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UBC Theses and Dissertations

Friction induced vibrations in a hydraulically driven system Johannes, Veikko Ilmari 1969

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FRICTION INDUCED VIBRATIONS I N A HYDRAULICALLY DRIVEN SYSTEM b y VEIKKO ILMARI JOHANNES B.A. S c . U n i v e r s i t y o f T o r o n t o , T o r o n t o , O n t a r i o , 19 6 4 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY 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 We a c c e p t t h i s t h e s i s as c o 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 B R I T I S H COLUMBIA A u g u s t , 1969 I n 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 l m e n t o f the r e -quirem 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 umbia, 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 t h e Head o f my Department o r 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 . VEIKKO ILMARI JOHANNES 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 A s t u d y h a s b e e n made o f t h e b e h a v i o u r o f a h y d r a u l i c -a l l y d r i v e n s y s t e m s u b j e c t t o 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 s . An e x p e r i m e n t a l a p p a r a t u s c o n s i s t i n g o f a h e a v y mass d r i v e n b y a h y d r a u l i c ram was c o n s t r u c t e d . A m a t h e m a t i c a l m o d e l f o r t h e s y s t e m was d e r i v e d , and i t s v a l i d i t y was shown by c o m p a r i s o n w i t h e x p e r i m e n t a l r e s u l t s . S t u d i e s w e r e s u b s e q u e n t l y c a r r i e d o u t w i t h t h e m a t h e m a t i c a l m o d e l t o d e t e r m i n e t h e e f f e c t s o f v a r y i n g s y s t e m p a r a m e t e r s and t o v e r i f y t h e v a l i d i t y o f s i m p l i f i c a t i o n s i n t h e g o v e r n i n g e q u a t i o n s . I t was f o u n d t h a t i n g e n e r a l t h e e f f e c t o f l e a k a g e a t t h e p i s t o n s h o u l d be n e g l i g i b l e , and t h a t t h e v a r i a t i o n i n b e h a v i o u r w i t h p i s t o n l o c a t i o n c a n be p r e d i c t e d b y a s i m p l e s y s t e m s t i f f n e s s c o r r e c t i o n . The e f f e c t o f c h a n g i n g m o s t p a r a m e t e r s i s t o change t h e r a t e o f t a n g e n t i a l l o a d a p p l i c a t i o n f o r a g i v e n a v e r a g e v e l o c i t y o f t r a v e r s e , and t h e r e s u l t i n g b e h a v i o u r c a n be p r e d i c t e d f r o m t h e s t a t i c f r i c t i o n r e l a t i o n -s h i p t h a t h a s b e e n p r o p o s e d . A d e t a i l e d s t u d y was made o f t h e n a t u r e o f t h e f r i c t i o n f o r c e s i n e f f e c t d u r i n g a s t i c k - s l i p p r o c e s s . A r e l a t i o n , f o r t h e v a r i a t i o n o f t h e 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 w h i c h a p p e a r s more a c c u r a t e t h a n p r e v i o u s t i m e o f c o n t a c t d e p e n d e n t f o r m u l a t i o n s h a s b e e n f o u n d . I n t h e c a s e o f t h e 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 , a g r a d u a l d e c r e a s e f r o m t h e s t a t i c v a l u e was found a t the i n c e p t i o n o f s l i p . T h i s has been observed b e f o r e b u t g e n e r a l l y i t has not been r e c o g n i z e d by i n v e s t i g a t o r s o f s t i c k - s l i p . Simple models have been g i v e n f o r b o t h the s t a t i c and k i n e t i c f r i c t i o n b e h a v i o u r . These may be u s e f u l as g u i d e i n f u r t h e r r e s e a r c h i n t o t h e fundamental n a t u r e o f f r i c t i o n . TABLE OF CONTENTS C h a p t e r Page 1 1.1 I n t r o d u c t i o n 1 1.2 B a c k g r o u n d 2 2 2.1 T h e o r y 12 3 3.1 E x p e r i m e n t a l A p p a r a t u s 23 3.2 M e a s u r e m e n t s 26 4 4.1 E x p e r i m e n t a l P r o c e d u r e . . . 30 4.2 M e a s u r e m e n t o f F r i c t i o n V a l u e s 33 5 5.1 F u r t h e r I n v e s t i g a t i o n s i n F r i c t i o n 40 5.2 M o d e l s f o r F r i c t i o n a l B e h a v i o u r 44 6 6.1 R e s u l t s f o r t h e C o m p l e t e S y s t e m 58 7 7.1 C o n c l u s i o n s 69 A p p e n d i x I 72 A p p e n d i x I I . . . . . . . . 78 A p p e n d i x I I I 84 R e f e r e n c e s 127 LIST OF FIGURES F i g u r e Page 1.1.1 Model o f a System S u b j e c t t o F r i c t i o n a l V i b r a t i o n 87 1.1.2 T y p i c a l S t i c k - S l i p D i s p l a c e m e n t Records . . . . 87 1.2.1 Model o f an A s p e r i t y J u n c t i o n 88 1.2.2 A r c h a r d ' s S u r f a c e P r o f i l e Model 89 1.2.3 Comparison of S t a t i c F r i c t i o n R e l a t i o n s . . . . 90 1.2.4 P l o t o f y k A g a i n s t S l i d i n g V e l o c i t y 91 2.1.1 Schematic o f E x p e r i m e n t a l System 92 2.1.2 Diagram f o r Leakage C a l c u l a t i o n 9 3 3.1.1 Diagram o f E x p e r i m e n t a l A p p a r a t u s 94 3.1.2 Photograph o f E x p e r i m e n t a l A p p a r a t u s 9 5 3.1.3 Lower F r i c t i o n S u r f a c e Attachment 96 3.1.4 Photograph o f S l i d i n g P l a t f o r m and S l i d e r s . . 97 3.1.5 Schematic o f H y d r a u l i c C i r c u i t 9 8 4.2.1 T y p i c a l K i n e t i c F r i c t i o n - Time Records . . . . 99 4.2.2 K i n e t i c F r i c t i o n as a F u n c t i o n o f D i s p l a c e m e n t 100 4.2.3 R e s u l t s of M a t s u z a k i and Hashimoto 101 4.2.4 E x p e r i m e n t a l and C a l c u l a t e d u, and V e l o c i t y Curves 102 4.2.5 S t a t i c F r i c t i o n v . s . Time o f S t a t i o n a r y C o n t a c t 10 3 4.2.6 S t a t i c F r i c t i o n v . s . 0 104 4.2.7 S t a t i c F r i c t i o n v . s . V e l o c i t y - P o t t e r . . . . 105 4.2.8 S t a t i c F r i c t i o n v . s . 0 - P o t t e r 106 v i F i g u r e Page 4.2.9 S t a t i c F r i c t i o n v . s . $ - Cameron, D a v i s . . . . 107 4.2.10 S t a t i c F r i c t i o n v . s . 0 - Dry and E t c h e d S u r f a c e s 108 5.1.1 Shape o f V e l o c i t y Curve a t I n c e p t i o n o f S l i p 109 5.1.2 Wear Mark on S t e e l B a l l - P l a i n S t e e l R a i l s . . 110 5.1.3 Wear Mark on P l a i n S t e e l R a i l 110 5.1.4 Wear Mark on S t e e l B a l l - Copper P l a t e d R a i l s I l l 5.1.5 Wear Mark on Copper P l a t e d R a i l s I l l 5.1.6 A s p e r i t y Temperature D u r i n g S l i p C y c l e 112 5.2.1 Diagram o f Model f o r S t a t i c F r i c t i o n 113 5.2.2 S t a t i c F r i c t i o n v . s . $ From Model 114 6.1.1 Comparison o f E x p e r i m e n t a l and Computed R e s u l t s 115 6.1.2 Comparison o f E x p e r i m e n t a l and Computed R e s u l t s 116 6.1.3 E f f e c t o f a Damping Term i n P i s t o n F r i c t i o n . . 117 6.1.4 E f f e c t o f V a r y i n g P i s t o n P o s i t i o n 118 6.1.5 E f f e c t o f V a r y i n g P i s t o n Rod t o Mass C o u p l i n g S t i f f n e s s 119 6.1.6 A m p l i t u d e o f S l i p as a F u n c t i o n o f $ 120 6.1.7 E f f e c t o f P i s t o n Leakage 121 A . I . I Diagram o f H y d r a u l i c C y l i n d e r C o n s t r u c t i o n . . . 122 A.I.2 Comparison o f T h i c k and T h i n S h e l l Theory . . . 123 A . I I . l Flow V a l v e C a l i b r a t i o n . 124 A . I I . 2 E x p e r i m e n t a l D e t e r m i n a t i o n o f H y d r a u l i c System S t i f f n e s s 125 A . I I . 3 Phase S h i f t i n g C i r c u i t s t o Match A c c e l e r o m e t e r 126 ACKNOWLEDGEMENT The a u t h o r i s g r a t e f u l t o Dr. C.A. B r o c k l e y f o r h i s a d v i c e and encouragement t h r o u g h o u t the r e s e a r c h program. Thanks a r e a l s o due t o the f a c u l t y and gradu a t e s t u d e n t s o f the Department 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 h e l p -f u l s u g g e s t i o n s and d i s c u s s i o n s , t o t h e t e c h n i c i a n s who a s s i s t e d i n c o n s t r u c t i n g the e x p e r i m e n t a l a p p a r a t u s , and t o Mr. E. Jones f o r i n v a l u a b l e a s s i s t a n c e w i t h the i n s t r u m e n t a t i o n . The e x p e r i m e n t a l work was c a r r i e d o u t i n the T r i b o l o g y L a b o r a t o r y o f the Department o f M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h 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 from the Defence Research Board o f Canada under G r a n t Number 7510-31. C H A P T E R 1 1.1 INTRODUCTION I f two b o d i e s i n f r i c t i o n a l c o n t a c t a r e s u b j e c t e d t o r e l a t i v e m o t i o n b y t h e a p p l i c a t i o n o f a f o r c e t h r o u g h an e l a s t i c medium, a d i s c o n t i n u o u s m o t i o n known as " s t i c k - s l i p " may o f t e n be o b s e r v e d . A s c h e m a t i c d i a g r a m o f s u c h a s y s t e m i s shown i n F i g u r e 1.1.1, a n d a t y p i c a l t i m e - d i s p l a c e m e n t r e c o r d f o r t h i s s y s t e m i s g i v e n i n F i g u r e 1.1.2. M o t i o n o f t h i s t y p e h a s l o n g b e e n o b s e r v e d , and i t s e x i s t e n c e h a s b e e n a t t r i b u t e d t o t h e n a t u r e o f t h e f r i c t i o n a l f o r c e s a t t h e c o n t a c t i n g s u r f a c e s . Thus we h a v e come t o u s e t h e t e r m " 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 " . T h i s k i n d o f v i b r a t i o n may o c c u r i n most m e c h a n i s m s i n w h i c h s l i d i n g f r i c t i o n i s p r e s e n t , and i n g e n e r a l i t i s u n d e s i r a b l e , g i v i n g undue n o i s e , a p o o r l y f i n i s h e d s u r f a c e i n m a c h i n i n g o p e r a t i o n s , o r a p o s i t i o n i n g e r r o r i n a s e r v o m e c h a n -i s m . Thus t h e r e i s an o b v i o u s e n g i n e e r i n g i n t e r e s t i n t h e phenomenon, and i n t h e p a s t few d e c a d e s numerous i n v e s t i g a t i o n s h a v e b e e n made i n t o t h e n a t u r e o f t h e s e o s c i l l a t i o n s . B e c a u s e o f t h e i n t i m a t e r e l a t i o n s h i p b e t w e e n t h i s p r o b l e m and f r i c t i o n a l f o r c e s , much o f t h e w o r k h a s b e e n -l a r g e l y c o n c e r n e d w i t h t h e b e h a v i o u r o f t h e c o e f f i c i e n t s .-of s t a t i c a n d k i n e t i c f r i c t i o n . As a r e s u l t , most p a s t i n v e s t i -g a t i o n s h a v e b e e n c a r r i e d o u t w i t h l i n e a r m e c h a n i c a l s y s t e m s s p e c i f i c a l l y d e s i g n e d f o r t h e m e a s u r e m e n t o f f r i c t i o n f o r c e s . The p r e s e n t r e s e a r c h was u n d e r t a k e n t o a n a l y z e t h e p r a c t i c a l 2 case o f a h y d r a u l i c ram d r i v i n g a mass s u b j e c t t o s l i d i n g f r i c t i o n . T h i s p a r t i c u l a r problem has n o t r e c e i v e d much a t t e n t i o n and e x i s t i n g t r e a t m e n t s a re i n c o m p l e t e i n d e t a i l s p e r t a i n i n g t o b o t h t h e g e n e r a l 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 s and the s p e c i f i c a p p l i c a t i o n i n v o l v i n g a h y d r a u l i c d r i v e . 1.2 BACKGROUND The background o f t h e p r e s e n t t o p i c i s l a r g e l y a h i s t o r y o f 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 a l f o r c e s g i v i n g r i s e t o " s t i c k - s l i p " v i b r a t i o n s . W h i l e a t t e m p t i n g t o measure k i n e t i c f r i c t i o n v a l u e s a t low s l i d i n g v e l o c i t i e s , W e l l s [1] i n 1929 ob s e r v e d some o c c a s i o n s o f s t i c k - s l i p . He p o s t u l a t e d t h a t t h e y were caused by a 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 l a r g e r t h a n t h e k i n e t i c c o e f f i c i e n t . S u b s e q u e n t l y much work has been done i n s t u d y i n g the phenomenon and d e f i n i n g more p r e c i s e l y the n a t u r e o f t h e s e s t a t i c and k i n e t i c f r i c t i o n c o e f f i c i e n t s . To a p p r e c i a t e the scope o f t h i s problem i t i s n e c e s s a r y t o l o o k a t some e x i s t i n g t h e o r i e s on f r i c t i o n . • The e l e m e n t a r y laws o f f r i c t i o n s e t f o r t h by Amonton s t a t e t h a t t h e f o r c e o f f r i c t i o n i s p r o p o r t i o n a l t o t h e normal l o a d and independent o f the a r e a o f c o n t a c t . Experiments have shown t h a t t h e s e statements are o n l y a p p r o x i m a t e l y t r u e and i n f a c t the c o e f f i c i e n t o f f r i c t i o n may v a r y w i t h l o a d , , c o n t a c t a r e a , s l i d i n g v e l o c i t y , and o t h e r a s p e c t s o f the dynamics o f the f r i c t i o n c o u p l e . 3 I n v e s t i g a t i o n s i n t o t h e n a t u r e a n d s o u r c e o f f r i c t i o n a l f o r c e s h a v e r e s u l t e d i n s e v e r a l i n t e r e s t i n g t h e o r i e s , some more p l a u s i b l e t h a n o t h e r s . Some o f t h e e a r l i e s t s p e c u l a t i o n was b a s e d on t h e a s s u m p t i o n o f r i g i d s o l i d s w h i c h w e r e known t o h a v e m i c r o s s c o p i c s u r f a c e r o u g h n e s s . P u r e l y g e o m e t r i c a l c o n s i d e r a t i o n s r e s u l t e d i n t h e p r o p o s a l t h a t t h e c o e f f i c i e n t o f f r i c t i o n was e q u a l t o t h e s l o p e o f t h e s u r f a c e i r r e g u l a r i t i e s , w i t h t h e f r i c t i o n f o r c e a r i s i n g f r o m s l i d i n g up t h i s f r i c t i o n l e s s ramp. I t i s e a s y t o d i s p o s e o f t h i s t h e o r y by n o t i n g t h a t on t h e a v e r a g e t h e two b o d i e s w i l l u n d e r g o no s e p a r a t i o n w h i l e s l i d i n g t a k e s p l a c e , r e s u l t i n g i n z e r o n e t f r i c t i o n a c c o r d i n g t o t h i s a p p r o a c h . B e s i d e s , e x p e r i m e n t s h a v e shown t h a t i n t h e c a s e o f v e r y smooth s u r f a c e s d e c r e a s i n g t h e b a s e a n g l e o f a s p e r i t i e s l e a d s t o h i g h e r v a l u e s o f f r i c t i o n [ 2 ] . A f t e r t h e m o l e c u l a r n a t u r e o f s o l i d s became known, t h e o r i e s b a s e d on m o l e c u l a r f o r c e s b e t w e e n c l o s e l y a p p r o a c h i n g a s p e r i t i e s h a v e b e e n p u t f o r w a r d . I n t h i s c a t e g o r y we may p l a c e t h e t h e o r y t h a t h a s b e e n t o d a t e t h e most w i d e l y a c c e p t e d , w h i c h i n v o l v e s t h e a d h e s i o n o f c o n t a c t i n g a s p e r i t i e s u n d e r p l a s t i c d e f o r m a t i o n . E r n s t and M e r c h a n t [3] a n d Bowden and T a b o r [4] h a v e among o t h e r s p r o p o s e d t h i s m e c h a n i s m . S i n c e r e a l s u r f a c e s a r e n o t p e r f e c t l y s m o o t h , c o n t a c t b e t w e e n them t a k e s p l a c e o n l y a t t h e p e a k s o f s u r f a c e a s p e r i t i e s o v e r an a r e a much s m a l l e r t h a n t h e a p p a r e n t a r e a o f c o n t a c t . A c c o r d -i n g t o t h e a d h e s i o n t h e o r y o f f r i c t i o n , t h e h i g h l o c a l i z e d s t r e s s e s a t t h e s e a s p e r i t i e s r e s u l t i n p l a s t i c f l o w and 4 m o l e c u l a r j o i n i n g o r w e l d i n g o f the a s p e r i t i e s as shown i n F i g u r e 1.2.1. Assuming p e r f e c t b o n d i n g a t the i n t e r f a c e , the c o e f f i c i e n t o f f r i c t i o n can be c a l c u l a t e d from the m a t e r i a l p r o p e r t i e s . W i t h the assumed p l a s t i c f l o w o f t h e a s p e r i t i e s , the a c t u a l c o n t a c t a r e a A can be g i v e n i n terms o f the f l o w p r e s s u r e o f the m a t e r i a l p and the normal W as ^ m^ A = | (1.2.1) pm I f t h e s h e a r s t r e n g t h o f t h e m a t e r i a l i s s, t h e c o -e f f i c i e n t o f f r i c t i o n i s g i v e n by _ t a n g e n t i a l f o r c e _ sA _ sA _ s_ (T o o\ y normal f o r c e W p A p U.^.^J m^ rm I n r e a l i t y the bond between th e a s p e r i t i e s may be i m p e r f e c t due t o the presence o f o x i d e s and o t h e r c o n t a m i n a n t s i n t h e s u r f a c e . T h i s would a f f e c t t h e v a l u e of s i n the,above e q u a t i o n , b u t i n any case t h e same mechanism would a p p l y . An e x t e n s i o n o f t h i s t h e o r y c o n s i d e r s t h e e f f e c t o f the t a n g e n t i a l f o r c e . As a t a n g e n t i a l s t r e s s i s a p p l i e d ; t o the j u n c t i o n , t h e combined s t r e s s a t the c o n t a c t r e g i o n r e s u l t s i n growth o f the j u n c t i o n a r e a , w i t h a r e s u l t a n t growth i n the v a l u e o f the c o e f f i c i e n t o f f r i c t i o n . E r n s t and Merchant a l s o c o n s i d e r e d t h e o r i e n t a t i o n o f the i n t e r f a c e s , b u t s i n c e a c t u a l s l o p e s o f s u r f a c e i r r e g u l a r i t i e s i n most m e t a l l i c , s u r -f a c e s used i n f r i c t i o n c o u p l e s are s m a l l , t h i s appears t o be o f secondary i m p o r t a n c e . 5 The a b o v e t h e o r y g i v e s a r e s u l t i n a g r e e m e n t w i t h Amonton's l a w , b u t i s n o t a c t u a l l y u s e f u l i n p r a c t i c e i n d e t e r m i n i n g f r i c t i o n c o e f f i c i e n t s as t h e s h e a r s t r e n g t h a s s i g n -a b l e t o t h e j u n c t i o n s i s unknown . T h i s o f c o u r s e i s due t o t h e f a c t t h a t i n p r a c t i c a l c a s e s p e r f e c t l y c l e a n s u r f a c e s w i t h no c o n t a m i n a n t f i l m s a r e n e v e r a c h i e v e d . I n t h i s d e r i v a t i o n , t h e a c t u a l a r e a o f c o n t a c t was p r o p o r t i o n a l t o t h e n o r m a l l o a d w i t h t h e a s s u m p t i o n o f p e r f e c t p l a s t i c i t y w i t h a c o n s t a n t f l o w p r e s s u r e p . However s i m i l a r r e s u l t s h a v e b e e n a c h i e v e d w i t h d i f f e r e n t m o d e l s , and i f i n t h e s e c a s e s t h e f r i c t i o n f o r c e i s t a k e n t o be p r o p o r t i o n a l t o t h e a c t u a l a r e a o f c o n t a c t , s i m i l a r r e s u l t s f o r t h e c o e f f i c i e n t o f f r i c t i o n a r e a c h i e v e d . A r c h a r d [5,6] h a s p r o p o s e d a m o d e l i n w h i c h t h e c o n -t a c t s a r e c o m p l e t e l y e l a s t i c s p h e r i c a l a s p e r i t i e s . A s i n g l e s u c h a s p e r i t y w o u l d by H e r t z g i v e t h e r e l a t i o n 2 A = W3 (1.2.3) However, i f t h e m o d e l i s e x t e n d e d t o many a s p e r i t i e s c o n s i s t i n g o f s m a l l e r s p h e r i c a l a s p e r i t i e s s u p e r i m p o s e d , a c o n f i g u r a t i o n as i n F i g u r e 1.2.2 y i e l d s t h e r e s u l t 44 A = W 4 5 (1.2.4) The e x p o n e n t h e r e i s as c l o s e t o u n i t y as e x p e r i m e n t a l r e s u l t s c a n j u s t i f y . 