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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 of 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 of Western Ontario, London Ontario 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MECHANICAL ENGINEERING We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA DECEMBER, 1971 In presenting 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 requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s representatives. I t i s understood that p u b l i c a t i o n , i n part or i n whole, or the copying of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my written permission. OMOJOLA OGUNLADE Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada i ABSTRACT Most of the r e s e a r c h done on the f r i c t i o n sawing of metals has been a p p l i c a t i o n o r i e n t e d and the o p e r a t i n g mechanism of f r i c t i o n sawing has not been f u l l y i n v e s t i g a -t e d . Hence the p r e s e n t study was d i r e c t e d towards g a i n i n g a b e t t e r u n d e r s t a n d i n g of f r i c t i o n sawing u s i n g a n a l y t i c and e x p e r i m e n t a l methods. The a n a l y t i c method c o n s i d e r e d the heat p a r t i t i o n a t the i n t e r f a c e and the temperature d i s t r i b u t i o n i n both the workpiece and the sawing d i s k . The c u t t i n g f o r c e s were measured e x p e r i m e n t a l l y by means of s t r a i n r i n g s t o which l i n e a r 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 . From thes e measured f o r c e s , the f r i c t i o n f o r c e a t the d i s k -workpiece i n t e r f a c e was o b t a i n e d and t h i s was used to c a l -c u l a t e the heat g e n e r a t e d d u r i n g the sawing o p e r a t i o n . The e x p e r i m e n t a l method attempted to measure the i n t e r f a c e temperature and the temperature d i s t r i b u t i o n i n the workpiece. Four d i f f e r e n t metals, which gave a wide range of p h y s i c a l p r o p e r t i e s , were used i n the experiments l e a d e d b r a s s , m i l d s t e e l , TI s t e e l and T1-6A1-4V a l l o y . The heat p a r t i t i o n between the sawing d i s k and a metal workpiece was o b t a i n e d by matching the temperatures a t the c o n t a c t zone. T h i s p a r t i t i o n f u r n i s h e d a temperature i i i d i s t r i b u t i o n i n the work-piece and d i s k when the t o t a l heat g e n e r a t e d a t the s l i d i n g c o n t a c t was known. U s i n g a PbS c e l l sensor and o p t i c g l a s s f i b r e t r a n s m i s s i o n u n i t s , an attempt was made t o measure the i n t e r f a c e temperature. However, owing t o the o b l i q u e c u t t i n g of the workpiece and the h i g h l y l o c a l i z e d n a t u r e of f r i c t i o n sawing p r o c e s s , the i n s t r u m e n t a t i o n developed c o u l d not g i v e an a c c u r a t e measurement of the temperature. N e v e r t h e l e s s , t h i s temperature measuring d e v i c e would f i n d good use i n h i g h temperature t r a n s i e n t heat t r a n s f e r s t u d i e s . Thermocouples embedded a t d i f f e r e n t d i s t a n c e s from the c u t t i n g zone were used to measure the temperature d i s -t r i b u t i o n i n the workpiece. The t h e o r e t i c a l r e s u l t s agree w e l l w i t h these e x p e r i m e n t a l measurements; the agreement b e i n g b e t t e r a t p o i n t s which were at l e a s t the o r d e r of magnitude of the k e r f width d i s t a n c e from the c u t t i n g zone. M e t a l l o g r a p h i c e xaminations were done on the a s -r e c e i v e d m a t e r i a l s , the k e r f m a t e r i a l s , and the c u t edge m a t e r i a l s . The heat treatment e v i d e n c e from both m i l d s t e e l and T l s t e e l i n d i c a t e d the k e r f m a t e r i a l and the c u t edge m a t e r i a l reached temperatures about 1600 deg F. T h i s was i n good agreement w i t h the t h e o r e t i c a l p r e d i c t i o n s . A l l the r e s u l t s o b t a i n e d i n t h i s study i n d i c a t e d n o n - f u s i o n c u t t i n g . F o r the l e a d e d b r a s s the f r i c t i o n sawing was s i m i l a r t o c o n v e n t i o n a l machining whereby d i s -c r e t e p a r t i c l e s of m a t e r i a l were formed d u r i n g sawing. F o r the s t e e l s , d u c t i l e f r a c t u r e t h e o r y e x p l a i n e d the c u t t i n g mechanism and f o r the t i t a n i u m a l l o y , b r i t t l e f r a c t u r e was observed a t slow f e e d speeds and b r i t t l e p l u some d u c t i l e f r a c t u r e s were observed a t h i g h e r f e e d speed TABLE OF CONTENTS CHAPTER PAGE I LITERATURE SURVEY, EVALUATION AND STATEMENT OF OBJECTIVE 1 1.1 B r i e f H i s t o r y of Saws 1 1.2 R e l a t e d S u b j e c t s t o F r i c t i o n Sawing 4 1.2.1 I n t r o d u c t i o n . . . . . . . 4 1.2.2 High Speed F r i c t i o n . . . 4 1.2.3 Heat D i s s i p a t i o n 10 1.2.4 Heat F l u x and Temperature D i s t r i b u t i o n • 11 1.2.5 Heat P a r t i t i o n 15 1.3 F r i c t i o n Sawing Mechanism and P r a c t i c e 15 1.4 A p p r a i s a l o f L i t e r a t u r e . . . . 18 1.5 Statement o f O b j e c t i v e 19 I I HEAT TRANSFER AND FORCE ANALYSES IN THE FRICTION 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 Saw Disk Heat T r a n s f e r A n a l y s i s 21 2.3 Workpiece 23 2.3.1 Heat T r a n s f e r 23 2.3.2 L a t e r a l Heat P e n e t r a t i o n . 31 2.3.2 P o s s i b l e Thermal Shock . . 32 v v i CHAPTER PAGE 2.4 The Heat P a r t i t i o n Problem . . . 33 2.4.1 Introduction . . . . . . . . 33 2.4.2 Jaeger's Treatment 34 2.4.3 Ling and Saibel's Approach . 34 2.4.4 Jaeger's Equivalence of Ling and Saibel's Approach . . . 36 2.4.5 Discussion of Jaeger's and, Ling and Saibel's Formulae . 37 2.4.6 Proposed Heat P a r t i t i o n . . 38 2.5 High Speed F r i c t i o n 40 2.6 F r i c t i o n Sawing Mechanism — Governing C r i t e r i a 42 2.6.1 Introduction . . . . . . . 42 2.6.2 Average Temperature C r i t e r i o n . . . . . . . . . 42 2.6.3 Maximum Temperature C r i t e r i o n 43 2.6.4 F r i c t i o n Studies and the C r i t e r i a 44 2.7 Summary . . 44 II I EXPERIMENTAL APPARATUS, INSTRUMENTATION AND METHODS 46 3.1 Apparatus . . . . . 46 3.1.1 Introduction 46 3.1.2 The Saw Disk and Shaft . . . 46 3.1.3 The Disk Motor and V-Belt . 47 3.1.4 The Table 47 3.1.5 The Workpiece Guides . . . . 48 3.1.6 Arrangement to Reduce F r i c t i o n a l Force between Workpiece and Table and Guide 48 v i i CHAPTER PAGE 3.1.7 Arrangement f o r Measuring Table Forces 49 3.1.8 The Feed Mechanism Drive and Transmission 49 3.1.9 The Carriage and the Power Screw 49 3.1.10The Push Rod and Thrust Measuring S t r a i n Ring . . . 50 3.2 Instrumentation 51 3.2.1 Introduction 51 3.2.2 The Disk Speed 51 3.2.3 The Table Force 51 3.2.4 The Feed Thrust 52 3.2.5 The Feed Speed 53 3.2.6 The Temperature D i s t r i b u t i o n Around the Cutting zone . . 53 3.3 Experimental Procedure and Data Treatment 57 3.3.1 Introduction 57 3.3.2 Specimen Preparation . . . . 58 3.3.3 Experimental Method . . . . 58 3.3.4 Treatment of Data 63 IV RESULTS AND DISCUSSIONS 65 4.1 Forces 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 67 4.2.1 T h e o r e t i c a l 67 4.2.2 Experimental 70 4.2.3 Comparison of T h e o r e t i c a l and Experimental Results . . 71 4.2.4 Comparison with Other Works 72 v i i i CHAPTER PAGE 4.3 Metallography 74 4.3.1 Mild Steel 74 4.3.2 TI S t e e l ( A l l o y S t r u c t u r a l Steel) 75 4.3.3 Brass 76 4.3.4 Ti-6Al-4V A l l o y 76 4.3.5 Conclusions Drawn from Metallographic Examinations 77 4.4 Materials from Kerf . . . . . . . 78 4.5 Explanation of the Cutting Process 79 V CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY 83 BIBLIOGRAPHY 85 APPENDIX I Estimation of the Heat Transfer C o e f f i c i e n t s . . . . . . . . . . . 91 APPENDIX II Force Analysis i n F r i c t i o n Sawing System 98 APPENDIX: III Estimation of Heat P a r t i t i o n F r a c t i o n 103 APPENDIX IV Stress Analysis on the Saw Disk . 106 APPENDIX V Table Force S t r a i n Ring C a l i b r a t i o n 119 APPENDIX VI Thrust Force S t r a i n Ring C a l i b r a t i o n . 121 LIST OF TABLES TABLE PAGE 1. Data and R e s u l t s f o r Bras s 122 2. Data and 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. Data and 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. Data and R e s u l t s f o r T l S t e e l 128 5. Data and R e s u l t s f o r Ti-6A1-4V A l l o y 133 6. Summary of F o r c e s 135 7. F u s i o n C u t t i n g Temperature D i s t r i b u t i o n a l o n g the C u t t i n g A x i s 136 8. P h y s i c a l P r o p e r t i e s o f the M a t e r i a l s Used i n E x p e r i m e n t a l Work 137 9. Comparison of Hauptmann and Ramsey's Disk S o l u t i o n w i t h M o d i f i e d Yu's I38 i x LIST OF FIGURES FIGURE PAGE 1 Thin Disk with Heat Input 140 2 F l a t Plate Configuration with Heat Input . 141 3 Plate with F i n i t e Width and a Plane Heat Source 142 4 Plate with F i n i t e Width and a Plane Heat Source along the Breadth of the Plate . . . 143 5 Jaeger's and, Ling and Saibel's Configura-t i o n s 144 6 Schematic Drawing of the General Arrangement i n F r i c t i o n Sawing Process 145 7 Sawing Disk and Table 146 8 General Arrangement of Equipment and Instrumentation 147 9 Guide Blocks 148 10 Workpiece and Workpiece Mounting Device . . 149 11 Mounted Workpiece 150 12 View Showing Modified Push Rod Head . . . . 151 13 S t r a i n Ring 152 14 View Showing Arrangement of Force Measuring Instrumentation 153 15 Block Diagram of Temperature Measuring Instrumentation 154 16(a) Thermocouple C i r c u i t 155 16(b) Thermocouple P o s i t i o n s during Experimental Run 156 x x i FIGURE PAGE 17 D e b r i s from F r i c t i o n Cut Leaded Brass . « « 157 18 M i c r o s t r u c t u r e from Leaded Brass . » . . . 158 19 L i g h t P ipe Response i n an Edge Cut « . . « 159 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 o f Workpiece Bottom (c) C l o s e - u p View of Workpiec Top . . » * . 160 21(a) Arrangement i n Edge Cut ? « . o o o . o e o 161 21(b) C l o s e - u p View o f "Worms" from Edge Cut o f M i l d S t e e l <>* o r o c o o o » < : < > • « • « • 162 22 T y p i c a l Table and T h u r s t Forces d u r i n g a l 6 S t Rlin « o © ft o * o e # > o e e o o « e X 6 3 23 Thermocouple Response Cor respond ing to the Forces i n F i g u r e 2 2 . e . » . . e . o . o e o 163 24 M i c r o s t r u c t u r e o f 8 A s Received* M i l d S t e e l 25 M i c r o s t r u c t u r e of 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 of 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 ide —• V i s i b l e Oxide Laye r (b) I n t e r i o r S t r u c t u r e . « . • . . . < . . . 165 2 7 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 from Sawing M i l d S t e e l 400 X — Bread th View (a) C lose to Edge S t r u c t u r e •— Shows Oxide Layer (b) I n t e r i o r .S t ructure * . < > • . „ . . . e « 166 28 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 the F r i c t i o n Cut 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 ' A s R e c e i v e d ' T l S t e e l , 4 0 0 Q « « o « « > 3 > e 9 o i t Q £ ' - e e o « » . C ' 3 30 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 Flame Cut Edge o f T l S t e e l 400 X . . . « . 168 31 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 FIGURE x i i PAGE 32 M i c r o s t r u c t u r e of Ke r f M a t e r i a l from Sawing TI S t e e l 400 X — Breadth View (a) C l o s e to One Edge (b) C l o s e to Other Edge «. . „ „ * » <, » » 169 33 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 —- Edge View „ * „ . * » * 170 34 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 the F r i c t i o n Cut Edge of TI S t e e l , 400 X . a . 170 35 M i c r o s t r u c t u r e of 'As Received' T1-6AI-4V A l l o y ^ 800 X «> o o o e o o * o • o • « o • X7X 36 M i c i ' o s t r u c t u r e o f Worm from Top of Ti-6A1-4V Workpiece, 800 X . . . » c o . . 171 37 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 9 Ti-6A1-4V A l l o y — Edge View (a) 400 X (i?) S00 a o o o o « o o * e e- o © o o o X-/2 38 M i c r o s t r u c t u r e of 'As R e c e i v e d ' T1-6A1-4V 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 39 Temperature D i s t r i b u t i o n i n Brass . » » e 174 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 41 Temperature 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 42 Temperature D i s t r i b u t i o n i n TI S t e e l „ „ * 177 43 Temperature D i s t r i b u t i o n i n TI S t e e l e „ » .178 44 Temperature D i s t r i b u t i o n i n TI S t e e l . « . 179 45 Temperature D i s t r i b u t i o n i n Ti-6A1~4V - <, * 180 46 Close-Up View of K e r f M a t e r i a l from M i l d ^ t ^ 6 6 X o e ^ o » o e o o o o o « * o o o Q « X t^X 47 P r o f i l e of 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 X4000. 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 of Metal i s C l e a r l y V i s i b l e [56, p 459] , «. « c „ c » 182 x i i i FIGURE PAGE 48 Schematic Representation of the Process of Cutting with a Saw [42] 182 II/-1 Ca) Table Forces (b) Workpiece Forces 183 IV-1 Sketch of Rotating Disk with Edge Load . . 184 IV-2 Equivalence of Figure IV-1 185 IV-3 S o l i d Disk with Edge Load 186 IV-4 Configuration Required to Modify Figure IV-3 to Get Sol u t i o n to Figure IV-2b 187 IV-5 Geometry fo r C a l c u l a t i n g r 188 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 189 VI-1 Arrangement f o r C a l i b r a t i n g Thrust S t r a i n Ring 190 ACKNOWLEDGEMENT In the course of my study, I have come i n contact with many h e l p f u l people: f a c u l t y , t e c h n i c a l s t a f f and fell o w graduate students. To a l l , I express my gratitude and appreciation. In p a r t i c u l a r , I thank Drs. Lund and Hawbolt of the Department of Metallurgy f o r t h e i r help i n the metallographic i n t e r p r e t a t i o n s , Drs. Hauptmann, Iqbal and Gartshore f o r t h e i r h e l p f u l discussions and suggestions, Dr. Hazel f o r his kindness i n lending us h i s l i g h t pipes and Mr. Jones of the Tribology Laboratory f o r h i s coopera-t i o n i n the experimental phase of the study. My sincere thanks to Dr. Brockley my research supervisor and f a c u l t y adviser f o r h i s sustained i n t e r e s t i n the study, h i s encouragement and guidance throughout the programme. Last, but not the l e a s t , I thank my wife, Olufunlayo, f o r her patience and endurance. The experimental work was done at the Tribology Laboratory of the Department of Mechanical Engineering, U n i v e r s i t y of B r i t i s h Columbia. F i n a n c i a l assistance was provided by the National Research Council of Canada under grant No. A -1065 and t h i s i s g r a t e f u l l y acknowledged. x i v NOMENCLATURE A Actual area of contact F F r i c t i o n force i n the c u t t i n g zone G Shear modulus of workpiece material K Thermal c o n d u c t i v i t y Thermal c o n d u c t i v i t y of moving member K g Thermal c o n d u c t i v i t y of stationary member L Latent heat of fu s i o n lij Parameter defined i n equation (34) N Normal force i n the c u t t i n g zone P Load applied at contact zone T Temperature at an a r b i t r a r y point T Melting point of s o l i d m T Ambient temperature U S l i d i n g speed V Cutting rate V c Peripheral v e l o c i t y of saw disk a Outer radius of saw disk b Inner radius of saw disk b^ Half width of workpiece plate c C h a r a c t e r i s t i c length associated with contact zone area c^ S p e c i f i c heat xv x v i Saw d i s k diameter = 2a F r a c t i o n o f heat g e n e r a t e d g o i n g i n t o s t a t i o n a r y member F r a c t i o n based on average temperature matching F r a c t i o n based on maximum temperature matching Heat t r a n s f e r c o e f f i c i e n t i n saw d i s k Average heat t r a n s f e r c o e f f i c i e n t o f the top and bottom o f the workpiece Length o f heat source i n x - d i r e c t i o n Lengths d e f i n e d i n Appendix I I Length of workpiece p l a t e Parameter d e f i n e d i n e q u a t i o n (12) Landau's parameter Mean f l o w p r e s s u r e T o t a l heat based on average temperature matching P o r t i o n o f heat g e n e r a t e d d i s s i p a t e d through the saw d i s k T o t a l heat based on maximum temperature matching T o t a l f r i c t i o n h eat g e n e r a t e d P o r t i o n of heat g e n e r a t e d d i s s i p a t e d through the workpiece r - c o o r d i n a t e i n p o l a r c o o r d i n a t e plane Time c o o r d i n a t e Saw d i s k t h i c k n e s s Workpiece t h i c k n e s s R a d i a l d i s p l a c e m e n t e X p x v i i v T a n g e n t i a l d i s p l a c e m e n t et L i n e a r thermal expansion ^ R e c i p r o c a l o f twice the thermal d i f f u s i v i t y , 1/aX Shear s t r a i n r e f e r r e d t o p o l a r c o o r d i n a t e s &Lj Kronecker d e l t a -^ Parameter d e f i n e d i n e q u a t i o n (42) £ r R a d i a l s t r a i n i n p o l a r c o o r d i n a t e s £jj S t r a i n t e n s o r elements £ e C i r c u m f e r e n t i a l s t r a i n i n p o l a r c o o r d i n a t e s Q Temperature r i s e above the i n i t i a l temperature of workpiece Angle i n p o l a r c o o r d i n a t e s Thermal d i f f u s i v i t y Thermal d i f f u s i v i t y of moving member k •s Thermal d i f f u s i v i t y of s t a t i o n a r y member ^ A parameter d e f i n e d i n e q u a t i o n (3a) ^ F r i c t i o n c o e f f i c i e n t . /Jrr, -6 10 meter - / U > 1 A parameter d e f i n e d i n e q u a t i o n Ql^a) Po i s s o n ' s r a t i o or k i n e m a t i c v i s c o s i t y ^ A b s c i s s a of the dynamic c o o r d i n a t e system P D e n s i t y o f m a t e r i a l ^ S t r e s s t e n s o r elements ^ r R a d i a l s t r e s s i n p o l a r c o o r d i n a t e s 6 C i r c u m f e r e n t i a l o r hoop s t r e s s i n p o l a r c o o r d i n a t e system ^ Temperature r i s e above the ambient temperature *C~S Shear s t r e n g t h of workpiece Thermal s t r e s s i n d u c e d i n the workpiece Shear s t r e s s i n p o l a r c o o r d i n a t e system Temperature f u n c t i o n A i r y s t r e s s f u n c t i o n Angle d e f i n e d i n F i g u r e I I - 2 A n g u l a r v e l o c i t y o f the saw d i s k CHAPTER I LITERATURE SURVEY, EVALUATION AND STATEMENT OF OBJECTIVE 1.1 A B r i e f H i s t o r y of Saws The use of saws may be r e g a r d e d as one of the most a n c i e n t a r t s known t o man. While no h i s t o r i c a l r e c o r d p o i n t s t o the e x a c t time of the f i r s t use of saws, t h e r e i s enough c i r c u m s t a n t i a l e v i d e n c e to make one c o n c l u d e t h a t saws, i n t h e i r most crude form, must have been used thousands of y e a r s b e f o r e the C h r i s t i a n e r a [ 1 ] . The o l d e s t saws known t o e x i s t a t the p r e s e n t time are i n the B r i t i s h Museum [ 2 ] . They bear the stamp of Thotmoth the second, k i n g of Egypt, who was b e l i e v e d t o have l i v e d about 2000 B. C. The E g y p t i a n s were b e l i e v e d t o have used t o o t h l e s s copper saw b l a d e s t o c u t stone b l o c k s u s i n g sand as a b r a s i v e s [ 3 ] . In the books o f Samuel, Kings and C h r o n i c l e s sawing of stone b l o c k s i s mentioned; t h e s e books were w r i t t e n about 1040, 580 and 460 B. C. r e s p e c t i v e l y [ 4 ] , The book of I s a i a h w r i t t e n about 732 B. C. conveyed the i d e a of the r e c i p r o c a t i n g a c t i o n of the saws used i n t h a t p e r i o d . The a n c i e n t Greek c i v i l i z a t i o n saw v e r y e x t e n s i v e use of saws. In 1 f a c t , the Greeks i g n o r a n t l y c r e d i t e d Daedalus or h i s d i s -c i p l e P e r d i x w i t h the i n v e n t i o n of the saw because these two men used saws i n t h e i r renowned a r c h i t e c t u r a l and s c u l p t u r a l works [ 2 ] , Between 1000 and 500 B. C , i r o n was i n t r o d u c e d i n t o man's c i v i l i z a t i o n . Thus the Romans used i r o n saws which, l o o k e d v e r y much l i k e our modern-day saws [ 3 ] . The c i r c u l a r saw, w i t h i t s c o n t i n u o u s c u t t i n g a c t i o n , d i d not seem to have appeared e a r l i e r than the l a t t e r h a l f of the e i g h t e e n t h c e n t u r y i n England. With r e s p e c t to woodworking machinery G e n e r a l S i r Samuel Bentham [2] was c r e d i t e d w i t h i n v e n t i n g and m a n u f a c t u r i n g c i r c u l a r saws, among o t h e r machines, b e f o r e 1800 A. D. but the e n g i n e e r s gave the c r e d i t of i n v e n t i n g c i r c u l a r saws to S i r Marc Isambard [ 5 ] , However, i n H o l l a n d , c i r -c u l a r saws had been used towards the end of the seven-t e e n t h c e n t u r y [ 5 ] , From these v a r i o u s s o u r c e s of i n f o r -mation, one may c o n c l u d e t h a t the c i r c u l a r saw was an i n v e n t i o n o f the i n d u s t r i a l r e v o l u t i o n i n Europe. F r i c t i o n sawing i s a much more r e c e n t p r a c t i c e than c o n v e n t i o n a l sawing p r o c e s s . A c c o r d i n g t o Lewis [6] the f i r s t r e p o r t on f r i c t i o n sawing was c o n t a i n e d i n a l e t t e r dated F e b r u a r y 3, 1823 which Rev. Herman Daggett of C o r n w a l l , C o n n e c t i c u t , wrote to P r o f e s s o r Benjamin S i l l i m a n of Y a l e C o l l e g e . The l e t t e r d i s c u s s e d the exp-e r i e n c e of a l o c a l c a b i n e t maker, Mr. Barnes, who wanted to r e p a i r a c r o s s c u t saw made of v e r y hard p l a t e . He f i x e d an a x l e to a c i r c u l a r i r o n p l a t e and put i t on h i s 3 l a t h e which gave i t a h i g h r o t a t i n g speed. He attempted to f i n i s h the edge of the p l a t e w i t h a f i l e but the f i l e was c u t i n t o two p i e c e s and no mark was observed on the edge of the p l a t e . He a p p l i e d a rock c r y s t a l p i e c e which was a l s o c u t . He then a p p l i e d h i s c r o s s saw p l a t e under the b l a d e and i t was n e a t l y and c o m p l e t e l y c u t through l o n g i t u d i n a l l y w i t h i n a few minutes. On s t o p p i n g the d i s k he found no wear marks and when he touched the edge t h e r e was no s e n s i b l e h eat. During the o p e r a t i o n t h e r e appeared a band o f i n t e n s e f i r e around the c u t t i n g zone which con-t i n u a l l y e m i t t e d sparks w i t h v i o l e n c e . Mr. Barnes a f t e r -wards marked the saw f o r the t e e t h and i n a s h o r t time c u t them out w i t h h i s new t o o l . I t seemed the d i s k never came i n a c t u a l c o n t a c t w i t h the p l a t e and so Mr. Daggett r a i s e d the q u e s t i o n whether the ' f i r e ' was the " e l e c t r i c f l u i d . " In 1824, D a r i e r and C o l l a d o n conducted experiments at Geneva, S w i t z e r l a n d w i t h a 7 1/2 i n c h e s diameter s o f t i r o n d i s k and found the p e r i p h e r a l speed r e q u i r e d to s t a r t f r i c t i o n sawing was around 35 f e e t per second. At t h i s speed and up to about 70 f e e t per second, the d i s k was s l i g h t l y damaged but. above 70 f e e t per second, the d i s k was u n a f f e c t e d . The f i r s t f r i c t i o n sawing machine b u i l t i n America was p r o b a b l y made by the C a r n e g i e Phipps Company L i m i t e d (now known as the C a r n e g i e S t e e l Company L t d . ) i n 1887 [ l ] . With the r a p i d r a t e of change of t e c h n o l o g y , i t appeared t h a t by the end of the n i n e t e e n t h c e n t u r y and 4 e a r l y i n the t w e n t i e t h c e n t u r y f r i c t i o n sawing was b e i n g used e x t e n s i v e l y . 1.2 R e l a t e d S u b j e c t s to F r i c t i o n Sawing 1.2.1 I n t r o d u c t i o n A f a i r a p p r a i s a l of f r i c t i o n sawing w i l l i n v o l v e u n d e r s t a n d i n g the f r i c t i o n a t h i g h s l i d i n g speeds; the e f f e c t 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 to heat genera-t i o n and d i s s i p a t i o n ; , heat t r a n s f e r i n the f r i c t i o n machine and the workpiece; the mode of c u t t i n g a c t i o n and the d i s p o s a l o f d e b r i s formed i n the k e r f . Most of these t o p i c s have been d i s c u s s e d i n the l i t e r a t u r e and t h i s s e c t i o n g i v e s a s u c c i n c t r e v i e w o f c u r r e n t knowledge. 