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Transitory static and kinetic boundary friction mechanisms Marion, Terence Lionel 1972

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TRANSITORY STATIC AND KINETIC BOUNDARY FRICTION MECHANISMS by TERENCE LIONEL MARION B.A.Sc, University of B r i t i s h Columbia, Vancouver, B r i t i s h Columbia, 1967 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering We accept t h i s t r e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1972 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the 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 study. 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 copying of 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 gran t e d by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n of 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 not be allowed without my w r i t t e n p e r m i s s i o n . TERENCE LIONEL MARION Department o f MECHANICAL ENGINEERING The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The purpose of the investi g a t i o n was to study, by analysis of friction-induced v i b r a t i o n , the underlying physical mechanisms of me t a l l i c boundary f r i c t i o n . Dynamic system response and i n t e r f a c i a l f r i c t i o n force data, from a pin-on-disc machine operating over a broad range of surface speeds, was e l e c t r o n i c a l l y monitored and photo-graphically recorded. Excellent agreement i n the response curve p r o f i l e s of recorded rate-sensitive s t a t i c f r i c t i o n data and a predictive curve developed by assumption of a p l a s t i c deformation model of contact area growth suggests strongly that p l a s t i c deformation i s indeed the c o n t r o l l i n g physical mechanism of m e t a l l i c s t a t i c f r i c t i o n . The existence of an upper asymptote of s t a t i c f r i c t i o n i n the presence of a lubricant, and the existence i n the " s l i p " f r i c t i o n curve of a transient which appears governed by the r e l a t i v e dynamic displacement of the surfaces, has been proven. Vibratory s l i p and quasi-harmonic o s c i l l a t i o n both exhibited simultaneous solid-contact and viscous f l u i d f i l m c h a r a c t e r i s t i c s . The "humped" form of f r i c t i o n force vs v e l o c i t y curve necessary for quasi-harmonic o s c i l l a t i o n was concluded to d i f f e r from that of non-oscillatory s l i p only because of thermal v a r i a t i o n i n the f l u i d v i s c o s i t y , s i m i l a r to that encountered i n elastohydrodynamic studies. In every i n s t a n c e r a t e e f f e c t s were found to determine or pro f o u n d l y i n f l u e n c e the p h y s i c a l mechanisms of m e t a l l i c boundary f r i c t i o n . i v ACKNOWLEDGEMENT Sincere thanks are extended to Mr. E. Jones, Tribology Laboratory technician, for his knowledgeable advice concerning p r a c t i c a l development of the experimental apparatus, as well as for the construction of the instrumen-tati o n c i r c u i t r y . His assistance was t r u l y invaluable. The author wishes to acknowledge the support and constructive c r i t i c i s m tendered him by Dr. C.A. Brockley, and to thank his associates i n the laboratory for the opportunity to pa r t i c i p a t e i n f r u i t f u l discussion. Equipment constructed by the author's predecessors, Dr. H.R. Davis and Dr. P.L. Ko, greatly reduced the task of preparing suitable experimental apparatus: the time so saved i s not unappreciated. The experimental investigation was performed i n the Tribology Laboratory, Department of Mechanical Engineering, University of B r i t i s h Columbia. F i n a n c i a l assistance, i n the form of NRC Bursaries, was received from the National Research Council of Canada. V TABLE OF CONTENTS CHAPTER PAGE I. INTRODUCTION 1 I I . HISTORICAL BACKGROUND 7 I I I . THEORY 19 3.1 S t a t i c F r i c t i o n 20 3.2 K i n e t i c Boundary F r i c t i o n . . 36 IV. APPARATUS AND EXPERIMENTAL PROCEDURE 47 4.1 Apparatus 48 4.2 Measurement of F r i c t i o n Forces 52 4.3 Instrumentation 56 4.4 Specimens 62 4.5 Testing Procedure 64 4.6 Lubricant • • 65 V. DISCUSSION OF RESULTS 67 5.1 S t a t i c F r i c t i o n 68 5.2 Kinetic Boundary F r i c t i o n 81 VI. CONCLUSION 105 APPENDICES i I. System Parameters 110 I I . Phase Plane Analysis of Vibratory Motion . . . 115 I I I . Viscous Squeeze Film Analysis 121 , IV. C a l i b r a t i o n and Scaling of Displacement, Ve l o c i t y , and F r i c t i o n Force Signals . . . . . 125 V. F r i c t i o n Surface Parameters 128 REFERENCES 130 v i LIST OF ILLUSTRATIONS FIGURE PAGE 1.1.1 Schematic System Required for Incidence of Friction-Induced O s c i l l a t i o n 5 1.1.2 Displacement vs Time Waveforms of the Two Forms of Friction-Induced O s c i l l a t i o n . . . 6 \ 2.1.1 Assumed Linearly Negative Ki n e t i c F r i c t i o n vs Velocity Relationship of Cameron, with Generated Phase Plane Behavioral Trace 14 2.1.2 Ki n e t i c F r i c t i o n vs Veloc i t y Relationship Recorded Experimentally by B e l l and Burdekin, with Generated Phase Plane Behavioral Trace 15 2.1.3 Ki n e t i c F r i c t i o n vs Veloc i t y Relationship for Quasi-Harmonic O s c i l l a t i o n 16 2.1.4 I n t e r f a c i a l Voltage Drop and Displacement Waveforms Recorded Simultaneously during S t i c k - S l i p O s c i l l a t i o n 17 2.1.5 E f f e c t on I n t e r f a c i a l Voltage Drop of Interrupted Tangential Load Application . . 18 3.1.1 Nature of M e t a l l i c Contact at Inter-f a c i a l Junction . . 21 3.1.2 Co-ordinate System Adopted for Develop-ment of Area Growth Equation 24 3.1.3 Poynting-Thomson Model for Analysis of Internal F r i c t i o n a l Dissipation 27 3.1.4 E f f e c t of Strain Rate on Ultimate Tensile Strength of Mild Steel and Copper 34 3.2.1 Free Vibration of a Linear Spring-Mass System Subject to Simple Damping Forms . . 37 3.2.2 Rol l i n g Contact Conditions 42 v i i FIGURE PAGE 3.2.3 Elastohydrodynamic F r i c t i o n Force vs V e l o c i t y Curves 43 4.1.1 Schematic Diagram of Experimental System 49 4.1.2 V i b r a t o r y System, with Instrumentation . . 53 4.1.3 Complete Experimental Apparatus 54 4.3.1 V e l o c i t y Transducer D e t a i l 58 4.3.2 S p o t - T r i g g e r i n g and S e q u e n t i a l - T r i g g e r i n g C i r c u i t r y 60 5.1.1 P o s t - T e s t Photomicrographs o f S l i d e r Surface and F r i c t i o n D i s c S u r f a c e ( S t e e l - o n - S t e e l ) 69 5.1.2 V a r i a t i o n of the C o e f f i c i e n t of S t a t i c F r i c t i o n with Load Rate V a r i a b l e cp, S t e e l -o n - S t e e l 70 5.1.3 Zone of Deformation beneath a L o n g i t u d i n a l Wedge I n d e n t a t i o n 7 4 5.1.4 P o s t - T e s t Photomicrograph of F r i c t i o n D i s c Surface (Brass-on-Steel) 76 5.1.5 V a r i a t i o n of the C o e f f i c i e n t of S t a t i c F r i c t i o n with Load Rate V a r i a b l e B r a s s - o n - S t e e l 79 5.2.1 Recorded H a l f - C y c l e S t i c k - S l i p Phase Plane Traces 8 2 5.2.2 Recorded M u l t i - C y c l e S t i c k - S l i p Phase Plane Traces 83 5.2.3 Normalized T r a n s i t o r y K i n e t i c F r i c t i o n vs Time 8 8 5.2.4 Normalized T r a n s i t o r y K i n e t i c F r i c t i o n vs R e l a t i v e V e l o c i t y 89 5.2.5 Normalized T r a n s i t o r y K i n e t i c F r i c t i o n vs R e l a t i v e Displacement 90 FIGURE v i i i PAGE 5.2.6 Recorded T r a n s i e n t S l i p Traces 92 5.2.7 One Form of Quasi-Harmonic F r i c t i o n Force Phase Plane Trace, w i t h Generated Displacement B e h a v i o r a l Curve . . . . . . . 96 5.2.8 Recorded Quasi-Harmonic Phase Plane Traces 98 A l . l Approximation of Composite Beam Employed f o r A n a l y s i s o f Beam P r o p e r t i e s I l l A2.1 Viscously-Damped Free V i b r a t i o n on Phase Plane 118 A2.2 Comparison o f Recorded and G r a p h i c a l l y Generated Phase Plane B e h a v i o r a l Traces ' 120 A3.1 Squeeze F i l m A n a l y s i s Co-ordinates . . . . 122 A4.1 C a l i b r a t i o n Curves f o r Composite Beam and S c a l e d Displacement A m p l i f i e r . . . . 127 ix NOTATION Description Units 2 t o t a l area of actual contact at the i n interface of two surfaces, area of actual contact at inception of s l i p , and rate of change of area of contact at inception of s l i p , respectively o modulii of e l a s t i c i t y of s t e e l and l b / i n aluminum, respectively t o t a l tangential load at interface of lb f r i c t i o n surfaces, i n t e r f a c i a l tangen-t i a l load at inception of s l i p , and rate of change of tangential load at inception of s|lip, respectively . 4 area moments of i n e r t i a of portions of i n composite cantilever beam work equivalent of heat in-lb/BTU constant of propo r t i o n a l i t y lengths of portions of composite i n cantilever beam 2 dynamic equivalent mass of s l i d e r lb-sec / i n and supporting structure normal load supported by f l u i d lb pressure normal load supported by s o l i d contact l b applied load lb v radius of s l i d e r i n 2 p r i n c i p a l stresses l b / i n temperature o^ ve l o c i t y of s l i d e r i n x d i r e c t i o n in/sec r e l a t i v e to lower surface, (x - v) X Symbol D e s c r i p t i o n U n i t s V v e l o c i t y of s l i d e r i n y d i r e c t i o n i n / s e c W t o t a l normal l o a d a t f r i c t i o n a l l b i n t e r f a c e 2 Y , y i e l d s t r e s s of d u c t i l e m a t e r i a l i n l b / i n u n i a x i a l t e n s i o n a, b constants of p r o p o r t i o n a l i t y 2 g a c c e l e r a t i o n of g r a v i t y i n / s e c h t h i c k n e s s of l i q u i d l u b r i c a n t l a y e r i n h(a) a c t i v a t i o n enthalpy BTU k s t i f f n e s s of composite c a n t i l e v e r l b / i n beam k^, kr, l i n e a r l y e l a s t i c s t i f f n e s s c o e f f i c i e n t s l b / i n m, n constants of p r o p o r t i o n a l i t y 2 p l o c a l s t a t i c p r e s s u r e i n l i q u i d l u b r i - l b / i n c ant r , r a c t u a l and c r i t i c a l v i s c o u s d i s s i p a t i o n I b - s e c / i n c c o e f f i c i e n t s f o r e l a s t i c a l l y - r e s t r a i n e d s l i d e r , r e s p e c t i v e l y t , t time, and time d u r a t i o n of s t i c k p o r t i o n sec s of s t i c k - s l i p c y c l e , r e s p e c t i v e l y u v e l o c i t y of l u b r i c a n t i n x d i r e c t i o n i n / s e c v v e l o c i t y of lower s u r f a c e i n / s e c • • • x, x, x displacement, v e l o c i t y , and a c c e l e r a -t i o n of s l i d e r , r e s p e c t i v e l y , i n plane of f r i c t i o n a l i n t e r f a c e y .normal to f r i c t i o n a l i n t e r f a c e z c o - o r d i n a t e normal to x-y plane D e s c r i p t i o n c o n s t a n t s of p r o p o r t i o n a l i t y c o e f f i c i e n t o f v i s c o s i t y v a r i a t i o n w i t h temperature l e n g t h of r e c t a n g u l a r l o a d - c a r r y i n g f i l m e l a s t i c e x t e n s i o n / d e f l e c t i o n from e q u i l i b r i u m l e n g t h / p o s i t i o n and r a t e of change of e x t e n s i o n / d e f l e c t i o n , r e s p e c t i v e l y r a t e of s t r a i n i n u n i a x i a l t e n s i o n v i s c o s i t y of l i q u i d l u b r i c a n t angular displacement width of r e c t a n g u l a r l o a d - c a r r y i n g f i l m 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 and va l u e of 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 a t which t r a n s i t o r y k i n e t i c f r i c t i o n curve meets s t a b l e k i n e t i c f r i c t i o n curve, r e s p e c t i v e l y c o e f f i c i e n t of f r i c t i o n a t i n c e p t i o n o f s l i p and r a t e o f change of c o e f f i c i e n t of f r i c t i o n a t i n c e p t i o n of s l i p , r e s p e c t i v e l y thermal c o n d u c t i v i t y r a d i a l c o - o r d i n a t e i n c y l i n d r i c a l c o - o r d i n a t e system compressive s t r e s s r e q u i r e d to produce complete p l a s t i c y i e l d i n g over t r u e area o f c o n t a c t normal s t r e s s a t f r i c t i o n a l i n t e r f a c e , W/A p l a s t i c shear s t r e n g t h of i n t e r f a c i a l j u n c t i o n , F s/A s, and p l a s t i c shear s t r e n g t h of b u l k m a t e r i a l , r e s p e c t i v e l y x i i Symbol D e s c r i p t i o n U n i t s x , t t a n g e n t i a l shear s t r e s s a t f r i c t i o n a l yx yx i n t e r f a c e , F/A, and r a t e of change of t a n g e n t i a l shear s t r e s s a t f r i c t i o n a l i n t e r f a c e -1 a) . l o a d r a t e v a r i a b l e , x /a sec yx y 2 c o e f f i c i e n t of v i s c o s i t y v a r i a t i o n i n / l b with p r e s s u r e u , u, undamped n a t u r a l frequency of v i b r a t i o n rad/sec n and damped n a t u r a l frequency of v i b r a t i o n , r e s p e c t i v e l y C H A P T E R I I. INTRODUCTION Boundary f r i c t i o n may be defined as resistance to motion which occurs when two s o l i d bodies are i n physical contact under the influence of an applied tangential stress. The nature of the opposing surfaces i s therefore of para-mount importance to any discussion of such f r i c t i o n a l action. No matter how well finished, a l l engineering surfaces are extremely rough on a microscopic scale. Thus, when two nominally f l a t surfaces are brought together they touch only at their extremities, and their f r i c t i o n a l behavior i s dominated by the properties of these small regions of contact. These microscopic extremities are commonly c a l l e d a s p e r i t i e s . Upon considering the present knowledge of physical and chemical properties of materials, one might suppose the nature of f r i c t i o n a l i n t e r a c t i o n between contacting surfaces to be well understood. The converse i s true. Not only i s the actual mechanism of f r i c t i o n i n doubt; pre-d i c t i o n s of boundary f r i c t i o n c o e f f i c i e n t values cannot be made, even empirically, with any notable degree of accuracy. So many variables a f f e c t f r i c t i o n a l behavior that investigators concur only i n that surfaces i n contact meet at opposing a s p e r i t i e s , and that these a s p e r i t i e s deform, either e l a s t i c a l l y or p l a s t i c a l l y , under load. 3 When one member o f a boundary f r i c t i o n p a i r i s e l a s t i c a l l y r e s t r a i n e d , and the oth e r member gi v e n a v e l o c i t y r e l a t i v e to the p o i n t o f r e s t r a i n t ( F i g . 1.1.1), o s c i l l a t i o n of the e l a s t i c a l l y - r e s t r a i n e d member, commonly c a l l e d " f r i c t i o n - i n d u c e d v i b r a t i o n " , f r e q u e n t l y o c c u r s . T h i s o s c i l l a t i o n i s of c o n s i d e r a b l e e n g i n e e r i n g i n t e r e s t , i n i t s own r i g h t , s i n c e i t c o n s i d e r a b l y i n c r e a s e s wear, and d e t r a c t s from the accuracy and r e l i a b i l i t y of mechanisms and measuring d e v i c e s . But, f o r purposes of the pr e s e n t study, the s i g n i f i c a n c e of such o s c i l l a t i o n i s t h a t i t may be used by ; i n v e s t i g a t o r s i n search of g r e a t e r i n s i g h t i n t o the phenomena of boundary f r i c t i o n . Two forms of f r i c t i o n - i n d u c e d v i b r a t i o n are r e c o g n i z e d ( F i g . 1.1.2). " S t i c k - s l i p v i b r a t i o n " i s c h a r a c t e r i z e d by a saw-tooth form of displacement vs time p l o t , whereas " q u a s i -harmonic v i b r a t i o n " e x h i b i t s a displacement vs time waveform t h a t i s n e a r - s i n u s o i d a l . During s t i c k - s l i p v i b r a t i o n the e l a s t i c a l l y - r e s t r a i n e d member " s t i c k s " to the d r i v e n member, causing the displacement o f the r e s t r a i n e d member to i n c r e a s e u n t i l the r e s t r a i n i n g f o r c e exceeds the maximum " s t a t i c " f r i c t i o n f o r c e which the i n t e r f a c i a l s u r f a c e i s capable of s u s t a i n i n g . The r e s t r a i n e d member then decreases i t s displacement, under the i n f l u e n c e of " k i n e t i c " f r i c t i o n f o r c e s , u n t i l i t once again achieves zero v e l o c i t y with r e s p e c t to the d r i v e n s u r f a c e . During quasi-harmonic . v i b r a t i o n , which occurs a t hi g h e r d r i v e n - s u r f a c e speeds 4 than s t i c k - s l i p v i b r a t i o n , r e l a t i v e v e l o c i t y between the surfaces always exceeds zero. Both forms of friction-induced v i b r a t i o n have received substantial documentation, but usually with the emphasis on d e f i n i t i o n of the vibratory behaviour. The f r i c t i o n mechanisms causing the v i b r a t i o n , which have general a p p l i c a b i l i t y to boundary f r i c t i o n as a whole, are s t i l l inadequately under-stood, lar g e l y because the presence of the f r i c t i o n a l o s c i l -l ations has obscured the form of the f r i c t i o n forces responsible for the o s c i l l a t i o n . Further investigation of these physical mechanisms of f r i c t i o n was consequently considered attention well directed. In the course of t h i s study the p l a s t i c deformation model of Tabor w i l l be extended to include the e f f e c t of rate of appli cation of tangential load, and the r e s u l t s compared to s t a t i c f r i c t i o n data c o l l e c t e d by monitoring f r i c t i o n -induced v i b r a t i o n . The dynamic f r i c t i o n force recordings , obtained by monitoring th,e v i b r a t i o n w i l l also be analyzed.. Minimized chemical effects w i l l be neglected; the only concern of the present inves t i g a t i o n w i l l be the physical phenomena relevant to f r i c t i o n of m e t a l l i c surfaces. . 