a r t i c u l a r form o f f r i c t i o n - v e l o c i t y c urve and the amount o f damping i n the system. He suggested t h a t the e s s e n t i a l c o n d i t i o n f o r the o c c u r r e n c e o f f r i c t i o n a l o s c i l l a t i o n s i s d e c r e a s i n g f r i c t i o n a l f o r c e f o r i n c r e a s i n g s l i d i n g v e l o c i t y . He e s t a b l i s h e d a q u a n t i t a t i v e c r i t e r i o n f o r t h e o n s e t o f s t i c k - s l i p t y p e o s c i l l a t i o n u s i n g a r e l a t i o n s h i p p l o t t e d between d i m e n s i o n l e s s parameters o f damping and s l i d i n g v e l o c i t y . F u r t h e r i n v e s t i g a t i o n o f the temperature f l a s h and the f r i c t i o n b e h a v i o u r d u r i n g the s l i p p o r t i o n o f t h e s t i c k - s l i p p r o c e s s was c a r r i e d out by Morgan, Muskat and Reed [10] and l a t e r by Sampson [ 1 1 ] . V a r i o u s k i n d s o f s l i d i n g s u r f a c e s and l u b r i c a n t s . were used by B r i s t o w [ 1 2 ] , [13] t o s t u d y 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 and t h e temperature e f f e c t s . I n h i s account o f the f r i c t i o n a l o s c i l l a t i o n s he s t a t e d t h a t the e x i s t e n c e o f a n e g a t i v 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 i s a n e c e s s a r y c o n d i t i o n f o r r e l a x a t i o n o s c i l l a t i o n s t o be 9 e x c i t e d i n an e l a s t i c system. He showed the e x i s t e n c e o f m i c r o - s l i p movement d u r i n g the s t i c k s t a g e . He a l s o s uggested t h a t the c o n d i t i o n f o r the e x i s t e n c e o f t h e q u a s i - s i n u s o i d a l o s c i l l a t i o n i s d e c r e a s i n g o f f r i c t i o n w i t h i n c r e a s e s o f v e l o c i t y i n the h i g h v e l o c i t y r e g i o n . The e x p l a n a t i o n i s not c o n v i n c i n g . In f a c t , i n the d i s c u s s i o n o f [13] S w i f t p o i n t e d o ut t h a t a g r a p h i c a l s o l u t i o n i n d i c a t e d no q u a s i - s i n u s o i d a l o s c i l l a t i o n s a t s u p e r - c r i t i c a l speeds i n a system w i t h a n e g a t i v 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 as suggested by B r i s t o w . E a r l i e r , a s i m p l e g r a p h i c a l method f o r d e t e r m i n i n g the v i b r a t i o n c y c l e from any f r i c t i o n - v e l o c i t y c u r v e was de v e l o p e d by Dudley and S w i f t [ 1 4 ] . The method makes d i r e c t 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 . They a p p l i e d the method t o s t u d y t h e growth and decay o f the s e l f - i n d u c e d v i b r a t i o n w i t h r e l a t i o n t o v a r i o u s f r i c t i o n -v e l o c i t y c u r v e s . They showed t h a t as the speed i n c r e a s e d d u r i n g t h e s t i c k - s l i p c o n d i t i o n s , the a m p l i t u d e o f the o s c i l l a t i o n s i n c r e a s e d . The r e s u l t s o f the i n c r e a s i n g a m p l i t u d e s d i d not agree w i t h the f i n d i n g o f some o t h e r • a u t h o r s . B r o c k l e y , Cameron and P o t t e r [15] showed t h a t the a m p l i t u d e o f o s c i l l a t i o n d e c r e a s e d as t h e d r i v i n g s u r f a c e v e l o c i t y was i n c r e a s e d . The d i f f e r e n c e would seem t o l i e i n the f a c t t h a t i n the g r a p h i c a l s o l u t i o n o f Dudley and S w i f t , t h e y have assumed t h a t the s t a t i c f r i c t i o n was the same as t h e k i n e t i c f r i c t i o n a t z e r o s l i d i n g v e l o c -i t y ; w h i l e t h i s may be t r u e i n some c a s e s , i t c e r t a i n l y cannot be c o n s i d e r e d as a g e n e r a l i s e d phenomenon. In h i s i n v e s t i g a t i o n o f a u t o m o b i l e brake s q u e a l , S i n c l a i r [16] a g a i n showed 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 caused by an i n v e r s e v a r i a t i o n o f 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 s l i d i n g v e l o c i t y . He a l s o s u g g ested t h a t t h e d e c r e a s e i n f r i c t i o n o b s e r v e d a t h i g h v e l o c i t y i s caused by the h i g h t e m p e r a t u r e s d e v e l o p e d a t h i g h v e l o c i t y . T h i s be-h a v i o u r was f u r t h e r d i s c u s s e d by R a b i n o w i c z [ 1 7 ] . He s t a t e d t h a t the n e g a t i v e f r i c t i o n - v e l o c i t y shape a t h i g h v e l o c i t y i s connected w i t h t h e r m a l s o f t e n i n g w h i c h produces a low shear s u r f a c e f i l m on a h a r d e r s u b s t r a t u m . He a l s o d e f i n e d the s t i c k - s l i p o s c i l l a t i o n s as b e i n g time c o n t r o l l e d and the q u a s i - s i n u s o i d a l o s c i l l a t i o n s as b e i n g v e l o c i t y c o n t r o l l e d . J a r v i s and M i l l s [18] i n v e s t i g a t e d the v i b r a t i o n caused by d r y f r i c t i o n i n a s i m u l a t e d d i s c brake system. T h e i r i n v e s t i g a t i o n shows t h a t unwanted v i b r a t i o n i n any system can p o s s i b l y be a v o i d e d by c a r e f u l c h o i c e o f d i m e n s i o n i n the d e s i g n . The s i g n i f i c a n c e o f t h e i r work i s the d e m o n s t r a t i o n o f the ' g e o m e t r i c a l l y i n d u c e d 1 i n s t a b i l i t y o f two e l a s t i c components i n t e r a c t i n g t h r o u g h the agency o f k i n e t i c d r y f r i c t i o n . However, i t i s d o u b t f u l t h a t the l i n e a r i s e d f r i c t i o n - v e l o c i t y c h a r a c t e r -i s t i c used i n t h e i r t h e o r e t i c a l a n a l y s i s a d e q u a t e l y r e p r e -s e n t e d the r e a l f r i c t i o n - v e l o c i t y r e l a t i o n o f t h e system. D e r j a g u i n , Push and T o l s t o i [ 1 9 ] , S i n g h [ 2 0 ] , Cook [21] and B r o c k l e y , Cameron and P o t t e r [15] a l l p r e s e n t e d t h e o r e t i c a l a n a l y s i s o f t h e s t i c k - s l i p t y p e f r i c t i o n -i n d u c e d v i b r a t i o n s . I n t h e a n a l y s e s the k i n e t i c f r i c t i o n c h a r a c t e r i s t i c was assumed t o be l i n e a r and t h e s t a t i c f r i c t i o n c h a r a c t e r i s t i c was c o n s i d e r e d t o be t i m e -dependent . A t s u s h i W a t a r i and Takanao Sugimoto [22] i n v e s t i -g a t e d t h e s e l f - e x c i t e d v i b r a t i o n s caused by d r y f r i c t i o n w i t h a d e c r e a s i n g 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 . I n the t h e o r e t i c a l i n v e s t i g a t i o n an attempt was made t o a n a l y s e the f r i c t i o n system u s i n g a n o n l i n e a r method as w e l l as l i n e a r methods. However, many assumptions were made t o s i m p l i f y t h e n o n l i n e a r f r i c t i o n - v e l o c i t y f u n c t i o n , thus r e s t r i c t e d t h e u s e f u l n e s s o f the a n a l y s i s . They showed t h a t the v i b r a t i o n a m p l i t u d e s i n c r e a s e w i t h t h e i n c r e a s e o f v e l o c i t y and t h a t t h e f r e q u e n c i e s a re n e a r l y e q u a l t o the system's n a t u r a l f r e q u e n c y . The v i b r a t i o n d i e d o u t a t a c e r t a i n v e l o c i t y . S i m i l a r r e s u l t s were o b s e r v e d by S h i z u o D o i and Shinobu Kato [23] , i n t h e i r i n v e s t i g a t i o n o f c h a t t e r v i b r a t i o n o f f l e x i b l e l a t h e t o o l s . They a l s o a t tempted t o c o r r e l a t e t h e growth and decay o f t h e v i b r a t i o n w i t h the shape o f t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c . The s t u d y o f machine t o o l s l i d e w a y s has been t h e . s u b j e c t o f c o n s i d e r a b l e e f f o r t r e c e n t l y . These i n v e s t i -12 g a t i o n s have u s u a l l y i n v o l v e d a s y s t e m a t i c s t u d y o f s l i d e -way f r i c t i o n and the s t u d y o f the s t i c k - s l i p t y p e o s c i l -l a t i o n s . Some i m p o r t a n t c o n t r i b u t i o n s r e g a r d i n g the i n v e s t i g a 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 c h a r a c t e r i s t i c have been r e p o r t e d . In h i s s t u d y of the s t a b i l i t y o f s l i d i n g m otion Stepanek [24] c l a r i f i e d the d i f f e r e n c e between f r i c t i o n t e s t s c a r r i e d out under s t e a d y - s t a t e ( z e r o a c c e l e r a t i o n ) c o n d i t i o n w hich y i e l d e d v a l u e s of k i n e t i c f r i c t i o n and t hose c a r r i e d out under dynamic c o n d i t i o n s where the r o l e o f a c c e l e r a t i o n was r e c o g n i s e d . However, i n the subse-quent t h e o r e t i c a l a n a l y s i s , o n l y a s i m p l i f i e d l i n e a r f r i c t i o n -v e l o c i t y c h a r a c t e r i s t i c w i t h a c o r r e c t i o n term f o r a c c e l e r -a t i o n was used. The paper f a i l s t o show the methods f o r o b t a i n i n g the two d i f f e r e n t t y p e s o f f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c . No e x p e r i m e n t a l r e s u l t s were p r e s e n t e d t o v e r i f y t h e t h e o r e t i c a l a n a l y s i s . Hunt, Torbe and Spencer [25] a p p l i e d the phase-plane t r a j e c t o r y a n a l y s i s t o i n v e s t i g a t e the s t i c k - s l i p m o tion a r i s i n g from machine t o o l p r a c t i c e . I n t h e i r a n a l y s i s t hey r e c o g n i s e d the r o l e o f t h e a c c e l e r a t i o n i n the f r i c t i o n a l v i b r a t i o n system, but f a i l e d t o o b t a i n a unique f r i c t i o n c u r v e . An i n g e n i o u s e x p e r i m e n t a l t e c h n i q u e t o d e t e r m i n e the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c was d e v e l o p e d by B e l l and B u r d e k i n [26] f o r e x a m i n i n g the dynamic a s p e c t s o f s l i d e w a y f r i c t i o n . In d e t e r m i n i n g th e f r i c t i o n c h a r a c t e r -i s t i c , t he a c c e l e r a t i o n o f the v i b r a t o r y movement as w e l l as t h e d i s p l a c e m e n t were r e c o r d e d . The f o r e g o i n g measure-ments p e r m i t t e d t h e d e t e r m i n a t i o n o f the dynamic f r i c t i o n -v e l o c i t y curve d u r i n g one c y c l e o f v i b r a t i o n . The method appears t o g i v e a r e a l i s t i c assessment o f dynamic f r i c t i o n -v e l o c i t y c u r v e . The fundamental o f the f r i c t i o n mechanism r e l a t i n g the s t a t i c f r i c t i o n c h a r a c t e r i s t i c has r e c e i v e d a wide range o f i n v e s t i g a t i o n . R a b i n o w i c z [27] e x p l a i n e d t h e h i g h s t a t i c c o e f f i c i e n t as b e i n g due t o t h e m e t a l l i c j u n c t i o n becoming s t r o n g e r a f t e r the s u r f a c e s have been i n s t a t i o n a r y c o n t a c t f o r some t i m e . He a l s o suggested t h a t t h e s t a t i c c o e f f i c i e n t u and the k i n e t i c c o e f f i c i e n t u, S JC are f u n c t i o n a l l y r e l a t e d , s i n c e u g f o r a c o n t a c t time t i s the same as f o r a s l i d i n g v e l o c i t y v, p r o v i d e d t h a t t = d/v where d i s t h e mean s i z e o f the j u n c t i o n formed a t the i n t e r f a c e [ 2 8 ] . A d e t a i l e d s t u d y o f f r i c t i o n mechanism o f s t e e l on s t e e l was p r e s e n t e d by V i n o g r a d o v , Korepova and Yu. Ya.• P o d o l s k y [ 2 9 ] . The study c o v e r e d a wide range o f v e l o c -i t i e s . A t h i g h s l i d i n g speeds the f r i c t i o n f o r c e reaches a s t a b l e v a l u e i n a v e r y s h o r t i n t e r v a l o f t i m e . A t v e r y low s l i d i n g speeds t h e f r i c t i o n f o r c e i n c r e a s e s s l o w l y w i t h t i m e , and the g r e a t e r the l o a d , the l o n g e r i t t a k e s t o r e a c h a s t e a d y f r i c t i o n v a l u e . They suggested t h a t a t h i g h s l i d i n g speeds, s e i z u r e r e s u l t e d from the s i m u l t a n e o u s 14 f o r m a t i o n o f b r i d g e s o f i n t e n s i v e a d h e s i o n between the m a j o r i t y o f t h e m i c r o - a r e a s o f t h e v i r g i n m e t a l l i c s u r f a c e ; whereas a t low s l i d i n g speeds the r h e o l o g i c a l p r o p e r t i e s o f the c o n t a c t i n g s o l i d s p l a y an i m p o r t a n t p a r t . S i m k i n s [30] showed t h a t d u r i n g the s t i c k p e r i o d o f a s t i c k - s l i p p r o c e s s t h e r e were m i c r o s l i p s p r e c e d i n g the g r o s s s l i p . He suggested t h a t t h e so c a l l e d ' s t a t i c f r i c t i o n f o r c e ' i s merely t h e ' l o c a l maximum' o f t h e v a l u e s a v a i l a b l e . The phenomenon o f m i c r o - s l i p d u r i n g s t i c k i s not new, i n f a c t , Papenhuyzen [7] had d i s c u s s e d the m i c r o -s l i p movement i n t h e s t i c k s t a g e . However, the more advanced e x p e r i m e n t a l t e c h n i q u e p e r m i t t e d S i m k i n s t o p r o -duce more s u b s t a n t i a l e v i d e n c e o f the phenomenon. A s u r v e y o f work i n f r i c t i o n , l u b r i c a t i o n and wear d u r i n g the l a s t decade was p r e s e n t e d by Bowden and Tabor [ 3 1 ] . The r e p o r t p r o v i d e d some u s e f u l r e f e r e n c e s f o r work i n f r i c t i o n s t u d i e s . The s u b j e c t o f c o n t r o l l i n g o r r e d u c i n g t h e f r i c t i o n by a p p l y i n g v i b r a t i o n t o the f r i c t i o n p a r t s has been r e c e i v e d some c o n s i d e r a t i o n f o r sometime. Houch [32] suggested t h a t a p r o p e r l y a p p l i e d v i b r a t i o n can be used t o e l i m i n a t e o r g r e a t l y r e l i e v e many o f t h e problems i n v o l v -i n g f r i c t i o n . G o d f r e y [33] found t h a t the v i b r a t i o n p e r i o d i c a l l y reduced m e t a l - t o - m e t a l c o n t r a c t due t o reduced l o a d . Thus, an apparent r e d u c t i o n o f the c o e f f i c i e n t of: f r i c t i o n was o b s e r v e d . G a y l o r d and Shu [34] obse r v e d t h a t s t a t i c c o e f f i c i e n t o f f r i c t i o n was lower under dynam-i c a l l y a p p l i e d l o a d s than under s t a t i c a l l y a p p l i e d l o a d s , a l t h o u g h no e x p l a n a t i o n f o r t h i s o b s e r v a t i o n was g i v e n . A more s y s t e m a t i c s t u d y o f the r e d u c t i o n o f s t a t i c f r i c t i o n by s o n i c v i b r a t i o n was r e p o r t e d by Fridman and Levesque [3 5 ] . The e f f e c t o f s o n i c v i b r a t i o n s on the s t a t i c co-e f f i c i e n t o f f r i c t i o n was measured f o r h i g h l y p o l i s h e d and f o r ground and sand b l a s t e d s t e e l s u r f a c e s . They suggested t h a t the s t a t i c c o e f f i c i e n t o f f r i c t i o n can v i r t u a l l y be reduced t o z e r o as a r e s u l t o f i n c r e a s e d v i b r a t i o n a t f r e -q u e n c i e s between 6-42 kHz, w i t h a peak t o peak v i b r a t i o n a m p l i t u d e o f 7.5 x 10 6cm. S t u d i e s o f v i b r a t i o n i n m e t a l w o r k i n g are r e p o r t e d by Wheeler [ 3 6 ] . The r e p o r t showed t h a t s u b s t a n t i a l r e d u c t i o n i n y i e l d s t r e s s was o b s e r v e d w i t h t h e a p p l i c a t i o n o f o s c i l l a t i o n a t 800 kHz. I n a n o t h e r c a s e , as the r e s u l t o f a p p l y i n g low f r e q u e n c y o s c i l l a t i o n , 16-40 Hz, t o the ram o f a h y d r a u l i c f o r g i n g p r e s s an apparent r e d u c t i o n i n f r i c t i o n o f up t o 60 p e r c e n t and c l o s e t o 50 p e r c e n t r e d u c t i o n i n f o r c e r e q u i r e d t o produce the d e f o r m a t i o n were a c h i e v e d . I n a d d i t i o n , the m e t a l was deformed more u n i f o r m l y . However, more s y s t e m a t i c i n v e s t i g a t i o n s as w e l l as t h e o r e t i c a l a n a l y s e s a re needed i n o r d e r t o g a i n a more g e n e r a l i z e d u n d e r s t a n d i n g o f the, u n d e r l y i n g p h y s i c a l r e a l i t i e s o f the problem. The problem o f f r i c t i o n - i n d u c e d v i b r a t i o n due t o a n o n l i n e a r 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 has s i m i l a r i t y t o many o t h e r e n g i n e e r i n g problems. Thus t h e a n a l y t i c a l methods d e v e l o p e d f o r t h e i n v e s t i g a t i o n o f f r i c t i o n - i n d u c e d v i b r a t i o n can be a p p l i e d t o o t h e r analogue problems. L e m p r i e r e [ 3 7 ] , i n a st u d y o f t e n s i l e t e s t i n g , showed t h a t a n o n l i n e a r s t r e s s - s t r a i n r a t e c u r v e would cause a u t o -o s c i l l a t i o n . The problem was a n a l y s e d by a p p l y i n g the phase-p l a n e t r a j e c t o r y method. C l a u s e r [38] p r e s e n t e d a r e v i e w o f n o n l i n e a r systems and showed the s i m i l a r i t y o f some a e r o n a u t i c a l problems w i t h the f r i c t i o n - i n d u c e d v i b r a t i o n phenomenon. The e f f e c t o f s t a t i c and s l i d i n g f r i c t i o n i n , feedback systems was r e p o r t e d by Tou and S c h u l t h e i s s [ 3 9 ] , A m a t h e m a t i c a l method was d e v e l o p e d t o c o r r e c t t h e non-l i n e a r i t y i n t h e feedback system due t o f r i c t i o n . I n a l l the p r e v i o u s work on f r i c t i o n - i n d u c e d v i b r a -t i o n , v e r y l i t t l e c o n s i d e r a t i o n has been g i v e n t o the q u a s i -harmonic type o s c i l l a t i o n . I t would appear t h a t the so c a l l e d q u a s i - s i n u s o i d a l v i b r a t i o n o b s e r v e d i n the h i g h v e l o c i t y r e g i o n i n a system w i t h d e c r e a s i n g 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 was i n f a c t s t i c k - s l i p o s c i l l a t i o n w i t h a v e r y s h o r t p e r i o d o f s t i c k so t h a t the x-x phase p l a n e appeared t o be al m o s t c i r c u l a r (Ref. F i g . 5.1.9). The above papers would seem t o i n d i c a t e t h a t an i m p o r t a n t q u e s t i o n f o r the u n d e r s t a n d i n g o f f r i c t i o n - i n d u c e d v i b r a t i o n and t h e p r a c t i c a l t r e a t m e n t o f m e c h a n i c a l systems i n v o l v i n g f r i c t i o n i s i t s dependence upon v e l o c i t y . However, l i t t l e was known about t h i s q u e s t i o n o t h e r than the g e n e r a l r u l e t h a t the c o n d i t i o n f o r the e x i s t e n c e o f f r i c t i o n - i n d u c e d v i b r a t i o n was t h a t the s t a t i c f r i c t i o n s h o u l d be g r e a t e r t h a n t h e k i n e t i c f r i c t i o n and/or a d e c r e a s i n g 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 e x i s t e d . The f r i c t i o n - v e l o c i t y c u r v e s as d e s c r i b e d i n the above papers were m o s t l y o f the l i n e a r form w i t h d e c r e a s i n g v a l u e as t h e s l i d i n g v e l o c i t y was i n -c r e a s e d . However, a c t u a l f r i c t i o n c o u p l e s d i s p l a y a v a r i e t y o f forms o f f r i c t i o n c u r v e . The l i n e a r form w i t h n e g a t i v e s l o p e i s o n l y one p o s s i b l e form. The p r e s e n t work, w h i c h r e l a t e s t o the t h e o r e t i c a l and 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 o f the mechanics o f t h e q u a s i - h a r m o n i c o s c i l l a t i o n , has p r o -duced an u n d e r s t a n d i n g o f the phenomenon. I l l THEORETICAL 3.1 I n t r o d u c t i o n F i g . 1.1.2 shows s c h e m a t i c a l l y t h e c o n f i g u r a t i o n s o f the t h r e e systems. B a s i c a l l y the system c o n s i s t s o f a s l i d e r o f mass m w i t h a normal f o r c e w a c t i n g t o impress the s l i d e r a g a i n s t a lower s u r f a c e which i s moving w i t h a c o n s t a n t v e l o c i t y v. The s l i d e r i s r e s t r a i n e d by a bond o f e l a s t i c i t y k and damper o f c o e f f i c i e n t r . L e t the c o e f f i c i e n t o f f r i c t i o n between t h e s l i d e r and t h e lower s u r f a c e be u^. The e q u a t i o n o f motion f o r the autonomous system can be w r i t t e n as The e q u a t i o n f o r the system s u b j e c t e d t o t r a n s v e r s e f o r c i n g i s mx + r x + kx = wu (3.1.1) mx + r x + kx = wu, + f s i n v t (3.1.2) I n the case o f normal v i b r a t i o n the e q u a t i o n i s mx + r x + kx = wu, (1 + B s i n v t ) (3.1.3) where f = pv e; p i s the mass of an o u t - o f - b a l a n c e w e i g h t and e i s the e c c e n t r i c i t y , B = n ~ ; n i s some c o n s t a n t , w Eq. (3.1.2) r e p r e s e n t s a system h a v i n g an e x t e r n a l e x c i t a t i o n f o r c e a c t i n g i n the d i r e c t i o n o f the f r i c t i o n -i n d u c e d v i b r a t i o n , whereas eq. (3.1.3) r e p r e s e n t s a system w i t h the e x t e r n a l e x c i t a t i o n b e i n g a p p l i e d v e r t i c a l l y a t the l o a d i n g end o f t h e beam thus s i m u l a t i n g the e f f e c t o f d y n a m i c a l l y a p p l i e d l o a d . E q u a t i o n s (3.1.1), (3.1.2) and (3.1.3) can be non-d i m e n s i o n a l i z e d by i n t r o d u c i n g a d i s p l a c e m e n t parameter h and l e t t i n g co = — ; X = r- ; V = — r ~ and x = cot . m h coh m, , * dX d t x -t it x Then we have X = -rr- = -r- and X = d t dx coh 2, co h M u l t i p l y i n g e q u a t i o n s ( 3 . 1 . 1 ) , (3.1.2) and (3.1.3) by 2 1/ (moj h) and s u b s t i t u t i n g the e x p r e s s i o n s f o r X, X e t c , we have the n o n - d i m e n s i o n a l i z e d e q u a t i o n s f o r the t h r e e systems. X + RX + X = - F(V-X) (3.1.4) hi X + RX + X = J F(V-X) + F s i n a x (3.1.5) JJ o X + RX + X = F(V-X) [1 + 3 s i n a x ] (3.1.6) where R = r / (raw) ; E = (mto h) ; a = v/w and F q = f/E The f r i c t i o n f o r c e f u n c t i o n f has been c o n v e r t e d i n t o a ^k f u n c t i o n o f the d i m e n s i o n l e s s s l i d i n g v e l o c i t y F ( V - X ) . E q u a t i o n s ( 3 . 1 . 4 ) , (3.1.5) and (3.1.6) can be i n -v e s t i g a t e d by t h e methods t o be d e s c r i b e d . Owing t o the n o n l i n e a r i t y o f t h e f r i c t i o n c h a r a c t e r -i s t i c , f r i c t i o n - i n d u c e d v i b r a t i o n has l o n g been c o n s i d e r e d one o f the c l a s s i c a l n o n l i n e a r problems o f m e c h a n i c a l systems. A l t h o u g h a v a r i e t y o f t e c h n i q u e s have been d e v e l o p e d f o r the approximate s o l u t i o n o f 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 s , v e r y l i t t l e has been done i n a p p l y i n g t h e s e methods t o a thorough a n a l y s i s o f f r i c t i o n - i n d u c e d v i b r a t i o n . I n g e n e r a l , most o f t h e s e a n a l y t i c a l methods are s i m p l e i n t h e sense o f q u a l i t a t i v e a n a l y s i s , b u t u s u a l l y become c o m p l i c a t e d once a q u a n t i t a t i v e s o l u t i o n i s sought. I n t h e p r e s e n t i n v e s t i g a t i o n a complete and thorough t h e o r e t i c a l a n a l y s i s o f the f r i c t i o n - i n d u c e d v i b r a t i o n b o t h w i t h and w i t h o u t e x t e r n a l e x c i t a t i o n w i l l be c a r r i e d o u t by a p p l y i n g the a v a i l a b l e methods. Sometimes i t i s n e c e s s a r y t o a p p l y more t h a n one method i n o r d e r t o i n v e s t i g a t e a problem, w i t h each method p r o v i d i n g p a r t o f the s o l u t i o n . The t o p o l o g i c a l method o f a n a l y s i s i s one o f the i m p o r t a n t means o f i n v e s t i g a t i n g v a r i o u s phenomena o f non-l i n e a r o s c i l l a t i o n s , and i t i s a p p l i c a b l e t o the st u d y o f autonomous systems. By t h i s method s o l u t i o n s a re sought as i n t e g r a l c u r v e s i n a phase p l a n e . P o i n c a r e [ 4 0 ] has shown t h a t l i m i t c y c l e s and s i n g u l a r p o i n t s form c e r t a i n t o p o -l o g i c a l c o n f i g u r a t i o n s . A s i m p l i f i e d s t a t e m e n t o f h i s theorem i s t h a t e v e r y l i m i t c y c l e c o n t a i n s a t l e a s t one s i n g u l a r p o i n t i n i t s i n t e r i o r o f s t a b i l i t y o p p o s i t e t o t h a t o f the c y c l e . Thus an u n s t a b l e s i n g u l a r p o i n t i s s u r r o u n d e d by a s t a b l e c y c l e and v i c e v e r s a , as shown i n F i g . 3.1.1a and F i g . 3.1.1b. S o f t s e l f - e x c i t a t i o n c o r r e s p o n d s t o the case i n which a system d e p a r t s from an u n s t a b l e s i n g u l a r i t y as i n F i g . 3.1.1a and a r r i v e s a t the s t a t i o n a r y s t a t e o f the l i m i t c y c l e C^. Hard s e l f -e x c i t a t i o n c o r r e s p o n d s t o the case as shown i n F i g . 3.1.1b i n w hich an i m p u l s e i s r e q u i r e d t o e n a b l e the system t o c r o s s t h e b a r r i e r r e p r e s e n t e d by t h e u n s t a b l e l i m i t c y c l e C^. In most a p p l i e d problems o f the autonomous t y p e the q u a l i t a t i v e f e a t u r e s o f the o s c i l l a t o r y p r o c e s s e s a r e , g e n e r a l l y , c o m p l e t e l y r e v e a l e d by t h e f i r s t a p p r o x i m a t i o n . For t h i s r e a s o n , many methods based on the a v e r a g i n g method are d e v e l o p e d . A l l t h e s e methods l e a d t o t h e s o l u t i o n by t h e f i r s t a p p r o x i m a t i o n . The methods o f van der P o l [41] and o f K r y l o v - B o g o l i u b o v [42] are v e r y c l o s e t o each o t h e r . In b o t h methods a s i m p l e harmonic s o l u t i o n i s ' f i t t e d ' i n t o the n e a r l y l i n e a r e q u a t i o n . The main d i f f e r e n c e between the two methods i s t h a t van d e r P o l t a k e s the harmonic s o l u t i o n i n the form (b^sintot + b 2cosu>t) whereas i n t h e K r y l o v - B o g o l i u b o v method t h i s s o l u t i o n i s ; 22 t a k e n i n the form, A cos (wt + <J>) , where A, b^, and <j> a r e s l o w l y v a r y i n g f u n c t i o n s o f t . F o r a nonautonomous system, t h a t i s the time t appears e x p l i c i t l y i n t h e n o n l i n e a r term o f the d i f f e r e n t i a l e q u a t i o n , t h e r e a r e two p r i n c i p a l cases t o be i n v e s t i g a t e d a c c o r d i n g t o whether the parameter a, the r a t i o between t h e h e t e r o p e r i o d i c and the a u t o p e r i o d i c f r e q u e n c i e s , i s a non-i n t e g e r o r an i n t e g e r . Here t h e a u t o p e r i o d i c o s c i l l a t i o n i s d e f i n e d as t h e f r e e o s c i l l a t i o n w i t h f r e q u e n c y e q u a l t o t h a t o f the s e l f - e x c i t a t i o n o s c i l l a t i o n , and the h e t e r o p e r i -o d i c o s c i l l a t i o n i s d e f i n e d as the f o r c e d o s c i l l a t i o n w i t h f r e q u e n c i e s e q u a l t o t h a t o f the e x t e r n a l e x c i t a t i o n and/ or m u l t i p l e s o f i t [ 4 3 ] . The f i r s t case i s r e l a t i v e l y s i m p l e and l e a d s t o the so c a l l e d non-resonance o s c i l l a t i o n . The second case c o r r e s p o n d s t o the resonance o s c i l l a t i o n . One c o u l d a l s o i n v e s t i g a t e a more g e n e r a l problem by con-s i d e r i n g a as a v a r i a b l e parameter. When a i s s u f f i c i e n t l y f a r away from a r a t i o n a l number, t h e r e may appear the so c a l l e d asynchronous a c t i o n , an a c t i o n i n which t h e h e t e r -o p e r i o d i c o s c i l l a t i o n sometimes m a n i f e s t s i t s e l f i n t h e appearance o f an a u t o p e r i o d i c o s c i l l a t i o n and sometimes i n the e x t i n c t i o n o f an e x i s t i n g a u t o p e r i o d i c o s c i l l a t i o n [ 4 4 ] . When a has i n t e g e r v a l u e s such as a = 2, 3, 4 t h e r e may be o c c a s i o n a l l y subharmonic s o l u t i o n s w i t h f r e q u e n c y r a t i o 1. Furthermore t h e r e i s the problem o f s y n c h r o n i z a t i o n when; a i s i n t h e neighbourhood o f 1. Thus t h e problem o f i n v e s -23 t i g a t i n g a nonautonomous system may become c o m p l i c a t e d . S e v e r a l q u a l i t a t i v e methods had been d e v e l o p e d r e -c e n t l y f o r the i n v e s t i g a t i o n o f t h e nonautonomous systems. These methods a r e r e l a t i v e l y s i m p l e t o a p p l y when o n l y a q u a l i t a t i v e a n a l y s i s i s r e q u i r e d . However, when q u a n t i -t a t i v e r e s u l t s a r e sought f o r an a p p l i e d problem, the c a l c u l a t i o n i s alm o s t always l o n g and c o m p l i c a t e d , p a r t i c u l a r l y when the n o n l i n e a r f u n c t i o n i n v o l v e s an ex-tended e x p r e s s i o n such as a h i g h o r d e r p o l y n o m i a l . Under thes e c i r c u m s t a n c e s n u m e r i c a l methods may have t o be used. In t he p r e s e n t i n v e s t i g a t i o n , the f i r s t approxima-t i o n o f the method o f K r y l o v and B o g o l i u b o f f was used i n the autonomous and nonautonomous c a s e s . However, i n the nonautonomous c a s e , i n a d d i t i o n t o the K and B method the method o f harmonic b a l a n c e was used. F o r the i n v e s t i g a t i o n o f the t r a n s i e n t s t a t e a n u m e r i c a l method was employed. 3.2 Form o f F r i c t i o n - V e l o c i t y F u n c t i o n R e q u i r e d f o r the  E x i s t e n c e o f Quasi-Harmonic F r i c t i o n - I n d u c e d  V i b r a t i o n In t he case o f t h e autonomous q u a s i - h a r m o n i c f r i c t i o n - i n d u c e d v i b r a t i o n , t h e r e i s no s t a t i o n a r y c o n t a c t between the s l i d i n g s u r f a c e s , t h e r e f o r e the motion i s governed by t h e v a r i a t i o n i n f r i c t i o n f o r c e w i t h s l i d i n g speed. In p a r t i c u l a r , i t w i l l be shown t h a t the e x i s t e n c e 24 o f the q u a s i - h a r m o n i c o s c i l l a t i o n i s c r i t i c a l l y dependent on the p a r t i c u l a r shape o f the dynamic f r i c t i o n c u r v e . The d e c r e a s i n g f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e as assumed i n much o f the p r e v i o u s work i s not the o n l y p o s s i b l e one, i n f a c t , a c t u a l f r i c t i o n c o u p l e s d i s p l a y a v a r i e t y o f forms o f dynamic f r i c t i o n c u r v e . F i g . 3.2.1 i l l u s t r a t e s two forms of t h e c u r v e commonly s u g g e s t e d . A l t e r n a t i v e l y , t h e r e i s e v i d e n c e t o s u p p o r t the type o f curve i l l u s t r a t e d by F i g . 3.2.2. K r a g e l s k i i [45] demon-s t r a t e s t h a t t h i s form o f c u r v e i s found f o r m e t a l s i n d r y s l i d i n g c o n t a c t . R e c e n t l y , the s t u d y o f the f r i c t i o n b e h a v i o u r o f r u b b e r [46] and o f polymers [47] has shown t h a t s i m i l a r humped f r i c t i o n - v e l o c i t y c u r v e s e x i s t f o r non-m e t a l l i c m a t e r i a l s . E x p e r i m e n t a l r e s u l t s [48] f o r the combined r o l l i n g and s l i d i n g c o n t a c t o f l u b r i c a t e d r o t a t i n g d i s c s r e v e a l t h e e x i s t e n c e o f f r i c t i o n t o r q u e v e r s u s s l i d -i n g v e l o c i t y c u r v e s o f the form o f F i g . 