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Friction sawing of metals Ogunlade, Omojola 1971

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FRICTION SAWING OF METALS by OMOJOLA  OGUNLADE  B.S.M.E. ( I l l i n o i s ) U n i v e r s i t y o f I l l i n o i s , Urbana I l l i n o i s 1965 M.E.Sc. (Western), U n i v e r s i t y o f Western London O n t a r i o 1966  Ontario,  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  i n t h e Department of MECHANICAL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d standard  THE  UNIVERSITY OF BRITISH COLUMBIA  DECEMBER, 1971  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 of the  r e q u i r e m e n t s 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 it  f r e e l y available for reference  agree t h a t for  and s t u d y .  permission f o r extensive copying of t h i s  publication,  my w r i t t e n  I t i s understood  i n p a r t o r i n whole, o r the c o p y i n g o f  t h i s thesis f o r f i n a n c i a l gain  s h a l l n o t be a l l o w e d w i t h o u t  permission.  OMOJOLA OGUNLADE  Department o f M e c h a n i c a l  The  thesis  s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my  Department o r by h i s r e p r e s e n t a t i v e s . that  I further  University  of B r i t i s h  Engineering  Columbia  Vancouver 8, Canada  i  ABSTRACT  Most o f of  metals  has  the r e s e a r c h  been a p p l i c a t i o n  mechanism o f f r i c t i o n ted.  the  analytic  the i n t e r f a c e workpiece  these  measured  workpiece culate  operating  not been f u l l y directed sawing  temperature  sawing  disk.  investiga-  towards using  gaining  analytic  forces,  the f r i c t i o n  was  temperature  brass,  Four  mild  partition was  contact  cutting  forces  strain  and  the  properties, steel, between  This  TI  the  this  rings  attempted  was  steel  and  sawing  matching  partition  the  to  From disk-  used  to  cal-  operation.  to measure  the  distribution  metals, which were u s e d  the  a t the  sawing  temperature  different  o b t a i n e d by  zone.  force  o b t a i n e d and  e x p e r i m e n t a l method  of p h y s i c a l  workpiece  The  i n both  d i s p l a c e m e n t t r a n s d u c e r s were f i x e d .  workpiece.  heat  partition  distribution  means o f  the heat generated d u r i n g  interface  leaded  the  interface  The  range  the  e x p e r i m e n t a l l y by  linear  the  the  method c o n s i d e r e d t h e h e a t  and  and  which  The  and  sawing  e x p e r i m e n t a l methods.  were m e a s u r e d  the  has  understanding of f r i c t i o n  The at  sawing  the f r i c t i o n  oriented  Hence t h e p r e s e n t s t u d y was  a better and  done on  gave  i n the  and  a wide  experiments  T1-6A1-4V disk  a  alloy. metal  temperatures  furnished  a  in  at  temperature  i i i distribution generated  i n t h e work-piece  a t the  Using transmission interface cutting of  a PbS  cell  units,  an  not give  Nevertheless, good use  an  disk was  s e n s o r and was  when t h e t o t a l  and  the  heat  known. optic  glass  fibre  made t o m e a s u r e  However, o w i n g  of the workpiece sawing  contact  attempt  temperature.  friction  could  sliding  and  t o the  highly  the  oblique  localized  nature  p r o c e s s , the i n s t r u m e n t a t i o n developed a c c u r a t e measurement o f t h e  this  temperature  i n high temperature  measuring  transient  temperature.  d e v i c e would  heat  find  transfer  studies. Thermocouples the c u t t i n g tribution well  zone  embedded  were u s e d  t o measure  i n the workpiece.  The  the  distances  from  temperature  dis-  theoretical  w i t h t h e s e e x p e r i m e n t a l measurements;  being  better  magnitude  at points  of  the k e r f  which  received  materials,  materials. steel  and  The  edge m a t e r i a l i n good All  agreement w i t h the the r e s u l t s cutting.  sawing  similar  crete  the k e r f  temperatures  particles  For  the c u t t i n g  and  about  deg  study  l e a d e d b r a s s the  were f o r m e d  and  zone. as-  edge  mild the c u t F.  This  predictions.  to c o n v e n t i o n a l machining  of m a t e r i a l  both  1600  theoretical  the  the c u t  material  obtained i n this the  agreement  were done on  materials,  agree  the order of  treatment evidence from  indicated  reached  non-fusion was  the k e r f  heat  Tl steel  examinations  from  results  the  were a t l e a s t  width distance  Metallographic  was  at d i f f e r e n t  indicated friction  whereby  during  dis-  sawing.  For  the s t e e l s ,  cutting fracture  ductile  mechanism was  some d u c t i l e  fracture  theory  and f o r t h e t i t a n i u m  observed fractures  a t slow were  feed  explained alloy,  speeds  observed  the  brittle  and b r i t t l e  at higher  feed  plu  speed  TABLE OF  CONTENTS  CHAPTER I  PAGE LITERATURE  SURVEY,  EVALUATION  AND  STATEMENT OF O B J E C T I V E  1  1.1 B r i e f  1  History  1.2 R e l a t e d  o f Saws  Subjects  to Friction  Sawing  4  1.2.1 I n t r o d u c t i o n  . . . . . . .  1.2.2 H i g h S p e e d F r i c t i o n  4  . . .  1.2.3 H e a t D i s s i p a t i o n 1.2.4 H e a t F l u x a n d T e m p e r a t u r e Distribution • 1.2.5 H e a t P a r t i t i o n 1.3 F r i c t i o n Practice 1.4 A p p r a i s a l  II  Sawing  Mechanism  4 10 11 15  and 15  of Literature  . . . .  18  1.5 S t a t e m e n t o f O b j e c t i v e  19  HEAT TRANSFER AND FORCE ANALYSES IN THE F R I C T I O N SAWING PROCESS . . . .  20  2.1 I n t r o d u c t i o n  20  2.2 F r i c t i o n Analysis  Saw D i s k  Heat  Transfer 21  2.3 W o r k p i e c e  23  2.3.1  23  Heat T r a n s f e r  2.3.2 L a t e r a l H e a t  Penetration  2.3.2 P o s s i b l e T h e r m a l S h o c k v  .  31  . .  32  vi CHAPTER  PAGE 2.4 The Heat 2.4.1  P a r t i t i o n Problem  Introduction  . . .  . . . . . . . .  33  2.4.2 J a e g e r ' s Treatment 2.4.3  34 .  34  2.4.4 J a e g e r ' s E q u i v a l e n c e o f L i n g and S a i b e l ' s Approach . . .  36  2.4.5  L i n g and S a i b e l ' s Approach  D i s c u s s i o n o f J a e g e r ' s and, L i n g and S a i b e l ' s Formulae  2.4.6  Proposed Heat P a r t i t i o n  2.5 High Speed F r i c t i o n 2.6 F r i c t i o n Sawing Mechanism Governing C r i t e r i a 2.6.1  Introduction  .  37  . .  38 40  —  42  . . . . . . .  42  2.6.2 Average Temperature Criterion . . . . . . . . .  42  2.6.3  Maximum Temperature Criterion  43  2.6.4 F r i c t i o n S t u d i e s and the Criteria 2.7 Summary III  33  44 . .  44  EXPERIMENTAL APPARATUS, INSTRUMENTATION AND METHODS 3.1 Apparatus  46 . . . . .  3.1.1 I n t r o d u c t i o n 3.1.2 The Saw Disk and S h a f t  46 46  . . .  3.1.3 The Disk Motor and V - B e l t  .  46 47  3.1.4 The T a b l e  47  3.1.5 The Workpiece Guides . . . . 3.1.6 Arrangement t o Reduce F r i c t i o n a l F o r c e between Workpiece and T a b l e and Guide  48 48  vii CHAPTER  PAGE 3.1.7 Arrangement f o r Measuring Table Forces  49  3.1.8 The Feed Mechanism D r i v e and Transmission  49  3.1.9 The C a r r i a g e and the Power Screw  49  3.1.10The Push Rod Measuring  and T h r u s t  S t r a i n Ring  . . .  3.2 I n s t r u m e n t a t i o n  51  3.2.1 I n t r o d u c t i o n  51  3.2.2 The Disk Speed  51  3.2.3 The T a b l e F o r c e  51  3.2.4 The Feed T h r u s t  52  3.2.5 The Feed Speed 3.2.6 The Temperature D i s t r i b u t i o n Around the C u t t i n g zone . .  53  3.3 E x p e r i m e n t a l Procedure Treatment  and Data  53 57  3.3.1 I n t r o d u c t i o n  57  3.3.2 Specimen P r e p a r a t i o n . . . .  58  3.3.3 E x p e r i m e n t a l Method  58  3.3.4 Treatment IV  50  . . . .  o f Data  63  RESULTS AND DISCUSSIONS 4.1 F o r c e s and F r i c t i o n C o e f f i c i e n t  65 .  4.2 Temperature D i s t r i b u t i o n  65 67  4.2.1 T h e o r e t i c a l  67  4.2.2 E x p e r i m e n t a l  70  4.2.3 Comparison o f T h e o r e t i c a l and E x p e r i m e n t a l R e s u l t s . .  71  4.2.4 Comparison w i t h Other Works  72  viii CHAPTER  PAGE 74  4.3 M e t a l l o g r a p h y 4.3.1 M i l d  Steel  4.3.2 T I S t e e l Steel)  (Alloy  74 Structural  75  4.3.3 B r a s s  76  4.3.4 T i - 6 A l - 4 V A l l o y  76  4.3.5 C o n c l u s i o n s Drawn from M e t a l l o g r a p h i c Examinations  77  4.4 M a t e r i a l s from K e r f  . . . . . . .  78  4.5 E x p l a n a t i o n o f the C u t t i n g P r o c e s s V  79  CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY  83  BIBLIOGRAPHY  85  APPENDIX I  E s t i m a t i o n of the Heat T r a n s f e r Coefficients . . . . . . . . . . . APPENDIX I I F o r c e A n a l y s i s i n F r i c t i o n Sawing System  91 98  APPENDIX: I I I E s t i m a t i o n o f Heat P a r t i t i o n Fraction APPENDIX IV S t r e s s A n a l y s i s on the Saw Disk T a b l e F o r c e S t r a i n Ring Calibration APPENDIX VI T h r u s t F o r c e S t r a i n Ring Calibration  103 .  106  APPENDIX V  119 .  121  L I S T OF  TABLES  TABLE 1. D a t a  PAGE and R e s u l t s  f o r Brass  122  2. D a t a a n d R e s u l t s  f o r 1/8 i n c h M i l d S t e e l  124  3. D a t a a n d R e s u l t s  f o r 1/4 i n c h M i l d S t e e l  126  4. D a t a  f o r T l Steel  128  and R e s u l t s  5. D a t a a n d R e s u l t s  f o r Ti-6A1-4V  Alloy  133  6. Summary o f F o r c e s 7. F u s i o n  135  C u t t i n g Temperature  Cutting  Distribution  along the  Axis  136  8. P h y s i c a l P r o p e r t i e s o f t h e M a t e r i a l s U s e d i n Experimental 9. C o m p a r i s o n Solution  Work  o f Hauptmann  with  Modified  137 a n d Ramsey's Yu's  ix  Disk I38  LIST OF FIGURES  FIGURE  PAGE  1  T h i n D i s k w i t h Heat Input  2  Flat  3  P l a t e with F i n i t e Source  4 5 6  140  P l a t e C o n f i g u r a t i o n w i t h Heat Input Width and a Plane  .  141  Heat 142  P l a t e w i t h F i n i t e Width and a Plane Heat Source a l o n g the Breadth o f the P l a t e . . .  143  Jaeger's tions  144  and, L i n g and S a i b e l ' s C o n f i g u r a -  Schematic Drawing of the G e n e r a l i n F r i c t i o n Sawing P r o c e s s  Arrangement 145  7  Sawing D i s k and T a b l e  146  8  G e n e r a l Arrangement o f Equipment and Instrumentation  147  9  Guide B l o c k s  148  10  Workpiece and Workpiece Mounting Device  . .  11  Mounted Workpiece  150  12  View Showing M o d i f i e d Push Rod Head . . . .  151  13  S t r a i n Ring  152  14  View Showing Arrangement o f F o r c e Measuring Instrumentation  153  15  B l o c k Diagram o f Temperature Measuring Instrumentation  154  16(a) Thermocouple C i r c u i t  149  155  16(b) Thermocouple P o s i t i o n s d u r i n g Run x  Experimental 156  xi FIGURE  PAGE  17  D e b r i s from F r i c t i o n C u t Leaded B r a s s  . « «  157  18  M i c r o s t r u c t u r e from Leaded B r a s s  . » . . .  158  19  L i g h t P i p e Response i n an Edge C u t  « . . «  159  20  Worms from F r i c t i o n C u t t i n g o f S t e e l (a) Edge View o f Workpiece showing "Worms" (b) C l o s e - u p View o f Workpiece Bottom ( c ) C l o s e - u p View o f W o r k p i e c Top . . » * .  160  21(a)  Arrangement  161  21(b)  C l o s e - u p View o f "Worms" from Edge C u t o f Mild Steel <>* o r o c o o o » <:<>•«•«•  162  22  T y p i c a l T a b l e and T h u r s t F o r c e s d u r i n g a l 6 S t Rlin « o © ft o * o e # > o e e o o « e  X63  23  Thermocouple Response C o r r e s p o n d i n g Forces i n Figure 2 2 . e . » . . e . o  163  i n Edge C u t  ?  «  .  o  o  o  .  o  e  o  t o the .  e  o  24  Microstructure of  25  M i c r o s t r u c t u r e o f Worm on Top o f M i l d S t e e l Workpiece 400 X » . . o e . . * «.«,«»«  164  26  M i c r o s t r u c t u r e o f K e r f M a t e r i a l from Sawing M i l d S t e e l 400 X Edge View (a) C l o s e o f One S i d e —• V i s i b l e O x i d e L a y e r (b) I n t e r i o r S t r u c t u r e . « . • . . . < . . .  165  M i c r o s t r u c t u r e o f K e r f M a t e r i a l from Sawing M i l d S t e e l 400 X — B r e a d t h View (a) C l o s e t o Edge S t r u c t u r e •— Shows O x i d e Layer (b) I n t e r i o r . S t r u c t u r e *.<>•. „ . . . e «  166  28  M i c r o s t r u c t u r e of M a t e r i a l Close to the F r i c t i o n C u t Edge o f M i l d S t e e l , 400 X .' .  167  29  M i c r o s t r u c t u r e of 'As Received' T l S t e e l ,  27  4 0 0  30 31  Q  «  «  o  «  «  >  8  3  A s Received* M i l d  o  >  e  9  o  i  t  Q  £  '  -  e  e  o  «  Steel  »  .  C  '  3  M i c r o s t r u c t u r e o f M a t e r i a l C l o s e to t h e Flame C u t Edge o f T l S t e e l 400 X . . . « .  168  M i c r o s t r u c t u r e o f Worm from Top o f T l S t e e l Workpiece 400 X c * . „ « , > , , . . , > » . • .  168  xii PAGE  FIGURE 32  33 34 35  36  37 9  M i c r o s t r u c t u r e o f K e r f M a t e r i a l from Sawing TI S t e e l 400 X — B r e a d t h V i e w (a) C l o s e t o One Edge (b) C l o s e t o O t h e r Edge «. . „ „ * » <, » »  169  M i c r o s t r u c t u r e o f K e r f M a t e r i a l f r o m Sawing TI S t e e l , 400 X — - Edge View „ * „ . * » *  170  Microstructure of M a t e r i a l Close to the F r i c t i o n C u t Edge o f T I S t e e l , 400 X .  .  a  170  M i c r o s t r u c t u r e o f 'As R e c e i v e d ' T 1 - 6 A I - 4 V A l l o y ^ 800 X «> o o o e o o * o • o • « o •  X7X  M i c i ' o s t r u c t u r e o f Worm f r o m Top o f T i - 6 A 1 - 4 V W o r k p i e c e , 800 X . . . »  171  Microstructure of Kerf Material Ti-6A1-4V A l l o y — Edge V i e w (a) 400 X (i?) S00 a o o o o « o o * e  c  o  from  e-  o  .  .  Sawing  ©  o  o  o  X-/2  38  M i c r o s t r u c t u r e o f 'As R e c e i v e d ' T1-6A1-4V A l l o y Heated w i t h O x y - A c e t y l e n e Torch t o Self-Burning; Allowed to Cool i n A i r (a) 400 X  39  Temperature  D i s t r i b u t i o n i n Brass  40  Temperature  D i s t r i b u t i o n i n 1/8" P l a t e  Mild  41  Temperature  D i s t r i b u t i o n i n 1/4" P l a t e  Mild  42  Temperature  D i s t r i b u t i o n i n TI S t e e l  43  Temperature  D i s t r i b u t i o n i n TI S t e e l  44  Temperature  D i s t r i b u t i o n i n TI Steel  . « .  179  45  Temperature  D i s t r i b u t i o n i n Ti-6A1~4V  - <, *  180  46  C l o s e - U p View ^  47  t  ^  6  6  X  o  e  of Kerf ^  o  »  o  e  Material o  o  o  o  o  . » »  „ „ *  *  o  „ »  e  from «  e  174  177 .178  Mild o  o  Q  «  P r o f i l e o f Wear Mark P r o d u c e d on C o p p e r H o r i z o n t a l M a g n i f i c a t i o n X400, V e r t i c a l X4000. The A r r o w I n d i c a t e s D i r e c t i o n o f Sliding. The D i s p l a c e m e n t o f M e t a l i s C l e a r l y V i s i b l e [ 5 6 , p 459] , «. « „ » c  c  X  182  t^X  xiii PAGE  FIGURE 48 II/-1  Schematic R e p r e s e n t a t i o n o f the P r o c e s s C u t t i n g w i t h a Saw [42] Ca) T a b l e  of 182  Forces  (b) Workpiece F o r c e s  183  IV-1  S k e t c h of R o t a t i n g Disk w i t h Edge Load  IV-2  E q u i v a l e n c e o f F i g u r e IV-1  185  IV-3  S o l i d Disk w i t h Edge Load  186  IV-4  C o n f i g u r a t i o n Required  IV-5 V-l VI-1  t o Modify F i g u r e  . .  184  IV-3  t o Get S o l u t i o n t o F i g u r e IV-2b  187  Geometry f o r C a l c u l a t i n g  188  r  Arrangement f o r C a l i b r a t i n g T a b l e S t r a i n Ring Arrangement f o r C a l i b r a t i n g T h r u s t S t r a i n Ring  189 190  ACKNOWLEDGEMENT  In the c o u r s e  o f my study, I have come i n c o n t a c t  w i t h many h e l p f u l p e o p l e :  faculty,  f e l l o w graduate  To a l l ,  and  students.  appreciation.  In p a r t i c u l a r ,  technical I express  s t a f f and my g r a t i t u d e  I thank Drs. Lund and  Hawbolt o f t h e Department o f M e t a l l u r g y f o r t h e i r h e l p i n the m e t a l l o g r a p h i c i n t e r p r e t a t i o n s , D r s . Hauptmann, I q b a l and  Gartshore  f o r t h e i r h e l p f u l d i s c u s s i o n s and s u g g e s t i o n s ,  Dr. H a z e l f o r h i s k i n d n e s s and  i n l e n d i n g us h i s l i g h t  Mr. Jones o f the T r i b o l o g y L a b o r a t o r y  t i o n i n the e x p e r i m e n t a l  pipes  f o r h i s coopera-  phase o f the s t u d y .  My s i n c e r e thanks t o Dr. B r o c k l e y my r e s e a r c h s u p e r v i s o r and f a c u l t y a d v i s e r f o r h i s s u s t a i n e d i n the study, h i s encouragement and guidance the programme.  interest  throughout  L a s t , b u t n o t the l e a s t , I thank my w i f e ,  O l u f u n l a y o , f o r h e r p a t i e n c e and endurance. The Laboratory  experimental  work was done a t the T r i b o l o g y  o f the Department o f M e c h a n i c a l  U n i v e r s i t y o f B r i t i s h Columbia.  Engineering,  F i n a n c i a l a s s i s t a n c e was  p r o v i d e d by the N a t i o n a l Research C o u n c i l o f Canada under g r a n t No. A -1065 and t h i s i s g r a t e f u l l y  xiv  acknowledged.  NOMENCLATURE  A  Actual  area o f contact  F  F r i c t i o n f o r c e i n the c u t t i n g  G  Shear modulus o f workpiece  K  Thermal  zone  material  conductivity  Thermal c o n d u c t i v i t y o f moving K  g  Thermal c o n d u c t i v i t y o f s t a t i o n a r y member  L  Latent  lij  Parameter d e f i n e d  N  Normal  P  Load a p p l i e d a t c o n t a c t  T  Temperature  T m  Melting  point of s o l i d  T  Ambient  temperature  U  Sliding  speed  V  Cutting  rate  V  c  heat o f f u s i o n i n e q u a t i o n (34)  f o r c e i n t h e c u t t i n g zone zone  a t an a r b i t r a r y  point  P e r i p h e r a l v e l o c i t y o f saw d i s k  a  Outer r a d i u s o f saw d i s k  b  Inner r a d i u s o f saw d i s k  b^  H a l f w i d t h o f workpiece p l a t e  c  C h a r a c t e r i s t i c length zone  c^  member  area  S p e c i f i c heat xv  associated  with  contact  xvi  Saw d i s k  diameter  = 2a  Fraction  of heat generated going i n t o  stationary  Fraction  based  on a v e r a g e  temperature  matching  Fraction  based  o n maximum t e m p e r a t u r e  matching  member  Heat  transfer  Average  heat  coefficient transfer  i n saw  coefficient  disk o f the t o p and  bottom o f the workpiece Length  o f heat  source i n x - d i r e c t i o n  Lengths defined Length  i n Appendix I I  of workpiece  Parameter Landau's  defined  plate  i n e q u a t i o n (12)  parameter  Mean f l o w p r e s s u r e Total  heat based  Portion saw  on a v e r a g e  temperature  o f heat generated d i s s i p a t e d  through the  disk  Total  heat based  Total  friction  Portion  o n maximum  temperature  of heat generated d i s s i p a t e d  r-coordinate  i n polar  coordinate  Saw d i s k Workpiece Radial  matching  heat generated through the  workpiece  Time  matching  thickness thickness  displacement  coordinate  plane  xvii v  Tangential  et  Linear  ^  Reciprocal Shear  displacement  thermal expansion o f twice  strain  the thermal d i f f u s i v i t y ,  referred to polar  &Lj  Kronecker  delta  ^-  Parameter  defined  i n equation  r  Radial  strain  i n polar  £jj  Strain  tensor  elements  £  Circumferential  £  e  Q  Temperature of  e  rise  above  (42)  coordinates  i n polar  coordinates  the i n i t i a l  temperature  workpiece  Angle  X  strain  coordinates  i n polar  Thermal  coordinates  diffusivity  p  Thermal  diffusivity  o f moving  member  Thermal  diffusivity  of stationary  k •s ^  A parameter  ^  Friction  i n equation  10  coefficient  .  meter  A parameter  /U>1  Poisson's  defined  ratio  i n equation  or kinematic  ^  Abscissa  P  Density  ^  Stress  tensor  r  Radial  stress i n polar  6  Circumferential  ^  (3a)  -6  /Jrr,  -  defined  member  o f t h e dynamic  Temperature  *C~  S  Shear  viscosity  coordinate  system  of material  coordinate ^  Ql^a)  elements  o r hoop  coordinates stress i n polar  system rise  strength  above  the ambient  of workpiece  temperature  1/aX  Thermal Shear  stress  stress  i n d u c e d i n the workpiece  i n polar  Temperature  function  Airy  function  Angle  stress  defined  Angular  coordinate  i n Figure  II-2  v e l o c i t y o f t h e saw  disk  system  CHAPTER  LITERATURE  SURVEY, EVALUATION  STATEMENT OF  1.1  A Brief  History  The ancient points is  use  arts  of  o f saws may  known t o man.  to the exact time  saws, i n t h e i r  thousands  are  oldest  i n the B r i t i s h  Thotmoth have  the  lived  second,  about  using  Samuel, K i n g s mentioned;  action Greek  732 of  evidence  king  2000 B.  sand  about  While  no  use  and  [2].  C.  The  copper  as a b r a s i v e s Chronicles  [3].  sawing  B. the  C.  [4],  conveyed  saws u s e d  civilization  saw  The  the  stamp  time  of  believed  to  believed stone  I n the books  of  stone b l o c k s i s  about  o f the  period.  v e r y e x t e n s i v e use 1  the  was  book o f  idea  i n that  at the present  who  of  used  era [1].  blades to cut  t h e s e b o o k s were w r i t t e n respectively  conclude  E g y p t i a n s were  saw  record  must h a v e b e e n  They b e a r  of Egypt,  most  o f saws, t h e r e  t o make one  form,  of the  historical  o f the f i r s t  Museum  blocks  C.  r e g a r d e d as one  saws known t o e x i s t  toothless  B.  be  most c r u d e  t o have used  460  OBJECTIVE  of y e a r s b e f o r e the C h r i s t i a n  The  AND  Saws  enough c i r c u m s t a n t i a l  that  I  1040, Isaiah  580  and  written  reciprocating The of  ancient saws.  In  fact,  the  ciple  Perdix  two  men  Greeks i g n o r a n t l y c r e d i t e d  used  sculptural was  with  iron  i n v e n t i o n of  saws i n t h e i r  works  introduced  used  the  [2],  into  Daedalus or the  renowned  Between  saw  1000  very  because  architectural and  500  man's c i v i l i z a t i o n .  saws which, l o o k e d  his  much  Thus  like  these and  C ,  B.  iron  the  our  dis-  Romans  modern-day  saws [ 3 ] . The action, latter  circular  d i d not half  of  saw,  seem t o h a v e the  t o woodworking  Bentham  [ 2 ] was  but  the  saws, among o t h e r  saws t o S i r M a r c cular  saws had  teenth  one  invention  the  [6] a of  the  first  letter  Silliman erience to  sawing  of  an  axle  end  various  the  friction 3,  saw  1823  D.  circular  i n Holland, of  the  sources  circular  saw  The  According s a w i n g was  which  wrote  cir-  sevenof  infor-  was  an  Rev.  letter  made o f v e r y iron  practice to  Herman  Barnes,  p l a t e and  in  Daggett  Benjamin  discussed  hard  Lewis  contained  to P r o f e s s o r  maker, Mr.  to a c i r c u l a r  A.  r e v o l u t i o n i n Europe.  process.  cabinet  a cross cut  the  1800  i s a much more r e c e n t  College.  a local  inventing  these  Connecticut,  of Yale  repair  fixed  February  credit  However,  With  manufacturing  before  of  the  S i r Samuel  machines,  that  sawing  r e p o r t on  dated  Cornwall,  From  than  i n England.  i n v e n t i n g and  [5],  industrial  conventional  century  towards  conclude  Friction than  the  Isambard  [5],  may  of  gave  been used  century  mation,  with  cutting  earlier  machinery General  credited  engineers  i t s continuous  appeared  eighteenth  respect  circular  with  the  exp-  who  wanted  plate.  He  put  i t on  his  3 lathe to  which  finish  was  gave  t h e edge o f t h e p l a t e  cut into  edge o f  two  also  the  b l a d e and  cut.  He  found no  a band  no  neatly  fire  actual  the  saw  contact  1824,  Geneva,  iron  disk  friction speed  was  and  up  (now [l].  was  'fire' and  the  was  70  plate  zone  Mr.  35  feet 70  feet per  under  s o Mr.  Barnes  after-  never  Daggett  per  came  fluid."  experiments  required  second.  