5 F i g u r e 1.1.1 Schematic System Required f o r Incidence o 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 6 X / l ! \ \ . i i . • i i i i i i i 111 1 1 i i i i 4*4-4--T" 1 ( 1 1 T n r t T i t , 1 1 1 l_ J i l l M M I I I I I t ( Q ) STICK-SLIP X 1 r / j 1 \ t y V / 1111 M M J M J M M 1 M T 1 1 1 1 t t T T fb) QUASI-HARMONIC Figure 1.1.2 Displacement vs Time Waveforms of the Two Forms of Friction-Induced O s c i l l a t i o n (Ko) C H A P T E R II I I . HISTORICAL BACKGROUND Although f r i c t i o n - i n d u c e d v i b r a t i o n i s a very common phenomenon, onl y i n r e c e n t years have the f r i c t i o n f o r c e s c a u s i n g the v i b r a t i o n been s u b j e c t e d to s e r i o u s i n v e s t i g a t i o n . In f a c t , i t was not u n t i l 1930 t h a t an i n v e s t i g a t o r , Thomas [1], r e p o r t e d an a n a l y t i c i n v e s t i g a t i o n o f f r i c t i o n - i n d u c e d v i b r a t i o n u s i n g g r a p h i c a l techniques. He e r r o n e o u s l y con-cluded t h a t the phenomenon we p r e s e n t l y c a l l quasi-harmonic o s c i l l a t i o n was simply t o t a l l y undamped simple harmonic motion, with the s t a t i c f r i c t i o n f o r c e as the i n i t i a l c o n d i t i o n . He d i d , however, even assuming a constant v a l u e f o r the 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 , c o r r e c t l y conclude t h a t s t i c k - s l i p o s c i l l a t i o n occurs o n l y i f the i n t e r f a c i a l 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 i s l e s s than 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 , and even then, o n l y i f the damping f o r c e s are o f l e s s than a c r i t i c a l magnitude. Papenhuyzen [2], i n 1938, i n v e s t i g a t i n g the s k i d d i n g of automobile t i r e s , was f i r s t to demonstrate t h a t s t i c k - s l i p o s c i l l a t i o n and quasi-harmonic o s c i l l a t i o n can occur under s i m i l a r f r i c t i o n a l c o n d i t i o n s , but a t d i f f e r e n t v e l o c i t i e s of the d r i v e n s u r f a c e . f Bowden and Leben [3], i n 1939, r e p o r t e d t h a t s l i d i n g v e l o c i t i e s d u r i n g the s l i p p o r t i o n of s t i c k - s l i p o s c i l l a t i o n were very h i g h , i n comparison to the d r i v e n s u r f a c e v e l o c i t y , 9 and suggested t h a t l o c a l i z e d welding c o u l d occur a t the t e r m i n a t i o n of the s l i p p o r t i o n of a c y c l e as a r e s u l t of high-temperature f l a s h e s a t opposing a s p e r i t i e s . T h i s s u g g e s t i o n was based on experiments by Bowden and R i d l e r [4] , r e p o r t e d i n 1936, i n which i n t e r f a c i a l temperature between two metal s u r f a c e s was i n v e s t i g a t e d as a f u n c t i o n of r e l a t i v e v e l o c i t y , both w i t h and without the a p p l i c a t i o n of l u b r i c a n t s . Peak l o c a l temperatures a t the s u r f a c e were, i n a l l cases,: fpund to r i s e w i t h speed, to an upper l i m i t equal numeric-a l l y to the m e l t i n g p o i n t of the s o f t e r metal. ; Bristow [5], i n 1945, s t a t e d t h a t a necessary c o n d i t i o n f o r the occurrence of s t i c k - s l i p o s c i l l a t i o n i s the e x i s t e n c e of a negative k i n e t i c f r i c t i o n f o r c e vs v e l o c i t y r e l a t i o n s h i p , but o f f e r e d no evidence to support t h i s h y p o t h e s i s . He d i d , however, demonstrate the e x i s t e n c e of micro-displacement d u r i n g the s t i c k p o r t i o n of the c y c l e . Dudley and S w i f t [6], i n 1949, r e p o r t e d the develop-ment of a simple g r a p h i c a l technique f o r determining the v i b r a t i o n c y c l e , on the phase plane, from any f r i c t i o n f o r c e vs v e l o c i t y curve, and the a p p l i c a t i o n of t h i s technique to analyze s t i c k - s l i p v i b r a t i o n . V a r i o u s forms of f r i c t i o n -f o r c e vs v e l o c i t y curves were e x p l o r e d , w i t h 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 assumed equal to the k i n e t i c f r i c t i o n f c o e f f i c i e n t e x t r a p o l a t e d to zero v e l o c i t y . They e r r o n e o u s l y concluded t h a t the amplitude of s t i c k - s l i p v i b r a t i o n should i n c r e a s e w i t h d r i v e n - s u r f a c e v e l o c i t y . 10 Bowden and Tabor [7], i n 1950, proposed that m e t a l l i c s t a t i c f r i c t i o n i s caused by l o c a l i z e d adhesion, r e s u l t i n g from p l a s t i c deformation, of contacting surfaces. The area of contact was thereby a function of both normal and tangential loading; i t would increase with either form of load. This theory, which offered creditable explanations for many observed me t a l l i c contact phenomena, was well received. Tabor subse-quently extended the theory [22] to an impressive mathematical treatment of s t a t i c f r i c t i o n , but the known influence of rate effects on observed c o e f f i c i e n t s of s t a t i c f r i c t i o n was ; s t i l l unexplained. Rabinowicz [8], i n 1957, concluded that the amplitude of s t i c k - s l i p o s c i l l a t i o n i s governed by the time duration of the s t i c k portion of the cycle, and that the amplitude of quasi-harmonic o s c i l l a t i o n i s governed by the v e l o c i t y of the driven surface. Courtney-Pratt and Eisner [9], i n 19 57, showed that the growth of actual contact area under the influence of increasing tangential load was unaffected by the addition of l i q u i d lubricants. The sole e f f e c t of the lubricants was to decrease the value of tangential load at which s l i p occurred. Potter [10], i n 1962, reported experimental results in which, contrary to Dudley and Swift's predictions, the amplitude of s t i c k - s l i p v i b r a t i o n decreased with increasing v e l o c i t y . 11 Cameron [11], i n 1963, r e p o r t e d an a n a l y s i s of s t i c k -s l i p o s c i l l a t i o n i n which he assumed a l i n e a r l y n e g ative f r i c t i o n f o r c e vs v e l o c i t y r e l a t i o n s h i p ( F i g . 2.1.1). E x p e r i -mentally-measured amplitudes were r e p o r t e d i n agreement w i t h those p r e d i c t e d by the a n a l y s i s . B e l l and Burdekin [12], i n 1966, r e p o r t e d t h a t by summation o f e l e c t r i c a l i n s t r u m e n t a t i o n s i g n a l s they were able to r e c o r d the i n t e r f a c i a l k i n e t i c f r i c t i o n f o r c e through-out the d u r a t i o n o f a s t i c k - s l i p c y c l e . Contrary to the v a r i o u s assumptions of a l l p r e v i o u s i n v e s t i g a t o r s , they found t h a t the f r i c t i o n f o r c e was not a s i n g l e - v a l u e d f u n c t i o n of the i n t e r f a c i a l v e l o c i t y ( F i g . 2.1.2). No e x p l a n a t i o n of . these r e s u l t s was o f f e r e d . Davis [13], i n 1966, r e p o r t e d t h a t the behaviour of e l e v e n metals, t e s t e d on a s t e e l d r i v e n s u r f a c e , was s i m i l a r , w i t h r e s p e c t to v a r i a t i o n of the c o e f f i c i e n t of s t a t i c f r i c t i o n with d r i v e n s u r f a c e v e l o c i t y . Johannes [14], i n 1969, demonstrated c o n c l u s i v e l y i t h a t the amplitude of s t i c k - s l i p v i b r a t i o n i s not governed by the time of c o n t a c t d u r i n g the s t i c k p o r t i o n of the c y c l e , a misconception popular w i t h many i n v e s t i g a t o r s a t t h a t time. He showed, r a t h e r , u s i n g the experimental data of P o t t e r [10], t h a t the c o e f f i c i e n t of s t a t i c f r i c t i o n can be c o r r e l a t e d to the r a t e of a p p l i c a t i o n of shear s t r e s s , d i v i d e d by the e x i s t i n g normal s t r e s s , a t the i n t e r f a c e . Johannes a l s o -( proposed a simple v i s c o - e l a s t i c mathematical model of a s p e r i t y 12 j u n c t i o n growth, but d i d not achieve good c o r r e l a t i o n between the model and h i s experimental data. Ko [15], i n 1969, r e p o r t e d t h a t quasi-harmonic o s c i l l a t i o n i s c h a r a c t e r i z e d by the presence of a "humped" form of s i n g l e - v a l u e d f r i c t i o n f o r c e vs v e l o c i t y curve ( F i g . 2.1.3). The i n v e s t i g a t i o n u t i l i z e d an improved v e r s i o n o f the e l e c t r o n i c summation techniques of B e l l and Burdekin [12]. Green [16], i n 1971, a f t e r s t u d y i n g s t i c k - s l i p v i b r a t i o n , r e p o r t e d s e v e r a l i n t e r e s t i n g r e s u l t s . Massive s e i z u r e , r a t h e r than s t i c k - s l i p o s c i l l a t i o n , would occur, i n a i r , when the major p o r t i o n of s u r f a c e contaminants was removed from the f r i c t i o n s u r f a c e s . The e l e c t r i c a l conduc-t i v i t y between s u r f a c e s was found to i n c r e a s e n o n - l i n e a r l y w i t h t a n g e n t i a l l o a d ( F i g . 2.1.4), and no s u b s t a n t i a l i n -creases i n c o n d u c t i v i t y were noted i f the a p p l i c a t i o n o f t a n g e n t i a l l o a d was a r r e s t e d d u r i n g the s t i c k p o r t i o n of a s t i c k - s l i p c y c l e ( F i g . 2.1.5). Of p a r t i c u l a r i n t e r e s t to t h i s study i s the d i v e r s i t y of r e l a t i o n s h i p s , proposed by former r e s e a r c h e r s , e x p r e s s i n g the c o e f f i c i e n t of s t a t i c f r i c t i o n achieved as a function, of the time d u r a t i o n o f the s t i c k p o r t i o n o f a s t i c k - s l i p c y c l e . The r e l a t i o n s h i p s are of three g e n e r a l forms. j ^s - Uk at s b + t Derjaguin, Push, and T o l s t o i [17] = 1 - e s Kosterin and K r a g e l s k i i [18] y - y. = at Rabinowicz [8] S S Davis [13] These relationships are a l l , at best, semi-empirical. The formulation proposed by Rabinowicz and Davis predicted i n f i n i t e c o e f f i c i e n t s of s t a t i c f r i c t i o n for i n f i n i t e duration of the s t i c k portion of a cycle; the other equations, though d i s s i m i l a r , predicted an upper asymptote for 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 . This basic difference in concept has endured for several years, because none of these investigators c o l l e c t e d data which could prove or disprove the presence of this upper asymptote. Obviously, the investigation of physical boundary f r i c t i o n phenomena applicable to friction-induced o s c i l l a t i o n has not, to date, achieved a high l e v e l of so p h i s t i c a t i o n . Even which variables are of significance to such phenomena i s not c e r t a i n . The purpose of the present study i s , by analysis of friction-induced v i b r a t i o n , to improve the current comprehension of the operative physical mechanisms of s t a t i c and k i n e t i c boundary f r i c t i o n of metals. 14 Figure 2.1.1 Assumed Linearly Negative Kinetic F r i c t i o n vs Velocity Relationship of Cameron, with Generated Phase Plane Behavioural Trace 15 F i g u r e 2.1.2 K i n e t i c F r i c t i o n v s V e l o c i t y R e l a t i o n s h i p R e c o r d e d E x p e r i m e n t a l l y b y B e l l a n d B u r d e k i n , w i t h G e n e r a t e d P h a s e P l a n e B e h a v i o u r a l T r a c e x y <^~\ H \ \ ++++ ++++ J M M \ '2-\ " ' ( -H-H- ++++ r y r — V •1 i j 1 -1 0 r to (/> UJ _) z g z Ul I i 2.0-UJ o or o u. z o u 1.0 8« DISC VELOCITY, v • 0.84 X 1.06 o 1.30 1.64 w = 5.4 lb 1.0 2.0 SLIDING VELOCITY, DIMENSIONLESS 3 . 0 Figure 2.1.3 Kinetic F r i c t i o n Force vs Velocity Relationship for Quasi-Harmonic O s c i l l a t i o n (Ko) 064-..054-o a CO .044-.034-.024-3.9 3A H 29 a g 24 > 3 1.9 | 1/H Q .9. . 4 . 0 . TIME (5 sec./div. ) F i g u r e 2.1.4 I n t e r f a c i a l V o l t a g e Drop and Displacement Waveforms Recorded Simultaneously d u r i n g S t i c k - S l i p O s c i l l a t i o n (Green) Upper Trace: Interfacial Voltage Drop Lower Trace: Displacement 4 A / / 1 / I i i i i i i i i . I I I 1 ! 1 Figure 2.1.5 E f f e c t on I n t e r f a c i a l Voltage Drop of Interrupted Tangential Load Application (Green) Upper Trace: Tangential Load Lower Trace: Interfacial Voltage Drop oo C H A P T E R III I I I . THEORY 3.1 S t a t i c F r i c t i o n When two m e t a l l i c surfaces are brought together under the influence of an applied normal load, they contact each other only at matching as p e r i t i e s (Fig. 3.1.1). Contact i s i n i t i a l l y e l a s t i c , but because the re a l area of contact i s a very small portion of the apparent area of contact, the de-f forming m e t a l l i c junctions might achieve a state of p l a s t i c i t y i. under minute loads, r e s u l t i n g i n growth of the junction contact area u n t i l the real area of contact i s just s u f f i c i e n t to support the applied load. Experimental evidence showing the area of contact to be d i r e c t l y proportional to applied perpendicular load was, for many years, thought to support this concept, proposed by Bowden and Tabor i n 1950 [7]. Recently, however, certain investigators have presented anal-yses demonstrating that this p roportionality of load and contact area might also be achieved under cert a i n conditions of e l a s t i c contact. The most notable of these analyses was performed by Greenwood and Williamson [19], who showed that, for an exponential d i s t r i b u t i o n of asperity heights, contact area could be proportional to load regardless of the mode,of deformation.: For most machined metal surfaces asperity ^heights have a near-Gaussian d i s t r i b u t i o n ; the Gaussian and 21 deformation F i g u r e 3.1.1 Nature of M e t a l l i c Contact a t I n t e r f a c i a l J u n c t i o n E x p o n e n t i a l d i s t r i b u t i o n s a r e s u f f i c i e n t l y s i m i l a r o v e r m o s t o f t h e p r o b a b i l i t y r a n g e t o p e r m i t a p p l i c a t i o n o f t h i s s t u d y t o c o n v e n t i o n a l s u r f a c e s . T h e f i r s t p o r t i o n o f t h e p r e s e n t i n v e s t i g a t i o n i s d e v o t e d t o a n a t t e m p t t o r e s o l v e t h e u n c e r t a i n t y c o n c e r n i n g t h e m o d e o f a s p e r i t y d e f o r m a t i o n d u r i n g s t a t i c c o n t a c t t h r o u g h d e v e l o p m e n t o f a p l a u s i b l e p l a s t i c d e f o r m a t i o n m o d e l a n d c o m p a r i s o n o f i t s p r e d i c t e d b e h a v i o u r w i t h e x p e r i m e n t a l s t a t i c f r i c t i o n d a t a . U s i n g V o n M i s e s ' y i e l d c r i t e r i o n , ( s 1 - s 2 ) 2 + ( s 2 - s 3 ) 2 + ( s 3 - s 1 ) 2 1 2 (3.1.1) w h e r e S ^ , S ^ , a n d a r e t h e p r i n c i p a l s t r e s s e s a n d Y i s t h e y i e l d s t r e s s i n u n i a x i a l t e n s i o n . F o r p l a n e ( t w o - d i m e n s i o n a l ) s t r e s s = s 2 + s 2 - s±s2 (3.1.2) w h e r e 2 3 S l S2 a / a 2 2~ 1 + Tyx ( 3 - 1 ' 3 ) i n the co-ordinate system designated i n Figure 3.1.2, and the y i e l d c r i t e r i o n may be reduced to Y 2 = o2 + 3T 2 . (3.1.4) y yx No a n a l y t i c solution exists for a r e a l three-dimen-sional contact, but empirical grounds exi s t [20,21] for a l t e r a t i o n of equation 3.1.4 to a 2 = a 2 + a x 2 , (3.1.5) 0 y yx ' where OQ, the amount of compressive stress required to induce p l a s t i c y i e l d i n g over the entire true area of contact i n the absence of tangential stress, i s approximately equal to 3Y [7]. Greenwood has shown [45] that a should have an approxi-mate numerical value of 25 by the following reasoning. , If a metal, possessing a c r i t i c a l shear strength x ^ , can be assumed free of work-hardening, then T0 S k °0 (3.1.6) - o r d i n a t e System Adopted f o r Development Area Growth Eq u a t i o n for an a x i a l l y symmetric contact [23] . Substitution i n equation 3.1.5 yields 25 x 2 = a 2 + a x 2 x . (3.1.7) I f , as i s the case with clean metals i n vacuum, the growth i n the area of contact i s very large, as a r e s u l t of the tangential load x , then 0 << x = x_, and a = 25. yx y yx u Attempts to measure a at the interface of two surfaces have resulted i n values of a = 3.3 for indium [20] and a = 12 for platinum [9]. Errors inherent i n the methods used for contact area measurement would r e s u l t i n conservative values of a, so that a numerical value of 25 does not seem unreasonable. So as not to be overly r e s t r i c t i v e , a w i l l be assumed to have a value between 10 and 25, a range of uncertainty which can be r e a d i l y tolerated. As w i l l be demonstrated, the present analysis i s f o r t u i t o u s l y i n s e n s i t i v e to v a r i a t i o n of this c o e f f i c i e n t . Substituting for a and x i n equation 3.1.5 y i e l d s y y x M2 + • W % = H H + « I I : , (3.1.8) where A i s the actual area of contact at the in t e r f a c e , F i s the applied tangential load, and W i s the normal load at the 26 interface; algebraic manipulation gives W 2 + aF 2 1 2 (3.1 .9) a r e s u l t derived by Tabor [22], provided the materials are i n a state of equilibrium, a basic assumption of Von Mises 1 y i e l d c r i t e r i o n . develop i n a r e l a t i v e l y short period of time [ 9 ] , a state of equilibrium cannot be supposed to e x i s t . A n e l a s t i c i t y , or inter n a l damping, of the materials must consequently be considered. dissipate v i b r a t i o n a l energy [24]. The mechanical model most successfully used i n the analysis of int e r n a l f r i c t i o n i s the Poynting-Thomson model (Fig. 3.1.3), commonly c a l l e d the "standard l i n e a r s o l i d " . For s t r u c t u r a l metals the contribution of to the behaviour of the model i s n e g l i g i b l e , except at rates of s t r a i n so great that the damper becomes a semi-r i g i d element. If such extreme rates are not achieved, the standard l i n e a r s o l i d i s e f f e c t i v e l y an e l a s t i c element (k^) and an energy-dissipating element (r) acting i n p a r a l l e l , governed by the approximate equation In contacting a s p e r i t i e s , where large shear strains A l l s olids display, in varying degree, the a b i l i t y to P (3.1.10) 27 I—CE—vW-i • — W W W F i g u r e 3.1.3 Poynting-Thonison Model f o r A n a l y s i s of I n t e r n a l F r i c t i o n a l D i s s i p a t i o n 28 If the standard l i n e a r s o l i d i s i n a state of equilibrium (6 = 0), i t s behaviour i s e n t i r e l y governed by the e l a s t i c element; i f a state of equilibrium i s not cl o s e l y approximated, the contribution to equation 3.1.10 of the energy-dissipating element cannot be ignored. The s i m i l a r i t y of form of equations 3.1.9 and 3.1.10 for the condition of equilibrium (6 = 0) i s obvious. If the analogy i s to continue aft e r a l t e r i n g equation 3.1.9 to include the effects of a n e l a s t i c i t y , the modified equation must assume the form ; 1 2 . (3.1.11) Since d i s s i p a t i o n i s due to deformation of the metal, one might reasonably postulate that the amount of d i s s i p a t i o n should increase with the volume of metal deformed. The area of contact, during indentation hardness t e s t s , has been found to be proportional to the progress of a hemispherical e l a s t i c -p l a s t i c boundary within the test material, for a l l indenters except those with a very sharp apex [25]. The damping > q o e f f i c i e n t r may, therefore, be expressed as a power function of the area of contact. r = r (A) = b A n (3.1.12) Substituting for r i n equation 3.1.11 y i e l d s the proposed general equation of contact area growth under varying r A + a 0 A 2 2 W + aF 29 t a n g e n t i a l l o a d , b A n A + a Q A 2 2 vr + a F 1 2 ( 3 . 