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Friction induced vibration Cameron, Roderick 1963

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FRICTION INDUCED VIBRATION  by RODERICK CAMERON Associate of the Royal College of Science,  Glasgow, 1959  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering  We accept t h i s thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA JUNE, 1963  i In presenting t h i s thesis i n . p a r t i a l fulfilment of the requirements for an advanced degree-at the University, of B r i t i s h Columbia, I.agree that the L i b r a r y . s h a l l make i t freely, available•for-reference-and study. I further agree that permission for extensive copying of t h i s thesis for scholarly, purposes, may "be granted-by. the'Head of my Department or b y . h i s representatives.  It i s understood that copying .or-publications  of t h i s thesis for f i n a n c i a l gain s h a l l not be-allowed-without my written permission.  Department of Mechanical Engineering, The University, of B r i t i s h Columbia, Vancouver 8, B . C .  June, 1963.  v  ii  ABSTRACT  F r i c t i o n a l vibrations have been>induced-in-a-system having ah . e l a s t i c a l l y suspended,.and viscously damped s l i d e r loaded onto a surface driven-at constant v e l o c i t y .  Exact mathematical analysis-of  the system reveals a unique value of the driven surface velocity, above which f r i c t i o n a l vibrations of the s l i d e r cannot e x i s t . Theory suggests that t h e • " c r i t i c a l " value-of the driven ..surface v e l o c i t y i s dependent upon the damping, load, and stiffness of the suspension, and the f r i c t i o n c h a r a c t e r i s t i c s - o f the rubbing surfaces. Using the approximation that the amplitude of s t i c k of the s l i d e r equals the maximum amplitude of v i b r a t i o n , a relationship .is developed which predicts the amplitude of v i b r a t i o n at any given value'of driven.surface -velocity.  The l i m i t i n g v e l o c i t y . o f t h i s function when the amplitude  tends to zero i s the c r i t i c a l v e l o c i t y .  Exact and approximate theories  are compared for specific p r a c t i c a l cases, . and reasonable agreement  is  found. Five systems were investigated experimentally, and displacementtime charts of the s l i d e r were obtained at different values of the driven surface v e l o c i t y .  Unstable regions were noted where the s l i d e r  fluctuated between smooth s l i d i n g and f r i c t i o n a l v i b r a t i o n s . The experimental data i l l u s t r a t e s the existence of a c r i t i c a l v e l o c i t y of the driven surface, and its. dependence upon the degree of damping i n the system.  The c o r r e l a t i o n between experimental data and  t h e o r e t i c a l curves indicates that the developed a n a l y t i c a l method could be used to predict the behaviour of systems, subject to f r i c t i o n , induced vibrations.  ix  ACKNOWLEDGEMENT  The author i s grateful for many.helpful  suggestions from  the faculty, and graduate students i n the Department of Mechanical Engineering. Special thanks are due to his research supervisor,  Dr. C i A .  Brockley, for sound d i r e c t i o n and constant encouragement i n the course of developing t h i s t h e s i s . Experimental work was c a r r i e d out i n the Lubrication 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 was received from the National Research Council of Canada under Grant Number A-1065-  iii  TABLE OF .CONTENTS Chapter  PAGE  I  1.1  Introduction  1  1.2  H i s t o r i c a l Background  2  Theoretical Analysis  6  Chapter  EE  11.1 11.2  Chapter  '  Velocity-Amplitude Curve  19  III  111.1  The Vibration Apparatus  111.2  Instrumentation  29  111.3  Preparation.of the S l i d i n g Surfaces  30  Experimental Results  3^+  Discussion  56  VI. 1  Conclusions  59  VI.2  Recommendations  62  C a l i b r a t i o n of the Loading System and Beam Stiffness  63  C a l i b r a t i o n of the Transducer  63  Chapter  IV  IV. 1 Chapter  V  V. l  Chapter  Appendix  Appendix  .23  VI  _A  B  iv  Appendix  Appendix  C a l i b r a t i o n of the Oscillograph  ,6k  Determination of the Degree of Damping Given by, the Eddy Current Damper:.  6k  A Comparison of the Exact and Approximate Theories  66  D  Appendix  LIST OF TABLES PAGE  TABLE  1  Range of Variables  28  2  Values of Parameters  33  3  Calculated Values of C r i t i c a l ' V e l o c i t y  .66  V  LIST OF FIGURES FIG.  PAGE  1  Diagram of the S l i d i n g System  6  2  Graph of Displacement of the S l i d e r Versus Time  7  Typical'Friction-Velocity-Curve  8  h  Junction Growth  9  5  Relationship Between Static F r i c t i o n . C o e f f i c i e n t •and Time of Stick  10  6  Linear F r i c t i o n - V e l o c i t y - Characteristic  10  .3  7  - Phase-Plane Diagram Illustrating., the' Effect  13  of V a r i a t i o n ..of Driven Surface Velocity 8  Phase-Plane Diagram  lh  9  Graph of Displacement Versus Time of Stick  17  10  Phase-Plane  Diagram  19  11  Phase-Plane Diagram  20  12  Diagram of the Apparatus  2k  ,13 'lk  Diagrammatic Layout -of the Apparatus Graph of Coefficient of F r i c t i o n Versus'Relative Velocity of Rubbing  26 35  15  Typical Traces-of the O s c i l l a t i o n for System 2  37  16  T y p i c a l Traces.of the O s c i l l a t i o n T f o r System -2  38  17  Graph of Maximum Displacement Versus Time•of Stick of the Surfaces  h0  18  Graph of (,7V \o-OSA- — X./ Versus Time of Stick of the Surfaces  5  •)  ( Q.QL8  \  hi  19  Graph of S l i d e r Displacement Versus Driven Surface V e l o c i t y - System 1  ~hh  20  Graph of S l i d e r Displacement Versus Driven Surface Velocity - System 2  • 45  VI  FIG.  PAGE  21  Graph of S l i d e r Displacement Versus Driven Surface Velocity - System -3  22  Graph of S l i d e r Displacement Versus Driven Surface Velocity - System k  23  Graph of S l i d e r Displacement Versus Driven Surface Velocity - System 5  2h  Graph of Amplitude of V i b r a t i o n Versus Driven Surface Velocity - System 1  : if9  25  Graph of Amplitude of V i b r a t i o n Versus Driven Surface V e l o c i t y - System 2  .50  26  Graph of Amplitude-of V i b r a t i o n Versus Driven Surface V e l o c i t y - System 3  51  27  Graph of Amplitude of Vibration Versus Driven • Surface Velocity - System h  .^2  28  Graph of Amplitude-of Vibration Versus Driven Surface V e l o c i t y -System 5  53  29  Graph of C r i t i c a l Velocity Versus- Damping Parameter  30  Graph of | Parameter  31  Traces of the Damped Natural V i b r a t i o n of the Slider  I — ' ) Versus Damping \Vc—0.0518/ O  K  Q  5  S  k6  kQ  55 65  vii LIST OF SYMBOLS UNITS Constant dependent upon f r i c t i o n properties . Stiffness  of the s l i d e r suspension  lb/in.  Equivalent mass of the v i b r a t i n g parts  lb sec / i n .  Damping c o e f f i c i e n t suspension  lb  sec/in,  lb  sec/in.  of the s l i d e r  Combined damping c o e f f i c i e n t , . ( r - +  Sp)  Slope of the p o s i t i v e section of the f r i c t i o n - v e l o c i t y curve • Time  sec.  Time of s t i c k of the surfaces Relative velocity, of s l i d i n g  sec. in/sec.  Driven . surface v e l o c i t y  .. i n / s e c  C r i t i c a l v e l o c i t y of the driven surface  .in/sec.  Normal load between .the surfaces  lb.  Displacement of the s l i d e r  in.  Displacement of the s l i d e r from i t s equilibrium p o s i t i o n at the instant of s l i p .  in.  Maximum displacement of the s l i d e r  in.  Minimum displacement of the s l i d e r  in.  Displacement of the s l i d e r at the instant of stick  in.  V e l o c i t y of the s l i d e r  in/sec.  Acceleration of the s l i d e r  in/sec.  2 Ratio of  v  c  sec.  viii SYMBOL  o<  °<s  . UNITS Amplitude-of s t i c k - s l i p  vibration  • Amplitude of stick of the v i b r a t i o n Maximum .amplitude of vibration  V  Damping r a t i o  A  Damping parameter  CO  in. in.  Natural.frequency.of: the v i b r a t i n g system  rad/sec  Damped natural frequency, of the v i b r a t i n g system  rad/sec  Coefficient:  of  friction  Minimum c o e f f i c i e n t of Static c o e f f i c i e n t of ^SoC  in.  friction friction  Static c o e f f i c i e n t of f r i c t i o n when time of stick approaches•infinity  -  1.  CHAPTER 1 1.1  INTRODUCTION When two s o l i d bodies are rubbed together i n t e r m i t t e n t r e l a t i v e  motion of some type often.occurs,.depending upon .the p h y s i c a l q u a l i t i e s of the system.  This i n t e r m i t t e n t motion .is r e f e r r e d to a s ' " s t i c k - s l i p "  or as a r e l a x a t i o n o s c i l l a t i o n .  Although man has been aware of the  phenomenon f o r hundreds of years, i t was. not u n t i l the time of the German, Hermann Helmholtz, . that s c i e n t i s t s became i n t e r e s t e d . The v i b r a t i o n can be u s e f u l or detrimental.  We can have the  v i o l i n i s t i n d u s t r i o u s l y , applying r o s i n to h i s bow,.while outside, the ;  caretaker i s o i l i n g a creaky door hinge i n order to prevent a disturbance during.the performance. Science-and modern :industry adhere more to the r o l e of the caretaker,•seeking to eliminate f r i c t i o n induced, v i b r a t i o n s from .lowspeed servo-mechanisms i n the automatic c o n t r o l of production p l a n t . The-application of preventative measures, depends, on -a-full.understanding of the p r i n c i p l e s of the v i b r a t i o n . I t . i s to t h i s end that workers i n recent years, have been applying a rigorous, s c i e n t i f i c method i n the ;  a n a l y s i s of f r i c t i o n induced.vibrations. The present i n v e s t i g a t i o n i s concerned with a dynamic a n a l y s i s of the v i b r a t i o n with p a r t i c u l a r - r e f e r e n c e to the•influence of damping on the value of the c r i t i c a l v e l o c i t y , of the-driven surface.  