soft  to  At  the d i s k  second,  cut  raised  inches diameter  per  con-  time  "electric  speed  disk  there  which  the d i s k  second,  feet  which  appeared  i n a short  the  the  through  there  C o l l a d o n conducted  peripheral  around  to about  and  w i t h a 7 1/2  damaged but. a b o v e  the  start  this was disk  unaffected. The  was  I t seemed  on  t h e edge  the c u t t i n g  and  file  s t o p p i n g the  around  with the p l a t e  found  sawing  and  slightly  On  touched  teeth  the  piece  completely cut  the o p e r a t i o n  Darier  but  saw  During  the  attempted  observed  his cross  when he  tool.  Switzerland  He  a rock c r y s t a l  minutes.  f o r the  q u e s t i o n whether In  at  mark was  emitted sparks with v i o l e n c e .  them o u t w i t h h i s new  the  speed.  with a f i l e  and  a few  heat.  of intense  no  applied  wear marks and  w a r d s marked  in  and  then applied  within  sensible  tinually  He  i t was  longitudinally  was  pieces  the p l a t e .  was  he  i t a high rotating  first  friction  p r o b a b l y made by known a s With  appeared  the  that  sawing  by  in  t h e C a r n e g i e P h i p p s Company  the C a r n e g i e S t e e l rapid  machine b u i l t  rate  t h e end  Limited  Company L t d . ) i n  o f change of  America  1887  technology, i t  of the n i n e t e e n t h c e n t u r y  and  4  early  i n the twentieth  century  friction  sawing  was  being  used e x t e n s i v e l y .  1.2 R e l a t e d 1.2.1  Subjects  understanding effect tion  appraisal of f r i c t i o n the f r i c t i o n  a t high  and d i s s i p a t i o n ; ,  heat  topics  and t h e w o r k p i e c e ;  have been d i s c u s s e d  section  gives  High  Speed  friction.  Using  m/sec.  between  field  at very  p a d s made o f o t h e r from  of current  versus For  a c t i o n and  Most o f t h e s e  i n the l i t e r a t u r e  and t h i s  knowledge.  [ 7 ] d i d e x t e n s i v e work o n h i g h  [ 8 ] , they bodies  apparatus were a b l e  three  sliding  bismuth  results  designed  by  t o measure  a t s p e e d s up t o 800  frequency  materials.  their  copper,  high  the b a l l  low-melting  They r e p o r t e d cient  i n the kerf.  an i n g e n i o u s  two s o l i d  pressing against  ranged  genera-  i n the f r i c t i o n  T h e y r o t a t e d a 1/2 i n c h d i a m e t e r  magnetic  t o heat  Friction  Beams a n d h i s c o - w o r k e r s friction  involve  speeds; the  t h e mode o f c u t t i n g  a s u c c i n c t review  Bowden a n d F r e i t a g speed  sliding  transfer  d i s p o s a l o f d e b r i s formed  1.2.2  sawing w i l l  of m a t e r i a l p r o p e r t i e s as r e l a t e d  machine  by  Sawing  Introduction A fair  the  to F r i c t i o n  steel  and then  ball  ina  decelerated  symmetrically  The p a d m a t e r i a l s to high-melting  i n graphs of f r i c t i o n  arranged used tungsten. coeffi-  speed. Bowden a n d F r e i t a g  found  that the  5 friction  coefficient-speed  affected  by t h e i n i t i a l  nounced tion the  coefficient speed  copper and  a t slow  speeds  was  was  to s t e e l  observed  a relatively  at high.  with  a violent  on f r i c t i o n  A t speeds  was  fric-  the  Load  coefficient  a n d a t 150  behaved  steel-  jerk.  g r e a t e r than  observed  layer  The  pro-  low b u t i n c r e a s e d a s  t o f l o w a n d when s l i d i n g  deep metal  like  were  100  m/sec,  m/sec,  was  prolonged,  a highly  vis-  material. Aluminum  copper  but t h e i r  m/sec.  At high  a lower  than  and d u r a l u m i n seizure speeds,  friction  duralumin.  alumin half  chipped.  except  that  of  that  transfer  While  o c c u r r e d a t about  than  Antimony behaved i t sfriction  40 copper  aluminum b u t h i g h e r  of both metals  a l u m i n u m was  observed  to  steel  to flow,  v e r y much l i k e  coefficient  low f r i c t i o n  b u t a t g r e a t e r speed  behaviour The  like  o f t h e o r d e r o f 700 m/sec,  had v e r y  Bowden a n d P e r s s o n speed  on s t e e l  v e r y much  was  dur-  about  duralumin.  Bismuth m/sec,  behaved  coefficient  Metal  considerable.  duralumin  that  effects  were  b e i n g more  120 t o 140 m/sec  seized  t o be i n s i g n i f i c a n t .  copper  was  at high  surface suddenly  transfer  had  than  A t about  curves  the e f f e c t  speeds  decreased.  copper  cous  speeds;  starting  s u r f a c e roughness  found  characteristic  results  friction  the f r i c t i o n  [9] t r e a t e d  of bismuth  i n more  as s l i d i n g  a t 200  increased rapidly.  the c h a r a c t e r i s t i c  f o r molybdenum  decreased  coefficient  friction-  detail. and t u n g s t e n speed  showed  i n c r e a s e d . Smeared  m e t a l o v e r the t e s t pads was f o u n d t o c o n s i s t m a i n l y o f i r o n b u t w i t h t r a c e s o f molybdenum and t u n g s t e n .  After  e t c h i n g t h e rubbed r e g i o n s on t h e p a d s ,  examination reveal  i r o n w h i c h Bowden and F r e i t a g s u g g e s t e d  t o be due t o h i g h  temperature  diffusion.  A n o t h e r s e t of e x p e r i m e n t s was done w i t h s t e e l plated with copper,  and s t e e l p l a t e d w i t h chromium  b a l l s rubbed a g a i n s t d i a m o n d .  The f r i c t i o n  coefficient-  speed c u r v e s showed sudden jumps o f f r i c t i o n values at c e r t a i n c r i t i c a l ball  speeds.  and chromium p l a t e d b a l l ,  coefficient  For both p l a i n  this critical  Above the c r i t i c a l  steel  speed was  about 200 m/sec and f o r the c o p p e r p l a t e d b a l l .was about 100 m / s e c .  steel,  the  speeds,  speed  the  c u r v e s l o o k e d s i m i l a r t o c u r v e s o b t a i n e d when each m e t a l was rubbed a g a i n s t i t s e l f the f r i c t i o n  b u t below t h e c r i t i c a l  coefficient fell  the e x p e r i m e n t s ,  speeds  t o a v e r y low v a l u e .  In  Bowden and F r e i t a g o b s e r v e d t h a t when  the s l i d i n g speed o f each m e t a l on diamond exceeded critical  the  speed the m e t a l was smeared o v e r the s u r f a c e o f  the diamond; t h u s r u b b i n g t o o k p l a c e between the  metal  and i t s e l f  smeared on the diamond and t h e diamond was n o t  affected.  However, when the s l i d i n g speed was b e l o w the  critical,  no a p p a r e n t  metal transfer  was o b s e r v e d on the  diamond.  The diamond s u r f a c e was e a s i l y abraded and  shed p r o b a b l y because o f h i g h - t e m p e r a t u r e  phase change  from diamond t o amorphous c a r b o n ; t h i s t r a n s f o r m a t i o n slow at  1000 deg C and a c c e l e r a t e s  as the  poli-  temperature  is  7 increases  a n d a t 1800 d e g C, i t p r o c e e d s  a t a high  Bowden a n d F r e i t a g ' s r e s u l t s a l s o  showed  iveness  diamond  their  melting  C) was of  of the metals  i n abrading  point:  chromium  (melting  100 t i m e s more e f f e c t i v e  that  the e f f e c t -  increased  point  than copper  with  o f 1615 d e g  (melting  point  1083 d e g C ) . Bowden a n d P e r s s o n  Bowden  and F r e i t a g e x c e p t  employed  an i m p a c t  technique and  such  [9] used  t h e same a p p a r a t u s a s  f o r the s l i d i n g  technique  instead  f o r large-scale melting  of s o l i d s  The p h y s i c a l p r o p e r t i e s  as t h e r m a l  c o n d u c t i v i t y and m e l t i n g  parameters  i n their  against  nitrate,  study.  bismuth,  copper,  tetrafluoroethylene),  rubber  melting  point  metals  melting  point  non-metal  acteristic certain  polymers  (nylon,  silver  nitrate  speeds.  increased  r a p i d l y with  melting.  For copper,  Above  tapers  and h i g h  these  macroscopic  o f f with  exhibited  was  increased  tin,  steel  silver poly-  F o r the lowPb, Sn) and l o w (AgNO^) t h e c h a r -  speeds,  increased melting  significant  speed.  useful  e x h i b i t minima v a l u e s  consequent  loads  were  r e s u l t s on  at  the f r i c t i o n  wear and  d i d not  However, t h e d i f f e r e n c e b e t w e e n t h e f r i c t i o n light  i n high-  terylene,  and g l a s s .  f r i c t i o n - s p e e d curves  critical  point  lead,  ( B i , Wood's a l l o y ,  wear,  of the s o l i d s  They r e p o r t e d  Wood's a l l o y ,  steel,  They  of the d e c e l e r a t i o n  friction.  rubbed  technique.  to i n v e s t i g a t e the types o f deformation,  criterion  speed  at  rate.  occur.  coefficients  a t low speeds and  Steel against  itself  s i m i l a r c h a r a c t e r i s t i c s as c o p p e r b u t h a d  lower  8 friction iour  coefficients.  was due t o t h e l o w e r  t h a n c o p p e r (K_ Cu rubbing  with  Even their  though  speed  their  the polymers  low.  i n raising  rather the  than  of  l o w wear o b s e r v e d  hard  ium  carbide  fused  silica  speeds. be  fied His  silicon  He r e a s o n e d  explained totally  first  proposed by W i l k s  results  solids  than a t  energy  was due  conductivity;  might  be e x p e n -  interface  zone and hence  that  carbide  by the thermal  and W i l k s  (SiC),  diamond and  mechanism  to crack  by  process,  [ 1 0 ] i n 1920 a n d l a t e r  that  cannot  proposed  some m e c h a n i c a l  showed  titan-  surfaces at high  i n 1959, must a l s o  and d i s c u s s i o n s  have a t e n d e n c y  (TiC^),  t h e a b r a s i o n o f diamond  by Tolkowsky  speeds.  rutile  on m e t a l  [7] b u t t h a t  giving  t h e a b r a s i v e mechanisms  (A^O^),  (SiC^) s l i d i n g  Bowden a n d F r e i t a g  ( P . T . F . E . ) was  f o r these m a t e r i a l s a t high  sapphire  (TiC),  as the s l i d i n g  a t h i g h speeds  the molten  points,  no m i n i m a v a l u e s ;  o f the molten  [10] i n v e s t i g a t e d  solids:  coefficient  low m e l t i n g  decreased  of the f r i c t i o n  i n expanding  Miller  have  and low t h e r m a l  the temperature  steel  speed.  coefficient  high melt-viscosity  behav-  steel  o f n y l o n and o f t h e polymers  a considerable part ded  of  the f r i c t i o n  curves exhibited  t o have h i g h e r f r i c t i o n  this  In both  Polytetrafluoroethylene  The b e h a v i o u r  their  that  conductivity  and copper,  friction  increased.  observed  thermal  increased sliding  friction-speed  rather,  suggested  = 0.9 v s K , , = 0.1). steel  against i t s e l f  decreased  to  Freitag  modi-  be c o n s i d e r e d .  some o f t h e s e  i n the presence  hard  of thermal  shock.  He  materials  explained that was  due  oped  i n the v e r y  That  diamond  at  the  by  Miller's  to the thin  So  failure  steep  temperature  reached  f a r , high  was  disk*  to perform on  of  speed  steel  speed,  friction  tests  rubbing  surfaces  amorphous [ 7 ] was  the  with  on  has  been  carbon  confirmed  studied i n  applications  speed  at  friction  [ I I ] c o n s t r u c t e d an s p e e d s up an  to  high  Giffen  steel.  [12]  They  coefficient  increasing  condi-  under  apparatus  750  feet  used  per  speeds  found  of f r i c t i o n  normal  load  this and  such  which  a i r turbine coupled  at various s l i d i n g  sliding  second. to  a  equipment normal  load  that: decreased  f o r any  high  asymptosliding  and  totically  with  coefficient increasing  of  friction  sliding  speed  decreased f o r any  asymp-  normal  applied. The  behaviour After the  high  c o n s i s t e d of  ( 2 ) the  load  study  W i l l i a m s and  (1) tically  gradient devel-  place i n o r d i n a r y atmospheric  attaining  apparatus  rotor  to  Williams  capable  This  takes  In order  conditions  these  experiments.,  cutting  tions.  to the  to g r a p h i t e or  vacuum w h e r e a s i n most p r a c t i c a l speed  o f most o f  temperature  region close  transformed  interfacial  the  asymptotes  of  the  correlating  following  w o u l d be  metals their  empirical u -  p r e d i c t e d by  i n v o l v e d i n the data,  the  sliding  liquid mechanism.  Williams and.Giffen obtained  formulae: K(UN)"  0  ,  4  5  10 where  u i s the  sliding force  speed  friction and  instead  of  N  where F i s t h e data  were  per  second.  spread  Earles  teristics to  655  and  and  normal  normal  =  over  the  modifying  range  of  80  to  Their  750  feet  studied  the  friction  charac-  steel  sliding  on  a function  normal  N  They found  sliding  0 (N  data c o r r e l a t e d  a constant.  [13]  was  Their  K'  appara-  coefficient  =  got  Giffen's  second.  y  friction  and  per  and  they  the  the  Williams  Kadhim  wear o f  Using  U  8 2  and  speed  feet  load  load,  force  K a constant,  load.  K'(UF)"°*  friction  Slightly tus,  the  the  u  coefficient,  of  steel  that  the  a parameter  at speeds  up  friction  involving  the  s p e e d , U, i . e .  1 / 2  U)  for different pin sizes  showed  that 1/2  the  friction  increased. Giffen's  1.2.3  Comparing  gave good  Heat  their  as  the  r e s u l t s with  parameter Williams  N  U  and  agreement.  [14]  extensively  d i s s i p a t i o n i n both  discussion  decreased  Dissipation  Blok heat  coefficient  followed  two  dealt  solid clearly  and  with fluid  defined  the  problem  friction. lines; v i z .  of His  primary the  dissipation  area close  dary  dissipation  surrounding direct As  dependent also  out on  on  defined  or  in  the the  the  paper,  nature only  of be  specified.  Closely heat  flux  solids. sisting  and  to  obtain  the  ring  the  slider  using  the  and  of  the  Blok's  the  Simkins  [16]  a disk  rubbing  against  (rotating  Green's  state  for  solved  disk)  function  contacting  surfaces  Therefore  and  using  at the  disk  and  is  in  model  is  dissipa-  the  the  subject  sector  convenient  They  able  positions  (ring  for  in  both  sector)  assumed  steady  con-  were  heat equation rider  of  sliding  a configuration  a ring  the  approach.  the  heat  heat  Distribution  and  They  lubrication.  lubrication.  distribution  sector.  found  strongly  heat d i s s i p a t i o n  profiles  the  is  discussed  hydrodynamic  to  into  m e a n i n g f u l when a w o r k i n g [15]  in  secon-  work h a s  hydrodynamic  interface.  Temperature  tied  and  dissipation  heat d i s s i p a t i o n  Crook  temperature  stationary  on  dissipation  generation  temperature  Ling of  heat  i n b o u n d a r y and his  heat  mechanism.  configuration  will  Heat F l u x  of  focussed  i n connection with  1.2.4  ring  source  which  media of  dissipation  tion  the  application  pointed  and  to  which c o n s i d e r e d  a  state  quasi-  for  the  sector. Ling  temperature configuration temperature  and  Pu  [17]  calculation they  approached  subject  theoretically.  introduced  jump a c r o s s  the  the  the  idea  interface;  For of a  a  of  surface  disk-ring  macroscopic temperature  12 difference surface  existing  between  temperatures.  temperature summarized  jump on their  On  the  the  friction  findings  as  disk  and  the  phenomenon o f heat  ring  sector  macroscopic  generation, Ling  and  Pu  follows:  "(a) In g e n e r a l , f r i c t i o n a l heat i s not genera t e d i n t h e s p a c e b e t w e e n two b o d i e s i n s l i d i n g contact. M o s t o f t h e h e a t i s g e n e r a t e d on t h e s u r f a c e and i n t h e s u r f a c e l a y e r i m m e d i a t e l y below the s u r f a c e of b o t h b o d i e s . This i s because f r i c t i o n d e r i v e s from e i t h e r the breaki n g of adhered j u n c t i o n s , o r from the thermodynamically i r r e v e r s i b l e process of p l a s t i c d e f o r m a t i o n o f a s p e r i t i e s and t h e b u l k b o d y . " " ( b ) Whenever t h e c a p a c i t y o f one o f t h e b o d i e s t o remove h e a t away f r o m t h e i n t e r f a c i a l z o n e i s l e s s t h a n t h e amount o f h e a t g e n e r a t e d on t h a t s u r f a c e , t h e r e w i l l be a t e m p e r a t u r e jump across the i n t e r f a c e . The s u r f a c e w i l l t h e n have a h i g h e r m a c r o s c o p i c temperature." A  steady-state solution  stationary cated was  f o r the  s u r f a c e s but  f o r the  transient  heat  solution  consisting  disk  were  and  flux  distribution  the heat  w h i c h was  moving over  impervious  to heat.  F o l l o w i n g up  flux  lend  support  based  distribution to the  on and  attained  theoretical  calculations.  a  and  solid  another  several  quasi-  this  for  lubri-  temperature  stochastic obtained  with  a  configuraa  slight  body w h i c h  work L i n g  stochastic  was  [18]  models  of  devised a simple experiment  transient  experiments  the  equation for a  protrusion  heat  assumed  of a s e m i - i n f i n i t e  made c a l c u l a t i o n s  ones,  Ling  of  Pu  s e c t o r , and  satisfactory  for unlubricated  time-dependent.  process  tion  state  f o r the r i n g  temperature  a reasonable  theory.  agreement w i t h  His the  to  13 Rabinowicz on  [19] developed  the s u r f a c e energy  flash  temperatures.  tively, close  of s l i d i n g  C a m e r o n and o t h e r s heat  equation i n r o l l i n g  three  situations,  existed  over  formula  based  materials to calculate  His formula  t o h i s measured  a simple  yielded  results  thermocouple  values.  [20] o b t a i n e d  sliding  compara-  solutions  contacts.  They  v i z . , when no t e m p e r a t u r e  t h e c o n t a c t zone f o r v e l o c i t y  t o the  treated  discontinuity ratios  lying  b e t w e e n -1 and +1; when a r e c t a n g u l a r s o u r c e moved o f f t h e surfaces built  a t v a r i o u s speeds;  up when  from  problems  band heat.  the f r e e  o f temperature  and a l s o  surface.  authors  Jaeger  the heat  [21] t r e a t e d  due t o a m o v i n g source of  [ 2 2 , 23, 24, 25] h a v e  t h e c o n t a c t i n terms o f f l a s h ,  plus  flash)  bulk  dealt  and t o t a l  (bulk  temperatures. [26] a n a l y z e d  disk  was d i s s i p a t e d state  and when  distribution  with  friction  temperatures  a moving r e c t a n g u l a r o r square  Several other  Yu  surface  t h e c o n t a c t was r e p e a t e d  was c o n v e c t e d the  t h e way  the heat  for a situation by t h e d i s k .  and j u s t i f i a b l y  transfer  problem  where a l l t h e h e a t He assumed  in a  generated  quasi-stationary  neglected radiative  heat  transfer.  Hauptmann and Ramsey [ 2 7 ] s o l v e d t h e same p r o b l e m  with a  more g e n e r a l m e t h o d .  heat  input to  over  the heat  periodic. Other  They c o n s i d e r e d a d i s k  with  an a r c o f i t s p e r i p h e r y a n d o b t a i n e d a diffusion  e q u a t i o n w h i c h was a s y m m e t r i c  This solution  w o r k e r s have a l s o  i s discussed further treated  the problem  solution and  i n Chapter  of the heat  IV.  transfer  from  rotating  Landau in  treated  a melting s o l i d .  liquid the  was  removed  e x i s t e n c e and  After in  [34]  m e l t i n g had  which  given  he  had  disks  He  [28,  c o n s i d e r e d the  immediately uniqueness started,  of  he  the  of heat problem  formed  33].  conduction whereby  and  solution  reduced  32,  the  assumed  to the  the problem  problem.  to a  form  parameter  by  melting latent  point, heat  extreme an  specific T  q  of  integral  discussed  dimensional  heat  of  the  solid, T , i t s ' m'  temperature  and  considered solutions infinity  on  this  f o r m u l a t i o n , H a m i l l and  or melting rates Their  He  z e r o and  the problem  dependent.  heat  , i t s original  of f u s i o n .  limits  c, (T - T ) h m o  L  where c, i s t h e h  [36]  i t was  31,  to consider only a single  m  ing  30,  the problem  1/2  Using  29,  o f upper  and  lower  when t h e b o u n d a r y  discussion  was  discussed various cases  of heat  f o r the  parameter.  Bankoff  [35]  b o u n d s on  conditions  limited  conduction equation.  L, i t s  to a  freezwere  time  one-  Carslaw transfer  and  Jaegar  where  phase  changes o c c u r r e d . Rosenthal  [37] c o n s i d e r e d b o t h  theoretical  aspects of  with  heat  moving  equation  the heat  sources.  for several  He  distribution  solved  geometries  the p r a c t i c a l  and  associated  the heat showed  and  the  diffusion practicality  15 of  such  solutions.  the mathematical and  flame  I n an e a r l i e r  theory of heat  cutting.  With  Heat  sliding was  a  practical  developed  also  worked out  The  on  the v a l u e of  pair.  also  The  the  a square  Friction  exotic and  associated  but  the  number.  the moving  sawing  for  with  The and  materials.  source  required  saw  of the  on  applied  practice  the range source members  are  method i n non-metals between  the h i g h  i n the former. sawing  f o r c o n v e n t i o n a l sawing  the  gen-  Jaeger.  main d i f f e r e n c e s  for friction  of  stationary  v a r i o u s metals,  employed  heat  moving h e a t the  Saibel  Practice  i s a widely  The  and  component  c o n s i d e r e d by  for cutting  toothless  that  the f r a c t i o n  v a l u e depended  c o n v e n t i o n a l sawing  pressure than  formulae  number, L i n g  S a w i n g M e c h a n i s m s and  many i n d u s t r i e s  and  partition  stationary  fraction  the F o u r i e r  Friction  tion  he d i s c u s s e d  geometry o f h i s moving h e a t  were t h e r e v e r s e o f t h o s e  and  welding  practice.  number  the F o u r i e r  w h i c h went i n t o  values of  1.3  heat  of the F o u r i e r  mechanisms.  sliding  was  during  examples,  in  considered  square.  erated  of  [21]  ranges  Based [39]  he  Partition-  Jaeger different  [38]  distribution  the v a r i o u s problems encountered  1.2.5  paper  The  i s somehow  [40],  fricspeed feed  less  16 In P.  V.  discussing  Vernon  carbon  piece  proposed  steels  taining  good  [41],  d i d not extend  from  the  that  sawed  saw  a s was  in  the c u t t i n g  sus i n the r a p i d anism  could  rates,  and  collected deduced  from  broken-off cutting size  industrial  deg  C)  chips.  different  distributions  governing In sawing  a serious  From  the c u t t i n g summary o f  mechanism,  a  demonstrated friction  consen-  data  Eshchenko  to high  the  strain  such  due  mech-  et a l .  to  brittle  temperatures  the f o r m a t i o n of  of s t e e l s , pointed  The  zone,  p r o c e s s was  by  phenomena  f r o m d a t a on  analyzing  heated  the  the c u t t i n g  A n a l y z i n g the p a r t i c l e s  which  used  in  sawing.  that  operations,  accompanied  grades  inch  supported  Harbord  investigated  steels  work-  penetra-  o f an  factor  con-  cutting.  indirectly  the main c u t t i n g of  of heat  theory.  p r o c e s s was  of c h i p s .  and  H i s work  p r e s s u r e i n the c u t t i n g  i n the case  t o 1000  not  studied  kerf  t o 1/100  high  article  s o h i s work c l e a r l y  i n flame  sawing  shape  that  fracture (800  and  the depth  i n high-speed hot  o n l y be  temperature  was  an  the d i s k ,  1/150  e t a l . [42]  zone  wrote  the workpiece.  b l a d e and  the case  Eshchenko  that  mechanism,  theory of c u t t i n g  taken of  the f u s i o n  penetration  sawing  Harbord  more t h a n  faces of  deformed  sawing  I n 1908,  arguments on  heat  fusion  They r e v e a l e d  tion  badly  the  photographs  bars.  Vernon's  the f r i c t i o n  collected  they found  from  similar  to a general p h y s i c a l  law  mechanism. the l i t e r a t u r e  the f r i c t i o n  heat  regarding  friction  generated at very high  17 speeds of  i s considered  the workpiece  weakening  Owing  pheral  heat  If  to the high  speed  i s dissipated  strong blade  the heat  workpiece  the f a s t fresh  believed  that the melting  action.  