6 T h i s view o f e l a s t i c c o n t a c t s i s s u p p o r t e d by O'Connor and Johnson [7] who found i n e x p e r i m e n t s on a r t i f i c i a l l y roughened s u r f a c e s t h a t p l a s t i c d e f o r m a t i o n took p l a c e o n l y on the f i r s t f o r c e a p p l i c a t i o n , w i t h t h e a s p e r i t i e s b e h a v i n g e l a s t i c a l l y upon subsequent f o r c e a p p l i c a t i o n . R e c e n t l y Greenwood [8,9] has c a r r i e d out some v e r y i n t e r e s t i n g work, c o n s i d e r i n g t h e geometry o f t h e s u r f a c e s i n c o n t a c t . He showed t h a t w i t h o u t making any assumptions as t o the t y p e o f d e f o r m a t i o n t a k i n g p l a c e a t the a s p e r i t i e s , i n the case o f e x p o n e n t i a l d i s t r i b u t i o n o f a s p e r i t y h e i g h t s t h e a c t u a l c o n t a c t a r e a w i l l be d i r e c t l y p r o p o r t i o n a l t o l o a d . The d i s t r i b u t i o n o f a s p e r i t y h e i g h t s i n most machined m e t a l s u r f a c e s i s c l o s e t o G a u s s i a n , and a l t h o u g h t h i s does not y i e l d the e x a c t p r o p o r t i o n a l i t y g i v e n by the e x p o n e n t i a l d i s t r i b u t i o n , t h e r e s u l t i s as c l o s e t o i t as t h e a c c u r a c y o f e x p e r i m e n t a l r e s u l t s . Y e t o t h e r p r o p o s a l s as t o the n a t u r e o f f r i c t i o n f o r c e s e x i s t . K r a g e l s k i i [2] i s o f the o p i n i o n t h a t a p l a s t i c p l o u g h -i n g a c t i o n o f i n t e r l o c k e d a s p e r i t i e s i s t h e major s o u r c e o f f r i c t i o n a l f o r c e s . R a binowicz [10] uses th e concept o f s u r f a c e e n e r g i e s t o e x p l a i n the phenomenon of f r i c t i o n and wear, and Schnurmann [11,12] contends t h a t w i t h a t h i n l u b r i c a n t f i l m , o f good d i e l e c t r i c p r o p e r t i e s , e l e c t r o s t a t i c f o r c e s a r e r e -s p o n s i b l e f o r the o r d e r o f 30 per c e n t o f t h e t o t a l f r i c t i o n f o r c e i n s e v e r e s t i c k - s l i p . The p r e c e d i n g d i s c u s s i o n has d e a l t w i t h t h e more fundamental n a t u r e o f f r i c t i o n . Of more d i r e c t c o n c e r n t o the 7 p r e s e n t problem i s the a c t u a l o b s e r v e d f r i c t i o n a l b e h a v i o u r under such c o n d i t i o n s as are met i 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 s . I t has been o b s e r v e d by a number o f workers t h a t i f the v e l o c i t y V o f F i g u r e 1.1.1 i s r e d u c e d , t h e maximum d i s -placement o f t h e mass becomes g r e a t e r as i n F i g u r e 1.1.2, i n d i c a t i n g a h i g h e r 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 . A number o f r e l a t i o n s h i p s have been d e r i v e d r e l a t i n g t h i s i n c r e a s e i n the f r i c t i o n f o r c e t o t h e t i m e o f s t a t i o n a r y c o n t a c t . Howe, Buddington and Benton [13] and K o s t e r i n and K r a g e l s k i i [14] propose t h a t y s = y k + ( y s o o - y k ) ( 1 " e " ° t s ) d.2.5) D e r j a g i n Push and T o l s t o i [15] found c t s U c = U v + — 2 - (1.2.6) S * k+t s and R abinowicz [10] and B r o c k l e y and D a v i s [16] g i v e t h e r e l a t i o n y s = y k + a t s b (1.2.7) F i g u r e 1.2.3 shows t h e above e q u a t i o n s 1.2.5 t o 1.2.7 p l o t t e d so t h a t a l l have t h e same v a l u e s o f f r i c t i o n a t t=0 and t = l . E q u a t i o n s 1.2.5 and 1.2.6 have t h e same asymptote whereas 1.2.7 g i v e s an i n f i n i t e v a l u e o f t h e c o e f f i c i e n t o f f r i c t i o n f o r an i n f i n i t e t i me o f c o n t a c t . 8 The 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 has g e n e r a l l y been r e l a t e d t o t h e s l i d i n g speed. W i t h r e s u l t s t a k e n from c o n s t a n t speed t e s t s , a s i n g l e v a l u e d f u n c t i o n o f the c o e f f i c i e n t o f f r i c t i o n v e r s u s v e l o c i t y o f s l i d i n g i s o b t a i n e d . T h i s approach has g e n e r a l l y l e d t o c o n s i d e r i n g t h e k i n e t i c c o e f f i c i e n t t o be c o n s t a n t o v e r t h e range o f v e l o c i t i e s e n c o u n t e r e d i n a g i v e n a p p l i c a t i o n , o r e l s e a l i n e a r v a r i a t i o n o f the c o e f f i c i e n t w i t h v e l o c i t y has been used. However, i n o b s e r v i n g the v a r i a t i o n o f t h e c o e f f i c i e n t o f f r i c t i o n d u r i n g the s l i p c y c l e , Sampson e t a l . [17] i n 1943 n o t i c e d a r a p i d drop i n f r i c t i o n f o r c e a t t h e b e g i n n i n g o f s l i d i n g , w i t h the. l o w e s t v a l u e b e i n g reached a t the end o f t h e c y c l e . Thus t h e y c o n c l u d e d t h a t the k i n e t i c f r i c t i o n was not a f u n c t i o n o f j u s t the s l i d i n g v e l o c i t y as i t was not a s i n g l e v a l u e d f u n c t i o n o f the v e l o c i t y . T h e i r r e s u l t s were not v e r y a c c u r a t e as t h e y were d e r i v e d by t w i c e g r a p h i c a l l y d i f f e r e n t i a t i n g d i s p l a c e m e n t r e c o r d s t o a r r i v e a t the a c c e l e r a t i o n . These f i n d i n g s were . not f o l l o w e d up by f u r t h e r i n v e s t i g a t i o n . The method used t o determine t h e f r i c t i o n v a l u e s d u r i n g s l i d i n g c o n s i s t e d o f a d d i n g the i n e r t i a f o r c e t o t h e s p r i n g f o r c e t o a r r i v e a t the f r i c t i o n f o r c e . I f t h e system i s . d e s c r i b e d by m x + cx + kx = F f (1.2.8) then i f the damping i s s m a l l we can combine i t w i t h t h e f r i c t i o n f o r c e w i t h o u t much e r r o r and we have 9 F f = m x + k x (1.2.9) Thus k n o w i n g t h e s y s t e m p a r a m e t e r s a s w e l l a s d i s p l a c e -ment and a c c e l e r a t i o n , we c a n compute t h e f r i c t i o n f o r c e F^. The same p r i n c i p l e was a p p l i e d by B e l l a nd B u r d e k i n [18] i n m e a s u r i n g f r i c t i o n u n d e r d y n a m i c c o n d i t i o n s . I n s t e a d o f t h e d i f f e r e n t i a t i o n o f Sampson e t a l . , h o w e v e r , t h e y u s e d an a c c e l e r o m e t e r and e l e c t r o n i c a l l y c o m b i n e d t h e i n e r t i a a nd s p r i n g f o r c e t e r m s t o a r r i v e a t t h e f r i c t i o n f o r c e . T hey a l s o f o u n d t h a t t h e u-v r e l a t i o n s h i p was n o t r e v e r s i b l e , b u t d u r i n g a s l i p c y c l e h a d t h e g e n e r a l f o r m shown i n F i g u r e 1.2.4. No a t t e m p t was made a t e x p l a i n i n g t h i s b e h a v i o u r . The p r e c e d i n g s u r v e y o f p r e v i o u s w o r k i n d i c a t e s t h a t • a c o n s i d e r a b l e amount o f i n v e s t i g a t i o n r e m a i n s t o be done i n t h e a n a l y s i s o f f r i c t i o n a l b e h a v i o u r as r e l a t e d t o s t i c k - s l i p v i b r a t i o n s . I n p a r t i c u l a r , i m p o r v e d t h e o r i e s and e x p e r i m e n t a l m ethods may e x p l a i n e x i s t i n g d i s c r e p a n c i e s . The p a r t i c u l a r c a s e o f a h y d r a u l i c a l l y d r i v e n s y s t e m 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 s h a s n o t b e e n t h e s u b j e c t o f much r e s e a r c h . The o p e r a t i o n o f a h y d r a u l i c ram was d e s c r i b e d by S h e a r e r [1 9 ] . He u s e d a s s u m p t i o n s o f o r i f i c e f l o w a t t h e c o n t r o l v a l v e , no f l u i d f r i c t i o n a nd i n e r t i a e f f e c t s , and z e r o v a l v e and p i s t o n l e a k a g e . A l t h o u g h he c o n s i d e r e d b o t h f l u i d c o m p r e s s i b i l i t y a n d e x p a n s i o n o f t h e h y d r a u l i c c y l i n d e r , i n t h e l a t t e r c a s e he o n l y c o n s i d e r e d t h e r a d i a l d e f o r m a t i o n o f t h e c y l i n d e r u n d e r p r e s s u r e , o m i t t i n g a x i a l d e f o r m a t i o n . M a t s u z a k i a n d H a s h i m o t o [20] u s e d S h e a r e r ' s a n a l y s i s w i t h t h e a d d i t i o n o f f l u i d f r i c t i o n and i n e r t i a e f f e c t s i n a n a l y z i n g a s y s t e m c o n s i s t i n g o f a h y d r a u l i c ram m o v i n g a s l i d i n g mass. T h e i r r e s u l t s do n o t a g r e e w i t h f i n d i n g s o f o t h e r s , as 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 i n c r e a s e d w i t h a v e r a g e s l i d i n g s p e e d , w h i c h i s u n l i k e t h e g e n e r a l f i n d i n g i n s t i c k -s l i p t h a t t h i s a m p l i t u d e d e c r e a s e s as t h e s p e e d i s i n c r e a s e d . I t seems p o s s i b l e t h a t r a t h e r t h a n o b s e r v i n g s t i c k - s l i p , t h e y w e r e o b s e r v i n g t h e o n s e t 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 s w h i c h c a n o c c u r i n a s y s t e m o f t h i s t y p e w i t h no a c t u a l t i m e o f s t i c k [ 2 1 ] . The r a n g e o f s l i d i n g v e l o c i t i e s u s e d was a l s o r a t h e r r e s t r i c t e d , v a r y i n g by a p p r o x i m a t e l y a f a c t o r o f f i v e , w h e r e a s v a r i a t i o n s b y a f a c t o r o f one h u n d r e d a r e common i n e x p e r i m e n t s , w i t h t h e p r e s e n t w o r k h a v i n g a v a r i a t i o n o f o v e r one t h o u s a n d . C o m p a r i s o n o f t h e M a t s u z a k i and H a s h i m o t o t h e o r y w i t h t h e i r e x p e r i m e n t a l d a t a i n d i c a t e s a d i s c r e p a n c y i n t h e s l i p v e l o c i t i e s b y a f a c t o r o f t h e o r d e r o f f o u r . Thus t h e n a t u r a l f r e q u e n c y o f t h e a c t u a l s y s t e m a p p e a r s t o h a v e b e e n c o n s i d e r a b l y l o w e r t h a n t h a t p r e d i c t e d by t h e t h e o r y . T h i s , may w e l l be due t o n e g l e c t o f e l a s t i c i t y i n t h e c o u p l i n g b e t w e e n t h e ram and t h e l o a d . A l s o t h e f r i c t i o n d a t a u s e d i n t h i s a n a l y s i s was u n s a t i s f a c t o r y , c o n s i s t i n g o f t h e a s s u m p t i o n o f a c o n s t a n t s t a t i c f r i c t i o n c o e f f i c i e n t a nd a k i n e t i c c o e f f i c i e n t f o r w h i c h a v e l o c i t y d e p e n d e n c e h a d b e e n d e r i v e d f r o m c o n s t a n t s p e e d t e s t s . The n e e d f o r f u r t h e r i n v e s t i g a t i o n i n b o t h t h e p r o b l e m 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 s i n g e n e r a l and t h e h y d r a u l i c ram type system i n p a r t i c u l a r i s apparent from t h i s s u r v e y o f p r e v i o u s work. F u r t h e r m o r e , an i n v e s t i g a t i o n o f the p r o c e s s e s r e l a t i n g t o s t a t i c and dynamic f r i c t i o n p r o v i d e s i n f o r m a t i o n o f fundamental v a l u e . C H A P T E R 2 2 .1 THEORY In F i g u r e 2.1.1 i s a s c h e m a t i c diagram o f the system under c o n s i d e r a t i o n a l o n g w i t h t h e d e f i n i t i o n s o f symbols used i n the a n a l y s i s . The complete system can be c o n s i d e r e d i n two s e p a r a t e p a r t s , namely the f l o w i n the h y d r a u l i c c i r c u i t , and the motion o f b oth the p i s t o n r o d and' the s l i d i n g mass which i s s u b j e c t e d t o f r i c t i o n a l f o r c e s . I n d e r i v i n g the e q u a t i o n g o v e r n i n g the f l o w , c o n s i d e r a t i o n must be g i v e n t o f l u i d c o m p r e s s i b i l i t y , c y l i n d e r e l a s t i c i t y , f l o w v a l v e c h a r a c t e r i s t i c s , p i s t o n l e a k a g e , and f r i c t i o n l o s s i n the p i p i n g . The e f f e c t o f each o f t h e s e w i l l be d e r i v e d i n d i v i d u a l l y and t h e n combined i n t h e complete f l o w e q u a t i o n : q. = A v + f l u i d c o m p r e s s i b i l i t y + l e a k a g e + system I p p e x p a n s i o n (2.V,1) q Q = A p v p ~ f l u i d c o m p r e s s i b i l i t y + l e a k a g e - system e x p a n s i o n where q^ and q are the f l o w s i n t o and out o f the c y l i n d e r r e s p e c t i v e l y and A^ i s t h e e f f e c t i v e p i s t o n a r e a . : F l u i d c o m p r e s s i b i l i t y : The e f f e c t o f c o m p r e s s i b i l i t y f o r a f l u i d o f b u l k modulus B i s g i v e n by dV = - | dp (2.1.2) where V i s the volume b e i n g compressed and p i s t h e a p p l i e d p r e s s u r e . Thus t h e c o n t r i b u t i o n o f t h e f l u i d c o m p r e s s i b i l i t y t o the f l o w i s dV. V. dp q 1 C O m P r d t B d t (2.1.3) dV V , dp_ o o ^2 xo compr d t B d t The volumes V\ and V o i n c l u d e the volume i n the p i p i n g as w e l l as c y l i n d e r volume. C y l i n d e r E x p a n s i o n : The e l a s t i c i t y o f the h y d r a u l i c c y l i n d e r and t h e p i p i n g c o n t r i b u t e s t o the f l o w terms. D e f i n i n g an e q u i v a l e n t c o m p l i a n c e per u n i t l e n g t h o f c y l i n d e r as k = ^ • i (2 1 4 ) e dp I we have q. = k "P v . dp. i e x p a n s i o n e A ^ (2.1.5) V dP~ . o 2 q . = - k — ^o e x p a n s i o n e ,. ,. ^ A d t P A d i s c u s s i o n on c a l c u l a t i n g t h i s s t i f f n e s s i s g i v e n i n Appendix I . Leakage: Laminar f l o w between p a r a l l e l p l a t e s one o f wh i c h i s moving has a v e l o c i t y p r o f i l e g i v e n by v Ap u = _ y + ( y - y d ) ( 2 . 1 . 6 ) A 2 Z U where u i s the v e l o c i t y and the o t h e r symbols as i n F i g u r e 2.1.2. The r e s u l t a n t l e a k a g e f l o w f o r an a n n u l a r s e c t i o n o f c i r c u m f e r e n c e D i s g i v e n by D u d r . I ™ [v- (2.1.7) A f o r c e can a l s o be a t t r i b u t e d t o the le a k a g e f l o w , and f o r the a n n u l a r s e c t i o n t h i s f o r c e F i s g i v e n as F = 7T D Z T Q ( 2 . 1 . 8 ) Where T q i s the f l u i d shear s t r e s s a t t h e s t a t i o n a r y w a l l ( f o r the h y d r a u l i c c y l i n d e r the s t a t i o n a r y w a l l r e p r e s e n t s the p i s t o n ) . E v a l u a t i n g T q from 15 g i v e s the l e a k a g e f o r c e as F]_ = v D [ Igv _ dAp_ ] (2.1.10) In a n a l y z i n g a h y d r a u l i c p i s t o n i t may be c o n v e n i e n t t o c o n s i d e r t h i s f o r c e as a p r e s s u r e drop Ap, g i v e n by F l r Ap = — (2.1.11) A P Flow V a l v e s : The v a l v e f l o w e q u a t i o n s are g i v e n f o r the case o f o r i f i c e f l o w . I n g e n e r a l i t may be n e c e s s a r y t o d e t e r m i n e the f l o w c h a r a c t e r i s t i c s by e x p e r i m e n t . I n t h e case o f a f o u r way v a l v e , note must be t a k e n o f o v e r l a p o r u n d e r l a p o f the s p o o l i n d e t e r m i n i n g the o r i f i c e a r e a . F o r o r i f i c e f l o w we have q. = C d a. 2 ( p s - p 4 ) P (2.1.12) q = C..a. 2 (p_ - p ) ° d l r 3 ^o ; P where the p r e s s u r e s are as shown i n F i g u r e 2.1.1, p i s the f l u i d d e n s i t y , C, the o r i f i c e c o e f f i c i e n t , and a., a the J d ' I o o r i f i c e a r e a s . 16 F r i c t i o n l o s s i n p i p i n g : A s s u m i n g l a m i n a r f l o w i n a c i r c u l a r p i p e o f d i a m e t e r D and l e n g t h I, t h e h e a d l o s s h i s g i v e n b y 2 I V I A p f h = f £• ~ = — - (2.1.13) D 2g pg wh e r e t h e f r i c t i o n f a c t o r f i s f = | | (2.1.14) Re b e i n g t h e R e y n o l d s number f o r t h e f l o w (Re = ^ — . The f l o w v e l o c i t y v ^ i n t h e p i p i n g i s g i v e n i n t e r m s o f t h e p i s t o n v e l o c i t y v ^ , p i s t o n a r e a A ^ , and p i p e a r e a A ^ by A v 7 = - £ v (2.1.15) 1 A p 1 C o m b i n i n g E q u a t i o n s (2.1.13, 2.1.14, and 2.1.15) y i e l d s f o r t h e f r i c t i o n a l p r e s s u r e d r o p 8-TTZUA p = v ^ = Rv^ (2.1.16) A P P F o r t h e f l o w v e l o c i t i e s e n c o u n t e r e d i n s t i c k - s l i p , t h i s t e r m becomes n e g l i g i b l e . F l o w E q u a t i o n : C o m b i n i n g e q u a t i o n s (2.1.3, 2.1.5, 2.1.7, and 2.1.12 i n t o 2.1.1) t h e e q u a t i o n s o f f l o w become V dp TTdD ( P l - P 2 ^ 2 1 + . V i d P l q. = A v + + Lv + J + k 1 P P B d t 2 P 6Zy e A d t P = c d a i / 2 ( P ^ " P 4 } (2.1.17) V Q d p 2 * d D ( P i - P z ) ^ 2 , V o d p 2 q = A v + [v + J - k ° P P B d t 2 P 6Zy e A d t p P where, u s i n g e q u a t i o n (2.1.16) p . - R v ^4 p P 2 = P 3 + R v p (2.1.18) These e q u a t i o n s a r e i n a v e r y u n w i e l d y form, and some s i m p l i f i c a t i o n s can be made. S i m p l i f y i n g A ssumptions: I f t h e f l o w s i n t o and out o f t h e c y l i n d e r a re v i r t u a l l y the same, we may p u t q ± * q 0 * q = + q Q ) / 2 ( 2 . 1 . 1 9 ) R e f e r r i n g t o e q u a t i o n s 2.1.18 i t can be seen t h a t under s t e a d y s t a t e f l o w b o t h q^ and q Q are i d e n t i c a l and 2.1.19 i s v a l i d . C o n s i d e r i n g t h e b e h a v i o u r o f t h e system under c o n d i t i o n s o f s t i c k - s l i p , i t may be noted t h a t i n t h e case o f no p i s t o n m o t i o n , dp^/dt and d p 2 / d t are o f o p p o s i t e 18 s i g n , t e n d i n g t o make and q Q e q u a l . F i n a l l y i f we have motion o f the p i s t o n , a g a i n the e f f e c t s on dp^/dt and d p 2 / d t are o p p o s i t e . Thus i n t h e case o f a s y m m e t r i c a l system, t h e assumption o f e q u a t i o n ( 2 . 1 . 1 9 ) appears t o be q u i t e r e a s o n a b l e . C o n s i d e r i n g t h e p i s t o n near t h e c e n t e r p o i n t o f i t s t r a v e l , we a c h i e v e symmetry w h i c h h e l p s t h e v a l i d i t y o f e q u a t i o n (2.1.19) and a l s o p r o v i d e s us w i t h a n o t h e r s i m p l i f i -c a t i o n , p u t t i n g jr | • ii V ± = V Q = V (2.1.20) O b v i o u s l y t h i s assumption l e a d s t o l i t t l e e r r o r i f t h e p i s t o n i s c l o s e t o the c e n t r e o f i t s s t r o k e . S u b s t i t u t i n g (2.1.19) and (2.1.20) i n t o (2.1.17) g i v e s 3 V 1 k d (p. -p ) TrDd TrDd q = _ [ _ + _£] + [ A + ] v + ( P i - P 2 ) 2 B A d t P 2 P 12ly 1 1 P = C d a v / p s - ( p 4 - p 3 ) (21.21) w i t h the f u r t h e r assumptions t h a t a. = a = a and t h a t r 1 o p = 0. *o The o r i f i c e a r e a s i n a f o u r way v a l v e w i t h no o v e r l a p o r u n d e r l a p w i l l be e q u a l , and a l t h o u g h t h e p r e s e n t equipment has i n d e p e n d e n t l y a d j u s t a b l e i n l e t and o u t l e t v a l v e s , b o t h were ke p t a t t h e same s e t t i n g s d u r i n g e x p e r i m e n t s , thus v a l i -d a t i n g the f i r s t o f t h e s e assumptions. S e t t i n g t h e o u t l e t 19 p r e s s u r e t o z e r o causes no l o s s i n g e n e r a l i t y . S u b t r a c t i n g t h e e q u a t i o n s (2.1.18), p l ~ p 2 = p 4 ~ p 3 " 2 R v p (2,1.22) 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 ti m e dt ( p l " P2> = It ( p4 - p3> - 2 R j f  ( 2 - 1 ' 2 3 ) dt Now s e t t i n g Ap = P 4 " P 3 V 1 k 2 B A p (2.1.24) y = A + ™ * P 2 12l\i and s u b s t i t u t i n g i n e q u a t i o n (2.1.21), t h e f l o w e q u a t i o n becomes dAp dv / a ^ r — - 2 R a ^ p + (y - 2 R6) V p + 6 A p = C d a p s ~ A p (2.1.25) ~ p F u r t h e r s i m p l i f i c a t i o n i s p o s s i b l e by l i n e a r i z i n g the o r i f i c e f l o w term. S i n c e i n most i n s t a n c e s t h e v a l u e o f Ap w i l l n o t change v e r y much compared t o p g - Ap, t h e assumption o f l i n e a r v a r i a t i o n o f f l o w i n t h e form 20 K ( p s - Ap) (2.1.26) w i l l be r e a s o n a b l e r e g a r d l e s s o f the e x a c t c h a r a c t e r i s t i c s o f the v a l v e . M e c h a n i c a l System: In c o n s i d e r i n g t h e motion o f t h e p i s t o n and mass, t h e i n e r t i a o f t h e f l u i d i n t h e c y l i n d e r and p i p e s may be s i g n i f i c a n t , Thus we d e f i n e an e q u i v a l e n t mass M g as M =• M. + M + p (V. + V ) + 2p£,A (2.1.27) e t p K i o £ P The l a s t term i s d e r i v e d by c o n s i d e r i n g t h e mass o f f l u i d I ^ n t w o e < 3 u a l l e n g t h s o f p i p e moving a t a v e l o c i t y as g i v e n by e q u a t i o n (2.1.15). The i n e r t i a f o r c e due t o t h i s f l o w i s dv^ dv F = 2p llAl = 2p I A — P (2.1.28) d t d t thus g i v i n g t h e e q u i v a l e n t mass as i n (2.1.27). The d r i v i n g f o r c e a c t i n g on the p i s t o n i s F = (p, - p.) A + i r D t i ^ + ^A p _ d R v ] (2.1.29) ^2 p d 2 P In most cases the second term w h i c h i s t h e f o r c e due t o p i s t o n l e a k a g e may be n e g l e c t e d . The r e m a i n i n g f o r c e s a c t i n g on t h e system are t h e f r i c t i o n a l f o r c e s a c t i n g a t t h e p i s t o n r o d s e a l s and a t t h e s l i d i n g mass. Assuming no r e l a t i v e m o tion between the p i s t o n and the mass, the e q u a t i o n o f m otion becomes dv M — 2 - + F. = A (Ap - 2 R V ) (2.1.30) e d t f P P t h e l e a k a g e f o r c e term h a v i n g been n e g l e c t e d . F^ i s the f r i c t i o n a l f o r c e a c t i n g on t h e system. D u r i n g s l i d i n g F^ = k i n e t i c f r i c t i o n f o r c e . D u r i n g s t i c k F^ <_ s t a t i c f r i c t i o n f o r c e . S i n c e many p r a c t i c a l systems do n o t have t h e d r i v e n mass d i r e c t l y on t h e end o f t h e p i s t o n r o d , a p p r e c i a b l e e l a s t i c i t y i n the c o n n e c t i o n may r e s u l t . I n the p r e s e n t apparatus t h i s e l a s t i c i t y e x i s t s i n t h e form o f a s t r a i n r i n g as shown i n F i g u r e 2.1.1. Thus i n t r o d u c i n g a s p r i n g o f s t i f f -ness k between the s l i d e r and p i s t o n r o d and d e f i n i n g M' = M + p (V. + V ) + 2p l7 A^ (2.1.31) P P 1 O L- p we have f o r the e q u a t i o n s o f m otion dv M 1 —2- + F_ + k (x - X ) = A (Ap - 2 R V ) P d t f p P m ? P dv M, — - + F_ + k ( x - x ) = 0 t -j. f m p d t m ^ (2.1.32) 22 The system b e h a v i o u r i s now d e s c r i b e d by t h e s i m u l t a n -eous s o l u t i o n o f e q u a t i o n s ( 2 . 