1 . 1 3 ) A l t h o u g h e q u a t i o n 3 . 1 . 1 3 i s o f f i r s t - o r d e r f o r m , n o s i m p l e s o l u t i o n e x i s t s f o r t h e v a r i a t i o n o f t h e c o n t a c t a r e a A , a s a f u n c t i o n o f t i m e , u n d e r t h e i n f l u e n c e o f a r a m p l o a d F = F Q + F t , s u c h a s e x i s t s d u r i n g t h e s t i c k p o r t i o n o f a s t i c k - s l i p c y c l e . F o r t u n a t e l y , t h e i n t e r e s t o f t h i s s t u d y i s i n t h e a r e a o f c o n t a c t a t t h e i n c e p t i o n o f s l i p , f o r w h i c h , a g e n e r a l s o l u t i o n i s n o t r e q u i r e d . T h e f r i c t i o n v a l u e a t t h e i n c e p t i o n o f s l i p i s t h e p o p u l a r l y - t e r m e d " 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 y g . I f g r o s s s l i d i n g o f t h e s u r f a c e s i s d e e m e d t o o c c u r a s a r e s u l t o f t h e s h e a r i n g o f i n t e r f a c i a l j u n c t i o n s , t h e n W T . A 1 s W ( 3 . 1 . 1 4 ) w h e r e t h e s u b s c r i p t s d e s i g n a t e s v a l u e s a t t h e i n c e p t i o n o f s l i p , a n d x^ i s t h e u l t i m a t e s h e a r s t r e n g t h o f t h e i n t e r f a c e . E q u a t i o n s 3 . 1 . 1 3 a n d 3 . 1 . 1 4 f o r m a p a i r o f s i m u l t a n e o u s d i f f e r e n t i a l e q u a t i o n s . A , a s g i v e n b y e q u a t i o n 3 . 1 . 1 4 , m a y s b e s u b s t i t u t e d i n t o e q u a t i o n 3 . 1 . 1 3 t o o b t a i n n A + a 0 x 2 2 W + aF 1 2 ( 3 . 1 . 1 5 ) which may be a l g e b r a i c a l l y manipulated to the form 30 A r 1 n 1-n -T . 1 W U b s _ 1 + ay - — °0 (3.1.16) As a consequence of entrapped contaminants, such as lubricant or oxide, the strength of the average i n t e r f a c i a l junction formed by ide a l p l a s t i c deformation of a s p e r i t i e s may approach, but not equal, the shear strength of the metal substrate. Suppose, for example, that T I = K T Q , (3.1.17) where K i s a constant less than unity. As Tabor points out [22], i f T y X = T i ' a = 25, and equation 3.1.7 i s assumed to describe the l i m i t i n g condition of s t i c k , gross s l i d i n g w i l l occur when x . _ i = . (3.1.18) ° Y 5(K" 2- 1) Substitution into the standard equation for the c o e f f i c i e n t of s t a t i c f r i c t i o n y i e l d s F x. A , y = 1 ? = X_ . (-3.1.19) s W a A c , -2 ., . y s 5 (K - 1) Examination of equation 3.1.19 suggests that, i f the i n t e r f a c i a l shear strength equals that of the metal substrate, i n f i n i t e c o e f f i c i e n t s of s t a t i c f r i c t i o n w i l l be observed. Such i s indeed the case w i t h degassed metals a t h i g h temperature i n h i g h vacuum [26] , where oxides and o t h e r contaminants are absent, i f a n e l a s t i c e f f e c t s are minimized. However, 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 , as determined by e q u a t i o n 3.1.19, f a l l s to u n i t y f o r a s m a l l r e d u c t i o n of K to 0.92, and to 0.5 i f K i s reduced to 0.85. For normal f r i c t i o n a l c o n t a c t s there i s consequently j u s t i f i c a t i o n f o r assuming t h a t the behaviour of the i n t e r -f a c i a l c o n t a c t s i s dominated by the p r o p e r t i e s of the m e t a l l i c s u b s t r a t e s . S p e c i f i c a l l y , the i n t e r f a c i a l u l t i m a t e shear s t r e n g t h x^ may be assumed to behave i n a manner not u n l i k e the s u b s t r a t e s ' u l t i m a t e shear s t r e n g t h , w i t h r e s p e c t to v a r i a t i o n with s t r a i n r a t e . I n t r o d u c t i o n of c}>, the r a t i o of r a t e of t a n g e n t i a l s t r e s s a p p l i c a t i o n d i v i d e d by normal s t r e s s a g a i n s t which Johannes found t h a t e x p e r i m e n t a l l y - d e t e r m i n e d v a l u e s f o r the c o e f f i c i e n t of s t a t i c f r i c t i o n c o u l d be c o r r e l a t e d , y i e l d s - yx _ j L d_ a W dt y T . A 1 s A c T . W s (3.1.20) I f the term c o n t a i n i n g x. i s s m a l l i n comparison t o ^ a c o n d i t i o n s a t i s f i e d u n l e s s the i n t e r f a c i a l shear s t r e n g t h x. e x h i b i t s s t r o n g r a t e dependence, equation 3.1.20 may be approximated by the r e l a t i o n s h i p 32 * = A s . (3.1.21) The v a l i d i t y o f t h i s approximation w i l l be f u r t h e r d i s c u s s e d i n s e c t i o n 5.1. S u b s t i t u t i n g f o r A g from equation 3.1,16 g i v e s the r e s u l t 1+n T . 1 ,,n , n W b y s / 1 + ay - — °0 x7 (3 .1.22) an e q u a t i o n d e s c r i b i n g tfye l o c u s o f y , the c o e f f i c i e n t of s t a t i c f r i c t i o n , as a f u n c t i o n of <j>, the lo a d r a t e v a r i a b l e . Examination of equation 3.1.22 w i l l r e v e a l t h a t , as the l o a d r a t e v a r i a b l e <$> approaches zero, 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 y must approach i n f i n i t y , the i n t e r f a c i a l shear s t r e n g t h x^ must approach zero, o r the p a r e n t h e s i z e d term must approach zero. In the presence of a l u b r i c a n t y cannot approach s i n f i n i t y . The i n t e r f a c i a l shear s t r e s s x^ may approach zero, due to m e t a l l i c creep c o n s i d e r a t i o n s , as <f> approaches zero, but c e r t a i n l y the two e f f e c t s are not p r o p o r t i o n a t e f o r any reasonable v a l u e of the s u p e r s c r i p t n; t h a t i s to say, tj) must be v i r t u a l l y zero before a s u b s t a n t i a l drop i n from common value s i s observed [27]. The i n e s c a p a b l e c o n c l u s i o n i s t h a t e q u a t i o n 3.1.22 can be s a t i s f i e d f o r a l l <j>, over . which s t i c k - s l i p o s c i l l a t i o n o c c u r s , o n l y i f the p a r e n t h e s i z e d term equals zero a t 4> = 0. There f o r e 1 + ay' + a cj) = 0 ( 3 . 1 . 2 3 ) <j) = 0 Both Von M i s e s 1 c r i t e r i o n and the l e s s exact maximum-s h e a r - s t r e s s theory show c l e a r l y t h a t T Q , the s t r e n g t h o f the d u c t i l e s u b s t r a t e i n pure shear, i s p r o p o r t i o n a l to C Q . S i n c e , f o r reasons a l r e a d y d i s c u s s e d , x^ may be c o n s i d e r e d approximately p r o p o r t i o n a l to x Q , i t l o g i c a l l y f o l l o w s t h a t the r a t i o °"Q/ t^ w i l l remain approximately c o n s t a n t , and i n -dependent of <f>. Use o f t h i s constancy permits rearrangement of equation 3 . 1 . 2 2 to the a l t e r n a t e form 1+n r 7n, n W b y s 1+n 1 + ay' °0 77 ^s i ( 3 . 1 . 2 4 ) C o n s i d e r a t i o n o f the nature o f the s u b s t r a t e t e n s i l e s t r e n g t h O Q , a t the i n c e p t i o n of s l i p , remains. The s i m p l e s t s o l u t i o n of equation 3 . 1 . 2 4 r e s u l t s from assuming O Q and, t h e r e f o r e , x^, to be a co n s t a n t . As data from Nadai and Majoine [ 2 7 ] , presented i n F i g u r e 3 . 1 . 4 , shows, 34 LTrfXT OF STRAIN RATE ON ULTIMATE STRESS OF MILD STEEL AT VARIOUS TEMPERATURES IQ-» IQ-« io» HT« KT' I W O* I0» HATE Of STRAIN PCR SECOND E F F E C T OP STRAIN R A T E ON ULTIMATE STRESS OF MILD S T E E L AT VARIOUS TEMPERATURES EFFECT OF STRAIN RATE ON ULTIMATE STRESS OF PURE COPPER AT VARIOUS TEMPERATURES E F F E C T OF STRAIN R A T E ON ULTIMATE STRESS OF PURF; COPPER AT VARIOUS TEMPERATURES Figure 3.1.4 E f f e c t of S t r a i n Rate on Ultimate Tensile Strength of Mild Steel and Copper (Nadai and Majoine) 3 5 t h i s c o n d i t i o n o f constancy i s approximately s a t i s f i e d , over a l a r g e range of s t r a i n r a t e , by m i l d s t e e l a t room tempera-t u r e . Most metals, u n f o r t u n a t e l y , have l e s s convenient behaviour. Copper, f o r example, a t room temperature appears to f o l l o w a r e l a t i o n s h i p of the form O Q = K e M ( 3 . 1 . 2 5 ) a t f r a c t u r e . Brass, whose l a t t i c e s t r u c t u r e i s unchanged from t h a t o f copper by the a d d i t i o n o f z i n c , and whose major component i s copper, might be expected to a l s o e x h i b i t be-h a v i o u r d e s c r i b e d by equation 3 . 1 . 2 5 . C o n s i d e r a t i o n of the e f f e c t of r a t e dependence of on e quation 3 . 1 . 2 4 w i l l be d e f e r r e d u n t i l s e c t i o n 5 . 1 , where i n d i v i d u a l f r i c t i o n a l examples are d i s c u s s e d . 36 3.2 K i n e t i c Boundary F r i c t i o n S e c t i o n 3.1 was devoted to o r i g i n a l development o f an equation p r e d i c t i n g the v a r i a t i o n i n 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 w i t h r a t e of l o a d i n g . In c o n t r a s t , t h i s s e c t i o n w i l l be concerned wi t h the b a s i c concepts r e q u i r e d f o r the d i s c u s s i o n of k i n e t i c r e s u l t s from the p r e s e n t study. The s i m p l e s t form of f r i c t i o n , under s t e a d y - s t a t e c o n d i t i o n s of s l i d i n g , c o n s i s t s of a r e t a r d a t i o n f o r c e , t o t a l l y independent of r e l a t i v e v e l o c i t y between s u r f a c e s , c a l l e d Coulombic f r i c t i o n . An example of a Coulombic f r i c t i o n f o r c e t r a c e on the phase plane, together w i t h a generated b e h a v i o u r a l t r a c e f o r a spring-mass system, s u b j e c t to t h i s f r i c t i o n f o r c e i n f r e e v i b r a t i o n , may be found i n F i g u r e 3.2.1. Coulombic f r i c t i o n i s approximated i n many i n s t a n c e s of nominally un-l u b r i c a t e d c o n t a c t of harder metals [28,29]. The e x p r e s s i o n "nominally u n l u b r i c a t e d " , r a t h e r than " u n l u b r i c a t e d " , i s used because crude l u b r i c a t i o n i s i n h e r e n t i n the presence of most m e t a l l i c o x i d e s . C o n t r a r y to i t s proven minimal e f f e c t on the governing mechanism of s t a t i c f r i c t i o n [9], i n which i t s s o l e f u n c t i o n appears to be the d e c r e a s i n g of the i n t e r f a c i a l shear s t r e n g t h , the presence of a l i q u i d l u b r i c a n t , even without chemical boundary l u b r i c a t i o n a d d i t i v e s , may have a profound e f f e c t on the observed f r i c t i o n a l behaviour under k i n e t i c c o n d i t i o n s . These chemical a d d i t i v e s are normally l o n g - c h a i n p o l a r molecules, which a t t a c h by one end to the metal s u r f a c e , or F i g u r e 3.2.1 Free V i b r a t i o n , of. a L i n e a r Spring-Mass System S u b j e c t t o Simple Forms of P r i c t i o n 38 compounds (organic or inorganic) which react with m e t a l l i c oxides to form a f i l m of low shear strength. Chemical additives which do not appreciably a f f e c t the l u b r i c a n t v i s c o s i t y have no e f f e c t on f r i c t i o n a l behaviour u n t i l opposing surfaces are s u f f i c i e n t l y close that the thickness of an additive layer i s a substantial portion of the gap between surfaces. If the surfaces do not achieve such close proximity, or i f chemical action between the lubricant and surfaces i s n e g l i g i b l e , f r i c t i o n a l behaviour i s governed by the viscous properties of the l u b r i c a n t . Linear viscous f r i c t i o n i s a simple form of energy d i s s i p a t i o n associated with the presence of a Newtonian f l u i d between two p a r a l l e l , plane surfaces. A Newtonian f l u i d i s a viscous f l u i d i n which the rate of shear i s d i r e c t l y proportional to the applied shear stress, as expressed by the equation x n ~ . (3.2.1) yx dy If two plane surfaces are separated by a Newtonian f l u i d layer of constant thickness, the v a r i a t i o n i n tangential f r i c t i o n force with r e l a t i v e v e l o c i t y of the surfaces i s l i n e a r . Figure 3.2.1 shows an example of l i n e a r l y viscous d i s s i p a t i o n action, together with a generated free v i b r a t i o n behavioural trace, on the phase plane. Another important e f f e c t associated with f l u i d v i s c o s i t y i s that of a squeeze f i l m . As examination of Appendix III w i l l reveal, a plane surface with a c i r c u l a r periphery, which approaches another, p a r a l l e l plane surface from which i t i s separated by a Newtonian l i q u i d of invariant v i s c o s i t y , has a load-bearing capacity M 3 ^4 1 dh , _ - „. N f = " 2 M R ' J d t ' (2.2.2) where h i s the gap between surfaces. The rate of change of dh the surface gap, -j^-, i s very small, for any normal load N f, i f the surface separation i s small. Note p a r t i c u l a r l y that equation 3.2.2, derived for the case of p a r a l l e l surfaces, i s applicable whether or not r e l a t i v e motion, and a consequent viscous tangential retardation force, exists between the surfaces i n t h e i r own plane. F u l l hydrodynamic l u b r i c a t i o n r e s u l t s from the pumping of a viscous f l u i d through a converging gap between s o l i d surfaces. These s o l i d surfaces do not touch, and f l u i d pressures are usually s u f f i c i e n t l y low that surface p r o f i l e s remain e s s e n t i a l l y unaltered. Observed f r i c t i o n a l resistance to motion of one surface tangential to the other i s a conse-quence of the energy expended i n overcoming viscous drag. Hydrodynamic analysis t r a d i t i o n a l l y considers bearing surfaces smooth, requiring that they be non-parallel i n order to generate load-carrying pressures i n the l u b r i c a n t . However, 40 the generation of hydrodynamic l i f t between closely-spaced p a r a l l e l surfaces i s well documented, i f not well understood [30]. Explanations advanced for this phenomenon have included thermal expansion of the lubricant, thermal deformation of the surfaces, and hydrodynamic l i f t , on a microscopic scale, as a consequence of surface rugosity. If hydrodynamic l i f t i s to occur as a consequence of surface rugosity l o c a l i z e d load-carrying pressures may be very much greater than the mean l i q u i d pressure between the surfaces. The r i g i d i t y of surfaces normally assumed i n hydrodynamic analysis becomes a questionable supposition. Of d e f i n i t e i n t e r e s t , then, i s the f i e l d of study c a l l e d elastohydro-dyanmic l u b r i c a t i o n . Elastohydrodynamic l u b r i c a t i o n analysis was developed because of the inadequacy of c l a s s i c a l hydro-dynamic analysis to explain the occurrence of e f f e c t i v e hydrodynamic l u b r i c a t i o n i n extreme-pressure point and l i n e contacts, the transmission of forces between gear teeth being one example. Elastohydrodynamic theory i s distinguished from c l a s s i c a l hydrodynamics by s p e c i f i c a t i o n of the lubricant f i l m thickness as a function of the e l a s t i c properties of the load-bearing s o l i d surfaces, as well as of the applied load, lubricant v i s c o s i t y , and i n i t i a l surface geometry. Elas t o -hydrodynamic theory, i n i t s simplest form, requires simultaneous solution of equations of hydrodynamics and e l a s t i c i t y . Exact a n a l y t i c solutions are not usually achieved . for even the simplest cases of elastohydrodynamic l u b r i c a t i o n . 