1.2.2 High Speed F r i c t i o n Bowden and F r e i t a g [7] d i d e x t e n s i v e work on h i g h speed f r i c t i o n . U s i n g an i n g e n i o u s apparatus d e s i g n e d by Beams and h i s co-workers [ 8 ] , they were a b l e t o measure f r i c t i o n between two s o l i d b o d i e s a t speeds up t o 800 m/sec. They r o t a t e d a 1/2 i n c h diameter s t e e l b a l l i n a magnetic f i e l d a t v e r y h i g h f r e q u e n c y and then d e c e l e r a t e d by p r e s s i n g a g a i n s t the b a l l t h r e e s y m m e t r i c a l l y arranged pads made of o t h e r m a t e r i a l s . The pad m a t e r i a l s used ranged from l o w - m e l t i n g bismuth to h i g h - m e l t i n g t u n g s t e n . They r e p o r t e d t h e i r r e s u l t s i n graphs of f r i c t i o n c o e f f i -c i e n t v e r s u s s l i d i n g speed. Fo r copper, Bowden and F r e i t a g found t h a t the 5 f r i c t i o n c o e f f i c i e n t - s p e e d c h a r a c t e r i s t i c c u r v e s were a f f e c t e d by the i n i t i a l speeds; the e f f e c t b e i n g more p r o -nounced a t slow s t a r t i n g speeds than a t h i g h . The f r i c -t i o n c o e f f i c i e n t a t h i g h speeds was low but i n c r e a s e d as the speed d e c r e a s e d . At about 120 t o 140 m/sec the s t e e l -copper s u r f a c e suddenly s e i z e d w i t h a v i o l e n t j e r k . Load and s u r f a c e roughness e f f e c t s on f r i c t i o n c o e f f i c i e n t were found t o be i n s i g n i f i c a n t . At speeds g r e a t e r than 100 m/sec, copper t r a n s f e r to s t e e l was observed and a t 150 m/sec, copper was observed t o f l o w and when s l i d i n g was p r o l o n g e d , a r e l a t i v e l y deep metal l a y e r behaved l i k e a h i g h l y v i s -cous m a t e r i a l . Aluminum and dur a l u m i n behaved v e r y much l i k e copper but t h e i r s e i z u r e on s t e e l o c c u r r e d a t about 40 m/sec. A t h i g h speeds, of the o r d e r of 700 m/sec, copper had a lower f r i c t i o n c o e f f i c i e n t than aluminum but h i g h e r than d u r a l u m i n . Metal t r a n s f e r o f both metals t o s t e e l was c o n s i d e r a b l e . While aluminum was observed to fl o w , d u r a l u m i n c h i p p e d . Antimony behaved v e r y much l i k e d ur-alumin e x c e p t t h a t i t s f r i c t i o n c o e f f i c i e n t was about h a l f t h a t o f d u r a l u m i n . Bismuth had v e r y low f r i c t i o n c o e f f i c i e n t a t 200 m/sec, but a t g r e a t e r speed the f r i c t i o n i n c r e a s e d r a p i d l y . Bowden and P e r s s o n [9] t r e a t e d the c h a r a c t e r i s t i c f r i c t i o n -speed b e h a v i o u r of bismuth i n more d e t a i l . The r e s u l t s f o r molybdenum and tu n g s t e n showed t h a t f r i c t i o n d e c r e a s e d as s l i d i n g speed i n c r e a s e d . Smeared meta l over the t e s t pads was found to c o n s i s t m a i n l y of i r o n but w i t h t r a c e s of molybdenum and t u n g s t e n . A f t e r e t c h i n g the rubbed r e g i o n s on the pads, e x a m i n a t i o n r e v e a l i r o n which Bowden and F r e i t a g suggested to be due to h i g h temperature d i f f u s i o n . Another se t of exper iments was done w i t h s t e e l , s t e e l p l a t e d w i t h 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 diamond. The f r i c t i o n c o e f f i c i e n t -speed cu rves showed sudden jumps o f f r i c t i o n c o e f f i c i e n t v a l u e s a t c e r t a i n c r i t i c a l speeds . Fo r bo th p l a i n s t e e l b a l l and chromium p l a t e d b a l l , t h i s c r i t i c a l speed was about 200 m/sec and f o r the copper p l a t e d b a l l the speed .was about 100 m/sec . Above the c r i t i c a l speeds, the c u r v e s l ooked s i m i l a r to cu rves o b t a i n e d when each meta l was rubbed a g a i n s t i t s e l f but below the c r i t i c a l speeds the f r i c t i o n c o e f f i c i e n t f e l l t o a v e r y low v a l u e . I n the e x p e r i m e n t s , Bowden and F r e i t a g observed t h a t when the s l i d i n g speed of each meta l on diamond exceeded the c r i t i c a l speed the meta l was smeared over the su r f ace o f the diamond; thus r u b b i n g took p l a c e between the meta l and i t s e l f smeared on the diamond and the diamond was not a f f e c t e d . However, when the s l i d i n g speed was below the c r i t i c a l , no apparent me ta l t r a n s f e r was observed on the diamond. The diamond su r f ace was e a s i l y abraded and p o l i -shed p robab ly because o f h igh - t empera tu re phase change from diamond to amorphous ca rbon ; t h i s t r a n s f o r m a t i o n i s s low a t 1000 deg C and a c c e l e r a t e s as the temperature 7 i n c r e a s e s and a t 1800 deg C, i t proceeds a t a h i g h r a t e . Bowden and F r e i t a g ' s r e s u l t s a l s o showed t h a t the e f f e c t -i v e n e s s o f the metals i n a b r a d i n g diamond i n c r e a s e d with t h e i r m e l t i n g p o i n t : chromium ( m e l t i n g p o i n t o f 1615 deg C) was 100 times more e f f e c t i v e than copper ( m e l t i n g p o i n t o f 1083 deg C ) . Bowden and P e r s s o n [9] used the same apparatus as Bowden and F r e i t a g e x cept f o r the s l i d i n g t e c h n i q u e . They employed an impact t e c h n i q u e i n s t e a d of the d e c e l e r a t i o n t e c h n i q u e t o i n v e s t i g a t e the types o f d e f o r m a t i o n , wear, and c r i t e r i o n f o r l a r g e - s c a l e m e l t i n g of s o l i d s i n h i g h -speed f r i c t i o n . The p h y s i c a l p r o p e r t i e s o f the s o l i d s such as thermal c o n d u c t i v i t y and m e l t i n g p o i n t were u s e f u l parameters i n t h e i r s t udy. They r e p o r t e d r e s u l t s on s t e e l rubbed a g a i n s t bismuth, Wood's a l l o y , l e a d , t i n , s i l v e r n i t r a t e , copper, s t e e l , polymers ( n y l o n , t e r y l e n e , p o l y -t e t r a f l u o r o e t h y l e n e ) , rubber and g l a s s . For the low-m e l t i n g p o i n t metals ( B i , Wood's a l l o y , Pb, Sn) and low-m e l t i n g p o i n t non-metal s i l v e r n i t r a t e (AgNO^) the c h a r -a c t e r i s t i c f r i c t i o n - s p e e d c u r v e s e x h i b i t minima v a l u e s a t c e r t a i n c r i t i c a l speeds. Above these speeds, the f r i c t i o n i n c r e a s e d r a p i d l y w i t h consequent i n c r e a s e d wear and m e l t i n g . For copper, macroscopic m e l t i n g d i d not o c c u r . However, the d i f f e r e n c e between the f r i c t i o n c o e f f i c i e n t s a t l i g h t and h i g h l o a d s was s i g n i f i c a n t a t low speeds and t a p e r s o f f w i t h i n c r e a s e d speed. S t e e l a g a i n s t i t s e l f e x h i b i t e d s i m i l a r c h a r a c t e r i s t i c s as copper but had lower 8 f r i c t i o n c o e f f i c i e n t s . F r e i t a g suggested t h a t t h i s behav-i o u r was due to the lower thermal c o n d u c t i v i t y of s t e e l than copper (K_ = 0.9 vs K , , = 0.1). In both s t e e l Cu s t e e l r u b b i n g a g a i n s t i t s e l f and copper, the f r i c t i o n c o e f f i c i e n t d e c r e a s e d w i t h i n c r e a s e d s l i d i n g speed. Even though the polymers have low m e l t i n g p o i n t s , t h e i r f r i c t i o n - s p e e d c u r v e s e x h i b i t e d no minima v a l u e s ; r a t h e r , t h e i r f r i c t i o n c o e f f i c i e n t d e c r e a s e d as the s l i d i n g speed i n c r e a s e d . P o l y t e t r a f l u o r o e t h y l e n e (P.T.F.E.) was observed t o have h i g h e r f r i c t i o n a t h i g h speeds than a t low. The b e h a v i o u r o f n y l o n and of the polymers was due to t h e i r h i g h m e l t - v i s c o s i t y and low thermal c o n d u c t i v i t y ; a c o n s i d e r a b l e p a r t of the f r i c t i o n energy might be expen-ded i n r a i s i n g the temperature o f the molten i n t e r f a c e r a t h e r than i n expanding the molten zone and hence g i v i n g the low wear observed f o r these m a t e r i a l s a t h i g h speeds. M i l l e r [10] i n v e s t i g a t e d the a b r a s i v e mechanisms o f h ard s o l i d s : s a p p h i r e ( A ^ O ^ ) , r u t i l e ( T i C ^ ) , t i t a n -ium c a r b i d e ( T i C ) , s i l i c o n c a r b i d e ( S i C ) , diamond and f u s e d s i l i c a ( S i C ^ ) s l i d i n g on metal s u r f a c e s a t h i g h speeds. He reasoned t h a t the a b r a s i o n o f diamond cannot be e x p l a i n e d t o t a l l y by the thermal mechanism proposed by Bowden and F r e i t a g [7] but t h a t some mechanical p r o c e s s , f i r s t proposed by Tolkowsky [10] i n 1920 and l a t e r modi-f i e d by W i l k s and W i l k s i n 1959, must a l s o be c o n s i d e r e d . H i s r e s u l t s and d i s c u s s i o n s showed t h a t some of these hard s o l i d s have a tendency t o c r a c k i n the presence of thermal shock. He e x p l a i n e d t h a t the f a i l u r e of most of these m a t e r i a l s was due to the s t e e p temperature g r a d i e n t d e v e l -oped i n the v e r y t h i n r e g i o n c l o s e to the r u b b i n g s u r f a c e s That diamond transformed to g r a p h i t e or amorphous carbon at the i n t e r f a c i a l temperature reached [7] was c o n f i r m e d by M i l l e r ' s experiments., So f a r , h i g h speed f r i c t i o n has been s t u d i e d i n vacuum whereas i n most p r a c t i c a l a p p l i c a t i o n s a t h i g h speed c u t t i n g takes p l a c e i n o r d i n a r y atmospheric c o n d i -t i o n s . In o r d e r to study h i g h speed f r i c t i o n under such c o n d i t i o n s W i l l i a m s [ I I ] c o n s t r u c t e d an apparatus which was c a p a b l e of a t t a i n i n g speeds up t o 750 f e e t per second. T h i s apparatus c o n s i s t e d of an a i r t u r b i n e c o u p l e d to a r o t o r d i s k * W i l l i a m s and G i f f e n [12] used t h i s equipment to perform t e s t s a t v a r i o u s s l i d i n g speeds and normal l o a d on s t e e l s l i d i n g on s t e e l . They found t h a t : (1) the c o e f f i c i e n t of f r i c t i o n d e creased asympto-t i c a l l y w ith i n c r e a s i n g normal l o a d f o r any h i g h s l i d i n g speed, and ( 2 ) the c o e f f i c i e n t of f r i c t i o n d e creased asymp-t o t i c a l l y w i t h i n c r e a s i n g s l i d i n g speed f o r any normal l o a d a p p l i e d . The asymptotes would be p r e d i c t e d by the l i q u i d b e haviour of the metals i n v o l v e d i n the s l i d i n g mechanism. A f t e r c o r r e l a t i n g t h e i r d a t a , W i l l i a m s a n d . G i f f e n o b t a i n e d the f o l l o w i n g e m p i r i c a l formulae: u - K ( U N ) " 0 , 4 5 10 where u i s the f r i c t i o n c o e f f i c i e n t , K a c o n s t a n t , U the s l i d i n g speed and N the normal l o a d . U s i n g the f r i c t i o n f o r c e i n s t e a d of the normal l o a d , they got u = K ' ( U F ) " ° * 8 2 where F i s the f r i c t i o n f o r c e and K' a c o n s t a n t . T h e i r d a t a were spread over the speed range of 80 t o 750 f e e t per second. S l i g h t l y m o d i f y i n g W i l l i a m s and G i f f e n ' s appara-t u s , E a r l e s and Kadhim [13] s t u d i e d the f r i c t i o n c h a r a c -t e r i s t i c s and wear of s t e e l s l i d i n g on s t e e l a t speeds up to 655 f e e t per second. They found t h a t the f r i c t i o n c o e f f i c i e n t was a f u n c t i o n of a parameter i n v o l v i n g the normal l o a d N and s l i d i n g speed, U, i . e . y = 0 ( N 1 / 2 U ) T h e i r d a t a c o r r e l a t e d f o r d i f f e r e n t p i n s i z e s showed t h a t 1/2 the f r i c t i o n c o e f f i c i e n t d e c r e a s e d as the parameter N U i n c r e a s e d . Comparing t h e i r r e s u l t s w i t h W i l l i a m s and G i f f e n ' s gave good agreement. 1.2.3 Heat D i s s i p a t i o n Blok [14] e x t e n s i v e l y d e a l t w i t h the problem of heat d i s s i p a t i o n i n both s o l i d and f l u i d f r i c t i o n . H i s d i s c u s s i o n f o l l o w e d two c l e a r l y d e f i n e d l i n e s ; v i z . p r i m a r y d i s s i p a t i o n which c o n s i d e r e d heat d i s s i p a t i o n i n the a r e a c l o s e to the source of heat g e n e r a t i o n and s econ-dary d i s s i p a t i o n which f o c u s s e d on the d i s s i p a t i o n i n t o the s u r r o u n d i n g media of the mechanism. Bl o k ' s work has found d i r e c t a p p l i c a t i o n i n boundary and hydrodynamic l u b r i c a t i o n . As p o i n t e d out i n h i s paper, heat d i s s i p a t i o n i s s t r o n g l y dependent on the c o n f i g u r a t i o n o f the c o n t a c t i n g s u r f a c e s and a l s o on the n ature of the i n t e r f a c e . T h e r e f o r e heat d i s s i p a t i o n w i l l o n l y be m e a n i n g f u l when a working model i s d e f i n e d or s p e c i f i e d . Crook [15] d i s c u s s e d heat d i s s i p a -t i o n i n c o n n e c t i o n w i t h hydrodynamic l u b r i c a t i o n . 1.2.4 Heat F l u x and Temperature D i s t r i b u t i o n C l o s e l y t i e d t o heat d i s s i p a t i o n i s the s u b j e c t of heat f l u x and temperature d i s t r i b u t i o n i n the s l i d i n g s o l i d s . L i n g and Simkins [16] u s i n g a c o n f i g u r a t i o n con-s i s t i n g of a d i s k r u b b i n g a g a i n s t a r i n g s e c t o r were a b l e to o b t a i n temperature p r o f i l e s a t c o n v e n i e n t p o s i t i o n s i n the r i n g s e c t o r . They s o l v e d the heat e q u a t i o n f o r both the s l i d e r ( r o t a t i n g d i s k ) and the r i d e r ( r i n g s e c t o r ) u s i n g Green's f u n c t i o n approach. They assumed a q u a s i -s t a t i o n a r y s t a t e f o r the d i s k and steady s t a t e f o r the r i n g s e c t o r . L i n g and Pu [17] approached the s u b j e c t o f s u r f a c e temperature c a l c u l a t i o n t h e o r e t i c a l l y . For a d i s k - r i n g c o n f i g u r a t i o n they i n t r o d u c e d the i d e a of macroscopic temperature jump a c r o s s the i n t e r f a c e ; a temperature 12 d i f f e r e n c e e x i s t i n g between the d i s k and the r i n g s e c t o r s u r f a c e temperatures. On the phenomenon of macroscopic temperature jump on f r i c t i o n heat g e n e r a t i o n , L i n g and Pu summarized t h e i r f i n d i n g s as f o l l o w s : "(a) In g e n e r a l , f r i c t i o n a l heat i s not gener-a t e d i n the space between two b o d i e s i n s l i d i n g c o n t a c t . Most of the heat i s g e n e r a t e d on the s u r f a c e and i n the s u r f a c e l a y e r immediately below the s u r f a c e of both b o d i e s . T h i s 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 break-i n g of adhered j u n c t i o n s , or from the thermo-d y n a m i c a l l y i r r e v e r s i b l e p r o c e s s of p l a s t i c d e f o r m a t i o n of a s p e r i t i e s and the b u l k body." "(b) Whenever the c a p a c i t y o f one o f the b o d i e s to remove heat away from the i n t e r f a c i a l zone i s l e s s than the amount of heat 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 temperature jump a c r o s s the i n t e r f a c e . The s u r f a c e w i l l then have a h i g h e r macroscopic temperature." A s t e a d y - s t a t e s o l u t i o n f o r the r i n g s e c t o r , and q u a s i -s t a t i o n a r y s t a t e f o r the d i s k were s a t i s f a c t o r y f o r l u b r i -c a t e d s u r f a c e s but f o r u n l u b r i c a t e d ones, the temperature was time-dependent. L i n g and Pu assumed a s t o c h a s t i c p r o c e s s f o r the heat f l u x d i s t r i b u t i o n and o b t a i n e d a t r a n s i e n t s o l u t i o n of the heat e q u a t i o n f o r a c o n f i g u r a -t i o n c o n s i s t i n g of a s e m i - i n f i n i t e s o l i d w i t h a s l i g h t p r o t r u s i o n which was moving over another body which was i m p e r v i o u s to h e a t . F o l l o w i n g up t h i s work L i n g [18] made c a l c u l a t i o n s based on s e v e r a l s t o c h a s t i c models of heat f l u x d i s t r i b u t i o n and d e v i s e d a simple experiment to l e n d s u p p o r t to the t r a n s i e n t temperature t h e o r y . H i s experiments a t t a i n e d a r e a s o n a b l e agreement w i t h the t h e o r e t i c a l c a l c u l a t i o n s . 13 Rabinowicz [19] developed a simple f o r m u l a based on the s u r f a c e energy of s l i d i n g m a t e r i a l s t o c a l c u l a t e f l a s h temperatures. H i s f o r m u l a y i e l d e d r e s u l t s compara-t i v e l y , c l o s e t o h i s measured thermocouple v a l u e s . Cameron and o t h e r s [20] o b t a i n e d s o l u t i o n s t o the heat e q u a t i o n i n r o l l i n g s l i d i n g c o n t a c t s . They t r e a t e d t h r e e s i t u a t i o n s , v i z . , when no temperature d i s c o n t i n u i t y e x i s t e d over the c o n t a c t zone f o r v e l o c i t y r a t i o s l y i n g between -1 and +1; when a r e c t a n g u l a r source moved o f f the s u r f a c e s a t v a r i o u s speeds; the way s u r f a c e temperatures b u i l t up when the c o n t a c t was r e p e a t e d and when the heat was c o n v e c t e d from the f r e e s u r f a c e . J a e g e r [21] t r e a t e d the problems of temperature d i s t r i b u t i o n due to a moving band and a l s o a moving r e c t a n g u l a r or square source o f heat. S e v e r a l o t h e r a u t h o r s [22, 23, 24, 25] have d e a l t w i t h the c o n t a c t i n terms of f l a s h , bulk and t o t a l ( bulk p l u s f l a s h ) t emperatures. Yu [26] a n a l y z e d the heat t r a n s f e r problem i n a f r i c t i o n d i s k f o r a s i t u a t i o n where a l l the heat g e n e r a t e d was d i s s i p a t e d by the d i s k . He assumed q u a s i - s t a t i o n a r y s t a t e and j u s t i f i a b l y n e g l e c t e d r a d i a t i v e heat t r a n s f e r . Hauptmann and Ramsey [27] s o l v e d the same problem w i t h a more g e n e r a l method. They c o n s i d e r e d a d i s k w i t h heat i n p u t over an a r c of i t s p e r i p h e r y and o b t a i n e d a s o l u t i o n to the heat d i f f u s i o n e q u a t i o n which was asymmetric and p e r i o d i c . T h i s s o l u t i o n i s d i s c u s s e d f u r t h e r i n Chapter IV. Other workers have a l s o t r e a t e d the problem of the heat t r a n s f e r from r o t a t i n g d i s k s [28, 29, 30, 31, 32, 33]. Landau [34] t r e a t e d the problem o f heat c o n d u c t i o n i n a m e l t i n g s o l i d . He c o n s i d e r e d the problem whereby the l i q u i d was removed immediately i t was formed and assumed the e x i s t e n c e and uniqueness of the s o l u t i o n t o the problem. A f t e r m e l t i n g had s t a r t e d , he reduced the problem to a form i n which he had to c o n s i d e r o n l y a s i n g l e parameter g i v e n by 1/2 c, (T - T ) h m o m L where c, i s the s p e c i f i c heat of the s o l i d , T , i t s h ' m' m e l t i n g p o i n t , T q , i t s o r i g i n a l temperature and L, i t s l a t e n t heat of f u s i o n . He c o n s i d e r e d s o l u t i o n s f o r the extreme l i m i t s o f zero and i n f i n i t y on t h i s parameter. U s i n g an i n t e g r a l f o r m u l a t i o n , H a m i l l and Bankoff [35] d i s c u s s e d the problem o f upper and lower bounds on f r e e z -i n g or m e l t i n g r a t e s when the boundary c o n d i t i o n s were time dependent. T h e i r d i s c u s s i o n was l i m i t e d t o a one-d i m e n s i o n a l heat c o n d u c t i o n e q u a t i o n . Carslaw and Jaegar [36] d i s c u s s e d v a r i o u s c a s e s of heat t r a n s f e r where phase changes o c c u r r e d . R o s e n t h a l [37] c o n s i d e r e d both the p r a c t i c a l and t h e o r e t i c a l a s p e c t s of the heat d i s t r i b u t i o n a s s o c i a t e d w i t h moving heat s o u r c e s . He s o l v e d the heat d i f f u s i o n e q u a t i o n f o r s e v e r a l geometries and showed the p r a c t i c a l i t y 15 of such s o l u t i o n s . In an e a r l i e r paper [38] he c o n s i d e r e d the mathematical t h e o r y of heat d i s t r i b u t i o n d u r i n g w e l d i n g and flame c u t t i n g . With p r a c t i c a l examples, he d i s c u s s e d the v a r i o u s problems encountered i n p r a c t i c e . 1.2.5 Heat P a r t i t i o n -J a e g e r [21] developed heat p a r t i t i o n formulae f o r d i f f e r e n t ranges of the F o u r i e r number a s s o c i a t e d w i t h s l i d i n g mechanisms. The geometry of h i s moving heat source was a square. Based a l s o on the F o u r i e r number, L i n g and S a i b e l [39] worked out the v a l u e of the f r a c t i o n of the heat gen-e r a t e d which went i n t o the s t a t i o n a r y component of the s l i d i n g p a i r . The f r a c t i o n v a l u e depended on the range of v a l u e s of the F o u r i e r number. The moving heat s o u r c e was a l s o a square but the moving and the s t a t i o n a r y members were the r e v e r s e of those c o n s i d e r e d by J a e g e r . 1.3 F r i c t i o n Sawing Mechanisms and P r a c t i c e F r i c t i o n sawing i s a w i d e l y a p p l i e d method i n many i n d u s t r i e s f o r c u t t i n g v a r i o u s metals, non-metals and e x o t i c m a t e r i a l s . The main d i f f e r e n c e s between f r i c -t i o n and c o n v e n t i o n a l sawing p r a c t i c e are the h i g h speed and the t o o t h l e s s saw employed i n the former. The f e e d p r e s s u r e r e q u i r e d f o r f r i c t i o n sawing i s somehow l e s s than t h a t f o r c o n v e n t i o n a l sawing [ 4 0 ] , 16 In d i s c u s s i n g the f r i c t i o n sawing mechanism, P. V. Vernon proposed the f u s i o n t h e o r y of c u t t i n g h i g h c a r b o n s t e e l s [ 4 1 ] , In 1908, Harbord wrote an a r t i c l e con-t a i n i n g good photographs taken of the d i s k , k e r f and work-p i e c e b a r s . They r e v e a l e d t h a t the depth o f heat p e n e t r a -t i o n d i d not extend more than 1/150 to 1/100 of an i n c h from the sawed f a c e s o f the workpiece. H i s work s u p p o r t e d Vernon's arguments on the f u s i o n t h e o r y . Harbord used a b a d l y deformed saw b l a d e and so h i s work c l e a r l y demonstrated t h a t heat p e n e t r a t i o n was not a s e r i o u s f a c t o r i n f r i c t i o n sawing as was the case i n flame c u t t i n g . Eshchenko e t a l . [42] i n v e s t i g a t e d the phenomena i n the c u t t i n g zone i n high-speed hot sawing. The consen-sus i n the r a p i d sawing p r o c e s s was t h a t the c u t t i n g mech-anism c o u l d o n l y be s t u d i e d i n d i r e c t l y from d a t a on the temperature and p r e s s u r e i n the c u t t i n g zone, s t r a i n r a t e s , and shape of c h i p s . From a n a l y z i n g such d a t a c o l l e c t e d from i n d u s t r i a l o p e r a t i o n s , Eshchenko e t a l . deduced t h a t the main c u t t i n g p r o c e s s was due to b r i t t l e f r a c t u r e i n the case of s t e e l s heated to h i g h temperatures (800 to 1000 deg C) accompanied by the f o r m a t i o n of b r o k e n - o f f c h i p s . A n a l y z i n g the p a r t i c l e s c o l l e c t e d from c u t t i n g d i f f e r e n t grades of s t e e l s , they found s i m i l a r s i z e d i s t r i b u t i o n s which p o i n t e d to a g e n e r a l p h y s i c a l law g o v e r n i n g the c u t t i n g mechanism. In summary of the l i t e r a t u r e r e g a r d i n g f r i c t i o n sawing mechanism, the f r i c t i o n heat generated a t v e r y h i g h 17 speeds i s c o n s i d e r e d t o be c o n c e n t r a t e d on a s m a l l a r e a of the workpiece and the b u i l d - u p o f t h i s heat r e s u l t s i n weakening of the m a t e r i a l immediately ahead of the r u b b i n g zone. Owing t o the h i g h speed o f the saw bla d e i t s p e r i -p h e r a l heat i s d i s s i p a t e d r a p i d l y and so i t s m a t e r i a l s t r e n g t h remains r e l a t i v e l y unchanged and t h e r e f o r e a r e l a t i v e l y s t r o n g b l a d e abrades the weakened workpiece. I f the heat was s u f f i c i e n t to melt the m a t e r i a l of the workpiece the f a s t moving b l a d e wipes o f f the melt and p r e s e n t s f r e s h m a t e r i a l f o r r u b b i n g . In some c a s e s , i t i s b e l i e v e d t h a t the m e l t i n g i s c o u p l e d w i t h some b u r n i n g a c t i o n . I n t h i s c a s e , the f r i c t i o n heat melts the mater-i a l i n the k e r f w h i l e the spaces between the t e e t h c a r r y oxygen to the k e r f f o r the b u r n i n g a c t i o n . In terms of p r a c t i c a l a p p l i c a t i o n H y l e r [43] d i s -c u s s e d v a r i o u s uses of f r i c t i o n sawing i n the s t e e l i n d u s -t r i e s and warehouses. One o u t s t a n d i n g advantage of t h i s p r o c e s s i s t h a t i t s e f f i c i e n c y i s independent o f the m a t e r i a l hardness. The f o u n d r i e s use f r i c t i o n saws t o t r i m g a t e s and r i s e r s on c a s t i n g s and i n the s t e e l m i l l s , "hot saws" are used to sever hot m a t e r i a l s coming out of the m i l l s [1, 43, 4 4 ] . The economic a s p e c t o f f r i c t i o n sawing i s one of the main reasons f o r i t s wide a p p l i c a t i o n s [45, 46, 4 7 ] . T h i s economic advantage i s due l a r g e l y t o the much f a s t e r c u t t i n g r a t e s o f f r i c t i o n saws than t h e i r c o n v e n t i o n a l c o u n t e r p a r t s [5, 40, 43, 48,49, 50, 51, 52]. A l s o , a t 18 h i g h c u t t i n g r a t e s the s u r f a c e s of the c u t p i e c e s tend t o be smooth thus making them ready f o r use wi t h o u t f u r t h e r p r o c e s s i n g by g r i n d i n g o r p o l i s h i n g . 