In this  ial  i n the k e r f while  oxygen  cussed tries  hardness.  gates  "hot  saws" a r e used mills  and r i s e r s  economic  main reasons economic  cutting  heat  some  i t is  burning  melts  the mater-  between the t e e t h c a r r y action.  sawing  Hyler  [43] d i s -  i n the s t e e l  indus-  advantage  this  i s independent  of  of the  f o u n d r i e s use f r i c t i o n  saws t o  o n c a s t i n g s and i n t h e s t e e l  to sever  and  h o t m a t e r i a l s coming  mills, out of  [ 1 , 43, 4 4 ] .  The  This  with  One o u t s t a n d i n g  The  workpiece.  I n some c a s e s ,  application  i s that i t s efficiency  trim  the  the spaces  and warehouses.  material  and t h e r e f o r e a  wipes o f f the melt  the f r i c t i o n  terms o f p r a c t i c a l  i t s peri-  the m a t e r i a l of the  i s coupled  v a r i o u s uses of f r i c t i o n  process  the  to melt  to the kerf f o r the burning In  results i n  and s o i t s m a t e r i a l  material f o r rubbing.  case,  area  o f the rubbing  t h e weakened  moving b l a d e  presents  ahead  unchanged  abrades  was s u f f i c i e n t  heat  o f t h e saw b l a d e  rapidly  remains r e l a t i v e l y  relatively  on a s m a l l  and t h e b u i l d - u p o f t h i s  of the m a t e r i a l immediately  zone.  strength  t o be c o n c e n t r a t e d  aspect  f o r i t s wide  advantage  sawing  applications  i s due l a r g e l y  rates of f r i c t i o n  counterparts  of f r i c t i o n  saws t h a n  [ 5 , 40, 43, 48,49,  i s one o f  [ 4 5 , 46, 4 7 ] .  t o t h e much  their  faster  conventional  50, 51, 5 2 ] .  Also, at  18 high be  cutting  rates  smooth t h u s  processing  The  the  making  them r e a d y  by g r i n d i n g  1.4 A p p r a i s a l  greatly  the s u r f a c e s of the c u t p i e c e s tend t o  or polishing.  work o n f r i c t i o n  scientific  investigation  subjects related  considered  quite  friction  work.  previous  heat  transfer faces.  transfer  solutions  results.  In addition,  sentative  speed  friction  sawing,  formulae  applicable clearly  the  assumption  geometry  than  However, have  been  with  quote  formulae  from  and s u c h  formulae  invar-  i s t r u e when  friction  sur-  specimens used i n  investigations  are not repre-  sawing.  transfer  by J a e g e r  understood.  heat  problems of s l i d i n g  the experimental  solutions  particularly  sawing  are applicable to  t h e work o f R o s e n t h a l .  and, L i n g  and S a i b e l  provided a l l their  dimensions  in friction  sawing  Par-  are also  The m a j o r i t y o f s o l u t i o n s  of i n i f n i t e  involved  associated  a r e n o t i n agreement w i t h the  practical  to f r i c t i o n  are  sawing  problems  The c o n v e r s e  of f r i c t i o n  Some h e a t  tition  to f r i c t i o n  Many o f t h e a u t h o r s transfer  high  rather  now,  have n o t been a d e q u a t e l y c o v e r e d i n  authors c i t e  previous  practice  o f t h e mechanism.  g i v e e s t i m a t i o n s which  experimental  h a d b e e n , up t i l l  extensively.  Most o f t h e heat  previous  sawing  towards i n d u s t r i a l  peripheral  iably  further  of the L i t e r a t u r e  oriented  high-speed  f o r use without  parameters make  but the f i n i t e and t h e d i f f e r e n t  19  shapes  t h a t might  problem  be encountered  considerably.  a l t e r the p a r t i t i o n  However, i f the f r i c t i o n  problems a r e h i g h l y l o c a l i z e d ,  these p a r t i t i o n  would lead to c o r r e c t temperature  sawing  heat  formulae  p r e d i c t i o n s i n b o t h the  saw b l a d e and the w o r k p i e c e . The  s u g g e s t i o n o f m e t a l l u r g i c a l e x a m i n a t i o n made i n  some o f the e a r l y work on f r i c t i o n  sawing  c o u l d be v e r y  complementary t o t h e r e s u l t s o b t a i n e d by heat t r a n s f e r and friction  force analyses.  1.5 Statement  of O b j e c t i v e  Even though  friction  sawing  i s w i d e l y used i n many  i n d u s t r i e s , i t s o p e r a t i n g mechanism i s n o t y e t f u l l y stood.  The o b j e c t i v e o f t h e p r e s e n t p r o j e c t i s to study  both a n a l y t i c a l l y and e x p e r i m e n t a l l y the f r i c t i o n p r o c e s s w i t h a view of  under-  sawing  to obtaining a greater understanding  the heat t r a n s f e r and o t h e r p r o c e s s e s  involved.  CHAPTER I I  HEAT TRANSFER AND FORCE ANALYSES IN THE FRICTION SAWING PROCESS  2.1  Introduction The c o n f i g u r a t i o n c o n s i d e r e d  friction  sawing p r o c e s s i s the d i s k - f l a t  i n t e r f a c e i s defined t a c t with a f l a t ted t a k e s p l a c e tion,  r e p r e s e n t a t i v e of  by the area  p l a t e type.  The  o f the d i s k edge i n con-  p l a t e workpiece and a l l the heat g e n e r a at this interface.  Using t h i s  configura-  the d i s k heat t r a n s f e r problem i s d i s c u s s e d  s o l u t i o n obtained  t o the heat d i f f u s i o n  and a  equation.  The workpiece i s t r e a t e d i n r e l a t i o n t o heat transfer, shock.  lateral  heat p e n e t r a t i o n  A uniform feed  and p o s s i b l e  thermal  speed was used i n the heat  f e r e q u a t i o n and t h i s i s j u s t i f i e d arrangement o f the f e e d i n g  trans-  by the e x p e r i m e n t a l  mechanism d i s c u s s e d i n  Chapter I I I . The heat p a r t i t i o n problem i s c o n s i d e r e d dependent on m a t e r i a l and  physical properties,  as  configuration  p h y s i c a l dimensions o f both the d i s k and the w o r k p i e c e .  The f r i c t i o n a l  b e h a v i o u r a t h i g h c u t t i n g speed i s i n v e s t i -  gated by a n a l y z i n g  the c u t t i n g f o r c e s i n v o l v e d 20  i n the  21 friction used  sawing  f o r the p a r t i c u l a r  With diffusion  „2  m  V  +  T  Saw  Disk  Heat T r a n s f e r A n a l y s i s  r e f e r e n c e t o F i g u r e 1, t h e g e n e r a l  , w 3T  1  2h  K 3 9 - K ?  CI-T.)  B  3T  f  - I  3r  a thin  (1)  9  9  r  2  plate,  9  r  r  80  2  9  2  <  2 9  the temperature  z  variation d  thickness time  i s negligible,  i s allowed  steady  heat  e q u a t i o n i s [27]  where  the  configuration  i n this research.  2.2 F r i c t i o n  For  process  state  is  and s o  —  distribution  ~ds  0.  9  ®*  =  a  )  a c r o s s the  2  If sufficient  f o r the o p e r a t i o n o f the system,  prevails  temperature  that  l  i s assumed  a  quasi-  Furthermore, i f symmetrical  about  3T  the diameter becomes  saw d i s k ,  by  the  the P e c l e t  number  i s very high.  This last  heat  Thus  of a disk statement  input to the disk  m e t e r 0 = 0, heat  -gg  to the high r o t a t i n g  one c o n s i s t i n g  ture.  TT , t h e n  =0  and t h e  problem  axisymmetric. Owing  system  0 = 0,  transfer  ir j u s t i f y problem.  speed  associated the system having combined  with may  be  friction  the c u t t i n g approximated  u n i f o r m r i m temperawith  i s symmetrical  the axisymmetric Chapter  of the  the f a c t about nature  I V c o n t a i n s more  that  the d i a of the  information  on t h e s e  assumptions.  With the f o r e g o i n g assumptions, problem reduces  the heat  transfer  to the s o l u t i o n of  V T - f£D  (T -  2  Tj = 0  ( ) 2  where „2 V  =  d T T dr  . 1 +  d  7  d7  (2a)  Putting = T - T  T  Equation  (2b)  ( 2 ) becomes  V T 2  - |£D  (3)  x = 0  K t  Equation  (3) i s a m o d i f i e d  Bessel equation  of order  zero  and i f  Hi  then,  equation  ( 3 ) may  V x 2  -  (3a)  be w r i t t e n a s  A T 2  = 0  For  the e s t i m a t i o n o f the heat  see  Appendix I .  (4)  transfer  coefficient  h,  23 The  general  solution  x(r) = C  ±  (4) i s  I (Ar) + ^ ( A r )  from  (4a)  Q  a n d C 2 a r e two a r b i t r a r y  where mined  of equation  constants  g i v e n boundary c o n d i t i o n s .  t o be  In t h i s  deter-  case,  the  boundary c o n d i t i o n s a r e :  where the  q  D  saw  a)  T  b)  - 2T Kt  heat  0  at  D  these  transfer  =  different  2.3  Workpiece  2irKt Aa  generated  going  into  to  f o r t h e saw d i s k i s  K (Ab)I (Ar) + I (Ab)K (Ar) o o o o I (Ab)K (Aa) + I (Aa)K (Ar) o 1 1 o  t o Yu's  [26] s o l u t i o n  boundary c o n d i t i o n s  (5)  except f o r  used.  Heat T r a n s f e r Rosenthal  flame  plate  at r = a  n  "D  boundary c o n d i t i o n s , the s o l u t i o n  (5) i s s i m i l a r  the  of  = q  problem  D  2.3.1  dr dr  a —  disk.  T(r) - T  Equation  r = b  i s the p o r t i o n o f the heat  Using the  =  cutting.  i s concerned,  [38] d e a l t  with  the mathematical  As f a r a s t h e h e a t h i s theory  distribution  i s a p p l i c a b l e to  theory i n the  friction  cutting. If  we  the  The  geometry of  assume t h a t  the  plate thickness,  tion  becomes  8x  2 9  Friction the  y  workpiece  tem  be  to preserve  W  shows t h e  the  heat  equa-  ( T  be  source  considered of heat  problem's o r i g i n a l  the  of  across  2(a).  a moving  transformed  uniformly  formulation  c u t t i n g p r o b l e m s may of  i s rectangular.  i s generated the  Kt  Thus the  2(b)  then  2  piece.  order  heat  [37], Figure  standpoint  may  the  into  further  along  static  dynamic c o o r d i n a t e  of  the  work-  coordinate  a dynamic c o o r d i n a t e  simplicity  the  from  problem.  system.  From  sys-  system  in  Figure this  Figure, K  - Vs  = x  (7) y' =  where V Using  H  In  i s the  the  2  cutting rate.  transformation  (7)  i n equation  2  35  3s  new  coordinate  system,  3 y  this  y  Kt  we  ( 6 ) , we  ( T  T  w  may  ~  obtain  )  (  assume  8  )  quasi-  3T stationary (8)  becomes  state, that  is,  ~%s  -*-  s  z  e  r  o  »  Then  equation  25  + i f l 2  _  =  V  9T  2hl  (  Kr  3E  2  9y  _  T = T + e  )  $ (£,y)  by d e f i n i n g [ 3 7 ] (10)  S(5,y)  ro  mined.  ( 9  w  E q u a t i o n ( 9 ) may be s i m p l i f i e d  where  } 00  i s a temperature  f u n c t i o n t o be  S u b s t i t u t i n g e q u a t i o n (10) i n t o e q u a t i o n  deter-  ( 9 ) , we  have  8  2  ^  $  8  +  9?  —  $  2  [, „ N 2  =  2  |^(Y  9y  2  I . V) + mj $  where 2hl W  m  (12)  Kt  G o i n g back t o the p h y s i c a l p r o b l e m and assuming the generated  i s s y m m e t r i c a l about the a x i s o f x , t h e n  be dependent  o n l y on the  r  „  / 2 5  heat $  will  parameter  +  y  2  '  (  1  3  )  or $ = $ (r)  (14)  26 In view  of equation  d$ 2  dr For  a thin  the  total  this the  heat  kind  heat  plate,  t o be  source,  From e q u a t i o n may be  =  following  this the  ^ - e "  ( 1 5 ) may be s o l v e d i f i s known.  Taking  , and c o n s i d e r i n g t h a t f a r f r o m  w  Y  i s t h e same a s a m b i e n t  of equation  V  (15) i s  K (pr)  5  /(yV)  =  2  (16) s o l u t i o n s  (16)  B e s s e l f u n c t i o n of the  second  + m  (16a)  f o r other  flat  plate  case  of i n t e r e s t width.  here  i s that of a rectangular  A solution  c a n be c o n s t r u c t e d  ( 1 6 ) b y m a k i n g u s e o f t h e image method solution  problem, section  geome-  generated.  c o n s i d e r s the plane  one e d g e o f t h e p l a t e  mirror  (15)  z e r o and  of f i n i t e  equation  and  equation  (pr) i s the modified  The  + m] $ = 0  the temperature  P  plate  q  the s o l u t i o n  of order  tries  2  a p p l i e d to the plate  T - T  o  - [(y V )  ( 1 1 ) may be w r i t t e n a s  2  temperature,  where K  £  infinite  total heat  + 1  (14), equation  only half  of plate  reflection  from  [ 3 7 ] , The  o f symmetry  y = 0  y = b^ as the m i r r o r p l a n e s . I n  o f the heat  s u p p l i e d goes  considered. • Taking  c o n t r i b u t i o n s , we  into  obtain  into  account  the  27  T - T  =  00  — 2irKt ^  e  y  v  ^  (17)  ^ K (pr ) ^—' o n n=l v  T  where r.  -  n  Equation [37]  ^ / ?+  ( 1 7 ) may  (y ±  (17a)  2nb )' 1  be t r a n s f o r m e d  Fourier  series  form  to get  e  T - T  ~  for  4b  l P  Kt  -CYV+PK  +  2  w  £ > 0  00  =  l  P  n  + V)C Y  ^  c o s  <r  1  (18)  n  , and  r  ^  -(p.y  n=,l  E  T - T  into  K t  ( V-p)? Y  +  2  -(YV - PU >5 n  c o s  —  Zny.)  (  1  n=l  w  (19)  K<0  for where  y  At  the heat  their  =  A  +  (  (19a)  pb^  source, equations  temperatures. in  n  ( 1 8 ) and  T h i s i s so because  derivation  i s that  (19) g i v e  the i n h e r e n t  of a l i n e  source  infinite  assumption  of heat.  28 Practically these size  speaking,  equations heat  may  has f i n i t e  size  be m o d i f i e d by c o n s i d e r i n g  a  and s o finite  source.  Consider infinitely from  the source  a plane  small sources  V = 0 to  Regarding  q"  V =  source  of heat  q"d£'  a s made, up o f  placed side  F i g u r e 3 shows t h i s  as c o n s t a n t ,  by  side  case.  then  W  q  " Now  imagine  (  a linear  V  from  t h e dynamic  at  the point  (£,y)  source origin;  of strength ~ the temperature  due t o t h i s  source  i a  -(YV+p )(?-£')  K> V  0  )  placed at  distribution  i s g i v e n by  -( V+py Y  ^ny cos Kr~) l  n  b  +  2  n=l  for  . d£'  2  n  (21)  , and  ( Y V - p y ^ ^  6(T-T ) - 4 b K t « , lP  - ( Y V - p )  1  ) ^ ^ b  +  2  l  n=l  w  (22)  for  € < 5 .  From  the e x p r e s s i o n s  three  1  cases  (21) and  arising.from  ( 2 2 ) , we  the heat  can d e r i v e the  transfer  problem:  29 Case 1:  %, > a  and  T-T  o°  V = a  =  integrate expression  to  W 4b.,pKt £ 1 W  (p+YV)£_  1}e  Y  (p+yV)  + 2  and  T-T  £ < 0 £' = A  =  (py  V)._  n + Y  S n=l  to  P  n + Y  V)  5 c o s ( F 1 Z )  1  (23) n  n  (22) between V = 0  get  4b,  ( p _ Y V H  }e  ( p  "  Y V K  (p-YV) - (  + 2  0<£<£  and e x p r e s s i o n  e  n  (22) between ( p + Y V ) ? 1  (p+YV)  l-e" P (  + 2 ^>.{ n-1  } <P V n  integrate expression  l_e" 4b,pKt £ • 1 W  Y V )£  l J  % P% (  +  + Y V )? n + Y V )  y  V  ) ?  cos ( J ^ ) ' l b  u (py - V)  n=l  =  -( M  y (py +YV)  {l-e P % "  T-T  l } e  integrate expression  {l-e"  Case 3 :  = 0  -(p+ V)C  { e  Case 2:  V  get  {e  l  (21) between  5  and  (24)  Y  (21) between 0 andC £  and t h e r e s u l t  is  _ -(p-YV)(£-5) e  (P-YV)  l-e" P * V (  y V )  y (py YV)  } cos Q  l  (25)  30 Equations  (23)  of  flame  is  generated  to  cutting by  (25) but  would  not  apply exactly  friction  sawing  the  rubbing  of  the  saw  friction  sawing,  the heat  to the  whereby  blade  on  problem the  the  heat work-  piece. In the b r e a d t h thickness q" dy'  of  of  the  the  plate  plate,  i s placed side is  say  from  t , see w  by  i s generated  side  t o +t^/2  F i g u r e 4.  from  considered constant;  within  y'  then  and  In view  = -t /2 D  of  the this,  to +t /2.  If  D  q sinf w  where  i  Since  i s d e f i n e d i n Appendix I I .  the  the  heat  i s generated  £ - axis, q sW  then  the  symmetrically with r e s p e c t to  heat  rate  to deal with  is  %^ > 2  w  i.e. q  Hence  =  a  q^dy'sin'/' 6  (  T  " »  for  T  )  =  -<YV+p)C  4b Kt t 1 W D l P  K > 0,  00  n  cos(  Tm(y-y-) b  +  2  l  n=l  n  (26)  and e  6(T-T ) =  -(pM +YV)  q dy'sinlf ^  l  P  B  Kt t  +(P-YV)5  +  2  u  (py -YV)  2: n=l  n  c o s (  ^n(y-y') b  l (27)  for from  £<  0.  T h e r e f o r e the  equations  (26)  and  temperature (27),  thus  distributions  follow  31  -(p+vV)C 4b,  -(p+yV)C 1  W  D. T  /  2b^  00  D  D  irnt  . ,  n=l  irny  N  b^  n  (28)  for  £ >  0,  and  irnt sinb;—)cos(r-^-) 2b b r  ,  T-T  L  ., 4b  .  D  m n  ir  °°  e'  r t  n  ny  n=l  n  (29)  for  £<  0.  Equations  (28)  and  (29)  give  the  maximum t e m p e r a t u r e ,  the  temperature  distribution  within  the  c u t t i n g zone  and  the  temperature  at  of  workpiece during  the  sawing  process 2.3.2  under  s o l u t i o n of  temperature  can  parallel be  conditions.  the  distribution  i f a plane  the  Penetration  under q u a s i - s t e a d y  temperature Thus,  point  quasi-steady  L a t e r a l Heat The  piece  any  heat  equation  f o r the  conditions furnishes at  any  to  p l o t t e d as  the  cross  the  s e c t i o n of  y-axis  worklateral the  plate.  i s considered,  a f u n c t i o n of  y.  The  the  most  32 critical origin; to  temperature K - 0 and  i s the  y = 0.  T-T  and  =  * V  K  equation  (28)  U-e-  Y  or  n=l  (29)  tion  (30)  d e p e n d s on  vergent. knowing  may  basis  workpiece  be  show t h a t  see  so  to a t h i n  Owing  being  )  Thermal  that  so h i g h as  i s of  Y V H  (30)  n  S  K  b  }  (31)  the  temperature  which  varia-  is rapidly  con-  terms of the  series,  temperature  rapidly  the  temperature  region  s u r r o u n d i n g the c u t t i n g  will zone.  Shock  to induce  the c a l c u l a t i o n  of  interest  Suppose  the  thermal  a  and  i t s shear  cut i s  O  to a high temperature  plate  becomes  }cos(^) l y (py -yv)  ( p y  n-  (24)  n  a few  the h i g h  layer  C  ny  inspecting  w i t h y and  Possible  stress  (31)  t h a t y - b-^ we  limited  2.3.3  2b  the c o s i n e s e r i e s  T h u s by  decreases  this  and  parallel  equation  n  n=l  Equations  plane  becomes  sin(-  =  source  b  4b, T-T  origin,  _ V)£  (p-YV)  V  at the  C o n s i d e r i n g the  the y - a x i s c o n t a i n i n g t h i s  ( p  be  temperature  the  g r a d i e n t , the cracking [10].  thermal  stress  thermal On i n the  here.  expansion  of  the m a t e r i a l  m o d u l u s i s G,  then,  the  33  dT shear s t r e s s due  t o the temperature  gradient  -j-  is  g i v e n by [ 1 0 ] .  = era  T  4?  ( 3 2 )  S i n c e T i s known from any of e q u a t i o n s (28) and Also  T  (29), T  max  can be c a l c u l a t e d from e q u a t i o n ( 3 2 ) .  can be c a l c u l a t e d  dT f o r (-77-) d£ max  compared w i t h the shear s t r e n g t h , m a t e r i a l and  >  T  (23) to (25) or  G  c s  w i l l c r a c k ahead of the c u t t i n g perature gradient occurs.  is  T  max  , of the  , the m a t e r i a l w i l l '  T  max  T  If  workpiece fail  and i t  zone where the l a r g e  tem-  T h i s mode of f a i l u r e by c r a c k -  i n g c o u l d be an i m p o r t a n t c r i t e r i o n f o r d e t e r m i n i n g the c u t t i n g mechanism i n f r i c t i o n F o r the metals shock  of ceramic  s e l e c t e d f o r the p r e s e n t study,  would not be  2.4 The  sawing  materials. thermal  applicable.  Heat P a r t i t i o n  Problem  2.4.1 I n t r o d u c t i o n Assuming no heat l o s s i n the c u t t i n g total saw  f r i c t i o n heat g e n e r a t e d  d i s k and the w o r k p i e c e .  i s dissipated  zone, through  I f t h i s t o t a l heat i s  then  where the s u b s c r i p t s D and W r e f e r  to the d i s k  and  the the q  T  workpiece  2.4.2  respectively.  Jaeger's  Treatment  Jaeger using  [21] d e a l t  the F o u r i e r  number  with  this  defined  heat  partition  problem  as  V c L  where V is  i s the p e r i p h e r a l  a characteristic  will  4K  J  speed  length  depend on t h e a c t u a l  of the contact area  saw a n d t h e w o r k p i e c e d u r i n g actual area  area of contact  contact, p  bodies  t h e mean f l o w  the sawing  zone.  pressure  Here c  between the The  the apparent  pressing  I f A i s the actual  pressing  and c  process.  from  of the load  together.  N the load  of contact  i s different  and i t i s a f u n c t i o n  tacting  o f t h e saw d i s k  t h e two b o d i e s  two  con-  area of together  o f t h e weaker m a t e r i a l ,  and  then  [53]  A  = £ P m  (35)  c = k/A~  (35a)  m  and  where k d e p e n d s on t h e s h a p e o f a s p e r i t y  used, f o r  example,  for a  k = 1/2  2.4.3 L i n g  and S a i b e l ' s  Based Ling  f o r a square  also  and S a i b e l  and l / / n ~  circle.  Treatment  on t h e F o u r i e r  [39] worked  number  out the value  as  parameter,  of the f r a c t i o n  o f the h e a t g e n e r a t e d  g o i n g i n t o the s t a t i o n a r y component  of  They d e f i n e d the F o u r i e r number i n  the s l i d i n g p a i r .  the  form  R =V8KcS—  (36)  m  F o r d i f f e r e n t r a n g e s o f R, t h e y o b t a i n e d e x p r e s s i o n s  for  the f r a c t i o n o f the h e a t g e n e r a t e d w h i c h i s d i s s i p a t e d t h r o u g h the s t a t i o n a r y member. fraction,  their results  I f f represents  a r e g i v e n by the  this  following:  (1) f o r R = 0  i (37) 1 + K m K s Where Km i s the t h e r m a l c o n d u c t i v i t y of the movinq member K i s t h a t o f the s t a t i o n a r y member. f  =  V J  ;  g  (2) f o r R > 5 f  =  (38)  1  K"m / TTT" ~K ~ s These two c a s e s were f i r s t d e r i v e d by B l o k who used a 2 1 +  /  square  R  source of heat of area c  to s i m p l i f y  m a t i c s i n v o l v e d w i t h o u t a f f e c t i n g the L i n g and S a i b e l m o d i f i e d (2)•  for R - 5 , f -  . K1  the  mathe-  fractions.  (2) t o r e a d (  3  9  )  36 They gap  also using  d e r i v e d the the  constitutes  fraction  same l i n e  f o r the remaining  o f r e a s o n i n g as  Blok.  uncovered  The  result  case  (3) f o r 0 - R -  5 (40) 1  + K K  2 T  m  s  [1 + 0 . 4 1 4 ( l - e  _ 1  *  3 R  )I(R) ]  where  eR  +1)R  (e  :  e  I(R) - R"  2  .  h  e -1)R  _n2 e  dri  de  2  2  (41)  and  e  I(R) 2.4.4  i s graphically Jaeger's The  (1) f o r f  =  /4K  m  (42)  S  evaluated.  Equivalence of Ling  and  corresponding derivations  L  T  < 0.1,  = 1 +  m K  L  Saibel's by  Jaeger  Approach [21]  are  small  (43)  (2) f o r L f  > 5,  T  L  large  T  =  1  (44)  K /I7R  1 + 0.5625/nr"  m  ~  K~ s  m (3) f o r 0.1 < L for  evaluating  2.4.5  large  developed  of Jaeger's  and L i n g  i n t e r e s t here  v a l u e s o f R.  suggests  that  a very  member w h i l e L i n g  considering Figure  large  the configurations  used  Jaeger's  may be n o t e d  here;  ing  the average  and  Saibel  by t h e s e  Jaeger  arrived  temperatures  ment o f e q u a t i o n  (44).  authors.  From t h i s  Figure, i t  point  of difference  a t the i n t e r f a c e whereas  on e q u a t i o n  temperature  calcu-  ( 3 9 ) and t h e c o m p l e -  These v a l u e s s h o u l d compare  to the c o r r e c t  Ling  temperatures.  comparable  i f the value of c i s accurately lead  by  a t h i s f o r m u l a by match-  made u s e o f t h e maximum  s h o u l d be b a s e d  resolved  f r a c t i o n s h o u l d be t h e c o m p l e Another  From t h e f o r e g o i n g ,  should  otherwise.  5 shows t h e s e c o n f i g u r a t i o n s . that  generated  or stationary  contradiction i s fully  and S a i b e l ' s .  and  predict  Formulae  derivation  o f the heat  the workpiece  and S a i b e l  ment o f L i n g  lations  used  f o r R > 5 or  case, Jaeger's  portion  through  apparent  becomes c l e a r  t o be  and S a i b e l ' s  i s the c o n d i t i o n  For this  w o u l d be d i s s i p a t e d  This  a graph  f.  Discussion Of  < 5, J a e g e r  calculated,  well  the values  temperature d i s t r i b u t i o n .  38 2.4.6  Proposed The  for  are b a s i c a l l y  the materials  in  friction  and  be  sawing,  rate.  accurately,  are developed  extent.  i n the s l i d i n g  configuration,  and t h e f e e d i n g  partition  formulae  on t h e t h e r m a l  many p a r a m e t e r s  Thus the properties  process.  are subject  t h e sawing  However,  to change  disk  thickness  In order t o evaluate the  some o f t h e s e p a r a m e t e r s  must  considered. Two c r i t e r i a  partition  may be u s e d  the workpiece  sawing  temperature  i n the c u t t i n g  equals the p e r i p h e r a l  temperature  zone of the  disk, (2) t h e a v e r a g e  the workpiece sawing  to evaluate the heat  formulae:  (1) t h e maximum in  dependent  involved  the workpiece size  heat  heat p a r t i t i o n  members o f s e m i - i n f i n i t e  of  viz.  