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Generalized modelling of flexible machining system with arbitrary tool geometry Kilic, Zekai Murat 2015

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GENERALIZED MODELLING OF FLEXIBLE MACHINING SYSTEM WITH ARBITRARY TOOL GEOMETRY  by  Zekai Murat Kilic  B.S., Middle East Technical University, 2006 M.S., Middle East Technical University, 2009  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  April 2015  © Zekai Murat Kilic, 2015 ii  Abstract The final shape of mechanical parts is mainly determined through turning, boring, drilling and milling operations. The prediction of the cutting forces, torque, and power of the machining process, and surface errors and vibration marks left on the parts is required to plan the machining operations and achieve shorter production cycle times while avoiding damage on the part, tool and machine. Past research has focused on developing dedicated mathematical models for each machining operation and tool type. However, the tool geometry and configuration of the machining set-up varies widely depending on the part geometry and application. This thesis presents a generalized mathematical model of machining operations carried out using geometrically defined cutting edges.  The mechanics of cutting between the tool edge and the work material are modelled to predict the friction and normal forces on the rake face of a single cutting edge. The combined static and dynamic chip thickness is modelled as a function of tool geometry, the kinematics of machining operation and the relative regenerative vibrations between the tool and workpiece. The cutting forces are transformed to process coordinates by considering the orientation of cutting edge and the kinematics of the machining operation, and are applied on the structural dynamics of the machine tool and workpiece by distribution along the cutting tool–workpiece contact zone. The cutting forces, vibrations, chatter stability and surface errors are simultaneously predicted in a semi-discrete time domain.  The geometry and force transformation models are unified in a parametric, mathematical model which covers all cutting operations. The application of the proposed model is demonstrated on turning, drilling and milling operations; multifunctional tools that combine iii  drilling-boring and chamfering in one operation; and two parallel face-milling cutters machining a plate from both sides. The proposed mathematical models are experimentally validated by comparing the measured forces, surface errors, vibrations and chatter stability charts against simulations.  The thesis shows the first unified, generalized mathematical modelling of metal cutting operations in the literature. The proposed model is expected to widen the application of science-based machining process simulation, planning and optimization methods in the virtual production of parts.    iv  Preface This Ph.D. dissertation proposes a generalized mathematical model which unifies turning, boring, drilling and milling operations, and is based on the papers that are already published, and on manuscripts currently prepared for publication in peer-reviewed journals. The presented work in these publications is carried out by the Ph.D. candidate in his Ph.D. research under the supervision of Dr. Yusuf Altintas. Relative contributions of the authors for each of the research articles are explained in this section.  The research in Chapter 3 is partially related to the study that is presented in Ref. [1]. The candidate is the co-author of this article focusing on a method for generalizing the machining operation by assuming two-dimensional insert geometry. Chapter 3 further describes the development and projection of the generalized method to the solid tools, and to the indexable cutters with three-dimensional insert geometry. Chapter 4 focuses on the basic mathematical model presented in Ref. [2]. The generalized method is extended to include turning, boring, drilling and milling applications. The results are presented in Chapter 6.  Chapter 5 advances the model published in Refs. [2,3]: • Ref. [2] sets up the general equation of motion for the end milling operation. The Ph.D. candidate contributed to the solution and experimental proof of the distributed dynamics of structure-milling process interaction which is described by the periodic, delay-differential equations. Caner Eksioglu, former M.A.Sc. student in Altintas lab, initiated the basic mathematical model required to represent the flexible end milling operation with distributed dynamics along the axial depth of cut. This contribution received ASME v  Blackall Machine Tool and Gage Award as the best journal paper in manufacturing engineering in 2013.   • Ref. [3] forms the concept of general equation of motion for all turning, boring, drilling and milling operations. The candidate is the main contributor of this study, and the supervisor presented the concept in the academy (CIRP) in a brief paper. The candidate advanced the general mechanics and dynamic model to include all tool and cutting operations in this thesis, which will be submitted for journal review.  • There will be a three part journal paper that shows: 1) The combined formulation of mechanics and dynamics of general cutting process, 2) adaptation of general model to predict the forces, torque, power and surface location errors in turning, drilling, boring and milling operations, 3) generalized stability analysis of all operations in semi-discrete time and frequency domains with experimentally supported case studies. The candidate is the primary contributor to all of the research here under the supervision of Dr. Altintas.  • Several applications of the proposed general modeling are given in Chapter 6. The basic modeling concept to predict the cutting forces for double-sided, parallel milling operation  covered in Chapter 6 was presented in a conference article [4]. The complete model including the dynamics and stability will be submitted as a journal article. The candidate contributed to the modeling, analysis and solution and received experimental support from Dr. Derek Luo of ITRI, Faby Feng of Apply Zeta, and Kevin Kao of KOVA cutter in Taiwan. Ref. [5] presents the force and stability prediction of multifunctional cutters. Dr. Min Wan analyzed the geometry of the multifunctional cutters and setup the equation of motion in the modal domain. The candidate made contributions to the experiments and the validation of results by applying the formulation as reported in Chapters 3-5.  vi  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ......................................................................................................................... vi List of Tables ................................................................................................................................ ix List of Figures .................................................................................................................................x List of Symbols and Abbreviations ......................................................................................... xvii Acknowledgements ................................................................................................................ xxxiv Dedication ............................................................................................................................... xxxvi Chapter 1: Introduction ................................................................................................................1 Chapter 2: Literature Survey  ......................................................................................................4 2.1 Overview ............................................................................................................................... 4 2.2 Mechanics of Metal Cutting .................................................................................................. 4 2.3 Basic Kinematics of Regenerative Machining Operation ..................................................... 5 2.4 Cutting Forces ....................................................................................................................... 8 2.4.1 Mechanistic Identification .............................................................................................. 9 2.4.2 Generalized Cutting Force Prediction .......................................................................... 10 2.5 Modelling of Generalized Tool Geometry .......................................................................... 11 2.6 Simultaneous Machining Modelling ................................................................................... 13 2.7 Dynamics of Machining Operations .................................................................................... 14 2.8 Surface Form Error .............................................................................................................. 16 2.9 Summary.............................................................................................................................. 17 vii  Chapter 3: Generalized Geometric Model of Cutting Tools  ...................................................19 3.1 Generalized Geometric Parameters of Arbitrary Tools ....................................................... 19 3.2 Generalized Modeling of Cutting Edge Coordinates on the Tool Body ............................. 32 3.2.1 Indexable Cutter ........................................................................................................... 32 3.2.2 Solid Tool ..................................................................................................................... 62 3.3 Computation of the Effective Tool Geometry ..................................................................... 67 3.3.1 Velocity of Cutting Edge of the Tool ........................................................................... 67 3.3.2 Tool Cutting Planes ...................................................................................................... 69 3.3.3 Effective Tool Angles .................................................................................................. 74 Chapter 4: Generalized Modelling of Cutting Mechanics  ......................................................81 4.1 Chip Geometry .................................................................................................................... 81 4.2 Modeling of Cutting Forces................................................................................................. 92 4.3 Force Transformations for Arbitrary Tools ......................................................................... 95 4.3.1 Rake face UV (friction-normal) to Cutting Edge RTA (radial-tangential-axial) .......... 96 4.3.2 RTA to Edge Reference Frame (xRyRzR), and to Process Frame (xyzθ) ........................ 98 4.4 Transformation of the Vibration Vector to the Dynamic Chip Thickness ........................ 105 4.5 Representation of General Force Vector ........................................................................... 107 4.5.3 Multiple-Level Tool with Lumped Force at Each Station/Level ............................... 113 4.6 Summary............................................................................................................................ 117 Chapter 5: Generalized Modelling of Cutting Dynamics  ......................................................118 5.1 General Equation of Motion .............................................................................................. 118 5.2 Solution of the Equation of Motion in Modal Domain ..................................................... 124 5.3 Simplified Solution Methods for Special Cases ................................................................ 149 viii  5.3.1 Low Radial Immersion with Free Vibration Condition (e.g. Double-Sided Milling) 150 5.4 Summary............................................................................................................................ 154 Chapter 6: Applications  ...........................................................................................................155 6.1 Overview ........................................................................................................................... 155 6.2 Application-1: Slender, Cylindrical Helical End Mill with Regular Pitch Angles [2] ...... 155 6.3 Application-2: Regular Pitch Serrated Cylindrical Helical End Mill [21] ........................ 180 6.4 Application-3: Two-Level Multifunctional Drilling/Boring Cutter [5] ............................ 191 6.5 Application-4: Drilling with a Multifunctional, Indexable Cutter .................................... 203 6.6 Application-5: Double-Sided Parallel Milling .................................................................. 210 6.7 Application-6: Prediction of Cutting Forces for a Two-Insert Indexable End Mill .......... 221 6.8 Application-7: Three Holders with the Same General Purpose Insert .............................. 221 6.9 Application-8: Parallel Turning with Two Tools .............................................................. 225 Chapter 7: Conclusions and Future Research Directions ......................................................230 7.1 Conclusions ....................................................................................................................... 230 7.2 Future Research Directions ............................................................................................... 232 References ...................................................................................................................................234 Appendix A Transforming the basis from Frame-1 to Frame-2  ..........................................245  ix  List of Tables Table 4.1 Inputs at the k=19-th axial segment of the 3-tooth serrated cylindrical end mill. ........ 85 Table 6.1 Information of the cylindrical end mill ....................................................................... 156 Table 6.2 Radial runout measurement at five axial locations using a dial gage with 5-micrometer tolerance. .............................................................................................................................. 156 Table 6.3 Mechanistic identification of the milling operation. Tool information in Table 6.1. . 167 Table 6.4 Identified modal parameters in x0 and y0 directions. .................................................. 171 Table 6.5 Information of the serrated end mill ........................................................................... 181 Table 6.6 Identified modal parameters in x and y directions (taken from Table 2 of Ref. [21]). 189 Table 6.7 Identified modal parameters for Sandvik and Kennametal multifunctional tools. ..... 200 Table 6.8 Measured angular location, pitch angle and radial runout of the left- and right-side cutters. .................................................................................................................................. 212 Table 6.9 Average forces derived from experiment and theoretical calculations. ...................... 215 x  List of Figures Figure 2.1 Schematics of chip removal at differential cutting edge element .................................. 5 Figure 2.2 Dynamic interaction between tool and workpiece. ....................................................... 6 Figure 3.1 Left hand turning side-view schematics ...................................................................... 21 Figure 3.2 Indexable drill schematics ........................................................................................... 22 Figure 3.3 Geometry of serrated end mill that is used for rough milling operations .................... 23 Figure 3.4 General end mill geometry schematics........................................................................ 24 Figure 3.5 Parameters for the drilling operation at initial time instant (t=0) ................................ 27 Figure 3.6 Rake face vectors at element Sjk .................................................................................. 31 Figure 3.7 Locating the local rake face frame in the insert reference frame ................................ 34 Figure 3.8 Locating the local cutting edge frame ......................................................................... 35 Figure 3.9 Step-1 of positioning the insert on the holder ............................................................. 37 Figure 3.10 Insert placement on the holder .................................................................................. 37 Figure 3.11 KOVA XNEX080608 insert...................................................................................... 38 Figure 3.12 Assigning the insert reference frame ......................................................................... 38 Figure 3.13 Modelling of the rake face of the insert..................................................................... 40 Figure 3.14 Highlighted two cutting edge sections (ces-5 and ces-7) with flat rake faces ........... 41 Figure 3.15 Rake face approximation of curved rake face ........................................................... 41 Figure 3.16 Local coordinates of the selected point e along the cutting edge section-1 .............. 42 Figure 3.17 Cutting edge-section-2 (Figure 3.13b) with curved rake face ................................... 44 Figure 3.18 Rake face geometry along the modelled cutting edge (bold curve): (a) Rake face vectors; (b) rake face normal vectors; (c) rake face and rake face normal vectors ............... 44 Figure 3.19 Indexable milling cutters ........................................................................................... 45 xi  Figure 3.20 Transformation from Frame-i to Frame-0 of the cylindrical face mill cutter ............ 46 Figure 3.21 Transformation from Frame-i-Right to Frame-0 of the double milling tool ............. 46 Figure 3.22 Transformation from Frame-i-Left to Frame-0 of the double milling tool ............... 47 Figure 3.23 Indexable U502 drilling cutter. (Courtesy of KOVA.).............................................. 47 Figure 3.24 Transformations from Frame-i to Frame-0 for the indexable drilling tool ............... 47 Figure 3.25 Sample two-insert end mill from Sandvik-Coromant ............................................... 49 Figure 3.26 Relative locations of the origins of cutter, insert and first rake face section of the first cutting edge section. .............................................................................................................. 50 Figure 3.27 Cutting edge and rake face sections of the insert ...................................................... 55 Figure 3.28 Insert edge geometry of the two-inserted end mill .................................................... 56 Figure 3.29 Chip length along the first 0.030-mm section of the cutting edge ............................ 57 Figure 3.30 Multifunctional cutters .............................................................................................. 58 Figure 3.31 Measured cutting edge geometry of Kennametal drill .............................................. 59 Figure 3.32 Modelled Kennametal cutter: (a) Axial location; (b) angular location. .................... 60 Figure 3.33 Modelled cutting edge geometry of the Sandvik TM880 cutter................................ 61 Figure 3.34 Locating the Frame-D on the solid tool starting from Frame-0 ................................ 63 Figure 3.35 Surface profilometer measurement data of the serrated end mill .............................. 64 Figure 3.36 Cutting edge angle of the serrated end mill ............................................................... 65 Figure 3.37 Resultant cutting directions for two consecutive edges at segment k of milling cutter moving in x0 direction. ........................................................................................................... 68 Figure 3.38 Tapered ball end mill to show Pf, Pfe and Pr planes ................................................... 70 Figure 3.39 Tapered ball end mill to show A , Pn, Ps and Pr planes ............................................ 72 Figure 3.40 Cutting edge angle calculation: (a) Right-hand milling; (b) Left-hand milling. ....... 76 xii  Figure 3.41 Cutting edge angle calculation of the drilling operation. .......................................... 77 Figure 3.42 Calculations of (a) Cutting edge inclination angle and (b) normal rake angle. ......... 78 Figure 3.43 Effective tool angles of the two-insert end mill. ....................................................... 80 Figure 3.44 Effective tool angles of U140 face mill ..................................................................... 80 Figure 3.45 Effective tool angles along the cutting edge (in one serration period) of the serrated end mill [21]........................................................................................................................... 80 Figure 4.1 Algorithm for predicting the actual static chip thickness and tooth passing delay period jkiT . ............................................................................................................................. 84 Figure 4.2 Serrated cylindrical end mill with the cross section at k-th segment. ......................... 85 Figure 4.3 Schematics for the 2-insert drill with axial runout. ..................................................... 89 Figure 4.4 Parallel turning schematics: (a) Radial discretization; (b) effect of axial offset and axial runout between cutters. ................................................................................................. 