1.2.  HISTORICAL BACKGROUND In .I85O Hermann Helmholtz ( l ) c a r r i e d out v i s u a l experiments with  bowed v i o l i n strings i n a study of the tone of musical.instruments. discovered.the  He  shape of the displacement-time curve.to be of a jagged  or-'"shark s f i n " type,.as compared to the sinusoidal curve produced by 1  a plucked s t r i n g .  Developing a Fourier-Series solution for the  amplitude of v i b r a t i o n , he made some interesting speculations on the relationship between the v e l o c i t i e s - o f - s l i p and stick i n one cycle of vibration.  He made-no mention of the idea that a high c o e f f i c i e n t -of  static f r i c t i o n i n comparison with the kinetic f r i c t i o n value may have been responsible for the s t i c k - s l i p v i b r a t i o n . It was not u n t i l 1929; when Wells • (2) observed s t i c k - s l i p while t r y i n g to measure kinetic f r i c t i o n coefficients present era of investigation began.  at low.speeds, that the  The v i b r a t i o n was l a t e r • studied  by Thomas (3) who used graphical and a n a l y t i c a l techniques to solve the d i f f e r e n t i a l equations..  Kaidanovsky and Haiken(U)  noted the  existence of a negative slope region i n the f r i c t i o n - v e l o c i t y , curve for rubbing surfaces undergoing f r i c t i o n .induced v i b r a t i o n s . agreed.that  Blok (5)  s t i c k - s l i p vibrations depended on the shape of the f r i c t i o n -  v e l o c i t y curve.  He l i n e a r i s e d the f r i c t i o n - v e l o c i t y curve-and treated  the problem a n a l y t i c a l l y .  His work suggested that a s u f f i c i e n t  increase  in the damping of the s l i d e r suspension c o u l d . r e s u l t i n elimination of the s t i c k - s l i p phenomenon, although his theory does not give an accurate prediction of actual system behaviour. Dudley., and Swift (6) used Lie"nard' s (7 ) graphical method for the solution of the d i f f e r e n t i a l equations of motion.  The Lienard method  makes d i r e c t use of the f r i c t i o n - v e l o c i t y curve irrespective of whether  or not i t may be expressed a n a l y t i c a l l y . Rabinowicz (8).found that micro-creep took place tangential to the surfaces during the s t i c k period, being of the order of UOyU.inches, , i n some cases.  This led t o - a more intensive exploration of the theory, of  junction growth between asperities on surfaces i n contact.  Rabinowicz  discovered that the value of the static c o e f f i c i e n t of f r i c t i o n increased to a maximum value as the time of stick of the surfaces increased. This increase of static f r i c t i o n with time could be represented.mathema t i c a l l y by an exponential r e l a t i o n s h i p . Derjagin, Push and T o l s t o i (9) published a t h e o r e t i c a l analysis of s t i c k - s l i p s l i d i n g i n 1957-  They assumed a constant value  of  s t a t i c f r i c t i o n , which was independent of the v e l o c i t y of the driven surface and hence independent of time of s t i c k of the surfaces. Conditions were defined where s t i c k - s l i p could not occur,. and the corresponding driven surface v e l o c i t y at t h i s point was c a l l e d the c r i t i c a l velocity.  The theory was l a t e r modified to be v a l i d for a  time-dependent static f r i c t i o n c o e f f i c i e n t .  Exclusion of phase-plane  diagrams i n the above t h e o r e t i c a l paper necessitates a lengthy development and there appears to be a c e r t a i n detachment from the phenomenon as i t e x i s t s .  The discontinuous slip, and stick stages were  combined into one period, . and ..velocity of s t i c k (or driven surface v e l o c i t y ) i s said to equal the average v e l o c i t y of s l i p . diagrams, would i l l u s t r a t e that t h i s i s not the case.  The driven .surface  velocity, i s usually of the order of twenty to f i f t y . t i m e s the-average velocity, of s l i p .  Phase-plane  smaller than  The t h e o r e t i c a l treatment of the static  f r i c t i o n versus time of contact relationship i s s i m i l a r l y detached from existing experimental data,.thus l i m i t i n g the usefulness of the paper.  Singh (10) used the general method of analysis employed by Derjagin, Push,.and T o l s t o i to predict the conditions necessary to suppress  stick-  s l i p motion when related .to the operation of p r e c i s i o n servo-mechanisms. The conditions most l i k e l y , to suppress - f r i c t i o n i n d u c e d v i b r a t i o n were.. :  found to be a reduction of the difference between the static and kinetic forces of f r i c t i o n , . a n d an increase of-damping and s t i f f n e s s - i n e r t i a ratios.  In Singh's experimental work,.paraffin o i l was used to  l u b r i c a t e the surfaces, , yet no attention was given to the possible v a r i a t i o n of the c o e f f i c i e n t time of stick of the  of f r i c t i o n with either -velocity, or with  surfaces.  Rabinowicz ( l l ) published a further study, of. the s t i c k - s l i p process i n ,1957-  Experimental curves of amplitude of v i b r a t i o n versus, driven  surface velocity, were i l l u s t r a t e d , . b u t no t h e o r e t i c a l argument was given . a s - t o - t h e i r shape..  The curves d i d - i l l u s t r a t e that.-the amplitude appeared  to die-out once a s u f f i c i e n t driven.surface v e l o c i t y was reached. Potter (12) made an experimental study, of f r i c t i o n induced vibrations at low driven surface v e l o c i t i e s .  Damping.was. not included  as a parameter,.and the value of the inherent structural.damping of the s l i d e r suspension was small enough to be neglected.in the theory.  On the  surfaces u s e d , . f r i c t i o n a l vibrations were observed even at the maximum speed of the apparatus ( 0 . 0 3 inches per second),thus preventing a study of c r i t i c a l v e l o c i t y . Previous t h e o r e t i c a l work on f r i c t i o n induced vibrations does, not give accurate or r e l i a b l e predictions of actual system behaviour. • The existence of a c r i t i c a l v e l o c i t y has been j u s t i f i e d i n theory, by Blok ( 5 ) and Derjagin et a l . ( 9 ) , but the respective theories do not define the shape of the amplitude-velocity relationship as the c r i t i c a l v e l o c i t y approached...  is  5-  In the present investigation i t was proposed to make a f u l l e r study of the dynamics of a t y p i c a l mechanical system,, which might "be subject to f r i c t i o n induced v i b r a t i o n s , . a n d . t o relate the p h y s i c a l behaviour of the rubbing surfaces with the c h a r a c t e r i s t i c s of the v i b r a t i n g ,system. In addition, the relationship between the•amplitude of any p a r t i c u l a r v i b r a t i n g system and the velocity, of i t s driven surface-was  investigated.  Attention was given to the existence of a c r i t i c a l velocity, and i t s value for any given degree of damping.in the v i b r a t i n g system.  CHAPTER II  II.1  THEORETICAL ANALYSIS Consider the case i l l u s t r a t e d i n F i g . 1 of a s l i d e r of mass-'m'  with a normal force *W* acting so as to impress the s l i d e r against a lower surface which i s moving with a v e l o c i t y V. by a bond of e l a s t i c i t y 'r'  The s l i d e r i s restrained  ' K ' . • A damper with viscous c o e f f i c i e n t  of value  i s also included.  w  X K  ^vWV /  Fig.  1.  Diagram of the S l i d i n g System  The absolute displacement •'x  1  i s measured from the unstrained  equilibrium p o s i t i o n of the e l a s t i c bond.  Let the c o e f f i c i e n t  f r i c t i o n between the lower surface and the s l i d e r be - ^JL .  It'is-assumed  1  that the lower surface i s i n f i n i t e l y , s t i f f , with the mass of the s l i d e r . s l i d e r , the c o e f f i c i e n t  of  and has a-large mass compared  I f , . a t the equilibrium position of the  of f r i c t i o n between the surfaces  is  s u f f i c i e n t l y large, the s l i d e r w i l l stick to the lower surface and move• along with i t at an absolute v e l o c i t y of value x=V.  During the s t i c k  7p e r i o d the motion of the s l i d e r may be w r i t t e n as:-  Kx <  TV+  >t  W/*s  s  <  (i)  w h e r e a s i s the s t a t i c c o e f f i c i e n t of f r i c t i o n . From r e s t , the s l i d e r w i l l be displaced from point A to point B as i n d i c a t e d on the displacement-time diagram i n Fig.2.  Up to point B, the  s t a t i c force of f r i c t i o n i s capable of withstanding the constant force rV plus the increasing spring force Kx. overcomes the s t a t i c f r i c t i o n f o r c e ,  WLL  s  damping  At B, ..the force (rV + Kx),  ,•and s l i p  occurs.  T I M E (SEC.)  F i g . • 2.  Graph of Displacement of S l i d e r versus Time  At the i n s t a n t of s l i p , the r e l a t i v e v e l o c i t y , "U. = ( v — X ) , between the two surfaces ceases•to be-zero.  