41 Consider, therefore, the increased d i f f i c u l t y of solution as changes i n lubricant v i s c o s i t y due to pressure, temperature, and shear rate s e n s i t i v i t y are included i n the equations. Numerical techniques, u t i l i z i n g d i g i t a l computer f a c i l i t i e s , are normally employed. One of these numerical solutions, v e r i f i e d experimentally [31], for the very simple case of two c y l i n d r i c a l r o l l e r s with i d e n t i c a l surface v e l o c i t i e s i n the contact zone, i s displayed i n Figure 3.2.2, together with comparable solutions for purely e l a s t i c and c l a s s i c a l hydro-dynamic cases. The elastohydrodynamic l u b r i c a t i o n property that i s of special i n t e r e s t to the present study i s the generation of f r i c t i o n forces when the opposing surfaces possess a non-zero tangential r e l a t i v e v e l o c i t y . A family of experimentally-obtained curves, displaying the v a r i a t i o n of f r i c t i o n force with r e l a t i v e v e l o c i t y of the surfaces for the case of p a r a l l e l c y l i n d r i c a l r o l l e r s , i s presented i n Figure 3.2.3. The lubricant employed was a commercial mineral turbine o i l , B r i t i s h Admiralty s p e c i f i c a t i o n OM 100, and the r o l l e r s were hardened s t e e l , diameter 3 inches. The i n i t i a l r i s e i n f r i c t i o n force with r e l a t i v e v e l o c i t y i s consistent with the properties of an isoviscous Newtonian lubricant; but as r e l -ative v e l o c i t y increases, f r i c t i o n forces become increasingly less than those predicted from isoviscous considerations, to the extent that measured f r i c t i o n force decreases with increasing r e l a t i v e v e l o c i t y at high r e l a t i v e v e l o c i t y values. Martin conditions—rigid solids, isoviscous lubricant Hertzian conditions—dry contact, elastic solids. ilrnln Elastohydrodynamic conditions—elastic solids, Newtonian lubricant F i g u r e 3.2.2 R o l l i n g Contact C o n d i t i o n s [42] 43 F i g u r e 3.2.3 Elastohydrodynamic F r i c t i o n Force vs V e l o c i t y Curves [43] U = 400 cm/sec. Normal loads (10 dynes/cm)--A:203 B:15, C:7.5. Values of u at U = 400 cm/sec shown at right of curves Experts i n the f i e l d c r e d i t t h i s r e d u c t i o n i n f r i c t i o n f o r c e to v i s c o s i t y decreases r e s u l t i n g from thermal e f f e c t s o f v i s c o u s l y - g e n e r a t e d heat. Crook [32] has developed, i n e x p l a i n i n g the curves of F i g u r e 3.2.3, an approximate r e l a t i o n s h i p o f the form F X n u A mm (3.2.3) A X n V i n l e n g t h of r e c t a n g u l a r l o a d - c a r r y i n g f i l m width of r e c t a n g u l a r l o a d - c a r r y i n g f i l m mean e f f e c t i v e l i q u i d v i s c o s i t y minimum f i l m t h i c k n e s s ( F i g . 3.2.2) i n which the mean e f f e c t i v e l i q u i d v i s c o s i t y i s i t s e l f expressed by an approximate h e a t - t r a n s f e r r e l a t i o n s h i p Y U- A A + 2 In U + In n s Y 2? f -I (3.2.4) thermal c o n d u c t i v i t y of l u b r i c a n t l u b r i c a n t v i s c o s i t y a t temperature o f bounding f r i c t i o n s u r f a c e s Y T - T P - P, In In c o e f f i c i e n t of l u b r i c a n t v i s c o s i t y v a r i a t i o n w i t h temperature c o e f f i c i e n t o f l u b r i c a n t v i s c o s i t y v a r i a t i o n with p r e s s u r e . In standard elastohydrodynamic s i t u a t i o n s h . i s a m m f u n c t i o n o f mean s u r f a c e speed, l u b r i c a n t p r o p e r t i e s , and l o a d . I f , however, f o r boundary l u b r i c a t i o n s i t u a t i o n s a n e a r l y constant, s u b s t a n t i a l p o r t i o n of the normal l o a d i s demonstrably supported by a s p e r i t y c o n t a c t , ^- m^ n i s , f o r p r a c t i c a l purposes, approximately c o n s t a n t . A s p e r i t y c o n t a c t does not mean t h a t h . n e c e s s a r i l y assumes the v a l u e zero, m m J as equation 3.2.3 might suggest. That e q u a t i o n , d e r i v e d from a two-dimensional model, makes no allowance f o r l a t e r a l flow of l u b r i c a n t , which must occur a t any p o i n t of c o n t a c t between the s u r f a c e s . When attempting to apply e q u a t i o n 3.2.3 to such circumstances the v a r i a b l e h . can no lon g e r m m ^ be i n t e r p r e t e d as t r u e minimum c l e a r a n c e ; i t must i n s t e a d assume some e f f e c t i v e p o s i t i v e v a l u e determined by summation of elastohydrodynamic e f f e c t s over the t o t a l l o a d - b e a r i n g f i l m . During t h i s study the l o a d and l o a d - b e a r i n g f i l m area and shape were u n a l t e r e d . I f , as a d d i t i o n a l c o n d i t i o n s , one assumes h . and n constant, Crook's f r i c t i o n a l r e l a t i o n -min s sh i p s may be reduced to the form F = U In U (3.2.5) which w i l l e x h i b i t a maximum a t some v a l u e of r e l a t i v e v e l o c i t y determined by the con s t a n t b. T h i s maximum i s r e l e v a n t to d i s c u s s i o n of quasi-harmonic o s c i l l a t i o n , which w i l l f o l l o w i n s e c t i o n 5.2. 46 Having reviewed the f o r e g o i n g concepts o f l u b r i c a t e d k i n e t i c f r i c t i o n , i t i s apparent t h a t even a c h e m i c a l l y i n e r t l i q u i d l u b r i c a n t can c o n t r i b u t e to or dominate the boundary f r i c t i o n behaviour of m e t a l l i c s u r f a c e s by p u r e l y v i s c o u s a c t i o n . The e f f e c t s of chemical a c t i o n between l u b r i c a n t and s u r f a c e s add to the e f f e c t s of v i s c o u s a c t i o n , making a n a l y s i s of a complex s i t u a t i o n even more i n v o l v e d . Chemical e f f e c t s have consequently, as much as p o s s i b l e , been e l i m i n a t e d from the p r e s e n t i n v e s t i g a t i o n so as to permit a more c e r t a i n a n a l y s i s of the p h y s i c a l f r i c t i o n mechanisms i n v o l v e d . C H A P T E R I V IV. APPARATUS AND EXPERIMENTAL PROCEDURE 4.1 Apparatus Because a certa i n portion of r e l a t i v e l y high-speed investigation was anticipated, the use of a linear-motion f r i c t i o n apparatus, of inherently limited track length, was discarded i n favour cf a rotary-motion apparatus, i n which the track length can be made e f f e c t i v e l y i n f i n i t e . The apparatus used by Ko [15], which required minimal modification to s u i t i t for the present study, was u t i l i z e d . A schematic diagram of this apparatus, as employed, may be found i n Figure 4.1.1. The f r i c t i o n couple consisted of a rotating s t e e l disc, of diameter 4 inches, on which rested a s l i d e r of diameter 3/8 inch. The s l i d e r was pressed into a hemispher-i c a l mount, which provided a means of i n i t i a l alignment of the s l i d e r with the disc surface. Load application and e l a s t i c r e s t r a i n t of the s l i d e r were provided by a compound cantilever beam passing across the disc, in order that the curvature of the s l i d e r path was i n the same sense as that of the f r i c t i o n track on the di s c . Applying load, as well as r e s t r a i n t , by means of the beam permitted independent v a r i a t i o n of load and the frequency of free v i b r a t i o n of the system. A compound, rather than simple, beam was used F i g u r e 4.1.1 Schematic Diagram of Experimental System VO 50 i n o r d e r t h a t f o r c e a p p l i c a t i o n c o u l d be a t the p r o j e c t e d n e u t r a l a x i s of the more e l a s t i c p o r t i o n of the beam, e l i m i n a t i n g u n d e s i r a b l e t o r s i o n a l e f f e c t s i n the beam due to f r i c t i o n f o r c e s a t the i n t e r f a c e . To minimize t o r s i o n a l e f f e c t s due to dynamic imbalance, l e a d weights were atta c h e d to the end o f the beam i n such a f a s h i o n as to a l s o p l a c e the c e n t e r of g r a v i t y of the v i b r a t i n g mass on the f o r e - , mentioned n e u t r a l a x i s . The " f i x e d " end of the c a n t i l e v e r beam was clamped to a r i g i d s t e e l s h a f t mounted i n s e l f - a l i g n i n g p i l l o w - b l o c k b e a r i n g s . These b e a r i n g s were end-loaded to e l i m i n a t e s l a c k , and mounted f i r m l y to an aluminum s h e l l . T h i s s h e l l , and the base on which i t r e s t e d , were both h e a v i l y weighted w i t h l e a d , so as to reduce, as much as p o s s i b l e , the n a t u r a l frequency of v i b r a t i o n of the f r i c t i o n mechanism's s u p p o r t i n g s t r u c t u r e . T h i s r e d u c t i o n i n the s u p p o r t i n g s t r u c t u r e ' s n a t u r a l frequency i s o l a t e d the f r i c t i o n mechanism from e x t e r n a l v i b r a t i o n to which i t would be most s e n s i t i v e by l o c a t i n g these "no i s e " f r e q u e n c i e s w e l l i n t o the s t r o n g l y -a t t e n u a t e d p o r t i o n of the s t r u c t u r e ' s t r a n s m i s s i b i l i t y spectrum. Such p r e c a u t i o n s were e x e r c i s e d because s e v e r a l i n v e s t i g a t o r s , the most no t a b l e among them being Fridman and Levesque [33], have found t h a t the a p p l i c a t i o n of e x t e r n a l f r e q u e n c i e s to a f r i c t i o n s u r f a c e s u b s t a n t i a l l y decreases the observed va l u e s of s t a t i c f r i c t i o n . 51 The drive t r a i n u t i l i z e d a 100:1 double-worm speed reducer immediately preceding the driven s t e e l d i s c . This reducer, as a r e s u l t of i t s sequential double-worm construction, negated the transmission of torque v a r i a t i o n s , at the f r i c t i o n surface, back to i t s power input shaft. That th i s input shaft exhibited a non-fluctuating reaction torque was esse n t i a l to obtaining a uniform driven-surface v e l o c i t y , since the remainder of the power transmission system consisted of j o i n t l e s s , s o f t rubber O-rings. The e l a s t i c i t y of these O-rings (manufactured for use as seals) made a d i r e c t , motor-to-disc, b e l t type of power transmission system impractical. Unfortunately, more common modes of power transmission were even more impractical. Spur gears generate v i b r a t i o n as a consequence of the meshing of teeth; conventional drive belts have j o i n t s , which generate force impulses when passing over drive pulleys. A drive t r a i n consisting solely of worm reducers would generate l i t t l e noise, but would be both clumsy and costly, as well as provid-ing an excellent noise path from the motor to the f r i c t i o n surface. The hybrid system, O-rings d r i v i n g a worm reducer, which provided uniform surface speeds, good noise i s o l a t i o n , and extreme f l e x i b i l i t y , was employed very successfully. The O-ring portion of the drive t r a i n was, i n r e a l i t y , a dual system. One portion of this system consisted simply of a variable-speed 3/16 hp. d.c. motor d i r e c t l y d r i v i n g the worm reducer through a single O-ring. This part of the system, though not s t r i c t l y necessary, was convenient for the production of the higher surface speeds employed i n this i n v e s t i g a t i o n . The other portion of the dual system consisted of a 1 hp. variable-speed d.c. motor dr i v i n g the worm reducer through a noise-free, m u l t i - r a t i o speed reducer, constructed of O-rings and pulleys, possessing a maximum 4 reduction c a p a b i l i t y of 10 . Surface speeds at the f r i c t i o n track radius ranged from 3 x 10 ^ in/sec. to 12 in/sec. Higher speeds could have been re a d i l y achieved, but were not required. Photographs of the system may be found i n Figures 4.1.2 and 4.1.3. System parameters may be found i n Appendix I. 4.2 Measurement of F r i c t i o n Forces Measurement of f r i c t i o n forces between the s l i d e r and the disc was required under dynamic conditions, when the s l i d e r and beam would be subject to non-negligible acceleration forces. Simple measurement of beam displacement, a procedure commonly employed in the past for both s t a t i c and dynamic investigations, would have been inadequate for obtaining such information because i t ignores these important i n e r t i a l forces. The e l e c t r i c a l summation methods of B e l l and Burdekin [12] were consequently employed. The equation of motion of the s l i d e r may be written i n the form Mx + r x + k x = F (4.2.1) where M i s the equivalent mass of the s l i d e r and i t s support-ing structure, and F i s the f r i c t i o n force experienced by the s l i d e r at i t s i n t e r f a c i a l surface. Rearranging t h i s equation to the form •• r • k 1 x M X M X p l a i n l y i l l u s t r a t e s that, i f the viscous d i s s i p a t i o n term r • J-J x i s of n e g l i g i b l e magnitude, with respect to the other terms, a scaled measure of the f r i c t i o n force may be obtained from the summation of the s l i d e r acceleration and the displacement signals, properly scaled. Examination of Appendix I w i l l confirm that ^ i s four orders of magnitude smaller than — . From phase plane analysis, and the knowledge that motion of the s l i d e r with respect to the f r i c t i o n surface i s performed under energy-d i s s i p a t i n g conditions, maximum v e l o c i t y x i s less than r ^ ' 1 max the maximum displacement x mul t i p l i e d by the natural max c J frequency of vi b r a t i o n of the s l i d e r and i t s supporting structure, to (Appendix I I ) . The damping term i s therefore two orders of magnitude less than the displacement term, , with respect to th e i r i n d i v i d u a l maxima, and may consequently be assumed to make a n e g l i g i b l e contribution to the equation i F M (4.2.2) 56 of motion, permitting equation 4.2.2 to be rewritten i n the form x + £ x = i P (4.2.3) M M V e c t o r i a l summation of the acceleration and displacement was achieved by transmitting the two signals to the oscilloscope d i f f e r e n t i a l a mplifier. The accuracy of a l l f r i c t i o n measurements was r e s t r i c t e d to that of the p r i n c i p a l data recording instrument, an o s c i l l o s c o p e . A more precise recording device would have been of no advantage because the instrument signals exhibited a superimposed high-frequency mechanical noise component, most prominent on the acceleration signal and l e a s t prominent on the displacement s i g n a l . The noise signals were most pronounced during quasi-harmonic o s c i l l a t i o n and following the inception of s l i p . Possible explanations for t h i s unwanted noise generation include v i b r a t i o n of portions of the cantilever beam's support structure and v i b r a t i o n of the beam i n non-, fundamental modes. 4.3 Instrumentation a. Displacement Two 350 ohm s t r a i n gages were mounted at the root of the cantilever beam, where bending moments in the beam were at a maximum. These gages, together with two i d e n t i c a l gages 57 used f o r temperature compensation purposes, were employed i n a four-arm b r i d g e c i r c u i t . B ridge output was a m p l i f i e d by an e l e c t r o n i c u n i t s i m i l a r to E l l i s A s s o c i a t e s 1 B r i d g e A m p l i f i e r Meter, model BAM-1. b. V e l o c i t y An e l e c t r o m a g n e t i c t r a n s d u c e r , c o n s i s t i n g o f a c o i l of enamelled wire p o s i t i o n e d i n a constant magnetic f i e l d , was s i t u a t e d such t h a t , d u r i n g beam v i b r a t i o n , the conductors moved t r a n s v e r s e l y to the d i r e c t i o n of the f i x e d magnetic f i e l d . Generated v o l t a g e was consequently p r o p o r t i o n a l to the instantaneous v e l o c i t y of the c o i l . This v o l t a g e s i g n a l was attenuated by a v a r i a b l e - o u t p u t v o l t a g e d i v i d e r c i r c u i t b e f o r e t r a n s m i s s i o n to the r e c o r d i n g instruments. D e t a i l s of the tra n s d u c e r may be found i n F i g u r e 4.3.1. c. A c c e l e r a t i o n A s e l f - c o n t a i n e d servo accelerometer, K i s t l e r model 305A, was mounted on the specimen h o l d e r . D i r e c t t r a n s m i s s i o n of the a c c e l e r a t i o n s i g n a l to the r e c o r d i n g instruments was p o s s i b l e because a 3 0 v o l t d.c. power source y i e l d e d a maximum accelerometer v o l t a g e of ±5 v o l t s . The l e v e l of a c c e l e r a t i o n corresponding to maximum s i g n a l , which c o u l d be a l t e r e d by changing an e x t e r n a l r e s i s t o r , was s e t a t 50 g, y i e l d i n g a s e n s i t i v i t y of 0.1 v o l t / g . Maximum r e s o l u t i o n o f the — 7 — 6 instrument was b e t t e r than 5 x 10 v o l t s , or 5 x 10 g. PERMANENT MAGNETS 1350 TURNS ENAMELLED WIRE Figure 4.3.1 Vel o c i t y Transducer D e t a i l 59 d. Recording Instruments Instruments used for recording data were a Brush dual channel r e c t i l i n e a r oscillograph, Mark. 842, and a Tektronix dual beam storage oscilloscope, Model 564. The oscilloscope was equipped with a dual-beam d i f f e r e n t i a l amplifier, Type 3A3, a single-beam d i f f e r e n t i a l amplifier, Type 2A63, and a time-base amplifier, Type 2B67. The s i n g l e -beam d i f f e r e n t i a l amplifier and the time-base amplifier were used interchangeably. The oscilloscope was modified so as to permit control of i t s beam-blanking c i r c u i t by an external voltage s i g n a l . e. One-Cycle Sequential-Triggering C i r c u i t Control of the oscilloscope's beam-blanking c i r c u i t was performed by an external c i r c u i t , s i m i l a r i n function to that used by Ko [15], constructed to permit the recording of a single cycle of v i b r a t i o n . This c i r c u i t was, i n turn, con-t r o l l e d by the displacement s i g n a l . The one-cycle c i r c u i t , which may be found i n Figure 4.3.2, e s s e n t i a l l y consisted of three relays surrounded by accessory c i r c u i t r y . The f i r s t and t h i r d relays were tripped by a given l e v e l of negative displacement signal to, respectively, unblank and reblank the oscilloscope beam. The second relay, tripped by a preset p o s i t i v e displacement signal l e v e l , served as an interlock, preventing "machine-gunning" of the f i r s t and t h i r d relays on the same negative displace-Figure 4.3.2 Spot-Triggering and Sequential-Triggering C i r c u i t r y o 61 ment s i g n a l . This t r i o of relays was energized by a key relay; a l l four relays were simultaneously reset by a manually-transmitted voltage s i g n a l . f. Spot Triggering To minimize inconsistency of results due to possible non-uniformity of the driven disc's surface, recorded data, throughout the duration of a t e s t , was obtained from a very r e s t r i c t e d portion of the disc's f r i c t i o n track. A l i g h t -discriminating r e s i s t o r (LDR) was used to signal the key i relay, i n the one-cycle sequential-triggering c i r c u i t , of the a r r i v a l of t h i s predetermined portion of the f r i c t i o n track at the s l i d e r ' s l o c a t i o n . Details of the spot-triggering c i r c u i t r y are included as an i n t e g r a l part of the sequential-tr i g g e r i n g c i r c u i t presented i n Figure 4.3.2. Each time a' f l a g , which could be fixed at any desired point on the disc's circumference, moved between a l i g h t source and the LDR a voltage pulse was transmitted to the key relay. If the relays had been previously reset the key relay would activate the b l a n k i n g - c i r c u i t control relays, r e s u l t i n g i n the record-ing of information over the duration of the succeeding cycle. The advantage of using a LDR t r i g g e r i n g system, rather than a simpler, mechanical t r i g g e r , lay i n the f a c t that no triggering-force reactions were experienced by the drive t r a i n . Mechanical devices, even those as e f f o r t l e s s i n operation as microswitches, had switching reactions which appeared as non-negligible acceleration signals. Any such acceleration spike could i n i t i a t e s l i p , i f the f r i c t i o n surfaces were approaching the l i m i t i n g conditions of s t i c k , and/or have a transient e f f e c t of s u f f i c i e n t duration to a f f e c t the recorded data. g. Speed Determination Belt stretch, and possible b e l t slippage, rendered i n d i r e c t measurement of f r i c t i o n disc speed, at some remote portion of the d r i v e t r a i n , t o t a l l y unfeasible. The reasons negating use of a mechanical spot-triggering device made employment of a mechanically-driven speed indicator equally unwise. A displacement in d i c a t i n g device, consisting of a precisely-machined strobe disc separating a l i g h t source and a LDR, was consequently constructed. The strobe disc was mounted on the shaft driving the f r i c t i o n d i s c , and the l i g h t source-LDR combination fixed to the apparatus frame. Signals from the LDR were registered by a counter operating i n conjunction with an elapsed-time indicator. Error i n speed determination, greatest at higher speeds, was less than ±2%. 