1.4 A p p r a i s a l o f the L i t e r a t u r e The work on f r i c t i o n sawing had been, up t i l l now, g r e a t l y o r i e n t e d towards i n d u s t r i a l p r a c t i c e r a t h e r than the s c i e n t i f i c i n v e s t i g a t i o n o f the mechanism. However, p e r i p h e r a l s u b j e c t s r e l a t e d to f r i c t i o n sawing have been c o n s i d e r e d q u i t e e x t e n s i v e l y . Most of the heat t r a n s f e r problems a s s o c i a t e d w i t h high-speed f r i c t i o n have not been a d e q u a t e l y c o v e r e d i n p r e v i o u s work. Many of the a u t h o r s quote formulae from p r e v i o u s heat t r a n s f e r s o l u t i o n s and such formulae i n v a r -i a b l y g i v e e s t i m a t i o n s which are not i n agreement w i t h the e x p e r i m e n t a l r e s u l t s . The converse i s t r u e when heat t r a n s f e r a u t hors c i t e p r a c t i c a l problems of s l i d i n g s u r -f a c e s . In a d d i t i o n , the e x p e r i m e n t a l specimens used i n p r e v i o u s h i g h speed f r i c t i o n i n v e s t i g a t i o n s are not r e p r e -s e n t a t i v e o f f r i c t i o n sawing. Some heat t r a n s f e r s o l u t i o n s are a p p l i c a b l e to f r i c t i o n sawing, p a r t i c u l a r l y the work of R o s e n t h a l . P a r -t i t i o n formulae by J a e g e r and, L i n g and S a i b e l are a l s o a p p l i c a b l e to f r i c t i o n sawing p r o v i d e d a l l t h e i r parameters are c l e a r l y u n d e r s t o o d . The m a j o r i t y of s o l u t i o n s make the assumption of i n i f n i t e dimensions but the f i n i t e geometry i n v o l v e d i n f r i c t i o n sawing and the d i f f e r e n t 19 shapes that might be encountered a l t e r the p a r t i t i o n problem considerably. However, i f the f r i c t i o n sawing heat problems are highly l o c a l i z e d , these p a r t i t i o n formulae would lead to co r r e c t temperature predictions i n both the saw blade and the workpiece. The suggestion of m e t a l l u r g i c a l examination made i n some of the ea r l y work on f r i c t i o n sawing could be very complementary to the r e s u l t s obtained by heat t r a n s f e r and f r i c t i o n force analyses. 1.5 Statement of Objective Even though f r i c t i o n sawing i s widely used i n many i n d u s t r i e s , i t s operating mechanism i s not yet f u l l y under-stood. The objective of the present project i s to study both a n a l y t i c a l l y and experimentally the f r i c t i o n sawing process with a view to obtaining a greater understanding of the heat t r a n s f e r and other processes involved. CHAPTER II 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 considered representative of f r i c t i o n sawing process i s the d i s k - f l a t plate type. The i n t e r f a c e i s defined by the area of the disk edge i n con-tac t with a f l a t plate workpiece and a l l the heat genera-ted takes place at t h i s i n t e r f a c e . Using t h i s configura-t i o n , the disk heat t r a n s f e r problem i s discussed and a s o l u t i o n obtained to the heat d i f f u s i o n equation. The workpiece i s treated i n r e l a t i o n to heat t r a n s f e r , l a t e r a l heat penetration and possible thermal shock. A uniform feed speed was used i n the heat trans-f e r equation and t h i s i s j u s t i f i e d by the experimental arrangement of the feeding mechanism discussed i n Chapter I I I . The heat p a r t i t i o n problem i s considered as dependent on material physical properties, c o n f i g u r a t i o n and p h y s i c a l dimensions of both the disk and the workpiece. The f r i c t i o n a l behaviour at high c u t t i n g speed i s i n v e s t i -gated by analyzing the c u t t i n g forces involved i n the 20 21 f r i c t i o n sawing p r o c e s s f o r the p a r t i c u l a r c o n f i g u r a t i o n used i n t h i s r e s e a r c h . 2.2 F r i c t i o n Saw Disk Heat T r a n s f e r A n a l y s i s With r e f e r e n c e t o F i g u r e 1, the g e n e r a l heat d i f f u s i o n e q u a t i o n i s [27] „ 2 m , w 3T 2h 1 3T V T + K 3 9 - K ? B CI-T.) - I f (1) where 9 9 9 3 r 2 r 9 r r 2 8 0 2 9 z 2 < l a ) F o r a t h i n p l a t e , the temperature v a r i a t i o n a c r o s s the d2 t h i c k n e s s i s n e g l i g i b l e , t h a t i s — 9 0. I f s u f f i c i e n t time i s a l l o w e d f o r the o p e r a t i o n of the system, a q u a s i -s t e a d y s t a t e p r e v a i l s and so ~ds = ®* Furthermore, i f the temperature d i s t r i b u t i o n i s assumed symmetrical about 3T the diameter 0 = 0, TT , then -gg = 0 and the problem becomes ax i s y m m e t r i c . Owing t o the h i g h r o t a t i n g speed of the f r i c t i o n saw d i s k , the P e c l e t number a s s o c i a t e d w i t h the c u t t i n g system i s v e r y h i g h . Thus the system may be approximated by one c o n s i s t i n g of a d i s k h a v i n g u n i f o r m r i m tempera-t u r e . T h i s l a s t statement combined w i t h the f a c t t h a t the heat i n p u t t o the d i s k i s symmetrical about the d i a -meter 0 = 0, ir j u s t i f y the axisymmetric n a t u r e of the heat t r a n s f e r problem. Chapter IV c o n t a i n s more i n f o r m a t i o n on these assumptions. With the f o r e g o i n g assumptions, the heat t r a n s f e r problem reduces to the s o l u t i o n of V 2T - f £ - (T - Tj = 0 ( 2 ) D where „2 d . 1 d V = T T + 7 d7 (2a) dr P u t t i n g T = T - T E q u a t i o n (2) becomes (2b) V 2 T - | £ - x = 0 (3) K t D E q u a t i o n (3) i s a m o d i f i e d B e s s e l e q u a t i o n of o r d e r z e r o and i f H i (3a) then, e q u a t i o n (3) may be w r i t t e n as V 2 x - A 2 T = 0 ( 4 ) For the e s t i m a t i o n of the heat t r a n s f e r c o e f f i c i e n t h, see Appendix I . 23 The g e n e r a l s o l u t i o n o f e q u a t i o n (4) i s x(r) = C± I Q ( A r ) + ^ ( A r ) (4a) where and C 2 are two a r b i t r a r y c o n s t a n t s t o be d e t e r -mined from g i v e n boundary c o n d i t i o n s . In t h i s c a s e , the boundary c o n d i t i o n s a r e : a) T = 0 at r = b dr b) - 2T Kt a — = q n at r = a D dr "D where q D i s the p o r t i o n o f the heat generated g o i n g i n t o the saw d i s k . U s i n g these boundary c o n d i t i o n s , the s o l u t i o n t o the heat t r a n s f e r problem f o r the saw d i s k i s T ( r ) - T = 2 i r K t DAa K (Ab)I (Ar) + I (Ab)K (Ar) o o o o I (Ab)K (Aa) + I (Aa)K (Ar) o 1 1 o (5) E q u a t i o n (5) i s s i m i l a r t o Yu's [26] s o l u t i o n e x c e p t f o r the d i f f e r e n t boundary c o n d i t i o n s used. 2.3 Workpiece 2.3.1 Heat T r a n s f e r R o s e n t h a l [38] d e a l t w i t h the mathematical t h e o r y of flame c u t t i n g . As f a r as the heat d i s t r i b u t i o n i n the p l a t e i s concerned, h i s t h e o r y i s a p p l i c a b l e to f r i c t i o n c u t t i n g . The geometry of the workpiece i s r e c t a n g u l a r . I f we assume t h a t the heat i s generated u n i f o r m l y a c r o s s the p l a t e t h i c k n e s s , then the f o r m u l a t i o n of the heat equa-t i o n becomes [ 3 7 ] , F i g u r e 2 ( a ) . 8 x 2 9 y 2 KtW ( T F r i c t i o n c u t t i n g problems may be c o n s i d e r e d f u r t h e r from the s t a n d p o i n t of a moving source of heat a l o n g the work-p i e c e . Thus the problem's o r i g i n a l s t a t i c c o o r d i n a t e s y s -tem may be t r a n s f o r m e d i n t o a dynamic c o o r d i n a t e system i n o r d e r to p r e s e r v e the s i m p l i c i t y of the problem. F i g u r e 2(b) shows the dynamic c o o r d i n a t e system. From t h i s F i g u r e , K = x - Vs (7) y' = y where V i s the c u t t i n g r a t e . U s i n g the t r a n s f o r m a t i o n (7) i n e q u a t i o n ( 6 ) , we o b t a i n H 2 3 y 2 35 3s K t w ( T T ~ ) ( 8 ) In t h i s new c o o r d i n a t e system, we may assume q u a s i -3T s t a t i o n a r y s t a t e , t h a t i s , ~%s -*-s z e r o » Then e q u a t i o n (8) becomes 25 + i f l = _ V9T 2hl ( _ } ( 9 ) 2 2 3 E Kr 0 0 9y w E q u a t i o n ( 9 ) may be s i m p l i f i e d by d e f i n i n g [ 3 7 ] T = Tro + e S(5,y) (10) where $ (£,y) i s a temperature f u n c t i o n to be d e t e r -mined. 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 equa t ion ( 9 ) , we have 8 2 $ ^ 8 2 $ [, „ N 2 I . + — 2 = |^(Y V) + mj $ 9? 2 9y where m 2hl (12) K t W Going back to the p h y s i c a l problem and assuming the heat genera ted i s symmet r i ca l about the a x i s of x , then $ w i l l be dependent o n l y on the parameter r „ / 5 2 + y 2 ' ( 1 3 ) or $ = $ (r) (14) 26 In view of e q u a t i o n (14), e q u a t i o n (11) may be w r i t t e n as d 2$ dr 2 + 1 £ - [(y V ) 2 + m] $ = 0 (15) F o r a t h i n i n f i n i t e p l a t e , e q u a t i o n (15) may be s o l v e d i f the t o t a l heat a p p l i e d to the p l a t e i s known. T a k i n g t h i s t o t a l heat t o be q w , and c o n s i d e r i n g t h a t f a r from the heat s o u r c e , the temperature i s the same as ambient temperature, the s o l u t i o n of e q u a t i o n (15) i s T - T = ^ - e " Y V 5 K (pr) (16) where K (pr) i s the m o d i f i e d B e s s e l f u n c t i o n of the second o k i n d o f o r d e r z e r o and P = / ( y V ) 2 + m (16a) From e q u a t i o n (16) s o l u t i o n s f o r o t h e r f l a t p l a t e geome-t r i e s may be g e n e r a t e d . The case of i n t e r e s t here i s t h a t of a r e c t a n g u l a r p l a t e of f i n i t e w i d t h . A s o l u t i o n can be c o n s t r u c t e d from e q u a t i o n (16) by making use of the image method [ 3 7 ] , The f o l l o w i n g s o l u t i o n c o n s i d e r s the plane o f symmetry y = 0 and one edge of the p l a t e y = b^ as the m i r r o r p l a n e s . In t h i s problem, o n l y h a l f o f the heat s u p p l i e d goes i n t o the s e c t i o n of p l a t e c o n s i d e r e d . • Taki n g i n t o account the m i r r o r r e f l e c t i o n c o n t r i b u t i o n s , we o b t a i n 27 T - T = — e y v ^ ^ K (pr ) 0 0 2irKt T ^—' o v n ^ n=l (17) where r . - / ? n ^ + (y ± 2nb 1)' (17a) E q u a t i o n (17) may be t r a n s f o r m e d i n t o F o u r i e r s e r i e s form [37] t o get T - T ~ 4 b l P K t w e-(p.y n + Y V ) C c o s ^ - C Y V + P K + 2 <r 1 n=,l n (18) f o r £ > 0 , and T - T = 0 0 ^ l P K t w E - ( Y V - P U n > 5 c o s (Zny.) r ( Y V - p ) ? + 2 — 1 n=l (19) f o r K < 0 where y n = A + ( p b ^ (19a) At the heat s o u r c e , e q u a t i o n s (18) and (19) g i v e i n f i n i t e t e m peratures. T h i s i s so because the i n h e r e n t assumption i n t h e i r d e r i v a t i o n i s t h a t of a l i n e source o f h e a t . 28 P r a c t i c a l l y s p e a k i n g , the source has f i n i t e s i z e and so these e q u a t i o n s may be m o d i f i e d by c o n s i d e r i n g a f i n i t e s i z e heat s o u r c e . C o n s i d e r a plane source of heat as made, up of i n f i n i t e l y s m a l l sources q"d£' p l a c e d s i d e by s i d e from V = 0 to V = F i g u r e 3 shows t h i s c a s e . Regarding q" as c o n s t a n t , then qW " ( 2 0 ) Now imagine a l i n e a r source of s t r e n g t h ~ . d£' p l a c e d a t V from the dynamic o r i g i n ; the temperature d i s t r i b u t i o n a t the p o i n t (£,y) due to t h i s source i s g i v e n by a-(YV+p )(?-£') + 2 -( YV+py ^ n y i n cos Kr~) b l n=l n (21) f o r K > V , and 6(T-T ) - 4b l P Kt w «, - ( Y V - p ) + 2 ( Y V - p y ^ ^ 1 ) ^ ^ b l n=l (22) f o r € < 5 1. From the e x p r e s s i o n s (21) and (22), we can d e r i v e the t h r e e c a s e s a r i s i n g . f r o m the heat t r a n s f e r problem: 29 Case 1: %, > a i n t e g r a t e e x p r e s s i o n (21) between V = 0 and V = a to get T - T = lW o° 4b.,pKt £ 1 W { e(p+YV)£_ 1 } e-(p+ YV)C (p+yV) { e ( p y n + Y V ) . _ l } e - ( P M n + Y V ) 5 c o s ( F 1 Z ) 1 + 2 S n=l y n(py n+YV) (23) Case 2: £ < 0 i n t e g r a t e e x p r e s s i o n (22) between V = 0 and £' = A to get T - T = 4b, { l - e " ( p _ Y V H } e ( p " Y V K (p-YV) + 2 n=l { l - e - ( P % " Y V ) £ } e <P V y V ) ? cos ( J ^ ) ' b l u n ( p y n - Y V ) (24) Case 3: 0<£<£ i n t e g r a t e e x p r e s s i o n (21) between 0 andC and e x p r e s s i o n (22) between 5 and £ and the r e s u l t i s T - T = 4b,pKt £ • 1 W + 2 ^>.{ l _ e " ( p + Y V ) ? 1_ e-(p -YV ) ( £ - 5 ) (p+YV) + (P-YV) l -e" ( P l J n + Y V ) ? l -e" ( P * V y V ) n-1 % ( P % + Y V ) y (py YV) } cos Q l (25) 30 E q u a t i o n s (23) to (25) would a p p l y e x a c t l y t o the problem of flame c u t t i n g but not f r i c t i o n sawing whereby the heat i s g e n e r a t e d by the r u b b i n g of the saw b l a d e on the work-p i e c e . In f r i c t i o n sawing, the heat i s generated w i t h i n the b r e a d t h o f the p l a t e say from t o +t^/2 and the t h i c k n e s s of the p l a t e , t w , see F i g u r e 4. In view of t h i s , q" dy' i s p l a c e d s i d e by s i d e from y' = - t D / 2 t o + t D / 2 . I f i s c o n s i d e r e d c o n s t a n t ; then q w s i n f where i i s d e f i n e d i n Appendix I I . S i n c e the heat i s g e n e r a t e d s y m m e t r i c a l l y w i t h r e s p e c t to the £ - a x i s , then the heat r a t e t o d e a l w i t h i s %^ 2> i.e. q = q w s W Hence q^dy'sin'/' 6 ( T " T » ) = 4 b l P K t t n 1 W D -<YV+p)C + 2 a-(pM n+YV) c o s (Tm(y-y-) b l n=l (26) f o r K > 0, and 6(T-T ) = q dy'sinlf 0 0 ^ l P K t u t B + ( P -YV ) 5 + 2 2: n=l e ( p y n - Y V ) c o s ( ^ n ( y - y ' ) b l (27) f o r £< 0. T h e r e f o r e the temperature d i s t r i b u t i o n s f o l l o w from e q u a t i o n s (26) and (27), thus 31 1 W D irnt T - ( p + y V ) C D - ( p + v V ) C . , D. /irny N 4b, 0 0 2b^ b^ n=l n (28) f o r £ > 0, and T-T irnt r ., m e ' r t n s i n b ; — ) c o s ( r - ^ - ) , . 4b n °° 2b b LD ir ny n=l n (29) f o r £< 0. E q u a t i o n s (28) and (29) g i v e the maximum temperature, the temperature d i s t r i b u t i o n w i t h i n the c u t t i n g zone and the temperature a t any p o i n t of the workpiece d u r i n g the sawing p r o c e s s under q u a s i - s t e a d y c o n d i t i o n s . 2.3.2 L a t e r a l Heat P e n e t r a t i o n The s o l u t i o n of the heat e q u a t i o n f o r the work-p i e c e under q u a s i - s t e a d y c o n d i t i o n s f u r n i s h e s the l a t e r a l temperature d i s t r i b u t i o n a t any c r o s s s e c t i o n of the p l a t e . Thus, i f a plane p a r a l l e l to the y - a x i s i s c o n s i d e r e d , the temperature can be p l o t t e d as a f u n c t i o n o f y. The most 32 c r i t i c a l temperature i s the temperature a t the source o r i g i n ; K - 0 and y = 0. C o n s i d e r i n g the plane p a r a l l e l to the y - a x i s c o n t a i n i n g t h i s o r i g i n , e q u a t i o n (24) becomes T-T = * V K V ( p _ Y V ) £ U - e - ( p y n - Y V H } c o s ( ^ ) b l (30) (p-YV) n=l y n ( p y n - y v ) and e q u a t i o n (28) or (29) becomes T-T = 4b, sin(-2b ) C O S K b } n=l ny (31) n E q u a t i o n s (30) and (31) show t h a t the temperature v a r i a -t i o n depends on the c o s i n e s e r i e s which i s r a p i d l y con-v e r g e n t . Thus by i n s p e c t i n g a few terms of the s e r i e s , knowing t h a t y - b-^  we see t h a t the temperature r a p i d l y d e c r e a s e s w i t h y and so the h i g h temperature r e g i o n w i l l be l i m i t e d to a t h i n l a y e r s u r r o u n d i n g the c u t t i n g zone. 2.3.3 P o s s i b l e Thermal Shock Owing to a h i g h temperature g r a d i e n t , the thermal s t r e s s may be so h i g h as to induce c r a c k i n g [ 1 0 ] . On t h i s b a s i s the c a l c u l a t i o n of the thermal s t r e s s i n the workpiece p l a t e i s of i n t e r e s t h e r e . Suppose the thermal expansion of the m a t e r i a l b e i n g c u t i s a and i t s shear modulus i s G, then, the 33 dT shear stress due to the temperature gradient - j - i s given by [ 1 0 ] . T = e r a 4? ( 3 2 ) Since T i s known from any of equations (23) to (25) or (28) and (29), T can be c a l c u l a t e d from equation ( 3 2 ) . dT Also T can be c a l c u l a t e d f o r (-77-) I f T i s max d£ max max compared with the shear strength, T G , of the workpiece material and T > T c , the material w i l l f a i l and i t max s ' w i l l crack ahead of the c u t t i n g zone where the large tem-perature gradient occurs. This mode of f a i l u r e by crack-ing could be an important c r i t e r i o n f o r determining the c u t t i n g mechanism i n f r i c t i o n sawing of ceramic materials. For the metals selected f o r the present study, thermal shock would not be a p p l i c a b l e . 2.4 The Heat P a r t i t i o n Problem 2.4.1 Introduction Assuming no heat l o s s i n the c u t t i n g zone, the t o t a l f r i c t i o n heat generated i s d i s s i p a t e d through the saw disk and the workpiece. I f t h i s t o t a l heat i s q T then where the subscripts D and W r e f e r to the disk and workpiece r e s p e c t i v e l y . 2.4.2 J a e g e r ' s Treatment Jaeger [21] d e a l t w i t h t h i s heat p a r t i t i o n problem u s i n g the F o u r i e r number d e f i n e d as V c L J 4K where V i s the p e r i p h e r a l speed of the saw d i s k and c i s a c h a r a c t e r i s t i c l e n g t h of the c o n t a c t zone. Here c w i l l depend on the a c t u a l a r e a of c o n t a c t between the saw and the workpiece d u r i n g the sawing p r o c e s s . The a c t u a l a r e a of c o n t a c t i s d i f f e r e n t from the apparent a r e a and i t i s a f u n c t i o n of the l o a d p r e s s i n g two con -t a c t i n g b o d i e s t o g e t h e r . I f A i s the a c t u a l a r e a of c o n t a c t , N the l o a d p r e s s i n g the two b o d i e s t o g e t h e r and p the mean f l o w p r e s s u r e of the weaker m a t e r i a l , then [53] A = £ P m (35) m and c = k/A~ (35a) where k depends on the shape of a s p e r i t y used, f o r example, k = 1/2 f o r a square and l / / n ~ f o r a c i r c l e . 2.4.3 L i n g and S a i b e l ' s Treatment Based a l s o on the F o u r i e r number as parameter, L i n g and S a i b e l [39] worked out the v a l u e of the f r a c t i o n of the heat genera ted go ing i n t o the s t a t i o n a r y component o f the s l i d i n g p a i r . They d e f i n e d the F o u r i e r number i n the form V c R = S— (36) 8K m For d i f f e r e n t ranges o f R, they o b t a i n e d e x p r e s s i o n s f o r the f r a c t i o n o f the heat genera ted which i s d i s s i p a t e d through the s t a t i o n a r y member. I f f r e p r e s e n t s t h i s f r a c t i o n , t h e i r r e s u l t s are g i v e n by the f o l l o w i n g : (1) f o r R = 0 f = i (37) 1 + K V J ; m K s Where K i s the thermal c o n d u c t i v i t y of the movinq member m K g i s t h a t of the s t a t i o n a r y member. (2) f o r R > 5 f = 1 (38) K 1 + "m / TTT" ~ K / ~ R s These two cases were f i r s t d e r i v e d by Blok who used a 2 square source o f heat o f a rea c to s i m p l i f y the mathe-ma 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 f r a c t i o n s . L i n g and S a i b e l m o d i f i e d (2) to read (2) • f o r R - 5 f - 1 ( 3 9 ) , . K 36 They a l s o d e r i v e d the f r a c t i o n f o r the r e m a i n i n g uncovered gap u s i n g the same l i n e o f r e a s o n i n g as B l o k . The r e s u l t c o n s t i t u t e s case (3) f o r 0 - R - 5 where 1 + K m 2 T K s [1 + 0 . 4 1 4 ( l - e _ 1 * 3 R ) I ( R ) ] (40) eR I(R) - R" (e +1)R: e 2 . h e -1)R 2 _n2 e dri de 2 (41) and e = / 4K S m (42) I(R) i s g r a p h i c a l l y e v a l u a t e d . 2.4.4 J a e g e r ' s E q u i v a l e n c e of L i n g and S a i b e l ' s Approach The c o r r e s p o n d i n g d e r i v a t i o n s by J a e g e r [21] are (1) f o r L T < 0.1, L s m a l l f = 1 + m K (43) (2) f o r L T > 5, L T l a r g e f = 1 (44) 1 + 0.5625/nr" Km/I7R ~ K~ m s (3) f o r 0.1 < L < 5, Ja e g e r developed a graph to be used f o r e v a l u a t i n g f . 2.4.5 D i s c u s s i o n of J a e g e r ' s and L i n g and S a i b e l ' s Formulae Of i n t e r e s t here i s the c o n d i t i o n f o r R > 5 or l a r g e v a l u e s of R. F o r t h i s c a s e , J a e g e r ' s d e r i v a t i o n suggests t h a t a v e r y l a r g e p o r t i o n o f the heat g e n e r a t e d would be d i s s i p a t e d through the workpiece or s t a t i o n a r y member w h i l e L i n g and S a i b e l p r e d i c t o t h e r w i s e . T h i s apparent c o n t r a d i c t i o n i s f u l l y r e s o l v e d by c o n s i d e r i n g the c o n f i g u r a t i o n s used by these a u t h o r s . F i g u r e 5 shows these c o n f i g u r a t i o n s . From t h i s F i g u r e , i t becomes c l e a r t h a t J a e g e r ' s f r a c t i o n s h o u l d be the comple-ment of L i n g and S a i b e l ' s . Another p o i n t o f d i f f e r e n c e may be noted here; J a e g e r a r r i v e d at h i s f o r m u l a by match-i n g the average temperatures a t the i n t e r f a c e whereas L i n g and S a i b e l made use of the maximum temperatures. From the f o r e g o i n g , comparable temperature c a l c u -l a t i o n s s h o u l d be based on e q u a t i o n (39) and the comple-ment of e q u a t i o n ( 4 4 ) . These v a l u e s s h o u l d compare w e l l and i f the v a l u e of c i s a c c u r a t e l y c a l c u l a t e d , the v a l u e s s h o u l d l e a d to the c o r r e c t temperature d i s t r i b u t i o n . 38 2.4.6 Proposed Heat P a r t i t i o n The f o r e g o i n g heat p a r t i t i o n formulae are developed f o r s t a t i o n a r y members o f s e m i - i n f i n i t e e x t e n t . Thus the formulae are b a s i c a l l y dependent on the thermal p r o p e r t i e s of the m a t e r i a l s i n v o l v e d i n the s l i d i n g p r o c e s s . However, i n f r i c t i o n sawing, many parameters are s u b j e c t to change v i z . the workpiece c o n f i g u r a t i o n , the sawing d i s k t h i c k n e s s and s i z e and the f e e d i n g r a t e . In o r d e r t o e v a l u a t e the heat p a r t i t i o n a c c u r a t e l y , some of these parameters must be c o n s i d e r e d . Two c r i t e r i a may be used to e v a l u a t e the heat p a r t i t i o n f ormulae: (1) the maximum temperature i n the c u t t i n g zone i n the workpiece e q u a l s the p e r i p h e r a l temperature of the sawing d i s k , the workpiece e q u a l s the p e r i p h e r a l temperature of the sawing d i s k . (2) the average temperature i n the c u t t i n g zone i n Based on these 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. D 1 + D n=l » s i n (• 2b 1 n (b) F o r d e r i v a t i o n s of e q u a t i o n s (a) and ( b j , see Appendix I I I Because of the s m a l l dimensions of the k e r f compared w i t h the whole workpiece, e q u a t i o n s (a) and (b) s h o u l d g i v e v e r y c l o s e r e s u l t s f o r the temperature d i s t r i b u t i o n s . Note t h a t the v a l u e s of f i n (a) and (b) are a f f e c t e d by the heat t r a n s f e r c o e f f i c i e n t ( r e : f o r m u l a f o r u and p ) . Hence f o r the v a l u e of h', the formulae n 1 i n Appendix I s h o u l d be used. In c a l c u l a t i n g temperature d i s t r i b u t i o n i n the hea t r a n s f e r a n a l y s i s s e c t i o n , the p o r t i o n of the heat genera-ted t r a n s m i t t e d to the workpiece depends on the c u t t i n g mechanism. I f f u s i o n c u t t i n g i s o p e r a t i n g , then qW " f q T " sintf where p i s the workpiece d e n s i t y , L, the workpiece l a t e n t 40 heat of f u s i o n and V, the c u t t i n g r a t e . Under t h i s c u t t i n g mechanism, T i s T , ,. and e q u a t i o n (31) w i l l ^ ' max m e l t i n g M be g i v e n by [fq n T - T = m <» s i n f ¥ B V D ] s i n ' / 4b x ^ s i n ( 2 b , D + ~ n^r n\ Trnt T (31' ) where T^ i s the m e l t i n g p o i n t of the workpiece. 