Partition  foregoing  stationary  formulae  Heat  temperature  equals the p e r i p h e r a l  i n the c u t t i n g temperature  zone i n  o f the  disk. Based  on t h e s e c r i t e r i a  the formulae o b t a i n e d are  (1) 1 ffnt.  D  (a)  and,  (2 )  1 irnt. »  2b 1  1 + D  For  derivations  Because  dimensions  the whole workpiece, very  close  results  Note affected  by  for  and  in  u  n  Appendix  ted  that  p).  where  p  (b) s h o u l d  coefficient  temperature  section,  f q  T  "  with  give  (b) a r e  (re: formula the  formulae  used.  the  cutting  distribution  portion  to the workpiece  "  III  distributions.  1  If fusion  W  compared  Hence f o r t h e v a l u e o f h ' ,  I s h o u l d be  q  of the k e r f  t h e v a l u e s o f f i n ( a ) and  analysis  mechanism.  (b)  ( b j , see A p p e n d i x  f o r the temperature  the heat t r a n s f e r  transmitted  n  e q u a t i o n s ( a ) and  In c a l c u l a t i n g transfer  n=l  o f e q u a t i o n s ( a ) and  of the small  D  s i n (•  i n the  of the heat  d e p e n d s on  i s operating,  the  hea  genera-  cutting  then  sintf  i s the workpiece  density,  L,  the workpiece  latent  40 heat  of  f u s i o n and  V,  the  c u t t i n g mechanism, T ^ ' max be  given  cutting  is T  rate.  , ,. melting  Under  and  equation M  Trnt  m  n  sinf  <»  i s the  2.5  Speed  4b  High  D  melting  will  p o i n t of  out  a  stress  disk, expressions  stresses  would  shear  obtained  be  friction  p  the  i  n  (  2 b ,  \  n  (31' )  workpiece.  the  these is  see  the  zone  IV.  area  expressions.  defined  the  normal  Appendix  contact  stress  then  a n a l y s i s of  f o r both  obtained,  f o r c e s at using  s  n^r  ~  +  ^  x  T  Friction  Carrying  of  sin'/  ¥ B V D  w h e r e T^  and  ]  =  sawing  (31)  by  [fq T -T  this  friction and The  shear normal  may  easily  be  The  coefficient  as  = L  u  ( 4 5 )  N  where F The with  and  value  N  are  obtained  experimental  shown i n A p p e n d i x  the for  shear y  results II.  The  and  from based  normal  forces  equation  (45)  on  results  the from  respectively. should  agree  force analysis Appendix  II  show  41 that  w[ q - & + 3 ) a - & ) + 3 ( £ - £ ) ] G  F =  1  1  2  2  + R£  (46)  and  N = F  T  sxvrf  -  (47)  1 atair/  + R£  where a l l t h e c h a r a c t e r s a r e shown i n A p p e n d i x different and  V N c  values  o f F a n d N, a p l o t  c a n be c o m p a r e d  experimental  results  with  whence  of  Williams they  y  I I . For  a g a i n s t V" F c  and G i f f e n ' s [ 1 2 ]  derived  the e m p i r i c a l  formulae:  y  =  -0.45 K(V N) c  (48)  and y  -0.82 = K'(V F) c  2.6 Friction 2.6.1  Sawing  Mechanism —  Governing  Criteria  Introduction The c u t t i n g  2.6.2  mechanism w i l l  be based on two  criteri  (1)  the average temperature i n the c u t t i n g  zone,  (2)  the maximum temperature i n the c u t t i n g  zone.  Average If  Temperature  Criterion  the average temperature over the c u t t i n g  zone i  e q u a l t o the m e l t i n g p o i n t o f the workpiece m a t e r i a l ,  then  l a r g e - s c a l e m e l t i n g must take p l a c e i n the k e r f and the c u t t i n g mechanism i s f u s i o n . cutting ted  Now  i f we  consider  zone a r e a , the average temperature can be  using equation ( 2 5 ) ;  the evalua-  thus  q s±ni  T -T av  =— »  ( p ^ V ) ' [_  4b pKt ^ 1  1_2  C <P+YV)a/21 j  C O S H  w  +  _ -3(p- vH/2  _J_ _[ _ -(p-yV)£/2 (p-YV) t  1  e  r  i  +  e  y (pu -YV) n  n  1-  Y  e  2 cosh[(py +YV)£/2] N  (pu +YV)£ n  -(py - V)£/2 n  Y  e  -3(py -YV)£/2 n  (py - V)Jl n  Y  x sin(-rr—) l 2  b  (49)  43 or  u s i n g e q u a t i o n (28) o r (29)  2/ D s i n (• 2b, n t  8b T  av~ co T  4b  ~  l P  Kt t w  C D  D  2  +  77  Equations the  *D n t l  2 n u  (50) n  (49) and (50) e n a b l e t h e average temperature i n  cutting  zone t o be d e t e r m i n e d .  known, we can p r e d i c t whether  I f this  temperature i s  the c u t t i n g mechanism i s  f u s i o n or not. If  the average temperature i s l e s s than the  m e l t i n g p o i n t o f the workpiece m a t e r i a l , t h r e e  possibili-  t i e s may account f o r the c u t t i n g mechanism; d u c t i l e ture, b r i t t l e bilities cutting material  f r a c t u r e or burning.  depend  two p o s s i -  on the s t r e s s d i s t r i b u t i o n around the  zone and the i n i t i a l [42].  The f i r s t  frac-  The t h i r d  temperature o f the workpiece  p o s s i b i l i t y depends on the r e a c -  t i v i t y o f t h e workpiece m a t e r i a l w i t h the s u r r o u n d i n g medium. 2.6.3 Maximum Temperature  Criterion  The maximum temperature c r i t e r i o n in  further d e t a i l .  i s considered  Even though the maximum temperature  may r e a c h t h e m e l t i n g p o i n t o f the workpiece  material,  t h e r e may n o t be l a r g e - s c a l e m e l t i n g i n the c u t t i n g  zone.  This  statement  temperature  in sliding  micro-melting effect  on  cutting  2.6.4  If this  may  r e p o r t s on  "flash"  high  friction,  speed  h a v e as  macro-melting.  much  Thus the  remain w e l l  bulk  below  maximum t e m p e r a t u r e i n d i c a t e s  i s the  case,  one  of  may  the  three  account f o r  be  that  This  c o u l d be  ties  of  the  chemical  the  possibilithe  Criteria  of  friction  the  above  behaviour.  p r e d i c t i o n of  the  associated with high  or  low  from  affects  that of  the  on  the  material.  In  interfacial  i t s solid  related  sawing the  solid-liquid  depending  molten workpiece of  a  is Thus  i n a f u s i o n c u t t i n g mechanism  reactivity  different  and  mechanism d i s c u s s e d  type  lead to  will  2.7  the  sawing  instance,  also  In  c u t t i n g z o n e may  Studies  a particular  For  mechanisms.  i n s e c t i o n 2.6.2  Friction  studies  various  mechanism.  Each to  by  f o r c e as  i n the  discussed  out  rubbing s u r f a c e s does not  point while  melting. ties  of  friction  temperature melting  i s borne  friction mechanism.  friction  interface. viscous  proper-  addition,  phase w i l l  counterparts  and  the  be this  friction.  Summary The  disk stant  was  heat  solved  thickness  assumption  was  t r a n s f e r problem f o r the by  assuming  a thin  whose p e r i p h e r y justified  by  the  annular  friction disk  i s uniformly high  saw  with  heated.  P e c l e t number  conThis  involved.  45  The ment by layer  Rosenthal  of heat  the c u t t i n g being  this  mechanism  dent of  on  heat  and,  difference  the  different  A proposed meters  average  i n the  flame  affected  was  develop-  cutting.  The  material  property-wise.  criterion  problem  physical  i n their  A  i t was  f o r the  noted cutting  c o n s i d e r e d as and  the  were c o m p a r e d .  results  partition  was  properties  Saibel  configurations  was  which  cutting  m e c h a n i s m was  and  maximum t e m p e r a t u r e s  formulae  The  in their  accounts  mechanism has  appar-  been based  over  of  papers.  f o r a l l the  para-  developed. on  two  criteria:  the c u t t i n g  show u n d e r what c o n d i t i o n s f u s i o n , fracture  depen-  e x p l a i n e d i n terms  they used  The  or b r i t t l e  of  materials.  and  sawing  the  c o n s i d e r e d and  important  partition  Ling  heat  These c r i t e r i a ductile  an  of ceramic  ent  and  followed a  i s good b e c a u s e  badly  possibility  the m a t e r i a l  Jaeger  this  n o t be  c o u l d be  The  transfer  f o r welding  z o n e and  shock  heat  p e n e t r a t i o n i s c o n f i n e d t o the v i c i n i t y  cut w i l l  thermal that  workpiece  mechanisms c a n  be  zone. burning,  predicted.  CHAPTER I I I  APPARATUS,  INSTRUMENTATION  AND  EXPERIMENTAL METHODS  3.1  Apparatus  3.1.1  Introduction The  a p p a r a t u s c a n be d i v i d e d  i n t o three  sub-  assemblies v i z : (i) (ii) (iii)  the sawing the table  of  hardness  periphery  between and  retained  mechanism.  (Figures  6, 7 a n d 8)  made o f SP p l a t e  R o c k w e l l C48.  had a smooth  diameter  collars  nut.  bearings.  SPS A t l a s  6 inches  steel  outside  The r i g h t  easy removal  46  bearing  o f t h e saw.  It  shaft  diameter  These c o l l a r s  The s h a f t was mounted  steel  outer  a n d a 1 1/4 i n . b o r e .  i n s i d e diameter.  by a l o c k  a way t o p e r m i t  The d i s k  diameter  o n a one i n c h  inches  self-aligned  guide,  saw was a 14-gauge d i s k  two s u p p o r t i n g  1 1/4  and i t s d r i v e ,  and S h a f t  o f 14 i n c h e s  was m o u n t e d  and  the feeding  3.1.2 The Saw D i s k The  disk  were  i n three  was i n s t a l l e d i n  47 3.1.3  The  Disk  A  15  induction two  was  and  used  V-belts. attached  arrangement,  the  A to  disk  minute,  16,920 f e e t  per  minute  Table  The long, four  16  inches  inches  tudinal  at  cut  table. each  flanged  and  the  support about  inches  supported  the  disk,  arm.  the  disk  stands.  disk  speed  Along  long  by  7.  spindle 7,  by  short  the  two  and  were f i x e d on  forces  by  spindles  axis.  arms  to  bearings  the  the  perwere  two  disk  moments  moments were  was  left  shaft  s i m p l i f i e d the  eliminating  when t h e  above  spindle  made h o l l o w  c o n c e n t r i c i t y of  inch  aluminum  between was  inches  mounted  3 inches  two  A  30  3/8  bearings  arm  of  i t s longi-  spindle  forces  shaft  4,620  The  The  pulley  to  shaft.  supporting  table  disk  The  up  I t was  t o emerge  Figure  shown i n F i g u r e  reaction  the  14  the  a l u m i n u m was  centre-line.  slot  through  m/sec).  thick.  was  table of  inch  table  mounting b a l l  culation  made o f  1/4  minute  9)  The  to  upper-guard  8 and  86  7)  diameter  by  a peripheral  disk  of  and  stepped  the  to each  access  7,  disk  outside  f t / s e c or  w h i c h was  the  disk  shaft  allow  hand b e a r i n g s , mit  disk  to  side  attached  the  inches  giving  w i d e and  above  shaft  (282  centre-line, a  w i d e was the  table  6 and  to d r i v e 9-3/8  (Figures  (Figures per  r o t a t i o n was  per  The  Drive  3,540 r e v o l u t i o n s  the  revolutions  3.1.4  V-Belt  horsepower,  m o t o r was  s i z e B,  pulley  Motor  calof  considered  48 3.1.5  The W o r k p i e c e G u i d e s On  sisting The  o f three  These  motion  block  sheets  be a d j u s t e d  thickness  3.1.6  Arrangement  Figure  path  whole  two  ite  lined  with  arrangement attached  t o i t and  and T a b l e  t o make room f o r w o r k p i e c e  moved o n two b r a s s  with  Teflon  strip. finish  contact  with  between  workpiece  the guide  was made s m a l l e r  except  the small  arrangement  thrust  that could  Figure  11. The  accommodated  roller  quite  than  small.  i t s mounting  any s u r f a c e  t o the upper  any p o s s i b l e  guide  side  on t h e w o r k p i e c e , s e e  p u s h r o d was a t t a c h e d any c l i m b  and t h e  were  was v e r y  with  attached  eliminated  be i m p o s e d  The r o l l e r  s i d e s and t h e a r b o r -  guides  a n d s o i t was n o t i n c o n t a c t steel  rollers.  on a l l s i d e s  these  f o r the side block  mechanism.  The two p r o n g s o f t h e  device  This  between  10 shows t h e w o r k p i e c e m o u n t i n g  The f r i c t i o n  The  Force  and G u i d e  mount were g r o u n d  linings  block.  walls  widths.  The u p p e r  vertically  edges i n s l i d i n g  smooth.  workpiece  friction.  t o Reduce F r i c t i o n a l  assembly  was l i n e d  workpiece  to the t a b l e .  variation.  Workpiece  The  inner  t o c u t down s l i d i n g roller  con-  c u t i n them a l l o w e d f o r  a l s o had t h e i r  had a s t e e l  could  slots  attached  t o accommodate d i f f e r e n t  side blocks  arborite  with  9)  o f t h e t a b l e was a g u i d e  aluminum b l o c k s  two s i d e b l o c k s  lateral  it  the front part  (Figure  to a rocker  arm w h i c h  o f t h e w o r k p i e c e o n t h e saw.  For  49 added  s a f e t y , a guide  through Figure  i tprovided  plate  was p r o v i d e d .  limitation  A slotted  hole  t o any p o s s i b l e r o d c l i m b ,  12. The  whole  arrangement  resulted  i n very  negligible  friction. 3.1.7  Arrangement f o r Measuring A  table  disk.  outside  from  The r i n g  the v e r t i c a l  centre-line  was made o f m i l d inches  under the  3 3/16 i n c h e s  A weight  l o a d i n g pan a t t h e r e a r o f t h e t a b l e  to the scale  ring,  Figure  motor  speed  shaft  output  Sufficient zero  F i g u r e 13. was  weights  l o a d on t h e t a b l e  6.  Drive  and T r a n s m i s s i o n ,  6 a n d 8)  current  (D. C.) v a r i a b l e  was c o n t r o l l e d was c o n n e c t e d  with  speed  a Variac.  through  horse-  motor.  The  The m o t o r  V - b e l t s t o a 40 t o 1  reducer.  T h e C a r r i a g e a n d t h e Power S c r e w , The  diameter,  f e e d m e c h a n i s m was d r i v e n b y a 1 1/2  power d i r e c t  worm s p e e d  the table.  pan t o g i v e  The Feed Mechanism  The  inside  steel  o f the  by t h r e e  (Figures  3.1.9  was mounted on a s t a n d  f o r counterbalancing  strain  Forces  diameter  were a d d e d  3.1.8  ring  13.95 i n c h e s  sawing  used  strain  Table  t r a n s m i s s i o n gear  coupled  v i a a shear  threads  p e r i n c h power  through  a split  reducer  p i n arrangement  nut.  screw which The s p l i t  ( F i g u r e 6)  output  shaft  to a single i n t u r n drove  was  start  four  a carriage  n u t l o c a t e d under t h e  50 carriage the  enabled  power  used  speed c o u l d  could  3.1.10 The  be  front  with  Rod  and  With  push  fixed  a pin.  the  arrangement  o f 0 t o 3»6 i n c h e s  Measuring S t r a i n  the s t r a i n ' r i n g Figure  to a rocker The  a steel  arm  l e v e r was  a l o c a t i n g hole  retained  the p u l l e y  by  I n a d d i t i o n , the speed  Thrust  r o d and  o f t h e p u s h r o d was  with  continuously  the c a r r -  per  could  Ring,  14)  of the c a r r i a g e .  rigidly  shows t h e p u s h r o d  connected to a l e v e r  pivoted  fixed  drilled  bolt  14  were mounted i n  a t the p i n .  to a small  flat  a t the c e n t r e .  connection  between  arm  The  other  faced  metal  This  hole  the push r o d  and  w o r k p i e c e mount. The  mounted  ring  was  c a r r i a g e as shown i n F i g u r e  14.  feed  thrust, F  push  r o d was  lever was  be c h a n g e d  obtained.  Push  The  the  s p e e d and h e n c e  by c h a n g i n g t h e p u l l e y s .  (Figure  the  angular  f o r the tests, a speed range  increased  end  screw  speed c o n t r o l V a r i a c .  minute be  power  linear  motor  from  screw.  The iage  i t s engagement t o or d i s e n g a g e m e n t  arm,  strain  , applied  transmitted  and  the s t e e l  directly The  ball  through the r o c k e r  arm,  t o the s t r a i n r i n g  c a l i b r a t e d t o measure t h e f e e d  to  r e a c t i o n o f the  through the s t e e l  ball  attached  thrust F  0  to the the which  i n pounds.  3.2 I n s t r u m e n t a t i o n 3.2.1  Introduction The  lated  following  from measured  quantities  were m e a s u r e d  or  calcu-  data:  the  disk  speed,  the  table  the  feed  thrust,  the  feed  speed,  the  temperature  the  workpiece  weight,  the  workpiece  centre  the  friction  the  c o e f f i c i e n t of  the  workpiece.  force,  d i s t r i b u t i o n around  the  cutting  zone,  saw  and  3.2.2  The  Disk  and  the p r e s e n t  3.2.3  The  table  13.95  i n c h e s from shaft.  axis  of the d i s k  shaft  the  strain  disk  table  used  steel strain  of the disk  the  sliding  and,  friction  between  the  to check  the  speed  of the  saw  Table Force  axis  at  forces  project.  A mild the  normal  gravity,  Speed  A S t r o b o t a c was in  of  r i n g by  centre  about  the  line axis.  The hence  r i n g was  the v e r t i c a l table  was  underneath  plane of  pivoted  the  about  m u l t i p l i c a t i o n of the  the distance gave  attached  load  between the r i n g  the r e s u l t a n t  moment on  the  and  the  52 The  strain  "Daytronic" read  calibration  the  done of  differential  inches  was  shown by  Since  the  during  sawing,  record  of  The  the  the  (Model  3.2.4  The  f o r c e was  a "Brush"  ring  was  C.  used  transmitted  through  a rocker  to the  picked  to read  Appendix  VI. As  the  ball  displacement  The  up  i n the  From  strain by  in  (Model  directly  case  the  done  the  arm,  A  milli-  14. was  moving a chart  output  was  BL-932).  force  This  the  oscillo-  record.  workpiece  rod,  the  a lever  ring.  another  transducer.  calibration,  transducer  BF).  indicator  t r a n s m i t t e d from  was  pounds.  V.  Thrust  ball.  calibrated  300  The  "Daytronic"  to d r i v e a "Brush"  a steel  linear  a  Figure  amplifier  by  was  i n pounds.  of g r a v i t y  transformer  push rod  deflection  to  obtained.  D.  t h r u s t was  steel  calibrated  deflection  centre  the  another  (Model  see  a  i n Appendix  RD-2622-02) f o r a c o n t i n u o u s  Feed  The  ring  by  f o r c e c o n t i n u o u s l y v a r i e d and  amplified voltage  graph  was  s u p p l i e d by  indicator,  differential  by  the  indicator  workpiece  the  was  strain  the  p i c k e d up  i s described  transducer  transformer  was  transducer  load a p p l i e d to  the  d i s p l a y of  The  The  "in situ"  visual  amplified  deflection  transducer.  directly  Excitation  ring  The  force  arm  strain  "Daytronic" transducer  to  and ring  103A-80 was  also  thrust force applied i n "in situ"  i s shown i n  of  the  measurement o f  excitation  was  f u r n i s h e d by  table  another  force,  53 "Daytronic" 300  BF)  ring  differential  which  (Model  was  3.2.5  amplified  BL-932).  another  The  which gave  The  Feed  method time  s p e e d was  gently  used  involved  3.2.6  on  a  The  timing  cutting  output feed  the  temperature  was  attempted  1 t o 3um  PbS  fibre steel  cell  band  two  14.  switch This those  "Brush"  chart  Another  methods  Optic  bundle  using  the  agreed  width of the  the C u t t i n g  distribution  fibre  and  around  Zone,  the  infrared responsive light  radiation  p i p e s were  to the  each  bonded  stepped  responds  around  using  were made o f g l a s s  i n mild  The  by  the i n f r a r e d  a glass  ends,  drive  hence  o f a c u t and  Distribution  Measuring  of  to  Figure  and  to the  speed.  These  16)  fibres  sisted  the  and  to transmit  These  used  limits.  photoconductive c e l l .  used  amplifier  a marker  of the d i s k  15  zone  C.  s h e a r p i n mount d i s k .  the speed.  Temperature  (Figures  was  o b t a i n e d by  the l e n g t h  very reasonable The  strain  transformer output  " B r u s h " D.  voltage  to c a l c u l a t e  to c a l c u l a t e  within  of the  "Brush" o s c i l l o g r a p h ,  the count of r e v o l u t i o n s  was  The  (Model  Speed  feed  rested  recorder  both  indication  another  amplified  t h e power s c r e w .  PbS  a visual  by  channel of the  The  the  gave  displacement i n m i l l i - i n c h e s .  voltage  of  also  transformer indicator  light  cell. pipe  t o g e t h e r and  conmounted,  sleeves.  to i n f r a r e d light  radiation  spectrum.  within  However,  the  glass fibres  lym  band width  Figure  have c u t o f f s  o f t h e s p e c t r u m may  15 shows t h e b l o c k  perature  measurement  ( a ) The L i g h t P i p e The in  i t , each  light  pipes  light hole  calibrated (b)  an The  pipe  carrier  into  the block  and S c a n n e r  PbS c e l l  switch  was  d i s k was  venient  d i s t a n c e from  and C a r r i e r block  steel  tubes  holes.  ground  finish  a 52-position  i t saxis, hole.  sensor  inch.  there  were  drilled  was  spaced.  the  holes  (5-6 and 10-1)  spaced  five  switch  was  A  was  mounted  was m a c h i n e d  aluminum d i s k in i t .  the c e n t r e  The  plugged  centres  of the  were  The o t h e r  positions apart.  cell  positions.  52 p o s i t i o n s o n t h e s c a n n e r ,  c o u l d n o t be e v e n l y  spaced  switch.  hole  a t the hole  holes  positions apart.  pipe  s h a f t and a t a c o n -  A stepped  ten  switch  light  t h e PbS c e l l  up e x a c t l y w i t h  whenever t h e c e l l Since  i n turn  The mounted d i s k c l e a r e d t h e  The d i s k had t e n h o l e s lined  mounted  on a l l s u r f a c e s .  step  o f t h e t o p o f t h e b o x and a l i p p e d  holes  The  drilled  s w i t c h were e n c l o s e d i n  out  these  Block  Switch  o f t h e b o x b y 1/16  of  o f t h e tem-  which  Each  top  i t .  )•  had e i g h t h o l e s  mounted o n t h e s w i t c h  a small d r i l l e d  in  (1 t o 2um  connection  and a s c a n n e r  micarta  under  only  adaptor.  a l u m i n u m b o x w h i c h was scanner  be u s e d  ; thus  saw a p o i n t o n t h e w o r k p i e c e .  i t s own  The PbS C e l l The  Transmitters  into  with  diagram  2um  instrumentation.  were f i t t e d  were i n s e r t e d  a t about  the  Hence two p a i r s o f to correspond  pairs  of holes  When t h e c e l l  to s i x were was  55 rotated,  i t was  ( c ) The BAM The control a four  PbS c e l l  unit.  The b r i d g e  device.  response  signal  number.  A threeway  ten light  positions  One  One  was  another  chronized  switch  calibration pipe  light  the c e l l  If  t h e s y s t e m was  and  through  a cam were  pipe  the gain.  as a  with  cell  battery.  was  used  Each  light  pipe  step  switch.  switch  into  a power  switch  mounted  pipe  step.  The  switch pipe  was  syn-  positions  synchronized. a switch step  b o x on w h i c h  was  pro-  switches  were  a marker  t o c o n t r o l the sweeping a v a i l a b l e ranges  position  accomplished  and s o l i g h t were  There  (stop-start).  was  This  pipe  to the ten  light  scanner  pipe  response.  a two-way s w i t c h  position  pipe  the l i g h t  a t each  and l i g h t  them b a c k  Thus,  two-channel  to designate  out of synchronization,  T h e r e were f o u r for  the  used f o r the l i g h t  box and t h e c o n t r o l u n i t  to b r i n g  powered  was  by t h e c e l l .  the c e l l  in  vided  to the  controlled  p o s i t i o n s corresponding  52-position with  was  a 22-1/2 v o l t  designed  channel  m i g h t be f r o z e n by t h r o w i n g  by  drift  a BAM  o f t h e arms was  and t h e o t h e r  scanned  The  hole.  f e d through  power was  control unit  zero,  each  Unit  read-out  station:  was  reference  arm D. C. b r i d g e .  The  up w i t h  Bridge  output  The BAM  The C o n t r o l  were  lined  and C o n t r o l  resistance. (d)  exactly  switch  time.  that could  be s e t  a wide range o f /response s i g n a l s  could  be accommodated;  ature  readings  f o r instance  i n the range  o f 500  we  a n t i c i p a t e d temper-  t o 3000 d e g F .  56 Corresponding  t o each  calibrated  ranges.  know w h i c h  range  of the f o u r  These  gain  calibrated  any p a r t i c u l a r  r a n g e s were  four  r a n g e s e n a b l e d us t o  light  pipe reading  was  taken. (e)  The C h a r t  Recorder  A two-channel the  light  number. of  The c a l i b r a t i o n  The L i g h t For  BAM, was  inserted  the  light  in the  paper  light  pipe.  pipe signals  attenuators  were u s e d . t o which  which  limited  The  the  attenuator  t h e o u t p u t end o f  plugged.  