4.4 Specimens As a consequence of the work of Davis [13] , the t e s t -ing of a range of d i f f e r e n t metals was deemed unnecessary. The i n i t i a l intention was, therefore, to test only one f r i c t i o n p a i r . A steel-on-steel f r i c t i o n couple was chosen. But when unique and in t e r e s t i n g r e s u l t s were obtained, outside the range of load rate encompassed by Davis' experiments, i t became prudent to confirm the s i m i l a r i t y of action of d i f f e r -ent metals over this extended range, so as to avoid the p o s s i b i l i t y of drawing general conclusions from anomalous behaviour. Confirmatory tests were performed using unleaded brass s l i d i n g on s t e e l , another common i n d u s t r i a l p a i r . The 3 / 8 inch diameter s l i d e r s and the 4 inch diameter s t e e l disc, whose periphery formed the f r i c t i o n path, were i n i t i a l l y prepared by grinding and lapping. F i n a l preparation consisted of wearing the specimens and f r i c t i o n disc, i n the presence of the lubricant used during testing, u n t i l t h e i r surfaces were as conformal as was thought possible under those conditions. The applied lubricant was exchanged < p e r i o d i c a l l y throughout the duration of this f i n a l prepara-tion, i n order to remove wear debris and ensure the presence of fresh o i l at the beginning of a test. The wear-in process was considered complete when i t s external evidence, the observed values of the c o e f f i c i e n t of s t a t i c f r i c t i o n , which increased during the wear-in process, appeared to reach an upper l i m i t , and when t h i s upper l i m i t existed over the entire periphery of the dis c . Incorporation of a wear-in process was a d i s t i n c t departure from the procedure of nearly a l l former i n v e s t i -gators of s t i c k - s l i p o s c i l l a t i o n , who have used f r e s h l y -prepared (ground or lapped) surfaces for th e i r tests. Wearing-i n of the surfaces, though tedious, offered c e r t a i n d e f i n i t e advantages. F r i c t i o n conditions were known to be very uniform over the e n t i r e f r i c t i o n track, and to remain stable through-out the duration of a t e s t . Uniformity of conditions, as evidenced by observed f r i c t i o n values, i s very unusual i n the case of freshly-prepared surfaces, and r e p e a t a b i l i t y of r e s u l t s i s impossible to achieve unless the surfaces are r e f i n i s h e d a f t e r each t r a v e r s a l . A d d i t i o n a l l y , i n d u s t r i a l f r i c t i o n surfaces are most commonly well run-in; numerical r e s u l t s from run-in surfaces have, therefore, greater a p p l i c a b i l i t y . Details of specimen surface parameters may be found i n Appendix V. 4 . 5 Testing Procedure Correct scaling of instrumentation signals (Appendix IV) was performed p r i o r to the wearing-in of f r i c t i o n surfaces. Immediately upon completion of the wear-in process data con-cerning v a r i a t i o n of f r i c t i o n a l properties with tangential load rate was recorded, with load rate application altered i n a systematic manner. Afj:er the desired speed range had been scanned i n this fashion, additional data recordings were made at the i n i t i a l load rates, i n order to ensure that f r i c t i o n a l conditions had not suffered any detectable change. At the completion of a test scaling of instrumentation signals was 65 checked to v e r i f y s t a b i l i t y of the instrumentation over the duration of the running-in and t e s t i n g . I t i s of p a r t i c u l a r s i g n i f i c a n c e to the conformability of the f r i c t i o n a l surfaces that the s l i d e r was not removed from the f r i c t i o n path, nor was i t s o r i e n t a t i o n i n any way. disturbed, from the beginning of the wear-in process to the completion of a t e s t . The major portion of the data was c o l l e c t e d by photo-graphing oscilloscope traces. Normal load at the interface was, i n a l l cases, 21.1 pounds. 4.6 Lubricant Repeated traversals of the f r i c t i o n track made use of a lubricant mandatory i f severe scoring of the surfaces was to be avoided. Chemical action between this lubricant and the m e t a l l i c surfaces was to be avoided, since such action could d i s t o r t r e s u l t s and obscure the physical mechanisms of boundary f r i c t i o n . A d d i t i o n a l l y , minimal viscous e f f e c t during periods of r e l a t i v e motion was desired, since s t i c k -s l i p o s c i l l a t i o n does not occur i f dynamic energy d i s s i p a t i o n exceeds a c r i t i c a l l e v e l . Greater dynamic energy d i s s i p a t i o n results i n more rapid decay, on the phase plane, of the s l i d e r ' s phasor amplitude. For persistence of o s c i l l a t i o n the s l i d e r must, afte r s l i p , at a lesser displacement re-achieve the v e l o c i t y of the other surface. Consequently, at any driven surface v e l o c i t y v some c r i t i c a l amount of 66 energy d i s s i p a t i o n w i l l e x i s t beyond which the phasor's decay rate i s too rapid to permit i t to again a t t a i n the v e l o c i t y v, r e s u l t i n g i n the continued decay of the s l i d e r ' s phasor to zero amplitude, a state corresponding to non-oscillatory motion of the s l i d e r over the driven surface. The f r i c t i o n surface, during a l l data runs, was flooded with Liquid Petrolatum (light) B.P., a medicinal-grade napthenic hydrocarbon, or "white mineral o i l " . This o i l does not, i n pure form, contain any polar groups, and was therefore, at le a s t nominally, i n e r t with respect to the meta l l i c f r i c t i o n surfaces. Measured v i s c o s i t y was 120 SSU at 80°F; for purposes of comparison, o i l rated at SAE 5W has a kinematic v i s c o s i t y of approximately 200 SSU at 80°F. To further avoid undesirable chemical e f f e c t s , lapping of f r i c t i o n surfaces was performed with a paste made of Liquid Petrolatum and alum powder. No l i q u i d , other than the l u b r i -cant, was permitted to contact the surfaces subsequent to the commencement of the lapping procedure. Chemical effects a r i s i n g from oxidation of the lubricant could not be eliminated, but the r e p e a t a b i l i t y of the results at the i n i t i a t i o n and. termination of each test suggests that lubricant oxidation, over the period of the few days required for the performance of a test, was not a s i g n i f i c a n t ' f a c t o r . C H A P T E R V V. DISCUSSION OF RESULTS 5.1 S t a t i c F r i c t i o n Two photomicrographs, o f f e r i n g clear evidence of simultaneous s o l i d contact and r e l a t i v e motion between opposing f r i c t i o n surfaces, are presented i n Figure 5.1.1. Confirmatory surface roughness measurements may be found i n Appendix V. That such s o l i d contact exists i n the absence of r e l a t i v e motion between surfaces l o g i c a l l y follows, since hydrodynamic ef f e c t s can only increase with r e l a t i v e v e l o c i t y . The assumption of a solid-contact model for the s t i c k portion of a s t i c k - s l i p cycle i s therefore vindicated, provided, as indicated by the re s u l t s of Courtney-Pratt and Eisner [9], the portion of the load supported by squeeze f i l m e f f e cts i s of a lesser order of magnitude than the portion supported by s o l i d contact. Experimental data showing the v a r i a t i o n of the co-e f f i c i e n t of s t a t i c f r i c t i o n with the load rate variable <j> i s presented, for the steel-on-steel f r i c t i o n couple, i n Figure 5.1.2, together with curves defined by the derived equation 3.1.24. Included i n Figure 5.1.2, for comparison purposes, i s data obtained by Potter [10] for a nominally unlubricated steel-on-steel f r i c t i o n couple using a l i n e a r , rather than rotary, apparatus. 69 (a) SLIDER SURFACE (b) DISC SURFACE Figure 5.1.1 Post-Test Photomicrographs of S l i d e r Surface and F r i c t i o n Disc Surface (Steel-on-Steel), X 7 5 Directional wear markings are -plain on both surfaces. Pre-test lapped surface f i n i s h is evident in right-hand upper corner of F r i c t i o n Disc Surface photomicrograph 0.6 0.5 0 .4 0.3 0.2 0.1 T T n TQ- ii+n o; = constant °o 1 = J a + 3.36 crc , % * j « + 4 . o Experimental Data • May 1971 • Now 1971 o Potter = 45 JL 5.0 4.0 3.0 2.0 , 0 9 , 0 '* 1.0 (see") 1.0 2.0 Figure 5.1.2 Va r i a t i o n # o f the Co e f f i c i e n t of S t a t i c F r i c t i o n with Load Rate Variable a ) , Steel-on-Steel o I 71 In the derivation of equation 3.1.24 the term A S " jj-p- was assumed n e g l i g i b l y small compared to the load rate variable <j>: t h i s assumption must now be v e r i f i e d . The postulated mechanism of asperity deformation and area growth, p l a s t i c deformation r e s u l t i n g from application of a ramp form of tangential load (in combination with a constant normal load), permits the assumption that the rate of s t r a i n i n an asperity i s approximately proportional to the load rate variable cj). A further assumption that s t r a i n at s l i p (fracture) i s approximately 1 i n / i n , both i n the present inves t i g a t i o n and i n the t e n s i l e experiments of Nadai and Majoine [27], i n combination with time of s t i c k data from the present i n v e s t i -gation,' y i e l d s the r e s u l t that $ and the t e n s i l e s t r a i n rate e, as presented i n Figure 3.1.4, are of the same order of magnitude. In other words, data from Nadai and Majoine, for a given magnitude of e, should be applicable to an approximately equivalent magnitude of cj). A y s The r a t i o i s i d e n t i c a l to — • Both quantities in this second r a t i o are r e a d i l y obtained from data. The time derivative may be approximately calculated, again assuming fracture s t r a i n to be 1 i n / i n , by computing T Q . In the,load -4 1 -1 A s • rate ranqe e = 10 to 10 sec pr- x. can be shown, for raild s t e e l , to be at least two orders of magnitude less than the corresponding s t r a i n rate e. Hence, over the same range of A ct, the term ~ T i s also two orders of magnitude less than w i i t s corresponding load rate $. The i n i t i a l assumption, 72 A s * * ^— i s negligibly small i n comparison to |, i s consequently j u s t i f i e d for low-strength s t e e l . In the f i t t i n g of equation 3.1.24 to the experimental data, 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 at zero load rate ^s | _ g was assumed to be the value of the apparent upper asymptote, y =0.55. Substitution i n equation 3.1.22 yi e l d s 25 (5.1.1) 10 The experimentally-deduced value of a o / / x i ^ s obviously strongly influenced by the assumed value of a. Interestingly, because the constant K i n equation 3.1.16 should not be subst a n t i a l l y less than unity, comparison of equations 3.1.6 and 5.1.1 favours choosing a to be approximately 25, r e i n f o r c i n g the conclusions of Greenwood [45]. For purposes of thi s study, the value of a, though valuable i f i t could be prec i s e l y determined, i s not of paramount importance. Examination of Figure 5.1.2 w i l l reveal that a change of a, within a fe a s i b l e range, has only a minor e f f e c t on the p r o f i l e of the s t a t i c f r i c t i o n response curve. A change i n a does, however, s h i f t the response curve to ri g h t or l e f t on the logarithmic load rate axis, necessitating a compensating change i n the location parameter a^~*n/V}nb ; determination of this parameter i s , therefore, contingent upon knowing the value of a. a + 3.36 5.3, 3.7, a a = The upper l i m i t of the power index, n = 1.5, would correspond to uniform deformation throughout the volume of material within the e l a s t i c - p l a s t i c boundary. The lower l i m i t , n = 1.0, would correspond to uniform deformation along some front advancing with the boundary of the contact; such a front might occur at the e l a s t i c - p l a s t i c boundary i f the p l a s t i c material was restrained i n the d i r e c t i o n normal to the free surface. Examination of Figure 5.1.3 w i l l reveal that n might be expected to have an e f f e c t i v e value equal to neither of these two l i m i t s , but rather, some intermediate value. Although determination of n could be of si g n i f i c a n c e to further understanding of the area growth phenomena, the i n s e n s i t i v i t y of the response curve p r o f i l e to changes i n n, as well as i n a , together with the scatter of the experimental data, prevents any statement more conclusive than that the previously-stated l i m i t s appear consistent with the available information. Var i a t i o n i n n, unfortunately, also causes l a t e r a l , s h i f t s i n the s t a t i c f r i c t i o n response curve. Determination of the value of the location parameter o^+n/Vlnh i s consequently doubly d i f f i c u l t , since i t r e l i e s on accurate knowledge of both n and a . Before terminating discussion of the information presented i n Figure 5.1.2, i t i s s i g n i f i c a n t to the general a p p l i c a b i l i t y of the present theory that the s t a t i c f r i c t i o n response curve p r o f i l e of equation 3.1.24 f i t s the data of Figure 5.1.3 Zone of Deformation beneath a Longitudinal Wedge Indentation [44] Note hemispherical contour of elastic-plastic boundary and non-uniform nature of deformation within boundary Potter, which exhibits both a wide range of s t a t i c f r i c t i o n values and the f i r s t indications of an upper asymptote, as capably as i t does data c o l l e c t e d during t h i s study, i f the l i m i t i n g s t a t i c f r i c t i o n value ^s ^ _ Q i s reduced to 0.50. The a b i l i t y of equation 3.1.24 to describe these data sets reinforces the i n i t i a l supposition that, during s t i c k , s o l i d contact i s the dominant load-bearing mechanism, and the consequent omission of squeeze-film effects from the a n a l y t i c a l model. The quantity of l i q u i d l ubricant displaced by the decrease i n mean surface separation during s t i c k must conse-quently have experienced very l i t t l e d i f f i c u l t y i n exiting' the i n t e r f a c i a l zone, at the rates required, through the flow passages between asperity contacts. During the testing of brass on s t e e l , m e t a l l i c transfer from the brass s l i d e r to the s t e e l disc occurred, as evidenced by the photomicrograph presented i n Figure 5.1.4. No evidence of material transfer from the s t e e l disc to the brass s l i d e r was detectable. One might consequently expect f r i c t i o n a l , behaviour to be dominated by the more ph y s i c a l l y active of; the two metals, brass. That brass was, i n f a c t , the more ph y s i c a l l y active of the two surfaces i s worthy of note. I f , for whatever reason, a contact junction should possess an ultimate shear strength i n excess of that of either metal matrix, fracture w i l l occur, not at the i n t e r f a c i a l junction, but i n the weaker matrix. Transfer might have been expected from the s t e e l , 7 6 DISC SURFACE Figure 5.1 .4 Post-Test Photomicrograph of F r i c t i o