2.5 High Speed F r i c t i o n C a r r y i n g out a s t r e s s a n a l y s i s of the f r i c t i o n sawing d i s k , e x p r e s s i o n s f o r both the normal and shear s t r e s s e s would be o b t a i n e d , see Appendix IV. The normal and shear f o r c e s a t the c o n t a c t zone a r e a may e a s i l y be o b t a i n e d u s i n g these s t r e s s e x p r e s s i o n s . The c o e f f i c i e n t of f r i c t i o n p i s then d e f i n e d a s u = L ( 4 5 ) N where F and N are the shear and normal f o r c e s r e s p e c t i v e l y . The v a l u e o b t a i n e d f o r y from e q u a t i o n (45) s h o u l d agree w i t h e x p e r i m e n t a l r e s u l t s based on the f o r c e a n a l y s i s shown i n Appendix I I . The r e s u l t s from Appendix I I show 41 t h a t F = w[ q G - & 1 + 3 ) a 1 - & 2 ) + 3 ( £ 2 - £ ) ] + R£ (46) and N = F T sxvrf - atair/ 1 + R£ (47) where a l l the c h a r a c t e r s are shown i n Appendix I I . For d i f f e r e n t v a l u e s of F and N, a p l o t o f y a g a i n s t V"cF and V cN can be compared w i t h W i l l i a m s and G i f f e n ' s [12] e x p e r i m e n t a l r e s u l t s whence they d e r i v e d the e m p i r i c a l f o r m u l a e : y = K(V cN) -0.45 and (48) y = K'(V F) c -0.82 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 The c u t t i n g mechanism w i l l be based on two c r i t e r i (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. 2 . 6 . 2 Average Temperature C r i t e r i o n If the average temperature over the c u t t i n g zone i equal to the melting point of the workpiece mat e r i a l , then la r g e - s c a l e melting must take place i n the kerf and the c u t t i n g mechanism i s f u s i o n . Now i f we consider the c u t t i n g zone area, the average temperature can be evalua-ted using equation ( 2 5 ) ; thus q s±ni T -T = — av » 4 b 1 p K t w ^ ( p ^ V ) ' [_1_2 C O S H C <P+YV)a/21 j + _ J _ t _ [ 1 _ e - ( p - y V ) £ / 2 _ e -3 (p- Y vH/2 (p-YV) + i r 2 cosh[(py N+YV)£/2] (pu +YV)£ n y n ( p u n - Y V ) e - ( p y n - Y V ) £ / 2 e - 3 ( p y n - Y V ) £ / 2 1 -(py n - Y V ) J l x sin(-rr—) 2 b l (49) 43 or using equation (28) or (29) T a v ~ T c o ~ 4 b l P K t w t D 8b sin (• 2/ n tD C D + 2 2b, 77 *D n t l 2 n u (50) n Equations (49) and (50) enable the average temperature i n the c u t t i n g zone to be determined. I f t h i s temperature i s known, we can pr e d i c t whether the c u t t i n g mechanism i s fu s i o n or not. If the average temperature i s less than the melting point of the workpiece material, three p o s s i b i l i -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 f r a c -ture, b r i t t l e f r a c t u r e or burning. The f i r s t two p o s s i -b i l i t i e s depend on the stress d i s t r i b u t i o n around the c u t t i n g zone and the i n i t i a l temperature of the workpiece material [42]. The t h i r d p o s s i b i l i t y depends on the reac-t i v i t y of the workpiece material with the surrounding medium. 2.6.3 Maximum Temperature C r i t e r i o n The maximum temperature c r i t e r i o n i s considered i n f u r t h e r d e t a i l . Even though the maximum temperature may reach the melting point of the workpiece material, there may not be large-scale melting i n the c u t t i n g zone. T h i s statement i s borne out by v a r i o u s r e p o r t s on " f l a s h " temperature i n s l i d i n g mechanisms. In h i g h speed f r i c t i o n , m i c r o - m e l t i n g of rubbing s u r f a c e s does not have as much e f f e c t on f r i c t i o n f o r c e as macro-melting. Thus the b u l k temperature i n the c u t t i n g zone may remain w e l l below m e l t i n g p o i n t w h i l e the maximum temperature i n d i c a t e s m e l t i n g . I f t h i s i s the c a s e , one of the t h r e e p o s s i b i l i -t i e s d i s c u s s e d i n s e c t i o n 2.6.2 may account f o r the c u t t i n g mechanism. 2.6.4 F r i c t i o n S t u d i e s and the C r i t e r i a Each sawing mechanism d i s c u s s e d above i s r e l a t e d t o a p a r t i c u l a r type of f r i c t i o n b e h a v i o u r . Thus f r i c t i o n s t u d i e s may l e a d t o the p r e d i c t i o n of the sawing mechanism. For i n s t a n c e , i n a f u s i o n c u t t i n g mechanism the f r i c t i o n w i l l be t h a t a s s o c i a t e d w i t h a s o l i d - l i q u i d i n t e r f a c e . T h i s c o u l d be h i g h or low depending on the v i s c o u s p r o p e r -t i e s of the molten workpiece m a t e r i a l . In a d d i t i o n , the c h e m i c a l r e a c t i v i t y of the i n t e r f a c i a l phase w i l l be d i f f e r e n t from t h a t o f i t s s o l i d c o u n t e r p a r t s and t h i s a l s o a f f e c t s f r i c t i o n . 2.7 Summary The heat t r a n s f e r problem f o r the f r i c t i o n saw d i s k was s o l v e d by assuming a t h i n a n n ular d i s k w i t h con-s t a n t t h i c k n e s s whose p e r i p h e r y i s u n i f o r m l y heated. T h i s assumption was j u s t i f i e d by the h i g h P e c l e t number i n v o l v e d . 4 5 The workpiece heat t r a n s f e r f o l l o w e d a d e v e l o p -ment by R o s e n t h a l f o r w e l d i n g and flame c u t t i n g . The l a y e r o f 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 o f the c u t t i n g zone and t h i s i s good because the m a t e r i a l b e i n g c u t w i l l not be b a d l y a f f e c t e d p r o p e r t y - w i s e . A thermal shock p o s s i b i l i t y was c o n s i d e r e d and i t was noted t h a t t h i s 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 the c u t t i n g mechanism of ceramic m a t e r i a l s . The heat p a r t i t i o n problem was c o n s i d e r e d as depen-dent on the m a t e r i a l p h y s i c a l p r o p e r t i e s and the formulae of J a e g e r and, L i n g and S a i b e l were compared. The appar-ent d i f f e r e n c e i n t h e i r r e s u l t s was e x p l a i n e d i n terms of the d i f f e r e n t c o n f i g u r a t i o n s they used i n t h e i r p a p e r s . A proposed heat p a r t i t i o n which accounts f o r a l l the p a r a -meters i n the sawing mechanism has been d e v e l o p e d . The c u t t i n g mechanism was based on two c r i t e r i a : average and maximum temperatures over the c u t t i n g zone. These c r i t e r i a show under what c o n d i t i o n s f u s i o n , b u r n i n g , d u c t i l e or b r i t t l e f r a c t u r e mechanisms can be p r e d i c t e d . CHAPTER I I I APPARATUS, INSTRUMENTATION AND EXPERIMENTAL METHODS 3.1 Apparatus 3.1.1 I n t r o d u c t i o n The apparatus can be d i v i d e d i n t o t h r e e sub-a s s e m b l i e s v i z : ( i ) the sawing d i s k and i t s d r i v e , ( i i ) the t a b l e and gu i d e , ( i i i ) the f e e d i n g mechanism. 3.1.2 The Saw Disk and S h a f t ( F i g u r e s 6, 7 and 8) The saw was a 14-gauge d i s k made of SP p l a t e s t e e l of hardness Rockwell C48. The d i s k had a smooth o u t e r p e r i p h e r y of 14 i n c h e s diameter and a 1 1/4 i n . b o r e . I t was mounted on a one i n c h diameter SPS A t l a s s t e e l s h a f t between two s u p p o r t i n g c o l l a r s 6 i n c h e s o u t s i d e diameter and 1 1/4 i n c h e s i n s i d e d i a m e t e r . These c o l l a r s were r e t a i n e d by a l o c k n ut. The s h a f t was mounted i n t h r e e s e l f - a l i g n e d b e a r i n g s . The r i g h t b e a r i n g was i n s t a l l e d i n a way t o permit easy removal o f the saw. 46 47 3.1.3 The Disk Motor and V - B e l t D r i v e ( F i g u r e s 6 and 7) A 15 horsepower, 3,540 r e v o l u t i o n s per minute i n d u c t i o n motor was used t o d r i v e the d i s k s h a f t through two s i z e B, V - b e l t s . A 9-3/8 i n c h e s o u t s i d e diameter p u l l e y was a t t a c h e d t o the d i s k s h a f t and by the p u l l e y arrangement, the d i s k r o t a t i o n was stepped up t o 4,620 r e v o l u t i o n s per minute, g i v i n g a p e r i p h e r a l d i s k speed of 16,920 f e e t per minute (282 f t / s e c or 86 m/sec). 3.1.4 The T a b l e ( F i g u r e s 7, 8 and 9) The t a b l e which was made of aluminum was 30 i n c h e s l o n g , 16 i n c h e s wide and 1/4 i n c h t h i c k . I t was mounted f o u r i n c h e s above the d i s k c e n t r e - l i n e . Along i t s l o n g i -t u d i n a l c e n t r e - l i n e , a s l o t 14 i n c h e s l o n g by 3/8 i n c h wide was c u t to a l l o w the d i s k t o emerge 3 i n c h e s above the t a b l e . The t a b l e was supported by two aluminum arms at each s i d e of the d i s k , F i g u r e 7. A s h o r t s p i n d l e was a t t a c h e d to each arm. The s p i n d l e between the two l e f t hand b e a r i n g s , shown i n F i g u r e 7, was made h o l l o w to p e r -mit a c c e s s to the d i s k s h a f t . The s p i n d l e b e a r i n g s were f l a n g e d mounting b a l l b e a r i n g s and were f i x e d on two upper-guard s t a n d s . The c o n c e n t r i c i t y o f the d i s k s h a f t and the t a b l e s u p p o r t i n g arm s p i n d l e s s i m p l i f i e d the c a l -c u l a t i o n of t a b l e f o r c e s by e l i m i n a t i n g the moments of s u p p o r t r e a c t i o n f o r c e s when the moments were c o n s i d e r e d about the d i s k s h a f t a x i s . 48 3.1.5 The Workpiece Guides ( F i g u r e 9) On the f r o n t p a r t o f the t a b l e was a guide con-s i s t i n g o f t h r e e aluminum b l o c k s a t t a c h e d to the t a b l e . The two s i d e b l o c k s w i t h s l o t s c u t i n them a l l o w e d f o r l a t e r a l motion t o accommodate d i f f e r e n t workpiece w i d t h s . These s i d e b l o c k s a l s o had t h e i r i n n e r w a l l s l i n e d w i t h a r b o r i t e sheets t o c u t down s l i d i n g f r i c t i o n . The upper b l o c k had a s t e e l r o l l e r arrangement a t t a c h e d to i t and i t c o u l d be a d j u s t e d v e r t i c a l l y to make room f o r workpiece t h i c k n e s s v a r i a t i o n . 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 F i g u r e 10 shows the workpiece mounting mechanism. The whole assembly moved on two b r a s s r o l l e r s . The r o l l e r path was l i n e d w i t h T e f l o n s t r i p . The two prongs of the workpiece mount were ground f i n i s h on a l l s i d e s and the two edges i n s l i d i n g c o n t a c t w i t h the guide were q u i t e smooth. The f r i c t i o n between these s i d e s and the a r b o r -i t e l i n i n g s f o r the s i d e b l o c k guides was ve r y s m a l l . The workpiece was made s m a l l e r than i t s mounting d e v i c e and so i t was not i n c o n t a c t w i t h any s u r f a c e e x c ept the s m a l l s t e e l r o l l e r a t t a c h e d to the upper guide b l o c k . T h i s arrangement e l i m i n a t e d any p o s s i b l e s i d e t h r u s t t h a t c o u l d be imposed on the workpiece, see F i g u r e 11. The push r o d was a t t a c h e d to a r o c k e r arm which accommodated any c l i m b o f the workpiece on the saw. F o r 49 added s a f e t y , a guide p l a t e was p r o v i d e d . A s l o t t e d h o l e through i t p r o v i d e d l i m i t a t i o n t o any p o s s i b l e r o d c l i m b , F i g u r e 12. The whole arrangement r e s u l t e d i n v e r y n e g l i g i b l e f r i c t i o n . 3.1.7 Arrangement f o r Measuring T a b l e F o r c e s A s t r a i n r i n g was mounted on a stand under the t a b l e 13.95 i n c h e s from the v e r t i c a l c e n t r e - l i n e o f the sawing d i s k . The r i n g was made of m i l d s t e e l 3 3/16 i n c h e s o u t s i d e diameter by t h r e e i n c h e s i n s i d e diameter, F i g u r e 13. A weight l o a d i n g pan a t the r e a r o f the t a b l e was used f o r c o u n t e r b a l a n c i n g the t a b l e . S u f f i c i e n t weights were added to the s c a l e pan t o g i v e zero l o a d on the t a b l e s t r a i n r i n g , F i g u r e 6. 3.1.8 The Feed Mechanism D r i v e and T r a n s m i s s i o n , ( F i g u r e s 6 and 8) The f e e d mechanism was d r i v e n by a 1 1/2 h o r s e -power d i r e c t c u r r e n t (D. C.) v a r i a b l e speed motor. The motor speed was c o n t r o l l e d w i t h a V a r i a c . The motor s h a f t o u t p u t was connected through V - b e l t s t o a 40 t o 1 worm speed r e d u c e r . 3.1.9 The C a r r i a g e and the Power Screw, ( F i g u r e 6) The t r a n s m i s s i o n gear r e d u c e r o u t p u t s h a f t was c o u p l e d v i a a shear p i n arrangement t o a s i n g l e s t a r t f o u r t h r e a d s per i n c h power screw which i n t u r n drove a c a r r i a g e through a s p l i t n u t . The s p l i t nut l o c a t e d under the 50 c a r r i a g e enabled i t s engagement t o or disengagement from the power screw. The power screw angular speed and hence the c a r r -i a g e l i n e a r speed c o u l d be changed c o n t i n u o u s l y by the motor speed c o n t r o l V a r i a c . With the p u l l e y arrangement used f o r the tests, a speed range of 0 to 3»6 i n c h e s per minute c o u l d be o b t a i n e d . In a d d i t i o n , the speed c o u l d be i n c r e a s e d by changing the p u l l e y s . 3.1.10 The Push Rod and T h r u s t Measuring S t r a i n Ring, ( F i g u r e 14) The push r o d and the s t r a i n ' r i n g were mounted i n f r o n t of the c a r r i a g e . F i g u r e 14 shows the push r o d r i g i d l y f i x e d to a r o c k e r arm connected to a l e v e r arm wit h a p i n . The l e v e r was p i v o t e d a t the p i n . The o t h e r end of the push rod was f i x e d to a s m a l l f l a t f a c e d metal wit h a l o c a t i n g h o l e d r i l l e d a t the c e n t r e . T h i s h o l e r e t a i n e d a s t e e l b o l t c o n n e c t i o n between the push r o d and the workpiece mount. The mounted s t r a i n r i n g was d i r e c t l y a t t a c h e d to the c a r r i a g e as shown i n F i g u r e 14. The r e a c t i o n o f the fe e d t h r u s t , F , a p p l i e d through the s t e e l b a l l to the push rod was t r a n s m i t t e d through the r o c k e r arm, the l e v e r arm, and the s t e e l b a l l to the s t r a i n r i n g which was c a l i b r a t e d to measure the f e e d t h r u s t F 0 i n pounds. 3.2 I n s t r u m e n t a t i o n 3.2.1 I n t r o d u c t i o n The f o l l o w i n g q u a n t i t i e s were measured or c a l c u -l a t e d from measured d a t a : the d i s k speed, the t a b l e f o r c e , the f e e d t h r u s t , the f e e d speed, the temperature d i s t r i b u t i o n around the c u t t i n g zone, the workpiece weight, the workpiece c e n t r e of g r a v i t y , the f r i c t i o n and normal f o r c e s and, the c o e f f i c i e n t of s l i d i n g f r i c t i o n between the saw and the workpiece. 3.2.2 The Disk Speed A S t r o b o t a c was used to check the speed of the saw i n the p r e s e n t p r o j e c t . 3.2.3 The T a b l e F o r c e A m i l d s t e e l s t r a i n r i n g was a t t a c h e d underneath the t a b l e 13.95 i n c h e s from the v e r t i c a l plane o f the a x i s of the d i s k s h a f t . The t a b l e was p i v o t e d about the a x i s of the d i s k s h a f t hence m u l t i p l i c a t i o n of the l o a d a t the s t r a i n r i n g by the d i s t a n c e between the r i n g and the d i s k c e n t r e l i n e gave the r e s u l t a n t moment on the t a b l e about the a x i s . 52 The s t r a i n r i n g d e f l e c t i o n was p i c k e d up by a " D a y t r o n i c " t r a n s d u c e r . The t r a n s d u c e r was c a l i b r a t e d to r e a d d i r e c t l y the l o a d a p p l i e d to the r i n g i n pounds. The c a l i b r a t i o n done " i n s i t u " i s d e s c r i b e d i n Appendix V. E x c i t a t i o n of the t r a n s d u c e r was s u p p l i e d by a " D a y t r o n i c " d i f f e r e n t i a l t r a n s f o r m e r i n d i c a t o r (Model 300 B F ) . A v i s u a l d i s p l a y of the s t r a i n r i n g d e f l e c t i o n i n m i l l i -i n c h e s was shown by the i n d i c a t o r , see F i g u r e 14. S i n c e the workpiece c e n t r e of g r a v i t y was moving d u r i n g sawing, the f o r c e c o n t i n u o u s l y v a r i e d and a c h a r t r e c o r d o f the f o r c e was o b t a i n e d . The d i f f e r e n t i a l t r a n s f o r m e r i n d i c a t o r o u t p u t was a m p l i f i e d by a "Brush" D. C. a m p l i f i e r (Model BL-932). The a m p l i f i e d v o l t a g e was used to d r i v e a "Brush" o s c i l l o -graph (Model RD-2622-02) f o r a c o n t i n u o u s f o r c e r e c o r d . 3.2.4 The Feed T h r u s t The t h r u s t was t r a n s m i t t e d from the workpiece to the push r o d by a s t e e l b a l l . From the r o d , the f o r c e was t r a n s m i t t e d through a r o c k e r arm, a l e v e r arm and another s t e e l b a l l t o the s t r a i n r i n g . The s t r a i n r i n g d e f l e c t i o n was p i c k e d up by another " D a y t r o n i c " 103A-80 l i n e a r d i s p l a c e m e n t t r a n s d u c e r . T h i s t r a n s d u c e r was a l s o c a l i b r a t e d t o r e a d d i r e c t l y the t h r u s t f o r c e a p p l i e d i n pounds. The c a l i b r a t i o n , done " i n s i t u " i s shown i n Appendix V I . As i n the case of the measurement of t a b l e f o r c e , the t r a n s d u c e r e x c i t a t i o n was f u r n i s h e d by another 53 " D a y t r o n i c " d i f f e r e n t i a l t r a n s f o r m e r i n d i c a t o r (Model 300 BF) which a l s o gave a v i s u a l i n d i c a t i o n of the s t r a i n r i n g d i s p l a c e m e n t i n m i l l i - i n c h e s . The t r a n s f o r m e r output v o l t a g e was a m p l i f i e d by another "Brush" D. C. a m p l i f i e r (Model BL-932). The a m p l i f i e d v o l t a g e was used to d r i v e another c h a n n e l of the "Brush" o s c i l l o g r a p h , F i g u r e 14. 3.2.5 The Feed Speed The f e e d speed was o b t a i n e d by a marker s w i t c h which r e s t e d g e n t l y on a shear p i n mount d i s k . T h i s gave the count of r e v o l u t i o n s of the d i s k and hence those of the power screw. The o u t p u t f e e d to the "Brush" c h a r t r e c o r d e r was used to c a l c u l a t e the speed. Another method i n v o l v e d t i m i n g the l e n g t h of a c u t and u s i n g the time to c a l c u l a t e the speed. These two methods agreed w i t h i n v e r y r e a s o n a b l e l i m i t s . 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, ( F i g u r e s 15 and 16) Measuring the temperature d i s t r i b u t i o n around the c u t t i n g zone was attempted by u s i n g i n f r a r e d r e s p o n s i v e PbS p h o t o c o n d u c t i v e c e l l . O p t i c f i b r e l i g h t p i p e s were used t o t r a n s m i t the i n f r a r e d r a d i a t i o n to the c e l l . These f i b r e s were made o f g l a s s and each l i g h t p i p e con-s i s t e d of a g l a s s f i b r e bundle bonded t o g e t h e r and mounted, both ends, i n m i l d s t e e l stepped s l e e v e s . The PbS c e l l responds to i n f r a r e d r a d i a t i o n w i t h i n the 1 t o 3um band w i d t h of the l i g h t spectrum. However, the g l a s s f i b r e s have c u t o f f s a t about 2um ; thus o n l y lym band width of the spectrum may be used (1 to 2um )• F i g u r e 15 shows the b l o c k diagram c o n n e c t i o n of the tem-p e r a t u r e measurement i n s t r u m e n t a t i o n . (a) The L i g h t P i p e T r a n s m i t t e r s and C a r r i e r B l o ck The l i g h t pipe c a r r i e r b l o c k had e i g h t h o l e s d r i l l e d i n i t , each h o l e saw a p o i n t on the workpiece. The mounted l i g h t p i p e s were f i t t e d i n t o s t e e l tubes which i n t u r n were i n s e r t e d i n t o the b l o c k h o l e s . Each l i g h t p i p e was c a l i b r a t e d w i t h i t s own ad a p t o r . (b) The PbS C e l l and Scanner Switch The PbS c e l l and a scanner s w i t c h were e n c l o s e d i n an aluminum box which was ground f i n i s h on a l l s u r f a c e s . The scanner s w i t c h was a 5 2 - p o s i t i o n s t e p s w i t c h . A m i c a r t a d i s k was mounted on the s w i t c h s h a f t and a t a con-v e n i e n t d i s t a n c e from i t s a x i s , the PbS c e l l was mounted under a s m a l l d r i l l e d h o l e . The mounted d i s k c l e a r e d the top of the box by 1/16 i n c h . A stepped hole was machined out of the top of the box and a l i p p e d aluminum d i s k plugged i n i t . The d i s k had t e n h o l e s d r i l l e d i n i t . The c e n t r e s of these h o l e s l i n e d up e x a c t l y w i t h the c e n t r e of the c e l l s e nsor whenever the c e l l was a t the hole p o s i t i o n s . S i n c e t h e r e were 52 p o s i t i o n s on the scanner, the ten h o l e s c o u l d not be e v e n l y spaced. Hence two p a i r s o f the h o l e s (5-6 and 10-1) were spaced t o c o r r e s p o n d t o s i x s w i t c h p o s i t i o n s a p a r t . The o t h e r p a i r s o f h o l e s were spaced f i v e s w i t c h p o s i t i o n s a p a r t . When the c e l l was 55 r o t a t e d , i t was e x a c t l y l i n e d up wi t h each h o l e . (c) The BAM and C o n t r o l B r i d g e The PbS c e l l o u t p u t was f e d through a BAM to the c o n t r o l u n i t . The BAM r e f e r e n c e d r i f t was c o n t r o l l e d w i t h a f o u r arm D. C. b r i d g e . One of the arms was the c e l l r e s i s t a n c e . The b r i d g e power was a 22-1/2 v o l t b a t t e r y . (d) The C o n t r o l U n i t The c o n t r o l u n i t was d e s i g n e d as a two-channel r e a d - o u t d e v i c e . One c h a n n e l was used f o r the l i g h t p i p e response s i g n a l and the o t h e r to d e s i g n a t e the l i g h t p i p e number. A threeway s w i t c h was used a t each l i g h t p i p e s t a t i o n : z e r o , c a l i b r a t i o n and l i g h t p i p e r e s p o n s e . There were t e n l i g h t p i p e p o s i t i o n s c o r r e s p o n d i n g t o the t e n p o s i t i o n s scanned by the c e l l . Each l i g h t p i p e p o s i t i o n might be f r o z e n by throwing a two-way s w i t c h ( s t o p - s t a r t ) . The l i g h t p i p e p o s i t i o n scanner was accomplished by another 5 2 - p o s i t i o n s t e p s w i t c h . T h i s s w i t c h was syn-c h r o n i z e d w i t h the c e l l s w i t c h and so l i g h t p i p e p o s i t i o n s i n the c e l l box and the c o n t r o l u n i t were s y n c h r o n i z e d . I f the system was out of s y n c h r o n i z a t i o n , a s w i t c h was p r o -v i d e d to b r i n g them back i n t o s t e p . The s t e p s w i t c h e s were powered through a power s w i t c h box on which a marker s w i t c h and a cam were mounted to c o n t r o l the sweeping time. There were f o u r a v a i l a b l e ranges t h a t c o u l d be s e t f o r the g a i n . Thus, a wide range of /response s i g n a l s c o u l d be accommodated; f o r i n s t a n c e we a n t i c i p a t e d temper-a t u r e r e a d i n g s i n the range of 500 to 3000 deg F. 56 C o r r e s p o n d i n g t o each of the f o u r g a i n ranges were f o u r c a l i b r a t e d r a n g e s . These c a l i b r a t e d ranges e n a b l e d us to know which range any p a r t i c u l a r l i g h t p i p e r e a d i n g was ta k e n . (e) The C h a r t Recorder A two-channel "Brush" r e c o r d e r was used to r e c o r d the l i g h t p i p e o u t p u t and the c o r r e s p o n d i n g l i g h t p i p e number. The c a l i b r a t i o n s i g n a l s e nabled the d e t e r m i n a t i o n of the c o n t r o l g a i n range of the l i g h t p i p e . ( f ) The L i g h t P i p e A t t e n u a t i o n F o r v e r y h i g h l i g h t p i p e s i g n a l s which l i m i t e d the BAM, t r a c i n g paper a t t e n u a t o r s were used. The a t t e n u a t o r was i n s e r t e d i n an adaptor tube t o which the o u t p u t end of the l i g h t p i p e was plugged. (g) The Thermocouples Two G/G-30-K chromel-alumel thermocouples were used. The c i r c u i t diagram of each thermocouple i s shown i n F i g u r e 16a. The thermocouple s i g n a l s were r e c o r d e d on the two c h a n n e l s of the "Brush" r e c o r d e r . The p r i n c i p l e o f the temperature d i s t r i b u t i o n r e g i s t e r e d by the thermocouples f o l l o w e d from the f a c t t h a t , as the workpiece moved i n t o the saw, each p o i n t was s u b j e c t to a range of temperatures. Hence the embedded thermocouples would sense d i f f e r e n t temperatures depending on t h e i r i n s t a n t a n e o u s d i s t a n c e from the c u t t i n g zone. S i n c e the workpiece moved s l o w l y , o n l y one i n c h per minute, the thermocouples' response gave r e a d i n g s v e r y c l o s e t o the a c t u a l temperature d i s t r i b u t i o n . For a p a r t i c u l a r t e s t r u n, F i g u r e 16(b) shows the p o s i t i o n s of the thermocouples i n the r i g . 