1 . 3 2 ) and e q u a t i o n s ( 2 . 1 . 1 7 ) o r some s i m p l i f i e d v e r s i o n t h e r e o f . The o r i g i n a l e q u a t i o n s a r e v e r y cumbersome, r e s u l t i n g i n a s i x t h o r d e r system, and even a f t e r t h e s i m p l i f i c a t i o n s l e a d i n g t o e q u a t i o n (2.1.25) we have a f i f t h o r d e r system. F u r t h e r c o m p l i c a t i n g t h e s o l u t i o n o f t h e e q u a t i o n s i s the p r e s ence o f t h e f r i c t i o n a l f o r c e terms F^. As d i s c u s s e d i n the background, t h e s e f o r c e s are not w e l l d e f i n e d , and a r e a s o n a b l e d e s c r i p t i o n o f t h e s e f o r c e s i s n e c e s s a r y t o p e r m i t a c c u r a t e s o l u t i o n s . A c l o s e d form s o l u t i o n o f e q u a t i o n s s i m i l a r t o (2.1.25) and (2.1.30) w i t h a l i n e a r i z e d f l o w t e r m , no l e a k -age, c o n s t a n t s t a t i c f r i c t i o n and k i n e t i c f r i c t i o n v a r y i n g l i n e a r l y w i t h v e l o c i t y was p r e s e n t e d by M a t s u z a k i and Hashimoto [2 0 ] . However, the r e s u l t s do n o t seem c o m p a t i b l e w i t h g e n e r a l o b s e r v a t i o n s on s t i c k - s l i p as may be e x p e c t e d w i t h t h e s i m p l i f -i c a t i o n s i n v o l v e d i n the t r e a t m e n t o f t h e f r i c t i o n f o r c e s a l o n e . I n the p r e s e n t work n u m e r i c a l s o l u t i o n s o f the e q u a t i o n s were c a r r i e d out u s i n g f r i c t i o n f o r c e d a t a from e x p e r i m e n t s . -T h i s approach a l s o p e r m i t t e d s i m p l i f i c a t i o n s i n t h e e q u a t i o n s t o be e v a l u a t e d f o r v a l i d i t y . C H A P T E R 3 3.1 EXPERIMENTAL APPARATUS F i g u r e 3.1.1 i l l u s t r a t e s t h e fundamentals 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 , and F i g u r e 3.1.2 shows a photograph o f the a c t u a l l a y o u t . The b a s i c system c o n s i s t s o f a h y d r a u l i c ram d r i v i n g a mass a l o n g s l i d e ways. The h y d r a u l i c c y l i n d e r has a 24 i n c h s t r o k e , 1.5 i n c h bore and 1 i n c h d i a m e t e r p i s t o n r o d . As seen i n t h e f i g u r e s the c y l i n d e r was mounted on a l a t h e bed by c e n t r a l l y l o c a t e d b a l l b e a r i n g p i l l o w b l o c k s . T h i s was done t o s i m p l i f y t h e problem o f a l i g n m e n t . A t t a c h e d t o t h e c y l i n d e r i s a p a n e l c a r r y i n g two f l o w c o n t r o l l i n g n e e d l e v a l v e s . The lower s l i d i n g s u r f a c e s whose p r e p a r a t i o n i s d i s -c u s sed l a t e r were .25 i n x 2 i n b a r s t o c k r a i l s w hich were h e l d i n p l a c e by clamps so t h e y c o u l d be e a s i l y r e p l a c e d ( F i g u r e 3.1.3). The r a i l bases were a r r a n g e d so t h a t the t o p s u r f a c e s o f the r a i l s were i n l i n e w i t h t h e c e n t e r l i n e o f the p i s t o n r o d , thus e l i m i n a t i n g a c o u p l e from a c t i n g on the s l i d -i n g p l a t f o r m when i n o p e r a t i o n . The d r i v e n mass c o n s i s t e d o f a 24 pound p l a t f o r m c a r r y -i n g up t o e i g h t 40 pound l e a d w e i g h t s . The p l a t f o r m was p r o -v i d e d w i t h t h r e e p o i n t s u s p e n s i o n on s e l f - a l i g n i n g specimen h o l d e r s ( F i g u r e 3.1.4). The s l i d i n g specimens were a t t a c h e d t o the h e m i s p h e r i c a l h o l d e r s by a s i n g l e screw, and were thus e a s i l y r e p l a c e a b l e . To ensure t h a t t h e p l a t f o r m had no motion i n the h o r i z o n t a l p l a n e p e r p e n d i c u l a r t o p i s t o n t r a v e l , b a l l b e a r i n g g u i d e s w e r e p r o v i d e d a s s e e n i n F i g u r e 3.1.4. T h e s e h o w e v e r p r o v e d t o be u n n e c e s s a r y as t h e c o n n e c t i o n t o t h e p i s t o n r o d was s u f f i c i e n t l y s t i f f t o p r e v e n t l a t e r a l m o t i o n when t h e l a t h e b e d was l e v e l . The d e t a i l e d t r e a t m e n t s g i v e n t o t h e s l i d i n g s u r f a c e s a r e g i v e n i n t h e e x p e r i m e n t a l p r o c e d u r e . I n g e n e r a l t h e r a i l s w e r e f i n i s h e d b y g r i n d i n g , and t h e s l i d e r s b y g r i n d i n g a nd l a p p i n g . The h y d r a u l i c c i r c u i t i s i l l u s t r a t e d s c h e m a t i c a l l y i n F i g u r e 3.1.5. The p r e s s u r e was s u p p l i e d b y a t i l t e d a x i s • p i s t o n t y p e c o n s t a n t d i s p l a c e m e n t pump d r i v e n b y an e l e c t r i c m o t o r . O r i g i n a l l y a c h a i n d r i v e was u s e d b u t t h i s was e v e n -t u a l l y r e p l a c e d b y a b e l t t o r e d u c e n o i s e and v i b r a t i o n . A p r e s s u r e r e l i e f v a l v e r e g u l a t e d t h e o u t l e t p r e s s u r e o f t h e pump. T h i s p r o v i d e d a r e a s o n a b l e c o n s t a n t p r e s s u r e s u p p l y , b u t some p u l s a t i o n s f r o m t h e pump w e r e s t i l l d e t e c t a b l e a t t h e h y d r a u l i c c y l i n d e r . To e l i m i n a t e t h e s e , an a c c u m u l a t o r was a d d e d t o t h e s y s t e m . B e s i d e s s m o o t h i n g a n y p r e s s u r e f l u c t u a t i o n s c a u s e d by t h e p u m p - r e l i e f v a l v e c o m b i n a t i o n , t h e a c c u m u l a t o r c o u l d be u s e d t o p r o v i d e a c o m p l e t e l y p u l s a t i o n f r e e c o n s t a n t p r e s s u r e f o r a s h o r t t r a v e r s e b y u s i n g i t on t h e blow-down p r i n c i p l e . ; F rom t h e a c c u m u l a t o r t h e o i l p a s s e d t o a f o u r way v a l v e w h i c h was u s e d s i m p l y f o r d i r e c t i o n a l c o n t r o l . S p e e d c o n t r o l was a c h i e v e d b y means o f .25 i n c h n e e d l e v a l v e s whose c h a r a c t e r i s t i c s a r e g i v e n i n A p p e n d i x I I . T h e s e v a l v e s w e r e l o c a t e d b e t w e e n t h e f o u r way v a l v e and t h e h y d r a u l i c c y l i n d e r as s e e n i n F i g u r e 3.1.1. 25 A l t h o u g h most s i m i l a r systems would use a f o u r way v a l v e f o r b o t h d i r e c t i o n and speed c o n t r o l , i t was d e c i d e d t h a t n e e d l e v a l v e s would be a s i m p l e r method o f f l o w c o n t r o l i n t h i s c a s e . V e r y low f l o w r a t e s were e a s i l y a c h i e v e d . With b o t h v a l v e s a d j u s t e d t o an e q u a l o p e n i n g , t h e c o n f i g u r -a t i o n i n v a l v i n g was e s s e n t i a l l y t h e same as i f a f o u r way v a l v e had been used, w i t h t h e o n l y d i f f e r e n c e b e i n g any p o s s i b l e v a r i a t i o n s i n the p r e s s u r e - f l o w c h a r a c t e r i s t i c s . As seen l a t e r , however, t h i s i s a t r i v i a l d i f f e r e n c e i f i t does e x i s t as o v e r the p r e s s u r e drop f l u c t u a t i o n s e n c o u n t e r e d , a l i n e a r i z e d p r e s s u r e - f l o w c h a r a c t e r i s t i c i s j u s t i f i a b l e . From the f o u r way v a l v e t h e o i l was r e t u r n e d d i r e c t l y t o the sump which was d e s i g n e d f o r a f i f t e e n g a l l o n c a p a c i t y i n o r d e r t o m i n i m i z e t e m p e r a t u r e f l u c t u a t i o n s i n t h e o i l . •, A l l t he h y d r a u l i c c o n n e c t i o n s were made w i t h f l e x i b l e hose w i t h the e x c e p t i o n o f t h e l i n e s from the n e e d l e v a l v e s t o the c y l i n d e r i n whi c h case s t e e l t u b i n g was used. O r i g i n a l l y the pump and motor assembly were l o c a t e d i n the l a b o r a t o r y n e x t t o t h e l a t h e bed. N o i s e and v i b r a t i o n from t h i s u n i t was s u f f i c i e n t t o be o b s e r v e d i n t h e i n s t r u m e n -t a t i o n , and thus the pump and motor were s u b s e q u e n t l y s i t u a t e d u nderneath the l a b o r a t o r y r e s t i n g on the ground, r e s u l t i n g i n v e r y good i s o l a t i o n . Freedom from e x t e r n a l v i b r a t i o n i s d e s i r a b l e i n s t u d i e s i n v o l v i n g s l i d i n g f r i c t i o n as t h e f r i c t i o n a l f o r c e s can be s i g n i f i c a n t l y a f f e c t e d by v i b r a t i o n [ 2 2 ] . The r e s e r v o i r , a c c u m u l a t o r , and r e l i e f v a l v e were l o c a t e d on the l a b o r a t o r y f l o o r near the main a p p a r a t u s . The f o u r way v a l v e was mounted on t h e l a t h e bed as i n F i g u r e 3.1.1, and p r o v i s i o n was made f o r s h u t t i n g t h e v a l v e a u t o m a t i c -a l l y a t any p o i n t i n the p i s t o n t r a v e r s e . T h i s was n e c e s s a r y s i n c e the p i s t o n s t r o k e was 24 i n c h e s and l a c k i n g a g r i n d e r w i t h such a l o n g t r a v e r s e , the r a i l s had t o be made o n l y 18 i n c h e s i n l e n g t h . T h i s r e s t r i c t i o n on t h e s t r o k e caused no d i f f i c u l t y t o the e x p e r i m e n t a t i o n , as any s e c t i o n o f the • s t r o k e c o u l d be used by r e p o s i t i o n i n g t h e r a i l s , and most o f the work was concerned w i t h low speed motion near t h e mid p o i n t o f p i s t o n t r a v e l . 3.2 MEASUREMENTS The s u p p l y p r e s s u r e w h i c h was a d j u s t a b l e w i t h t h e r e l i e f v a l v e was m o n i t o r e d by a 0-2000 pound p e r square i n c h d i a l t y p e p r e s s u r e gauge. T h i s gauge was mounted on t h e l a t h e bed as seen i n t h e i l l u s t r a t i o n s , w i t h the p r e s s u r e t a p b e i n g l o c a t e d between the ac c u m u l a t o r and t h e f o u r way v a l v e . The h y d r a u l i c p r e s s u r e a t each end o f the c y l i n d e r was measured by a K i s t l e r Model 601A q u a r t z p r e s s u r e t r a n s d u c e r c a p a b l e o f r e s o l v i n g t o .1 pounds p e r square i n c h w i t h a range o f 0 t o 3000 pounds p e r square i n c h . The c a p a c i t a n c e changes i n t he t r a n s d u c e r s were c o n v e r t e d i n t o v o l t a g e s i g n a l s p r o -p o r t i o n a l t o the p r e s s u r e s by u s i n g K i s t l e r Charge A m p l i f i e r s , model 504. With t h i s c o m b i n a t i o n a s e n s i t i v i t y up t o 1 pound p e r v o l t was a t t a i n a b l e , and l i n e a r i t y d e v i a t i o n a c c o r d i n g t o s p e c i f i c a t i o n s d i d n ot exceed .1 p e r c e n t . The n a t u r a l f r e -27 q u e n c y o f t h e t r a n s d u c e r s was 140,000 H e r t z , a n d t h e f r e q u e n c y r e s p o n c e o f t h e c h a r g e a m p l i f i e r s was f l a t f r o m n e a r D.C. t o 100,000 H e r t z . Thus t h e f r e q u e n c i e s e n c o u n t e r e d i n t h e e x p e r i m e n t s w h i c h r a n g e d f r o m n e a r D.C. t o l e s s t h a n 100 H e r t z w e r e e a s i l y accommodated. E a c h p r e s s u r e c o u l d be i n d e p e n d e n t l y d i s p l a y e d on an o s c i l l o s c o p e o r by u s i n g a d i f f e r e n t i a l a m p l i f i e r t h e p r e s s u r e d i f f e r e n c e a c r o s s t h e p i s t o n c o u l d be o b s e r v e d . 4 A s t r a i n r i n g o f 1.9 x 10 p o u n d s p e r i n c h s t i f f n e s s was mounted b e t w e e n t h e p i s t o n and t h e s l i d i n g mass. T h i s i n t r o d u c e d a d d i t i o n a l e l a s t i c i t y t o t h e s y s t e m , m a k i n g i t more g e n e r a l s i n c e many c o n f i g u r a t i o n s i n v o l v i n g a h y d r a u l i c ram and a s l i d i n g mass do n o t h a v e a d i r e c t r i g i d c o n n e c t i o n b e -twe e n t h e p i s t o n r o d and mass a s was p o s s i b l e i n t h e p r e s e n t e x p e r i m e n t a l a p p a r a t u s . I t may w e l l be t h a t i g n o r i n g e l a s t i c i t y i n t h i s c o u p l i n g was one o f t h e r e a s o n s f o r t h e d i s c r e p a n c y b e t w e e n e x p e r i m e n t a l a n d t h e o r e t i c a l r e s u l t s o f M a t s u z a k i a n d H a s h i m o t o . The f o r c e e x e r t e d o n t h e s t r a i n r i n g was c o n v e r t e d t o a v o l t a g e b y u s e o f a b r i d g e a m p l i f i e r . Thus t h e f o r c e b e t w e e n t h e p i s t o n r o d and t h e s l i d i n g p l a t f o r m c o u l d be e a s i l y m o n i t o r e d . The u s e o f t h i s s t r a i n r i n g a l s o made i t p o s s i b l e t o s e p a r a t e t h e f r i c t i o n f o r c e s a t t h e p i s t o n and p i s t o n r o d s e a l s f r o m t h e f o r c e s a t t h e s l i d e r s . The s e a l a n d p i s t o n f r i c t i o n i s s i m p l y t h e d i f f e r e n c e b e t w e e n t h e p r e s s u r e f o r c e ; on t h e p i s t o n and t h e f o r c e on t h e s t r a i n r i n g , u n l e s s t h e i n e r t i a f o r c e s i n t h e p i s t o n r o d become s i g n i f i c a n t . 28 The s l i d e r motion was p r i m a r i l y m o n i t o r e d by a Sanborn LVSyn v e l o c i t y t r a n s d u c e r model 6-LV-4. The t r a n s d u c e r i s o f the type i n w h i c h a magnet moving th r o u g h a c o i l i n d u c e s a v o l t a g e w h i c h i s p r o p o r t i o n a l t o t h e magnet v e l o c i t y . W ith a u s e a b l e s t r o k e o f 4 i n c h e s and a s e n s i t i v i t y o f 600 m i l l i v o l t s p e r i n c h p e r second, t h i s t r a n s d u c e r was q u i t e s u i t a b l e f o r • the s l o w , u n i d i r e c t i o n a l motion b e i n g o b s e r v e d . A l t h o u g h a d i s p l a c e m e n t t r a n s d u c e r was a v a i l a b l e , i t was r a r e l y used, s i n c e the c u m u l a t i v e n a t u r e o f t h e d i s p l a c e -ment made i t p o s s i b l e t o observe o n l y a few c y c l e s b e f o r e . r e p o s i t i o n i n g o f t h e t r a n s d u c e r became n e c e s s a r y . I n t e g r a t i o n o f the v e l o c i t y s i g n a l s c o u l d be used t o det e r m i n e d i s p l a c e -ments when d i r e c t measurements were not made. :, As mentioned i n the background, B e l l and B u r d e k i n o b s e r v e d k i n e t i c f r i c t i o n v a l u e s d u r i n g non s t e a d y s l i d i n g by e l e c t r o n i c a l l y a d d i n g the i n e r t i a f o r c e o f the s l i d e r t o the a p p l i e d s p r i n g f o r c e . As d i s c u s s e d i n t h e r e s u l t s , s o l u t i o n o f the g o v e r n i n g e q u a t i o n s i n d i c a t e d t h a t t h e k i n e t i c f r i c t i o n v a l u e s o b t a i n e d from s t e a d y s t a t e s l i d i n g were not a p p l i c a b l e d u r i n g the s l i p c y c l e . Thus a d i f f e r e n t i a t i n g c i r c u i t was d e s i g n e d t o be used i n c o n j u n c t i o n w i t h the v e l o c i t y t r a n s d u c e r t o p r o v i d e the a c c e l e r a t i o n term n e c e s s a r y f o r t h i s method o f measuring f r i c t i o n f o r c e s under unsteady s l i d i n g c o n d i t i o n s . A l t h o u g h some r e a s o n a b l e r e s u l t s were a c h i e v e d , t h e e l e c t r o n i c n o i s e p i c k up o f t h e v e l o c i t y t r a n s d u c e r was s u f f i c i e n t t h a t a f t e r d i f f e r e n t i a t i o n t h e h i g h e r f r e q u e n c y components r e s u l t e d i n e x c e s s i v e l y l a r g e n o i s e l e v e l s on the a c c e l e r a t i o n s i g n a l . I n o r d e r t o pursue t h i s approach f u r t h e r , s i n c e i t appeared p r o m i s i n g , an a c c e l e r o m e t e r was a c q u i r e d and mounted on t h e s l i d i n g p l a t f o r m . (Appendix I I c o n t a i n s a d i s c u s s i o n o f t h e a c c e l e r o m e t e r and i t s u s e ) . T h i s t e c h n i q u e p e r m i t t e d much b e t t e r measurements o f f r i c t i o n f o r c e s under t r a n s i e n t con-d i t i o n s , and a l t h o u g h l i t t l e a t t e n t i o n seems t o have been g i v e n t o t h i s approach i n the l i t e r a t u r e , i n f o r m a t i o n d e r i v e d from measurements o f t h i s t y p e i s o f fundamental importance i n s t u d i e s i n v o l v i n g f r i c t i o n . The i n f o r m a t i o n from the v a r i o u s t r a n s d u c e r s was . d i s p l a y e d on a T e k t r o n i x Type 564 s t o r a g e o s c i l l o s c o p e . Thus s i n g l e c y c l e s o f s h o r t d u r a t i o n c o u l d be ob s e r v e d a t l e i s u r e , o r p h o t o g r a p h i c r e c o r d s c o u l d be made when s u f f i c i e n t i n f o r m a t i o n c o u l d n o t be c o n v e n i e n t l y r e a d d i r e c t l y from t h e s c r e e n . By u s i n g an a m p l i f i e r r a t h e r t h a n a time base f o r the h o r i z o n t a l d e f l e c t i o n , i t was a l s o p o s s i b l e t o o b t a i n p l o t s o f d i f f e r e n t v a r i a b l e s a g a i n s t each o t h e r . An example o f t h i s i s t h e 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 s l i d i n g v e l o c i t y . C H A P T E R H 4.1 EXPERIMENTAL PROCEDURE A f t e r t h e e x p e r i m e n t a l a p p a r a t u s was i n i t i a l l y assembled, the f i r s t phase o f e x p e r i m e n t a t i o n c o n s i s t e d o f c h e c k i n g t h e c a l i b r a t i o n o f the i n s t r u m e n t a t i o n , t h e f l o w v a l v e s , and the system s t i f f n e s s . T h i s c a l i b r a t i o n p r o c e d u r e i s d e s c r i b e d i n Appendix I I . System V e r i f i c a t i o n : The f i r s t o p e r a t i o n o f t h e system was a q u a l i t a t i v e t e s t t o see i f 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 s would a c t u a l l y o c c u r . T h i s p r e l i m i n a r y t e s t i n g was done u s i n g r a i l s o f unprepared SAE 1020 c o l d r o l l e d s t e e l and s l i d e r s o f A t l a s Nutherm hardened t o 53 R o c k w e l l C and f i n i s h e d by g r i n d i n g . The e x p e c t e d form o f s t i c k - s l i p was o b s e r v e d , and t h e d i s p l a c e -ment-time r e c o r d was found t o be s u r p r i s i n g l y c o n s i s t e n t o ver a l a r g e number o f c y c l e s . S i m i l a r b e h a v i o u r was o b s e r v e d d u r i n g subsequent e x p e r i m e n t a t i o n . I n many o t h e r systems used f o r t h e i n v e s t i g a t i o n o f s t i c k - s l i p , c o n s i s t e n c y o v e r o n l y a few c y c l e s has been a c h i e v e d , and t h e r e s u l t s w i t h t h e p r e s e n t a p p a r a t u s are p r o b a b l y a t t r i b u t a b l e t o t h e l a r g e s i z e o f the s l i d e r s i n comparison t o t h e d i s p l a c e m e n t p e r c y c l e (about 200:1), r e s u l t i n g i n an i n t e g r a t e d e f f e c t o f any s l i g h t v a r i -a t i o n s i n s u r f a c e p r o p e r t i e s . S u r f a c e P r e p a r a t i o n : I n o r d e r t o o b t a i n u s e f u l d a t a from t h e equipment, t h e f r i c t i o n s u r f a c e s had t o be p r e p a r e d by a r e p e a t a b l e p r o c e d u r e . 