n Disc Surface (Brass-on-Steel), X250 M e t a l l i c transfer of brass to the st'eel f r i c t i o n disc surface is evident 77 the weaker of the two materials at room temperature (steel 55 kpsi UTS, brass 75 kpsi UTS, approximately), to the brass. That transfer occurred i n an inverse manner suggests that, at the time of shear, the strength of the brass at the points of transfer was less than that of the s t e e l . Perhaps thermal e f f e c t s , such as were reported by Bowden and Ridler [4 ] , would provide an explanation for this apparent reversal of r e l a t i v e strengths. The brass material, as received and employed, was highly cold-worked; heat generated at the i n t e r -f a c i a l surfaces, whether during s t i c k or s l i p , might well have p a r t i a l l y annealed the relevant surface portions of the brass specimen, r e s u l t i n g i n l o c a l regions wherein the shear Strength of the brass was less than that of the s t e e l or the i n t e r f a c i a l junction. Conversely, cold-working of the surface of the annealed s t e e l might have s u f f i c i e n t l y increased i t s shear strength, i n these surface regions, so that the e f f e c -t i v e shear strength of the s t e e l exceeded that of the cold-worked brass. Whatever the explanation, the properties of the brass surface may be expected to govern the observed f r i c t i o n a l behaviour. f S p e c i f i c data on the fracture strength of brass, under conditions of varied stress rate, or even varied s t r a i n rate, has not been located. The most v a l i d comparative data available i s that presented, for copper, i n Figure 3.1.3. As has been stated, the response of the ultimate strength of brass to varying s t r a i n rate might, i n the absence of better 7 8 information, be assumed described by equation 3.1.25, but the rate of shearing s t r a i n i n an asperity junction i s unknown, precluding d i r e c t use of thi s equation. The metal at the asperity junction i s , for the postu-lated mechanism, marginally p l a s t i c . For this condition of p l a s t i c i t y the rate of shearing s t r a i n , at the inte r f a c e , may be considered proportional to the rate of horizontal load application, at lea s t as a f i r s t approximation, so that m C Q = . K <j> . (5,1.2) Substitution of equation 5.1.2 into equation 3.1.23 resu l t s i n the graphical plots exhibited i n Figure 5.1.5/ together with the data obtained for v a r i a t i o n of 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 with^ load rate. The p l o t for which m equals zero (a^ = constant) i s quite inadequate to describe the data, whereas, i f m i s of the order 0.1 a very respectable matching of derived plo t and data i s achieved. Examination of Figure 3.1.3 w i l l reveal that, for copper, the power index m has a value of the order of .02,. approximately o n e - f i f t h of the value required to match the p r o f i l e of equation 3.1.24 to that of.the data for the brass-on-steel f r i c t i o n p a i r . This difference i n magnitude i s not beyond c r e d i b i l i t y when one considers how d r a s t i c a l l y the physical properties of a metal can change when i t i s alloyed. F i g u r e 5.1.5 V a r i a t i o n . o f the C o e f f i c i e n t of S t a t i c F r i c t i o n w i t h Load Rate V a r i a b l e <J>, Bra s s - o n - S t e e l 80 S p e c i f i c a l l y , t h e d i f f e r e n c e i n m a g n i t u d e o f t h e power i n d e x m c o u l d be a t t r i b u t e d t o t h e f a c t t h a t d i s l o c a t i o n movement - i n b r a s s , w h i c h has s u b s t i t u t i o n a l z i n c atoms i n a c o p p e r l a t t i c e , i s more r e s t r i c t e d t h a n d i s l o c a t i o n movement i n p u r e c o p p e r . F o r t h e c o p p e r d a t a p r e s e n t e d i n F i g u r e 3.1.4, t h e A s • t e r m ^ — x^ i n e q u a t i o n 3.1.20 c a n be shown t o be, t h r o u g h o u t t h e r a n g e o f l o a d r a t e s s p a n n e d by t h i s i n v e s t i g a t i o n , t h r e e o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h e l o a d r a t e v a r i a b l e <j). The a s s u m p t i o n s and method o f a n a l y s i s a r e i d e n t i c a l t o t h o s e e mployed i n d e t e r m i n i n g t h e r e l a t i v e m a g n i t u d e o f t h o s e terms f o r t h e c a s e o f s t e e l - o n - s t e e l f r i c t i o n a l c o n t a c t . The d i f f e r e n c e i n m a g n i t u d e o f t h e power i n d i c e s m, f o r c o p p e r and b r a s s , i n d i c a t e s t h a t , f o r b r a s s , <J> w o u l d be o n l y two A o r d e r s o f m a g n i t u d e g r e a t e r t h a n ~ x . , b u t t h a t d i f f e r e n c e W i A i s s t i l l e n t i r e l y a d e q u a t e f o r a s s u r a n c e t h a t _JLx. i s W i n e g l i g i b l e t h r o u g h o u t t h e e x p e r i m e n t a l l o a d r a n g e . T h i s a s s u m p t i o n i s t h e r e f o r e v a l i d a t e d f o r t h e b r a s s - o n - s t e e l , as w e l l as t h e s t e e l - o n - s t e e l , f r i c t i o n p a i r . I n summary, two d u c t i l e m e t a l s , b r a s s and s t e e l , have b e e n t e s t e d u n d e r b o u n d a r y f r i c t i o n c o n d i t i o n s on a s t e e l s u r f a c e . B o t h m a t e r i a l c o m b i n a t i o n s showed d e f i n i t e u p p e r l i m i t s o f s t a t i c f r i c t i o n i n t h e p r e s e n c e o f a l u b r i c a n t , and t h e b e h a v i o u r o f b o t h f r i c t i o n p a i r s , when s u b j e c t e d t o v a r i e d r a t e s o f t a n g e n t i a l s t r e s s a p p l i c a t i o n , a p p e a r s a d e q u a t e l y d e s c r i b e d by t h e d e v e l o p e d e q u a t i o n 3.1.24. 81 5.2 Kine t i c Boundary F r i c t i o n a. S l i p The established form of the f r i c t i o n force curve during s l i p , , as determined by B e l l and Burdekin [12], i s represented schematically i n Figure 2.1.2. The maximum value of f r i c t i o n force, which determines the c o e f f i c i e n t of s t a t i c f r i c t i o n , corresponds to conditions at the incidence of s l i p . Once s l i p was i n i t i a t e d , the f r i c t i o n force magnitude f e l l while r e l a t i v e surface v e l o c i t y increased to a maximum; as the r e l a t i v e v e l o c i t y decreased from maximum to zero, the * f r i c t i o n force remained univalued. The form of f r i c t i o n force trace reported by B e l l and Burdekin has been recorded during the course of thi s study, and a representative data recording of thi s type, for the steel-on-steel f r i c t i o n pair, may be found i n Figure 5.2.1. The lower surface v e l o c i t y during generation of thi s record--2 ing was 2.8 x 10 in/sec; the achieved c o e f f i c i e n t of s t a t i c f r i c t i o n was 0.26. This form of trace i s not, however, invariant. As the lower surface v e l o c i t y was reduced, i n order to achieve greater c o e f f i c i e n t s of s t a t i c f r i c t i o n , the phase plane displacement and f r i c t i o n traces a l t e r e d pro-gressively to the forms displayed i n Figure 5.2.2. Unlike the traces displayed i n Figure 5.2.1, these recordings were made sequentially; s t a t i c f r i c t i o n values achieved d i f f e r e d s l i g h t l y , but there i s no doubt concerning the r e p e a t a b i l i t y and mutual compatibility of the f r i c t i o n force and d i s -placement traces. 82 X Figure 5.2.1 Recorded Half-Cycle S t i c k - S l i p Phase Plane Traces X F F i g u r e 5.2.2 Recorded M u l t i - C y c l e S t i c k - S l i p Phase Plane Traces ( S t e e l - o n -S t e e l , 10 0 mv/div) CO 84 S e v e r a l s i g n i f i c a n t o b s e r v a t i o n s r e s u l t from examin-a t i o n of the t r a c e s of F i g u r e 5.2.2. T r a d i t i o n a l s t i c k - s l i p theory p o s t u l a t e s t h a t the s l i p p o r t i o n of the displacement t r a c e should be a h a l f - c y c l e because the s u r f a c e s would r e -a t t a c h as soon as t h e i r r e l a t i v e v e l o c i t y f e l l t o z e r o . This re-attachment was supposed to occur because the r e s t o r i n g f o r c e a t the time when the s u r f a c e v e l o c i t i e s matched, a f t e r an e n e r g y - d i s s i p a t i n g h a l f - c y c l e , should be l e s s than t h a t r e q u i r e d to overcome s t a t i c f r i c t i o n . T h i s e x p l a n a t i o n , which ignores r a t e e f f e c t s , i s somewhat o v e r s i m p l i f i e d ; i t i s sometimes a p p l i c a b l e , but F i g u r e 5.2.2 p l a i n l y shows t h a t i t s a p p l i c a t i o n i s f a r from u n i v e r s a l , s i n c e the r e s t o r i n g , f o r c e a t the end of two c y c l e s , which corresponds to a f r i c t i o n a l c o e f f i c i e n t of 0.12, s t i l l exceeds i n magnitude the f o r c e s f a v o r i n g re-attachment. Perhaps the most s i g n i f i c a n t r e v e l a t i o n of F i g u r e 5.2.2 i s t h a t the upper p o r t i o n of B e l l and Burdekin's d u a l -valued f r i c t i o n curve i s a t r a n s i e n t which e x i s t s o n l y b e f o r e the system achieves s t a b l e s l i p c o n d i t i o n s , as d e f i n e d by the lower p o r t i o n of the f r i c t i o n curve. T h i s s t a b l e lower p o r t i o n of the f r i c t i o n curve t r a d i t i o n a l l y e x h i b i t s a Coulombic form, but t h a t of F i g u r e 5.2.2 shows both Coulombic and v i s c o u s c h a r a c t e r i s t i c s , s u g g e s t i n g t h a t both mechanisms of f r i c t i o n are a c t i v e . S p e c i f i c a l l y , the curve suggests the presence of both s o l i d - c o n t a c t and v i s c o u s modes of energy d i s s i p a t i o n . That the s u r f a c e s should have s o l i d c o n t a c t 85 has been demonstrated by the c o n d u c t i v i t y s t u d i e s o f Green [16]; t h a t v i s c o u s e f f e c t s should be added t o the s o l i d - c o n t a c t e f f e c t s i n the presence of a l i q u i d l u b r i c a n t i s not un-expected . The v a l u e of the observed f r i c t i o n f o r c e a t zero r e l a t i v e v e l o c i t y , where v i s c o u s drag i s zero, corresponds to a f r i c t i o n c o e f f i c i e n t o f approximately 0.03. S t e e l - o n - s t e e l c o n t a c t s , u s i n g m i l d s t e e l , have been found t o have a minimum 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 , a t low speeds and i n the presence o f h i g h l y e f f e c t i v e boundary l u b r i c a n t s , o f 0.053 [34]. Because the l u b r i c a n t used i n t h i s study was chosen f o r i t s poor boundary l u b r i c a t i o n p r o p e r t i e s , some p h y s i c a l mechanism other than s o l i d c o n t a c t was o b v i o u s l y r e s p o n s i b l e f o r the r e d u c t i o n of the k i n e t i c f r i c t i o n to the l e v e l observed, a r e d u c t i o n which c o u l d occur only i f some p o r t i o n of the normal l o a d was not supported by s o l i d c o n t a c t . That p a r t of the normal l o a d was supported by the v i s c o u s l i q u i d f i l m , even a t zero r e l a t i v e v e l o c i t y , appears i n e s c a p a b l e . The e x i s t e n c e of a l o a d - b e a r i n g squeeze f i l m a t the primary matching, a f t e r s l i p i n i t i a t i o n , of s u r f a c e v e l o c i t i e s suggests t h a t t h i s f i l m must have been e s t a b l i s h e d d u r i n g the f i r s t h a l f - c y c l e o f s l i p . That l u b r i c a n t i s pr e s e n t between the s u r f a c e s d u r i n g the t e r m i n a l stages of s t i c k , i n t e r s p e r s e d i n the f r e e volume between a s p e r i t y c o n t a c t s , i s c e r t a i n , but because the i n t e r f a c i a l gap i s g r e a t e r (area of c o n t a c t s m a l l e r ) d u r i n g s l i p than d u r i n g s t i c k , t h i s 86 residual volume of lubricant i s i n s u f f i c i e n t to form the viscous squeeze f i l m observed during s l i p . An additional volume of lubricant, greater than that to be expected i f the normal load was e n t i r e l y supported by s o l i d contact, c o l l e c t e d between the f r i c t i o n surfaces during the f i r s t h a l f - c y c l e of s l i p . The source of this additional lubricant i s no mystery, since the f r i c t i o n surface over which the s l i d e r moved was flooded with o i l . The source of the l i f t i n g forces experienced by the f l a t s l i d e r i s less obvious. Hydrodynamic l i f t , which nominally requires the existence of a convergent surface gap, would i n i t i a l l y appear to be inoperative, i n t h i s i n v e s t i -gation, as a r e s u l t of the carefully-maintained p a r a l l e l i s m of the s l i d e r and disc faces. Recall, though, that hydro-dynamic l i f t does exi s t |n just such cases, and i s c e r t a i n l y the most pla u s i b l e source of l i f t i n g forces under the experimental conditions of this study. If present during the f i r s t h a l f - c y c l e of s l i p , hydrodynamic l i f t must also have been present during the remainder of s l i p . The s t a b i l i t y of Figure 5.2.2, for a duration of two cycles, would suggest that the l i f t i s due to surface rugosity, and not to a transit o r y thermal expansion of the lubricant or short _term thermal deformation of the surfaces. Although i n s u f f i c i e n t to form a squeeze f i l m , the residual lubricant between the surfaces during s t i c k would be s u f f i c i e n t to generate hydrodynamic l i f t between closely approaching a s p e r i t i e s , since such l i f t generation 87 requires only lubricant i n the microscopic regions of con-vergency. Such l i f t would e f f e c t the entrance of more lubricant beneath the s l i d e r , allowing the continuation of the process to a state of equilibrium. Note also that the magnitude of f r i c t i o n values over the stable portion of the f r i c t i o n force trace was reduced i f the s l i d e r t r a v e l l e d a greater distance over the lower surface.before stable conditions were achieved, a fa c t compatible with the concept that hydrodynamic l i f t causes the establishment of an increasingly thicker f l u i d f i l m during the transient portion of s l i p . In order to further investigate t r a n s i t o r y character-i s t i c s of s l i p , t r a n s i t i o n curves recorded at a variety of lower surface v e l o c i t i e s were graphically compared. The curves were normalized with respect to the value of f r i c t i o n at which they met the stable k i n e t i c f r i c t i o n curve, i n order that only t r a n s i t o r y behaviour might be studied, and plotted as functions of time, r e l a t i v e v e l o c i t y , and displacement of the s l i d e r with respect to the lower surface. These p l o t s , which may be found i n Figures 5.2.3 to 5.2.5, suggested strongly that distance t r a v e l l e d by the s l i d e r was the dominant variable governing the rate of decay of the transitory k i n e t i c f r i c t i o n . Further confirmation of t h i s finding was provided by increas-ing the system's natural frequency of v i b r a t i o n to 152 rad/sec by removing the lead weights from the end of the beam. Un-desirable dynamic imbalance and reduction of v i b r a t i n g mass Figure 5.2.3 Normalized Transitory Kinetic F r i c t i o n vs Time (Steel-on-Steel, w, =102 rad/sec) Figure 5.2.4 Normalized Transitory Kinetic F r i c t i o n vs Relative V e l o c i t y m (Steel-on-Steel, co^ = 102 rad/sec) u> Figure 5.2.5 Normalized Transitory Kinetic F r i c t i o n vs Relative Displacement (Steel-on-Steel, cu, = 10 2 rad/sec) o concentration resulted i n imperfect phase lags between the instrumentation signals, as well as introducing the p o s s i b i l i t y of non-parallel dynamic orien t a t i o n of the f r i c t i o n surfaces, but the information gained provided valuable support for the findings of the graphical t r a n s i t o r y p l o t s . Comparison of the f r i c t i o n force curve (Fig. 5.2.6) with that presented i n F i g u r e 5.2.1 again indicates, despite small out-of-phase ef f e c t s evident i n the recording of a s l i g h t l y p o s i t i v e v e l o c i t y during the f i r s t quarter-cycle of s l i p , that there i s l i t t l e basis for considering the rate of decay of the t r a n s i t i o n to be d i r e c t l y governed by either v e l o c i t y or time. The recorded curves of f r i c t i o n force versus displacement, however, show s t a r t l i n g s i m i l a r i t y for the two natural f r e -quencies. This s i m i l a r i t y , together with the s i m i l a r i t y of the normalized f r i c t i o n curves of Figure 5.2.5, forces the assertion that the decay rate of the t r a n s i t o r y portion of the k i n e t i c f r i c t i o n curve i s apparently governed by the distance t r a v e l l e d by the s l i d e r over the lower surface. The distance t r a v e l l e d at the termination of the transient portion of the s l i p cycle i s many orders of magnitude greater than the size of any p r a c t i c a l asperity. The t r a n s i t i o n cannot, therefore, correspond to progressive fracture of the asperity contacts formed during s t i c k , as former investigators have suggested. Remember, however, that the s l i d e r and i t s supporting structure possess i n e r t i a i n the normal, as well as tangential, plane and that the s t r a i n during s t i c k i s such F i g u r e 5.2.6 Recorded T r a n s i e n t S l i p Traces ( S t e e l - o n - S t e e l ) vo to 93 t h a t the ce n t e r o f mass of the s l i d e r i s d i s p l a c e d toward the lower s u r f a c e . T h i s i n e r t i a , w i t h r e s p e c t t o pe r p e n d i c -u l a r movement, ensures t h a t although the a s p e r i t y j u n c t i o n s formed d u r i n g s t i c k might a l l have f r a c t u r e d , the area of c o n t a c t a t the i n c e p