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Mechanics and dynamics of ballend milling Lee, Peter Pak Wing 1995

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M E C H A N I C S A N D D Y N A M I C S O F B A L L E N D M I L L I N G By Peter Pak Wing Lee B. A. Sc. (Mechanical Engineering) University of British Columbia , 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R O F A P P L I E D S C I E N C E in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the requirecbstandard THE UNIVERSITY OF BRITISH COLUMBIA August 1995 © Peter Pak Wing Lee, 1995 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Vl&fUMcftL k v l l l f < « £ U G , The University of British Columbia Vancouver, Canada Date OctoegR Ik .199b DE-6 (2/88) Abstract B a l l end mil l ing has been used extensively in current manufacturing industry in pro-ducing parts with sculptured surfaces. Due to its complex cutter geometry, ball end mi l l ing mechanics and dynamics have not been studied until recently. In this research, the mechanics and dynamics of cutting with a special helical ball end cutter are mod-eled. A unified mathematical model, which considers the true rigid body kinematics of mi l l ing , static deformations and forced and self excited vibrations, is presented. The ball end m i l l attached to the spindle is modeled by two orthogonal structural modes in the feed and normal directions at the tool tip. For a given cutter geometry, the process dependent cutting coefficients are obtained by applying oblique tool geometry to the fundamental properties such as shear yield stress, shear angle and average friction angle measured from orthogonal cutting tests. The three dimensional surface finish generated by the helical flutes is digitized using the true kinematics of ball end mil l ing process. The dynamically regenerated chip thickness, which consists of rigid body motion of the cutter and structural vibrations, is evaluated at discrete time intervals by comparing the present and previous tooth marks left on the finish surface. The process is simulated in time domain, by considering the instantaneous regenerative chip load, local cutting force coefficients, structural transfer functions and the geometry of ball end mi l l ing process. The proposed model predicts cutting forces, finished surface and chatter-free condition charts, and is verified experimentally under both static and dynamic cutting conditions. The model allows process planners to select cutting conditions to minimize dimensional surface errors, shank failure and chatter vibrations for end mil l ing operations. 11 Table of Contents Abstract ii Table of Contents iii List of Tables vi List of Figures vii Acknowledgement xi 1 Introduction 1 2 Literature Review 4 2.1 Overview 4 2.2. M i l l i n g Force Models 7 2.2.1 Mechanistic Model 9 2.2.2 Mechanics of Mi l l i ng Model 12 2.3 Dynamic Cut t ing 13 3 Geometric Modelling of Mill ing Cutters 17 3.1 Introduction 17 3.1.1 Modell ing of Cyl indrical End M i l l 18 3.1.2 Modell ing of Ballend/Bullriose M i l l 20 3.1.3 Modell ing of Tapered Ballend M i l l 24 3.1.4 Cut t ing Geometry and Chip Thickness Calculation . . . . . . . . 25 iii 3.2 Conclusions 28 4 Modell ing of Mill ing Forces 30 4.1 Introduction 30 4.2 Milling Force Models 32 4.2.1 Mechanistic approach with Average Force Model 32 4.2.2 Mechanics of Milling Model 36 4.2.3 Prediction of Cutting Force Coefficients from an Oblique Cutting Model 42 4.2.4 Simulation and Experimental Verification 53 4.3 Conclusions 58 5 Dynamic Mill ing and Chatter Stability 63 5.1 Introduction 63 5.2 Modelling of Dynamic Milling 68 5.2.1 Dynamic Milling Implementation 69 5.2.2 Structural Dynamic Model 74 5.3 Chatter Stability 77 5.3.1 Prediction of Chatter Stability from Time Domain Simulation . . 80 5.4 Simulations and Experimental Verification 86 5.4.1 Verifications with Cylindrical End Mill 87 5.4.2 Verifications with Ballend M i l l 101 5.4.3 Summary 138 5.5 Conclusions 138 Conclusions 140 Bibliography 143 iv Appendices 147 A Ballend Milling Force Coefficients 147 List of Tables 4.1 Orthogonal cutting data 51 4.2 Cut t ing conditions for static ballend tests and simulations 55 5.1 Time Domain Simulation Conditions for Dynamic Bal l End M i l l i n g Tests I 119 5.2 Cut t ing Conditions for Dynamic Bal l End Mi l l i ng Tests 11 120 A.1 Edge Force Coefficients, Rake = 0 degree . . . 148 A.2 Cut t ing Force Coefficients, Rake = 0 degree 148 A.3 Results from Orthogonal Cutt ing Experiments 149 v i List of Figures 2.1 Common mil l ing geometries: a) Up mill ing, b) Down mil l ing 5 2.2 a) Orthogonal cutting geometry, b) Oblique cutting geometry 8 2.3 Force diagram in chip formation 10 2.4 Dynamic Cut t ing and Wave Regeneration in Orthogonal Cut t ing 14 3.1 a) Cyl indr ica l End M i l l , b) Ballend M i l l , c) Bullnose M i l l , d) Tapered Ballend M i l l 19 3.2 a-d Geometric model of Ballend cutter 22 3.3 Geometric parameters along ballend cutter 23 3.4 Common mil l ing geometry and chip formation 27 3.5 Chip Thickness Profile at two cutter locations, solid line - approximation, dotted line - exact kinematics 29 4.1 Average Forces vs Feed rate for different axial depth of cuts in ballend mil l ing 37 4.2 Average Edge Forces vs axial depth of cut in ballend mil l ing 38 4.3 Ballend M i l l i n g Force coefficients estimated from Average Force Model : a) vs st} b) vs a 39 4.4 Oblique Cut t ing Geometry of a Ballend M i l l 41 4.5 Experimental Setup - Orthogonal Turning test 45 4.6 Edge Force Extrapolation from Orthogonal Turning test 47 4.7 Cut t ing Ratio identified from Orthogonal Turning test 48 4.8 Shear Stress identified from Orthogonal Turning test 49 vii 4.9 Frict ion angle identified from Orthogonal Turning test 50 4.10 Generalized Mechanics of Cutt ing Approach to Mi l l i ng Force Predict ion . 52 4.11 Experimental Setup for Mi l l i ng Test 54 4.12 Statistical Evaluation of Model and Data 57 4.13 Measured and predicted cutting forces for slot cutting tests, ballend m i l l , spindle speed 269rev/rmn, R0 = 9.525mm, z 0 = 30°, st = 0.0508[mm///ute] (a)- a — 1.27 [mm] (b)- a — 6.35[mm], rake=0 59 4.14 Measured and predicted cutting forces for slot cutting tests, ballend m i l l , spindle speed 269rev/min, i0 — 30°. (a) a=3.81 [mm], i? 0=9.525 m m , s (=0.1016[mm/flute], rake=5 (b) a=3.048 [mmj, # 0=6.35 m m , s f=0.0762 [mm/flute], rake^lO 60 4.15 Measured and predicted cutting forces for half radial immersion cutting tests, ballend m i l l , aTL = 0°, a=6.35mm, spindle speed 269rev/rnin,io = 30°, (a) 5 t=0.0508 [mm/flute], (b)- a t=0.102 [mm/flute] 61 4.16 Measured and predicted cutting forces for slot mil l ing with a 3 fluted bal-lend m i l l , a = 8.9mm, i 0 = 30°, (a) spindle speed 480rew/mm, an = 15°, 5 t=0.0889 [mm/flute], (b) spindle speed Q15rev/min, an = 0°, ^ t=0.127 [mm/flute] 62 5.1 Schematic Representation of a S D O F System 65 5.2 Frequency Domain characteristic curve for a S D O F Dynamic System . . 67 5.3 Evaluation of chip thickness from the dynamically generated surfaces at present and previous tooth periods 70 5.4 Dynamic model and Chip thickness regeneration mechanism in mi l l ing . 73 5.5 Analyt ica l and time domain stability limit predictions for a case analyzed by Smith and Tlusty 78 V111 5.6 Phase difference between the current and previous surfaces resulting from tool vibration 79 5.7 T ime Domain Stability Limi t Calculation Algori thm 82 5.8 Experimental and Simulated Cutt ing Forces, cylindrical end mi l l ing . . . 88 5.9 Stability Lobe Diagram for Cylindrical End mil l ing. Week's Experiments 89 5.10 Cut t ing Force Simulation I, cylindrical cutter, Week's experiments . . . . 91 5.11 Vibrations Simulation I, cylindrical cutter, Week's experiments 92 5.12 Peak To Peak Graphs from Time Domain Simulation I, cylindrical cutter. Week's Experiments 94 5.13 Cut t ing Forces and Vibrations Simulation II, Week's Experiments . . . . 95 5.14 Fourier Spectrums from Simulations II, Week's Experiments 96 5.15 Simulated Surface under Chatter Condition, from Simulations II 97 5.16 Simulated Surface Mark at z = 2.0mm under Chatter Condit ion, from Simulations II . . . 98 5.17 Analyt ica l and Time Domain Predicted Stability Lobes, cylindrical cutter 100 5.18 Model Verification I - Chip Thickness Profile 102 5.19 Model Verification Tl - Cutt ing Force Patterns 103 5.20 Model Verification III - Surface Feed Marks: a) * = 0° (up mil l ing side), b) * = 90° (down mill ing side) 105 5.21 Experimental Setup - Dynamic Test 107 5.22 Measured Transfer Function at Tool T i p , Ballend M i l l i n g Test I 108 5.23 Static Stiffness Measurements, Ballend Mi l l i ng Test I 109 5.24 Measured Workpiece Transfer Function, Ballend M i l l i n g Test I 110 5.25 Dynamic Ballend Mi l l i ng Test 1, Speed = 115-430 R P M I l l 5.26 Dynamic Ballend M i l l i n g Test I, Speed = 730-1450 R P M 112 5.27 Fourier Spectrum of F x from Ballend Mi l l i ng Test I 113 ix 5.28 Measured and Simulated Cutting Forces, N=115 R P M , Ballend Milling Test I 115 5.29 Fourier Spectrum from Ballend Milling Test I, N=115 R P M 116 5.30 Measured and Simulated Cutting Forces, N=1000 R P M , Ballend Milling Test I 117 5.31 Fourier Spectrum from Ballend Milling Test I, N=1100 R P M 118 5.32 Transfer Function Measured at tool tip. Ballend Test II 121 5.33 Ballend Milling Experiments II, a = 2mm 122 5.34 Ballend Milling Experiments II, a = 3mm 123 5.35 Experimental and Simulated P T P results. Ballend Milling Test II . . . . 126 5.36 Simulated Stability Lobe Diagram for Ballend milling Test II 127 5.37 Simulated Cutting Forces and Vibrations. Ballend Milling Test II . . . . 128 5.38 Simulated Cutting Forces and Vibrations. Ballend Milling Test II . . . . 129 5.39 Wave generation on different cutting geometries. Ballend Milling 130 5.40 Simulated Stability Lobe Diagram for Ballend milling III 131 5.41 Simulations showing the waviness on cut surface at different cutting speeds 132 5.42 Simulations of cutting forces at 2 different speeds, Ballend milling . . . . 134 5.43 Simulations of vibrations at 2 different speeds, Ballend milling 135 5.44 F F T of the simulated cutting forces at different speeds, Ballend milling . 136 5.45 Simulations of finished surfaces at different speeds, Ballend milling . . . . 137 x Acknowledgement I would like to express my sincere gratitude to my research supervisor Dr. Yusuf Altmta§, who provided me with valuable instructions and supports throughout the course of my graduate work. I would also like to extend special thanks to Dr. Erhan Budak, as well as to all my colleagues in the Manufacturing research group at U B C , for their patience, assistances and more importantly, their precious friendships. I am thankful to my girl friend, Phoebe, and my brothers and sisters in Thy family, for their continuous supports and encouragements during the toughest time of my research. My deepest gratitude is extended to my family, who are always behind me and have confident in me. I dedicate this work to them. "The fear of the LORD is the beginning of wisdom, and knowledge of the Holy One is understanding." Proverbs 9:10 xi Nomenclature a a x i a l d e p t h of cut a;im l i m i t i n g a x i a l d e p t h of cut for chat ter s t a b i l i t y o.xx,o,yy d i r e c t i o n a l d y n a m i c m i l l i n g coefficients b r a d i a l d e p t h of cut db di f ferential c u t t i n g edge l e n g t h i n the d i r e c t i o n p e r p e n d i c u l a r to the c u t t i n g v e l o c i t y dz di f ferential height i n a x i a l d i r e c t i o n dFtj, dFTj, dFaj d i f ferential c u t t i n g forces i n t a n g e n t i a l , r a d i a l , a n d a x i a l d i rect ions for t o o t h j h u n c u t chip thickness n o r m a l to c u t t i n g edge i n m i l l i n g io h e l i x angle at b a l l s h a p e d flute a n d shank m e e t i n g p o i n t l o c a l h e l i x angle on the flute rt c u t t i n g chip r a t i o in o r t h o g o n a l c u t t i n g st feed per t o o t h t,tc u n c u t a n d cut chip thickness i n o r t h o g o n a l c u t t i n g x i i d i r e c t i o n a l d y n a m i c m i l l i n g coefficient m a t r i x Fc, Ff c u t t i n g a n d feed force i n o r t h o g o n a l c u t t i n g FXJ) Fyj, Fzj m i l l i n g forces i n C a r t e s i a n coordinates on flute j Gx,Gy Transfer funct ions i n x a n d y d i r e c t i o n s , l u m p e d at t of c u t t e r Ktc, KTC-, Kac t a n g e n t i a l , r a d i a l , a n d a x i a l c u t t i n g force coefficients m i l l i n g KteyKTe,Kae t a n g e n t i a l , r a d i a l , a n d a x i a l edge force coefficients m i l l i n g Kz n u m b e r of a x i a l e lements Nf n u m b e r of flutes N s p i n d l e speed i n r e v o l u t i o n per m i n u t e Ro b a l l radius R(ip) t o o l radius in x-y plane at a p o i n t defined by ip x m a r , a n r a d i a l a n d n o r m a l rake angles e phase between succesive waves i n chatter v i b r a t i o n s 7 flank c learance angle K, angle i n a v e r t i c a l p lane between a p o i n t o n the flute a n d the z axis <f),/3 shear a n d average f r i c t i o n angles i n o r t h o g o n a l c u t t i n g 4>n,f3n n o r m a l shear a n d n o r m a l f r i c t i o n angles i n o b l i q u e c u t t i n g rjc c h i p flow angle o n the rake face UJC chat ter frequency ip lag angle between the t i p (z=0) a n d a p o i n t o n the h e l i c a l flute at height z ipo m a x i m u m lag angle between the tip(z=0) a n d u p p e r m o s t c u t t i n g p o i n t ( z = a ) (f)p p i t c h angle,of the cut ter ( = 2TT/Nf) T shear stress at the shear plane 6 t o o l r o t a t i o n angle, measured f r o m +ve y-axis cw \F/ l o c a l i m m e r s i o n angle in g loba l coordinates , m e a s u r e d f r o m + y axis ( C W ) Q, s p i n d l e speed i n r a d i a n per second xiv Chapter 1 Introduction M i l l i n g r e m a i n s one of the most c o m m o n m a c h i n i n g o p e r a t i o n s in m a n u f a c t u r -i n g i n d u s t r y m a i n l y due to its f l e x i b i l i t y in p r o d u c i n g a wide range of shapes such as s t a m p i n g dies a n d pockets found in die and m o l d i n d u s t r y , h i g h p e r f o r m a n c e parts en-c o u n t e r e d i n aerospace c o m p o n e n t s , a n d compressor i m p e l l e r s . C u r r e n t m a n u f a c t u r i n g technologies have focused on increas ing the p r o d u c t i v i t y a n d the q u a l i t y of the f in ished p a r t s w h i l e m a i n t a i n i n g a sufficient process s t a b i l i t y a n d r e l i a b i l i t y . T h e m o s t c o m m o n m e t h o d s i n v o l v e se lect ing conservat ive c u t t i n g c o n d i t i o n s , d e t e r m i n i n g the a p p r o p r i a t e m a c h i n i n g c o n d i t i o n s by the t r i a l and error m e t h o d s , or e m p l o y i n g m u l t i p l e f i n i s h i n g cuts to achieve bet ter accuracy, w h i c h result i n lower p r o d u c t i v i t y a n d h i g h e r cost. Sat-i s factory p r o d u c t i o n levels c a n be achieved if the process layout considers a l l aspects of t h e m a c h i n i n g o p e r a t i o n s , such as the pre-specif ied surface to lerance , m a c h i n e t o o l v i -b r a t i o n s , a n d t o o l breakage. F u r t h e r m o r e , these issues are b e c o m i n g m o r e c r i t i c a l d u e t o the recent advances i n h i g h speed m i l l i n g technology e m p l o y e d i n die a n d m o l d m a k i n g i n d u s t r y . T h e d e v e l o p m e n t of re l iable m i l l i n g process m e t h o d s is therefore i m p o r t a n t for i n c r e a s i n g m e t a l r e m o v a l rates w h i l e m a i n t a i n i n g the process s t a b i l i t y a n d the r e q u i r e d accuracy . T h i s s t u d y a i m s at a n a l y z i n g the physics of a m a c h i n i n g process, i n p a r t i c u l a r , t h e m e c h a n i c s a n d d y n a m i c s of m i l l i n g w i t h b a l l e n d m i l l s used m a i n l y i n p r o d u c i n g scu lp-t u r e d surfaces i n die a n d m o l d a n d aerospace i n d u s t r y . T w o key p r o b l e m s assoc ia ted w i t h th is process, the c u t t i n g force p r e d i c t i o n a n d chat ter s t a b i l i t y analys is i n b a l l e n d 1 Chapter 1. Introduction 2 m i l l i n g , are s t u d i e d i n the f o l l o w i n g chapters. A c o m p r e h e n s i v e a n d i m p r o v e d m i l l i n g force m o d e l , based o n the exact k i n e m a t i c s m o d e l presented b y M o n t g o m e r y [1], is ap-p l i e d to the b a l l e n d m i l l i n g for the s i m u l a t i o n of the process. T h e m o d e l p r e d i c t s the m i l l i n g forces, d i m e n s i o n a l surface f in ish, a n d chatter-free m a c h i n i n g c o n d i t i o n s u s i n g shear stress, shear angle a n d f r i c t i o n coefficient of the c u t t i n g process, o b l i q u e t o o l g e o m -e t r y a l o n g the h e l i c a l b a l l ended flutes a n d s t r u c t u r a l d y n a m i c p a r a m e t e r s of t h e m a c h i n e t o o l s y s t e m . H e n c e f o r t h , the thesis is organized as follows: C h a p t e r 2 provides necessary b a c k g r o u n d a n d l i t e r a t u r e rev iew on m i l l i n g a n a l y s i s . T h e f u n d a m e n t a l s of m e t a l c u t t i n g analys is , o r t h o g o n a l a n d o b l i q u e c u t t i n g m e c h a n i c s , g e o m e t r y of m i l l i n g , past m o d e l s for the p r e d i c t i o n of c u t t i n g forces, v i b r a t i o n s a n d m a c h i n e t o o l chat ter , are briefly reviewed. C h a p t e r 3 presents the geometr ic m o d e l s of several c o m m o n m i l l i n g cut ters . T h e g e o m e t r i c m o d e l s of c y l i n d r i c a l , b u l l nose, bal l end m i l l s a n d t a p e r b a l l e n d m i l l s are d e v e l o p e d . T h e m e c h a n i c s of m i l l i n g , i n p a r t i c u l a r , w i t h b a l l e n d cut ters , are i n v e s t i g a t e d i n C h a p t e r 4. T w o different m i l l i n g force models , the m e c h a n i s t i c a n d the m e c h a n i c s of m i l l i n g m o d e l s , are presented. T h e m e c h a n i s t i c m o d e l requires d irect e s t i m a t i o n of cut-t i n g constants f r o m m i l l i n g e x p e r i m e n t s . T h e mechanics of m i l l i n g is based o n the o b l i q u e c u t t i n g m e c h a n i c s m o d e l , w h i c h requires the m i l l i n g c u t t e r geometry , the shear stress, shear angle a n d average f r i c t i o n coefficient for the p r e d i c t i o n of c u t t i n g forces. T h e m a t h -e m a t i c a l m o d e l s have been e x p e r i m e n t a l l y verif ied a long w i t h t h e i r s t a t i s t i c a l e v a l u a t i o n . I n C h a p t e r 5, the d y n a m i c s of the m i l l i n g process is m o d e l e d . T h e general m o d e l considers s t r u c t u r a l v i b r a t i o n s , chip thickness regenerat ion, exact r i g i d b o d y k i n e m a t i c s Chapter 1. Introduction 3 of m i l l i n g . T h e m e c h a n i c s m o d e l presented i n chapter 4 is i n t e g r a t e d to t h e d y n a m i c m o d e l . T h e c o m b i n e d m o d e l is able to p r e d i c t c u t t i n g forces, surface f i n i s h a n d v i b r a -t ions d u r i n g b a l l e n d m i l l i n g operat ions . T h e m o d e l p r e d i c t i o n s are c o m p a r e d w i t h the e x p e r i m e n t a l results . T h e thesis is c o n c l u d e d w i t h a short s u m m a r y of the c o n t r i b u t i o n s a n d f u t u r e research w o r k . C h a p t e r 2 L i t e r a t u r e R e v i e w 2 . 