During the s l i p period,  the motion .of the s l i d e r can be described by.the f o l l o w i n g equation:-  7 Y I X + - rx  4-KX  =  W/*  (2)  JJL i s the apparent c o e f f i c i e n t slip.  of f r i c t i o n "between the surfaces, during  H o w e v e r , . m a y not be constant and.may vary, with v e l o c i t y ,  for any given set of surfaces.  It must i n fact decrease from the  static  value i f s t i c k r ' s l i p vibrations, are to occur. ' A t y p i c a l f r i c t i o n - v e l o c i t y curve-is shown i n F i g . 3* :  o I  U=( -X) V  Fig. yUm.  3-  Typical F r i c t i o n - V e l o c i t y curve  may be defined as the minimum value of the k i n e t i c f r i c t i o n  coefficient. Experiments by-Rabinowicz (8) suggest that the high value of the static-coefficient  of f r i c t i o n , JJL$ , can .be attributed to -junction  growth of asperities, between.surfaces-in contact.  The existence of a  cold welding effect when contacting asperities come within molecular distances of each other was postulated.  At such close-distances,  . d i f f u s i o n and Van der Waal's forces could e x i s t .  Furthermore,  . increase of the junction area with increase i n time of s t i c k was explained by.the p l a s t i c behaviour of contacting asperities as. they  9-  deformed i n a way.analogous to creep ( F i g . 4 ) . I f the cold welded junction had a s u f f i c i e n t shear strength, the s t a t i c f r i c t i o n would be observed to•increase with time of stick of the surfaces.  Oxide coated.metals may b u i l d up junction area,.but may  not have a shear s t r e n g t h a t the junction which is. comparable to the shear strength of the parent metals.  Instant of contact of two asperities  After time t. sees. F i g . h.  After time t sees. z  Junction growth  In summary, two separate factors were-found to control the s t a t i c f r i c t i o n , . t h e s e being the creep rate of compression of the asperities r e s u l t i n g i n increase of the junction areas, • and the shear strength of the junctions, which formed. • Howe, Benton,.and Puddington (13) made investigations .into the b u i l d .up ,of a t t r a c t i v e forces, between two bodies i n contact.  The  r e s u l t s of t h e i r work suggest .that the .increase of s t a t i c f r i c t i o n with time could take the shape shown .in F i g . 5- JUsoc i - ^ s  of the static c o e f f i c i e n t of f r i c t i o n .as  t  s  n e  maximum value  approaches i n f i n i t y .  The relationship shown . i n F i g . 5 can be expressed mathematically, as.' : -  10. c-, i s a constant dependent on the rate of b u i l d up of s t a t i c f r i c t i o n .  o Time of  Fig.  5-  stick,,t  g  Relationship between 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 , u. , s  and time of s t i c k , t  s  ,  Sampson, Morgan, Reed,.and Muskat (ik)  compared velocity, and  amplitude traces of the s l i p portion of the s t i c k - s l i p process with t h e o r e t i c a l curves :obtained by assuming a f r i c t i o n - v e l o c i t y curve similar i n shape to the curve i n Fig.6.  Relative v e l o c i t y , " U F i g . 6.  Linear f r i c t i o n - v e l o c i t y  =  (v-x).  characteristic  11. Correlation was very close,, suggesting that i f a negative slope exists (Fig.3) i n the apparent f r i c t i o n - v e l o c i t y characteristic during the s l i p stage,. i t can.be approximated to an.instantaneous kinetic value,JJLm-  drop to the  This approximation does, not appear to invalidate  l i n e a r d i f f e r e n t i a l equation solutions as Sampson et  al..illustrate.  The assumption leads to the - important result that a n a l y t i c a l  solutions  can be obtained using the f r i c t i o n - v e l o c i t y shape of Fig.6. Equation (2) governing the motion,of the s l i d e r during s l i p i s ;  a second order d i f f e r e n t i a l equation,, and may. be represented on a phaseplane p l o t of v e l o c i t y versus  displacement:-  i r n x -f- Y°X + K X =  = WyU —  (2)  -  (3) Setting  ~~rZ,~®  dX  ,-equation (3) ;  becomesr-  The zero slope i s o c l i n e i s given by e q u a t i o n ( h ) .  From a trace  of the curve of equation. (U), plotted on .the phase-plane,  the i n t e g r a l  curve,, or phase trajectory, can be drawn using Menard's graphical construction ( 1 5 ) -  The zero slope i s o c l i n e  (equation (h)) contains JJ*.  For a graphical solution, . i t i s not necessary to define jU. a n a l y t i c a l l y . The values from the^U— t l curve can be simply transferred onto the phase-plane.  However a \ 1  relationship similar to that on^Fig.4  can be dealt with a n a l y t i c a l l y by expressing y U L as the k i n e t i c value:-  12.  jx =  s (v-x)  (5)  P  ^Jl may be. included as a. boundary, condition. 6  Equation (5). substituted  into Equation. (2) gives:-  m x + ( r H - W S ^ X - r - KX = W ( ^ - | - S t > V )  (6)  Equation (6) i s now a l i n e a r second order d i f f e r e n t i a l equation,,which can.be solved a n a l y t i c a l l y .  Nonetheless,.a phase-plane solution w i l l  be q u a l i t a t i v e l y , demonstrated,, as i t helps i l l u s t r a t e boundary conditions as the c r i t i c a l condition i s approached. Referring to the zero i s o c l i n e , Equation (h) i s plotted on the phaseplane (Fig. (7))> and the l i m i t cycle trajectory drawn for various driven surface v e l o c i t i e s , . assuming that the value of static f r i c t i o n does not a l t e r i n each case. at a low v e l o c i t y .  Fig. 7  a  shows a large l i m i t cycle obtained  F i g . 7^ shows a l i m i t cycle at the c r i t i c a l v e l o c i t y  condition, where the t a i l end of the trajectory touches the • x=V.'.line tangentially at the point known .as the'•"knee" of the zero slope  isocline.  I f the v e l o c i t y V i s increased further, the trajectory w i l l miss the v e l o c i t y V . l i n e and s p i r a l i n to an equilibrium p o s i t i o n where the zero slope i s o c l i n e crosses the x axis. Not only i s the driven surface velocity•important, but also the p o s i t i o n reached by, the s l i d e r , ^ ^ l i L X X ^ w h e n s l i p occurs. point i s i l l u s t r a t e d i n F i g . , 8 .  This  Increasing the damping c o e f f i c i e n t  tends to a l t e r the slope o f the zero slope i s o c l i n e . i n the d i r e c t i o n shown, about a centre given by. the intersection of the x axis and the zero slope i s o c l i n e .  Greater damping (Fig.81) therefore tightens  the  trajectory so that i t could miss•intercepting the driven surface  13-  F i g . 7.  Phase-plane  diagram i l l u s t r a t i n g the effect of v a r i a t i o n of driven surface v e l o c i t y .  ih. v e l o c i t y l i n e , , and hence approach a stable displacement as time approaches infinity.  X  1  o  V  • F i g . 8..  Phase-plane diagram  With the reservation that v a r i a t i o n of static f r i c t i o n with time of s t i c k is. a most important factor i n determining the c r i t i c a l velocity, at which the vibrations- die out,.consider i n i s o l a t i o n the c r i t i c a l p o s i t i o n of the t r a j e c t o r y on the phase plane. condition i s shown i n F i g . Jb.  The c r i t i c a l  The trajectory i s tangent to the driven  surface v e l o c i t y V . l i n e where-it i s intersected by the zero slope isocline  (Point ( 3 ) ) -  Further increase of V w i l l cause the trajectory  to s p i r a l i n to an equilibrium p o s i t i o n .  It i s required to know the  p o s i t i o n of the point of s l i p ' ( P o i n t (o)).such that the t r a j e c t o r y ends up at Point ( 3 ) -  For the given driven.surface velocity, and  damping, Point ( 0 ) w i l l be the minimum displacement away from the equilibrium p o s i t i o n that t h e - s l i d e r must have,in order to achieve s t i c k - s l i p vibrations.  Any-point of s l i p nearer to the equilibrium  p o s i t i o n of the s l i d e r w i l l result i n a stable s l i d i n g . p o s i t i o n .  15Restating equation (6) :-  (6)  4-(r+-W5j»)X 4-Kx = w ( / ^ + - S V ) P  and defining:-  (7)  (8)  U]\-IS -  63d =  2  A damping parameter ^  (9)  can:be defined:-  A=  (10)  A standard solution to equation;(6) i s given .in .equation.(11):  (11) D i f f e r e n t i a t i n g equation ( l l )  gives:-  (12) D i f f e r e n t i a t i n g equation (12) gives:  x = — dJd e  A 0 - X ) + 2 X B cosCOdt + B O - X n - 2 - X A  Define t=0 at the instant of s l i p . Referring to F i g . Jbzx.  =  XQ  x = x-j_  at t = 0 at  t = tj  sine  (13)  16.  X  = tr  =  For a unique v e l o c i t y , . V and at x^ at  ,.the trajectory i s i n i t s c r i t i c a l case,  there must.exist  the following boundary.conditions:-  t = t-  *  (l»0  -]<  r  3  V  X* =  X = 3  c  (15)  0  (16)  It i s required to find XQ. under these conditions at t = 0. at  t =  0,,x.= V.  These  boundary  conditions give  A  — C O S 6Jdta — \ SinCJA-ta —  H  GJA(| + X * )  e  sincjata  Note that,  the expressiDn:-  (17)  where  co, tj = tan  (18)  A^(i+A*)-ik(i-A ) l  Let  _ A  Z=  In addition, t i s known:-  (20)  K Regarding the physical properties of the two rubbing surfaces, experiments  (8) suggest that the following relationship holds:•C.t-6'  (21)  17I f equation (2l) i s m u l t i p l i e d by  — i t becomes an expression K  for x.  (22) The straight l i n e graph shown . i n F i g . 9 indicates the rate of increase of restoring force displacement with time of s t i c k , at the c r i t i c a l condition.  