t i o n of s l i p cannot be i n s t a n t a n e o u s l y reduced, d e s p i t e the encouragement of such area r e d u c t i o n by the decreased t o t a l s u r f a c e s t r e s s and the growing f l u i d f i l m . U n f o r t u n a t e l y , the t r a n s i t i o n a l decay r a t e cannot be c o r r e l -ated w i t h time, as i t would be i f i n e r t i a normal to the s u r f a c e dominated the behaviour; removing the l e a d weights from the end of the beam a l t e r e d the system's moment of i n e r t i a f o r t h a t plane by l e s s than 3%. The i n t e r a c t i o n o f f o r c e s d u r i n g t h i s important p o r t i o n of the s l i p c y c l e i s o b v i o u s l y complex, and would be worthy of much c l o s e r a t t e n t i o n than the scope of t h i s study p e r m i t s . In summary, the s l i p p o r t i o n of the s t i c k - s l i p c y c l e begins when, d u r i n g s t i c k , the a p p l i e d t a n g e n t i a l shear s t r e s s exceeds the f r a c t u r e s t r e n g t h of the r e a l c o n t a c t a r e a . S l i p p r ogresses through a complex t r a n s i t o r y regime, c o r r e l a t a b l e with r e l a t i v e displacement of the s u r f a c e s , d u r i n g which the area o f c o n t a c t i s d r a s t i c a l l y reduced and, i n the presence of a l i q u i d l u b r i c a n t , a l o a d - b e a r i n g f l u i d f i l m i s e s t a b l i s h e d by hydrodynamic a c t i o n on a m i c r o s c o p i c s c a l e . The t r a n s i t o r y s l i p c o n d i t i o n decays t o a s t a b l e s t a t e of s l i p wherein the normal l o a d i s supported p a r t i a l l y by s o l i d c o n t a c t , p a r t i a l l y by the f l u i d f i l m . Because e f f e c t i v e s u r f a c e s e p a r a t i o n i s of the order of the height of a surface asperity, squeeze f i l m e f f e c t s assure the persistence of th i s f l u i d f i l m f or the short periods (milliseconds) when hydrodynamic l i f t i s ne g l i g i b l e due to inadequate r e l a t i v e v e l o c i t y . The s l i p portion of a vibratory cycle terminates at the f i r s t matching of surface v e l o c i t i e s a f t e r f r i c t i o n a l energy d i s s i p a t i o n has reduced the corresponding displacement, whether p o s i t i v e or negative, s u f f i c i e n t l y that the sum of the forces favouring reattachment of the surfaces exceeds the e l a s t i c restoration force. Upon reattachment of the surfaces another period of s t i c k commences. If the v e l o c i t y of the driven surface i s s u f f i c i e n t l y great that the forces favouring reattachment do not exceed the e l a s t i c restoration force before the maximum pos i t i v e v e l o c i t y of the o s c i l l a t i n g surface becomes less than the driven surface v e l o c i t y , reattachment cannot occur, and s t i c k - s l i p o s c i l l a t i o n disappears. Non-oscillatory s l i d i n g of the elastically-mounted surface, at a fixed displacement proportional to the magnitude of the k i n e t i c t r a c t i o n forces, ensues. b. Quasi-Harmonic O s c i l l a t i o n S l i p and quasi-harmonic motion are d i s t i n c t i v e e n t i t i e s because, although the f r i c t i o n forces encountered during s l i p are e n t i r e l y d i s s i p a t i v e , dynamic f r i c t i o n forces causing quasi-harmonic o s c i l l a t i o n are not. The "humped" form of f r i c t i o n force vs v e l o c i t y curve reported by Ko [15] to be c h a r a c t e r i s t i c of quasi-harmonic o s c i l l a t i o n , one form being 95 that shown i n Figure 2.1.3, may be energy-additive, energy-d i s s i p a t i v e , or both (Appendix I I ) , depending on the v e l o c i t y of the driven surface. An example of a humped f r i c t i o n curve, together with a generated behavioural trace, i s presented on the phase plane i n Figure 5.2.7. Inherent i n the balancing of additive and d i s s i p a t i v e energy effects to achieve a stable, quasi-harmonic l i m i t cycle i s the necessity for the zero axis of absolute v e l o c i t y of the driven surface to f a l l i n the immediate v i c i n i t y of the "hump" of the curve. If this hump and the zero v e l o c i t y axis are too widely separated the d i s s i p a t i v e and additive energy e f f e c t s w i l l not balance, negating the achievement of a stable l i m i t cycle. A related point of sign i f i c a n c e i s that the p o s i t i v e l y and negatively sloped portions of the f r i c t i o n - v e l o c i t y curve need not have the r e l a t i v e positions i l l u s t r a t e d i n Figure 3.3.1 for achievement of a stable l i m i t cycle. I f , instead, the f r i c t i o n - v e l o c i t y curve has a minimum, rather than maximum, value i n the v i c i n i t y of the zero v e l o c i t y axis, due to reversal of the r e l a t i v e positions of the sloped portions of the curve, a stable l i m i t cycle i s s t i l l achieved. The sole c r i t e r i o n for the endurance of such a l i m i t cycle i s that the cycle must encompass balanced proportions of energy-additive and energy-dissipative zones on the f r i c t i o n - v e l o c i t y curve. 96 F i g u r e 5.2.7 One Form of Quasi-Harmonic F r i c t i o n F orce Phase Plane Trace, w i t h Generated D i s p l a c e -ment B e h a v i o u r a l Curve 97 Ko has proven t h a t the form of the f r i c t i o n f o r c e vs v e l o c i t y curve recorded d u r i n g quasi-harmonic o s c i l l a t i o n may be e i t h e r convex upward or convex downward. This b a s i c change i n f r i c t i o n curve p r o f i l e was accomplished simply by changing l u b r i c a n t s . Other r e s e a r c h e r s have found t h a t s i m i l a r d r a s t i c changes can be made i n the n o n - v i b r a t p r y boundary f r i c t i o n f o r c e vs v e l o c i t y curve simply by changing the a d d i t i v e s i n an otherwise homogeneous base o i l [35,36]. I t i s of importance t h a t h i g h l y - r e f i n e d m i n e r a l o i l has been found to e x h i b i t only a convex upward primary i n f l e c t i o n , s i m i l a r to the example o f F i g u r e 5.2.7, both i n the work of Ko and i n the present study. The i n v e r t e d , convex downward form of the f r i c t i o n f o r c e vs v e l o c i t y curve would t h e r e f o r e appear to r e s u l t from the presence of o i l a d d i t i v e s . A r e p r e s e n t a t i v e s e t of quasi-harmonic displacement and f r i c t i o n f o r c e phase plane t r a c e s , recorded d u r i n g the p r e s e n t study, may be found i n F i g u r e 5.2.8. Included are zero r e f e r e n c e l i n e s f o r displacement and f o r c e , s i n c e l o c a t -i n g zero f o r both curves a t the o r i g i n of the o s c i l l o s c o p e g r i d , as was done f o r a l l s t i c k - s l i p r e c o r d i n g s , would have r e s u l t e d i n a p a r t i a l o v e r l a y of the t r a c e s . The s t r i k i n g resemblance of the quasi-harmonic f r i c t i o n f o r c e curve to Crook's p l o t s of elastohydrodynamic 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 versus v e l o c i t y ( F i g . 3.2.3) i s immediately apparent. T h i s resemblance i s h a r d l y s t a r t l i n g ; the l u b r i c a n t s were s i m i l a r , and l o c a l p r e s s u r e s a t the l o a d - b e a r i n g extrem-98 F i g u r e 5.2.8 Recorded Quasi-Harmonic Phase Plane Traces ( S t e e l - o n - S t e e l , 50 mv/div, to , = 102 rad/ s e c , v = 5.3 i n / s e c ) i t i e s of the quasi-harmonic boundary f r i c t i o n surfaces were of the same order as the f i l m pressures of Crook's experiments. Important differences do, however, e x i s t i n the r e s u l t s . The near-horizontal portion of the quasi-harmonic f r i c t i o n force curve corresponds to a 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 of approximately 0.15, with the force curve maximum occurring at a r e l a t i v e surface v e l o c i t y of 1 in/sec. The near-horizontal portions of the elastohydrodynamic k i n e t i c f r i c t i o n curves have an approximate c o e f f i c i e n t magnitude of 0.03, with maxima occurring at 20 in/sec. The most s i g n i f i c a n t of the differences between the f r i c t i o n curves of the two studies i s t h e i r c o e f f i c i e n t magnitude r a t i o of 5. The elastohydrodynamic f r i c t i o n curves of Crook have magnitudes which suggest l i t t l e , i f any, s o l i d contact between opposing surfaces. Hydrodynamic l u b r i c a t i o n i s also apparent i n the way that the f r i c t i o n forces approach zero with r e l a t i v e v e l o c i t y . These hydrodynamic conditions resulted from the maintenance of high surface v e l o c i t y , during a l l t e s t s , on at lea s t one surface, i n conjunction with the obvious l i q u i d wedge e f f e c t of p a r a l l e l cylinders with surface v e l o c i t i e s , i n the load-bearing region, of the same sense. The magnitude of the quasi-harmonic f r i c t i o n c o e f f i c i e n t s , however, together with the d i s t i n c t l y p o s i t i v e f r i c t i o n force as r e l a t i v e v e l o c i t y approaches zero [14], indicates that the majority of the load i s supported by s o l i d contact during this form of vib r a t i o n , and that the e f f e c t i v e minimum clearance 100 h . f o r quasi-harmonic o s c i l l a t i o n i s much l e s s than i t s min ^ elastohydrodynamic c o u n t e r p a r t . The v a r i a t i o n i n the q u a s i -harmonic 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 would appear to be the r e s u l t o f a minor p o r t i o n of elastohydrodynamic a c t i o n superimposed on the dominant s o l i d c o n t a c t e f f e c t s . A t any g i v e n v e l o c i t y , an i n c r e a s e i n the c o e f f i c i e n t of f r i c t i o n r e s u l t s i n g r e a t e r heat g e n e r a t i o n . How much the l u b r i c a n t v i s c o s i t y i s a f f e c t e d by the i n c r e a s e d r a t e o f heat g e n e r a t i o n depends g r e a t l y on the a b i l i t y o f the f r i c t i o n s u r f a c e s to conduct the heat away from the l u b r i c a n t , but one can c o n f i d e n t l y s t a t e t h a t , s i n c e the maximum of an e l a s t o -hydrodynamic f r i c t i o n curve i s f i x e d by v i s c o u s temperature e f f e c t s , t h i s maximum w i l l occur a t a lower s u r f a c e v e l o c i t y f o r i n c r e a s e d f r i c t i o n c o e f f i c i e n t magnitudes. F r i c t i o n f o r c e s r e s u l t i n g from simultaneous v i s c o u s and m e t a l l i c sources are much h i g h e r , w i t h p r o p o r t i o n a t e l y g r e a t e r heat g e n e r a t i o n , than those caused by v i s c o u s sources a l o n e . The occurrence of the f r i c t i o n f o r c e peak a t a s u r f a c e v e l o c i t y , i n q u a s i -harmonic o s c i l l a t i o n , one-twentieth t h a t a t which i t o c c u r r e d i n Crook's i n v e s t i g a t i o n i s thus c o n s i s t e n t w i t h the concept t h a t the quasi-harmonic f r i c t i o n curve i s the r e s u l t of superimposed elastohydrodynamic and m e t a l l i c c o n t a c t e f f e c t s . To summarize, i t i s p o s s i b l e t h a t the form of f r i c t i o n f o r c e vs v e l o c i t y r e l a t i o n s h i p found to cause quasi-harmonic o s c i l l a t i o n when u s i n g h i g h l y r e f i n e d m i n e r a l o i l s as l u b r i -cants r e s u l t s from the s u p e r p o s i t i o n o f s o l i d c o n t a c t and > 101 elastohydrodynamic f r i c t i o n f o r c e vs v e l o c i t y c h a r a c t e r i s t i c curves to y i e l d the r e q u i r e d "humped" form of f r i c t i o n a l behaviour, w i t h the maximum de s i g n a t e d as the "hump" a t t r i -b u t a b l e , i n the absence o f chemical l u b r i c a n t a d d i t i v e s , to v a r i a t i o n i n l u b r i c a n t v i s c o s i t y as a consequence of thermal s e n s i t i v i t y . c. Quasi-Steady V i s c o u s Thermal E f f e c t s The s i m i l a r i t i e s i n the observed f r i c t i o n a l c h a r a c t e r -i s t i c s o f elastohydrodynamic l u b r i c a t i o n and the v i s c o u s c o n t r i b u t i o n to quasi-harmonic behaviour have been o u t l i n e d , and the observed d i f f e r e n c e s r e s o l v e d by c o n s i d e r a t i o n o f the e f f e c t t h a t superimposed m e t a l l i c s o l i d c o n t a c t would have on the elastohydrodynamic f r i c t i o n curve. One apparent anomaly i s y e t unexplained. I f , indeed, the maximum of the q u a s i -harmonic f r i c t i o n f o r c e curve, which occurs a t a r e l a t i v e v e l o c i t y o f 1 i n / s e c , i s caused by thermal v a r i a t i o n o f l u b r i c a n t v i s c o s i t y , as i s t h a t of elastohydrodynamic l u b r i -c a t i o n , why i s th e r e no s i m i l a r maximum i n the s t a b l e f r i c t i o n curve of s l i p , d u r i n g which s u r f a c e v e l o c i t i e s g r e a t l y exceed 1 i n / s e c ? Comparison of the r e s u l t s d i s p l a y e d i n F i g u r e s 5.2.1, 5.2.2, and 5.2.8 r e v e a l s t h a t , as the lower s u r f a c e v e l o c i t y v, the average r e l a t i v e speed of the two f r i c t i o n s u r f a c e s , i s i n c r e a s e d , n o n - t r a n s i e n t f r i c t i o n c o e f f i c i e n t magnitudes i n c r e a s e from values s u g g e s t i n g l i t t l e s o l i d c o n t a c t t o valu e s i n d i c a t i n g predominant m e t a l l i c c o n t a c t . For the p h y s i c a l 102 system of t h i s i n v e s t i g a t i o n the t r e n d means p r o g r e s s i v e l y l e s s v i s c o u s a c t i o n w i t h i n c r e a s i n g average speed. The reason f o r t h i s e f f e c t might be found i n the work of Bowden and R i d l e r [4] and of Jaeger [ 3 7 ] , who has performed a thermo-dynamic a n a l y s i s o f the s u r f a c e temperatures to be expected a t a s p e r i t y c o n t a c t s under k i n e t i c c o n d i t i o n s . Jaeger's a n a l y s i s , which ign o r e d the thermal c a p a c i t y o f any l u b r i c a n t p r e sent, y i e l d e d the r e s u l t y k v N s g b i . 9 J A 1 / 2 [ i . i s2b + 0.7 e ^ / V 2 ] (5.2.1) where N g = normal l o a d c a r r i e d by s o l i d c o n t a c t g = a c c e l e r a t i o n of g r a v i t y T^ ~ temperature a t i n f i n i t y J = work e q u i v a l e n t of heat ^ 1 ' ^2 = thermal c o n d u c t i v i t i e s of s l i d e r and lower s u r f a c e , r e s p e c t i v e l y b = a r a t i o of m a t e r i a l c onstants o f the s l i d e r . For a g i v e n p a i r o f f r i c t i o n s u r f a c e s t h i s e q u a t i o n may be reduced to XT V 2 ^k V N s T - T = K . (5.2.2) 1/2... 1/4 a + v ' N ' OO s 103 Jaeger's r e s u l t s , derived for non-oscillatory s l i d i n g con-d i t i o n s , may not be d i r e c t l y applied to the present study, but general trends would s t i l l be analagous. Note that, for the present study, the average temperature of an asperity contact would therefore have increased with both driven surface v e l o c i t y and the proportion of normal load supported by s o l i d contact, which i t s e l f increased with driven surface v e l o c i t y . Peak temperatures achieved can be very high [4,37]; the average temperature of the continuously-exposed upper surface, and with i t the average temperature of the lubricant i n the load-bearing region, could increase s u b s t a n t i a l l y with driven surface v e l o c i t y , providing an explanation for the observed quasi-steady v i s c o s i t y decreases. Consider now the e f f e c t of increased surface tempera-tures on the elastohydrodynamic f r i c t i o n a l t r a c t i o n curve (Figure 3.2.2). The conditions for which Crook's a n a l y t i c a l expression, equation 3.2.4, was derived are not s a t i s f i e d i n t h i s i n vestigation, preventing an appeal to that established r e s u l t . Heat was not generated exclusively i n the lubricant, as i n the elastohydrodynamic case; the lubricant may even have contributed to the pooling, rather than heating, of the s o l i d surfaces. One can only again point out that the decrease i n elastohydrodynamic f r i c t i o n force with increasing r e l a t i v e surface v e l o c i t i e s i s attributed to a thermal decrease i n lubricant v i s c o s i t y . That the maximum of an elastohydrodynamic f r i c t i o n force curve would occur at a 104 lower s u r f a c e v e l o c i t y f o r i n c r e a s e d o v e r a l l o p e r a t i n g temperatures t h e r e f o r e o f f e r s an e x p l a n a t i o n f o r the appear-ance of such a maximum onl y w i t h the h i g h d r i v e n s u r f a c e v e l o c i t i e s of quasi-harmonic o s c i l l a t i o n . C H A P T E R VI V I . CONCLUSION The i n t e r e s t of the p r e s e n t study was the i n f l u e n c e of r a t e e f f e c t s on s t a t i c and dynamic boundary f r i c t i o n . An equation p r e d i c t i n g the v a r i a t i o n i n the observed c o e f f i c i e n t of s t a t i c f r i c t i o n w i t h l o a d r a t e was developed from con-s i d e r a t i o n o f p l a s t i c flow, and the p r e d i c t e d behaviour com-pared to e x p e r i m e n t a l l y - o b t a i n e d data. The dynamic f r i c t i o n a l a c t i o n encountered d u r i n g quasi-harmonic o s c i l l a t i o n and du r i n g the s l i p p o r t i o n o f the s t i c k - s l i p v i b r a t i o n c y c l e was recor d e d and analyzed. In every case r a t e e f f e c t s were found to determine o r profoundly i n f l u e n c e the observed f r i c t i o n a l behaviour. In s p e c i f i c d e t a i l , the f o l l o w i n g c o n c l u s i o n s may be l i s t e d : 1. The assumption of a p l a s t i c deformation model, f o r the growth i n c o n t a c t area of opposing s u r f a c e s s u b j e c t to normal and t a n g e n t i a l l o a d i n g , p e r m i t t e d development of an e q u a t i o n p r e d i c t i n g the v a r i a t i o n of the c o e f f i c i e n t of s t a t i c f r i c t i o n w i t h r a t e o f s t r e s s a p p l i c a t i o n . W i t h i n the l i m i t s of s c a t t e r of experimental data, the p r o f i l e of the equation's p l o t t e d curve matches the p r o f i l e of the p l o t t e d d ata. The c o m p a t i b i l i t y of these p r o f i l e s suggests s t r o n g l y t h a t p l a s t i c deformation i s indeed the governing mechanism of c o n t a c t area growth between metal s u r f a c e s . 