1 O v e r v i e w M i l l i n g is used extens ive ly i n the m a n u f a c t u r i n g i n d u s t r y where b o t h p r e c i s i o n a n d eff iciency are c r i t i c a l . In this process the c o m p o n e n t shape, size a n d its surface f i n i s h is generated by s y s t e m a t i c a l l y r e m o v i n g or ' c u t t i n g ' the excess m a t e r i a l f r o m the o r i g i n a l w o r k p i e c e b y a c u t t i n g t o o l . T h e m i l l i n g cut ter possesses a n u m b e r of c u t t i n g edges. It is p r o v i d e d w i t h a r o t a r y m o t i o n a n d the work is g r a d u a l l y fed. C h i p s are r e m o v e d by each c u t t i n g edge d u r i n g r e v o l u t i o n a n d a surface is p r o d u c e d . M i l l i n g is one of the m o s t i m p o r t a n t m a n u f a c t u r i n g processes because it can produce a w i d e v a r i e t y of c o m p o n e n t shapes a n d sizes w i t h h i g h d i m e n s i o n a l accuracy a n d g o o d surface f i n i s h , a n d it can be c o m p u t e r c o n t r o l l e d and a u t o m a t e d . In m a n y of these operat ions the c u t t i n g force is a n i m p o r t a n t p a r a m e t e r w i t h respect to e i ther the deflect ion ( d i m e n s i o n a l t o l e r a n c e ) a n d breakage of the c u t t e r , or the s t a b i l i t y of the process. M o r e o v e r , knowledge of t h e m a g n i -t u d e a n d v a r i a t i o n of the c u t t i n g force w i t h respect to the c u t t i n g c o n d i t i o n s is essent ia l for t h e process p l a n n i n g of m i l l i n g operat ions . C o m m o n m i l l i n g geometries i n c l u d e c o n v e n t i o n a l ( u p ) , c l i m b ( d o w n ) , a n d slot m i l l i n g . I n the u p m i l l i n g process (figure 2.1 a) , the chip thickness varies f r o m zero to m a x i m u m . I n d o w n m i l l i n g , the chip thickness starts f r o m the m a x i m u m a n d decreases t o zero at ex i t (f igure 2.1 b ) . D o w n m i l l i n g usual ly produces better surface f i n i s h since t h e s h e a r i n g 4 Figure 2.1: Common mil l ing geometries: a) Up mill ing, b) Down mi l l ing Chapter 2. Literature Review 6 process ends at the finished surface. H o w e v e r , the direct ions of the r e s u l t i n g c u t t i n g forces are o r i e n t a t e d i n such a way that the cut ter tends to be p u s h e d away f r o m t h e w o r k p i e c e , l e a v i n g e x t r a m a t e r i a l on the surface, i.e. u n d e r c u t t i n g . M o r e o v e r , d o w n m i l l i n g produces h igher shocks at the e n t r y of the cut due to t h e a b r u p t change i n c h i p t h i c k n e s s , thus r e q u i r i n g m o r e g r i p p i n g power f r o m b o t h the s p i n d l e a n d the c l a m p s . I n u p m i l l i n g , the shear ing a c t i o n begins at the f in ished surface, the c u t t e r is a lways p u s h e d i n towards the w o r k p i e c e , r e m o v i n g excess m a t e r i a l f r o m the desired surface d i m e n s i o n s , i .e. o v e r c u t t i n g . T h e m e c h a n i c s of m i l l i n g can be best u n d e r s t o o d by e x a m i n i n g the c h i p f o r m a t i o n d u r i n g the process. A n ear ly d e t a i l e d work by M a r t e l l o t t i [2] showed t h a t , due to the c o m b i n e d r o t a t i o n a l a n d l inear feeding m o t i o n s of the c u t t e r towards the w o r k p i e c e , t h e t r u e p a t h of a c u t t e r t o o t h is a n arc of a t r o c h o i d , ra ther t h a n a c i rc le , w h i c h c o m p l i c a t e s t h e m a t h e m a t i c s i n the analys is . However , M a r t e l l o t t i c l a i m e d t h a t , i n m o s t p r a c t i c a l c u t t i n g c o n d i t i o n s where the cut ter radius is much larger t h a n the feed per t o o t h , t h e c i r c u l a r t o o t h p a t h a s s u m p t i o n is v a l i d and the error i n t r o d u c e d is i n s i g n i f i c a n t . A s t h e c u t t e r rotates , the i m m e r s i o n angle ( v t ) varies, hence the c h i p load changes s i n u s o i d a l l y a n d g i v e n by M a r t e l l o t t i as: t = sLsin($) (2.1) where t = instantaneous c h i p load st = feed per t o o t h \1/ = i m m e r s i o n angle of c u t t i n g p o i n t Chapter 2. Literature Review 7 2.2 M i l l i n g Force Models The simplest form of machining operation is reviewed in this section. In the early 1940's Ernst and Merchant [3] presented a detailed analysis of a basic 2-D cutting process, often referred to as orthogonal cutting. Such an operation is realized when the cutting edge is straight and the relative velocity of the work and the tool is perpendicular to the cutting edge (figure 2.2 a). On the other hand, the term oblique cutting is used when the relative velocity of the work and the tool is not perpendicular to the cutting edge (figure 2.2 b). The similarity between orthogonal and oblique cutting and their applications to milling will be detailed in the later sections. Some basic assumptions proposed by Mer-chant's model are: 1. A Type 2 (continuous) chip is formed. 2. The deformation is two dimensional without side spread of the chip. 3. Sharp cutting edge, (no edge force) 4. The thin shear zone is idealized by a 'shear plane'. 5. The shear stress r in the 'shear plane' is uniform. Merchant's analysis, whose diagrammatic representation is shown in figure 2.3, was based on the equilibrium of the chip under the action of friction forces at the rake face and shear forces in the 'shearing zone'. The tool face, with a rake angle a and clearance angle 7, penetrates into the work material at an uncut chip thickness t .The material shears along the shear plane and results in a cut chip with thickness tc. The cutting ratio, r c , is defined as: rc = t/tc (2.2) The shear plane is inclined at an shear angle (f> and is determined by the rake angle er 2. Literature Review Figure 2.2: a) Orthogonal cutting geometry, b) Oblique cutting geometry Chapter 2. Literature Review 9 a r of the t oo l and by the f r i c t ion between the ch ip and the too l face. rccosar iancf) — : (2.3) 1 — rcsinaT T h e components of force found in the t o o l / c h i p in te rac t ion p lane are the shear force F3 and the n o r m a l force Fn. In prac t ice , these forces are resolved in to d i rec t ions p a r a l l e l (Fc) and pe rpend icu la r (Ff) to the work veloci ty , Fc —-FaCoscj) + Fnsin<j> Ft = — F.sind) + Fr,cos6 f * (2.4) F ~ — FcsinaT — FjcosaT N = —Fccosar -\- FfsinaT where the shear ing force is assumed to be p ropor t iona l to the ch ip area: F = Ktb (2.5) where K and b are specific c u t t i n g pressure and w i d t h of ch ip respect ive ly . T h e above general re la t ionsh ip between the c u t t i n g force and the ch ip area was fur ther e x p l o r e d and a p p l i e d by m a n y researchers i n m i l l i n g analys is . T h e i r analyses a t t e m p t e d to relate the forces and power i n m i l l i n g to the p r a c t i c a l var iables such as the t o o l geomet ry and cut geometry as we l l as work m a t e r i a l and de fo rmat ion proper t ies . T w o c o m m o n l y used m i l l i n g force models are exp la ined i n the fo l lowing : 2.2.1 Mechanistic Model A n ea r ly analys is of m i l l i n g forces p roduced by a he l i c a l flute was presented by K o e n i g s -berger et a l . [4, 5, 6]. T h e y re la ted the e lementa l ch ip load ing to r i g i d c u t t i n g forces ( in Figure 2.3: Force diagram in chip formation Chapter 2. Literature Review LL tangential and radial directions) through experimentally calibrated mil l ing force coeffi-cients KT and KR. In general, these coefficients are identified through empirical curve fitting technique and provide little physical insight in terms of the nature of the pro-cess. The cutting forces were expressed as a function of cutting pressure exerted on the instantaneous uncut chip area: F, = KT * a * stsin(ty) (2.6) Fr = Kft * Ft Later, other researchers employed the same model with improvements such as in-cluding the effect of run out and tool deflections [7, 8, 9, 10, 11]. Tlusty and M a c N e i l [12], Yellowley [13], Armarego and Budak et al. [14] further improved the accuracy in predicting the mi l l ing force coefficients for cylindrical helical end mills by introducing an edge force component to their linear model. The limitations of the mechanistic approach became apparent in recent research on mil l ing with complex cutters such as ballend mills which have variable geometry in the axial direction. As a result, Lirn and Feng [15] approximated the cutting force coefficients with a third degree polynomial which depends on the axial depth of cut. They then estimated the coefficients by iteratively curve fitting using average run-out values on the cutters. Similarly, Yucesan and Altintas [16] presented a semi-mechanistic, model which predicts the shear and friction load distribution on. the rake and flank faces of the helical bal l end m i l l flutes. They used a complex mathematical model and identified the mi l l ing force coefficients through an empirical curve fit to measure average mil l ing forces. The result is a set of cutter geometry and cutting condition dependent mi l l ing force coefficients, which limits the mechanistic model's applicability. W i t h the mechanistic model, prior evaluation of the milling force coefficients is neces-sary for any analysis and prediction. Moreover, a large amount of mil l ing tests is required Chapter 2. Literature Review 12 to be run at different feed rates, cutting speeds, and constant cutting geometry, and the results are applicable to a particular workpiece material-cutter pair only. Although the model allows quick calibration of existing cutters, it is not practical and useful in the case of complex cutter such as ball end mill due to its variable geometry such as helix and rake angles. 2.2.2 Mechanics of M i l l i n g M o d e l Armarego [14, 17, 18] was among a few researchers to apply the orthogonal cutting theory on milling force predictions. The basic idea is to represent the milling geometry by segments of oblique cutting processes. Based on Merchant's thin shear zone theory, Armarego determined the milling force coefficients for cylindrical helical end mills without calibration tests. From Merchant's theory, By measuring the forces in turning, Budak, Altintas and Armarego [19] identified the traditional material properties such as shear strength and friction from a set of standard orthogonal turning test. They showed an accurate method of transformation between the orthogonal cutting parameters (i.e. shear stress, shear and friction angles) and local milling force coefficients, taking into consideration the cutter geometry such as helix angle and rake angle. Yang and Park [20] took a similar approach and applied the orthogonal theory on ballend mills. They obtained the fundamental cutting parameters from orthogonal turning tests and simplified the analysis by approximating the oblique cutting on each ballend flute by infinitesimal orthogonal processes. Later they considered the variation in the chip loads due to static tool deflections. Tai and Fuh [21] used a similar cutting model but they represented the cutting edge geometry as intersections Chapter 2. Literature Review 13 between skew planes and spherical ball end mill surface. Ramarji [22) applied a similar theory on taper shaped cutter by obtaining the shear stress from the stress-strain curve. The mechanics of milling approach is more versatile to apply on cutters with complex geometries such as ballend and tapered ballend mills. Moreover, it eliminates the need for new calibration tests required for each new cutter geometry, it also gives more physical insight to the real process. 2.3 D y n a m i c C u t t i n g The milling models considered above deal with steady-state, static cutting, which is an ideal machining situation - where the structural vibrations during cutting are ignored. A cutting system is often subjected to some type of external dynamic forces. This forcing function can be time-dependent, harmonic, periodic, or random in nature. In dynamic milling, under some conditions, force induced vibrations may be inherent into the process, at the tooth passing frequency. For other conditions the vibration may cause the cutting process to vary so that it supplies a positive input of energy to maintain the vibration, which is known as self-induced or self-excited vibration in machining. Chatter is a self-excited type of vibration which occurs in metal cutting due to the lack of stiffness in the process, or can be best explained by a phenomenon called "regeneration of waviness". "'Regeneration of waviness' is a mechanism by which the input force is modulated by system dynamics so as to produce force variations and vibration." [23]. Figure 2.4 shows an orthogonal cutting process where the relative vibration between the tool and the workpiece produces a "wavy" machining surface therefore influences the uncut chip thickness variation at the chatter frequency. This undulation affects the cutting forces which in turn excite the structure to produce vibrations and modulating the chip thickness more. Chatter vibrations occur around the natural frequency of the Figure 2.4: Dynamic Cutt ing and Wave Regeneration in Orthogonal Cut t ing Chapter 2. Literature Review 15 most flexible part of the machine tool-workpiece structure which is closest to the cutting point. Chatter in machining requires in-depth analysis as the machine tool, the workpiece structure and the cutting process form a complex system. Severe vibrations in milling can lead to imperfections on the work surface as the tool impacts on the work and produces heavy marks or rough gouges. Chatter may also increase the rate of wear of the tool, the likelihood of machine-tool break down, and it may cause a high frequency sound which is unpleasant and can be physically harmful to nearby personnel. Some of the main factors affecting the occurrence of chatter include: the structure of the machine tool and workpiece, the overall system stiffness, damping, material hardness, tool geometry, and orientation of tool vibration modes, and cutting conditions such as speed, depth and width of cut. In general, chatter can be prevented by stiffening the relative compliance between the tool/workpiece, or by selecting conservative cutting width and depth of cuts, but at the expense of productivity. The use of non-uniform pitch cutters in chatter suppression was also subjected to many investigations [24, 25]. The main idea is to disturb the normal wave regeneration mechanism which leads to self-excitation chatter. Tobias [26, 27], and Tlusty [28] were among the few early researchers to study dynamic cutting. They explained the fundamental mechanism of chatter during turing and derived the stability theory which analytically predicts chatter free cutting conditions. Their stability theory was based on the orthogonal cutting model where the directions of cutting forces and the structural dynamics of the tool workpiece system were assumed constant. It was later adopted by many investigators and resulted in more understanding in chatter prediction and avoidance. Chatter in milling, however, is different from turning in which the cutter rotates with Chapter 2. Literature Review 16 respect to the structure/workpiece system. The directions of the cutting force compo-nents and chip thickness are no longer constant but changed with respect to the cutter's coordinate, hindering the direct application of the time invariant chatter stability theory developed in turning. Sridhar [29] presented a mil l ing stability model which requires nu-merical solution of the mil l ing equation. Later Minis [30] proposed an improved method of solving the stability equation analytically by iteration. Recently, Budak et al. [31] have solved the chatter stability in milling for helical end mills. They represented the time varying mil l ing force coefficients in Fourier series and were able to solve the stability equation analytically and more efficiently, taking into account the change in the mi l l ing force directions. However, the applicability of their theory has yet to be justified with other complex cutters such as ballend mills. A l l of the above analytical models target towards a "chatter stability chart" which allows chatter-free cutting speed, axial and radial depth of cuts to be selected accordingly. Tlusty et al. [32, 33] also took a different approach in understanding the dynamics and the stability of the process by introducing the time domain simulations of chatter vibrations in mi l l ing . They simulated the real machining process numerically by considering the speed dependent process damping, structural dynamics, and the nonlinearities signified by the tool jumping in and out of the workpiece due to excessive vibrations which cannot be modelled in the analytical stability theory. Altintas and Montgomery [1] presented an improved simulation model, including the true kinematics of the tool and workpiece motions and the tool interference with the workpiece. In general, the time domain chatter simulation provides a useful and reliable tool in predicting chatter free cutting conditions. Chatter vibrations in ball end mil l ing, however, have not received a similar attention due to its complex geometry and dynamic chip load generation mechanism. Chapter 3 Geometric Modelling of Mill ing Cutters 3.1 I n t r o d u c t i o n A large var iety of m i l l i n g tools are avai lable and used i n die a n d m o l d m a n u f a c t u r i n g . E a c h c u t t i n g edge on the too l is careful ly designed to ensure m a x i m u m c u t t i n g perfor-m a n c e w h i l e m i n i m i z i n g the r e s u l t i n g c u t t i n g forces. A d e t a i l e d analys is of t h e c u t t i n g edge g e o m e t r y is also necessary for the deve lopment of the force m o d e l . T h e i m p o r t a n t character i s t i cs of m i l l i n g tools are the m a c r o geometry ( c y l i n d r i c a l , b a l l - e n d , bul l -nose , t a p e r e d b a l l - e n d m i l l s , e tc . ) , the m i c r o geometry ( h e l i x angle, rake angle, re l ief angle , b a l l r a d i u s , e tc . ) , the c u t t i n g m a t e r i a l (carbide , HSS a n d P C B N ) , a n d c o a t i n g m a t e r i a l (e.g. T i N , T i A L N ) . I n genera l , several aspects are considered c o n c e r n i n g the c u t t i n g edge g e o m e t r y : - A p o s i t i v e rake angle i m p r o v e s the shearing a n d c u t t i n g act ions a n d decreases the forces, b u t weakens the wedges a n d m o r e suscept ible to wear a n d breakage. - H i g h h e l i x angle " s m o o t h e n s " a n d reduces the i m p a c t force d u r i n g e n t r y c u t t i n g . - Presence of c learance angle reduces the r u b b i n g on the m a c h i n e surface thus i m p r o v e s the surface f i n i s h , b u t at the expense of weakening the wedge a n d m o r e i m p o r t a n t l y , reduces process d a m p i n g a n d increases the l i k e l i h o o d to chat ter v i b r a t i o n s . A second c learance angle m a y be p r o v i d e d to reduce the a m o u n t of g r i n d i n g d u r i n g s h a r p e n i n g . O n e c o m m o n approach i n m o d e l l i n g c o m p l e x m i l l i n g cutters is by n u m e r i c a l m e t h o d . 17 Chapter 3. Geometric Modelling of Milling Cutters 18 The tool and its cutting edge geometry are discretized and recorded by a coordinate mea-suring device, and the data is used to construct the curve through fitting. This method requires extensive computational time and vast amount of data to be stored. Moreover, the same data cannot be applied to any other design or in the case of regrinding of the cutting edge shape. Therefore, simple mathematical models are desirable which would increase the accuracy of the mechanistic model. Such models would also eliminate the need for extensive testing to determine the functional relationship of model parameters. The following section will begin by examining the geometric modelling of four practically used milling cutters, as shown in Figure 3.1. 3.1.1 Mode l l i ng of Cy l ind r i ca l E n d M i l l Figure 3.1.a shows the common nomenclature for a helical end mill with Nf teeth. Each flute is inclined at a constant helix angle io and the immersion angle for flute j changes along the axial direction. For cylindrical cutters which have a constant radius and helix angle, the immersion angle varies linearly with the height. At a height of z, the angular position \J> of a cutting point on flute j can be described as: Vj(z) = 8 + (j- l ) ^ p - 4" t a « *o (3.1) where Ro is the cutter radius, cf)p = 2%/Nf is the cutter pitch angle. 