31 As p r e v i o u s l y m e n t i o n e d , t h e s l i d e r s w e r e made o f A t l a s N u t h e r m h a r d e n e d t o 53 R o c k w e l l C. They w e r e f i n i s h e d i n a s u r f a c e g r i n d e r w i t h a r e s u l t i n g s u r f a c e r o u g h n e s s o f a p p r o x i m a t e l y 6 m i c r o i n c h e s RMS a c r o s s t h e d i r e c t i o n o f g r i n d i n g a n d 4 m i c r o i n c h e s RMS a l o n g t h e g r i n d . The r a i l s o f SAE 1020 w e r e s t r e s s r e l i e v e d a t 1100. d e g r e e s F a h r e n h e i t f o r one h o u r a nd t h e n f i n i s h e d b y g r i n d -i n g . The h a r d n e s s a f t e r s t r e s s r e l i e v i n g was 64-65 R o c k w e l l B, and t h e s u r f a c e r o u g h n e s s was t h e same as f o r t h e s l i d e r s . B e f o r e e x p e r i m e n t a l r u n s , b o t h r a i l s a n d s l i d e r s w e r e w a s h e d t h o r o u g h l y w i t h t r i c h l o r o e t h y l e n e . N o r m a l l o a d s u n d e r 200 pounds w i t h t h e a b o v e s u r f a c e s g a v e v i r t u a l l y no s t i c k - s l i p . U n d e r h i g h e r l o a d s , s t i c k - s l i p b e g a n t o o c c u r , b u t t h e f o r m d i d n o t r e m a i n c o n s i s t e n t , and t h e c o e f f i c i e n t o f f r i c t i o n was o b s e r v e d t o i n c r e a s e c o n s i d e r -a b l y as t h e m o t i o n p r o g r e s s e d . T h i s i n c r e a s e a p p l i e d t o b o t h t h e s t a t i c a n d k i n e t i c f r i c t i o n c o e f f i c i e n t s as e v i d e n c e d by. t h e f a c t t h a t t h e c o m p l e t e s t r a i n r i n g f o r c e r e c o r d s h i f t e d t o a h i g h e r v a l u e . I n s p e c t i o n r e v e a l e d t h a t t h i s r i s e i n f r i c t i o n was a c c o m p a n i e d b y s e v e r e s c o r i n g o f t h e r a i l s . Sub-s e q u e n t t e s t s w i t h t h e same r a i l s r e f i n i s h e d t o 8 m i c r o i n c h e s by.22 m i c r o i n c h e s RMS r o u g h n e s s a l o n g a n d a c r o s s t h e g r i n d r e s p e c t i v e l y a n d t h e s l i d e r s l a p p e d a f t e r g r i n d i n g t o 18 m i c r o i n c h e s RMS r o u g h n e s s r e s u l t e d i n s i m i l a r b e h a v i o u r . A l t h o u g h g r e a t c a r e was t a k e n t o c l e a n t h e s u r f a c e s , and t h e edges, o f t h e s l i d e r s w e r e r o u n d e d s l i g h t l y t o l e s s e n t h e s t r e s s c o n -c e n t r a t i o n s a t t h e c o n t a c t , t h i s c o m b i n a t i o n o f m e t a l s c o n -s i s t e n t l y g a ve c o n s i d e r a b l e s c o r i n g e x c e p t u n d e r t h e l i g h t e s t l o a d s . T h i s r e s u l t e d i n n o n - r e p r o d u c i b l e s u r f a c e c o n d i t i o n s and was u n s a t i s f a c t o r y f o r q u a n t i t a t i v e e x p e r i m e n t s . S i n c e the s c o r i n g was p r i m a r i l y i n the s o f t e r r a i l s , w i t h t h e s l i d e r s showing v e r y l i t t l e damage, i t was d e c i d e d t o use a h a r d e r s e t o f r a i l s . Thus r a i l s o f A t l a s Chromoloy o f hardness o f 9 0 R o c k w e l l B were p r e p a r e d and f i n i s h e d by g r i n d i n g t o a roughness o f about 10 t o 20 m i c r o i n c h e s RMS roughness. U s i n g t h e s e r a i l s w i t h the l a p p e d s l i d e r s , numerous t r a v e r s e s c o u l d be made b e f o r e the f r i c t i o n f o r c e showed a r i s e , w h ich was a g a i n a t t r i b u t a b l e t o t h e commencement o f s c o r i n g . A l t h o u g h the s c o r i n g a g a i n i n v a l i d a t e d f u r t h e r t e s t s b e f o r e r e f i n i s h i n g , i t was not as s e v e r e as w i t h t h e s o f t e r r a i l s . I t was p o s s i b l e t o o b t a i n d a t a u s i n g t h i s c o m b i n a t i o n , b u t t h e e v e n t u a l change i n c o n d i t i o n s was u n d e s i r e a b l e . To a l l e v i a t e t h i s problem, l i q u i d p e t r o l a t u m (U.S.P.) was e v e n t u a l l y s u p p l i e d t o the s u r f a c e s as a l u b r i c a n t . T h i s v i r t u a l l y e l i m i n a t e d s c o r i n g , and s i n c e the g e n e r a l b e h a v i o u r o f the system was not a l t e r e d e x c e p t f o r a d e c r e a s e i n the magnitude o f t h e f r i c t i o n f o r c e s , t h e m a j o r i t y o f t h e d a t a was t a k e n u s i n g t h i s l u b r i -c a n t . An a d d i t i o n a l advantage o f t h e l u b r i c a n t was a f u r t h e r improvement i n the c o n s i s t e n c y o f t h e r e s u l t s . Not o n l y was the s t i c k - s l i p c o n s t a n t i n a m p l i t u d e and p e r i o d f o r a l a r g e , number o f c y c l e s , b u t a l s o t e s t s performed on d i f f e r e n t o c c a s i o n s w i t h r e f i n i s h e d s u r f a c e s were i n good agreement. T h i s a g a i n i s a r e s u l t o f t e n d i f f i c u l t t o a c h i e v e i n s t i c k -s l i p e x p e r i m e n t s . 4.2 MEASUREMENT OF FRICTION VALUES 33 V a l u e s o f 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 were de t e r m i n e d from t h e maximum f o r c e measured by t h e s t r a i n r i n g d u r i n g a s t i c k - s l i p c y c l e . The e a r l i e r e x p e r i m e n t s suggested t h a t t h i s c o e f f i c i e n t a p p r o x i m a t e l y f o l l o w e d an e q u a t i o n o f the form o f e q u a t i o n (1.2.7). S u i t a b l e c o n s t a n t s were e v a l u a t e d , and u s i n g t h i s r e l a t i o n w i t h a c o n s t a n t v a l u e o f k i n e t i c f r i c t i o n as d e r i v e d from s t e a d y s l i d i n g t e s t s , t h e g o v e r n i n g e q u a t i o n s were s o l v e d n u m e r i c a l l y . S i n c e the c o n s t a n t s f o r e q u a t i o n (1.2.7) were d e r i v e d from e x p e r i m e n t a l d a t a , n a t u r a l l y the p e r i o d and maximum f o r c e from the computed r e s u l t s were i n agreement w i t h e x p e r i m e n t . However, i t appeared t h a t t h e 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 used d i d not p r o v i d e a p p r o p r i a t e d i s s i p a t i o n , as t h e c a l c u l a t e d s o l u t i o n s r e s u l t e d i n s l i p a m p l i t u d e s and f o r c e f l u c t u a t i o n s g r e a t e r t h a n t h o s e i n the e x p e r i m e n t s . There seemed t o be l i t t l e j u s t i f i c a t i o n i n u s i n g a c o n s t a n t v a l u e o f k i n e t i c f r i c t i o n h i g h enough t o cause the a p p r o p r i a t e r e d u c t i o n i n a m p l i t u d e , as t h e v a l u e s d e r i v e d ' from s t e a d y s t a t e runs were q u i t e c o n s i s t e n t . An a l t e r n a t e approach was t o assume some v i s c o u s damping b e h a v i o u r a t t h e s l i d e r , c a u s i n g an i n c r e a s e i n t h e f r i c t i o n f o r c e . However, w i t h the slow s l i d i n g v e l o c i t i e s i n v o l v e d (of the o r d e r o f .1 i n c h e s per s e c o n d ) , a damping term s u f f i c i e n t l y l a r g e t o cause a : s i g n i f i c a n t change was c o m p l e t e l y out o f p r o p o r t i o n , b e i n g o f the o r d e r o f t e n s o f pounds p e r i n c h p e r second. I f such 34 damping e x i s t e d , e x t r e m e l y h i g h f r i c t i o n v a l u e s would have been found i n h i g h e r speed s t e a d y s t a t e s l i d i n g e x p e r i m e n t s , whereas no v e l o c i t y dependence was measurable. The s e a r c h f o r an e x p l a n a t i o n f o r t h e a p p a r e n t l y h i g h d i s s i p a t i o n d u r i n g s l i d i n g e v e n t u a l l y l e d t o the i n s t r u m e n -t a t i o n d e s c r i b e d p r e v i o u s l y 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 unsteady s l i d i n g . The r e s u l t s o f t e s t s thus performed were as shown i n F i g u r e 4.2.1. I t i s apparent t h a t the f r i c t i o n f o r c e does not i n f a c t drop i n s t a n t a n e o u s l y t o some lower k i n e t i c one, b u t d e c r e a s e s o v e r a c o n s i d e r a b l e l e n g t h o f t i m e , e x p l a i n i n g a h i g h e r average v a l u e o f the • f r i c t i o n a l d i s s i p a t i o n . U s i n g t h i s e x p e r i m e n t a l l y o b s e r v e d k i n e t i c f r i c t i o n i n the n u m e r i c a l c a l c u l a t i o n s r e s u l t e d i n good agreement w i t h e x p e r i m e n t . I t s h o u l d be noted t h a t a l t h o u g h t h e n a t u r e of the s t a t i c f r i c t i o n e q u a t i o n f o r c e d agreement between c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s f o r p e r i o d and s t a t i c f r i c t i o n f o r c e , t h i s k i n e t i c f r i c t i o n was i n d i r e c t l y measured, and does not f o r c e agreement i n a s i m i l a r f a s h i o n . A l s o the o b s e r v e d r e s u l t i s i n q u a l i t a t i v e agreement w i t h t h e o b s e r v a t i o n s o f Sampson e t a l . [17] and B e l l and B u r d e k i n [18]. A problem i n d e a l i n g w i t h t h i s o b s e r v e d k i n e t i c f r i c t i o n c u rve arose i n t r y i n g t o f i n d a s i m p l e m a t h e m a t i c a l r e p r e s e n -t a t i o n f o r c o m p u t a t i o n a l p u r p o s e s . O b v i o u s l y t h e c o e f f i c i e n t o f f r i c t i o n c o u l d not be e x p r e s s e d as a f u n c t i o n o f t h e s l i d i n g v e l o c i t y o r a c c e l e r a t i o n as i t would not y i e l d a s i n g l e v a l u e d e x p r e s s i o n . The time dependent r e l a t i o n w hich was e a s i l y o b s e r v e d on t h e o s c i l l o s c o p e d i d n o t appear t o have a s i m p l e 35 f o r m , a n d as d i s c u s s e d l a t e r , i t i s d i f f i c u l t t o d e f i n e t h e t i m e a t t h e i n c e p t i o n o f s l i p i n t h e e x p e r i m e n t s . I n t e r e s t -i n g l y a p l o t o f t h e f r i c t i o n f o r c e v e r s u s d i s p l a c e m e n t i n t h e c y c l e r e s u l t e d i n an a p p r o x i m a t e l y e x p o n e n t i a l r e l a t i o n s h i p ( F i g u r e 4 . 2 . 2 ) , a nd t h i s was t h e one u s e d i n n u m e r i c a l c o m p u t a t i o n s . I n c o m p a r i n g n u m e r 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 t , i s n o t s u f f i c i e n t t o h a v e a g r e e m e n t i n d i s p l a c e m e n t s p e r c y c l e t o c o n f i r m t h e v a l i d i t y o f t h e m a t h e m a t i c a l m o d e l . I n t h e r e s u l t s o f M a t s u z a k i a n d H a s h i m o t o ( F i g u r e 4.2.3) i t i s o b v i o u s t h a t t h e r e i s p o o r c o r r e l a t i o n i n t h e v e l o c i t i e s . F o r t h e p r e s e n t c a s e , F i g u r e 4.2.4 shows a c o m p a r i s o n o f v e l o c i t i e s and f r i c t i o n f o r c e s f r o m e x p e r i m e n t and n u m e r i c a l c a l c u l a t i o n s . The e x p e r i m e n t a l c y c l e u s e d i n t h e c o m p a r i s o n was c h o s e n o n t h e b a s i s o f h a v i n g t h e same 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 r i n o t h e r w o r d s , i n t h e c a s e o f s t e a d y s t i c k - s l i p , t h e same p e r i o d o f s t i c k as t h e c a l c u l a t e d c y c l e . I n t h e f i g u r e t h e v e l o c i t y c u r v e s w e r e a l i g n e d on t h e t i m e b a s e s o as t o a g r e e a p p r o x i m a t e l y d u r i n g t h e r i s i n g v e l o c i t y p a r t o f t h e c y c l e , s i n c e t h e t i m e o f i n i t i a t i o n o f s l i p i s d i f f i c u l t t o d e f i n e as m e n t i o n e d b e f o r e . B e f o r e a s s e s s i n g t h e p r o b l e m s i n v o l v e d i n d e f i n i n g t h e i n c e p t i o n o f s l i p , we must f i r s t d i s c u s s f u r t h e r t h e o b s e r v a t i o n s on t h e s t a t i c f r i c t i o n c o e f f i c i e n t . As s t a t e d p r e v i o u s l y , a v a r i a t i o n i n t h i s v a l u e was f o u n d a s t h e ram v e l o c i t y was c h a n g e d . T h i s was r e l a t e d t o t h e t i m e o f s t a t i o n a r y c o n t a c t w h i c h i n c r e a s e d w i t h s l o w e r v e l o c i t i e s as was shown i n F i g u r e 36 1.1.2. Under c o n d i t i o n s o f u n i f o r m s t i c k - s l i p t h i s c o r r e l a t i o n o f the f r i c t i o n v a l u e w i t h t h e time o f s t i c k was r e a s o n a b l e . However, when at t e m p t s were made t o o b t a i n v a l u e s f o r l o n g t i m e s o f s t a t i o n a r y c o n t a c t by s t o p p i n g t h e ram f o r a l e n g t h o f time and t h e n r e s t a r t i n g i t , t h e r e s u l t s became i n c o n s i s t e n t as shown i n F i g u r e 4.2.5. From t h i s i t became o b v i o u s t h a t merely t h e time o f s t a t i o n a r y c o n t a c t was i n s u f f i c i e n t t o d e s c r i b e the b e h a v i o u r o f the s t a t i c f r i c t i o n c o e f f i c i e n t . I n v e s t i g a t i o n o f t h e r e s u l t i n d i c a t e d t h a t the r a t e o f a p p l i -c a t i o n o f t h e t a n g e n t i a l f o r c e was o f g r e a t e r s i g n i f i c a n c e i n d e t e r m i n i n g the l i m i t i n g f r i c t i o n f o r c e . The r e s u l t s o f F i g u r e 4.2.5 are p l o t t e d on t h i s b a s i s i n F i g u r e 4.2.6. I n t h i s p l o t , t h e c o e f f i c i e n t o f f r i c t i o n i s p l o t t e d a g a i n s t 0 , t h e r a t e o f i n c r e a s e o f the t a n g e n t i a l f o r c e c o e f f i c i e n t . The t a n g e n t i a l f o r c e c o e f f i c i e n t i s d e f i n e d as where F i s t h e a p p l i e d t a n g e n t i a l f o r c e and W t h e normal l o a d . C o n s i d e r i n g F i g u r e 1.1.1, t h e r a t e o f i n c r e a s e o f t h e t a n g e n t i a l f o r c e i s g i v e n by 0 = F W (4.2.1) F kV (4.2.2) and thus 0 = kV W (4.2.3) 37 T h i s d e f i n i t i o n i s used i n comparing t h e r e s u l t s o f o t h e r w o r k e r s . I n t h e p r e s e n t work t h e r a t e o f l o a d a p p l i c a t i o n was measured d i r e c t l y from r e c o r d s o f s t r a i n r i n g f o r c e v e r s u s t i m e . The c o r r e l a t i o n i n t h e p l o t o f F i g u r e 4.2.6 i s con-s i d e r a b l y b e t t e r than t h a t i n F i g u r e 4.2.5, a l t h o u g h a s l i g h t dependence on t h e time o f s t i c k appears t o e x i s t a t the h i g h e r l o a d i n g r a t e s w h i c h i n ' s t e a d y s t a t e s t i c k - s l i p would c o r r e s p o n d t o a v e r y s h o r t t i m e o f s t i c k . Thus one may c o n c l u d e t h a t the v a l u e o f 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 i s p r i m a r i l y r e -l a t e d t o the n a t u r e o f the t a n g e n t i a l f o r c e a p p l i c a t i o n , with perhaps a secondary dependence on the o v e r a l l t ime o f s t a t i o n a r y c o n t a c t . Comparison w i t h d a t a o f o t h e r workers i s not s i m p l e i n most c a s e s , as o n l y t h e tim e s o f s t a t i o n a r y c o n t a c t a re g i v e n , w i t h no i n f o r m a t i o n on t h e l o a d i n g r a t e s . One case w h i c h c o u l d be compared e a s i l y was t h e work o f P o t t e r [22]. He gave h i s r e s u l t s f o r t h e 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 as a f u n c t i o n o f the v e l o c i t y o f l o a d i n g (V i n F i g u r e 1.1.1). H i s r e s u l t s a r e shown i n F i g u r e 4.2.7. U s i n g the g i v e n v a l u e s o f s t i f f n e s s and normal l o a d , the p l o t o f F i g u r e 4.2.8 was d e r i v e d , and t h i s i s a l s o shown i n F i g u r e 4.2.6. Reasonable c o r r e l a t i o n i s a g a i n e v i d e n t , and the v a r i -a t i o n s i n some o f the r e s u l t s may be a t t r i b u t a b l e t o t h e f a c t t h a t the d i f f e r e n t t e s t s were performed w i t h r e f i n i s h e d s u r f a c e s , and i t has been g e n e r a l l y o b s e r v e d t h a t i d e n t i c a l r e s u l t s i n f r i c t i o n measurements a re d i f f i c u l t t o achieve- a f t e r 38 s u r f a c e r e f i n i s h i n g . F u r t h e r comparison was done w i t h t h e r e s u l t s o f D a v i s [ 2 3 ] . I n t h i s case the l o a d i n g r a t e s were n ot a v a i l a b l e , but a p p r o x i m a t e l y c a l c u l a t e d on t h e b a s i s o f an assumed c o n -s t a n t 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 (see Appendix I I I ) . These r e s u l t s a r e shown i n F i g u r e 4.2.9. A t h i r d s o u r c e o f comparison was the work o f Cameron [2 4 ] . • These r e s u l t s w h i c h a re a l s o shown i n F i g u r e 4.2.9. were the o n l y ones found t o be i n poor agreement w i t h t h e t y p e o f b e h a v i o u r o b s e r v e d i n the o t h e r c a s e s . T h i s may be a t t r i b u t e d t o the f a c t t h a t f o r t h e l o n g e r t i m e s o f s t i c k Cameron d i d not make t h e f r i c t i o n measurements under s t i c k - s l i p c o n d i t i o n s , b u t l e f t t he system s t a t i o n a r y f o r a p e r i o d o f time b e f o r e a p p l y i n g t a n g e n t i a l l o a d . S i n c e l o a d i n g r a t e d a t a f o r the s e t e s t s was u n a v a i l a b l e , and t h e method o f Appendix I I I g i v e s i n v a l i d r e s u l t s f o r t h i s t y p e o f t e s t s , a p r o p e r com-p a r i s o n c o u l d n o t be made. Based on the p r e s e n t f i n d i n g s , t h e c o n s t a n t f r i c t i o n v a l u e g i v e n by Cameron f o r l o n g time o f s t i c k would be the e x p e c t e d r e s u l t i f a c o n s t a n t l o a d i n g r a t e was used a f t e r v a r i o u s t i m e s o f s t a t i o n a r y c o n t a c t . To c o n f i r m the g e n e r a l v a l i d i t y o f t h e o b s e r v e d data> t e s t s were performed w i t h d r y s u r f a c e s , and as shown i n F i g u r e 4.2.10, t h e r e s u l t s were s i m i l a r t o thos e w i t h l u b r i c a t e d -s u r f a c e s w i t h t h e o n l y d i f f e r e n c e s b e i n g a r i s e i n t h e a b s o l u t e v a l u e o f the f r i c t i o n f o r c e . I n a d d i t i o n , a t e s t was c a r r i e d out u s i n g a s u r f a c e p r e p a r a t i o n i d e n t i c a l t o t h e one used by Cameron. I n t h i s c a s e , the SAE 1020 r a i l s were l i g h t l y e t c h e d w i t h a 4 per c e n t s o l u t i o n o f c o n c e n t r a t e d n i t r i c a c i d i n e t h a n o l , and they were th e n k e p t immersed i n e t h a n o l d u r i n g use. The s l i d e r s were s i m i l a r l y e t c h e d . A g a i n t h e r e s u l t s agreed w i the o t h e r f i n d i n g s as seen i n F i g u r e 4.2.10. A b r i e f s e t o f e x p e r i m e n t s on the e t c h e d s u r f a c e s w i t h o u t immersion i n e t h a n