3.2.2. A humped form o f f r i c t i o n f o r c e - v e l o c i t y c urve was a l s o r e p o r t e d by Muskat and Morgan [ 4 9 ] , i n t h e i r i n v e s t i g a t i o n o f b e a r i n g s w i t h v a r i o u s t y p e o f l u b r i c a n t s . I n a n o t h e r s t u d y o f b e a r i n g l u b r i c a n t s , Hagg [50] showed a humped form f o r the s h e a r i n g s t r e s s v e r s u s j o u r n a l v e l o c i t y c u r v e . The s t u d y o f f r i c t i o n c h a r a c t e r i s t i c s o f a u t o m a t i c t r a n s m i s s i o n f l u i d components [51] has shown t h a t humped f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c e x i s t s f o r c e r t a i n t y p e s o f a d d i t i v e com-b i n a t i o n s . In t h e i r i n v e s t i g a t i o n o f s l i d e w a y f r i c t i o n , B e l l and B u r d e k i n [26] ob s e r v e d t h a t the humped type f r i c t i o n -v e l o c i t y c u r v e e x i s t s w i t h p o l a r l u b r i c a n t s . M a t e r i a l s such as l o n g c h a i n n a p h t h e n i c a c i d s , f a t t y a c i d s , l o n g c h a i n a l k y l p h o s p h a t e s e t c . , g e n e r a l l y p o s s e s s h i g h p o l a r a c t i v i t y l e v e l [52] and a r e u s u a l l y used as f r i c t i o n m o d i f i e r s f o r a u t o m a t i c t r a n s m i s s i o n f l u i d s [ 5 3 ] . Thus i t i s o f i n t e r e s t t o note t h a t the humped f r i c t i o n - v e l o c i t y c urve e x i s t s i n a wide v a r i e t y o f f r i c t i o n a l s i t u a t i o n s and t h a t i n many cases v i b r a t i o n o f t h e q u a s i - h a r m o n i c form may o c c u r . I t i s p r o b a b l e t h a t the humped form i s a s s o c i a t e d w i t h t h e v i s c o u s component whose v i s c o s i t y i s s e n s i t i v e t o temperature.. curve i s n e c e s s a r y . V a r i o u s m a t h e m a t i c a l e x p r e s s i o n s have been used t o r e p r e s e n t the f r i c t i o n f o r c e f u n c t i o n [ 5 4 ] . From some e a r l y work on r a i l w a y t r a n s p o r t a t i o n , t h e f r i c t i o n f o r c e was r e p r e s e n t e d by the e m p i r i c a l e x p r e s s i o n F o r an a n a l y t i c a l s t u d y , an e q u a t i o n f o r the f r i c t i o n = n + 1+qV L a t e r , the r e l a t i o n s h i p was e x p r e s s e d i n the form = f e o -sV where n, p, q and s are some c o n s t a n t s , f and V are the M k f r i c t i o n f o r c e and v e l o c i t y r e s p e c t i v e l y . Owing t o the s i m p l i c i t y o f t h e s e e x p r e s s i o n s , t h e i r a c c u r a c y f o r f i t t i n g e x p e r i m e n t a l c u r v e s i s l i m i t e d . I n t h e p r e s e n t i n v e s t i g a t i o n , the f r i c t i o n f o r c e f u n c t i o n was e x p r e s s e d i n t h e form o f an e x p o n e n t i a l th. f u n c t i o n as w e l l as i n the form o f a n o r d e r p o l y n o m i a l . [ C 1 + C 2 ( v - x ) ] e _ C 3 ( v ~ x ) + C 4 ( v - x ) + C 5 (3.2.1) C ( v - x ) n + C _ ( v - x ) n - 1 + + Crt (3.2.2) n n-1 0 where (v-x) i s the s l i d i n g v e l o c i t y and C Q, e t c . are c o n s t a n t s which may be a d j u s t e d t o f i t the e q u a t i o n t o measured f r i c t i o n v a l u e s . When c a r r y i n g o u t t h e t h e o r e t i c a l a n a l y s i s , t h e f r i c -t i o n f o r c e f u n c t i o n e x p r e s s e d i n the p o l y n o m i a l form has c e r t a i n advantages over the e x p o n e n t i a l form. I n p a r t i c u l a r , the p o l y n o m i a l e x p r e s s i o n r educes t o a power s e r i e s i n x which upon i n t e g r a t i o n y i e l d s a n o t h e r power s e r i e s o f the am p l i t u d e o f v i b r a t i o n , thus the s t a t i o n a r y s t a t e a m p l i t u d e can be r e a d i l y o b t a i n e d by s o l v i n g the a m p l i t u d e p o l y n o m i a l . On the o t h e r hand, the e x p o n e n t i a l e x p r e s s i o n y i e l d s a t r a n s c e n d e n t a l e q u a t i o n f o r the a m p l i t u d e o f v i b r a t i o n . In o r d e r t o s o l v e t h e e q u a t i o n some i n i t i a l e s t i m a t i o n s had t o be s u p p l i e d . Another advantage i s t h a t the complete • p r o c e s s o f t h e t h e o r e t i c a l a n a l y s i s can be g e n e r a l i s e d f o r a n^*1 o r d e r p o l y n o m i a l . However, the e x p o n e n t i a l e x p r e s s i o n has t h e advan-tage o f b e i n g a b l e t o p r o v i d e a more a c c u r a t e r e p r e s e n t a t i o n o f the e x p e r i m e n t a l f r i c t i o n f o r c e c u r v e , p a r t i c u l a r l y o f the humped form. T h i s i s p a r t i c u l a r l y u s e f u l when a p p l i e d t o t he n u m e r i c a l method. The hump i n t h e f r i c t i o n c urve u s u a l l y appears v e r y c l o s e t o t h e zer o v e l o c i t y end o f t h e curve and under t h e s e c i r c u m s t a n c e s t h e cu r v e f i t t e d by p o l y n o m i a l s o f t e n p r e s e n t s d i f f i c u l t i e s ( r e f . F i g . 5.1.4), In a d d i t i o n , t h e p o l y n o m i a l f i t t e d c urve may show d r a s t i c change once o u t s i d e the range o f f i t , thus g r e a t l y l i m i t i n g i t s u s e f u l n e s s . The e x i s t e n c e o r n o n - e x i s t e n c e o f s e l f - e x c i t e d v i b r a t i o n o f t h e q u a s i - h a r m o n i c form may be i n v e s t i g a t e d f o r t he v a r i o u s dynamic f r i c t i o n c u r v e s p r oposed. The phase-plane g r a p h i c a l method o f L i e n a r d [55] p r o v i d e s a u s e f u l t e c h n i q u e f o r a s e m i - q u a l i t a t i v e i n v e s t i g a t i o n . I n o r d e r t o a p p l y t h e method, eq. (3.1.4) i s m o d i f i e d by l e t t i n g X = Y 1. A f t e r m a n i p u l a t i o n , t h e s e m o d i f i c a t i o n s y i e l d , dY' = F (V-Y')/E - RY 1 - X ( 3 2 3 ) dX y 1 dY 1 I f ^ x i s s e t t o z e r o i n eq. (3.2.3) th e n i t i s found t h a t X = F(V-Y')/E - RY' (3.2.4) Eq. (3.2.4) d e s c r i b e s the l o c u s o f a l l p o i n t s o f maximum v e l o c i t y on a phase p l a n e diagram. I t i s e v i d e n t t h a t t h e l o c u s i s s i m p l y a m o d i f i e d f r i c t i o n - v e l o c i t y • c h a r a c t e r i s t i c c u r v e , where the RY 1 term i s the system v i s c o u s damping w h i c h i s u s u a l l y v e r y s m a l l . A c c o r d i n g l y , e m ploying t h i s e q u a t i o n and L i e n a r d method, the diagrams of F i g . 3.2.3 were p r e p a r e d u s i n g F i g . 3.2.1a and F i g . 3.2.3a i l l u s t r a t e s t h e case o f a l i m i t c y c l e o f t h e s t i c k -s l i p t y p e produced by e n t r a i n m e n t o f the phase t r a j e c t o r y i n t o the s t a t i c f r i c t i o n a x i s . F i g . 3.2.3b shows a s i t u a t i o n whereby t h e mass a c h i e v e s a p o s i t i o n o f s t a b l e dynamic e q u i l i b r i u m and l i m i t c y c l e m o t i o n does n o t o c c u r . In g e n e r a l , i t may be o b s e r v e d t h a t t h e f r i c t i o n c h a r a c t e r -i s t i c o f F i g . 3.2.1a w i l l g i v e r i s e t o s t i c k - s l i p v i b r a t i o n o r s t a b l e d i s p l a c e m e n t , depending on t h e system p a r a m e t e r s . In any e v e n t , o s c i l l a t i o n s o f q u a s i - h a r m o n i c form do not o c c u r f o r t h i s p a r t i c u l a r 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 . However, l i m i t c y c l e m otion i s p o s s i b l e i n t h e case o f the humped f r i c t i o n - v e l o c i t y c u r v e o f F i g . 3.2.2 and the phase p l a n e s o l u t i o n o f F i g . 3.2.3c i l l u s t r a t e s t h a t near-harmonic o s c i l l a t i o n o c c u r s . Hence, th e hump i n the f r i c t i o n -v e l o c i t y c u r v e appears t o be one o f the c o n d i t i o n s n e c e s s a r y f o r t h e e x i s t e n c e o f t h i s form o f o s c i l l a t i o n . 3.3 Autonomous System 3.3.1 Method o f F i r s t A p p r o x i m a t i o n by K r y l o v -B o g o l i u b o f f For an autonomous system, eq. (3.1.4) can be w r i t t e n i n the form X + X + y G ( X ) = 0 (3.3.1) I f y = 0/ ecJ» (3.3.1) reduces t o a s i m p l e l i n e a r d.e. w i t h s o l u t i o n X = a s i n (T + cf>) (3.3.2) where a and cb are c o n s t a n t s . F o r y ¥• 0 t> ut s m a l l , eq. (3.3.2) can be used as a g e n e r a t i n g s o l u t i o n f o r the f i r s t a p p r o x i m a t i o n , p r o v i d e d t h e q u a n t i t i e s a and cb are c o n s i d e r e d n o t as c o n s t a n t s b u t as c e r t a i n f u n c t i o n s o f time t o be d e t e r m i n e d . Thus we have X = a ( r ) s i n [T + < | > ( T ) ] (3.3.3) T h i s i s t h e b a s i c i d e a o f t h e method f o r t h e f i r s t approx-i m a t i o n o f t h e s o l u t i o n o f the d.e. by K r y l o v - B o g o l i u b o v [56](K.& B. Method). I n d e v e l o p i n g the method an a d d i t i o n a l c o n d i t i o n was imposed t h a t X s h o u l d be o f t h e form X = a cos (T + cb) (3.3.4) From e q u a t i o n s (3.3.3) and (3.3.4) one o b t a i n s the d.e. a s i n (T + <f>) + a<J> cos (x + cf>) = 0 (3.3.5) A second d.e. i s o b t a i n e d by s u b s t i t u t i n g t h e e x p r e s s i o n s f o r X, X and X ( d i f f e r e n t i a t i n g w i t h r e s p e c t t o a and <$>) i n eq. (3.3.1). S o l v i n g t h e s e two e q u a t i o n s g i v e s ex-p r e s s i o n s f o r a and <f> as p e r i o d i c f u n c t i o n s o f t i m e . Owing t o the s m a l l n e s s o f y, a and <f> can be c o n s i d e r e d as s l o w l y v a r y i n g f u n c t i o n s o f x . Hence we can assume a and cf> remain c o n s t a n t o v e r the i n t e r v a l x t o x + 2ir. I n t e g r a t i o n o f the e q u a t i o n s f o r a. and <f> between the l i m i t s x t o x + 2TT , shows t h a t a l l t r i g o n o m e t r i c terms drop out [ 5 7 ] , and o n l y the c o n s t a n t terms K Q ( a ) and H (a) r emain. We have ^ = - Y K (a) dx o R e p l a c i n g K q ( a ) and H q ( a ) by t h e i r F o u r i e r e x p a n s i o n s [58] g i v e s t h e u s u a l form o f the f i r s t a p p r o x i m a t i o n by K r y l o v -B o g o l i u b o v . 2TT G(a costy) costydty = $(a) (3.3.7) dcj> dx = Z H (a) (3.3.6) 31 i ± = i + r IT 2aTr / Jo ,, ,'2T\ §T  1 + o f c / G < a cosip) sinijJdiD = fi (a) (3.3.8) The c o n d i t i o n f o r a s t a t i o n a r y o s c i l l a t i o n o r a l i m i t c y c l e i s $ ( a ) = 0 [ 5 9 ] . T h a t i s $ (a^) = 0 i s t h e c o n d i t i o n f o r a l i m i t c y c l e w i t h a m p l i t u d e a^. To i n v e s t i g a t e t h e s t a b i l i t y o f t h e f i r s t a p p r o x i -m a t i o n , we c o n s i d e r a s l i g h t l y p e r t u r b e d a m p l i t u d e ( a ^ + 6a) w h e r e 6a i s an a b s o l u t e v a l u e o f d e p a r t u r e . I t c a n be shown b y t h e v a r i a t i o n a l e q u a t i o n s t h a t t o t h e f i r s t o r d e r cl Thus i f $ (a,) < 0 we h a v e - 5 — (6a) < 0, i n o t h e r w o r d s t h e a 1 d f ' i n i t i a l d e p a r t u r e 6a h a s a t e n d e n c y t o d i s a p p e a r f o r $ (a.) < 0. Thus $ (a,) < 0 i s t h e c o n d i t i o n f o r a s t a b l e a 1 a 1 l i m i t c y c l e and t h e c o n d i t i o n $ (0) > 0 i s e q u i v a l e n t t o c l t h e e x i s t e n c e o f an u n s t a b l e s i n g u l a r i t y , i . e . t h e e x i s t e n c e o f s o f t s e l f - e x c i t a t i o n . F o r h i g h e r a p p r o x i m a t i o n s t h e A s y m p t o t i c m e t h o d d e v e l o p e d by B o g o l i u b o v , K r y l o v a n d M i t r o p o l s k y [60] c a n be a p p l i e d . H owever, t h e s m a l l amount o f a c c u r a c y g a i n e d f r o m t h e h i g h e r a p p r o x i m a t i o n s u s u a l l y d o e s n o t j u s t i f y t h e v e r y l o n g c a l c u l a t i o n s i n v o l v e d i n c a r r y i n g o u t t h e h i g h e r a p p r o x i m a t i o n s . The f i r s t a p p r o x i m a t i o n u s u a l l y g i v e s s u f f i c i e n t i n f o r m a t i o n i n a p r a c t i c a l p r o b l e m . 32 3.3.2 A p p l i c a t i o n o f the E x p o n e n t i a l E x p r e s s i o n  t o t h e K and B Method I t has been i n d i c a t e d e a r l i e r t h a t the f r i c t i o n f o r c e f u n c t i o n c o u l d be e x p r e s s e d i n the form o f an e x p o n e n t i a l . S u b s t i t u t i o n o f e x p r e s s i o n (3.2.1) i n t o eq. (3.1.4) g i v e s a f t e r m a n i p u l a t i o n ro)h+C . C +C V • C . • C 4V+C x + ( i ) X - e 3 + — X e L 3 x + x = -3 i E E e C 3 V E e C 3 V E (3.3.9) The c o n s t a n t term on t h e r i g h t hand s i d e o f eq. (3.3.9) c o n s t i t u t e s t h e s t a t i c d i s p l a c e m e n t o f t h e v i b r a t i o n , s i n c e we are i n t e r e s t e d o n l y i n t h e a m p l i t u d e o f the o s c i l l a t i o n t h e r e f o r e t h i s c o n s t a n t term i s o m i t t e d i n t h e a n a l y s i s . Thus from eq. (3.3.1) we have D • YG(X) = — (X - D e C 3 X + D X e C 3 X ) (3.3.10) E C 1 + C 2 V C 2 where D, = rwh + C ., D„ = / D 4' 2 ^ C^ V ' 3 _ CoV D^e 3 D-^ e J S u b s t i t u t i o n o f eq. (3.3.10) i n t o eq. (3.3.7) and eq. (3.3.8) y i e l d s e x p r e s s i o n s f o r a and ty. (Appendix I) D a = {a - 2 D 2 I 1 ( C 3 a ) + D 3 a [ I 2 ( C 3 a ) + I Q (C 3a) ] }= $ (a) 2E (3.3.11a) 33 D l D 3 o r a = {a - 2 ( D 2 + — ) l 1 ( C ; J a ) + 2 D 3 a I Q (C 3a) }= *(a) 2E C 3 (3.3.11b) where I Q (C 3a) , I ^ t ^ a ) and I^iC^a) are m o d i f i e d B e s s e l f u n c t i o n s o f t h e f i r s t k i n d . and ^ = 1 ; (3.3.12) s i n c e the i n t e g r a l o f eq. (3.3.8) v a n i s h e s . G e n e r a l l y , i n t e r e s t c e n t r e s on s t a t i o n a r y v a l u e s f o r a. I f $ (a) = 0 i n eq. (3.3.11b), then D 3 2 (D 2 + — ) I 1 ( C 3 a ) - 2 D 3 a I 0 (C 3a) - a = 0 (3.3.13) C 3 Eq. (3.3.13) can be s o l v e d by a computer f o r d i s -c r e t e V v a l u e s i f t h e system c o n s t a n t s are known. C a r r y i n g out the d i f f e r e n t i a t i o n o f eq. (3.3.11b), we have * a ( a ) = - ^~ { 1 - 2 C 3 D 2 I 0 ( C 3 a ) + 2 E ^ [D 3 + C 3 D 2 + ( C 3 a ) 2 D 3 J I ^ a ) } (3.3.14) Thus the s t a b i l i t y o f the a m p l i t u d e o b t a i n e d from eq. (3.3.13) can be i n v e s t i g a t e d by s u b s t i t u t i n g i t i n t o eq. (3.3.14) and f o l l o w i n g the c o n d i t i o n s f o r s t a b i l i t y d e s c r i b e d e a r l i e r . Eq. (3.3.9) can be f u r t h e r a n a l y s e d by o m i t t i n g t h e v i s c o u s damping term (D^X/E). T h i s r e p r e s e n t s a system w i t h a f r i c t i o n c h a r a c t e r i s t i c as shown i n F i g . 3.3.1 and w i t h n e g l i g i b l e damping. F o r t h i s c o n d i t i o n eq. (3.3.11b) reduces t o n C + C V C.a a = {2 — I±(C3a) - — — [ I 2 (C 3a) + I o ( C 3 a ) ] } e C 3 V e C 3 V F o r the s t a t i o n a r y s t a t e v a l u e o f 'a' we s e t a = 0 and have C C I (C a) + I (C a) C V + - i - i = C a — - - - — (3.3.15) C 2 2 I l ( C 3 a ) A p p l i c a t i o n o f v a l u e s f o r 1^ (C^a) , I 2 ( C 3 a ) and I Q ( C 3 a ) p e r m i t s t h e c o n s t r u c t i o n o f t h e s o l u t i o n d i s p l a y e d by F i g . 3.3.2. T h i s s o l u t i o n i n d i c a t e s t h a t v i b r a t i o n w i l l commence a t a v e l o c i t y c o r r e s p o n d i n g t o t h e peak o f the f r i c t i o n -v e l o c i t y c u r v e (See F i g . 3.3.1). I n t h e undamped case th e v i b r a t i o n a m p l i t u d e i n c r e a s e s w i t h o u t l i m i t as the l o w e r s u r f a c e v e l o c i t y i n c r e a s e s . However a c t u a l systems p o s s e s s some damping which s u g g e s t s t h a t a m p l i t u d e l i m i t a t i o n w i l l e x i s t . Indeed, th e s t a b i l i t y a n a l y s i s t o be d i s c u s s e d i n s e c t i o n 3.3.4 w i l l show t h a t w i t h damping p r e s e n t the v i b r a t i o n w i l l be l i m i t e d a t some upper v e l o c i t y boundary. 35 3.3.3 A p p l i c a t i o n o f t h e P o l y n o m i a l E x p r e s s i o n t o  the K and B Method S u b s t i t u t i n g e x p r e s s i o n (3.2.2) i n eq. (3.1.4) we have a f t e r some m a n i p u l a t i o n , 1 r / , . T , x ~ ,-,2 . , ,,n„ -n, . „ P 0 X + ~ [ (rwh + P.) X - P_X + - (-1) P X ] + X ih x. z n E (3.3.16) 2 where E = mu) h ; and n P n = C n + C,V + + C V 0 0 1 n P k = C k + + n J k C n V n _ k f o r k = 0, 1, 2,...,n where the b i n o m i a l c o e f f i c i e n t s are g i v e n by J, = ^ 1 n k ( n - k ) ! k l The c o n s t a n t term on the r i g h t hand s i d e o f eq. (3.3.16) c o n s t i t u t e s o n l y t h e s t a t i c d i s p l a c e m e n t t h e r e f o r e i t can be o m i t t e d i n the a m p l i t u d e a n a l y s i s . Eq. (3.3.16) can be f u r t h e r s i m p l i f i e d by l e t t i n g P k 0, = rtoh + P, and Q, = — , k = 2 / 3 , . . . , n 1 Q l X + — [X - Q„X 2 + - ( - l ) n Q X n ] + X = 0 (3.3.17) E n The a p p l i c a t i o n o f the method s p e c i f i e d i n [56] g i v e s t h e i n t e g r a l s k< n+1 Q l _ 2 2 k - l k „ 2 k - l — l K„, , A 2E , (2k-2) Z K 1 k = l $ (a) (3.3.18) where = 1 and R^ . = , K — 2 , 3 , . n and \b = 1 The c o n d i t i o n f o r a s t a t i o n a r y s t a t e a m p l i t u d e i s $(a) = 0. From eq. (3.3.18) we have 1 fri^t) R 2 k - i a 2 k " 2 = 0 ( 3 - 3 - 1 9 ) k = l D i f f e r e n t i a t i n g eq. (3.3.18) w i t h r e s p e c t t o 'a' we have k<S±i Q 2 ( 2 k - l ) J V a> = = 2 ( 2 k - 2 ) * 2 k - l a < 3- 3' 2 0> k= l Eq. (3.3.19) i s a p o l y n o m i a l i n 'a* which can be e a s i l y s o l v e d w i t h t h e a i d o f a computer. The r e g u l a r i t y ' o f t h e c o e f f i c i e n t s , w h i c h are a l l formed by some s e r i e s i n n and are r e l a t e d t o the power o f the v a r i a b l e X, p e r m i t t e d a c o m p l e t e l y g e n e r a l i z e d computer programme t o be s e t up f o r c a r r y i n g o u t the a n a l y s i s w i t h the f r i c t i o n f o r c e f u n c t i o n r e p r e s e n t e d by a n*"*1 o r d e r p o l y n o m i a l . 37 3.3.4 S t a b i l i t y by S i n g u l a r P o i n t A n a l y s i s A s i n g u l a r p o i n t a n a l y s i s p r o v i d e s f u r t h e r i n f o r -m a t ion c o n c e r n i n g the c o n d i t i o n s n e c e s s a r y f o r t h e e x i s t e n c e o f the o s c i l l a t i o n and i t s s t a b i l i t y [ 6 1 ] . F or t h e case where t h e f r i c t i o n f o r c e f u n c t i o n i s r e p r e s e n t e d by e x p r e s s i o n ( 3 . 2 . 1 ) , t h e c o n d i t i o n f o r a s t a b l e system can be o b t a i n e d by s u b s t i t u t i n g the e x p r e s s i o n f o r F ( V - Y 1 ) i n t o eq. (3.2.3) d y l [ C 1 + C 2 ( V - Y ' ) ] e C 3 ( V " Y , ) + (C 4V+C 5) - (C 4+R)Y' - X (3.3.21) Fo r s m a l l p e r t u b a t i o n s o f Y 1 about a s i n g u l a r p o i n t , e xpanding C Y 1 i 2 e 3 and d e l e t i n g terms i n Y and h i g h e r powers g i v e s dY' [ ( C l C 3 + C 2 C 3 V - C 2 ) e " C 3 V " C 4 " R ] Y ' dX Y' (3.3.22) (C,+C,,V) e _ C 3 V + C.V + C c - X 1 2 4 b_ Y' Assessment o f t h i s e q u a t i o n shows t h a t the s i n g u l a r p o i n t o c c u r s a t Y' = 0 and X = (C^+C^V) e " C 3 V + (C 4V+C 5) = e The a n a l y s i s proceeds by t r a n s f e r i n g the c o o r d i n a t e system t o X = X' + 0 , y i e l d i n g 38 D Y , [ ( C ^ + C ^ V - C ^ ) e ~ C 3 V -C 4 - R] Y* - X' (3.3.23) dX' Y' T h i s l a s t e q u a t i o n i s o f t h e form dY' gx' + oY' dX1" £X' + £Y' (3.3.24) where g = -1 ; a = ( C ^ ^ C ^ V - C ^ ) e " C 3 V - - R ; e = 0 ; K = 1. F o l l o w i n g the c r i t e r i a f o r d i s c r i m i n a t i n g between d i f f e r e n t t y p e s o f s i n g u l a r i t i e s [61] whi c h was d e r i v e d from the c h a r a c t e r i s t i c e q u a t i o n f o r eq. (3.3.24). The c h a r a c t e r i s t i c e q u a t i o n has t h e form x 2 - (a + e) x - ( e£ - ae) = 0 (3.3.25) Eq. (3.3.25) has s o l u t i o n s x l ' x 2 = I { ( C T + £ ) ± H a - e ) 2 + 4 ? U 2 > 2 L e t t i n g N = ( a - e ) + 4z;£ , t h e c r i t e r i a shows t h a t f o r N < 0, t h e n / N i s i m a g i n a r y and i f (a + e) < 0, x^, X £ are b o t h complex c o n j u g a t e , b o t h h a v i n g n e g a t i v e r e a l p a r t s , and the s i n g u l a r i t y i s a s t a b l e s p i r a l p o i n t . Thus from eq. (3.3.24) we have ( C ^ + C ^ V - C ^ ) < (R+C 4) e C 3 V as t h e c o n d i t i o n f o r a s t a b l e system o r n o n - v i b r a t i n g system. A t t h e boundary between s t a b i l i t y and i n s t a b i l i t y we have C 1 C 3 + C 2 C 3 V ~ C 2 = ( R + C 4 ) (3.3.26) L e t t i n g y 1 = C 1 C 3 + C^^V - C 2 and y 2 = (R+C 4) e C 3 V , a p l o t o f y^ and y 2 as f u n c t i o n s o f V i s shown i n F i g . 3.3.3. I t i s q u i t e a p p arent t h a t t h e r e i s i n s t a b i l i t y f o r < V < V 2 and t h a t eq. (3.3.26) has two r o o t s . V i b r a t i o n can be a v o i d e d by a d j u s t i n g R i n o r d e r t h a t y^ becomes , t a n g e n t t o y 2 . A t t h i s p o i n t we have dy^/dV = dy 2/dV, t h e r e f o r e i t can be shown t h a t the c r i t i c a l v e l o c i t y a t t h e t a n g e n t p o i n t i s 1 C 2 V = ~- I n ( - ) (3.3.27) C U 3 CA + R S u b s t i t u t i n g V c i n eq. (3.3.26) we o b t a i n t h e damp-i n g R c r e q u i r e d f o r a c o m p l e t e l y s t a b l e system, thus (C 1 C 3 " 2 C2 ) R c = C 2e C 2 - C 4 (3.3.28) 40 3.4 Non-Autonomous Systems In the pr e s e n c e o f t h e t r a n s v e r s e e x t e r n a l e x c i t -a t i o n , t h e e q u a t i o n o f motion t a k e s t h e form o f eq. (3.1.5). I f t h e f r i c t i o n f o r c e f u n c t i o n i s absent t h e n we have a l i n e a r second o r d e r d.e. s u b j e c t e d t o harmonic f o r c -i n g . The s t e a d y s t a t e s o l u t i o n o f t h e e q u a t i o n i s [62] F X = - ^ (3.4.1) [(1 - a 2 ) 2 + ( R a ) 2 ] 2 2 2 where F Q = (pv e)/(nuo h) , p i s t h e mass o f an o u t - o f - b a l a n c e w e i g h t e t h e e c c e n t r i c i t y , v the f r e q u e n c y o f t h e e x t e r n a l e x c i t a t i o n , a t h e f r e q u e n c y r a t i o and R = r/mo> . A c o n t i n u o u s c u r v e o f X v s . a can be o b t a i n e d by v a r y i n g the f r e q u e n c y r a t i o . However, when th e f r i c t i o n f o r c e f u n c t i o n i s p r e s e n t we have a n o n l i n e a r non-autonomous system. Under t h e s e c o n d i t i o n s t h e e x i s t e n c e o f s t a b l e a m p l i t u d e v a l u e s becomes of i n t e r e s t . In the a n a l y s i s o f t h e non-autonomous system the f r i c t i o n f o r c e f u n c t i o n i s r e p r e s e n t e d by a n o r d e r p o l y n o m i a l . F o l l o w i n g the same p r o c e d u r e as c a r r i e d out f o r t h e autonomous system, we have 4 1 X + X = yG(X) + F q s i n a x ( 3 . 4 . 2 ) where yG(X) = - — [X-Q X 2+ - ( - l ) n CD X n ] ( 3 . 4 . 3 ) E n where Q k and E have the same meaning as i n t h e autonomous case (See p. 3 5 ) . Eq. ( 3 . 4 . 2 ) can be t r a n s f o r m e d by i n t r o d u c i n g a new v a r i a b l e [ 6 3 ] F X = Y + S i n a X ( 3 . 4 . 4 ) 1 - a S u b s t i t u t i n g t h e above e x p r e s s i o n and i t s c o r r e s p o n d i n g e x p r e s s i o n s f o r X and X i n t o eq. ( 3 . 4 . 2 ) , we have Y + Y = yG(Y, ax) ( 3 . 4 . 5 ) where Y G ( Y , ax) has the form as eq. ( 3 . 4 . 3 ) e x c e p t t h a t t h e F a • * o v a r i a b l e X i s r e p l a c e d by (Y + ^ cos a x ) . 1 - a I n g e n e r a l , the n o n l i n e a r f u n c t i o n i n eq. ( 3 . 4 . 5 ) i s c o n s i d e r e d t o have t h e form [ 6 4 ] G(Y,ax) = g Q ( Y ) + £ [ g n l ( Y ) c o s nax + g n 2 ( Y ) s i n nax] ( 3 . 4 . 6 ) n=o where g Q , g n ^ and g ^ a r e n o r m a l l y e x p r e s s e d as p o l y n o m i a l s i n Y. When y = 0, we have as s o l u t i o n 42 Y = a s i n ( x + ty) ; Y = a cos (T + ty) , and ty = x + ty when y ^ 0 b u t s m a l l , s o l u t i o n f o r eq. (3.4.5) can be sought i n t h e form Y = a s i n ty + Y U ^ a , ty, a x ) + (3.4.7) The f u n c t i o n u\ ( a , ty, a x ) are p e r i o d i c i n b o t h t h e a n g u l a r v a r i a b l e s ty and ax w i t h a p e r i o d 2 T T , and t h a t ty = ax + ty. 3.4.1 Non-Resonance Case In the absence o f r e s o n a n c e , t h e r e i s no s t a t i o n a r y phase r e l a t i o n s h i p between the e x t e r n a l f r e q u e n c y and the f r e q u e n c y o f t h e s e l f - i n d u c e d v i b r a t i o n , t h e r e f o r e t h e phase does not e x e r t any i n f l u e n c e e i t h e r on the a m p l i t u d e o r on the f u l l phase o f the o s c i l l a t i o n . Thus t h e q u a n t i t i e s . a and ty can be d e f i n e d as [65] a = Y A 1 ( a ) + y2 A 2 ( a ) + (3.4.8a) ty = 1 + Y B 1 (a) + (3.4.8b) The p r o c e d u r e o f f i n d i n g e x p r e s s i o n s f o r U\ , ; and B. are s i m i l a r t o t h a t o f the autonomous system [ 6 6 ] , 43 e x c e p t t h a t f u n c t i o n U\ depends on ax as w e l l as a and ij;, A c c o r d i n g l y , t h e e x p r e s s i o n s f o r A^, and a r e : 2TT /*2TT A, = / / G (a, \b, ax) cos ip d(ax)dijj (3.4.9a) 1 4TT' H i Jo Jo .1 . 2TT_ 2TT B., = - 0 1 j G (a, IJJ, ax) s i n ip d ( a x ) d ^ (3.4.9b) 'o •'o n = 1 y cos (pax + qip) 1 2 2 2TT p,q 1 - (pa + q) [ p 2 + ( q 2 - l ) 2 ^ 0 ] n2fT G (a, ifi, ax) cos (pax + qip)d(ax) dip s i n (pax + q^) 2 1 - ( p a + q) n2fr G (a, if;, ax) s i n (pax + q^)d(ax) dip (3.4.9c) where G ( a , ip, ax) = G (acos ip, ax) S u b s t i t u t i o n o f eq. (3.4.6) i n t o e q u a t i o n s (3.4.9a), (3.4.9b) f o r A^ and B^, a l l the terms b e h i n d t h e summation s i g n o f eq. (3.4.6) w i l l d i s a p p e a r upon i n t e g r a t i o n between the i n t e r v a l 2TT. Thus A^ and B^ can be w r i t t e n as J/.27T g o 2TT (a costy) costy dty (3.4.10a) o 2^ , . . i f 1 2a7T / Jo B i = ~ 9 ^ 7 / g (a.costy) sinty dty (3.4.10b) I t f o l l o w s t h a t i n the e q u a t i o n s o f the f i r s t a p p r o x i m a t i o n t h e r e appears o n l y t h e f r e e term g Q ( a c o s t ) o f the e x p a n s i o n o f t h e p e r t u r b i n g f o r c e G (a costy, ax). The e f f e c t o f t h e e x t e r n a l p e r i o d i c e x c i t a t i o n i s f e l t o n l y i n t he second a p p r o x i m a t i o n . The f i r s t a p p r o x i m a t i o n d e t e r m i n e s t h e e x i s t e n c e and the s t a b i l i t y o f t h e a u t o -p e r i o d i c o s c i l l a t i o n . The f u n c t i o n g Q ( Y ) o f eq. (3.4.6) can be o b t a i n e d by d e v e l o p i n g the f u n c t i o n G(Y, ax) and c o l l e c t i n g terms not depending e x p l i c i t l y on x. Thus, we have Y g o ( Y ) = - EQ + E^Y - E 2 Y 2 + - ( - l ) N E N Y N (3.4.11) k+2m<n j j R L 2 m , „ _ . „ r2m-l m k+2m 2m k+2m •, ._ . n i . where E, = R, + L [ j (3.4.11a) k = 0 , 1, 2, . . . , n and R Q = 0, R^ = 1, R k = Q^ . f o r k = 2, 3, . . . , n ; F a L = _ 0 _ k 1 where t h e b i n o m i a l c o e f f i c i e n t s a r e g i v e n by = (k-m)'ml S u b s t i t u t i o n o f eq. (3.4.11) i n eq. (3.4.10a) y i e l d s k < n ± i Q — 2 J a = 2 2 k " 1 k ' E 2 k - 1 a 2 k " 1 = A ( a ) (3.4.12) k = 1 2 ( 2 k _ 2 ) and ty = 1 The s t a t i o n a r y s t a t e i s reached when A(a) = = 0 (3.4.13) D i f f e r e n t i a t i n g eq. (3.4.13) w i t h r e s p e c t t o 'a' y i e l d s k<£±i V a ) = ~ T i l F I ) E 2 k - 1 ( 2 ^ 2 ) a 2 k - 3 (3.4.14) 2 E k = 1 2 The c o n d i t i o n f o r the s t a b i l i t y o f the a m p l i t u d e a^ a c c o r d i n g t o K r y l o v - B o g o l i u b o f f [67] i s t h a t the s t a t i o n a r y s t a t e i s s t a b l e i f A (a,) > 0 ; and t h a t A (0) < 0 i s the a 1 c o n d i t i o n f o r s e l f - e x c i t a t i o n . When t h e a u t o p e r i o d i c o s c i l l a t i o n i s a b s e n t , t h a t i s when a = 0, i n o r d e r t o i n v e s t i g a t e t h e e f f e c t o f the e x t e r n a l p e r i o d i c e x c i t a t i o n t h e second a p p r o x i m a t i o n must be employed. We have from eq. (3.4.9c) 46 (0,0,ax) = i [ c o s a T 2 I G(0,0,ax) cosax d ( a x ) ] (3.4.15) 1 - a I and from eq. (3.4.3) yG(0, ax) = [Lcosax - Q L 2 c o s 2 a x + . . . - ( - l ) n Q L n c o s n a x ] E n S u b s t i t u t i o n o f t h e e x p r e s s i o n f o r yG (0, ax) i n eq. (3.4.15) y i e l d s Q-Lcosax - 2 2 k - l J k 2 k - l Y U ( 0 , 0 , ax) = - - i ^ E R L 2 k 1 1 E ( l - a ) k = x 2 U J C Z ) Z K l (3.4. J16) where Q^, R^ and L have the same meaning as i n t h e p r e v i o u s c a s e s . The f o r e g o i n g e q u a t i o n r e p r e s e n t s a p u r e l y h e t e r o -p e r i o d i c o s c i l l a t i o n w i t h f r e q u e n c i e s e q u a l t o t h o s e o f the e x t e r n a l e x c i t a t i o n . A c c o r d i n g l y , f o r a system w i t h o n l y h e t e r o p e r i o d i c o s c i l l a t i o n , we have k < n ± l Q K _ 2 J Y - l y r 2 k - l k T 2 k - 1 l c o s a T n 4 17^ Y - — L L ( 2 k _ 2 ) R 2 k _ 1 L ] — 2- (3.4.17) 3.4.2 Fundamental Resonance Case The fundamental resonance case o c c u r s when t h e e x t e r n a l e x c i t a t i o n i s a p p l i e d a t f r e q u e n c i e s c l o s e t o the f r e q u e n c y o f t h e a u t o p e r i o d i c o s c i l l a t i o n , t h a t i s when a - 1. The s o l u t i o n can s t i l l be sought i n the form o f eq. (3.4.7). However, owing t o t h e presence o f r e s o n a n c e , the phase d i f f e r e n c e between the f r e e o s c i l l a t i o n and t h e e x t e r n a l e x c i t a t i o n may e x e r t a v i t a l i n f l u e n c e on the change i n the a m p l i t u d e and t h e f r e q u e n c y o f the o s c i l l a -t i o n . T h e r e f o r e t h e q u a n t i t i e s a and ty are d e f i n e d as f u n c t i o n s o f ty as w e l l as 'a'. a = y A 1 ( a , ty) + y 2 A 2 ( a , ty) + .. ty = 1 + yB1(a, ty) + . F o l l o w i n g t h e a s y m p t o t i c method by K.B.M. [68] , e x p r e s s i o n s are o b t a i n e d f o r f i n d i n g t h e q u a n t i t i e s U^, A^ and . The method o f e q u i v a l e n t l i n e a r i z a t i o n [69] p r o -v i d e s a s i m p l e r p r o c e d u r e f o r d e r i v i n g the e x p r e s s i o n s o f the f i r s t a p p r o x i m a t i o n w h i c h w i l l r e v e a l s u f f i c i e n t i n f o r m a t i o n f o r t h e fundamental r e s o n a n c e . The s o l u t i o n i s sought i n t h e form X = a sinty ; ty = a x + ty The n o n l i n e a r e x c i t i n g f o r c e o f eq. (3.4.5) can be r e p l a c e d by t h e e q u i v a l e n t l i n e a r one, thus 48 YG(ax, X) = - k-jX - XX The d.e. i n the e q u i v a l e n t l i n e a r i z e d form becomes X + X-^ X + X = 0 f o r the f i r s t a p p r o x i m a t i o n , a c c o r d i n g t o M i n o r s k y [69] , we have a = - ~ X 1 ; (3.4.18a) and i = (1 + k j 2 - a * i - (1 + k n - a 2 ) 1 2a 1 f o r k x << 1 ; a - 1 (3.4.18b) E x p r e s s i o n s f o r the e q u i v a l e n t parameters k^ and X^ are o b t a i n e d by e q u a t i n g the fundamental harmonic o f t h e non-l i n e a r f o r c e term t o the l i n e a r i z e d terms [ 6 9 ] . A c c o r d i n g l y , we have from eq. (3.4.2) 1 afra I 2TT [G(X) + F Q sin(ty-c}>)] cos tydty iTT a I "o o 2TT F G(aacosty) costydty + — s i n <}> (3.4.19) ai  i aa ,2rr k 1 = - [G(aacos^) + F Q s i n ( ^ - c b ) ] s i n tydib = - ( F Q / a ) coscb (3.4.19b) S u b s t i t u t i n g k.^  and \^ t o the e x p r e s s i o n s f o r a and cb, eq. (3.4.18), and l e t t i n g 2-n 6 = - _ ± — | G(aacos^) cos ibdty 2ai\a — f we have the e x p r e s s i o n s o f the f i r s t a p p r o x i m a t i o n : F a = - 6 e a - sincb ( 3 . 4 . 2 0 a ) 2a<j> = 1 - a 2 - (F /a) coscb (3.4.20b) o r K 1 ( a , cb) = - 2 a a 6 e - F q sincb = 2aa (3.4.21a) K„(a, cb) = (1 - a 2 ) a - F coscb = 2aacb (3.4.21b) Cm O The s t a t i o n a r y s t a t e i s g i v e n by the e q u a t i o n s \ K 1(a,cb) = 0 ; K 2 ( a , cb) = 0 (3.4.22) Thus, e q u a t i n g e q u a t i o n s (3.4.21a) and (3.4.21b) t o z e r o , s q u a r i n g and a d d i n g we have 5 0 K ( a , of) = 4 a 2 d 2 6 2 + (1 - a 2 ) 2 a 2 - F 2 = 0 (3.4.23) 6 O S u b s t i t u t i n g eq. ( 3 . 4 . 3 ) , t h e e x p r e s s i o n f o r G ( X ) , y i e l d s i n t o the e x p r e s s i o n f o r &Q and c a r r y i n g o u t the i n t e g r a t i o n , * Q l v 2 k - l J k _ 2 k - l 2 k - l 6 = 2 (7k-2) R2k-1 a a e 2aEa . . . 2 U K l ) Z K 1 k = 1 2 In o r d e r t o o b t a i n an e x p r e s s i o n f o r , we l e t ,2k-l Jk ' 2k-1 a2k-lv2 Z k , k " ( 0(2k^2) R2k-1 a 3 ' where k = 1, 2, ...,<_ „ , _ 0 , 2 i - l J i _ 2 i - l 2 i - l , a n d Z i , j = 2 (7T2i^2) R 2 i - 1 a a > (2r l J3 R a 2 ^ " 1 a 2 ^ 1 ) l,(2j-2) K 2 j - 1 a a J where i = 1, 2, £ ~ i and j = i + l , i + 2 , , <_ the n l e t S = E Z. . i = 1 i<_j <_—2~- and l + j = m where m = 2, 3, nn ; and nn i s an even number and <(n+1) 51 f i n a l l y we have nn Q N N 4 a2 6 2 = (-i)2 E S = (-i) I T a 2 m " 2 a 2 m " 2 m=2 m=2 where T a r e t h e c o e f f i c i e n t terms o f t h e e x p r e s s i o n S . m c m 2 2 S u b s t i t u t i o n o f the e x p r e s s i o n f o r 4a 6 g i n eq. (3.4.23) y i e l d s Q N N K ( a , a) = ( — ) 2 [ S S ] + (1 - a 2 ) 2 a 2 - F Q 2 = 0 (3.4.24) E m=2 Eq. (3.4.24) g i v e s t h e c o n d i t i o n f o r the e x i s t e n c e o f s t a t i o n a r y s t a t e a m p l i t u d e 'a' f o r d i s c r e t e v a l u e s o f a and F . o The s t a b i l i t y o f t h e s t a t i o n a r y s t a t e can be i n v e s t i g a t e d from t h e v a r i a t i o n a l e q u a t i o n [ 7 0 ] . 2 a ^ a - = K n 6a + K, 6cp dx 1 l j . a cp 2aa^- = K~ 6a + K 0 6cp dx 2 2, r a cp where 6a and 6cp are p e r t u r b a t i o n s i n a and cp. The c h a r a c t e r i s i t c e q u a t i o n o f the system i s a S 2 - (aK1 + K 2 ) S + ( K 1 K 2 - K 1 K 2 ) = 0 a cp a <p cp a 52 The c o n d i t i o n s f o r s t a b i l i t y a r e a K l + K < 0 ; K x K - K K 2 > 0 a tp a cp cp a The above c o n d i t i o n s f i n a l l y reduce t o [70] ^ > 0 i f a < 1 da ^ < 0 i f a > 1 da dK da dK , 8K From = T T — + T T ~ = 0 da da da 9a we have /^1\2 r n ? n i \ m 2m-3 2m-2, „ „ , 2 . 2 (—) [ E 2(m-l)T a a ] - 4 a ( l - a ) a da E m=2 m (3.4.25) da / 1\ 2 r „ ~ , - i \ m 2m-2 2m-3, , „ ... 2.2 (—) [ E 2(m-l)T a a ] + 2 a ( l - a ) „ „ m E m=2 (3.4.26) da Thus the s t a t i o n a r y s t a t e a m p l i t u d e a 1 i s s t a b l e i f g—• < 0 da f o r a > 1 o r i f g^- > 0 f o r a < 1. The s t a b i l i t y a n a l y s i s d e t e r m i n e s whether o r not the v i b r a t i o n i s a t f r e q u e n c y a o r a t o t h e r f r e q u e n c i e s . 