G/G-30-K c h r o m e l - a l u m e l  16a.  registered  diagram  of each  The t h e r m o c o u p l e  principle  to a range  thermocouples their  would  moved  thermocouples'  thermocouple  signals  followed  moved  i s shown  were r e c o r d e d o n  different from  slowly,  distribution from  the f a c t  t h e saw, e a c h  of temperatures. sense  were  recorder.  into  instantaneous distance  the workpiece  thermocouples  of the temperature  by t h e t h e r m o c o u p l e s  as t h e w o r k p i e c e  subject  the  pipe  Attenuation  two c h a n n e l s o f t h e " B r u s h "  Since  to record  enabled the d e t e r m i n a t i o n  of the l i g h t  i n an a d a p t o r t u b e  The c i r c u i t  Figure  that,  used  Thermocouples  The  on  Pipe  p i p e was  Two used.  signals  gain range  very high  tracing  ( g ) The  was  p i p e o u t p u t and t h e c o r r e s p o n d i n g l i g h t  the c o n t r o l  (f)  "Brush" recorder  point  was  Hence t h e embedded temperatures the c u t t i n g  only  one i n c h  depending zone. per minute,  r e s p o n s e gave r e a d i n g s v e r y c l o s e  to the  actual run, in  temperature  Figure  distribution.  16(b)  shows t h e  a particular test  p o s i t i o n s of  the  thermocouples  the r i g .  3.3  Experimental  3.3.1  Procedure  t o be  cut.  copper, had  and  Data  Treatment  Introduction Preliminary The  brass,  a wide  coat  the  the  t e s t s were r u n  tests consisted steel  range of  From  at  For  these  edge o f  and  Ti-6A1-4V  melting tests,  the  saw  of  blade.  were c o n s i d e r e d  aluminum,  alloy.  These  1220  This  t o be  materials  cutting  a l u m i n u m and  i n v i o l e n t v i b r a t i o n of  materials  s e l e c t the  points:  s a w - w o r k p i e c e i n t e r f a c e and  resulted  to  to  materials  3000 deg  c o p p e r were f o u n d resulted in  the  the  F.  slipping  saw.  for  slipping  action  Thus  unsuitable  to  these friction  cutting. B r a s s was burr at  existed  the  top  at  of  cut the  the  bottom  of  rolled  thin  sparks, metal  Observation  showed v i s i b l e  the  zone,  cutting  blade. black was  of  The oxide the  the  material and  from  type.  no  very  little  burr  at  top  the  dull  the  red  sparks,  small  Cutting  w i d t h was  i t was  fusion  and  workpiece.  a bright cloud worm o f  smoothly w i t h  of  was  the the  that  observed  attended  bottom  and  by a  workpiece.  material  about  small  t e a r was  steel at  a  just  of  the  ahead  of  saw  kerf  was  heavily coated  conjectured  the  cutting  with  mechanism  58 A previously run test the  material  the  test  blade was  be c u t w i t h  c o n d i t i o n s were  was  damaged.  cut after The  was  could  made a s b r a s s ,  3.3.2  Specimen  severe  mild  three  plates  the  inches  T l steel  (a)  specimen,  four  Experimental  and T i - 6 A 1 - 4 V  alloy.  inch  long.  and l e n g t h were  holes  were  see F i g u r e  plates cut to Two  1/8  drilled  mild  inch  to hold  steel  thick. i t to  11.  Method  General All  instrumentation  m e a s u r e m e n t s were period light for  1/4  w i d e b y 12 i n c h e s  specimen c a r r i e r ,  3.3.3  f o r the experiment  Preparation  o f t h e same w i d t h  On e a c h  alloy  finished.  M o s t o f t h e s p e c i m e n s were size  However  t e s t s the t i t a n i u m  of materials steel,  showed  and t h e edge o f t h e saw  t e s t s were  choice  alloy  the equipment.  In subsequent  a l l other  final  on T i - 6 A 1 - 4 V  block  the c u t .  determined driving  thirty  carrier  A rough  from  checked  minutes. were  after  feed  and  force  a warm-up  The s p e c i m e n  installed  "a p r i o r i "  the s e t t i n g  temperature  and t h e  i n p o s i t i o n ready  speed  of the Variac  estimate  was  c o n t r o l of the  motor. Just  checked ranges  thoroughly  of at l e a s t pipe  f o r both  before  each  test  r u n , t h e BAM  and r e s e t i f n e c e s s a r y . f o r each  perature  light  control unit.  pipe  reference  The e x p e c t e d  p o s i t i o n were  was  temperature  s e t on t h e tem-  59 (b)  Brass The  minute. The  from The  was  was  s e t a t about  o p e r a t e d f o r an e i g h t  was  the k e r f  were c o l l e c t e d  temperature  were  or just  therefore  successfully  inside  inch per minute c u t .  A t the end o f the  t a k e n o u t f o r e x a m i n a t i o n a n d some  record  unknown w h e t h e r t h e t r u e  record  1/4  p i p e s c a n n e r s were o p e r a t e d .  the specimen  debris  by  speed  The saw was  light  cut,  feed  the"cell  for observation.  showed much d r i f t  temperature  thermal d r i f t  made t o e l i m i n a t e isolating  was  the c e l l  the scanner  t h e box t o o u t s i d e ,  the c e l l  and i t  distribution obtained.  Efforts  thermal d r i f t  switch c o i l  and  from  thermal d r i f t  was  eliminated. The the  force  experiment  o n b r a s s was  m e a s u r e m e n t s were r e c o r d e d o n t h e c h a r t .  temperature  record  was  o b t a i n e d from  instrumentation  sensitivity  temperature  registered.  into  was  was  the t h e o r e t i c a l  tion.  From t h i s ,  was  from  cutting.  Hence  for  t h e i n s t r u m e n t a t i o n was  The mechanism  the temperatures  material  from  f o r b r a s s was  machining, examination  Figure showed  17.  and s t i l l  f o r temperature that  were f e d  the f r i c t i o n  force  the theory f o r f u s i o n fusion  s e t up. showed  t h e same a s t h a t In a d d i t i o n  no  distribu-  were much b e l o w  the k e r f  No  The  The m e a s u r e d f o r c e s  programmes  predicted  the t e s t .  increased  i t became o b v i o u s  much b e l o w t h a t  which  r e p e a t e d making sure  that  the c u t t i n g  i n conventional  the m e t a l l o g r a p h i c  no h i g h t e m p e r a t u r e  effect  on t h e  60 friction-cut (c)  edge, F i g u r e  Recalibration  of  18.  Temperature  Instrumentation  for  Higher  Sensitivity An make t h e ever, 2ym  elaborate  temperature  owing  to  the  wavelength  calculated below  650  to  be  deg  (d)  Force  and  screwed  650 not  deg be  push r o d  onto  i t and  t h i s piece  to  retain  up  or  up  expected  forces.  (e)  Steel  operated. minute.  The  point.  to  plate  The  a  feed  light No  to  an  100  mild  about was  temperature  This  reading  roll  a  ground  flat  The so  disturbing the  on  thick  small  a steel ball.  agreed  with  the  Teflon  metal  recessed pushed  that the  table  revised.  the  l b f w h i c h was  s t e e l was set  focussed was  beyond  installed at on  about the  registered.  1/4  highest The  of rod  force.  thrust  the  and  hole  push  table  and  piece  end  i n s i g n i f i c a n t f r a c t i o n of  s p e e d was pipe  to  small  had  arrangement reduced  forces,  1/4"  had  down w i t h o u t  interaction  Mild  Hence any  designed  thrust  This  total  at  temperature  i n s t r u m e n t a t i o n was  the  ride  forces  cut-off  registered.  measuring  s p e c i m e n c a r r i e r was  could  F.  cut-off  How-  observations.  force  i t s centre  ture  about  the  to  sensitive.  transmission  infrared,  s p e c i m e n c a r r i e r was  path  the  pipe  more  followed  Measurement  The  at  light  F could  experimental  new  instrumentation  i n the  the  A  c a l i b r a t i o n p r o c e d u r e was  the  range  the  saw  inch  per  of  tempera-  forces  were  61  t a k e n and f e d i n t o the  theoretical  found t h a t t h e s e f o r c e s  programme  was  gave r i s e t o t e m p e r a t u r e d i s t r i -  b u t i o n s below the m e l t i n g p o i n t o f s t e e l zone,  and i t  f o r w h i c h the p i p e was  i n the  cutting  calibrated.  A second t e s t was c a r r i e d o u t whereby one o f more s e n s i t i v e expected a short  the  l i g h t p i p e s was put i n the p o s i t i o n o f  highest  temperature.  the  A r e a d i n g was r e g i s t e r e d  time o n l y , F i g u r e 19.  for  T h i s b e h a v i o u r was due  to  the  "worm" o f t h i n m e t a l on t o p o f the w o r k p i e c e b l o c k i n g  the  l i g h t p i p e , F i g u r e 20.  tered  The h i g h e s t  temperature  regis-  was 1050 deg F . S i n c e we c o u l d n o t e l i m i n a t e the worm due t o  •angular cuts.  the  sawing p o s i t i o n , we d e c i d e d t o make some edge F i g u r e 21(a)  shows the edge c u t c o n f i g u r a t i o n .  The  w o r k p i e c e was shaped and mounted on the c a r r i e r i n such a way t h a t the w i d t h o f the m a t e r i a l f e d t o the same as the b l a d e t h i c k n e s s .  saw was  the  Thus, a l i g h t pipe f i x e d  to  the w o r k p i e c e c a r r i e r saw an a r e a o f the edge b e i n g  cut.  T h i s arrangement e n a b l e d  the  r a d i a t i o n from the we o b t a i n e d  the  the  l i g h t p i p e to t r a n s m i t  a r e a to the PbS c e l l  sensor  and  thereby  temperature d i s t r i b u t i o n along the c u t  The edge c u t s gave maximum t e m p e r a t u r e s o f the magnitude  as the t e m p e r a t u r e o b t a i n e d when the l i g h t  was f o c u s s e d on t o p o f the p l a t e ture area.  a t the h i g h e s t  I t was o b s e r v e d t h a t the k e r f  p l a c e d s i d e w a y s i n s t e a d o f g o i n g under the k e r f  m a t e r i a l p a r t i a l l y b l o c k e d the  Figure 2 K b ) .  edge. same  pipe  tempera-  m a t e r i a l was d i s -  the w o r k p i e c e and so l i g h t pipe  entry,  62 The  forces  to  the heat  in  the c u t t i n g  o b t a i n e d from  transfer zone  theory indicated was  Metallographic cut  e d g e and  the k e r f  microstructures edge  indicated  order  as  that  temperature,  The obtained. cutting right (f)  predicted  the  cutting  light  observation  was  Distribution obvious from  temperature zone  zone  and  cutting  axis  on  was  i n . i n size  plate. size)  The fitted Both  plates  The cut  same This  times  the  t h e same r e s u l t colour  was  near  the  p i p e r e a d i n g s were o f  the  the  Using  Thermocouples  the t e s t s  was  might  at d i f f e r e n t  h o l e was  of the  a half  of the s t e e l  the l i g h t  were d r i l l e d  each  t o 28.  F.  the  measurements.  and  repeated but  easily  pre-selected test  (using  drill)  1/8  inch  localized picked  #76  and  of the  thermocouple  and  1/4  were c u t w i t h C h r o m e l - A l u m e l  inch  that in  by  the  holes  from Each  the depth  the hole  of  specimen  (0.01  the h o l e s d r i l l e d  thick  up  distances  specimens.  midway t o t h e t h i c k n e s s  30-gauge C r / A l  above  T h e r e f o r e thermocouple  subsequent  snugly i n t o  reported  very highly  n o t be  pipe instrumentation.  .02  one  deg  the f r i c t i o n  r e a c h e d was  about  o f 1600  magnitude.  Temperature  high  24  and  the f o r c e  temperature  temperature.  e d g e c u t was  suggested  the  applied  t h e n made o f  Figures  material  from  measured  order of  the  materials,  h o w e v e r , was  zone  It  e x a m i n a t i o n s were  of the k e r f  Visual  that  when  i n the neighbourhood  the temperature  experimentally  these tests  i n . wire  i n the  thick  workpiece.  mild  thermocouples  steel  embedded  63 in  the d r i l l e d  curves 22  Thus t e m p e r a t u r e  were o b t a i n e d d u r i n g  shows  the c u t t i n g  distributions Tl also  holes.  shown  steel,  temperature  the c u t t i n g  forces  associated  Ti-6A1-4V  process.  Figure  with the temperature  23.  i n Figure  c u t with thermocouple  distribution  alloy  and l e a d e d b r a s s were  attachments  to r e g i s t e r  the  distributions,  (g) M e t a l l o g r a p h y Microstructural mild  steel,  T l steel,  Photographs material  showing  at either  .For  brass Figure  the  friction  the  microstructures  3.3.4  18 shows  of chart  distances  from  plates,  the time  allowed  for i n interpreting  the  workpiece.  done.  i n each  were  taken.  structure  24 t o 38  and  show  were o b t a i n e d f r o m  and t h e t e m p e r a t u r e were i n s e r t e d  some a l l o w a n c e was  c u t i n a workpiece.  since  were  materials.  records  initial  required  Figures  f o r the other  specimen,  cuts  of interest  t h e "as r e c e i v e d "  the thermocouples  of each  subsequent  parts  400 o r 800 m a g n i f i c a t i o n  run; the f o r c e s  Since  longitudinal  different  alloy  from the  o f Data  sets  experimental  plane  b r a s s and Ti-6A1-4V  c u t edge s t r u c t u r e .  Treatment Two  examinations of materials  t o c u t 1/8  on a t e s t t h e saw was  the c u t t i n g  to the mid-  given  f o r the  i n 1/4  o f the workpiece  the temperature  piece,  distribution.  zone f o r e v e r y  For instance, inch  no t i m e  already  each  inch was  records.  allowance  i n the k e r f ;  For  was  well  into  64 It reached. the of  shows t h a t some t i m e  forces built  up  to  were  the be  t o be  force  tailend  friction  subroutine  theoretical  gramme and  hence  b a s e d on  programme was  the  light  1 to  5  the  data  the  the  friction  measured  easily  table  converted  for generating  to  the  distribution.  f o r c e from  i n the  the  this  cutting  temperature  subroutine  zone  to the  furnished  main  distribution  pro-  i n the  work-  obtained. temperature  t h e r m o c o u p l e s were  mental  w r i t t e n to c a l c u l a t e  This  friction  The  y-axis  before  a n a l y s i s of  i n a m a i n programme u s e d  generated  direction  In  included i n Tables  coefficient  temperature  The  was  elapsed  be  done.  thrust forces.  piece  state to  state value.  i f statistical  programme was  and  heat  steady  results  disregarded  A  the  seconds f o r steady  22  should  a  a few  Figure  this  and  took  of  used  results.  records  tabulated against  feed with as  distribution  points along  parameters.  Tables  from  the  the  distance  in  lines  parallel  to  1 to  5 give  the  the  experi-  CHAPTER I V  RESULTS AND  4.1 F o r c e s  and F r i c t i o n  Table in  vary  considerably  material,  Coefficient  6 contains  the experiments.  t h e summary  from  test  to test.  the c a l c u l a t e d f r i c t i o n  whereas the normal  trend  was  obtained  experiments  4340 s t e e l  were a b o u t  4340  but the transverse  steel  remained  essentially  obtained  decreased  cities about  increased. 50 p e r c e n t  90,000 f t / m i n  three  for cutting velocity  t o 120,000 f t / m i n . i s much g r e a t e r  velocity  o f 16920 f t / m i n of r e s u l t s  the heat  used  i s only  generation  cases.  force) They  as t h e c u t t i n g  these  forces  also  velo-  decreased  increase  from  Vaughn and K r u e c k ' s  than  the f r i c t i o n  i n the present valid  saw  study  and so  i f a p h y s i c a l law  i n the c u t t i n g zone. 65  treated  of the annealed  (friction  i n both  F o r example,  range  governs  forces  cutting forces  velocity  comparison  those  same  techniques.  f o r c u t t i n g heat  times  t h e same  This  con-  [54] i n t h e i r  machining  recorded  forces  f o r c e s were n e a r l y  by Vaughn and Krueck  they  obtained  B u t f o r t h e same  forces fluctuated.  on u l t r a - h i g h - s p e e d  normal f o r c e s  o f the f o r c e s  The m e a s u r e d t a b l e and t h r u s t  stant  The  DISCUSSION  Vaughn  66 and  Krueck  proposed  such  a theory  which  i s dealt  with  below. From T a b l e affect  both  6, w o r k p i e c e  the t a b l e  and t h r u s t  much e f f e c t on t h e f r i c t i o n inch in  mild  close  steel  workpieces  agreement.  for  the TI s t e e l  for  t h e same f e e d  tially steel than  Brass  the t a b l e  but Ti-6A1-4V  alloy  friction  of f r i c t i o n  those  forces  of the mild forces  forces  were  steel, essen-  were l e s s  r e g i s t e r e d higher  forces  than f o r  friction  forces  c o e f f i c i e n t s showed d e p e n d e n c e on  and m a t e r i a l  were a l s o  slightly  thickness.  thick  speed  per minute,  o f 1/4 i n c h  similar  to those  minute.  arose  from  constant  steel  speed.  was c u t a t a  that  the feed  the f r i c t i o n  speed.  coefficient  columns  9 a n d 10 f r o m  the  literature.  the  present  values  speed  i s nearly  d i f f e r e n t authors  D i r e c t comparison  friction  studies,  some  given i n  as r e p o r t e d i n  of these  greatly.  forces  6 shows  f o r the conditions  differ  inch  whereas t h e c u t t i n g  Column 8 i n T a b l e  conditions  were  coefficient  force  i n v e s t i g a t i o n i s of l i t t l e  experimental  feed  c u t a t a b o u t one  The v a r i a t i o n i n t h e f r i c t i o n the f a c t  For instance,  the c u t t i n g forces  o f the TI s t e e l  friction  high  mild  f o r t h e same m a t e r i a l  with  Friction coefficients  a f f e c t e d by f e e d  when t h e 1/4 i n c h  of  and 1/4  steel.  material  vary  b u t does n o t g i v e  and t h r u s t  the f r i c t i o n friction  c a n be s e e n t o  The 1/8 i n c h  gave v a l u e s  While  speed,  forces  force.  were a b o u t h a l f  t h e same.  The  per  thickness  value  values  with  because the  I n most  cases  the normal f o r c e s a r e  67 much s m a l l e r  than  present  showing  work The  lower tion  forces  f a c t o r s o f 10  involved  than those of  Shaw  1.85  and  [55] r e p o r t e d friction  cutting  1020  steel  materials  i n this  under  and  over  0.4  lower.  sawing  o f 281  the f r i c t i o n  the h i g h e s t  much  the  fric-  Finnie  c o e f f i c i e n t s o f 0.88  w i t h 18-4-1HSS t o o l .  study,  are  friction  t o 775 l b f F o r a l l the  coefficient  force  to  was  was  slightly  13 l b f .  friction  independent cal  difference.  For instance  i n the range  From t h e r e s u l t s h e r e 56]  t o 50  i n the  m e t a l c u t t i n g and  friction  forces  reported  in friction  low-speed  c o e f f i c i e n t s are also  and  in  the c u t t i n g f o r c e s  coefficient of  Amonton's  load  and  laws.  and  at high  elsewhere speed  friction  hence d e v i a t e s  The  parameter  [13,  from  54,  23,  i s not  the  classi-  o f c o r r e l a t i o n used  by  1/2 some a u t h o r s normal  load  parameter vailing normal  and  load of  i s the q u a n t i t y  the  sliding  i n the  literature  U where W i s t h e The  range  i s below  of  that  this pre-  i n v e s t i g a t i o n whereby W i s t h e  i n the c u t t i n g the  speed.  W  zone  and  U  the  peripheral  saw.  Distribution  Theoretical Two  ature  U,  reported  Temperature  4.2.1  23]  i n the present  velocity  4.2  [13,  approaches  distribution.  were f o l l o w e d  In the f i r s t  to obtain  approach  a  the  temper-  friction  68 force  was  assumed  assumed.  The  and  former  i n the  p r o v i d e d the  the c u t t i n g  zone which  temperature  distribution  t o be  carried  imposed  out.  a limit  calculation  enabled  The  on  the  of the  second,  fusion  total  latter,  the  saw  and  assuming  of  the  fusion  maximum t e m p e r a t u r e force  was  heat generated  the c a l c u l a t i o n  i n both  friction  cutting  required  in  the workpiece  cutting,  and  led to  the  for  fusion  and  assuming  cutting. Using uniform  heat  the f u s i o n transfer  cutting  coefficient  temperature  distributions  brass,  steel  The  mild  brass  titanium  alloy  especially  close  these  view  the h i g h  the  low  of  the  total  dissipated fractions  of  heat  titanium Table  and  3 per c e n t  respectively.  The  partition  perature zone.  of  the  saw  Hauptmann and  transfer  blade  equation with  whereas  and  brass  7 that  about  Table  alloy  10  lay  and  per  It cent  be  corresponding are  6 per  on m a t c h i n g  the workpiece  the  in  8.  b r a s s would  whereas the  based  [27]  expected  of the  7.  gradients  mild steel  see  titanium  was  Ramsey  The  alloy,  i n sawing  the workpiece  f o r mild steel  heat  zone.  the  shown i n T a b l e  T h i s w o u l d be  generated  through  are  a  axis f o r  temperature  conductivity  from  the workpiece,  steep temperature  cutting  thermal  t o see  alloy  uniform  extremes.  conductivity  interesting  and  fairly  over  the c u t t i n g  Ti-6A1-4V  to the  two  is  along  showed f a i r l y  between of  and  displayed  approach,  the  i n the  s o l v e d the d i s k  a v e r y g e n e r a l method.  tem-  cutting  heat  Yu's  cent  69 solution  o f t h e same e q u a t i o n a s s u m e d u n i f o r m  temperature using  f o r the d i s k .  uniform peripheral  peripheral Hauptmann criteria.  Table  matchings.  forces  ments i n v o l v e d  in  this  cutting  disk,  from  was  particular  were u s e d  temperature  cent for  length  heat  partition  o f t h e saw b l a d e  than  T l steel  width  c  9.  effect  results  give  f o r the  very  solution,  w o u l d be  argumuch  preferred  calculations. partition  heat  formulae  alloy  s o u r c e s . The  (35a) gave  fairly  compared w i t h t h e of mild  i t t o be a b o u t  (the apparent  and T i - 6 A 1 - 4 V  Thus,  In the case  showed  5 per cent but f o r brass The  Table  of  fractions  u s i n g values of the F o u r i e r  i n equation  formula.  these  to o b t a i n the numerical  f o r square  distribution  the c a l c u l a t i o n  both  c  matching  o f the complex  and S a i b e l ' s  f o r some c a l c u l a t i o n s  characteristic  steel,  solutions  and,  predicted  the  other.  for practical  and, L i n g  f o r the  temperature  a n d Ramsey's  number w h i c h were c a l c u l a t e d  proposed  each  required  problem  solution  solutions,  However, b e c a u s e  time  of uniform  o b t a i n e d from  Hence Yu's m o d i f i e d s o l u t i o n  Jaeger's  good  zone  t h e two  i n Hauptmann  computer  results.  the r e s u l t s  slightly  rotating  instead  were u s e d  t h e same f o r t h e two  agreeable r e s u l t s .  greater  solution  9 shows  differ  high-speed  flux  m o d i f i e d by-  The m o d i f i e d Y u ' s  The a v e r a g e  were e x a c t l y and  heat  temperature. and Ramsey's  Y u ' s method was  peripheral  contact  5 per  length);  t h e v a l u e s were  less  t h e v a l u e was g r e a t e r .  o f f e e d speed  F o r t h e same m a t e r i a l ,  c o u l d be o b s e r v e d the heat  generated  from was  70 only  m i l d l y i n f l u e n c e d by  good  agreement with  Because brass, for  of  this,  mild  the  4.2.2  titanium  and  TI  This  Krueck's  speed  steel  was  but  is in  very  observations  employed  h a l f the  for  speed  [54].  sawing was  used  alloy.  Experimental 19  temperature  i n the  calibration  chart  was  1050  observed  deg  i n the  shows a  f o r the F.  region  ted  that  temperatures  Figure ced.  the 20  This kerf  right  shows how  recorded.  explained  ducting  kerf  heat  glow  In  less  the  a dull  the  behaved  the  workpiece  the the  temperaglow and  colour  was from  indica-  instrumentation  recorded  magnitude.  kerf  than  red  c u t t i n g zone this  of  From  maximum  m a t e r i a l was  the  light  650  deg  pipe  F no  like  entry  and  the  interval light  pipe  a f i n r a p i d l y con-  c u t t i n g zone. disappeared  displa-  s i g n a l would  short response  a d d i t i o n to blocking  material  away f r o m  i n the  of  the  be  the  the  order  was  entry,  to  trace  edge c u t .  cut,  measuring  i t s temperature  19.  