9 is the base angle of the reference flute in global coordinate system. It should be noticed that, the cutting action on a milling cutter with straight edges is equivalent to orthogonal cutting with a tool having the same rake angle. Similarly, a helical cutting edge on the cutter is equivalent to an oblique tool with an inclination angle io. The application of the geometric similarity between the two processes will become apparent in the derivation of the milling force coefficients in chapter 4. Chapter 3. Geometric Modelling of Milling Cutters 19 Figure 3.1: a) Cylindrical End Mi l l , b) Ballend M i l l , c) Bullnose M i l l , d) Tapered Ballend M i l l Chapter 3. Geometric Modelling of Milling Cutters 20 3.1.2 Modelling of Ballend/Bullnose Mill W i t h ballend and bullnose end mills, the exact description of the cut geometry is com-plicated because the points along the cutting edge generate trochoid curves. Thus, there are complex relationships between the machining parameters (feed per tooth, depth and width of cut) and technological parameters (chip thickness and cutting cross section). In most cases, those relationships can only be found using numerical methods. Another important difference between using ballend mills and end mills in machining is the speed of material removal or the cutting speed. A ball end m i l l cuts at a portion of the sphere near the axis of rotation and thus has various cutting speeds. W i t h end-mil ls , however, the material is always cutting at the periphery of the cutter at a constant cutting speed. A bullnose mil l ing cutter (shown in Figure 3.1.c) has vertical cutting edges and each flute lies on the envelope of the hemisphere. Due to its zero helix angle, any cutting point on the same flute j has a constant angular position ip, tf^fl + f j - 1 ) 6 , (3.2) and the height z can be expressed in terms of the angle K between the cutting point and the z-axis in the vertical plane as, z = RQ(1 — COSK) (3-3) The parametric equation of cutting edges with nonzero helix angle is more compli-cated (figure 3.1.b). The detailed geometry of a helical ball end mi l l ing cutter is shown in Figure 3.2 a-d. Each flute lies on the surface of the hemisphere, and is ground with a constant helix lead. The flutes have a helix angle of io at the ball-shank meeting bound-ary (figure 3.2.a). Due to the reduction of radius at (x — y) planes towards the ball t ip Chapter 3. Geometric Modelling of Milling Cutters 21 in axial (z) direction, the local helix angle i(ip) along the cutting flute varies for constant helix-lead cutters. The expression for the envelope of the ball part is given by, x2 + y2 + (Ro - z)2 = Rl (3.4) where R0 is the ball radius of the cutter measured from the center of the sphere (C) . The cutter radius in x — y plane at axial location z is, R2(z) = x2 -\-y2 (3.5) and it is zero at the ball tip. The z coordinate of a point located on the cutting edge is, Roi> z — (3.6) taniQ where ip is the lag angle between the tip of the flute at z=0 and at axial location z, and it is due to the helix angle, ip is measured clockwise from + y axis, see Figure 3.2.b. The center of the local coordinate system coincides with the global coordinate system X - Y - Z on the dynamometer shown by ball tip point 0 in the figure. For cutters which have constant lead length, the local helix angle is scaled by a radius factor and can be expressed as, tan i(4>) — —-^j-^ tan i0 (3-7) Ro From the equations given above, the cutter radius in x — y plane, which touches a point on the helical and spherical flute located at angle </», can be expressed as, R(ij>) = Ro^l - ( V > c o t a i 0 - 1 ) 2 (3-8) A vector r is drawn from the cylindrical coordinate center (C) to a point on the cutting edge and defined by, Chapter 3. Geometric Modelling of Milling Cutters 22 i Nf = 2 F I G U R E b F I G U R E d Figure 3.2: a-d Geometric model of Ballend cutter Chapter 3. Geometric Modelling of Milling Cutters 23 Variation of geometric parameters along axial direction, 30deg helix, lOdeg normal rake 30 T 25 20 a> 5S 15 d) a 10 5 0 • ^ ^ ^ - " " " ^ L o c a l he l ix a n g l e R a d i a l r a k e a n g l e / N o r m a l r a k e a n g l e 4 6 A x i a l L o c a t i o n a l o n g c u t t e r [ m m ] 10 Figure 3.3: Geometric parameters along ballend cutter Chapter 3. Geometric Modelling of Milling Cutters 24 f(tp) = R(tp)(cos tpi + sintpj) -f R0ip cot i 0A; (3.9) The length of an infinitesimal curved cutting edge segment, dS along the ball part is computed from, dS =| | df ||= y/{R'(ip))2 + R2{rp) + R20 cot 2 i0dij> (3.10) where R'(tjj) is the derivative of R(ip) Ro(tp cot i0 — 1) cot io (3-11) ^ l - ^ c o t z o - l ) 2 From Figure 3.2.c, the radial rake a r and relief angle ctf are both defined in x-y plane and the normal rake angle a n is measured on a plane passing through the cutting point and center of the bal l , see Figure 3.2.a. Figure 3.3 shows the variation of helix angle, normal and radial rake angle in the axial direction along a flute. As formulated, the local helix angle i approaches the nominal helix angle i0 at the ball-shank meeting point, i.e. z — R0. A point on the flute j at height z is referenced by its angular position \I/ in global coordinate system, where Ro is the ball radius, <f>p = 2-7T/Nj is the pitch angle of the cutter, and 8 is the rotation angle of reference flute j — 1. 6 is measured clockwise from y—axis, and from the center point 0 . 3.1.3 M o d e l l i n g of Tapered Bal lend M i l l Tapered ballend mills (Figure 3.1.d) are widely used in manufacturing components wi th low curvature surfaces such as those encountered with turbine and compressor blades. #,•(*) = 0+ (] ~l)cj>p-— t anz 0 (3.12) -o Chapter 3. Geometric Modelling of Milling Cutters 25 There are two commonly used tapered end m i l l , each possesses slightly different cutting edge geometry. A constant lead tapered mil l has constant pitch but variable helix angle along the cutting edge due to the change in radius, while a constant helix tapered m i l l has a smaller pitch at the tip but larger pitch closer to the shank. The parametric equation describing the cutting edge on a constant lead tapered mi l l is given by, R(ip) = R0( \ -I- rf/cotiotawy) (3.13) Similar to the equations developed for ball end mills, the position angle ijj defines the position of any point on a cutting edge. R0 is the radius of the cutter at the t ip, i0 is the helix angle at the tip and 7 is the half-apex tapered angle. These cutters are usually combined wi th ball shaped flutes and referred to as tapered ballend m i l l . The equation of the cutting edge on the taper part which starts at ipa0 is, R(il>) = RQ\\ + f> - il>ao)cotiQtanT\ (3.14) where if}a0, representing the ball/taper intersection point, is obtained by substituting (z — R0) into E q . (3.6), resulting in tpa0 — tani0 (3.15) 3.1.4 Cutting Geometry and Chip Thickness Calculation During cutting, the tool edge penetrates into the workpiece material and removes a chip section which runs along the rake face. The chip thickness prediction is a key to mi l l ing analysis as the instantaneous cutting forces are proportional to the cross-sectional chip thickness at each tooth edge and its maximum value is always a good indication of the process load. Unlike in turning where the chip thickness is always constant and can be Chapter 3. Geometric Modelling of Milling Cutters 26 set directly, the chip thickness varies in mil l ing and, in general, depends on the workpiece geometry, the tool location and geometry, and the cutting parameters such as the axial depth of cut and the feed direction simultaneously. The following section summarizes the solutions to the chip thickness prediction of simple mil l ing operations. Later in chapter 5, a more accurate numerical model which incorporates the exact kinematics of the process wi l l be compared against the closed form solutions. For example, Figure 3.4 shows a typical mil l ing operation where the cutter periphery is in full contact with the workpiece, and the workpiece is fed perpendicularly towards a rotating cutter in the feed (x) direction, the actual shape of an undeformed chip is rather complex as the cutting edge traverses a trochoidal path proven by Marte l lo t t i [2]. Host{ ^—TB—Tf }sin[ rrrs—rr tn = , rag+flo-s* ^ + * Q - S * ( 3 1 6 ) K^hY + Ro' + ^Rocosl The above expression, although it accounts for the continuous motions of both the cutter and the workpiece, is too complex to be applied in practice. Various approxima-tions, such as assuming a circular tooth path, or neglecting the effect of the table feeding term (V/2TTN)2 in the expression, have been made to give the most commonly used chip thickness expression in mil l ing analyses, i.e. t(V>,0) = stsin($) (3.17) In practice, the cutting speed is much greater than the table feeding speed and the above equation gives good prediction of the chip thickness in the x-y plane, normal to the cutting edge in the case of cylindrical cutters. In ballend/bullnose mil l ing, the chip thickness normal to the cutting edge on the spherical surface, denoted by tn, is a function of both the radial position angle 0 and the axial angle K, as shown in Figure 3.2 d, Chapter 3. Geometric Modelling of Milling Cutters Figure 3.4: Common mil l ing geometry and chip formation Chapter 3. Geometric Modelling of Milling Cutters 28 tn(ip, 6, K) = stsin(^)sin(hi) (3.18) where •* = « n - ^ (3.19) Ro The two chip thickness formulations (Eq. 3.16 and E q . 3.17) are plotted in Figure 3.5 at two cutter locations [z = O.lRo and z = 0.25i£o). The radius of the cutter is lOmra and the feed rate is 0.127'mm/tooth. From the figure, it can be seen that, the error introduced by the above analytical chip thickness expression is significant only in areas around the ball t ip. Hence, the use of exact kinematic model of chip thickness formulated by Martelott i wi l l not significantly improve the results when the axial depth of cut is not too small. The chip thickness in other unconventional mill ing operations, for instance, when the workpiece geometry and the cutter orientation vary, requires a much involved analyses. Ana ly t ica l solution is usually not possible as the tool/part engagement l imit can become irregular and has to be identified with the available solid modelling system [34]. 3.2 Conclusions In this chapter several common mil l ing cutter geometries, in particular, ballend mil ls , were thoroughly investigated. A n analytical model of the spiral edge on the ballend was developed. It considers the chip thinning effect, variable helix angle and changing cutting speeds which are unique in ballend mil l ing. The use of simplified chip thickness formulation was justified and the error was found to be insignificant except at very low axial depth of cut. The derived cutter geometries and the helical cutting edge orientation w i l l be used to formulate the cutting forces during the dynamic mi l l ing process in the following chapter. Chapter 3. Geometric Modelling of Milling Cutters 29 Figure 3.5: Ch ip Thickness Profile at two cutter locations, solid line - approximation, dotted line - exact kinematics Chapter 4 Mode l l ing of M i l l i n g Forces 4.1 Introduct ion In the machining of complex and costly parts such as those involved in aircraft or die and mold industries, prior knowledge of the mil l ing forces, surface form errors and chatter vibrations is essential to assist engineers in designing machine tools, jigs and fixtures. It can also helps them in selecting economic cutting conditions to reduce tool deflection and chatter vibrations which affect the surface quality of the finished parts. For instance, if the chip load is selected to be too large, the forces cause chipping of the cutting edge or even tool breakage, while a small chip load wi l l increase the percentage of deformation and friction action in the process which leads to higher temperatures, abrasive wear, and poor surface quality. The basic difference between predicting cutting forces from empirical testing and metal cutting theory is that in the former, tests are required for the specific cutting operations and the approach usually considers the influence of practical rather than fundamental variables. Therefore, it may be difficult to apply data from one process to another. The second approach, which involves a more detailed analysis and testing procedure, aims at determining the fundamental machining parameters which may be useful in solving various cutting processes. Many investigations have been made into the nature of the cutting forces during mi l l ing [4, 12, 35, 34]. Cutt ing forces are evaluated in the following way, a complete representation of the chip load on the end m i l l at any instant is obtained by integrating 30 Chapter 4. Modelling of Milling Forces 31 the chip area on the cutter by considering thin disc-like sections along the axis of the cutter. The location of each section is first determined and for each flute engaged in the cut, the chip thickness times the chip width yields the chip load. As the angular position of the cutter changes, the chip load is recomputed. Traditionally, cutting force coefficients are calibrated experimentally for a particular tool/workpiece set through empirical curve fitting techniques in which polynomial regression models are developed to relate the average chip load to measured average forces. The mil l ing force coefficients are then applied in an analytical model which predicts the cutting forces produced by the same cutter in other machining conditions. The mechanistic approach described above usually results in simple formulation and accurate predictions in the cases with cutters of simple geometry such as cylindrical end mills or face mills. However, it suffers a few shortcomings. For cutters with complex geometry, such as ballend mills where the cutting edge geometry changes in the axial direction, the applicability and flexibility of the mechanistic approach will be greatly reduced. More calibration tests are required since the mil l ing force coefficients become functions of multiple geometry factors. A second approach deals with the unified mechanics of cutting was first presented by Armarego [14, 17, 18]. Each cutter tooth is partitioned into a series of axial elemental oblique tools. Then the cutting geometry of each in-cut tooth element is identified and thin shear zone mechanics of cutting analyses are applied to obtain the resultant force. This mechanics of mil l ing approach, which results in the estimation of the mil l ing force coefficients from orthogonal machining data, will be studied, and applied in this chapter, to mil l ing force prediction with ballend cutters. Chapter 4. Modelling of Milling Forces 32 4.2 Milling Force Models Two mil l ing force models wi l l be presented here. The average force model, which involves calibration of each mil l ing cutter prior to the predictions, and the mechanics of mi l l ing approach, which is more suitable to apply on cutters with complex geometries. 4.2.1 Mechanistic approach with Average Force Model It has been shown by Sabberwal [4, 35] that force predictions may be made by considering the tangential cutting force to be proportional to the chip load and the radial force proportional to the tangential force, where dFt is the tangential force on a disc element, dFr is the radial force on a disc element, daz is the thickness of the axial discs, tc is the chip thickness from Eq . (3.17), and KT, KR are mil l ing force coefficients. One common method used in estimating the mil l ing force coefficients is to assume average values at all the cutting points. This allows the above integrations to be evaluated analytically. Average force expressions are obtained and the values for KT and KR may then be solved by relating the expressions to the measured average forces. The mi l l ing force coefficients, in general, vary with the machining conditions. For cutters with simple dFt = KTtcdaz (4.1) dFT = KRdl<\ (4.2) Chapter 4. Modelling of Milling Forces 33 geometry such as cylindrical end mills, it has been shown that, they are strong functions of the average chip thickness [36]. Therefore, empirical models are developed to predict them as functions of the average chip thickness. The above model is modified and adapted to ballend mi l l ing in the following section. The geometry of a cutting edge on the ballend cutter was shown in Figure 3.2 c. A set of curvilinear coordinate system normal to the ball envelope is used to specify the resultant cutting forces acting on the flute. The elemental tangential, radial, and axial cutting forces dFt, dFT, dFa acting on the cutter are given by dFt(9,z) = KtedS + Klctn(6,i>,K)db dFr(6,z) = KredS + KTCtn(6^,K)db (4.3) dFa(9,z) = KaJS + K a c t n { e ^ ^ ) d b where tn(9,ip,n) is the uncut chip thickness normal to the cutting edge, and varies wi th the position of the cutting point. The cutting forces are separated as edge (e) and shear cutting (c) components. The edge force coefficients (Kte, Kre, Kae) are in [N/mm], constant and lumped at the edge of the flutes. dS is the differential length of the curved cutting edge segment given by Eq . (3.10). In the mechanistic approach, the shearing coefficients Ktc, KTC, Kac are identified from a set of standard mi l l ing tests wi th the same cutter and workpiece. The resultant forces in Cartesian coordinates are obtained by introducing the trans-formation matrix T { d F x y z } = [T]{dF r t a } (4.4) Chapter 4. Modelling of Milling Forces 34 dFx —cos(ty) — cos(K,)sin(^) dFr dFy = — 5zn(/c)cos(^') sin^ty) —COS(K)COS{^) dFt dFz COS(K) 0 -sin(n) dFa The total cutting forces acting on one flute w i t h an ax i a l depth of cut z: {F} = j ' dF (4.5) where the differential force components are dependent on the engaged flute segment length dS, instantaneous chip load tn(ip, z , K), the- local rotation (6) and lag angle (•0). FX][6{z)\ = j£( -dFrjsm(Kj)sm(^j) -dFtj cos(#j) - d F a j COS(KJ) s i n ( $ j ) ) d z Fyj[6(z)] = / / 2 ( - dF r jS in ( / c J - ) cos (* j ) +dFljsin('$j) -dFajcos{Kj)cos(^j))dz F.j[0{*)} = J2{ dFrj COBM -dF^sm^dz (4.6) When the helix angle is zero, i.e. straight cutting edges as in bullnose cutters, the lag angle (tp) is constant for each flute and it is possible to integrate the above expressions analytically to obtain a close form solution for the cutting forces. However, in that case, constant mi l l ing coefficients are assumed. The integrations given above are calculated numerically by evaluating the contribution of each discrete cutting edge element at dz intervals. However, geometric considerations must be taken when the edge segment is outside the immersion zone. i.e. If the edge is not in contact with the workpiece, the contribution to the cutting forces is zero. \ Chapter 4. Modelling of Milling Forces 35 Assuming average mi l l ing force coefficients Ktc, Kac, KTC along the cutter axis, the above integral can be evaluated and averaged over one revolution of the cutter. Fx ~ FTeITex ~\- KTCITCX -) KteItex f- KtcItcx \~ KaeIaex -j- KacIa Fy FTeIT£y ~\- KrClTCy ~\ Kl^llQy f~ F^llcy ~\ Kag IaQy "f" KaCI( acy Fz KreIrez -\- FrcITCZ -j- Kieliez [- Kicl\cz ~\~ FaeIaez -\- FacIa (4.7) where Irei and Irci (i — xyy,z) are functions of geometric variables such as local radius and immersion angle. They have to be evaluated numerically. The above expressions for the average forces are assumed to be constant over a complete revolution of the cutter. As proved by Whitfield [37], the helix angle does not have any effect on the average forces since the total amount of chip load remains constant in every revolution. Thus the mi l l ing force coefficients can be estimated, by applying least squares minimizat ion technique, for each axial depth of cut, and as functions of average chip load and cutting speed. D e t e r m i n a t i o n of Edge Fo rce Coef f ic ien ts : The average forces are obtained from experiment and plotted against the chip thickness and axial depth of cut as shown in F ig -ure 4.1. Then the edge forces are obtained by extrapolating the measured average forces to zero chip thickness (Figure 4.2). A finite average force value at zero chip thickness reflects the existence of a secondary 'rubbing' process. The average edge force coefficients Kte, KTe) Kae represent the rubbing forces per unit width, normalized with respect to the flute length 1 . Past research show that the edge forces do not depend heavily on feed rate [16]. 