53 3.4.3 E n t r a i n m e n t o f F r e q u e n c i e s The phenomenon o f f r e q u e n c y e n t r a i n m e n t o c c u r s when a p e r i o d i c f o r c e i s a p p l i e d t o a system whose f r e e o s c i l l a t i o n i s o f t h e s e l f - e x c i t e d t y p e [ 7 1 ] . I f a, the f r e q u e n c y r a t i o o f the e x t e r n a l e x c i t a t i o n t o t h a t o f the f r e e o s c i l l a t i o n , i s s u f f i c i e n t l y f a r away from u n i t y , t h e r e i s u s u a l l y t h e phenomenon o f i n t e r f e r e n c e o r 'beat' o f the two f r e q u e n c i e s . I f however, a approaches s u f f i c i e n t l y near t o u n i t y the b e a t s d i s a p p e a r s u d d e n l y and t h e r e remains o n l y one f r e q u e n c y thus s u g g e s t i n g t h a t t h e f r e q u e n c y o f t h e a u t o -p e r i o d i c o s c i l l a t i o n has been e n t r a i n e d by t h e e x t e r n a l f r e q u e n c y . The e n t r a i n m e n t o f f r e q u e n c y may a l s o o c c u r when the f r e q u e n c y r a t i o a i s i n the neighbourhood o f an i n t e g e r , o r a f r a c t i o n . Under t h e s e c o n d i t i o n s , t h e f r e q u e n c y o f f r e e o s c i l l a t i o n i s e n t r a i n e d by a f r e q u e n c y w h i c h i s an i n t e g r a l m u l t i p l e o r s u b m u l t i p l e o f the e x t e r n a l f r e q u e n c y . Van d e r P o l gave a t h e o r y f o r t h i s phenomenon [72] by assuming a s o l u t i o n o f t h e form X(T) = b 1 ( x ) s i n a i + b 2 ( r ) cos a i (3.4.27) i n which the f u n c t i o n b ^ ( r ) a r e assumed t o be " s l o w l y v a r y i n g f u n c t i o n s ' o f t i m e . From eq. (3.4.27) i t i s p o s s i b l e t o reduce the o r i g i n a l d.e. t o a system o f the form 54 b, = M,(b , b~) and b 2 = M ^ , b 2 ) (3.4.28) t h u s t h e c o n d i t i o n s f o r a s t a t i o n a r y o s c i l l a t i o n , b^ = c o n s t , and b 2 = c o n s t . , reduce t o M-^b^ b 2 ) = 0, and M 2 ( b 1 , b 2 ) = 0. based on the above t h e o r y was deve l o p e d by Andronov and W i t t [ 7 3 ] , i n which t h e system o f e q u a t i o n s (3.4.28) i s t r a n s f o r m e d i n t o t h e form I n t h e p r e s e n t i n v e s t i g a t i o n , owing t o t h e c o m p l e x i t y o f the n o n - l i n e a r f u n c t i o n , the stu d y i s limited to the cases where the a m p l i t u d e and f r e q u e n c y o f the e x t e r n a l e x c i t a t i o n a r e e i t h e r i n s i d e o r o u t s i d e the r e g i o n s o f e n t r a i n m e n t b u t n o t near the boundary o f the r e g i o n o f e n t r a i n m e n t . Under such c i r c u m s t a n c e s a s i m p l e r method c o u l d be employed by a p p l y i n g t h e p r i n c i p l e o f harmonic b a l a n c e . The method gave s u f f i c i e n t i n f o r m a t i o n f o r the ste a d y s t a t e c o n d i t i o n . In a p p l y i n g the method, the harmonic s o l u t i o n o f eq. ( 3 . 4 . 2 ) , when a-1, may be w r i t t e n t o the f i r s t approx-i m a t i o n , as A p u r e l y t o p o l o g i c a l t h e o r y o f s y n c h r o n i z a t i o n M 1 ( b 1 , b 2 ) M 2 ( b l f b 2 ) 55 X(T) s i n a x + b~ cosax (3.4.29) where the a m p l i t u d e s and b 2 a r e e x p r e s s e d as c o n s t a n t s s i n c e we a r e i n t e r e s t e d o n l y w i t h the p e r i o d i c s o l u t i o n . I f t h e e x t e r n a l f o r c e i s p r e s c r i b e d o u t s i d e the r e g i o n s o f e n t r a i n m e n t , one may e x p e c t th e o c c u r r e n c e o f an a l m o s t p e r i o d i c o s c i l l a t i o n . Under t h e s e c o n d i t i o n s t h e method would s t i l l f u r n i s h a f a i r l y good d e s c r i p t i o n o f t h e a l m o s t p e r i o d i c o s c i l l a t i o n [74] . However, i f t h e e x t e r n a l f o r c e i s p r e s c r i b e d c l o s e t o the boundary o f t h e r e g i o n s o f e n t r a i n m e n t , th e a n a l y s i s does n o t a d e q u a t e l y a c c o u n t f o r i t s i n c e the waveform o f t h e a l m o s t p e r i o d i c o s c i l l a t i o n d i f f e r s c o n s i d e r a b l y from t h a t o b t a i n e d as a sum o f two s i m p l e harmonic o s c i l l a t i o n s . The a m p l i t u d e s b^ and b 2 i n eq. (3.4.29) have t o be c o n s i d e r e d as f u n c t i o n s v a r y i n g s l o w l y w i t h t i m e , as i n eq. (3.4.27). S u b s t i t u t i n g eq. (3.4.2) and e q u a t i n g the co-e f f i c i e n t s o f s i n a x and cosax from b o t h s i d e s o f the e q u a t i o n , we have, a f t e r l o n g a l g e b r a i c m a n i p u l a t i o n , two e q u a t i o n s i n terms o f b, and b_. b r ( l a 2) - b 0aZ ( a 2 ) = F (3.4.30) b 2 (1 - a 2) + b ^ a Z ( a 2 ) = 0 5 6 2 2 2 2 2 where a = + b 2 and Z(a ) i s a f u n c t i o n o f a D i v i d i n g eq. (3.4.30) t h r o u g h by a and l e t t i n g 2 a 1 = (1 - a )/a we have from eq. (3.4.30) 2 h^a1 - b 2 Z ( a ) = F Q/a (3.4.31) b 2 a ' + b±Z(a2) = 0 S q u a r i n g b o t h e q u a t i o n s o f (3.4.31) and a d d i n g we f i n a l l y have an e q u a t i o n i n terms o f a, a', and F a 2 [ a ' 2 + Z 2 ( a 2 ) ] = ( F Q / a ) 2 (3.4.32) 2 The f u n c t i o n Z(a ) o f eq. (3.4.32) has t h e g e n e r a l i z e d form f o r a f r i c t i o n f o r c e f u n c t i o n e x p r e s s e d i n the form o f a n t h o r d e r p o l y n o m i a l , k<2±i z < * 2 ) = ~ 7T2T^) R 2 k - l * 2 k " 2 * 2 k " 2 ( 3 - 4 ' 3 3 ) E k = 1 2 where Q 1 = rwh + P , = 1, R^ = f o r k = 2, 3, n~f* 1 — ""2 ' s a m e a s ^ n ^ e P r e v i ° u s c a s e s . S u b s t i t u t i n g eq. (3.4.33) i n t o eq. (3.4.32) we have a 2 a 2 Z 2 ( a 2 ) + (1 - a 2 ) 2 a 2 - F Q 2 = 0 (3.4.34) 57 Eq. (3.4.34) has a s i m i l a r form as eq. (3.4.23) o f the fundamental resonance case by the f i r s t a p p r o x i m a t i o n method o f K r y l o v - B o g o l i u b o f f . I n f a c t the method o f e q u i v a l e n t l i n e a r i z a t i o n i s based on the p r i n c i p l e o f harmonic b a l a n c e . 2 a v a l u e s can be o b t a i n e d from eq. (3.4.34) f o r d i s c r e t e v a l u e s o f a and F . The a m p l i t u d e s b.. and b~ o f o c 1 2 eq. (3.4.29) can be o b t a i n e d by s o l v i n g the s i m u l t a n e o u s 2 e q u a t i o n s o f eq. (3.4.31) w i t h t h e a v a l u e s o b t a i n e d from eq. (3.4.34), w h i c h g i v e s 2 F (1-a ) F Za 1 2r72J_,, 2>2 ' 2 2 1 72.,, 2,2 u ^ , J 3 j a Z + ( l - a ) a Z + ( l - a ) When a i s i n t h e neighbourhood o f 2, 3 e t c . sub-harmonic e n t r a i n m e n t may e x i s t i n t h e system. An approx-i m a t i o n s o l u t i o n f o r eq. (3.4.2) may have t h e form [ 7 5 ] . F X ( T ) = b ^ s i n cax + cos c a r + ^ s i n a T (3.4.36) 1-a where c = J' 3  etc" A g a i n , two e q u a t i o n s i n terms o f b^ and can be o b t a i n e d by s u b s t i t u t i n g eq. (3.4.36) and e x p r e s s i o n s f o r X and X i n t o eq. (3.4.2) and e q u a t i n g t h e c o e f f i c i e n t s o f s i n cax and cos c a r s e p a r a t e l y t o z e r o . However, due t o the e x t r e m e l y c o m p l i c a t e d form o f the e q u a t i o n s , a g e n e r a l -i z e d e q u a t i o n such as eq. (3.4.32) cannot be o b t a i n e d f o r t h i s c a s e . 58 A l t e r n a t i v e l y , an approximate s o l u t i o n f o r eq. (3.4.2) may be o b t a i n e d from a s l i g h t l y d i f f e r e n t approach, i n w hich the s o l u t i o n i s sought i n the form [ 7 6 ] . X(T) = a s i n x + a g s i n a x (3.4.37) I n a d d i t i o n , a phase a n g l e i s i n t r o d u c e d t o the e x t e r n a l f o r c i n g term o f eq. (3.4.2). Thus eq. (3.4.2) can be w r i t t e n as X + X = yG(X) + F q s i n ( a x + cb) (3.4.38) Three e q u a t i o n s can be o b t a i n e d by s u b s t i t u t i n g t h e s o l u t i o n o f eq. (3.4.37) i n t o eq. (3.4.38) and e q u a t i n g t h e c o e f f i c i e n t s o f s i n a x and cosax from b o t h s i d e s o f the e q u a t i o n , a l s o e q u a t i n g the c o e f f i c i e n t s o f cosx t o z e r o . Thus we have 2 (1-a )a s i n a x + H.. (a,a ) cosx + H„ (a,a ) cosax e 1 e 2 ' e F s i n a x coscb + F cosax s i n cb (3.4.39) o o Y and (1 - a 2 ) a e = F q coscb (3.4.40a) H 2 ( a , a e ) = F q sincb (3.4.40b) H 1 ( a , a e ) = 0 (3.4.40c) where H,(a,a ) and H„(a,a ) a r e f u n c t i o n s o f a and a , t h e X 6 Z 6 6 a m p l i t u d e s o f t h e a u t o p e r i o d i c and h e t e r o p e r i o d i c o s c i l l a t i o n r e s p e c t i v e l y . The f u n c t i o n s are o b t a i n e d by c o l l e c t i n g terms o f C O S T and c o s a i r e s p e c t i v e l y from t h e e x p a n s i o n o f the p o l y n o m i a l G (X) i n terms o f X = a C O S T + a a g cosax. S o l v i n g t h e t h r e e e q u a t i o n s o f eq. (3.4.40) g i v e s v a l u e s o f a, a Q and cp. Owing t o the presence o f t h e a g s i n a x term, a g e n e r a l -i z e d s o l u t i o n e q u a t i o n as g i v e n i n e q u a t i o n s (3.4.33) and (3.4.34) cannot be o b t a i n e d f o r the e q u a t i o n s o f eq. ( 3 . 4 . 4 0 ) . F o r d i f f e r e n t o r d e r s o f p o l y n o m i a l , d i f f e r e n t s e t s o f e q u a t i o n s have t o be d e r i v e d . Appendix ( I I ) shows an e x p a n s i o n o f a f r i c t i o n f o r c e f u n c t i o n i n terms o f X = a sinT + a sinaT e t h f o r a 7 o r d e r p o l y n o m i a l . The p r e s e n t i n v e s t i g a t i o n i s m a i n l y concerned w i t h f i n d i n g the e f f e c t o f the e x t e r n a l e x c i t a t i o n upon the a m p l i t u d e o f the f r i c t i o n - i n d u c e d v i b r a t i o n i n t h e st e a d y s t a t e c o n d i t i o n ; whereas t h e phase r e l a t i o n s h i p among t h e s e a m p l i t u d e s i s n o t o f p a r t i c u l a r i n t e r e s t . F o r t h i s r e a s o n , t h e approximate s o l u t i o n o f eq. (3.4.37) p r o v i d e s a s i m p l e b u t e f f e c t i v e t o o l f o r g a i n i n g some i n s i g h t o f t h e non-autonomous system w i t h f r i c t i o n - i n d u c e d v i b r a t i o n . Approx-imate s o l u t i o n s f o r a ^ 1 and f o r a = 2, 3, e t c . can be o b t a i n e d by a p p l y i n g the approximate s o l u t i o n o f eq. (3.4.37) t o eq. ( 3 . 4 . 3 8 ) . When a-1, e i t h e r the f i r s t a p p r o x i m a t i o n o f the K and B method o r the harmonic b a l a n c e method u s i n g 60 eq. (3.4.29) can be a p p l i e d . The K and B method f o r the non-resonance case p r o v i d e s a u s e f u l t e c h n i q u e f o r p r e -l i m i n a r y i n v e s t i g a t i o n o f the system w i t h p a r t i c u l a r r e f e r -ence t o t h e e x t e r n a l e x c i t a t i o n p a r a m e t e r s . N e e d l e s s t o s a y , a l l t h e s e methods i n v o l v e t r i g o n m e t r i c m a n i p u l a t i o n , and t h e f i n a l n u m e r i c a l s o l u t i o n can o n l y be o b t a i n e d w i t h the a i d o f a computer. 3.4.4 D y n a m i c a l l y Loaded System The e q u a t i o n o f m o t i o n f o r the t h i r d system i s shown by eq. (3.1.6) i n the n o n - d i m e n s i o n a l i z e d form. I n e s t a b l i s h -i n g t h i s e q u a t i o n t h e f r i c t i o n f o r c e v e r s u s v e l o c i t y c h a r a c -t e r i s t i c was c o n s i d e r e d unchanged d u r i n g dynamic l o a d i n g , i . e . the same f r i c t i o n - v e l o c i t y c u r v e as i n the autonomous case was a p p l i e d . T h i s may not be t h e case i n p r a c t i c e ; t h e p resence o f a normal p e r i o d i c e x c i t a t i o n may cause e a r l y breakdown o f the a s p e r i t y c o n t a c t . The e x t e r n a l e x c i t a t i o n was a p p l i e d a t the l o a d i n g end w h i c h was connected t o the e l a s t i c beam system by a massive s t e e l clamp and two heavy s t e e l beams. T h e r e f o r e normal o s c i l l a t i o n a t the s l i d e r end due t o the e x t e r n a l e x c i t a t i o n would be n e g l i g i b l y s m a l l and was n o t c o n s i d e r e d i n the e q u a t i o n . Eq. (3.1.6) can be r e - w r i t t e n t o g i v e X + y d + Bsinax) G(X) + X = C Q + 6C Q s i n a x (3.4.41) where C n, a c o n s t a n t , i s t h e s t a t i c d i s p l a c e m e n t o f the f r i c t i o n - v e l o c i t y c u r v e , and G(X) i s a n o n l i n e a r f u n c t i o n i n X. Eq. (3.4.41) i s a n o n l i n e a r d.e. w i t h p e r i o d i c c o e f f i c i e n t s . There a r e methods f o r s o l v i n g t h i s t y p e o f e q u a t i o n s such as t h e S t r o b o s c o p i c method [77] o r t h e WKBJ method [ 7 8 ] . However, t h e a n a l y s i s i n v o l v e s t e d i o u s a l g e b r a i c m a n i p u l a t i o n even f o r a l i n e a r system o r f o r a system w i t h o n l y one n o n l i n e a r term. I n the p r e s e n t i n v e s t i g a t i o n , eq. (3.4.41) was i n v e s t i g a t e d by n u m e r i c a l methods u s i n g a computer. 3.5 Summary In a p p l y i n g the t h e o r e t i c a l a n a l y s i s , e x p e r i m e n t a l t h f r i c t i o n f o r c e f u n c t i o n s were computer f i t t e d by a n o r d e r p o l y n o m i a l w i t h c o e f f i c i e n t s CQ, C^, e t c . as shown i n eq. (3.2.2). The c o e f f i c i e n t s were t r a n s f o r m e d i n t o c o e f f i c i e n t s P^, and R k r e s p e c t i v e l y as shown i n e q u a t i o n s (3.3.16), (3.3.17) and (3.3.18). F i n a l l y a s o l u -t i o n e q u a t i o n f o r t h e autonomous case was o b t a i n e d as shown i n eq. (3.3.19). S o l v i n g eq. (3.3.19), t h e s t e a d y s t a t e a m p l i t u d e 'a' was o b t a i n e d . The s t a b i l i t y o f the am p l i t u d e was i n v e s t i g a t e d by a p p l y i n g eq. (3.3.20). The c o e f f i c i e n t s were f u r t h e r t r a n s f o r m e d t o g i v e c o e f f i c i e n t s E^ a c c o r d i n g t o the e x p r e s s i o n o f eq. (3.4.11a). An e q u a t i o n i n terms o f a m p l i t u d e 'a' f o r the non-autonomous non-resonance case was o b t a i n e d as shown i n eq. (3.4.12). The s t a t i o n a r y s t a t e a m p l i t u d e and i t s s t a b i l i t y c o u l d be i n v e s t i g a t e d by a p p l y i n g e q u a t i o n s (3.4.13) and (3.4.14). F o r t h e fundamental resonance c a s e , the c o e f f i c i e n t s were a p p l i e d t o t h e e x p r e s s i o n s f o r Z^ . ^, and Z^ . The c o e f f i c i e n t s T were o b t a i n e d by c o l l e c t i n g terms o f Z. . m J 3 I , j f o r i + j = m. F i n a l l y eq. (3.4.24) and eq. (3.4.26) f o r t h e a m p l i t u d e and f r e q u e n c y o f the fundamental resonance o s c i l l a t i o n and t h e s t a b i l i t y o f the a m p l i t u d e w i t h f r e -quency a were o b t a i n e d . I n the subharmonic e n t r a i n m e n t c a s e s , e q u a t i o n s were d e r i v e d f o r a f r i c t i o n f o r c e f u n c t i o n t h e x p r e s s e d by a 7 o r d e r p o l y n o m i a l (Appendix I I ) . The e x p o n e n t i a l e x p r e s s i o n o f eq. (3.2.1) was used i n t h e t h e o r e t i c a l a n a l y s i s o f the autonomous case as w e l l as i n a l l t r a n s i e n t s t a t e i n v e s t i g a t i o n s by n u m e r i c a l methods. I n t h e autonomous c a s e , the e x i s t e n c e and n o n - e x i s t e n c e o f the s e l f - e x c i t e d v i b r a t i o n as w e l l as the growth and decay o f the s t a b l e o s c i l l a t i o n i s p r e d i c t e d f o r v a r i o u s forms o f f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e . The q u a s i - h a r m o n i c t y p e f r i c t i o n - i n d u c e d v i b r a t i o n may be e x t i n g u i s h e d by the a p p l i c a t i o n o f e x t e r n a l v i s c o u s damping i t s magnitude i s r e l a t e d t o the c o e f f i c i e n t s o f t h e e x p o n e n t i a l e x p r e s s i o n r e p r e s e n t i n g the form o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e . 63 When the quasi-harmonic type friction-induced v i b r a t i o n i s under the influence of external transverse ex c i t a t i o n , there exists two d i s t i n c t i v e cases of p a r t i c -ular i n t e r e s t , depending on whether the frequency r a t i o a i s close to an integer or far away from i t . When a i s not equal or close to an integer, the amplitude of the autoperiodic o s c i l l a t i o n i s decreased and i s eventually completely extinguished as the external force magnitude i s increased. When the frequency of the external e x c i t a t i o n i s equal or close to a multiple of the frequency of the autonomous case, subharmonic entrainment i s predicted, whereby the system vibrates at a frequency equal or close to the frequency of the autonomous case. The amplitude of vibr a t i o n during subharmonic entrainment, p a r t i c u l a r l y i n the case of ~ harmonic, i s increased as the magnitude of the external e x c i t a t i o n i s increased, u n t i l a c r i t i c a l value i s reached whereby the amplitude of the vib r a t i o n drops o f f rapidly and only the heteroperiodic o s c i l l a t i o n remains at the frequency of the external e x c i t a t i o n , usually of small amplitude due to the comparatively high frequency. IV EXPERIMENTAL 4.1 I n t r o d u c t i o n The v a r i a t i o n o f f r i c t i o n f o r c e w i t h s l i d i n g v e l o c i t y has been o b s e r v e d and d i s c u s s e d e x t e n s i v e l y i n t h e l i t e r a t u r e . V a r i o u s t y p e s of a p p a r a t u s have been employed f o r the i n v e s t i g a t i o n o f 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 s . P i n on d i s c machines, c r o s s e d c y l i n d e r s , f o u r b a l l t r i b o m e t e r s and o t h e r c o n f i g u r a t i o n s have been used. I n g e n e r a l , f r i c t i o n f o r c e i s r e l a t e d t o some form o f d i s p l a c e m e n t measurement. However, the assessment of dynamic f r i c t i o n f o r c e by r e c o r d i n g the d i s p l a c e m e n t a l o n e does not always g i v e a t r u e r e p r e s e n t a t i o n o f the f r i c t i o n v a l u e s and c a u t i o n must be e x e r c i s e d i n t h e i n t e r -p r e t a t i o n o f r e c o r d s . Dynamic f r i c t i o n measurements are s u b j e c t t o v a r i o u s e r r o r s w h i c h may be c l a s s i f i e d as f o l l o w s : ( i ) E x t e r n a l v i b r a t i o n : V i b r a t i o n from the d r i v e mechanism o f t h e f r i c t i o n machine o r from e x t e r n a l s o u r c e s can d i s t u r b f r i c t i o n measurements. G o d f r e y [33] o b s e r v e d v a r i a t i o n s o f the measured c o e f f i c i e n t o f f r i c t i o n p r o -duced by e x t e r n a l v i b r a t i o n s . A c c o r d i n g l y , the a c c u r a t e measurement o f f r i c t i o n would appear t o demand t h a t e x t r a n e o u s v i b r a t i o n be removed from the a p p a r a t u s and the s u r r o u n d i n g s . ( i i ) S e l f - I n d u c e d V i b r a t i o n : Many measuring systems are s u b j e c t t o f r i c t i o n - i n d u c e d v i b r a t i o n o f the s t i c k -s l i p t y p e . A t low s l i d i n g v e l o c i t i e s i n p a r t i c u l a r , s t i c k -s l i p v i b r a t i o n i s l i a b l e t o o c c u r and the dynamic f r i c t i o n f o r c e s o b t a i n e d by a v e r a g i n g d i s p l a c e m e n t s o r f o r c e s d u r i n g v i b r a t i o n may be g r o s s l y i n e r r o r . I n some systems, q u a s i -harmonic f r i c t i o n - i n d u c e d v i b r a t i o n may e x i s t . The v i b r a t i o n waveform i s n e a r - s i n u s o i d a l and f r i c t i o n f o r c e s o b t a i n e d by a v e r a g i n g d i s p l a c e m e n t s are u s u a l l y r e a s o n a b l y a c c u r a t e . In some cases b o t h s t i c k - s l i p and q u a s i - h a r m o n i c v i b r a t i o n may be p r e v e n t e d o r l i m i t e d by t h e use o f s u i t -a b l e damping d e v i c e s i n t h e measuring system. However, i t w i l l be shown t h a t dynamic f r i c t i o n f o r c e s can be a c c u r a t e l y measured even i n the p r e s e nce of f r i c t i o n e x c i t e d v i b r a t i o n . ( i i i ) S u r f a c e , L u b r i c a t i o n and E n v i r o n m e n t a l F a c t o r s : In many cases f l u c t u a t i o n s i n f r i c t i o n f o r c e s a r e produced by v a r i a t i o n s i n t h e p h y s i c a l q u a l i t y o f s u r f a c e s , l u b r i -c a t i o n c o n d i t i o n s , c h e m i c a l and e n v i r o n m e n t a l f a c t o r s . However, many o f t h e s e f l u c t u a t i o n s i n f r i c t i o n can be removed by c a r e f u l s u r f a c e p r e p a r a t i o n , c o n t r o l o f l u b r i -c a t i o n c o n d i t i o n s and c h e m i c a l s u r f a c e f a c t o r s and by t h e use o f i n e r t o r h i g h vacuum e n v i r o n m e n t s . 4.2 App a r a t u s As i t had been i n d i c a t e d e a r l i e r t h a t t h e p r i m a r y o b j e c t i v e o f t h e p r e s e n t i n v e s t i g a t i o n was t o s t u d y the q u a s i - h a r m o n i c f r i c t i o n - i n d u c e d v i b r a t i o n and i t s r e l a t i o n -s h i p w i t h the dynamic f r i c t i o n c h a r a c t e r i s t i c . A secondary o b j e c t i v e was t o s t u d y t h e e f f e c t on t h e f r i c t i o n - i n d u c e d v i b r a t i o n produced by e x t e r n a l harmonic e x c i t a t i o n s , e i t h e r t r a n s v e r s l y o r n o r m a l l y . The ap p a r a t u s was d e s i g n e d t o meet t h e s e o b j e c t i v e s . Some e a r l y i n v e s t i g a t i o n s i n t h e p r e s e n t r e s e a r c h showed t h a t f o r c e r t a i n c o m b i n a t i o n s o f f r i c t i o n m a t e r i a l s a r u n - i n p e r i o d was r e q u i r e d f o r t h e q u a s i - h a r m o n i c f r i c t i o n -i n d u c e d v i b r a t i o n t o o c c u r . The c h o i c e of' a r o t a t i n g d i s c as t h e lower s u r f a c e o f the f r i c t i o n a p p a r a t u s p r o v i d e d a c o n v e n i e n t means when r u n - i n i s r e q u i r e d . The r a t i o o f the speed r e d u c e r and c o n s e q u e n t l y the speed range o f the r o t a t i n g d i s c was d e t e r m i n e d t o p r o v i d e a s u i t a b l e com-promise o f maximum and minimum speeds such t h a t i t would be low enough f o r t h e appearance o f some s t i c k - s l i p t y pe f r i c t i o n - i n d u c e d v i b r a t i o n and s t i l l h i g h enough t o show the decay o f t h e q u a s i - h a r m o n i c v i b r a t i o n i n some c a s e s . C o n s i d e r a t i o n was g i v e n a l s o t o t h e n a t u r a l f r e -quency o f t h e f r i c t i o n system which was b a s i c a l l y d e t e r m i n e d by t h e v i b r a t i n g mass and 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. I t was i n t e n d e d t o s t u d y the sub-harmonic 1 1 1 resonances o f o r d e r , and i n the e x t e r n a l e x c i t -67 a t i o n c a s e s , t h e r e f o r e t h e n a t u r a l f r e q u e n c y o f t h e f r i c -t i o n system s h o u l d n o t exceed a c e r t a i n l i m i t so t h a t i t s t h 4 harmonic would s t i l l be w i t h i n the range o f t h e ex-t e r n a l e x c i t a t i o n g e n e r a t o r . F i g . 4.2 .1 and F i g . 4.2.2 show two views o f the e x p e r i m e n t a l s e t - u p which i n c l u d e s the f r i c t i o n a p p a r a t u s and r e c o r d i n g i n s t r u m e n t s . The a p p a r a t u s shown i n F i g . 4.2.3 was d e s i g n e d f o r the s t u d y o f f r i c t i o n and f r i c t i o n - i n d u c e d v i b r a t i o n between two s l i d i n g s u r f a c e s . Three major v a r i a b l e s were i n v o l v e d i n the i n v e s t i g a t i o n , namely; s u r f a c e v e l o c i t y , s t i f f n e s s o f the s u p p o r t i n g system and the normal l o a d . Thus, the e s s e n t i a l p a r t s o f the a p p a r a t u s c o n s i s t e d o f a r o t a t i n g d i s c d r i v e n by a v a r i a b l e speed d r i v i n g u n i t , a c a n t i l e v e r beam and a l o a d system w h i c h d i d n o t a l t e r the magnitude o f the v i b r a t i n g mass. The f r i c t i o n c o u p l e was formed by a 4 i n c h d i a m e t e r s t e e l d i s c as t h e lower s u r f a c e and a s l i d e r as t h e upper s u r f a c e . The s l i d e r was a t t a c h e d t o a h e m i s p h e r i c a l shaped s l i d e r mount ( F i g . 4.2.3b) wh i c h p r o v i d e d t h e s e l f - a l i g n i n g a c t i o n f o r the s l i d e r thus e n s u r i n g u n i f o r m c o n t a c t . The s l i d e r mount was made from a s t e e l b a l l b e a r i n g w h i c h rode i n a h e m i s p h e r i c a l shaped r e t a i n i n g cup i n t h e specimen h o l d e r . The s l i d e r , t h e s l i d e r mount and t h e specimen h o l d e r t o g e t h e r w i t h a p r o p o r t i o n o f t h e s u p p o r t i n g beam formed th e v i b r a t i n g mass. The c e n t r e o f t h e s l i d e r i s 68 w i t h i n i i n . t h e C.G. o f t h e v i b r a t i n g mass. The beam was a r r a n g e d such t h a t i t s n e u t r a l a x i s i s almost i n l i n e w i t h t h e p l a n e o f t h e c o n t a c t i n g s u r f a c e s , so t h a t the t o r s i o n a l e f f e c t on the beam o f t h e f r i c t i o n t o r q u e was n e g l i g i b l e . A c a n t i l e v e r s u p p o r t i n g beam p r o v i d e d t h e e l a s t i c i t y o f the system. The o t h e r end o f t h e s t e e l beam was f i x e d s o l i d l y i n a s t e e l clamp and s h a f t assembly which was p i v o t e d by two low f r i c t i o n b e a r i n g s . The l e n g t h o f t h e c a n t i l e v e r beam was a d j u s t a b l e and p r o v i s i o n f o r i n t e r c h a n g e -a b i l i t y o f beams was made so t h a t t h e s p r i n g s t i f f n e s s o f the system c o u l d be v a r i e d . Load was a p p l i e d by means o f w e i g h t s t h r o u g h a p u l l e y arrangement which imposed a moment on the system. The system p e r m i t t e d the normal l o a d t o be v a r i e d w i t h o u t c h a n g i n g the v i b r a t i n g mass. I n o r d e r t o m i n i m i s e the c u r v a t u r e e f f e c t d u r i n g v i b r a t i o n , t h e specimen h o l d e r l o c a t i o n was a r r a n g e d such t h a t the r a d i u s o f c u r v a t u r e f o r t h e s l i d e r v i b r a t i o n was i n the same sense -as t h e r u n n i n g t r a c k on the r o t a t i n g d i s c . 3 The d r i v i n g u n i t c o n s i s t e d o f a v a r i a b l e speed jg"vh.p. d.c. motor and a 100: 1 double worm speed r e d u c e r . I n t h e e a r l y s t a g e s o f t h e development o f the d r i v e system a ig - h.p. s e r v o motor d r i v i n g a c h a i n o f spur gears were used t o d r i v e the r o t a t i n g d i s c t h r o u g h a s e t o f b e v e l , g e a r s . However i t was found t h e s e elements i m p a r t e d un-wanted v i b r a t i o n t o the d i s c and s l i d e r and they were sub-s e q u e n t l y a b a n d o n e d . I n o r d e r t o e l i m i n a t e t h e e f f e c t s o f m o t o r v i b r a t i o n on f r i c t i o n p r o c e s s e s c a r e f u l i s o l a t i o n t e c h n i q u e s w e r e e m p l o y e d . Power was t r a n s m i t t e d b y g r o o v e d a l u m i n u m s h e a v e s a nd a s o f t r u b b e r 0 - r i n g . T h i s t y p e o f t r a n s m i s s i o n r e d u c e d a l i g n m e n t p r o b l e m s and p r o v i d e d g o o d v i b r a t i o n i n s u l a t i o n b e t w e e n t h e m o t o r a nd t h e s p e e d r e d u c e r . F i n a l l y , i n o r d e r t o g i v e g o o d v i b r a t i o n i s o l a t i o n t h e f r i c t i o n a p p a r a t u s was p l a c e d on a m a s s i v e t a b l e w h i c h r e s t e d on a r i g i d f o u n d a t i o n . 3 The h.p. d.e. m o t o r h a d an e f f e c t i v e s p e e d r a n g e f r o m 50 rpm t o 3,000 rpm. Two t w o - s t e p s h e a v e s w e r e u s e d , t h e s e g a v e f o u r s p e e d r a t i o s b e t w e e n t h e m o t o r s h a f t and t h e s p e e d r e d u c e r . The r a t i o s w e r e .41, .65, 1.55 and 2.45. T h e s e t o g e t h e r w i t h t h e 100:1 s p e e d r e d u c e r p r o v i d e d a r e a s o n a b l e r a n g e o f d i s c v e l o c i t y . T h u s , on a i n . d i a . r u n n i n g t r a c k , t h e t r a c k v e l o c i t y h a d a r a n g e f r o m 0.038 i n / s e c t o 13.5 i n / s e c . The d.e. m o t o r was s o o r i e n t e d s u c h t h a t a minimum amount o f m a g n e t i c f i e l d i n f l u e n c e w o u l d be p i c k e d up by t h e m e a s u r i n g i n s t r u m e n t s . D u r i n g l o w s p e e d t e s t s a c o n s t a n t t o r q u e was a p p l i e d t o t h e r o t a t i n g d i s c t h r o u g h a c o r d a n d p u l l e y a r r a n g e m e n t . T h i s t e c h n i q u e p e r m i t t e d t h e c a n c e l l a t i o n o f t h e s m a l l amount o f r e s i d u a l b a c k l a s h i n t h e worm g e a r s p e e d r e d u c e r . 4.2.1 E x t e r n a l H a r m o n i c V i b r a t i o n G e n e r a t o r : The s e c o n d and t h i r d p h a s e s o f t h e i n v e s t i g a t i o n r e q u i r e d t h e p r o v i s i o n o f e x t e r n a l e x c i t a t i o n , t h u s h a r m o n i c 70 v i b r a t i o n g e n e r a t o r s were added t o t h e f r i c t i o n a p p a r a t u s . The v e r y n a t u r e o f t h e i n v e s t i g a t i o n , w h i c h i n v o l v e d f r i c -t i o n - i n d u c e d v i b r a t i o n , p r o h i b i t e d t h e a p p l i c a t i o n o f t r a n v e r s e e x c i t a t i o n t h r o u g h l i n k a g e s o r any o t h e r mechanism h i n g e d t o a f i x e d l o c a t i o n . The e x c i t i n g f o r c e c o u l d have been a p p l i e d e i t h e r by e l e c t r o m a g n e t s o r by unbalanced r o t a t i n g masses. A l t h o u g h a d e v i c e u s i n g e l e c t r o m a g n e t s would have g i v e n a-b e t t e r f r e q u e n c y and a m p l i t u d e c o n t r o l , i t u n a v o i d a b l y r e q u i r e d more e l e c t r o n i c i n s t r u m e n t s t h u s t h e s i m p l e r con-c e p t u s i n g u n b a l a n c e d r o t a t i n g masses was adopted. T h i s system c o n s i s t e d o f one o r two r o t a t i n g u n b a l a n c e d masses. The b e a r i n g s s u p p o r t i n g t h e s e masses were f i x e d t o the v i b r a t i n g mass. In F i g . 4.2.4a was shown a v i b r a t i o n g e n e r a t o r d e v e l o p e d i n t h e e a r l y s t a g e o f t h e i n v e s t i g a t i o n . The g e n e r a t o r c o n s i s t e d o f a m i n i a t u r e permanent magnet d.c. motor and two 1 i n . d i a . d i s c s r o t a t i n g i n o p p o s i t e d i r e c -t i o n s . The motor drove t h e upper d i s c t h r o u g h m i n i a t u r e c o u p l i n g s w h i l e t h e lower d i s c was d r i v e n by the upper d i s c s h a f t t h r o u g h a c h a i n o f p r e c i s i o n spur g e a r s . The un-b a l a n c e d masses were a r r a n g e d such t h a t t h e f o r c e was o b t a i n e d i n the same d i r e c t i o n as t h e s e l f - i n d u c e d v i b r a -t i o n . The motor was housed i n a c y l i n d r i c a l shaped motor h o u s i n g which p e r m i t t e d the motor t o move f r e e l y i n the v e r t i c a l d i r e c t i o n . The complete d e v i c e weighed o n l y 14 oz was a t t a c h e d t o the s l i d e r h o l d e r o f t h e f r i c t i o n a p p a r a t u s . V a r i o u s o u t - o f - b a l a n c e masses were p r e p a r e d which p r o v i d e d v a r i a b l e f o r c e a m p l i t u d e s a t a s e t f r e q u e n c y . However, the unwanted h i g h f r e q u e n c y o s c i l l a t i o n s w h i c h were a s s o c i a t e d w i t h g e a r s and b a l l b e a r i n g s i n t h e u n i t showed up i n s i g n i f i c a n t l y h i g h a m p l i t u d e s i n t h e a c c e l e r a t i o n s i g n a l , thus r e n d e r i n g the g e n e r a t o r u n u s a b l e when a c c e l e r a t i o n and v e l o c i t y were t o be r e c o r d e d . The v i b r a t i o n g e n e r a t o r was s u b s e q u e n t l y r e d e s i g n e d . The b a l l b e a r i n g s and spur gears were e l i m i n a t e d and r e p l a c e d by b e a r i n g f e l t bushes and a r u b b e r O - r i n g . F i g . 4.2.4b shows the r e d e s i g n e d harmonic v i b r a t i o n g e n e r a t o r . I t c o n s i s t e d o f a ^ h.p. s e r v o motor and a 2 i n . d i a . d i s c w i t h 4 h o l e s d r i l l e d a t d i f f e r e n t r a d i i . The motor was l o c a t e d away from the m easuring i n s t r u m e n t s so t h a t the f i e l d i n t e r f e r e n c e would not be p i c k e d up by the i n s t r u m e n t s . A s o f t r u b b e r O - r i n g and a s e t o f aluminum sheaves were used t o d r i v e the d i s c w hich r a n on a p a i r o f f e l t bushes. T h i s t e c h n i q u e p e r m i t t e d smooth and q u i e t o p e r a t i o n even a t r o t a t i n g speeds e x c e e d i n g 5,000 rpm. The s o f t O - r i n g showed no measurable e f f e c t s on f r i c t i o n r e c o r d i n g d u r i n g f r i c t i o n - i n d u c e d v i b r a t i o n . The 4 h o l e s a t d i f f e r e n t r a d i i t o g e t h e r w i t h s e v e r a l d i f f e r e n t o u t - o f -b a l a n c e masses p r o v i d e d a wide range o f f o r c e a m p l i t u d e s over a range of f r e q u e n c i e s . U n l i k e the e a r l i e r d e s i g n , o n l y one r o t a t i n g d i s c was used i n the f i n a l d e s i g n , 72 t h e r e f o r e no p r o v i s i o n was made t o r e s t r i c t t h e r e s u l t a n t f o r c e t o one d i r e c t i o n . However, t h e e f f e c t o f the com-ponent on the l o n g i t u d i n a l d i r e c t i o n was n e g l i g i b l e due t o the r i g i d i t y o f the beam system a l o n g t h i s d i r e c t i o n . A s i m i l a r arrangement (Ref. F i g . 