o l i n d i c a t e d t h a t the r e s u l t was v i r t u a l l y t he same as f o r t h e d r y ground s u r f a c e s , w i t h p e r haps s l i g h t l y h i g h e r f r i c t i o n a l v a l u e s . C H A P T E R 5 5.1 FURTHER INVESTIGATION OF THE NATURE OF FRICTION FORCES Exper i m e n t s by C o u r t n e y - P r a t t and E i s n e r [25] showed t h a t when two s u r f a c e s were l o a d e d n o r m a l l y and s u b j e c t e d t o a t a n g e n t i a l f o r c e t o o s m a l l t o cause s l i d i n g , a s m a l l r e l a t i v e d i s p l a c e m e n t o c c u r r e d , accompanied by a growth i n the a r e a o f c o n t a c t . They a l s o found t h a t a l u b r i c a n t does not e s s e n t i a l l y a f f e c t t h e d e f o r m a t i o n p r o c e s s f o r t a n g e n t i a l f o r c e s l e s s than t h a t r e q u i r e d f o r s l i p , b u t o n l y l o w e r e d t h e f o r c e a t which s l i p took p l a c e . T h i s l a t t e r f i n d i n g i s i n agreement w i t h o b s e r v a t i o n s made i n t h e p r e v i o u s c h a p t e r i n t h a t t h e g e n e r a l f r i c t i o n b e h a v i o u r under l u b r i c a t e d c o n d i t i o n s was the same as w i t h d r y s u r f a c e s w i t h o n l y a change i n t h e magnitude o f t h e f r i c t i o n f o r c e s . To check f o r the e x i s t e n c e o f t a n g e n t i a l m o t i o n d u r i n g s t i c k , a D a y t r o n i c s d i s p l a c e m e n t t r a n s d u c e r c a p a b l e o f r e s o l v -i n g t o the o r d e r o f 10 m i c r o i n c h e s was used t o ob s e r v e t h e 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 t h e s t i c k p o r t i o n o f t h e c y c l e . Measurements under slow l o a d i n g showed t h a t t h e s l i d e r d i d undergo a g r a d u a l d i s p l a c e m e n t w h i c h r e a c h e d a magnitude of the o r d e r o f 50 m i c r o i n c h e s p r i o r t o g r o s s s l i p . The r e s u l t s o f C o u r t n e y - P r a t t and E i s n e r on s t e e l s u r f a c e s gave a p p r o x i m a t e l y a 40 m i c r o i n c h d i s p l a c e m e n t f o r a c o e f f i c i e n t o f f r i c t i o n o f .4. T h i s agreement i n t h e m i c r o - d i s p l a c e m e n t s s u g g e s t s t h a t t h e p r e s e n t system a l s o had a r e a growth d u r i n g 41 the time o f s t i c k , a l t h o u g h no atte m p t s were made t o measure the a c t u a l a r e a o f c o n t a c t . I n i n v e s t i g a t i n g t h e k i n e t i c f r i c t i o n f o r c e s , d i s p l a c e -ment measurements were i n c o n v e n i e n t , and thus t h e f o r c e v a r i a t i o n d u r i n g t h e c y c l e was d i s p l a y e d as a f u n c t i o n o f t i m e . T h i s n e c e s s i t a t e d t r i g g e r i n g t h e o s c i l l o s c o p e sweep a t t h e i n s t a n t when s l i p commenced. C a r e f u l i n v e s t i g a t i o n o f t h i s t r a n s i t i o n showed t h a t t h e i n c e p t i o n o f s l i p was not i n s t a n -taneous. The presence o f motion t h r o u g h o u t the s t i c k p o r t i o n o f the c y c l e would l e a d one t o e x p e c t t h i s r e s u l t . The g r a d u a l n a t u r e o f t h e i n c e p t i o n o f s l i p i s o b s e r v a b l e from t h e v e l o c i t y p l o t o f F i g u r e 4.2.4. The l o w e s t p o i n t on the e x p e r i m e n t a l v e l o c i t y c urve shows a t y p i c a l l e v e l o f t h e t r i g g e r i n g v a l u e used i n t h e f r i c t i o n f o r c e - t i m e c u r v e s o f F i g u r e 4.2.1. The s l o p e o f the v e l o c i t y c urve a t t h i s p o i n t i s s m a l l and i n c r e a s -i n g r a p i d l y . 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 v e l o c i t y i n the e a r l y s t a g e s o f s l i p gave t h e r e s u l t o f F i g u r e 5.1.1. Here i t can be seen t h a t the t r a n s i t i o n i s i n f a c t v e r y g r a d u a l , . and the problem o f d e f i n i n g t h e p o i n t when s l i p commences, becomes o b v i o u s . The time v a r i a t i o n i n t h e l e n g t h s o f the f r i c t i o n - t i m e p l o t s as i n F i g u r e 4.2.1 i s e x p l a i n a b l e by t h i s v e l o c i t y c u r v e , as a v e r y s l i g h t v a r i a t i o n i n t h e t r i g g e r i n g l e v e l used can cause a l a r g e v a r i a t i o n i n the l e n g t h o f c y c l e o b s e r v e d . The computed curve o f F i g u r e 5.1.1 i s from compu-t a t i o n u s i n g the d i s p l a c e m e n t dependent k i n e t i c f r i c t i o n ; r e l a t i o n w i t h no p r o v i s i o n f o r m i c r o s l i p . I t may be n o t e d . 42 t h a t a d d i n g a s l i g h t v e l o c i t y t o the computed curve b r i n g s i t t o c l o s e r agreement w i t h e x p e r i m e n t . T h i s a d d i t i o n a l v e l o c i t y may be a t t r i b u t e d t o the m i c r o - d i s p l a c e m e n t which i n a c t u a l i t y t a k e s p l a c e d u r i n g s t i c k . A l t h o u g h the time d u r i n g w h i c h t h e v e l o c i t y i s s l o w l y i n c r e a s i n g i s c o n s i d e r a b l e , the d i s p l a c e m e n t d u r i n g t h i s p e r i o d i s q u i t e s m a l l , and thus when the k i n e t i c f r i c t i o n f o r c e i s p l o t t e d on a d i s p l a c e m e n t b a s i s the cu r v e s a r e r a t h e r i n s e n -s i t i v e t o the e x a c t p o i n t t a k e n as the i n c e p t i o n o f s l i p . I n s e e k i n g an e x p l a n a t i o n f o r the g r a d u a l l y d e c r e a s i n g k i n e t i c f r i c t i o n f o r c e , e x p e r i m e n t s were c a r r i e d o u t t o a s s e s s the p o s s i b l e r o l e o f a p l o u g h i n g component o f f r i c t i o n . The s l i d e r s were r e p l a c e d by s t e e l b e a r i n g b a l l s , and a t r a v e r s e o f the r a i l s was made a t v a r y i n g f e e d speeds. The w i d t h o f r the r e s u l t i n g grooves was measured w i t h the r e a s o n i n g t h a t v a r i a t i o n s i n the f r i c t i o n f o r c e due t o p l o u g h i n g would be caused by v a r i a t i o n s i n the p l o u g h i n g a r e a , and t h e r e f o r e t r a c k w i d t h . No s i g n i f i c a n t v a r i a t i o n i n t r a c k w i d t h was . obs e r v e d , even a f t e r s t i c k - s l i p where the c o e f f i c i e n t o f f r i c -t i o n f l u c t u a t e d by 50 per c e n t . An i n t e r e s t i n g o b s e r v a t i o n on the b e h a v i o u r o f the s u r f a c e s was made i n i n s p e c t i n g the wear marks on the s t e e l b a l l s . W i th s t e e l on s t e e l a zone o f b u i l t up m e t a l appeared on the l e a d i n g edge, t a p e r i n g o f f t o an are a showing some wear marks b u t l i t t l e m e t a l l i c t r a n s f e r . , When the r a i l s were f i r s t p l a t e d w i t h copper u s i n g Stead's r e a g e n t , no such l e a d i n g edge b u i l d up was o b s e r v e d , and con-s i d e r a b l y lower f r i c t i o n v a l u e s were measured. F i g u r e s 5.1.2 43 t h r o u g h 5.1.5 show photographs o f t h e s e r e s u l t s . A n o t h e r p o s s i b l e cause o f t h e o b s e r v e d f r i c t i o n b e h a v i o u r was c o n s i d e r e d t o be a v a r i a t i o n i n t h e c o n t a c t temperature d u r i n g s l i d i n g . I f l a r g e temperature r i s e s o c c u r r e d a t the c o n t a c t i n g a s p e r i t i e s , s o f t e n i n g o f t h e j u n c t i o n i n t e r f a c e s c o u l d r e s u l t i n lower f r i c t i o n v a l u e s i n a way analogous t o the p r e v i o u s l y mentioned copper p l a t i n g where a low shear s t r e n g t h i n t e r f a c e on h a r d e r b a c k i n g m a t e r i a l caused a d e c r e a s e i n t h e f r i c t i o n f o r c e . The work o f J a e g e r [26] was extended t o c o n s i d e r a m a t r i x o f square a s p e r i t i e s , and the peak tem p e r a t u r e o f a c e n t r a l l y l o c a t e d a s p e r i t y was computed d u r i n g a t y p i c a l s l i p c y c l e . F o r the slow v e l o c i t i e s e n c o u n t e r e d d u r i n g s l i p c y c l e s , the temperature r i s e was found t o be o f the o r d e r o f o n l y a few d e g r e e s , and i t c l o s e l y f o l l o w e d the v e l o c i t y c u r v e , showing l i t t l e l a g a t the end o f t h e c y c l e . F or example, c o n s i d e r i n g t h e a s p e r i t i e s t o be s t e e l , and u s i n g a m a t r i x o f 81 a s p e r i t i e s .001 c e n t i m e t e r s square w i t h c o n t a c t o v e r 10 per c e n t o f the apparent a r e a , the maximum temperature r i s e o f the c e n t r a l a s p e r i t y was o n l y a p p r o x i m a t e l y 1 degree F a h r e n -h e i t , w i t h a r e s i d u a l temperature r i s e o f about .2 degrees a t the end o f the s l i p c y c l e . T h i s r e s u l t i s shown g r a p h i c a l l y i n F i g u r e 5.1.6. These s m a l l t emperature changes were c o n s i d e r e d u n l i k e l y t o i n f l u e n c e j u n c t i o n s t r e n g t h s , and the temperature v a r i a t i o n d u r i n g a c y c l e would not p r e d i c t the m o n o t o n i c a l l y d e c r e a s i n g f o r c e w i t h d i s p l a c e m e n t which was o b s e r v e d i n the e x p e r i m e n t . 44 Having d i s c a r d e d t h e s e mechanisms as the cause o f t h e obs e r v e d b e h a v i o u r , a model based on g e o m e t r i c a l c o n s i d e r a t i o n s o f the c o n t a c t a r e a s was d e r i v e d and i s p r e s e n t e d i n the n e x t s e c t i o n . A l s o a s i m p l e model f o r the s t a t i c f r i c t i o n i s p r e -s e n t e d , and a l t h o u g h t h e r e i s no e x p e r i m e n t a l p r o o f o f t h e v a l i d i t y o f some o f the assumptions i n the two models, t h e y do g i v e a r e s u l t i n g e n e r a l agreement w i t h e x p e r i m e n t and s h o u l d p r o v i d e some gu i d e l i n e s toward more a c c u r a t e d e s c r i p t i o n o f the phenomenon o f m e t a l l i c f r i c t i o n . 5.2 MODELS FOR THE OBSERVED FRICTIONAL BEHAVIOUR S t a t i c F r i c t i o n I t has been seen 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 i n c r e a s e s w i t h s l o w e r r a t e s o f t a n g e n t i a l f o r c e a p p l i c a t i o n , w i t h some i n d i c a t i o n o f dependence on t h e time o f s t a t i o n a r y c o n t a c t i n the case o f h i g h l o a d i n g r a t e s . I f we c o n s i d e r the i n c r e a s e i n the f r i c t i o n f o r c e t o be the r e s u l t o f j u n c t i o n growth, some type o f v i s c o - e l a s t i c model f o r t h i s a r e a growth appears n e c e s s a r y . The s i m p l e s t such model i s i l l u s t r a t e d i n F i g u r e 5.2.1. In t h i s model the a p p l i c a t i o n o f t h e f o r c e F q + F t causes t h e c o n t a c t i n g a s p e r i t y t o i n c r e a s e i n a r e a , w i t h t h i s growth b e i n g r e s i s t e d by m a t e r i a l p r o p e r t i e s d e s c r i b e d by the s p r i n g o f s t i f f n e s s k and a v i s c o u s damper o f damping c. The model shows an i n i t i a l c o n t a c t a r e a w h i c h i s p r o p o r t i o n a l t o the di m e n s i o n X q, w i t h the i n c r e a s e i n a r e a b e i n g r e p r e s e n t e d by an i n c r e a s e Ax i n t h e l e n g t h o f the j u n c t i o n . The i n i t i a l c o n t a c t a r e a i s d e t e r m i n e d by the normal l o a d and m a t e r i a l 45 p r o p e r t i e s . A l i m i t i n g v a l u e may be o b t a i n e d from e q u a t i o n (1.2.1). S i n c e the o r i g i n a l a r e a i s p r o p o r t i o n a l t o X q, and the f i n a l a r e a i s p r o p o r t i o n a l t o X q + Ax, i n t h e f o l l o w i n g d e r i v a t i o n a u n i t a s p e r i t y w i d t h i s c o n s i d e r e d . The a r e a growth o f a r i g i d p e r f e c t l y - p l a s t i c wedge has been d e r i v e d by Johnson [27] and t h i s may be used as a gui d e t o t h e v a l u e s o f a r e a growth t o be e x p e c t e d from o t h e r models. I n the p r e s e n t measurements, v a r i a t i o n s i n t h e s t a t i c f r i c t i o n c o e f f i c i e n t i n the range o f .18 t o .38 were o b s e r v e d . I f a r e a growth were t h e s o l e cause o f t h e i n c r e a s e i n t h i s f r i c t i o n f o r c e , one would e x p e c t o v e r 100 per c e n t i n c r e a s e i n a r e a a t a c o e f f i c i e n t o f f r i c t i o n o f .38. However, even, t h e p e r f e c t l y - p l a s t i c model o f Johnson g i v e s o n l y about 75 p e r c e n t a r e a growth f o r t h i s f r i c t i o n l e v e l , and i t appears r e a s o n a b l e t h a t i n r e a l m e t a l l i c c o n t a c t the a r e a growth would be sub-s t a n t i a l l y l e s s . Thus we may p o s t u l a t e t h a t as j u n c t i o n growth t a k e s p l a c e , the shear s t r e n g t h o f t h e j u n c t i o n s i n c r e a s e s . T h i s c o u l d be the r e s u l t o f t h e s q u e e z i n g o ut o r t h i n n i n g o f weaker s u r f a c e f i l m s , r e s u l t i n g i n s t r o n g e r m e t a l t o m e t a l c o n t a c t , as w e l l as some i n t e r l o c k i n g o f s u r f a c e c o n t o u r s as the deformed a s p e r i t y conforms t o the o p p o s i t e s u r f a c e . With the above p r e m i s e s and the model o f F i g u r e 5.2.1, we may p e r f o r m computations t o g i v e some i n s i g h t i n t o t h i s t y p e o f f r i c t i o n a l b e h a v i o u r . Summarizing the assumptions i n v o l v e d i n the f o l l o w i n g d e r i v a t i o n f o r t h e s t a t i c f r i c t i o n f o r c e : 1) The a r e a growth o f j u n c t i o n s i s v i s c o e l a s t i c . 2) The a p p l i e d t a n g e n t i a l l o a d i s o f the form F q + F t . 3) The j u n c t i o n shear s t r e n g t h v a r i e s w i t h t h e a r e a growth. F o r t h i s a s i m p l e l i n e a r model i s t a k e n , g i v i n g t h e shear s t r e n g t h T as T q + T'Ax. 4) The growth i n a r e a i s p r o p o r t i o n a l t o Ax, w i t h no growth p e r p e n d i c u l a r t o the a p p l i e d f o r c e . The d i f f e r e n t i a l e q u a t i o n g o v e r n i n g the model o f F i g u r e 5.2.1 i s cAx + kAx = F + F t (5.2.1) o S o l v i n g f o r the a r e a growth Ax, we g e t _k F CF F t Ax = ae c + - 2 _ _ + — (5.2.2) k k^ k The f o r c e r e q u i r e d t o break the j u n c t i o n i s 2 (x + Ax)(T + T'Ax) = x T + T'x Ax + T Ax + T 1 ( A x ) o o o o o o (5.2.3) The l a s t term may be n e g l e c t e d i f Ax i s t a k e n as s m a l l . I n t h a t case s l i p w i l l o c c u r when x T + T'x Ax + T Ax = F + F t (5,2.4) o o o o o Case I T a k i n g the i n i t i a l c o n d i t i o n Ax = 0 a t t = 0 (5.2.5) g i v e s CF F _k Ax = ( - -° ) ( e c - 1) + £ t (5.2.6) ]< k K S u b s t i t u t i n g (5.2.6) i n (5.2.4) g i v e s cF F k X T + (T +T'x ) [ ( _ - . • - _£) ( e ~ c fc - 1) +| t ] = F^ + F t (5.2.7) R e a r r a n g i n g a nd m u l t i p l y i n g b y k/ W ( T + T ' X Q ) , F k - kx^T k c F F^ > k F t _2 9_°_ + _ ( O ) ( 1 _ e - ^ t j + ( 1 } _ = Q W(T + T'x ) W k 2 k T + T'x W o o o o (5.2.8) F F F t W D e f i n i n g 0 = - , y = - ° , Ay = — ' Pm ~ — W W W x o we h a v e k T c k A y o, ( ) (u - — ) + ( -0 - u ) (1 - e & ) + T + T ' x 0 p k ° c ^ o o ^m k ( 1 ) Ay = 0 (5.2.9) T + T'x o o C a s e I I T a k i n g t h e i n i t i a l c o n d i t i o n w h e r e t h e c o n s t a n t f o r c e F q h a s b e e n a p p l i e d f o r a l o n g t i m e b e f o r e t h e a d d i t i o n o f t h e ramp l o a d F t 48 F Ax = — a t t = 0 (5.2.10) we f i n d c F i t 1 Ax = (e c - 1) + - (F +Ft) (5.2.11) k 2 k ° S u b s t i t u t i n g (5.2.11) i n (5.2.4) and p r o c e e d i n g as above g i v e s k T c. _ k Ay k ( ) ( y _ _ ° ) + _ 0 ( i _ e c i ) + ( 1) Ay-T+T'x p k ^ T+T'x o o cm o o U Q = • 0 (5.2,12) As e x p e c t e d e q u a t i o n s (5.2.9) and (5.2.12) are i d e n t i c a l f o r t h e case where y Q = 0. As a f i r s t s t e p i n e x a m i n i n g t h e s e e q u a t i o n s c o n s i d e r the l i m i t i n g cases as 0 >> 1 and 0 << 1; For Case I : a) 0 >> 1 i n e q u a t i o n (5.2.9) g i v e s k T c kAy k ( ) ( U Q - -O) + — j2f ( — ) ( 1) Ay = 0 (5.2.13) T +T1 x p k C0 T +T1 x O ' O M i l  r O O o r k T ( ) (u + Ay - -°-) = 0 (5.2.14) T +T'x p o o m^ From w h i c h we f i n d ^min = % + ^ = — ( 5 ' 2 - 1 5 ) Pm b) 0 << 1 i n e q u a t i o n (5.2.9) g i v e s k T k ( ) (u - — ) - u + ( 1) Au = 0 (5.2.16) T+T'x p T+T'x o o m^ o o y i e l d i n g kT W = P 0 + Ay = 2 (5.2.17) p [k-(T +T'x )] Mtr o o For Case I I : a) 0 » 1 g i v e s s i m i l a r l y T u (T +T'x ) u . = _° + _ 2 _ o °_ (5.2.18) mm . v . . / P™ k b) 0 » 1 g i v e s kT % a x = 2 ( 5 ' 2 ' 1 9 1 P m [ k - ( T o + T ' X o ) ] I n s p e c t i o n o f t h e s e l i m i t i n g v a l u e s shows t h a t 1) F o r b o t h c a s e s , u approaches an a s y m p t o t i c v a l u e f o r ' both 0 » 1 and 0 << 1. 2) F o r Case I t h e asymptotes are independent o f u . 50 3) F o r Case I I t h e v a l u e o f u depends on u Q f o r l a r g e 0 . Comment 3 i s a r e s u l t w h i c h may have been e x p e c t e d as t h e i n i t i a l c o n d i t i o n s would be more predominant a t h i g h e r l o a d i n g r a t e s . To observe the b e h a v i o u r o f t h e s e s o l u t i o n s f o r o t h e r 0, the parameters o f e q u a t i o n ( 5 . 2 . 