1 The in-cut flute length is obtained by integrating dS in Eq. (3.10) numerically Chapter 4. Modelling of Milling Forces 36 The effect of cutting speed has on the edge force coefficients is investigated by con-ducting tests at similar cutting conditions but with different spindle speeds. Al though the cutting speed varies in each test due to varying cutter radius, results show that [31], the edge forces do not vary significantly with cutting speeds for the particular workpiece material (Ti6Al4V) used here. The cutting components of the average forces are used in the evaluation of the cutting force coefficients. Results are plotted and shown in Figure 4.3. It can be seen that, due to variable cutting geometry along the axis of the cutter, the cutting force coefficients are strong functions of both the chip load and the axial immersion of the cutter. As a result, high order polynomial curve fitting technique is required to calibrate the coefficients for different axial depths. Due to the l imited use of the mechanistic approach, the identification procedure will be kept outside the scope of this work. Similar works can be found in references [15, 16]. It can be seen that, the mechanistic approach requires vast amount of cutting data for each new cutter geometry and more importantly, the data cannot be generalized to be applied on other cutters as there is no explicit relationship between the tool geometry, cutting conditions, and the cutting force coefficients. Moreover, the empirical polynomial fitting does not give physical insight, for instance, the shearing resistance and the friction, in the process. 4.2.2 Mechanics of Milling Model The mechanistic approach used in the previous section involves empirical curve fitting and assumes constant and average values which may not be suitable to be applied on cutters with complex geometry as the cutting force coefficients (K t c, Kac, KTC) may be dependent on the local cutter geometry and cutting geometry (i.e. local chip thickness). In this section a mechanics of cutting analysis for the mil l ing operation with complex Chapter 4. Modelling of Milling Forces 37 0 . 1 2 0 0.02 0.04 0.06 0.08 0.1 0.12 350 T 0 0.02 0.04 0.06 0.08 0.1 0.12 S t [mm/tooth] a = 1.27 mm a = 2.54 mm — o — a = 3.81 mm — £ — a = 6.35 mm —x— a = 8.89 mm — s Figure 4.1: Average Forces vs Feed rate for different axial depth of cuts in ballend mi l l ing Chapter 4. Modelling of Milling Forces Axial depth of cut a [mm] Figure 4.2: Average Edge Forces vs axial depth of cut in ballend mi l l ing Chapter 4. Modelling of Milling Forces 39 2500 T 2000 £ 1500 s 2 1000 500 0 800 T 700 600 + V 500 Q. 3. 400 £ * 300 200 -f 100 0 002 0.04 006 0.08 st [mm/tooth] — I — 0.1 0.12 H 1 h- 1 1 1 0.02 0.04 0.06 0.08 0.1 0.12 "t [mm/tooth] »t [mm/tooth] a = 1.27 mm a = 2.54 mm a = 3.81 mm a = 6.35 mm a) -2r -X-2500 2000 + »" 1500 s 5 1000 500 4-800 j 700 --600 --500 -• a. s 400 --300 --200 -100 -• 4 a [mm] 4 a [mm] a [mm] st = 0.0254 mm/tooth |st = 0.0508 mm/tooth st = 0.0762 mm/tooth st = 0.1016 mm/tooth b) - + -Figure 4.3: Ballend M i l l i n g Force coefficients estimated from Average Force Mode l : a) vs st, b) vs a Chapter 4. Modelling of Milling Forces 40 ballend cutters is developed and experimentally tested. The section wi l l begin by ex-amining some mil l ing variables such as tooth helix angle, normal rake angle, and local chip thickness and their importances to the mil l ing force analysis. The mil l ing process is then examined theoretically, drawing emphasis particularly on its relationship wi th the fundamental theories of orthogonal and oblique cutting. It wi l l be shown how the funda-mental two dimensional orthogonal cutting model can be applied on the mi l l ing analysis and more importantly, how the required mil l ing force coefficients can be estimated from the orthogonal machining data through oblique transformation. Figure 4.4 shows the instantaneous geometry of a helical cutter during a mi l l ing operation with one tooth engaged in cutting at a given cutter orientation 6. The material approaches the flute at a speed approximately equal to the tangential cutting speed so that the cutting edge is inclined to the resultant cutting speed by the helix angle. Thus the cutting action may be considered equivalent to an oblique cutting process wi th a variable cut thickness and constant width of cut equal to a. Unlike orthogonal cutting which has a two dimensional cutting geometry, oblique cutting is a more general case of the cutting operation and is the appropriate model to represent mil l ing with helical tool. As in orthogonal cutting, the analysis is aimed at developing equations relating the three force components Fp,F'Q,Fji to the cutting chip geometries, b and t, the tool ge-ometry an,i, the deformation geometry </>„,77c, the workmaterial property r , and friction characteristic /3 at the tool/chip interface. For oblique cutting, the primary shear and friction forces lie on an orientated plane affected by the local helix angle and rake angle. Therefore, a geometric transformation has to be performed to obtain the resultant forces normal and parallel to the cutting edge. Other important parameters required in the elemental oblique force prediction model are the inclination angle and the normal rake angle, or the velocity rake angle. As mentioned above, the inclination angle is the local helix angle on the cutter and the Chapter 4. Modelling of Milling Forces Figure 4.4: Oblique Cut t ing Geometry of a Ballend M i l l Chapter 4. Modelling of Milling Forces 42 normal rake angle is defined in the plane perpendicular to the cutting velocity. Thus the working angles for the elemental oblique cuts can all be related to the local tool angles given in the cutter geometrical specifications. 4.2.3 Prediction of Cutting Force Coefficients from an Oblique Cutt ing Mode l In this section, the mechanics of mil l ing approach wi l l be applied on ballend mil ls . The pressure applied on the cutter due to ploughing and cutting is related to the total forces wi th edge force coefficients Kte, KTe, Kae, and cutting force coefficients Ktc, KTC, Kac and they wi l l be estimated from basic machining parameters. B a l l end m i l l flutes are treated as combination of series of oblique cutting edge seg-ments. A n oblique cutting geometry based on the thin shear zone model is shown in Figure 4.4. The chip velocity Vc is inclined at an acute angle i to the plane Pn normal to the cutting edge. A l l the following equations wi l l be written on this plane Pn which also has its orientation changes along the cutting flute due to the change in radius of the cutter. The elemental width db of the active cutting edges on all teeth in the axial direction depends on the axial depth of cut a and the number of elements selected. Due to helix, the uncut chip thickness seen by each tooth element is different and can be approximated by E q . (3.17). From Figure 4.4, db is the projected length of an infinitesimal cutting flute in the direction along the cutting velocity. It should be noted that db is consistent wi th the chip width defined in the classical oblique cutting theory. The relationship between db and dz is given by: (4.8) Chapter 4. Modelling of Milling Forces 43 App ly ing Merchant's theory, resolving the resultant forces due to 'cutt ing' on the shear plane into three mutually perpendicular components yields ([19], [38]), where F F0 = FT rbt cos(/3w - a n ) + tan7? g sm/3n tanz ] sin cj)n c rbt si.n(/3T1 - a „ ) sin (f>n cos i c rbt cos(/3n — Q!n) tan i — tan ?yc sin /3 n sin c ycos 2 {4> n -V (3n - ctn) + tan 2 T?c s in 2 /3ri (4.9) .Fp, Fq, are the power, thrust and radial forces acting on the oblique cutting edge segment, respectively. The normal friction angle, /3 n , is defined as, tan/3 n = tan/3cosr/ c (4-10) where j3 is the average friction angle at the rake face of an orthogonal cut, and T]C is the chip flow angle(angle between a perpendicular to cutting edge and the direction of chip flow over rake face, as measured in the plane of tool face). The normal rake angle is constant, and set during cutter grinding. The normal shear angle, </>n, is obtained from the cutting ratio Tt cos a.,, , . t a n ^ = — ^ — 4.11) 1 — rt sin a n r t is the chip thickness ratio in oblique cutting and is related to orthogonal parameter r bv TV = r C Q S ^ . This relation is'obtained from the mass continuity equation of the J 1 cosi chip before and after the cut. To further simplify the model, Stabler rule [39] is applied Chapter 4. Modelling of Milling Forces 44 in which the chip flow angle rjc is approximated by the local inclination angle i. F rom Figure 4.4 in which a small segment on the ball end cutter is shown, the tangential, radial and axial force components dFt,dFr,dFa are compatible with the power, thrust, and radial force components Fp, Fq, FT in oblique cutting when the elemental cut thick-ness t and width of cut b are given by the instantaneous chip thickness tnj(ip,Q,K.) and length dz/sin^K,) [17]. Thus the mil l ing force component coefficients due to 'cutt ing' , i.e. Ktc,KTC and Kac, can be expressed in terms of the transformed cutting coefficients. „ _ T cos(/3n — an) + tan ryc sin f3n tan i K t c = r rsin(A,- O n ) ( 4 . 1 2 ) sin (f)n cos I c v — r cos(/3n — an) tan i — tan r/c sin/3 n The edge force coefficients, which represent the rubbing force on a unit length basis, wi l l be identified from orthogonal cutting experiments and can be directly used on the cutter edge, taking into consideration the orientation factors. Cut t ing pressures in tan-gential, radial and axial directions are calculated locally along each cutting flute segment, and elemental forces are summed along each cutting edge to evaluate the total cutting forces. Orthogonal Cutt ing Parameters: A set of turning experiments resembling or-thogonal cutting was conducted on Ti tan ium tubes (TiQAliV) with tools of different rake angles over different feeds and cutting speeds (See Figure 4.5). The diameter of the tube was 100 m m (O.D.) and the thickness was t = 3.81mm. The cutting speed range was from 2.6 to 47 m / m i n . Resultant forces in the power (Fc) and thrust (Ff) directions were measured with force dynamometer. Small steps in cutting conditions were used to Chapter 4. Modelling of Milling Forces Figure 4.5: Experimental Setup - Orthogonal Turning test Chapter 4. Modelling of Milling Forces 46 increase the reliability of the measured forces. Since the measured forces may include both the forces due to shearing and a secondary process "ploughing" or "rubbing" at the cutting edge, the edge force model given in section 3.2 is applied, and the measured force components are expressed as: (4.13) Fpt = FPe + FPC Fot = FQ£ + FQC The edge forces are obtained by extrapolating the measured forces to zero chip thick-ness. From Figure 4.6, it can be seen that the edge forces do not vary significantly wi th cutting speeds and rake angles for the particular workpiece material (TiSAlAV) used here. The average edge force coefficients Kte, Kre, Kae represent the rubbing forces per unit chip width. The Kae value is usually small and taken as zero [17]. The chip compression ratio (r), shear stress r , shear angle <f>, and friction angle /3 are calculated from the measured 'cutting' component of the forces and the cutting ratio by applying the orthogonal cutting theory presented by Merchant [3]. Fpc cos </> — FQC sin </>) sin <f> I T tan <f> tan 13 = r cos a 1 — r sin a FQC + Fpc tan a Fpc — FQC tan a (4.14) The fundamental machining parameters, the shear stress ( r ) , average friction angle /3, chip compression ratio r , are estimated by employing least square curve fitting technique to the measured orthogonal test results. Results are shown in figures 4.7, 4.8, 4.9. Chapter 4. Modelling of Milling Forces Measured total cutting forces from orthogonal tests 900 0 0.02 0.04 0.06 0.06 0.1 0.12 Measured total cutting forces from orthogonal tests 900 0 0.02 0.04 0.06 0.08 0.1 0.12 Figure 4.6: Edge Force Extrapolation from Orthogonal Turning test Chapter 4. Modelling of Milling Forces 48 Figure 4.7: Cut t ing Ratio identified from Orthogonal Turning test Chapter 4. Modelling of Milling Forces 49 Figure 4.8: Shear Stress identified from Orthogonal Turning test Chapter 4. Modelling of Milling Forces 50 Figure 4.9: Frict ion angle identified from Orthogonal Turning test Chapter 4. Modelling of Milling Forces 51 The identified relationships are summarized in Table 4.1. From the results, it is ob-served that: 1. The shear stress does not vary significantly with either the rake angle or the cutting speed. It is higher than that found from conventional materials tests due to high shear strains, shear strain rates and temperatures involved in machining. 2. The cutting ratio, however, depends slightly on the uncut chip thickness. 3. The average friction angle on the rake face is expressed as a function of the rake angle. These relationships are valid only for the particular workpiece material (TiQAlAV) tested here, and other materials may exhibit different relationships. Table 4.1; Orthogonal cutting data T = 613 M P a P = 19.1 +0.29a r = r0ta To = 1.755 - 0.028a a = 0.331 - 0.0082a Kte = 24 N / m m KTe • = 43 N / m m After the above data base is completed, mil l ing force calculation can be made for any mi l l ing cutter geometry by using the same orthogonal data. As an example, consider the ballend m i l l geometry shown in Figure 3.2. Cut t ing velocity and helix angle vary from 0° at the bal l t ip to finite values at the ball/shank intersection, resulting in variable cutting conditions along each cutting flute. The in-cut portion of every cutting edge is first divided into small sections, from which each section is treated as an oblique cutting tool with a different cutting velocity and helix angle. For this reason, mi l l ing force coefficients have to be evaluated locally according to E q . (4.12), using the un-cut Chapter 4. Modelling of Milling Forces 52 Define: Tool ( Cuttin Work 3eometry g Conditions Diece Material \ / Orthogonal Cutting Data <l>, p\ r t \ / Prediction of Chip Flow angle \ / Milling Force Coefficients t^e '^ re >^ ae >^ tc >^ rc '^ ac \ / Integrate: dFt = dFr = dFa = K t e dS +Ktc tc db K r e dS +Krc tc db KaedS +K.dCtc db \ Transformation Matrix : [T] Fx >FZ Figure 4.10: Generalized Mechanics of Cut t ing Approach to Mi l l i ng Force Predict ion Chapter 4. Modelling of Milling Forces 53 chip thickness identified from the instantaneous immersion of each section. Total mi l l ing forces at any cutter location can then be obtained by integrating al l the force components dFt, dFT, and dFa and transforming into the global coordinate system ( X , Y and Z). The flow chart of the mil l ing force prediction procedure based on the generalized mechanics of cutting approach can be found in Figure 4.10. 4.2.4 Simulation and Experimental Verification The proposed cutting force models have been assessed by means of numerical simulation and experimental testing of the forces over a wide range of conditions encountered in mi l l ing . More than 60 cutting tests were conducted on a vertical S A J O C N C mi l l ing machine with 30 degrees nominal helix, single flute, carbide ball end mil ls . The experimental set-up is shown in Figure 4.11. During mi l l ing tests, the cut-ting forces were measured with a Kist ler Mode l 9257A three-component dynamome-ter and recorded by a P C equipped with a 16 channel data collection board, through charge/voltage conversion amplifiers. A microphone was also set up to measure any sig-nificant sound pressure fluctuation during machining caused by chatter vibrations. Slot cutting experiments were selected on Ti tan ium alloy [TiftAlAV) at different feeds and axial depth of cuts with cutter rake angles ranging from 0 to 15 degrees. T i t an ium alloy, which has high strength to weight ratio and low thermal conductivity, was selected as the workpiece material. Its application can be found mainly in the aerospace industry where high strength, high temperature resistance materials are desired on engine components. Ti6Al4V remains one of the most difficult material to be machined due to its high shear stress (more power in shearing the material) and low temperature gradient during cutting (thus low cutting speed). Cutters with two different radius were used (6.35 and 9.525 mm). The tests were Chapter 4. Modelling of Milling Forces 54 Figure 4.11: Experimental Setup for M i l l i n g Test Chapter 4. Modelling of Milling Forces 55 conducted without lubricant. Cutt ing conditions for the tests and simulations are sum-marized in Table 4.2. Table 4.2: Cut t ing conditions for static ballend tests and simulations Ro = 6.35, 9.525 m m an = 0 - 1 5 degrees' io = 30 degrees Nf = 1 st = 0.0127 - 0.127 mm/ too th a = 1.27 - 8.89 m m N = 269 R P M The solution of E q . (4.6) is obtained by applying numerical integration technique. The cutter is first discretized axially into small discs with different radii , the lag angle of the flute j at height z in global coordinate is given by E q . (3.12). The forces for mult iple fluted cutters are obtained by summing the forces acting on the individual flutes in cut. The immersion of the cutter in the workpiece is an important factor in determining the cutting forces. In general, the contact zone between the cutter and the workpiece can be found by examining the instantaneous immersion angle of al l cutting points on the cutter digitally against the entry and exit angles. A t any tool rotation angle 9, it is first checked that whether the flute is in or out of cut. For example, in half immersion up mi l l ing the flute segment k is in cut if (tpk)j > 0 and (ipk)j < 7 r / 2 . Using the predictive force model with classical orthogonal cutting machining param-eters, the simulation and experimental results were compared. The percentage difference between the predicted and experimental results has been used to evaluate the statistical reliability of the model. Chapter 4. Modelling of Milling Forces 56 Figure 4.12 shows the histograms of the percentage errors and the mean values of the average forces. It can be seen that the model provides, both qualitatively and quan-titatively, sound predictions of the cutting forces in all three components, with mean deviation of -0.93 % for average F x , 1.36 % for average Fy, and 3.03 % for average F z . The corresponding ranges in percentage deviation are -30 % to 25 % for average F x , -25 % to 15 % for average Fy, and -35 % to 40 % for average Fz . In Figure 4.13, a sample simulation of cutting forces for a slotting test with axial depth of cuts a = 1.27mm and 6.35mm at feed-rate — 0.0508mrnpertooth are shown for a single fluted cutter with zero rake angle. The cutting force pattern in the axial direction is dominated by the ploughing action between the flute and the material. As soon as the flutes enter into cut, the flank forces develop rapidly while the chip thickness is s t i l l small. In Figure 4.14 , predicted and measured cutting forces for a slotting test with different sizes of cutters are shown. The predicted and simulated cutting forces are in good agreement. In a separate case, half immersion up-milling tests with axial depth of cut a = 6.35mm but with two different feed-rates of fj ~ 0.0508mm/rev and = 0.1016mm/rei; were conducted. Again, the predicted and measured cutting forces, as seen in Figure 4.15, are in good agreement except with slight phase shifts at the exit due to difficulty in selecting the reference point in the half immersion test data. To further validate the developed geometric and cutting force models, simulations and experiments were performed on multi-fluted ballend cutters. A three fluted ballend m i l l wi th a zero rake angle and angular grinding tolerance of ( + 2 ° , —2°) was used in the tests. For the experiments conducted, the tool holder and spindle were both very stiff, making the runout level quite small, (the maximum radial runouts measured at the ball-shank meeting region of the flutes were 0.015mm, 0.01mm and 0.0mm), therefore the effect of Chapter 4. Modelling of Milling Forces 25 c 20 0 1 15 ~<fi _ £t 10 O 4 8 5 Fx average Mean = - 0.93 % •— <— CN % Deviation 30 25 l 2 0 a 15 Q o 10 Fy average Mean = T36 % -25 -20 -15 -10 -5 0 % Devaition 10 15 14 12 10 8 6 4 2 CO Fz average Mean = 3.03 % O O CM .