4.2.2) was used f o r t h e i n v e s t i g a t i o n o f the e f f e c t o f h a r m o n i c a l l y e x c i t e d l o a d i n g . I n t h i s case t h e v i b r a t i o n g e n e r a t o r was a t t a c h e d a t the l o a d i n g end o f the c a n t i l e v e r beam arrangement. 4.3 I n s t r u m e n t a t i o n The i n v e s t i g a t i o n r e q u i r e d t h e d e t e r m i n a t i o n o f t h e a m p l i t u d e o f f r i c t i o n - i n d u c e d v i b r a t i o n o v e r a range o f d i s c speeds under the i n f l u e n c e o f v a r i o u s t y p e s o f f r i c t i o n c h a r a c t e r i s t i c . The e f f e c t o f e x t e r n a l e x c i t a t i o n s on t h e system was a l s o s t u d i e d . The i n s t r u m e n t a t i o n was d e s i g n e d t o meet t h e above r e q u i r e m e n t s . I n p a r t i c u l a r , t h e de t e r - r m i n a t i o n o f the f r i c t i o n c h a r a c t e r i s t i c d u r i n g v i b r a t i o n w h ich was c o n s i d e r e d t o be v e l o c i t y dependent, was o f major i m p o r t a n c e . The c o n v e n t i o n a l method o f d e t e r m i n i n g t h e f r i c t i o n c h a r a c t e r i s t i c by a v e r a g i n g d i s p l a c e m e n t s o r f o r c e s d u r i n g v i b r a t i o n does n o t g i v e a c c u r a t e i n f o r m a t i o n . I n f a c t , t h e i n f o r m a t i o n thus o b t a i n e d i s u s u a l l y g r o s s l y i n e r r o r i f the v i b r a t i o n i s o f t h e s t i c k - s l i p t y p e . D u r i n g f r i c t i o n - ^ i n d u c e d v i b r a t i o n t h e r e l a t i v e speed o f t h e f r i c t i o n c o u p l e 73 was v a r i e d and the a c c e l e r a t i o n had t o be c o n s i d e r e d as w e l l . C o n s i d e r i n g t h e d.e. f o r a damped f r e e v i b r a t i o n o f a mass-spring-damper system: rax + r x + kx = 0 t h i s c o u l d be w r i t t e n as r x = - (mx + k x ) . I f (mx + kx) was p l o t t e d a g a i n s t the a b s o l u t e v e l o c i t y x, a s t r a i g h t l i n e was o b t a i n e d w h i c h had s l o p e r . D u r i n g f r i c t i o n a l v i b r a t i o n the f o l l o w i n g d.e. a p p l i e s mx + r x + kx = f thus a p l o t o f (mx + kx) a g a i n s t x r e p r e s e n t e d t h e dynamic f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c w hich include'd t h e v i s c o u s damping f o r c e . I n p r a c t i c e s c a l e d a c c e l e r o m e t e r and d i s -placement s i g n a l s were f e d t o t h e d i f f e r e n t i a l a m p l i f i e r o f an o s c i l l o s c o p e , and the v e l o c i t y s i g n a l was i n t r o d u c e d t o t h e h o r i z o n t a l d i s p l a y a m p l i f i e r . The method gave a •• r e a l i s t i c assessment o f dynamic f r i c t i o n f o r c e . The g e n e r a l arrangement o f the c i r c u i t r y i s shown i n F i g . 4.3.1. The d e t a i l s o f the v a r i o u s measurements are as f o l l o w s : 74 a. D i s p l a c e m e n t I n t h e e a r l y s t a g e s o f t h e d e v e l o p m e n t o f t h e m e a s u r i n g s y s t e m a l i g h t d i s c r i m i n a t i n g r e s i s t o r (LDR) s y s t e m was u s e d . The d e v i c e c o n s i s t e d o f a l i g h t s t r i p w i t h a s l o t f o r t h e l i g h t beam t o p a s s t h r o u g h , a l i g h t s o u r c e a n d a LDR. The s t r i p was a t t a c h e d t o one s i d e o f t h e s t e e l c a n t i l e v e r beam. The l i g h t beam and t h e LDR w e r e a r r a n g e d s u c h t h a t a p r o p o r t i o n a l amount o f l i g h t w h i c h d e p e n d e d o n t h e d e f l e c t i o n o f t h e s t e e l beam was t r a n s m i t t e d t h r o u g h t h e s l o t and i m p i n g e d on t h e f a c e o f t h e LDR. The o u t p u t f r o m t h e LDR was d i s p l a y e d on t h e o s c i l l o s c o p e . The r e s p o n s e o f t h e LDR was f a i r l y l i n e a r b u t t h e r e was a c o n s i d e r a b l e p h a s e l a g w h i c h d i d n o t p o s e any p r o b l e m s when d i s p l a c e m e n t a l o n e was r e c o r d e d . H owever, i f v e l o c i t y a n d / o r a c c e l e r a t i o n w e r e i n v o l v e d a t t h e same t i m e p h a s e c o r r e c t i o n h a d t o be a p p l i e d w h i c h was n o t a l w a y s a s a t i s f a c t o r y s o l u t i o n . The LDR s y s t e m was s u b s e q u e n t l y r e p l a c e d b y s t r a i n g a u g e s . A t t h e r o o t o f t h e s t e e l beam 350 ohm s t r a i n g a u g e s w e r e c e m e n t e d , one o n e a c h s i d e o f t h e beam. Two i d e n t i c a l dummy g a u g e s w e r e u s e d f o r c o m p e n s a t i o n p u r p o s e s . The s t r a i n g a u g e s f o r m e d a f o u r arm b r i d g e c i r c u i t a n d w e r e c o n n e c t e d t o a b r i d g e a m p l i f i e r . The o u t p u t f r o m t h e a m p l i f i e r was c h a n n e l e d t o v a r i o u s r e c o r d i n g i n s t r u m e n t s . b, V e l o c i t y An e l e c t r o m a g n e t i c t y p e t r a n s d u c e r was u s e d t o r e c o r d t h e v e l o c i t y o f t h e v i b r a t i n g mass F i g . 4.3.2a. The moving p a r t o f t h e t r a n s d u c e r c o n s i s t e d o f 1,350 t u r n s of f i n e e n a m e l l e d w i r e and was a t t a c h e d t o t h e s t e e l beam, The c o i l w h i c h had a r e s i s t a n c e o f 6 30 ohms and an i n d u c -t a n c e o f 0.1 h e n r y s was d e s i g n e d f o r low mass and h i g h e l e c t r i c a l o u t p u t . Two horseshoe-shaped magnets housed i n an aluminum box were p o s i t i o n e d such t h a t d u r i n g beam v i b r a t i o n t h e c o n d u c t o r s c u t a c r o s s t h e magnetic f i e l d between th e p o l e s o f the magnets. The v o l t a g e g e n e r a t e d was p r o p o r t i o n a l t o the v e l o c i t y o f o s c i l l a t i o n o f t h e c o i l . The o u t p u t o f t h i s t r a n s d u c e r was f e d t o a s i n g l e s t a g e d.e. a m p l i f i e r F i g . 4.3.2b b e f o r e i t was c h a n n e l l e d t o v a r i o u s r e c o r d i n g i n s t r u m e n t s . C a p a c i t o r c o u p l i n g had t o be a v o i d e d i n the d.e. a m p l i f i e r so as t o ensure t h a t no phase s h i f t would o c c u r a f t e r t h e a m p l i f i c a t i o n s t a g e . The v e l o c i t y a m p l i f i e r had an o u t p u t o f 0.875 v o l t / i n / s e c . c. A c c e l e r a t i o n A Model 305A K i s t l e r s e r v o a c c e l e r o m e t e r was a t t a c h e d a t t h e t o p o f t h e specimen h o l d e r . The a c c e l e r o m e t e r weighed 3 oz and had a d i m e n s i o n o f 1 i n . d i a . x 2 i n . The a c c e l -erometer was a s e l f - c o n t a i n e d u n i t ; no a m p l i f i e r was r e -q u i r e d and the o u t p u t was f e d d i r e c t l y t o the r e c o r d i n g i n s t r u m e n t s . F u l l s c a l e o u t p u t o f the a c c e l e r o m e t e r was-5 v o l t s . The f u l l s c a l e range was n o r m a l l y s e t a t ± 50 g., t h u s g i v i n g a v o l t a g e s e n s i t i v i t y o f 0.1 v o l t / g . and a r e s o l u t i o n o f l e s s than 5 m i c r o , g. The f u l l s c a l e range c o u l d be v a r i e d by v a r y i n g an e x t e r n a l r e s i s t o r . The 76 d i s p l a c e m e n t and the a c c e l e r a t i o n s i g n a l s were w i t h i n 1° o f the e x p e c t e d 180° phase s h i f t . 4.3.1 R e c o r d i n g System The r e c o r d i n g i n s t r u m e n t s i n c l u d e d a Model 13-6624-00, Mark 842 Brush d u a l c h a n n e l r e c t i l i n e a r o s c i l l o g r a p h , a model 56 4 T e k t r o n i x d u a l beam s t o r a g e o s c i l l o s c o p e and a model 502A T e k t r o n i x d u a l beam o s c i l l o s c o p e . The d i s p l a c e m e n t and a c c e l e r o m e t e r s i g n a l s were f e d t o one d i f f e r e n t i a l ampli - r f i e r o f the s t o r a g e o s c i l l o s c o p e , and the v e l o c i t y s i g n a l was i n t r o d u c e d t o the h o r i z o n t a l d i s p l a y a m p l i f i e r . The c a l i b r a t i o n o f t h e s e s i g n a l s i s shown i n Appendix IV. The combined t r a c e from t h e o s c i l l o s c o p e r e p r e s e n t e d t h e dynamic f r i c t i o n f o r c e v e r s u s v e l o c i t y . S i m i l a r f r i c t i o n - d i s p l a c e -ment and f r i c t i o n - a c c e l e r a t i o n t r a c e s c o u l d be o b t a i n e d by i n t r o d u c i n g the d i s p l a c e m e n t o r a c c e l e r a t i o n s i g n a l s t o the h o r i z o n t a l d i s p l a y a m p l i f i e r . The o s c i l l o g r a p h was used f o r r e c o r d i n g t ime based s i g n a l s . I n a d d i t i o n t o r e c o r d i n g o f v a r i o u s s i g n a l s t h e r e c o r d i n g system o f F i g . 4.3.1 was d e s i g n e d t o a c h i e v e two main o b j e c t i v e s . The f i r s t o b j e c t i v e was t o r e c o r d d a t a a t the same p o i n t on t h e d i s c r u n n i n g t r a c k t h r o u g h o u t a t e s t s e r i e s . T h i s method m i n i m i s e d t h e i n c o n s i s t e n c y a r i s i n g from p o s s i b l e n o n - u n i f o r m i t y o f t h e d i s c s u r f a c e . The second o b j e c t i v e was t o o b t a i n a s i n g l e phase p l a n e p l o t on the s t o r a g e o s c i l l o s c o p e . The s t o r a g e o s c i l l o s c o p e d i d not p r o v i d e a ' b u i l t - i n ' means o f o b t a i n i n g t h i s t y p e o f d i s p l a y and an u n d e s i r e d c o n t i n u o u s t r a c e appeared on the s c r e e n . I n o r d e r t o overcome t h i s d i f f i c u l t y t he one c y c l e sequency t r i g g e r i n g system o f F i g . 4.3.3 was d e v e l o p e d . In a d d i t i o n , a m o d i f i c a t i o n t o t h e tube beam b l a n k i n g c i r -c u i t was n e c e s s a r y . The purpose o f the t r i g g e r i n g system was t o unblank and b l a n k the s t o r a g e tube beam a t t h e de-; s i r e d i n s t a n t s o f t i m e . 4.3.2 Spot T r i g g e r i n g U n i t A l i g h t d i s c r i m i n a t i n g r e s i s t o r (LDR) was used f o r the s p o t t r i g g e r i n g . Each time a s h i e l d , w h i c h c o u l d be a t t a c h e d a t any d e s i r e d p o i n t around the d i s c c i r c u m f e r e n c e moved between the l i g h t beam and t h e LDR i t a c t i v a t e d a r e l a y which s i m u l t a n e o u s l y t r i g g e r e d the event marker o f the o s c i l l o g r a p h , the s i n g l e sweep t r i g g e r o f t h e time base a m p l i f i e r o f the o s c i l l o s c o p e and t h e s p e c i a l l y de-s i g n e d sequence c i r c u i t o f F i g . 4.3.3. The sequence t r i g g e r p e r m i t t e d t h e d i s p l a y o f a s i n g l e x - x phase p l a n e d i s p l a y on t h e o s c i l l o s c o p e s c r e e n . P r o v i s i o n was made a l s o t o u t i l i s e t he sequence c i r c u i t f o r a c t i v a t i n g the event marker and s i n g l e sweep t r i g g e r t h u s e n s u r i n g s y n -chronous r e c o r d i n g when a comparison between v a r i o u s com-b i n a t i o n s such as f r i c t i o n - v e l o c i t y o r f r i c t i o n - t i m e t r a c e s was r e q u i r e d d u r i n g one c y c l e o f v i b r a t i o n . 4.3.3 One C y c l e Sequence T r i g g e r i n g System The one c y c l e sequence t r i g g e r i n g system used the, d i s p l a c e m e n t s i g n a l produced by the s t r a i n gauges i n t h e 7 8 f r i c t i o n system. I t was i m p o r t a n t t h a t s w i t c h i n g l o a d e f f e c t s from t h e t r i g g e r i n g system would i n no way a f f e c t the d i s p l a c e m e n t s i g n a l s b e i n g r e c e i v e d and r e c o r d e d . A vacuum tube d.c. a m p l i f i e r w i t h a 10 megohms i n p u t impedance b e f o r e t h e t r i g g e r i n g u n i t e l i m i n a t e d s w i t c h i n g l o a d e f f e c t . The o u t p u t impedance o f t h i s a m p l i f i e r had a low v a l u e and was w e l l s u i t e d t o t h e i n p u t impedance o f the t r i g g e r i n g u n i t . The e s s e n t i a l p a r t s o f the sequence c i r c u i t con-s i s t e d o f t h r e e r e l a y s w h i c h were a c t i v a t e d by t h e d i s p l a c e -ment s i g n a l . The n e g a t i v e e l e c t r i c a l d i s p l a c e m e n t s i g n a l was chosen as t h e s t a r t p o i n t . To o b t a i n a one c y c l e d i s -p l a y a l s o meant t h a t the n e g a t i v e d i s p l a c e m e n t s i g n a l must a l s o s e r v e as an end p o i n t s i g n a l . I t was n e c e s s a r y t h e r e -f o r e t o i n t e r l o c k t h i s s w i t c h i n g c i r c u i t so t h a t t h e s t a r t and end p o i n t r e l a y s c o u l d n e i t h e r o p e r a t e s i m u l t a n e o u s l y nor o p e r a t e c l o s u r e b e f o r e s t a r t . F i g . 4 . 3 . 3 a shows a diagram o f t h e s i g n a l . The use o f a t h i r d r e l a y t o the system p r o v i d e d t h e means o f s e q u e n c i n g e v e n t s t o t h e d e s i r e d o p e r a t i o n . T h i s a d d i t i o n a l r e l a y was e n e r g i s e d by t h e p o s i t i v e e l e c t r i c a l d i s p l a c e m e n t s i g n a l and a g a i n had t o be sequenced so t h a t a n e g a t i v e s t a r t s i g n a l had t o o c c u r b e f o r e t h i s r e l a y would e n e r g i s e . A f o u r t h r e l a y p r o v i d e d t h e means f o r synchronous t r i g g e r i n g o f the r e c o r d i n g i n s t r u m e n t s . 79 The a m p l i t u d e o f t h e e l e c t r i c a l d i s p l a c e m e n t s i g n a l was n o t c o n s t a n t and was dependent on the degree o f d i s -placement o f the m e t a l beam. I t was n e c e s s a r y t h e r e f o r e t o p r o v i d e v a r i a b l e c o n t r o l s f o r a d j u s t i n g the e l e c t r i c a l s i g n a l i n the t r i g g e r i n g u n i t so as t o encompass the v a r y i n g s i g n a l l e v e l s thus e n s u r i n g r e l a y c l o s u r e . E l e c -t r i c a l n o i s e had t o be e l i m i n a t e d so as t o p r e v e n t a c c i d e n t a l t r i g g e r i n g o f t h e u n i t . 4.3.4 . Speed C o u n t i n g System The speed o f the r o t a t i n g d i s c was measured by a . c o u n t i n g d i s c and LDR system. The c o u n t i n g system r e -q u i r e d no d r i v i n g mechanism thus t h e c o u n t e r u n i t c o u l d be p l a c e d i n a remote a r e a away from the f r i c t i o n a p p a r a t u s . T h i s p r o c e d u r e m i n i m i s e d the p o s s i b i l i t y o f m e c h a n i c a l n o i s e b e i n g p i c k e d up by t h e r e c o r d i n g i n s t r u m e n t s . The c o u n t i n g d i s c had two rows o f -^ i n . d i a . h o l e s . The o u t e r row w i t h 60 e q u a l l y spaced h o l e s was used f o r low speed c o u n t i n g and t h e i n n e r row w i t h 12 e q u a l l y spaced h o l e s was used f o r h i g h speed c o u n t i n g . A s t r o b o s c o p e was n o t s u i t -a b l e f o r the low speed range o f 100 rpm. The c o u n t i n g d e v i c e c o n s i s t e d o f two e l e c t r i c a l p u l s e c o u n t e r s . One c o u n t e r r e g i s t e r e d t h e number o f h o l e s scanned by the LDR and t h e o t h e r r e g i s t e r e d t h e e l a p s e d time i n m i n u t e s . The c o u n t i n g p r o c e d u r e was a c t u a t e d by a m u l t i p l e i n t e r v a l t i m e r d r i v e n by a one rpm sychronous motor. In a d d i t i o n , a 60 0 rpm synchronous motor t o g e t h e r w i t h a LDR c i r c u i t was used as a b u i l t - i n c a l i b r a t i o n u n i t . L i g h t s o u r c e s f o r the LDR c i r c u i t s were s u p p l i e d from a b u i l t - i n v a r i a b l e d.c. power s u p p l y . P r o v i s i o n was a l s o made t o t r a n s f o r m the o n - o f f s i g n a l s r e c e i v e d from a LDR c i r c u i t i n t o near s i n e wave o u t p u t w h i c h c o u l d be d i s p l a y e d on t h e o s c i l l o s c o p e s c r e e n . T h i s method was p a r t i c u l a r l y u s e f u l f o r p r o v i d i n g an i n s t a n t c l o s e e s t i m a t e o f t h e speed. 4.4 Specimens P r e l i m i n a r y i n v e s t i g a t i o n i n d i c a t e d t h a t s e v e r a l ;• m a t e r i a l and l u b r i c a n t c o m b i n a t i o n s gave the d e s i r e d q u a s i - h a r m o n i c v i b r a t i o n c h a r a c t e r i s t i c . The f o l l o w i n g c o m b i n a t i o n s were s t u d i e d . ( i ) S t e e l s l i d e r r u n n i n g on s t e e l d i s c : Many methods f o r p r e p a r i n g the s u r f a c e s were t r i e d . I t was found t h a t the most s a t i s f a c t o r y s u r f a c e s were o b t a i n e d by g r i n d i n g and t h e n l a p p i n g w i t h f i n e l a p p i n g compound. I n i t i a l l y a s t e e l s l i d e r specimen p r e p a r e d from Keewatin s t e e l (Appendix V) hardened t o R 55 and a d i s c o f A t l a s Nutherm s t e e l c (Appendix V) hardened and annealed t o R c 53.5 were used. I n c o n s i s t e n c y i n f r i c t i o n r e s u l t s was o b s e r v e d , and u n i f o r m q u a s i - h a r m o n i c o s c i l l a t i o n c o u l d n o t be a c h i e v e d w i t h t h i s c o m b i n a t i o n o f f r i c t i o n m a t e r i a l s . L a t e r , the Keewatin s t e e l s l i d e r was r e p l a c e d by a m i l d s t e e l s l i d e r and more s a t i s f a c t o r y r e s u l t s w e r e o b t a i n e d a l t h o u g h some i n c o n s i s -t e n c y s t i l l e x i s t e d . I t i s l i k e l y t h a t t h e n o n u n i f o r m i t y o f t h e d i s c s u r f a c e was f e l t w i t h l e s s i n f l u e n c e f r o m a s l i d e r o f s o f t e r m a t e r i a l r e s u l t i n g i n more u n i f o r m l y d e f o r m e d c o n t a c t i n g a r e a s . I n g e n e r a l , s t i c k - s l i p o s c i l - r l a t i o n was o b s e r v e d f o r t h i s c o m b i n a t i o n o f f r i c t i o n m a t e r i a l s a f t e r a r u n - i n p e r i o d , o f t e n w i t h t h e f o r m a t i o n o f a f i n e b l a c k d e p o s i t o n t h e d i s c t r a c k . The b e h a v i o u r was p r o b a b l y due t o t h e f i n e l a p p i n g compound b e i n g embedded on t h e t o p l a y e r o f t h e s u r f a c e s t r u c t u r e d u r i n g t h e l a p p i n g p r o c e s s . The m i l d s t e e l s l i d e r on h a r d e n e d s t e e l d i s c com-b i n a t i o n was a l s o t e s t e d i n t h e p r e s e n c e o f p e t r o l a t u m (U.S.P.) as t h e l u b r i c a n t . The r e s u l t s w e r e f o u n d t o be more c o n s i s t e n t a l o n g t h e d i s c t r a c k . ( i i ) B l o t t i n g p a p e r s l i d e r r u n n i n g on t h e same d i s c a s ( i ) w i t h a u t o m a t i c t r a n s m i s s i o n f l u i d a s l u b r i c a n t : T h i s p a r t i c u l a r c o m b i n a t i o n s i m u l a t e s t h e b e h a v i o u r o f some a u t o m a t i c t r a n s m i s s i o n s . The o r d i n a r y c o m m e r c i a l f a c i n g f o r a u t o m a t i c t r a n s m i s s i o n c l u t c h p l a t e s i s u s u a l l y some k i n d o f p a p e r m a t e r i a l w h i c h i s r e s i l i e n t , s p o n g y and r e a d i l y a b s o r b s l u b r i c a n t . The m a t e r i a l u s u a l l y c o n s i s t s o f c e l l u l o s i c f i b r e s w h i c h a r e c o a t e d w i t h p h e n o l i c r e s i n . I t was f o u n d t h a t b l o t t i n g p a p e r a g a i n s t s t e e l g a v e f r i c t i o n r e s u l t s s i m i l a r t o t h o s e o b t a i n e d f r o m c l u t c h f a c i n g m a t e r i a l s ( A p p e n d i x V ) . The c o m b i n a t i o n w i t h c e r t a i n t y p e s o f a u t o m a t i c t r a n s m i s s i o n f l u i d o r t h e i r n e u t r a l f l u i d s gave q u a s i - h a r m o n i c o s c i l l a t i o n . I t was d e c i d e d t h a t a l l t e s t s would be c a r r i e d o u t u s i n g b l o t t i n g paper on the s l i d e r s u r f a c e a g a i n s t a s t e e l d i s c w i t h a u t o m a t i c t r a n s -m i s s i o n f l u i d s as the l u b r i c a n t s . T h i s d e c i s i o n was governed by t h e f a c t t h a t u n i f o r m v i b r a t i o n r e s u l t s were o b t a i n e d . The l u b r i c a n t s were s u p p l i e d by C i t i e s S e r v i c e O i l Corp. o f New J e r s e y . These i n c l u d e d some n e u t r a l o i l s and some t r a n s m i s s i o n f l u i d s o f v a r i o u s v i s c o s i t i e s (Appendix V ) . The s t e e l d i s c used i n b o t h cases was made from A t l a s Nutherm s t e e l hardened and ann e a l e d t o R 5 3.5. The c d i s c i s 4 i n . d i a . and 1 i n . t h i c k . The normal r u n n i n g t r a c k had a mean d i a m e t e r o f 3^ - i n . The s l i d e r was ^ i n . d ( i i i ) Polymer s l i d e r r u n n i n g on the same d i s c as ( i ) : The s l i d e r were p r e p a r e d by the C e n t r e f o r M a t e r i a l S c i e n c e U.B.C. The f i r s t s l i d e r was p r e p a r e d from p o l y e s t e r r e s i n m a t e r i a l . The s l i d e r s u r f a c e measured j i n . d i a . A second s l i d e r o f carbon f i b r e s c omposite o f the r e s i n had t h e same d i m e n s i o n as t h e r e s i n s l i d e r . Carbon f i l a m e n t s o f a p p r o x i m a t e l y 6 ym d i a m e t e r were grouped i n bu n d l e s o f a p p r o x i m a t e l y 0.25 mm d i a m e t e r . These b u n d l e s o f carbon f i b r e s were a r r a n g e d p e r p e n d i c u l a r t o the s l i d i n g s u r -f a c e as shown i n F i g . 5.1.11a. W i t h i t s l i g h t w e i g h t and h i g h s t r e n g t h , carbon f i b r e composites have r e c e i v e d wide a t t e n t i o n i n r e c e n t y e a r s , p a r t i c u l a r l y i n t h e a i r -c r a f t i n d u s t r y w h e r e t u r b i n e b l a d e s h a v e b e e n made f r o m c a r b o n f i b r e m a t e r i a l . ( i v ) R u b b e r s l i d e r r u n n i n g on same d i s c a s ( i ) : The s l i d e r was p r e p a r e d by c e m e n t i n g a t h i n p i e c e o f N e o p r e n e , j i n d i a . x i n . t h i c k o n t o a s t e e l s l i d e r . The h e m i s -p h e r i c a l s h a p e d s l i d e r h o l d e r a n d t h e r e t a i n i n g c u p were r e p l a c e d b y a c y l i n d r i c a l s h a p e d s l i d e r h o l d e r . T h i s a l t e r n a t i o n p r e v e n t e d t h e s l i d e r h o l d e r f r o m r o l l i n g o v e r due t o t h e h i g h f r i c t i o n f o r c e . 4.5 E x p e r i m e n t a l M e t h o d 4.5.1 P r e l i m i n a r y I n v e s t i g a t i o n o f S y s t e m T e s t s w e r e c o n d u c t e d t o s t u d y t h e e f f e c t o f v a r i o u s p a r a m e t e r s ; t h e s e i n c l u d e d d i s c s p e e d , l o a d , l u b r i c a n t s , e x t e r n a l f o r c i n g a m p l i t u d e a n d e x t e r n a l f o r c i n g f r e q u e n c y . The i n f l u e n c e o f t h e s t i f f n e s s o f s u p p o r t i n g s y s t e m was i n v e s t i g a t e d i n t h e e a r l y s t a g e s o f t h e r e s e a r c h . T h r e e c a n t i l e v e r s u p p o r t i n g beams w e r e t e s t e d , t h e y w e r e -^ i n . , 3 1 i n . , and j i n . t h i c k r e s p e c t i v e l y . The beams w e r e made f r o m A t l a s N u t h e r m s t e e l a n d w e r e a n n e a l e d , h a r d e n e d a n d t e m p e r e d t o o b t a i n t h e b e s t c o m b i n a t i o n f o r t o u g h n e s s a n d h a r d n e s s . I t was f o u n d t h a t a ^ i n . t h i c k x 1 i n . deep beam p r o v i d e d s u f f i c i e n t l y h i g h f r i c t i o n f o r c e w i t h a d i s p l a c e m e n t s t i l l l o w enough n o t t o h a v e s i g n i f i c a n t c u r v a t u r e e f f e c t . The s t a n d a r d f r i c t i o n s y s t e m , u s i n g t h e j i n . t h i c k beam h a d an e q u i v a l e n t v i b r a t i o n w e i g h t o f 1.2 l b . , t h e e q u i v a l e n t beam s t i f f n e s s a t t h e s l i d e r was 60 l b / i n . and t h e v i s c o u s d a m p i n g c o e f f i c i e n t was i n t h e r e g i o n o f 0.01 l b / i n / s e c . ( A p p e n d i x I I I ) . The d a m p i n g c o e f f i c i e n t was o b t a i n e d by p e r f o r m i n g a damped f r e e v i b r a t i o n t e s t w i t h t h e s l i d e r c l e a r o f t h e s u r f a c e . A d i s p l a c e m e n t - t i m e c u r v e was o b t a i n e d f r o m s u c h a t e s t and t h e c o e f f i c i e n t was c a l c u l a t e d b y t h e l o g a r i t h m i c d e c r e m e n t m e t h o d . The s y s t e m d a m p i n g c o u l d be a l s o o b t a i n e d f r o m t h e (mx + k x ) v s x t r a c e i n w h i c h t h e c o e f f i c i e n t was o b t a i n e d d i r e c t l y f r o m t h e s l o p e o f t h e t r a c e ( A p p e n d i x I I I ) . As i t c o u l d be o b s e r v e d f r o m t h e o s c i l l o s c o p e t r a c e i n F i g . A2 t h a t t h e t r a c e was a l m o s t l i n e a r t h e r e f o r e t h e v a l u e c o u l d be c o n s i d e r e d as v i s c o u s d a m p i n g c o e f f i c i e n t . The f o r g o i n g v a l u e s g a v e a damped n a t u r a l f r e q u e n c y o f 138 r a d / s e c . f o r t h e s l i d e r s u p p o r t i n g s y s t e m . Owing t o t h e s m a l l n e s s o f t h e s y s t e m d a m p i n g t h e n a t u r a l f r e q u e n c y was v i r t u a l l y t h e same as t h e damped c a s e . The beam s t i f f n e s s a n d t h e s y s t e m n a t u r a l f r e q u e n c y c o u l d be v a r i e d b y v a r y i n g t h e l e n g t h o f t h e s u p p o r t i n g c a n t i l e v e r beam. The c a l i b r a t i o n o f t h e s y s t e m i s d e s c r i b e d i n A p p e n d i x I V . 4.5.2 S p e c i m e n P r e p a r a t i o n The s t e e l s u r f a c e s w e r e p r e p a r e d b y f i r s t g r i n d i n g and t h e n l a p p i n g o n a c a s t i r o n l a p p i n g p l a t e w i t h 85 pe t r o l a d u m (U.S.P.) and alum powder as a c u t t i n g agent. The r e s u l t a n t f i n i s h was 25-30 m i c r o i n c h AA. P r i o r t o each t e s t the s u r f a c e s were r e l a p p e d and t h o r o u g h l y c l e a n e d w i t h hexane. 4.5.3 T e s t P r o c e d u r e The s t a n d a r d p r o c e d u r e f o r the i n v e s t i g a t i o n o f the f r i c t i o n - i n d u c e d v i b r a t i o n c o n s i s t e d o f r u n n i n g t h e d i s c a t i t s l o w e s t speed and i n c r e a s i n g the speed g r a d u a l l y t o i t s maximum o r u n t i l v i b r a t i o n decay o c c u r r e d . D u r i n g each speed i n c r e m e n t a s e t o f o s c i l l o s c o p e t r a c e s and c h a r t r e c o r d s were t a k e n , t h e s e u s u a l l y i n c l u d e d t h e f r i c t i o n -v e l o c i t y and d i s p l a c e m e n t - v e l o c i t y t r a c e s from the s t o r a g e o s c i l l o s c o p e , and v e l o c i t y - t i m e and a c c e l e r a t i o n - t i m e or. d i s p l a c e m e n t - t i m e t r a c e s from the c h a r t r e c o r d e r . A l l t h e r e s u l t s were t a k e n a t the same p o i n t on the d i s c . P r i o r t o the c o m p l e t i o n o f each t e s t , s e v e r a l random d i s c speeds were r e - r u n and r e c o r d s were a g a i n t a k e n a t t h e s e speeds. These r e s u l t s were compared w i t h t h e r e s u l t s o b t a i n e d i n ; the f i r s t t i m e . The f o r e g o i n g p r o c e d u r e p e r m i t t e d the o b s e r v a t i o n o f any s i g n i f i c a n t changes due t o e n v i r o n m e n t a l f a c t o r s o r s u r f a c e wear d u r i n g a t e s t . F o r each s e t o f t e s t s o n l y one parameter was v a r i e d . P r i o r t o each t e s t , the s l i d e r was s e t a t i t s n e u t r a l p o s i t i o n and a l l i n s t r u -ments were checked f o r r e f e r e n c e l e v e l . i 86 S i m i l a r t e s t s were c a r r i e d out f o r d i f f e r e n t combin-a t i o n s o f f r i c t i o n m a t e r i a l s and f o r d i f f e r e n t l u b r i c a n t s . However, t e s t s f o r the i n v e s t i g a t i o n o f the e f f e c t o f l o a d were c a r r i e d o u t o n l y i n two c a s e s , #9 o i l and # 4 o i l as the l u b r i c a n t . I t was f e l t t h a t t h e s e t e s t s would g i v e s u f f i c i e n t i n f o r m a t i o n t o show the e f f e c t o f the l o a d on the f r i c t i o n and on the f r i c t i o n - i n d u c e d v i b r a t i o n . As i t has been d e s c r i b e d e a r l i e r t he e x t e r n a l e x c i t a t i o n was a p p l i e d by means o f unbalanced r o t a t i n g mass. 2 The f o r c e thus o b t a i n e d f o l l o w i n g t he r e l a t i o n s h i p mto r / g ; thus v a r y i n g the fr e q u e n c y a l s o v a r i e d t h e e x t e r n a l f o r c e a m p l i t u d e . T h e r e f o r e i t would seem i m p o s s i b l e t o v a r y e i t h e r t h e e x t e r n a l f r e q u e n c y o r t h e e x t e r n a l f o r c e a m p l i -tude a l o n e . However, by a d j u s t i n g the two parameters m and r i n the r e l a t i o n s h i p i t was p o s s i b l e t o o b t a i n a r e a s o n a b l e range o f v a l u e s f o r e i t h e r t h e f r e q u e n c y o r t h e a m p l i t u d e , a l t h o u g h t h i s t y p e o f v a r i a t i o n d u r i n g the con-d u c t o f a t e s t was u s u a l l y v e r y cumbersome, and was not always s a t i s f a c t o r y . A f o r c e a m p l i t u d e ranged from 0.00011 2 N l b . u s i n g the s m a l l e s t mass and t h e s m a l l e s t r a d i u s , t o 2 0.00065 N l b . u s i n g the l a r g e s t mass and the l a r g e s t r a d i u s , c o u l d be o b t a i n e d from t h e p r e s e n t arrangement, where N i s t h e e x t e r n a l e x c i t a t i o n f r e q u e n c y i n rev./sec> T e s t s were performed by v a r y i n g t h e f r e q u e n c y w h i l e k e e p i n g the d i s c v e l o c i t y c o n s t a n t . I n some cases a t t e m p t s were made t o keep the e x t e r n a l f o r c e magnitude c o n s t a n t by a d j u s t i n g t h e parameters m and r . Photographs o f the d i s p l a c e m e n t - v e l o c i t y phase p l a n e t r a c e on the s t o r a g e o s c i l l o s c o p e and c h a r t r e c o r d s o f d i s p l a c e m e n t - t i m e and v e l o c i t y - t i m e were t a k e n f o r each f r e q u e n c y s e t t i n g . The complete p r o c e s s was r e p e a t e d f o r f o u r d i f f e r e n t t y p e s o f l u b r i c a n t , namely; #5, #1, #9 and #7. These f o u r l u b r i c a n t s d i s p l a y e d t h r e e d i s t i n c t l y d i f f e r e n t f r i c t i o n c h a r a c t e r i s t i c s . V RESULTS AND DISCUSSION 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 t r a c e s o b t a i n e d from the o s c i l l o s c o p e were f i t t e d by e x p o n e n t i a l e x p r e s s i o n s t h and by n o r d e r p o l y n o m i a l s . These m a t h e m a t i c a l e x p r e s -s i o n s were used i n the t h e o r e t i c a l v e r i f i c a t i o n o f the e x p e r i m e n t a l r e s u l t s . Owing t o the c o m p l i c a t e d a l g e b r a i c m a n i p u l a t i o n i n v o l v e d , even w i t h a low o r d e r p o l y n o m i a l i n the t h e o r e t i c a l a n a l y s i s o f the non-autonomous c a s e s , the p o l y n o m i a l s e x p r e s s i o n s d i d n o t exceed 9 o r d e r . T h i s somewhat l i m i t e d t h e a c c u r a c y o f the p o l y n o m i a l f i t t e d c u r v e s , however, t h e a n a l y s i s o f t h e non-autonomous cases was supplemented by a p p l y i n g the more a c c u r a t e e x p o n e n t i a l e x p r e s s i o n t o t h e e q u a t i o n o f m o t i o n and s o l v i n g t h e e q u a t i o n by n u m e r i c a l methods. 5.1 Autonomous Case 5.1.1 B l o t t i n g Paper on S t e e l As i t had been i n d i c a t e d e a r l i e r the b l o t t i n g paper on s t e e l c o m b i n a t i o n t o g e t h e r w i t h a u t o m a t i c t r a n s m i s s i o n f l u i d as l u b r i c a n t s i m u l a t e d the b e h a v i o u r o f some a u t o m a t i c t r a n s m i s s i o n s . S e v e r a l a u t o m a t i c t r a n s m i s s i o n f l u i d s and t h e i r n e u t r a l f l u i d s (Appendix V) were used i n o r d e r t o o b t a i n v a r i o u s forms o f f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c . Three g e n e r a l forms o f f r i c t i o n - v e l o c i t y c u r v e s as i n d i c a t e d i n F i g . 3.2.1 and F i g . 3.2.2 o f the t h e o r e t i c a l s e c t i o n were i n v e s t i g a t e d . I n the d i s c u s s i o n t h r o u g h o u t t h i s c h a p t e r , 8 9 the f r i c t i o n - v e l o c i t y c u r v e s h a v i n g the g e n e r a l forms r e s e m b l i n g t h o s e o f F i g . 3.2.2, F i g . 3.2.1b and F i g . 