90 Figure 4.5 Transformation of rake face forces (Fu and Fv) into the forces along the R,T and A axes of the cutting edge plane Ps. (Adapted from [1,127].) ................................................... 96 Figure 4.6 Transformations from RTA to xRyRzR, and from xRyRzR to x0y0z0 for basic machining operations. (Images adapted from Sandvik and Ref.[3].) .................................................... 102 Figure 4.7 Right hand turning and parallel turning schematics .................................................. 103 Figure 4.8 Double-sided milling schematics. ............................................................................. 105 Figure 4.9 Double-sided milling tool geometry [4]: (a) Left-side cutter; (b) right-side cutter. .. 105 Figure 4.10 Modelling of cutting forces ..................................................................................... 111 Figure 4.11 Radial force distribution on the flexible tool-workpiece zone ................................ 113 Figure 4.12 Sandvik TM880 multifunctional tool with D=2 dynamic levels. ............................ 115 xiii  Figure 4.13 The double-sided milling cutter with forces applied at the opposite sides of the workpiece. ............................................................................................................................ 116 Figure 4.14 Cutting forces of parallel turning operation. ........................................................... 117 Figure 5.1 Interpolation of the delay state using five neighboring states. .................................. 130 Figure 5.2 The neighboring states and the Lagrange- 4-th order interpolation .......................... 133 Figure 5.3 Surface location error for the milling operation ........................................................ 147 Figure 6.1 Geometry along cutting edge of cylindrical end mill: (a) Angular location; (b) relative angular location, (c) radial runout; (d) radius. ..................................................................... 158 Figure 6.2 Modelled 3D locations in the Frame-0 of cylindrical end mill. ................................ 159 Figure 6.3 Tool entry and exit parameters for milling operation ................................................ 161 Figure 6.4 (a) Schematics of 5% down milling. (b) In-cut and out-of-cut regions..................... 161 Figure 6.5 Effects of radial tunout for 5% down milling of cylindrical helical end mill, on (a) tool entry angle; (b) Maximum effective chip load along the cutting edge. ............................... 162 Figure 6.6 Zoomed tool-workpiece contact zone (from Figure 6.4b) at 1.2-mm axial depth, for tooth/edge numbers: (a) #1; (b) #2; (c) #3; (d) #4. .............................................................. 164 Figure 6.7 Effect of runout on: (a) effective chip load; (b) actual instantaneous pitch angle. ... 165 Figure 6.8 Dynamometer direct force transfer function: ............................................................ 168 Figure 6.9 Experiment vs. predicted forces with runout effect. .................................................. 169 Figure 6.10 FRF measurements along the tool length: (a) Test locations; (b) Feed direction magnitude plot; (c) Normal direction magnitude plot. ........................................................ 171 Figure 6.11 Sketch of Figure 6.10a. ............................................................................................ 171 Figure 6.12 Results of discrete-time simulation ......................................................................... 178 Figure 6.13 Discrete time force simulation for the experiments (A) and (B) of Figure 6.12a ... 179 xiv  Figure 6.14 Serrated end mill flute geometry ............................................................................. 182 Figure 6.15 Serrated end mill: (a) angular location; (b) relative angular location. .................... 182 Figure 6.16 Serrated end mill: (a) Modelled 3D geometry; (b) close-up 3D view. .................... 183 Figure 6.17 Computed tool geometry of serrated end mill ......................................................... 183 Figure 6.18 (a) Radial runout for the first 8-mm depth; (b-f) Maximum effective chip load along the cutting edge for five cases.............................................................................................. 184 Figure 6.19 Force simulations compared to Cutpro [118] .......................................................... 186 Figure 6.20 Force simulation compared to the experiment plots of [21] .................................... 187 Figure 6.21 Comparison of the stability prediction to the results presented in [21]. .................. 190 Figure 6.22 Sandvik multifunctional tool: (a) 3D geometry; (b) Radial depth of the midpoints of the segments along the cutter body. ..................................................................................... 192 Figure 6.23 Angular location (a) and relative angular location (b) of element Sjk. .................... 193 Figure 6.24 Axial location: (a) Level-1; (b) Level-2. ................................................................. 193 Figure 6.25 (a) Axial runout: (a) Level-1; (b) Level-2. .............................................................. 193 Figure 6.26 Pitch angle: (a) Feedrate neglected (fr≈0); (b) fr=0.1-mm. ..................................... 195 Figure 6.27 Effective tool geometry calculated from the rake face orientation ......................... 195 Figure 6.28 Cutting force coefficients (a,b,c) and edge coefficients (d,e,f) in RTA directions. 196 Figure 6.29 Element-available-to-cut (a) and missed-cut (b,c,d,e) conditions for Sandvik multifunctional tool.............................................................................................................. 198 Figure 6.30 Predicted cutting forces for Sandvik multifunctional tool ...................................... 199 Figure 6.31 Sandvik multifunctional tool stability predictions compared to experiments ......... 202 Figure 6.32 3D locations of the segments on the Kennametal cutter body ................................ 204 Figure 6.33 Angular location (a) and relative angular location (b) of element Sjk. .................... 205 xv  Figure 6.34 (a) Axial location (height), and (b) corresponding axial runout. ............................. 205 Figure 6.35 Pitch angle: (a) Norunout/zero feedrate; (b) fr=0.1-mm/rev .................................... 205 Figure 6.36 Effective tool geometry ........................................................................................... 206 Figure 6.37 Effect of the rubbing of the pilot drill on the cutting forces of Kennametal drill. .. 207 Figure 6.38 Predicted and measured cutting forces at period-B of Figure 6.37. ........................ 207 Figure 6.39 Cutting forces without rubbing effect. Pilot hole diameter is larger than 10-mm. .. 208 Figure 6.40 Predicted and measured cutting forces (close-up of Figure 6.39). .......................... 208 Figure 6.41 Stability test with Kennametal drill: (a) Stability chart compared to experiments; (b) Forces with the resulting surface finish photo of the selected unstable cutting test [5]. ..... 209 Figure 6.42 Measurement of radial runouts on the double-sided milling tool............................ 210 Figure 6.43 Double-sided milling tool: (a) In operation; (b) with brazed inserts; (c) with detachable inserts. ................................................................................................................ 211 Figure 6.44 Double-sided milling tool: (a) Components‒see Figure 4.8; (b) operation is ~20% down milling. ....................................................................................................................... 212 Figure 6.45 Supporting edge line (tool holder reference line) shown on: (a) Left-side and (b) right-side cutters with labeled insert numbers. .................................................................... 213 Figure 6.46 Double-sided milling experiment setup: (a) Part cross section; (b) 3D view. ......... 213 Figure 6.47 Predicted (solid line) and experimental (broken line) cutting forces ...................... 214 Figure 6.48 Experiment setup with flexible workpiece .............................................................. 216 Figure 6.49 Pre-set spacer width of the double-sided milling tool. ............................................ 216 Figure 6.50 Engagement condition of double-sided cutting tests. .............................................. 216 Figure 6.51 Tool entry (a) and exit (b) angles for the indexable double-sided milling tool. ...... 217 Figure 6.52 First two vibration mode shapes from finite element simulation ............................ 219 xvi  Figure 6.53 Measurement at selected  point (at A and B distances) on the steel part. ............... 219 Figure 6.54 Real and imaginary parts of measured and fitted FRF. ........................................... 219 Figure 6.55 Predicted limit depth of cut per side for stable double sided milling. ..................... 220 Figure 6.56 Identified cutting edge of the two-insert end mill ................................................... 220 Figure 6.57 Predicted forces compared to the experimental results reported by Eynian [75]. ... 221 Figure 6.58 KOVA U140 face mill geometry. (Courtesy of KOVA.) ....................................... 222 Figure 6.59 Face milling operation cutting force prediction. ..................................................... 223 Figure 6.60 KOVA U502 drill geometry. (Courtesy of KOVA.) ............................................... 224 Figure 6.61 Drilling operation cutting force predictions. ........................................................... 224 Figure 6.62 Parallel turning setup ............................................................................................... 228 Figure 6.63 Comparison of the predicted stability chart (modelled) to the literature................. 229  xvii  List of Symbols and Abbreviations ( ),,N qj kinput∑ : Function of double summation from k=1..q for each j=1..N       ( )lumped⋅ : Lumped force vectors at one point along tool-workpiece contact zone       ( ) ( ) ( )x y z, ,⋅ ⋅ ⋅ : Magnitude components of a vector       ( ) jk⋅ : Matrix/vector related to element Sjk       ( ) ( )r nP P,⋅ ⋅ : Projection of vector on Pr-plane and on Pn-plane       ( ) ( ) ( )external bored-hole drilled-hole, ,⋅ ⋅ ⋅ : Refer to external radius of workpiece, inner radius of bored-hole and inner radius of drilled-hole       ( )[ x ]rows cols⋅ : Way to show the dimensions of a matrix/vector. rows and cols refer to number of rows and columns       ( ) ( )exact approx,⋅ ⋅ : Exact and approximate matrices       ( )11 11e e,0,x z : Location of point-e in rake face section-1 in Figure 3.27b       [ ]specificgeometry.. : Matrix values computed for specific geometry       ⋅ : Absolute value of the elements of the input vector       ⋅  : Euclidean norm (magnitude) of vector       0 : Zeros matrix/vector       2D, 3D : Two- and three-dimensional       A, B: Dimensions showing the modal test location on flexible workpiece       A0 : Constant state-transition matrix       xviii  Ajk : Local axial location of element Sjk       ap: Depth of cut       ap1 , ap2 : Depth of cut at tool#1 and tool#2       arccos ( )⋅ , arcsin ( )⋅ : Trigonometric inverse-cosine and inverse-sine functions       ast,jk , aex,jk : Entry and exit distance at element Sjk       atan2 ( ),inputX inputY : Trigonometric atan2 function       Aγ , αA : Rake face and clearance face (flank)       b : Largest multiple of modified delay term ( *n )       B1(t) , B2,jk(t) : Time dependent current and delayed state matrices       paaki kiki, ,c c γ : Arrays of actual feed per edge, chip load and decrease in chip load at segment-k at specific time instant       pajkic : Instantaneous chip load of element Sjk       ajkic : Instantaneous feed per edge of element Sjk       jkc : Nominal feed per edge of element Sjk       C, c : Number of cutting edge sections and index of selected cutting edge section       C1,i , C2,i , C3,jki : Matrices that represent dynamic forces       C4,i : Vector that represents static forces       CAD : Computer-Aided Design       ceil ( )⋅ : Operator to round the input number to its closest larger integer value       ces: Cutting edge section       cj(t) 0, j d,j, ( ), ( )c t c t  Instantaneous regenerative, static and dynamic chip loads of tooth-j       xix  cr,jki : Numerator coefficient of weight term for interpolating instantaneous delay of element Sjk       CW, CCW: Clockwise, counterclockwise       D : Total number of dynamic stations/levels       D1,i-3 , D2,jk,i-3 : Merged matrices of current and delayed dynamic forces within one spindle period      de : Last radial segment index of the e-th dynamic level       DF , Df : State transition matrices for forced and free vibration periods       dr : Denominator coefficient of weight term for interpolating instantaneous delay of element Sjk       ds: Differential thickness along flute direction of serrated end mill       dz: Constant thickness of each differential cutting edge segment-k (1..N)       e : Index for dynamic station/level       e : Index of selected point on the rake face section-c,r       ( )e  , exp( )⋅ ⋅ : Matrix exponential       Ei-3 : Merged static force vectors within one spindle period       eig ( )⋅ : Eigenvalues of the input matrix       djkie : General transformation vector from tool reference coordinates to the negative-R       ek: Surface location error at axial segment-k       cx,jkiF ,cy,jkiF ,cz,jkiF : Shear force components in Frame-0       djkiF : Dynamic force multiplier of the dynamic chip load       edjkiF : Dynamic ploughing (process damping) force vector       f t,j : Feedrate per tooth-j       i i, ( )tF F : Force at i-th time instant (ti)       xx  djkif : Local delayed dynamic force term of element Sjk element at i-th time instant (ti)       pjkiF : Process damping force multiplier of the velocity       c c cr,jki t,jki a,jki, ,F F F : Shear force components in radial, tangential and axial directions       c crta,jki uv,jki,F F : Shear force vectors in RTA and UV frames       cs cdjki jki,F F : Static and dynamic shear force vectors       es es esr,jki t,jki a,jki, ,F F F : Static ploughing force components in radial, tangential and axial directions       esrta,jkiF : Static ploughing force vector in RTA frame       cs cd es edi i i i, , ,f f f f : Static shear, Dynamic shear, dynamic, static ploughing and dynamic ploughing force terms of equation of motion at i-th time instant (ti)       t wjki jki,F F : Total cutting force on tool and workpiece in process coordinates       trta,jkF : Total cutting force on tool in RTA       L Ri i,F F : Total force vector of left- and right-side cutters       st ssjki jki,F F : Total static and steady state cutting forces       F(t), c es( ), ( )F t F t : (Chapter 2) Instantaneous total static, shear and static ploughing forces       f(xr) : Neighbouring state values for interpolation       Fjki : Instantaneous cutting force       c esjki jk,F F : Shear and static ploughing components of cutting force       floor ( )⋅ : Function to round the input down to closest integer       fr, ft : Translational feed per revolution, and feed per tooth       xxi  Frame-2, Frame-3 : Coordinate frames defined for mechanics of oblique cutting (Figure 4.5)       Frame-c,r : Local rake face frame       Frame-D , Frame-c,r,e : Local cutting edge design frame       Frame-i : Insert reference frame       Frame-R : Tool reference frame       Frame-s : Insert pocket (seat) frame       FRF : Frequency response function or Transfer function       Fu, Fv: Rake face forces in friction and normal directions       c cu,jki v,jki,F F : Shear force in friction and normal directions       g1,jk : Element-available-to-cut condition of element Sjk       L L L L1,jk 2,jki 3,jki 4,jki,r, , ,g g g g : Binary in-cut condition parameters related to left-side cutter       R R R R1,jk 2,jki 3,jki 4,jki,r, , ,g g g g : Binary in-cut condition parameters related to right-side cutter       g1,k , g3,ki : Element-available-to-cut and missed-cut arrays of segment-k at specific time instant       g2,jki : Instantaneous edge-in-cut condition for element Sjk       g3,jki : Instantaneous missed-cut condition parameter of the element Sjk based on the runout       g4,jki,r : Binary parameter to decide if r-th radial zone is in-cut       st djki jki,h h : Instantaneuos static and dynamic components of chip thickness of element Sjk       h(τ) : Intermediate function for shortening integral form of Eq.       ( )stjki zerorunouth : Static chip thickness when runout is zero at segment-k       h, lc: Uncut chip thickness, projected chip length       hjki , Ac,jki : Instantaneous uncut chip thickness and uncut chip area of element Sjk       xxii  I : Identity matrix       i : Index for time instants       i0 0 0, ,j k : Local unit axes vectors of Frame-0       i2 , j2 , k2 : Axis vectors of Frame-2       1 1 12 2 2, ,i j k : Axis vector representation of Frame-2 relative to Frame-1       i i ic,r c,r c,r, ,i j k : Local unit axis vectors of Frame-c,r defined in Frame-i       i i iD D D, ,i j k  (i i ic,r,e c,r,e c,r,e, ,i j k ): Local unit axis vectors of Frame-D (Frame-c,r,e) defined in Frame-i       0 0 0D D D, ,i j k : Local unit axes vectors of Frame-D defined in Frame-0       id1, id2, id3: Index points used for separating free and force vibration zone during one spindle rotation       ii i i, ,j k : Local unit axes vectors of Frame-i       0 0 0i i i, ,i j k : Local unit axes vectors of Frame-i defined in Frame-0       int ( )⋅ : Operator to round the input number to its closest integer value       0 0 0R R R, ,i j k : Local unit axes vectors of Frame-R (tool reference frame) defined in Frame-0       is s s, ,j k : Local unit axes vectors of Frame-s       J : Total number of paired free-forced vibration periods       j: Index of current edge/tooth/flute       ja : Index of edge in chip thickness calculation algorithm       jout: Selected output edge for the chip thickness calculation algorithm       xxiii  c c cr,jki t,jki a,jki, ,K K K : Cutting force coefficients in radial, tangential and axial directions       e e er,jki t,jki a,jki, ,K K K : Edge coefficients in radial, tangential and axial directions       k: Index of axial or radial segment-k       K: Total number of axial or radial segments of tool       Kc(p1,..pL): Cutting force coefficient as a function of L parameters (p1,..pL)       c ejki jk,K K : Cutting force and edge coefficients       Ksp : Material-dependent specific contact force       c cu,jki v,jki,K K : Cutting force coefficients in friction and normal directions       t tx,n ,t y,n ,t,k k : Modal stiffnesses in x and y directions for nt-th tool vibration mode       L1: Length of cutting edge section-1 in Figure 3.16       lc,jk : Projected uncut chip length along T direction of element Sjk       Lw : Flank wear land of tool       m, mt, mw: Total number of vibration modes, number of tool modes and number of workpiece modes       ( ),maxj i⋅ : Maximum function among edges (j) and time instants (i)       ( )max ⋅ : Maximum scalar value of the inputted matrix       Mf , Kf , Cf : Mass, stiffness, damping matrices of full model       ( ) ( )min ... , max ...j j: Minimum and maximum value functions for the input parameter of all j=1..N teeth at the segment k       mjki : Delay state index of element Sjk at time ti       xxiv  mod(input1,input2): Common residue function       mt1 , mt2: Number of vibration modes of tool#1 and tool#2 in Section 6.9       n : The maximum delay state index value       n*: Modified delay term       N: Total number of teeth/edges/flutes of tool       Na , Na,new : Actual and updated edges-in-cut array at segment-k at specific time instant       Na,k: Actual number of cutting edges that remove material from the workpiece at segment-k       njki : Instantaneous number of edges skipping the cut after the element Sjk       np11: Number of points along cutting edge section-1 in Figure 3.16       ns : Number of discrete time steps in one spindle period       ns: (Appendix A) Number of rotation steps       ns: Number of rotation steps to locate the cutting edge       nt : Index for tool vibration mode       nw : Index for workpiece vibration mode       P : Number of simulated spindle rotations       P, μ : Process damping multiplier vector and Coulomb friction coefficient       Pf: Assumed working plane at the selected location on the cutting edge       Pfe,d, Pfe,m: Working planes of drilling and milling tools       Pfe: Working plane at the selected location on the cutting edge       Pn: Cutting edge normal plane at the selected location on the cutting edge       Pr: Tool reference plane at the selected location on the cutting edge       Ps: Cutting edge plane at the selected location on the cutting edge       xxv  Q(s), F(s) : Displacement and force vectors in Laplace domain       Q(t), F(t) : Displacement and force vectors in time domain       q: Number of cutting edge segments along the tool-workpiece contact zone       Qf(t), Ff(t) : Displacement and force vectors for full model in time domain       qki k i jki, ( )t T−q : Relative physical displacement vector and delayed vibration vector       kiq : Relative physical velocity vector       12R  : Rotation matrix from Frame-2 to Frame-1       1 2 1 2 1 2x y z, ,r r r : Cartesian components of position vector of origin of Frame-2 relative to Frame-1       ( )( )1 2test⋅r : Frame-1 representations of test vector when procedures A1, A2, B1, B2, C1 and C2 used       2testr : Frame-2 test vector to check accuracy of rotations in Appendix A       sjkR : Local radius measured along flute direction of serrated end mill       0 ir : Local translation vector from origin of Frame-0 to origin of Frame-i       i Dr : Local translation vector from origin of Frame-i to origin of Frame-D       i ir : Local translation vector from origin of Frame-i to origin of Frame-i       0 D 0 D 0 Dx y z, ,r r r : Magnitude components of position vector in Frame-D       0 i 0 i 0 ix y z, ,r r r : Magnitude components of position vector in Frame-i       0 0x y,R R : Magnitude components of radius vector       j j j, ,r j κ : Parameters for locating insert on multifunctional cutter (Figure 3.31)       1 2r : Vector of origin of Frame-2 represented in Frame-1       xxvi  R, r : Number of rake face sections in the selected cutting edge section-c and index of selected rake face section       r, s: Refers to selected point and total number of points along periphery of cutter contacting flexible workpiece       R: Calculated number to decide the quality of curve fitting (maximum value 1)       R0: Nominal radius of the tool       0rD, 0DR , 0DT  Translation vector, rotation matrix and homogeneous transformation matrix from Frame-D to Frame-0       0ri, 0iR , 0iT : Translation vector, rotation matrix and homogeneous transformation matrix from Frame-i to Frame-0       0rR, 0RR , 0RT  Translation vector, rotation matrix and homogeneous transformation matrix from Frame-R to Frame-0       r2 : Vector of origin of Frame-2 represented in Frame-2       c,rrc,r,e, c,rc,r,eR , c,rc,r,eT : Translation vector, rotation matrix and homogeneous transformation matrix from Frame-c,r to Frame-i       Ri, Rf: Initial and machined workpiece radii       irc,r, ic,rR , ic,rT : Translation vector, rotation matrix and homogeneous transformation matrix from Frame-c,r to Frame-i       0jkR : Radius vector of Sjk element in Frame-0       Rjk: Local radius (scalar) of element Sjk in Frame-0       max,kR : Maximum radius at segment-k       xxvii  Rotx ( )⋅ , Roty ( )⋅ , Rotz ( )⋅ : Coordinate rotation functions defined in Appendix A       Rt,jk : Moment arm at element Sjk       RTA: Cutting edge coordinate frame       RTA: Radial-tangential-axial       S(t) : Static force vector with process damping       s: Laplace-domain operator       sameNotSame: Parameter to check if actual edges-in-cut array updated during chip thickness calculation algorithm       Sjk: Cutting edge element of edge-j at segment-k       SLE: Surface location error       ( )( )12⋅T  : Homoegeneous transformation matrices procedures A1, A2, B1, B2, C1 and C2 used       12T  : Homoegeneous transformation matrix from Frame-2 to Frame-1       ic,r,eT , iDT : Homogeneous transformation matrix from Frame-c,r,e (Frame-D) to Frame-i       cjkiT : Cutting torque       RI, jk,L 0R, jki L RI, jk,R 0R, jki R, , ,, ,T T T T : Transformation matrices for left- and right- side cutters       RI, jkT : Transformation matrix from RTA to Frame-R       IU, jkT : Transformation matrix from UV to RTA       t, ti : Continuous time parameter, and discrete time at instant i       T0M : Transformation matrix from process to tool coordinate system       T0R,jk : Transformation matrix from RTA to Frame-0       t1, t2, r1, r2, r3: Labels for representing new frames in Figure 3.10 and Figure 3.34       xxviii  Tj: Delay period that is the time passed since the cut of previous tooth-(j+1) until the cut of current tooth-j (s)       Tjk(t) : Instantaneous ime delay of element Sjk       tjk, 1, jkn , 1c, jkn , 2, jkn : Local unit vectors of element Sjk in tangent, rake face, corrected rake face and rake face normal directions       Tjki : Instantaneous delay period of element Sjk       Tmax : Maximum delay period along the tool’s edges during one spindle rotation       Trans ( )⋅ : Coordinate translation function defined in Appendix A       Ts : Period (in [s]) of one spindle rotation       U, Uf : Mode shape matrices of reduced-order model and of full model       Ue , Fe(t) , Qe(t): Extended mode shape matrix, force vector and displacement vector covering both cutting and non-cutting zone       tk,n ,tu : Mode shape vector contribution at segment-k on tn -th (tool) mode       wk,n ,wu : Mode shape vector contribution at segment-k on wn -th (workpiece) mode       Unc , Fnc(t) : Mode shape matrix and force vector covering only non-cutting zone       Ut, Uw : Mode shape matrices of tool and of workpiece       Ut1 , Ut2: Mode shape matrices of tool#1 and tool#2 in Section 6.9       UV: Rake face friction-normal       UV: Rake face friction-normal coordinate frame       t t t tx,k,n ,t y,k,n ,t z,k,n ,t θ,k,n ,t, , ,u u u u : Components of tk,n ,tu        w w w wx,k,n ,w y,k,n ,w z,k,n ,w θ,k,n ,w, , ,u u u u : Components of wk,n ,wu        xxix  fv , fv : Translational feed vector and its magnitude       f,dv , f,mv : Translational feed vectors for drilling and milling       0c,jkv : Cutting velocity of element Sjk on tool relative to workpiece       0c,jkv : Local cutting speed of tool relative to workpiece       0r,jkv : Resultant cutting vector of element Sjk       vc,jk : Cutting speed at element Sjk       wh,jki , whr,jki: Half-size weight matrices (r=1..5) for delay approximation in the time domain cutting force simulation       wjki , wr,jki , wr,jki : Sub-components (r=1..5) of weight matrix of the Lagrange interpolation polynomial       Wjki : Weight function matrix of the Lagrange interpolation polynomial       2 2 2, ,x y z  : Coordinates of a point in Frame-2       1 2 1 2 1 2, ,x y z  : Coordinates of a point in Frame-2 relative to origin of Frame-1       x0, y0, z0: Cartesian coordinates in tool reference frame (Frame-0)       xF : Parameter for nonlinear cutting force       xjki : Relative time of delay state       Ijkix : x-component of relative physical vibration vector in radial direction       xki, yki, zki, θki: Components of relative physical vibration vector       xr , xs : Relative times of neighbouring state       xxx  xR,j(t), xR,j j( )t T−  Instantaneous relative tool/workpiece current displacement at current tooth-j, and delay displacement at previous tooth-(j+1)       xyz: Coordinates of process coordinate frame       ya,k : Actual surface distance at axial segment-k       yd,k : Desired surface distance at axial segment-k       yL4(xjki) : Approximate value of the delayed state       Zk, 0kz : Local position of the midpoint of the k-th segment       zt1, zt2, zw : z0-displacement of tool#1, tool#2 and workpiece       serrβ : Helix angle of serrated end mill       βj: Helix angle of tooth-j       f p r, ,γ γ κ : Euler angles to locate cutting edge on tool       jkiγ : Instantaneous decrease in chip load of element Sjk       Γ(s) : Modal displacement vector in Laplace domain       Γ(t) , ( ), ( )t t Γ Γ : Modal displacement, velocity and acceleration vectors in time domain       Γc(t-Tjki) : Delay combined modal displacement       Γc,i c,i c,i i i i, , (or ( ), ( ), ( ))t t t   Γ Γ Γ Γ Γ : Combined modal displacement, velocity and acceleration vectors at i-th time instant (ti)       γn n, , ,sλ η α : (Chapter 2) Normal rake, cutting edge inclination, chip flow and clearance angles       Γt(t) t t, ( ), ( )t t Γ Γ : Tool’s modal displacement, velocity and acceleration vectors in time domain       Γw(t) w w, ( ), ( )t t Γ Γ : Workpiece’s modal displacement, velocity and acceleration vectors in time domain       xxxi  Δ : Delta sign to account for chip regeration on the component       Δqjki : Instantaneous dynamic chip load of element Sjk       Δt : Sampling time of time domain simulation       4jε : Axial runout of tooth-j relative to fourth insert       measr,jkε : Measured radial runout       measr,kε : Measured radial runout vector at segment-k       r,kε : Normalized radial runout vector at segment-k       εr,j: Radial runout at tooth-j       εr,jk , a,jkε : Radial and axial runouts of element Sjk       ζ, ωn : Modal damping ratio and natural frequency matrices       ζc, ωn,c : Combined modal damping ratio and natural frequency matrices       t tn ,t n,n ,t,ωζ : Modal matrices of tn -th tool mode       w wn ,w n,n ,w,ωζ : Modal matrices of wn -th workpiece mode       ζt, ωn,t : Tool’s modal damping ratio and natural frequency matrices       ζw, ωn,w : Workpiece’s modal damping ratio and natural frequency matrices       ηjk kin, j, β : Chip flow angle and projected friction angle       Θ(t) : Modal state vector of the tool-workpiece system       Θ(ti - Tjk(ti)) , jkii,TΘ : Delayed modal state vector       Θfh : Full time history vector of modal states       Θi : Modal state vector of the tool-workpiece system at i-th time instant       xxxii  serr,sr,jkκ ,serrr,jkκ : Cutting edge angles of serrated end mill, measured along flute and mapped along tool axis       *r,jk s, jk n, jk, ,κ λ γ : Effective tool angles (true cutting edge angle, cutting edge inclination and normal rake angle) of element Sjk       κrj , κr,jk : Approach angles for tooth-j and for element Sjk       μ: (Chapter 2) Overlap factor       ( )p i jkit Tσ − : Elements of approximation of delayed modal state vector (p=1..2m)       ( )p tσ : Elements of modal state vector (p=1..2m)       τ : Integral time operator       τs,jki n,jki a,jki, ,φ β : Shear stress, shear angle and friction angle of cutting mechanics model       ϒi : Merged modal state vectors within one spindle period       Φ(s) : Transfer function matrix in Laplace domain       ϕj(t): Instantaneous angular position of tooth-j measured from normal axis (rad)       ϕjki : Instantaneous angular position of element Sjk in [rad]       ϕjki,L jki,jk R,L jk,R, , ,ψφψ : Angles of left- and right- side cutters       Φp : Period-transition matrix of the disrete system       ( )multiplepregionsΦ : Period-transition matrix when there are more than one free and forced vibration region       ϕp,j: Pitch angle of tooth-j (rad)       ϕp,jk , Tjki : Time invariant pitch angle and delay period for zero-runout case       xxxiii  ϕp,jki : Instantaneous pitch angle of element Sjk       ϕp,ki : Array of instantaneous pitch angle at segment-k       ϕpd,j: Designed pitch angle of tooth-j       ϕst,jk , ϕex,jk : Local entry and exit angles of element Sjk       ϕst,r , ϕex,r : Entry and exit angles of r-th radial zone       Ψ(ti) , ( )itΨ : Relative displacement and velocity vectors       Ψi , Ψd,jki : Overall displacement and delay vectors at segment-k       ψjkreljk,ψ : Angular and relative angular locations of Sjk element       Ψki , Ψk(ti-Tjki) : Current and delayed relative displacement vectors at segment-k       ψr,krelk,ψ  : Arrays of angular location and relative angular location at segment-k       0ω : Rotation vector of tool relative to workpiece       ω(t) : Time varying angular spindle speed       Ω: Angular velocity of tool relative to workpiece in [rev/min]       ω: Constant angular spindle speed (rad/s)       Ωw: Angular velocity of workpiece       11eθ : Slope at point-e in rake face section-1 in Figure 3.27b          xxxiv  Acknowledgements I am sincerely grateful to Dr.Yusuf Altintas, my research supervisor, for his patient guidance and motivational support of this research work. In Manufacturing Automation Laboratory (MAL), he has created a comfortable, family-like environment which is very important while studying far away from homes and families. I am happy that he encouraged me to do internship, which helped me to see the practical aspects of my research work. I am thankful to Dr.Derek Luo, director of ITRI-IMTTC in Taiwan, for hosting me during my internship on the double-sided milling project and for his encouragement. I am also grateful to Faby Feng (Apply Zeta Technology, Taiwan) for his continuous technical support on the experiments that are conducted in Taiwan and for his contributions on this research. I would also like to thank Eason Lai and Yiming Chen, engineers of ITRI-IMTTC, for helping in design of double-sided milling experiment setup and for their tireless help with the experiments. I am also thankful to Dr.Darjen Peng, Kenzi Liao, and  to all brilliant engineers of ITRI-IMTTC for their support and discussions.  I am very thankful to Dr.Min Wan, Dr.Keivan Ahmadi, Caner Eksioglu and to all previous and current members of MAL for all great discussions and shared knowledge. I will miss the times that I spent with the friendly, world-class researchers of MAL.  I would like to express my thanks to the industrial supporters of my laboratory; MTTRF Foundation for donating the Mori Seiki NMV5000DCG machine tool; Sandvik and Kennametal for generously donating the cutting tools. I am also grateful to the president, Kevin Kao, and staff of KOVA company for supplying tools for my experiments in Taiwan and for helping me understand the geometry of different tools.  I am thankful to the NSERC-CANRIMT for the support during my internship.  xxxv  Thank you, Dr.Michelle Tan, for thoroughly checking the grammar of my thesis. I would also like to thank Ozge Goktepe and Dr.Orkun Ozsahin for their inputs on my writing.  I am indebted to my mother, Dr.Nedret Kilic, my father, Dr.S.Engin Kilic, and my sister, Dr.Pelin Kilic, for all their guidance and for always supporting me.  I would like to thank Yuchien Hsu for her love and invaluable support during the last two years. I am truly thankful to all my Vancouver friends for all the nice memories.  I am very happy that it was possible to be actively involved in the board of directors of ASME-UBC Student Section (2011-2014), in the board of directors of Turkish-Canadian Society (2014) and in Vancouver's Tango community. These volunteering experiences helped me develop professional skills and let me meet great people.    xxxvi  Dedication          This thesis is dedicated to my family.  1  Chapter 1:  Introduction Machining operations are widely used in industry in producing the final shapes on the parts. A typical metallic part may need metal cutting operations with geometrically defined cutting edges (i.e., turning, boring, drilling, milling and tapping), electrochemical metal removal, or abrasive operations (e.g., grinding). The objective of this thesis is to develop generalized mathematical models which can be used to simulate and optimize metal cutting operations ahead of currently practiced, high-cost physical trials.  Turning, boring, drilling, and milling are most commonly used operations in the metal cutting industry. Depending on the shape of the workpiece, each operation may need tool geometries which vary widely. For example: a solid end mill may have tapered body with ball end, and serrated, helical flutes spaced at varying pitch angles. An indexed cutter may have a unique body shape with distributed inserts both in radial and axial directions. A boring bar may have one insert at its tip, or may have a boring head with distributed inserts in heavy duty finishing of cylinder bores. A multifunctional tool may drill, bore and chamfer large holes in a single operation in an automotive application. The current industry mostly relies on the past experience in designing tools and their use in manufacturing. This is an unproductive and costly trial and error approach. Researchers have been developing mathematical models of each operation individually, and allowing the prediction of metal cutting forces, vibrations, torque and power ahead of costly physical trials. However, the past research mainly focused on fixed operation type (i.e. turning or milling) with fixed tool geometries. Since the part may need hundreds of different tools and variety of operations, there is a strong need to develop a unified, generalized mathematical model that can be adapted to cover all tool geometries and machining operations.  2  This thesis starts with introducing the modelling of metal cutting mechanics between the cutting edge and chip removed from the workpiece, which is common for all metal cutting operations. The metal cutting process forces at the cutting edge is later transformed to tool-workpiece and machine tool coordinate systems by considering the kinematics of each operation type (i.e. turning, boring, drilling, milling) and tool geometry. The relative vibrations between the tool and workpiece are also considered in the generalized model, which can be used to simulate any machining operation with tools having arbitrary geometry. The proposed model can be used to predict the performance of various tool geometries, machine tool configurations and tool path strategies ahead of costly physical trials.  Henceforth the thesis is organized as follows.  The previous literature in modelling of metal cutting mechanics and machining vibrations is reviewed in Chapter 2. The mechanics of orthogonal cutting theory and oblique transformation to predict cutting forces in three dimensional applications are briefly surveyed, while most relevant literature used in the model development is cited in dedicated chapters in extensive detail.   The generalized geometric model of cutting edges, tools with solid body or cutters with inserts is presented in Chapter 3. A differential cutting edge is oriented to form oblique geometry, i.e., rake, inclination and approach angles. The differential cutting edge elements are distributed along the cutter body by considering its local geometry such as radius, radial and axial runouts, and axial location along the tool axis. Insert orientations on the cutter body is modelled in accordance with ISO standards. Several turning, boring, drilling and milling geometries are presented to illustrate the applicability of proposed general model on a variety of machining applications.  3  The generalized mechanics of cutting is modelled in Chapter 4. The chip thickness distribution along the cutting edge is modeled by considering both the tool geometry and kinematics of specific machining operation. The regenerative vibrations between the tool and workpiece and the time delay terms determine the chip thickness distribution, and are determined by the tool geometry and process kinematics. The generalized mechanics model is extended to consider the interaction between the machining operation and the structural dynamics of machine tool/workpiece in Chapter 5. Linear equation of motion at a differential cutting zone is modelled. The external load is modelled as a cutting force, which can have periodic amplitude and affected by the present and past vibration amplitudes. Past vibration amplitudes may bring multiple delays to the equation of motion due to varying pitch angle or runouts. The dynamics of the process is modeled in semi-discrete time domain. The process states, such as force, vibration, surface form error, torque, and power are predicted by solving the semi–discrete time domain model of the machining process which may be defined by a set of time varying, periodic differential equations with a single or multiple time delays.  The proposed, generalized model is experimentally proven in Chapter 6. Turning and boring with single cutting edge; boring with multiple inserts; drilling with inserted or solid body; milling with solid tools and indexable cutters; and complex operations such as multifunctional tools, parallel turning and milling applications are presented. The experimental verification includes the prediction of cutting forces, chatter stability and surface location errors. The thesis is concluded with a summary of contributions and suggestions for future research directions in Chapter 7 .  4  Chapter 2: Literature Survey  2.1 Overview The literature and the state-of-art in modeling machining operations are reviewed in this chapter. Mechanics of cutting, prediction of forces and surface error, and the solution of cutting dynamics and chatter stability are covered. Further relevant literature is presented in main chapters where specific contributions in generalized mathematical modeling of machining operations are presented. 2.2 Mechanics of Metal Cutting  The metal chip is plastically sheared from the workpiece material in shear zone, and the deformed chip slides on the rake face of the tool experiencing both sticking and sliding friction as shown in Figure 2.1 [6]. The shear angle, average shear stress and average friction between the moving chip and stationary tool surface are used to predict the cutting forces, torque and power in orthogonal cutting where the cutting edge is perpendicular to the cutting velocity, (i.e., 0sλ =  in Figure 2.1a). When the cutting edge is inclined relative to the cutting velocity, the process becomes oblique with cutting forces in radial, tangential and axial (RTA) directions (i.e., 0sλ ≠  in Figure 2.1a). While orthogonal cutting process is used to identify the fundamental process parameters such as shear stress, shear angle and friction angle (shown by aβ  in Figure 2.1a), most cutting tools used in practice have oblique tool geometry.   5   Figure 2.1 Schematics of chip removal at differential cutting edge element: (a) Oblique cutting with rake face forces shown; (b) orthogonal plane view. 2.3 Basic Kinematics of Regenerative Machining Operation When there are relative vibrations between the workpiece and tool structures, the chip thickness oscillates at a frequency close to one of the natural modes of the structure (Figure 2.2) . The phase or time delay between the waves on the upper and lower surface of the chip leads to machining instability, referred as “chatter”.  6   Figure 2.2 Dynamic interaction between tool and workpiece. Since Taylor [7] explained the chatter of the tool as one of the most difficult problems facing the machinist; there have been numerous research papers focusing on understanding and the solution of vibrations of machine tool and chatter. Ehmann et al. [8] gave an extensive review of the research done until 1997.  