— .— CN CO o % Deviation Figure 4.12: Statistical Evaluation of Model and Data Chapter 4. Modelling of Milling Forces 58 runout on the resultant cutting forces, as seen in Figure 4.16, was min imum. 4.3 Conclusions A generalized cutting force model was developed for common mil l ing operations in this chapter. The cutting forces were modelled by separating the shearing and edge friction components on the edge. Mi l l i ng force coefficients were obtained from an orthogonal ma-chining data base using oblique transformation, which eliminates the need of calibrating for each cutter geometry. The data base was applied on mi l l ing with ballend cutters in which the model was experimentally verified, both qualitatively and quantitatively. The results should help the process planners in designing tools and in selecting opt imum cutting conditions prior to each operation to increase the productivity. Chapter 4. Modelling of Milling Forces 59 300 T F x Rotation Angle [deg] Rotation Angle [deg] Figure 4.13: Measured and predicted cutting forces for slot cutting tests, ballend m i l l , spindle speed 269rev/mm, R0 = 9.525mm,i0 = 30°, st = 0.050S[mm/flute] (a)- a = 1.27 [mm] (b)- a — 6.35[mm], rake=0 Chapter 4. Modelling o[ Milling Forces 60 Rotation Angle [deg] Figure 4.14: Measured and predicted cutting forces for slot cutt ing tests, bal-lend m i l l , spindle speed 269rev/min, i0 = 30°. (a) a=3.81 [mm], #o=9.525 m m , 3 t=0.1016[mm/flute], rake=5 (b) a=3.048 [mm], R0=6.35 m m , ^,=0.0762 [mm/flute], rake=10 Chapter 4. Modelling of Milling Forces 61 800 T Rotation Angle [Deg.] 1500 -1000 - Fy z 500 - Fz • — ting Force: n • [ \ \ \ ting Force: \j ( —g^^- 1 — 1 40 60 80 100 120 u -500 • -1000 -Fx ^ — -1500 -Rotation Angle [Deg] Figure 4.15: Measured and predicted cutting forces for half radial immersion cutt ing tests, ballend m i l l , a n = 0°, a=6.35mm J spindle speed 269rev/mm, i0 = 30°, (a) 5 t =0.0508 [mm/flute], (b)- a t=0.102 [mm/flute] Chapter 4. Modelling of Milling Forces 62 2000 1500 „ 1000 z Fy -1500 -1- Fx Rotation Angle [deg] Rotation Angle [deg] b) Figure 4.16: Measured and predicted cutting forces for slot mi l l ing wi th a 3 fluted bal-lend m i l l , a = 8.9mm,z 0 = 30°, (a) spindle speed 4&Qrev/min, an = 15°, 5 t=0.0889 [mm/flute], (b) spindle speed Ql5rev/min, a n = 0°, st=0A27 [mm/flute] Chapter 5 Dynamic Mill ing and Chatter Stability 5.1 Introduction A vibration system can be represented, in the most simplest case, in terms of a single degree of freedom system with a mass m on a restoring spring of stiffness A; and a dashpot wi th damping c, as shown in Figure 5.1. The equation of motion can be obtained using Newton's second law: mq(t) + cq\t) -f kq'(t) — Fq(t) (5.1) This form of equation is useful in cases such as free vibration, (Fq(t) — 0) and forced vibration, when the forcing functions come from external sources or from the cutting process. However, there are systems for which the exciting force is a function of the motion parameters of the system, such as displacement, or velocity. Such systems, as occurred in dynamic mil l ing, are called self-excited vibrations or chatter since the motion itself produces the exciting force. Al though very few practical structures could realistically be modelled by a single degree of freedom system, the properties of such a system are very important since those for a more complex but linear multidegree of freedom system can always be represented as the linear superposition of a number of single degree of freedom systems. The solutions to the above equation of motion usually come in two different forms. The time domain solution which is well explored in the literature gives the response in 63 Chapter 5. Dynamic Milling and Chatter Stability 64 discrete time domain while the frequency transfer function considers the solution in terms of magnitude and phase of the system transfer function in frequency domain. These two solution forms w i l l be briefly reviewed in the following: Consider a S D O F system as shown in Figure5.1, the equation of motion is: mxx'(t) + cxx\t) -I- kxx'(t) = Fx(t) (5.2) The above equation can be evaluated at each discrete time interval 5t with numerical techniques such as recursive, discrete equivalent of the continuous differential equation. Taking Laplace transform of Eq . (5.2) gives: ms2x(s) + csx(s) + kx(s) — Fx(s) ;5.3) or x{s) = — ^ — - . 5.4 v ' ms2 + cs + k K 1 where s is the Laplace operator. The discrete time equivalent of the Laplacian operator can be approximated using the bilinear transformation (trapezoidal integration) method [40] by letting, 5 - T I T T I ^ ( 5 - 5 ) where z is the Z-transform operator and (T) is the time step. Substituting E q . (5.5) into E q . (5.3) and after simplification gives the response x in discrete time, Chapter 5. Dynamic Milling and Chatter Stability 65 Stiffness k Damping c Figure 5.1: Schematic Representation of a S D O F System Chapter 5. Dynamic Milling and Chatter Stability 66 where C\ = —^Tnx -\- 2KX C2 = ^rnx - j,cx + kx (5.7) This numerical solution procedure is a rapid and robust integration scheme which is numerically stable for reasonably small time step T, making it ideal to be used on digital computers. The structural parameters in the equations above are usually identified by experimental modal analysis. Moda l analysis is defined as the process which is "involved in testing components or structures with the objective of obtaining a mathematical de-scription of their dynamic or vibration behavior [41]." Common methods include exciting the structure with a known source (e.g. impact hammer or shaker) and measuring its responses (e.g. displacement, velocity, or acceleration) at different locations. The trans-fer function is then obtained in the form known as frequency response function which is computed from the Fourier transforms of the excitation and the response t ime histories. The mathematical representation for a frequency domain transfer function is found by substituting (s = ju) into E q . (5.4): G { U J ) = = ( l b - W ) + > c u / ( 5 - 8 ) which can be represented graphically in terms of i) modulus and phase angle against frequency, ii) real and imaginary components of response with varying frequency, or i i i ) vector diagram of the real component versus the imaginary component of the response. For a S D O F mechanical structures with light damping, a sample transfer function is shown in Figure 5.2. As described by Eq . (5.3), it gives a complete physical description of the dynamic characteristics such as the natural frequency, damping ratio, and the stiffness of the system based on some well-established mathematical relationships: Chapter 5. Dynamic Milling and Chatter Stability Figure 5.2: Frequency Domain characteristic curve for a S D O F Dynamic System Chapter 5. Dynamic Milling and Chatter Stability 68 un = Jk/m~ ( = ujd = unJl-£2 (5.9) 2vkm The system parameters can be used to predict the response to various excitations or to improve the dynamic behavior of the system by design modifications. In the modal analysis performed in this study, the structure is assumed to be linear and the parameters to be time-invariant. This chapter presents a comprehensive simulation model for dynamic mi l l ing in which the mi l l ing cutter attached to the spindle is modelled by two orthogonal structural modes in the feed and normal directions at the tool tip. For a given cutter geometry, the cutting coefficients are transformed from an orthogonal cutting data base using an oblique cutting transformation model as presented in the previous chapter. The surface finish is digitized using the true trochoidal kinematics of mil l ing process at each tooth period. The dynamically regenerated chip thickness, which consists of rigid body motion of the tooth and structural displacements, is evaluated at discrete time intervals by comparing the present and previous tooth marks left on the finish surface. The process is simulated by considering the instantaneous regenerative chip load, local cutting force coefficients, structural transfer functions and the geometry of mil l ing process. The proposed model is capable of providing information about the severity of any resulting vibration, the magnitude of the cutting forces, and the surface finish by combining the kinematics of both chatter free static and dynamic cutting with chatter vibrations in a unified mathematical model. 5.2 M o d e l l i n g of Dynamic M i l l i n g The simulation model presented in this section is similar to the earlier dynamic face mi l l ing work presented by Montgomery and Altintas [1] with modifications being made Chapter 5. Dynamic Milling and Chatter Stability 69 for ballend mil l ing analysis. The combination of the translating motion of the workpiece and the rotatory motion of the cutter results in a non-circular tooth path, or described as trochoidal. In previous investigations of the mil l ing process [2, 12], it has been customary, for the sake of simplicity, to assume that the path generated by the cutter tooth is circular, which results in simple chip thickness expression. However, as shown in chapter 3, the error introduced by the approximation becomes significant when the feed is high, or for cutters with small radii , such as ballend mills which have variable radii . As the axial depth of cut and feed per tooth are usually very small in ball end mi l l ing operations, especially when machining hard die and tool steels, the circular tooth path assumption may be insufficient to be applied in the analysis. Therefore, the true kinematics of mi l l ing presented by Martel lot t i [2, 42] are used to obtain accurate cutting force predictions. It should be noted that, although emphasis has been placed mainly on ballend mi l l ing , the model presented in this chapter is general enough to be used in the prediction of dynamic mi l l ing process for other cutter geometries outlined in chapter 3 and other cutting conditions. 5.2.1 Dynamic M i l l i n g Implementation In order to model the dynamic motion of the cutting edges in discrete time domain, the process is digitized and simulated at At[s] time intervals. Both the cutter and the workpiece kinematics are modelled as follows: Geometry of Tool Motion: The geometry of dynamic mill ing with ballend mi l l is shown in Figure 5.3. The cutter is digitized into kz slices axially with a step size of 8z. The center of the coordinate system is selected as the stationary spindle axis at (0,0,0), coincides with the tip of the cutter initially. The cutter rotates with a speed u clockwise. Each point on the flute j at height z is referenced by its immersion angle \& in global coordinate. During dynamic milling, the center of the cutter shifts by an amount Chapter 5. Dynamic Milling and Chatter Stability 70 Y 4 Workpiece Figure 5.3: Evaluation of chip thickness from the dynamically generated surfaces at present and previous tooth periods Chapter 5. Dynamic Milling and Chatter Stability 71 determined by the structural vibrations in the feed and normal directions: xc(t) = x\t) & yc(t) = y'{t) (5.10) where [x (t),y (t)] are the displacements of the cutter center relative to the stationary spindle axis due to structural vibrations at time t. Therefore, at any time t, the coordinates of a cutting edge point are calculated for each axial depth of cut kz as, x(kz,t) = R(ip)sin(t>) + x'(t) y(kZit) = R(iP)cos(V) + y'(t) > (5.11) z(kz,t) = kzAz The radial distance between the center of the spindle and the cutting edge point is, Rc(kz,t) = y/x(kt,ty + y(kz,ty. (5.12) The corresponding discrete angular rotation interval becomes AO = u>At. In the beginning of the simulation, the cutter is assumed to reach its steady state immersion ( 0 a ) without any vibrations, therefore the static and dynamic displacements are not considered and a smooth surface is generated by the rigid body motion of the mi l l ing system. One factor which influences the accuracy of the model is cutter runout, first studied in detail by Kl ine and Devor [10]. The effect of cutter runout is often induced by the tightening action of a set screw which causes the cutter to offset the tool holder. Other sources of runout include spindle runout, and cutter errors result from grinding. As a result, the teeth on the offset side of the holder have a larger effective radius than those on the other side. The presence of runout increases the average chip thickness (or uneven chip load distribution) which leads to higher force fluctuations, hence greater surface Chapter 5. Dynamic Milling and Chatter Stability 72 error. The general modelling technique is to replace the radius of each tooth, which can be a function of the axial depth of cut, by' an effective radius. Although the effect of the runout wi l l not be investigated in this study to simplify the analysis, it can be easily included in the proposed cutter model. Another possible modification to the cutter model is to include the effect of gash generation for some complex cutters. In the case of multiple fluted ballend mil ls , one of the flute is always grounded up to the ball tip while the rest of the flutes are grounded both radially and axially away from the tip to eliminate the finished surface marks. Again , this effect wi l l be ignored in the following study for simplicity, although it can be considered in the general model. Geometry of Workpiece Motion: The workpiece surface is digitized by a number of points and stored in Cartesian coordinates format in an array, see Figure 5.4. The center of the coordinate system is the same as the cutters' and it is initialized to the workpiece geometry. The digitized surface is represented by M points at each axial layer, each having the coordinates SURF(P(rn), z^), where point P(rn) — {x(m), y(m)} is located at axial depth Zk = kzAz. For example, the in plane coordinates of the digitized surface point at an axial depth of cut z = kzAz is stored in computer as : (5.13) x3(m) = SURF(rn, Zk,x) ya(m) = SURF(m,zk,y) The surface at axial layer z^ is represented by points m = 0,1,2, . . . M , and their coordinates are updated at each time interval At as a result of the feed motion. Unlike in the cutter model where structural vibrations are superimposed onto the tool motion, the workpiece used in mil l ing, in particular die machining with ballend mil ls , usually has shallow depth, tight clamping and is more rigid than the machine tool/cutter assembly. Therefore, although vibrations can be included in the workpiece model, for the sake of Chapter 5. Dynamic Milling and Chatter Stability 73 SURF(m,x) Figure 5.4: Dynamic model and Chip thickness regeneration mechanism in mi l l ing Chapter 5. Dynamic Milling and Chatter Stability 74 simplifying the analysis, it is reasonable to assume that the structural vibrations from the workpiece are negligible compared to the tools'. 5.2.2 Structural Dynamic Model Since the depth of cut is small in ball end milling operations, the spindle-cutting tool assembly can be represented by two orthogonal degrees of freedom in the the feed (x) and normal (y) directions lumped at the tool tip, see Figure 5.3. The cutter-spindle assembly is assumed to be rigid in the axial direction. The modal parameters of the structure are experimentally identified, and represented by two mutually perpendicular orthogonal modes in the x and y directions. x'(t) + 2(xunxx(i) + LJ2nxx'(t) = *£Fx(t) (5-14) y(t) + 2(yL,nvy'(t) + u>lvy'(t) = ^Fy[t) where (u}nx,Wny), {(x, Cv) a n ( i (kx,ky) are the natural frequencies, structural damping ratios and stiffness in feed (x) and normal (y) directions, respectively. The Fx(t) and Fy(t) are the dynamic cutting forces calculated according to the model presented in the previous section. The equations of motion (Eq. 5.14) are evaluated separately at each discrete time interval A i using recursive, discrete equivalent of the continuous differential equation. The discrete time domain solution of Eq. (5.14) was given earlier in Eqs. (5.6) and (5.7). Chip Thickness Estimation: The arc of cut is divided into M = Q3/A8 angular seg-ments, and the corresponding points on the arc of cut are stored in an array for each discrete axial element, see Figure 5.4. In the subsequent revolutions, the structural dis-placements are considered. The cutting edge penetrates into the workpiece due to the Chapter 5. Dynamic Milling and Chatter Stability. 75 rigid body feed motion combined with the structural vibrations of the mi l l ing system. At each time interval t, the cutter undergoes a rotation angle of 6(t) = u x t , where uj[rad/s] = 2ITN is the angular speed and N\revj a\ is the spindle speed. The table moves in the negative x direction with a feed speed of f\mm/s}. Therefore, the x coordinates of all points on each axial layer z^ of the surface, represented by points rn = 0,1, 2, ...M, are updated at each time interval A i by amount of incremental feed motion / X A i . SURF{m, zk>x) = SURF(rn, z{k),x) - f At (5.15) for all m , and k Depending on the magnitude and velocity of the vibration, the cutting edge point may be anywhere in the x — y plane of the cut. The two points, which are generated in the previous tooth period and are closest to the new cutting edge location, are identified by searching the previous surface array, see Figure 5.4. Using a linear interpolation between the two points, the point P3(m) = {xB(m), ys(m)}, which lies on the radial vector Rc(kz,t) but on the previously generated surface is evaluated. The radial distance between the point Ps(m) and the spindle center is given by, Rs(kztm,t) = ^x3(kz,m,t)2 + ys(kz,m,t)2 (5.16) The actual chip thickness is found by projecting the difference in the radial distances on the line which is passing through the ball center (figures 5.3 and 5.4), h(t) = \Rc(kz,t) - R„(kz,m,t)} sinK (5.17) where K has been given in chapter 3 as: K = s i r r 1 ^ (3.19) Chapter 5. Dynamic Milling and Chatter Stability 76 If the value of the chip thickness is positive, the tooth is in cut and the mi l l ing force model presented in chapter 4 is used to determine the cutting force. Otherwise, the tooth has jumped out of the cut due to excessive vibrations and the cutting force in that case wi l l be zero. , The chip thickness evaluation geometry is given in figures 5.3 and 5.4. As the cutter rotates at discrete (Ad) intervals, the material, which is swept by the cutting edge due to its rigid body and vibration motions, is identified and the surface points are updated. Thus, at any time, the surface array SURF has M number of points per axial level which represent the instantaneous arc of cut. Since the axial depth of cut is divided into Kz number of layers, M X Kz number of points represent the entire, semi-spherical shape cut surface. The simulation strategy is summarized as follows: • The cutter is divided into equal number of elements in the axial direction. • The radial planes of all axial elements, the arc of cut at each level, are digitized by equal number of points. • Using the exact kinematic of mil l ing the rigid body motion of each point on the cutting edge is calculated, and subtracted from the surface point left in the previ-ous tooth period to calculate the chip thickness. B y including static or dynamic displacements of the cutting edge point, the regenerative chip thickness can be calculated as well. o The cutting force coefficients are evaluated for each local chip thickness from or-thogonal cutting data base using oblique transformation model. • The cutting forces for all cutting edge points which are in contact wi th the material are evaluated one by one, and summed and resolved in the x, y and z directions. Chapter 5. Dynamic Milling and Chatter Stability 77 • The cutting forces are applied on the structure, vibrations are obtained, (or static deflections if the mass and damping are neglected). • The surface is updated, the cutter is rotated one angular increment, and the solution is repeated. 