3.3 E x p e r i m e n t a l Procedure and Data Treatment 3.3.1 I n t r o d u c t i o n P r e l i m i n a r y t e s t s were run to s e l e c t the m a t e r i a l s t o be c u t . The t e s t s c o n s i s t e d of c u t t i n g aluminum, copper, b r a s s , s t e e l and Ti-6A1-4V a l l o y . These m a t e r i a l s had a wide range o f m e l t i n g p o i n t s : 1220 to 3000 deg F. From these t e s t s , aluminum and copper were found to c o a t the edge of the saw b l a d e . T h i s r e s u l t e d i n s l i p p i n g a t the saw-workpiece i n t e r f a c e and the s l i p p i n g a c t i o n r e s u l t e d i n v i o l e n t v i b r a t i o n o f the saw. Thus these m a t e r i a l s were c o n s i d e r e d to be u n s u i t a b l e f o r f r i c t i o n c u t t i n g . B r a s s was c u t smoothly w i t h no s p a r k s , a s m a l l b u r r e x i s t e d a t the bottom and v e r y s m a l l t e a r was o b served a t the top of the workpiece. C u t t i n g s t e e l was a t t e n d e d by a b r i g h t c l o u d of s p a r k s , l i t t l e b u r r a t the bottom and a r o l l e d worm of t h i n metal a t the top of the workpiece. O b s e r v a t i o n showed v i s i b l e d u l l r e d m a t e r i a l j u s t ahead of the c u t t i n g zone, the width was about t h a t of the saw b l a d e . The m a t e r i a l from the k e r f was h e a v i l y c o a t e d w i t h b l a c k o x i d e and i t was c o n j e c t u r e d the c u t t i n g mechanism was of the f u s i o n t y p e . 58 A p r e v i o u s l y r u n t e s t on Ti-6A1-4V a l l o y showed the m a t e r i a l c o u l d be c u t w i t h the equipment. However the t e s t c o n d i t i o n s were severe and the edge of the saw b l a d e was damaged. In subsequent t e s t s the t i t a n i u m a l l o y was c u t a f t e r a l l o t h e r t e s t s were f i n i s h e d . The f i n a l c h o i c e o f m a t e r i a l s f o r the experiment was made as b r a s s , m i l d s t e e l , T l s t e e l and Ti-6A1-4V a l l o y . 3.3.2 Specimen P r e p a r a t i o n Most of the specimens were 1/4 i n c h p l a t e s c u t to s i z e t h r e e i n c h e s wide by 12 i n c h e s l o n g . Two m i l d s t e e l p l a t e s o f the same width and l e n g t h were 1/8 i n c h t h i c k . On each specimen, f o u r h o l e s were d r i l l e d to h o l d i t to the specimen c a r r i e r , see F i g u r e 11. 3.3.3 E x p e r i m e n t a l Method (a) G e n e r a l A l l i n s t r u m e n t a t i o n f o r both temperature and f o r c e measurements were t h o r o u g h l y checked a f t e r a warm-up p e r i o d of a t l e a s t t h i r t y minutes. The specimen and the l i g h t p i p e b l o c k c a r r i e r were i n s t a l l e d i n p o s i t i o n ready f o r the c u t . A rough "a p r i o r i " f e e d speed e s t i m a t e was determined from the s e t t i n g o f the V a r i a c c o n t r o l of the d r i v i n g motor. J u s t b e f o r e each t e s t run, the BAM r e f e r e n c e was checked and r e s e t i f n e c e s s a r y . The expected temperature ranges f o r each l i g h t p i p e p o s i t i o n were s e t on the tem-p e r a t u r e c o n t r o l u n i t . 59 (b) Brass The f e e d speed was s e t a t about 1/4 i n c h per minute. The saw was o p e r a t e d f o r an e i g h t minute c u t . The l i g h t p i p e scanners were o p e r a t e d . At the end of the c u t , the specimen was taken out f o r ex a m i n a t i o n and some d e b r i s from the k e r f were c o l l e c t e d f o r o b s e r v a t i o n . The temperature r e c o r d showed much d r i f t and i t was unknown whether the t r u e temperature d i s t r i b u t i o n r e c o r d o r j u s t t h e " c e l l thermal d r i f t was o b t a i n e d . E f f o r t s were t h e r e f o r e made t o e l i m i n a t e the c e l l thermal d r i f t and by s u c c e s s f u l l y i s o l a t i n g the scanner s w i t c h c o i l from i n s i d e the box to o u t s i d e , the c e l l thermal d r i f t was e l i m i n a t e d . The experiment on b r a s s was r e p e a t e d making sure the f o r c e measurements were r e c o r d e d on the c h a r t . No temperature r e c o r d was o b t a i n e d from the t e s t . The i n s t r u m e n t a t i o n s e n s i t i v i t y was i n c r e a s e d and s t i l l no temperature was r e g i s t e r e d . The measured f o r c e s were f e d i n t o the t h e o r e t i c a l programmes f o r temperature d i s t r i b u -t i o n . From t h i s , i t became ob v i o u s t h a t the f r i c t i o n f o r c e was much below t h a t p r e d i c t e d from the t h e o r y f o r f u s i o n c u t t i n g . Hence the temperatures were much below f u s i o n f o r which the i n s t r u m e n t a t i o n was s e t up. The m a t e r i a l from the k e r f showed t h a t the c u t t i n g mechanism f o r b r a s s was the same as t h a t i n c o n v e n t i o n a l machining, F i g u r e 17. In a d d i t i o n the m e t a l l o g r a p h i c e x a m i n a t i o n showed no h i g h temperature e f f e c t on the 60 f r i c t i o n - c u t edge, F i g u r e 18. (c) R e c a l i b r a t i o n o f Temperature I n s t r u m e n t a t i o n f o r Higher S e n s i t i v i t y An e l a b o r a t e c a l i b r a t i o n procedure was f o l l o w e d to make the temperature i n s t r u m e n t a t i o n more s e n s i t i v e . How-ev e r , owing t o the l i g h t p i p e t r a n s m i s s i o n c u t - o f f a t about 2ym wavelength i n the i n f r a r e d , the c u t - o f f temperature was c a l c u l a t e d to be about 650 deg F. Hence any temperature below 650 deg F c o u l d not be r e g i s t e r e d . T h i s agreed w i t h the e x p e r i m e n t a l o b s e r v a t i o n s . (d) F o r c e Measurement The f o r c e measuring i n s t r u m e n t a t i o n was r e v i s e d . A new specimen c a r r i e r was d e s i g n e d to r o l l on the T e f l o n path and the t h r u s t push r o d had a s m a l l t h i c k metal p i e c e screwed onto i t and t h i s p i e c e had a s m a l l r e c e s s e d h o l e a t i t s c e n t r e to r e t a i n a s t e e l b a l l . The pushed end of the specimen c a r r i e r was ground f l a t so t h a t the push rod c o u l d r i d e up or down w i t h o u t d i s t u r b i n g the t a b l e f o r c e . T h i s arrangement reduced the t a b l e and t h r u s t f o r c e s i n t e r a c t i o n to an i n s i g n i f i c a n t f r a c t i o n of the t o t a l f o r c e s , up to 100 l b f which was beyond the range of e x p e c t e d f o r c e s . (e) M i l d S t e e l 1/4" p l a t e m i l d s t e e l was i n s t a l l e d and the saw o p e r a t e d . The f e e d speed was s e t a t about 1/4 i n c h per minute. The l i g h t p i p e f o c u s s e d on the h i g h e s t tempera-t u r e p o i n t . No r e a d i n g was r e g i s t e r e d . The f o r c e s were 61 t aken and fed i n t o the t h e o r e t i c a l programme and i t was found t h a t these f o r c e s gave r i s e to temperature 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 i n the c u t t i n g zone, f o r which the p ipe was c a l i b r a t e d . A second t e s t was c a r r i e d out whereby one of the more s e n s i t i v e l i g h t p i p e s was put i n the p o s i t i o n of the expec ted h i g h e s t t empera tu re . A r e a d i n g was r e g i s t e r e d f o r a s h o r t t ime o n l y , F i g u r e 19. Th i s behav iour was due to the "worm" of t h i n meta l on top o f the workpiece 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 . The h i g h e s t temperature r e g i s -t e r e d was 1050 deg F . S i n c e we c o u l d not e l i m i n a t e the worm due to the •angular sawing p o s i t i o n , we dec ided to make some edge c u t s . 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 workp iece 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 fed to the saw was the same as the b l ade t h i c k n e s s . Thus, a l i g h t p ipe f i x e d to the workp iece c a r r i e r saw an a r ea of the edge be ing c u t . T h i s arrangement enabled the l i g h t p ipe to t r a n s m i t the r a d i a t i o n from the area to the PbS c e l l sensor and thereby we o b t a i n e d the temperature d i s t r i b u t i o n a long the c u t edge. The edge c u t s gave maximum temperatures of the same magnitude as the temperature o b t a i n e d when the l i g h t p ipe was focussed on top o f the p l a t e a t the h i g h e s t tempera-t u r e a r e a . I t was observed t h a t the k e r f m a t e r i a l was d i s -p l a c e d s ideways i n s t e a d of go ing under the workpiece and so 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 l i g h t p ipe e n t r y , F i g u r e 2 K b ) . 62 The f o r c e s o b t a i n e d from these t e s t s when a p p l i e d t o the heat t r a n s f e r t h e o r y i n d i c a t e d t h a t the temperature i n the c u t t i n g zone was i n the neighbourhood of 1600 deg F. M e t a l l o g r a p h i c e x a m i n a t i o n s were then made o f the c u t edge and the k e r f m a t e r i a l s , F i g u r e s 24 to 28. The m i c r o s t r u c t u r e s of the k e r f m a t e r i a l and the f r i c t i o n c u t edge i n d i c a t e d the temperature reached was of the same o r d e r as t h a t p r e d i c t e d from the f o r c e measurements. T h i s temperature, however, was about one and a h a l f times the e x p e r i m e n t a l l y measured temperature. The edge c u t was r e p e a t e d but the same r e s u l t was o b t a i n e d . V i s u a l o b s e r v a t i o n of the s t e e l c o l o u r near the c u t t i n g zone suggested the l i g h t p i p e r e a d i n g s were of the r i g h t o r d e r of magnitude. ( f ) Temperature D i s t r i b u t i o n U s i n g Thermocouples I t was o b v i o u s from the t e s t s r e p o r t e d above t h a t the h i g h temperature zone was v e r y h i g h l y l o c a l i z e d i n the c u t t i n g zone and might not be e a s i l y p i c k e d up by the l i g h t p i p e i n s t r u m e n t a t i o n . T h e r e f o r e thermocouple h o l e s were d r i l l e d a t d i f f e r e n t p r e - s e l e c t e d d i s t a n c e s from the c u t t i n g a x i s on the subsequent t e s t specimens. Each h o l e was .02 i n . i n s i z e ( u s i n g #76 d r i l l ) and the depth of each h o l e was midway t o the t h i c k n e s s of the specimen p l a t e . The 30-gauge C r / A l thermocouple (0.01 i n . wire s i z e ) f i t t e d s n u g l y i n t o the h o l e s d r i l l e d i n the workpiece. Both 1/8 i n c h t h i c k and 1/4 i n c h t h i c k m i l d s t e e l p l a t e s were c u t w i t h Chromel-Alumel thermocouples embedded 63 i n the d r i l l e d h o l e s . Thus temperature d i s t r i b u t i o n c u r v e s were o b t a i n e d d u r i n g the c u t t i n g p r o c e s s . F i g u r e 22 shows the c u t t i n g f o r c e s a s s o c i a t e d w i t h the temperature d i s t r i b u t i o n s shown i n F i g u r e 23 . T l s t e e l , Ti-6A1-4V a l l o y and le a d e d b r a s s were a l s o c u t w i t h thermocouple attachments to r e g i s t e r the temperature d i s t r i b u t i o n s , (g) M e t a l l o g r a p h y M i c r o s t r u c t u r a l examinations of m a t e r i a l s from the m i l d s t e e l , T l s t e e l , b r a s s and Ti-6A1-4V a l l o y were done. Photographs showing d i f f e r e n t p a r t s o f i n t e r e s t i n each m a t e r i a l at e i t h e r 400 or 800 m a g n i f i c a t i o n were taken. .For b r a s s F i g u r e 18 shows the "as r e c e i v e d " s t r u c t u r e and the f r i c t i o n c u t edge s t r u c t u r e . F i g u r e s 24 to 38 show the m i c r o s t r u c t u r e s f o r the o t h e r m a t e r i a l s . 3.3.4 Treatment of Data Two s e t s of c h a r t r e c o r d s were o b t a i n e d from each e x p e r i m e n t a l run; the f o r c e s and the temperature d i s t r i b u t i o n . S i n c e the thermocouples were i n s e r t e d to the mid-plane of each specimen, some allowance was g i v e n f o r the l o n g i t u d i n a l d i s t a n c e s from the c u t t i n g zone f o r e v e r y i n i t i a l c u t i n a workpiece. For i n s t a n c e , i n 1/4 i n c h p l a t e s , the time t o c u t 1/8 i n c h of the workpiece was a l l o w e d f o r i n i n t e r p r e t i n g the temperature r e c o r d s . F o r subsequent c u t s on a t e s t p i e c e , no time allowance was r e q u i r e d s i n c e the saw was a l r e a d y i n the k e r f ; w e l l i n t o the w o r k p iece. 64 I t took a few seconds f o r steady s t a t e t o be r e a c h e d . F i g u r e 22 shows t h a t some time e l a p s e d b e f o r e the f o r c e s b u i l t up to steady s t a t e v a l u e . In the l i g h t of t h i s the t a i l e n d r e s u l t s i n c l u d e d i n T a b l e s 1 to 5 s h o u l d be d i s r e g a r d e d i f s t a t i s t i c a l a n a l y s i s of the d a t a were to be done. A programme was w r i t t e n to c a l c u l a t e the f r i c t i o n f o r c e and f r i c t i o n c o e f f i c i e n t based on the measured t a b l e and t h r u s t f o r c e s . T h i s programme was e a s i l y c o n v e r t e d to a s u b r o u t i n e i n a main programme used f o r g e n e r a t i n g the t h e o r e t i c a l temperature d i s t r i b u t i o n . The f r i c t i o n f o r c e from t h i s s u b r o u t i n e f u r n i s h e d the heat generated i n the c u t t i n g zone to the main p r o -gramme and hence the temperature d i s t r i b u t i o n i n the work-p i e c e was o b t a i n e d . The temperature d i s t r i b u t i o n r e c o r d s from the thermocouples were t a b u l a t e d a g a i n s t the d i s t a n c e i n d i r e c t i o n of f e e d w i t h p o i n t s a l o n g l i n e s p a r a l l e l to the y - a x i s used as parameters. T a b l e s 1 to 5 g i v e the e x p e r i -mental r e s u l t s . CHAPTER IV RESULTS AND DISCUSSION 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 T a b l e 6 c o n t a i n s the summary of the f o r c e s o b t a i n e d i n the ex p e r i m e n t s . The measured t a b l e and t h r u s t f o r c e s v a r y c o n s i d e r a b l y from t e s t to t e s t . But f o r the same m a t e r i a l , the c a l c u l a t e d f r i c t i o n f o r c e s were n e a r l y con-s t a n t whereas the normal f o r c e s f l u c t u a t e d . T h i s same t r e n d was o b t a i n e d by Vaughn and Krueck [54] i n t h e i r e xperiments on u l t r a - h i g h - s p e e d machining t e c h n i q u e s . The normal f o r c e s they r e c o r d e d f o r c u t t i n g heat t r e a t e d 4340 s t e e l were about t h r e e times those of the annealed 4340 s t e e l but the t r a n s v e r s e f o r c e s ( f r i c t i o n f o r c e ) remained e s s e n t i a l l y the same i n both c a s e s . They a l s o o b t a i n e d d e c r e a s e d c u t t i n g f o r c e s as the c u t t i n g v e l o -c i t i e s i n c r e a s e d . F o r example, these f o r c e s d e c r e a s e d about 50 per c e n t f o r c u t t i n g v e l o c i t y i n c r e a s e from 90,000 f t / m i n t o 120,000 f t / m i n . Vaughn and Krueck's v e l o c i t y range i s much g r e a t e r than the f r i c t i o n saw v e l o c i t y of 16920 f t / m i n used i n the p r e s e n t study and so comparison of r e s u l t s i s o n l y v a l i d i f a p h y s i c a l law governs the heat g e n e r a t i o n i n the c u t t i n g zone. Vaughn 65 66 and Krueck proposed such a t h e o r y which i s d e a l t w i t h below. From T a b l e 6, workpiece t h i c k n e s s can be seen to a f f e c t both the t a b l e and t h r u s t f o r c e s but does not g i v e much e f f e c t on the f r i c t i o n f o r c e . The 1/8 i n c h and 1/4 i n c h m i l d s t e e l workpieces gave v a l u e s o f f r i c t i o n f o r c e s i n c l o s e agreement. While the t a b l e and t h r u s t f o r c e s f o r the TI s t e e l were about h a l f those of the m i l d s t e e l , f o r the same f e e d speed, the f r i c t i o n f o r c e s were e s s e n -t i a l l y the same. B r a s s f r i c t i o n f o r c e s were l e s s than f o r s t e e l but Ti-6A1-4V a l l o y r e g i s t e r e d h i g h e r f r i c t i o n f o r c e s than s t e e l . The f r i c t i o n c o e f f i c i e n t s showed dependence on m a t e r i a l and m a t e r i a l t h i c k n e s s . F r i c t i o n c o e f f i c i e n t s were a l s o s l i g h t l y a f f e c t e d by f e e d speed. F o r i n s t a n c e , when the 1/4 i n c h t h i c k m i l d s t e e l was c u t a t a f e e d speed of 1/4 i n c h per minute, the c u t t i n g f o r c e s were s i m i l a r to those o f the TI s t e e l c u t a t about one i n c h per minute. The v a r i a t i o n i n the f r i c t i o n c o e f f i c i e n t a rose from the f a c t t h a t the f r i c t i o n f o r c e i s n e a r l y c o n s t a n t f o r the same m a t e r i a l whereas the c u t t i n g f o r c e s v a r y w i t h the f e e d speed. Column 8 i n T a b l e 6 shows some f r i c t i o n c o e f f i c i e n t v a l u e s f o r the c o n d i t i o n s g i v e n i n columns 9 and 10 from d i f f e r e n t a u t hors as r e p o r t e d i n the l i t e r a t u r e . D i r e c t comparison of these v a l u e s w i t h the p r e s e n t i n v e s t i g a t i o n i s of l i t t l e v a l u e because the e x p e r i m e n t a l c o n d i t i o n s d i f f e r g r e a t l y . In most c a s e s of h i g h speed f r i c t i o n s t u d i e s , the normal f o r c e s are 67 much s m a l l e r than the c u t t i n g f o r c e s r e p o r t e d i n the p r e s e n t work showing f a c t o r s of 10 t o 50 d i f f e r e n c e . The f o r c e s i n v o l v e d i n f r i c t i o n sawing are much lower than those of low-speed metal c u t t i n g and the f r i c -t i o n c o e f f i c i e n t s are a l s o lower. For i n s t a n c e F i n n i e and Shaw [55] r e p o r t e d f r i c t i o n c o e f f i c i e n t s o f 0.88 to 1.85 and f r i c t i o n f o r c e s i n the range of 281 to 775 l b f i n c u t t i n g 1020 s t e e l w i t h 18-4-1HSS t o o l . F o r a l l the m a t e r i a l s i n t h i s study, the f r i c t i o n c o e f f i c i e n t was under 0.4 and the h i g h e s t f r i c t i o n f o r c e was s l i g h t l y o v e r 13 l b f . From the r e s u l t s here and elsewhere [13, 54, 23, 56] f r i c t i o n c o e f f i c i e n t a t h i g h speed f r i c t i o n i s not independent of l o a d and hence d e v i a t e s from the c l a s s i -c a l Amonton's laws. The parameter o f c o r r e l a t i o n used by 1/2 some authors [13, 23] i s the q u a n t i t y W U where W i s the normal l o a d and U, the s l i d i n g speed. The range o f t h i s parameter r e p o r t e d i n the l i t e r a t u r e i s below t h a t p r e -v a i l i n g i n the p r e s e n t i n v e s t i g a t i o n whereby W i s the normal l o a d i n the c u t t i n g zone and U the p e r i p h e r a l v e l o c i t y of the saw. 4.2 Temperature D i s t r i b u t i o n 4.2.1 T h e o r e t i c a l Two approaches were f o l l o w e d t o o b t a i n the temper-a t u r e d i s t r i b u t i o n . In the f i r s t approach a f r i c t i o n 68 f o r c e was assumed and i n the second, f u s i o n c u t t i n g was assumed. The former p r o v i d e d the t o t a l heat g e n e r a t e d i n the c u t t i n g zone which enabled the c a l c u l a t i o n of the temperature d i s t r i b u t i o n i n both the saw and the workpiece to be c a r r i e d o u t . The l a t t e r , assuming f u s i o n c u t t i n g , imposed a l i m i t on the maximum temperature and l e d t o the c a l c u l a t i o n o f the f r i c t i o n f o r c e r e q u i r e d f o r f u s i o n c u t t i n g . U s i n g the f u s i o n c u t t i n g approach, and assuming a u n i f o r m heat t r a n s f e r c o e f f i c i e n t over the workpiece, the temperature d i s t r i b u t i o n s a l o n g the c u t t i n g a x i s f o r b r a s s , m i l d s t e e l and Ti-6A1-4V a l l o y are shown i n T a b l e 7. The b r a s s d i s p l a y e d f a i r l y u n i f o r m temperature whereas the t i t a n i u m a l l o y showed f a i r l y s t e e p temperature g r a d i e n t s e s p e c i a l l y c l o s e to the c u t t i n g zone. The m i l d s t e e l l a y between these two extremes. T h i s would be e x p e c t e d i n view o f the h i g h thermal c o n d u c t i v i t y of the b r a s s and the low c o n d u c t i v i t y of t i t a n i u m a l l o y , see T a b l e 8. I t i s i n t e r e s t i n g t o see from T a b l e 7 t h a t about 10 per c e n t o f the t o t a l heat g e n e r a t e d i n sawing b r a s s would be d i s s i p a t e d through the workpiece whereas the c o r r e s p o n d i n g f r a c t i o n s f o r m i l d s t e e l and t i t a n i u m a l l o y are 6 per c e n t and 3 per c e n t r e s p e c t i v e l y . The heat p a r t i t i o n was based on matching the tem-p e r a t u r e of the saw b l a d e and the workpiece i n the c u t t i n g zone. Hauptmann and Ramsey [27] s o l v e d the d i s k heat t r a n s f e r e q u a t i o n w i t h a v e r y g e n e r a l method. Yu's 69 s o l u t i o n of the same e q u a t i o n assumed u n i f o r m p e r i p h e r a l temperature f o r the d i s k . Yu's method was m o d i f i e d by-u s i n g u n i f o r m p e r i p h e r a l heat f l u x i n s t e a d o f u n i f o r m p e r i p h e r a l temperature. The m o d i f i e d Yu's s o l u t i o n and, Hauptmann and Ramsey's s o l u t i o n were used f o r the matching c r i t e r i a . T a b l e 9 shows the r e s u l t s o b t a i n e d from these matchings. The average c u t t i n g zone temperature p r e d i c t e d were e x a c t l y the same f o r the two s o l u t i o n s , the f r a c t i o n s and f o r c e s d i f f e r s l i g h t l y from each o t h e r . Thus, f o r the high-speed r o t a t i n g d i s k , the two s o l u t i o n s g i v e v e r y a g r e e a b l e r e s u l t s . However, because of the complex argu-ments i n v o l v e d i n Hauptmann and Ramsey's s o l u t i o n , much g r e a t e r computer time was r e q u i r e d t o o b t a i n the n u m e r i c a l r e s u l t s . Hence Yu's m o d i f i e d s o l u t i o n would be p r e f e r r e d i n t h i s p a r t i c u l a r problem f o r p r a c t i c a l c a l c u l a t i o n s . J a e g e r ' s and, L i n g and S a i b e l ' s p a r t i t i o n formulae were used f o r some c a l c u l a t i o n s u s i n g v a l u e s of the F o u r i e r number which were c a l c u l a t e d f o r square heat s o u r c e s . The c h a r a c t e r i s t i c l e n g t h c i n e q u a t i o n (35a) gave f a i r l y good temperature d i s t r i b u t i o n r e s u l t s compared w i t h the proposed heat p a r t i t i o n f o r m u l a . In the case o f m i l d s t e e l , the c a l c u l a t i o n o f c showed i t t o be about 5 per c e n t of the saw bl a d e width (the apparent c o n t a c t l e n g t h ) ; f o r b o t h T l s t e e l and Ti-6A1-4V a l l o y the v a l u e s were l e s s than 5 per c e n t but f o r b r a s s the v a l u e was g r e a t e r . The e f f e c t o f f e e d speed c o u l d be observed from T a b l e 9. F o r the same m a t e r i a l , the heat g e n e r a t e d was 70 o n l y m i l d l y i n f l u e n c e d by f e e d speed. T h i s i s i n v e r y good agreement w i t h Vaughn and Krueck's o b s e r v a t i o n s [ 5 4 ] . Because of t h i s , a c o n s t a n t speed was employed f o r sawing b r a s s , m i l d s t e e l and TI s t e e l but h a l f the speed was used f o r the t i t a n i u m a l l o y . 