9 ) were f i t t e d t o t h e e x p e r i m e n t a l d a t a o f F i g u r e 4.2.6 w i t h y o t a k e n as 0. T h i s gave the v a l u e s c kT = 17 ; - - 1 second ; :—- * .275 (5.2.20) T +T 1x k p (T +T 1x ) o o o U s i n g t h e s e v a l u e s a l o n g w i t h v a r i o u s v a l u e s o f u , the e q u a t i o n s (5.2.9) and (5.2.12) were s o l v e d t o g i v e t h e graph o f F i g u r e 5.2.2. I t i s now o f i n t e r e s t t o i n v e s t i g a t e the s i g n i f i c a n c e o f the v a l u e s o f t h e parameters as d e r i v e d from f i t t i n g t o e x p e r i m e n t a l d a t a . T h i s f i t does n o t p e r m i t t h e e v a l u a t i o n o f i n d i v i d u a l p a r a m e t e r s , b u t the f o l l o w i n g d i s c u s s i o n g i v e s some i n s i g h t t o t h e i r r o l e . F o r t h e d e f o r m a t i o n o f a p l a s t i c wedge, Johnson [27] ux f i n d s t h e v a l u e o f v a r y i n g from a p p r o x i m a t e l y 18 f o r s m a l l Ax , c c Ax , , _ c Ax ^ , — t o .5 f o r — = 1 and .2 f o r — >> 1. x o o x o In the p r e s e n t c a s e , uW = kAx + cx 51 o r x kx cxx k cxx y-°- = — + ° = _ + 2. (5.2.21) Ax W WAx p WAx F o r 0 << 1, x << 1, g i v i n g ux„ k H o _ Ax p (5.2.22) From e q u a t i o n (5.2,20) th e p r e s e n t case gave \ k T ( °__) = .275 m T +T'x I f the assumption o f e q u a t i o n (5.2.4) t h a t Ax << X Q i s v a l i d , we s h o u l d e x p e c t —>.8. T h i s would i m p l y t h a t Pm T o 1 2 T o Ax — < o r T' > . Thus f o r an a r e a growth — , we T 0 + T ' x 0 X o X o would e x p e c t a shear s t r e n g t h 2T Ax 2Ax T > T Q + — - = T (1 + — ) (5.2.23) x ° x o o The v a l u e s o f t h e o t h e r parameters i n e q u a t i o n (5.2.20) v a r y w i t h t h e minimum and maximum v a l u e o f the f r i c t i o n f o r c e s , k -1 but the time c o n s t a n t — i s o f t h e o r d e r o f 1 second f o r a l l c o b s e r v e d d a t a . T h i s i m p l i e s t h a t t h e mechanism c a u s i n g t h e i n c r e a s e d f r i c t i o n f o r c e i s not the same as the c r e e p d e f o r -m ation found i n i n d e n t a t i o n hardness t e s t s such as the work o f A t k i n s , S i l v e r i o and Tabor [28] s i n c e t h e i r t i me c o n s t a n t would be o r d e r s o f magnitude l a r g e r . The b e h a v i o u r o f the f r i c t i o n f o r c e s as a f u n c t i o n o f the parameters assumed i n the p r e s e n t model i s summarized i n the f o l l o w i n g : 1 } ymax' y m i n a n d Vax " % i n i n c r e a s e w ^ h i n c r e a s i n g T Q 2 ) ymax' y m i n a n d W " V i n d e c r e a s e w i t h i n c r e a s i n g p m 3) y . i s independent o f k o r T'x mm c o 4) y „ d e c r e a s e s w i t h i n c r e a s i n g k and w i t h d e c r e a s i n g max J 3 T'x o . The above model y i e l d s a b e h a v i o u r w h i c h i n g e n e r a l agrees w i t h the e x p e r i m e n t a l r e s u l t s . The time c o n s t a n t i s based on the f a c t t h a t i n a l l cases t h e lo w e r l i m i t i n g f r i c t i o n v a l u e o c c u r r e d when the l o a d i n g r a t e 0 was o f t h e o r d e r o f one. The asymptote f o r 0 << 1 c o u l d n ot be c o n f i r m e d by ex p e r i m e n t s due t o the e x t r e m e l y slow f e e d r a t e s r e q u i r e d t o go down an o r d e r o f magnitude from 0 = .001. A d i f f e r e n t mechanism w i t h a l o n g e r time c o n s t a n t c o u l d appear a t such low r a t e s , as might be e x p e c t e d from the above mentioned c r e e p hardness t e s t s . I t may be noted t h a t the e f f e c t o f y Q i n Case I I o n l y appears f o r 0 > .1, and i t i s o n l y f o r t h e s e v a l u e s t h a t t h e 1 e x p e r i m e n t a l d a t a o f F i g u r e 4.2.6 show any n o t i c e a b l e v a r i a t i o n due t o e x p e r i m e n t a l p r o c e d u r e c o r r e s p o n d i n g t o Case I I . K i n e t i c F r i c t i o n From t h e d i s c u s s i o n o f the s t a t i c f r i c t i o n i t appears t h a t e i t h e r the j u n c t i o n a r e a o r j u n c t i o n shear s t r e n g t h o r b o t h undergo an i n c r e a s e d u r i n g s t a t i c c o n t a c t . S i n c e t h e 53 c y c l e o f s t a t i c f r i c t i o n r i s e i s r e p e t i t i v e , s l i p must r e s t o r e the s u r f a c e s t o t h e same i n i t i a l c o n d i t i o n s , i m p l y i n g t h a t a c o r r e s p o n d i n g d e c r e a s e i n t h e s e v a l u e s must o c c u r . A q u e s t i o n t h a t a r i s e s i s t h a t i f t h e c o n t a c t a r e a has been i n c r e a s e d i n the s t a t i c case by p l a s t i c d e f o r m a t i o n , why does not t h i s i n c r e a s e d a r e a p e r s i s t d u r i n g f u t u r e t i m e . The f a c t t h a t i t does not has been shown by p r o l o n g e d t e s t s over the same s l i d i n g s u r f a c e , w h i c h r e s u l t e d i n no change i n the a m p l i t u d e o f the s t i c k - s l i p . I f some c o n s t a n t s t a t e o f a s p e r i t y d e f o r m a t i o n were r e a c h e d , the f r i c t i o n v a l u e s h o u l d a l s o become c o n s t a n t and the s t i c k - s l i p b e h a v i o u r would d i s a p p e a r . T h i s c o n t i n u o u s r e t u r n t o i n i t i a l c o n d i t i o n s may be e x p l a i n e d by t h e random n a t u r e and o r i e n t a t i o n o f a s p e r i t y • c o n t a c t s . Whenever an a s p e r i t y t h a t has deformed t o conform t o the o p p o s i n g s u r f a c e b e g i n s t o move, i t w i l l n o t conform t o the n e x t a s p e r i t y i t c o n t a c t s and thus a c t s as a brand new c o n t a c t , whether o r not e i t h e r o f the newly m a t i n g a s p e r i t i e s had p r e v i o u s l y been deformed. A n o t h e r r e a s o n f o r t h i s r e t u r n t o p r e v i o u s c o n t a c t a r e a o r f r i c t i o n f o r c e may be t h e e x i s -t ence o f m e t a l l i c t r a n s f e r and wear. As wear p a r t i c l e s are formed o r m e t a l p a r t i c l e s a r e t r a n s f e r r e d from one s u r f a c e t o the o t h e r , a f r e s h s u r f a c e i s c r e a t e d , r e s u l t i n g i n c o n t i n u o u s l y c h anging s u r f a c e geometry. In c o n s i d e r i n g the d e c r e a s e o f the j u n c t i o n a r e a s and shear s t r e n g t h s w i t h s l i p , i t seems l o g i c a l t o r e l a t e them t o t h e o r i g i n a l a r e a s t i l l i n c o n t a c t and t h e new a r e a formed. . From c o n s i d e r a t i o n o f the d i s t r i b u t i o n o f c o n t a c t s i z e s , we can see t h a t t h e f i r s t i n c r e m e n t o f m o t i o n d e s t r o y s a l l c o n t a c t s as s m a l l as o r s m a l l e r t h a n t h e d i s p l a c e m e n t . I n a d d i t i o n a f r a c t i o n o f t h e l a r g e r c o n t a c t s may be d e s t r o y e d by t h i s d i s p l a c e m e n t . From t h i s i t i s o b v i o u s t h a t t h e d e s t r u c t i o n o f o r i g i n a l c o n t a c t a r e a o c c u r s a t a d e c r e a s i n g r a t e w i t h d i s p l a c e m e n t . A s i m p l e model f o r t h e a r e a d i s t r i b u t i o n h a v i n g t h i s p r o p e r t y i s an e x p o n e n t i a l d e crease i n t h e o r i g i n a l a r e a w i t h d i s p l a c e m e n t . Thus we may make the f o l l o w i n g a s s u m p t i o n s : 1) The f r a c t i o n o f o r i g i n a l a r e a i n c o n t a c t a f t e r a r e l a t i v e d i s p l a c e m e n t z i s g i v e n by e s. 2) The s h e a r s t r e n g t h o f t h e o r i g i n a l j u n c t i o n s i s g i v e n by T = T + T'Ax . s o 3) The shear s t r e n g t h o f the new j u n c t i o n s formed i s t h e o r i g i n a l s t r e n g t h a t t h e b e g i n n i n g o f t h e s t i c k c y c l e , T o Now the f r a c t i o n o f o r i g i n a l a r e a d e s t r o y e d f o r a _z_ g i v e n d i s p l a c e m e n t i s 1 - e s . A p a r t o f t h i s a r e a i s r e -p l a c e d by new c o n t a c t s , and s i n c e we know t h a t once a l l t h e o r i g i n a l a r e a X q + Ax has been d e s t r o y e d , the new a r e a o f con-t a c t must a g a i n be X q, we may g i v e the i n s t a n t a n e o u s magnitude o f the new a r e a formed by X X. Z X z x, = ( i - e ~ s) = ( °—) [ ( X +Ax) ( l - e " s ) ] (5.2.24) x, x +Ax The f r i c t i o n f o r c e o v e r the t o t a l c o n t a c t may now be t a k e n as t h e sum o f the f o r c e s a t the i n d i v i d u a l c o n t a c t s , and s i n c e g r o s s s l i d i n g i s t a k i n g p l a c e so a l l c o n t a c t s a r e i n the p r o c e s s o f f a i l u r e , t h i s f o r c e i s g i v e n by F = E a i T i ' I n t h e p r e s e n t model t h e r e are two lumped area s w i t h d i f f e r e n t shear s t r e n g t h s . The t o t a l f r i c t i o n f o r c e i s g i v e n by T x + T x. (5.2.25) s s o k where x s = x l e s = ^ xo + A x ^ e s (5.2.26) g i v i n g F = T x n e s + T x (1-e s) = T x + (T x. -T x )e s (5.2.27) o l o o o o s l o o U s i n g the f a c t t h a t F = F . . f o r z = 0 and F = F. . 3 s t a t k i n f o r z -*- °° F = T x, and F. . = T x (5.2.28) s t a t s 1 k i n o o v ' S u b s t i t u t i n g e q u a t i o n (5.2.28) i n e q u a t i o n (5.2.27) and d i v i d i n g by the normal l o a d W t o c o n v e r t the f o r c e s i n t o f r i c t i o n c o e f f i c i e n t s , y k + ( y s ~ y k J e _ s (5.2.29) 56 T h i s e q u a t i o n may be compared t o the one d e r i v e d from e x p e r i m e n t a l d a t a , F i g u r e 4.2.2 3 y = uk + ( y s " V e ~ 5 X 1 0 2 (5.2.30) The form o f the e q u a t i o n s i s i d e n t i c a l . I t i s now -4 i n t e r e s t i n g t o see what an s o f 2 x 10 i n e q u a t i o n (5.2.29) shows about c o n t a c t a r e a s : D i s p l a c e m e n t z m i c r o - i n c h e s P e r c e n t A r e a d e s t r o y e d z (1 - e s) 21 10 57 25 138 50 460 90 600 99 1380 99 .9 1840 99.99 These v a l u e s appear r e a s o n a b l e i n comparison w i t h e s t i m a t e s o f the s i z e o f a c t u a l a s p e r i t y c o n t a c t . R a b i n o w i c z [10] measured wear p a r t i c l e s from s l i d i n g copper on low carbon s t e e l . From t h i s he found j u n c t i o n d i a m e t e r s from 370 m i c r o -i n c h e s t o 2500 m i c r o - i n c h e s , w i t h the s m a l l e r d i a m e t e r j u n c t i o n s b e i n g most numerous. I t may be n o t e d t h a t a l t h o u g h one might e x p e c t v a r i -a t i o n s i n s w i t h changes i n normal l o a d o r w i t h changes i n 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 , the work o f Greenwood [8, 9] s u g g e s t s t h a t the j u n c t i o n s i z e d i s t r i b u t i o n does n o t i n f a c t v a r y s i g n i f i c a n t l y w i t h changes i n the c o n t a c t a r e a . C H A P T E R 6 6.1 RESULTS FOR THE COMPLETE SYSTEM The i n i t i a l n u m e r i c a l c a l c u l a t i o n s on the system were performed u s i n g the complete e q u a t i o n s (2.1.17) and (2.1.32), n e g l e c t i n g o n l y the p i s t o n l e a k a g e term. T h i s l e a k a g e was d i s r e g a r d e d s i n c e t e s t s o v er p e r i o d s up t o 45 minutes w i t h 400 pounds per square i n c h . p r e s s u r e a c r o s s the p i s t o n gave no measurable l e a k a g e . The measurement was made by o b s e r v i n g t h e o i l l e v e l i n a s m a l l d i a m e t e r g l a s s tube a t the o u t l e t , and -5 leakage o f a p p r o x i m a t e l y 2 x 10 c u b i c c e n t i m e t e r s p e r minute o r g r e a t e r would have been d e t e c t e d . L a t e r c a l c u l a t i o n s i n c l u d i n g the l e a k a g e term showed t h a t even a t slow o p e r a t i o n , l e a k a g e o f l e s s than 10 c u b i c c e n t i m e t e r s per minute had an i n s i g n i f i c a n t e f f e c t on the r e s u l t s . These f i r s t c a l c u l a t i o n s r e s u l t e d i n a s t i c k - s l i p motion which i n i t i a l l y had the d i s c r e p a n c y d i s c u s s e d under k i n e t i c f r i c t i o n measurement ( S e c t i o n 4.2). S u b s e q u e n t l y ; w i t h the f r i c t i o n c h a r a c t e r i s t i c s d e s c r i b e d i n S e c t i o n 4.2, the agreement between the n u m e r i c a l and e x p e r i m e n t a l r e s u l t s became q u i t e s a t i s f a c t o r y . S i n c e e x p e r i m e n t a l v a l u e s o f s t a t i c f r i c t i o n c o e f f i c i e n t s were used, n a t u r a l l y t h e maximum f o r c e and c y c l e p e r i o d were always i n good agreement when the time o f s t i c k dependent r e l a t i o n from F i g u r e 4.2.5 was used. The l o a d i n g r a t e dependent v a r i a t i o n o f s t a t i c f r i c t i o n was a l s o used i n computation by a p p r o x i m a t i n g the d a t a o f F i g u r e 4.2.6 by a s t r a i g h t l i n e , r a t h e r t h a n u s i n g the cumbersome e q u a t i o n s (5.2.9) o r ( 5 . 2 . 1 2 ) . As e x p e c t e d , t h i s d i d n o t a l t e r t h e r e s u l t s s i g n i f i c a n t l y , s i n c e a g a i n t h e same e x p e r i m e n t a l d a t a was u s e d . The o n l y s o u r c e o f d i s c r e p a n c y was t h e d i f f e r e n c e i n t h e f i t s a p p l i e d t o t h e d a t a p o i n t s . The c a l c u l a t i o n s p e r -f o r m e d w i t h t h i s f i t c a u s e d l e s s t h a n 5 p e r c e n t v a r i a t i o n i n t h e 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 f o r a g i v e n t i m e o f s t i c k as c o m p a r e d t o t h e c a l c u l a t i o n s u s i n g t h e t i m e o f s t i c k d e p e n d e n t r e l a t i o n . A c o m p a r i s o n o f e x p e r i m e n t a l and n u m e r i c a l r e s u l t s i s p r e s e n t e d i n F i g u r e s 6.1.1 and 6.1.2. A n o t h e r more c r i t i c a l c o m p a r i s o n o f t h e s e r e s u l t s i s t h e c o m p a r i s o n shown i n F i g u r e 4.2.4, s i n c e t h e b e h a v i o u r d u r i n g s l i p i n v o l v e s t h e d y n a m i c s o f t h e c o m p l e t e s y s t e m , w h e r e a s t h e s t i c k p h a s e i s m o s t l y , g o v e r n e d b y t h e s t a t i c f r i c t i o n r e l a t i o n s h i p . The c o m p u t a t i o n s w e r e n o t d i r e c t l y m a t c h e d t o the. v a l v e s e t t i n g s u s e d i n t h e e x p e r i m e n t s s i n c e t h e s e w e re n o t r e p e a t -a b l e e nough t o w a r r a n t a c c u r a t e c a l i b r a t i o n as s e e n f r o m A p p e n d i x I I . Thus f o r e x p e r i m e n t a l p u r p o s e s t h e v a l v e s w e r e m e r e l y a d j u s t e d t o g i v e e q u a l p r e s s u r e d r o p s d u r i n g u n i f o r m f l o w w i t h c o n s t a n t s p e e d p i s t o n m o t i o n w h i l e t h e l o a d was d i s -c o n n e c t e d . F o r f r i c t i o n m e a s u r e m e n t s o f c o u r s e t h i s p r o c e d u r e was u n n e c e s s a r y . The n u m e r i c a l c o m p u t a t i o n s w e r e c a r r i e d o u t f o r a r a n g e o f a r b i t r a r y v a l v e o p e n i n g s . The r e s u l t s c o u l d t h e n be m a t c h e d f o r e q u a l f l o w r a t e s by c o m p a r i n g a v e r a g e v e l o c i t i e s . T h i s o f c o u r s e i s a c h i e v e d i n h a v i n g t h e same t i m e o f s t i c k and t h e same d i s p l a c e m e n t p e r c y c l e . The m a i n d i s c r e p a n c y b e t w e e n t h e c a l c u l a t e d and e x p e r i m e n t a l r e s u l t s was t h e s h a p e o f t h e s t r a i n r i n g f o r c e 60 curve d u r i n g t h e e a r l y p o r t i o n o f t h e s t i c k c y c l e . S i n c e t h e p r e s s u r e f o r c e d i d n o t behave s i m i l a r l y , t h i s c u r v a t u r e had t o be due t o i n s t r u m e n t a t i o n , o r the n a t u r e o f the p i s t o n m otion had t o be n o n - l i n e a r d e s p i t e l i n e a r l y i n c r e a s i n g d r i v i n g f o r c e . The response o f t h e b r i d g e a m p l i f i e r was f l a t from 0 t o 20,000 H e r t z , w h i c h was r a p i d enough t o e l i m i n a t e the p o s s i b l i t i y o f e l e c t r o n i c l a g , and the o s c i l l o s c o p e response was even h i g h e r . A m e c h a n i c a l p o s s i b i l i t y f o r e r r o n e o u s r e a d -i n g s was t h e p i v o t i n g o f t h e h y d r a u l i c c y l i n d e r i n the b a l l b e a r i n g mounts, r e s u l t i n g i n a t w i s t i n t h e s t r a i n r i n g . T h i s seemed u n l i k e l y s i n c e t h e p i s t o n was n e a r l y b a l a n c e d d u r i n g the e x p e r i m e n t s a t the c e n t r e o f t h e s t r o k e , and t h e s t r a i n r i n g was always under c o n s i d e r a b l e t e n s i o n . A l s o manual p i v o t i n g o f the c y l i n d e r from i t s normal a l i g n m e n t caused an i n c r e a s e i n the f o r c e r e a d i n g , and t h e o b s e r v e d b e h a v i o u r c o u l d o n l y o c c u r from an e r r o n e o u s l y low r e a d i n g a t t h e end o f s l i p . To f u r t h e r a s c e r t a i n t h a t t h i s was not t h e c a u s e , t h e c y l i n d e r was b r a c e d a g a i n s t p i v o t i n g a t t h e ends, and no n o t i c a b l e change i n the waveform o c c u r r e d . P o s s i b l e causes f o r n o n - l i n e a r p i s t o n m o t i o n were ;.next sought. The s i m p l e s t method of a l t e r i n g t h e p i s t o n motion i n the c a l c u l a t i o n s was by the a d d i t i o n o f a damping term t o t h e p i s t o n and s e a l f r i c t i o n f o r c e . T h i s approach l e d t o r e s u l t s r e s e m b l i n g the observed c a s e , as shown i n F i g u r e 6.1.3. The damping used i n t h i s case was 100 pounds p e r i n c h p e r second. A p o s s i b l e s o u r c e o f a damping term would a r i s e from p i s t o n l e a k a g e , b u t a l t h o u g h computations i n v o l v i n g l e a k a g e showed some s l i g h t t e n d e n c y f o r a n o n - l i n e a r r i s e i n t h e s t r a i n r i n g f o r c e , t h i s n o n - l i n e a r i t y a l s o a p p e a r e d i n t h e p r e s s u r e f o r c e , w h i c h was n o t t h e c a s e i n t h e e x p e r i m e n t s . I f d a m p i n g o f t h i s m a g n i t u d e i s i n f a c t t h e c a u s e o f t h e o b s e r v e d b e h a v i o u r , i t i s o n l y e x h i b i t e d d u r i n g s t i c k -s l i p c y c l e s . A number o f c o n s t a n t s p e e d t e s t s w e r e made t o m e a s u r e t h e s e a l f r i c t i o n a t s p e e d s up t o a b o u t 3 i n c h e s p e r s e c o n d , and a l t h o u g h some r e p e a t a b l e v a r i a t i o n i n t h e f o r c e was a p p a r e n t w i t h t h e p o s i t i o n o f t h e p i s t o n , no v e l o c i t y d e p e n d e n c e was o b s e r v e d . To e l i m i n a t e t h i s p o s i t i o n a l v a r i -a t i o n , m o s t e x p e r i m e n t s w e r e p e r f o r m e d o v e r a b o u t one i n c h o f s t r o k e w h e r e t h e f o r c e r e m a i n e d c o n s t a n t w i t h i n a p p r o x i m a t e l y one p o u n d . I n any c a s e t h e v a r i a t i o n o f t h i s f o r c e was g r a d u a l and s m a l l e nough ( o f t h e o r d e r o f ± 5 p o u n d s ) t h a t t h e b e h a v i o u r o f t h e s y s t e m a t d i f f e r e n t p o i n t s a l o n g t h e t r a v e r s e w o u l d n o t be s i g n i f i c a n t l y a f f e c t e d . The p r e s s u r e d r o p a c r o s s t h e p i s -t o n w o u l d change b y o n l y a b o u t 5 p e r c e n t . A n o t h e r p o s s i b l e c a u s e f o r t h e o b s e r v e d b e h a v i o u r swas t h a t t h e f r i c t i o n a t t h e s e a l s m i g h t d r o p d u r i n g t h e more , r a p i d m o t i o n o f t h e s l i p c y c l e , a nd due t o t h e i r f l e x i b i l i t y t h e d e f o r m a t i o n o f t h e s e a l s d u r i n g t h e i n i t i a l s t a g e s o f s t i c k m i g h t be c a u s i n g t h e v a r i a b l e p i s t o n s p e e d . An a t t e m p t t o i n v e s t i g a t e t h i s was made b y r e m o v i n g t h e r o d w i p e r s . T h i s r e s u l t e d i n a d e c r e a s e i n f r i c t i o n f o r c e f r o m a b o u t 140 p o u n ds t o a b o u t 70 p o u n d s , b u t t h e s h a p e o f t h e s t r a i n r i n g f o r c e u n d e r s t i c k - s l i p was n o t a l t e r e d . A f t e r p r o l o n g e d use w i t h o u t w i p e r s , t h e s e a l f r i c t i o n appeared t o become s l i g h t l y e r r a t i c w i t h o c c a s i o n a l tendency towards d i s c o n t i n u o u s motion o f t h e p i s t o n r o d when o p e r a t e d w i t h no mass a t t a c h e d t o t h e s t r a i n r i n g . T h i s was p r o b a b l y caused by d u s t p a r t i c l e s h a v i n g worked i n t o t h e s e a l s , and the presence o f d i r t as w e l l as the accompanying wear on t h e s e a l s may have l e d t o t h e e r r a t i c f r i c t i o n f o r c e . S i n c e t h e n u m e r i c a l s o l u t i o n s o f t h e g o v e r n i n g e q u a t i o n s gave good agreement w i t h e x p e r i m e n t w i t h o u t m a t c h i n g t h e e x a c t shape o f the s t r a i n r i n g f o r c e c u r v e , a number o f computer s t u d i e s were made t o dete r m i n e t h e e f f e c t o f l i n e a r i z i n g t h e system o f e q u a t i o n s and v a r y i n g system p a r a m e t e r s . , The s i m p l i f i c a t i o n i n v o l v i n g the assumption o f symmetry which l e d t o e q u a t i o n (2.1.25) was found v a l i d f o r a c e n t r a l l y l o c a t e d p i s t o n . The p l o t s from u s i n g t h i s s i m p l i f i e d e q u a t i o n were i n d i s t i n g u i s h a b l e from t h o s e d e r i v e d from u s i n g the o r i g i n a l e q u a t i o n s w i t h a c e n t r a l l y l o c a t e d p i s t o n . The e f f e c t o f l i n e a r i z i n g the f l o w v a l v e c h a r a c t e r i s t i c as i n e q u a t i o n (2.1.6) was t e s t e d and a g a i n the r e s u l t s were v i r t u a l l y t he same as w i t h t h e o r i g i n a l s o l u t i o n s t o an a c c u r a c y c l o s e r t h a n w a r r a n t e d by comparison w i t h e x p e r i m e n t s . Thus t h