3.2.1a w i l l be r e f e r r e d t o as Type A, B and C r e s p e c t i v e l y . Type A, the humped form o f F i g . 3.2.2 i s d i s c u s s e d f i r s t , f o l l o w e d by Type B, t h e cu r v e w i t h i n i t i a l n e g a t i v e s l o p e , and f i n a l l y Type C, t h e al m o s t l i n e a r c u r v e w i t h p o s i t i v e s l o p e i s c o n s i d e r e d . (a) F r i c t i o n - V e l o c i t y Curve o f Type A l F i g . 5.1.1 i l l u s t r a t e s a phase p l a n e o s c i l l o s c o p e t r a c e o f d i s p l a c e m e n t x v e r s u s v i b r a t i o n v e l o c i t y k w i t h #7 o i l as the l u b r i c a n t , a t a l o a d o f 5.4 l b . and a d i s c speed o f 1.05 i n / s e c . A p l o t o f f r i c t i o n f o r c e v e r s u s v e l o c i t y i s d i s p l a y e d i n t h e same diagram. The f o r e g o i n g r e s u l t s were o b t a i n e d d u r i n g one c y c l e o f the v i b r a t i o n . S i m i l a r t r a c e s were o b t a i n e d f o r a sequence o f d i s c v e l o c -i t i e s and t h e r e s u l t s o f F i g . 5.1.2 were o b t a i n e d by p l o t t i n g v i b r a t i o n a m p l i t u d e v e r s u s d i s c v e l o c i t y . A p l o t o f v i b r a t i o n f r e q u e n c y as a f u n c t i o n o f d i s c v e l o c i t y i s a l s o shown i n F i g . 5.1.2. The e x p o n e n t i a l and p o l y n o m i a l f u n c t i o n s o f eq. (3.2.1) and eq. (3.2.2) were computer f i t t e d t o 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 c u r v e o f F i g . 5.1.2 w h i c h was r e p r o d u c e d from the o s c i l l o s c o p e t r a c e o f F i g . 5.1.1 a l l o w i n g f o r a s m a l l c o r r e c t i o n f o r t h e system damping. The e x p o n e n t i a l e q u a t i o n was a p p l i e d t o t h e t h e o r y d e v e l o p e d e a r l i e r eq. (3.3.13), t o g i v e the t h e o r e t i c a l a m p l i t u d e - v e l o c i t y c u r y e d i s p l a y e d i n F i g . 5.1.2. T h e o r e t i c a l a m p l i t u d e v a l u e s d e r i v e d from eq. (3.3.13) were i n s e r t e d i n t o eq. (3.3.14) i n o r d e r t o check the s t a b i l i t y . A s i m i l a r p r o c e d u r e was f o l l o w e d f o r t h e p o l y n o m i a l a p p r o x i m a t i o n employing eq. (3.3.19) and eq. (3.3.20). I n F i g . 5.1.2 the a m p l i t u d e v e l o c i t y c u r v e f o r t h e p o l y n o m i a l t h e o r y i s shown f o r the r e g i o n A t o B o n l y f o r t h e r e a s o n s t a t e d e a r l i e r i n Cha p t e r I I I , namely t h a t the p o l y n o m i a l e x p r e s s i o n was a p p l i c a b l e o n l y w i t h i n the v e l o c i t y r e g i o n f o r whi c h the curve was o r i g i n a l l y f i t t e d . F i g . 5.1.3 i l l u s t r a t e s f o u r o s c i l l o s c o p e t r a c e s o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e i n c l u d i n g system damping r e c o r d e d a t d i s c v e l o c i t i e s o f 0.84, 1.06, 1.30 and 1.64 i n / s e c r e s p e c t i v e l y . A comparison o f the c u r v e s i s a l s o shown i n the same diagram. The humped shape appears t o be v e r y c o n s i s t e n t . (b) F r i c t i o n - V e l o c i t y Curve o f Type A2 The c u r v e s o f F i g . 5.1.4 were o b t a i n e d u s i n g #9 o i l as t h e l u b r i c a n t . The r e s u l t s were o b t a i n e d f o l l o w i n g the same pr o c e d u r e as o u t l i n e d e a r l i e r . The f r i c t i o n -v e l o c i t y c h a r a c t e r i s t i c c u r v e o b t a i n e d from t h i s combin-a t i o n has a s i m i l a r form as the Type A l e x c e p t t h a t the hump i s l e s s pronounced. The a m p l i t u d e - v e l o c i t y c u r v e s o f F i g . 5.1.2 and F i g . 5.1.4 i l l u s t r a t e t h a t t h e e x p e r i m e n t a l r e s u l t s and the p r e d i c t i o n s by the e x p o n e n t i a l and p o l y n o m i a l t h e o r i e s a re i n r e a s o n a b l e agreement. I t s h o u l d be noted t h a t the e x p e r i m e n t a l p o i n t s t o the l e f t o f A i n F i g . 5.1.4 r e p r e s e n t s t i c k - s l i p a m p l i t u d e v a l u e s . I n t h e o r y and by e x p e r i m e n t the q u a s i - h a r m o n i c o s c i l l a t i o n commences a t a d i s c r e t e v e l o c i t y A and the a m p l i t u d e o f v i b r a t i o n i n c r e a s e s i n a n e a r - l i n e a r f a s h i o n w i t h i n c r e a s i n g v e l o c i t y u n t i l sudden decay o c c u r s a t an upper c r i t i c a l v e l o c i t y , B. Theory p r e d i c t s t h a t t h e v i b r a t i o n i s s t a b l e between the v e l o c i t y l i m i t s A and B. To t h e r i g h t o f B i n s t a b i l i t y i s p o s s i b l e , and i n f a c t two v i b r a t i o n a m p l i t u d e s a r e p r e d i c t e d a t each v e l o c i t y . The v i b r a t i o n o f l a r g e r a m p l i -tude i s s t a b l e whereas the l o w e r a m p l i t u d e c u r v e r e p r e s e n t s an u n s t a b l e c o n d i t i o n . I f t h e system i s p e r t u r b e d t o an a m p l i t u d e between th e two a m p l i t u d e c u r v e s , v i b r a t i o n w i l l grow u n t i l i t s a m p l i t u d e reaches t h e s t a b l e a m p l i t u d e c u r v e . A l t e r n a t i v e l y , i f the p e r t u r b a t i o n d i s p l a c e m e n t i s below the u n s t a b l e a m p l i t u d e c u r v e , v i b r a t i o n w i l l d i e o u t . V i b r a t i o n i n the r e g i o n A t o B c o r r e s p o n d s t o s o f t s e l f - e x c i t a t i o n whereby th e system d e p a r t s from an u n s t a b l e s i n g u l a r i t y and a r r i v e s a t a s t a t e o f s t e a d y s t a t e v i b r a -t i o n w i t h a s t a b l e a m p l i t u d e [ 4 0 ] . To the r i g h t o f B the s i n g u l a r i t y i s s t a b l e and h a r d s e l f - e x c i t a t i o n p r e v a i l s . Under t h e s e c o n d i t i o n s t h e b a r r i e r p r e s e n t e d by an u n s t a b l e l i m i t c y c l e must be c r o s s e d b e f o r e a s t a b l e v i b r a t i o n can e x i s t (Ref. F i g . 3.1.1). E x p e r i m e n t a l l y , t h e v i b r a t i o n tended t o e x i s t f u r t h e r t o t h e r i g h t o f p o i n t B l a r g e l y because i n c o n s i s t e n c i e s i n f r i c t i o n a l o n g t h e d i s c t r a c k 92 g e n e r a t e d s m a l l p e r t u r b a t i o n s which c a r r i e d the s l i d e r o v e r t h e b a r r i e r formed by t h e u n s t a b l e a m p l i t u d e c u r v e . How-e v e r , a t s t i l l h i g h e r d i s c v e l o c i t i e s , s m a l l p e r t u r b a t i o n s are n ot s u f f i c i e n t t o c a r r y t h e s l i d e r o v e r the b a r r i e r , and v i b r a t i o n would not o c c u r . (c) D i s t i n c t i o n Between S t i c k - S l i p and Q u a s i -Harmonic O s c i l l a t i o n s The s c a l e s o f F i g u r e s 5.1.1, 5.1.2 and 5.1.4 are d i m e n s i o n l e s s . The n o n - d i m e n s i o n a l i z e d parameters are r e s p e c t i v e l y : x = Xh, x = Xcoh and t = f/to, where ooh i s e q u a l t o u n i t y . 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 v e r s u s v e l o c i t y c urve o f F i g . 5.1.2 and F i g . 5.1.4 t h e dimen-s i o n l e s s s c a l e s have th e advantage o f d i s t i n g u i s h i n g the d i f f e r e n c e between the s t i c k - s l i p type v i b r a t i o n and the q u a s i - h a r m o n i c form, s i n c e the 45° l i n e i n the a m p l i t u d e -v e l o c i t y diagram i s a c l o s e a p p r o x i m a t i o n t o the boundary between th e s t i c k - s l i p and the q u a s i - h a r m o n i c forms o f v i b r a t i o n . The d i s t i n c t i o n between the two t y p e s o f v i b r a t i o n can be e a s i l y v i s u a l i z e d from the x-x phase p l a n e diagrams o f F i g . 3.2.3. In t h e case o f q u a s i - h a r m o n i c o s c i l l a t i o n , the p l o t does n o t r e a c h the z e r o s l i d i n g v e l o c i t y a x i s , thus the a m p l i t u d e o f v i b r a t i o n i s l e s s t h a n the d i s c v e l o c i t y i n terms o f d i m e n s i o n l e s s s c a l e s , p r o v i d e d the phase p l a n e i s v e r y n e a r l y c i r c u l a r . Thus, i f t h e a m p l i t u d e o f v i b r a t i o n appeared above th e 45° l i n e i n the a m p l i t u d e - v e l o c i t y diagram t h e v i b r a t i o n would be-o f t h e s t i c k - s l i p t y p e . 93 (d) C o r r e l a t i o n B e t w e e n t h e A m p l i t u d e o f V i b r a t i o n and t h e f -V C u r v e y k An e x a m i n a 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 c u r v e and t h e a m p l i t u d e - v e l o c i t y c u r v e o f F i g . 5.1.2 and F i g . 5.1.4 r e v e a l s t h a t t h e u p p e r c r i t i c a l v e l o c i t y o c c u r r e d n e a r a p o i n t w here t h e s l o p e 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 b e g a n t o c hange f r o m n e g a t i v e t o p o s i t i v e . I t was a l s o n o t e d t h a t t h e v e l o c i t y A w h e r e q u a s i - h a r m o n i c o s c i l l a t i o n commenced, a p p e a r e d n e a r t h e p e a k o f t h e hump. No q u a s i - h a r m o n i c o s c i l l a t i o n was o b s e r v e d i n t h e r e g i o n t o t h e l e f t o f t h e p e a k . However, s t i c k - s l i p t y p e o s c i l l a t i o n c o u l d o c c u r i n t h i s r e g i o n d e p e n d i n g on t h e s l o p e 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 o v e r t h i s r e g i o n and t h e s t a t i c f r i c t i o n c h a r a c t e r -i s t i c s o f t h e f r i c t i o n c o m b i n a t i o n . I f t h e f r i c t i o n m a t e r i a l c o m b i n a t i o n h a s s m a l l r i s e i n s t a t i c f r i c t i o n o r t h e s l o p e 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 o v e r t h i s r e g i o n i s v e r y s t e e p t h u s g i v i n g h e a v y s u r f a c e d a m p i n g , t h e n u n d e r s u c h c o n d i t i o n s s t i c k - s l i p o s c i l l a t i o n may n o t o c c u r . I n f a c t , s t i c k - s l i p o s c i l l a t i o n was n o t o b s e r v e d w i t h t h e Type A l f r i c t i o n - v e l o c i t y c u r v e . (e) E f f e c t o f E x t e r n a l Damping w i t h Type A2 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 s t a n t s C^ t o C^ f o r t h e e x p o n e n t i a l f u n c 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 c u r v e o f F i g . 5.1.4 w e r e s u b s t i -t u t e d i n t o e q . (3.3.28) i n o r d e r t o o b t a i n an e s t i m a t e o f t h e d a m p i n g c o e f f i c i e n t r e q u i r e d f o r c o m p l e t e e x t i n c t i o n o f v i b r a t i o n o v e r the e n t i r e range o f s l i d i n g v e l o c i t i e s . The damping c o e f f i c i e n t o b t a i n e d from t h i s c a l c u l a t i o n was 0.055. A t e s t was performed u s i n g a permanent magnet as damper wh i c h had a damping c o e f f i c i e n t o f a p p r o x i m a t e l y 0.08. With t h i s damper o n l y s t i c k - s l i p o s c i l l a t i o n was • obser v e d i n the low v e l o c i t y r e g i o n and q u a s i - h a r m o n i c o s c i l l a t i o n d i d n o t o c c u r . The a m p l i t u d e - v e l o c i t y c u r v e o b t a i n e d from t h i s t e s t i s a l s o shown i n F i g . 5.1.4. Hence i t i s p o s s i b l e t o i n t r o d u c e c o n t r o l l e d damping i n t o the system i n o r d e r t o p r o h i b i t q u a s i - h a r m o n i c o s c i l l a t i o n s . T h i s f i n d i n g c o u l d be o f u t i l i t y i n the d e s i g n o f p r a c t i c a l systems where v i b r a t i o n i s u n d e s i r e d . (f) E f f e c t o f V a r y i n g the Normal Load F i g . 5.1.5 shows t h r e e e x p e r i m e n t a l a m p l i t u d e -v e l o c i t y c u r v e s under d i f f e r e n t normal l o a d s , 5.4 l b , 3.85 l b and 2.4 l b r e s p e c t i v e l y . The f r i c t i o n - v e l o c i t y c u r v e has t h e form o f Type A2. The r e s u l t s were o b t a i n e d by v a r y i n g the d i s c v e l o c i t y and d u r i n g each v e l o c i t y s e t t i n g the normal l o a d was v a r i e d . I t s h o u l d be noted t h a t the a m p l i t u d e o f v i b r a t i o n i n c r e a s e d a l m o s t l i n e a r l y as the d i s c v e l o c i t y was i n c r e a s e d i n a l l t h r e e c a s e s . Very l i t t l e change 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 was o b s e r v e d when t h e normal l o a d was v a r i e d . However, t h e r e was a d i s t i n c t d i f f e r e n c e i n r e l a t i o n t o the upper c r i t i c a l v e l o c i t y , the s m a l l e r t h e normal l o a d the e a r l i e r the decay commenced. I t w i l l be noted l a t e r i n F i g . 5.1.7 t h a t a s i m i l a r phenomenon was a l s o o b s e r v e d i n t h e c a s e w i t h t h e Type B2 f r i c t i o n - v e l o c i t y c u r v e , A s t u d y o f t h e one c y c l e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c t r a c e s r e v e a l e d t h a t t h e r e was a s l i g h t c h a n g e i n t h e s l o p e 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 as t h e n o r m a l l o a d was v a r i e d . I t was n o t e d t h a t t h e n e g a t i v e s l o p e became l e s s s t e e p as t h e n o r m a l l o a d was r e d u c e d t o 2.4 l b . The f o r e g o i n g o b s e r v a t i o n p r o v i d e s an e x p l a n a t i o n o f t h e b e h a v i o u r . I t i s a p p a r e n t t h a t when t h e v i s c o u s d a m p i n g o f t h e s y s t e m was a d d e d t o s u r f a c e d a m p i n g p r o v i d e d by 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 p o i n t w h e r e t h e s l o p e c h a n g e d f r o m n e g a t i v e t o p o s i t i v e w o u l d o c c u r e a r l i e r i f t h e i n i t i a l n e g a t i v e s l o p e was s m a l l e r . T h i s a l s o e x p l a i n s t h e s l i g h t c h ange i n a m p l i t u d e o f v i b r a t i o n w h i c h i s shown i n F i g . 5.1.5. However, f u r t h e r s t u d y w o u l d be r e q u i r e d i n o r d e r t o e x p l a i n t h e s l i g h t v a r i a t i o n i n t h e f r i c t i o n - v e l o c i t y c u r v e due t o n o r m a l l o a d . (g) F r i c t i o n - V e l o c i t y C u r v e o f Type B l The e x p e r i m e n t a l c u r v e s o f F i g . 5.1.6 w e r e o b t a i n e d u s i n g #1 o i l a s t h e l u b r i c a n t . The f o r m 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 d i c a t e s t h a t t h e hump i s a l m o s t i n s i g n i f -i c a n t , i t i s v e r y c l o s e t o t h e z e r o s l i d i n g v e l o c i t y a x i s and t h a t t h e n e g a t i v e s l o p e o f t h e c u r v e i s s t e e p e r t h a n t h e two p r e v i o u s c a s e s . A n a l y t i c a l l y , t h i s f o r m o f f r i c t i o n - v e l o c i t y c u r v e s u g g e s t s t h a t o v e r a l a r g e v e l o c i t y range th e a m p l i t u d e o f v i b r a t i o n would f a l l on the l e f t s i d e o f t h e 4 5 ° l i n e o f a d i m e n s i o n l e s s a m p l i t u d e - v e l o c i t y p l o t and o n l y i n t h e h i g h e r v e l o c i t y r e g i o n under th e combined e f f e c t o f t h e hump and of t h e system v i s c o u s damping does th e a m p l i t u d e o f v i b r a t i o n g r a d u a l l y f a l l back t o t h e r i g h t o f the 4 5 ° l i n e . E x p e r i m e n t a l l y , t h o s e a m p l i t u d e s on the l e f t s i d e o f the l i n e would appear i n t h e form o f s t i c k - s l i p v i b r a t i o n . Under such c i r c u m s t a n c e s , the q u a s i - h a r m o n i c t h e o r y would n o t p r o v i d e a c c u r a t e p r e -d i c t i o n s , s i n c e the t h e o r y does not t a k e i n t o account the s t a t i c f r i c t i o n c h a r a c t e r i s t i c o f t h e f r i c t i o n c o u p l e . N e v e r t h e l e s s , the a m p l i t u d e - v e l o c i t y c u r v e s o f F i g . 5 . 1 . 6 i l l u s t r a t e t h a t t h e e x p e r i m e n t a l r e s u l t s and t h e p r e d i c -t i o n s by the q u a s i - h a r m o n i c t h e o r y are s t i l l i n r e a s o n a b l e agreement. The l a r g e r d i s c r e p a n c y i n t h e h i g h v e l o c i t y ; r e g i o n i s l i k e l y due t o the f a c t t h a t when the amplitude, o f v i b r a t i o n becomes l a r g e , t h e system damping may become n o n l i n e a r . The damping may be p r o p o r t i o n a l t o some power of the v e l o c i t y i n s t e a d o f b e i n g i n d i r e c t p r o p o r t i o n t o the v e l o c i t y . (h) F r i c t i o n - V e l o c i t y Curve o f Type B2 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 o b t a i n e d from u s i n g #4 o i l as the l u b r i c a n t i s shown i n F i g . 5 . 1 . 7 . The c u r v e was r e p r o d u c e d from a one c y c l e o s c i l l o s c o p e t r a c e a t a d i s c v e l o c i t y o f 0.8 i n / s e c and w i t h a normal 97 l o a d o f 3.85 l b . The cu r v e appears t o be almost l i n e a r w i t h s l i g h t n e g a t i v e s l o p e i n the low v e l o c i t y r e g i o n and l e v e l s o f f r a p i d l y . Due t o t h e absence o f t h e hump i n the f r i c t i o n - v e l o c i t y c u r v e , q u a s i - h a r m o n i c o s c i l l a t i o n was not p r e d i c t e d by the t h e o r y . I n f a c t , the a m p l i t u d e -v e l o c i t y c u r v e s o f F i g . 5.1.7 show t h a t a l l the o s c i l l a t i o n s a re o f the s t i c k - s l i p type and t h a t t h e upper c r i t i c a l v e l o c i t y where decay commences i s re a c h e d as soon as the f r i c t i o n - v e l o c i t y c u r v e l e v e l s o f f . The a m p l i t u d e - v e l o c i t y c u r v e s o f F i g . 5.1.7 were o b t a i n e d w i t h normal l o a d s o f 5.4 l b , 3.85 l b , 2.4 l b and 1.0 l b r e s p e c t i v e l y . The c u r v e s show t h a t the normal l o a d n o t o n l y a f f e c t e d the upper c r i t i c a l v e l o c i t y a t wh i c h decay commences, b u t a l s o the a m p l i t u d e o f the s t i c k - s l i p o s c i l l a t i o n near the low v e l o c i t y end where the s t i c k -s l i p o s c i l l a t i o n i s m a i n l y under the i n f l u e n c e o f the s t a t i c f r i c t i o n c h a r a c t e r i s t i c . Under such c i r c u m s t a n c e s , i t i s q u i t e a p parent t h a t the l o w e r the normal l o a d the lower the maximum s t a t i c f r i c t i o n f o r c e and t h e r e f o r e the s m a l l e r t h e a m p l i t u d e o f s t i c k - s l i p o s c i l l a t i o n . ( i ) Frequency o f t h e F r i c t i o n - I n d u c e d V i b r a t i o n The f r e q u e n c y o f the f r i c t i o n - i n d u c e d v i b r a t i o n was p l o t t e d as a r a t i o v e r s u s the d i s c v e l o c i t y i n F i g u r e s 5.1.2, 5.1.4 and 5.1.6. I t i s t o be n o t e d t h a t t h e f r e -quency o f s t i c k - s l i p o s c i l l a t i o n was g e n e r a l l y l o w e r t h a n 98 the damped n a t u r a l f r e q u e n c y o f the e l a s t i c system, and the f r e q u e n c y o f o s c i l l a t i o n approached the damped n a t u r a l f r e q u e n c y as t h e o s c i l l a t i o n t r a n s f o r m e d t o t h e q u a s i -harmonic t y p e . In F i g . 5.1.2 where s t i c k - s l i p t y pe o s c i l -l a t i o n was not r e c o r d e d , the f r e q u e n c y - v e l o c i t y p l o t appears t o be c o n s t a n t a t a f r e q u e n c y v e r y n e a r l y e q u a l t o the damped n a t u r a l f r e q u e n c y o f t h e system. ( j ) F r i c t i o n - V e l o c i t y Curve o f Type C In F i g . 5.1.8 i s shown two f r i c t i o n - v e l o c i t y c u r v e s q u i t e d i f f e r e n t from t h o s e d e s c r i b e d e a r l i e r . The f r i c t i o n -v e l o c i t y c urve o b t a i n e d by u s i n g #5 o i l as the l u b r i c a n t shows a h o r i z o n t a l s e c t i o n i n the v e r y low v e l o c i t y r e g i o n which i s f o l l o w e d by a s t e e p p o s i t i v e s l o p e a t h i g h e r v e l o c i t i e s ; t h e p o s i t i v e s l o p e s e c t i o n i s almost l i n e a r . The f r i c t i o n - v e l o c i t y c urve u s i n g #6 o i l as the l u b r i c a n t shows a curve w i t h a s t e e p p o s i t i v e s l o p e i n the v e r y low v e l o c i t y r e g i o n w i t h the s l o p e g r a d u a l l y d i m i n i s h i n g a t h i g h e r v e l o c i t i e s ; t h i s t y pe o f f r i c t i o n - v e l o c i t y c urve p r o v i d e s good performance i n some a u t o m a t i c t r a n s m i s s i o n c l u t c h e s [ 5 3 ] . Quasi-harmonic o s c i l l a t i o n was not o b s e r v e d f o r th e s e c a s e s , w h i c h i s p r e d i c t a b l e because o f t h e form 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 which show no n e g a t i v e s l o p e r e g i o n . However, some s t i c k - s l i p o s c i l l a t i o n was o b s e r v e d i n t h e v e r y low v e l o c i t y r e g i o n i n the case o f t h e #5 o i l , w h ich i s no doubt due t o the presence o f a s m a l l h o r i z o n t a l s e c t i o n i n the f r i c t i o n - v e l o c i t y c u r v e . I t s h o u l d be 99 n o t e d t h a t t h e f r i c t i o n - v e l o c i t y c u r v e s o f F i g . 5.1.8 were o b t a i n e d by p l o t t i n g the f r i c t i o n f o r c e v a l u e s from a sequence o f d i s c v e l o c i t i e s a t a normal l o a d o f 5.4 l b . A one c y c l e t r a c e was n o t o b t a i n e d s i n c e t h e r e was no q u a s i - h a r m o n i c o s c i l l a t i o n . However, as i t has been i n d i c a t e d e a r l i e r t h a t i n t h e absence o f f r i c t i o n -i n d u c e d v i b r a t i o n the f r i c t i o n - v e l o c i t y c u r v e o b t a i n e d by measuring d i s p l a c e m e n t a l o n e g i v e s a c c u r a t e r e s u l t s . 5.1.2 S t e e l on S t e e l The n a t u r e o f the t e s t s r e q u i r e d r e p e a t e d runs o v e r the same t r a c k a t v a r i o u s speeds, t h i s u s u a l l y r e s u l t e d i n e a r l y damage o f the t e s t t r a c k w i t h the d r y s t e e l - o n -s t e e l c o m b i n a t i o n . I n o r d e r t o m i n i m i s e the damage, p e t r o l a t u m was used as t h e l u b r i c a n t f o r the s t e e l - o n -s t e e l c o m b i n a t i o n . The c o m b i n a t i o n had t h e p r o p e n s i t y t o ex e c u t e s t i c k - s l i p o s c i l l a t i o n . However, n e a r - c i r c u l a r phase p l a n e disgrams were o b t a i n e d on t h e o s c i l l o s c o p e a t h i g h e r v e l o c i t i e s . (a) S t i c k - S l i p O s c i l l a t i o n F i g . 5.1.9a i l l u s t r a t e s a t y p i c a l x-x phase p l a n e o s c i l l o s c o p e t r a c e o f a s t i c k - s l i p type o s c i l l a t i o n o b t a i n e d f o r s t e e l - o n - s t e e l s u r f a c e s w i t h p e t r o l a t u m (U.S.P.) as the l u b r i c a n t . The normal l o a d was 3.85 l b and the d i s c speed was 0.08 i n / s e c . A s l i g h t amount o f r i p p l e d u r i n g the e a r l y p o r t i o n o f the s t i c k p o r t i o n o f t h e c y c l e was' a t t r i b u t e d t o t h e i n e r t i a f o r c e a s s o c i a t e d w i t h the r e s i - •. d u a l a c c e l e r a t i o n o f t h e v i b r a t i n g mass whi c h appears i n the e a r l y p o r t i o n o f the s t i c k p e r i o d . The f r e q u e n c y o f t h i s secondary o s c i l l a t i o n i s c o n s i d e r e d t o be some c o m b i n a t i o n o f t h e f r e q u e n c y o f t h e d r i v i n g system and o f ncl the 2 mode o f t h e e l a s t i c s u p p o r t i n g system. A computer s i m u l a t i o n was made u s i n g t h e o b s e r v e d f r e q u e n c y and a r o u g h l y e s t i m a t e d mass f o r t h e d r i v i n g system. The r e s u l t s o f t h i s s i m u l a t i o n s t u d y appeared t o agree w i t h t h e e x p e r i m e n t a l o b s e r v a t i o n s . F i g . 5.1.9b i l l u s t r a t e s a n e a r -c i r c u l a r x-x phase p l a n e o s c i l l o s c o p e t r a c e o b t a i n e d a t a d i s c speed o f 2.45 i n / s e c , a s l i g h t t r a c e o f s t i c k s t i l l e x i s t s . The s m a l l l o o p appears near the z e r o s l i d i n g v e l o c i t y a x i s i s due t o the s m a l l amount o f r e s i d u a l back-l a s h o f the d r i v i n g system. F o r p r a c t i c a l r e a s o n s , the b a c k l a s h takeup d e s c r i b e d i n t h e a p p a r a t u s s e c t i o n was used f o r t e s t s a t low d i s c v e l o c i t i e s o n l y . The one c y c l e t r a c e o f the f r i c t i o n - v e l o c i t y c u r v e o f F i g . 5.1.9b was r e p r o d u c e d i n F i g . 5.1.10. I t would be n o t e d t h a t t h e curve has the form as t h a t o f t h e Type B e x c e p t i t i s a l m o s t l i n e a r i n t h i s c a s e ; t h e average s l o p e i s e q u i v a l e n t t o a n e g a t i v e damping o f 0.045 l b / i n / s e c . In t h e low v e l o c i t y r e g i o n the e q u i v a l e n t n e g a t i v e damping i s a p p r o x i m a t e l y 0.075 l b / i n / s e c and t h e curve l e v e l s o f f a t v e l o c i t i e s above 6 i n / s e c . The q u a s i - h a r m o n i c theory, d i d not g i v e an a c c u r a t e p r e d i c t i o n f o r t h i s t y p e o f f r i c t i o n - v e l o c i t y c urve s i n c e some s t i c k - p o r t i o n e x i s t e d 101 i n a lmost a l l v i b r a t i o n s t h r o u g h o u t the e n t i r e v e l o c i t y range. I n t h e h i g h e r v e l o c i t y r e g i o n t h e s t i c k p o r t i o n was s m a l l so t h a t the phase p l a n e was a l m o s t c i r c u l a r . The e x p e r i m e n t a l a m p l i t u d e - v e l o c i t y c u r v e i s shown i n F i g . 5.1.10 and i t can be seen t h a t a l l the a m p l i t u d e s are on the l e f t s i d e o f the 45° l i n e w hich i s an i n d i c a t i o n o f the s t i c k - s l i p t y p e o s c i l l a t i o n . i (b) E f f e c t o f E x t e r n a l Damping The e f f e c t o f heavy e x t e r n a l damping was s t u d i e d u s i n g a p o w e r f u l permanent magnet which was a p p l i e d t o the v i b r a t o r y system d u r i n g f r i c t i o n - i n d u c e d v i b r a t i o n . The system damping was i n c r e a s e d from a p p r o x i m a t e l y 0.009 t o 0.0 8 l b / i n / s e c . The e x p e r i m e n t a l a m p l i t u d e - v e l o c i t y r e s u l t s o b t a i n e d w i t h the heavy e x t e r n a l damping are a l s o shown i n F i g . 5.1.10. I t s h o u l d be n o t e d t h a t some s t i c k - s l i p t y p e o s c i l l a t i o n s t i l l e x i s t s i n the low v e l o c i t y r e g i o n , how-e v e r , an upper c r i t i c a l v e l o c i t y a l s o appears. V i b r a t i o n was not o b s e r v e d as the d i s c v e l o c i t y was i n c r e a s e d beyond 2.5 i n / s e c . Furthermore the a m p l i t u d e of s t i c k - s l i p o s c i l -l a t i o n i n the v e r y low v e l o c i t y r e g i o n w i t h the presence o f e x t e r n a l damper are s u b s t a n t i a l l y l o w e r than t h o s e o b t a i n e d w i t h o u t e x t e r n a l damping; i n f a c t the a m p l i t u d e -v e l o c i t y c u r v e appears t o be almost l i n e a r t h r o u g h o u t •; the e n t i r e l e n g t h . One c y c l e o s c i l l o s c o p e t r a c 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 s o b t a i n e d w i t h the heavy e x t e r n a l damping showed t h a t t h e r e was s i g n i f i c a n t m o d i f i c a t i o n on the cu r v e due t o t h e presence o f the heavy damping. The n e g a t i v e s l o p e became l e s s s t e e p i n the h i g h e r v e l o c i t y r e g i o n and the c u r v e l e v e l e d o f f a t a d i s c v e l o c i t y o f about 3 i n / s e c . I t s h o u l d be noted t h a t the one c y c l e method i n c l u d e s the damping term and t h a t the e x t e r n a l damping i s v i s c o u s i n n a t u r e t h e r e f o r e i t has l e s s e f f e c t i n the low v e l o c i t y r e g i o n . Due t o the h i g h damping c o e f f i c i e n t , phase r e l a t i o n s h i p s between the a c c e l e r a t i o n , v e l o c i t y and d i s p l a c e m e n t s i g n a l s became s i g n i f i c a n t , t h e r e f o r e the c u r v e so o b t a i n e d was not c o n s i d e r e d as an a c c u r a t e r e p r e s e n t a t i o n o f the f r i c t i o n - v e l o c i t y c h a r a c t e r -i s t i c w i t h heavy damping. 5.1.3 Polymer M a t e r i a l on S t e e l M i c r o p h o t o g r a p h s o f one o f the b u n d l e s o f carbon f i b r e s a re shown i n F i g u r e s 5.1.11a, b, c and d, the m a g n i f i c a t i o n s a re 100, 200, 400 and 800 r e s p e c t i v e l y . The p i c t u r e s were t a k e n from the s l i d e r s u r f a c e a f t e r t h e t e s t was completed. I t was obse r v e d t h a t the f r i c t i o n r e s u l t s were c r i t i c a l l y dependent on t h e presence o f r e s i n m a t e r i a l on the d i s c s u r f a c e . T h e r e f o r e g r e a t c a r e was e x e r c i s e d d u r i n g each t e s t and f r i c t i o n r e s u l t s were o b t a i n e d a t i n c r e a s i n g d i s c v e l o c i t i e s as w e l l as d e c r e a s i n g d i s c v e l o c i t i e s . S e v e r a l s i m i l a r t e s t s were performed. Owing t o t h e absence o f f r i c t i o n - i n d u c e d v i b r a t i o n the two c u r v e s o f F i g . 5.1.12 103 w e r e o b t a i n e d b y p l o t t i n g t h e f r i c t i o n f o r c e a g a i n s t t h e d i s c v e l o c i t y . The n o r m a l l o a d s w e r e 5.4 l b and 3.85 l b r e s p e c t i v e l y . I t was f o u n d t h a t w h i l e t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e showed a c o n s i s t e n t p o s i t i v e s l o p e i n e v e r y t e s t , v a r i a t i o n o f as much as 15% i n s t a t i c d i s -p l a c e m e n t was o b s e r v e d f r o m t e s t t o t e s t . T h i s v a r i a t i o n was b e l i e v e d t o be due t o t h e p r e s e n c e o f r e s i n m a t e r i a l on t h e d i s c s u r f a c e . The c u r v e s i l l u s t r a t e t h a t t h e f r i c t i o n f o r c e i n c r e a s e s as t h e d i s c v e l o c i t y i s i n c r e a s e d a n d l e v e l s o f f a t d i s c v e l o c i t y a r o u n d 2.0 i n / s e c . T h i s t y p e o f f r i c t i o n -v e l o c i t y c u r v e s u g g e s t s h e a v y s u r f a c e d a m p i n g and no q u a s i -h a r m o n i c o s c i l l a t i o n w o u l d b e e x p e c t e d . I n f a c t , f r i c t i o n a l o s c i l l a t i o n o f any f o r m was n o t o b s e r v e d i n t h e e x p e r i m e n t a l t e s t s . I t w o u l d be o f i n t e r e s t t o n o t e t h a t i n t h e i r • s t u d i e s o f s t a t i c c o e f f i c i e n t o f f r i c t i o n o f p o l y m e r m a t e r i a l w i t h r e l a t i v e t o t i m e , W e i t e r and S c h m i d t [79] showed t h a t w h i l e PTFE ( T e f l o n ) i s h i g h l y t i m e d e p e n d e n t w h e r e a s t e f l o n -g r a p h i t e c o m p o s i t e i s n o t t i m e d e p e n d e n t . A s e c o n d s l i d e r p r e p a r e d f r o m r e s i n m a t e r i a l w i t h o u t t h e c a r b o n f i b r e s was u s e d i n a t e s t t o s t u d y t h e b e h a v i o u r o f r e s i n - s t e e l c o m b i n a t i o n . The c o e f f i c i e n t o f f r i c t i o n ; o f t h i s c o m b i n a t i o n was f o u n d t o be much h i g h e r t h a n t h a t o f t h e c a r b o n - r e s i n c o m p o s i t i o n . A g a i n , t h e r e s u l t s w e r e f o u n d t o be g r e a t l y i n f l u e n c e d b y t h e p r e s e n c e o f r e s i n •, d e p o s i t on t h e d i s c s u r f a c e . R e s u l t s w e r e i n c o n s i s t e n t even w i t h i n t h e c o n f i n e s o f one t e s t . The bro k e n l i n e curve o f F i g . 5.1.12 was o b t a i n e d by a s s e s s i n g the r e s u l t s o f s e v e r a l s i m i l a r t e s t s . I n g e n e r a l , t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c shows a s l i g h t p o s i t i v e s l o p e a t the low v e l o c i t y end and l e v e l s o f f a t d i s c v e l o c i t i e s around 1.0 i n / s e c . Quasi-harmonic o s c i l l a t i o n t o g e t h e r w i t h h i g h •,• f r e q u e n c y o s c i l l a t i o n was obs e r v e d a t d i s c v e l o c i t i e s above 1 i n / s e c . T h i s b e h a v i o u r was a t t r i b u t e d t o the presence of r e s i n d e p o s i t on the d i s c s u r f a c e . S i n c e a l a r g e p a r t o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c i s almost l i n e a r , and l e v e l and i s o n l y preceded by a s m a l l amount o f p o s i t i v e s l o p e , the system i s i n some k i n d o f n e u t r a l s t a t e a t d i s c v e l o c i t i e s above 1 i n / s e c . A c c o r d i n g l y any o u t s i d e d i s t u r b a n c e such as when the r e s i n s l i d e r rode o v e r s p o t s o f r e s i n d e p o s i t on the d i s c s u r f a c e would change t h e system i n t o an u n s t a b l e s t a t e and r e s u l t e d i n q u a s i - h a r m o n i c s e l f - i n d u c e d v i b r a t i o n . The carbon f i b r e s and s t e e l c o m b i n a t i o n w i t h i t s s u b s t a n t i a l p o s i t i v e s l o p e i n the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c has a d e f i n i t e advantage i n many e n g i n e e r i n g a p p l i c a t i o n s where v i b r a t i o n i s t o be a v o i d e d . F o r i n s t a n t , the carbon f i b r e s would p r o v i d e g r e a t e r r e s i s t a n c e t o wear and low f r i c t i o n when used as b e a r i n g m a t e r i a l s . 