5.3 Chatter Stability The previous section provides an effective tool in predicting the onset of chatter in various mi l l ing operations and forms a basis to the "stability lobe" diagram, where the values for chatter free depth of cuts are shown like the one in Figure 5.5 analyzed by Smi th and Tlusty [23]. The diagram is plotted on axes of axial depth of cut versus spindle speed. The curve shown in the diagram represents the l imit of stability and separates the unstable machining conditions from the stable conditions. The stability chart provides useful information for a range of mil l ing operations, indicating which combinations of axial depth of cut and spindle speed will be stable. The stability lobe diagram can be best explained by examining the regeneration mech-anism, in particular, the phase difference between the inner and outer modulations on the surface. As shown by Tlusty [23], the "regeneration of waviness" is the dominant factor in self-excited vibration. In vibratory mil l ing, each cutting tooth removes a wavy surface created by the preceding tooth, and due to excessive vibration, creates another wavy surface. The phasing e between these subsequent undulations is determined as: Z-ir I\ * J\f where Nw and e are the integral and fractional number of waves between current and previous surfaces, and fc is the chatter frequency [Hz]. Chapter 5. Dynamic Milling and Chatter Stability 78 3 o CL CD Q o 0.012 0.01 0.008 0.006 0.004 0.002 + Analytical k=1 J\ k=0 • Simulation (Smith & Tlusty) Stable 0 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 Spindle Speed /10000 (rpm) Figure 5.5: Analytical and time domain stability limit predictions for a case analyzed by Smith and Tlusty Chapter 5. Dynamic Milling and Chatter Stability 79 Figure 5.6: Phase difference between the current and previous surfaces resulting from tool vibration Chapter 5. Dynamic Milling and Chatter Stability 80 A s s h o w n i n F i g u r e 5.6, m a x i m u m v a r i a t i o n i n c h i p thickness occurs w h e n t h e phase shift between two surfaces is IT, w h i c h causes c u t t i n g force v a r i a t i o n to e x c i t e v i b r a t i o n . O n t h e other h a n d , constant c h i p th ickness(s table c u t t i n g ) is o b t a i n e d w h e n t h e r e is zero phase shift , a n d at c e r t a i n values of e, m a x i m u m sel f -exci tat ion, resul ts . F o r each i n t e g r a l n u m b e r of wave n u m b e r Nw, due to different poss ib le values of e, t h e s t a b i l i t y of t h e s y s t e m changes, as reflected i n the c h a n g i n g height of each " l o b e " . A t l o w c u t t i n g speeds, t h e s t a b i l i t y is h i g h due to the presence of process d a m p i n g associated w i t h t h e t o o l f lank face/workpiece interference [43]. E x t e n s i v e efforts have been spent o n t h e e s t i m a t i o n of the process d a m p i n g . M o n t g o m e r y [1] m o d e l l e d the process d a m p i n g b y e s t i m a t i n g the interference between the v i b r a t i n g tool ' s flank face a n d t h e cut sur face . . O t h e r s [44, 45, 46, 47] have t r i e d to ident i fy the d a m p i n g d u r i n g t h e c h i p f o r m a t i o n process b y u s i n g the D y n a m i c C u t t i n g Force Coeff icients ( D C F C ) , w h i c h are d e t e r m i n e d -m a i n l y f r o m e x p e r i m e n t s . In general , the process gains a d d i t i o n a l s t a b i l i t y at low speeds w h e r e there are m o r e steeper a n d shorter v i b r a t i o n waves a n d hence larger d a m p i n g . O n , t h e o ther h a n d , as the c u t t i n g speed increases, the s t a b i l i t y lobes " e x p a n d " , i n d i c a t i n g a t r e n d of i n c r e a s i n g s t a b i l i t y . M a t h e m a t i c a l l y , th is c a n be e x p l a i n e d by t h e larger change i n s p i n d l e speed r e q u i r e d to p r o d u c e the same change i n e a c c o r d i n g t o E q . (5.18). T h e r e f o r e , t h e stable reg ion between each lobe becomes w i d e r as t h e s p i n d l e speed increases. 5.3.1 Prediction of Chatter Stability from Time Domain Simulation T h i s sect ion presents a useful m e t h o d for d e r i v i n g the chatter s t a b i l i t y d i a g r a m f r o m t h e resul ts o b t a i n e d by u s i n g the t i m e - d o m a i n s i m u l a t i o n p r o g r a m d e v e l o p e d w h i c h has t h e c a p a b i l i t y to p r e d i c t the occurrence a n d severity of chat ter v i b r a t i o n s u n d e r a n y p r a c t i c a l m i l l i n g c o n d i t i o n s . I n order to develop the s t a b i l i t y charts as e x p l a i n e d above, P T P ( P e a k - t o - P e a k ) graphs are used. A s i n d i c a t e d by S m i t h [33], the peak to peak values , Chapter 5. Dynamic Milling and Chatter Stability 81 calculated by scanning the maximums and mini mums of the vibration and cutting force signals, provide a useful and convenient way to interpret the results obtained from the simulations. A closed examination of the P T P values gives a good indication of the process's stability, which can be used to construct a stability diagram. The stability algorithm is set up as follows: • Specify milling operation, tool geometry and cutting conditions. • Select a range of spindle speeds and spindle speed step size. • At each speed, a simulation with a small axial depth of cut (e.g. base a = 0.5mm in our simulations) is performed initially to obtain the P T P data for chatter-free conditions. • Normalize the P T P values (no chatter) for unit axial depth of cut. • Run simulation program, stores maximum and minimum values of cutting forces and vibrations when the program reaches its steady state. Evaluate normalized P T P for each new axial depth of cut. • Compare normalized P T P values with those obtained in the based case (without chatter). If they exceed the chatter-free values by a wide margin (25 percents in our simulations), the axial depth of cut is reduced, and vice versa. • Simulations are then re-run with the new axial depth of cut. • Simulation stops when the borderline axial depth of cut is found. Resets and re-'t. starts for new cutting speed. A detailed break down of the algorithm is provided in Figure 5.7. The simulation time step should be kept constant for the selected range of speeds, therefore longer Chapter 5. Dynamic Milling and Chatter Stability 82 Select: Spindle Speed Based axial depth of cut (base a) Initial axial step size, min astep Set: Stability Criteria: tolerance % Iphase = False ± Run simulation for base a, Compute PTP values per unit depth of cut, F. = -EJEf-b a s e base a Store old_a = a a = a + A a V Run Time Domain Simulation Compute new Nomalized Ptp values NormF = PtpF Yes Iphase = True a = olda A a = . A a 2 F i g u r e 5.7: T i m e D o m a i n S t a b i l i t y L i m i t C a l c u l a t i o n A l g o r i t h m Chapter 5. Dynamic Milling and Chatter Stability 83 time is required in computing the stability lobes at low cutting speed. Furthermore, it was noticed that, during the simulations, there is always a transient region before the simulation reaches a steady state, and that part of the signal should not be used in the P T P computation. The length of each simulation run can be shorten considerably, for the interest of efficiency, or experience, by reducing the number of cycles. However, in general, a simulation period of at least 25-30 cutter rotations should be used, to allow the simulation to reach a steady state condition. Al though the time domain model provides a clear and rather precise picture of each mi l l ing operation such as the structural displacements, chip thickness and cutting force variations, and the cut surface generation during the machining process, it often requires multiple runs, and each run sometimes involves extensive computation load, as in the case of simulating the cutting at low speed range, which makes it sometimes unacceptable for practical use. Recently, Budak [48, 31] presented a new method for the analytical prediction of stability lobes in mill ing. The method is based on the formulation of dynamic mi l l ing wi th regeneration in chip thickness, time varying directional factors and the interaction with the machine tool structure. A brief discussion of the method is given in the following: The dynamic mil l ing model is first expressed in terms of the time varying directional dynamic mi l l ing force coefficients [A(t)j as, •{F(t)} = ±aKt[A(t)]{W)} (5-19) where A ( t ) represent the dynamic displacements of the cutter structure at the present and previous tooth periods respectively. Kt is the constant cutting force coefficient. Due to the variable force vector in mil l ing, the time varying directional matrix [A(i)] is periodic at tooth passing frequency o> = NfQ or tooth period T — 2ir/u. It depends on the Chapter 5. Dynamic Milling and Chatter Stability 84 immersion conditions and the number of teeth in cut. In a most simplistic approximation, Budak considered only the average component of the Fourier series expansion of i.e. r = 0, [-4o] = | JjA(t)\dt (5.20) Transfer function matrix of the machine tool system ([G(ZOJ)]) is then experimentally identified at the cutter/workpiece contact zone. B y replacing the dynamic displacement term in E q . (5.19) by the vibrations in frequency domain using harmonic functions, i.e. {A(icoc)} = [ l - e - ^ r ] e ^ [ G K ) ] { F } where u>cT is the phase delay between the vibrations at successive tooth periods T. Substituting {A(iu>c)} into the dynamic mil l ing Eq . (5.19) gives, {F}e^ = ±aKt[l - e - ^ T ] [ A 0 ] [ G ( i o ; c ) ] { F } e 1 ^ (5.21) To be able to obtain a non-trivial solution, the determinant of E q . (5.21) has to be zero, i.e. det[[I] + A[G0(iu>c)]] = 0 (5.22) The eigenvalue of the above equation can easily be solved for a given chatter frequency uc, static cutting factors (Kt)KT) which can be stored as a material dependent quantity for any mil l ing cutter geometry [19], radial immersion (4>st, <f>ex) and transfer function of the structure. Assuming only the two orthogonal modes are considered in [G(icj)] (i.e. Gxy = Gyx — 0.0), the eigenvalue A is obtained as: Chapter 5. Dynamic Milling and Chatter Stability 85 A = -7T-(ai ± Ja2 - 4a0) (5.23) zao where ao and a\ are constants and known for a given cutting geometry and structure. The crit ical axial depth of cut at chatter frequency uc is obtained by taking the real part of the eigen value, A T T M = ~~NJKS1 + } ( 5 - 2 4 ) Therefore, given the chatter frequency (UJC), the chatter l imit in terms of the axial depth of cut can directly be determined from equation (5.24). From E q . (3.19), the angular distance travelled by the tooth due to chatter frequency uc at tooth period T is found as K2 — 1 LJCT = c o s - 1 = - c o s - 1 2ip (5.25) np + 1 Note that KP — A / / A ^ = tan ^ and ip is the phase shift of the eigenvalue. Thus if k is the integer number of full vibration waves (i.e. lobes) imprinted on the cut arc, tjcT = 7T - 2ip + 2A;TT = e + 2A;TT (5.26) where ip — tan~xnp and e = 7r — 2ip is the phase shift between inner and outer modulations (present and previous vibration marks). The spindle speed N(rev/min) is simply calculated by finding the tooth passing period T(s), r = i ( £ + 2 f a ) -,N = ^ (5.27) Therefore, by approximating the time varying mil l ing directional factors by an average value, the stability l imit can be solved analytically. For each chatter frequency, the eigen Chapter 5. Dynamic Milling and Chatter Stability 86 value and the critical depth of cut are first obtained, then the corresponding spindle speed is computed for each stability lobe. The analytical formulation presented in this section was used as an additional tool to verify the predictions obtained from the time domain simulation. Preliminary evaluations of this method gave reliable and promising results. However, its adaptation to other complex cutter, such as ballend mills where there is a strong variation in the cutter and cutting geometries (hence variable directional coefficients) in the axial direction, is still unknown and wi l l be kept out of the scope in this study. 5.4 Simulations and Experimental Verification The dynamic mil l ing model has been implemented in C programming language on S U N / S P A R C workstation. The input data to the program includes the cutter geometry, workpiece material and dimensions, cutting conditions (axial depth of cut, radial immer-sion, chip load, spindle speed), machine tool dynamics, force model and coefficients, and other miscellaneous variables such as number of discretizing points on the cut surface and simulation time step. The simulation time step (sampling frequency) is selected to capture the highest vibration frequency in interest. As noted by Tlusty [23], the sampling frequency should be at least ten times larger than the highest natural frequency of the system encountered. Each simulation runs in a series of small time steps for the chosen duration. At each instant, the cutting forces and tool deflections are recomputed, the surface geometry is updated and stored. The output of the simulation program contains mi l l ing force prediction, dynamic displacements of the machine tool structure, and the finished surface profile. In the following sections, the proposed simulation model wi l l be tested on two different mil l ing cutters under various cutting conditions. The simulated and experimental results wi l l be provided in each case. Chapter 5. Dynamic Milling and Chatter Stability 87 5.4.1 Verifications wi th Cy l indr i ca l E n d M i l l Static C u t t i n g Simulations The simulation model is first tested on a simple cylindrical cutter geometry. In the first run, rigid mil l ing forces are simulated for a slotting operation and compared wi th the experimental results, see Figure 5.8. The work material TiQALAV was mil led by a single fluted carbide end m i l l (R0 = 9.525mm) with 30 degree helix angle. Different feed rates were chosen at an axial depth of cut a = 7.62mm and a spindle speed of 269RPM. The cutting force pattern is periodic at a tooth passing frequency. In a slotting operation, the tooth, due to helix, has a total immersion period of n -)- tan(io) * a/Ro = 206.5degrees. The axial forces are usually small in endmilling due to negligible cutting velocity in the axial direction. It can be seen that there is good agreement between the predictions and the experiments. D y n a m i c C u t t i n g Simulations and Verifications To verify the dynamic model established, the chatter simulation program is tested in an end mil l ing experiment with a cylindrical cutter investigated by Week, Alt intas and Beer [49] at the Machine Tool Laboratory of Technical University of Aachen. The workpiece material is aluminum alloy A l Z n M g C u 1.5 which is machined with a 30 m m diameter, three fluted helical end mil l with 30° helix angle and 110 mm gauge length. The measured mil l ing force coefficients are Kt = 6 0 0 M P a and KT = 0.07. The tests were conducted under a wide range of spindle speeds, axial depth of cuts, and radial width of cuts. The dynamic parameters of the end mi l l , which has one dominant mode in each direction, were determined from modal tests as: kx = 5590 N/mm (x = 0.039 CJX = 603#z ky = 5715N/mm £y = 0.035 uy = 666Hz (5.28) Chapter 5. Dynamic Milling and Chatter Stability 88 Figure 5.8: Experimental and Simulated Cut t ing Forces, cylindrical end mi l l ing Chapter 5. Dynamic Milling and Chatter Stability 89 2000 3000 Spindle speed N [RPM] 5000 w = 1/4 D Data by Week, Altintas and Beer, 1993 Simulation o no chatter -> © light chatter Experimental • chatter J Figure 5.9: Stabili ty Lobe Diagram for Cyl indr ica l End mil l ing. Week's Experiments Chapter 5. Dynamic Milling and Chatter Stability 90 Their results, summarized in the form of stability curves, are reproduced and shown in Figure 5.9 for two up-milling cases. A fixed feed rate of st = 0.07mm/tooth was used in al l tests and those stable and unstable cutting conditions are identified. Each small circle represents one experimental observation at a particular cutting speed and axial depth of cut. Whi le an empty circle implies that a stable cutting condition was achieved, a solid circle represents a clear occurrence of chatter under that particular cutting condition, characterized by a rapid and significant increase in the energy level (both cutting force and vibration amplitudes) concentrated at the chatter frequency. A series of chatter simulations were then run to verify Week's experimental results. The spindle speed was chosen as 3000RPM and the occurrence of chatter was monitored by increasing the axial depth of cut in small steps (8a = 0.5mm). A l l other cutting parameters are the same as those employed in Week's experiments. The results of the simulated cutting forces and vibrations are shown in figures 5.10 and 5.11. A close examination of the simulation results reveals that chatter has developed substantially at approximately a = 2.5mm to a = 3mm, signifies by a more than 100 percent jump in the magnitudes of both the cutting forces and vibrations. After this cr i t ical axial depth of cut (a = 3mm), the cutting forces continue to increase in a non-linear fashion and become unreal!stically high. The system finally appears to have saturated and is no longer increasing with time. After obtaining positive feedback from the ini t ial simulations, a series of half immer-sion up-mill ing simulations were set up to generate a stability lobe chart. Figure 5.12 shows the P T P values recorded for different axial depth of cuts, ranging from 1mm to 3.25mm, and spindle speeds. Other cutting conditions are kept constant. Each sim-ulation was allowed to run long enough to assure that steady state has been reached before the P T P are being evaluated. The P T P values for the last 5 revolutions (a total of 25 are used in all simulations) are shown here for a selected speed and axial depth Chapter 5. Dynamic Milling and Chatter Stability Figure 5.10: Cut t ing Force Simulation I, cylindrical cutter, Week's experiments apter 5. Dynamic Milling and Chatter Stability Figure 5.11: Vibrations Simulation I, cylindrical cutter, Week's experiments Chapter 5. Dynamic Milling and Chatter Stability 93 range. It can be seen that, the peak amplitude varies both axially and horizontally, implying that there exists some combination of axial depth of cut and spindle speed in which the cutting process is stable. Although the P T P graph gives good indication on the magnitude of the forces and vibrations for each cutting condition, it does not give direct information (interpolation of data required) to the process planners on selecting the stable axial depth of cuts and spindle speeds. Therefore, the algorithm presented in section 5.3.1 is tested on the P T P information for various speeds starting from 1 0 0 0 R P M up to 10000RPM in steps of 1 0 0 R P M . The result is plotted on top of the experimental results observed by Week in Figure 5.9. Discrepancy occurs due to the difference in the stability criteria employed i n two cases. For instance, Week's simulation algorithm "as-sumes that the process is unstable when the peak values of the vibration grow in thir ty consecutive oscillation periods, otherwise the process is assumed to be stable" [49], while the method of P T P identification technique, which was proven to be a reliable tool, was used in this study. Furthermore, there exists no precise definition of chatter to be relied upon in conducting the experimental observations. In other