Altintas and Weck [9] discuss the significant research (until 2004) in modelling the stability of metal cutting. Tlusty and Polacek [10] and Tobias et al. [11,12] modelled the chip regeneration which then results in chatter. The basic mechanism of chatter vibration is explained as the present tooth leaving a material on the workpiece, and next tooth removing the remaining material. Tobias [13] proposes the overlap factor which takes the possibility when material left by present tooth cannot be completely removed by the next tooth. This overlap factor is especially important for cutters with high corner radius used at high feed operations [14]. Merritt [15] proposed a chatter feedback loop model for defining the regenerative chip thickness behavior.  The regenerative chip load of tooth-j ( j( )c t ) is as follows:  j 0,j R R j( ) ( ) ( ) ( )c t c t x t x t Tµ= + − ⋅ − , (2.1) 7  where the static chip load of tooth-j is 0, j( )c t . Dynamic chip load ( d,j( )c t ) depends on the relative tool/workpiece displacement at current tooth-j ( R,j( )x t ), and the delay displacement of previous tooth-(j+1) ( R,j j( )x t T− ):  d,j R,j R,j j( ) ( ) ( )c t x t x t Tµ= − ⋅ −   (2.2) where µ  is the overlap factor [13]. It is assumed in this thesis that present tooth removes all material left by previous tooth, therefore µ =1. Tj is defined as the delay period that is the time passed since the cut of previous tooth-(j+1) until the cut of current tooth-j. Although the static chip load is constant for turning, boring and drilling operations, it is time varying in milling operations. Martellotti [16] approximated the actual trochoid milling tool path by a series of circular arcs (cycloid); then, static chip thickness is related to feedrate by a simple equation:  0, j t,j j( ) sin ( )c t f tφ= ⋅ , (2.3) where t,jf  is the feedrate per tooth-j of the tool and j( )tφ  is the angular position of tooth-j measured from normal axis. Long et al. [17] and Faassen et al. [18] computed the chip thickness for trochoid path analytically; and Montgomery and Altintas [19] digitized the part surface to calculate the exact dynamic chip thickness numerically.  Regular tools have equal tooth spacing whereas the general irregular tools have unequal tooth spacing. Pitch angle of tooth-j is the angle between the current tooth-j and the previous tooth-(j+1):  p,j j+1 j( ) ( )t tφ φ φ= − , (2.4) and it is directly related to the delay period for the constant angular spindle speed (ω ): 8   pj,jTφω=  . (2.5) The radial or axial runout of the tool depends on insert/flute size variation, spindle out-of-roundness, improper tool balancing, dimensional tolerance of insert seats (pockets) [20]; on designed location of inserts and pockets [5]; on designed variation of the edge geometry [21]. In milling, for instance, radial runout of previous tooth-(j+1) and radial runout of present tooth-j affect the static chip load of Equation (2.3) as follows [20]:  0, j t,j j r,j r,j+1( ) sin ( )c t f tφ ε ε= ⋅ + −  . (2.6) For small radial runout r,jε  and r,j+1ε  for present and previous teeth, respectively. For high runout values the tooth-j would not be in cut if the radial runout of previous tooth is enough to remove the chip load of the present tooth:  r,j+1 t,j j r,jsin ( )f tε φ ε> ⋅ +  . (2.7) A special algorithm for calculating the chip load is given in Refs. [20,22] to compute the chip load for high runout tools.  2.4 Cutting Forces The chip load that is found in Equation (2.1) is projected onto the chip thickness direction based on the kinematics of the operation. Fu et al. [20] presented the geometrical transformations for the milling kinematics. Similarly, Armarego [23] has formed the kinematics of the oblique geometry of  twist drill.  Cutting force has been an area of extensive research (see Figure 4 of Ref.[8]). Merchant [24] introduced the widely used thin-zone shear model that considers the tool with sharp edge, i.e., the cutting force is assumed only by the shearing action. As noted by Hitomi [25] and Shaw [26], Masuko (1953) derived an additional force due to indentation and rubbing at the edge of the tool. 9  Albrecht [27] also reported the same mechanism and called it ploughing force. Ploughing force is practically negligible for the sharp tools [26]. Then, cutting force is composed of two components: Shearing and ploughing,   c es( ) ( ) ( )F t F t F t= +   (2.8) Koenigsberger and Sabberwal [28] validated the dependence of the cutting force to area of the chip section (chip area) by repeating the cutting tests with 5 different cutting edge angles (or corner angles). Instantaneous shear force is related to the instantaneous chip area ( c ( )A t ) with the following function [28]:  c c 1 2 L c( ) ( , ,.. ) ( )F t K p p p A t= ⋅  (2.9) where the cutting coefficient or specific cutting pressure (Kc) is a function of different L number of cutting parameters ( 1p  to Lp ). It is also reported by Koenigsberger and Sabberwal [28] that the cutting coefficient increases with decreasing chip thickness (also called feedrate).  Static ploughing forces are modeled using the chip length and the so called edge force coefficients [29]. In general, ploughing forces account both for the static indentation of the tool into the uncut chip [25] and the rubbing of the flank of the tool on the wavy machined surface. First component is called the static ploughing forces or edge forces, and second component is called the dynamic ploughing forces or the process damping forces [30]. More literature related to ploughing is presented in Chapter 4.  2.4.1 Mechanistic Identification Mechanistic identification is a curve fitting of cutting force coefficients  to the experimentally measured cutting force data collected with a specific tool. Kline et al. [31] fitted tangential and radial miling force coefficients to a second-order polynomial function of radial 10  depth of cut, axial depth of cut and feed. Altintas [6] modelled the cutting force coefficients as a linear function of feed; this model is more convenient in solving chatter stability than the nonlinear models. Chandrasekharan et al. [32] presented a mechanistic method to identify the cutting coefficients for drilling by collecting the cutting force data during the entry to cutting. A detailed survey of mechanistic cutting force coefficient methods are given in [33]. 2.4.2 Generalized Cutting Force Prediction Armarego (Equation (11) in [34]) correlated the cutting force to tool geometry and to cutting conditions defined by the depth of cut, the feedrate and the speed. Based on the thin-zone shear theory of Merchant [24], the orthogonal and oblique cutting forces can be represented as functions of shear stress, shear angle, friction angle (shown by aβ  in Figure 2.1a) and tool geometry [35]. Armarego [36] explains how the cutting force is related to the shear plane parameters using the oblique thin shear zone analysis.  Armarego and Whitfield [29] introduced the concept of single edge oblique machining operation, and stated that all practical machining operations can be modeled using this model. They also added the edge force into the force model and correlated it to the chip length. Armarego and Deshpande [37] set the goal to unify the cutting force model for the practical machining operations such as turning, drilling and milling; their study gave the idea of computing the oblique cutting forces based on the rake face along the cutting edge of arbitrary tool. They also introduced the idea to build up a common database for the shear parameters that are the shear stress, shear angle and friction angle. Based on Refs. [29,36], Budak et al. [35] generalized the milling force using the mechanics of oblique cutting and predicted the cutting forces on the orthogonal cutting plane (RTA plane) along the cutting edge of the milling tool. Modelling the shear cutting forces this way sets the basic idea of general machining model. This 11  model takes the oblique geometry and cutting conditions as input and predicts the cutting force coefficients along the cutting edge using an available shear parameter database for the tool-workpiece material combination. Yucesan and Altintas [38] modelled the rake face and flank geometry of ball end mill. They applied the kinematics of ball end mill to represent the forces, then they identified the cutting coefficients from the experiments. Furthermore, Engin and Altintas [39] modelled the cutting edge geometry of general end mill, and mechanistically identified the cutting coefficients. They also modelled the indexable end mill by rotating the 2D-geometry inserts to locate on tool holders [40]. Altintas and Lee [41] used the theory of Ref. [35] to estimate the cutting forces and stability of ball end milling operation, without any need of mechanistic identification tests. Subsequently, they developed a similar model for helical end mills [42].  2.5 Modelling of Generalized Tool Geometry Armarego [34] discussed about the generalized or unified model in his extensive review of different operations. Figure 15 of Ref. [34] shows the goal to unify all the turning, boring, drilling, milling and tapping operations into one model. There are two important studies in modelling the general geometry: Chandrasekharan et al. [43] proposed a general model for computing the drilling force with arbitrary drill geometry. Their model determines the effective tool geometry (normal rake, cutting edge inclination and cutting edge angles) at each discrete oblique segment along cutting edge. Yucesan and Altintas [38] modeled the rake face and clearance face (flank) of ball end mill tool using five vectors at each point along the cutting edge: Cutting edge tangent, rake face normal, rake face, clearance face normal and clearance face vectors. Although these two models give a great insight into modeling the cutting operation for drilling and ball end milling, still there is no mathematical model that combines multiple 12  machining operations. This thesis models the geometry of arbitrary cutter, and determines the cutting forces at each segment using the mechanics of oblique cutting for all metal cutting operations with varying tool geometries.  Lee and Lin [44] also used a similar model to Ref. [38] but only with the rake face geometry. They only used this geometric model to improve the tool path design for the ball end milling operations. Fontaine et al. [45] modeled the ball end mill geometry and predicted cutting force for an arbitrary feed direction; they used thermo-mechanical process model.  Tsai and Liao [46] also proposed a cutting force model for ball end mill geometry considering vertical and horizontal feed directions. Unlike dividing the cutting edge into discrete disk segments, their method needs to compute chip areas along the cutting edge (see also [47]). This type of cutting force prediction is harder to generalize for other cutting operations.  Recently, Tuysuz et al. [48] proposed a model for ball end milling based on the five-axis milling model of Ozturk and Budak [49], and additionally included the indentation effect at the tool tip. They used disk elements to discretize the cutting edge. Thus, their method of predicting indentation forces is more applicable for general cutting operation.  The first study combining the geometric and force modeling is from Kaymakci, Kilic and Altintas [1]. They included the standard indexable cutter geometry set by ISO standards [50,51]. Their model is only limited to two-dimensional rake face geometry, and only applied to indexable cutters. Campocasso et al. [52] modelled the geometry of a general turning insert. They used homogeneous transformation matrices to locate the insert on tool holder. In [53], they further extended their method to generalize mechanics for turning and drilling operations. Altintas and Kilic [3] developed a generalized dynamics model to include the regenerative 13  vibrations for all metal cutting operations. Otto et al. [54] introduced a fast unified stability prediction method for the machining operations.  Wan et al. [5,55] modelled the multifunctional and thread milling operations using the similar geometric modeling and force prediction concept. Tunc et al. [56] modelled the five-axis milling operation using an arbitrary geometry tool and predicting the force by thermo-mechanical process model. Recently, Ahmadi and Savilov [57] modeled the arbitrary drilling cutter based on the concept developed by [1,3,43]. This thesis takes their model one step forward to include arbitrary tool geometry that is used for general machining operation.  2.6 Simultaneous Machining Modelling Multiple tools are used in simultaneous machining of features on a single part. Modelling of dynamic chip thickness for serial machining operation (only one tool in cut at a time) is done by using single pair of relative tool-workpiece vibration. However, simultaneous machining operations involve more than one tool, thus the dynamic chip thickness depends on multiple pairs of relative tool-workpiece vibrations. Lazoglu et al. [58] modelled the dynamics of simultaneous (parallel) turning operation in modal domain, such that they could consider the flexibility at different parts on workpiece; they presented a general model for two turning tools, and included the flexibility of both tools and workpiece; they solved the equation of motion in time-domain. Budak and Ozturk [59] also solved the same system in frequency domain based on the method of Altintas and Budak [60]. They observed an increase in stability for specific depth of cut values; this is later supported by Ding et al. [61]. Budak and Ozturk applied the same methodology to parallel milling operations as well [62]; they solved the equation using conventional time-domain solution. Budak et al. [63] solved the same system in frequency 14  domain. They used a similar modal model to [58] and [64], to consider the flexible dynamics along different sections of tool-workpiece contact zones.  A specific application of parallel milling double-sided milling operation, where two face milling tools simultaneously cut from each side of a flexible part. Shamoto et al. [65] designed a system of two spindles and set a difference between rotation speeds of dual spindles and eliminated the forces and vibrations along flexible direction of the workpiece. They used the same idea as in Ref. [66], where the authors designed tool pitch angles to eliminate the unsable vibrations. Shamoto et al. extended the stability model to general multitool systems in [67]. Although the flexibility of the part changes along the tool-workpiece contact, they averaged the dynamics along the contact. Olgac and Sipahi [68] developed an algorithm to solve the equivalent algebraic form of the characteristic equation of simulatenous machining operation. Brecher et al. [69] simulated the dual-spindle milling system in tim-domain; they investigated the effects of relative angular offset and the speed-difference, and validated by experiments.  2.7 Dynamics of Machining Operations Machining operation involves the interaction of tool-workpiece structural dynamics with the chip generation process (Figure 2.2). Vibrations of the structure creates a regeneration cycle of chip thickness. Chatter vibration amplitudes are against the presumed small vibration assumption. Solution by Tlusty and Ismail [70] considers the case when the tool instantaneously jumps out-of-cut which happens when the vibration amplitude exceeds the instantaneous chip load. Non-linear solution methods can be replaced by linear solutions if only the stable or marginally stable solutions are sought. Minis and Yanushevsky [71] proposed a frequency domain method using Fourier analysis and Floquet’s theory; they solved the equation numerically. Altintas and Budak [60] used a similar method, and introduced an analytical frequency domain solution to predict 15  the stability lobes of machining for the cylindrical milling cutters. They represented the cutting force with its average value and assumed single chatter vibration frequency. In addition to being very fast, this method is accurate for milling operations with regular pitch angles and high radial depth of cut to diameter ratio. For low radial depth of cut to diameter ratio, an improved method is presented by Budak and Altintas [72] and by Merdol and Altintas [73]. Altintas et al. [66] presented a method for stability analysis with unequal pitch tools. Li et al. [74] proposed a fast, analytical solution to obtain exact solution for stability of boring bars. Similar concept is used by Eynian and Altintas [75] to model milling tools with rotating dynamics.  Zero order analytical frequency domain solution of Budak and Altintas gives a fast analysis of stability. Due to the time-varying nature of dynamic chip thickness, higher order methods increase the accuracy of predictions [76]. Bachrathy and Stepan [77] introduced an efficient method for higher order frequency domain solution. Insperger and Stepan [78,79] developed a semi discrete-time domain solution (semi-discretization method) which can approximate the delay-differential equations of milling by a series of ordinary-differential equations [80]; only delayed states and time-periodic coefficients of the milling equation are approximated. The resulting simulation accuracy is higher than the conventional time-domain simulation of delay-differential equations. Ding et al. [81] isolated the major component of the system dynamics matrix, and approximated the rest of the terms in the integral part of the approximate difference equation; they obtained a faster solution. Sims et al. [82] discuss the accuracy of semi-discretization method for variable pitch milling tools. Asl and Ulsoy [83] solved linear delay-differential equations using the matrix Lambert function. Yi and Ulsoy [84] further extended the method to general system of delay-differential equations, and they extended the method to simulate stability of machining operation [85] by considering multiple vibration modes. Butcher 16  et al. solved the milling dynamics equation using the Chebyshev polynomials and collocation method [86,87]. Ding et al. [88] also used Chebyshev points while interpolating the delay term using barycentric Lagrange interpolation polynomials.  2.8 Surface Form Error Stability and surface form error (called surface location error, or surface error in the literature) are usually considered during the design of machining operations. Surface error can be defined as the difference between commanded and actual locations of the machined surface; it results from the relative vibrations of the flexible tool-flexible workpiece structure under the influence of dynamic cutting forces. For end milling operation, Kline et al. predicted the surface errors using static deflection model for the tool and workpiece [89]; they also used the cutting force model of Kline and DeVor [22] that considers radial runout of the tool. Ko et al. [90] included the runout effect in the cutting force model, and used the experimental cutting force data to predict the runout parameters. Wan et al. [91] also showed a model on how to obtain the runout parameters form the cutting tests. In another study, Wan et al. compared different runout models [92], and found out that radial runout model of Kline and DeVor [22] is accurate despite its simplicity. Sutherland and DeVor presents an improved method for predicting the surface error considering the tool runout [93]. Smith and Tlusty [94] discussed  the evolution of the cutting force models and their effect on surface error prediction. Li and Shin [95] used a modal model with complex mode shapes obtained at 5 measurement points along the flexible end mill-flexible workpiece, and solved the equation of motion in time domain using Duhamel integration. They successfully predicted the surface errors under the cutting forces. Schmitz and Mann [96] predicted surface location error in frequency domain. They applied Fourier analysis to the cutting force and obtained the surface error while also including helix angle effect of the end 17  mill. Bachrathy et al. [97] also used Fourier transformation to analyze the surface error for helical end mills. Pirtini and Lazoglu [98] constructed a static drilling force model to simulate machined hole profile by considering forced vibrations. Bayly et al. [99] developed a time-domain model to predict the lobed holes; they solved the equations in discrete-time domain. Roukema and Altintas [100] introduced an extensive time-domain simulation model for drilling; their model is also capable of predicting machined hole shape. In their paper, they give a detailed literature review on modelling of drilling operations. Time-domain analyses are used for simulating the dynamic forces, relative vibrations of tool-workpiece system, and surface errors [19,101,102]. Bachrathy et al. [103] introduced a nonlinear solution method for milling operations with high amplitude; they also argued about two-delay contact at the tool rake face. Insperger et al. [104] predicted surface error with conventional time-domain simulation, while Mann et al. [105] developed a fast algorithm to predict the surface error in time-domain.They used the steady state component of the solution method proposed by Bayly et al. [106]. Ding et al. [107] used the steady state part of the state transitions. Eksioglu, Kilic and Altintas [2] used semi-discretization, and simulated general end milling operation by simultaneously predicting the forces, vibrations and surface errors.  2.9 Summary Almost all of the presented relevant literature in machining modelling is on milling operations. The method presented in this thesis utilizes the extensive literature in milling, and combine it with turning, boring and drilling studies. The generalized model presented in the remaining chapters of this thesis is capable of not only covering all individual modelling results given in the literature, but without needing to reconstruct the model for each specific machining 18  process. The model developed by the author is general, and can be parametrically configured to solve all machining operations with varying tool geometry and configurations.  