5.1.4 Rubber on S t e e l The v i s c o e l a s t i c p r o p e r t i e s o f ru b b e r a r e s u b j e c t t o v a r i a t i o n w i t h t emperature [ 4 7 ] , [ 8 0 ] , t h e r e f o r e r e l i a b l e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e s can o n l y be o b t a i n e d a t low d i s c v e l o c i t i e s . F i g . 5.1.13 shows a f r i c t i o n -v e l o c i t y c u r v e o b t a i n e d a t a d i s c v e l o c i t y o f 0.39 i n / s e c . The curve i l l u s t r a t e s a d i s t i n c t l y humped form w i t h . t h e maximum o c c u r r i n g a t d i s c v e l o c i t i e s around 0.4 i n / s e c . I n g e n e r a l , f r i c t i o n a l o s c i l l a t i o n was n o t ob s e r v e d a t d i s c v e l o c i t i e s below 0.3 i n / s e c . Some o s c i l l a t i o n was o b s e r v e d i n some t e s t s a t d i s c v e l o c i t i e s between 0.3 i n / s e c . t o 0.4 i n / s e c . The f r i c t i o n a l o s c i l l a t i o n became c o n s i s t e n t a t d i s c v e l o c i t i e s above 0.4 i n / s e c a l t h o u g h t h e funda-. m e n t a l o s c i l l a t i o n was u s u a l l y accompanied by h i g h f r e q u e n c y o s c i l l a t i o n s and under t h e s e c i r c u m s t a n c e s a c l e a r one c y c l e t r a c e c o u l d n o t be o b t a i n e d . I t i s b e l i e v e d t h a t t h e h i g h f r e q u e n c y o s c i l l a t i o n i s r e l a t e d t o the temperature v a r i a t i o n o f the r u b b e r s l i d e r caused by the h i g h s l i d i n g v e l o c i t y . I t was a l s o o b s e r v e d t h a t the f r i c t i o n f o r c e was g e n e r a l l y h i g h e r as the system s t a r t e d from a s t a t i o n a r y s t a t e and g r a d u a l l y d e c r e a s e d t o a lower f r i c t i o n v a l u e as s l i d i n g c o n t i n u e d a t a c o n s t a n t d i s c v e l o c i t y . The h i g h i n i t i a l f r i c t i o n v a l u e a p p l i e d t o b o t h s t a t i c and k i n e t i c f r i c t i o n f o r c e s . T h i s f i n d i n g s u g gests t h a t the tempera-t u r e e f f e c t as w e l l as t h e v e l o c i t y s h o u l d be i n v e s t i g a t e d i n the s t u d i e s o f the f r i c t i o n a l c h a r a c t e r i s t i c s o f the rubb e r on s t e e l c o m b i n a t i o n . \ 10 5.1.5 Summary By u s i n g v a r i o u s t y p e s o f f r i c t i o n c o u p l e s and \ l u b r i c a n t s , i t was p o s s i b l e t o show a v a r i e t y o f f r i c t i o n -v e l o c i t y c h a r a c t e r i s t i c s . However, the fundamental r e a s o n s f o r the shape o f the v a r i o u s f r i c t i o n - v e l o c i t y c h a r a c t e r -i s t i c c u rve were n o t i n v e s t i g a t e d . A n o t h e r f u l l i n v e s t i -g a t i o n c o u l d be devoted t o t h i s t o p i c . N e v e r t h e l e s s , t h e a c c u r a c y o f the measuring t e c h n i q u e i n the p r e s e n t i n v e s t i g a t i o n has made i t p o s s i b l e t o c a t e g o r i s e the v a r i o u s forms o f dynamic f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e s i n r e l a t i o n t o f r i c t i o n - v i b r a t i o n b e h a v i o u r . The i n v e s t i g a t i o n r e v e a l e d t h a t the d y n a m i c a l be-h a v i o u r o f a s l i d i n g c o u p l e i s c r i t i c a l l y dependent on the shape of the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e . The c u r v e forms employed were d e s i g n a t e d as type A, B and C f o r the humped, d e c r e a s i n g and i n c r e a s i n g c h a r a c t e r i s t i c s r e s p e c t i v e l y (Ref. F i g u r e s 3.2.2, 3.2.1b and 3.2.1a). G e n e r a l l y , a f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e o f Type A i s the major c o n d i t i o n f o r the e x i s t e n c e o f q u a s i -harmonic type f r i c t i o n - i n d u c e d v i b r a t i o n , a l t h o u g h s t i c k -s l i p type o s c i l l a t i o n may e x i s t i n the low v e l o c i t y r e g i o n f o r t h i s t y p e o f curve depending on the s t a t i c f r i c t i o n c h a r a c t e r i s t i c . When q u a s i - h a r m o n i c o s c i l l a t i o n o c c u r s , the a m p l i t u d e o f v i b r a t i o n i n c r e a s e s as the d r i v i n g v e l o c i t y i s i n c r e a s e d . However, the system becomes u n s t a b l e when a c r i t i c a l v e l o c i t y i s r e a c h e d ; the v i b r a t i o n may decay o r 1 0 7 grow depending on whether the s l i d e r d i s p l a c e m e n t and v e l o c i t y c o o r d i n a t e s are i n s i d e o r o u t s i d e the u n s t a b l e l i m i t c y c l e on t h e phase p l a n e . The b l o t t i n g paper-s t e e l c o m b i n a t i o n w i t h a u t o m a t i c t r a n s m i s s i o n f l u i d as the l u b r i c a n t as w e l l as p o l y m e r - s t e e l and r u b b e r - s t e e l c o m b i n a t i o n s g i v e t h i s t y p e o f f r i c t i o n - v e l o c i t y c h a r a c t e r -i s t i c . Good agreement was o b t a i n e d between the 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 . Pure s l i d i n g was o b s e r v e d i n f r i c t i o n c o u p l e s e x h i b i t i n g i n c r e a s i n g f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c (Type C ) . T h i s type o f c h a r a c t e r i s t i c was o b s e r v e d i n the b l o t t i n g p a p e r - s t e e l c o m b i n a t i o n w i t h c e r t a i n t y p e o f a u t o m a t i c t r a n s m i s s i o n f l u i d as the l u b r i c a n t . F r i c t i o n c o u p l e s such as the s t e e l - s t e e l c o m b i n a t i o n u s u a l l y e x h i b i t f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c s o f Type B. In g e n e r a l , the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c w i t h a c o n t i n u o u s n e g a t i v e s l o p e has the p r o p e n s i t y t o e x e c u t e s t i c k - s l i p o s c i l l a t i o n . I t s h o u l d be n o t e d t h a t t h e p r e -s e n t i n v e s t i g a t i o n does not t a k e i n t o c o n s i d e r a t i o n the s t a t i c f r i c t i o n c h a r a c t e r i s t i c which i s g e n e r a l l y t h e g o v e r n i n g f a c t o r f o r s t i c k - s l i p t y pe f r i c t i o n - i n d u c e d v i b r a t i o n . However, when s t i c k - s l i p o s c i l l a t i o n does o c c u r , the form o f the dynamic f r i c t i o n c h a r a c t e r i s t i c does i n f l u e n c e the a m p l i t u d e o f t h e o s c i l l a t i o n and t h e t r a n s -i t i o n t o the q u a s i - h a r m o n i c o s c i l l a t i o n . Thus th e a m p l i t u d e v a r i a t i o n w i t h d r i v i n g v e l o c i t y o f the s t i c k - s l i p t y pe 108 o s c i l l a t i o n depends on the shape o f t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c as w e l l as t h e s t a t i c f r i c t i o n c h a r a c t e r -s i t i c . G e n e r a l l y , i n t h e h i g h e r v e l o c i t y r e g i o n the v i b r a t i o n becomes near s i n u s o i d a l a l t h o u g h t h e r e s t i l l e x i s t s a s m a l l p o r t i o n o f s t i c k p e r i o d . The a p p l i c a t i o n o f the e x t e r n a l v i s c o u s damping has the e f f e c t o f e x t i n g u i s h i n g the f r i c t i o n - i n d u c e d v i b r a t i o n , e x c e p t i n t h e v e r y low v e l o c i t y r e g i o n when the o s c i l l a t i o n i s m a i n l y under th e i n f l u e n c e o f the s t a t i c f r i c t i o n c h a r a c t e r i s t i c . Under t h e s e c i r c u m s t a n c e s the e x t e r n a l • damping reduces t h e a m p l i t u d e o f the s t i c k - s l i p t y p e o s c i l l a t i o n . The f r e q u e n c y o f the q u a s i - h a r m o n i c type f r i c t i o n -i n d u c e d v i b r a t i o n i s c l o s e t o the damped n a t u r a l f r e q u e n c y o f the v i b r a t o r y system, whereas the f r e q u e n c y o f t h e s t i c k -s l i p o s c i l l a t i o n depends on the s t i c k p e r i o d and i s g e n e r a l l y l ower t h a n the damped n a t u r a l f r e q u e n c y o f the system. 5.2 Non-Autonomous Cases •; The b l o t t i n g paper on s t e e l c o m b i n a t i o n was used f o r a l l f r i c t i o n a l t e s t s w i t h t r a n s v e r s e e x t e r n a l e x c i t a t i o n . Three t y p e s o f t r a n s m i s s i o n f l u i d w h i c h gave t h r e e d i s t i n c t forms of f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c s , namely Type A, B, and C, were used. The r e s u l t s show the t r a n s i t i o n from a l i n e a r c h a r a c t e r i s t i c t o a n o n - l i n e a r c h a r a c t e r i s t i c . I n i t i a l l y , f o r c e d v i b r a t i o n w i t h the s l i d e r c l e a r o f the lower s u r f a c e was s t u d i e d . The p o s i t i v e s l o p e f r i c t i o n -109 v e l o c i t y c h a r a c t e r i s t i c , Type C, was i n v e s t i g a t e d n e x t w h i c h c o u l d be c o n s i d e r e d as e q u i v a l e n t t o a l i n e a r system w i t h heavy damping. T h i s s t u d y was f o l l o w e d by an i n v e s t i -g a t i o n o f the f r i c t i o n a l c h a r a c t e r i s t i c w i t h the al m o s t l i n e a r c u r v e h a v i n g n e g a t i v e s l o p e , and w i t h t h e hump c l o s e t o t h e z e r o s l i d i n g v e l o c i t y a x i s . F i n a l l y , d e t a i l e d i n v e s t i g a t i o n s which i n c l u d e d the non-resonance, fundamental resonance and subharmonic e n t r a i n m e n t c a s e s , were conducted w i t h t h e humped t y p e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c , Type A. 5.2.1 F o r c e d V i b r a t i o n ( l i n e a r case) F i g . 5.2.1 shows a p l o t o f the m a g n i f i c a t i o n f a c t o r , X/Xq v e r s u s f r e q u e n c y r a t i o v/to. The r e s u l t s were o b t a i n e d w i t h the s l i d e r c l e a r o f t h e d i s c s u r f a c e and by v a r y i n g the r o t a t i o n a l speed o f the unbalanced w e i g h t . The a m p l i -tudes o f v i b r a t i o n x were r e c o r d e d from t h e o s c i l l o s c o p e . , and the s t a t i c d i s p l a c e m e n t s x were c a l c u l a t e d from the unbalanced f o r c e s . A t h e o r e t i c a l c u r v e u s i n g t h e l i n e a r v i b r a t i o n t h e o r y w i t h a damping r a t i o r / r o f 0.01, where c r c i s the c r i t i c a l damping c o e f f i c i e n t , i s a l s o shown i n F i g . 5.2.1. I n g e n e r a l , the 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 a re i n r e a s o n a b l y good agreement. These r e s u l t s i n d i c a t e t h a t the system p a r a m e t e r s , p a r t i c u l a r l y the system damping e s t i m a t e d from the f r e e v i b r a t i o n t e s t were o f the c o r r e c t o r d e r . 110 5.2.2 F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve o f  T y P e c The e x p e r i m e n t a l r e s u l t s o f F i g . 5.2.2 were o b t a i n e d u s i n g the #5 o i l as the l u b r i c a n t , i t s f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c has the form as shown i n F i g . 5.1.8. The f r i c t i o n c u r v e has a c o n t i n u o u s p o s i t i v e s l o p e and i s almost l i n e a r i n the r e g i o n shown wh i c h s u g g e s t s t h a t t h e i c u r v e c o u l d be c o n s i d e r e d s i m p l y as v i s c o u s damping h a v i n g an e q u i v a l e n t damping c o e f f i c i e n t o f 0.20 l b / i n / s e c i n c l u d -i n g the system damping. The t h e o r e t i c a l curve o f F i g . 5.2.2 was o b t a i n e d by a p p l y i n g the e q u i v a l e n t damping c o e f f i c i e n t o f 0.2 t o eq. (3.4.1) o f the l i n e a r v i b r a t i o n t h e o r y . E x p e r i m e n t a l r e s u l t s were t a k e n a t c o n s t a n t d i s c v e l o c i t i e s o f 1.085 i n / s e c and 1.325 i n / s e c and a t a normal l o a d o f 5.4 l b . The r e s u l t s a re i n good agreement w i t h t h e t h e o r y . F r i c t i o n - i n d u c e d o s c i l l a t i o n does not e x i s t i n the system due t o t h e presence o f the c o n t i n u o u s p o s i t i v e s l o p e o f -j the f r i c t i o n - v e l o c i t y c u r v e ; the o s c i l l a t i o n i s due s o l e l y t o the e x t e r n a l e x c i t a t i o n . The good agreement between the 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 a g a i n i l l u s t r a t e s the a c c u r a c y and t h e s e l f - c o n s i s t e n c y o f the ap p a r a t u s and t h e measuring i n s t r u m e n t a t i o n . 5.2.3 G e n e r a l D i s c u s s i o n o f a Non-Autonomous F r i c t i o n -Induced V i b r a t i o n System A n o n l i n e a r system s u b j e c t t o f o r c e d v i b r a t i o n was o b t a i n e d when the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c has t h e I l l form o f a Type B l curve as i l l u s t r a t e d i n F i g . 5 . 1 . 6 . I n t h i s c a s e , the l i n e a r v i b r a t i o n t h e o r y c o u l d no l o n g e r be a p p l i e d ; i n f a c t , as i t would be n o t e d l a t e r even th e n o n l i n e a r t h e o r y c o u l d not g i v e a c c u r a t e p r e d i c t i o n s f o r c e r t a i n t y p e s o f f r i c t i o n - v e l o c i t y c u r v e s , p a r t i c u l a r l y when the f r e q u e n c y o f t h e e x t e r n a l e x c i t a t i o n i s c l o s e t o the fundamental r e s o n a n c e . Around t h i s r e g i o n t h e a m p l i t u d e o f o s c i l l a t i o n tends t o exceed t h e z e r o s l i d i n g v e l o c i t y a x i s i n the x-x phase p l a n e p l o t . The p o s i t i o n o f the z e r o s l i d i n g v e l o c i t y a x i s i n the x-x phase p l a n e i s d e t e r -mined by t h e d i s c v e l o c i t y . In t h e o r y , when the t r a j e c t o r y goes beyond t h i s a x i s , i t s p a t h w i l l be under the i n f l u e n c e o f a f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c w i t h r e v e r s e d s i g n . In the autonomous c a s e , when the t r a j e c t o r y r e a c h e s t h e , z e r o s l i d i n g v e l o c i t y a x i s s t i c k may o c c u r and t h e o s c i l l a t i o n t r a n s f o r m s i n t o some form o f s t i c k - s l i p t y pe o s c i l l a t i o n . However, i n t h e case where the magnitude o f the e x t e r n a l e x c i t a t i o n p r e d o m i n a t e s , the t r a j e c t o r y may be f o r c e d t o t h e o t h e r s i d e o f the a x i s and r e s u l t i n some d i s t o r t e d form o f o s c i l l a t i o n . I n F i g . 5 .2.3 two o s c i l -l o s c o p e t r a c e s are shown, one o f w h i c h i l l u s t r a t e s t h e d i s t o r t e d form o f o s c i l l a t i o n . A p l o t o f t h e m a g n i f i c a t i o n f a c t o r v e r s u s f r e q u e n c y r a t i o i s shown i n F i g . 5 . 2 . 4 . The r e s u l t s were o b t a i n e d a t d i s c v e l o c i t i e s o f 1 .85, 1.6 and 1.06 i n / s e c , the normal l o a d was 5.4 l b . No attempt was made t o i n v e s t i g a t e t h i s case i n d e t a i l . N e v e r t h e l e s s , the cu r v e shows t h a t sub-harmonic e n t r a i n m e n t e x i s t s i n t h e system due t o t h e non-l i n e a r i t y o f t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c . Sub-harmonic e n t r a i n m e n t o c c u r s when the f r e q u e n c y r a t i o a i s v e r y c l o s e o r e q u a l t o 2, 3, e t c . , the system v i b r a t e s a t a f r e q u e n c y w h i c h i s a s u b - m u l t i p l e o f the e x t e r n a l e x c i t a t i o n f r e q u e n c y and i s c l o s e t o the f r e q u e n c y o f t h e a u t o p e r i o d i c o s c i l l a t i o n . F i g . 5.2.5 shows f i v e c u r v e s o b t a i n e d a t v a r i o u s d i s c v e l o c i t i e s and w i t h d i f f e r e n t e x t e r n a l e x c i t a t i o n f o r c e s . The r e s u l t s were o b t a i n e d w i t h #9 o i l as t h e l u b r i c a n t and t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c has a Type A2 cu r v e as i l l u s t r a t e d i n F i g . 5.1.4. The normal l o a d was 5.4 l b . The c u r v e s o f F i g . 5.2.5 i l l u s t r a t e a v e r y s i m i l a r p a t t e r n a l t h o u g h the o c c u r r a n c e o f the peak v a r i e s o v er a l a r g e range. A comparison o f c u r v e s (1) and ( 2 ) , and c u r v e s (3) and (5) shows t h a t f o r the same d i s c v e l o c i t y , the h i g h e r the e x t e r n a l e x c i t a t i o n f o r c e the lower the peak and a l s o the e a r l i e r t he peak o c c u r s . I t s h o u l d be n o t e d t h a t the c u r v e s were p l o t t e d as m a g n i f i - , c a t i o n f a c t o r v e r s u s f r e q u e n c y r a t i o . I n t h e l i n e a r c a s e , the curve would be independent o f the e x t e r n a l e x c i t a t i o n f o r c e . The v a r i a t i o n w i t h the e x t e r n a l e x c i t a t i o n f o r c e s u g g ests t h a t t h e f r i c t i o n f o r c e i s n o n l i n e a r . The v a r i -a t i o n w i t h the e x t e r n a l e x c i t a t i o n f o r c e would be due t o the r e a s o n i n d i c a t e d e a r l i e r whereby the d i s c v e l o c i t y , 113 which d e t e r m i n e s t h e l o c a t i o n o f the z e r o s l i d i n g v e l o c i t y a x i s i n the x-x phase p l a n e , e x e r t s a c e r t a i n r e s t r i c t i o n on the a m p l i t u d e o f o s c i l l a t i o n whenever i t s t r a j e c t o r y tends t o grow beyond t h i s a x i s i n t h e x-x phase p l a n e . Any i n c r e a s e i n a m p l i t u d e e x c e e d i n g t h e l i m i t would be p a r t i a l l y r e s t r i c t e d and t h e r e s u l t i n g v i b r a t i o n would be o f the form shown i n F i g . 5.2.3b which i s somewhat s i m i l a r t o s t i c k - s l i p t y p e o s c i l l a t i o n s . In f a c t , a comparison o f c u r v e s 2, 4, and 5 f u r t h e r v e r i f i e s t h i s p o i n t . The c u r v e s were o b t a i n e d a t t h r e e d i f f e r e n t d i s c v e l o c i t i e s w i t h t h e same e x t e r n a l f o r c e h a v i n g a magnitude l a r g e enough such t h a t the a m p l i t u d e o f v i b r a t i o n i n t h e r e g i o n near t h e fundamental resonance f r e q u e n c y tended t o exceed the l i m i t . The r e s u l t s i l l u s t r a t e t h a t i n c r e a s i n g t h e d i s c v e l o c i t y a l s o i n c r e a s e s the peak o f t h e c u r v e and d e l a y s i t s o c c u r r a n c e . I t i s apparent t h a t i n c r e a s i n g the d i s c v e l o c -i t y extends the l i m i t where th e r e s t r i c t i o n would a p p l y . Under t h e s e c i r c u m s t a n c e s t h e a m p l i t u d e o f o s c i l l a t i o n would be a b l e t o grow f u r t h e r b e f o r e i t r e a c h e s t h e b a r r i e r . F i g . 5.2,6 shows f o u r diagrams i l l u s t r a t i n g t h e e f f e c t o f the e x t e r n a l e x c i t a t i o n on the average d i s p l a c e -ment a t z e r o a b s o l u t e v e l o c i t y . I n each diagram b o t h the a m p l i t u d e o f v i b r a t i o n and average d i s p l a c e m e n t were p l o t t e d a g a i n s t t h e f r e q u e n c y r a t i o . The approximate l i m i t o f t h e q u a s i - h a r m o n i c o s c i l l a t i o n i s a l s o shown i n the diagrams. A s t a r shown near the v e r t i c a l a x i s i n d i -114 c a t e s t h e average d i s p l a c e m e n t when the e x t e r n a l e x c i t a t i o n was ab s e n t . I t i s i m m e d i a t e l y c l e a r t h a t whenever t h e a m p l i t u d e o f v i b r a t i o n exceeds t h e l i m i t f o r q u a s i - h a r m o n i c o s c i l l a t i o n t he c o r r e s p o n d i n g average d i s p l a c e m e n t b e g i n s t o drop and t h a t t h e h i g h e r t h e a m p l i t u d e v a l u e the lower t h e average d i s p l a c e m e n t becomes. The average d i s p l a c e m e n t r e t u r n s t o i t s normal v a l u e once the a m p l i t u d e o f o s c i l - . l a t i o n i s w i t h i n the l i m i t . I n F i g . 5.2.6d where the a m p l i t u d e o f o s c i l l a t i o n nowhere exceeds the l i m i t , t he average d i s p l a c e m e n t remains c o n s t a n t o v e r the f u l l f r e -quency range. The r e s u l t s o f (a) and (b) were o b t a i n e d h a v i n g a Type B l f r i c t i o n - v e l o c i t y c u r v e w h i l e (c) and (d) were o b t a i n e d h a v i n g Type A2 and Type C f r i c t i o n - v e l o c i t y c u r v e r e s p e c t i v e l y , i n a l l f o u r cases t h e normal l o a d was 5.4 l b . F u r t h e r i n v e s t i g a t i o n o f t h e x-x phase p l a n e diagrams o b t a i n e d from t h e o s c i l l o s c o p e r e v e a l e d t h a t when t h e q u a s i -harmonic o s c i l l a t i o n l i m i t was exceeded the x-x phase p l a n e resembled s t i c k - s l i p t y pe o s c i l l a t i o n , a l t h o u g h the s t i c k p o r t i o n was not w e l l d e f i n e d due t o the presence o f the e x t e r n a l e x c i t a t i o n . The p o s i t i v e d i s p l a c e m e n t o r t h e upper h a l f o f the phase p l a n e seemed t o be under t h e i n -f l u e n c e o f t h e s t i c k p o r t i o n which i n t u r n was due t o the e x t e r n a l e x c i t a t i o n , whereas t h e n e g a t i v e d i s p l a c e m e n t o r the lo w e r h a l f o f the phase p l a n e was f r e e from t h i s k i n d o f r e s t r i c t i o n . Thus i t would appear t h a t the i n c r e a s e 115 i n a m p l i t u d e was m a i n l y a c c o m p l i s h e d by t h e i n c r e a s e i n the l o w e r p o r t i o n o f the phase p l a n e p l o t thus g i v i n g an apparent d e c r e a s e i n the average d i s p l a c e m e n t . The above r e s u l t s i l l u s t r a t e t h e importance o f c h o o s i n g a p r o p e r range o f e x t e r n a l e x c i t a t i o n f o r c e and o f d i s c v e l o c i t y so t h a t m e a n i n g f u l r e s u l t s can be o b t a i n e d from a f u l l s c a l e i n v e s t i g a t i o n o f a non-autonomous system w i t h f r i c t i o n - i n d u c e d v i b r a t i o n . 5.2.4 I n v e s t i g a t i o n o f a Non-Autonomous F r i c t i o n -Induced V i b r a t i o n System Having a Type A l  F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve The f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e o f Type A l as i l l u s t r a t e d i n F i g . 5.1.2 was chosen f o r t h e d e t a i l e d s t u d i e s o f a non-autonomous system, f o r i t s w e l l d e f i n e d humped form which 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 i s -t e n c e o f q u a s i - h a r m o n i c o s c i l l a t i o n . The Type A l curve a l s o p r o v i d e s a f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c from ; whi c h t h e a m p l i t u d e o f t h e s e l f - e x c i t e d v i b r a t i o n was w e l l w i t h i n the l i m i t f o r q u a s i - h a r m o n i c o s c i l l a t i o n . I n t h e t h e o r e t i c a l a n a l y s i s t h e f r i c t i o n f o r c e was e x p r e s s e d as a s e v e n t h o r d e r p o l y n o m i a l because the a l g e -b r a i c m a n i p u l a t i o n was c o m p a r a t i v e l y easy whereas the e x p o n e n t i a l form o f e x p r e s s i o n would i n e v i t a b l y i n v o l v e d B e s s e l f u n c t i o n s and would become e x t r e m e l y c o m p l i c a t e d i n the non-autonomous c a s e . However, t h e e x p o n e n t i a l e x p r e s s i o n was used f o r o b t a i n i n g more a c c u r a t e s o l u t i o n s when a n u m e r i c a l method was used. (a) Non-Resonance Case The c u r v e s o f F i g . 5.2.7 were p r e p a r e d by a p p l y i n g eq. (3.4.13) and eq. (3.4.14) o f t h e non-resonance c a s e . 2 By v a r y i n g t h e e x t e r n a l parameters L = F a / ( l - a ) t h e st e a d y s t a t e a m p l i t u d e o f t h e a u t o p e r i o d i c o s c i l l a t i o n was v a r i e d , and as a c r i t i c a l v a l u e o f L was r e a c h e d , the a m p l i t u d e p o l y n o m i a l had no p o s i t i v e r e a l r o o t which i n d i c a t e d t h a t t h e a u t o p e r i o d i c o s c i l l a t i o n was no l o n g e r p r e s e n t . I f the a b s o l u t e v a l u e o f L was f u r t h e r i n c r e a s e d , o n l y h e t e r o p e r i o d i c o s c i l l a t i o n a t t h e f r e q u e n c y o f the e x t e r n a l e x c i t a t i o n e x i s t e d i n the system. When L = 0, the s t e a d y s t a t e a m p l i t u d e o f the v i b r a t i o n i s same as t h a t o b t a i n e d from t h e autonomous t h e o r y . Thus the c u r v e s o f F i g . 5.2.7 i n d i c a t e whether p u r e l y h e t e r o p e r i o d i c o s c i l -l a t i o n w i t h e x t e r n a l f r e q u e n c y o r combined a u t o p e r i o d i c and h e t e r o p e r i o d i c o s c i l l a t i o n w i t h b e a t f r e q u e n c y e x i s t e d i n t h e system. I t may be ob s e r v e d t h a t t h e c u r v e s o f F i g . 5.2.7 were e x p r e s s e d i n a g e n e r a l i s e d form where L c o n t a i n s b o t h t h e e x t e r n a l e x c i t a t i o n f o r c e magnitude and f r e q u e n c y . I n f a c t , a whole s e r i e s o f c u r v e s can be r e -p l o t t e d from any s i n g l e c u r v e o f F i g . 5.2.7. I n F i g . 5.2.8 t h e c u r v e was p l o t t e d f o r a c o n s t a n t f r e q u e n c y o f 1.79 and t h e c o r r e s p o n d i n g v a l u e s o f F were 117 d e r i v e d from t h e e x p r e s s i o n o f L. Curve (1) o f F i g . 5.2.8 was r e p l o t t e d from curve (3) o f F i g . 5.2.7 r e p l a c i n g t h e L - a x i s by t h e F q - a x i s . Curve (3) o f F i g . 5.2.8 was p r e -p a r e d from eq. (3.4.17) when t h e a u t o p e r i o d i c o s c i l l a t i o n was absent and i t r e p r e s e n t s the h e t e r o p e r i o d i c o s c i l l a t i o n i n t h e system. When the a u t o p e r i o d i c o s c i l l a t i o n i s p r e -s e n t , t h e e f f e c t o f the e x t e r n a l f o r c e i s f e l t o n l y i n t h e second a p p r o x i m a t i o n w h i c h i s n o r m a l l y s m a l l when F q i s s m a l l . However, when F^ becomes s u f f i c i e n t l y l a r g e , t h e e f f e c t o f the e x t e r n a l f o r c e i s s i g n i f i c a n t ; the d o t t e d l i n e o f c u r v e (3) i n F i g . 5.2.8 was o b t a i n e d from eq. (3.4.17) by c o n s i d e r i n g t h e a u t o p e r i o d i c o s c i l l a t i o n t o be absent. I n the non-resonance a n a l y s i s t h e o r i g i n a l d.e..: i n terms o f v a r i a b l e X was t r a n s f o r m e d by i n t r o d u c i n g a new v a r i a b l e as shown i n eq. ( 3 . 4 . 4 ) , t h e r e f o r e i n a d d i t i o n t o t h e a u t o p e r i o d i c and h e t e r o p e r i o d i c o s c i l l a t i o n s as ; shown by c u r v e s (1) and ( 3 ) , the a m p l i t u d e o f t h e a c t u a l o s c i l l a t i o n i n terms o f v a r i a b l e X s h o u l d be o b t a i n e d from eq. (3.4.4) w i t h t h e a d d i t i o n a l f o r c i n g term w h i c h was i n t r o d u c e d i n the o r i g i n a l d.e. The magnitude o f t h i s f o r c i n g term was p l o t t e d as curve ( 2 ) . Thus i n c o n s i d e r i n g t h e s e c u r v e s , i t s h o u l d be noted t h a t c l o s e t o t h e l e f t hand s i d e a x i s the a u t o p e r i o d i c o s c i l l a t i o n p r e d o m i n a t e s ; 'beat' s t a r t e d t o show as the e x t e r n a l f o r c e F q was i n c r e a s e d ; around F q = 0.75 where the a m p l i t u d e o f the a u t o p e r i o d i c o s c i l l a t i o n e q u a l s the a m p l i t u d e o f t h e e x t e r n a l e x c i t a t i o n t h e b e a t w o u l d become most s i g n i f i c a n t . As F was f u r t h e r o i n c r e a s e d o n l y h e t e r o p e r i o d i c o s c i l l a t i o n a t t h e e x t e r n a l f r e q u e n c y e x i s t e d . F i g . 5.2.9 shows s i x s e t s o f c o m p u t e r p l o t t e d d i s -p l a c e m e n t - t i m e and v e l o c i t y - t i m e t r a c e s . The t r a c e s w e r e o b t a i n e d by a p p l y i n g t h e e x p o n e n t i a l e x p r e s s i o n f o r t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c o f F i g . 5.1.2 (Type A l ) i n e q . (3.4.2) and s o l v i n g t h e e q u a t i o n by a n u m e r i c a l m e t h o d . I t s h o u l d be n o t e d t h a t f o r F = 0 . 2 t h e o s c i l l a t i o n was o a l m o s t a u t o p e r i o d i c w i t h a f r e q u e n c y e q u a l t o t h a t o f t h e autonomous s y s t e m . A t F = 0 . 4 and 0.6 t h e o s c i l l a t i o n s -o w e r e o f t h e c o m b i n e d a u t o p e r i o d i c and h e t e r o p e r i o d i c , as F Q r e a c h e d 0.8 t h e o s c i l l a t i o n t u r n e d i n t o a w e l l d e f i n e d ' b e a t ' s h o w i n g t h a t t h e a m p l i t u d e s o f t h e a u t o p e r i o d i c and o f t h e h e t e r o p e r i o d i c c a s e s w e r e a b o u t e q u a l . The l a s t two s e t s o f t r a c e s a t F q = 1.05 and 2.5 showed p u r e l y h e t e r o - . . p e r i o d i c o s c i l l a t i o n a t a f r e q u e n c y e q u a l , t o t h a t o f t h e e x t e r n a l e x c i t a t i o n . C o m p a r i n g t h e t r a c e s o f F i g . 5.2.9 and t h e c u r v e s o f F i g . 5.2.8, some s l i g h t d i s c r e p e n c y i s a p p a r e n t w h i c h i s p r o b a b l y a s s o c i a t e d w i t h t h e d i f f e r e n c e b e t w e e n t h e two e x p r e s s i o n s f o r t h e f r i c t i o n f o r c e f u n c t i o n . (b) F u n d a m e n t a l R e s o n a n c e C a s e The a n a l y s i s i n t h e f o r e g o i n g s e c t i o n i s a p p l i c a b l e o n l y when t h e e x t e r n a l f r e q u e n c y i s n o t t o o c l o s e t o t h e r e s o n a n c e f r e q u e n c y . 119 The s o l i d c u r v e o f F i g . 5.2.10 was o b t a i n e d from eq. (3.4.24) o f t h e fundamental resonance t h e o r y . I t was noted t h a t the lower s e c t i o n s o f the c u r v e , a t f r e q u e n c y r a t i o s below 0.9 and above 1.1, were m e r e l y p a r t o f the c u r v e s o f the non-resonance case (Ref. Curve ( 2 ) , F i g . 5.2.8). Thus F i g . 5.2.10 i l l u s t r a t e s t h e t r a n s i t i o n from the non-resonance case t o fundamental resonance and back t o the non-resonance case a g a i n . The two dashed l i n e c u r v e s are the a m p l i t u d e s o f a u t o p e r i o d i c o s c i l l a t i o n o b t a i n e d from t h e non-resonance t h e o r y (Ref. Curve ( 1 ) , F i g . 5.2.8). The s o l u t i o n s o b t a i n e d by the n u m e r i c a l method u s i n g the e x p o n e n t i a l e x p r e s s i o n are a l s o shown i n the same diagram. S i x s e t s o f t h e s e computer p l o t t e d s o l u t i o n s as w e l l as s i x s e t s o f c o r r e s p o n d i n g e x p e r i m e n t a l t r a c e s are shown i n F i g . 5.2.11 and F i g . 5.2.12 r e s p e c -t i v e l y . I t s h o u l d be n o t e d t h a t w h i l e i n t h e o r y t h e ex- , t e r n a l e x c i t a t i o n f o r c e can be k e p t c o n s t a n t f o r v a r i o u s f r e q u e n c i e s , whereas i 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 t h i s was n o t always p o s s i b l e . The v a r i a b l e parameters p and e, namely th e o u t - o f - b a l a n c e mass and the e c c e n t r i c i t y , a re i n s t e p s i z e s r a t h e r t h a n c o n t i n u o u s v a r i a t i o n , hence some d i s c r e p e n c y between th 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 i s t o be e x p e c t e d . I t would be noted from F i g . 5.2.10 t h a t w i t h an e x t e r n a l e x c i t a t i o n f o r c e F q = 0.2, a l m o s t p e r i o d i c o s c i l l a t i o n o r beat o s c i l l a t i o n would o c c u r a t f r e q u e n - . c i e s b e l o w 0.87 and a b o v e 1.17. W i t h i n t h e r e g i o n o f f r e -q u e n c i e s o f 0.87 and 1.17, h a r m o n i c o s c i l l a t i o n s c o u l d e x i s t i n t h e s y s t e m . The n u m e r i c a l s o l u t i o n s o f F i g . 5.2.11 s u b s t a n t i a t e s t h e f i n d i n g s o f t h e a n a l y t i c a l m ethods i, A g a i n , a s l i g h t d i s c r e p e n c y e x i s t s due t o t h e two d i f f e r e n t e x p r e s s i o n s w h i c h w e r e u s e d i n t h e two m e t h o d s . F i g . 5.2.13 shows c u r v e s p l o t t e d w i t h m a g n i f i c a t i o n f a c t o r a g a i n s t f r e q u e n c y r a t i o . The e x p e r i m e n t a l r e s u l t s w e r e o b t a i n e d a t a d i s c v e l o c i t y o f 1.05 i n / s e c and a n o r m a l l o a d o f 5.4 l b . I t s h o u l d be n o t e d t h a t g o o d a g r e e -ment e x i s t s b e t w e e n t h e e x p e r i m e n t a l c u r v e and t h e c u r v e o b t a i n e d by t h e n u m e r i c a l m e thod u s i n g t h e e x p o n e n t i a l e x p r e s s i o n . R e a s o n a b l y g o o d a g r e e m e n t a l s o e x i s t e d b e t w e e n t h e e x p e r i m e n t a l c u r v e and t h e c u r v e o b t a i n e d f r o m t h e f u n d a m e n t a l r e s o n a n c e t h e o r y u s i n g t h e p o l y n o m i a l e x p r e s s i o n . The g r e a t e r amount o f d i s c r e p e n c y i s l i k e l y r e l a t e d t o t h e i n a c c u r a c y o f t h e p o l y n o m i a l c u r v e f i t t i n g . I n f a c t , F i g . 