Again, the c r i t i c a l condition means that the t r a j e c t o r y on  the phase, plane touches tangentially the driven surface v e l o c i t y value at the "knee" of the zero slope i s o c l i n e .  RAOIENT V  "t. Fig. 9.  Graph of X =  W  In other words' stick-occurs at x^ where : ^  _  WyUvA  —  >rV  c  K  X3 i s therefore the point at which the restoring force displacement l i n e crosses the x axis ( i n F i g . . 9 o n l y ) . .  At the c r i t i c a l condition the  18. equation for t h i s straight l i n e can be written:-  X- Vets +  w^^rVc  (23)  At the intersection of t h i s l i n e and the curve represented by.equation (22) s l i p w i l l occur.  This intersection i s the s l i p point X Q and to f i n d  i t we can solve equation (23) for t „ and substitute t h i s value into equation ( 2 2 ) , g i v i n g : -  v  — W  Ac -  1-  K  e  (2k)  Equation (2k) may be manipulated to separate V g i v i n g : -  (25)  CY K  1  n  Both equation (25) for the surfaces, and equation (20) for the system must be s a t i s f i e d .  Simultaneous solution y i e l d s one unique value  of c r i t i c a l velocity, for.the p a r t i c u l a r combination of system and surface q u a l i t i e s .  Therefore, eliminating X Q by substituting  equation (20) into equation (25) gives:-  (26)  w K  K  19II. 2.  VELOCITY - AMPLITUDE CURVE ;  Consider the case when V-^- 0  Fie.•10.  On the phase-plane X Q = X ^ and  X£=x^  Also as V - ^ 0 , SpV-*-0  t -*-o© ,  and  s  yAa,—-yu 5 o c  • at t = 0  A.-  at  K  -  — ^ —  "U, 60dt — TT  °^ way  a  —  O^s =  X|—X  t  -'XTT  if e  (27)  An .exact approach for defining the relationship between.driven • surface velocity, and amplitude of v i b r a t i o n . l e a d s to transcendental expressions.  20. To allow the amplitude v e l o c i t y relationship to he expressed i n a usable form assumer-  x  0  = x  .1  and  x 2  .= x -3  The error involved i n t h i s assumption i s very small.  The driven , surface  v e l o c i t i e s near the c r i t i c a l condition .for, the systems used,, ( i . e . 0.05 - 0.1 i n / s e c . ) when drawn to • scale on the phase-j-plane are hardly distinguishable from the x axis.  • With this, assumption we can.say, the  apparent amplitude equals-the-amplitude of stick ( i . e . o(  Fig.  = 0^3 ) .  -1.1  It i s known that:JX* =  yLl  w  -  -r- ( | U ~  and on the assumption that XQ .=  e  C  ' ) tS  and x^-= ^  (21)  , , where  and X2 are  separated.in phase by TT radians  •*-£(/»»-M+e ) l  XTT  (28)  21.  Referring to F i g . 1 1 : -  5—  °k V  (29)  V  Substitute equation (21) and equation (29) into equation ( 2 8 )  (30)  This gives  V  =  In  e) m  * (j*.  (31)  I f we c a l l  =¥  (2?)  we can express equation ( 3 1 ) as:-  v= If  V—0,  CM  1  o<>rvax  (32)  —  °K  °<  To f i n d the l i m i t of V when o^—r-O i . e . . t h e c r i t i c a l v e l o c i t y , . use L H o s p i t a l ' s Rule which states:1  Lt v  Lt &(<:••<) W:  ^CKYVAX — C<  22. giving:-  Limit V = C-jO<f. ot—O  or  V = c  C °<^^  (33)  ±  From equation-. (28) i t can be s a i d : -  °<=(x.-*H0+e ") X  (34)  Substituting equation (3^) for o( i n equation (32) gives:-  V  (35)  =  In Equation (35) i s the top curve of the amplitude envelope. Knowing cX = x-j_ - X 2 and  (3*0  (37) Equation (37) can be used to p l o t the lower curve of the amplitude envelope. A comparison of the exact and approximate theories i s included in Appendix E .  23CHAPTER III III.l  THE VIBRATION APPARATUS Maximum s l i p v e l o c i t i e s of the order of 5 i n . / s e c . . h a d been  noted i n experimental work done by- Potter ( 1 2 ) . - Since i t was hoped to extend h i s work, using the same order of parameters, an apparatus was designed that would give driven surface v e l o c i t i e s in excess of the maximum s l i p v e l o c i t i e s previously encountered. velocity variable,.tests  With an extended  to determine the f r i c t i o n - v e l o c i t y character-  i s t i c s could be c a r r i e d out on the actual surfaces -which were used for the v i b r a t i o n experiments.  It was f e l t that a driven surface of 5 feet  in length would be required to give a minimum of twelve seconds to record data before the s l i d e r reached the end of the driven specimen t r a v e l l i n g at 5 i n s . / s e c  The s l i d e r and supporting unit used-in  Potter's work was incorporated i n a .revised apparatus having the following  details:-  With reference to Figure 1 2 , . t h e driven surface specimen : (A), measuring 6 f t .  i n length by 1 i n . . i n width was clamped onto a 6 foot  t r o l l e y (B).made from heavy 6 i n . s t e e l channel.and supported by.seven sets of b a l l bearing wheels (c).. The t r o l l e y ran upon a set of r i g i d l y mounted r a i l s 11 f t .  long (D) made from ground steel bar.  The design  adopted gave the driven specimen n e g l i g i b l e r o l l i n g f r i c t i o n , great transverse r i g i d i t y , . and a large mass. 8 ft.  The t r o l l e y was driven by an  screw ( E ) , and a r e c i r c u l a t i n g b a l l bearing nut.  The f r i c t i o n of  the assembled unit was such that, when pushing the t r o l l e y by hand,. the lead screw.would revolve e a s i l y .  The incorporation .of r o l l i n g elements  in the traverse mechanism l i m i t e d the - s t i c k - s l i p phenomenon to the specimen surfaces  only.  FIG. 12.  rilAGKRM  OF  THE  APPARATUS  25Control of the, v e l o c i t y of traverse was effected by. d r i v i n g .the t r o l l e y by, an hydraulic transmission unit ( F i g . 1 3 , ( L ) ) through a system of worm: gears ' (Fig. 1 3 , ( M ) ) . 'The hydraulic transmission was driven by a 1 H P e l e c t r i c motor ( F i g . 1 3 , ( H ) ) at 1750 rpm.  The output  speed of the hydraulic unit could be varied from 0—550 rpm forward or reverse.  -By changing the reduction gear r a t i o , , a continuous range of  driven surface! v e l o c i t i e s from 0.000 - 9 i n . / s e c .  (+^  was available.  The large: mass of the t r o l l e y ensured that..intermittent  tangential  forces on .its surface produced by s t i c k - s l i p vibrations would have n e g l i g i b l e effect i n a l t e r i n g the value of the driven surface velocity.  A micrometer screw and handwheel ( F i g . 13>- (o)) ,on the  stroke c o n t r o l of the hydraulic unit permitted accurate,,reproducible settings of the driven surface  velocity.  The s l i d e r with i t s v i b r a t i n g system was mounted on a transverse beam, which was secured r i g i d l y . t o the r a i l framework.  Referring again  to F i g . 1 2 , , t h e s l i d e r , which measured 1^ i n . longitudinally, b y , 5 / 8 i n . wide, was mounted on a cantilevered beam (F) by means of a universal . j o i n t coupling , (G).  This coupling allowed the s l i d e r ( H ) to bear  evenly on the driven surface,- despite-any.minor surface i r r e g u l a r i t i e s of a wavelength greater than the length of the s l i d e r .  The'length of  the cantilevered beam could be altered by. s l i d i n g i t back into a heavy block ( I ) ,  thus changing i t s s t i f f n e s s .  The s l i d e r was gravity loaded  against the lower surface, the load being applied by. a pivoted cantilevered beam with weights hanging on one end. The pivoted cantilever assembly: was mounted on b a l l - b e a r i n g journals which reduced f r i c t i o n and increased the s e n s i t i v i t y of the loading system.  This method of load .application kept the mass of the  v i b r a t i n g system independent of the normal load between the two surfaces.  S P E E C H C H A N G E R (6)^  HYDRAULIC  h AGNET  •RAIL.  TRftHSMlSSION^  m  14-- B A L L - B E A R I N G , IWHEELS  Z.  [3/1  I  TRAMSTiU(ZEJZ INDICATOR  o  B A L L BEAMING  SURFACE  o  5  CANTILEVER  BEAM  TRANSDUCER OSCILLOGRAPH  CLAMPING  A  SI  FlG.13  IDiAcSR A M M A T I C  LAVOLlT  OP  THE.  BLOCk \  APPARATUS  1 HP ELECTRIC MOTOR T  W O R M  0 uj  r  SCREW  G E A R ( M )  27Damping was provided by an aluminum plate attached to the beam so as to move i n the magnetic f i e l d provided by.two Magnetron permanent magnets (K) which had a rating of 3,300 gauss each.  Different damping  r a t i o s were obtained by/introducing iron pole pieces onto.the ends, of the permanent pole pieces, thus reducing the flux gap. The mass of the v i b r a t i n g system was made up of the s l i d e r , universal joint,.aluminum p l a t e , cantilevered beam,. and the amplitude transducer arm (mentioned l a t e r i n the instrumentation section). v i b r a t i o n .traces, were taken for various beam lengths. s t i f f n e s s coefficient  Free  Knowing the  of the beam, K,.the equivalent mass at the s l i d e r :  could be calculated from the r e l a t i o n : -  The free v i b r a t i o n traces indicated that the s t r u c t u r a l damping inherent in the system was e s s e n t i a l l y . v i s c o u s the value of the magnetic damper.  i n nature and so could be added to  In addition,•the s t r u c t u r a l damping  was. s m a l l . i n comparison with the surface.and eddy current damping. Determination of the degree of damping.in the system i s outlined i n Appendix D. The following parameters can be controlled on the apparatus:a)  Driven surface v e l o c i t y , "V"  b) .Stiffness  "K" of the  suspension  c)  Normal .load "W" between .the s l i d e r and the driven surface  d)  The damping coefficient  "r" of the v i b r a t i n g system.  Table 1 gives the range of variables controllable i n the apparatus.  28.  Table 1 .. INCREMENT  SYMBOL  RANGE  Load  W  0-301b.  Stiffness  K  12-i801b/in.  