words, the choice between assigning a "light chatte" and "chatter" to the observation, as indicated in Figure 5.9, is rather a subjective decision and remains to be justified. Although there are differences between the two, Figure 5.9 indicates that, the values predicted by the analysis are at least consistent and reasonably closed to those obtained by the experiments. A separate simulation run, from which the results are shown in Figure 5.13, was per-formed at a lower speed oi N = 1000RPM to provide more understanding of machining under unstable conditions. A very high axial depth of cut (a = 4mm which exceeds the stability l imit based on the experimental results from Week) was selected. Three char-acteristic regions can be identified in the tool's responses and the cutting force patterns. The first few revolutions represent the signals prior to the onset of chatter, where the amplitude of oscillation is small, increasing slowly with time. Its Fourier spectrum is Chapter 5. Dynamic Milling and Chatter Stability 94 Figure 5.12: Peak To Peak Graphs from Time Domain Simulation I, cyl indr ical cutter. Week's Experiments Chapter 5. Dynamic Milling and Chatter Stability 95 Figure 5.13: Cut t ing Forces and Vibrations Simulation II, Week's Experiments Chapter 5. Dynamic Milling and Chatter Stability 200 400 600 800 1000 1200 1400 1600 1800 2000 frequency (Hz) 0 200 400 600 800 1000 1200 1400 1600 1800 2000 frequency (Hz) Figure 5.14: Fourier Spectrums from Simulations II, Week's Experiments Chapter 5. Dynamic Milling and Chatter Stability 97 (3D) || Print || 18 Aug 1995 || flatsurf.pitTf Figure 5.15: Simulated Surface under Chatter Condit ion, from Simulations II Chapter 5. Dynamic Milling and Chatter Stability 98 |(20)[| Prim || 18Augl995 | | chatmark.pH || -5 -4 -3 -2 -1 Feed Direction [mm] Figure 5.16: Simulated Surface Mark at z = 2.0mm under Chatter Condi t ion , from Simulations II Chapter 5. Dynamic Milling and Chatter Stability 99 distributed in a wide bandwidth around the tooth's passing frequency (333Hz). During the next several revolutions, vibrations increase which modulate the chip thickness dis-tr ibution and influence the cutting forces. This finally results in fully developed chatter in which the tool oscillates with a very high amplitude and at a frequency (chatter fre-quency) close to the structure's dominant natural frequency (600 Hz) , see Figure 5.14. Figure 5.15 shows the simulated finished surface. As seen, the surface roughness is dom-inated by severe vibration marks which leave undulations of approximately 10 microns on the surfaces. The vibration marks left on the surface: has a wave length A of approxi-mately 0.428mm. Since the workpiece is moving at a speed of 3.5rnm/s, and the tooth period is Tc = 0.02sec, that is, the wavy surface has a period of Twb — A / / = 0.122.sec., which is much greater than tooth period Tc or the vibration periods in either direction (wx = 594i/z(0.00168.sec.), wy — 675Hz(0.00148.sec.)). This phenomena is consistent wi th the observation made by Montgomery [l], and referred to as washboarding, which exists due to the dynamic interaction between the vibrating (and rotating) cutter and the moving workpiece. As a result, there is an integral number plus a fraction of vibrat ion cycles between successive tooth periods. The fractional waves accumulate in every Nw tooth periods, which leads to the low frequency wave left on the surface. In general, if the vibration frequency is different in the two directions, the resultant vibration frequency has to be known for accurate Nw estimation. The time domain stability lobe simulation is further verified with the analytical chat-ter prediction outlined in section 5.3.1. The analytical formulation was implemented and tested under the cutting conditions similar to Week's experiments. Figure 5.17 shows the results of two mil l ing cases obtained from the analytical predictions and time domain simulations respectively. The excellent agreement confirms the accuracy and reliability of the time domain simulation model. Chapter 5. Dynamic Milling and Chatter Stability 100 Stability Lobes - 90 degrees immersion up-milling 0 -I 1 1 1 1 1 1 1 1 H 1 0 1000 2000 3000 4000 5000 • 6000 7000 8000 9000 10000 Spindle speed [RPM] a) Stability Lobes - 60 degrees immersion up-milling 7 - - - - Time Domain, 30 deg. helix 6 • I of cut 5 -Analytical prediction, 0 deg. helix :ial depth [mm] 4 3 -eg a 2 -o O 1 -0 -( 1 ' > i ' i i i I | ) 1 1 1 i ..... j i i i 1000 2000 3000 4000 5000 6000 7000 8000 Spindle speed [RPM] 9000 10000 b) Figure 5.17: Analytical and Time Domain Predicted Stability Lobes, cylindrical cutter Chapter 5. Dynamic Milling and Chatter Stability 101 5.4.2 Verifications with Ballend Mil l Static Cutt ing Simulations In this section, rigid mil l ing forces with ballend cutters are simulated and compared wi th the experimental tests shown in the previous chapter. The results of both the simulations and measured cutting forces are plotted for two different cutting geometries. Figure 5.18 shows the chip thickness profile at the cutting edge for a slotting operation. It can be seen that, during the first revolution, the chip thickness has not developed fully (st = 0.0508mm/tooth) as the cut surface has been initialized to the tool's geometry. The profile for the subsequent revolutions verify the correctness of the chip thickness prediction by the simulation. In figures 5.19, simulations of mil l ing force wi th ballend cutter are shown separately for a slot cutting and a half immersion up mi l l ing operations. Comparison of the two shows good agreement between the predicted and the experimental cutting force patterns, except the slight numerical instability arises when the tooth leaves or enters the cut abruptly such as in the half-immersion case shown in the bottom diagram of Figure 5.19. This problem can be solved by increasing the integration step size. The model is furthered tested by examining the simulated surface profile in r igid mi l l ing . Since there is no vibration in the process, the expected magnitude of the surface feed mark should be close to the analytical prediction derived by Marte l lo t t i [2]: k = 8[J2f» + 3tN,M ( 5 - 2 9 ) where h is the height of the tooth mark above point of lowest level. It is a function of the feed rate st, the number of flutes Nj, and the radius of cutter R. For cutters which have variable radius in the axial direction, the height of the feed mark changes axially, as shown i n Figure 5.20. Here, the variation of the feed marks at two axial locations (zi — 0.27mm, z2 = 1.08mm) resulting from a single fluted ballend mil l ing operation Chapter 5. Dynamic Milling and Chatter Stability 102 Static Chip Thickness Variation at tooth tip 0.06 -10 0 10 20 30 40 SO 60 70 80 90 100 110 120 130 140 150 160 170 180 190 Immersion Angle [Degrees) Figure 5.18: Model Verification I - Chip Thickness Profile Chapter 5. Dynamic Milling and Chatter Stability -800 • L Rotation Angle [Deg.] Figure 5.19: Model Verification II - Cutt ing Force Patterns Chapter 5. Dynamic Milling and Chatter Stability 104 are shown. The feed rate, which is 0.0508mm/tooth, can be seen between the peaks of the marks. The value of h, calculated with Eq . 5.29, gives a value of 0.0001272mm at R(z = 0.127) = 2.52mm and 0.00007296mm at R(z = 1.08) = 4.405mm, both showing good agreements with the predicted values. The accurate prediction of vibration free, static mil l ing forces on both cylindrical and ballend cutters verify the following facts about the model; • The kinematic model of chip thickness calculation is correct. • The oblique transformation method used in calculating the mil l ing force coefficients from a general orthogonal cutting parameters is sufficiently accurate. Dynamic Test and Simulation I Bal lend mil l ing tests at different speeds were conducted to verify the dynamic simula-tions. A single fluted carbide ball end mi l l with 9.525 radius, 30 degree nominal helix and zero degree rake angle was used in slot mil l ing of the t i tanium alloy with a = 2.54mm axial depth of cut and feed rate of st = 0.0508mm/tooth/rev. The following sections out-line the experimental procedure in identifying the dynamic characteristics of the ball m i l l attached to the spindle assembly of a S A J O vertical mil l ing machine as well as presenting the experimental and simulation results. Experimental Set-up: A P C B accelerometer is attached onto the structure close to the tip of the ballend cutter (see Figure 5.21). Transfer function at the tool tip is obtained by exciting the structure with an impact hammer and recording its force response through a H P 3562A dynamic signal analyzer. The identified dynamic parameters and cutting conditions are given in 5.1. kx = 5590 N/mm £x = 0.039 ux = 593.75Hz ky = 5710 N/mm (y = 0.035 uy = 675Hz (5.30) Chapter 5. Dynamic Milling and Chatter Stability 105 |r(z=0.27 r(z=1. -0.55 -0.5 -0.4 -0.35 - 0.3 0.25 -0.2 -0.15 Distance X In Cutting Direction [mm] 0.00012 0.00008 | o 0.00006 o o | 0.00004 « r(z=0.27) \ r(z= 1.08)1 -0.55 -0.4 -0.35 -0.3 -0.25 -0.2 -0.15 Distance X in Cutting Direction [mm) 0.05 0.00014 0.00012 0.0001 f 0.00008 I o 0.00006 » u € 0.00004 w Figure 5.20: Model Verification III - Surface Feed Marks: a) $ = 0° (up mi l l ing side), b) * = 90° (down mil l ing side) Chapter 5. Dynamic Milling and Chatter Stability 106 Due to the large static error results from integrating the signal measured by the ac-celerometer, the static stiffness of the system is measured by performing a static stiffness test and the result is shown in Figure 5.23. To verify the rigid workpiece assumption, the transfer function of the Ti tan ium workpiece (150 X 100 X 80mm) is also measured and shown in Figure 5.24. In die machining, the workpiece is usually stiffer than the tool. Therefore, as proven by the measurement (about 3 times higher than the tool), only the dynamics of the tool is considered. The performance of the model in predicting the dynamic mil l ing process is illustrated by two sample time domain simulations and mil l ing experiments. The tests were per-formed on a S A J O mil l ing machine retrofitted with an in-house developed research C N C , and cutting forces were recorded on a P C through Kist ler dynamometer and charge am-plifiers. The cutting conditions for all cases were identical except the spindle speed and damping coefficients (see table 5.1) 1 . First the experimental results are shown in figures 5.25 and 5.26, together with the Fourier spectrums (see Figure 5.27). The experiments were repeated at spindle speeds N=115 rev /min to N=1100 rev /min in roughly 6 equal steps, at identical cutting conditions. The measured and simulated cutting forces during low and higher speed mi l l ing tests are shown in figures 5.28 to 5.30 respectively. It was observed from the experimental results that the chatter vibrations are quite evident during the higher speed machining (figure 5.30), whereas there is almost no chatter at N=115 r ev /min (figure 5.28). Since all the cutting conditions are identical except the cutting speed, the absence of chatter vibrations is attributed to the process damping which is known to be quite effective when the tooth passing frequency (i.e. cutting speed) is significantly lower than the vibrat ion frequency. The process damping reflects the combined effects of tool flank penetrating, 1Note: Damping Ratios of ( x •— ( y = 0.15 are used at the low spindle speed (JV = 115.RP.M) in milling simulation. Chapter 5. Dynamic Milling and Chatter Stability 107 Accffileromete impulse hammer . D o w e r u n i t ^ digital signal analyzer \ O O O O C D O O oooo ooo oooo ooo ooo ooo ooo ooo Figure 5.21: Experimental Setup - Dynamic Test Chapter 5. Dynamic Milling and Chatter Stability 108 0.00006 0.00005 0.00004 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] 0.00006 0.00005 0.00004 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency [Hz] Figure 5.22: Measured Transfer Function at Tool T i p , Ballend M i l l i n g Test I Chapter 5. Dynamic Milling and Chatter Stability 109 0.07 » i = 1 1 ! 1 1 1 0 200 400 600 800 1000 1200 Applied Force [N] Figure 5.23: Static Stiffness Measurements, Ballend M i l l i n g Test I Chapter 5. Dynamic Milling and Chatter Stability 110 o u c a a. <p o a> OC 8.00E-05 7.00E-05 6.00E-05 5.00E-05 4.00E-05 3.00E-05 2.00E-05 1.00E-05 O.OOE+00 500 1000 1500 Frequency [Hz] 2000 2500 Figure 5.24: Measured Workpiece Transfer Function, Ballend M i l l i n g Test I Chapter 5. Dynamic Milling and Chatter Stability Speed = 115 RPM \t—1+. 0.2 0.4 T 0.6 0.8 I V !.2 Time [sec] Speed - 265 RPM Time [sec.I Speed = 430 RPM Time [sec] Figure 5.25: Dynamic Ballend M i l l i n g Test I, Speed = 115-430 R P M Chapter 5. Dynamic Milling and Chatter Stability 112 Speed = 730 RPM •800 1 Time [sec] Speed = 1100 RPM 0.25 Time [sec] Speed =1450 RPM 800 j Time [sec] Figure 5.26: Dynamic Ballend M i l l i n g Test I, Speed = 730-1450 R P M Chapter 5. Dynamic Milling and Chatter Stability 113 Speed = 430 RPM 2 120 « 100 1 80 t . 60 S 40 200 400 600 800 Frequency [Hz| 1000 1200 Speed = 730 RPM — 120 V 100 1 80 f 60 | 40 t 2 0 u. o •[ 500 1000 Frequency [Hz] 1500 2000 Speed = 1100 RPM 120 — 100 OB ? 80 f 60 | 40 t 20 iky] — • . A. . 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency [Hz) Speed = 1450 RPM 120 5oo o •§80 f 60 140 t 20 u. 0 i . . ll . 'I i W f WW 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Frequency [Hz| Figure 5.27: Fourier Spectrum of F x from Ballend M i l l i n g Test I Chapter 5. Dynamic Milling and Chatter Stability 114 contacting and rubbing against finished wavy surface of the workpiece. Ana ly t ica l mod-elling of such a contact and rubbing phenomenon has been proven to be difficult [1], and it is usually obtained by comparing the cutting forces or vibrations obtained from time domain simulations and measurements [50]. The analysis of the test results indicated that the damping ratio at the low speed cutting test is 15%, which is approximately four times larger than the measured structural damping ratio of 4%. Consequently 15% and 4% damping ratios were used in the simulations of mil l ing tests at N=115rev/min and N=1100rev/min, respectively. The agreement between the amplitudes and chatter behavior of the mil l ing tests and simulations is quite reasonable. The Fourier spectrums of the simulated and experimentally measured cutting forces are also provided in figures 5.29 and 5.31. The spectrum of the low speed test is dominated by the average value of the cutting forces (0Hz), and the strength of the force at tooth passing frequency (1.91Hz) and its first harmonic (3.82Hz). Since there are no chatter vibrations, the spec-t rum is dominated by the static cutting forces generated by the rigid body motion of the mi l l ing system. At N=1100 rev /min however, in addition to the static force compo-nents at tooth passing frequency range (18.33Hz), the spectrum indicates the presence of chatter vibrations at 600 to 700 Hz range which corresponds to the structural modes in feed (x) and normal (y) directions. Again, the agreement between the simulation and experimental results are quite satisfactory. The dynamic mil l ing simulation model can be used to prepare stability lobes for the generation of chatter free N C tool paths, as suggested by Week et al. [49]. Chapter 5. Dynamic Milling and Chatter Stability 115 Figure 5.28: Measured and Simulated Cut t ing Forces, N=115 R P M , Bal lend M i l l i n g Test I Chapter 5. Dynamic Milling and Chatter Stability 116 200 180 + 160 140 120 100 +: 80 -J-l 60 40 20 +| 0 40 60 80 100 200 180 160 + 140 120 100 80 60 40 20 0 kL /^y^ -^ AAjU, ^_ 20 40 60 80 100 200 T jjjeo •geof a 40 o &20 CO £ 0 0 § 8 0 + o "-60 cn #40 4-j O20 --j 20 40 60 Frequency [Hz] 100 a) Experimental Results 200 j z 180 -• ¥ 160 •• 3 0 140 -at a CO 120 •• >. Li. 100 •• O U 80 -O U . 60 • Ol C 40 •• Cut 20 - ' 0 --200 5 80 060 § 4 0 -f at 820 ifioo at « 80 0 01 020 \ ' \ \ 20 40 60 80 100 20 40 60 Frequency [Hz] 80 100 b) Time Domain Simulations' Figure 5.29: Fourier Spectrum from Ballend Mi l l i ng Test I, N = 115 R P M Chapter 5. Dynamic Milling and Chatter Stability 117 Figure 5.30: Measured and Simulated Cut t ing Forces, N=1000 R P M , Ballend Mi l l i ng Test I Chapter 5. Dynamic Milling and Chatter Stability 118 120 j 2 •J300 -3 | 80 -a. tn £ 6 0 • I 40 J 200 400 600 800 1000 1200 1400 1600 Frequency [Hz] 120 j 2 rroo •• ? I g80 -a co 200 400 600 800 1000 1200 1400 1600 Frequency [Hz] Experimental Ffr8i|lffi Time Domain Simntotinnc Figure 5.31: Fourier Spectrum from Ballend Mi l l ing Test I, N = U0O R P M Chapter 5. Dynamic Milling and Chatter Stability 119 Table 5.1: T ime Domain Simulation Conditions for Dynamic B a l l E n d M i l l i n g Tests I Workpiece material Mi l l i ng mode Ti tan ium Ti^AlV^ Slotting Spindle speeds (N) Feed rate (st) A x i a l depth (a) Test 1: 115 - 1100 [rev/min] 0.0508 mm/ too th 2.54 m m Cutter material B a l l radius (Ro) Rake angle (an) Helix angle (i0) Number of flutes Solid carbide 9.525 m m 0 degree 30 degree 1 Chapter 5. Dynamic Milling and Chatter Stability 120 D y n a m i c Test and Simulation II The stability method is applied to a practical ballend mil l ing operation. Exper i -ments were conducted on a F A D A L C N C machining center. The mil l ing system under consideration is similar to the one shown in Figure 5.3. It is represented by a single degree-of-freedom system with the following parameters identified from impact test (see Figure 5.32): kx = 5725Af/mm (x = 0.04 OJX = 1150tfz ky = 5740N/mm £y = 0.05 uy = 880#z (5.31) Table 5.2: Cut t ing Conditions for Dynamic B a l l E n d M i l l i n g Tests II Workpiece material M i l l i n g mode Ti tan ium Ti^AlV^ Half immersion Upmil l ing Spindle speeds (N) Feed rate (st) W i d t h of cut (w) A x i a l depth (a) 1000 - 6000 [rev/mini in 1000 R P M steps 0.0508 mm/tooth 6.35 mm (fixed) Variable Cutter material B a l l radius (Ro) Rake angle ( a n ) Helix angle (iQ) Number of.flutes Solid carbide 6.35 m m 0 degree 30 degree 1 A single fluted ballend cutter was used in an up mil l ing operation wi th T i t an ium alloy workpiece, which is assumed to remain rigid in all simulations. A range of spindle speeds and axial depth of cuts were selected and the measured cutting forces of two particular cases are showed in figures 5.33 and 5.34. At each spindle speed, all the cutting conditions (see table 5.2) except the axial depth of cut are kept constant. Increasing increments of axial depth of cut a are taken until the point of chatter onset occurs. Then the procedure is repeated for a new spindle speed. Chatter onset was determined by Chapter 5. Dynamic Milling and Chatter Stability 1 Impact Hammer Signal (X-direction) 0.002 0.004 0.006 0.006 0.01 Time [sec.) Impact Hammer Signal (Y-direction) 0.002 0.004 0.006 0.008 Time [sec] 0.01 Accelerometer Signal (X-direction) 40 S 20 1 0 2 -20 -40 |002 0.004 0.006 0.008 0.01 O.d 12 Time [sec] Accelerometer Signal (Y-direction) Time [sec] 500 1000 1500 2000 Frequency [Hz] 2500 3500 4000 1000 1500 2000 Frequency [Hz] 3500 Measured Dynamic Response at Tool Tip, X-direction Measured Dynamic Response at Tool Tip, Y-direction Figure 5.32: Transfer Function Measured at tool tip. Ballend Test II Chapter 5. Dynamic Milling and Chatter Stability 122 Single flute Ballend cutter, b=6.35mm, speed=1000RPM, a=2mm 400 T -400 J-Time [sec] Single flute Ballend cutter, b=6.35mm, speed=2000RPM, a=2mm 300 -Time [sec] Single flute Ballend cutter, b=6.35mm, speed=3000RPM, a=2mm 300 -400 1 Time [sec] Single flute Ballend cutter, b=6.35mm, speed=4000RPM, a=2mm 300 T -400 - 1 Time [sec] Figure 5.33: Ballend M i l l i n g Experiments II, a = 2mm Chapter 5. Dynamic Milling and Chatter Stability 123 Single flute Ballend cutter, b=6.35mm, speed=1000RPM, a=3mm 400 T •400 Time [sec] Single flute Ballend cutter, b=6.35mm, speed=2000RPM, a=3mm 400 T -400 Time [sec] Single flute Ballend cutter, b=6.35mm, speed=3000RPM, a=3mm -600 Time [sec] Single flute Ballend cutter, b=6.35mm, speed=4000HPM, a=3mm 600 -800 1 Time [sec] Figure 