5.2.13 d e m o n s t r a t e s t h e i m p o r t a n c e o f an a c c u r a t e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c . B o t h t h e o r e t i c a l c u r v e s o f F i g . 5.2.13 show h i g h e r m a g n i f i c a t i o n v a l u e s t h a n t h e e x p e r i m e n t a l c u r v e , p a r t i c -u l a r l y n e a r a f r e q u e n c y r a t i o o f 1. The e x p e r i m e n t a l v i b r a t i o n a m p l i t u d e s l i g h t l y e x c e e d e d t h e l i m i t f o r q u a s i -h a r m o n i c o s c i l l a t i o n a r o u n d t h e r e s o n a n c e f r e q u e n c y r e g i o n , r e s u l t i n g w i t h s l i g h t r e s t r i c t i o n on t h e a m p l i t u d e o f v i b r a t i o n as i t s t r a j e c t o r y t e n d e d t o g r o w b e y o n d t h e 121 z e r o s l i d i n g v e l o c i t y a x i s i n t h e x-x phase p l a n e . On the o t h e r hand, i n the t h e o r e t i c a l a n a l y s i s a r e s t r i c t i o n o f t h i s k i n d was n o t i n c o r p o r a t e d . N e v e r t h e l e s s , i n b o t h t h e o r e t i c a l and 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 the e x t e r n a l e x c i t a t i o n parameters were chosen such t h a t the v i b r a t i o n a m p l i t u d e would be w i t h i n o r a t l e a s t n e a r t h e q u a s i -harmonic l i m i t . (c) Sub-harmonic E n t r a i n m e n t s When the f r e q u e n c y o f t h e e x t e r n a l e x c i t a t i o n f o r c e was near o r e q u a l t o a m u l t i p l e o f t h e f r e q u e n c y o f the autonomous system, subharmonic e n t r a i n m e n t was ob s e r v e d i n the e x p e r i m e n t s , d u r i n g w h i c h the system o s c i l l a t e d a t a f r e q u e n c y e q u a l o r c l o s e t o t h e f r e q u e n c y o f t h e autonomous system. The a m p l i t u d e o f v i b r a t i o n was h i g h e r t h a n t h e a m p l i t u d e o f v i b r a t i o n w i t h o u t the e x t e r n a l f o r c e . I n the non-resonance case the a m p l i t u d e o f v i b r a t i o n i s n o r m a l l y l o w e r than t h a t o f the s e l f - e x c i t e d v i b r a t i o n a l o n e . F i g u r e s 5.2.14 t o 5.2.17 i l l u s t r a t e the,wave forms 1 1 1 and x-x phase p l a n e s o f the 2 ' 3 ' a n d 4" har m o n i c s . On the l e f t hand s i d e o f each f i g u r e the e x p e r i m e n t a l t r a c e s from the o s c i l l o s c o p e and o s c i l l o g r a p h a re shown. The computer p l o t t e d t r a c e s on t h e r i g h t hand s i d e were o b t a i n e d by a p p l y i n g the e x p o n e n t i a l e q u a 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 c u r v e t o eq. (.3.4.2)- and s o l v i n g t h e e q u a t i o n n u m e r i c a l l y . A l l t h e e x p e r i m e n t a l r e s u l t s were o b t a i n e d a t a d i s c v e l o c i t y o f 1.05 i n / s e c and a normal l o a d o f 5.4 l b . The s c a l e s o f t h e computer p l o t t e d t r a c e s are d i m e n s i o n l e s s . The d i m e n s i o n l e s s parameter u)h i s e q u a l t o u n i t y and co i s 138 r a d . / s e c . The s c a l e s o f t h e e x p e r i m e n t a l t r a c e s are shown w i t h u n i t s o f 0.001 i n , i n / s e c , and second f o r d i s p l a c e m e n t , v e l o c i t y and time r e s p e c t i v e l y . The x-x phase p l a n e o f t h e j harmonic resembles a c a r d i a c p a t t e r n was shown i n F i g . 5.2.14 and F i g . 5.2.15. The e x t e r n a l e x c i t a t i o n f o r c e s were 0.245 and 0.895 ( d i m e n s i o n l e s s ) r e s p e c t i v e l y . E x t r e m e l y good s i m u l a t i o n s o f the e x p e r i m e n t a l t r a c e s were o b t a i n e d from the n u m e r i c a l method. I n c r e a s e i n a m p l i t u d e was n o t e d when the e x t e r n a l e x c i t a t i o n f o r c e was i n c r e a s e d . The f r e q u e n c y o f the o s c i l l a t i o n was e q u a l t o the f r e q u e n c y o f the autonomous system. F i g u r e s 5.2.16 and 5.2.17 i l l u s t r a t e two d i f f e r e n t x-x phase p l a n e p a t t e r n s f o r the ^ a n c ^ j harmonic c a s e s . The d i m e n s i o n l e s s e x t e r n a l f o r c e magni-tude was 1.35 f o r the harmonic case and 1.00 f o r t h e ^ harmonic c a s e . A g a i n good agreement between th e e x p e r i m e n t a l t r a c e s and the computer p l o t t e d t r a c e s was o b t a i n e d . F i g . 5.2.18 shows a summary o f the subharmonic cases and t h e non-resonance c a s e s . The c u r v e s were p r e -pared u s i n g the harmonic b a l a n c e method. The s e v e n t h o r d e r p o l y n o m i a l e x p r e s s i o n r e p r e s e n t i n g the f r i c t i o n - v e l o c i t y c u r v e o f Type A l was a p p l i e d t o t h e e q u a t i o n s d e r i v e d from the harmonic b a l a n c e method (Appendix I I ) . The c u r v e f o r the — harmonic case shows a r i s e i n a u t o p e r i o d i c v i b r a t i o n a m p l i t u d e as the magnitude o f t h e e x t e r n a l ex-c i t a t i o n f o r c e was i n c r e a s e d . The a m p l i t u d e o f t h e a u t o p e r i o d i c o s c i l l a t i o n s t a r t e d t o d e c l i n e as t h e e x t e r n a l f o r c e r eached a c e r t a i n v a l u e , i n f a c t , two r e a l r o o t s were o b t a i n e d f o r each e x t e r n a l f o r c e magnitude near the end o f the c u r v e , t h i s s u g g e sted t h a t the a u t o p e r i o d i c o s c i l l a t i o n became u n s t a b l e j u s t p r i o r t o i t s b e i n g com-, p l e t e l y quenched by the i n c r e a s i n g e x t e r n a l f o r c e . F u r t h e r i n c r e a s e o f t h e e x t e r n a l f o r c e magnitude r e s u l t e d i n harmonic o s c i l l a t i o n a t the f r e q u e n c y o f the e x t e r n a l e x c i t a t i o n . The a m p l i t u d e curve o f the h e t e r o p e r i o d i c term i s shown as d o t t e d l i n e s i n t h e same diagram. A comparison of the r e s u l t s from F i g u r e s 5.2.14 and 5.2.15 w i t h the t h e o r e t i c a l c u r v e o f F i g . 5.2.18 shows t h e y are i n good agreement. The c u r v e s f o r the -j and j harmonic are a l s o shown i n F i g . 5.2.18. The a m p l i t u d e s o f a u t o p e r i o d i c o s c i l l a t i o n v a r y s l i g h t l y w i t h t h e e x t e r n a l e x c i t a t i o n f o r c e and drops o f f r a p i d l y as t h e e x t e r n a l f o r c e exceeds a c e r t a i n l i m i t . A g a i n , t h e r e s u l t s from F i g u r e s 5.2.16 and 5.2.17 are i n ; r e a s o n a b l y good agreement w i t h the -j and j harmonic c u r v e s o f F i g . 5.2.18. I t s h o u l d be n o t e d from F i g . 5.2.18 t h a t the a m p l i t u d e o f o s c i l l a t i o n X c o mprises two terms o f -; d i f f e r e n t f r e q u e n c y , t h e r e f o r e a d i r e c t comparison between the 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 i s d i f f i c u l t . 124 A comparison o f the non-resonance c u r v e s o f F i g . 5.2.18 w i t h t h o s e o f the F i g . 5.2.8 shows t h e y are i n v e r y good agreement. S i m i l a r l y , eq. (3.4.34) o f the harmonic b a l a n c e method i s a l m o s t t h e same as t h a t o f eq. (3.4.23) o f the fundamental resonance c a s e . Thus i n t h e f i r s t a p p r o x i m a t i o n the K and B method f o r t h e non-resonance case and fundamental resonance case g i v e v e r y n e a r l y t h e same r e s u l t as the harmonic b a l a n c e method. The c u r v e o f F i g . 5.2.19 was o b t a i n e d by p l o t t i n g the a m p l i t u d e o f the e x t e r n a l e x c i t a t i o n f o r c e magnitude a t w h i c h a g o f eq. (3.4.37) became z e r o , a g a i n s t t h e f r e q u e n c y r a t i o a. O s c i l l a t i o n o c c u r r i n g i n t h e r e g i o n above the s o l i d l i n e would be o f a harmonic o s c i l l a t i o n a t the f r e q u e n c y o f t h e e x t e r n a l e x c i t a t i o n . Below the s o l i d l i n e , o s c i l l a t i o n would have e i t h e r a combined f r e q u e n c y o r e n t r a i n e d f r e q u e n c y . Thus th e c u r v e i l l u s -t r a t e s the l i m i t s o f the e x t e r n a l e x c i t a t i o n f o r c e w h i c h s e p a r a t e s t h e o s c i l l a t i o n from t h a t o f t h e resonance o r o f the combined f r e q u e n c y t y p e t o t h a t o f t h e p u r e l y harmonic t y p e . S t a b i l i t y i n v e s t i g a t i o n was not c a r r i e d o u t i n t h i s c a s e , t h e r e f o r e i t i s not p o s s i b l e t o d e f i n e e x a c t l y the subharmonic r e g i o n s . S i x s e t s o f computer p l o t t e d d i s p l a c e m e n t - t i m e and v e l o c i t y - t i m e t r a c e s f o r a = 4 u s i n g the n u m e r i c a l method are shown i n F i g . 5.2.20. The t r a c e s i l l u s t r a t e the g r a d u a l t r a n s f o r m a t i o n o f t h e subharmonic: o s c i l l a t i o n t o the harmonic o s c i l l a t i o n as t h e e x t e r n a l 125 e x c i t a t i o n f o r c e magnitude i s i n c r e a s e d . Thus, i t can be c o n c l u d e d from F i g . 5.2.19 t h a t subharmonic e n t r a i n m e n t o c c u r s w i t h i n a narrow range o f t h e e x t e r n a l f r e q u e n c y whereas t h e harmonic e n t r a i n m e n t o c c u r s a t any e x t e r n a l f o r c e f r e q u e n c y p r o v i d e d t h e e x t e r n a l f o r c e magnitude F Q i s s u f f i c i e n t l y l a r g e . No attempt was made t o p e r f o r m s i m i l a r sequence o f t e s t s f o r v a r i o u s d i s c v e l o c i t i e s . I t i s b e l i e v e d t h a t f o r d i s c v e l o c i t i e s w i t h i n t h e range o f q u a s i - h a r m o n i c o s c i l l a t i o n , t h e r e s u l t s would f o l l o w a s i m i l a r p a t t e r n e x c e p t the a m p l i t u d e o f o s c i l l a t i o n a t and around f r e q u e n -c i e s o f resonance o r subharmonic resonance would be l a r g e r f o r h i g h e r d i s c v e l o c i t i e s . F i g . 5.2.21 shows two s e t s o f e x p e r i m e n t a l r e s u l t s p l o t t e d and computer p l o t t e d x-x phase p l a n e s o b t a i n e d a t a d i s c v e l o c i t y o f 1.35 i n / s e c f o r f r e q u e n c y r a t i o s o f 3 and 4. A comparison o f F i g . 5.2.21 and F i g . 5.2.17 i l l u s t r a t e s t h a t the p a t t e r n s a r e t h e same bu t the a m p l i t u d e s are l a r g e r f o r t h e case a t a d i s c v e l o c i t y e q u a l t o 1.35 i n / s e c . (d) Subharmonic Resonance Case U s i n g a 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 F i g . 5.2.22 i l l u s t r a t e s two s e t s o f computer p l o t t e d d i s p l a c e m e n t - t i m e and v e l o c i t y - t i m e t r a c e s u s i n g a l i n e a r -i s e d f r i c t i o n - v e l o c i t y c u r v e . I t shows t h a t the o s c i l -l a t i o n would e i t h e r decay i n t o a harmonic o s c i l l a t i o n a t the f r e q u e n c y o f t h e e x t e r n a l e x c i t a t i o n o r grow i n a m p l i -126 tude u n t i l i t exceeds t h e l i m i t f o r q u a s i - h a r m o n i c o s c i l l a t i o n depending on whether the l i n e a r i s e d f r i c t i o n -v e l o c i t y c u r v e has a s l i g h t l y p o s i t i v e o r s l i g h t l y n e g a t i v e s l o p e . I n b o t h cases t h e o s c i l l a t i o n n e ver reached a s t e a d y s t a t e subharmonic e n t r a i n m e n t such as shown by F i g . 5.2.18. The i l l u s t r a t i o n f u r t h e r proves t h a t the humped t y p e f r i c t i o n - v e l o c i t y c u r v e o f F i g . 5.1.2 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 i s t e n c e o f q u a s i - h a r m o n i c o s c i l l a t i o n . 5.2.5 Summary When under t h e i n f l u e n c e o f e x t e r n a l t r a n s v e r s e e x c i t a t i o n , f r i c t i o n a l systems h a v i n g an i n c r e a s i n g f r i c t i o n -v e l o c i t y c h a r a c t e r i s t i c (Type C) behaved i n a manner s i m i l a r t o a l i n e a r v i b r a t o r y system w i t h heavy damping; whereas subharmonic e n t r a i n m e n t as w e l l as resonance and non-resonance o s c i l l a t i o n s were o b s e r v e d i n systems h a v i n g \ f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e s o f Type A o r Type B. In t h e case o f Type B f r i c t i o n - v e l o c i t y c u r v e , owing t o the c o n t i n u o u s d e c r e a s i n g c h a r a c t e r i s t i c , t h e r e was a tendency f o r the t r a j e c t o r y o f t h e x-x phase p l a n e t o o v e r s h o o t the z e r o s l i d i n g v e l o c i t y a x i s when the f r e q u e n c y r a t i o a s s o c i a t e d w i t h the e x t e r n a l e x c i t a t i o n was about u n i t y . Under t h e s e c i r c u m s t a n c e s t h e form o f the o s c i l l a t i o n was d i s t o r t e d . The e f f e c t o f e x t e r n a l t r a n s v e r s e e x c i t a t i o n on a f r i c t i o n a l system s u b j e c t t o q u a s i - h a r m o n i c t y p e o s c i l l a t i o n was demonstrated by u s i n g the #7 o i l as t h e l u b r i c a n t on a b l o t t i n g p a p e r - s t e e l c o m b i n a t i o n . The e x t i n g u i s h i n g o f the a u t o p e r i o d i c o s c i l l a t i o n by t h e e x t e r n a l e x c i t a t i o n f o r c e and the o c c u r r e n c e o f t h e harmonic and subharmonic e n t r a i n m e n t as p r e d i c t e d ' by t h e t h e o r e t i c a l a n a l y s i s were o b s e r v e d . The 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 are i n r e a s o n a b l y good agreement. 5.2.6 D y n a m i c a l l y Loaded System Two t y p e s o f i n v e s t i g a t i o n have been conducted. The f i r s t e m p loying the s t e e l - o n - s t e e l c o m b i n a t i o n as f r i c t i o n m a t e r i a l s e x e c u t e d s t i c k - s l i p t y p e f r i c t i o n - i n d u c e d v i b r a t i o n when the e x t e r n a l e x c i t a t i o n was a b s e n t . The second i n v e s t i g a t i o n was performed i n t h e presence o f q u a s i - h a r m o n i c t y p e f r i c t i o n - i n d u c e d v i b r a t i o n . (a) S t i c k - S l i p Type O s c i l l a t i o n The r e s u l t s o f F i g . 5.2.23 were o b t a i n e d from a s t e e l s l i d e r on s t e e l d i s c c o m b i n a t i o n w h i c h has a type C f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c urve and has the p r o p e n s i t y o f e x e c u t i n g s t i c k - s l i p t y p e o s c i l l a t i o n . The d i s c v e l o c i t y was 0.15 i n / s e c and the normal l o a d ( s t a t i c ) was 6.4 l b . The c u r v e o f F i g . 5.2.23 was p l o t t e d w i t h the a m p l i t u d e of the s t i c k - s l i p o s c i l l a t i o n v e r s u s 8, the r a t i o o f the e x t e r n a l e x c i t a t i o n f o r c e magnitude ( o s c i l - -l a t i n g l o a d ) t o the s t a t i c l o a d . The f r e q u e n c y o f t h e c o r r e s p o n d i n g e x t e r n a l e x c i t a t i o n f o r c e i s a l s o shown i n the same diagram. The v a l u e s o f 3 were t a k e n i n random o r d e r , t h e number i n s i d e the p o i n t s i n d i c a t e s t h e sequence the r e s u l t was t a k e n . T h i s has t h e s i g n i f i c a n c e o f showing t h e change i n a m p l i t u d e was not due t o the e f f e c t o f the number o f t r a v e r s e s o f t h e d i s c t r a c k . The r e s u l t s show t h a t t h e a m p l i t u d e o f the s t i c k - s l i p o s c i l l a t i o n d e c r e a s e s as the e x t e r n a l e x c i t a t i o n f o r c e magnitude i s i n c r e a s e d . As a c r i t i c a l v a l u e o f 3 i s re a c h e d , pure s l i d i n g e x i s t s i n t h e system. F i g . 5.2.24 shows f o u r o s c i l l o g r a p h t r a c e s i l l u s t r a t i n g f i r s t t he s t i c k - s l i p o s c i l l a t i o n i n the absence o f e x t e r n a l e x c i t a t i o n , n e x t t h e d i m i n i s h i n g s t i c k - s l i p o s c i l l a t i o n when the ex-t e r n a l e x c i t a t i o n i s p r e s e n t and f i n a l l y t h e pure s l i d i n g w i t h the t r a c e s showing h i g h f r e q u e n c y o s c i l l a t i o n due t o the e x t e r n a l e x c i t a t i o n . I t appears t h a t t h e f r e q u e n c y o f the e x t e r n a l e x c i t a t i o n ( w i t h i n t h e range 20-60 cps) has no s i g n i f i c a n t e f f e c t on the a m p l i t u d e o f t h e s t i c k - s l i p o s c i l l a t i o n , r a t h e r , the magnitude o f t h e e x t e r n a l e x c i t a t i o n i s the major f a c t o r . I t i s o f i n t e r e s t t o note t h a t t h e c r i t i c a l v a l u e o f 3 f o r the complete quenching o f the s t i c k - s l i p o s c i l l a t i o n i s o n l y 0.15, when t h e normal l o a d ( s t a t i c ) i s 6.4 l b . I t i s b e l i e v e d t h a t t h e c r i t i c a l 3 v a l u e v a r i e s w i t h t h e s t a t i c normal l o a d , a t h i g h e r s t a t i c normal l o a d s , the c r i t i c a l 3 v a l u e may become s m a l l e r . However, a more d e t a i l e d s t u d y o f t h e f r i c t i o n mechanism i s r e q u i r e d i n o r d e r t o r e v e a l the r e l a t i o n s h i p between t h e normal l o a d and t h e c r i t i c a l 3 v a l u e . 129 A c u r v e o f maximum s t a t i c f r i c t i o n f o r c e v e r s u s 8 i s a l s o shown i n F i g . 5.2.23. I t shows t h a t the maximum s t a t i c f r i c t i o n f o r c e d e c r e a s e s as B i s i n c r e a s e d u n t i l a pure s l i d i n g f r i c t i o n v a l u e i s r e a c h e d . I t i s b e l i e v e d t h a t t h e de c r e a s e i n maximum s t a t i c f r i c t i o n i s a s s o c i a t e d w i t h t h e breakdown o f j u n c t i o n s between the c o n t a c t i n g s u r f a c e s . (b) Quasi-Harmonic Type O s c i l l a t i o n The t e s t was c a r r i e d o u t u s i n g t h e #7 o i l as the l u b r i c a n t and t h e f r i c t i o n m a t e r i a l c o m b i n a t i o n was b l o t t i n g paper on s t e e l which has a f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e o f Type A l i n t h e autonomous c a s e . I t was found t h a t the r e s u l t s are v e r y s i m i l a r t o t h o s e o f t h e t r a n s -v e r s e e x t e r n a l e x c i t a t i o n c a s e , b e a t f r e q u e n c i e s as w e l l as subharmonic e n t r a i n m e n t were o b s e r v e d , a l t h o u g h the t h e o r e t i c a l r e s u l t s do n o t g i v e as good agreement t o the e x p e r i m e n t a l r e s u l t s as i n the t r a n s v e r s e v i b r a t i o n c a s e . The t h e o r e t i c a l r e s u l t s were o b t a i n e d by a p p l y i n g the e x t e r n a l e x c i t a t i o n p a r a m e t e r s , namely 8 and a, t o eq. (3,4.41) and s o l v i n g t h e e q u a t i o n by a n u m e r i c a l method. I n f a c t , eq. (3.4.41) i s v e r y s i m i l a r t o eq. (3.4.2) o f the t r a n s v e r s e e x t e r n a l e x c i t a t i o n c a s e , e x c e p t the non-* l i n e a r f u n c t i o n y G ( x ) n a s e x t r a p e r i o d i c c o e f f i c i e n t terms. I t i s apparent t h a t when B i s s m a l l , t h a t i s when the e f f e c t o f the p e r i o d i c c o e f f i c i e n t terms o f t h e n o n l i n e a r f u n c t i o n i s n o t s i g n i f i c a n t , eq. (3.4.41) would behave as 130 i n the non-autonomous case w i t h t r a n s v e r s e e x c i t a t i o n . However, when 3 becomes l a r g e , t h e e f f e c t o f t h e p e r i o d i c c o e f f i c i e n t terms becomes s i g n i f i c a n t , the system would not f o l l o w a s i m i l a r p a t t e r n as the t r a n s v e r s e c a s e . F i g . 5.2.25 shows s i x s e t s o f e x p e r i m e n t a l r e s u l t s o f the d i s p l a c e m e n t - t i m e and v e l o c i t y - t i m e t r a c e s i l l u s t r a t i n g t h e s i m i l a r b e h a v i o u r as i n t h e t r a n s v e r s e v i b r a t i o n c a s e . I t shows the combined a u t o p e r i o d i c and h e t e r o p e r i o d i c o s c i l l a t i o n w i t h b e a t f r e q u e n c y when a i s n o t c l o s e t o u n i t y o r m u l t i p l e s o f one; t h e resonance when a - 1 and t h e subharmonic e n t r a i n m e n t when a * 2, 3 e t c . The l a s t s e t o f t r a c e s ( F i g . 5.2.25f) i l l u s t r a t e s the pure harmonic o s c i l l a t i o n a t t h e f r e q u e n c y o f the e x t e r n a l e x c i t a t i o n when 3 i s s u f f i c i e n t l y l a r g e . The r e s u l t s were o b t a i n e d a t a d i s c v e l o c i t y o f 1.05 i n / s e c and w i t h a normal l o a d o f 6.4 l b . F i g . 5.2.26 shows a x-x phase p l a n e diagram and the d i s p l a c e m e n t - t i m e and v e l o c i t y - t i m e t r a c e s f o r a = 3 and 3 = 0.23. The diagrams on t h e l e f t a re e x p e r -i m e n t a l o s c i l l o s c o p e and o s c i l l o g r a p h t r a c e s and t h o s e on the r i g h t a r e computer p l o t t e d t r a c e s o b t a i n e d by a p p l y i n g t h e e x t e r n a l e x c i t a t i o n parameters t o eq. (3.4.41). I t i s b e l i e v e d t h a t w i t h h i g h (3 v a l u e s , t h e e x t e r n a l e x c i t a t i o n may have caused v a r i a t i o n i n the c o n t a c t c o n d i t i o n s between the s l i d i n g s u r f a c e s ; under t h e s e c o n d i t i o n s , the f r i c t i o n -v e l o c i t y c h a r a c t e r i s t i c may be q u i t e d i f f e r e n t from t h a t ; o b t a i n e d i n the autonomous case and eq. (3.4.41) w i l l n o t g i v e c o r r e c t p r e d i c t i o n s . (c) Summary The r e s u l t s o b t a i n e d from a system h a v i n g normal e x t e r n a l e x c i t a t i o n show t h a t w i t h a s u f f i c i e n t l y h i g h e x t e r n a l f o r c e magnitude, b o t h the s t i c k - s l i p t y p e and the q u a s i - h a r m o n i c t y p e f r i c t i o n - i n d u c e d v i b r a t i o n may be e x t i n g u i s h e d . The normal e x t e r n a l e x c i t a t i o n reduces the maximum s t a t i c f r i c t i o n i n t h e s t i c k - s l i p t y p e v i b r a t i o n c a s e . However, r e d u c t i o n i n s l i d i n g f r i c t i o n o r average d i s p l a c e m e n t d u r i n g q u a s i - h a r m o n i c o s c i l -l a t i o n was not o b s e r v e d . VI CONCLUSION I n o v e r a l l summary o f t h e r e s e a r c h , the f o l l o w i n g c o n c l u s i o n s may be l i s t e d : 1. A r e l i a b l e a p p a r e t u s f r e e from unwanted e x t e r n a l v i b r a t i o n and n o i s e was de v e l o p e d f o r the i n v e s t i g a t i o n o f q u a s i - h a r m o n i c t y p e f r i c t i o n - i n d u c e d v i b r a t i o n . The ap p a r a t u s t o g e t h e r w i t h t h e i n s t r u m e n t a t i o n t e c h n i q u e s d e v e l o p e d p e r m i t t e d e x p e r i m e n t a l r e s u l t s o f r e a s o n a b l e a c c u r a c y t o be o b t a i n e d . 2 . The advantage of t h e one c y c l e method ( A c c e l e r a t i o n -v e l o c i t y - d i s p l a c e m e n t ) i n d e t e r m i n i n g t h e f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e i s c l e a r l y demonstrated by t h e f a c t t h a t a s i n g l e c y c l e o f o s c i l l a t i o n a c r o s s t h e s u r f a c e c o m p l e t e l y d e f i n e s t h e e n t i r e dynamic c h a r a c t e r i s t i c c u r v e . F u r t h e r m o r e , the presence o f the hump i n the low v e l o c i t y r e g i o n c o u l d n o t be v e r i f i e d u n t i l t he one c y c l e method was used. 3. The r e s u l t s from the i n v e s t i g a t i o n o f the autonomous case demonstrated t h a t a humped shape f r i c t i o n - v e l o c i t y • c u r v e i s a c o n d i t i o n n e c e s s a r y f o r t h e e x i s t e n c e o f q u a s i -harmonic o s c i l l a t i o n p r o v i d i n g t h a t t h e n e g a t i v e s l o p e d s e c t i o n o f t h e c u r v e i s n o t t o o s t e e p . The i n v e s t i g a t i o n a l s o c l e a r l y d e f i n e s t h e d i s t i n c t i o n between the q u a s i -harmonic and t h e s t i c k - s l i p forms o f f r i c t i o n - i n d u c e d v i b r a t i o n . 133 4. I n the humped f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c , s t i c k -s l i p o s c i l l a t i o n may o c c u r i n t h e low v e l o c i t y r e g i o n . I n g e n e r a l , q u a s i - h a r m o n i c o s c i l l a t i o n would s t a r t a t a d i s c v e l o c i t y near the peak o f t h e hump. The a m p l i t u d e o f the q u a s i - h a r m o n i c o s c i l l a t i o n i n c r e a s e s as t h e d r i v i n g v e l o c i t y i s i n c r e a s e d . A t a d r i v i n g v e l o c i t y near t h e p o i n t o f i n f l e c t i o n from n e g a t i v e s l o p e t o p o s i t i v e s l o p e o f the f r i c t i o n - v e l o c i t y c urve the v i b r a t i o n becomes u n s t a b l e and s e l f - e x c i t e d v i b r a t i o n cannot s t a r t from r e s t . 5. Other t y p e s o f f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c g i v e r i s e t o s t i c k - s l i p t y pe f r i c t i o n - i n d u c e d v i b r a t i o n o r s t a b l e d i s p l a c e m e n t (pure s l i d i n g ) , depending on whether the c u r v e has a d e c r e a s i n g c h a r a c t e r i s t i c o r an i n c r e a s i n g c h a r a c t e r -i s t i c . I n the low v e l o c i t y r e g i o n the a m p l i t u d e o f the s t i c k - s l i p o s c i l l a t i o n i s m a i n l y governed by t h e s t a t i c f r i c t i o n c h a r a c t e r i s t i c . However, i n the h i g h e r v e l o c i t y r e g i o n , th e shape o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c c u r v e becomes i m p o r t a n t . 6. The a p p l i c a t i o n o f t h e e x t e r n a l v i s c o u s damping d i m i n i s h e s o r e x t i n g u i s h e s t h e f r i c t i o n - i n d u c e d v i b r a t i o n . I n t h e low v e l o c i t y r e g i o n when the o s c i l l a t i o n i s m a i n l y under the i n f l u e n c e o f the s t a t i c f r i c t i o n c h a r a c t e r i s t i c , t h e e f f e c t o f the e x t e r n a l damping i s l e s s s i g n i f i c a n t . The amount of e x t e r n a l v i s c o u s damping r e q u i r e d f o r e x t i n g u i s h i n g the q u a s i - h a r m o n i c type f r i c t i o n - i n d u c e d v i b r a t i o n i s r e l a t e d t o t he c o e f f i c i e n t s o f the f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c e q u a t i o n and can be de t e r m i n e d from t h e t h e o r y . 7. The dynamic f r i c t i o n c h a r a c t e r i s t i c c u r v e s were th. r e p r e s e n t e d by n o r d e r p o l y n o m i a l s and by e x p o n e n t i a l e x p r e s s i o n s . These e x p r e s s i o n s were a p p l i e d t o t h e f i r s t a p p r o x i m a t i o n method o f K r y l o v and B o g o l i u b o f f . The t h e o r -e t i c a l p r e d i c t i o n s were i n good agreement w i t h t h e e x p e r i -m ental r e s u l t s . 8. T r a n s v e r s e v i b r a t i o n s o b t a i n e d by a p p l y i n g r o t a t i n g o u t - o f - b a l a n c e masses were a p p l i e d t o t h e f r i c t i o n a l system i n the d i r e c t i o n o f the f r i c t i o n f o r c e . The harmonic b a l a n c e method as w e l l as t h e f i r s t a p p r o x i m a t i o n methods o f K r y l o v and B o g o l i u b o f f were used i n t h e t h e o r e t i c a l a n a l y s i s . The e x p e r i m e n t a l r e s u l t s and the t h e o r e t i c a l p r e d i c t i o n f o r a p a i r o f f r i c t i o n m a t e r i a l s h a v i n g a type A f r i c t i o n - v e l o c i t y c h a r a c t e r i s t i c curve i n d i c a t e d t h a t subharmonic e n t r a i n m e n t c o u l d o c c u r , and when t h e e x t e r n a l e x c i t a t i o n magnitude was s u f f i c i e n t l y h i g h , harmonic e n t r a i n m e n t a t the e x t e r n a l e x c i t a t i o n f r e q u e n c y e x i s t e d i n t he system. T h i s f i n d i n g s u g g e s t s t h a t h i g h f r e q u e n c y e x t e r n a l e x c i t a t i o n w i t h a s m a l l v i b r a t i o n a m p l i t u d e , may be used as a means o f e x t i n g u i s h i n g the unwanted q u a s i -harmonic type f r i c t i o n - i n d u c e d v i b r a t i o n . 9. By a p p l y i n g the e x t e r n a l e x c i t a t i o n v e r t i c a l l y a t the l o a d i n g end, the e f f e c t o f dynamic l o a d i n g was s i m u l a t e d . 135 The e x p e r i m e n t a l r e s u l t s showed t h a t s t i c k - s l i p t y p e o s c i l l a t i o n may be e x t i n g u i s h e d w i t h a dynamic t o s t a t i c l o a d r a t i o o f 0.15. The normal e x t e r n a l e x c i t a t i o n has t h e e f f e c t o f r e d u c i n g the maximum s t a t i c f r i c t i o n o f the s t i c k - s l i p t y p e v i b r a t i o n . I n the q u a s i - h a r m o n i c c a s e , the n o n l i n e a r f u n c t i o n has v a r i a b l e c o e f f i c i e n t s . R e d u c t i o n i n s l i d i n g f r i c t i o n o r average d i s p l a c e m e n t d u r i n g q u a s i - h a r m o n i c o s c i l l a t i o n was n o t o b s e r v e d . How-e v e r , q u a s i - h a r m o n i c o s c i l l a t i o n may s t i l l be e x t i n g u i s h e d w i t h a s u f f i c i e n t l y h i g h e x t e r n a l f o r c e magnitude. The s i m u l a t e d t h e o r e t i c a l r e s u l t s d i d not g i v e a c c u r a t e p r e -d i c t i o n s i n t h i s case s u g g e s t i n g t h a t t h e f r i c t i o n mechan-isms between the s l i d i n g s u r f a c e s may have been changed d u r i n g dynamic l o a d i n g . REFERENCES 1. Bowden, F.P. and Tabor, D. "The F r i c t i o n and L u b r i -c a t i o n o f S o l i d s " , O x f o r d , V o l . I I , 1965. 2. Johannes, V . I . " F r i c t i o n - I n d u c e d V i b r a t i o n s i n a H y d r a u l i c a l l y D r i v e n System", Ph.D. T h e s i s , Dept. o f Mech. Eng., U.B.C., 1969. 3. Coulomb, C A . "Memoir'es de Mathematique e t de P h y s i q u e de l'Academie", Royale des S c i e n c e s , p. 161, 1785. 4. W e l l s , J.H. " K i n e t i c Boundary F r i c t i o n " , The E n g i n e e r (London), V o l . 147, p. 454, 1929. 5. Thomas, S. " V i b r a t i o n s Damped by S o l i d F r i c t i o n " , The P h i l o s o p h i c a l Magazine, S e r i e s 7, V o l . 9, p.329, 1930 . 6. K a i d a n o v s k y , N.L. and H a y k i n , S.E. Z e i t s . f . t e c h . P h y s i k 3, p. 91, 1933. 7. Papenhuyzen, P . J . " W r i j v i n g s Proeven i n Verbandmet h e t S l i p p e n Van Autobanden", De I n g e n i e u r , 53, p. 75, 1938. 8. Bowden, F.P. and Leben, L. "The Nature o f S l i d i n g and The A n a l y s i s o f F r i c t i o n " , P r o c e e d i n g o f the R o y a l S o c i e t y , A 169, p. 371, 1939. 9. B l o k , H. "Fundamental A s p e c t s o f Boundary L u b r i c a t i o n " , J o u r n a l o f Soc. o f Automotive E n g i n e e r s , V o l . 46, p. 54, 1940. 10. Morgan, F., Muskat, M. and Reed, D.W. " F r i c t i o n Phen-omena and the S t i c k - S l i p P r o c e s s " , J o u r n a l o f A p p l i e d P h y s i c s , V o l . 12, p. 743, 1941. 11. Sampson, J.B., Morgan, F., Muskat, M. and Reed, D.W. . " F r i c t i o n B e h a v i o u r d u r i n g the S l i p P o r t i o n o f the S t i c k - S l i p P r o c e s s " , J o u r n a l o f A p p l i e d P h y s i c s , V o l . 14, p. 689, 1943. 12. B r i s t o w , J.R. " K i n e t i c Boundary F r i c t i o n " , P r o c e e d i n g o f the R o y a l S o c i e t y , V o l . 189, p. 88, 1945. 1 3 7 1 3 . B r i s t o w , J.R. "The Measurement o f K i n e t i c Boundary-F r i c t i o n o r 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 o f ' O i l i n e s s ' " P r o c e e d i n g s o f the I n s t i t u t i o n o f M e c h a n i c a l E n g i n e e r s , V o l . 1 6 0 , p. 2 5 9 , 1 9 4 9 . 1 4 . Dudley, B.R. and S w i f t , H.W. " 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 " , The P h i l o s o p h i c a l Magazine, S e r i e s 7 , V o l . 4 0 , p. 8 4 9 , 1 9 4 9 . 1 5 . B r o c k l e y , C.A., Cameron, R. and P o t t e r , A.F. " F r i c t i o n Induced V i b r a t i o n " , J o u r n a l o f L u b r i c a t i o n T e chnology, S e r i e s F. V o l . 9 0 , p. 3 5 , 1 9 6 8 . 1 6 . S i n c l a i r , D. " F r i c t i o n a l V i b r a t i o n " , J o u r n a l o f A p p l i e d M e c h a n i c s , p. 2 0 7 , June, 1 9 5 5 . 1 7 . R a b i n o w i c z , E. "A Study o f S t i c k - S l i p P r o c e s s " , P r o -c e e d i n g s of The Symposium on F r i c t i o n and Wear, D e t r o i t , 1 9 5 7 . 1 8 . J a r v i s , R.P. and M i l l s , B. " V i b r a t i o n s Induced by Dry F r i c t i o n " , P r o c e e d i n g s o f the I n s t i t u t i o n o f M e c h a n i c a l E n g i n e e r s , V o l . 1 7 8 , P a r t 1 , No. 3 2 , 1 9 6 3 . 1 9 . D e r j a g u i n , B.V., Push, V.E. and T o l s t o i , D.M. "A Theory o f S t i c k - S l i p S l i d i n g s o f S o l i d s " , P r o c e e d i n g s o f The Conference on L u b r i c a t i o n and Wear, p. 2 5 7 , O c t . , 1 9 5 7 . 