VARIABLE  Driven Surface Velocity Damping Coefficient  -V r  .  0-9in./sec.  21b. 131b/in. O.OOlin./sec.  0.01-0.. 591b. sec/in. 0. l i b . s e c / i n .  29III.2  INSTRUMENTATION Deflection of the s l i d e r from the equilibrium p o s i t i o n was  measured by a l i n e a r displacement electro-mechanical  transducer,  (Make:-Brush, Type i ) which r e l i e d upon electro-magnetic i t s operation.  properties for  The transducer was mounted on the side of the  cantilever  beam three inches from the fixed end to i s o l a t e i t from the damper magnets.  Displacement of the beam at t h i s point was found to be l i n e a r  and in proportion to the displacement of the s l i d e r .  The transducer  was c a l i b r a t e d to read d i r e c t l y . t h e displacement of the s l i d e r .  The  method of c a l i b r a t i o n i s described i n Appendix B. A Daytronic d i f f e r e n t i a l transformer indicator (Model coupled to the transducer.  3 0 0 B F )  was  This unit' supplied the excitation to the  transducer, and analysed the output.  S l i d e r displacement was  indicated v i s u a l l y on a d i a l incorporated i n the d i f f e r e n t i a l transformer i n d i c a t o r .  Chart records of the s l i d e r displacement versus time  were obtained by coupling an Edin modulator amplifier ' (Model 8 l 0 8 A ) , and a Brush oscillograph (Type BL20l) to the d i f f e r e n t i a l transformer indicator.  The transducer arm was mechanically zeroed in the  electrical  n u l l p o s i t i o n by mounting the transducer body on a s l i d e r positioned by a micrometer screw.and graduated handwheel. In summary, i t i s possible  to measure the displacement of the  s l i d e r , . a n d knowing the s t i f f n e s s "K" of the suspension, forces may be calculated.  the f r i c t i o n  30. III.3  PREPARATION OF THE SLIDING SURFACES Meaningful results can .only "be obtained i f the  characteristics  of the rubbing surfaces, are maintained constant throughout the experiments.  Many methods of preparing .the surface of the driven  specimen and the s l i d e r were tried-.before a successful method was found. It was proposed to determine the f r i c t i o n - v e l o c i t y , curve from the same surfaces, as. were to be used.for the s t i c k - s l i p t e s t s . rubbing the surfaces together at r e l a t i v e - v e l o c i t i e s 6 in./sec.  This necessitated ranging from 0 to  As. t h i s was to be-a dynamic study of f r i c t i o n - induced  v i b r a t i o n s , . t h e surfaces were simply,required to have a consistent friction-velocity characteristic. The f i r s t t r i a l s for consistency were made using cold r o l l e d C1020 steel for the- driven surface-and .the s l i d e r .  Both surfaces were finished  .by grinding then honing to a f i n i s h of kmicro-inches 12 micro-inches, across the specimen.  longitudinally:and  Unlubricated rubbing was t r i e d  but heavy tearing resulted, even with a l i g h t load.of six pounds. Better results were obtained by.using .rougher surfaces, with a ground f i n i s h of l 8 micro-inches l o n g i t u d i n a l l y . b y ; 6 0 micro-inches across the specimen.  At t h i s stage no means of protecting the specimen from  atmospheric contamination was -being used.  It was thought that rough  surfaces were l i a b l e to have-less damage because foreign matter such as dust from the atmosphere or sheared of asperities could l i e i n the deeper v a l l e y s , whereas contact was made only on the ridges. Conversely, hard foreign matter which could not be completely/accommodated i n the valleys of the smooth finishes could gouge out long tear marks i n the specimen surfaces.  This i r r e g u l a r phenomenon could a l t e r  the f r i c t i o n c h a r a c t e r i s t i c s of the  surfaces.  31The freedom from tearing could also be attributed to a protective skin on rough surfaces.  Grinding, such as was .used on the driven specimen, ;  could produce a protective surface. •• Although tearing .was minimised with the rougher surfaces,.the  s l i d e r soon became smooth since i t was always  rubbing, whereas only, a small section of the driven specimen was. i n contact with the s l i d e r at any.instant.  The change ,in surface f i n i s h of  the s l i d e r gave variable f r i c t i o n c h a r a c t e r i s t i c s . After further experiment i t was found that a hardened.steel  slider  with a smooth f i n i s h of k micro-inches along the specimen and 8 microinches, .across operating on a .soft driven surface gave a minimum of damage. However, tests revealed that when tearing was a b s e n t , . f r i c t i o n c h a r a c t e r i s t i c s were inconsistent,  even when the m i c r o - f i n i s h appeared  to be constant throughout..the length of the driven specimen. :  It was  thought that contamination from the atomsphere was responsible for the inconsistency.  The surfaces-were  treated-in the following way and  improved.results were obtained. The lower specimen was made.from cold r o l l e d C1020 s t e e l measuring .1 i n . wide,  i n . thick and 6 f t .  f l a t on a surface grinder with a 6 f t .  long.  flat  This bar was ground  t r a v e l , . t h e n finished with.a  carborundum wet stone,, using Varsol as a cutting l u b r i c a n t , u n t i l the surface f i n i s h was 3 micro-inches, along the specimen by/12 micro-inches •across.  The specimen was afterwards cleaned with tri-chloroethylene,  followed b y . e t h y l - a l c o h o l .  The surface was then etched . l i g h t l y , b y  swabbing with cotton wool soaked i n a solution of k^> concentrated n i t r i c acid dissolved i n e t h y l - a l c o h o l .  After 30 minutes the specimen was  washed with pure ethyl-alcohol and immediately placed on the apparatus while the surface was s t i l l wet.  The design of the t r o l l e y was such that  3 2 .  a shallow trough was formed with the specimen at the bottom.  This  trough was f i l l e d with ethyl-alcohol so that the lower surface was always protected from the atmosphere by. a f i l m of f l u i d .  When the  specimen..had to be l e f t for long periods between ..tests, i t was covered with i n e r t l i q u i d p a r a f f i n .  Upon commencing the t e s t s , . t h e  specimen  was c a r e f u l l y washed to remove the p a r a f f i n , • and .fresh ethyl-alcohol was applied.  The s l i d e r was .prepared from ground hardened steel,-' (Atlas  Nutherm, . hardness:-Rc • 55-5) •  The surf ace-was -.U micro-inches along the ;  specimen and 8 micro-inches across.  It was cleaned with t r i - c h l o r e t h y l e n e ,  then ethyl-alcohol and..immediately.placed onto the apparatus.  Ethyl-  alcohol was chosen for.'its low v i s c o s i t y , . and steadying effect on the friction  characteristics.  Using .the.above method, the whole series of tests was, accomplished with one surface treatment.  It was f e l t that t h i s gave -a better  standardization of f r i c t i o n c h a r a c t e r i s t i c s than ..ref inishing the :  surfaces, after each pass.  After a l l five systems had been tested the  driven surface specimen was inspected,, and no surface damage could be detected by eye.  Table 2  SYSTEM No.  r  K l b / in.  lbsec/in.  rad/sec.  rad/sec.  W lb.  1  119  0.02  0.01  I83  ,183  16  2  119  0.51  0-59  183  163  16  0.453  3  119  0;26  0.323  I83  .171  16  0.249  k  119  0.1U5  O.I87  I83  181  16  5  119  0.108  0.139  183  182  •16  •0.01  •O.lkk  0.107  1-3  &  3h.  CHAPTER IV IV.  1 EXPERIMENTAL RESULTS The F r i c t i o n V e l o c i t y Curve With the s l i d e r and lover specimen .surfaces, prepared as described  i n . the previous section, . the s l i d e r was • loaded i n .the equilibrium ;  p o s i t i o n of the beam ..and.allowed to rest for periods of 10 minutes, 5 minutes, and .2 minutes.  After each rest period, • s t a t i c  friction  c o e f f i c i e n t tests were c a r r i e d out by. driving the lower specimen at a very low v e l o c i t y ( . 0 0 0 5 . i n . / s e c u n t i l s l i p .occurred.  The maximum  value of s l i d e r displacement was noted on the oscillograph.  From t h i s  i displacement, . x^^, the f r i c t i o n c o e f f i c i e n t could be evaluated knowing that:11 . .  K XwxAy W  It was found that f o r . a rest period greater than 2 minutes,.the value of the static c o e f f i c i e n t of f r i c t i o n d i d not noticeably increase.  The  result of t h i s s t a t i c f r i c t i o n test was taken as the value of yUaoo • Once smooth s l i d i n g was, achieved, , the value of the k i n e t i c  coefficient  of friction.was found to be constant up to a v e l o c i t y of 6.in./sec. covering the range of v e l o c i t i e s that would be experienced: by.the - s l i d e r during s l i p .  Rather than t r y i n g to obtain experimental values of the  k i n e t i c c o e f f i c i e n t of f r i c t i o n i n the range of v e l o c i t i e s where s t i c k s l i p .occurred,.the k i n e t i c value of the f r i c t i o n c o e f f i c i e n t for. smooth s l i d i n g was extrapolated to zero, and the-'linear f r i c t i o n - v e l o c i t y relationship shown by F i g . ity was used i n - t h e - c a l c u l a t i o n s . The f r i c t i o n - v e l o c i t y curve gave the following values:-  Jls*•= 0.U01;  y t U = 0.193;  S .=0 p  .60  E X P E R T 1ENTAL  .50  O  X N  )  r o u  b FS .2.0 u  P) 0 0  7 c  u_  U-  >  c  *b  5  f\ 1  ;  r  <c  V  9-  LU  0  U -IO  PELATIYE  VELOCITY  OF R U B B I N G  "'U=V-X*  (llt/sEC.)  