19  Chapter 3: Generalized Geometric Model of Cutting Tools  This chapter generalizes the geometric model of indexable and solid tools that are used in machining operations. First, the geometric parameters are identified. 15 parameters are chosen to define geometry of an arbitrary tool. Then, cutting edge tangent, rake face tangent and rake face normal vectors are defined along the cutting edge of the cutter. Finally, effective tool geometry is derived for general operation. The method is demonstrated on example indexable and solid tools.  It is noted that there are various terminologies used for tools in the literature and industry. In general, the term “tool” is used to refer to a body that holds cutting edge(s). Correspondingly, single point tools have only one cutting edge or tooth, and multi-point tools have multiple cutting edges. While tools with multiple cutting edges ground on one body are called solid tools (e.g., twist drills, solid end mills with fluted cutting edges), tools with added “inserts” are called indexable cutters (e.g., indexable milling cutters and boring heads).  3.1 Generalized Geometric Parameters of Arbitrary Tools The proposed mechanics and dynamics model developed in the thesis is based on the generalized, parametric definition of cutter geometries. While single point turning and boring operations have one tooth, there may be multiple layers of teeth or varying flute geometry on multi-point tools such as drills, indexable cutters, fluted end mills and boring heads.  Discretization of cutting edges: As shown in Figure 3.1, Figure 3.2, Figure 3.3 and Figure 3.4, the cutting edge of the cutter is discretized into K number of segments with a constant differential thickness dz for turning, drilling, roughing end mill and general milling tools, respectively. Geometry of a milling tool is discretized in its axial direction; turning tool geometry is discretized in radial direction of workpiece; geometry of boring and drilling tools are 20  discretized in their radial direction. Thus, for milling tools, these segments look like discs with axial thickness, whereas turning, boring and drilling cutters are discretized by segment-like rings with radial thickness. Each discrete cutting edge element is coded as Sjk for edge-j at k-th axial/radial segment of a tool with j=1..N number of teeth and k=1..K number of segments.  The axial (for milling) or radial (for turning/boring/drilling) location of each differential cutting edge segment-k is measured from the zero depth of cut point, i.e., measured from the bottom side of the turning (Figure 3.1a) and milling (Figure 3.3a and Figure 3.4a) tools, and measured from the outer diameter of the bored or drilled hole (Figure 3.5c). Each differential cutting edge segment has a constant thickness of dz with constant geometry.  The total number of cutting edges on the tool is N, while the actual number of edges that remove material from the workpiece at axial or radial segment-k is Na,k. As seen in Figure 3.2a,c, element S21 of the insert-2 at radial segment-1 of the drilling cutter do not exist. Although the total number of inserts is 2, the actual number of cutting edges at radial segment-1 is 1. Edge index , j (j=1..N) is used for numbering the edges (also called teeth, flutes or inserts) of the cutter. Edge-j increases from the bottom to upwards in axial direction for milling tools; and from outside towards the cutter center for drilling tools. At the same axial or radial segment-k, for right hand cutter with clockwise (CW) rotation when looked from drive side, the edge number-j is increased in CW direction (see, e.g., Figure 3.3b or Figure 3.4c). For left hand cutter, j increases in counterclockwise (CCW) direction. Thus, if the current edge in cut is j, then the previous edge in cut is j+1 and the next edge in cut is j-1. 21   Figure 3.1 Left hand turning side-view schematics: (a) Edge is discretized in radial direction from workpiece axis; (b) general side view during cutting; (c) Close-up of element S1k . The geometric parameters of turning tools are stored as a function of radial location starting from the machined workpiece radius (Rf in Figure 3.1c). The boring and drilling tool parameters are derived as a function of radial location starting from the outside circle of the hole, and the cutting edge is discretized at dz intervals along the radius of the tool (Figure 3.2c). The discretization is carried out along the tool axis (or rotation axis), i.e., along the axial direction of the tool in milling (Figure 3.3a, Figure 3.4a). Each cutting edge element Sjk at each discrete segment-k is defined by 15 geometric parameters: • Element available-to-cut condition, • Location in Cartesian coordinates (x0, y0 and z0), • Axial depth and axial runout, 22  • Radius, radial depth and radial runout, • Location angle, relative location angle and designed pitch angle,  • True cutting edge angle, cutting edge inclination angle and normal rake angle,  These 15 parameters are computed for each tool-machining operation (i.e., turning, drilling, boring and milling) for the generalized mathematical model developed in the thesis.   Figure 3.2 Indexable drill schematics: (a) Side view; (b) bottom view with insert numbers; (c) bottom  with the radial discretization. 23   Figure 3.3 Geometry of serrated end mill that is used for rough milling operations: (a) Axial segment numbering; (b) radial locations (including runout) at k-th axial segment.  Contact conditions between the work material-cutting edges: Depending on the engagement geometry of the part (workpiece) with the tool, there are three parameters defined for the contact condition: 1- Element-available-to-cut is related to the physical availability of the element Sjk; 2- edge-in-cut condition depends on the engagement boundaries; 3- missed-cut condition depends on the tool runout, i.e., on the chip load variation between the edges at same segment-k.  24   Figure 3.4 General end mill geometry schematics. (a) Side view with the axial segments; (b) side view with cutting forces and chip geometry; (c) top view with edge numbers and angular parameters; (d) top view with radial runouts. (Adapted from Eksioglu et al. [2].)  The availability to cut the work material: It is assumed that all edges (j=1..N) of the tool would engage the workpiece.When the cutter is discretized into K number of segments, there may not be any geometry parameter assigned for some element Sjk (see Figure 3.2c). Segment-k 25  of tooth j is flagged by time invariant binary parameter g1,jk, i.e., if the contact exists g1,jk=1; and if it does not g1,jk=0.  Actual number of cutting edges (Na,k) is found by summing up the element-available-to-cut array at each segment-k:  a,k 1,jkNjN g= ∑ . (3.1)  Coordinates of cutting edge on the tool: Cartesian coordinate locations (x0y0z0) are assigned for each element Sjk in tool reference frame. z0 is set as tool axis for milling (Figure 3.3a) and drilling/boring (Figure 3.2a) operations, and it is set as workpiece rotation axis for turning (Figure 3.1a) operation. x0 and y0 are located radially using the radius and angular location of the element Sjk.  The axial/radial position (Zk) of the midpoint of the edge segment k is:  k( 1)d2kZ z−= . (3.2) The number of differential cutting edge segments that are in contact with the work material is q; and it is smaller than the total number of segments: q K≤ . The number of the contacting cutting edge segments (q) depends on the type of operation. For milling, segments are aligned in z0 (tool axis) direction. For turning and boring, the tool is radially immersed into the workpiece, thus the cutting edge segments are aligned with radial direction of the workpiece. For turning (Figure 3.1c), the depth of cut is the difference of the initial and final external radii of the workpiece, ( )p i f externala R R= − . For boring, the depth of cut is the difference of the initial and final radii of the bored hole in the workpiece, ( )p f i bored-holea R R= − . Drilling operation (Figure 3.5a) is similar to the boring process, and the cutting edge segments are aligned with radial 26  direction of the tool. The depth of cut is the difference of the initial and final radii of the drilled hole, ( )p f i drillled-holea R R= − . The number of cutting edge segments in the tool-workpiece contact zone is:  pdaqz=   (3.3) The cutting edge segment k on the cutting edge j is referred as element Sjk with a unique geometry, and there are N q× amount of cutting edge elements on the tool.   Angular location of cutting edge: Angular location ( jkψ ) of the element Sjk is independent of the time and the rotation of the cutter or workpiece. As shown in drilling cutter of Figure 3.5b, it is defined as the angle measured from +y0 axis to the radius vector  ( 0jkR ), which is defined on x0y0 plane. Positive direction of the jkψ angle is CW around the rotation axis (z0) of tool or workpiece axis, depending on which one is rotating. Relative angular location reljkψ  is measured from radius vector ( 011R ) of the S11 element or the first segment (k=1) on the first edge (j=1), which leads to rel11 0ψ = . The relative angular location of element Sjk, defined relative to element S11, becomes reljk jk 11ψ ψ ψ= − .  Local radius of cutting edge: The local radius Rjk of the cutting edge element Sjk is defined as the distance from the edge-segment to the tool center in milling; from the edge-segment to the workpiece center in single point turning (Figure 3.1c) and boring tools; as the distance from the edge-segment to the hole center for drill (Figure 3.5a) and indexable boring heads. The local radius of the tool does not change during the operation.  27    Figure 3.5 Parameters for the drilling operation at initial time instant (t=0). (a) Side view with the radius vector ( 01kR ) of segment-k on insert-1; (b) top view with the angular and relative angular locations.  Local entry and exit angles of cutting edge: Local entry angle ( st,jkφ ) is the angle of the tool at which the element Sjk engages with the workpiece and starts to cut. Similarly, the local exit angle ( ex,jkφ ) is the angle of the tool at which the element Sjk leaves the cut. For conventional turning, boring and drilling operations of the tool is always engaged with the workpiece, thus st,jk 0φ =  and ex,jk 2πφ = . For general milling operation (Figure 3.4c), these angles depend on actual radius (including the tool’s radial runout) and the translational feed per revolution of the cutter [22]. Although not included in this thesis, the tool may experience multiple local entry and exit angles during one rotation if the part geometry changes.  28  Feed per revolution of the tool: Translational feed per revolution (fr) of the tool is used in determining the instantaneous chip load (i.e. thickness) at time t. The tool runout and feedrate determine the chip load and the delay period between the subsequent tooth at each time instant. Also, the entry and exit angles of each element Sjk are found using the tool runout and the feedrate. As a special case, if there is no runout (e.g., if all r,jk 0ε =  in Figure 3.4d) and if all elements are available-to-cut (i.e., all g1,jk=1), feed per tooth (ft) is calculated as  ft= fr/N.  Instantaneous edge-in-cut condition parameter: The cutting edge elements become in- and out-of-cut especially during interrupted cutting operation like milling. In cut conditions are determined by the time-dependent binary parameter g2,jki of the cutting edge element Sjk, as a function of local entry and exit angles as :  st,jk jki ex,jk2,jki1 ,0 , otherwisegφ φ φ≤ ≤=   (3.4) For the uninterrupted turning, boring and drilling operations, the g2,jki is always 1.   Instantaneous angular position of the cutting edge segment: The instantaneous angular position ( jkiφ ) of the element Sjk is evaluated as a function of tool-workpiece contact and spindle’s angular position (see Figure 3.4c), as:   jki jk i  (milling, boring and drilling)tφ ψ ω= + ⋅ , (3.5) where ω  (rad/s) and Ω (rev/min) is the spindle speed; ti (s) is the time at discrete time interval counter i. Instantaneous angular position is generally different than angular location which is 29  independent of tool rotation. However, if the cutter does not rotate (as in turning and boring), the instantaneous position and angular location become equal for all time instants,  jki jk  (turning, boring)φ ψ= . (3.6)  Tool runout: The radial runout (see Figure 3.3b) r,jkε  of the element Sjk is modeled as part of the local radius of tool, and defined as ( )r,jk jk jkminjR Rε = − . For milling, other than 3- and more axis cutting, the axial runout is only useful for deciding which segments of the edge are in-cut for the current depth of cut. For turning with multiple tools (Figure 4.4b), boring and drilling (Figure 4.3), the feed direction is aligned with the tool axis, hence the the axial runout a,jkε  of the element Sjk becomes important. The local axial location Ajk of the element Sjk is used for deciding the axial runout at each segment, ( )a,jk jk jkmaxjA Aε = − .   Instantaneous missed-cut condition parameter: The radial runout of the element Sjk compared to the previous edge Sj+1,k may result in zero chip generation (missed-cut condition [108]), since the previous edge might have already removed a higher amount of chip (negative chip thickness means no cutting). The zero chip occurrence at segment-k may be caused by several successive cutting edges njki after the edge jkij+n ,kS , i.e., no cutting occurred for the past “njki-1” edge-passes [22]. The limits of njki is jki1 n N≤ ≤ . A binary time dependent parameter g3,jki is introduced to account for the runout effect on the chip removal. If chip generation occurs at the element Sjk at the i-th time instant, g3,jki is set as 1, otherwise 0. 30  Instantaneous pitch angle: The pitch angles are important to model the cutting edge geometry for the cutters with multiple cutting edges. Helical or indexable cutters may have axial cutting edge segments that may be positioned at unequal radius around the body of the tool, due to the design or presence of radial runout. Due to the edge-in-cut condition, the pitch angle may be quite different at each cutting edge segment along the axis of the cutter. The general term for the pitch angle is defined as jkip,jki j+n ,ki jkiφ φ φ= −  [22]. The maximum possible value of the pitch angle is p,jki 2πφ =  and the minimum possible value is the no runout condition, p,jki j+1,ki jkiφ φ φ= −  (assuming that the previous edge j+1 is in-cut, i.e. g1,j+1,k=1). Parameter njki is iteratively computed by considering the runout and the feed per revolution of the cutter. Details are given in Chapter 4.   Instantaneous delay period: The delay period Tjki of the cutting edge element Sjk at each time instant is derived from the rotational speed of the tool or workpiece and the instantaneous pitch angle as:  p,jkijkiTφω=  (3.7) which may vary periodically in addition to varying along the cutting edge.  Cutting speed: Relative cutting speed ( 0c,jkv ) at element Sjk of the tool relative to workpiece is defined using the rotational speed of the tool or workpiece and the local radius:  [ ]0c,jk jkm/min 2π [m] [rev/min]v R= ⋅ ⋅Ω . (3.8)  31  Geometric definition of the cutting edge: The cutting edges are usually not straight for the roughing tools, hence they can be represented by local, unit tangent vector jkt  at the midpoint of the element Sjk. The cutting edge tangent vector jkt  lies on the rake face ( γA  in Figure 3.6b) which can be defined by a vector (not parallel to the cutting edge) 1, jkn  on the rake face, or by the rake face normal vector 2, jkn . In case of vector 1, jkn , its direction should be corrected to 1c, jkn  and made perpendicular to the cutting edge as:  ( )( )jk 1, jk jk1c, jkjk 1, jk jk× ×=× ×t n tnt n t  (3.9) Crossing the cutting edge and corrected rake face plane vectors gives the rake face normal vector which is directed into the rake face (similar to the models in Refs. [38,44]):  2, jk 1c, jk jk= ×n n t . (3.10) As shown in Figure 3.6 rake face vectors are the axes of the cutting edge frame (Frame-D), which is described in Figure 3.8.   Figure 3.6 Rake face vectors at element Sjkn arbitrary tool; (b) close-up view of local cutting edge frame. 32  A note on the modelled milling operations: In pocket milling operation (2+1-axis cutting) [109], the feed direction is always perpendicular to the tool axis. Thus, the tool moves only in x or y directions of the machine tool. In this thesis, additional movement in z or rotational axes (A,B,C) are not taken into account. However, the same developed methods can be applied to 3-, 4- and 5-axis tools moving in arbitrary feed directions [48,49]. In this thesis, milling cutter discretization is conventionally done in tool axis (z0) direction. The geometry of each of the discrete cutting edge is also referred as local cutting-edge geometry or simply, local geometry. The local geometry of the milling cutter can be stored as a function of the axial height. For the standard milling cutters, for example, Engin and Altintas derived analytical functions [39]. For general/arbitrary cutter it is assumed that the local geometry is available only numerically [21,56,110]. At each axial height for the k-th (k=1..K) segment of the j-th edge/teeth/insert/flute (j=1..N) of the milling cutter the geometrical parameters are stored.  3.2 Generalized Modeling of Cutting Edge Coordinates on the Tool Body The generalized geometric model of tools is built first by defining rake face of individual cutting edge or insert geometry in local insert frame. Then, origin of local insert frame is translated to the position on the tool holder. Finally, new frame is rotated to obtain the designed orientation of the cutting edge. The method is applicable to the kinematics of turning, boring, drilling and milling operations.  3.2.1 Indexable Cutter There are two steps for locating the cutting edge of the indexable cutter: 1. Transformation from the insert reference coordinate frame (Frame-i) to the cutting edge coordinate frame; 2. Insert is translated and rotated to match the desired orientation on the cutter. The first 33  transformation locates the cutting edge relative to the local insert frame, whereas the latter transformation locates the insert on the holder.   Locating the cutting edge on the insert: The conventional inserts have relatively simple geometry and can be approximated as 2D flat face with constant rake face plane geometry [1]. The modern insert designs have non-straight cutting edges, to reduce the vibrations and forces, and to improve the surface quality. Thus, the edge geometry of modern inserts are not on the same plane on the entire edge [111]. For complicated or unknown geometry, if the 3D model is not available, a contact-type instrument (e.g., a Coordinate Measurement Machine/CMM) or tool microscope may be used to measure the edge geometry of the insert.  With the local cutting edge tangent vector and the local rake face ( γA ) known at the selected point along the cutting edge, the transformations are in two steps as follows:   Step-1: The rake face may or may not be constant along the cutting edge, even when the cutting edge is on the same plane (see example insert geometry in Figure 3.15). As shown in Figure 3.7a, the 3D cutting edge of the insert is divided into C number of cutting edge section planes, and the selected section-c (c=1..C) is divided into R number of flat-rake face sections. Thus, the number of the rake face sections is equal or larger than the number of cutting edge sections on the insert. Each selected point in the rake face section-c,r (r=1..R) lies on the same local flat rake face γA  (Figure 3.7b). The orientation of the local rake face frame (Frame-c,r) of the selected point in the insert reference frame (Frame-i) is determined by the following homogeneous transformation: 34   Figure 3.7 Locating the local rake face frame in the insert reference frame. (a) Dividing the cutting edge into flat rake face sections. (b) Close-up view of the selected rake face section on the cutting edge   i i c,ri c,rc,r1x3 4x41 =    R rT0  (3.11) This is done by translation vector (irc,r shown in Figure 3.7b) and rotation transformation (ic,rR ). Rotation transformation is found directly by the unit axis vectors (shown in Figure 3.7b) of the local rake face frame. Tangent vector ( i c,ri ), rake face normal vector (ic,rj ) and rake face vector ( i c,rk ) are expressed relative to Frame-i. Then, the rotation matrix of Equation (3.11) is:  i i i ic,r c,r c,r c,r =  R i j k . (3.12) Thus, Equation (3.11) is rewritten by inserting Equation (3.12):  i i i i c,ri c,r c,r c,rc,r 0 0 0 1 =    i j k rT   (3.13)  35   Figure 3.8 Locating the local cutting edge frame: (a) Highligted rake face section of Figure 3.7a with the selected point circled; (b) close-up view of the rake face section with selected point-e.   Step-2: Step-1 transformation from the insert reference frame to the local rake face frame is shown in Figure 3.7b. Step-2 is to transform to local cutting edge design frame (Frame-D). The transformation from Frame-c,r to the selected point-e in section-c,r is shown in Figure 3.8. General form of the transformation matrix is:  c,r c,r c,r,ec,r c,r,ec,r,e1x3 4x41 =    R rT0. (3.14) The transformation is only translational if the cutting edge is a straight line in the rake face section (e.g., see Figure 3.11); it is both translational and rotational if the cutting edge has at least two lines or at least one arc (e.g., see Figure 3.27b). Rotation transformation within Frame-c,r is on local rake face ( γA ), which is defined by local tangent (ic,ri ) and rake face (ic,rk ) vectors (Figure 3.7b). Therefore, rotation of the Step-2 transformation is only around the rake face normal vector ( i c,rj ).  