2 0 . S i n g h , B.R. " S e n s i t i v i t y o f Slow S h i f t i n g Under S t i c k -S l i p C o n d i t i o n s " , The E n g i n e e r , p. 1 8 7 , J u l y , 1 9 6 0 . -•• 2 1 . Cook, N.H. "A Study o f Dynamic F r i c t i o n " , A.S.M.E. paper 5 8 - A - 2 5 7 , 1 9 5 8 . 2 2 . A t s u s h i , W a t a r i and Takanao, Sugimoto. " V i b r a t i o n s Caused by Dry F r i c t i o n " , B u l l e t i n o f Japan S o c i e t y o f M e c h a n i c a l E n g i n e e r s , V o l . 7 , No. 2 5 , . p. 4 0 , 1 9 6 4 . 2 3 . S h i z u o D o i and Shinobu K a t o . "On C h a t t e r V i b r a t i o n due t o F l e x i b l e L a t h e T o o l " , T r a n s , o f Japan S o c i e t y o f M e c h a n i c a l E n g i n e e r s , V o l , 1 9 , No. 8 6 , p. 2 8 , 1 9 5 3 . 2 4 . Stepanek, K. " S t a b i l i t y o f S l i d i n g M o t i o n " , Czech. Heavy I n d . , No. 3 , p. 3 8 , 1 9 5 7 . 2 5 . Hunt, J.B., Torbe, I . and Spencer, G.C. "The Phase-P l a n e A n a l y s i s o f S l i d i n g M o t i o n " , Wear, V o l . 8 , p. 4 5 5 , 1 9 6 5 . 2 6 . B e l l , R. and B u r d e k i n , M. "Dynamic B e h a v i o u r o f P l a i n S l i d e w a y s " , P r o c e e d i n g s o f t h e I n s t i t u t i o n o f M e c h a n i c a l E n g i n e e r s , V o l . 1 8 1 , P a r t 1 , No. 8 , p. 1 6 9 , 1 9 6 6 . 138 27. R a b i n o w i c z , E. "The Nature o f t h e 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 o f F r i c t i o n " , J o u r n a l o f A p p l i e d P h y s i c s , V o l . 22, p. 1373, 1951. 28. R a b i n o w i c z , E. "The I n t r i n s i c V a r i a b l e s A f f e c t i n g the S t i c k - S l i p P r o c e s s " , P r o c e e d i n g o f the P h y s i c s S o c i e t y , London, V o l . 71, p. 668, 1958. 29. V i n o g r a d o v , G.V., Korepova, I.V. and P o d o l s k y , Yu, Ya. " S t e e l - t o - S t e e l F r i c t i o n Over a V e r y Wide Range o f S l i d i n g Speeds", Wear, V o l . 10, p. 338, 1967. 30. S i m k i n s , T.E. "The M u t u a l i t y o f S t a t i c and K i n e t i c F r i c t i o n " , L u b r i c a t i o n E n g i n e e r i n g , p. 26, J a n u a r y , 1967. 31. Bowden, F.P. and Tabor, D. " F r i c t i o n , L u b r i c a t i o n and Wear: A Survey o f Work D u r i n g the L a s t Decade", J o u r n a l o f A p p l i e d P h y s i c s , V o l . 17, p. 1521, 1966. 32. Houch, F. " V i b r a t i o n " , M e c h a n i c a l E n g i n e e r i n g , p. 48, Sept . , 1966. 33. G o d f r e y , D. " V i b r a t i o n Reduces M e t a l t o M e t a l C o n t a c t and Causes an Apparent R e d u c t i o n i n F r i c t i o n " , T r a n s . American S o c i e t y o f L u b r i c a t i o n E n g i n e e r s , V o l . 10, No. 2, p. 183, 1967. 34. G a y l o r d , E.W. and Shu, H. " C o e f f i c i e n t s o f S t a t i c F r i c t i o n Under S t a t i c a l l y and D y n a m i c a l l y A p p l i e d Loads", Wear, V o l . 4, p. 401, 1961. 35. Fridman, H.D. and Levesque, P. " R e d u c t i o n o f S t a t i c F r i c t i o n By S o n i c V i b r a t i o n s " , J o u r n a l o f A p p l i e d P h y s i c s , V o l . 30, No. 10, p. 1572, 1959. 36. Wheeler, F, " V i b r a t i o n Eases the M e t a l w o r k e r ' s Load", New S c i e n t i s t , V o l . 7, p. 302, Nov., 1968. 37. L e m p r i e r e , B.M. " O s c i l l a t i o n s i n T e n s i l e T e s t i n g " , I n t . J . Mech. S c i . , V o l . 4, p. 171, 1962. 38. C l a u s e r , F.H. "The B e h a v i o u r o f N o n l i n e a r Systems", J o u r n a l o f the A e r o n a u t i c a l S c i e n c e s , V o l . 23, p. 411, 1956. 39. Tou, J . and S c h u l t h e i s s , P.M. " S t a t i c and S l i d i n g F r i c t i o n i n Feedback Systems", J o u r n a l o f A p p l i e d P h y s i c s , V o l . 24, No. 9, p. 1210, 1953. 139 4 0 . M i n o r s k y , N. " N o n l i n e a r O s c i l l a t i o n s " , D. Van N o r t r a n d Company, I n c . , pp. 7 4 - 7 7 , 1 9 6 2 . 4 1 . M i n o r s k y , N. " N o n l i n e a r M e c h a n i c s " , Edwards B r o t h e r s , I n c . , p. 1 6 7 , 1 9 4 7 . 4 2 . I b i d . , 4 1 , p. 1 8 6 . 4 3 . I b i d . , 4 1 , p. 2 9 5 . 44 . I b i d . , 4 1 , p. 2 9 8 . 4 5 . K r a g e l s k i i , I.V. " F r i c t i o n and Wear", B u t t e r w o r t h s and Company, L i m i t e d , pp. 1 8 2 - 1 8 3 , 1 9 6 5 . 4 6 . G r o s c h , K.A. "The R e l a t i o n Between the F r i c t i o n and V i s c o - e l a s t i c P r o p e r t i e s o f Rubber", P r o c e e d i n g s o f the R o y a l S o c i e t y , A, V o l . 2 7 4 , p. 2 1 , 1 9 6 3 . 4 7 . Ludema, K.C. and Tabor, D. "The F r i c t i o n and V i s c o -E l a s t i c P r o p e r t i e s o f P o l y m e r i c S o l i d s " , Wear, V o l . 9, pp. 3 2 9 - 3 4 8 , 1 9 6 6 . 4 8 . J e f f e r i s , J.A. and Johnson, K.L. " S l i d i n g F r i c t i o n Between L u b r i c a t e d R o l l e r s " , P r o c e e d i n g s I n s t i t u t i o n o f M e c h a n i c a l E n g i n e e r s , London, V o l . 1 8 2 , P a r t 1 , No. 1 4 , p. 2 8 1 , 1 9 6 8 . 4 9 . Muskat, M. and Morgan, F. "Temperature B e h a v i o u r o f J o u r n a l B e a r i n g Systems", J o u r n a l o f A p p l i e d P h y s i c s , V o l . 1 4 , p. 2 3 4 , 1 9 4 3 . 5 0 . Hagg, A.E. "Heat E f f e c t s i n L u b r i c a t i n g F i l m s " , T r a n s . A.S.M.E., 6 6 , A, p. 7 2 , 1 9 4 2 . 5 1 . Rodgers, J . J . and G a l l o p o u l o s , N.E. " F r i c t i o n C h a r a c t e r -i s t i c s o f Some A u t o m a t i c T r a n s m i s s i o n F l u i d Components", A.S.L.E. paper No. 6 6 , L C - 1 1 , 1 9 6 6 . 5 2 . D o y l e , W.P., Henry, C.J. and Thomas, P.R. "A Fundamental Study o f Antisquawk A g e n t s " , SAE paper 7 7 4 B , The SAE , J o u r n a l , p. 9 4 , D e c , 1 9 6 3 . 5 3 . Nann, N.A. and P i n c h b e c k , F.H. " T a i l o r i n g A u t o m a t i c T r a n s m i s s i o n F l u i d S h i f t Q u a l i t y i n the L a b o r a t o r y " , SAE paper No. 6 5 0 4 6 6 , The SAE J o u r n a l , p. 1 6 2 , J u l y , 1 9 6 5 . 5 4 . I b i d . , 4 5 , pp. 1 7 8 - 1 8 4 . 5 5 . L i e n a r d , A. "Etude des O s c i l l a t i o n s E n t r e t e n u e s " , Rev. Gen. d. E l e c t . , V o l . 2 3 , p. 9 0 1 , 1 9 2 8 . 140 56 , 57 , 58, 59 , 60 , 61, 62, 63 , 64, 65, 66 , 67, 68, 69 . 70, 71, 72, 73, 74 , 75, 76 , 77, 78, K r y l o f f , N. and B o g o l i u b o f f , N. " I n t r o d u c t i o n t o N o n l i n e a r M e c h a n i c s " , P r i n c e t o n U n i v e r s i t y P r e s s , P r i n c e t o n , N.J., pp. 8-12, 1943. I b i d . , 56, p. 12. I b i d . , 56, p. 11. I b i d . , 56, pp. 25-26. I b i d . , 40, p. 329. M a c l a c h l a n , N.W. " O r d i n a r y 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 s " , O x f o r d U n i v e r s i t y P r e s s , 2 n E d i t i o n , pp. 180-221, 1955. Thomson^W.T. " M e c h a n i c a l V i b r a t i o n " , P r e n t i c e - H a l l , I n c . , 2 E d i t i o n , p. 65, 1953. I b i d . , 41 I b i d . , 41 I b i d . , 40 I b i d . , 40 I b i d . , 41 I b i d . , 40 I b i d . , 41 I b i d . , 40 p. 293. p. 289. p. 360. p. 329. p. 292. p. 368. pp. 313-319 p. 377. C h i h i r o H a y a s h i , " N o n l i n e a r O s c i l l a t i o n s i n P h y s i c a l Systems", M c G r a w - H i l l Book Company, I n c . , p. 309, 1964. S t o k e r , J . J . " N o n l i n e a r V i b r a t i o n " , p. 149. I b i d . , 72, p. 153. I b i d . , 71, p. 309. I b i d . , 71, p. 286. Cunningham, W.J. " N o n l i n e a r A n a l y s i s " , M c G r a w - H i l l Book Company, I n c . , p. 214, 195 8. I b i d . , 40, p. 390. I b i d . , 76, p. 253. 141 79. Schmidt, A.O. and W e i t e r , E . J . " C o e f f i c i e n t o f F l a t -S u r f a c e F r i c t i o n " , M e c h a n i c a l E n g i n e e r i n g , V o l . 79, pp. 1130-1136, D e c , 1957. 80. B a r t e n e v , G.M. and E l ' k i n , A . I . " F r i c t i o n o f High E l a s t i c M a t e r i a l s " , Wear, V o l . 8, No. 1, pp. 8-21, 1965. 81. G r a d s h t e y n , I.S. and R y z h i k , I.M. "Table o f I n t e g r a l s , S e r i e s and P r o d u c t s " , Academic P r e s s , N.Y., p. 488. 82. M c L a c h l a n , N.W. "Theory o f V i b r a t i o n s " , Dover P u b l i c a t i o n I n c . , p. 12, 1951. APPENDIX I DERIVATION OF STEADY STATE AMPLITUDE AND PHASE EQUATIONS FROM THE EXPONENTIAL EXPRESSION S u b s t i t u t i o n o f e q . ( 3 . 3 . 1 0 ) i n t o e q . ( 3 . 3 . 7 ) a n d e q . ( 3 „ 3 . 8 ) y i e l d s D 1 f 2 , , , T. f C * a c o s ^ . . . _ ( C.acost j j 2, , a • -Typ- I a c o s ijjdijj - D~ I e 5 cos^di j j + D I ae 3 r c o s ip&ty ( A . 1 ) D f 2 7 r r 2 7 r i f - 1 + J o f s i n Z ^ d ^ - D 2 J o e C 3 a c o s ^ s i n ^ d ^ + D , l ae 5 8 ^ c o s ^ s i n ^ d i j ; ( A . 2 ) "0 F rom t h e T a b l e o f I n t e g r a l s , S e r i e s and P r o d u c t s [ 8 1 ] , we h a v e ( 2 r e ( p c o s x + q 8 i n x ) B i n ( a c o 8 x + b B . n x _ m x ) d x J O - i u [ ( b - p ) 2 + ( a + q ) 2 ] " 2 [ ( A + i B ) 2 I m ( C - i D ) * - ( A - i B ) 7 i J C + i D ) * ] f 2 7 r e ( p c o 8 X + q s i n x ) c o s ( a c o 8 x + b s i n x _ m x ) d x J O - T I [ ( b - p ) 2 + ( a + q ) 2 ] [ ( A + i B ) 7 i ^ C - i D ) * + ( A - i B ) ? I ^ C + i D ) * ] w h e r e ( b - p ) 2 + ( a + q ) 2 > 0 , m - 0 , 1 , 2 , 143 2 2 2 2 and A » p -q +a -b , B « 2(pq+ab) , C - p 2+q 2-a 2-b 2 , D - -2(ap+bq). Application of the in t e g r a l s to eq.(A.l) and noting that I 2 ( C 3 a ) = I D ( C 3 a ) - 2 1 / 0 ^ ) 7 0 ^ y i e l d s a - - 2 l [ a + 2 a D 3 Io ( c 3 a ) " 2 ( V c y W * ] (A-5) S i m i l a r l y , we have from eq.(A .2) ij - 1 (A.U) D i f f e r e n t i a t i n g eq.(A.j) with respect to a gives f t - - | s ^ 2 D 3 c 5 a I o ( C 3 a ) + 2 V o ( C 3 f t ) " 2 C 3 ( V ^ I i ( C 3 a ) ] Noting I l j ( C 5 a ) - ^(C^a) - T ^ ^ a ) and I ^ C ^ a ) - I ^ C ^ ) we f i n a l l y have * a O ) = - f|[1 - 2 C 3 D 2 I o ( C 3 a ) + ^  ^ ( C ^ ) 2 + + D ? I ^ C y ) ] (A.5) APPENDIX I I HARMONIC BALANCE METHOD A solution of eq .(3.U .38) can be sought i n the form X > asin T + a g s i n a x where a and a are considered to be constants, e For a seventh order polynomial YG(X) - | 1 ( X - D 2X 2 + + D 7X 7) "2 Expansion of the X, X etc, and wri t i n g A <= a a y i e l d s 6 « X = acosT + Acosax *2 1 2 2 1 2 1 2 X = — (a +A ) + -^ a cos2 T + —A cos2 ax + aA [cos( 1+a)x+cos( 1-a) t ] X^ m -^(a '+2aA 2)cosx + -^a 'cos3x + -^A 'cos3ax + 2a 2A+A^)cosax + -^a2A [cos(2+a) x+cos (2-a) x] +-^aA2 [cos( 1+2a )T +cos(l - 2 a)x] Xk » -|(aU+4a2A2+A^) + •|(a \ - 3 a 2A 2 ) c o s 2 x+ •£( 3 » 2 A 2 + A 4 ) c o s 2 ax + ^-a^cos^ x + -^A^cos a T + •^(a^A+aA^) [ c o s ( 1+a) x + c o B ( l - a ) x] 1 5 1 3 " + —a^A [cos(3+a) x +cos(3-a) x] + ^ aA^ [cos( 1 + 3 a ) T +cos( 1 - 3 a ) x j + 4 a 2 A 2 t c o s 2 ( l + a ) x + c o s 2 ( l - a ) x ] 145 •g-(a +6a^ A +3aA )cos T + -jg-(a +4a^A )cos3 x + yg"a cos5 x 5/ 5 2 3 4 \ 5 / 5 2 5\ 1 5 + •g'(A^ +6a A +^3a A)cos ax + -^(A +4a A^)cos3 ai + -ygAycos5ax + g(2a A+3a A ) [cos(2+a)T +cos(2-a)x J + -yg-a A [cos( 4+a)x+cos( 4-a ) T ] + "ilaA^ fcos(4a+l) x +cos(4a-1 )x] + -|( 3a3A2+2aA^) jcos( 1+2a) x+ cos(l -2a) T] + -|a3A2 [cos(3+2a) x +cos(3-2a) xj + fcos(2+3a) x +cos ( 2 - 3 a H ] •^(9a^+a^A2+a2A^+9A^) + a^+8a^A2+6a2A**) cos2 x + "j^a^cos6 x + * ^ ( a +5a A 2)cos4 T + •l|(A 6+8a 2A +6aV)cos2ax + TJ|(A +5a 2A 4) cos4ax + ~ A ^ c o s 6 ax+ -^(a^A+3a3A3+aA^) [cos( 1*t )x +cos(l-a)x] + ~^(a^A+2a 3A 3) [cos(3+a)x +cos(3-a)x] + rj^a^A {cos( 5+a) x +cos(5-a)x] + j^aA^ [cos(il + 5a)x +cos(l-5a) x] + -^-(aA^+2a3A3) [coB(l+3a)x + COS (1 -3«)T] + -i|(a\ 2+a 2A^) [COS2(14U)X +COB2 (1-O;)XJ + -||aUA2 Icos2(2+a) x+cos2(2-a) xj + "^a A jcos2( 1+2 a) x +cos2( 1-2a) x J + ^ a3A 3Icos3(l+a)x +cos3(l-a)xJ ||(a 7+12a 5A 2+18a 5A 4+4aA 6)cosx + |£(a 7+10a 5A 2 +10aV*)cos3 T 7 7 5 2 1 7 1 7 7 7 2 5 + T-f(a +6a A )cos5x + -rra cos7x + T T A COS7 ax + T H A +6a A ?)cos5ax + |^(A 7+10a 2A 5+10a UA 3)cos3ax + ||(A 7+12a 2A 5+18a UA 5+4a 6A)cosax 146 + ~|(a 6A +i4aV+2a 2A 5) [cos(2+a)T +cos(2-a)xJ + aA6+*aV + 5 2 21 6 2a^ A ) [C O S(1+2X ) T + C O S ( 1 - 2 O , ) T J + -j^a. A[cos(4+ a) x +cos(4-a) x] 216 7 6 + ^ a A [cos(l + 4<x)x + c o s ( l - 4 A ) T ] + "g^a A[cos(6+ a) x +cos(6- a) x] + -g|uA6 [cos( 1+6a)x +co8(l-6a)x] + ^ ( 3a^A2+4a^A^) [cos( 3+2a) T + cos(3-2a )T ] + ||(3a 2A 5+Ua\ 5) [cos(2+3a)T +cos(2-3a)x] 2 1 5 2 212 5 + -gj^rA [cos(5+2a)x +co8(5-2a)x] + -g^a A 3 [cos(2+5a)x +cos(2-5a) TJ + -f^aV3 [cos(4+3a) x +cos(4-3a) x] + J^a 5A^ [cos( 3+4a) x+cos( 3-4a) T ] + -^r^a^A 3 [cos(4+a)x +cos(4-a)x] + -^^a^A^ [cos(l+U a) x +cos(l-4a) TJ For a=1 but £ 1 , we have H 1 ( a , a Q ) = — [ a + - ^ ( a ^ a A 2 ) + ^ ( s ^ a V - O a A * * ) + ||D 7(a 7+l2a 5A 2+18aV+4aA 6)J H 2(a,a e) - ^-[A + - ^ ( A ^ A a 2 ) + |]> 5(A 5 +6A 3a 2 +3Aa 4) + -||D 7(A 7+12A 5a 2+18A 5a +Ua )] APPENDIX I I I (1) System Parameters For the cantilever beam as shown i n Fi g . A.3, the deflec t i o n and slope at a due to a force P acting at b are 0 a " 3EI 2EI a 2EI P1211 EI i k2l-, + 5 l i ; .'. s t a t i c deflection at b due to P acting at b i s 6 b - «; - V 1 + ' i f + 5(T;) 2 ] The estimated equivalent s t i f f n e s s of the e l a s t i c system = = 62.3 l b / i n 1 b 11 m 1 2 A. J Steel Beam £ " x 1 " E = 30 x 1 0 6 l b / i n 2 b - 1", d » i n 311 V ? 8 V 5 1 " 6 a " 3EI So53 p / 5 Q - 282 l b / i n m « 1 .225 l b 0 ^ = Mass of Steel Beam = 0 . 4 7 5 lb The natural frequency of the system can be estimated from the equation given i n [82 J co n xm sec The experimentally recorded s t i f f n e s s and damped natural frequency of the system are respectively = 60 l b / i n , co = 138 rad./sec 148 (2) D e t e r m i n a t i o n o f System Damping C o e f f i c i e n t a) L o g a r i t h m i c decrement method I n o r d e r t o det e r m i n e system damping, the e l a s t i c beam was 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 and th e n r e l e a s e d i n f r e e v i b r a t i o n w i t h t h e s l i d e r c l e a r o f t h e lower s u r f a c e . O s c i l l o g r a p h c h a r t r e c o r d s of t h e f r e e v i b r a t i o n s were o b t a i n e d f o r f i v e s i m i l a r t e s t s . The l o g a r i t h m o f the v i b r a t i o n a m p l i t u d e s were p l o t t e d a g a i n s t the c y c l e numbers. The cu r v e was found t o be alm o s t l i n e a r thus s u g g e s t i n g t h a t t h e damping c o e f f i c i e n t o f the system was p r o p o r t i o n a l t o t h e v e l o c i t y o f the v i b r a t i o n . From each t e s t the a m p l i t u d e s o f v i b r a t i o n o f e v e r y t e n t h c y c l e were r e c o r d e d and the r a t i o between c o n s e c u t i v e a m p l i t u d e s was c a l c u l a t e d . The average a m p l i t u d e r a t i o and t h e f r e q u e n c y from t h e f i v e t e s t s were used f o r d e t e r m i n i n g t h e damping c o e f f i c i e n t . F i g . A.2a shows an o s c i l l o g r a p h t r a c e o f the f r e e v i b r a t i o n wave form. The e q u a t i o n o f motion i s mx + r x + kx = 0 the s o l u t i o n o f t h e d.e. i s x = e ^ [d..cosa)t + d„sinwt] k r 1_ where A = r/2m and co = [ — - — j ] 2 m 4m F o r any two maxima i n the d e c a y i n g s i n e wave the a m p l i t u d e o f the v i b r a t i o n d i m i n i s h e s from e ^ t o e - ^ ^  + ~ ^ . T h e r e f o r e f o r t e n i n t e r v a l s , we have -At x e A n _ 20TT-e co Xn+10 e-A (t+20u/co) x or A = — l o g (— ) 20TT e xn+10 ^ 2 2 ^ . 2 and co = co + A n x From e x p e r i m e n t a l r e s u l t s we have = 1.925 and xn+10 co = 22.1 cps A = 1.45 r a d . / s e c , and co = 138.5 r a d . / s e c ' n ' 2 The e q u i v a l e n t mass o f t h e system m = k/to = 1.205 l b , The damping c o e f f i c i e n t o f the system r = 2mA = 0.009 l b / i n / s e c . The c o e f f i c i e n t f o r c r i t i c a l damping r c = 2mcon = 0.86 l b / i n / s e c . 150 b) D i r e c t measurement from o s c i l l o s c o p e E x t e r n a l l y a p p l i e d v i s c o u s damping as w e l l as the system damping can be c a l i b r a t e d d i r e c t l y from t h e o s c i l l o s c o p e . By a p p l y i n g s c a l e d a c c e l e r o m e t e r and d i s -placement s i g n a l s t o t h e d i f f e r e n t i a l a m p l i f i e r o f an o s c i l l o s c o p e , and t h e v e l o c i t y s i g n a l t o t h e h o r i z o n t a l d i s p l a y a m p l i f i e r , a s t r a i g h t l i n e was o b t a i n e d . The s l o p e * * o f the s t r a i g h t l i n e i s — ^ = r . F i g . A. 2b shows an example o f t h i s t r a c e o b t a i n e d w i t h t h e heavy permanent magnet as e x t e r n a l damper. APPENDIX IV CALIBRATION AND SCALING OF DISPLACEMENT, VELOCITY AND ACCELERATION SIGNALS 1) D i s p l a c e m e n t A depth micrometer was r i g i d l y mounted w i t h i t s s p i n d l e l y i n g h o r i z o n t a l l y and imposed p e r p e n d i c u l a r t o the s i d e o f the s l i d e r mount. The beam was th e n g r a d u a l l y d e f l e c t e d by the micrometer. The v a r i a t i o n i n o u t p u t s i g n a l from the o s c i l l o s c o p e was r e c o r d e d . A c a l i b r a t i o n c u r v e was o b t a i n e d by p l o t t i n g t h e beam d e f l e c t i o n i n 1/1000 i n c h a g a i n s t o u t p u t s i g n a l i n m i l l v o l t . F o r d e f l e c -t i o n s w i t h i n 50/1000 i n c h e s the curve i s l i n e a r . 2) V e l o c i t y F o r a damped f r e e v i b r a t i o n system the a m p l i t u d e of v i b r a t i o n i s x = Ae -At cos (qt + cp) x = Ae -At [-A cos (qt + cp) - q s i n ( q t + cp)] X= Ae" A t [ ( A 2 - q 2 ) cos (qt + cp) + 2 A q s i n (qt + cp) ] where A = r/2m and q = / From Appendix I I I q >> A T h e r e f o r e f o r c a l i b r a t i o n p u r p o s e , i t i s s u f f i c i e n t t o c o n s i d e r X = qX and X = q 2X, where X, X and X a r e the peak v a l u e s o f x, x and x. The c a l i b r a t i o n o f t h e v e l o c i t y s i g n a l was f u l f i l l e d by p e r f o r m i n g a f r e e o s c i l l a t i o n t e s t o f the v i b r a t o r y system. A x-x phase p l a n e p l o t was o b t a i n e d from the o s c i l l o s c o p e . The d.c. o u t p u t s i g n a l s o f X and X were r e c o r d e d . The a c t u a l d i s p l a c e m e n t i n terms o f i n c h e s was o b t a i n e d from the c a l i b r a t i o n curve o f 1. Knowing q from Appendix I I I , the a c t u a l v e l o c i t y i n terms o f i n / s e c was o b t a i n e d from X = qX. Comparing the v e l o c i t y i n i n / s e c w i t h i t s c o r r e s p o n d i n g d.c. o u t p u t s i g n a l , a r e l a t i o n s h i p i n terms o f d.c. o u t p u t per i n / s e c c o u l d be o b t a i n e d . 3) S c a l i n g o f D i s p l a c e m e n t and A c c e l e r a t i o n S i g n a l s The a c c e l e r o m e t e r i s a commercial u n i t and has a p r e - c a l i b r a t e d o u t p u t o f 0.1 v o l t per g. C a l i b r a t i o n o f the a c c e l e r o m e t e r i s t h e r e f o r e n o t r e q u i r e d . However, s c a l i n g o f t h e a c c e l e r a t i o n f o r c e term and the s p r i n g f o r c e term was r e q u i r e d so t h a t when x = 0, mx + kx = 0. The s c a l i n g p r o c e d u r e was c a r r i e d o u t by a p p l y i n g the d i s p l a c e -ment and a c c e l e r o m e t e r s i g n a l s t o the d i f f e r e n t i a l a m p l i -f i e r o f the o s c i l l o s c o p e and the v e l o c i t y s i g n a l t o the h o r i z o n t a l d i s p l a y a m p l i f i e r d u r i n g f r e e v i b r a t i o n t e s t and a d j u s t i n g the g a i n c o n t r o l o f the b r i d g e a m p l i f i e r meter u n t i l t he combined d i s p l a c e m e n t and a c c e l e r o m e t e r s i g n a l formed a s t r a i g h t l i n e . A l t e r n a t i v e l y , the time based d i s p l a c e m e n t and a c c e l e r o m e t e r s i g n a l s c o u l d be f e d t o s e p a r a t e a m p l i f i e r s o f the o s c i l l o s c o p e and the g a i n con-t r o l o f the b r i d g e a m p l i f i e r meter was a d j u s t e d u n t i l the d i s p l a c e m e n t s i g n a l e q u a l e d the a c c e l e r o m e t e r s i g n a l . APPENDIX V SPECIMEN COMPOSITION A. A t l a s Nutherm S t e e l C 0.7 % Mn 2.0 % S i 0,3 % P 0.014 % S 0.010 % Cr 1.0 % Mo 1.35 % B. A t l a s Keewatin S t e e l C . 0.9 % Mn 1 . 2 ' % S i 0.3 % Cr 0.5 % V 0.2 % 1 W 0.5 % C. A u t o m a t i c T r a n s m i s s i o n F l u i d The f o l l o w i n g o i l samples and i n f o r m a t i o n were r e c e i v e d from the C i t i e s S e r v i c e O i l Company, C r a n b u r y , New J e r s e y . (1) C i t i e s S e r v i c e 100 N e u t r a l O i l (2) C i t i e s S e r v i c e 200 N e u t r a l O i l 155 (3) C i t i e s S e r v i c e 350 N e u t r a l O i l (4) C i t i e s S e r v i c e 650 N e u t r a l O i l (5) C i t i e s S e r v i c e 150 B r i g h t Stock (6) C i t i e s S e r v i c e 100 N e u t r a l O i l + 1% V i c t a b l u b e 5810 (7) CITGO A u t o m a t i c T r a n s m i s s i o n F l u i d Type "A" S u f f i x "A" (8) CITGO A u t o m a t i c T r a n s m i s s i o n F l u i d Type F (9) C i t i e s S e r v i c e 100 N e u t r a l O i l + 1% S u l f u r i z e d Sperm O i l The f i r s t f i v e a re base s t o c k s w i t h no a d d i t i v e s . T h e i r v i s c o s i t i e s are' 100 , 200, 350 , 670 and 2650 S a y b o l t Seconds a t 100°F. i n t h e o r d e r o f t h e sample numbers and the v i s c o s i t i e s o f t h e A u t o m a t i c T r a n s m i s s i o n F l u i d s a re 190 Seconds f o r Type A and 180 Seconds f o r the Type F. The Kodak B l o t t e r g i v e s f r i c t i o n d a t a s i m i l a r t o even as v a s t l y d i f f e r e n t a m a t e r i a l as the R-3681-22 c l u t c h f a c i n g m a t e r i a l a g a i n s t s t e e l . O i l Sample 8 shows the hump i n t h e f r i c t i o n c urve and behaves i n a c h a t t e r y and squawky manner d i s p i t e t h e f a c t t h a t s t a t i c f r i c t i o n i s lower t h a n k i n e t i c f r i c t i o n i n most c a s e s . 156 X • 1 I I 1 k 1 1 i i i i i i i i M M 1 1 1 1 III +™ V t \Jr\ 1 1 II M M .1 1 1 1 -' / JH I t 1 M 1 i i i i I t ( a ) STICK-SLIP x : , t (b) QUASI-HARMONIC F i g u r e 1.1.1 Displacement-Time Waveforms o f F r i c t i o n -Induced V i b r a t i o n w 157 /SWsinvt 0<\ (O F i g u r e 1.1.2 Schematic Diagrams o f Three F r i c t i o n a l Systems 158 ( b ) F i g u r e 3.1.1 T o p o l o g i c a l Diagrams I l l u s t r a t i n g S o f t and Hard S e l f - E x c i t a t i o n s 159 F i g u r e 3.2.1 P o s s i b l e Forms o f F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curves SLIDING VELOCITY F i g u r e 3.2.2 Humped Form o f a F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve F i g u r e 3.2.3 x-x Phase P l a n e Diagrams Figure'"3 .3.1 Humped "Form o f a " F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve Represented by a S i m p l i f i e d E x p o n e n t i a l E x p r e s s i o n CTl to 163 F i g u r e 3.3.2 T h e o r e t i c a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y Curve From t h e Humped F r i c t i o n -V e l o c i t y Curve o f F i g . 3.3.1 164 F i g u r e 3.3.3 Y-,, y 2 vs V e l o c i t y P l o t Showing Regions o f s t a b i l i t y and i n s t a b i l i t y F i g u r e 4.2.3 I s o m e t r i c Diagram o f A p p a r a t u s H F i g u r e 4.2 ...4 D i a g r a m of. E x t e r n a l E x c i t a t i o n G e n e r a t o r C7\ 00 POWER S U P P L Y T A B L E P O S I T I O N T R I G G E R S T R A I N G A U G E S " V E L O C I T Y T R A N S D U C E R B R I D G E A M P L I F I E R A C C E L E R O M E T E R - V E L O C I T Y D.C. A M P L I F I E R A C C E L E R O M E T E R P O W E R S U P P L Y 8 C O N T R O L D .C . A M P L I F I E R D U A L B E A M S T O R A G E O S C I L L O S C O P E I L A M P P O W E R S U P P L Y i 2 LDR O O N E C Y C L E S E Q U E N C E T R I G G E R L . D U A L B E A M O S C I L L O S C O P E R E C T I L I N E A R C H A R T R E C O R D E R r r 3/16 H.P. P O W E R S U P P L Y D.C. M O T O R 8 C O N T R O L F i g u r e 4.3.1 B l o c k Diagram o f I n s t r u m e n t a t i o n C i r c u i t r y F i g u r e 4.3.2 V e l o c i t y T r a n sducer and d.e. A m p l i f i e r DISPLACEMENT SIGNAL TABLE POSITION SWITCH » » -D.C AMPLIFIER TRIGGERING LEVEL o o X o Z Xo-A3 Yo- yh Zo- I —ao B2 4 4 p |TR IM Bl B RELAY ENERGISES READIES CIRCUIT FOR BLANKING A l Q O — | A RELAY \ ENERGISES. \ UN BLANKS OSCILLOSCOPE BEAM I—P ^ Q—| J RESET L BUTTON A.C C RELAY ENERGISES, BLANKS OSCILLOSCOPE BEAM A4 READY LAMP ON/OFF c/cZ 15 V. Dl ( ^ o — — o RECORDER l _ D 2 ^ EVENT MARKER OSCILLOSCOPE TRIGGER D3 D4 A2 -o -o SPARE -o SPARE TO STORAGE| OSCILLOSCOPEI C2 ^ -o -»> RELAY LAMP (a) F i g u r e 4 . 3 . 3 One C y c l e Sequence T r i g g e r C i r c u i t • H-d n CD UI PJ O PJ O X o I I—1 T J O cn o> o cn H-CD M •tf o H cn 0) o 3 O CD CD a H- i-3 CU t-i H O CD 3 O H i H H-O rt-H-O I < (D I—1 O a p-rr n < CD m z CO o m co co FRICTION FORCE, DIMENSIONLESS r- ro ro O J C M cn O cn O cn T T O cn CO r g < m r~ o o —I -< o cn ro b ro cn r- o ° P o cn cn O AMPLITUDE OF VIBRATION,DIMENSIONLESS ZLl < U - l 3.0r -+++ i i i i I I I ! F I 1 1 1 1 1 1 1 1 1 1 1 j I i i m m / / \ \ / / i r h i \ t i i i • i • • i in i t i i i I 1111 till Tt t 1 1 III 1 1— -v 4 CO CO LU g CO •z. UJ Q e A A a 88 2.0 LU o or o o or a X O DISC VELOCITY, v 0.84 in/sec 1.06 1.30 1.64 w = 5.4 lb •°o F i g u r e 5.1.3 1.0 " 2.0 •• SLIDING VELOCITY, DIMENSIONLESS 3.0 One C y c l e O s c i l l o s c o p e T r a c e s a t V a r i o u s D i s c V e l o c i t i e s - Type A l F i g u r e 5.1.5 1.0 1.5 2.0 2.5 3.0 - " V E L O C I T Y , DIMENSIONLESS Graph o f E x p e r i m e n t a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y Curves a t V a r i o u s Normal Loads - Type A2 i—1 i-3 *< CD 1 1 1 1 ft0.0 0.5 1.0 l.S 20 FREQUENCY RATIO LLT a cn' in nj' cn o I—< to LU 5^in O D I — 1 I— a o cn" in CM ' CO COo o t—I CO GQ_ O Q_in \ x x x E X P E R I M E N T A L — : — E X P O N E N T I A L F I T P O L Y N O M I A L F I T FRICTION FORCE-7 0.0 0.5 x x T 1.0 V E L O C I T Y . w = 3.85 lb n 1 1.5 2.0 D I M E N S I O N L E S S 2.5 3.0 F i g u r e 5.1.7 Graph o f E x p e r i m e n t a l A m p l i t u d e o f V i b r a t i o n v e r s u s V e l o c i t y -Curves a t V a r i o u s Normal Loads - Type B2 oo FRICTION FORCE, DIMENSIONLESS AMPLITUDE OF VIBRATION, DIMENSIONLESS 6LT 180 X 1 1 1 1 1 1 1 1 1 1 1 1 i i i i 11 i i 1 t II l i l t 1 1 II i i i i M M I 1 y — 1 H — V (Q) v = 0 .08 in/sec , w = 3 .85 lb x x - -II 1 1 1 1 1 1 1 1 1 1 1 ? t t T ? t t t ? r t t 1 « t t t s — V (b) v= 2.45 in/sec, w = 5.4 lb F i g u r e 5.1.9 x-x Phase P l a n e Diagrams o f the S t i c k - S l i p Type in in in CO &>. ZP: O «—i CO z. LUin LU O Of m O • U_ — U_ a a ' in CO CO a ,n o ' — I CO CD ,_in CL_r-—X—X—X-X— -o-o-o-o- EXPERJMENTRL POLYNOMIAL FIT AMPLITUDE FRICTION FORCE * * * * * " X X K FREQUENCY RATIO < x — * -X - o - o - O - EXTERNALLY DAMPED 0.0 0.75 V E L O C I T Y . ^ F M E N S I O N L E S S 3.75 4.5 a in w.cr z UJ n i n 0 a : a ' a 5.25 co F i g u r e 5.1.10 Graph o f E x p e r i m e n t a l Curves Showing E f f e c t o f E x t e r n a l Damping - S t e e l on S t e e l X 4 0 0 F i g u r e 5.1.11 M i c r o p h o t o g r a p h s X 2 0 0 H -H CD to 0 3 ri CO O f+ r t CD CD O I < CD h- 1 O » o P> r t o H co p> r t O CD CD r t CD H H -CO r t H -O n ri < CD tn n PJ ri cr o H -cr n CD I CD cn H -3 ro FRICTION FORCE, DIMENSIONLESS. O J -£> D O C.F b FRICTION FORCE, DIMENSIONLESS. RESIN 8.5r-LU 8.4 O oo | 3.3 Q a 82 o Li_ • - 8 . o on. 8.0 1 v = 0.39 in/sec w= 5.4 lb 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 SLIDING VELOCITY, DIMENSIONLESS F i g u r e 5.1.13 F r i c t i o n - V e l o c i t y C h a r a c t e r i s t i c Curve Reproduced From A One C y c l e O s c i l l o s c o p e Trace - Rubber on S t e e l 185 F i g u r e 5.2.1 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency . R a t i o - L i n e a r System 3.0 o X \ X 01 I ! I I 1 1 0 1.0 2.0 3.0 F R E Q U E N C Y RATIO, a F i g u r e 5.2.2 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency-R a t i o - Type C F i g u r e 5.2.3 x-x Phase P l a n e Diagram D u r i n g F o r c e d V i b r a t i o n - Type B l 7.0 F i g u r e 5.2.4 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type B l 189 6.0 * 5.0 or o i54.0 L? o P3.0 < o 3 2.0 1.0-F0 V (1) 0.83a2 0.98 (2) 0.42 a 2 0.98 (3) 0.83 a 2 0.7 (4) 0.42 a 2 0.67 (5) 0.42 a 2 0.46 0 0.5 J J L J I I I I L J L 1.0 1.5 FREQUENCY RATIO, a 2.0 F i g u r e 5.2.5 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type A2 AVERAGE DISPLACEMENT, x 0.001 in O M * O AVERAGE DISPLACEMENT, x 0.001 in O N> i» 0> oo o ro AMPLITUDE OF VIBRATION, x 0.001 in O fV> -t» 0"> AMPLITUDE OF VIBRATION, x 0.001 in AVERAGE DISPLACEMENT, x 0.001 in AVERAGE DISPLACEMENT, x 0.001 in ro £ 55 co to J> o~> CD AMPLITUDE OF VIBRATION, x 0.001 in ro * o) oo AMPLITUDE OF VIBRATION, xOOOl in 061 « 191 F i g u r e 5.2.7 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Parameters a t V a r i o u s D i s c V e l o c i t i e s - Non-Resonance 192 v =1.05 in/sec 01 = 1.79 (I) AMPLITUDE a (2 ) EXTERNAL FORCE TERM -(3) SECOND APPROX. TERM (a = 0 ) 0.2 0.4 0.6 0.8 1.0 EXTERNAL FORCE F 0, DIMENSIONLESS F i g u r e 5.2.8 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Magnitude - Non-Resonance F i g u r e 5.2.9 194 11 i I i i I : I l _ i 0.7 0.8 0.9 1.0 I.I 1.2 1.3 FREQUENCY RATIO, a F i g u r e 5.2.10 Graph o f A m p l i t u d e o f V i b r a t i o n vs Frequency R a t i o - Fundamental Resonance 195 EX.F • Vf! -- I.G5C FRE .575 "1 VEl. - 1.05C FRE i.GGC VEL " 1.G5C FRE :. . 9G0 VEl '-- 1.G5G FRE -- .975 VEL. 1.050 FRE i . 075 MN I l i i mmmm VEL - 1.050 FRE 1.200 f'LPCEMENl VELOCITY ~i 1 r~ r ISPLflCEMENT |j VELOCITY ~i 1— r~ DISPLACEMENT i M j l M M i l i W ' EL 0CITv F i g u r e 5.2.11 Displacement-Time and V e l o c i t y - T i m e T r a c e s a t V a r i o u s E x t e r n a l E x c i t a t i o n F r e q u e n c i e s V = l .05 F 0 == 0 .2 196 . . I A A A A A A A A A A - A A A A A A A / 1 -^ v " v "v y ii v v \i \i v v v v v wv v a = 0.875 a = i.oo WHI a = 0.900 IftM M M * Oi = 1.075 i M ' i1; 11 /' /1 /' /'/''' /1A A A A • \' i /; •' a =0.975 i ! i mm i/ a = 1.20 ill ^Hp^fHP F i g u r e 5.2.12 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s a t V a r i o u s E x t e r n a l E x c i t a t i o n F r e q u e n c i e s 197 V =1.05 0.9 O E X P E R I M E N T A L A N U M E R I C A L ( E X P O N ^ ) x F U N D A M E N T A L R E S O N A N C E T H E O R Y ( P O L Y t ) .0 I.I FREQUENCY RATIO, a F i g u r e 5.2.13 Graph o f M a g n i f i c a t i o n F a c t o r vs Frequency R a t i o - Type A l Displacement x 0.001 in. Velocity in/sec Displacement 0.001 in cn H-3 I CD 0 (_i 002 Displacement x 0.001 in. Velocity in/sec Displacement 0.001 in. c TOZ i . 4 r EXTERNAL FORCE F0, DIMENSIONLESS F i g u r e 5.2.18 Graph o f A m p l i t u d e o f V i b r a t i o n vs E x t e r n a l F o r c e Magnitude 3.0 FREQUENCY RATIO, a F i g u r e 5.2.19 Graph o f E x t e r n a l E x c i t a t i o n Magnitude f o r the E x t i n c t i o n o f A u t o p e r i o d i c O s c i l l a t i o n vs E x t e r n a l E x c i t a t i o n Frequency < p o i Figure- 5^2.21 D i s p l a c e m e n t - V e l o c i t y Phase P l a n e Diagrams VEL ~ 1 .050 FRE = 3.000 r e =O.OI5 Ib/in/feec 20 ISPLRCEMENT L J c: -j • -•CD-C D ' VELOCITY i — 2 S . 0 — 1 5 0 . 0 —I 1 2 0 0 . 0 2 2 5 . 0 0 . 0 I 7 5 . 0 1 0 0 . 0 TIME 1 2 5 . 0 1 5 0 . 0 1 7 5 . 0 VEL = 1.050 FRE = 3.000 re=-o.OI5 lb/in/feec DISPLACEMENT o cr >—><C . a 1 h h h W W \l 'ELOCITY 0.0 - 1 — 25.0 S O . O - 1 7 5 . 0 I 1 1 0 0 . 0 1 2 5 . 0 T I M E T T 1 5 0 . 0 1 7 5 . 0 — 1 1 2 0 0 . 0 2 2 5 . 0 F i g u r e 5.2.22 Displacement-Time and V e l o c i t y - T i m e T r a c e s - L i n e a r i s e d F r i c t i o n - V e l o c i t y Curve 207 3 . 5 r NORMAL LOAD = 6.4 lb 17.0 FREQ. RATIO a CD 1.58 1.50 ® 2.0 2.0 2.64 2.42 (2) 2.0 ® 1.82 ® 2.19 2.27 2.42 0.9 MAXIMUM STATIC FRICTION AMPLITUDE OF VIBRATION 0.1 0 2 L O A D R A T I O , /S 6 . 0 C/5 t o LU -J z o CO 5 . 0 Q LsJ O O 4 . 0 u-o H O LT 3 . 0 2 . 0 00 1.0 x < 0 F i g u r e 5.2.23 Graph o f A m p l i t u d e o f S t i c k - S l i p V i b r a t i o n and Maximum S t a t i c F r i c t i o n F o r c e vs Load R a t i o DISPLACEMENT VELOCITY , OH IO ! i /3= 0 . 0 5 6 „,..jZ[...|...".".I..!"..|r| j.. ! . ; . . ; ; w 1 1 I ; ! i ! 1 | i 1 j , — 1 !- : j t t 1 -+ — ! — — f-!----4- -fi = o . i o y6 = 0 . 1 6 O 00 F i g u r e 5.2.24 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s I l l u s t r a t i n g t h e E x t i n c t i o n o f S t i c k - S l i p V i b r a t i o n Due To Normal E x c i t a t i o n s 01 = o /5 = o CX =0.84 ^ = 0 . 0 3 a«i.o ^S«o.04 209 1.05 a =1.18 £ = 0.03 Oi=l.99 /Q=0.I6 O»»3.0 £ = 0 . 3 7 F i g u r e 5.2.25 E x p e r i m e n t a l O s c i l l o g r a p h T r a c e s a t V a r i o u s Normal E x c i t a t i o n - Type A l Displacement x 0.001 in Velocity in/sec Displacement 0.001 in 0 1 2 F i g u r e A . l C a l i b r a t i o n Curves o f E l a s t i c Beam 212 F i g u r e A.2 L o g a r i t h m i c Decrement O s c i l l o g r a p h Trace and One C y c l e O s c i l l o s c o p e Trace For The D e t e r m i n -a t i o n o f System Damping C o e f f i c i e n t 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0102151/manifest

Comment

Related Items