4.2.2 E x p e r i m e n t a l F i g u r e 19 shows a l i g h t p i p e response t r a c e of the temperature i n the c u t t i n g zone of an edge c u t . From the c a l i b r a t i o n c h a r t f o r the l i g h t p i p e , the maximum tempera-t u r e was 1050 deg F. During the c u t , a d u l l r e d glow was o b s e r v e d i n the r e g i o n c l o s e to the c u t t i n g zone and from the c o l o u r - t e m p e r a t u r e s c a l e f o r s t e e l t h i s c o l o u r i n d i c a -t e d t h a t the temperature measuring i n s t r u m e n t a t i o n r e c o r d e d temperatures of the r i g h t o r d e r of magnitude. F i g u r e 20 shows how the k e r f m a t e r i a l was d i s p l a -c e d . T h i s k e r f m a t e r i a l b l o c k e d the l i g h t p i p e e n t r y and i f i t s temperature was l e s s than 650 deg F no s i g n a l would be r e c o r d e d . T h i s e x p l a i n e d the s h o r t response i n t e r v a l seen i n F i g u r e 19. In a d d i t i o n t o b l o c k i n g the l i g h t pipe e n t r y , the k e r f m a t e r i a l behaved l i k e a f i n r a p i d l y con-d u c t i n g heat away from the c u t t i n g zone. Hence the d u l l r e d glow i n the workpiece d i s a p p e a r e d a f t e r a s h o r t i n t e r v a l . T a b l e s 1 to 5 show the r e s u l t s of temperature d i s -t r i b u t i o n as measured by thermocouples compared w i t h the p r e d i c t e d r e s u l t s based on the c u t t i n g f o r c e s . These r e s u l t s were a l s o p l o t t e d i n F i g u r e s 39 to 45. 71 4.2.3 Comparison of 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 F i g u r e 39 shows the t h e o r e t i c a l p l o t s of the r e s u l t s f o r l e a d e d b r a s s with the e x p e r i m e n t a l p o i n t s imposed. Remote from the c u t t i n g zone; y > 0.2 i n . , the e x p e r i m e n t a l p o i n t s were c l o s e to the t h e o r e t i c a l p r e d i c t i o n s . The f i r s t one or two e x p e r i m e n t a l p o i n t s were o b t a i n e d when the c u t t i n g f o r c e s were s t i l l unsteady (see F i g u r e 22) so these p o i n t s s h o u l d be d i s r e g a r d e d i n making comparisons. In F i g u r e s 40 and 41 the agreement a t y = 0.1 i n . , i s not as good as a t y = 0.2 i n . , 0.3 i n . , 0.5 i n . , and 1.0 i n . ; thus the m i l d s t e e l temperature d i s t r i b u t i o n agrees w e l l w i t h t h e o r y when the p o i n t s are not too c l o s e to the c u t t i n g zone. The same t r e n d s are shown by TI s t e e l , F i g u r e s 42 to 44 and Ti-6A1-4V a l l o y , F i g u r e 45. The g r e a t e s t d i s c r e p a n c i e s between the t h e o r y and the experiment are g e n e r a l l y noted near the c u t t i n g zone. In the case of s t e e l , the worms formed behaved l i k e f i n s c o n d u c t i n g away some heat from the c u t t i n g zone. T h i s might lower the temperature i n the c u t t i n g zone r e s u l t i n g i n the lower e x p e r i m e n t a l temperatures r e c o r d e d . The agreement of the e x p e r i m e n t a l and t h e o r e t i -c a l r e s u l t s a t p o i n t s remote from the c u t t i n g zone suggests something s i m i l a r to the S a i n t Venant's p r i n c i -p l e i n e l a s t i c i t y t h e o r y , t h a t i s , the heat f l u x d i s t r i -b u t i o n a t the c u t t i n g zone does not s u b s t a n t i a l l y a f f e c t the temperature d i s t r i b u t i o n a t p o i n t s remote from the zone. 72 In F i g u r e 43 where a thermocouple was i n s e r t e d i n the c u t t i n g zone, i t was i n t e r e s t i n g t o note t h a t the h i g h e s t temperature r e g i s t e r e d was o n l y 1116 deg F. T h i s v a l u e i s i n the same range as the 1050 deg F measured w i t h the l i g h t p i p e . The g e n e r a l agreement between the e x p e r i m e n t a l r e s u l t s and the t h e o r e t i c a l p r e d i c t i o n s suggests t h a t the t h e o r e t i c a l assumptions are s u b s t a n t i a l l y j u s t i f i e d . 4.2.4 Comparison w i t h Other Works Heat d i s t r i b u t i o n or temperature d i s t r i b u t i o n i n the f r i c t i o n sawing zone had not been worked out by p r e -v i o u s i n v e s t i g a t o r s . However, some comparisons w i t h r e l a t e d machining p r o c e s s e s are p o s s i b l e . S a v i t s k i i [57] measured the temperature d i s t r i b u -t i o n i n a g r i n d i n g p r o c e s s . He used c y l i n d r i c a l t e s t p i e c e s of 4 mm (.1575 i n ) diameter p r e s s e d a g a i n s t a r o t -a t i n g a b r a s i v e wheel under a l o a d o f 2 kg (4.4 l b f ) . The r e l a t i v e speed used was 5.8 m/sec. He used embedded thermocouples f o r measuring the temperature d i s t r i b u t i o n i n the t e s t p i e c e and the p l o t t e d r e s u l t s showed the same shape as would be o b t a i n e d by j o i n i n g the e x p e r i m e n t a l p o i n t s i n F i g u r e 43. H i s maximum temperature of 1292 deg F compared w e l l w i t h the 1116 deg F measured i n the p r e s e n t s tudy. S a v i t s k i i o b t a i n e d s t e e p e r temperature g r a d i e n t s than those shown i n F i g u r e 43. T h i s might be e x p l a i n e d p a r t i a l l y by the d i f f e r e n c e i n t e s t specimen geometry. In the c o n c l u s i o n of h i s work, S a v i t s k i i 73 stated that the grinding temperature of hardened s t e e l s was higher than that of untreated s t e e l s . This agreed well with Vaughn and Krueck's [54] observation. In the present study, the T l s t e e l attained a s l i g h t l y higher temperature at the c u t t i n g zone than the mild s t e e l , Figures 41 and 43. The T l s t e e l has lower thermal cond u c t i v i t y than mild s t e e l (Table 8) and t h i s might account f o r the higher heat concentration i n the T l s t e e l c u t t i n g zone which would r e s u l t i n a higher temperature. ^ Boothroyd [58] obtained temperature d i s t r i b u t i o n contours i n a metal c u t t i n g operation by using i n f r a r e d photographic techniques. The maximum contour was 700 deg C or 1292 deg F. This temperature i s the same as the maximum obtained i n S a v i t s k i i ' s work. Boothroyd preheated the work to between 350 and 500 deg C. This made c u t t i n g e a s i e r but the maximum temperature was s t i l l 700 deg C. This f i g u r e i s lower than the t h e o r e t i c a l l y predicted maximum i n t h i s work but agreed well with the thermocouple measured value of 1116 deg F i n Table 4. I t should be expected that temperature involved i n low-speed machining should be lower than i n high-speed sawing. Parker and Marshall [59] worked with a view to c u t t i n g down the d e t e r i o r a t i o n of brake shoes i n the railway brake-wheel system. They studied the temperature attained at the s l i d i n g i n t e r f a c e during the braking operation. Using a PbS c e l l pyrometer, they recorded temperatures of 1470 deg F or greater at the s l i d i n g 74 i n t e r f a c e . This temperature l i e s i n the range of the predicted f r i c t i o n c u t t i n g zone temperatures. 4.3 Metallography 4.3.1 Mild S t e e l [60, 61] Figure 24 shows the f e r r i t e - p e a r l i t e microstructure of the "as received" mild s t e e l . The material i s i n the normalized ( a i r cooled) co n d i t i o n . Figure 25 i s the micro-structure of the "worm" which came from the top of the work-piece i n the c u t t i n g zone. The microstructure i n the f i g u r e suggests that the material underwent process annealing with the annealing temperature being about 1200 deg F. The material was maintained f o r a s u f f i c i e n t time to spheroidize the cementite plates o r i g i n a l l y present i n the p e a r l i t e . Figures 26 and 27 are the microstructures of the kerf material viewed edgewise and breadthwise r e s p e c t i v e l y . Figure 26(a) taken close to one side of the edge shows an oxide f i l m l a y e r . Figure 26(b) i s the i n t e r i o r s t r u c t u r e . Figure 27(a) taken close to one edge of the width also shows an oxide f i l m l a y e r . The very fine-grained f e r r i t e - p e a r l i t e structure was the r e s u l t of r a p i d l y heating the o r i g i n a l mild s t e e l above the AC 1 temperature for a short i n t e r v a l such as to produce f i n e austenite. This on cooling transformed to a f i n e f e r r i t e - p e a r l i t e s t r u c t u r e . The material was 75 probably heated only s l i g h t l y above AC 1 or between AC^ and AC^; that i s about 1600 deg F. Figure 28 shows the microstructure of the material close to the f r i c t i o n cut edge. A very narrow band of the material at the edge underwent the same treatment as the kerf material. This depth of material i s only about 0.00246 i n . 4.3.2 T l Steel (Alloy S t r u c t u r a l Steel) [60,61,63,66] -Figure 29 shows the microstructure of the "as received" T l s t e e l which consists of f e r r i t e and p e a r l i t e . Figure 30 shows the structure of the material.close to the flame cut edge of the T l s t e e l . This f i g u r e reveals a f u l l y martensitic s t r u c t u r e . Hence during the flame c u t t i n g of the edge, the material must have been heated above the ACj temperature, that i s , above 1670 deg F and probably to around 1700 deg F and then r a p i d l y cooled by the mass of the adjacent metal. Figure 31 shows the microstructure of the "worm" from the top of the workpiece i n the c u t t i n g zone. The structure i s not martensitic, but i s f i n e r than that of the o r i g i n a l metal. This would imply the material was heated s l i g h t l y above the AC^ temperature to produce fine-grained austenite but was not cooled r a p i d l y enough to form martensite. The temperature reached would probably be around 1600 deg F. Figures 32 and 33 show the structure i n the "edge view" and "width view" of the kerf material. These f i g u r e s reveal f i n e martensitic structures.These structures are 76 c h a r a c t e r i s t i c of heating TI s t e e l above the AC^ tempera-ture and very r a p i d l y c o o l i n g i t down to below 600 deg F. The temperature reached would be between 1600 and 1700 deg F. -Figure 34 i s the microstructure of the material close to the f r i c t i o n cut edge. Like the mild s t e e l case, a t h i n layer of material about 0.00197 i n . deep shows the same c h a r a c t e r i s t i c heat treatment as the kerf m a t e r i a l . This t h i n layer must have reached between 1600 and 1700 deg F during the sawing process and then been r a p i d l y cooled down to below 600 deg F. 4.3.3 Brass [66] Figure 18 shows the microstructures of the "as received" leaded brass and the material c l o s e to the f r i c t i o n cut edge. This f i g u r e shows no evident e f f e c t of high temperature on the cut edge st r u c t u r e . 4.3.4 Ti-6A1-4V A l l o y [66, 67] Figure 35 shows the microstructure of the "as received" titanium a l l o y . This structure i s character-i s t i c of a r a p i d l y cooled product. The two-phase s t r u c -ture i s not evident and therefore the structure i s non-e q u i l i b r i u m , f i n e and a c i c u l a r . Figure 36 i s the micro-structure of the small worm o f f the top of the workpiece i n the c u t t i n g zone. The structure i s s i m i l a r to that of the "as received" metal but with coarser grains. Com-parison of Figures 35 and 36 suggests that the material of Figure 36 was quenched from the 0 -phase region at a 77 slower c o o l i n g r a t e than the m a t e r i a l of F i g u r e 35. F i g u r e 37 shows the m i c r o s t r u c t u r e of the mater-i a l from the k e r f a t 400 and 800 m a g n i f i c a t i o n s . The photographs show f i n e r s t r u c t u r e s than the "as r e c e i v e d " . W i t h i n the l a r g e i r r e g u l a r shaped g r a i n appear e q u i a x e d d i r e c t i o n a l g r a i n s or s u b g r a i n s . The o r i e n t e d i n t e r n a l g r a i n s t r u c t u r e s u ggests t w i n n i n g t o which the hexagonal c l o s e packed a-phase i s s u b j e c t . These twins i m p l y heavy d e f o r m a t i o n . The i r r e g u l a r boundary of the l a r g e g r a i n i s not r e p r e s e n t a t i v e o f the 3 -phase ( h i g h temper-a t u r e p h a s e ) . Hence the heat treatment h i s t o r y i s not o b v i o u s from the m i c r o s t r u c t u r e . F i g u r e 38 shows the m i c r o s t r u c t u r e o b t a i n e d from h e a t i n g the "as r e c e i v e d " metal to v e r y h i g h temperature ( s e l f - b u r n i n g ) w i t h an o x y - a c e t y l e n e t o r c h and a l l o w e d to c o o l i n a i r . The s t r u c t u r e i s an e q u i l i b r i u m two-phase s t r u c t u r e . I f the "as r e c e i v e d " metal was l i k e t h i s , then the heat treatment h i s t o r y might have y i e l d e d b e t t e r i n f o r m a t i o n than i n the p r e c e d i n g paragraphs. 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 E x a m i n a t i o n s The f o l l o w i n g may be deduced from the m i c r o -s t r u c t u r e e x a m i n a t i o n of the d i f f e r e n t m a t e r i a l s i n v o l v e d i n the c u r r e n t i n v e s t i g a t i o n : (1) B r a s s d i d not r e a c h h i g h temperature; measured c u t t i n g zone temperature was about 600 deg F. 78 (2) The s t e e l reached temperatures i n the c u t t i n g zone w i t h i n the range of 1200 to 1700 deg F. (3) The heat a f f e c t e d zone i s v e r y t h i n ; hence f r i c t i o n sawing i s a l o c a l i z e d s u r f a c e heat phenomenon. (4) For the f o u r m a t e r i a l s e v i d e n c e o f b u l k m e l t i n g i n the k e r f i s absent s u g g e s t i n g t h a t f u s i o n c u t t i n g d i d not o c c u r . (5) The Ti-6A1-4V a l l o y y i e l d s v e r y l i t t l e i n f o r -mation as to i t s temperature h i s t o r y i n the k e r f . 4.4 M a t e r i a l s from the K e r f F o r l e a d e d b r a s s , the k e r f m a t e r i a l i s i n the form of s m a l l c h i p s , F i g u r e 17. These c h i p s are c h a r a c t e r i s t i c of c h i p s formed i n c o n v e n t i o n a l machining of le a d e d b r a s s . F o r m i l d s t e e l and TI s t e e l , the k e r f m a t e r i a l s are r e p r e s e n t e d i n F i g u r e 46. The m a t e r i a l i s a bead w i t h c u r v e d s t r a n d s i n the d i r e c t i o n o f sawing. The edges are q u i t e rough. From the e x p e r i m e n t a l o b s e r v a t i o n s , the t h i c k n e s s o f t h i s k e r f m a t e r i a l appeared independent o f the workpiece t h i c k n e s s and the c u t t i n g r a t e . F o r the t i t a n i u m a l l o y , the k e r f m a t e r i a l was dependent on the f e e d r a t e . When the m a t e r i a l was c u t a t about 1/4 i n c h per minute, t h e r e was no "worm" formed i n e i t h e r the k e r f o r the top of the workpiece; a l l t h e r e was was b l a c k p o w d e r - l i k e d e b r i s c h a r a c t e r i s t i c o f b u r n i n g . When the f e e d was i n c r e a s e d to about 0.54 i n c h per minute, s m a l l "worms" were formed but most of the d e b r i s was s t i l l 79 powder. 4.5 E x p l a n a t i o n of the C u t t i n g P r o c e s s From the r e s u l t s o b t a i n e d i n t h i s work, no unique t h e o r y would e x p l a i n the f r i c t i o n sawing mechanism. Never-t h e l e s s , s u f f i c i e n t i n f o r m a t i o n has been o b t a i n e d to e x p l a i n the c u t t i n g mechanism of s p e c i f i c m a t e r i a l s w i t h i n the con-t e x t of the c o n d i t i o n s of the experiments. The main f i n d i n g s a r e : (1) Leaded b r a s s f r i c t i o n - c u t s i n the same way as i t would machine i n c o n v e n t i o n a l machining p r a c t i c e . (2) S t e e l ( m i l d and T l ) y i e l d s v e r y i n t e r e s t i n g r e s u l t s . In d i s c u s s i n g s l i d i n g mechanism at h i g h speed, Bowden and Tabor [56] p o s t u l a t e d " s u r f a c e f l o w " to e x p l a i n the e x p e r i m e n t a l r e s u l t s o b t a i n e d f o r s t e e l - c o p p e r s l i d i n g p a i r . T h i s mechanism was deduced on the b a s i s of m i c r o -g r a p h i c examinations which r e v e a l e d metal d i s p l a c e m e n t by s u r f a c e f l o w w i t h h a r d l y any l o s t m a t e r i a l . F i g u r e 47 shows a p r o f i l e s e c t i o n taken a c r o s s a wear mark i n the d i r e c t i o n of s l i d i n g . Furthermore, i n the e x p e r i m e n t a l works r e p o r t e d by Bowden and Tabor, the f o l l o w i n g were obs e r v e d : ( i ) High temperature was always p r e s e n t i n the s l i d i n g i n t e r f a c e , even when th e r e was no e v i d e n c e of h i g h s c a l e m e l t i n g , the r e c r y s t a l l i z a t i o n s t r u c t u r e of v e r y f i n e g r a i n s i z e c h a r a c t e r i z e d by the absence of p r e f e r r e d o r i e n t a t i o n such as would be formed a f t e r p l a s t i c f l o w i n 80 c o l d working would suggest a h i g h temperature e f f e c t . ( i i ) The s h a r p l y d e f i n e d boundary between the f i n e - g r a i n e d equiaxed s t r u c t u r e and the c o a r s e - g r a i n e d s u b s t r a t e supported the view t h a t a t h i n s u r f a c e l a y e r melted by r u b b i n g . ( i i i ) The hig h d i f f u s i o n r a t e observed when s t e e l was s l i d a g a i n s t hard and h i g h m e l t i n g p o i n t m a t e r i a l s such as t u n g s t e n and molybdenum would a l s o imply h i g h c o n t a c t t e m p e r a t u r e s . ( i v ) The energy b a l a n c e d i d not support l a r g e -s c a l e m e l t i n g when h i g h m e l t i n g metals were i n v o l v e d i n the s l i d i n g mechanism, f o r example, s t e e l v e r s u s 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 , m e l t i n g on a m i c r o - s c a l e might s t i l l be o b t a i n e d . Eshchenko e t a l . [42] p o s t u l a t e d a b r i t t l e f r a c -t u r e mechanism t o e x p l a i n the c u t t i n g o f hot s t e e l s (800 t o 1000 deg C) w i t h a h i g h speed toothed saw. They e x p l a i n e d the two types of f a i l u r e , d u c t i l e and b r i t t l e , which c o u l d be observed i n high-speed c u t t i n g . F i g u r e 48 shows the s t e p by st e p p r o c e s s o f the f r a c t u r e s . At the l a s t stage a f t e r the s t r e s s reached the y i e l d s t r e s s the pr o c e s s i s e i t h e r d or e. They remarked t h a t a t room temperature c u t t i n g speeds i n the range of 1500 to 1700 m/sec would be r e q u i r e d to produce b r i t t l e f r a c t u r e i n s t e e l . Hence the b r i t t l e f r a c t u r e o b t a i n e d i n t h e i r work must be a s s o c i a t e d w i t h the h i g h temperature of the m a t e r i a l b e i n g c u t , s i n c e the hot saw speed ranged from 81 80 to 125 m/sec; the same speed range f o r f r i c t i o n saws. Vaughn and Krueck [54] proposed an a d i a b a t i c shear mechanism f o r metal removal i n t h e i r u l t r a h i g h speed machining t e c h n i q u e s f o r 4340 s t e e l (annealed and heat t r e a t e d ) . T h i s mechanism i m p l i e s t h a t as the c u t t i n g v e l o c i t y i n c r e a s e s , an a d i a b a t i c c o n d i t i o n i s approached i n which thermal energy i s r e s t r i c t e d to p r e f e r r e d s l i p p l a n e s and due to weakening i n these p l a n e s , a d d i t i o n a l s l i p o c c u r s and complete shear r e s u l t s . The r e s u l t s o b t a i n e d i n the p r e s e n t study c o u l d b e s t be e x p l a i n e d by the d u c t i l e f r a c t u r e t h e o r y i n the c ase of s t e e l s . The Bowden and Tabor s u r f a c e f l o w t h e o r y i n h i g h speed s l i d i n g i s very much l i k e d u c t i l e f r a c t u r e but not i n i t s complete form. F i g u r e 46 c o u l d be e x p l a i n e d i n terms of the d u c t i l e f r a c t u r e . The a d i a b a t i c shear t h e o r y l e n d s credence to the e x p e r i m e n t a l o b s e r v a t i o n s made i n the p r e s e n t work w i t h r e s p e c t to the c u t t i n g of m i l d s t e e l at d i f f e r e n t f e e d r a t e s . The heat generated appeared independent of the c u t t i n g r a t e and m a t e r i a l t h i c k n e s s . The t h e o r e t i c a l p r e d i c t i o n s show a l s o t h a t f r i c t i o n f o r c e s do not v a r y much w i t h the f e e d r a t e and t h i s s u p p o r t s the a d i a b a t i c shear t h e o r y . T h i s i n t e r e s t i n g phenomenon c o u l d imply t h a t the o n l y d i f f e r e n c e i n c u t t i n g the same mater-i a l of d i f f e r e n t t h i c k n e s s e s i s t h a t the t h i n n e r m a t e r i a l can be c u t f a s t e r than the t h i c k e r one under the same i n t e r f a c e f o r c e c o n d i t i o n s . 82 (3) Ti-6A1-4V A l l o y Vaughn and Krueck's o b s e r v a t i o n on Ti-6A1-4V a l l o y i n u l t r a h i g h speed machining showed v e r y good agreement w i t h the c u r r e n t i n v e s t i g a t i o n . They found t h a t b r i t t l e f r a c t u r e was the mechanism of i t s machining. In t h i s s tudy, a t slow f e e d r a t e s , say below 1/2 i n c h per minute, b r i t t l e type f r a c t u r e o c c u r r e d w i t h v e r y f i n e b l a c k powder d e b r i s . The b r i g h t white spark c l o u d observed i n the c u t t i n g zone suggested b u r n i n g of the k e r f m a t e r i a l because the c o l o u r was the same as was o b t a i n e d when oxy-a c e t y l e n e t o r c h was used to heat the a l l o y t o s e l f - b u r n i n g . However, a t h i g h e r speeds, above 1/2 i n c h per minute, the mechanism became p a r t i a l l y b r i t t l e and p a r t i a l l y d u c t i l e . Hence a t 0.536 i n . per minute, s m a l l "worms" were o b t a i n e d i n both the k e r f and on the top of the c u t specimen. CHAPTER V CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE STUDY The f o l l o w i n g c o n c l u s i o n s may be made from the p r e s e n t study: (1) Temperature measurements and m e t a l l o g r a p h i c e v i d e n c e c o n f i r m e d t h a t f r i c t i o n sawing i s a h i g h l y l o c a l i z e d heat phenomenon whereby steep temperature g r a d -i e n t s e x i s t i n the neighbourhood of the c u t t i n g zone; the heat a f f e c t e d zona on the m i l d s t e e l workpiece i s of the o r d e r ox 0.003 i n . w h i l e t h a t of T l s t e e l i s o f the o r d e r of 0.002 i n . f o r the c u t edge. (2) For both m i l d s t e e l and TI s t e e l , t h e r e e x i s t e d a r e l a t i v e l y h i g h temperature i n the c u t t i n g zone which was not near the bulk f u s i o n p o i n t of the k e r f m a t e r i a l . (3) The f r i c t i o n c u t t i n g mechanism f o r the s t e e l s was d u c t i l e f r a c t u r e r a t h e r than f u s i o n . (4) The f r i c t i o n c u t t i n g mechanism f o r T1-6A1—4V a l l o y was a mixture o f b r i t t l e f r a c t u r e and b u r n i n g at slow fe e d r a t e s and a mixture o f b r i t t l e f r a c t u r e , b u r n i n g and soma d u c t i l e f r a c t u r e a t h i g h e r f e e d r a t e s . 8 3 84 (5) The f r i c t i o n c u t t i n g mechanism f o r l e a d e d b r a s s was s i m i l a r t o i t s c o n v e n t i o n a l machining p r o c e s s whereby p a r t i c l e s of m a t e r i a l were removed from the kerf» (6) The s o l u t i o n of the heat t r a n s f e r e q u a t i o n s f o r the f r i c t i o n sawing system agree w e l l w i t h e x p e r i m e n t a l r e s u l t s o b t a i n e d a t r e g i o n s away from the c u t t i n g zone; f o r example at d i s t a n c e s g r e a t e r than or equal to 0.2 i n . from the c u t t i n g zone on the workpiece the agreement between the s o l u t i o n and the experiment i s good, (7) W i t h i n the c o n t e x t of the experiments conducted i n the c u r r e n t study, the r e s u l t s agree w e l l w i t h o t h e r r e l a t e d work r e p o r t e d i n the l i t e r a t u r e . (8) F r i c t i o n sawing, u s i n g a s t e e l d i s k saw, i s not s u i t a b l e f o r c u t t i n g aluminum and copper<. (9) The l i g h t p i p e temperature measurement i n s t r u -mentation developed i n the c o u r s e of t h i s study may f i n d u s e f u l a p p l i c a t i o n s i n t r a n s i e n t high temperature s t u d i e s . Recommendations f o r F u t u r e Study (1) The study of the s t r e s s e s i n v o l v e d i n f r i c t i o n sawing would p r o v i d e complementary approach to the p r e s e n t studyo (2) In o r d e r to a c c u r a t e l y c a l c u l a t e the heat t r a n s f e r c o e f f i c i e n t over the d i s k and the workpiece, the f l o w p a t t e r n should be known. 