36  Combining the two transformations of Equations (3.11) and (3.14) gives ic,r,eT , which orients the local cutting edge frame (Frame-c,r,e or generally, Frame-D) at the selected point-e to the insert frame (Frame-i):   i i c,rc,r,e c,r c,r,e= ⋅T T T  (3.15) Then, the unit axis vectors at the rake face of the selected point are found by expressing Equation (3.15) similar to the form of Equation (3.13):  i i i i Di D D DD 0 0 0 1 =    i j k rT  (3.16) where the local tangent ( i Di  or ic,r,ei  ), rake face normal vector (iDj  or ic,r,ej ) and rake face        ( i Dk  or ic,r,ek ) vectors are expressed in insert reference coordinates (Frame-i).   Locating the inserts on the tool holder: The cutting edge frame orientation is obtained by transforming from the insert frame (Frame-i) to tool coordinate frame (Frame-0) by using homogeneous transformation matrices as follows:  • The origin of the fixed tool reference frame (Frame-0) is selected. Then, the desired location for insert j is set. As shown in Figure 3.9a, for simplicity, the desired orientation of the insert is taken same as the pocket (seat) frame (Frame-s). Initially, the insert is aligned with the fixed frame. After the transformations, the insert frame (Frame-i) is represented with respect to Frame-0 (Figure 3.9b).  • The origin of the insert frame is translated to the seat location of insert j on the cutter frame. This is done by the translation vector ( 0 ir ) as shown in Figure 3.10a.  37  • The insert frame is rotated around its axes for the angular orientation of the insert in the insert seat of the tool body [1]. This is shown in Figure 3.10b.  The details of the transformation matrices are given in Appendix A. Transformations in this section are independent of the insert type, and only depend on the tool holder type and its seat orientation.  Figure 3.9 Step-1 of positioning the insert on the holder: (a) Desired insert orientation is selected; (b) Final orientation of the insert frame (Frame-i) matches the pocket (seat) frame (Frame-s).  Figure 3.10 Insert placement on the holder: (a) by translation in step-2; (b) by rotation  in step-3. t1 is to label new frame after first translation. r1 and r2 are labels for the new frames after the first and second rotations. 38  If the unit axes vectors (Figure 3.9b) of the final orientation are identified from CAD drawing, the transformation matrix (from Frame-i to Frame-0) is directly obtained:   0 0 i0 ii1x3 4x41 =    R rT0 (3.17) where the rotation matrix is a combination of the 0 ii , 0ij  and 0ik  vectors (Figure 3.9b) represented in Frame-0:   0 0 0 0i i i i =  R i j k . (3.18)  Figure 3.11 KOVA XNEX080608 insert: (a) General view; (b) cutting edge highlighted; (c) rake face highlighted; (d) flank highlighted. (Courtesy of KOVA.)  Figure 3.12 Assigning the insert reference frame: (a) Front plane (b) insert reference plane; (c) insert coordinate frame; (d) front view. (Courtesy of KOVA.)      39  Case studies for indexable cutter model:  Case-1: Insert used with different tool holders: KOVA product insert (Figure 3.11a) with XNEX080608 code is used for various cutting operations. Assuming the top side of the insert is blocked by the tool holder, the bottom-side of the cutting edge of the insert is highlighted (Figure 3.11b). Depending on the type of the operation, any section of the highlighted section may be in-cut. The purpose of the geometry model is to locate the rake face (Figure 3.11c): First on the insert, and then on the cutter body. Thus, the insert reference frame is decided first. Flank is also shown in Figure 3.11d, and its geometry can be modelled similarly to rake face [38].  Insert reference coordinate frame: As shown in Figure 3.12c, the geometric center of the insert is selected as the center of the insert’s reference frame, with axis directions (Figure 3.12d) consistent with the insert shape: Front plane (Figure 3.12a) is selected parallel to the reference plane of the insert; reference plane (Figure 3.12b) of the insert coincides its mid-plane.  Locating the rake face on the insert: Seven cutting edge sections (C=7 in Figure 3.7a) are defined on this insert (Figure 3.13a-b). Top-left side edge is labelled 1, and the rest is labelled in CCW direction (Figure 3.13b). Totally 21 approximate rake face sections are defined on the seven cutting edge sections, and the rake face geometry is interpolated along the cutting edge (Figure 3.13c). Cutting edge sections 1, 3, 5 and 7 (ces-1, ces-3, ces-5 and ces-7) do not have curved rake face; thus, each of them is defined by single rake face section (see Figure 3.14). 40  Cutting edge sections 2, 4 and 6 (labeled ces-2, ces-4 and ces-6 in Figure 3.13b) have curved rake face geometry, thus local flat rake face sections are used for the approximation. Rake face approximation of cutting edge section-6 (Figure 3.15a) is shown in Figure 3.15b. Each cutting edge section with curved rake face geometry is approximated using six rake face section planes. Rake face approximation of cutting edge section-2 (labeled ces-2 in Figure 3.13b) is shown in Figure 3.19 with the six corresponding rake face coordinate frames.  Cutting edge section-1 (c=1 in Figure 3.6a) is defined at the left-side of the insert (as in Figure 3.16). Since the rake face includes the cutting edge at all points, only one rake face section is defined in this section. The coordinate frame of the rake face section-1 of the cutting edge section-1 is named Frame-1,1. The relative position vector i 1,1r  of the origin of “Frame-1,1” is defined in Frame-i of the insert, and is found from the CAD model as:  [ ]Ti 1,1 5.797 2.978 3.089 (mm)= − −r . (3.19)    (a) (b) (c) Figure 3.13 Modelling of the rake face of the insert: (a) General front view (b) labelled cutting edge sections (ces-1 to ces-7); (c) local rake face vectors along the cutting edge. 41   Figure 3.14 Highlighted two cutting edge sections (ces-5 and ces-7) with flat rake faces (a)  (b)  Figure 3.15 Rake face approximation of curved rake face: (a) ces-6 plane; (b) local flat rake faces. The relative axis unit vectors are defined in Frame-i:  [ ][ ][ ]Ti1,1Ti1,1Ti i i i1,1 1,1 1,1 1,10.257 0.122 0.959 ;0.862 0.419 0.284 ;; 0.436 0.900 0.00252 .= − − −= −= × = −ikj k i j  (3.20) Here, the i i1,1 1,1,i j  and i1,1k  correspond to the cutting edge tangent vector, the rake face normal vector and the rake face vector of the rake face section-1 of the cutting edge section-1, respectively. The transformation of the Frame-1,1 to Frame-i is carried out by using the located origin of the Frame-1,1 given from Equation (3.13): 42   i i i i 1,1i 1,1 1,1 1,11,1 0 0 0 1 =    i j k rT   (3.21) To define point-e on the cutting edge, additional translation Trans(0,0,x1,1,e) is needed for the length of the straight edge:   ( )i i i i 1,1 i i i i 1,1,e1,1,e1,1 1,1 1,1 1,1,e 1,1,e 1,1,eTrans 0,0,0 0 0 1 0 0 0 1x   ⋅ =         i j k r i j k r,  ( )i i i i 1,1,ei 1,1,e1,1,e 1,1,e 1,1,e1,1 Trans 0,0,0 0 0 1x = ⋅   i j k rT . (3.22) where the index for selected point along the cutting edge is e. If the number of points along cutting edge section-1 is np11, and the total length of the section-1 is L1, then the point along the edge is (Figure 3.16):  1,1,e 1p11 1Lx en= ⋅−  (3.23) The transformation from the “Frame-1,1,e” to the Frame-i becomes,  ( )i i 1,1,e1,1,e 1,1 Trans 0,0, x= ⋅T T   (3.24)  Figure 3.16 Local coordinates of the selected point e along the cutting edge section-1 43  The measured position and rake face vectors along all the 21 points are measured and stored in  i i i i c,rc,r c,r c,r: : :  i j k r  (3.25) for each of the r-th rake face section of the c-th cutting edge section ( {1,2...7}c∈ , {1,2...6}r∈ ).  Then, the transformation matrices are found throughout the entire edge. The transformation matrix for the r-th rake face section of the c-th cutting edge section is:   i i i i c,ri c,r c,r c,rc,r , {1,2...7}, {1,2...6}0 0 0 1c r = ∈ ∈   i j k rT . (3.26) Similar translational transformations are performed at each rake face section. Then, each point along the cutting edge is identified by applying Equation (3.24) at each section. Identified rake face vectors (Figure 3.18a) and rake face normal vectors (Figure 3.18b) of the cutting edge section (highlighted in Figure 3.11b). When the insert rake face geometry is fully identified (Figure 3.18c), it can be located on different kinds of tool holders depending on the operation type.  44   Figure 3.17 Cutting edge-section-2 (Figure 3.13b) with curved rake face  (a)  (b)  (c)  Figure 3.18 Rake face geometry along the modelled cutting edge (bold curve): (a) Rake face vectors; (b) rake face normal vectors; (c) rake face and rake face normal vectors.   Indexable face milling tool: A cylindrical right-handed face milling cutter (Figure 3.19a), a double right-hand milling cutter (Figure 3.19b) and a double left-hand milling cutter (Figure 45  3.19c) are used to demonstrate the placement of inserts on the tool body. In Figure 3.20a, Coordinate transformation from insert reference frame (Frame-i) to tool reference frame (Frame-0) is shown for an example insert on the tool body. As shown in Figure 3.20b-c, the transformation of coordinate frame involves both translation and rotation depending on the orientation of the pocket geometry.  Transformation for right-hand side cutter of the double milling tool, from Frame-i-Right (insert frame of right-side holder) to Frame-0, is shown in Figure 3.21a-b. Transformation for left-hand side cutter of the double milling tool, from Frame-i-Left (insert frame of left-side holder) to Frame-0, is shown in Figure 3.22a. Transformation bor both cutters of double-milling tool is shown in Figure 3.22b. (a)  (b)  (c)  Figure 3.19 Indexable milling cutters: (a) U140 cylindrical face mill; (b) double milling right-hand side cutter; (c) double milling left-hand side cutter. (Courtesy of KOVA.) 46   Figure 3.20 Transformation from Frame-i to Frame-0 of the cylindrical face mill cutter: (a) Top view; (b) front view; (c) isometric view. (Courtesy of KOVA.)  Figure 3.21 Transformation from Frame-i-Right to Frame-0 of the double milling tool: (a) Top view; (b) 3D view. (Courtesy of KOVA.)  Indexable drilling tool: A right-hand drilling tool (Figure 3.23) with the same type of inserts is used as a case study. As shown in Figure 3.24a-b, both translation distance and rotation angles are different for peripheral (insert#1) and central (insert#2) inserts of the drill cutter. 47   Figure 3.22 Transformation from Frame-i-Left to Frame-0 of the double milling tool: (a) Top view; (b) 3D view. (Courtesy of KOVA.)  Figure 3.23 Indexable U502 drilling cutter. (Courtesy of KOVA.)   Figure 3.24 Transformations from Frame-i to Frame-0 for the indexable drilling tool: (a) Top view at the insert cross section; (b) 3D view. (Courtesy of KOVA.)  48   Case-2: Indexable two-insert end mill: A sample indexable R390-020A20L-11L end mill (with R390-11 T3 04M inserts) from Sandvik-Coromant1 is used as a case study, see Figure 3.25. The cutter frame (Frame-0) is the fixed coordinate frame of the tool holder, and named as x0y0z0 frame. The translational feed motion of the cutter is aligned with i0 unit axis vector (positive x0-direction), and the tool rotation axis is the k0 unit axis vector (positive z0-direction). The velocity vector is always on x0y0 plane. The plane including the velocity and feed vectors, defined at the selected point on the cutting edge, is the assumed working plane Pf. This plane is also parallel to the x0y0 plane at the zero depth of cut position of the cutter, as is shown in Figure 3.25.  The first step is to determine the desired location and orientation of the insert on the holder (as shown in Figure 3.9a and Figure 3.25a-b). Insert pocket (seat) frame (Frame-s) on the holder is assumed to match the desired orientation of the insert frame (Frame-i), as is shown in Figure 3.9b and Figure 3.25b-c. This assumption is to avoid an extra transformation from insert pocket frame to desired location of the insert frame. The unit axis vectors of the insert frame, Frame-i, are 0ii, 0ji, and 0ki (or 0is, 0js, and 0ks of pocket/seat frame, Frame-s) with the origin positioned on the tool holder, are shown in Figure 3.26. Initial orientation of the insert frame (Frame-i) is the same as the tool reference frame (Frame-0), as is shown in Figure 3.9a.                                                   1  For example: http://www.sandvik.coromant.com/en-gb/products/pages/productdetails.aspx?c=R390-020A20L-11L#/?active=detail 49   Figure 3.25 Sample two-insert end mill from Sandvik-Coromant (a) Position of the insert on the tool holder; (b) Frame-s on the tool; (c) Frame-i from the back; (d) Frame-i from the front. The second step is to place the insert origin at the insert pocket. In Figure 3.10a and Figure 3.26, 0 ir  is the position vector of the origin of the Frame-i relative to the origin of the Frame-0 expressed in Frame-0:   0 i 0 i 0 i 0 ix 0 y 0 z 0r r r= ⋅ + ⋅ + ⋅r i j k , (3.27) where 0 0 0, ,i j k  are the axis unit vectors represented in Frame-0 in 0 0 0, ,x y z  directions. 0 i 0 i 0 ix y z, andr r r  are the magnitude components of the position vector 0 ir . In homogeneous vector form, the position vector 0 ir  of Equation (3.27) can be written as:  T0 i 0 i 0 i 0 ix y zr r r =  r. (3.28) Third step is the rotation of the insert frame (Figure 3.10b). The Euler angle rotations for the insert frame matching the pocket frame orientation are carried out in the following order [1]:  • Negative- fγ  rotation around the zt1-axis of the new frame after the first translation; as shown in Figure 3.10a, since there is no rotation at the first translation step, the new frame (Frame-t1) has the same unit axis vectors as Frame-0; 50   Figure 3.26 Relative locations of the origins of cutter, insert and first rake face section of the first cutting edge section. • negative- pγ  rotation around the xr1-axis of the new frame after the first rotation;  • negative- rκ  rotation around the yr2-axis of the new frame after the second rotation.  The order of rotations is generally important, but for small angles (less than 30 [deg]), this effect is negligible [53]. The transformation from Frame-i to the Frame-0 is:  0 0 0 0 i i i0 i 0 i 0 ii i i i i ix y z f p rTrans Rotz Rotx Roty( , , ) ( ) ( ) ( )0 0 0 1 0 0 0 1r r r γ γ κ   = − − −         i j k r i j k r (3.29) The position vector of the origin of the Frame-i in Frame-i is zero:  [ ]Ti i 0 0 0=r . (3.30) The axis vectors are represented in Frame-i as:  [ ] [ ] [ ]T T Ti i i1 0 0 ; 0 1 0 ; 0 0 1 .= = =i j k   (3.31) 51  Although there is research to find the analytical expression with large rotation angles [112], approximation is possible with multiple ns step rotations as:  ( ) s0 0 0 0 i n0 i 0 i 0 ii i ix y z f s p s r sTrans( , , ) Rotz( )Rotx( )Roty( ) .0 0 0 1r r r n n nγ γ κ = − − −   i j k r (3.32) Generally the transformation is performed by 0iT  matrix:  0 0 0 0 i i i0i i i i i ii0 0 0 1 0 0 0 1   =         i j k r i j k rT , (3.33) where   0 0 i 0 i 0 ii x y z f p rTrans( , , )Rotz( )Rotx( )Roty( )r r r γ γ κ= − − −T . (3.34) The negative rotations are due to the inverse rotations which start from the original tool frame orientation (see Figure 3.10b). The form of the homogeneous transformation matrix in Equation (3.34) is:  0 0 i0 ii1x3 1 =    R rT0 (3.35) where 0iR is the rotation matrix from the insert frame to the tool frame, and is given as:  r f r p f p f r f r p fr f rf f r r0i f f p pr rp pcos cos sin sin sin cos sin sin cos cos sin sin0 cos sin sin sicos sin 0 1 0 0 cos 0 sinsin cos 0 0 cos sin 0 1 0 ,0 0 1 sin 0 cos0 sin cosκ γ κ γ γ γ γ κ γ κ γ γκ γ κγ γ κ κγ γ γ γκ κγ γ+ − +− + −       = −        −     =RR p f p f r f r p fr p p r pin cos cos cos sin sin cos sin cossin cos sin cos cos.γ γ γ γ κ γ κ γ γκ γ γ κ γ+−       (3.36) The reverse rotation matrix (from Frame-0 to Frame-i) is given by: 52   ( )r f r p f r f r p f r pp f p f pr f r p f r f r p f r pT0 0i icos cos sin sin sin cos sin sin sin cos sin cosi cos sin cos cos sin0sin cos cos sin sin( sin sin cos sin cos cos cos;.κ γ κ γ γ κ γ κ γ γ κ γγ γ γ γ γκ γ κ γ γ κ γ κ γ γ κ γ+ − +−− + +=  =    R RR (3.37)  The inverse of the rotation matrix is normally equal to its transpose. However, this may not hold true for an arbitrary rotation matrix. In that case, an approximation is needed based on the identified Euler angles. The axis vectors of the insert coordinate frame (Frame-i) are used to for the approximation. From the 3D drawing of the cutter (Figure 3.26), the exact vector axis representation in Frame-0 are:   0 0i i i i i0 0 0 0i i i i1 0 0 0.9874 00 ; 1 ; 0 ; 0.1580 ; 0.1742 ;0 0 1 0.00344 0.98470.1550; 0.9723 .0.1720                  = = = = =                           −  = × =   − i j k i kj k i j  (3.38) The position vector 0 ir  is already given by manufacturer or obtained by measurements.The exact transformation matrix is:  ( )( )( )0 0 0 0 i i i0i i i i i iiexact0 ix0 0 i0 iiy exact0 i1x3z0ie.0 0 0 1 0 0 0 10.9874 0.1550 0 1 0 0 00 1 0 00.1580 0.9726 0.1742;0 0 1 010.003440 0.1720 0.98470 0 0 10 0 0 1rrr   = ⋅          −          = ⋅     −         i j k r i j k rTR r0Rxact0.9878 0.1549 00.1580 0.9726 0.1741 .0.003442 0.1719 0.9847−  =   −   (3.39) 53  The 3-unknown Euler angles are fitted by matching 5-equations of the parametric matrix (Equation (3.36)) and the numerical matrix (Equation (3.39)):  ( )( )p pr prr pp ffp fsin( ) 0.1720; 9.9035[deg];sin( )cos( ) 0.00344atan2 0.00344,0.9847 0.2002 [deg];cos( )cos( ) 0.9847cos( )sin( ) 0.1550atan2 0.1550,0.9723 9.0544 [deg].cos( )cos( ) 0.9723γ γκ γκκ γγ γγγ γ− = − ==  = == = −  = − = −=   (3.40)  Although a non-linear iterative fitting method with 9-equations of Equation (3.36) and 3-unknown Euler angles may also be used, it is observed that using the method in Equation (3.40) gives satisfyingly accurate result. The approximate rotation matrix compares well to the exact one:  ( )( )0i f p rapprox0iapproxRotz( ) Rotx( ) Roty( )0.9874 0.1550 0.03050.1580 0.9728 0.1693 .0.00344 0.1720 0.9851γ γ κ= − ⋅ − ⋅ −− −  =   − RR  (3.41) The homogeneous transformation matrix is thus reassigned using the approximate rotation matrix:  ( )0 0 ii0 approxi1x3 1  =  R rT0  (3.42)  Locating the edge on the insert: Although exact dimensions of the insert’s edge and tool would be preferred for accurate analysis, a 2D model could still give a good estimation for process prediction [1]. Since the transformation from the Frame-0 to Frame-i is done, the transformation from the Frame-i to the local cutting edge needs to be carried out. Since the insert 54  of Figure 3.25 does not have a flat-face, either an approximation is needed or every selected point along the cutting edge is treated individually. For the design of this insert, the assumption of three (C=3) cutting edge sections with flat rake faces (R=1) is valid (Figure 3.27a), i.e., each cutting edge section has single rake face section. Then, each rake face section is formulated within itself. Before transforming to the local cutting edge, transformation to the reference of the local rake face is needed. An example close-up view of the first local rake face section is shown in Figure 3.27b. Cutting edge is approximated by line-arc-line function series in this section. The reference of the first section of the rake face (r=1) on the first cutting edge section (c=1) of the insert is named as Frame-1,1 (Figure 3.27b). The transformation from the Frame-1,1 to the Frame-i is:  i i i i 1,1 1,1 1,1i1,1 1,1 1,1 1,1 1,1 1,11,1 .0 0 0 1 0 0 0 1   =         i j k r i j k rT   (3.43) Similar to Equation (3.35), the transformation i1,1T  has the following form:  i i 1,1i 1,11,11x3 1 =    R rT0. (3.44) Here, the relative position vector i 1,1r  is defined in Frame-i. It is the vector from the origin of the Frame-i to the first rake face frame (Frame-1,1). i1,1R  is the rotation matrix of the first rake face frame after the translation. Similar to locating the insert frame on the cutter (Equation (3.42)), an Euler angle identification is needed here for each rake face section along the cutting edge.  ( )i i 1,11,1i approx1,11x3 1  =  R rT0. (3.45)  55   Figure 3.27 Cutting edge and rake face sections of the insert: (a) Flat rake face sections along the cutting edge; (b) example close-up view of the first local rake face section. Obtaining the transformation matrix i1,1T  usually requires information about the three dimensional geometry of the tool. Since the local rake face plane is flat, there is no more transformation in the local y-axis (j1,1) of the first rake face frame of the first cutting edge section. The rotation from Frame-1,1 to Frame-D (and for other rake faces) is inherently an Euler angle rotation. The transformation to the cutting edge frame, Frame-D, of the local cutting edge is a 2D transformation. The cutting edge section on a constant rake face plane is assumed to consist of a combination of the 2D lines and arcs. For example, rake face section-1 is defined as the combined line-arc-line segments, as is shown in Figure 3.27b. Thus, for point-e along the cutting edge in rake face section-1, the translation and rotation are combined using the local coordinates ( )11 11e e,0,x z  and slope 11eθ :   ( ) ( )1,1 11 11 11e e eD Trans ,0, Rotyx z θ= ⋅ −T . (3.46)  56  (a) (b) (c) (d)     Figure 3.28 Insert edge geometry of the two-inserted end mill: (a) Modelled 3D cutting edge; (b) modelled rake face geometry; (c) radius along the cutting edge; (d) relative location angle along the cutting edge. Then, the transformation from the Frame-D to the Frame-0 is the series multiplication of all transformation matrices. The transformation matrix from Frame-D to Frame-0 is:  0 0 i 1,1D i 1,1 D .