O  FlG. 14. GRAPH  O F  COEFFICIENT O F F  R I C T I O N  V E R S U S  RELATIVE.  V E . L O C 1 T Y  OF RUBBING  36. F r i c t i o n a l V i b r a t i o n a l Results The main point of study i n the series of experiments was that of c r i t i c a l velocity, and.its r e l a t i o n to the degree of damping i n the vibrating system.  The magnetic damper was adjusted to give  different values of damping.  five  Singh (13) had noted that reduction of  W the value.of  rp decreased the amplitude-of v i b r a t i o n , the-load and  stiffness were therefore held constant,-and the effect of v a r i a t i o n of damping was experimentally observed. • A normal load of l 6 lbs.- and a stiffness .of 119 l b s . / i n . were selected for the basic system.  The  case of zero damping could not be investigated since the system always had some inherent s t r u c t u r a l damping. At the start of every.test  involving a change of v e l o c i t y , care  was taken to ensure that the slider, was..always at the same starting point on the lower specimen.  This procedure ensured that the same  section of the driven surface was used i n each t e s t , . t h u s p a r t i a l l y guaranteeing consistency of the f r i c t i o n c h a r a c t e r i s t i c s .  To find the  best section of the lower surface,. a consistency p l o t was made along the whole length,of the lower surface at some constant table v e l o c i t y which gave s t i c k - s l i p vibrations.  The section with the most consistent  amplitude was taken as that having the most c o n s i s t e n t - f r i c t i o n characteristics. Before each test the s l i d e r was allowed to stand for five minutes under load.  A trace containing several cycles was taken for each driven  surface velocity, setting. sample oscillograph traces.  Figs.-15 and.l6 show photographic copies of The value of the maximum and minimum  displacement -of the s l i d e r at a given v e l o c i t y was determined by summing these values i n every cycle on the trace and c a l c u l a t i n g the  DRIVEN  SURFACE VELOCITY  V=  V=Q02IM./SEC C H A R T VELOC I T 0 . 2 FIG. 15  0.00181M./SETC  V »  0.005 IM./SEC  V * O.Oi IN/SEC s  IK/SEC  TYPICAL TRACES OF THE OSCILLATION  SCALE:  FOR  SYSTEM  2  1 DIVISION =  0.00  v= 0:039  V= O.O 2 9 IM./SEC  in./SEC  V - 0 . 0 7 IN./SEC C H A R T V E L O C I T Y - 0 < 2 IM./SEC FIG. 16  TYPICRL T R A C E S  S C A L E : 1 DIVISION-Ct002 IN.  O F T H E OSCILLATION  FOR SYSTEM 2.  I § 1  39mean..  In F i g s . 15 and .16 one-division i s equivalent to two thousandths  of an i n c h , . and the zero equilibrium p o s i t i o n .is on the sixth l i n e from the-left-hand edge :of the chart. • Daring the f r i c t i o n - v i b r a t i o n tests the'lower surface-velocity-was•increased from zero by., increments of approximately. 0;01 i n . / s e c .  and a recording was made o f . s e v e r a l cycles  of the v i b r a t i o n at each v e l o c i t y setting.  Vibrations tended-to die  out with.increase of driven surface v e l o c i t y , i n a l l cases. :  The exact  v e l o c i t y at which vibrations died out and smooth s l i d i n g commenced was impossible to define exactly, because-of minor-variations, i n f r i c t i o n values. An, e f f o r t was made to define a band of v e l o c i t i e s for each system where conditions were unstable.  The-term unstable i s used i n the sense  that s t i c k - s l i p would occur,.then die out,.and occur again.  Continual  increase - i n the driven surface v e l o c i t y f i n a l l y . r e s u l t e d i n smooth sliding.  It was considered more r e a l i s t i c to p l o t the range of unstable  table velocities.-and the maximum amplitude of s t i c k - s l i p that occurred i n the unstable condition, rather than t r y to f i n d one unique experimental value-of c r i t i c a l v e l o c i t y for each system. The smallest defined c r i t i c a l v e l o c i t y range was found in System ( l ) with a V value ranging from .0;O98 - 0.10 , i n . / s e c . , - with a Q  maximum amplitude of 0.006 i n . within t h i s r a n g e . - A c r i t i c a l v e l o c i t y of 0.10 i n . / s e c . was selected as being representative of t h i s system. Theory p r e d i c t e d : -  V = C, W c  (33)  m a x  and substitution of values gave C = I.85. 1  This value of Cj_ should hold  for a l l systems tested with the same surfaces,  so C-j_. = I.85 was used  .060  UJ  § .020  r x  <.  .010  :  z  ol  I  1  1  2.  3  _i 4  I  5  L  6  1  7  —1  S  I  9  — I  IO  TIME O F STICK t s (SEC) FIS.I7  GRRPH O F MAXIMUM D 5 P L A C E M E N T VERSUS TIME OF STICK O F T H E S U R F A C E S —SYSTEM 1  U2. throughout the.-theoretical c a l c u l a t i o n s .  With the above, experimental  values inserted, equation (22) becomes:-  X  = 0 . 0 5 4 — G02&e  Q  = C 5 4 In l°22*  t  or  (38)  \  (39)  . 0 3be 4-X 0/ With the assumption that XQ ..= X]_ i tI 0can stated that:5  (Ul) Equation-(38) is-represented'by the s o l i d l i n e i n Fig.17. :  Experimental values of the maximum displacement of stick are superimposed on t h i s graph.  The time of s t i c k for each of these experimental values  •is calculated from the expression:"f~ L s  —  ° ^—  X~m&.<—  ~ V "  v  XfY\'m  Equation (39) i s plotted.-in F i g . . 18 on semi-log axes,, giving a t h e o r e t i c a l straight l i n e relationship-with the same experimental points inserted,. as are i n F i g . 17- Figs. 19-23 show the t h e o r e t i c a l outlines of the envelopes .of v i b r a t i o n with change i n driven surface v e l o c i t y V. •of  In each case,, the upper l i n e of the .envelope .is a plot  equation (35)-  For. any-X^ , • X2 may. be found from equation (37)>  enabling the lower l i n e of the envelope to be p l o t t e d . -Experimental values of  X  M  A  X  ,  in Figs. • 26-30.  . and'X j_ m  The  X  M  noted from the chart records are-inserted  n  A  X  and ^ ± mean .values taken .from the chart m  n  records for any given lower surface v e l o c i t y V constitute the values  .of  X]_ and Xp • The difference between-3^^ and X^^-gives the  experimental value of the.-amplitude of v i b r a t i o n . Figs. 2k-28 show the experimental values of amplitude-compared with the t h e o r e t i c a l p l o t of equation ( 3 2 ) .  If equation (33) i s  expanded to g i v e : -  V.-c.£(yw-/*-Xl+e ) X T  and with C , W, K, yUsoo (  X  constant, V" can be p l o t t e d against c  With experimental, values inserted, equation ( 3 3 ) becomes:-  Vc= or Fig.  , . and JX^  (33)  A  O.O5»B(I+  e ) ATT  TT LA \ V c - 0 . 0 5 \ 8 /  (^2)  (^3)  29 shows a .theoretical plot of equation (1+2) with the unstable  bands of driven surface v e l o c i t y , i n s e r t e d .  F i g . 30 shows the same  •points and relationship plotted..on semi-log axes using . equation . (U3) •  .060  THEORETICAL. EXPERIMENTAL  O  .050  .040  O<:<:O-OO6IN.  h r  uJ  Z  LU 0 < OL  <n  .oao  fl  (y uJ  .010  o (A  •oi  FIG. 19  GRAPH  •03  -04DRIVEN  O F SLIDER DISPLACEMENT  -OS SURFACE  -07  06 VELOCITY  V E R S U S DRIVEN  -OQ  .09  •IO  * V * ClN./SEc)  SURFACE  VELOCITY—SYSTEM I  060 THEORETICAL.  ~  EXPERIMENTAL  •Ol  •OZ  -03  -OS  -04-  DRIVEN  FIG.ZO  G R A P H OF SLIDER DISPLACEMENT  SURFACE  VERSUS DRIVEN  -Ob  -07  VELOCITY  -08  O  -09  •IO  °V~(\K./5Ec)  SURFACE VELOCITY  SYSTEM Z.  DRIVEN  SURFACE  VELOCITY  (iN./SEc)  FlG.2.1 G R A P H O F SLIDER D I S P L A C E M E N T V E R S U S D R I V E N S U R F A C E V E L O C I T Y  SYSTEM  3  -060  •050  r  — - .040  h  z  ul  •O30  u  'J J  a a  .oeo  LID  uJ  .OiO  10  •03  04-  -os  -ofo  DRIVEM SURFACE VELOCITY V FIG. ^"3  or  -os  (|N./SEC)  GRAPH OF SLIDER DISPLACEMENT VERSUS DRIVELM S U R F A C E VELOCITY— SYSTEM 5  .060 THEORET CAL.  <3  EXPERI  O  O  —  MENTAL  o  ^  ,O50  .040  )  o  .cso  X K  1  UNSTABLE  X \ ©< < o-oc  56 IN.  .os>r> !  \ .  0  AMPLITUDE  cQ > LL O  C  o  r-  • Ol  .02.  -03  -04-  -05  -Ofc  -07  \ -08  -O?  -IO  THRIVEM SURFACE VELOCITY "V* ClN./SEc) FfG.2.4-  G R A P H O F AMPLITUDE O F VIBRATION  VERSUS  DRIVEN  SURFACE  VELOCITY  SYSTEM  1  .060  THEC)RETICAL ELXPII R I M L M T / \ L  O  .OSD  * . 0 4 0  Y  1 < )0  <C . 0 3 0  °  C  v.  >  UNSTABLE 0 o  c«-OIZlN —*c  O  AMPLITUDE  1  LL  -Ol  -02L  03  -04-  DRIVEN FIG.2S  G R A P H O F AMPLITUDE  O F VVBRATIOH  OS  SURFACE  *—  \  -Ob  VELCX1ITY  .07 'V  V E R S U S "DRIVEN S U R F A C E  O S  -09  (iN./SEC) VELOCITY  S Y S T E M Z.  ./O  •060  •O50  Y •O40 z  OO  o  i  ,0"50  >  o u  <  • O20  'OlO  DRIVEN  SURFACE VELOCITY "V  (lN./5Ec)  FIG.Z6 G R A P H OF AMPLITUDE OF VIBRATION VERSUS DRIVEN SURFACE V E L O C i T Y -  ^Y5TE^3  •Ob DRIVEN SURFACE V E L O C I T Y FIG. 2.7  G R A P H O F AMPLITUDE  OF  .VIBRATION  -07 -08 "V* (iM./SEC)  VERSUS DRIVEN S U R F A C E . VELOCITY  SYSTEM  4  . 1rHEORET I C A L  —  E X P E R I M ENTAL. C) .D50  ) O  r  ^ 7'———_^  c)  V >°  TO  0 .040  p  (y .030 y U0  UJ H -J 2; <  STABLE 1 )  -OIO  •Oi  -02.  OS  DRIVEN RS.ZS  GRAPH  -04-  -OS  SURFACE  OF AMPLITUDE OF V\BRAT!ON  -Ofr  VELOCITY  VERSUS  -Cf7  V  -os  1 'CM4IN  -09  -to  (\H./SEC)  "DRIVEN S u R l P A C E VELOCITY  SYSTEM  5  FlG.29  G R A P H OF  CRITICAL VELOCITY  VERSUS DAMPING  PARAMETER  56. .CHAPTER V  V. 1. .DISCUSSION The main • object of the experimental work was to  study.the  existence of a so-called " c r i t i c a l velocity" at which the - s t i c k - s l i p vibrations appear to die out,.and to relate the value of t h i s v e l o c i t y to the degree of damping in.the s l i d e r suspension Consequently,.five systems were .investigated^using  system.  