e e q u a t i o n s g o v e r n i n g t h e system were f i n a l l y s i m p l i f i e d t o the f o l l o w i n g form: V 1 k d(p -p ) _ [_ + _E] ± £_ + A v = Ca [p - ( p , - p 9 ) ] (6.1.1) 2 B A d t P P s J. / P dv % -zf + F f + k ( x P " V = V pi - P 2 } ( 6 - 1 - 2 ) ^ d t p * ^ dv M — + F f + k (x - x ) = 0 (6.1.3) r d t m m p where C i s a c o n s t a n t and i s the k i n e t i c f r i c t i o n f o r c e rm d u r i n g s l i p . F o r the s t i c k c y c l e , e q u a t i o n (6.1.3) becomes k ( x - x ) < F- (6.1.4) p m — f ^ s w i t h s l i p commencing when the two s i d e s o f the e q u a t i o n become e q u a l . A n a l y t i c a l s o l u t i o n s have not been a c h e i v e d , as even c o n s i d e r i n g e q u a t i o n (6.1.3) a l o n e w i t h t h e complex f r i c t i o n f u n c t i o n s and t a k i n g x - x = C . t , s i m p l e s o l u t i o n s are n o t ^ m p 1 ' * a v a i l a b l e . As was p o i n t e d o u t , the above e q u a t i o n s g i v e r e s u l t s v i r t u a l l y i n d i s t i n g u i s h a b l e from t h o s e w i t h the complete ; e q u a t i o n s f o r a c e n t r a l l y l o c a t e d p i s t o n . To d e t e r m i n e th e e f f e c t o f p i s t o n l o c a t i o n , a n u m e r i c a l i n v e s t i g a t i o n was c a r r i e d o u t . F i g u r e 6.1.4 shows the v a r i a t i o n o f t h e t ime o f s t i c k , a m p l i t u d e o f s l i p , and p e r i o d o f the s l i p c y c l e f o r a c o n s t a n t f l o w v a l v e opening a g a i n s t t h e p i s t o n l o c a t i o n . The e f f e c t o f p o s i t i o n i n g the p i s t o n o t h e r t h a n c e n t r a l l y was t o s t i f f e n the h y d r a u l i c system. The second a b s c i s s a o f F i g u r e 6.1.4 shows the " e q u i v a l e n t s t i f f n e s s " k ^ o f t h e c y l i n d e r and h y d r a u l i c f l u i d c o m b i n a t i o n c a l c u l a t e d from eq + I L-Z 64 (6.1.5) where k c i s t h e s t i f f n e s s f o r a c e n t r a l l y l o c a t e d p i s t o n , I the d i s t a n c e from one end, and L t h e s t r o k e o f t h e c y l i n d e r . To determine the v a l i d i t y o f t h i s c o m p u t a t i o n f o r an e q u i v a l e n t s t i f f n e s s , t h e e q u a t i o n s were s o l v e d f o r a symmet-r i c a l c y l i n d e r o f t h e s e s t i f f n e s s v a l u e s , and t h e r e s u l t s a re a l s o g i v e n i n F i g u r e 6.1.4. The p o i n t s a re v i r t u a l l y i d e n t i c a l , p r o v i n g t h a t t h e b e h a v i o u r o f a c e n t r a l l y l o c a t e d p i s t o n w i t h a s t i f f n e s s as g i v e n by e q u a t i o n (6.1.5) i s t h e same as t h a t f o r t he c o r r e s p o n d i n g n o n - c e n t r a l l y l o c a t e d one. As would be e x p e c t e d , the r e s u l t o f s t i f f e n i n g the h y d r a u l i c system was t o i n c r e a s e the f r e q u e n c y o f o c c u r r e n c e o f s l i p c y c l e s w i t h a c o r r e s p o n d i n g d e c r e a s e i n t h e motion p e r c y c l e . The average v e l o c i t y t h r o u g h o u t the range remained c o n s t a n t a t .0032 i n c h e s p e r second. S i n c e the compliance o f the c y l i n d e r and f l u i d as seen from e q u a t i o n (6.1.1) i s g i v e n by ; V 1 k - [ - + — ] (6.1.6) 2 B A P the s t i f f e n i n g e f f e c t d emonstrated p r e v i o u s l y can be a c h i e v e d by any c o m b i n a t i o n o f d e c r e a s i n g the c y l i n d e r Volume V, i n c r e a s i n g the f l u i d b u l k modulus B, i n c r e a s i n g p i s t o n a r e a Ap w h i l e m a i n t a i n i n g c o n s t a n t volume, o r d e c r e a s i n g t h e com-p l i a n c e o f the c y l i n d e r k . ' 65 V a r y i n g t h e s t i f f n e s s o f t h e c o u p l i n g between the p i s t o n r o d and the d r i v e n mass g i v e s the r e s u l t s shown i n F i g u r e 6.1.5. A g a i n t h e c y c l i n g f r e q u e n c y i n c r e a s e s and d i s p l a c e m e n t p e r c y c l e d e c r e a s e s w i t h i n c r e a s e d s t i f f n e s s . I n b o t h cases t h i s can be d i r e c t l y r e l a t e d t o t h e i n c r e a s e i n 0, the r a t e o f i n c r e a s e o f t h e t a n g e n t i a l f o r c e c o e f f i c i e n t . I n t h e case o f ch a n g i n g t h e c o u p l i n g s t i f f n e s s a l a r g e r v a r i a t i o n i n t h e s l i p c y c l e p e r i o d i s e v i d e n t as would be e x p e c t e d s i n c e t h e coupling-mass c o m b i n a t i o n s h o u l d be t h e main d e t e r m i n a n t o f t h e n a t u r a l f r e q u e n c y u n l e s s t h i s c o u p l i n g i s v e r y much s t i f f e r t han t h e h y d r a u l i c system. The e f f e c t s from s t i f f e n i n g the c o u p l i n g d e c r e a s e w i t h i n c r e a s e d s t i f f n e s s s i n c e the compliance o f the c y l i n d e r - f l u i d system becomes more predominant. The above two s e t s o f r e s u l t s i n v o l v e d a change i n t h e n a t u r a l f r e q u e n c y as w e l l as 0. To observe the v a r i a t i o n o f the system b e h a v i o u r w i t h v a r y i n g 0, computations were c a r r i e d out f o r a range o f v a l v e o p e n i n g s . F i g u r e 6.1.6 shows t h e r e s u l t s a l o n g w i t h some e x p e r i m e n t a l p o i n t s . F o r comparison a p l o t o f the d i f f e r e n c e between t h e s t a t i c f r i c t i o n c o e f f i -c i e n t s from F i g u r e 4.2.6 and t h e 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 (taken as .18) i s g i v e n . T h i s i l l u s t r a t e s t h e f a c t t h a t t h e a m p l i t u d e o f s l i p i s p r i m a r i l y a f u n c t i o n o f t h e d i f f e r e n c e between the s t a t i c and k i n e t i c c o e f f i c i e n t s o f f r i c t i o n . (The r e l a t i o n f o r an undamped case w i t h a c o n s t a n t c o e f f i c i e n t o f f r i c t i o n i s g i v e n i n Appendix I I I , e q u a t i o n ( A . I I I . 1 2 ) ' 66 Curve 1 i n F i g u r e 6.1.6 i s s i m p l y a s t r a i g h t l i n e a p p r o x i m a t -i n g t he s t a t i c f r i c t i o n p l o t . E q u a t i o n ( A . I I I . 1 2 ) was used t o draw cu r v e 2. I n the e q u a t i o n the s t r a i n r i n g s t i f f n e s s was used f o r k and the s l i d e r w e i g h t f o r W. I t i s apparent t h a t the a m p l i t u d e o f motion i s g r e a t e r t h a n t h a t c a l c u l a t e d , which i s o f c o u r s e due t o t h e s o f t e n i n g o f t h e system by t h e h y d r a u l i c c y l i n d e r . However t h i s c l e a r l y shows t h e c l o s e r e l a t i o n between s t i c k - s l i p a m p l i t u d e and the s t a t i c f r i c t i o n c o e f f i c i e n t . I n s p e c t i o n o f t h e r e s u l t s o f v a r i o u s w o r k e r s i n the p l o t s o f s e c t i o n 4.2 shows t h a t i n a l l cases t h e minimum s t a t i c f r i c t i o n v a l u e was reached f o r a v a l u e o f 0 o f approx-i m a t e l y one second 1 . Thus a c r i t i c a l v e l o c i t y above wh i c h s t i c k - s l i p d i s a p p e a r s can e a s i l y be e s t i m a t e d f o r a system o f the t y p e shown i n F i g u r e 1.1.1 by (6.1.7) In g e n e r a l , s t i c k - s l i p can be a v o i d e d by h a v i n g t h e -1 v a l u e o f 0 exceed one second Other workers have shown t h a t t h e i n t r o d u c t i o n o f damping a t the d r i v e n mass reduces the a m p l i t u d e o f s l i p c y c l e s , and r e s u l t s i n a lower c r i t i c a l v e l o c i t y [24] . The e f f e c t o f some damping v a l u e s i s shown i n F i g u r e 6.1.6. C o n s i d e r i n g o n l y the s t r a i n r i n g and s l i d e r , the c r i t i c a l damping f o r t h e case shown i s 190 pounds p e r i n c h p e r second. Based on t h i s i t can 67 be s e e n t h a t a d a m p i n g c o e f f i c i e n t o f a b o u t .25 h a l v e s t h e c r i t i c a l v e l o c i t y . I n p r a c t i c a l s y s t e m s o f h i g h s t i f f n e s s and. mass, s u c h l a r g e d a m p i n g w o u l d seem i m p r a c t i c a l . A s m e n t i o n e d p r e v i o u s l y , c o m p u t a t i o n s w e r e a l s o c a r r i e d o u t t o e v a l u a t e t h e e f f e c t o f l e a k a g e on t h e s y s t e m b e h a v i o u r . A number o f d i f f e r e n t l e a k a g e r a t e s w e r e a p p l i e d w h i l e m a i n t a i n i n g a c o n s t a n t f l o w v a l v e o p e n i n g , a nd t h e r e s u l t s a r e g i v e n i n F i g u r e 6.1.7. The a b s c i s s a i s an a p p r o x -i m a t e v a l u e o f t h e a v e r a g e l e a k a g e as a p e r c e n t a g e o f t h e a v e r a g e f l o w t h r o u g h t h e s y s t e m . The p e r i o d o f t h e s l i p c y c l e i s n o t a f f e c t e d , b u t t h e d i s p l a c e m e n t p e r c y c l e i n c r e a s e s somewhat w h i l e t h e t i m e o f s t i c k i n c r e a s e s c o n s i d e r a b l y . The o v e r a l l e f f e c t o f c o u r s e i s a d e c r e a s e i n $ s i n c e t h e e f f e c t i v e f l o w r a t e i s d e c r e a s e d . F o r c o m p a r i s o n a p l o t i s shown o f t h e r e s u l t s c o n s i d e r i n g no l e a k a g e , b u t u s i n g a v e r a g e f l o w v a l u e s e q u a l t o t h e t o t a l f l o w m i n u s l e a k a g e . T h i s i n d i c a t e s t h a t t h e v i s c o u s f o r c e s due t o l e a k a g e c a n h a v e a n o t i c e a b l e e f f e c t on t h e s y s t e m b e h a v i o u r , b u t t h i s i s n o t s i g n i f i c a n t u n t i l a v e r y l a r g e amount o f l e a k a g e t a k e s p l a c e . I n t h e l e a k a g e c a l c u l a t i o n s a p i s t o n o f 1.5 i n c h e s d i a m e t e r a nd 1 i n c h l o n g was u s e d . F o r 8 p e r c e n t l e a k a g e , a d i a m e t r a l c l e a r a n c e o f .001 i n c h e s was n e c e s s a r y , i n d i c a t i n g t h a t i n most p r a c t i c a l a p p l i c a t i o n s t h e l e a k a g e s h o u l d be s m a l l e n o u g h t o be i g n o r e d . F i n a l l y some c o m p u t a t i o n s w e r e p e r f o r m e d t o f i n d s o l u t i o n s by u s i n g a c o n s t a n t v a l u e f o r t h e 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 . A l t h o u g h t h e t o t a l d u r a t i o n o f the s l i p c y c l e was c o n s i d e r a b l y s h o r t e r w i t h t h i s a p praoch, i n c r e a s i n g t h e v a l u e o f t h e k i n e t i c f r i c t i o n c o e f f i c i e n t by t h e o r d e r o f 20 p e r c e n t o f t h e d i f f e r e n c e between the s t a t i c c o e f f i c i e n t and the minimum k i n e t i c c o e f f i c i e n t gave r e s u l t s a g r e e i n g ; i n a m p l i t u d e and f o r c e f l u c t u a t i o n . C H A P T E R 7 7.1 CONCLUSION The b e h a v i o u r o f a h y d r a u l i c a l l y d r i v e n system under-g o 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 has been a n a l y z e d from t h e p o i n t o f view o f t h e h y d r a u l i c c i r c u i t and t h e f r i c t i o n c o u p l e a t the d r i v e n mass. N u m e r i c a l c o m p u t a t i o n has shown t h a t the b e h a v i o u r o f the h y d r a u l i c ram i s a d e q u a t e l y d e s c r i b e d by t h e e q u a t i o n s d e r i v e d , and t h a t c e r t a i n s i m p l i f i c a t i o n s i n t h e o r i g i n a l d e r i v a t i o n s may be c a r r i e d out w i t h n e g l i g i b l e e f f e c t on t h e s o l u t i o n s . Some sour c e o f damping o r o t h e r f o r c e a f f e c t i n g the motion o f t h e p i s t o n appears t o e x i s t d u r i n g t r a n s i e n t c o n d i t i o n s , b u t i s not apparent i n s t e a d y s t a t e m o t i o n . How-eve r t h i s does n o t c o n t r i b u t e s u f f i c i e n t l y t o the g e n e r a l b e h a v i o u r t o i n v a l i d a t e r e s u l t s d e r i v e d by n e g l e c t i n g i t . S i n c e t h e a m p l i t u d e o f s t i c k - s l i p c y c l e s i s approx-i m a t e l y p r o p o r t i o n a l t o the d i f f e r e n c e between the s t a t i c and the f i n a l k i n e t i c v a l u e s o f f r i c t i o n as was shown i n Appendix I I I , a p l o t o f t h e ty p e shown f o r the s t a t i c f r i c t i o n f o r c e can be used t o a p p r o x i m a t e l y d e t e r m i n e c y c l e a m p l i t u d e s . I t i s a l s o seen t h a t t h e o s c i l l a t i o n s cease when t h e v a l u e o f the r a t e o f i n c r e a s e o f the t a n g e n t i a l f o r c e c o e f f i c i e n t r eaches a v a l u e o f about u n i t y , s i n c e t h e r e i s no l o n g e r a , s i g n i f i c a n t growth i n the v a l u e o f t h e 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 . Thus f o r a s i m p l e undamped system such as shown i n F i g u r e 1.1.1, a l i m i t i n g v e l o c i t y can be e a s i l y computed. 70 I n v e s t i g a t i o n o f v a r y i n g system parameters showed t h a t i n g e n e r a l t h e h y d r a u l i c system behaved l i k e a s p r i n g a c t e d on by a ramp l o a d i n g from th e f l u i d p r e s s u r e , and t h e n a t u r e o f t h e o b s e r v e d s t i c k - s l i p m o tion was m a i n l y a t t r i b u t -a b l e t o the f r i c t i o n c h a r a c t e r i s t i c s a t t h e s l i d i n g mass. The e f f e c t o f l e a k a g e was found t o be n e g l i g i b l e , and the v a r i a t i o n i n b e h a v i o u r w i t h t h e l o c a t i o n o f t h e p i s t o n i n the c y l i n d e r was seen t o become s i g n i f i c a n t m a i n l y i n t h e l a s t 25 p e r c e n t o f the s t r o k e . V e ry d e t a i l e d a t t e n t i o n was g i v e n t o t h e f r i c t i o n a l , f o r c e s p r e s e n t a t t h e s l i d i n g mass d u r i n g s t i c k - s l i p , and ,new and unique d e s c r i p t i o n s f o r t h e i r b e h a v i o u r were found. In d e a l i n g w i t h the s t a t i c f r i c t i o n f o r c e , t h o s e p r e -v i o u s workers who c o n s i d e r e d any v a r i a t i o n from a c o n s t a n t v a l u e , r e l a t e d t h i s v a r i a t i o n t o t h e t i m e o f s t a t i o n a r y con-t a c t . I t has been shown t h a t t h i s approach i s v a l i d o n l y f o r a ramp l o a d i n g w i t h no d w e l l , and t h a t i n f a c t t h e s t a t i c f r i c t i o n c o e f f i c i e n t depends on the n a t u r e o f t h e t a n g e n t i a l l o a d a p p l i c a t i o n . A number o f models f o r t h i s t y p e o f be-^ h a v i o u r were i n v e s t i g a t e d , and t h e s i m p l e one p r e s e n t e d gave r e a s o n a b l e c o r r e l a t i o n w h i l e p r o v i d i n g some i n s i g h t i n t o t h e o r d e r o f magnitude o f parameters g o v e r n i n g s t a t i c f r i c t i o n . T h i s i n f o r m a t i o n s h o u l d be u s e f u l t o f u t u r e i n v e s t i g a t o r s as a s t a r t i n g p o i n t i n d e r i v i n g a more a c c u r a t e model o r i n s u g g e s t i n g e x p e r i m e n t a l p r o c e d u r e s t o f u r t h e r c l a r i f y t h e mechanics of s u r f a c e c o n t a c t s . . The f r i c t i o n f o r c e s e x i s t i n g a t t h e o n s e t o f g r o s s s l i d i n g have a l s o been i n v e s t i g a t e d i n g r e a t e r d e t a i l t h a n i n the p a s t , and f o r the f i r s t t i me has t h i s t y pe o f f r i c t i o n a l b e h a v i o u r been used i n a n a l y z i n g s t i c k - s l i p . A r e a s o n a b l e mechanism t o account f o r t h e obser v e d k i n e t i c f r i c t i o n be-r h a v i o u r i s p r e s e n t e d . To t h e a u t h o r ' s knowledge t h i s i s t h e f i r s t a ttempt e v e r made t o d e s c r i b e t h i s l i t t l e s t u d i e d phenomenon. A P P E N D I X I APPENDIX I CALCULATION OF CYLINDER STIFFNESS In d e t e r m i n i n g t h e e l a s t i c d e f o r m a t i o n o f the h y d r a u l i c c y l i n d e r M a t s u z a k i and Hashimoto l i k e S h e a r e r c o n s i d e r e d a t h i c k s h e l l c y l i n d e r w i t h u n r e s t r a i n e d ends. A more a c c u r a t e d e r i v a t i o n s h o u l d c o n s i d e r end f o r c e s as w e l l , and i n the f o l l o w i n g some d e r i v a t i o n s w i l l be made, u s i n g t h e symbols o f F i g u r e A.1.1. C o n s i d e r a t i o n w i l l be g i v e n o n l y t o f i n d i n g s t r a i n s i n each case as the compliance can be found as f o l l o w s : F o r d i a m e t r a l e x p a n s i o n — = TT (a-j^ - a ) = 2Tra Ae Q ( A . I . I ) where a n i s the c y l i n d e r i n s i d e d i a m e t e r a f t e r a s t r a i n A e Q . 1 u S i n c e the s t r a i n can be g i v e n as a c o n s t a n t t i m e s the p r e s s u r e , and the compliance as d e f i n e d i n e q u a t i o n (2.1.4) becomes Ae f i = k„Ap (A. 1. 2) k = 2-rra k 0 (A.I.3) F o r l e n g t h w i s e e x p a n s i o n , AV I = A Ae P z (A.I.4) A g a i n t h e s t r a i n i s p r o p o r t i o n a l t o p r e s s u r e , g i v i n g A e z = k z A p (A.1.5) and thus k = A k (A.I.6) e z p z When bo t h d i a m e t r a l and l e n g t h w i s e e x p a n s i o n are con-s i d e r e d , t h e compliance s i m p l y becomes k = k + k (A.I.7) e e 9 e z Case I : T h i c k c y l i n d e r , f r e e ends 2, 2 1 2 a b p a s r = [~72—— T + 7T~2] P b -a r b -a 2, 2 , 2 a b p 1 a 72 2 ~2 + 72 2 ] p b -a r b -a (A.I.8) s = 0 z U s i n g t h e s e s t r e s s e s e v a l u a t e d a t r = a, we g e t 1 b +a e 6 = ~ [ ~2 2 + v ] p (A.1.9) 1 E b -a where E i s t h e modulus o f e l a s t i c i t y and v t h e P o i s s o n ' s r a t i o . 74 Case I I : T h i n s h e l l , f r e e ends pa s Q = s r = s z = 0 (A.1.10) b-a g i v i n g 1 a e = [ ] p (A.1.11) 2 E b-a E q u a t i o n s (A.I.9) and(A.I.11) a r e f u n c t i o n s o f o n l y the r a t i o b/a i f v i s c o n s t a n t . F i g u r e A.I.2 compares t h e s o l u t i o n s f o r v = .3. Case I I I : T h i n s h e l l , ends as i n F i g u r e A . I . I s. A, = -s A + pA (A. 1.12) b b z z c p o r s. A, +pA = b b p (A.I.13) A S i n c e the end p a r t s must remain i n c o n t a c t , s = s, , c ' z b' y i e l d i n g s b = E e b = E e z (A.I.14) t h e r e f o r e 75 -Ee A, + pA s z = (A.1.15) A U s i n g Hooke's Law and s Q f r o m e q u a t i o n (A.I.10) y i e l d s pA A a A s = 2_ [ _P_ _ v ]+ p _E (A.i.16) A +A A b - a A z p z z R e p l a c i n g s z i n e q u a t i o n (A.I.8) b y t h i s v a l u e , t h e s t r a i n s become p a vA, A a A ^ = - { £_ [ _ £ - v ] - v - £ } (A.1.17) 3 E b - a A +A, A b - a A z b z z p A, A v a A v a e = - { — [ -£ - ] + -E } (A.1.18) E A +A, A b - a A b - a z b z z Now t h e e q u a t i o n f o r s t i f f n e s s i s no l o n g e r j u s t a f u n c t i o n o f t h e r a t i o b / a b u t a l s o d e p e n d s on p i s t o n a r e a and e x t e r n a l a s s e m b l y b o l t s i z e . Thus a s i m p l e c o m p a r i s o n i s n o t p o s s i b l e w i t h e q u a t i o n s (A.I.9) o r ( A . I . 1 1 ) . However i t may be s e e n t h a t c o n s i d e r i n g s z h a s a s t i f f e n i n g e f f e c t : 1) F o r r a d i a l e x p a n s i o n e 9 = I ( S 9 - V S z } (A.I.19) f r o m w h i c h i t becomes e v i d e n t t h a t e„ d e c r e a s e s as s i s 0 z i n c r e a s e d . 2) A x i a l e x p a n s i o n i s i n g e n e r a l n e g a t i v e as shown by t h e f o l l o w i n g d e r i v a t i o n : 76 C o n s i d e r i n g f o r s i m p l i c i t y the case where = 0, ( e q u a t i o n A.I.18) becomes 2 2 p a - r va e z = - [ ~ 2 2 ~ ] (A.1.20) E b - a b-a For t h i s e x p r e s s i o n t o be n e g a t i v e , a/(b-a) ( a 2 - r 2 ) / ( b 2 - a 2) . 