5.34: Ballend M i l l i n g Experiments II, a = 3mm Chapter 5. Dynamic Milling and Chatter Stability 124 observing the output of the cutting forces, as well as the sound pressure recorded by the microphone. The audible sound associated with high amplitude oscillation can be used as a further check of an unstable machining operation. It should be noted that, due to the extremely high strength property of TzQALiV, difficulties were experienced especially during those tests at high cutting speeds (above 4 0 0 0 R P M or 160 m / m i n ) . During high speed machining, the cutting tool deteriorates and wears out very fast, which may introduce inconsistency and inaccuracy into the data collected. Therefore, each tool was monitored closely throughout the tests to report any significant tool wear problem, a small chip load was chosen in all the tests to prolong tool life, and replacement of tools was done after every 20-30 tests, depending on the cutting conditions and the condition of that particular tool. The experimental results are presented in a way similar to Weeks' (Figure 5.9) in Figure 5.36 where the classical stability chart obtained from the experiments is plotted, on top of the simulation results. Fine step size was chosen in the simulations: the min imum step size of the axial depth of cut was 0.0125mm and the spindle speeds were run from 1000 to 5000 .RPM in 100RPM steps. For the 1mm axial depth of cut experiments, all the tests and simulations showed no significant vibrations and cutting force fluctuations, implying stable cutting was achieved. As the depth of cut increases, the onset of chatter was recognized and recorded, as indicated in the chart. A t the same time, the simulation program searches for the l imit of stability by applying the P T P algorithm given above. For example, both the measured and simulated P T P forces in the feed direction are plotted in Figure 5.35 for three different axial depth of cuts. F rom the experiments, chatter was noticeable at a few spindle speeds (A^ = 3000,4000, 5000RPM) when a = 3mm. A further examination of Figure 5.35 reveals that at the l imi t of stability, the P T P forces have jumped significantly (more than 150 percents) wi th a 50 percent increase in the depth of cut, as noted by Tlusty [23]. Although force vibrations w i l l Chapter 5. Dynamic Milling and Chatter Stability 125 always present in the process, it does not significantly influence the DC component of the cutting forces, as the waves (resulting from force vibrations) left on the surface are repeating at the tooth passing frequency, thus making "the chip thickness independent of the amplitude of the forced vibration" [33J as long as the cut is stable. From Figure 5.35, it can be seen that the process becomes unstable at N — 3Q0QRPM and a = 3mm in which there is a substantial jump in the cutting forces. Although the agreement between the chatter behavior of the milling tests and simu-lations is quite reasonable, the simulated stability lobes are all "squeezed" and "packed" together and have a rather flat shape especially at low cutting speeds, which deserves some further explanation. Firstly, the simulation model predicts a lower stability limit especially at low speeds because it doesn't take into account the effect of process damp-ing. Furthermore, due to the high chatter frequency to tooth passing frequency ratio (fc/N * Nj) at low speeds, there are numerous number of vibration waves left on the surface, which requires an extremely high sampling frequency to capture the "regener-ation of waviness" mechanism (vibration waves left on surface). In the simulation, the sampling frequency was chosen as 35 times the natural frequency of the system and the predictions appeared to be more stable. The third reason can be explained by the chip thinning effect which is an unique characteristic in ballend milling. At low axial depth of cut, the stiffness of the system is high due to the increased cutting pressures at small chip thickness. The immersion condition also influences the stability lobe prediction. Two different cutting modes are shown in Figure 5.39. Figure 5.39a shows the cutting geometry used in the ballend experiments II and in the stability simulation. The width of cut is constant (ii; = 6.35mm). A close examination of the wave regeneration mechanism along the axial direction reveals that, due to the changing immersion angle, the number of vibration waves left on the surface changes along the axial direction. There is no longer a constant Chapter 5. Dynamic Milling and Chatter Stability 126 1200 500 1000 1500 2000 2500 3000 Spindle Speed [RPM] 3500 4000 4500 5000 11 Simulation, a = 2,3 mm (3 — Experimental, a = 2,3,4 mm Figure 5.35: Experimental and Simulated P T P results. Bal lend M i l l i n g Test II Chapter 5. Dynamic Milling and Chatter Stability CL m TJ O 5 4 5 4 3.5 3 2.5 2 1.5 1 0.5 0 A * T r I k A c r ^ -^v- A / \ r \ ( —fc If ) V. 1- -f> -1 1000 2000 3000 Spindle Speed [RPM] 4000 5000 Simulation o no chatter -i • light chatter Experimental • chatter J Figure 5.36: Simulated Stability Lobe Diagram for Ballend mi l l ing Test Chapter 5. Dynamic Milling and Chatter Stability 128 j(20)|| Pnnt || 28 Aug 1995 || s3000a2.plt || 600 500 400 300 8 100 o LL -300 0.00 Cutting Forces, a = 2 mm L _J_ 0.10 0.20 0.30 Time [sec] 0.40 Fx _i L _ _ l 0.50 (2D) |i Print || 28 Aug 1995 || s3000a2v.plt || 0.10 0.05 h c E 8 o.oo -0.05 0.00 0.10 Dynamic Displacements, a = 2 mm 0.20 0.30 Time [sec] 0.40 0.50 Figure 5.37: Simulated Cut t ing Forces and Vibrations. Ballend M i l l i n g Test II Chapter 5. Dynamic Milling and Chatter Stability 129 l(2D) || Print || 28 Aug 1995|| s3000a3f pit || 1500 1000 Z. 500 -500 Cutting Forces, a = 3 mm (2D) || Print || 28 Aug 1995 || s3000a3v.plt || 0.2 Dynamic Displacements, a = 3 mm .,..1 l_ 0.00 0.10 0.20 0.30 Time (sec] 0.40 Dy Dx 0.50 Figure 5.38: Simulated Cut t ing Forces and Vibrations. Ballend Mi l l i ng Test II Chapter 5. Dynamic Milling and Chatter Stability 130 a) b) e / z=z1 c) wave surface / \ ± / z=z2 d) wave surface Figure 5.39: Wave generation on different cutting geometries. Bal lend M i l l i n g Chapter 5. Dynamic Milling and Chatter Stability Figure 5.40: Simulated Stabili ty Lobe Diagram for Ballend mil l ing III Chapter 5. Dynamic Milling and Chatter Stability 132 7 JL 0 1 2 3 4 5 6 7 X - Feed direction [mm] 7 T 0 20 40 60 80 100 Immersion angle [deg.] N = 3000 RPM 7 T 0 1 2 3 4 5 6 7 X - Feed direction [mm] 7 r 0 20 40 60 80 100 Immersion angle [deg.] N = 6300 RPM Figure 5.41: Simulations showing the waviness on cut surface at different cutting speeds Chapter 5. Dynamic Milling and Chatter Stability 133 integral number of waves along the axial direction. Depending on the cutting conditions, the stability of the process may change substantially, especially at low speeds, when the process damping is high due to shorter wave length cycles. O n the other hand, if the radial immersion angle is constant (i.e. variable width of cut), as shown in Figure 5.39b, there is always an identical number of vibration waves left on the surface along the axial direction. This phenomena is tested in a half i m -mersion ballend upmill ing simulation, in which R0 = 9.525mm, kx = 19300iv"/mm, ky = 14200A/ r/mm. Other cutting conditions are the same as in the second simulation. F ig -ure 5.40 shows the simulated stability lobes. The influence of spindle speed is demon-strated by examining the simulated results at N = 3000RPM and N = 6300RPM. A t N — 3000RPM, the integral number of waves imprinted on the surface per tooth spacing is, according to E q . 5.18, Nw = 4. (see Figure 5.41a) The self excitation mechanism takes place and resulting in unstable process. Whi le keeping other conditions constant except for a higher spindle speed (N — 6300RPM), as shown in diagram 5.41b, the number of waves between each teeth is lower, i.e., Nw = 2 and both the cutting forces and the tool vibrations reduce significantly. This change in spindle speeds has a significant im-pact on the stability of the process, which are reflected in Figures 5.42, 5.43, 5.44, and 5.45. 2 The resultant cutting forces, vibrations, and surface finish demonstrate clearly the advantages of regulating under a stable machining condition using the stability chart developed. 2 The curvature of the surface is filtered out for better visualization Chapter 5. Dynamic Milling and Chatter Stability 134 (20)|| Print || 8 Jun 1995 || force.plt | O) c O 1000 500 -500 -1000 0.00 Cutting Forces, 3 fluted Ballend cutter, half immersion up milling fd=0.0508mm/tooth,a=2.85mrn N = 6300 R P M 0.05 0.10 Time [seal 0.15 0.20 Fy F z Fx (2D) || Prim || 6Jun1995|| force .pH || Cutting Forces, 3 fluted Ballend cutter, half immersion up milling | I fd=0.0508mnvtooth, a=2.85mm Time [sec] Figure 5.42: Simulations of cutting forces at 2 different speeds, Ballend milling Chapter 5. Dynamic Milling and Chatter Stability 135 (2D) |[ Print || 8Junl995| | vibrateplt | E E 5 0.20 0.15 0.10 0.05 0.00 -0.05 -0.10 -0.15 -0.20 0.00 Dynamic displacement, 3 fluted Ballend cutter, half immersion up milling fd=0.0508mrrvtooth,a=2.85mm 0.05 0.10 Time [sec] N = 6300 R P M 0.15 0.20 Dy Dx (2D) || Print || 6 Jon 1995 || vibrate pit [ 0.20 0.10 E E 0.00 g 2? .o > -0.10 -0.20 0.00 Dynamic displacement, 3 fluted Ballend cutter, half immersion up milling fd=0.0508mm/tooth, a=2.85mm 0.05 0.10 Time [sec.| 0.15 Dy Dx 0.20 Figure 5.43: Simulations of vibrations at 2 different speeds, Ballend mi l l ing Chapter 5. Dynamic Milling and Chatter Stability 136 (20)|| Print || 8 Jun 1995|| fftfx.plt [ Figure 5.44: F F T of the simulated cutting forces at different speeds, Ballend milling Chapter 5. Dynamic Milling and Chatter Stability 1 3 7 Figure 5.45: Simulations of finished surfaces at different speeds, Ballend mi l l ing Chapter 5. Dynamic Milling and Chatter Stability 138 5.4.3 Summary This chapter presented a general milling model in which both the cutter and the gener-ation of cut surface by the true kinematics of milling system were accurately modelled. Simulations and experimental verifications showed that, the combined model is capable of predicting the cutting forces and generated surface finish with or without the presence of chatter vibrations during ball end milling operations. The model has several advantages: • When the vibrations are neglected, the static milling can be simulated. • The finish surface is predicted by keeping the points which have zero immersion, \& = 0. The finish surface generated represents feed marks, static form errors or chatter vibration errors depending on whether rigid, statically or dynamically flexible milling system structure is assumed, respectively. • In chatter analysis, the resulting chip thickness h(t) represents the true regenerated value. There is no need for iterative evaluation of multiple tooth periods to find the true chip load [51]. • The nonlinearity in the chatter, i.e. the tool jumping out of cut, is accounted for by imposing the condition : if h{t) < 0 then h(t) = 0. • Cutter runout and flank face/cut surface interference(i.e. process damping) can be included in the model easily. Chapter 5. Dynamic Milling and Chatter Stability 139 5.5 Conclusions B o t h s t a t i c a n d d y n a m i c m i l l i n g w i t h two different cutters have been e x a m i n e d i n t h i s c h a p t e r b y s i m u l a t i o n a n d e x p e r i m e n t a l studies. C h a t t e r s t a b i l i t y was e x p e r i m e n t a l l y e x a m i n e d , t h o u g h i n a ra ther low speed range. H o w e v e r , the t rends f r o m s i m u l a t i o n s suggested t h a t h i g h l y rea l i s t ic results c o u l d be o b t a i n e d i f a m o r e a c c u r a t e e x p e r i m e n t a l p r o c e d u r e is i m p l e m e n t e d , i n w h i c h the p r e d i c t e d s t a b i l i t y lobe d i a g r a m c a n be f u l l y u t i l i z e d . Conclusions The objective of the thesis has been to develop methods and algorithms to understand and predict the behavior of milling operations. Special emphasis has been placed on the ball end milling operations which are widely used in machining dies, molds, and hydro-dynamic and aerospace parts with sculptured surfaces. The results of this study provide prediction of cutting power and torque required from the machine tool; the direction, amplitude and density of cutting forces applied on the cutter and workpiece structures; static and dynamic deformation errors left on the finish surface; chatter vibration free stable cutting conditions. The prediction of milling performance is used to plan ball end milling operations in industry. First, geometric models of helical, bull nose and ball end cutters are developed in Cartesian coordinates. For a given cutter diameter, radial rake angle, helix angle, the number of flutes, bull nose radius and the ball end radius, a general geometric model of the cutter is defined. The milling operation consists of a rotating cutter and linearly feeding the workpiece. The rigid body motion of a cutting point on the tool edge is described by a trochoidal path. Previous researches assumed a linear shift of circular paths, which produce significant errors when the axial depth of cut and radial immersion are significantly small as in the case of ball end milling operations, fn this thesis, the true kinematics of ball milling are modelled. The instantaneous positions of all points on the flutes are expressed as a function of time using the stationary spindle as a coordinate center. The intersection of rotating and linearly moving end mill with the workpiece sur-face is computed, and the finished surface is stored at time intervals. The chip thickness is calculated by finding the radial distance between the two subsequent surfaces digitized 140 Conclusions 141 in two successive tooth passages. The static and dynamic displacements of the cutter are integrated to the kinematic model, hence allowing the influences of vibrations and static deflections on the chip thickness and the finish surface. The time varying cutting forces are proportional to the chip thickness predicted by the geometric and kinematic models of mil l ing. The force amplitudes are dependent on the oblique geometry of the end mi l l and plastic deformation properties of the workpiece material during cutting. The machining properties of the material are identified from two dimensional orthogonal cutting tests, which resembles plain strain deformation. The average yield shear stress, shear angle and friction coefficient between the tool and work-piece material are measured at a range of rake angle, cutting speed and chip thickness. The pressure and friction load on the oblique cutting edge of helical end m i l l and ball end m i l l are predicted using orthogonal to oblique cutting transformation method. It has been experimentally verified that the ball end mil l ing forces in three directions can be predicted with sufficient accuracy using the kinematic and cutting mechanics approach presented. The dynamic interaction between the tool and workpiece, which involves tool vibra-tions due to excessive cutting forces and structural flexibility in the system, including the effects such as tool jumping in and out of cut due to excessive vibrations are imple-mented in a time domain simulation program. The structural properties of the end m i l l attached to the spindle is considered to predict the chatter vibrations. The simulation program has the capability and versatility to predict mill ing forces, both statically and dynamically, surface form errors and chatter marks, under different mi l l ing operations, wi th different cutter geometries. The model is able to predict chatter vibration free ax-ial depth of cuts and spindle speeds, i.e. stability lobes, for a given workpiece material , tool geometry and the frequency response characteristics of the end mi l l ing system. The Conclusions 142 s t a b i l i t y lobe d iagrams have been developed and e x p e r i m e n t a l l y ver i f ied for b o t h c y l i n -d r i c a l a n d ba l l end cu t te r geometries. Process d a m p i n g has been found to be a d o m i n a n t factor at low speeds, w h i c h in t roduces discrepancies in to the m o d e l . F u r t h e r m o r e , the t i m e d o m a i n s i m u l a t i o n results have been c o m p a r e d w i t h a n a l y t i c a l l y p r e d i c t e d cha t te r s t a b i l i t y lobes. It has been observed that whi le the a n a l y t i c a l s t a b i l i t y p r e d i c t i o n m e t h o d is suff icient ly accurate at large w i d t h of cuts , the t i m e d o m a i n s i m u l a t i o n m o d e l gives m o r e accura te p red ic t ions at low w i d t h of cuts due to its h a n d l i n g of non l inea r i t i e s i n the d y n a m i c c u t t i n g sys tem. T h e k i n e m a t i c and geometr ic models developed i n th is thesis are genera l , a n d ap-p l i c a b l e to o ther workpiece mater ia l s ra ther t h a n t i t a n i u m al loy (TiQAlVA) used here. However , the o r thogona l to obl ique c u t t i n g mechanics t r ans fo rma t ion m o d e l m a y not be app l i cab l e when the shear ing is not cont inuous. Hence , further i nves t iga t ion is re-q u i r e d to ana lyze the v i a b i l i t y of the m e t h o d i n m a c h i n i n g hardened die and m o l d steels, where the ch ip is segmented and fracture and shearing m a y be coup led i n the p r i m a r y de fo rma t ion zone of cu t t i ng . T h e in teg ra t ion of proposed a lgor i thms to the N C too l p a t h genera t ion i n die a n d m o l d m a c h i n i n g is a na tu ra l extens ion of the work . O p t i m i z a t i o n of dep th of cu t , w i d t h of cu t , c u t t i n g speed and feed d u r i n g N C too l p a t h p l a n n i n g m a y s ign i f ican t ly i m p r o v e the p r o d u c t i v i t y and accuracy of m i l l i n g dies and molds . Bibliography [1] D . Montgomery and Altintas Y . Mechanism of Cut t ing Force and Surface Gen-eration in Dynamic Mi l l i ng . Trans. ASME Journal of Engineering for Industry, 113:160-168, 1991. [2] M . E . Martel lot t i . A n Analysis of the M i l l i n g Process. 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International Journal of Machine Tool Design and Research. (accepted for publication, 1993). [50] S. Smith and J . Tlusty. Update on High Speed Mi l l i ng Dynamics. Trans. ASME, Journal of Engineering Industry, 112:142-149, 1990. [51] J . Tlusty and F . Ismail. Basic Nonlinearity in Machinig Chatter. Annals of the CIRP, 30:21-25, 1981. A p p e n d i x A Ballend M i l l i n g Force Coefficients Table A . l : Edge Force Coefficients, Rake - 0 degree KTE = 37.205 N / m m KRB = 21.04 N / m m KAE = 0.64 N / m m Table A.2 : Cutt ing Force Coefficients, Rake = 0 degree R a fd K r c K t c Kac 1.27 0 0254 513.96747 1888.6835 -3.953168 1.27 0 0508 678 2344,5276 -70 1.27 0 0762 685.91083 2238.9265 0.706506 1.27 0 1016 535.4541 2111.4316 -91.45192 2.54 0 0254 244.91734 1988.1965 -242.039 2.54 0 0508 790.64752 2101.0342 -247.3838 2.54 0 0762 722.81268 2053.55 -200.5565 2.54 0 1016 669.86633 1992.5186 -259.0192 3.81 0 0254 291.18442 1800.4019 -435 3.81 0 0508 529.13086 1833.5325 -375.5063 3.81 0 0762 636.08765 1921.2904 -342.0091 3.81 0 1016 554.21527 1818.9087 -363.1614 6.35 0 0254 196.4164 1651.0255 -481.531 6.35 0 0508 630.36206 1820.4677 -335.1002 6.35 0 0762 647.32898 1534.0551 -337.677 6.35 0 1016 246.82477 1163.6793 -206.1739 147 Appendix A. Ballend Milling Force Coefficients 148 CD O 0 0) o> O * (^  O) c o c o o a |co co o TO V. 0 CD O O 0 CD U o B "a u a c o at o a « O o * lO ao co S3 9 o o d o 00 lO CM —' o o "ii o 0> c lO CO o o o' lO — — o © 0) c o c o 4= o o a 5 CD ke o 0 « * CO r> •O O o O d O >o CN co •5 o CO r-1 d o c 0 ec TO c 1 5 5 CO LO o' CD c o o a o a Z •si -o CM CN CN .— o o o » O D TO c o a c o c. u o .fc 5 o 0) a a d CO d 0 D 01 c i 2 Z CN LO a o 5 o TO •o 0) c LO 1 3 CO c o c o o a 4> 5 CD . CO * co • O O c O o P d o . ' co O l O CN co CN O o a « o 5 CO TO 9 Table A.3: Results from Orthogonal Cutt ing Experiments 

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