107 2. The e x i s t e n c e of an upper asymptote f o r s t a t i c f r i c t i o n o f m e t a l l i c s u r f a c e s f a t slow r a t e s of l o a d i n g and i n the presence of a l u b r i c a n t , has been proven. 3. The k i n e t i c f r i c t i o n f o r c e vs v e l o c i t y curve f o r s t i c k -s l i p o s c i l l a t i o n o f m e t a l l i c s u r f a c e s has been proven to c o n s i s t of two regimes, t r a n s i e n t and s t e a d y - s t a t e . 4 . Decay o f the t r a n s i e n t p o r t i o n of the s t i c k - s l i p f r i c t i o n vs v e l o c i t y curve to the s t e a d y - s t a t e k i n e t i c f r i c t i o n curve appears to be governed by the d i s t a n c e t r a v e l l e d , subsequent to the i n c e p t i o n of s l i p , by one s u r f a c e over the o t h e r . 5. The s t e a d y - s t a t e p o r t i o n of the k i n e t i c f r i c t i o n curve f o r s l i p e x h i b i t s d e f i n i t e c h a r a c t e r i s t i c s o f both v i s c o u s and m e t a l l i c s o l i d - c o n t a c t behaviour. 6 . The v i s c o u s c o n t r i b u t i o n to the dynamic f r i c t i o n a l behaviour became i n c r e a s i n g l y s m a l l e r as the average r e l a t i v e v e l o c i t y of the f r i c t i o n s u r f a c e s i n c r e a s e d , a p p a r e n t l y because the temperature of the l u b r i c a n t i n the l o a d - b e a r i n g i n t e r f a c i a l r e g i o n i n c r e a s e d , and i t s v i s c o s i t y decreased, as average s u r f a c e v e l o c i t y i n c r e a s e d . 7. Comparison of r e c o r d e d k i n e t i c f r i c t i o n vs r e l a t i v e v e l o c i t y curves from quasi-harmonic o s c i l l a t i o n and elastohydrodynamic l u b r i c a t i o n i n v e s t i g a t i o n s i n d i c a t e s 108 t h a t the observed f r i c t i o n a l behaviour c a u s i n g q u a s i -harmonic o s c i l l a t i o n i s the r e s u l t of superimposed m e t a l l i c s o l i d - c o n t a c t and elastohydrodynamic e f f e c t s . 8. N o n - o s c i l l a t o r y s l i p and quasi-harmonic o s c i l l a t i o n would t h e r e f o r e , i n a p h y s i c a l system, appear to be d i s t i n c t e n t i t i e s only because, i n the case of q u a s i -harmonic o s c i l l a t i o n , instantaneous v i s c o u s thermal e f f e c t s are s u f f i c i e n t l y severe to cause a "humped" form of f r i c t i o n f o r c e vs r e l a t i v e v e l o c i t y curve. A P P E N D I C E S APPENDIX I SYSTEM PARAMETERS A l . 1 System S t i f f n e s s For the composite beam of F i g u r e A l . l , which a p p r o x i -mates the c a n t i l e v e r beam employed i n the experimental system, d e f l e c t i o n and s l o p e a t p t . a due to a f o r c e P a c t i n g a t p t . b are [38] 1 P L 1 3 E l H + 2 V l 1 P L 1 E 1 I 1 3 ^2 2 L, 1 P L 1 2 E 1 I 1 P L 2 L 1  E 1 I 1 1 P L 1 3 E 1 I 1 L 2 2 L l L 2 D e f l e c t i o n a t p t . b due t o a f o r c e P a c t i n g a t p t . b i s , f o r small d e f l e c t i o n s , 6 + a 1 P L 2 9 a L 2 + 3 ETT 2 2 1 P L 1 3 E l I x 2 i J2. 2 J l J 1 _ 3 E P L ; 2 I2 I l l * b < i b * m \ 1 L 2 ^ L, - rtf E, = 30 x IO6 % 2 I. = 1.30 x IO"3 In4 L 2 = 53/8" E 2 10 x |0 6 l b/ i n2 I2 = 3.38 x id" 2 In4 m. 0.48 lb g m 1.90 lb 9 F i g u r e A l . l A p p r o x i m a t i o n o f C o m p o s i t e B e a m E m p l o y e d f o r A n a l y s i s o f B e a m P r o p e r t i e s 112 The s t i f f n e s s of the composite cantilever beam was therefore estimated to be k = ^- = L 3 1 L l 3 E ^ 1 + 3 ~ + 3 -L l L 2 i 'A 2 3 E„I 2 2 -1 64 ±*> i n The undamped natural frequency of the system may be estimated from the equation [39] n - 1 3 3ra. k m 2 1 - b 280 m 110 rad sec Measured values of system s t i f f n e s s and frequency of free v i b r a t i o n were, respectively, 59.2 lb i n 102 rad sec Al.2 System Damping The equation of motion for a l i n e a r system i n free v i b r a t i o n i s Mx + r x + k x = 0 113 A solution for this d i f f e r e n t i a l equation i s [40] x = x„ e s i n (oo,t + 0) 0 d . where x^ i s the i n i t i a l condition. If the time axis i s oriented such that maximum positi v e displacement peaks occur TT when (w^t + 0) = (1 + 4 n) , then for these displacement peaks where T i s the period of one cycle of v i b r a t i o n , and conse-quently 2M , X0 r = In — nT x n Measurement of vi b r a t i o n amplitudes several cycles apart w i l l therefore y i e l d the value of the lin e a r damping c o e f f i c i e n t , but the value of M, the equivalent mass of the system, must f i r s t be determined by use of the equation Combination of the two previous equations y i e l d s 114 M = 1 I n X ° HT l n X~ n T2 enabling the determination of M d i r e c t l y , since a l l variables on the right-hand side of the equation were found by measure-ments performed on the system. Recorded curves of displacement versus time were found to be only s l i g h t l y non-linear, with an averaged decay c o e f f i c i e n t 1 x0 -1 — In — = 0.87 sec nT x n Substitution i n the previous equations y i e l d s M 5.70 x 10 -3 lb-sec" 2.20 lb xn o no i n ~ 2 l b - S e C r = 0.99 x 10 i •. in For purposes of comparison, the c r i t i c a l damping c o e f f i c i e n t and the undamped natural frequency of the vibratory system are, respectively, = 2 (k M , , , lb-sec l.ib : , in to n ]j M 102 rad sec APPENDIX I I PHASE PLANE ANALYSIS OF VIBRATORY MOTION The p o p u l a r l y - d e s i g n a t e d "phase p l a n e " i s a p l o t o f a d i f f e r e n t i a l equation's z e r o t h d e r i v a t i v e of the dependent v a r i a b l e versus the f i r s t d e r i v a t i v e o f the dependent v a r i a b l e . . The c h i e f advantage of employing such a p l o t i s t h a t i n the case of autonomous second-order d i f f e r e n t i a l e q u a t i o ns, i n which the independent v a r i a b l e found i n the. denominator of a d e r i v a t i v e (eg. time) appears o n l y i n the denominator of a d e r i v a t i v e , t h i s independent v a r i a b l e i s not e x p l i c i t l y expressed on the phase plane; r a t h e r , the i n t e r a c t i o n of the z e r o t h and f i r s t d e r i v a t i v e s of the dependent v a r i a b l e i s e x p l o r e d . A l e s s e r b e n e f i t d e r i v e d i s t h a t c o n v e r s i o n of a second-order equation t o phase plane form reduces i t to s i m p l e r , f i r s t - o r d e r , form. Consider, f o r example, the d i f f e r e n t i a l equation governing the motion of a mass r e s t r a i n e d by an i d e a l s p r i n g . M x + k x = (J Conversion of the eq u a t i o n to phase plane form i s accomplished through use of the f o l l o w i n g i d e n t i t y . 116 d , • > dx dx • dx X = d t ( x ) = dx" d t = X dx" S u b s t i t u t i o n i n the equation of motion y i e l d s dx + k x _ Q dx M x which may be i n t e g r a t e d by s e p a r a t i o n o f v a r i a b l e s to o b t a i n the s o l u t i o n 2 , M • 2 . . x + j- x = cons t a n t k where the v a l u e o f the constant i s determined by i n i t i a l c o n d i t i o n s . T h i s s o l u t i o n , i f p l o t t e d on x and x co-r k i V 2 M o r d i n a t e s , i s o b v i o u s l y an e l l i p s e . However, s i n c e d e f i n e s the n a t u r a l frequency o f v i b r a t i o n o f the system, the s o l u t i o n becomes a c i r c l e when p l o t t e d on the normalized c o - o r d i n a t e s x and — n I n c l u s i o n of a v i s c o u s damping term i n the equ a t i o n of motion g i v e s i t the form M x + r x + k x = 0 , which when converted to phase plane form becomes 117 with the viscous damping c o e f f i c i e n t serving as a negative forcing function. The solution of t h i s equation i s a c i r c u l a r , decaying s p i r a l when plotted on normalized phase plane co-ordinates. Of p a r t i c u l a r i n t e r e s t i s the l i n e c a l l e d the "zero i s o c l i n e " , defined by the simultaneous v a l i d i t y of the phase plane form of the equation of motion and the condition $k = o dx U For the case of free v i b r a t i o n with viscous damping the zero i s o c l i n e i s a s t r a i g h t l i n e through the o r i g i n , with slope r when plotted on the normalized phase plane. Generally the zero i s o c l i n e i s a l i n e which i s neither st r a i g h t nor passing through the o r i g i n . Its importance l i e s i n the f a c t that the phase plane behavioural curve described by the equation of motion can be r e a d i l y generated, by graphical means, from the zero i s o c l i n e [41]. An example of a phase plane curve, for the case of free v i b r a t i o n with viscous damping, generated graphically from the zero i s o c l i n e , i s shown i n Figure A2.1. A l i n e drawn through the i n t e r s e c t i o n of the zero• i s o c l i n e and the zero v e l o c i t y axis, perpendicular to the zero v e l o c i t y axis, subdivides the phase plane into energy-additive and energy-dissipative quadrants. For the axes presented i n t h i s study, any portion of the zero i s o c l i n e F i g u r e A2.1 Viscously-Damped Free V i b r a t i o n on Pha Plane 119 i n the r i g h t upper or l e f t lower e n e r g y - t r a n s f e r quadrants i s d i s s i p a t i v e ; any p o r t i o n of the zero i s o c l i n e i n the l e f t upper or r i g h t lower quadrants i s e n e r g y - a d d i t i v e . The phase plane example of F i g u r e A2.1 d i s p l a y s a zero i s o c l i n e which i s e n t i r e l y e n e r g y - d i s s i p a t i v e , because t h i s i s o c l i n e i s t o t a l l y c o n t a i n e d i n the upper r i g h t and lower l e f t energy-t r a n s f e r quadrants. For the p r e s e n t study, as c o n s u l t a t i o n of e quation 4.2.3 w i l l show, the z e r p i s o c l i n e was the graph of the f r i c t i o n f o r c e e x p e r i e n c e d by the s l i d e r a t the f r i c t i o n i n t e r f a c e , d i v i d e d by the system s t i f f n e s s , p l o t t e d as a f u n c t i o n of — . Simultaneous r e c o r d i n g of the zero n i s o c l i n e and the phase plane b e h a v i o u r a l t r a c e t h e r e f o r e p e r m i t t e d the v a l i d i t y of the r e c o r d e d f r i c t i o n curves to be t e s t e d ( F i g . A2.2). More important, r e c o r d i n g data i n phase plane form has the advantage t h a t the s l i p p o r t i o n of a s t i c k - s l i p c y c l e has c o n s i s t e n t l y e x c e l l e n t r e s o l u t i o n of d e t a i l on the phase pl a n e . As a consequence of t e c h n i c a l d i f f i c u l t i e s , such r e s o l u t i o n i s v i r t u a l l y u n a t t a i n a b l e i n the time domain. F i g u r e A2.2 Comparison of Recorded and G r a p h i c a l l y Generated Phase Plane B e h a v i o u r a l Traces to o APPENDIX I I I VISCOUS SQUEEZE FILM ANALYSIS Consider the g e n e r a l form of Reynolds' equation, 3_ 3x h 33p n 3x 9_ 3z h 39p ri 3z , r t 3 h . , , 9U . , 0 dh 6 U ~— + 6 h T T — + 12 - T - r -9x 9x d t where the c o - o r d i n a t e s are as d e f i n e d i n F i g u r e A3.1. For p a r a l l e l s u r f a c e s separated by a c o n s t a n t - v i s c o s i t y f l u i d , so t h a t h and n are independent o f x and z, a 2 a 2 9x 2 9 z 2 12n dh , 3 d t For convenience, s i n c e i n t h i s stuay a c i r c u l a r s l i d e r f a ce 2 was employed, the L a p l a c i a n o p e r a t o r V may be expressed i n c y l i n d r i c a l c o - o r d i n a t e s , so t h a t the e q u a t i o n i n orthogonal components i s a l t e r e d to 1 9_ P 9P i E 9p 2 S O 2 p 90 i 2 9y 12n dh , 3 d t 3 2 9 2p The q u a n t i t i e s — a n d — § • may be equated to zero from con-30^ 9y^ s i d e r a t i o n s of symmetry and t h i n - f i l m approximations, respec-122 F i g u r e A3.1 Squeeze F i l m A n a l y s i s Co-ordinates 123 t i v e l y , leaving the r e l a t i o n s h i p 1 d_ P dp dp 12n dh h 3 dt Integrating twice, and solving for the boundary conditions p = 0 , a ? - ». p = R p = 0 y i e l d s 3n , 2 n2. dh P = J T ( p ~ dt Integrating the pressure d i s t r i b u t i o n over the surface area of the s l i d e r r esults i n an expression for the normal load supported by f l u i d pressure, N, p=0 p=R p • 2irp dp 3 ^ n 4 1 dh 2 i r n R * 73 dt h Note that the form of t h i s expression i s such that, for a constant normal load, the rate of decrease of the f i l m thickness diminishes with the t h i r d power of the f i l m thick-ness. For plane s u r f a c e s the f i l m t h i c k n e s s can be reduced to zero o n l y as time or normal l o a d assumes an i n f i n i t e v a l u e . Real s u r f a c e s , which are not p l a n a r , cannot reduce the mean f i l m t h i c k n e s s to zero, but the l o a d - b e a r i n g c a p a c i t y of the f i l m i s s t i l l governed by the e x p r e s s i o n f o r i f the p l a n a r f i l m t h i c k n e s s h i s r e p l a c e d by i t s e f f e c t i v e c o u n t e r p a r t . APPENDIX IV CALIBRATION AND SCALING OF DISPLACEMENT, VELOCITY, AND FRICTION FORCE SIGNALS A4.1 Scaling of Displacement and Velocit y Signals The accelerometer employed was a commercial unit with a fixed output voltage equal to 0.1 vol t s per u n i t gravity. This output, which was re a d i l y checked at any time, was used as a standard c a l i b r a t i o n reference for the displacement and ve l o c i t y s i g n a l s . The displacement signal was scaled simply by subject-ing the cantilever beam, complete with a l l instrumentation, s l i d e r , and s l i d e r mount, to conditions of free v i b r a t i o n , and adjusting the magnitude of the displacement signal u n t i l , at zero v e l o c i t y , the sum of the displacement and acceleration signals equaled zero. Because the viscous damping e f f e c t was so small, t h i s procedure resulted i n the summed signal taking the form of a str a i g h t l i n e whose slope, on the face of an oscilloscope, was indistinguishable from that of the zero displacement. Upon achieving the proper magnitude of displacement si g n a l , the v e l o c i t y signal strength was adjusted u n t i l the system's phase plane trace, as recorded by an oscilloscope, was a c i r c u l a r , as opposed to e l l i p s o i d a l , s p i r a l . That the 126 t r a c e was a m u l t i - c y c l e s p i r a l , r a t h e r than a r e p e t i t i v e loop, i s due to the s m a l l amount of n e a r - l i n e a r v i s c o u s damping p r e s e n t under c o n d i t i o n s o f f r e e v i b r a t i o n . A4.2 C a l i b r a t i o n of Displacement, V e l o c i t y , and F r i c t i o n  Force S i g n a l s A f t e r the displacement s i g n a l was p r o p e r l y s c a l e d a depth micrometer was r i g i d l y mounted w i t h i t s s p i n d l e i n the h o r i z o n t a l plane and p e r p e n d i c u l a r to the s i d e of the s l i d e r mount. V o l t a g e output from the s t r a i n gage b r i d g e was p l o t t e d a g a i n s t displacements imposed at the s l i d e r mount by the micrometer to o b t a i n a c a l i b r a t i o n curve ( F i g . A4.1). De-f l e c t i o n s r e c o r d e d on the o s c i l l o s c o p e were then r e a d i l y converted to s l i d e r d i s placements. V e l o c i t y c a l i b r a t i o n was i n h e r e n t i n the adjustment of v e l o c i t y s i g n a l s t r e n g t h to achieve a c i r c u l a r form of phase plane t r a c e i n f r e e v i b r a t i o n . One l i n e a r u n i t on the " v e l o c i t y " a x i s i s then equal to the displacement r e p r e -sented by t h a t u n i t on the displacement a x i s m u l t i p l i e d by the n a t u r a l frequency of v i b r a t i o n of the system, w . C a l i b r a t i o n of the o s c i l l o s c o p e f r i c t i o n f o r c e t r a c e was a simple matter of m u l t i p l y i n g the o s c i l l o s c o p e d i s -placement c a l i b r a t i o n by the system s t i f f n e s s k. 127 Figure A4.1 C a l i b r a t i o n Curves for Composite Beam and Scaled Displacement Amplifier APPENDIX V FRICTION SURFACE PARAMETERS Surface Roughness Surface Roughness Along D i r e c t i o n a l Marks Roughness Across Di r e c t i o n a l Marks Lapped Surface Measurements Steel S l i d e r 18 y i n . CLA Steel Disc 18 y i n . CLA Post-Test Measurements Steel S l i d e r 1-l^j- yin. CLA 3 y i n . CLA Steel Disc 13 y i n . CLA 14 y i n . CLA (Measurements made with Talysurf 4 equipment) 129 Surface Composition and Hardness Composition and C o n d i t i o n I n d e n t a t i o n Hardness ( ~ n d i a . b a l l ) S t e e l D i s c m i l d s t e e l , AISI C1020 f u l l y annealed B r i n e l l 120 S t e e l S l i d e r a l l o y s t e e l (C 1.05%, Mn 0.20%, S i 0.20%) as d e l i v e r e d B r i n e l l 235 Brass S l i d e r unleaded brass (Cu 70%, Zn 30%) as d e l i v e r e d B r i n e l l 165 R E F E R E N C E S REFERENCES Thomas, S. "Vibrations Damped by S o l i d F r i c t i o n " , The  Philosophical Magazine, Series 7, Vol. 9, p. 329, 1930. Papenhuyzen, P.J. "Wrijvings Proeven i n Verbandmet het Slippen Van Autobanden", De Ingenieur, 53, p. 75, 1938. Bowden, F.P. and Leben, L. "The Nature of S l i d i n g and the Analysis of F r i c t i o n " , Proceedings of the Royal  Society (London), A169, p. 371, 1939. Bowden, F.P. and Ridler, K.E.W. 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