an  pipe,  material blocked  This  response  scale for steel  if  seen i n F i g u r e  of  the  close  temperature of  pipe  light  During  colour-temperature the  light  c u t t i n g zone  the  red  speed.  V a u g h n and  a constant  steel  Figure  ture  feed  Hence  after  a  the  dull  short  interval. Tables  1 to  tribution  as  predicted  results  results  5 show t h e  m e a s u r e d by based  results  of  temperature  t h e r m o c o u p l e s compared on  the  cutting forces.  were a l s o p l o t t e d i n F i g u r e s  39  to  45.  with These  disthe  71 4.2.3 C o m p a r i s o n Figure for  leaded  one  the c u t t i n g  f o r c e s were should In  In  in  the case  results  22) s o t h e s e  comparisons.  distribution  the p o i n t s are not too c l o s e  T h e same t r e n d s  a r e shown b y T I alloy,  F i g u r e 45.  near  the c u t t i n g  t h e worms f o r m e d b e h a v e d from  the temperature experimental  the c u t t i n g  temperatures  a t p o i n t s remote  something  similar theory,  a t the c u t t i n g  zone  from  like  zone.  i n the c u t t i n g  zone  fins  This resulting  and  the c u t t i n g  theoretizone  to the S a i n t Venant's t h a t i s , the heat  flux  does n o t s u b s t a n t i a l l y  distribution  zone.  recorded.  agreement o f t h e e x p e r i m e n t a l  temperature  zone.  first  when t h e  (see Figure  temperature  are g e n e r a l l y noted  in elasticity  bution the  zone.  when  away some h e a t  the lower  suggests ple  steel  theory  of s t e e l ,  lower  The cal  unsteady  The  g r e a t e s t d i s c r e p a n c i e s between t h e t h e o r y and  experiment  might  predictions.  p o i n t s were o b t a i n e d  F i g u r e s 42 t o 44 and T i - 6 A 1 - 4 V  conducting  imposed.  a s a t y = 0.2 i n . , 0.3 i n . , 0.5 i n . , and  well with  The the  of the r e s u l t s  z o n e ; y > 0.2 i n . , t h e e x p e r i m e n t a l  still  the mild  the c u t t i n g  steel,  Results  F i g u r e s 40 and 41 t h e a g r e e m e n t a t y = 0.1 i n . ,  n o t as good  agrees  plots points  be d i s r e g a r d e d i n m a k i n g  1.0 i n . ; t h u s  to  the experimental  were c l o s e t o t h e t h e o r e t i c a l  cutting  is  with  o r two e x p e r i m e n t a l  points  and E x p e r i m e n t a l  39 shows t h e t h e o r e t i c a l  brass  Remote f r o m points  of Theoretical  princidistriaffect  a t p o i n t s remote from t h e  72 In F i g u r e the  cutting  highest value the  zone,  registered  4.2.4  The  general  and  the t h e o r e t i c a l  Heat  sawing  deg  deg  between  F.  This with  the e x p e r i m e n t a l suggests that  zone  Works  or temperature had  not been  However,  distribution  worked  o u t by  some c o m p a r i s o n s  the temperature  process.  speed used  thermocouples the t e s t  be  i n Figure  F compared study.  under was  5.8  and  m/sec.  o b t a i n e d by  well  than those  explained  partially  used  test  results  joining  w i t h t h e 1116  deg  rot-  lbf).  The  distribution  showed t h e  same  the e x p e r i m e n t a l of  F measured  1292 i n the  obtained steeper temperature  shown i n F i g u r e by  a  embedded  H i s maximum t e m p e r a t u r e  Savitskii  gradients  He  (4.4  the temperature  the p l o t t e d  43.  o f 2 kg  pre-  distribu-  used c y l i n d r i c a l  a load  f o r measuring  piece  as would  He  in  with  (.1575 i n ) d i a m e t e r p r e s s e d a g a i n s t wheel  the  justified.  [57] measured  geometry.  in  the  F measured  Savitskii  abrasive  present  1116  possible.  o f 4 mm  points  that  p r o c e s s e s are  relative  shape  1050  inserted  machining  i n a grinding  ating  only  predictions  with Other  investigators.  pieces  to note  are s u b s t a n t i a l l y  distribution  the f r i c t i o n  related  the  agreement  assumptions  Comparison  vious  as  was  was  pipe.  theoretical  deg  interesting  i s i n t h e same r a n g e  results  in  where a t h e r m o c o u p l e  i t was  temperature  light  tion  43  the d i f f e r e n c e  In the c o n c l u s i o n  43.  This  i n test  o f h i s work,  might  be  specimen  Savitskii  73  stated  t h a t the g r i n d i n g  temperature o f hardened  s t e e l s was  h i g h e r than t h a t o f u n t r e a t e d s t e e l s .  T h i s agreed w e l l w i t h  Vaughn and Krueck's  In the present  [54] observation.  s t u d y , t h e T l s t e e l a t t a i n e d a s l i g h t l y h i g h e r temperature a t the c u t t i n g  zone t h a n t h e m i l d s t e e l , F i g u r e s 41 and 4 3 .  The T l s t e e l has lower thermal c o n d u c t i v i t y than m i l d steel  ( T a b l e 8) and t h i s might account f o r the h i g h e r heat  c o n c e n t r a t i o n i n the T l s t e e l c u t t i n g result ^  zone which would  i n a higher temperature. Boothroyd  [ 5 8 ] o b t a i n e d temperature  distribution  c o n t o u r s i n a metal c u t t i n g o p e r a t i o n by u s i n g photographic techniques. o r 1292 deg F.  infrared  The maximum c o n t o u r was 700 deg C  T h i s temperature i s the same as t h e maximum  o b t a i n e d i n S a v i t s k i i ' s work.  Boothroyd p r e h e a t e d the  work t o between 350 and 500 deg C.  T h i s made c u t t i n g  e a s i e r b u t t h e maximum temperature was s t i l l T h i s f i g u r e i s lower than the t h e o r e t i c a l l y  700 deg C. predicted  maximum i n t h i s work but agreed w e l l w i t h t h e thermocouple measured v a l u e o f 1116 deg F i n T a b l e 4. e x p e c t e d t h a t temperature  I t s h o u l d be  i n v o l v e d i n low-speed  s h o u l d be lower than i n h i g h - s p e e d P a r k e r and M a r s h a l l  machining  sawing.  [ 5 9 ] worked w i t h a view t o  c u t t i n g down t h e d e t e r i o r a t i o n o f brake shoes i n t h e r a i l w a y brake-wheel  system.  a t t a i n e d a t the s l i d i n g operation.  They s t u d i e d t h e temperature  i n t e r f a c e during the braking  U s i n g a PbS c e l l  pyrometer,  they r e c o r d e d  temperatures o f 1470 deg F o r g r e a t e r a t t h e s l i d i n g  74 interface.  This  temperature l i e s i n the range o f  p r e d i c t e d f r i c t i o n c u t t i n g zone  4.3  the  temperatures.  Metallography  4.3.1  M i l d S t e e l [60, Figure  o f the  24  shows the f e r r i t e - p e a r l i t e  "as r e c e i v e d " m i l d  normalized  steel.  The  ( a i r cooled) c o n d i t i o n .  s t r u c t u r e o f the piece  61]  material i s i n  Figure  "worm" which came from the  i n the c u t t i n g zone.  The  microstructure  25  i s the  the micro-  top o f the work-  microstructure  i n the  figure  suggests t h a t the m a t e r i a l underwent p r o c e s s a n n e a l i n g the a n n e a l i n g m a t e r i a l was  temperature b e i n g  about 1200  Figures  26 and  present  taken c l o s e to one  oxide f i l m l a y e r . Figure  27(a)  Figure  s t r u c t u r e was steel  the  above the AC  t o produce f i n e a fine  26(b)  t a k e n c l o s e t o one  an o x i d e f i l m l a y e r .  The  result 1  i n the  very  The spheroidize  pearlite.  27 are the m i c r o s t r u c t u r e s  k e r f m a t e r i a l viewed edgewise and 26(a)  F.  m a i n t a i n e d f o r a s u f f i c i e n t time t o  the c e m e n t i t e p l a t e s o r i g i n a l l y  Figure  deg  of  s i d e o f the edge shows an i s the i n t e r i o r  structure.  edge of the w i d t h a l s o shows fine-grained  ferrite-pearlite the o r i g i n a l  temperature f o r a s h o r t i n t e r v a l  ferrite-pearlite  the  breadthwise r e s p e c t i v e l y .  of r a p i d l y heating  austenite.  with  such  T h i s on c o o l i n g t r a n s f o r m e d  structure.  The  material  was  mild  to  as  75 p r o b a b l y heated o n l y and  slightly  above A C  1  o r between AC^  AC^; t h a t i s about 1600 deg F. Figure  28 shows the m i c r o s t r u c t u r e  c l o s e t o the f r i c t i o n c u t edge.  A very  o f the m a t e r i a l  narrow band o f  the m a t e r i a l a t t h e edge underwent the same t r e a t m e n t as the k e r f m a t e r i a l .  T h i s depth o f m a t e r i a l  i s only  about  0.00246 i n . 4.3.2 T l S t e e l ( A l l o y S t r u c t u r a l S t e e l ) Figure  [60,61,63,66] -  29 shows the m i c r o s t r u c t u r e  o f t h e "as  r e c e i v e d " T l s t e e l which c o n s i s t s o f f e r r i t e and p e a r l i t e . Figure  30 shows the s t r u c t u r e o f t h e m a t e r i a l . c l o s e  flame c u t edge o f the T l s t e e l . f u l l y martensitic  structure.  t o the  This figure reveals a  Hence d u r i n g  the flame  c u t t i n g o f the edge, t h e m a t e r i a l must have been heated above the ACj temperature, t h a t i s , above 1670 deg F and p r o b a b l y to around 1700 deg F and then r a p i d l y c o o l e d by the mass o f the a d j a c e n t  metal.  Figure  31 shows the m i c r o s t r u c t u r e  of the "worm" from the t o p o f the workpiece i n the c u t t i n g zone.  The s t r u c t u r e i s not m a r t e n s i t i c , but i s f i n e r  that of the o r i g i n a l metal.  than  T h i s would imply t h e m a t e r i a l  was heated s l i g h t l y above the AC^ temperature t o produce fine-grained austenite t o form m a r t e n s i t e . be  but was not c o o l e d  r a p i d l y enough  The temperature reached would p r o b a b l y  around 1600 deg F. Figures  32 and 33 show the s t r u c t u r e i n the "edge  view" and "width view" o f the k e r f m a t e r i a l . reveal fine martensitic  structures.These  These f i g u r e s  s t r u c t u r e s are  76 c h a r a c t e r i s t i c of h e a t i n g t u r e and The  TI s t e e l above the AC^  tempera-  v e r y r a p i d l y c o o l i n g i t down t o below 600  temperature reached would be between 1600 Figure  c l o s e t o the  34 i s the m i c r o s t r u c t u r e  f r i c t i o n c u t edge.  and  o f the  L i k e the m i l d  deg  F.  1700  deg  F. -  material steel  case,  a t h i n l a y e r o f m a t e r i a l about 0.00197 i n . deep shows the same c h a r a c t e r i s t i c heat t r e a t m e n t as the k e r f  material.  T h i s t h i n l a y e r must have r e a c h e d between 1600  and  deg  F during  the  sawing p r o c e s s and  down t o below 600 4.3.3  Brass  then been r a p i d l y  cooled  F.  [66]  Figure received"  deg  1700  18  leaded  shows the m i c r o s t r u c t u r e s brass  f r i c t i o n c u t edge.  and  o f the  the m a t e r i a l c l o s e to  T h i s f i g u r e shows no e v i d e n t  "as  the e f f e c t of  h i g h temperature on the c u t edge s t r u c t u r e . 4.3.4  Ti-6A1-4V A l l o y [66, Figure  35 shows the m i c r o s t r u c t u r e  received" titanium istic  alloy.  of a r a p i d l y c o o l e d  t u r e i s not e v i d e n t  and  e q u i l i b r i u m , f i n e and s t r u c t u r e of the in  product.  therefore  acicular.  The  of F i g u r e  36 was  The the  35 and  36  two-phase s t r u c -  structure i s  Figure  36 i s the  top of the  with coarser  nonmicro-  workpiece that  grains.  suggests t h a t the  quenched from the  "as  character-  s t r u c t u r e i s s i m i l a r to  "as r e c e i v e d " metal but  p a r i s o n of F i g u r e s  of the  This structure i s  s m a l l worm o f f the  the c u t t i n g zone.  o f the  67]  Com-  material  0 -phase r e g i o n a t a  77 slower  cooling Figure  ial  from  37  the  than  a t 400  show f i n e r large  directional  the m a t e r i a l  and  irregular  grains  or  close  packed  heavy  deformation.  grain  i s not r e p r e s e n t a t i v e  ature  phase).  a-phase  Figure heating  the  38  structure. the heat  4.3.5  structure  internal  the  hexagonal  twins of  3 -phase  imply the  (high  history  to very h i g h  oxy-acetylene torch  structure  i s an  history  have y i e l d e d  might  i n the p r e c e d i n g  may  be  was  from  allowed  equilibrium  metal  examination  zone  temper-  i s not  like  two-phase this,  then  better  paragraphs.  Metallographic Examinations deduced  from  of the d i f f e r e n t  the  micro-  materials  involved  investigation:  (1) B r a s s d i d n o t r e a c h h i g h t e m p e r a t u r e ; cutting  large  temperature  and  "as r e c e i v e d "  following  the c u r r e n t  oriented  treatment  metal  C o n c l u s i o n s Drawn f r o m The  in  an  The  treatment than  equiaxed  boundary  the  received".  shows t h e m i c r o s t r u c t u r e o b t a i n e d  I f the  information  of  The  appear  These  irregular  "as r e c e i v e d "  in air.  The  mater-  microstructure.  (self-burning) with to c o o l  grain  i s subject.  The  the  "as  t w i n n i n g t o which  Hence t h e h e a t the  than the  subgrains.  suggests  35.  magnifications.  shaped  structure  from  800  structures  grain  obvious  of F i g u r e  shows t h e m i c r o s t r u c t u r e o f  the k e r f  photographs Within  rate  temperature  was  about  600  deg  F.  measured  78 (2)  The s t e e l  reached  zone w i t h i n  the range  o f 1200 t o 1700 d e g F .  (3) friction  The h e a t  sawing (4)  i s a localized  For the four  i n the kerf  cutting  d i d not occur.  mation  i s absent  leaded  brass,  the kerf  chips,  Figure  of  chips  formed  i n conventional  are  represented  curved are  quite  thickness the  was  small  46.  infor-  i n the d i r e c t i o n  appeared  i s a bead  with  The e d g e s  observations, the independent o f  and t h e c u t t i n g r a t e .  the titanium  alloy,  rate.  per minute,  the kerf  When there  m a t e r i a l was  the m a t e r i a l  debris  was i n c r e a s e d  "worms" were f o r m e d  was c u t a t  was no "worm" f o r m e d i n  o r the t o p o f the workpiece;  powder-like  brass.  materials  o f sawing.  material  thickness  of leaded  The m a t e r i a l  kerf  When t h e f e e d  little  the kerf  of this  the kerf  black  and T I s t e e l ,  i n Figure  d e p e n d e n t on t h e f e e d  either  fusion  are c h a r a c t e r i s t i c  machining  the experimental  1/4 i n c h  that  very  These c h i p s  From  For  of bulk  m a t e r i a l i s i n t h e form  rough.  workpiece  about  17.  steel  strands  yields  phenomenon.  the Kerf  small  mild  evidence  hence  h i s t o r y i n the k e r f .  of  For  thin;  heat  suggesting  alloy  as t o i t s temperature  For  surface  materials  The T i - 6 A 1 - 4 V  4.4 M a t e r i a l s f r o m  i n the c u t t i n g  a f f e c t e d zone i s v e r y  melting  (5)  temperatures  characteristic t o about  a l l there of  0.54 i n c h  was  burning. per minute,  b u t most o f t h e d e b r i s  was  still  79 powder.  4.5  Explanation From  theory  would e x p l a i n the  friction  sawing  sufficient  of  mechanism o f  specific  c o n d i t i o n s of  The  main f i n d i n g s a r e :  machine (2)  results.  brass  the  ( m i l d and  T a b o r [56]  experimental This  m e c h a n i s m was  graphic  examinations  surface  flow  with  shows a p r o f i l e direction  of  by  machining  Tl) yields  on  lost  the  across  at high  speed,  to e x p l a i n  b a s i s of  sliding micro-  displacement Figure  a wear mark i n i n the  Tabor,  the  as  interesting  material.  Furthermore,  Bowden and  con-  same way  for steel-copper  deduced  any  the  practice.  very  mechanism  obtained  s e c t i o n taken  sliding.  works r e p o r t e d  i n the  which r e v e a l e d metal  hardly  Never-  to e x p l a i n  materials within  postulated "surface flow"  results  unique  mechanism.  been o b t a i n e d  friction-cuts  i n conventional  Steel  work, no  experiments.  In d i s c u s s i n g s l i d i n g  Bowden a n d  pair.  i n f o r m a t i o n has  the  would  the  Process in this  (1) L e a d e d it  Cutting  obtained  cutting  text  the  the  results  theless, the  of  by  47 the  experimental  f o l l o w i n g were  observed: (i) sliding scale fine  High  interface,  melting,  the  grain size  orientation  such  temperature  e v e n when t h e r e  would  be  the  formed  always  was  recrystallization  c h a r a c t e r i z e d by as  was  no  present  evidence  s t r u c t u r e of absence after  in of  the high  very  of p r e f e r r e d  plastic  flow  in  80 cold  working  would  (ii) fine-grained  suggest  equiaxed  supported  melted  rubbing. (iii)  steel  was s l i d  materials imply  high  the  sliding  that a thin  diffusion  as t u n g s t e n  contact  melting  the view  The h i g h  (iv) scale  effect.  s t r u c t u r e and t h e c o a r s e - g r a i n e d  against hard  such  temperature  The s h a r p l y d e f i n e d b o u n d a r y b e t w e e n t h e  substrate by  a high  surface  layer  r a t e observed  and h i g h  melting  when  point  and molybdenum w o u l d  also  temperatures.  The e n e r g y  when h i g h  mechanism,  balance  melting  d i d not support  metals  f o r example,  large-  were i n v o l v e d i n  steel  versus  copper.  However, when c o n s i d e r i n g m i c r o a s p e r i t y c o n t a c t s , on  a micro-scale  might  still  be  melting  obtained.  Eshchenko e t a l . [42] p o s t u l a t e d a b r i t t l e ture to  mechanism t o e x p l a i n t h e c u t t i n g  1000 d e g C) w i t h  explained which  a high  t h e two t y p e s  c o u l d be o b s e r v e d  shows t h e s t e p b y s t e p last  stage  process  after  i s either  temperature  speed  the s t r e s s d o r e.  cutting  speeds  saw.  ductile  i n high-speed process  of hot steels  toothed  of f a i l u r e ,  cutting.  They remarked i n the range  steel.  obtained  fracture  work must be a s s o c i a t e d w i t h material  being  Figure  the high  48  At the  s t r e s s the  o f 1500 t o 1700  brittle  the b r i t t l e  brittle,  t h a t a t room  m/sec w o u l d be r e q u i r e d t o p r o d u c e Hence  and  the y i e l d  (800  They  o f the f r a c t u r e s .  reached  frac-  fracture i n i n their  temperature  c u t , s i n c e t h e h o t saw s p e e d  ranged  of the from  81 80  to  125  m/sec;  Vaughn  the  and  Krueck  mechanism  f o r metal  machining  techniques  treated).  This  same s p e e d  removal  mechanism  increases,  in  thermal energy  planes slip  and  due  occurs  results  be  explained  case  of  steels.  but in  high not  speed  lends  in  present  the  steel  independent The do  of  vary  adiabatic imply  that  only  of  different  can  be  cut  feed This  the  conditions.  approached slip  additional  study  46  The  to  could  heat  and  material  explained shear  appeared  thickness. forces  supports  phenomenon same  thinner  under  made  mild  friction  this  the  t h i c k e r one  be  generated  interesting  i s that  theory  c u t t i n g of  and  the  observations  The  rate  in  fracture  adiabatic  the  could  flow  ductile  d i f f e r e n c e i n c u t t i n g the  than  heat  cutting  surface  experimental  rates.  thicknesses  faster force  Tabor  respect  the  theory.  ial  interface  present  p r e d i c t i o n s show a l s o t h a t  shear the  the  planes,  Figure  cutting rate  much w i t h  and  f r a c t u r e theory  fracture. the  speed  to p r e f e r r e d  much l i k e  form.  to  feed  the  theoretical not  ductile  work w i t h  at d i f f e r e n t  i n the  i s very  ductile  credence  as  shear  results.  Bowden and  sliding  theory  shear  the  The  the  i n these  obtained  by  (annealed  that  saws.  adiabatic  ultrahigh  is restricted  i n i t s complete  terms of  implies  an  adiabatic condition i s  complete  best  in  in their  to weakening  and  The  an  for friction  proposed  f o r 4340 s t e e l  velocity which  [54]  range  the  the could  mater-  material same  82 (3)  Ti-6A1-4V  Alloy  V a u g h n and alloy  in ultrahigh  agreement w i t h brittle this  study,  black  the  speed  at  the  type  the  cutting  because  the  acetylene  zone  colour  torch  However, a t  The  was  the  used  to  speeds,  m e c h a n i s m became p a r t i a l l y Hence a t in  both  0.536 i n . p e r the  kerf  and  on  They  below  1/2  spark of  was  the  the  alloy  above  1/2  inch  top  of  inch  that  and  to per  kerf  observed material oxy-  self-burning. minute,  partially  cut  per  when  "worms" were  the  In  fine  cloud  obtained  heat  small  found  with very  white  brittle  good  i t s machining.  burning  minute, the  of  occurred  same as  Ti-6A1-4V  showed v e r y  say  bright  suggested  was  higher  rates,  fracture  powder d e b r i s .  on  investigation.  mechanism  slow f e e d  brittle  observation  machining  current  f r a c t u r e was  minute,  in  Krueck's  the  ductile. obtained  specimen.  CHAPTER V  CONCLUSIONS  AND  RECOMMENDATIONS  FOR FUTURE  The present  conclusions  may be made f r o m t h e  study: (1)  evidence  Temperature  confirmed  localized ients  following  STUDY  heat  exist  the  heat  the  order  m e a s u r e m e n t s and m e t a l l o g r a p h i c  that  friction  sawing  phenomenon w h e r e b y  is a  highly  steep temperature  i n the neighbourhood of the c u t t i n g  affected  zona  on t h e m i l d  ox 0.003 i n . w h i l e  zone;  s t e e l workpiece  that  grad-  i s of  of T l s t e e l i s o f the  o r d e r o f 0.002 i n . f o r t h e c u t e d g e . (2) existed  F o r both mild  a relatively  w h i c h was n o t n e a r  high  steel  and T I s t e e l ,  temperature  the bulk  fusion  there  i n the c u t t i n g  point  zone  of the kerf  material. (3) was d u c t i l e (4) alloy slow and  The f r i c t i o n fracture  feed  rather  The f r i c t i o n  was a m i x t u r e rates  soma d u c t i l e  cutting than  cutting  of b r i t t l e  and a m i x t u r e fracture  mechanism f o r t h e s t e e l s fusion. m e c h a n i s m f o r T1-6A1—4V  fracture of brittle  at higher 83  feed  and b u r n i n g a t fracture, rates.  burning  84 (5) The was  similar  particles  for  the  friction  cutting  mechanism f o r l e a d e d  to i t s c o n v e n t i o n a l machining  of m a t e r i a l  were r e m o v e d  (6) The  solution  friction  sawing  of the system  from  heat agree  results  o b t a i n e d a t r e g i o n s away f r o m  example  at d i s t a n c e s g r e a t e r than  the c u t t i n g the  zone on  solution  and  the  well  whereby  kerf»  transfer  equations  with  experimental  the c u t t i n g  or equal  the workpiece  the experiment  process  zone;  t o 0.2  the agreement  the c u r r e n t s t u d y ,  related  between  i s good,  (8) F r i c t i o n not  suitable  mentation  light  developed  applications  Recommendations (1) The sawing  would  sawing,  for cutting  (9) The  useful  the r e s u l t s  work r e p o r t e d i n t h e  agree  using a steel  with  other  i n the course in transient  disk  saw,  is  copper<.  pipe temperature  study  well  conducted  literature.  a l u m i n u m and  f o r Future  for  i n . from  (7) W i t h i n t h e c o n t e x t o f t h e e x p e r i m e n t s in  brass  measurement  of t h i s high  instru-  s t u d y may  find  temperature  studies.  involved  friction  Study  o f the  stresses  p r o v i d e complementary  approach  in  to the  present  studyo (2) I n o r d e r transfer flow  calculate  over  and  p a t t e r n s h o u l d be  pattern a better over  coefficient  to a c c u r a t e l y  of the basis  the d i s k  friction  the d i s k  known. sawing  for evaluating s u r f a c e s and  the  Thus,  the  heat  the workpiece,  a study  of the  c o n f i g u r a t i o n would the  heat  transfer  workpiece.  the  flow provide  coefficient  84a (3) so  The  i t w o u l d be  various  saw  saw  speed  interesting  speeds.  was  constant i n this  to study  the sawing  project  and  process  at  BIBLIOGRAPHY  1.  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Metals  Jet" Fluid  Mechanics,  vol. 1  Addison-Wesley  Engineering Processes  Handbook V o l I pp 529, 1154.  