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"On the Mechanism of C o n t a c t Between M e t a l S u r f a c e s : P a r t 2 — The R e a l A r e a and the Number o f the C o n t a c t P o i n t s " , ASME Trans J . L u b r i c a t i o n Technology Paper No. 6 7 — L U B — 1 1 , 1967. 54. Vaughn, R. L. and Krueck, R. T. " U l t r a h i g h - S p e e d M a c h i n i n g " , T o o l and Mfg E n g i n e e r , v 44 No. 4 A p r i l 1960 pp 95-100. 55. F i n n i e , I a i n and Shaw, M. C. "The F r i c t i o n P r o c e s s i n M e t a l C u t t i n g " , Trans ASME Nov 1956. 56. Bowden, F. P. and Tabor, D. "The F r i c t i o n and L u b r i c a -t i o n of S o l i d s " , P a r t I I O x f o r d a t the C l a r e n d o n P r e s s , 1964 pp 451-478. 57. Kuznetsov, V. D. "Metal T r a n s f e r and B u i l d - U p i n F r i c -t i o n and C u t t i n g " , E d i t . E. H. F r e i t a g , Pergamon P r e s s 1966 pp 30-35. 58. Boothroyd, G. " P h o t o g r a p h i c Technique f o r the Determin-a t i o n o f M e t a l C u t t i n g Temperatures", B r i t . J . A p p l . Phys. V o l 12 May 1961. 59. P a r k e r , R. C. and M a r s h a l l , P. R. "The Measurement o f the Temperature of S l i d i n g S u r f a c e s , w i t h P a r t i c u l a r R e ference t o Railway Brake B l o c k s " , P r o c . I n s t n Mech Eng 1948 pp 209-229. 60. Lund, J . A. " P r i v a t e D i s c u s s i o n s " , Dept. of M e t a l l u r g y , U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada, 1971. 61. "Heat Treatment and P r o p e r t i e s o f I r o n and S t e e l " , U.S. Dept of Commerce NBS Monograph 88. 62. Gordon, P a u l e t a l . " K i n e t i c s o f A u s t e n i t e Decom-p o s i t i o n i n High Speed S t e e l " , Trans ASM v o l 31, No. 1, March 1943 pp 161-217. 63. " A t l a s o f I s o t h e r m a l T r a n s f o r m a t i o n Diagrams", U. S. S t e e l , 1953 Supplement p 366. 64. G i l l , J . P. e t a l . " T o o l S t e e l s " ASM C l e v e l a n d Ohio 1944 pp 438-528. 65. Payson, P. and K l e i n , J . L. "The Hardening o f T o o l S t e e l s " , Trans AMS, v o l 31, 1943 pp 218-256. 66. Hawbolt, E. B. " P r i v a t e D i s c u s s i o n s " , Dept of M e t a l l u r g y , U n i v e r s i t y of B r i t i s h Columbia, Vancouver,Canada, 1971. 90 67. "The S c i e n c e , Technology and A p p l i c a t i o n o f T i t a n i u m " , E d i t . R. I . J a f f e e & N. E. P r o m i s e l 1968. 68. K r e i t h , F. " P r i n c i p l e s of Heat T r a n s f e r " , 2nd E d i t . I n t . Textbook Co., S c r a n t o n , P e n n s y l v a n i a 1969 pp 353-357; 311-318. 69. I q b a l , M. " P r i v a t e D i s c u s s i o n s " Dept o f M e c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada, 1971. 70. Gregory, N., S t u a r t , J . T. and Walker, W. S. "On the S t a b i l i t y o f T h r e e - D i m e n s i o n a l Boundary L a y e r s w i t h A p p l i c a t i o n t o the Flow due to a R o t a t i n g D i s k " , P h i l . Trans.1946 v 248, 155, S e c t i o n I . 71. Cochran, W. G. "The Flow due t o a R o t a t i n g D i s c " , P r o c . Camb. P h i l . Soc. 30, 365 1934. 72. Hannah, Miss D. M. " F o r c e d Flow a g a i n s t a R o t a t i n g D i s c " , Rep. Memor. Aero. Res. Comm., London 2772, 1952. 73. S q u i r e , H. B. " R a d i a l J e t s " 50Tahre G r e n z s c h i c h t f o r -schung 1954, 47. 74. G a r t s h o r e , I . S. " P r i v a t e D i s c u s s i o n s " , Dept o f Me c h a n i c a l E n g i n e e r i n g , U n i v e r s i t y o f B r i t i s h Columbia, Vancouver, Canada, 1971. 75. G l a u e r t , M. B. "The W a l l J e t " F l u i d Mechanics, v o l . 1 1956 pp 625-643. 76. Timoshenko, S. and Goo d i e r , J . N. "Theory o f E l a s t i c i t y " , M cGraw-Hill Book Company I n c . 2nd e d i t . 1951, pp 107-111. 77. D'Isa, F. A. "Mechanics o f M e t a l s " , Addison-Wesley Pub. Co. 1968. 78. Marks* M e c h a n i c a l E n g i n e e r s ' Handbook e d i t . Theodore Baumeister, 6th e d i t . McGraw-Hill, I n c . 1958 p 13-107. 79. ASME Handbook: M e t a l s E n g i n e e r i n g P r o c e s s e s p 359. 80. Parkus, Heinz, " T h e r m o e l a s t i c i t y " , B l a i s d e l l P u b l i s h i n g Company, Waltham, M a s s a c h u s e t t s , 1968 p 5. 81. ASM Me t a l s Handbook V o l I pp 529, 1154. APPENDIX I ESTIMATION OF HEAT TRANSFER COEFFICIENTS The Sawing Disk Cobb and Saunders [68] d i s c u s s e d the problem of heat t r a n s f e r around a t h i n r o t a t i n g d i s k . In t h e i r e x p e r i m e n t s , they found the f l o w p a t t e r n around the d i s k changed from l a m i n a r t o t u r b u l e n c e a t a c r i t i c a l r a d i u s , r . F o r the l a m i n a r r e g i o n , r < r , the average heat t r a n s f e r c o e f f i c i e n t was g i v e n by wr 2 h ( r ) = 0,35( — ) 1 / 2 — (1-1) r c where w i s the a n g u l a r v e l o c i t y o f the d i s k , K the thermal c o n d u c t i v i t y of the f l u i d s u r r o u n d i n g the d i s k and v the k i n e m a t i c v i s c o s i t y of the f l u i d . F o r r > r c , the l o c a l heat t r a n s f e r c o e f f i c i e n t i s g i v e n by 2 h ( r ) = 0.0195( - i l £ _ ) 0 - 8 _JL_ ( i _ 2 ) v r In the heat t r a n s f e r s o l u t i o n f o r the d i s k , depending on the v a l u e o f r , e q u a t i o n s (1-1) or (1-2) w i l l be used t o e s t i m a t e the heat t r a n s f e r c o e f f i c i e n t . F o r the sawing 92 d i s k a n g u l a r v e l o c i t y used i n the p r e s e n t i n v e s t i g a t i o n r i s 3.64 i n c h e s , c The Workpiece The f l o w p a t t e r n i n the neighbourhood of a r o t a t i n g d i s k i s v e r y complex and has not been c o m p l e t e l y s o l v e d [69, 70, 29, 71, 72, 73]. I n f o r m a t i o n on f l o w p a t t e r n s over a s u r f a c e l o c a t e d near a d i s k i s not a v a i l -a b l e i n the l i t e r a t u r e . However, s i n c e the c u t t i n g zone i n the workpiece i s l o c a t e d a t the p e r i p h e r y o f the d i s k and the d i s k r o t a t e s a t h i g h speed (282 f t / s e c ) , i t i s r e a s o n a b l e t o assume t u r b u l e n c e over the workpiece. The v e l o c i t y d i s t r i b u t i o n over the f l a t p l a t e would be con-s i d e r e d t o a r i s e from the source o f d i s t u r b a n c e due t o the saw d i s k d r a g g i n g a i r and i m p i n g i n g i t on the p l a t e [ 7 4 ] . C o n s i d e r the l a y e r o f a i r dragged by the r o t a t i n g d i s k as a j e t w i t h u n i f o r m v e l o c i t y e q u a l t o the p e r i p h e r a l v e l o c i t y o f the d i s k , V c . The v e l o c i t y d i s t r i b u t i o n a r i s i n g from t h i s j e t i m p i n g i n g on the f l a t p l a t e may be thought of as t h a t due t o a t u r b u l e n t r a d i a l w a l l j e t . A c c o r d i n g t o G l a u e r t [ 7 5 ] , the v e l o c i t y d i s t r i b u t i o n , u s i n g P r a n d t l ' s h y p o t h e s i s , i s of the form u = P - j ) f ' ( n ) (1-3) U r c where ^ i s a constant, v the kinematic v i s c o s i t y , U c a c h a r a c t e r i s t i c v e l o c i t y , r the distance from the j e t axis ( c u t t i n g zone) and f ' ( n ) a dimensionless v e l o c i t y d i s -t r i b u t i o n , n i s defined as i • ( I - 4 > where z i s the coordinate normal to the plane of the wall, 1/4 From Table 1 of Glauert's paper, K R X / = 0.0215 when a = 2 and since equation (7-14) of the paper i s i n the form <R 1 / 4 - A < V ' > - 3 / 4 (1-5) t om X may be determined when n . and f ' are known. From t om Figure 3 of the paper these values can be read to be n . = 1.67 and f • = 0.32. t om X = (0.0215) (1.67 x 0.32) = 0.0135 From Figure 1 of Glauert's paper f ' ( n ) may be represented as a polynomial, f'(n) = a.n 1 (1-6) 1=1 Because of the symmetry of the problem, only the odd powers need be considered i n equation ( 1 - 6 ) . Taking the f i r s t two terms, we have 94 f • ( TI ) = + a 3 n 3 (I-6a) From Figure 1 of Glauert's paper, take two non-zero points to c a l c u l a t e a.^  and a^» f'( n ) = 0.1 when n = 0.5 or 4.3 and f'( n ) = 0.315 when n = 2.1 whence a x = 0.2002 and a 3 = -0.0132, and so f'( n) = 0.2002n - 0.0132n3 (I-6b) But n = 49.5z/r, therefore f ' ( - ) = 9.93(-) - 0.654(-) 3 (I-6c) r r r Hence u = 4 9 . 5 ( - ^ 0 1 / 3 [9.93- - 0.654(f) 3] (1-7) U c r 4 For the value of U cequation (4.6) of Glauert's paper gave 3 v ^ U = — U c 40F where F = 1/2(typical v e l o c i t y ) (volume flow per radian) The t y p i c a l v e l o c i t y here may be termed as that character-i s t i c of the impinging j e t , V"c and the volume flow per 2 radian may be expressed as V c t n 5 / 71 where t Q and <5 give the c r o s s - s e c t i o n a l dimensions of the impinging j e t and 2V"c gives the t o t a l length of the j e t i n two minutes (average time of run f o r each experiment). Thus F = 1/2V (V t n )2 c C D or 2 3 2 2 F = l/*1 \ \ $ Hence u = 3 i r v C 2 0 V c 3 t D 2 6 2 S u b s t i t u t i n g f o r U"c i n equation (1-7) gives A 2 62 Nl/3 u = 4 3 . 5 v J ~ - 3 -)[9.93(f)-0. 6 5 4 ( f ) 3 ] (1-8) According to K r e i t h [68], the l o c a l Nusselt number f o r a f l a t plate i n a turbulent flow region i s given by But Nu v = 0.0288Pr 1 / 3[Re ] ° ' 8 (1-9) h c x x Nu x . - f S - ( I.9a) therefore 96 h - 0 . 0 2 8 8 P r ^ ^ R e X l 0 , 8 ( I - 1 0 ) Equation (1-10) may be used to c a l c u l a t e the heat trans-f e r c o e f f i c i e n t over the f l a t plate workpiece with some modification: replace x with r and use a c o e f f i c i e n t which w i l l give the best temperature d i s t r i b u t i o n agree-able with the experimental r e s u l t s . Thus we may write h» = C P r 1 / 3 — [ R e ] 0 * 8 (1-11) cr r r where Pr = 0.72 for a i r , the a i r thermal c o n d u c t i v i t y and C i s a constant determined on the basis of experimen-t a l r e s u l t s . For the experiments i n the present study C = 0.0096. With the value of u given by equation (1-8), the heat t r a n s f e r c o e f f i c i e n t over the workpiece can be estimated. I t should be noted however that equation (1-11) does not apply to the region where the j e t impinges on the plate or, i n t h i s study, the c u t t i n g zone. For t h i s region the heat p a r t i t i i o n assumes zero heat trans-f e r . On the other hand the heat t r a n s f e r c o e f f i c i e n t may be estimated as that due to a stagnation point region which i s estimated to be h' = 18.8Btu/hr/ft 2/°F (1-12) co 97 The c u t t i n g zone temperature i s v e r y i n s e n s i t i v e t o the change i n the heat t r a n s f e r c o e f f i c i e n t and so the heat p a r t i t i o n assumption i s j u s t i f i e d . APPENDIX II FORCE ANALYSIS IN FRICTION SAWING SYSTEM Forces on Table and Workpiece [Figure I I - l ] Consider the l i n e a r dimensions: t, , I = f (x) 1' w I f L i s the length of the workpiece, then I = L-xt ( I I - l ) w where k i s the rate at which the t a i l end of the workpiece approaches the c u t t i n g zone, i n other words, the c u t t i n g r a t e . I f x = V = constant, then and I = L-Vt (TI-2) w l1 = L + a s i n ^ - Vt (II-3) From equations (II-2) and (II-3) L - I = a s i n ^ (II-4) 1 w r Now l e t us consider the table f o r c e s , (Figure I I - l ) : I f we take moments about the point 0, we obtain R i(^ 1-3) - Rt - R2l2 = 0 • (II-5) 98 C o n s i d e r the f o r c e s on the workpiece: f o r e q u i l i b r i u m o f the f o r c e s "2lFx = ^ F y = 0, t h a t i s F N s i n ^ + F f c o s ^ - F T = 0 ( I I - 6 ) F N c o s ^ f - F f s i n " Y + R X - R 2 - W = 0 ( I I - 7 ) A l s o ^- M A = ^ ' ^ ' E * F„£ cos^p - F-l sin'S' + N w ' f w 3 R 1 - R 2 ( i - 1 - V - -w£ g =o ( I I - 8 ) R e a r r a n g i n g e q u a t i o n s ( I I -5) to ( I I -8) F N F f R l R 2 = N ( I I - 5 ) 0 0 - l 2 / l R ( I I - 6 ) sin'/' cos'f 0 0 F T ( I I - 7 ) cos'f' - s i n ^ 1 -1 W ( I I - 8 ) c o s ^ - s i n ^ 3 T~ w I w W ^w U s i n g Cramer's r u l e , e q u a t i o n s ( I I - 5 ) t o ( I I - 8 ) may be s o l v e d e a s i l y . U s i n g e q u a t i o n s ( I I - 2 ) and ( I I - 3 ) 0 0 L - Vt •*• a s i n ^ f -3 s i n ^ c o s ' Y 0 cos-^p -sin'Y 1 c o s ^ - s i n ^ 3 L-Vt - i2/Jt 0 -1 L-V t-£ £ + a s i n L-Vt 100 N N D asin'f (L-Vt+asin^ -3 -l-2) /(L-Vt) R 0 F T cos^p W - s i r i " ^ L -Vt+asin^-3 o I L-Vt - s i n ^ L-Vt - l 2 / l 0 - 1 L - V t + a s i n ^ - i 2 L-Vt "F.T = F m s i n ^ f , „ N T ' atan^p D x r (L-Vt+asirvy-^) +3+Vt-L-asin^) + 3 ( ^ 2 [W- L-Vt+asin^|/-3-X, -Rt] (II - 9 ) and s i m i l a r l y F = F _ c o s ^ + i [ w (L-Vt+asin^-^ 0) (I +3+Vt-L-asin'f) + 3 ( ^ ~ - i ) +R£] (11-10) (L-Vt+asin R^ and R2 may be obtained i n a s i m i l a r way. In using equations (II - 9 ) and (11-10) i t must be noted that the value of t i s l i m i t e d by 0 - t - L/V (11-11) The angle ^ i s dependent on the thickness of the material being cut as well as the height above the h o r i z o n t a l 101 c e n t r a l plane of the sawing disk. The following shows the c a l c u l a t i o n of ^ as a f u n c t i o n of the workpiece thickness t and the p o s i t i o n of the table H. For added workpiece w support, i t i s s u f f i c i e n t to add the height of the support to the value of H. I Figure I I - 2 102 = l/2( « ) (11-12) c o s A = ~ - (11-13) ' a . , COSCxl H + fcW (11-14) From (11-12), (11-13) and (11-14); H + t = l/ 2 [ c o s ~ 1 ( - ) + c o s " 1 ( -)] (11-15) For d i f f e r e n t workpiece thicknesses, p o s i t i o n s of the table and even saw disk r a d i i , equation (11-15) furnishes the value of . I f V R i s the r e l a t i v e s l i d i n g speed of the saw disk to the workpiece, the t o t a l heat generated can be calculated. In t h i s case V < < V and so V D ^ V . C K C F f V c q T = -3-^- (11-16) where J i s the mechanical equivalent of heat. This t o t a l heat i s d i s s i p a t e d through the saw disk and the workpiece i f heat loss i n the c u t t i n g zone i s neglected. APPENDIX I I I CALCULATION OF HEAT PARTITION FRACTION E q u a t i o n (33) s t a t e s q T - q D • q w <IH-1> I f the f r a c t i o n o f the heat g e n e r a t e d d i s s i p a t e d through the workpiece i s f , then and so, q w = f q T ( I I I - 2 ) q D = ( l - f ) q T ( I I I - 3 ) E q u a t i o n (5) may then be w r i t t e n as ( l - f ) q T X (Ab)I (Ar) + I (Ab)K (A r ) L K T J <*> 2TTK r Aa 1 I (A b ) K (A a) + I . ( A a ) K (Ab) J ( I I I - 4 ) D o 1 1 o The p e r i p h e r y temperature i s T ( a ) , where a i s the d i s k r a d i u s ( l - f ) q t r K (Ab)I (Aa) + I (Ab)K (Aa) T(a) T = — f o o 2 o . 0 0 ~ 2TTK t Aa 1 I (Ab)K (Aa) + I, (Aa)K (Ab) J D O 1 1 O ( I I I - 4 a ) 103 104 For the disk, a = 7" and b = 3/4". From Bessel function tables, the term i n the bracket i s c a l c u l a t e d to be 1.055, hence T(a) - T = 1.055(l-f)q„ 2TTK t DAa (III-5) From equation (28), when 5 = 0 and y = 0 T(0,0)-T = fq Tsiiv/ - 4 b l P K V D 4 b ] S l n ( 2 b 7 _ ) n=l ny (III-6) Equation (50) gives the average temperature i n the c u t t i n g zone; fq Tsin7! Tav " T c o _ A b l P K t w t D 8b' D n=l sxn ( 2 ^ - ) 2 n y_ (III-7) Applying the matching technique, we equate equation (III-6) to equation (III-5) f o r the maximum temperature matching and equation (III-7) to equation (III-5) f o r the average matching. The former gives 105 max » , 14.05 . i b l p t W Ks 4b. n=l sin (2^-) nu ( I I I and the l a t t e r g i v e s av 1 + 14.05 \ sinf 8b^  tD + D sxn ( — ) n=l 2 n u n ( I I I 106 APPENDIX IV STRESS ANALYSIS ON THE SAW DISK S t r e s s A n a l y s i s on the Disk In the s t r e s s problem, two assumptions w i l l be made. F i r s t , the d i s k i s t h i n and so the s t r e s s i s i n d e -pendent of z, i n o t h e r words, the problem i s two-dimensional, Second, the problem i s l i n e a r and so s u p e r p o s i t i o n p r i n c i p l e may be u s e f u l l y employed. Based on the f o r e g o i n g assumptions, the g e n e r a l s t r e s s e q u a t i o n s are g i v e n by _ I M + i _ A2! ° r = r 9 r r 2 30 2 9 20 r0 9r v r 90 ' (IV-1) where 0 i s the A i r y s t r e s s f u n c t i o n . The c o m p a t i b i l i t y e q u a t i o n i s g i v e n by ( » r * + * I ? ' ^ + 7 ^ > = 0 ( I V . 2 ) 107 The k i n e m a t i c r e l a t i o n s are 3u £ r 9r u 1 3v £ 0 r r 30 _ _1 3ju 3jv Y r0 r 90 3r F o r f u l l c i r c l e s o l u t i o n s , e q u a t i o n s (IV-3) reduce t o 3u e = T— r 9r e 0 = 7 (IV-3a) Y n = 0 r0 The c o n s t i t u t i v e e q u a t i o n s a r e : e = Tr (a - vo~ ) r E r 6 1 (IV-4) ee = I ( 0o " v a - } EZ = " E ( a r + ° 6 ) The problem t o be s o l v e d i s shown i n F i g u r e IV-1 and because of the l i n e a r i t y assumption, an e q u i v a l e n t problem i s shown i n F i g u r e IV-2. C o n s i d e r F i g u r e I V - 2 ( a ) , the boundary c o n d i t i o n s used are (a) o = 0 a t r = a r (b) u = 0 a t r = b (IV-3) 108 For t h i s problem, the s t r e s s f u n c t i o n i s g i v e n by [76] whence « A i B 3 + v 2 3 0 = Ar + — - — ~ — pu> r (IV-5) r o . . B 3 + v 2 2 " r = 2 8 ~ P W r r B 1 + 3v 2 2 G0 = 2 8 P U r r (IV-6) T = 0 r0 U s i n g the boundary c o n d i t i o n s , A and B are determined and the s o l u t i o n i s F , G 3 + v 2 2 T R = - + — 2 g p U r Dr F G 1 + 3v 2 2 °n = n 9 ~ — « p u r 0 D Dr 2 8 (IV-7) T = 0 r0 where 2 , . F = - - e — T [2(l-v)b 4 - (l+v)(3+v) a^] 8aZb 2 G = £|- [2(l-v)b 2 + (l-v)(3+v) a 2] ( I V _ 8 ) D = [(1+v) a 2 + (l-v)b 2] a b Now c o n s i d e r F i g u r e I V - 2 ( b ) . T h i s problem was t r e a t e d by Timoshenko and Goodier [ 7 6 ] , The f o r c e on the d i s k w i l l be t r e a t e d as a f u n c t i o n o f -uv . F o l l o w i n g the n o t a t i o n s 109 i n F i g u r e IV-3, the s o l u t i o n i s o b t a i n e d by s u p e r p o s i n g the f o l l o w i n g s t r e s s e s on the simple r a d i a l s t r e s s d i s -t r i b u t i o n Ttr / \ 1) Normal s t r e s s u n i f o r m l y d i s t r i b u t e d a l o n g the boundary ^ ) P(<f) sin (0 1 + 0 2 ) d t \ 2) Shear s t r e s s e s of i n t e n s i t y P('i) cos (G^ . ^ 2 i ^ f Ci) 0, + 0 9 ) dif 3) S t r e s s whose normal and shear components are r e s p e c t i v e l y 1_ ird A. X P(/) s in (0 2 - 9 ^ di " i, and 110 From geometry, we note the f o l l o w i n g ; F i g u r e IV - 3 0 c j f ( « Q l + 0 2 = I " * (IV - 1 0 ) 0 - 0. = £ - 0 1 2 2 c whence 0 i = f -1 « + V i (IV - 1 1 ) e 2 = | (0 c - i ) S u b s t i t u t i n g these i n the e x p r e s s i o n s g i v e n above, we g e t 4< a = r f r f |^ I P('f)sin | CRQ c)-+ ^ ) P a c e s ' / d1< cos0 r> rf ;os0 c p *d J a„ = 0 TTd 1± ' f i - J PC/0 cosf df T r0 i l i r^f s i n 9 rf ^ J p t f v sin? df + — ^ / P W <tf 1 (IV - 1 2 ) C o n s i d e r i n g the case when i s c o n s t a n t , say P(10 = P , e q u a t i o n s (IV-11) reduce to D 4cos(0 + '*) . cosG o = - Cff - *.) [ £ + + j r Tr f f 1 r d d J 'e - fd (*iV c o s* T r0 = fd ( * f ~ V ( s i n * + 8 i n 9 c ) (IV-12a) where i> = ~ (ff+1f^) and 0£ f- i±) i s c o n s i d e r e d s m a l l . E q u a t i o n s (IV-12) and (IV-12a) r e p r e s e n t s o l u t i o n s f o r a s o l i d d i s k . For an a n n u l a r d i s k , the s o l u t i o n s must be m o d i f i e d by superimposing the s o l u t i o n o f the problem r e p r e s e n t e d i n F i g u r e IV-4 on the s o l u t i o n s i n e q u a t i o n s ( I V - 1 2 ) . In t h i s c a s e , the g e n e r a l i z e d p o l a r c o o r d i n a t e two-dimensional s o l u t i o n g i v e n i n Timoshenko and Goodier [76] i s f o l l o w e d . E q u a t i o n s (IV-12) g i v e the s o l u t i o n f o r a s o l i d d i s k whereby o r and x r Q can be c a l c u l a t e d f o r any p o i n t l y i n g on the boundary of a c o n c e n t r i c d i s k o f r a d i u s b i n s i d e the s o l i d d i s k o f r a d i u s a. I f these s t r e s s e s a r e d e s i g n a t e d as a r b and rrQb » the boundary c o n d i t i o n s f o r the problem t o be s o l v e d w i l l be g i v e n by a t t h e • i n n e r boundary 112 2) ar = 0 / a t the o u t e r boundary t _ = 0 (IV-13) r0 1 The boundary c o n d i t i o n a t the i n n e r boundary may be expanded i n t o F o u r i e r s e r i e s , t h u s : o , = A + > A_cos n 0 + B sin n 0 rb o ^ - -^ n c n c n=l n=l - T = C + C cos n 6 + >~ D sin n 0 r0D o n c ^ n c (IV-14) From geometry by the c o s i n e r u l e , F i g u r e IV-5 r = A 2 + b 2 - 2ab cos (0 - i) (IV-15) c U s i n g e q u a t i o n s (IV-12) and d e f i n i n g the f o l l o w i n g : 4* . I i = j f P C f ) s i n 2 d r p f P W cos I d^ 1>. I f 'I. I2 = J PC/)- cos * d1> 1 f f I 3 = J P W sin ^ di i f (IV-16) i 4 = J p ( / ) di 113 then, 0 , „ . 0 2(T1 C O S ~2 + 1 • 2"^ 12 A V 0 8 ^ O" , = H T H j rb A= y : Trd Trd /a +b -2ab cos(0 -1) (IV-17) T _i V I 4 s l n 0 c (IV - 1 8 ) r0b = - j f j — ird Trd U s i n g the f u n c t i o n s f o r <rr^ and from e q u a t i o n s (IV-17) and (IV-18) the F o u r i e r c o e f f i c i e n t s A Q, A n, B n, C Q , C n and D i n e q u a t i o n s (IV-13) can be e v a l u a t e d w i t h i n the n l i m i t s o f 0 c from z e r o t o 2TT . Thus the boundary c o n d i -t i o n a t the i n n e r boundary i s w e l l d e f i n e d . We now t u r n t o the g e n e r a l p o l a r c o o r d i n a t e s e x p r e s s i o n f o r the s t r e s s f u n c t i o n 0 g i v e n by M i c h e l l [ 7 6 ] . 0 = a log r + b r 2 + c r 2 log r + d r 2 0 + a 1 0 o ° o o o o a l 3 -1 + Y~ r 0 s in 0 + (b 1 r + a | r + b^ r log r) cos 0 °1 3 ' -1 - y - r 0 cos 0 + (d^ r + c^ r + d^ r log r) sin 0 OP , " < \ , n . , n+2 . . -n . , . -n+2. + >- ( a r + b r + a r + b ' r ) cos n 0 ^ r.-1 n n n n n=2 00 _i_ ' /„ n n+2 . „, -n , , , - n + 2 . + ^> ( c r + d r + c ' r + d ' r ) sm n 0 ^ - V n n n n n=2 (IV-19) 114 S i n c e the problem here i s t h a t o f s t r e s s d i s t r i b u t i o n due to i n n e r boundary s t r e s s e s of a f u l l c i r c l e , i t i s s u f f i -c i e n t t o c o n s i d e r the s t r e s s f u n c t i o n 0 , 2 v. ' , n , , n+2 , , -n , , . -n+2>. = b r + "> ( a r + b r + a ' r + b ' r ) o ^-TC- 1 n n n n—Z 00 • / n , , n+2 . , -n . ,, -n+2. + ;> ( c r + d r + c r + d r ) n n n n n=2 (IV-20) whence from e q u a t i o n (IV-1) oo a = 2b + X [(n+n2)a r n ~ 2 ) a r 1 1 " 2 + (n+2+n2) b rn + (n 2-n)c ? r ~ n ~ 2 r o . - n n n n n=2 + (n2-n+2) b^ r ~ n J cos n 0 + ^  [(n+n2) c rn~2 + (n+2+n2) d xn + (n2-n) c' r " n ~ 2 9 - n (IV-21) + (n -n+2) d' r ] sin n 0 n T n ^Tn{(n-l)a vn~2 + (n+l)b r n - (n+1) a' r n 2 r0 =x n n n n=2 - (n-l)b' r " n } sin n 0 n n{(n-l)c n vn~2 + (n+l)d n r n - (n+l)c^ r " n " 2 n=2 - (n-l)d' r n } cos n 0 n (IV-22) 115 From e q u a t i o n (IV-11) e « f - | (e c + i) In o r d e r t o a v o i d h a l f - a n g l e s , put n =» 2m (IV-23) i n e q u a t i o n s (IV-21) and ( I V - 2 2 ) . N o t i n g t h a t s i n ( 0 + a) = s i n O c o s a + c o s O s i n a and cos(© + a) = cosOcosa - s i n G s i n a we can w r i t e e x p r e s s i o n s (IV-21) and (IV-22) i n the form of ( I V - 1 4 ) . Thus = t> + F. (r)cos m 0 r o / Im c m=T + J> G.. ( r ) s i n m 0 ^ — z r - J lm c m=l (IV-24) and T = F„ (r)cos m 0 rO ^ • 2m c m=o + ^>\ G„ ( r ) s i n m 0 — 2 m m=l (IV-25) 116 where Flm ( r ) = D l m S i n m i - e i m c o s Glm ( r ) = D l m C O S ^ + C l m s i n m ^ f2m ( r ) = C 2 m S i n m * + D 2m C O S m * (IV-26) G2m ( r ) = C 2 m C 0 S m * " D 2 m s i n ^ and Dlm = ( 2 m 2 + m ) c 2 m r 2 m ~ 2 + ( 2 m 2 " h n + 1 ) d 2 m ^ + (2m2-m)c; r " 2 m ~ 2 + (2m2-m+l)d; r~ 2 m C = (2m2+m)a0 r 2 m ~ 2 + (2m2+m+l)b r~ 2 m lm 2m v y 2 m + (2m2-m)a9 r ~ 2 m _ 2 + (2m2-m+l)b; r " 2 m 2m ' 2m C2m = m U 2 m-l)a 2 m r ^ 2 + (2*4.1)1, ^  r 2 m - (2m+l)a 2 m r - 2 " 1 - 2 - ( i n-Db^ r ~ 2 m } D 2 m = m{(2m-l)c2m r 2 m " 2 + Um+Dd^ r 2 m " ( 2 m + 1 ) c 2 m r _ 2 m " 2 " ( 2 m - 1 ) d 2 m r " 2 m (IV-27) For any given value of r , the c o e f f i c i e n t s b . a 0 , b 0 , 2 ^ * o' 2m' 2m' c o > d o ~ t d o m a n d d ' o can be evaluated by using (IV-14), 2m m^' <L dm J 3 7 (IV-24) and (IV-25) and the second boundary c o n d i t i o n i n (IV-13). Noting that r may be zero at point A and the stresses at t h i s point are f i n i t e . a' 0 , b * 0 . c'~ and 1 dm1 dm1 dm d'2 m may be eliminated. I f P( i>) i s constant, the equa-tions involved w i l l be much s i m p l i f i e d f o r numerical c a l -c u l a t i o n s . Thermo-Elastic Stresses i n Disk Following D'Isa's treatment [77] the modified Hooke's law equations become du ^ . r 0 e = - T — = at + — — r dr E E dw ^ v a r v a 0 (IV-28) e z - d i " = a t - 1 r f o r a plane stres s (a = 0) z taken as a p l a i n s t r e s s case, i s given by problem. The disk may be The e q u i l i b r i u m equation da -r1- + - (a - aj = 0 dr r r 0 Equation (IV-29) may be expressed i n terms of u. The s o l u t i o n of the r e s u l t a n t equation gives: 118 = U+v r >7 / t r d r + l r + (IV-29b) where t i s the temperature d i s t r i b u t i o n i n the d i s k . U s i n g e q u a t i o n ( I V - 2 9 ) , the s t r e s s e s can be e x p r e s s e d as and aE ( r < C.jE C 2E t r d r + r . 