=T T T T   (3.47) Again similar to the Equation (3.35), the transformation 0DT  has the following form:  0 0 D0 DD1x3.1 =    R rT0 (3.48) Here, the relative position vector 0 Dr  (Figure 3.27a) of the origin of the cutting edge frame at the selected point-e is defined in Frame-0, and is defined as the vector from the origin of the Frame-0 to the local cutting edge frame (Frame-D). 0DR  is the rotation matrix to represent the design frame axes in the cutter frame axes. Modelled 3D geometry of the cutting edge is given in Figure 3.28a. Rake face and rake face normal vectors are shown in Figure 3.28b at each sampled point along the cutting edge of the insert. Radius (Figure 3.28c) and angular location (Figure 3.28d) are defined from the 3D geometry. Wiper edge geometry (Figure 3.28b) is also included in the model. wiper edge 57  (a)  (b)  Figure 3.29 Chip length along the first 0.030-mm section of the cutting edge: (a) 0.001-mm segment thickness; (b) 0.010-mm segment thickness. Cutting edge length is derived from the 3D geometry (Figure 3.28a). For small depths of cut, the effect of the cutting edge length is given in Figure 3.29. For a 0.030-mm axial depth of cut, if the depth increment (segment thickness) is taken at dz=0.001-mm intervals, the total cutting edge length is 1.064-mm. Figure 3.29a shows the chip length distribution along the first 0.030-mm section of the cutting edge for dz=0.001-mm thickness. But, if the depth increment is taken to be dz=0.010-mm, the total cutting edge length becomes 0.2480-mm. Figure 3.29b shows the chip length distribution for dz=0.010-mm thickness. Prediction of the cutting edge length is reduced to less than half the value of the more accurate value (1.064-mm) when 10 times finer segment thickness is used. This significant reduction shows the signinificant effect of the wiper edge section (see Figure 3.28b) underneath the insert (the initial line section in Figure 3.27b). Although the wiper section makes a small contribution to the chip thickness, it has a large contribution to the chip length. The wiper edge has a major effect for the first 0.010-mm depth (see Figure 3.29a) which dominates the chip length distribution.   Case-3: Multifunctional single level indexable cutter: Multifunctional cutters combine at least two conventional operations in one operation. Figure 3.30a shows a single-level multi-step 58  drilling cutter from Kennametal with HTSR043R025M tool holder, DFR030204GD-KC7020 inserts, and with a twist drill (pilot drill) at center; Figure 3.30b shows a two-level combined drilling, step-boring, countersinking and chamfering cutter from Sandvik with TM880 tool holder and 5513-020 series inserts.  Since there is no available CAD geometry of the inserts and cutters, the measured data is used for identification of the rake face geometry. Optical microscope is used outside the machine, and the tool holder is measured with the inserts installed on it. The cutting edge along the indexable section (without the pilot drill) is measured. The sample 3D geometry of the inserts located on the tool holder body is given in Figure 3.31a-b. Since the rake angle of the inserts is zero, the same figures also represent the rake face geometry.    Figure 3.30 Multifunctional cutters: (a) Multi-step drilling tool from Kennametal; (b) Two-level cutter from Sandvik. (Adapted from Wan et al. [5].)          ZX(a)Bottom viewBott m view132 4Bottom levelZX(b)Bottom viewBottom view1324Bottom levelTop level59   Figure 3.31 Measured cutting edge geometry of Kennametal drill: (a) Top and front views ofeach insert; (b) transformation of insert frames to their locations on tool frame. The insert geometry is formed from the optical microscope measurement. As shown in Figure 3.31a, the origin of the insert frame coordinates is set at the insert tip point. Radial location of the origin is measured by a microscope, and the axial location at the tip of each insert is measured by a dial gage instrument (as in measurement of axial runout of inserts). Transformation of the insert #1 is shown in Figure 3.31b, and the other three inserts are located on the cutter in a similar way. Transformation steps are as follows:  • Translation Trans(0,0, 4jε ) in tool z0-axis for the axial runout relative to the fourth insert (44 0ε = ); • Rotation Rotz( jj ) by the location angle jj  around tool z0-axis. jj  is measured from the +x0-axis in CCW direction; • Translation Trans(rj,0,0) by the radial offset rj from the tool center in the x-axis of the new frame;  60  • Rotation Roty( jκ− ) by the jκ  rotation angle which is measured around the y-axis of the new frame in CCW direction. This jκ  angle corresponds to α  angle of Figure 2a of [5].  The resulting Frame-i (insert frame) is found by multiplying the relative transformations in the order:  0 0 0 0 i i i0i i i i i ii0 0 0 1 0 0 0 1   =         i j k r i j k rT   (3.49) The transformation matrix is found by the kinematic chain:  ( ) ( ) ( ) ( )0 4i j j j jTrans 0,0, Rotz Trans ,0,0 Rotyrε j κ= ⋅ ⋅ ⋅ −T . (3.50) The transformation ( )iDjkT  at each element Sjk of the cutting edge is known from the measurements. The representation of the cutting edge tangent, rake face normal, rake face and position vectors in Frame-0 is: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 D DD D D D D Djk jk jk0 ijk jk jk jk jki Djk0 0 0 1 0 0 0 1      =         i j k r i j k rT T .  (3.51)    Figure 3.32 Modelled Kennametal cutter: (a) Axial location; (b) angular location. 61  (a)  (b)  Figure 3.33 Modelled cutting edge geometry of the Sandvik TM880 cutter: (a) Drawing; (b) schematics. Cutting edge tangent ( )D jki , rake face normal ( )D jkj , rake face ( )D jkk  and position ( )D jkr  vectors of the element Sjk are given as follows:  ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]T T T TDD D Djk jk jk jk1 0 0 ; 0 1 0 ; 0 0 1 ; 0 0 0= = = =i j k r   (3.52) The cutting edge is represented in the Frame-0 by the cutting edge tangent ( )0 Djki , rake face normal ( )0 Djkj , rake face ( )0 Djkk  and position ( )0 Djkr  vectors of the element Sjk.  Axial location (Figure 3.32a) and angular location (Figure 3.32b) are identified from the 3D geometry (Figure 3.31b).   Case-4: Multi-functional two-level cutter: Sandvik TM880 multifunctional tool with drilling, boring, countersinking and chamfering functions is shown in Figure 3.30a. It has two levels (see Figure 3.33a) to give the depth to the machined countersunk hole. The mathematically modelled geometry (Figure 3.33b) is obtained by measuring the cutting edge under an optical 62  microscope. The axial runout of the insert #3 is measured by a dial gage when the cutter installed on the spindle.  3.2.2 Solid Tool Unlike the indexable cutters which require the definition of the inserts and their placements on the tool body, the cutting edges are an integral part of the solid tool geometry. The cutting edge is defined relative to the tool reference coordinate frame. The transformation matrices are given in Appendix A. The method of locating the cutting edge is as follows: • The location of the fixed tool reference coordinate frame, e.g., center of the tool’s bottom geometry (Figure 3.34a), is selected for the cutter; • The cutting edge element Sjk ( { }1,2,..k K∈ , { }1,2,..j N∈ ) is placed at the axial segment (Zk=kdz) as shown in Figure 3.34b; radial location (x0,y0) is defined by using rotational (Figure 3.34c) and translational (Figure 3.34d) transformations; • The cutting edge element Sjk is rotated using local helix angle (Figure 3.34e), rake angle (Figure 3.34f) and cutting edge angle (Figure 3.34g) to obtain the local rake face geometry (Figure 3.34h).  It may be difficult to represent some of the tools’ geometry analytically (as in Ref. [39]), but they can be digitized and represented numerically [110].   63   Figure 3.34 Locating the Frame-D on the solid tool starting from Frame-0: Steps (a)-(d) serve to locate the x0y0z0 coordinates; steps (e)-(h) are for rotational alignment of the local cutting edge frame. 64  Case-5: Serrated cylindrical, helical end mill: Serration waves, which are constructed from spline points, are ground on the helical flutes of roughing end mills [21]. The sample serrated end mill is Sandvik R216.33-20040-AC38U H10F. The serration profile of its cutting edge is measured by a surface profilometer. The stylus is set to zero at the bottom of the flute and the radial variation is traced upwards along the flute at ds intervals. Surface profilometer measurement data for one serration period is given in Figure 3.35. Radius ( sjkR ) is measured at each point along the flute (Figure 3.35a).  The corresponding sampling distance dz along the cutter axis is given by:  serrddcoszsβ=   (3.53) where serrβ  is the helix angle of the serrated end mill. The radial variation ( sjkR ) of flutes are mapped in the tool coordinate frame; radius ( jkR ) at each segment along tool axis is given in Figure 3.35b.    Figure 3.35 Surface profilometer measurement data of the serrated end mill: (a) Radius along the flute; (b) axial and radial distance from the tool tip along the tool axis. 65    Figure 3.36 Cutting edge angle of the serrated end mill: (a) Along the flute; (b) mapped along the tool axis. Cutting edge angle ( serr,sr,jkκ  of Figure 3.36a) along the flute is interpolated from the measured profile. It is mapped along tool axis using Equation (3.53) to obtain serrr,jkκ  of Figure 3.36b. Each cutting edge segment, with differential height (dz) is placed on the tool body using Denavit-Hartenberg (also called “moving axes approach” [113]) transformations, as follows:  • Translation Trans(0,0,kdz) in the tool axis (z0 of the Frame-0) at the axial height kdz.  • Rotation Rotz( jkπ 2 ψ− ) around z0-axis. If the helix and pitch ( pφ ) angles are constant, and R0 is the nominal radius of the tool, the angular location angle jkψ of flute j and edge segment k is given as:   ( )serrjk p0tan1 dj k zRβψ φ= ⋅ − − ⋅   (3.54) • Translation Trans(Rjk,0,0) along the radius vector: Tranforms the coordinate frame to the position along the cutting edge (see Figure 3.34e). • Rotation Rotx( serrβ− ) around x-axis of the new frame: Aligns the cutting edge with the helical flute of the tool.  66  • Rotation Rotz( serrγ− ) around z-axis of the new frame: Aligns the coordinate frame with the rake face of the cutting edge. • Rotation Roty( serrr,jkκ− ) around y-axis of the new frame: This final transformation aligns the coordinate frame with the cutting edge design frame (Frame-D) of the tool.  Transformation from Frame-D to Frame-0 (tool reference frame) is thus (see Figure 3.34h):  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )0 0 0 0 D DD D D D D Djk jk jkDjk jk jk jk jk0jk0 0 0 1 0 0 0 1      =         i j k r i j k rT . (3.55) The complete transformation matrix to place the edge element Sjk on the tool body is found by the following kinematic chain:  ( ) ( ) ( ) ( )( ) ( ) ( )D0 jk jkjkserr serr serrjkπTrans 0,0, d Rotz Trans ,0,02Rotx Roty Rotzk z Rψβ γ κ= ⋅ − ⋅ ⋅− ⋅ − ⋅ −T  (3.56) Cutting edge tangent ( )D jki , rake face normal ( )D jkj , rake face ( )D jkk  and position ( )D jkr  vectors of the element Sjk are:  ( ) [ ] ( ) [ ] ( ) [ ] ( ) [ ]T T T TDD D Djk jk jk jk1 0 0 ; 0 1 0 ; 0 0 1 ; 0 0 0= = = =i j k r   (3.57) Thus, the cutting edge of the element Sjk is represented in the Frame-0 (tool reference coordinates) by the cutting edge tangent ( )0 Djki , rake face normal ( )0 Djkj , rake face ( )0 Djkk  and position ( )0 Djkr  vectors.  67  3.3 Computation of the Effective Tool Geometry Based on the geometry definition supplied by ISO standards [51], the local tool angles (or effective geometry) are found by using the feed and velocity vectors of the cutting edge segment [1]. They are defined in the order of relevance as follows:  3.3.1 Velocity of Cutting Edge of the Tool Velocity of the element Sjk is defined by cutting velocity and feed vectors. Their superposition gives the resultant cutting vector.  Cutting velocity ( 0cv ): It is defined as the tool velocity relative to the workpiece. The position vector 0 Dr  of the origin of the local cutting edge coordinates relative to the cutter frame is similar to Equation (3.28):  T0 D 0 D 0 D 0 Dx y zr r r =  r. (3.58) 0 Dr  is used for describing the cutting velocity direction. The rotation axis is the tool axis for milling, drilling and boring operations. The workpiece axis is the rotation axis for turning operation. For right-hand milling (Figure 3.3a), drilling (Figure 3.5a) or boring operation, the relative rotation is CW when looked from the tool side. For left-hand turning (Figure 3.1a) operation, the relative rotation is also CW when looked from the tool side. Thus, the rotation vector 0ω  defined in the tool frame (Frame-0) as:  000(right-hand cutting)(left-hand cutting)ωω− ⋅=  ⋅kkω  . (3.59) where ω (rad/s) (or Ω [rev/min]) is the relative angular speed of the tool with respect to the workpiece. The magnitude of the cutting velocity vector is given in Equation (3.8). The local radius vector 0jkR  of the element Sjk is defined in Frame-0, and evaluated as:  68   ( )0 0 Djk 0 0 .= × ×R k r k  (3.60)  The cutting velocity of the tool is realized from the radius vector and the rotation vector as:  0 0 0c jk= ×v Rω . (3.61) where the radius vector 0jkR  is  in the x0y0 plane:  T0 0 0jk x y 0R R =  R  (3.62)  Feed vector: The direction of the feed vector fv  is defined in Frame-0 (tool coordinate frame) with magnitude fv  (m/min) which is product of rf  (mm/rev) and Ω  (rev/min). The feed is assumed to be in positive-x0 direction in milling, in negative-z0 direction in turning, boring and drilling operations, as follows:   ( )( )-3r 0f-3r 010 if milling10 if turning,boring or drillingff Ω ⋅= − Ω ⋅ivk  (3.63)    Figure 3.37 Resultant cutting directions for two consecutive edges at segment k of milling cutter moving in x0 direction.  69  Resultant cutting vector: The definition of the resultant relative velocity of the tool includes the feed direction. As shown in Figure 3.37, if the feed motion is comparable to the cutting velocity, then the resultant cutting direction is not same for the two consecutive cutters. Generally, with the exception of thread cutting operation, the magnitude of the feed motion is small compared to the cutting speed. Thus, the resultant cutting direction is assumed to be the same as the cutting velocity. This implies that the planes defining the tool-in-hand and tool-in-use systems are the same, except for the working plane (Pfe) [51]. Since the working plane (Pfe) includes the directions of cutting velocity and feed vectors, its orientation depends on the feed direction. For thread cutting tools, the feed amplitude has the same order as the cutting speed. Hence, the assumption of equal tool-in-hand and tool-in-use systems is no longer valid for thread cutting application.  3.3.2 Tool Cutting Planes  Tool cutting planes are demonstrated by using tapered ball end mill (Figure 3.38a) which is capable of 5-axis machining. As shown by green circle (Figure 3.38a-b), the selected point along cutting edge is at its spherical (bottom) section.  Assumed working plane (Pf) and working plane (Pfe): Although Pf and Pfe planes both include the cutting velocity and feed vectors, they are in general different planes [51]. Assumed working plane is shown in Figure 3.38b. It is parallel to x0y0 plane with an offset in tool axis (z0). Working plane is parallel to tool axis for drilling operation, and it is perpendicular to the tool axis for 2-axis milling operation (Figure 3.38b). For 3-axis milling, the working plane orientation becomes position-dependent due to the z0 component of the feed velocity. 70   Figure 3.38 Tapered ball end mill to show Pf, Pfe and Pr planes: (a) General view; (b) working planes for milling and drilling operations; (c) Tool reference plane (Pr) with Frame-R. While the Pf assumes that the feed direction is only in the x0y0 plane of Frame-0, Pfe takes the actual feed direction into account. For 2-axis milling operation, the translation feed vector ( f,mv  in Figure 3.38a-b) is in x0y0 plane. Thus, as shown in Figure 3.38b, Pf and Pfe,m are the same planes for 2-axis milling. For turning, boring and drilling, feed vector ( f,dv  in Figure 3.38a-b) is towards the negative-z0 direction. Thus, as seen in Figure 3.38b, assumed working plane (Pf) does not represent the working plane (Pfe,d) of drilling. Therefore, the more general Pfe plane is used for calculating the effective tool geometry.  First defining vector of the working plane (Pfe) is the tool axis z0 direction as it is perpendicular to the cutting velocity. For 2-axis (x0y0 plane) milling, the radius vector 0jkR  of the element Sjk is the first defining vector. For turning, boring and drilling, the radius vector is relative to the workpiece rotation axis or the center of the drilled/bored hole. The second defining vector is the feed vector. Pfe plane is perpendicular to the tool reference plane. 71  For 2-axis milling, the working plane includes the cutting velocity, radius vector and xR-axis of the tool reference plane frame (Frame-R in Figure 3.38c). For turning, boring and drilling operations, the working plane includes the cutting velocity, radius vector and negative zR-axis of the tool reference plane (Pr in Figure 3.38c) frame. 5-axis milling can also be analyzed similarly. Its working plane includes the cutting velocity, radius vector and the feed axis direction on the Pr frame (Frame-R).   Tool reference plane (Pr): Pr plane (see Figure 3.38c and Figure 3.39e) is perpendicular to the cutting velocity vector which is defined by the radius vector 0jkR and the unit tool axis 0k vector in the z0 direction. The corresponding axis unit vectors ( 0 Ri , 0Rj  and 0Rk ) of the Pr coordinate frame (Frame-R) represented in Frame-0 are shown in Figure 3.38 by xR,yR and zR directions.   0 0jk0 0 0cR R R 00 0jk c; ; .= = − =R vi j k kR v (3.64) Similar to Equation (3.39) the transformation matrix 0RT  from Frame-R to Frame-0 is obtained using the axis unit vectors of Equation (3.64):  0 0 0 0 R0 R R RR .0 0 0 1 =    i j k rT   (3.65) The position vector 0 Rr  of the origin of the Frame-R is the same as the position vector 0 Dr  of the origin of the Frame-D, because their origins coincide (compare Figure 3.38a and Figure 3.39a):   T0 R 0 D 0 R 0 D 0 D 0 Dx y z; .r r r = =  r r r (3.66) 72   Figure 3.39 Tapered ball end mill to show Aγ, Pn, Ps and Pr planes: (a) General view same as in Figure 3.38a; (b) Cutting edge frame (Frame-D); (c) Cutting edge normal plane (Pn) and Frame-3; (d) related rake face normal view; (e) cutting edge plane (Ps). Then, the rotation matrix can be extracted from 0RT  by the analysis as in Equation (3.35):  0 0 RR0R T 1  =  R rT0  (3.67)  Rake face (Aγ): The local cutting edge frame, Frame-D, coincides with the rake face (Aγ) plane, as is shown in Figure 3.39b, where clarance face (flank, Aα) is also shown. Di  is the unit cutting edge tangent vector; Dj  is the unit rake face normal vector (directed into the rake face); 73  Dk  is the unit rake face vector, which is tangent to the rake face but perpendicular to the cutting edge:  [ ] [ ] [ ]T T TD D D1 0 0 ; 0 1 0 ; 0 0 1 .= = =i j k  (3.68) The corresponding coordinate system is named as Frame-3 (Figure 4.5), with the following mapping of the axis directions:   3 D 3 D 3 D; ; .x y y x z z≡ ≡ − ≡   (3.69) The cutting edge tangent Di  vector is the same as jkt  of Equation (3.9). The rake face normal Dj  direction is also linked to Equation (3.9). The rake face direction Dk  is the same as Equation (3.10).   Cutting edge normal plane (Pn): As shown in Figure 3.39d, Pn plane is defined as perpendicular to the cutting edge tangent Di  vector (see Figure 3.39c), thus it includes the rake face normal Dj  and rake face Dk  vectors. Pn is equivalent to the x2z2 plane of the Frame-2 (Figure 4.5). y3 and x3 directions of Frame-3 (Figure 4.5) are perpendicular to the Pn plane. The yDzD components of the cutting edge design coordinates can be used for projecting vectors on the Pn plane.  Cutting edge plane (Ps): Ps includes the cutting edge tangent vector and it is perpendicular to tool reference plane (Pr). It includes the cutting velocity vector 0cv . The representation of the cutting edge tangent vector in the cutter reference Frame-0 0 Di  is found using the Equation (3.55): 74   0 0D D D= ⋅i T i . (3.70) The cutting edge tangent unit vector Di  represented is from Equation (3.68).   3.3.3 Effective Tool Angles Local true cutting edge angle ( *r,jkκ ): Tool cutting edge angle (*r,jkκ ) is defined as the angle between the tool cutting edge plane Ps and the assumed working plane Pfe measured in the tool reference plane Pr [51]. Calculation of the cutting edge angle depends on the Pfe plane orientation, and thus dependent on the feed vector direction. Since for all operations, the cutting velocity is perpendicular to the Pr plane, the Pfe plane intersects the Pr plane at the feed vector direction. Thus, the corresponding method for milling is to project the cutting edge tangent xD-axis vector on the Frame-R. Subsequently, find the angle that the projected vector makes with the feed direction.  The representation of the cutting edge tangent vector on Frame-0, 0 Di  is found in Equation (3.70). This vector is in xD direction in Figure 3.39e. For its Frame-R representation ( R Di ), the transformation from Frame-D to Frame-R is done:  R R 0D 0 D= ⋅i T i  (3.71) where, the inverse transformation from Frame-0 to Frame-R is found from Equation (3.65),  ( ) 1R 00 R .