five- different  degrees of damping for constant load and s t i f f n e s s . A t h e o r e t i c a l analysis•suggested.that.the amplitude of v i b r a t i o n would decrease and cease -altogether for three d i s t i n c t reasons.  First,  with damping and static f r i c t i o n constant,.increasing the driven surface velocity, would eventually result i n the s l i d e r being unable-to catch up .-to the-driven surface,  so ;that s t i c k becomes impossible.  conditions the s l i d e r would o s c i l l a t e .of smooth s l i d i n g .  Under these  to a stable equilibrium p o s i t i o n  Second, with the v e l o c i t y and static  friction  constant,,increasing the damping results i n a decrease of the. amplitude of v i b r a t i o n . • Under c r i t i c a l damping conditions the s l i d e r should d r i f t back asymptotically. t o ' i t s equilibrium condition. k i n e t i c f r i c t i o n i s absolutely constant,  Unless  s t i c k i n g couid occur before  the s l i d e r reaches • i t s equilibrium p o s i t i o n , . and so i n practice • s t i c k s l i p of an i r r e g u l a r nature can s t i l l . o c c u r , . e v e n with c r i t i c a l damping. T h i r d , with damping constant i n the system, varying the time of s t i c k tends t o • a l t e r the value of the static c o e f f i c i e n t  of f r i c t i o n ,of the  two surf aces,,'which means that the value of maximum displacement of the s l i d e r w i l l be altered accordingly. by. equation . (20).  The v a r i a t i o n i s described  Time of stick and.lower surface v e l o c i t y are'not  ts  —  cVs "V~  57As V i s i n c r e a s e d , t g i s decreased,.the p o s i t i o n of the s l i d e r ;  :  from equilibrium i s decreased,.and o ^ ' i s decreased,.so that the vibrations tend to die out.  In the t h e o r e t i c a l analysis,. an.attempt  was made to correlate -the three c r i t e r i a into one unique mathematical ;  expression, which would define-a c r i t i c a l v e l o c i t y above which s t i c k s l i p v i b r a t i o n s could not occur. With one minor assumption,;an expression was. developed which :  related the amplitude of v i b r a t i o n ,to the driven .surface'Velocity (equation (32)).  The l i m i t of t h i s expression when, the amplitude  -tended to zero was taken •.as being .the t h e o r e t i c a l value of the - c r i t i c a l velocity.  From t h i s expression of c r i t i c a l  velocity.the  t h e o r e t i c a l relationship between c r i t i c a l v e l o c i t y and degree of damping" was " derived (equation  (Uo).).  From the experimental r e s u l t s , the - c r i t i c a l v e l o c i t y was most accurately :  noted on System (l).where the unstable band extended over a range of 0.002 i n . / s e c .  - This system was used to determine-the-value of C^.  The use of alcohol as-a protective shield from the atmosphere, and.as a weak lubricant during the s l i p portion .of the vibration-was -a major help .in s t a b i l i z i n g the f r i c t i o n properties. One i n t e r e s t i n g aspect of the presence of a t h i n l i q u i d f i l m between .the surface i s that -.it does not necessarily, suppress the -sticks l i p v i b r a t i o n s . •On :the contrary, the presence-of quasi-hydrodynamic l u b r i c a t i o n during the s l i p portion actually, increases the amplitude of v i b r a t i o n .  On the other hand, ,-the alcohol, did not seem to hinder  the build-up of the high values of s t a t i c f r i c t i o n ;  normally experienced i n d r y . s l i d i n g .  coefficient  The persistency of s t i c k - s l i p was  evident even when f l u i d s as viscous as. l i q u i d p a r a f f i n coated the surfaces.  , 8. 5  It i s not generally true to say. that ,a l i q u i d f i l m .on the rubbing :  surfaces has no effect or. the build-up of s t a t i c f r i c t i o n .  • Bowden,and  Tabor '(16).found that f a t t y - a c i d lubricants acted on the rubbing surfaces in, a way that suppressed the build-up .of static f r i c t i o n with time of stick. Each time s l i p .occurred, cold-welded ..junctions between-the metal surfaces were sheared,, and the face .of the s l i d e r , e s p e c i a l l y , ;• became-a l i t t l e more work-hardened.  Its p h y s i c a l c h a r a c t e r i s t i c s became•slightly  different from what they, had been at the start of the t e s t . ;  Behaviour  of t h i s nature makes the f r i c t i o n induced v i b r a t i o n i n h e r e n t l y . h a r d to control.  This may explain why. a precise value of c r i t i c a l v e l o c i t y  could not be found i n the t e s t s , . b u t o n l y . a range of v e l o c i t i e s ,  where  the v i b r a t i o n sometimes died out.and sometimes returned. When vibrations i n the unstable regions occurred after a smooth s l i d i n g state, they rarely occurred-in .one-or two cycles and then died away again.  In general, they, appeared small.at f i r s t , . g r a d u a l l y  increased,-then gradually died o u t ' ( F i g . 16).  Bearing .in.mind the  nature of the v i b r a t i o n , c o r r e l a t i o n between the t h e o r e t i c a l curves •and the experimental points strengthens the theory.that the amplitude does generally..-follow a - ' b u l l e t '  shaped envelope of v i b r a t i o n (Figs. 19-  23), when the driven surface v e l o c i t y . i s increased.  Experimental  results also showed that increasing the damping .in-the - system .reduced bpth. the amplitude of v i b r a t i o n , .• (Figs..-. 2U-28) ,and the value of the c r i t i c a l v e l o c i t y (Fig. 29).  59CHAPTER VI  VI. 1.  CONCLUSIONS  A t h e o r e t i c a l analysis of f r i c t i o n induced vibrations has been developed to determine the existence of a c r i t i c a l v e l o c i t y of the driven surface.  The value of t h i s c r i t i c a l velocity, i n r e l a t i o n to the  degree of damping in the s l i d e r v i b r a t i n g system was linked-to the timedependent f r i c t i o n c h a r a c t e r i s t i c s of the rubbing surfaces,.and a transcendental expression was found.  The equation could be s a t i s f i e d by one  unique value of driven surface-velocity.  This unique  velocity,was-the  c r i t i c a l v e l o c i t y for the system having that p a r t i c u l a r degree of damping and . f r i c t i o n c h a r a c t e r i s t i c s . By.using the assumption that the-amplitude of stick was equal to the maximum amplitude,.an approximate expression-was  developed which  related.amplitude of v i b r a t i o n to the driven surface v e l o c i t y ,  and the  c r i t i c a l velocity, was found by.taking the l i m i t of the function as the amplitude tended towards zero.  T h i s , l i m i t of driven surface  was taken as the c r i t i c a l v e l o c i t y .  velocity  The t h e o r e t i c a l values of c r i t i c a l  v e l o c i t y for two systems were calculated, using both the-exact  theory,  and the theory which used one assumption. . A maximum of 6.9$ difference •was found, i n comparing the c r i t i c a l v e l o c i t i e s worked out by, the two different-methods.  It was thought that the approximate theory.was of  sufficient-accuracy to be used to p l o t t h e o r e t i c a l curves of the relationship -between table v e l o c i t y and the -displacement of the s l i d e r . F r i c t i o n a l v i b r a t i o n tests were c a r r i e d out on five  separate  systems,designed to i l l u s t r a t e the behaviour of the v i b r a t i o n with different degrees-of  damping.in the s l i d e r suspension.  Oscillograph  .60. traces were recorded.of the displacement of the. s l i d e r at different lower surface v e l o c i t i e s .  Graphic i l l u s t r a t i o n , w a s given of the  v a r i a t i o n of the envelope and.amplitude of v i b r a t i o n with increase i n driven surface v e l o c i t y , . a n d of the v a r i a t i o n of c r i t i c a l v e l o c i t y with increase of damping. Correlation between the theory developed e a r l i e r i n the thesis, and the experimental data recorded i s considered close enough to suggest that the amplitude of f r i c t i o n a l v i b r a t i o n and driven surface v e l o c i t y can .be defined.by an expression of the form:-  V= — ^ provided:a)  The driven surface v e l o c i t y , i s constant at .any p a r t i c u l a r value.  b)  The mass of the lower surface i s • l a r g e compared with the mass of the s l i d e r .  c)  The k i n e t i c f r i c t i o n c h a r a c t e r i s t i c s are consistent at any.value-of r e l a t i v e v e l o c i t y of movement between the surfaces.  d)  The rate of increase 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  i s dependent on  the time of stick of the s u r f a c e s , . i n a way. similar to the relationship expressed by equation ( 2 l ) . The theory revealed that the degree of damping i n the system not only tended to reduce the amplitude of v i b r a t i o n at any given lower surface v e l o c i t y , but also lowered the value of the c r i t i c a l v e l o c i t y , tending therefore to suppress the t o t a l effect ,of s t i c k - s l i p . Experimental data substantiated t h i s e f f e c t , . b u t  i n t r y i n g to determine  the value of driven surface velocity, at which the vibrations appeared to die out, no unique value could be found.  Instead a band of driven  6i. surface v e l o c i t y values was found within the l i m i t s of which the amplitude of v i b r a t i o n was subject to random v a r i a t i o n . 