p 359.  APPENDIX I  ESTIMATION  The  Sawing  OF HEAT TRANSFER  Disk  Cobb and S a u n d e r s heat  COEFFICIENTS  transfer  around  [68] d i s c u s s e d the problem o f  a thin  In their  they  changed  laminar  to turbulence  at a c r i t i c a l  F o r the laminar  region, r < r  , the average  r  .  transfer  coefficient  the flow  disk.  experiments, from  found  rotating  = 0,35(  —  where w i s t h e a n g u l a r  kinematic  )  1  /  —  2  heat  velocity  of the f l u i d  viscosity  transfer  radius, heat  2  (1-1) r  conductivity  the disk  was g i v e n b y  wr h(r)  p a t t e r n around  c o f the d i s k , K the thermal  surrounding  of the f l u i d .  coefficient  the d i s k and v t h e  For r > r  c  , the l o c a l  i s g i v e n by 2  h(r)  = 0.0195( - i l £ _ ) 0  v  In the  the heat value  estimate  transfer  solution  of r , equations the heat  transfer  8  _JL_ r  ( i _  2  )  f o r the d i s k , depending  (1-1) o r (1-2) w i l l coefficient.  be u s e d  on to  F o r t h e sawing  92 disk r  c  angular  i s 3.64  The  i n the present  investigation  inches,  Workpiece The  rotating solved  able  flow  disk  pattern  i s very  over  the disk  velocity  rotates  t o assume  t o a r i s e from  saw d i s k  dragging  near  a disk  since  speed  turbulence  the source  on  flow  i s not  availzone  o f the  (282 f t / s e c ) ,  over  the f l a t  completely  the c u t t i n g  a t the periphery  at high  d i s t r i b u t i o n over  sidered  Information  However,  i s located  of a  and h a s n o t b e e n  located  i n the l i t e r a t u r e .  reasonable  the  complex  a surface  the workpiece  and  i n the neighbourhood  [ 6 9 , 70, 29, 71, 72, 7 3 ] .  patterns  in  v e l o c i t y used  disk  i t is  the workpiece. p l a t e would be  of disturbance  a i r and i m p i n g i n g  The con-  due t o  i t on t h e p l a t e  [74]. Consider disk  t h e l a y e r o f a i r dragged by t h e r o t a t i n g  as a j e t with u n i f o r m  velocity arising  o f the disk, from  this  t h o u g h t o f as t h a t According using  u  c  on t h e f l a t  due t o a t u r b u l e n t  radial  [75], the v e l o c i t y  plate  may  wall j e t .  distribution,  hypothesis,  i s o f the form  = P - j )  f'(n)  U r c  t o the p e r i p h e r a l  The v e l o c i t y d i s t r i b u t i o n  j e t impinging  to Glauert  Prandtl's  V .  v e l o c i t y equal  (1-3)  be  where  ^ i s a constant, v  characteristic (cutting  velocity,  zone) and  tribution,  the k i n e m a t i c  viscosity, U  r the d i s t a n c e from  f ' ( n ) a dimensionless  n i s defined  i  a  c  the j e t a x i s  velocity  dis-  as  •  ( I  - > 4  where z i s the c o o r d i n a t e normal to the plane o f the  wall,  1/4  From T a b l e a  1 o f G l a u e r t ' s paper,  = 2 and  since equation  KR  X /  = 0.0215 when  (7-14) of the paper i s i n the  form <R  1/4  - A<V' t  X may Figure  be determined  >  (1-5)  - 3 / 4  om  n . and f ' are known. From t om 3 of the paper these v a l u e s can be r e a d t o be and  n . = 1.67  f•  t  when  =  0.32.  om X  =  (0.0215)  (1.67  x 0.32)  = 0.0135  From F i g u r e 1 of G l a u e r t ' s paper f ' ( n ) may r e p r e s e n t e d as a  polynomial,  f'(n)  =  1=1  a.n  (1-6)  1  Because o f the symmetry o f the problem, o n l y the powers need be c o n s i d e r e d i n e q u a t i o n first  two  be  terms, we  have  (1-6).  odd  Taking  the  94  f • ( TI ) =  + a n  (I-6a)  3  3  From F i g u r e 1 o f G l a u e r t ' s paper, take two non-zero p o i n t s t o c a l c u l a t e a.^ and a^» f ' ( n ) = 0.1 when  n = 0.5 o r 4.3 and  f ' ( n ) = 0.315 when n = 2.1 whence a  x  = 0.2002 and a  = -0.0132, and so  3  f ' ( n) = 0.2002n - 0.0132n  (I-6b)  3  But n = 49.5z/r, therefore f ' ( - ) = 9.93(-) - 0 . 6 5 4 ( - ) r r r  (I-6c)  3  Hence  u = 49.5(-^0 U r  [9.93- - 0 . 6 5 4 ( f ) ]  1 / 3  3  (1-7)  4  c  F o r the v a l u e o f U e q u a t i o n (4.6) o f G l a u e r t ' s c  paper  gave 3v ^ U U  c  = —  40F 2  where F = 1/2(typical velocity) The  (volume f l o w per r a d i a n )  t y p i c a l v e l o c i t y here may be termed as t h a t  istic  character-  o f the i m p i n g i n g j e t , V" and the volume f l o w per c  r a d i a n may be e x p r e s s e d as V t c  5 n  /  71  where t  Q  and <5  g i v e the c r o s s - s e c t i o n a l dimensions o f the i m p i n g i n g j e t and  2V"  c  g i v e s the t o t a l  l e n g t h o f the j e t i n two minutes  (average time o f r u n f o r each e x p e r i m e n t ) .  F = 1/2V  (V t  c  Thus  )  2  C D n  or F = l/*  1  2  3  \  2  \  $  2  Hence  u = C  3 i r  2 0 V  c  v  D  3 t  2  6  2  S u b s t i t u t i n g f o r U" i n e q u a t i o n c  (1-7) g i v e s  2 2 Nl/3  A  6  u = 43.5vJ~-3  -)[9.93(f)-0. 6 5 4 ( f ) ] 3  According  t o K r e i t h [ 6 8 ] , the l o c a l  number f o r a f l a t  (1-8)  Nusselt  p l a t e i n a t u r b u l e n t flow r e g i o n i s  g i v e n by  Nu  v  = 0.0288Pr  1 / 3  [Re  ]°'  8  (1-9)  But h  Nu therefore  x  cx  x  . -fS-  (I  .9a)  96  h  Equation  -  0.028  8 P  (1-10) may be used  r^^  which w i l l  X  = CPr  1 / 3  temperature  —[Re ] * r r  where P r = 0.72 f o r a i r ,  0  -  1 0 )  distribution  agree-  Thus we may w r i t e  (1-11)  8  the a i r thermal c o n d u c t i v i t y  and C i s a c o n s t a n t determined tal results.  ( I  r e p l a c e x w i t h r and use a c o e f f i c i e n t  g i v e the b e s t  cr  0 , 8  p l a t e workpiece w i t h some  able with the experimental r e s u l t s .  h»  l  t o c a l c u l a t e t h e heat t r a n s -  f e r c o e f f i c i e n t over t h e f l a t modification:  R e  on the b a s i s of experimen-  F o r the experiments  i n the p r e s e n t  study  C = 0.0096. With the v a l u e o f u g i v e n by e q u a t i o n  ( 1 - 8 ) , the  heat t r a n s f e r c o e f f i c i e n t o v e r the workpiece c a n be estimated.  I t s h o u l d be noted however t h a t e q u a t i o n  (1-11) does n o t a p p l y t o the r e g i o n where the j e t  impinges  on t h e p l a t e o r , i n t h i s  For  study, t h e c u t t i n g  zone.  t h i s r e g i o n t h e heat p a r t i t i i o n assumes z e r o heat t r a n s fer.  On t h e o t h e r hand the heat t r a n s f e r  coefficient  may be e s t i m a t e d a s t h a t due t o a s t a g n a t i o n p o i n t r e g i o n which i s e s t i m a t e d t o be h'  = 18.8Btu/hr/ft /°F 2  co  (1-12)  97 The  cutting  change  zone  temperature  i n the heat  partition  transfer  assumption  is  i s very insensitive coefficient  justified.  and  to  so the  the heat  APPENDIX I I  FORCE ANALYSIS IN FRICTION SAWING SYSTEM  Forces  on T a b l e  and  Consider  Workpiece [ F i g u r e I I - l ]  the  linear  dimensions:  t, , I = f (x) 1' w I f L i s the I  w  l e n g t h o f the workpiece, then = L-xt  (II-l)  where k i s the r a t e a t which the t a i l  end  of the  workpiece  approaches the c u t t i n g zone, i n o t h e r words, the c u t t i n g rate. If x = V = constant, I  w  then  = L-Vt  (TI-2)  = L + a s i n ^ - Vt  (II-3)  and l  1  From e q u a t i o n s L  1  Now  - I  ( I I - 2 ) and  w  = asin^  l e t us c o n s i d e r  I f we  (II-3) (II-4)  r  the t a b l e f o r c e s , ( F i g u r e  take moments about the p o i n t 0, we R ( ^ - 3 ) - Rt - R l i  1  2  2  98  = 0  II-l):  obtain •  (II-5)  Consider the  t h e f o r c e s on t h e w o r k p i e c e :  f o r equilibrium of  forces  "2lFx = ^ F y  that i s  F cos^ - F  F sin^  +  F cos^f  - F sin"Y  N  N  ^- A  Also  = 0,  M  =  f  ^'  + R  = 0  T  f  -  X  R  (II-6)  2  -  W = 0  (II-7)  ^' * E  -w£ =o F „ £ cos^p - F-l sin'S' + 3 R - R ( i - N w ' f w V 1  Rearranging F  equations  N  f  F  ( I I -5) R  2  (II-8)  g  1  t o ( I I - 8) R  l  N R  (II-5)  0  (II-6)  sin'/'  cos'f  0  0  F  (II-7)  cos'f'  - s i n ^  1  -1  W  (II-8)  cos^  - s i n ^  3  Using solved  0  =  2  Cramer's easily.  0  -l /l 2  rule, Using  0  equations  ^w  w (II-5)  t o ( I I - 8 ) may  (II-2)  L - V t •*• a s i n ^ f -3  and  -  (II-3)  i /Jt 2  cos'Y  0  0  cos-^p - s i n ' Y  1  -1  sin^  W  I  T~ w equations  T  L-V t-£ £ a s i n +  cos^ - s i n ^  3 L-Vt  L-Vt  be  100 -l- )  a s i n ' f ( L - V t + a s i n ^ -3  2  /(L-Vt)  N  N D  L-Vt+asin^-3  0  R F  T  W  -  -l / l 2  cos^p  o  siri"^  I  0 -  1  L-Vt+asin^-i  -sin^ L-Vt  L-Vt  2  L-Vt  D  x  "F. = F s i n ^ f N T ' T  r  , „[Watan^p  m  (L-Vt+asirvy-^)  +3+Vt-L-asin^) + 3 ( ^  L-Vt+asin^|/-3-X,  (II-9)  -Rt]  and  F  similarly  = F_cos^  +  i[w  ( L - V t + a s i n ^ - ^ ) (I 0  +3+Vt-L-asin'f) + 3 ( ^ ~ - i )  (L-Vt+asin +R£]  R^ and R  2  (11-10)  may be o b t a i n e d i n a s i m i l a r way.  In u s i n g e q u a t i o n s  (II-9)  and (11-10)  i t must be noted  that  the v a l u e o f t i s l i m i t e d by 0 - t - L/V The  (11-11)  angle ^ i s dependent on t h e t h i c k n e s s o f the m a t e r i a l  b e i n g c u t as w e l l as the h e i g h t above the h o r i z o n t a l  2  101  c e n t r a l p l a n e of the c a l c u l a t i o n of ^ t  and  sawing d i s k .  The  following  shows  as a f u n c t i o n of the workpiece  the p o s i t i o n of  the  t a b l e H.  For  the  thickness  added workpiece  w s u p p o r t , i t i s s u f f i c i e n t t o add to the v a l u e of  the  H.  I Figure  II - 2  height  of the  support  102 = l/2( « cos A  =  )  (11-12)  ~ -  (11-13)  a.,  '  H  +  fc  COSCxl  From  (11-12),  (11-13) and  W  (11-14)  (11-14); H + t  = l/2[cos~ (-)+ cos" ( 1  F o r d i f f e r e n t workpiece  If V  R  I n t h i s case  e q u a t i o n (11-15) f u r n i s h e s  .  i s the r e l a t i v e  the w o r k p i e c e ,  (11-15)  t h i c k n e s s e s , p o s i t i o n s o f the  t a b l e and even saw d i s k r a d i i , the v a l u e o f  -)]  1  sliding  speed o f the saw d i s k t o  the t o t a l heat g e n e r a t e d can be c a l c u l a t e d . V < < V  and so C  T  D  K  ^  V  . C  f c = -3-^F  q  V  V  (11-16)  where J i s the m e c h a n i c a l e q u i v a l e n t o f h e a t .  This  total  h e a t i s d i s s i p a t e d through the saw d i s k and the workpiece if  heat l o s s i n the c u t t i n g  zone i s n e g l e c t e d .  APPENDIX I I I  CALCULATION  Equation q If  workpiece  and  • q  D  <IH-1>  w  of the heat  generated d i s s i p a t e d  through  i s f , then q  w  = f q  (III-2)  q  D  = (l-f)q  T  so,  Equation  ( 5 ) may  (l-f)q L  The  FRACTION  (33) s t a t e s - q  T  the f r a c t i o n  the  OF HEAT PARTITION  K  T  J  <*>  t h e n be w r i t t e n  as  X ( A b ) I (Ar) + I (Ab)K (A r )  T  2TTK r Aa D  periphery  (III-3)  T  I  1  (A b ) K (A a) + o 1  temperature  I . ( A a ) K (Ab) 1 o  (III-4)  J  i s T ( a ) , where a i s t h e d i s k  radius  T(a)  T 0 0  =  ~  (l-f)  —  q t r  2TTK t Aa D  f 1  K (Ab)I (Aa) + I (Ab)K (Aa) I  o o ( A b ) K (Aa) + O 1  2 o I, ( A a ) K (Ab) 1 O  .  J  (III-4a) 103  104 For the d i s k ,  a = 7" and b = 3/4".  From B e s s e l f u n c t i o n t a b l e s , calculated  the term i n the b r a c k e t i s  t o be 1.055, hence  1.055(l-f)q„  T(a) - T =  2TTK  D  From e q u a t i o n ( 2 8 ) , when  T  cutting  4 lP VD b  T  l  n  (  2 b 7  _  )  ny  (III-6)  (50) g i v e s t h e average temperature i n the zone;  _  Ab Kt t l P  w  sxn ( 2 ^ - )  8b'  T  av" co  S  b ]  n=l  K  fq sin7! T  and y = 0  =  -  Equation  5=0  4  fq siiv/ T(0,0)-T  (III-5)  t Aa  2 D  D n=l  Applying  n y_  t h e matching t e c h n i q u e , we equate  ( I I I - 6 ) t o e q u a t i o n ( I I I - 5 ) f o r the maximum  (III-7)  equation temperature  matching and e q u a t i o n ( I I I - 7 ) t o e q u a t i o n ( I I I - 5 ) f o r the average matching.  The former g i v e s  105  max » , 14.05 l W b  and the  p t  latter  K  sin ( 2 ^ - )  4b.  . i  s  n=l  1 +  nu  gives  av 14.05  (III  \  sinf  sxn  8b^ t  D  +  D  n=l  (—)  2 n u  n  (III  106  APPENDIX  STRESS ANALYSIS ON  Stress  Analysis In  made.  problem,  the disk  pendent  o f z, i n o t h e r  Second,  the problem  may  be u s e f u l l y Based  stress  THE SAW  DISK  on t h e D i s k  the stress  First,  IV  i s thin words,  two a s s u m p t i o n s  will  and s o t h e s t r e s s the problem  be  i s inde-  i s two-dimensional,  i s l i n e a r and so s u p e r p o s i t i o n  principle  employed.  on t h e f o r e g o i n g  equations are given  assumptions,  the general  by  _I M + i_ A! 2  °  r  =  r  9  r  r  2  30  2  90 2  r0  9r  v  r  90 '  (IV-1) where 0  i s the A i r y The  (  » r *  +  *  stress  compatibility  I?'  function. equation  i s given  ^ 7 ^ +  >=0  by  (IV  .  2)  107 The  kinematic r e l a t i o n s are  3u  r  9r  £  0  u r  Y  r0  £  For  full  1 3v r 30  _ _1 3ju r 90  circle  3jv 3r  solutions,  e e  r 0  Yr0n The  (IV-3)  constitutive  =  3u  =  7  e q u a t i o n s (IV-3) reduce t o  T—  9r  =  (IV-3a)  0  equations are:  e  = Tr (a - vo~ ) E r  r  6  (IV-4)  1  e  e  E  = Z  The  problem  because is  I  =  ( 0  o "  " E  of the l i n e a r i t y  +  -  }  °6  )  i s shown i n F i g u r e assumption,  IV-1 and  an e q u i v a l e n t  problem  shown i n F i g u r e I V - 2 . Figure  used are  (b)  r  t o be s o l v e d  Consider  (a)  ( a  v a  o = 0 a t r = a r u = 0 at r = b  IV-2(a),  the boundary  conditions  108 For  this  problem,  the s t r e s s f u n c t i o n  A i B 0 = Ar + — r «  whence "r  =  0  =  G  . .B 2 r B 2 r  T  the  3 + v — ~ — o  2 3 pu> r  3 + v 8~  by [76]  (IV-5)  2 2 P W  r  1 + 3v 8  2 2 P  U  r  (IV-6)  0  =  r0  Using  -  i s given  the boundary  conditions,  A and B a r e d e t e r m i n e d and  solution i s T  F ,G  = -  R  3 + v g  + — 2  p  U  2 2 r  Dr F  °n 0  T  r0  G Dr  D  2 2  1 + 3v  n  =  9 ~ —« 2  p u  r  (IV-7)  8  0  =  where 2 -e—T 8a b  F = -  , . [ 2 ( l - v ) b - (l+v)(3+v) a^] 4  Z  G =  2 £|-  D =  [ 2 ( l - v ) b + (l-v)(3+v) a ] 2  [(1+v) a  2  2  ( I V  _ ) 8  + (l-v)b ] 2  a b Now  consider  Figure  IV-2(b).  T i m o s h e n k o and G o o d i e r be  treated  [76],  as a f u n c t i o n  This  problem  The f o r c e  o f -uv .  was  treated  on the d i s k  Following  the  by  will  notations  109 in the  Figure  IV-3,  following  the s o l u t i o n  stresses  on  i s obtained  the simple  by  superposing  radial stress  dis-  tribution  /  Ttr  \ 1) N o r m a l  stress  ^  2) S h e a r  distributed  P(<f) sin ( 0 + 0 )  )  1  \  stresses  ^  uniformly  f  of  2  along  the  dt  intensity  P('i) PCi) cos (G^ (0, +. 0^ ) dif 92  i  3) S t r e s s  whose n o r m a l  and  s h e a r components  respectively  1_  P(/) sin (0 - 9 ^ di  ird " and  2  A. i, X  are  boundary  110 From geometry,  0 Q  j  c  l  0  we n o t e  +  0  Figure IV-3  thefollowing;  f(«  2  I " *  =  (IV-10)  - 0. = £ - 0 2 2 c  1  whence  0  e  i  f  =  Substituting  +  V (IV-11)  i |  =  2  «  -1 (0  these  c  - i)  i n theexpressions  given  4< f a  r  |^  =  I  )  CRQ )-+ ^ c  r> rf  J  ' f J PC/0 cosf df 1± r^f J p t f v sin? df + — s  r0  ^  P a c e s ' / d1<  p  c  *d  T  we g e t  r f P('f)sin |  cos0 ;os0  a„0 = TiTd  above,  1  i  i  n  i  9  ^  /  r  f  P W <tf (IV-12)  l  Considering  the case  = P ,  P(10  when  equations  = - Cf - *.) Tr f 1  'e  . +  £  f  r  T  [  f  say  (IV-11) reduce t o  4cos(0 + '*)  D  o  i s constant,  cosG +  d  r  j  d  J  - fd * i V * (  r0  fd  =  (  cos  *f ~ V  ( s i n  *  +  8 i n 9  c  )  (IV-12a) where  i> = ~ (f +1f^) f  Equations solid  by  (IV-12).  i n Figure  In t h i s  boundary  as  a  r b  t o be  t h e s o l u t i o n s must  be  on t h e s o l u t i o n s i n e q u a t i o n s  the g e n e r a l i z e d  s o l u t i o n given  (IV-12) g i v e  and x  solid  solutions for a  the s o l u t i o n o f the problem  IV-4  case,  disk,  small.  polar  coordinate  i n Timoshenko and  Goodier  followed.  Equations r  i s considered  ±  (IV-12a) r e p r e s e n t  superimposing  two-dimensional  o  f  F o r an a n n u l a r  represented  [76] i s  0£ - i )  ( I V - 1 2 ) and  disk.  modified  and  c a n be  r Q  the s o l u t i o n f o r a s o l i d  c a l c u l a t e d f o r any  of a c o n c e n t r i c  disk  of radius  and solved  r  Q  r  b  will  a.  disk  of radius  I f these  » the boundary be  given  point b  lying  the•inner  on  the  designated  f o r the  by at  whereby  i n s i d e the  stresses are  conditions  disk  boundary  problem  112 a  2)  = 0  r  /  t _ = 0 r0  The  boundary  expanded  o ,  rb  -  (IV-13)  c o n d i t i o n a t the i n n e r  into Fourier  series,  = A + > A_cos n 0 + o ^--^ n n=l  c  n  From geometry  by  r =  c  the cosine  A  + b  2  may  be  B sin n 0 c  n=l  n  D sin n 0  ^  n  rule,  - 2ab cos  2  boundary  thus:  C cos n 6 + >~  o  r0D  boundary  1  = C +  T  at the outer  c  Figure  (0  - i)  (IV-14)  IV-5  (IV-15)  c  Using  equations  I  i  j  =  ( I V - 1 2 ) and  .  4* f  d e f i n i n g the f o l l o w i n g :  P  C  f  )  s  i  n  2  d r p  P W cos I d^  f  1>.  I  I  2  3  I  f  = J  'I.  f =J  PC/)- cos * d1>  1  f  P W sin ^ di  i  (IV-16)  f i  4  =  J  p ( / )  di  113 then,  2( 1 T  O"  =  ,  rb  A=  C O S  0 ~2  ,„ +  . 0 • 2"^  1  T _i V r0b = - j f ird the f u n c t i o n s  I  4  tion We for  c  now  turn  t o 2TT .  boundary  the s t r e s s f u n c t i o n 2  from equations  coefficients  t o the general  0 = a log r + b r o ° o  Trd  A ,  A ,  Q  n  ( I V - 1 3 ) c a n be e v a l u a t e d  from zero  at the inner  j  (IV-18)  r  D  of 0  ^  8  (IV-17)  f o r <r ^ a n d  and  limits  Trd  -1)  0  c j — Trd  (IV-18) the F o u r i e r i n equations  V  T H  A  s l n 0  and  n  2  :  y  /a +b -2ab cos(0  Using  1  H  + c  o  r  0 given 2  n  C , Q  condi-  expression  by M i c h e l l [ 7 6 ] . 2  0 + a 0 o 1  l 3 -1 + Y~ r 0 s i n 0 + (b r + a | r + b^ r log r) cos 0 a  1  °1 3 ' -1 - y - r 0 cos 0 + (d^ r + c^ r + d^ r log r) sin 0 OP <  +, " >-\ ^  r.-  1  (, a rn +. b, r n+2 +. a . r -n +. b, '. r -n+2.) cos n 0 n n n n  n=2 00  _i+_ ^> ' ^-V n=2  n  defined.  coordinates  log r + d r o  C  w i t h i n the  Thus t h e boundary  i s well polar  B ,  (IV-17)  /„ ( nc rn + d nr n+2 +. c„,n' r -n +, d,,n' r - n +) 2s .m n 0 (IV-19)  114 Since to  the problem here i s that  inner  cient  boundary  stresses  to consider  0  of stress  of a f u l l  the s t r e s s  distribution  circle,  due  i t is suffi-  function  , 2 v. ' , n ,, n+2 , , -n , , . -n+2>. = bo r + ^ "> ( a r + b r +a'r + b 'n r ) -TCn n n—Z  00  • + ;>  n=2  1  / n , , n+2 . , -n . ,, -n+2. ( c r + d r + c r + d r ) n n n n (IV-20)  whence f r o m e q u a t i o n  (IV-1)  oo  a  = 2b + X [(n+n )a r ~ ) a r o .n n n=2 2  r  n  2  1 1  " + (n+2+n ) b r n 2  2  n  + (n -n)c r ~ ~ n 2  ?  + (n -n+2) b^ r ~ J cos n 0 2  + ^  [(n+n ) c  n  r~  2  n  + (n+2+n ) d  2  x  2  n  + (n -n) c' r " ~ 2  n  9 -n + (n -n+2) d' r ] s i n n 0 n  T  ^Tn{(n-l)a v ~ r0 = x n n  + (n+l)b  2  n  n  r  n  - (n+1) a' r n  2  (IV-21)  n  2  n=2 - (n-l)b' r " } s i n n 0 n n  n{(n-l)c  v~ n  n  + (n+l)d  2  n  r  n  - (n+l)c^ r " " n  2  n=2 - (n-l)d' r n  n  } cos n 0  (IV-22)  n  2  115 From e q u a t i o n  (IV-11)  e «f In  order  c+  i)  =» 2m  equations  Noting  (e  to avoid h a l f - a n g l e s , put n  in  - |  (IV-23)  (IV-21) and  (IV-22).  that sin(0  + a) = s i n O c o s a  cos(©  + a)  + cosOsin a  and  we  = cosOcosa -  can write expressions  form  of  (IV-14).  r  =  sinGsina  (IV-21) and  (IV-22) i n the  Thus  t> + o /  F. Im  m=T  (r)cos m 0 c  (IV-24) +  J> ^—z m=l  J  r  G..  lm  (r)sin m 0  c  and T  rO  =  ^  m=o  •  +  F„  2m  (r)cos m 0 c  ^ > \ G„ — 2 m m=l  (r)sin m 0  (IV-25)  116 where lm  (  lm  ( r ) =  F  G  f  2m  2m  G  r  ) =  D  D  lm  S i n  lm  C O S  m  -  i  ^  +  i  e  l  C  c  o  i  n  s  m  m  (  r  )  =  C  2m  S i n  m  *  (  r  )  =  C  2m  C0S  m  * " 2m  +  s  m  D  2m  ^  COS m  D  *  (IV-26)  ^  s i n  and  D  lm  =  ( 2 m 2 + m ) c  2m  ~  r 2 m  2  + (2m -m) ; c  = (2m +m)a 2  2m 0  r  ( 2 m 2  r"  2  C lm  +  2 m  ~  D  2 m  r ^  2 m  = m{(2m-l)c  "  ( 2 m + 1 ) c  r~  2  y  r~  m  2 m  2 m  2  m  - (2m+l)a  + (2m -m+l)d; 2  2m r ~  2m = U 2 m - l ) a  2 m  v  2 m _ 2  9  C  2  ^  h n + 1 ) d  + (2m +m+l)b  2  + (2m -m)a 2  ~  2 m  "  r  2m  2m  2m  r  2 m  "  "  r _ 2 m  ' 2m r "  2  + (2*4.1)1, ^  2  - 2  + (2m -m+l)b;  1 - 2  2  2 m  - (in-Db^ r~  + Um+Dd^ r  2  "  r  1  d  2 m }  2 m  " ( - ) 2m " 2 m  2 m  r  2 m  (IV-27)  F o r any g i v e n v a l u e o f r , the c o e f f i c i e n t s b . a , b , ^ * o' 2m' 2m' 2  c  0  o2m > o^m' ~ t o <L d  d  m a  n  d  d  ' o dm can be e v a l u a t e d by u s i n g J  3  0  (IV-14),  7  (IV-24) and (IV-25) and the second boundary c o n d i t i o n i n (IV-13).  N o t i n g t h a t r may be z e r o a t p o i n t A and the  s t r e s s e s a t t h i s p o i n t are f i n i t e .  a'  d'2  m  may be e l i m i n a t e d .  tions involved w i l l  , b*  0  dm  1  1  . c'~  0  dm  and  dm  1  I f P( i>) i s c o n s t a n t , t h e equa-  be much s i m p l i f i e d f o r n u m e r i c a l  cal-  culations.  Thermo-Elastic S t r e s s e s i n Disk F o l l o w i n g D'Isa's treatment  [77] the m o d i f i e d  Hooke's law e q u a t i o n s become  e  r  du ^ . r = - T — = at + — dr E  dw e  for  z - di"  ^ =a t  v  a  r  0 — E  v a  0  r  - 1  a plane s t r e s s (a = 0) z  t a k e n as a p l a i n s t r e s s  (IV-28)  case,  problem. The  The d i s k may be  equilibrium  equation  i s g i v e n by da -r - + - (a - aj dr r r 0 1  Equation  =0  (IV-29) may be e x p r e s s e d  i n terms o f u.  s o l u t i o n o f the r e s u l t a n t e q u a t i o n g i v e s :  The  118  r  = U+v > 7  /  lr  trdr +  +  (IV-29b)  where  t i s the temperature  Using  equation (IV-29), the stresses aE r  distribution  C.jE  (  trdr +  c a n be e x p r e s s e d a s  C E 2  < -">  2  < r.  i n the d i s k .  (l+v)r  1  2  1  and  C  = - aEt + ^  cx  e -  r  2 J  J  trdr +  1  E  C  -zn—+ 2d-)  2  E  ( 1 + v ) r  2 (IV-29c)  where  the c o n s t a n t terms  the  boundary  conditions  and  o u t e r edges, ^  and  r  Q  or  7  are the inner  and C  2  c a n be c a l c u l a t e d  of the d i s k : cr r  stress free  = 0 a t r = r . and r i o  and o u t e r r a d i i  from  inner  where r . i  of the disk  respectively. Since problem,  t i s known f r o m  the thermal stresses  calculated.  the temperature i n the disk  distribution  c a n be  APPENDIX V  TABLE STRAIN RING CALIBRATION  F i g u r e V - l shows the arrangement f o r the t a b l e strain ring  calibration.  An aluminum h o l l o w c y l i n d e r was p l a c e d on the table at a l o c a t i o n corresponding strain ring.  t o the c e n t r e of the  The i n d i c a t o r was warmed up f o r a t l e a s t  t h i r t y minutes.  The z e r o was s e t and the t a b l e was  slightly  pushed up and down t o make sure the t r a n s d u c e r was working properly. S t a r t i n g w i t h s m a l l l o a d s , the i n d i c a t o r v i t y was s e t a t h i g h e s t l e v e l .  Loads were added i n i n c r e -  ments o f 2 l b up t o 10 l b and t h e i n d i c a t o r was a d j u s t e d t o r e a d d i r e c t l y was then lowered  i n pounds.  displacement  The  sensitivity  t o r e a d 25 l b maximum and the l o a d s were  added i n 5 l b increments tivity  sensiti-  was s w i t c h e d  up t o 25 l b . F i n a l l y  the s e n s i -  t o the 100 l b maximum range and the  c a l i b r a t i o n was done up t o 60 l b . The expected  table force  was l e s s than 40 l b and so c a l i b r a t i o n up t o 60 l b was sufficient. Each time  the t a b l e f o r c e i n d i c a t o r was r e a d , the  c o r r e s p o n d i n g d e f l e c t i o n on the t h r u s t s t r a i n r i n g was 119  read.  120 I t happened t o be z e r o up to 60 l b .  The t a b l e was  pushed down by hand t o r e a d up t o 100 l b and the s t r a i n r e a d i n g was s t i l l the  cutting  forces  was  zero.  then  thrust  Hence the i n t e r a c t i o n o f  negligible.  /  APPENDIX VI  THRUST STRAIN RING  CALIBRATION  Two s t e e l r o d s 1/4 i n c h diameter by 12 i n c h e s long were notched a t p o i n t s end.  about h a l f i n c h from each  The second r o d was notched a t the c e n t r e . The r o d s were s t r i n g e d as shown i n F i g u r e V I - 1 .  The c e n t r a l n o t c h i n the second r o d was s t r i n g e d t o a l o a d i n g pan, F i g u r e  VI-1.  over a p u l l e y f o r l o a d i n g .  The l o a d i n g pan was passed The arrangement made i t easy  t o l o a d the t h r u s t s t r a i n r i n g h o r i z o n t a l l y . The  s t r a i n r i n g c a l i b r a t i o n proceeded as f o r the  table strain r i n g . the  I t was found t h a t no i n t e r a c t i o n o f  t h r u s t f o r c e on the t a b l e f o r c e was r e c o r d e d by the .  table force indicator.  121  T A B L E 1 EXPERIMENTAL RESULTS  I (in. )  Temperature  y (in. ) T/C  Expt.  Pos n 1  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0*125 0 .0.125 X).,250 0.375 0.500 0.625 0.750 0.875 1.000  0.0  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.1875  FOR  314 346 375 386 606 408 364 296 165  +  305 328 341 364 386 341 318 262 154  Theory T 0 maximum matching  VS.  THEORETICAL  BRASS  CF) (y=0.0) A average matching T  183 197 215 238 268 310 373 488 899 473 352 286 242 212 188 170 156  187 202 221 245 276 319 385 505 934 490 363 294 249 217 193 174 159  181 194 211 232 258 290 332 379 402 368 314 268 233 206 185 168 155  185 199 217 238 265 299 343 392 416 380 324 276 240 211 189 172 158  E x p t ' l and Theoretical Conditions  t  = 1/4 i n . w F,,, = 10.00 l b f l R „ = -2.500 l b f l V = 1.035 i n . / m i n t F  = 1.25 mir  = 7.7859 l b f  N = 22.2828 l b f y  +  122  , elapse  = 0.3494  ytheory  " °'  2  i  n  123  TABLE 1 (Cont'd)  5 (in.)  Temperature  y (in. ) T/C  Expt.  Theory T u  (°F)  (y=0.0) T  A A  Pos' n  maximum matching  average matching  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 t).250 0.375 0.500 0.625 0.750 0.875 1.000  0.375*  162 172 184 197 213 229 246 258 261 252 234 214 194 177 163 151 141  166 176 188 202 218 236 253 266 269 259 240 219 199 181 166 154 143  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.875  230 230 221 199 186 165 138  +  195 195 190 186 182 173 160 147  148 153 158 163 168 172 175 176 175 172 168 162 156 149 142 136 130  .  151 156 162 167 172 176 179 180 179 176 171 166 159 15 2 145 138 132  E x p t ' and Theoretical Conditions  t  = 1/4 i n . W F™ = 10.00 l b f 1  R_, = -2.500 l b f l  V  = 1.035 i n . / m i n  Elapse = 2  2 5  m  i  n  F  = 6.7340 l b f  N  = 21.2058 l b f  /  = 0.3176  ^theory = °'  4  i  ^theory  9  i  = °*  n  r  ' u  TABLE 2 EXPERIMENTAL VS. THEORETICAL RESULTS FOR MILD STEEL  I (in.)  Temoerature (°F) Expt. Theory E x p t ' l and T Theoretical A A Conditions O average maximum matching matching  y (in.) T/C  T  Pos' n -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.234*  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.327  240 251 274 296 318 364 408 475 508 475 397 274 208  206 226 250 280 317 365 426 493 518 449 356 283 230 193 165 145 130  213 234 259 290 33 0 380 445 515 541 469 371 294 238 198 170 148 132  t ^ = 1/8 i n . F  T  = 12.500 l b f  R  T  = -5.833 l b f  V  = 1.035 i n . / m i n  t , elapse  = 1 . 7 0 min  F  = 10.2733 l b f  N  = 27.9059 l b f  /J  = 0.3681  •y,, = 0.2 i n . theory 1  +  ^ t h e o r y = 0.3 i n . +  240 251 262 285 307 341 375 408 420 386 282 251 173  202 220 241 267 297 334 372 405 408 370 313 260 218 185 161 142 128  124  209 227 250 277 309 347 388 422 425 385 325 270 225 191 165 145 131  125 TABLE 2 (Cont'd)  5 (in.)  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  Y (in. ) T/C  Temperature( •F) Expt.  Theory T  o  Pos 'n  maximum matching  0.500  190 204 219 236 .•• 253 271 284 291 286 269 243 215 190 167 149 135 123  310 305 300 287 260 204 99  A average matching T  196 210 226 244 262 281 296 303 297 279 252 222 195 172 153 138 125  E x p t ' l and Theoretical Conditions  i_  \  =  1/8 i n .  F  T = 11.292 l b f  R  T = -5.000 l b f  V  = 1.035 i n / m i n  Elapse - ° -  7 5 0  m  F  = 10.2489 l b f  N  = 26.1752 l b f  r  -  0.3196  i  n  TABLE 3 EXPERIMENTAL VS. THEORETICAL RESULTS FOR MILD STEEL  I  (in. )  Temperature  Y (in.)  Expt.  Theory E x p t ' l and T Theoretical A Conditions 0 average maximum matching matching T  T/C Pos • n -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  (°F)  0.1094*  408 453 519 638 756 585 464 352 230  304 328 356 391 436 497 584 715 836 647 482 377 306 256 219 191 170  315 339 369 405 453 516 607 745 872 674 501 391 317 264 226 196 174  t  = 1/4 i n . W F,- = 5 6 . 5 4 2 l b f i R„ = - 2 1 . 1 0 4 l b f i V = 1.035 i n / m i n t , ^ =1.017 min elapse F N fA  *y,, = 0.1 i n . •'theory +  y,, = 0.3 i n . theory J  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.3294  +  375 408 453 497 552 497 397 296 197  296 316 339 367 399 436 474 504 503 459 394 333 282 242 211 186 166  306 327 351 380 414 452 493 524 523 477 409 345 292 250 217 191 170 126  = 11.4909 l b f = 92.6846 l b f = 0.1240  127 TABLE 3 (Cont'd)  I  (in.)  Temperature ( *F) Theory Expt.  y (in. ) T/C  T  o  Pos' n  maximum matching  A average matching  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.549*  292 308 325 344 363 381 394 398 388 364 331 294 260 229 204 182 164  301 318 336 356 376 395 409 413 403 378 342 304 268 236 209 187 168  -1.000 -0.875 -0.750 -0.625 -0.500 ^0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  1.019  365 372 382 372 359 350 274 199  T  E x p t ' l and Theoretical Conditions  t  = 1/4 i n .  W F_ = 66.061 l b f l R = -25.333 l b f i V = 1.035 i n . / m i n m  t , =2.034 min elapse F  = 12.0729 l b f  N  = 106.9034 l b f  yU  = 0.1129  *y,, = 0.5 i n . theory 1  +y,, = 1.0 i n . theory 1  +  296 287 282 278 260 251 186 134  253 259 264 267 269 269 266 261 252 241 228 214 200 186 172 160 149  261 267 272 276 278 278 275 269 260 249 235 220 205 190 176 164 152  TABLE 4 EXPERIMENTAL VS. THEORETICAL RESULTS FOR T l STEEL  (in.)  y (in.) T/C Pos'n  -1.000 0.578* -0.875 -0.750 -0.625 -0.500 -0.375 ,-0.250 ,-0.125 0 , 0.125 ' 0.250 0.375 0.500 0.625 0.750 0.875 1.000  Temperature (°F) Expt,  310 314 314 314 305 287 262 218 165  Theory maximum matching  average matching  299 309 320 331 339 345 344 335 316 288 257 225 196 171 151 135 122  309 321 332 343 353 358 357 348 328 299 266 232 202 176 155 138 125  E x p t ' l and Theoretical Conditions  t  w  = 1/4 i n .  F  T  = 29.271 l b f  R  T  = 12.354 l b f  V  = 1.035 i n / m i n  t , =2.504 min elapse F  = 12.3748 l b f  N  = 54.5607 l b f  )A  = 0.2268  •y,, = 0.6 i n . theory 1  +y,, = 1.0 i n theory J  •1.000 1.078 •0.875 •0.750 •0.625 •0.500 -0.375 •0.250 •0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  282 278 264 260 238 216 197 154 132  255 257 257 256 253 248 239 229 216 202 186 171 157 143 132 122 113  128  263 265 266 265 262 256 248 237 223 208 192 176 161 147 135 124 115  129 TABLE 4  I  (in. )  y (in.) T/C  Temperature Expt  -1.000 0.2 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  maximum matching  1116 852 606 395 221  661 596 475 199  CF)  Theory  0  Pos' n  -1.000 0.0 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  (Cont'd)  E x p t ' l and Theoretical Conditions  A average matching  351 377 408 447 500 571 680 882 1623 737 482 350 268 215 178 152 133  365 392 425 466 521 596 710 928 1704 771 503 363 2 78 222 183 156 136  344 367 395 429 471 522 584 647 651 545 419 323 255 208 173 150 132  357 380 411 447 490 545 611 676 680 569 436 335 264 214 179 153 134  w -  fc  F  T  R  T  V t F  =  1/4 i n . 26.667 l b f  = -10.833 l b f = 1.035  in/min  , = 1.250 min elapse = 12.5343 l b f  N = 50.9972 l b f }J = 0.2458  TABLE  I (in.)  • 1.000 -0.875 •0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.4  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  0.9  O maximum matching  269 269 269 264 256 247 225 182 132  Expt'l  CF)  Theory  Expt  282 282 296 300 307 318 330 296 218  130  (Cont'd)  Temperature  Y  (in.) T/C Pos' n  4  and  Theoretical  A average matching T  281 295 312 329 348 366 380 383 365 328 281 237 200 171 148 132 119  291 306 323 342 362 381 395 408 380 340 291 245 206 175 152 134 121  232 235 237 238 237 233 222 218 206 193 178 163 149 137 126 117 109  240 243 245 246 244 241 234 225 213 198 183 167 153 140 128 118 110  Conditions  = 1/4 i n .  -  F  T  R  T  V  =  = 28.333 l b f •- -10.833 l b f '  == 1.035  Elapse  =  2  -  in/min 2 5  m  i  n  F N  = 10.2837 l b f = 51.0773 l b f  ^  =  0.2013  TABLE  I  y  (in.)  (in.) T/C Pos' n  • 1.000 •0.875 • 0.750 •0.625 •0.500 •0.375 •0.250 •0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  4  (Cont'd)  Temperature  251 256 260 264 264 264 264 264 251 225 195 143  0  maximum matching  A average matching  238 246 254 261 268 271 271 264 251 231 207 184 163 145 131 119 110  246 254 263 270 277 281 281 277 259 238 214 190 168 149 133 121 112  T  and  Theoretical  Theory  Expt,  0.5625*  Expt' 1  (°F)  Conditions  t  = 1/4 i n . W  F  m  R  m  = 25.000 l b f  i  = -9.167 l b f  i  V t  = 1.035  in./min  , =2.59 elapse  F  min  = 8.9879 l b f  N = 44.9807 l b f y »  0.1998  •v.. = 0.7 i n . theory 1  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  1.0625 247 247 247 242 238 230 216 204 186 160 134 108  206 207 208 207 205 201 195 187 178 168 156 145 135 125 117 110 103  131  212 214 214 213 211 207 201 193 183 172 160 149 138 128 119 111 105  + i r  theory=  1.0 i n .  TABLE 4  I  (in.)  y (in.)  Tern o e r a t u r e ( °F) Theory Expt.  T/C Pos' n  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  132  (Cont'd)  T  o  maximum matching  0.39062!  386 408 408 419 442 408 341 240 143  316 333 352 373 395 416 433 435 415 371 316 264 221 187 161 141 126  A average matching T  328 346 366 387 411 433 450 453 432 386 328 274 228 192 163 144 128  E x p t ' 1 and Theoretical Conditions  t F  M  W 1 T  >= 1/4 i n . = 27.667 l b f  T == -10.833 l b f i. V = 1.035 i n . / m i n R  elapse F  = 12.0582 l b f  N = 51.9409 l b f 0.2322 •v. . = 0.4 i n . -tneory  TABLE 5 EXPERIMENTAL  VS. THEORETICAL  RESULTS FOR TJL-6A1-4V  I  Temperature (°F) E x p t ' l and Expt. Theory Theoretical T Conditions A 0 average maximum matching matching  y , (in.) (in. ) T/C Pos • n  -1.000 -0.875 -0.750 -0.625 -0.500 . -0.375 -'0.250 -0.125 0 0.125 .0.250 0.3 75 0.500 0.625 0.750 0.875 1.000  0.09375  T  •  649 671 606 397  333 361 395 437 492 565 671 830 972 724 515 387 303 246 205 175 153  345 375 411 455 513 590 701 869 1019 758 535 40 3 315 254 211 180 157  t W  = 1/4 i n .  F  T  = 16.667 l b f  R  T  =-8.333 l b f  V t  = 0.536 i n . / m i n  elapse=  -  0.34375'  638 606 552 364  321 345 372 405 442 485 530 562 554 493 411 349 276 230 196 170 149  333 358 387 421 461 506 553 586 578 515 428 366 286 238 202 174 153  133  i  n  F  = 13.252 l b f  N  = 37.4205 l b f  /-< = 0.3541 '^theory  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  m  =  °'  1  ^theory = °*  i n  3  *  i n  *  134  TABLE 5 (Cont'd)  (in.)  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  y (in.) T/C Pos' n  Temperature (°F) E x p t ' l and Theory Expt. Theoretical T A Conditions average maximum matching' matching T  0.48437 5* 274 307 330 352 375 431 431 386 305 186  286 302 320 339 357 374 386 387 372 343 305 266 230 199 175 155 139  297 314 332 352 371 389 401 402 387 356 316 275 237 206 180 159 142  <w  +  266 276 287 296 304 309 309 304 291 272 249 224 201 180 161 146 133  275 286 297 307 316 321 321 315 302 281 257 232 207 185 166 149 136  i  r  u  1  R  m  l  = -8.333 l b f  V = 0.536 i n . / m i n t , = 2.38 min elapse F  = 12.3645 l b f  N  = 36.5119 l b f = 0.3386  •y, , = 0.5 i n . •'theory +  0.734375 260 282 305 323 341 386 377 341 256 147  1 / 4  F_, = 16.667 l b f  y 1  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  =  . =0.7 i n . theory  TABLE 6 SUMMARY OF FORCES  Material  T (lbf)  F  T (lbf)  R  N  F  (lbf)  (lbf)  Friction Coefficient p  W^U (lbf  1 / 2  ft/sec)  Friction Coefficient p from L i t  U from L i t . from Lit.  (ft/sec)  1/8" Mild Steel  13 11  -6 -5  28 26  10 10  0.4 0.4  1445 1525  0.07[13]+ 0.12[23]  1250 980  £ 650 150  1/4" M i l d Steel  57 66  -21 -25  93 107  11 12  0.1 0.1  2730 2920  0.14[23] 0.09[ 9 ] *  980 980  150 300  1/4" TI Steel  29 27 28 28 25  -12 -11 -11 -11 - 9  55 51 51 51 45  12 13 10 12 9  0.2 0.2 0.20.2 0.2  2080 1995 2020 2030 1890  1/4" Brass  10 10  - 2 - 2  22 21  8 7  0.3 0.3  1335 1300  1/4" T 1 6A1-4V  17 17  - 8 - 8  37 37  13 12  0.4 0.3  1725 1705  x  0.2 [ 9 ] °  300  +En 1A s t e e l xSAE 1113 s t e e l *1% C 1.5% Cr s t e e l °Copper s l i d i n g on s t e e l 00 cn  TABLE  7 TEMPERATURE  DISTRIBUTION  ALONG THE CUTTING FUSION Feed  speed  Friction f o r c e (1 D f ) Friction into Workpiec 2  p e r minute  T e m p e r a t u r e (°F) Mild Steel  (h=4.5B/h/ft /°F)  -1.000 -0.875 -0.750 -0.625 -0.500 -0.375 -0.250 -0.125 0 0.125 0.250 0.375 0.500 0.625 0.750 0.875 1.000  CUTTING  = 2 inches  Brass (in. )  AXIS:  2  (h=7.54B/h/ft /°F) 2  1080 1104 1130 1161 1198 1244 1308 1415 1750 1413 1305 1239 1191 1153 1121 1093 1068  1133 1187 1250 1323 1411 1524 1684 1953 2800 1945 1671 1507 1390 1298 1222 1157 1099  14.450  22.357  0.096127  0.059668  136  Ti-6A1-4V (h=4.5B/h/ft /°F) 2  759 823 899 991 1106 1256 1472 1843 3035 1820 1436 1211 1053 933 837 758 692  22.261  0.027451  T A B L E 8 P H Y S I C A L PROPERTIES OF MATERIALS USED I N THE  Material  Brass Mild  Tl  Stee:  Steel  Ti-6A1-4V  Thermal Conductivity (B/h/ft/°F)  EXPERIMENTS  ComDensity Thermal D i f - S p e c i f i c fusivity Heat posi(ft2/h) (B/lbm/°F) C l b / i n 3 ) t i o n  70.0  1.490  0.090  0.308  26.0  0.480  0.110  0.284  19.10  0.272  0.120  0.313  8.0  0.184  0.157  0.160  137  0.20 t o 0.25% C B a l Fe o i - o-2£C  0-6-l-o%Mn  s*i. 4 ° 6% A l , 4%V, Bal T i  M o  TABLE 9 HAUPTMAN AND RAMSEY'S DISK SOLUTION VERSUS MODIFIED YU'S Material  Feed V e l o c i t y (ipm)  ^0 (BTU/HR) xl0~ 5  Brass  (BTU/HR) xlO-5  Heat" Generated going to workpiece . f  o  Average C u t t i n g Temperature  Friction F  0  Forces F  A  (°F)  (lbf)  (lbf)  1666.531 1666.531  17.076 17.485  16.429 16.844  0.25  *MY +HR  0.22282 0.22816  0.21438 0.21979  0.24252 0.23684  A 0.25206 0.24586  0.50  MY HR  0.22294 0.22828  0.21450 0.21991  0.24294 0.23725  0.25249 0.24629  1666.508 1666.508  17.085 17.494  16.438 16.853  0.75  MY HR  0.22307 0.22841  0.21463 0.22004  0.24339 0.23770  0.25296 0.24675  1666.470 1666.470  17.095 17.505  16.448 16.863  1.00  MY HR MY HR MY HR  0.22322 0.22856 0.22337 0.22871 0.22353 0.22887  0.21477 0.22018 0.21492 0.22032 0.21507 0.22048  0.24387 0.23818 0.24439 0.23869 0.24495 0.23923  0.25347 0.24724 0.25401 0.24777 0.25458 0.24834  1666.417 1666.417 1666.347 1666.347 1666.260 1666.260  17.106 17.516 17.118 17.527 17.131 17.540  16.459 16.873 16.470 16.884 16.482 16.896  MY HR MY HR MY HR MY HR MY HR MY HR  0.30708 0.31578 0.30733 0.31603 0.30765 0.31634 0.30802 0.31672 0.30844 0.31714 0.30890 0.31760  0.29359 0.30240 0.29379 0.30261 0.29404 0.30285 0.29432 0.30313 0.29463 0.30343 0.29496 0.30376  0.10479 0.10191 0.10552 0.10262 0.10644 0.10351 0.10752 0.10457 0.10874 0.10576 0.11007 0.10706  0.10961 0.10641 0.11039 0.10717 0.11136 0.10813 0.11252 0.10925 0.11383 0.11053 0.11527 0.11194  2666.480 2666.480 2666.061 2666.061 2665.378 2665.378 2664.473 2664.473 2663.369 2663.369 2662.095 2662.095  23.533 24.200 23.552 24.219 23.576 24.243 23.605 24.271 23.637 24.304 23.673 24.339  22.499 23.174 22.515 23.190 22.534 23.209 22.555 23.230 22.579 23.254 22.605 23.279  1.25 1.50 Mild Steel  Heat Generated 4A  0.25 0.50 0.75 1.00 1.25 1.50  *MY = M o d i f i e d Yu's +HR - Hauptmann & Ramsey's  f  TABLE 9 Material  Feed V e l o c i t y (ipm)  Heat Generated  (Cont'd)  (BTU/HR) xlO  ^A (BTU/HR) xlO"  Heat Generated going to workpiece fn fA  % - 5  5  Average C u t t i n g Temperature ( °F)  Friction  Forces  FO  FA  (lbf)  Cibf)  Ti-6A14V  *MY +HR  0.25  "MY +HR  0.32909 0.33854  0.31629 0.32593  0.09249 0.08991  0.09623 0.09338  2908.425 2908.425  25.220 25.944  24.239 24.978  0.50  MY HR  0.32950 0.33895  0.31659 0.32623  0.09364 0.09103  0.09745 0.09458  2907.254 2907.254  25.252 25.976  24.262 25.001  0.75  MY HR  0.33007 0.33952  0.31698 0.32661  0.09520 0.09255  0.09913 0.09621  2905.466 2905.466  25.295 26.019  24.292 25.030  1.00  MY HR  0.33075 0.34020  0.31743 0.32705  0.09706 0.09436  0.10113 0.09816  2903.196 2903.196  25.347 26.071  24.326 25.064  1.25  MY HR  0.33151 0.34096  0.31793 0.32754  0.09911 0.09636  0.10334 0.11031  2900.644 2900.644  25.405 26.129  24.364 25.101  1.50  MY HR  0.33230 0.34175  0.31845 0.32805  0.10126 0.09846  0.10566 0.10257  2897.936 2897.936  25.466 26.190  24.404 25.140  - M o d i f i e d Yu's - Hauptmann & Ramsey's  CO  .  •-  •• 140  --  i  Figure  1 Thin  Disk  with  Heat  Input  Figure  2 F l a t P l a t e C o n f i g u r a t i o n w i t h Heat  Input  Plane Source.  0  *3'  -i  i  i  Figure  3 P l a t e with F i n i t e Width and a P l a n e Heat Source  gure 4 P l a t e w i t h F i n i t e Width a Plane Heat Source a l o n g the B r e a d t h o f the P l a t e  and  144  (b)  F i g u r e 5(a) J a e g e r ' s C o n f i g u r a t i o n (b) L i n g and S a i b e l ' s C o n f i g u r a t i o n  Ai>APToR.  PUSH R«>r>  SAWIN<S  p>! SK MOTOR  <=-AR£iACse P O W E R  S H E A R  PINJ  D E V I C E  &PEED  REDUCER.  TRANSMISSION P V I U C Y  VARIABLE SPm D C . MOTOR  SYSTEM  I-*  F i g u r e 6 Schematic Drawing o f the G e n e r a l Arrangement i n F r i c t i o n Sawing P r o c e s s  146  IT  1  iTTt,  UPPER. G U A R D  /  UPPER GUARD STAND  BELT PUL.L.EY  -TAe.L.E A « . i^t  T A B L E  sp/io»i_e  SUPPOilT C_OCK  C O L L A R S  L E F T  sioe  HT<WD  ttouo  S I D E  rm  Tar Figure  7 Sawing  Disk and Table  Figure  8 General  Arrangement  o f Eguipment  and  Instrumentation  F i g u r e 9 Guide B l o c k s  149  F i g u r e 10 Workpiece and Workpiece Mounting Device  150  Figure  11 Mounted  Workpiece  151  F i g u r e 12 View Showing M o d i f i e d  Push Rod Head  152  For Table Support 1  ih'S pari  is  S/rcr/h Ring  fhreodecf  Materia/:  Figure  13  S t r a i n Ring  MifJ  vT/ee./  g u r e 14 View Showing A r r a n g e m e n t Force Measuring Instrumentation  LiariT  PiPts BCi  ose.  N^l £ T £  B^usH  R  ( B A M )  TWO-criANwet-  ® <§> ® d § (> <)i <s> © & &  7  cn  F i g u r e 15 Block Diagram o f Temperature M e a s u r i n g I n s t r u m e n t a t i o n  F i g u r e 16(a) Thermocouple  Circuit  156  F i g u r e 16(b)  Thermocouple P o s i t i o n s E x p e r i m e n t a l Run  during  157  F i g u r e 17 D e b r i s from F r i c t i o n Cut Leaded  Brass  158  F i g u r e 18 M i c r o s t r u c t u r e from Leaded (a) 'As R e c e i v e d , 400 X (b) F r i c t i o n Cut Edge; 400  Brass  1  X  159  L4w  —i  4-  •M4  —_  — I  t  r  •  j  —  _—  _—  -  E;  -4—  — — — 1..  —  _  1  •—•  4  i  —J—f-~J^L ]  —  B R U S H  -11  "1  1I  r  — —  •  —• L  7-1 —  —h  ,..„  —i  -  I-  Mr  — L  _  —  — — _  — .....  F i g u r e 19 L i g h t Pipe Response i n an Edge Cut  — i  1  160  F i g u r e 20 Worms from F r i c t i o n C u t t i n g of S t e e l (a) Edge View o f Workpiece showing "worms" (b) C l o s e - u p View of Workpiece Bottom (c) Close-up View of Workpiece Top  WORKPIEXE  pb-S*  Ceuu  F i g u r e 21(a) Arrangement i n Edge Cut  SENSOR  162  Figure  2Kb) C l o s e - U p V i e w s o f "Worms" f r o m Edge C u t o f M i l d S t e e l  163  F i g u r e 22 T y p i c a l T a b l e and T h r u s t F o r c e s d u r i n g a T e s t Run  F i g u r e 23 Thermocouple Response Corresponding to the F o r c e s i n F i q u r e 22  164  F i g u r e 2 5 M i c r o s t r u c t u r e o f Worm on of M i l d S t e e l Workpiece 400X  Top  165  (a)  F i g u r e 26 M i c r o s t r u c t u r e o f Kerf M a t e r i a l from Sawing M i l d S t e e l Edge View (a) C l o s e t o one s i d e - V i s i b l e O x i d e L a y e r -4.OOX (b) I n t e r i o r S t r u c t u r e 8°ox  166  F i g u r e 27 M i c r o s t r u c t u r e o f K e r f M a t e r i a l f r o m Sawing M i l d S t e e l 400X - B r e a d t h View ( a ) C l o s e t o edge S t r u c t u r e - Shows Oxide Layer (b) I n t e r i o r S t r u c t u r e  167  F i g u r e 28 M i c r o s t r u c t u r e of M a t e r i a l C l o s e to the F r i c t i o n Cut Edge of M i l d S t e e l , 400 X  168  Ca)  F i g u r e 32 M i c r o s t r u c t u r e of K e r f M a t e r i a l f r o m Sawing T l S t e e l 400X - B r e a d t h View (a) C l o s e (b) C l o s e  t o One Edge t o O t h e r Edge  F i g u r e 33 M i c r o s t r u c t u r e from Sawing TI S t e e l ,  F i g u r e 34 M i c r o s t r u c t u r e t o t h e F r i c t i o n C u t Edge  of Kerf M a t e r i a l 400X - Edge View  of Material Close o f T I S t e e l , 400X  Figure  3 5 M i c r o s t r u c t u r e o f 'As R e c e i v e d ' T i - 6 A 1 - 4 V A l l o y , 800X  F i g u r e 3 6 M i c r o s t r u c t u r e c f Worm f r o m Top o f T i - 6 A 1 - 4 V W o r k p i e c e , 800X  172  Figure  37 M i c r o s t r u c t u r e of Kerf M a t e r i a l from Sawing Ti-6A1-4V A l l o y - Edge View (a) 400 X (b) 800 X  173  Figure  38 M i c r o s t r u c t u r e of 'As Received' Ti-6A1-4V A l l o y Heated w i t h Oxy-Acetylene Torch t o S e l f - B u r n i n g ; Allowed to C o o l i n A i r (a) 400 X (b) 800 X  THEORY EXPERIMENT y  900  o • • •  800  o o in -188 -375 -875  700  I  LU  AV. TEMR MATCH  600  D  I— < • • 5 0 0 L-  MAX- TEMR MATCH  LU  a,  LU  400  300  200  100"  -•75  1  1  -•50  -25  DISTANCE Figure  1  0  1 -25  -50  ALONG CUTTING A X I S - i n  39 T e m p e r a t u r e D i s t r i b u t i o n  i n Brass  175  -1-0  -75  --50  --25  0  -25  DISTANCE ALONG CUTTING A X I S - i n F i g u r e 40 Temperature D i s t r i b u t i o n i n 1/8" P l a t e M i l d S t e e l  -50  o1 -75  i  i  --50  —25  i  1  r  I  l  I  I  0  -25  -50  75  DISTANCE ALONG CUTTING A X I S Figure  41 T e m p e r a t u r e D i s t r i b u t i o n i n 1/4" P l a t e M i l d S t e e l  in  177  -75  "-50  DISTANCE Figure  -.2 5  0  ALONG  CUTTING A X I S — in  42 T e m p e r a t u r e  Distribution  -25  i n TI  -50  Steel  '75  0  "75  --50  DISTANCE Figure  ALONG  -25  0  CUTTING  43 T e m p e r a t u r e D i s t r i b u t i o n  -25  AXIS— i n TI  Steel  179  0,1  10  1  =  -75  1  1  1  --50  -25  0  DISTANCE ALONG Figure  44 T e m p e r a t u r e  I -25  CUTTING AXIS— in Distribution  i n TI Steel  I -50  -10  -75  DISTANCE Figure  T  I  --50  -25  1  0  ALONG CUTTING  45 T e m p e r a t u r e  Distribution  T  -25  A X I S — in i n Ti-6A1-4V  -50  F i g u r e 46 C l o s e - U p View o f K e r f M a t e r i a l from M i l d S t e e l  182  F i g u r e 47 P r o f i l e o f Wear Mark Produced on Copper. H o r i z o n t a l M a g n i f i c a t i o n X400, V e r t i c a l X 4000. The arrow i n d i c a t e s d i r e c t i o n o f s l i d i n g . The Displacement o f M e t a l i s C l e a r l y V i s i b l e . [56, p. 459]  A  B  llM  1 F i g u r e 48 Schematic  r e p r e s e n t a t i o n o f the p r o c e s s  o f C u t t i n g w i t h a Saw. [42]  Figure  I I - l ( a ) Table Forces (b) W o r k p i e c e F o r c e s  Figure  iv-1  Sketch of R o t a t i n g Disk Edge L o a d  with  185  Figure  IV-2  Equivalence of Figure  IV-1  ! J  Figure  IV-3  Solid  Disk  with  Edge  Load  186  187  F i g u r e IV-4 C o n f i g u r a t i o n R e q u i r e d t o M o d i f y F i g u r e IV-3 to Get S o l u t i o n to F i g u r e IV-2(b)  188  F i g u r e IV-5  Geometry f o r C a l c u l a t i n g  r  189  Figure  V - l Arrangement f o r C a l i b r a t i n g Table S t r a i n Ring  Figure  VI-1 Arrangement Thrust Strain  for Calibrating Ring  

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