1 2<1-"> (l+ v ) r 2 C 1 E C 2 E cx = - aEt + ^ J t r d r + -zn—+ e - r2 J 2 d - ) ( 1 + v ) r 2 (IV-29c) where the c o n s t a n t terms and C 2 can be c a l c u l a t e d from the boundary c o n d i t i o n s of the d i s k : s t r e s s f r e e i n n e r and o u t e r edges, or cr = 0 a t r = r . and r where r . ^ 7 r i o i and r Q are the i n n e r and o u t e r r a d i i of the d i s k r e s p e c t i v e l y . S i n c e t i s known from the temperature d i s t r i b u t i o n problem, the t h e r m a l s t r e s s e s i n the d i s k can be c a l c u l a t e d . APPENDIX V TABLE STRAIN RING CALIBRATION Figure V - l shows the arrangement f o r the table s t r a i n r i n g c a l i b r a t i o n . An aluminum hollow c y l i n d e r was placed on the table at a l o c a t i o n corresponding to the centre of the s t r a i n r i n g . The i n d i c a t o r was warmed up f o r at l e a s t t h i r t y minutes. The zero was set and the table was s l i g h t l y pushed up and down to make sure the transducer was working properly. S t a r t i n g with small loads, the i n d i c a t o r s e n s i t i -v i t y was set at highest l e v e l . Loads were added i n i n c r e -ments of 2 l b up to 10 l b and the i n d i c a t o r displacement was adjusted to read d i r e c t l y i n pounds. The s e n s i t i v i t y was then lowered to read 25 l b maximum and the loads were added i n 5 l b increments up to 25 l b . F i n a l l y the s e n s i -t i v i t y was switched to the 100 l b maximum range and the c a l i b r a t i o n was done up to 60 l b . The expected table force was les s than 40 l b and so c a l i b r a t i o n up to 60 l b was s u f f i c i e n t . Each time the table force i n d i c a t o r was read, the corresponding d e f l e c t i o n on the thrust s t r a i n r i n g was read. 119 120 I t happened to be zero up to 60 l b . The table was then pushed down by hand to read up to 100 l b and the thrust s t r a i n reading was s t i l l zero. Hence the i n t e r a c t i o n of the c u t t i n g forces was n e g l i g i b l e . / APPENDIX VI THRUST STRAIN RING CALIBRATION Two s t e e l rods 1/4 inch diameter by 12 inches long were notched at points about h a l f inch from each end. The second rod was notched at the centre. The rods were stringed as shown i n Figure VI-1. The c e n t r a l notch i n the second rod was stringed to a loading pan, Figure VI-1. The loading pan was passed over a p u l l e y f o r loading. The arrangement made i t easy to load the thrust 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 s t r a i n r i n g . I t was found that no i n t e r a c t i o n of the thrust force on the table force was recorded by the . table force i n d i c a t o r . 121 TABLE 1 EXPERIMENTAL VS. THEORETICAL RESULTS FOR BRASS I ( i n . ) y ( i n . ) T/C Pos 1 n Temperature C F ) E x p t ' l and T h e o r e t i c a l C o n d i t i o n s E x pt. Theory (y=0.0) T 0 maximum matching T A average matching -1.000 0.0 183 187 -0.875 197 202 -0.750 215 221 t = 1/4 i n . -0.625 238 245 w -0.500 314 268 276 F,,, = 10.00 l b f -0.375 346 310 319 l -0.250 375 373 385 R„ = -2.500 l b f -0*125 386 488 505 l 0 606 899 934 V = 1.035 i n . / m i n .0.125 408 473 490 X).,250 364 352 363 t , = 1.25 mir 0.375 296 286 294 e l a p s e 0.500 165 242 249 F = 7.7859 l b f 0.625 212 217 0.750 188 193 N = 22.2828 l b f 0.875 170 174 1.000 156 159 y = 0.3494 -1.000 0.1875 + 181 185 + y t h e o r y " ° ' 2 i n -0.875 194 199 -0.750 211 217 -0.625 232 238 -0.500 305 258 265 -0.375 328 290 299 -0.250 341 332 343 -0.125 364 379 392 0 386 402 416 0.125 341 368 380 0.250 318 314 324 0.375 262 268 276 0.500 154 233 240 0.625 206 211 0.750 185 189 0.875 168 172 1.000 155 158 122 TABLE 1 (Cont'd) 123 5 y Temperature (°F) (in.) ( i n . ) Expt. Theory (y=0.0) Expt' and T/C T T A T h e o r e t i c a l Pos' n u maximum matching A average matching Conditions -1.000 0.375* 162 166 t = 1/4 i n . -0.875 172 176 W -0.750 184 188 F™ = 10.00 l b f -0.625 197 202 1 -0.500 213 218 R_, = -2.500 l b f -0.375 229 236 l -0.250 230 246 253 V = 1.035 in./min -0.125 0 230 221 258 261 266 269 Elapse = 2 - 2 5 m i n 0.125 t).250 199 186 252 234 259 240 F = 6.7340 l b f 0.375 165 214 219 N = 21.2058 l b f 0.500 0.625 138 194 177 199 181 / = 0.3176 0.750 0.875 163 151 166 154 ^ t h e o r y = °'4 i n ' 1.000 141 143 ^ t h e o r y = °* 9 i r u -1.000 0.875 + 148 . 151 -0.875 153 156 -0.750 158 162 -0.625 163 167 -0.500 168 172 -0.375 172 176 -0.250 195 175 179 -0.125 195 176 180 0 190 175 179 0.125 186 172 176 0.250 182 168 171 0.375 173 162 166 0.500 160 156 159 0.625 147 149 15 2 0.750 142 145 0.875 136 138 1.000 130 132 TABLE 2 EXPERIMENTAL VS. THEORETICAL RESULTS FOR MILD STEEL I y T emoerature (°F) (in.) (in.) Expt. Theory Ex p t ' l and T/C T O maximum matching T A T h e o r e t i c a l Pos' n A average matching Conditions -1.000 0.234* 240 206 213 t ^ = 1/8 i n . -0.875 251 226 234 -0.750 -0.625 274 296 250 280 259 290 F T = 12.500 l b f -0.500 -0.375 318 364 317 365 33 0 380 R T = -5.833 l b f -0.250 408 426 445 V = 1.035 in./min -0.125 475 493 515 0 0.125 508 475 518 449 541 469 t , = 1 . 7 0 min elapse 0.250 397 356 371 F = 10.2733 l b f 0.375 0.500 274 208 283 230 294 238 N = 27.9059 l b f 0.625 193 198 / J = 0.3681 0.750 0.875 165 145 170 148 •y,, = 0.2 i n . 1 theory 1.000 130 132 + ^theory = 0.3 i n . -1.000 0.327 + 240 202 209 -0.875 251 220 227 -0.750 262 241 250 -0.625 285 267 277 -0.500 307 297 309 -0.375 341 334 347 -0.250 375 372 388 -0.125 408 405 422 0 420 408 425 0.125 386 370 385 0.250 282 313 325 0.375 251 260 270 0.500 173 218 225 0.625 185 191 0.750 161 165 0.875 142 145 1.000 128 131 124 TABLE 2 (Cont'd) 125 5 Y Temperature( •F) (in.) ( i n . ) Expt. Theory Expt'l and T/C Pos 'n T o maximum matching T A average matching T h e o r e t i c a l Conditions -1.000 0.500 190 196 i_ 1/8 i n . -0.875 204 210 \ = -0.750 -0.625 219 236 226 244 F T = 11.292 l b f -0.500 -0.375 310 .•• 253 271 262 281 R T = -5.000 l b f -0.250 -0.125 305 300 284 291 296 303 V = 1.035 in/min 0 0.125 287 260 286 269 297 279 Elapse - ° - 7 5 0 m i n 0.250 204 243 252 F = 10.2489 l b f 0.375 0.500 99 215 190 222 195 N = 26.1752 l b f 0.625 167 172 r -0.3196 0.750 149 153 0.875 135 138 1.000 123 125 TABLE 3 EXPERIMENTAL VS. THEORETICAL RESULTS FOR MILD STEEL I ( i n . ) Y (in.) T/C Pos • n Temperature (°F) E x p t ' l and Th e o r e t i c a l Conditions Expt. Theory T 0 maximum matching T A average matching -1.000 0.1094* 304 315 t = 1/4 i n . -0.875 328 339 W -0.750 356 369 F,- = 56.542 l b f -0.625 391 405 i -0.500 408 436 453 R„ = -21.104 l b f -0.375 453 497 516 i -0.250 519 584 607 V = 1.035 in/min -0.125 638 715 745 t , ^ =1.017 min elapse 0 756 836 872 0.125 585 647 674 0.250 464 482 501 F = 11.4909 l b f 0.375 352 377 391 N = 92.6846 l b f 0.500 230 306 317 fA = 0.1240 0.625 256 264 0.750 219 226 *y,, = 0.1 i n . 0.875 191 196 •'theory 1.000 170 174 + y,, = 0.3 i n . J t h e o r y -1.000 0.3294 + 296 306 -0.875 316 327 -0.750 339 351 -0.625 367 380 -0.500 375 399 414 -0.375 408 436 452 -0.250 453 474 493 -0.125 497 504 524 0 552 503 523 0.125 497 459 477 0.250 397 394 409 0.375 296 333 345 0.500 197 282 292 0.625 242 250 0.750 211 217 0.875 186 191 1.000 166 170 126 TABLE 3 (Cont'd) 127 I y Temperature ( *F) (in.) ( i n . ) Expt. Theory Ex p t ' l and Th e o r e t i c a l Conditions T/C To TA Pos' n maximum matching average matching -1.000 0.549* 292 301 t = 1/4 i n . W -0.875 308 318 -0.750 325 336 F_ = 66.061 l b f -0.625 344 356 l -0.500 365 363 376 Rm = -25.333 l b f -0.375 372 381 395 i -0.250 382 394 409 V = 1.035 in./min -0.125 372 398 413 t , =2.034 min elapse 0 359 388 403 0.125 0.250 350 274 364 331 378 342 F = 12.0729 l b f 0.375 199 294 304 N = 106.9034 l b f 0.500 0.625 260 229 268 236 yU = 0.1129 0.750 204 209 *y,, = 0.5 i n . 1 theory 0.875 182 187 1.000 164 168 +y,, = 1.0 i n . 1 theory -1.000 1.019 + 253 261 -0.875 259 267 -0.750 264 272 -0.625 267 276 -0.500 296 269 278 ^0.375 287 269 278 -0.250 282 266 275 -0.125 278 261 269 0 260 252 260 0.125 251 241 249 0.250 186 228 235 0.375 134 214 220 0.500 200 205 0.625 186 190 0.750 172 176 0.875 160 164 1.000 149 152 TABLE 4 EXPERIMENTAL VS. THEORETICAL RESULTS FOR T l STEEL (in.) y (in.) T/C Pos'n Temperature (°F) Expt, Theory maximum matching average matching Ex p t ' l and Th e o r e t i c a l Conditions -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 •1.000 •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 0.578* 1.078 310 314 314 314 305 287 262 218 165 282 278 264 260 238 216 197 154 132 299 309 320 331 339 345 344 335 316 288 257 225 196 171 151 135 122 255 257 257 256 253 248 239 229 216 202 186 171 157 143 132 122 113 309 321 332 343 353 358 357 348 328 299 266 232 202 176 155 138 125 263 265 266 265 262 256 248 237 223 208 192 176 161 147 135 124 115 t w = 1/4 i n . F T = 29.271 l b f R T = 12.354 l b f V = 1.035 in/min 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 . 1 theory +y,, = 1.0 i n J t h e o r y 128 129 TABLE 4 (Cont'd) I ( i n . ) y ( i n . ) T/C Pos' n Temperature C F ) Expt Theory 0 maximum matching A average matching E x p t ' l and T h e o r e t i c a l C o n d i t i o n s -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 -1.000 -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 0.0 0.2 1116 852 606 395 221 661 596 475 199 351 377 408 447 500 571 680 882 1623 737 482 350 268 215 178 152 133 344 367 395 429 471 522 584 647 651 545 419 323 255 208 173 150 132 365 392 425 466 521 596 710 928 1704 771 503 363 2 78 222 183 156 136 357 380 411 447 490 545 611 676 680 569 436 335 264 214 179 153 134 fcw - 1/4 i n . F T = 26.667 l b f R T = -10.833 l b f V = 1.035 in/mi n t , = 1.250 min e l a p s e F = 12.5343 l b f N = 50.9972 l b f }J = 0.2458 TABLE 4 (Cont'd) 130 I ( i n . ) Y ( i n . ) T/C Pos' n Temperature C F ) Expt Theory O maximum matching T A average matching E x p t ' l and T h e o r e t i c a l C o n d i t i o n s • 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 -1.000 -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 0.9 282 282 296 300 307 318 330 296 218 269 269 269 264 256 247 225 182 132 281 295 312 329 348 366 380 383 365 328 281 237 200 171 148 132 119 232 235 237 238 237 233 222 218 206 193 178 163 149 137 126 117 109 291 306 323 342 362 381 395 408 380 340 291 245 206 175 152 134 121 240 243 245 246 244 241 234 225 213 198 183 167 153 140 128 118 110 -= 1/4 i n . F T = = 28.333 l b f R T ' •- -10.833 l b f V = = 1.035 in/min Elapse = 2 - 2 5 m i n F = 10.2837 l b f N = 51.0773 l b f ^ = 0.2013 TABLE 4 (Cont'd) I ( i n . ) y ( i n . ) T/C Pos' n Temperature (°F) Expt, Theory T 0 maximum matching A average matching Expt' 1 and T h e o r e t i c a l C o n d i t i o n s • 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 -1.000 -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 0.5625* 251 256 260 264 264 264 264 264 251 225 195 143 1.0625 247 247 247 242 238 230 216 204 186 160 134 108 238 246 254 261 268 271 271 264 251 231 207 184 163 145 131 119 110 206 207 208 207 205 201 195 187 178 168 156 145 135 125 117 110 103 246 254 t = 1/4 i n . W 263 270 F m = 25.000 l b f 277 i 281 R m = -9.167 l b f 281 i 277 V = 1.035 i n . / m i n 259 238 t , =2.59 min e l a p s e 214 190 F = 8.9879 l b f 168 149 N = 44.9807 l b f 133 121 y » 0.1998 112 •v.. = 0.7 i n . 1 t h e o r y 212 214 214 + i r t h e o r y = 1.0 i n . 213 211 207 201 193 183 172 160 149 138 128 119 111 105 131 TABLE 4 (Cont'd) 132 I y Tern o e r a t u r e ( °F) Expt'1 and ( i n . ) ( i n . ) Expt. Theory T h e o r e t i c a l T/C T o T A C o n d i t i o n s Pos' n maximum average matching matching -1.000 0.39062! 316 328 -0.875 333 346 t M > = 1/4 i n . -0.750 352 366 W -0.625 373 387 F T = 27.667 l b f -0.500 386 395 411 1-0.375 408 416 433 R T = = -10.833 l b f -0.250 408 433 450 i. -0.125 419 435 453 V = 1.035 i n . / m i n 0 442 415 432 0.125 408 371 386 e l a p s e 0.250 341 316 328 0.375 240 264 274 F = 12.0582 l b f 0.500 143 221 228 N = 51.9409 l b f 0.625 187 192 0.750 161 163 0.2322 0.875 141 144 1.000 126 128 •v. . = 0.4 i n . - t n e o r y TABLE 5 EXPERIMENTAL VS. THEORETICAL RESULTS FOR TJL-6A1-4V I y Temperature (°F) Expt'l and , (in.) ( i n . ) Expt. Theory T h e o r e t i c a l T/C Pos • n T 0 maximum matching TA average matching Conditions -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 • 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 = 1/4 i n . W F T = 16.667 l b f R T =-8.333 l b f V = 0.536 in./min telapse= m i n F = 13.252 l b f N = 37.4205 l b f /-< = 0.3541 '^theory = °' 1 i 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 0.34375' -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 ^ t h e o r y = °* 3 i n * 133 TABLE 5 (Cont'd) 134 y Temperature (°F) (in.) (in.) Expt. Theory E x p t ' l and T/C T TA T h e o r e t i c a l Pos' n maximum matching average matching' Conditions -1.000 0.48437 5* 286 297 -0.875 274 302 314 <w = 1 / 4 i r u -0.750 307 320 332 -0.625 330 339 352 F_, = 16.667 l b f -0.500 352 357 371 1 -0.375 375 374 389 R m = -8.333 l b f -0.250 -0.125 431 431 386 387 401 402 l V = 0.536 in./min 0 386 372 387 t , = 2.38 min elapse 0.125 305 343 356 0.250 186 305 316 F = 12.3645 l b f 0.375 0.500 266 230 275 237 N = 36.5119 l b f 0.625 199 206 = 0.3386 0.750 0.875 175 155 180 159 •y, , = 0.5 i n . •'theory 1.000 139 142 + y . =0.7 i n . 1 theory -1.000 0.734375+ 266 275 -0.875 260 276 286 -0.750 282 287 297 -0.625 305 296 307 -0.500 323 304 316 -0.375 341 309 321 -0.250 386 309 321 -0.125 377 304 315 0 341 291 302 0.125 256 272 281 0.250 147 249 257 0.375 224 232 0.500 201 207 0.625 180 185 0.750 161 166 0.875 146 149 1.000 133 136 TABLE 6 SUMMARY OF FORCES M a t e r i a l F T ( l b f ) R T ( l b f ) N ( l b f ) F ( l b f ) F r i c t i o n C o e f f i -c i e n t p W ^ U ( l b f 1 / 2 f t / s e c ) F r i c t i o n C o e f f i -c i e n t p from L i t from L i t . U from L i t . ( f t / s e c ) 1/8" 13 -6 28 10 0 .4 1445 0.07[13]+ 1250 £ 650 M i l d 11 -5 26 10 0 .4 1525 0 . 1 2 [ 2 3 ] x 980 150 S t e e l 1/4" M i l d 57 -21 93 11 0.1 2730 0 .14[23] 980 150 S t e e l 66 -25 107 12 0.1 2920 0 .09[ 9]* 980 300 1/4" TI 29 -12 55 12 0.2 2080 S tee l 27 -11 51 13 0.2 1995 28 -11 51 10 0.2- 2020 28 -11 51 12 0 .2 2030 25 - 9 45 9 0 .2 1890 1/4" 10 - 2 22 8 0 .3 1335 0 .2 [ 9 ] ° 300 Brass 10 - 2 21 7 0 .3 1300 1/4" T1- 17 - 8 37 13 0.4 1725 6A1-4V 17 - 8 37 12 0 .3 1705 +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 AXIS: FUSION CUTTING Feed speed = 2 i n c h e s per minute Temperature (°F) Brass M i l d S t e e l Ti-6A1-4V ( i n . ) (h=4.5B/h/ft 2/°F) (h=7.54B/h/ft 2/°F) (h=4.5B/h/ft 2/°F) -1.000 1080 1133 759 -0.875 1104 1187 823 -0.750 1130 1250 899 -0.625 1161 1323 991 -0.500 1198 1411 1106 -0.375 1244 1524 1256 -0.250 1308 1684 1472 -0.125 1415 1953 1843 0 1750 2800 3035 0.125 1413 1945 1820 0.250 1305 1671 1436 0.375 1239 1507 1211 0.500 1191 1390 1053 0.625 1153 1298 933 0.750 1121 1222 837 0.875 1093 1157 758 1.000 1068 1099 692 F r i c t i o n f o r c e (1 D f ) 14.450 22.357 22.261 F r i c t i o n i n t o Workpiec 2 0.096127 0.059668 0.027451 136 TABLE 8 PHYSICAL PROPERTIES OF MATERIALS USED IN THE EXPERIMENTS Thermal Con- Thermal D i f - S p e c i f i c D e n s i t y Com-M a t e r i a l d u c t i v i t y f u s i v i t y Heat C l b / i n 3 ) p o s i -(B/h/ft/°F) ( f t 2 / h ) (B/lbm/°F) t i o n B r a s s 70.0 1.490 0.090 0.308 M i l d Stee: 26.0 0.480 0.110 0.284 0.20 t o 0.25% C B a l Fe T l S t e e l 19.10 0.272 0.120 0.313 o i - o-2£C Ti-6A1-4V 8.0 0.184 0.157 0.160 0-6-l-o%Mn s*i. 4 ° M o 6% A l , 4%V, B a l T i 137 TABLE 9 HAUPTMAN AND RAMSEY'S DISK SOLUTION VERSUS MODIFIED YU'S Ma t e r i a l Feed V e l o c i t y (ipm) Heat Generated Heat" Generated Average Cutting F r i c t i o n Forces ^0 (BTU/HR) x l 0 ~ 5 4A (BTU/HR) xlO-5 going to work- Temperature F 0 (l b f ) FA (lb f ) piece . (°F) fo fA Brass 0.25 *MY 0.22282 0.21438 0.24252 0.25206 1666.531 17.076 16.429 +HR 0.22816 0.21979 0.23684 0.24586 1666.531 17.485 16.844 0.50 MY 0.22294 0.21450 0.24294 0.25249 1666.508 17.085 16.438 HR 0.22828 0.21991 0.23725 0.24629 1666.508 17.494 16.853 0.75 MY 0.22307 0.21463 0.24339 0.25296 1666.470 17.095 16.448 HR 0.22841 0.22004 0.23770 0.24675 1666.470 17.505 16.863 1.00 MY 0.22322 0.21477 0.24387 0.25347 1666.417 17.106 16.459 HR 0.22856 0.22018 0.23818 0.24724 1666.417 17.516 16.873 1.25 MY 0.22337 0.21492 0.24439 0.25401 1666.347 17.118 16.470 HR 0.22871 0.22032 0.23869 0.24777 1666.347 17.527 16.884 1.50 MY 0.22353 0.21507 0.24495 0.25458 1666.260 17.131 16.482 HR 0.22887 0.22048 0.23923 0.24834 1666.260 17.540 16.896 M i l d 0.25 MY 0.30708 0.29359 0.10479 0.10961 2666.480 23.533 22.499 Steel HR 0.31578 0.30240 0.10191 0.10641 2666.480 24.200 23.174 0.50 MY 0.30733 0.29379 0.10552 0.11039 2666.061 23.552 22.515 HR 0.31603 0.30261 0.10262 0.10717 2666.061 24.219 23.190 0.75 MY 0.30765 0.29404 0.10644 0.11136 2665.378 23.576 22.534 HR 0.31634 0.30285 0.10351 0.10813 2665.378 24.243 23.209 1.00 MY 0.30802 0.29432 0.10752 0.11252 2664.473 23.605 22.555 HR 0.31672 0.30313 0.10457 0.10925 2664.473 24.271 23.230 1.25 MY 0.30844 0.29463 0.10874 0.11383 2663.369 23.637 22.579 HR 0.31714 0.30343 0.10576 0.11053 2663.369 24.304 23.254 1.50 MY 0.30890 0.29496 0.11007 0.11527 2662.095 23.673 22.605 HR 0.31760 0.30376 0.10706 0.11194 2662.095 24.339 23.279 *MY = Modified Yu's +HR - Hauptmann & Ramsey's TABLE 9 (Cont'd) M a t e r i a l Feed V e l o c i t y (ipm) Heat Generated Heat Generated Average Cutting F r i c t i o n Forces % (BTU/HR) x l O - 5 ^A (BTU/HR) x l O " 5 going to work- Temperature FO (l b f ) FA Cibf) piece ( °F) fn fA Ti - 6 A 1 -4V 0.25 "MY 0.32909 0.31629 0.09249 0.09623 2908.425 25.220 24.239 +HR 0.33854 0.32593 0.08991 0.09338 2908.425 25.944 24.978 0.50 MY 0.32950 0.31659 0.09364 0.09745 2907.254 25.252 24.262 HR 0.33895 0.32623 0.09103 0.09458 2907.254 25.976 25.001 0.75 MY 0.33007 0.31698 0.09520 0.09913 2905.466 25.295 24.292 HR 0.33952 0.32661 0.09255 0.09621 2905.466 26.019 25.030 1.00 MY 0.33075 0.31743 0.09706 0.10113 2903.196 25.347 24.326 HR 0.34020 0.32705 0.09436 0.09816 2903.196 26.071 25.064 1.25 MY 0.33151 0.31793 0.09911 0.10334 2900.644 25.405 24.364 HR 0.34096 0.32754 0.09636 0.11031 2900.644 26.129 25.101 1.50 MY 0.33230 0.31845 0.10126 0.10566 2897.936 25.466 24.404 HR 0.34175 0.32805 0.09846 0.10257 2897.936 26.190 25.140 *MY - Modified Yu's +HR - Hauptmann & Ramsey's CO . • - -- •  140 i F i g u r e 1 T h i n D i s k w i t h Heat Input Figure 2 F l a t Plate Configuration with Heat Input Plane Source. 0 *3' - i i i Figure 3 Plate with F i n i t e Width and a Plane Heat Source gure 4 Plate with F i n i t e Width and a Plane Heat Source along the Breadth of the Plate 144 (b) Figure 5(a) Jaeger's Configuration (b) Ling and Saibel's Configuration A i > A P T o R . PUSH R«>r> SAWIN<S p>! SK MOTOR <=-AR£iACse S H E A R PINJ P O W E R D E V I C E & P E E D R E D U C E R . TRANSMISSION V A R I A B L E SPm P V I U C Y D C . M O T O R SYSTEM I-* Figure 6 Schematic Drawing of the General Arrangement i n F r i c t i o n Sawing Process I T 146 B E L T P U L . L . E Y / UPPER GUARD STAND 1 iTTt, U P P E R . G U A R D A «. i^t s p / i o » i _ e rm Tar L E F T H T < W D sioe C _ O C K - T A e . L . E T A B L E SUPPOilT C O L L A R S ttouo S I D E F i g u r e 7 Sawing Disk and T a b l e F i g u r e 8 G e n e r a l Arrangement of Eguipment and I n s t r u m e n t a t i o n Figure 9 Guide Blocks 149 Figure 10 Workpiece and Workpiece Mounting Device 150 F i g u r e 11 Mounted Workpiece 151 Figure 12 View Showing M o d i f i e d Push Rod Head 152 For Table Support1- S/rcr/h Ring ih'S pari is fhreodecf Materia/: MifJ vT/ee./ F i g u r e 13 S t r a i n Ring gure 14 View Showing Arrangement Force Measuring I n s t r u m e n t a t i o n L i a r i T P i P t s B C i ose. N^l £ T £ R ( B A M ) B^usH T W O -criANwet-® <§> ® d (§> <i) <s> © & & 7 cn Figure 15 Block Diagram of Temperature Measuring Instrumentation Figure 16(a) Thermocouple C i r c u i t 156 F i g u r e 16(b) Thermocouple P o s i t i o n s d u r i n g Expe r imen ta l Run 157 F i g u r e 17 Debr i s from F r i c t i o n Cut Leaded Brass 158 Figure 18 Microstructure from Leaded Brass (a) 'As Received 1, 400 X (b) F r i c t i o n Cut Edge; 400 X 159 •M4 4 -—i L4w 4 Mr j r -4— —_ t • _ E; — —i ,..„ — I — -_— — 1  — •—• — —h _ — ] 1.  i —J—f-~J^ L B R U S H — 7-1 — — — r -11 1 I-"1 I —• L -L _ — • _ — — . . . . . — — — i 1 Figure 19 Light Pipe Response i n an Edge Cut 160 Figure 20 Worms from F r i c t i o n Cutting of Steel (a) Edge View of Workpiece showing "worms" (b) Close-up View of Workpiece Bottom (c) Close-up View of Workpiece Top W O R K P I E X E p b - S * Ceuu S E N S O R Figure 21(a) Arrangement i n Edge Cut 162 F i g u r e 2Kb) Close-Up Views of "Worms" from Edge Cut of M i l d S t e e l 163 Figure 22 Ty p i c a l Table and Thrust Forces during a Test Run Figure 23 Thermocouple Response Corresponding to the Forces i n Fiqure 22 164 F i g u r e 2 5 M i c r o s t r u c t u r e of Worm on Top of M i l d S t e e l Workpiece 400X 165 (a) F i g u r e 2 6 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 S a w i n g M i l d S t e e l Edge V i e w ( 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 ° o x 166 F i g u r e 27 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 M i l d S t e e l 400X - Breadth View (a) C l o s e to 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 Figure 28 Microstructure of Material Close to the F r i c t i o n Cut Edge of Mild Steel, 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 Kerf M a t e r i a l from Sawing T l S t e e l 400X - Breadth View (a) C l o s e to One Edge (b) C l o s e to Other Edge F i g u r e 33 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 from Sawing TI S t e e l , 400X - Edge View F i g u r e 34 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 TI S t e e l , 400X F i g u r e 35 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 36 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 Microstructure of Kerf Material from Sawing Ti-6A1-4V A l l o y - Edge View (a) 400 X (b) 800 X 173 Figure 38 Microstructure of 'As Received' Ti-6A1-4V Al l o y Heated with Oxy-Acetylene Torch to Self-Burning; Allowed to Cool i n A i r (a) 400 X (b) 800 X 900 800 700 I 600 LU D I— < ••500 LU a, LU 4 0 0 300 THEORY EXPERIMENT y o oo in • -188 • -375 • -875 AV. TEMR MATCH 2 0 0 L- MAX- TEMR MATCH 1 1 1 1 100" -•75 - • 5 0 - 2 5 0 -25 -50 DISTANCE ALONG CUTTING A X I S - i n F i g u r e 39 Temperature D i s t r i b u t i o n i n B r a s s 175 -1-0 - 7 5 - -50 --25 0 -25 -50 DISTANCE ALONG CUTTING A X I S - i n Figure 40 Temperature D i s t r i b u t i o n i n 1/8" Plate Mild Steel i 1 r o 1 i i I l I I - 7 5 - - 5 0 —25 0 -25 -50 75 DISTANCE ALONG CUTTING A X I S - in F i g u r e 41 Temperature 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 177 - 7 5 " -50 -.2 5 0 -25 -50 '75 DISTANCE ALONG CUTTING A X I S — in F i g u r e 42 Temperature D i s t r i b u t i o n i n TI S t e e l 0 "75 --50 -25 0 -25 D I S T A N C E A L O N G C U T T I N G A X I S — F i g u r e 43 Temperature D i s t r i b u t i o n i n T I S t e e l 179 0,1 1 = 1 1 1 I I 10 - 7 5 --50 - 2 5 0 -25 -50 DISTANCE ALONG CUTTING AXIS— in F i g u r e 44 Temperature D i s t r i b u t i o n i n TI S t e e l T I 1 T -10 -75 --50 -25 0 -25 -50 DISTANCE ALONG CUTTING A X I S — in F i g u r e 45 Temperature D i s t r i b u t i o n i n Ti-6A1-4V F i g u r e 46 Close-Up View of Kerf M a t e r i a l from M i l d S t e e l 182 Figure 47 P r o f i l e of Wear Mark Produced on Copper. Horizontal Magnification 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 of s l i d i n g . The Displacement of Metal i s C l e a r l y V i s i b l e . [56, p. 459] A B l l M 1 Figure 48 Schematic representation of the process of Cutting with a Saw. [42] F i g u r e I I - l ( a ) T a b l e F o r c e s (b) Workpiece F o r c e s F i g u r e iv-1 S k e t c h o f R o t a t i n g Disk w i t h Edge Load 185 F i g u r e IV-2 E q u i v a l e n c e o f F i g u r e IV-1 ! 186 J F i g u r e IV-3 S o l i d Disk w i t h Edge Load 187 Figure IV-4 Configuration Required to Modify Figure IV-3 to Get S o l u t i o n to Figure 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 F i g u r e V - l Arrangement f o r C a l i b r a t i n g T able S t r a i n Ring F i g u r e VI-1 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 

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