'However,.when the value of driven surface v e l o c i t y was increased s l i d i n g resulted.  sufficiently,.smooth  The bands of velocity, values,.or'"elongated points",  were plotted on a graph comparing them with the t h e o r e t i c a l relationship between c r i t i c a l v e l o c i t y and damping parameter.  Correlation was. such  that the actual relationship could be that prescribed by the theory, but i t need not have-been,. as the bands could accommodate several different curves. In the course of the experimental work, i t was. noticed that the u  presence of a l i q u i d f i l m of ethyl-alcohol between the rubbing surfaces did not suppress the s t i c k - s l i p i n a way similar to the f a t t y acid lubricants investigated by Bowden and Tabor ( l 6 ) .  It may be concluded  that l i q u i d s with the same-action as ethyl-alcohol on rubbing surfaces would be poor lubricants on bodies s l i d i n g at low v e l o c i t i e s . The amplitude-velocity curves agreed i n shape with similar tests perf ormed'by Rabinowicz ( l l ) .  'No mathematical relationship was given  by Rabinowicz for the amplitude-velocity.curves.  62.  VI. 2 .  RECOMMENDATIONS  The main d i f f i c u l t y i n experimental work dealing with f r i c t i o n induced v i b r a t i o n ,is in. defining the f r i c t i o n .characteristics  and,.once  defined,.maintaining the consistency of the measured values.  Two new  approaches may possibly.be of value. The f i r s t concerns t h e . f r i c t i o n - v e l o c i t y curve,.and whether a curve derived from separate tests is-.the actual one that, influences f r i c t i o n a l vibrations during the s l i p process. or electronic amplitude d i f f e r e n t i a t o r ,  A velocity, transducer,  could be•added to the equipment,  so that the phase—plane trajectory could be displayed on an oscilloscope screen.  Photographic results..could be used g r a p h i c a l l y , . b y performing  a process i n reverse <of that used.in .the Lienard method to f i n d the apparent f r i c t i o n - v e l o c i t y . r e l a t i o n s h i p . The second recommendation i s for a new. design of apparatus. This should.be-a t o r s i o n a l system, the rubbing surfaces being two discs • ;  mounted on the same centre l i n e , . s o each other face-wise.  as to have two annuli bearing against  One disc could be mounted r i g i d l y , and be the  driven surface rotating at V r a d i a n s / s e c  The other disc could be  mounted so as to have a t o r s i o n a l stiffness and damping. The advantages would b e : a)  Compactness  b)  Unlimited range of v e l o c i t i e s  :C.)  Greater ease of i s o l a t i n g the contacting surfaces from atmospheric contamination  • d)  Surfaces-could be well run i n together tending to render the f r i c t i o n characteristics, more consistent than i n the present l i n e a r system.  63APPENDIX A  Calibration of the Loading System and Beam Stiffness C a l i b r a t i o n of the.- loading system and beam stiffness had been c a r r i e d out e a r l i e r by Potter (12) i n .his preliminary-work on s t i c k slip.  A s t r a i n ring was used i n conjunction with a Brush Strain  Analyser.  Since the same vibratory system was used i n the present  apparatus, his c a l i b r a t i o n could be used,.after  the beam had been  re-balanced to compensate for the slight addition of weight due to the aluminum p l a t e .  APPENDIX B Calibration of the Transducer A depth micrometer was r i g i d l y mounted with i t s spindle l y i n g in the horizontal plane,.and the end of the spindle imposing.on the end of the s l i d e r .  With the beam i n i t s equilibrium p o s i t i o n and the  micrometer spindle just touching the s l i d e r , . t h e d i f f e r e n t i a l transformer indicator was set to zero.  The micrometer was then given a .0.025 i n .  increment,.and the gain of the indicator unit increased u n t i l the d i a l showed a f u l l scale deflection.  The-micrometer was used to relax the  beam i n 0.005 i n . increments thus checking the d i a l reading for l i n e a r i t y of response. of 0.100 i n .  L i n e a r i t y . e r r o r was less than  over a.range  Attenuation switches permitted several ranges to be used,  giving f u l l scale deflections -initial calibration.  of,.0.010 to 0.U0 i n . , . u s i n g the same  With the transducer c a l i b r a t e d , , t h e i n t e r n a l  c a l i b r a t i o n signal could be noted for future resettings.  6k. APPENDIX C C a l i b r a t i o n of the Oscillograph. The o s c i l l o g r a p h could be calibrated.separately, from the transducer. With the d i f f e r e n t i a l transformer i n d i c a t o r d i a l at zero,,the modulator a m p l i f i e r p o s i t i o n c o n t r o l could be adjusted to send a D.C.  b i a s voltage  i n t o the o s c i l l o g r a p h which e f f e c t i v e l y . a l t e r e d the e q u i l i b r i u m p o s i t i o n of the pen.  I t was accordingly, adjusted to read zero-in the middle l i n e  of the chart.  The d i f f e r e n t i a l transformer i n d i c a t o r d i a l was then given  a'deflection.corresponding t o , . s a y , . 0 . 0 2 0 . i n .  The gain of the modulator  a m p l i f i e r could then be adjusted to give-any convenient 1 div. = - 0 i 0 0 2 - i n .  scale, say,  Having set the scale f o r the-whole system,.the  e q u i l i b r i u m p o s i t i o n of the s l i d e r was adjusted on the dial, so that the zero e q u i l i b r i u m p o s i t i o n on the chart was. located on the s i x t h l i n e from the l e f t hand edge of the paper. APPENDIX D Determining  the Degree of Damping Given by the Eddy Current Damper  With :the recording system switched, on,,the beam,;adjusted so as to v i b r a t e f r e e l y , could be given a displacement suddenly, released.  of about 0 . 2 i n . and  The o s c i l l o g r a p h would record the d e f l e c t i o n of the  s l i d e r as i t v i b r a t e d a t i t s n a t u r a l frequency.  This procedure was  •carried out, w i t h no e x t e r n a l damping, then with the damper i n p o s i t i o n , and i t s pole pieces i n s e r t e d one at a time,.taking readings of free damped v i b r a t i o n s .  Five traces were recorded with f i v e d i f f e r e n t  degrees of damping (see F i g . 3 l ) . The amplitudes of each successive cycle of any one trace were p l o t t e d on semi-log axes and found to be nearly l i n e a r , suggesting that damping of a viscous nature was i n f a c t being exercised on the system.  The best s t r a i g h t l i n e s through  66. the points were drawn, . and ..the five respective damping coefficients found.  APPENDIX E A Comparison of the Exact and Approximate Theories The value of c r i t i c a l velocity.was calculated for two systems, using both the exact theory, and .the•approximate theory. are shown, below i n Table-3-  The results  The values of the parameters of Systems ( 2 )  and ( 3 ) were used i n the c a l c u l a t i o n .  These values are l i s t e d i n  Table 2.  TABLE 3  System  Exact V  c  Approximate V  c  Error  in./sec.  in./sec.  • 2  O.O58  0.062  .6-9  3  .O.O78  O.O75  3.85  (  BIBLIOGRAPHY  ,Helmholtz, H. "Sensations of Tone". Second e d i t i o n , 1862, p 8 2 . Wells, J . H . "Kinetic Boundary F r i c t i o n " The Engineer, (London) . V o l . 1^7, 1929, P.U5U. Thomas, S. "Vibrations Damped by S o l i d F r i c t i o n " The-Philosophical Magazine (London) Series 7 , V o l . 9 , March 1 9 3 0 , " p, 329Kaidanovsky, N . L . , Haiken, S . E . Journal of Technical Physics, U . S . S . R . V o l . 3 , - 1 9 3 3 , .'.P. 9 1 . Blok, H. "Fundamental Aspects of Boundary F r i c t i o n " Journal Soc. Automotive Engineers V o l . \G, 19U0, p . 2 7 5 . Dudley, B . R . , Swift, H.W. • " F r i c t i o n a l Relaxation O s c i l l a t i o n s " The Philosophical Magazine, (London) Series 7, Vol. k0, I9U9, ; p. 8U9. Lie"nard, A. •"Etude des O s c i l l a t i o n s Rev. Gen. E l e c t . V o l . 2 3 , p. 901-  Entretenues"  Rabinowicz, E . "The Nature of Static and Kinetic Coefficients Friction" Journal of Applied Physics •No. 1 2 , V o l . 2 2 2 , . 1 9 5 1 , p. 1373-  of  Derjagin, B . V . , Push, V . E . , T o l s t o i , D.M. "A Theory of S t i c k - S l i p S l i d i n g of Solids" Proceedings of the Conference on Lubrication and Wear, London :  October, 1957. Singh, B.R. . "A Study of C r i t i c a l V e l o c i t y of S t i c k - S l i p Trans. A . S . M . E . Journal of Engineering for Industry November, i 9 6 0 , p, 393-  Sliding"  Rabinowicz, E . "A Study of the S t i c k - S l i p Process" Proceedings of the Symposium of F r i c t i o n and Wear Detroit, 1 9 5 7 , p. 1U9. .Potter, • A . F . "A Study of F r i c t i o n Induced Vibrations" M.A.Sc. Thesis i n Mechanical Engineering The University of B r i t i s h Columbia, 1962. Howe, P . G . , Benton, D . P . P u d d i n g t o n , I . E . "London-Van der Waal s- Attractive Forces Between Glass Surfaces" Canadian Journal of Chemistry V o l . 3 3 , 1 9 5 5 , P. 13751  Sampson, J . P . , Morgan, F . , Reed, D.W., Muskat, M. ' " F r i c t i o n Behaviour During the S l i p Portion of the S t i c k - S l i p Process" Journal of Applied Physics V o l . Ik,.19k3, >p. 689. Lauer, H . , Lesnick, R.,-Matson, L . E . "Servomechanism Fundamentals" McGraw-Hill Book Co. Second .edition, : i 9 6 0 , . p, 375Bowden, F . P . , Tabor, D. "The F r i c t i o n and Lubrication of Solids" Clarendon Press Oxford, 1950, ; p. l&k.  

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