1 > — v r e w r i t i n g g i v e s (b/a) + 1 x 1 - ( r / a ) 2 ( r / a ) 2 v (b/a) + 1 (A.I.21) > - o r < v (A.1.22) C o n s i d e r i n g the w o r s t case o f b/a = 1 and t a k i n g t h e v a l u e v = .3 g i v e s the r e q u i r e m e n t ~ >.63 (A.I.23) a Both the p r e s e n t system and t h a t o f M a t s u z a k i and Hashimoto s a t i s f i e d t h i s c r i t e r i o n . The f o l l o w i n g t a b u l a t i o n shows the v a l u e s o f c y l i n d e r c o mpliance f o r the p r e s e n t system u s i n g each o f the above methods. Method: I I I I I I . i n c h e s 4 e pounds 5.74 x 1 0 ~7 4.7 x 1 0 ~ 7 4.3 x 1 0 " 7 Thus i t appears t h a t c o n s i d e r i n g the end f o r c e has s u f f i c i e n t e f f e c t on the s t i f f n e s s t h a t the e x c e s s i v e l y s t i f f r e s u l t o f Case I I may w e l l be as a c c u r a t e as the r e s u l t o f Case I i n d e s c r i b i n g the c y l i n d e r s t i f f n e s s . I n c o n s i d e r i n g the complete system, the e x p a n s i o n o f the p i p i n g must a l s o be c o n s i d e r e d , and t h i s f a c t o r added;to t h a t f o r the c y l i n d e r . A P P E N D I X I I APPENDIX I I CALIBRATION OF INSTRUMENTS AND DETERMINATION OF SYSTEM PARAMETERS 1. V e l o c i t y t r a n s d u c e r : The v e l o c i t y t r a n s d u c e r had a m a n u f a c t u r e r ' s c a l i b r a -t i o n o f 596 m i l l i v o l t s p e r i n c h p e r second. T h i s c a l i b r a t i o n was checked by s t e a d y speed runs where t h e speed was d e t e r m i n e d by a d i a l gauge and s t o p watch. The f a c t o r y c a l i b r a t i o n was found t o be c o r r e c t . * 2. P r e s s u r e t r a n s d u c e r s : The p r e s s u r e t r a n s d u c e r s were s u p p l i e d w i t h a f a c t o r y c a l i b r a t i o n and the charge a m p l i f i e r s were a d j u s t a b l e t o the s e n s i t i v i t y o f the t r a n s d u c e r s . Comparison o f the p r e s s u r e t r a n s d u c e r r e a d i n g s w i t h t h e Bourdon p r e s s u r e gauge c o n f i r m e d the a c c u r a c y and l i n e a r i t y o f the t r a n s d u c e r - c h a r g e a m p l i f i e r c o m b i n a t i o n i n the range 0 t o 1000 pounds per square i n c h . * 3. Needle v a l v e s : The n e e d l e v a l v e c a l i b r a t i o n was performed by measuring f l o w r a t e s a t v a r i o u s p r e s s u r e drops and v a l v e In a l l the above c a l i b r a t i o n c h e c k s , one r e a d o u t was on the o s c i l l o s c o p e s c r e e n , which p r o v i d e s an a c c u r a c y o f ±3 per c e n t . Thus a l t h o u g h agreement w i t h f a c t o r y c a l i b r a t i o n was always found w i t h i n the c a p a b i l i t i e s o f the t e s t a p p a r a t u s , these c a l i b r a t i o n s c o u l d n ot be c o n f i r m e d t o t h e number o f s i g n i f i c a n t f i g u r e s p r o v i d e d by t h e m a n u f a c t u r e r . openings. The f l o w r a t e was d e t e r m i n e d by t i m i n g the o i l f l o w i n t o a g r a d u a t e d c y l i n d e r , and t h e p r e s s u r e drop was s i m p l y the s u p p l y p r e s s u r e as shown by the p r e s s u r e gauge. F i g u r e A . H . l shows t h e t y p e o f r e s u l t s o b t a i n e d . The s c a t t e r i s due t o d i f f i c u l t y i n r e p r o d u c i n g e x a c t l y t h e v a l v e s e t t i n g s . I t can be seen t h a t the p r e s s u r e - f l o w r e l a t i o n i s a p p r o x i m a t e l y l i n e a r , e s p e c i a l l y i f we c o n s i d e r i t i n a range o f i n t e r e s t such as a t p r e s s u r e s o f 200 ± 20 pounds per square i n c h . 4. System s t i f f n e s s : The s t r a i n r i n g s t i f f n e s s was i n i t i a l l y d e t e r m i n e d by measuring i t s d e f l e c t i o n under a g i v e n l o a d . However the r e s u l t was not v e r y p r e c i s e as a l o a d o f 50 pounds r e s u l t e d i n a d e f l e c t i o n o f o n l y .003 i n c h e s , and the measurement o f t h i s d i s p l a c e m e n t w i t h an i n s i d e micrometer would be a c c u r a t e t o o n l y about 10 per c e n t . Very l a r g e l o a d s were n o t used i n o r d e r t o a v o i d p l a s t i c d e f o r m a t i o n i n the r i n g . The above t e s t y i e l d e d 4 a s t i f f n e s s o f 1.7 x 10 pounds per i n c h . A b e t t e r v a l u e f o r the s t i f f n e s s was o b t a i n e d by measuring the n a t u r a l f r e q u e n c y o f the m a s s - s t r a i n r i n g system w i t h the mass mounted on b e a r i n g b a l l s t o e l i m i n a t e f r i c t i o n . 4 A s t i f f n e s s v a l u e o f 1.9 x 10 pounds per i n c h g i v e s the f o l l o w i n g t a b u l a t i o n : MASS pounds 24 65 105 147 f e x p e r i m e n t a l n ^ 88 52 43 38 f c a l c u l a t e d n 88 53 42 36 80 The s t i f f n e s s o f the h y d r a u l i c c y l i n d e r was c a l c u l a t e d i n Appendix I . To c o n f i r m the c a l c u l a t e d s t i f f n e s s , an e x p e r -i m e n t a l measurement was made by d e t e r m i n i n g the m o t i o n o f t h e p i s t o n r o d under a g i v e n p r e s s u r e a p p l i e d a t one end w h i l e the v a l v e a t the o t h e r end remained c l o s e d . A graph o f t h e r e s u l t s i s shown i n F i g u r e A . I I . 2 f o r t h e case when the p i s t o n was c e n t r a l l y l o c a t e d . T e s t s a t o t h e r p o s i t i o n s showed a l i n e a r dependence on the d i s t a n c e from t h e p i s t o n t o the c l o s e d end. 4 The graph shows a s t i f f n e s s o f 1.25 x 10 pounds p e r 5 5 i n c h , which on a per u n i t l e n g t h b a s i s i s 1.5 x 10 pounds 4 p e r i n c h . The c a l c u l a t e d v a l u e f o r t h i s s t i f f n e s s i s g i v e n i n terms o f the f l u i d b u l k modulus B and t h e c y l i n d e r c o m p l i a n c e k g o f Appendix I by i = | + k e ( A . I I . l ) c y l -7 4 U s i n g the above v a l u e f o r k , and 4.7 x 10 i n c h e s c y l p e r pound f o r k^ (Case I I , Appendix I ) , and s o l v i n g f o r B we 4 g e t B = 16 x 10 pounds per square i n c h . T h i s i s a r e a s o n a b l e v a l u e f o r t h e b u l k modulus, a l -though the e x a c t v a l u e was not a v a i l a b l e from S h e l l O i l f o r comparison. In g e n e r a l the b u l k modulus o f h y d r a u l i c o i l s i s 4 o f the o r d e r o f 20 x 10 pounds per square i n c h w i t h t h i s f i g u r e b e i n g f o r a r a t h e r h i g h b u l k modulus o i l . 5. A c c e l e r o m e t e r : The s e n s i t i v i t y o f t h e a c c e l e r o m e t e r was v a r i a b l e b y e x t e r n a l c i r c u i t r y , a n d c o u l d e a s i l y be c h e c k e d b y s t a n d i n g t h e a c c e l e r o m e t e r on i t s e n d s t o g i v e i n p u t s o f ± l g . The c o n f i g u r a t i o n u s e d was t h e one w i t h t h e l e a s t g a i n s i n c e t h i s r e s u l t e d i n t h e l e a s t p h a s e s h i f t . The s e n s i t i v i t y s p e c i f i e d f o r t h i s c a s e was .1 v o l t p e r g and was f o u n d t o be a c c u r a t e . * The a c c e l e r o m e t e r h a d a n o m i n a l n a t u r a l f r e q u e n c y o f 1000 H e r t z . Compared t o t h e s l i p c y c l e f r e q u e n c y o f 10 t o 50 H e r t z , t h i s w o u l d a p p e a r t o p r o v i d e s u f f i c i e n t r e s p o n s e t o t r a n s i e n t s d u r i n g t h e s l i p c y c l e t o g i v e an a c c u r a t e m e a s u r e o f t h e i n e r t i a f o r c e s i n v o l v e d . The m a i n p r o b l e m i n e l e c t r o n -i c a l l y summing t h e s t r a i n r i n g and a c c e l e r o m e t e r s i g n a l s was t h e f a c t t h a t t h e a c c e l e r o m e t e r h a d an i n h e r e n t p h a s e l a g . To c o m p e n s a t e f o r t h i s , a p h a s e s h i f t i n g c i r c u i t was u s e d i n t h e b r i d g e a m p l i f i e r o u t p u t . F i g u r e A . I I . 3 shows a c o m p a r i s o n o f e x p e r i m e n t a l r e s u l t s on t h e a c c e l e r o m e t e r , t h e r e s p o n s e o f a s e c o n d o r d e r s y s t e m w i t h t h e same n a t u r a l f r e q u e n c y , and t h e b e h a v i o u r o f two s i m p l e p h a s e s h i f t i n g c i r c u i t s w i t h two d i f f e r e n t t i m e c o n s t a n t s . I t c a n be s e e n t h a t f o r a l a r g e r a n g e o f f r e q u e n c i e s , t h e p h a s e d i s c r e p a n c y b e t w e e n t h e a c c e l e r o m e t e r s i g n a l and t h e s t r a i n r i n g s i g n a l c a n be k e p t q u i t e s m a l l . The a t t e n u a t i o n c h a n g e s n e g l i g i b l y w i t h t h e f r e q u e n c y i n t h e r a n g e o f f r e q u e n c shown. * I b i d . 82 I n o r d e r t o add t h e s p r i n g f o r c e and t h e i n e r t i a f o r c e , t h e c o r r e s p o n d i n g s i g n a l s must be a p p r o p r i a t e l y s c a l e d . I n t h e e x p e r i m e n t a l s y s t e m u s e d , t h e g a i n o f t h e a c c e l e r o m e t e r was k e p t f i x e d as i t s p h a s e s h i f t i n c r e a s e d w i t h i n c r e a s e d g a i n , w h e r e a s t h e b r i d g e a m p l i f i e r g a i n c o u l d be e a s i l y v a r i e d w i t h no e f f e c t on t h e s i g n a l p h a s e . Thus i f t h e f o r c e s e n s i t i v i t y o f t h e a c c e l e r o m e t e r i s known, t h e s t r a i n r i n g o u t p u t m e r e l y h a s t o be m a t c h e d t o t h a t s e n s i t i v i t y . C o n s i d e r an a c c e l e r o m e t e r w i t h an a c c e l e r a t i o n s e n -s i t i v i t y o f x v o l t s p e r g. Then f o r an a c c e l e r a t i o n o f a g , t h e o u t p u t o f t h e a c c e l e r o m e t e r i s x a v o l t s . The c o r r e s p o n d i n g i n e r t i a f o r c e i s ma. D i v i d i n g t h e a c c e l e r o m e t e r o u t p u t by t h e i n e r t i a f o r c e we g e t t h e f o r c e s e n s i t i v i t y . U s i n g t h e f e e t pounds s e c o n d s s y s t e m f o r t h e d i m e n s i o n s , „ . . . . , x a ( v o l t s ) x ( v o l t s ) F o r c e s e n s i t i v i t y = 5 - ~ r - = 3 — r * ma (pounds) mg (pounds) £ ^ 2 i t s ( A . I I . 2 ) W pounds Thus t h e s c a l i n g o f t h e s p r i n g f o r c e h a s t o be s e t t o a s e n s i t i v i t y o f x/W v o l t s p e r p o u n d , and t h e n summing t h e s p r i n g f o r c e and t h e a c c e l e r o m e t e r s i g n a l s w i l l g i v e t h e f r i c t i o n f o r c e . I n t h e p r e s e n t c a s e W i s t h e w e i g h t o f t h e s l i d i n g p l a t f o r m p l u s l e a d w e i g h t s . I n o t h e r c o n f i g u r a t i o n s i t may be n e c e s s a r y t o f i n d m t h r o u g h t h e n a t u r a l f r e q u e n c y and s p r i n g 83 s t i f f n e s s o f the system. T h i s l a t t e r method i n g e n e r a l a p p l i e s when t h e s p r i n g mass i s a l a r g e enough p o r t i o n o f t h e t o t a l moving mass t h a t i t s d i s t r i b u t e d mass must be t a k e n i n t o a c count. A P P E N D I X I I I APPENDIX I I I RATE OF INCREASE OF TANGENTIAL FORCE COEFFICIENT FROM TIME OF STICK AND FRICTION COEFFICIENTS C o n s i d e r t h e c a s e o f an undamped s e c o n d o r d e r s y s t e m ( F i g u r e 1.1.1) w i t h a c o n s t a n t 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 V The g o v e r n i n g e q u a t i o n i s m x + k x = Wy k ( A . I I I . l ) The c o m p l e t e s o l u t i o n o f t h i s e q u a t i o n i s g i v e n b y Wy, x = A s i n (wt + 0) + — - (A. I I I . 2) W i t h t h e i n i t i a l c o n d i t i o n x = x a t t = 0, we h a v e o ' Wy x = A s i n 0 + — - ( A . I I I . 3 ) o ^ When t h e c y c l e i s c o m p l e t e , t h e v e l o c i t y w h i c h i s g i v e n by x = Aw c o s (wt + 0) ( A . I I I . 4 ) r e t u r n s t o i t s i n i t i a l v a l u e o f x = Aw c o s 0 ( A . I I I . 5 ) 0 f T h i s happens when wt + 0 = 2TT - 0 ( A . I I I . 6 ) A t t h a t time the d i s p l a c e m e n t i s g i v e n by Wy, x = A s i n (2TT - 0) + — - ( A . I I I . 7 ) k o r wy k x f = -A s i n 0 + — - ( A . I I I . 8 ) Thus the d i f f e r e n c e between the i n i t i a l and f i n a l d i s p l a c e m e n t s i s g i v e n by Ax = x - x , = 2A s i n 0 ( A . I I I . 9 ) o r I f the i n i t i a l d i s p l a c e m e n t x i s due t o the s t a t i c o c o e f f i c i e n t o f f r i c t i o n y , i t i s g i v e n as wy„ x = — - ( A . I I I . 1 0 ) o , S u b s t i t u t i n g t h i s i n e q u a t i o n ( A . I I I . 3 ) , we g e t W A s i n 0 = (u - u. ) ( A . I I I . 1 1 ) k S K The d i s p l a c e m e n t over which t h e r e i s s t a t i o n a r y con-t a c t , Ax, may be g i v e n by 2W Ax = 2 A s i n 0 = — (u - y, ) ( A . I I I . 1 2 ) k S J C The time o f s t a t i o n a r y c o n t a c t t and the v e l o c i t y V g i v e us the f o l l o w i n g r e l a t i o n : 2W or Ax = t V = — (y -. u. ) ( A . I I I . 1 3 ) s k s K kv 2 (y - y v ) — = § — ^ _ ( A . I I I . 1 4 ) w t s Thus t h i s e q u a t i o n g i v e s the r a t e o f i n c r e a s e o f t h e t a n g e n t i a l f o r c e c o e f f i c i e n t , g i v e n y , y, , and t . S K S 87 v V\A-m P i g . l o l e l . Model of a System Subject to F r i c t i o n a l V i b r a t i o n time P i g . l . l o 2 T y p i c a l S t i c k - S l i p Displacement Records, V , > V 88 P i g . 1.2.1 Model of an Asperity Junction 89 P i g . 1.2.2 Archard's Surface P r o f i l e Model X e q u a t i o n 1.2.6 B( e q u a t i o n 1.2.5 ^ e q u a t i o n 1.2.7 i r 1 — i i — 3 4 5 6 t ime - seconds Compar i son of S t a t i c F r i c t i o n R e l a t i o n e velocity Pig. 1.2.4 Kinetic Coefficient of F r i c t i o n Against Velocity P3 P4 £_ P2_ Pi V V P I- In" «^ l i H / ///////////// F i g . 2.1.1 Schematic of Experimental System 93 f l u i d of v i s c o s i t y ji / / / / / / / p + A p P i g . 2.1.2 Diagram f o r Leakage C a l c u l a t i o n Pig. 3 o l . l Diagram of Experimental Apparatus Pig. 3.1.3 Lower Friction Surface Attachment P i g . 3.1.4 S l i d i n g Platform and S l i d e r s X u ACC L -0 . 3.1.5 Schematic of Hydraulic C i r c u i t .30 .28-displacement - m i l l i - i n c h e s P i g . 4.2.2 K i n e t i c F r i c t i o n as a Function of Displacement o o 101 20 T .5 o o o •*0 *-0 «o o o O O <b° ° ° ° ° -o-0 cP o o X X *x X r a m s t o p p e d d u r i n g s t i c k c y c l e o c o n t i n u o u s s t i c k - s l i p ^ c o n t i n u o u s s t i c k - s l i p d i s p l a c e d one c y c l e on t ime s c a l e 10 T 1 1000 .1 t ime - seconds 100 P i g . 4.2.5 S t a t i c P r i c t i o n v . s . Time of S t a t i o n a r y Ci.nt.ict o 0 0 0 0 0 0 0 0 8 • • a 0 000 ^ O 0 c c$ o o , ° o o a ^ X v «4 9) X o O <90 X X o O X 0 0 0 o x X ram s topped d u r i n g s t i c k c y c l e 0 o cont inuous s t i c k - s l i p 0 P o t t e r 0-1 l 1 1 1 1 3 1 1 1 1 1 .001 .002 .005 .01 . 02* , . 05 .1 .2 . 5 1 2 5 10 j3 -sec i . P i g . 4 . 2 . 6 S t a t i c F r i c t i o n v . s . 0 .5-.4-o O •H IH <H o +» C! CO •rl o •H <H V i o -O o X O + O k W 38.4 6 41.4 6 62.6 6 85 10 119 10 155 10 ,1-0- i ,01 .02 1^ ,03 T" .04 —T" .05 To6 Pig. v e l o c i t y - inches per second 4.2.7 S t a t i c F r i c t i o n v.s. Velocity - Potter o 901 .6 X X • Davis - gold )4 Cameron - ateei <J Davis - steel & Davis - indium O O O 0 O . o o O 0 0 0 o 0 0 0 .001 .01 _p_ .1 b - sec"1 Pig. 4.2.9 S t a t i c F r i c t i o n v.s. $ - Cameron, Davis i — 1 o -N3 • 5T 00*0 o o 9 o <bo o OO oo o o o o oo o tf a 0 O d r y s t e e l s u r f a c e s f e t c h e d s t e e l s u r f a c e s Immersed In a l c o h o l - I 10 .001 .01 0 - sec - 1 .1 I 1 . P i g . 4.2 .10 S t a t i c F r i c t i o n v . s . # - Dry and Stched S u r f a c e s o CO time - m i l l i s e c o n d s P i g . 5.1.1 Shape of Velocity Curve at Inception of S l i p 110 Fig. 5.1.^ Wear Mark on Steel B a l l - Copper Plated R a i l s F i g . 5.1.5 Wear Mark on Copper Plated R a i l s 112 time - milliseconds P i g . 5.1.6 Asperity Temperature During S l i p Cycle 113 w 7 k it F 0 + p-t ax P i g . 5.2.1 Model f o r S t a t i c F r i c t i o n 0~ o2 A .6 „8 time - seconds (1) piston displacement (2) s l i d e r displacement (3) pressure force (3)* pressure force - 100 pounds (4) s t r a i n r i n g force 1 o 0.0 0.4 0.8 1.2 ! . B 2.0 TIME F i g . 6.1.1 Comparison of Experimental and Calculated Results 116 time - seconds (1) piston displacement (2) s l i d e r displacement (3) pressure force (3)* pressure force - 100 pounds (4) s t r a i n r i n g force (1) (2) Fig. 6.1.2 Comparison of Experimental and Computed Results 117 (1) piston displacement (2) s l i d e r displacement (3) pressure force (k) s t r a i n r i n g force Damping = 100 pounds per inch per second F i g . 6.1.3 Effect of a Damping Term i n Piston F r i c t i o n X oequivalent s t i f f n e s s Xvarying position time of s t i c k p e r i o d of s l i p c y c l e d i s p l a c e m e n t per c y c l e 1 1 I 1 1 0 20 40 60 80 100 p i s t o n displacement from one end - per cent of s t r o k e i—r 8 1 I I 2 1.5 2 s t i f f n e s s - (pounds/inches^) x 10~^ T P i g . 6.1.4 E f f e c t of V a r y i n g P i s t o n P o s i t i o n 119 121 P i g . 6.1.7 E f f e c t of P i s t o n Leakage P i g . A . i . l Diagram of Hydraulic Cylinder Construction 50-, VO P i g . A.1.2 Comparison of Thick and Thin S h e l l Theory pressure - pounds per square inch o H ' H> H< « O CTi SS O > « M M c+ o CD H 1 * J o t—1 C 03 6 CT O -H-O P O CD < ft) c+ H-o S CO M l—j c+ C O -H- CO O cr hf i-J 03 c+ •a CD O 4 a B H- I—1 c o c+ CO t—1 O ' o 4 CD 1—1 O P3 •a c+ o CO H-P <!. o CD y (Ki < 03 fO i—' CO +721 cd r-l CU <n ca xi <x 12 10 co 8 cu Q)cu T3 6 -4 -2 H R n A A n c Ml—1 R R T T T O- T27 (2) - RC = 3.7 x 10~ 4 . (2) - RC = 3.2 x 10" 4 ( l ) - RC = 5.0 x 10" 4 second order system, damping r a t i o accelerometer response ~ ~ r 20 15 30 40 50 60 7 0 8 0 —1 r 90 100 frequency - Hertz P i g . A.II.3 Phase S h i f t i n g C i r c u i t s to Match Accelerometer ro ON R E F E R E N C E S REFERENCES 1. W e l l s , J .H., " K i n e t i c B o u n d a r y F r i c t i o n " , The E n g i n e e r ( L o n d o n ) , V o l . 1 4 7 , 1 9 2 9 , p. 454. 2. K r a g e l s k i i , I . 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T h e s i s i n 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 , 1965. 22. P o t t e r , A.F., "A S t u d y 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 " , M.A. S c . T h e s i s i n 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 , 1962. 23. D a v i s , H.R., "The D e p e n d e n c e o f t h e 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 on t h e Time o f S t a t i o n a r y C o n t a c t " , M.A. S c . T h e s i s i n 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 , 1966. 24. C a m eron, R., " F r i c t i o n I n d u c e d V i b r a t i o n " , M.A. S c . T h e s i s i n 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 , 1963. 25. C o u r t n e y - P r a t t , J . S . , E i s n e r , E., "The E f f e c t o f a T a n g e n t i a l F o r c e on t h e C o n t a c t o f M e t a l l i c B o d i e s " , P r o c e e d i n g s o f t h e R o y a l S o c i e t y ( L o n d o n ) , A, 2 3 8 , 1957, p. 529. " 129 26. J a e g e r , J . C , " M o v i n g S o u r c e s o f H e a t and t h e T e m p e r a t u r e a t S l i d i n g C o n t a c t s " , P r o c e e d i n g s o f t h e R o y a l S o c i e t y o f  New S o u t h W a l e s , 5 6 , 1 9 4 2 , p. 203. 27. J o h n s o n , K.L., " D e f o r m a t i o n o f a P l a s t i c Wedge by a R i g i d F l a t D i e U n d e r t h e A c t i o n o f a T a n g e n t i a l F o r c e " J o u r n a l o f M e c h a n i c s and P h y s i c s o f S o l i d s , V o l . 1 6 , 1968, p. 395. 28. A t k i n s , A.G., S i l v e r i o , A., T a b o r , D., " I n d e n t a t i o n H a r d n e s s and t h e C r e e p o f S o l i d s " , J o u r n a l o f t h e  I n s t i t u t e o f M e t a l s , V o l . 94, 1966, p. 369. 

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