−=T T  (3.72) Or, from the homogeneous matrix algebra [114],  ( ) ( )T T0 0 0 RR RR0T 1 − =    R R rT0  (3.73) 75  Pr plane is formed by the xRzR components of the Frame-R. So the projected vector ( ) rPR Di  is obtained by setting the y-component of the transformed vector R Di  to zero. If the transformed vector is  ( ) ( ) ( )R R R RD D D Dx y z0 , =   i i i i   (3.74) then, the projected vector is  ( ) ( ) ( )rPR R RD D Dx z0 0 . =   i i i   (3.75)  Two-axis Milling: It is shown in Figure 3.38b that if the working plane (Pfe) is parallel to x0y0 plane for 2-axis milling, the feed vector direction can be assumed as the positive xR direction. Figure 3.40a is a 2D representation of the Pr plane in Figure 3.39e. Cutting edge tangent vector in xD direction (Figure 3.39e) is projected onto Pr plane in Equation (3.75). As shown in Figure 3.40a, the cutting edge angle is defined as the angle between the projected cutting edge tangent vector and the xR-axis:  ( ) rP* Rr R Darccosκ  = ⋅  i i , (3.76) where, [ ]TR 1 0 0=i . Thus,  ( )( ) ( )RD* xr2 2R RD Dx zarccos.κ   = +  ii i. (3.77) The range of arccos function is [0,π], so Equation (3.76) covers all the possible *rκ  values. But, if R Di  has negative zR-axis component, the cutting edge physically makes contact with the 76  back side, thus it is not in-cut. It is thus more convenient to use atan22 function for a general cutter case:  ( ) ( )* R Rr D Dz xatan2 ,κ  =   i i . (3.78) Calculation of the cutting edge angle in Equation (3.78) only applies for right-hand milling (Figure 3.40a), where the tool axis rotation is in CW direction. In the case of left-hand milling (Figure 3.40b), the tool rotates in the CCW direction. As a result, the calculation of the cutting edge angle is based on the negative cutting edge tangent vector.   ( ) ( )* R Rr D Dz xatan2 ,κ  = − −  i i . (3.79)  Figure 3.40 Cutting edge angle calculation: (a) Right-hand milling; (b) Left-hand milling.                                                   2 More information at: http://www.mathworks.com/help/matlab/ref/atan2.html 77   Figure 3.41 Cutting edge angle calculation of the drilling operation. Turning, Boring and Drilling: For turning, boring and drilling operations, the feed direction is the negative zR direction. The definition of the cutting edge angle is the angle between the negative cutting edge tangent ( )R D− i  vector and the negative zR direction. Hence, as shown in Figure 3.41, the cutting edge angle formula is a rotated version of Equation (3.78):   ( ) ( )* R Rr D Dx z3πatan2 ,2κ  = − − −  i i  (3.80)  Local tool cutting edge inclination angle ( s, jkλ ): Tool cutting edge inclination angle ( sλ ) is measured on the cutting edge plane Ps. It is the angle between the cutting edge and the tool reference plane Pr [51]. The cutting edge tangent and the cutting velocity vectors are both included in the Ps plane, and the cutting velocity is perpendicular to the Pr plane. As shown in Figure 3.42a, the cutting edge inclination angle can be defined as the complementary angle (adding up to 90-deg in total) of the angle between negative cutting edge tangent ( 0 D− i ) and the cutting velocity directions.  78   Figure 3.42 Calculations of (a) Cutting edge inclination angle and (b) normal rake angle. The dot product between the negative cutting edge tangent ( 0 D− i ) and the cutting velocity directions gives the complementary angle of the cutting edge inclination angle:  ( )0 0D cs 0cπarccos .2λ − ⋅ = −    i vv (3.81) Alternatively, same angle can be found by taking the dot product of the cutting edge tangent vector and the yR-axis vector of the Frame-R:  ( )0 0s D Rπ arccos2λ = − ⋅i j . (3.82) Since the range of the arccos function is [0,π], the range for Equation (3.81) is [-π/2,π/2], which encompasses all the possible values.   Local tool normal rake angle ( n, jkγ ): Tool normal rake angle ( nγ ) is defined as the angle between the rake face γA  and the tool reference plane Pr as measured in the cutting edge normal 79  plane Pn [51]. Alternatively, the same angle is obtained if the cutting velocity is projected onto Pn plane (Frame-D); the angle that the projected velocity makes with the rake face normal is found as shown in Figure 3.42b. The cutting velocity can be first transformed to the Frame-D as inverse operation of Equation (3.70):  D 0 D 0c 0 c= ⋅v T v , (3.83) where, the inverse transformation from Frame-0 to Frame-D is found from Equation (3.47):  ( ) 1D 00 D .−=T T  (3.84) Pn plane is formed by the yDzD components of the Frame-D. So the projected vector ( ) nPD 0cv  is obtained by setting the x-component of the transformed vector D 0cv  to zero. If the transformed cutting velocity is:  ( ) ( ) ( )D 0 D 0 D 0 D 0c c c cx y z, =   v v v v   (3.85) then, the projected vector is:  ( ) ( ) ( )nPD 0 D 0 D 0c c cy z0 . =   v v v   (3.86) The positive normal rake angle is defined as the angle between the negative of the projected cutting velocity and the rake face normal Dj  vector. It is found by taking the inverse-sine of the negative z-component:  ( )( ) ( )D 0czn2 2D 0 D 0c cy zarcsinγ  − =   +  vv v. (3.87) 80  The range of arcsin function is [-π/2,π/2], so Equation (3.87) covers all the possible nγ  values.  Sample effective geometry data are given in Figure 3.43, Figure 3.44 and Figure 3.45 for indexable cutter (two-insert end mill and U140 face mill) and solid tool (serrated end mill), respectively.    Figure 3.43 Effective tool angles of the two-insert end mill.   Figure 3.44 Effective tool angles of U140 face mill  Figure 3.45 Effective tool angles along the cutting edge (in one serration period) of the serrated end mill [21]. 81  Chapter 4: Generalized Modelling of Cutting Mechanics  The generalized tool geometry presented in the previous chapter is used to model the chip geometry and corresponding cutting forces in this chapter. The kinematics of turning, boring, drilling and milling operations are developed to transform the forces and vibrations at the desired tool and part locations.  4.1 Chip Geometry The chip thickness and length are the basis for predicting the cutting forces. The cutting forces are then used to determine resulting vibrations and the machining errors. Chip thickness (jkih ) is defined as the material removed in the radial R direction of the cutting edge element Sjk; it consists of static ( stjkih ) and dynamic (djkih ) components:  st djki jki jkih h h= + . (4.1) The static chip thickness is formed by the relative rigid body motion of cutting edge and workpiece elements, and evaluated from the feed per edge ( ajkic ) of the axial or radial segment (k) of tooth (j) at time instant (i), with true cutting edge angle ( *r, jkκ ) as:  ( )( )a *jki jki r, jkstjki a *jk jk rjk, jiksin sin if millingsin if turning,boring or drillingchcγ κγ κφ ⋅ − ⋅=  − ⋅, (4.2) where the actual feed per edge ( ajkic ) and decrease in chip load ( jkiγ ) depend on the runout of the cutting edges. In milling, the chip load depends on the angular location of the edge due to the feed movement in radial direction; whereas turning, boring and drilling operations have constant chip load, due to the feed motion in axial direction. Hence, ajkc  and jkγ  are time invariant as is noted in Equation (4.2). 82   Static chip thickness and tooth passing delay with zero runout: If there is no runout, the actual feed per edge for the Sjk element becomes time invariant and takes the value of the nominal feed per edge, ajki jkc c= . Since there is no deviation from the chip engagement due to runout, variation in the chip load becomes jki 0γ = , and the static chip thickness is rewritten using Equation (4.2):  ( )*jk r, jkstjki zero *runout jk r, jkjkisin sin if millingsin if turning,boring or drillingchcφ κκ ⋅= ⋅⋅. (4.3) The tooth passing delay period jkiT  for multi-point tools also becomes time invariant and only depends on the designed pitch angle p,jkφ  of the edge element Sjk:  p,jkjk 2π60Tφ= Ω  . (4.4)  Static chip thickness and tooth passing delay with runout effect: A block diagram is prepared to show the influence of the axial runout (for turning, boring and drilling) or radial runout (for milling) on the chip thickness and the time delay term. Similar to tooth period, the delay period of element Sjk is defined as the time passed after the last edge jkij+n ,kS cut, which is at the same axial or radial segment. Here, njki-1 is the number of edges skipping the cut before tooth-j at segment-k and time instant-i. If njki=Na,k (Equation (3.1)), then only one edge is cutting at that segment instantaneously. Hence, the chip is removed like in a “no-runout” case. The prediction of actual static chip thickness and tooth passing delay period ( jkiT ) from the runout 83  geometry is based on the previous studies [20,22,108,115]. Due to different kinematic configuration, milling operation and turning, boring and drilling operations are presented separately. The resultant feed direction of milling is on the x0y0-plane. Thus, only the radial runout is affecting the chip thickness. If the tool is also moving in the z0-axis, the axial runout is additionally required to evaluate a resultant runout in the feed direction. The feed direction of turning, boring and drilling operations are in the z0-axis. Thus, the axial runout affects the actual chip thickness.  Chip thickness derivation is based on the translational feed motion of the cutter per one spindle revolution. Cutting edge elements at each discrete segment-k of the tool share the removed chip section according to the runout at the edges.  Example case with a radial runout: The algorithm presented in Figure 4.1 is demonstrated on a cylindrical serrated end mill, as shown in Figure 4.2 (taken from Ref.[21]). The tool geometry is sampled in axial direction (z0) with dz thick segments. Each depth of cut geometry is approximated at its middle axial distance (0.5kdz-0.5dz). For this case, 1.9-mm depth of cut is selected (Figure 4.2). Segment thickness dz is 0.1-mm, thus k=19. The k-th axial segment information is given as in Table 4.1.  Initial observation at k-th segment (Figure 4.2) is as follows: Since the 1st edge has the most runout ( r,1k 0.1766-mmε = ), it will remove most of the material. The 3rd edge has the least runout ( r,3k 0-mmε = ), it removes the least material among the 3 edges.  84   Figure 4.1 Algorithm for predicting the actual static chip thickness and tooth passing delay period jkiT . 85    Figure 4.2 Serrated cylindrical end mill with the cross section at k-th segment.   Table 4.1 Inputs at the k=19-th axial segment of the 3-tooth serrated cylindrical end mill. Element-available-to-cut condition array [ ]1,k 1 1 1=g   Radial runout vector [ ]r,k 0.1766 0.1688 0.0 mm=ε   Relative angular locations vector (measured from 1st edge in CW direction) [ ]relki 0 120 240 deg=ψ   Actual number of edges in cut at k-th segment 3a,k 1, jk11 1 1 3NjN g=== = + + =∑   Spindle speed 1000 rev/minΩ = −   Feedrate per one spindle revolution r 0.120 mmf = −   Edge for which the output is requested out 2j =  Instantaneous angular position of the output edge (measured from +y0 axis in CW direction) 2ki 114.6 degφ = −   86  Initial case: Since all the cutting edge elements (S1k, S2k and S3k) are available-to-cut (all 1, jk 1g =  for j=1,2,3), initial instantaneous missed cut condition is set to 1 for all teeth: [ ]3,ki 1 1 1=g . Edges in cut array aN  is assumed to include all edges: [ ]a 1 2 3=N . No edge is assigned into the new edges in cut array a,newN  yet: a,new [ ]=N . It is assumed in the beginning that the assigned arrays are correct and will not be updated during the loop, and there is a binary flag parameter is initially set to zero: “sameNotSame=0”. If this “sameNotSame“ changes to 1 during the first loop, then the computation will continue to the second loop (see Figure 4.1).   First loop: Starting with the first edge ( a a1..j N= ) in the edges in-cut list, the pitch angles for each edge are calculated from the angular locations of the edges in cut:  a a a a a ap, ( j ),ki ( j +1),k ( j ),k a afor 1..j Nφ ψ ψ= − =N N N , (4.5) which yields [ ]p,ki 1x120 120 120 N=φ  (deg), as expected for regular pitch cutter. Actual feed per edge array akic  is found using the pitch angle array:   p,kiaki r2πf=cφ, (4.6) which leads to [ ]aki 0.040 0.040 0.040 mm= −c . Actual chip load ( pakic ) at the inputted instantaneous angular position ( 2kiφ ) is:  pa aki 2kiki sinφ= ⋅c c , (4.7) with [ ]paki 36.4e-3 36.4e-3 36.4e-3=c -mm. Reduction in chip load array ( kiγ ) is found by considering the effect of the runout of the previous in-cut tooth (a ar, ( j +1),kε N ) to the current one (a ar, ( j ),kε N ): 87   a a a a a a( j ),ki r, ( j +1),k r, ( j ),kγ ε ε= −N N N , (4.8) which gives [ ]ki 0.0147 0.1618 0.1766= − −γ -mm. Negative value of jkiγ  means that element Sjk removes more chip than the actual value. If the reduction in chip load ( jkiγ ) is more than the actual chip load for element Sjk, then its missed cut condition parameter ( 3, jkig ) is set to zero:  pajki 3, jki 3, jkijkicheck ; if : set 0; if : keep 1c yes g no gγ ≥ = = . (4.9) This updates the missed cut array: [ ]3,ki 1 1 0=g . Thus, the third edge is not able to cut at this time step. Edges-in-cut array is updated according to the missed cuts, a,new [1 2]=N , i.e., only edges j=1 and j=2 are in cut. Since the updated and original 3,kig  are different, the flag parameter is updated: “sameNotSame=1”. The “sameNotSame“ parameter changed to 1 during the first loop, then the computation continues to the second loop (see Figure 4.1).  Second loop: The calculation of the second loop starts with setting the actual edges in cut array to the previously updated edges-in-cut array: a [1 2]=N . Number of edges in cut is a 2N = . Since edge number 3 (j=3) is not in the aN  list, parameters related to j=3 have no meaning, but the vector sizes are still kept at 3 for consistency. Pitch angles for each edge in-cut ( a 1j =  and a 2j = ) are found similar to the first loop:  a a aa a ap, (1),ki (2),k (1),k a p,1kip, (2),ki (1),k (2),k a p,2kifor 1, 120 deg,2π for 2, 240 deg,jjφ ψ ψ φφ ψ ψ φ= − = = −= − + = = −N N NN N N (4.10) which leads to the updated pitch angle array: [ ]p,ki 1x120 240 0 N=φ  deg. The 0 value in the array is not used in the calculations since the 3rd edge is not cut ( 3,3ki 0g = ). Using Equation (4.6)88  , the actual feed per edge array is: [ ]aki 0.040 0.080 0 mm= −c . Actual chip load is found from Equation (4.7) at the instantaneous location angle: [ ]paki 36.4e-3 72.7e-3 0.0=c -mm. The decrease in chip load is calculated similar to Equation (4.8) using the actual edges in cut:  a a aa a a(1),ki (2),k (1),k a 1ki(2),ki (2),k (1),k a 2kifor 1, 0.0147 mm,for 2, 0.0147 mm,jjγ ε ε γγ ε ε γ= − = = −= − = =N N NN N N  (4.11) which leads to the updated array: [ ]ki 0.0147 0.0147 0= −γ -mm. Carrying out a similar evaluation  as in Equation (4.9) updates the missed cut array as:  [ ]pajki 3, jki 3, jkijki3,kicheck ; if : set 0; if : set 1;1 1 0 .c yes g no gγ ≥ = ==g (4.12) The updated missed cut array is the same as the previous one (result of the first loop in Equation (4.9)), and the flag parameter for the convergence of iteration is set to sameNotSame=0.  At the given angle ( 2ki 114.6 degφ = − ), the algorithm gives the following output at segment-k for the selected edge (jout=2):  a2ki2kipa a2ki 2ki2kipa2ki2ki3,2kiActual feed per edge: 0.080-mm;Decrease in chip load: 0.0147-mm;Actual chip load: sin 0.0727-mm;Effective chip load: 0.0580-mm;Missed cut condition: 1;Actual cc ccgγφγ=== ⋅ =− ==p,2kip,2ki2kipitch angle: 240-deg;π240 180Actual delay period: 0.04-s.2π 2π 100060 60Tφφ== = =Ω ⋅  (4.13) 89   Figure 4.3 Schematics for the 2-insert drill with axial runout.  90  Figure 4.3 shows the schematics, radial geometry discretization and the axial runout definition for the drill. Axial runout definition is similar to the radial runout definition of the milling cutter (Figure 4.2). Radial geometry discretization for turning, boring and drilling cutter is equivalent to the axial geometry discretization for the milling cutter. The algorithm for determining the actual chip load is exactly the same for the milling tool, except the calculation in Equation (4.7). The chip load is constant for all rotation angles due to the feed motion in the z0 axis:   pa akiki =c c   (4.14)  Figure 4.4 Parallel turning schematics: (a) Radial discretization; (b) effect of axial offset and axial runout between cutters. 91  Parallel turning operation shown in Figure 4.4 has a similar structure as the drilling operation, except that the radial geometry discretization (Figure 4.4a) starts from the diameter of the machined part of the workpiece (smaller diameter) and increases towards to the nonmachined diameter (larger diameter). The chip load with runout effect (Figure 4.4b) is the same as the drilling operation.   Chip length: When the axial segment height is fixed to dz, then for the true cutting edge angle *r,jkκ  at the element Sjk, the chip length is defined as the projected length of the cutting edge along the tangential (T) direction (Figure 2.1a):   c,jk *r,jkdsinzlκ= . (4.15)  Chip thickness ( jkih ): The static (stjkih ) and dynamic (djkih ) chip thickness terms are time-dependent due to the runout. The static component depends on the feed amount and the geometry of the tool, whereas the dynamic component depends on the flexibility of the system and the geometry, as presented in later chapters. The instantenous chip thickness is evaluated as:  st djki jki jkih h h= + . (4.16)  Chip Area: The time dependent chip area at the element Sjk is found from the product of the chip thickness and the chip length:   c,jki jki c,jkA h l= ⋅ . (4.17)  92  4.2 Modeling of Cutting Forces Cutting force is separated into chip shearing and ploughing components [25,27,116]. If all the tool angles are identified, then the shearing forces can be predicted using the shear parameters that are identified from orthogonal cutting experiments [35,36]. The angles derived in Chapter 3: are used to predict the rake face cutting force coefficients. While the shearing forces are defined in UV coordinate frame, the ploughing forces are calibrated in RTA frame. The mechanism of ploughing is related to the tool indentation into the workpiece in cutting edge plane (Ps), which includes the radial (R) and tangential (T) directions. Thus, it is assumed that there is no ploughing force component in axial (A) direction. Huang et al. [30] separated the ploughing forces into static (edge force [34]) and dynamic (process damping [117]) parts. Edge forces (static ploughing forces) in radial and tangential directions are assumed to be linearly proportional to the chip length [29] with edge force coefficients as multipliers. One way to predict the edge coefficients is to create a generic database [34,118] for the tool-workpiece material combination and tool coating. A second way is to calibrate for each individual cutting operation. For example, Atabey et al. [47] calibrated the edge forces of boring operation for fixed depth of cut, fixed spindle speed and varying feedrate. Dynamic component of the ploughing force is linear to the extruded workpiece volume under the tool flank [119]. The dynamic part (process damping) is analytically modeled by Chiou and Liang [120] for small amplitude vibrations of tool relative to workpiece. Recent methods are available for large edge radius tool under large amplitude vibrations [121,122].  Cutting forces without process damping: Shearing force is constituted by a combination of the friction (Fu) and the normal (Fv) forces on the rake face. Without the process damping 93  (dynamic ploughing) forces, the rake face forces are separated into shearing ( cjkiF ) and static ploughing ( esjkiF ) components, as:  ( ) ( )c es c ejki jki jki jki c,jk n,jk c,jki c,jk n,jk c,jk, , ,F F F K h v A K v lγ γ= + = ⋅ + ⋅ , (4.18) where the shear component ( cjkiF ) is expressed as a linear function of the chip area:  c c es e c esjki jki jki c,jk jk jk c,jk jki jki jk; ;F K h l F K l F F F= ⋅ ⋅ = ⋅ = + , (4.19) where chip length ( c,jkl ) is given in Equation (4.15). Due to the variation of cutting velocity, the normal rake angle and chip thickness along the geometry, the cutting coefficient ( )c jki c,jk n,jk, ,K h v γ , and the edge coefficient ( )e c,jk n,jk,K v γ  are geometry-dependent. Moreover, ( )c jki c,jk n,jk, ,K h v γ  is also time-variant due to the time dependence of the chip thickness (especially for the milling operation). Although the chip size effect is not significant when using the linear force models, the effect would be more pronounced when nonlinear force models are used, such as the Kienzle model (see [123]):  ( ) ( ) ( )Fc ejki c,jk n,jk jki c,jk c,jk n,jk c,jk, ,xF K v h l K v lγ γ= ⋅ ⋅ + ⋅ , (4.20) Range of parameter Fx  is typically: F0 1x< ≤ . The linear force model in Equation (4.18) is obtained, if F 1x =  in Equation (4.20). Equations (4.18) and (4.20) are nonlinear due to their dependence on the total chip thickness, but they can be linearized if the dynamic component (djkih ) of the chip thickness is dropped for a rigid case, leaving only the static term (stjki jkih h≈ ). The cutting forces in Equations (4.18) and (4.20) are thus linearized around static part of instantaneous chip thickness:  94   ( ) ( )( ) ( ) ( )Fc st ejki jki c,jk n,jk jki c,jk c,jk n,jk c,jk1c st ejki c,jk n,jk F jki jki c,jk c,jk n,jk c,jk, , , ;, , .xF K h v h l K v lF K v x h h l K v lγ γγ γ−= ⋅ ⋅ + ⋅ = ⋅ ⋅ ⋅ ⋅ + ⋅   (4.21) Cutting coefficients can be identified mechanistically by fitting the coefficients specifically for each cutting operation empirically from cutting tests, which is costly [6,32,124]. A more effective method is to design an orthogonal cutting database with shear angle, shear stress and friction coefficients specific to each tool-workpiece material combination [34]. The orthogonal parameters can be transformed into the oblique geometry of any machining operation and the tool geometry. For example, for AISI 1045 steel, the fitted equations for the local (of element Sjk at time ti) shear stress s,jkiτ  (MPa), shear angle n,jkiφ  (rad) and average friction angle a,jkiβ  (rad), as functions of static chip thickness stjkih  (mm) and cutting velocity c,jkv  (m/min) are [118]:  ( )( )[ ] [ ] [ ]s,jki c,jknstjkistjkistjkist,jki c,jka,jki c,jkc,jk n,i jj kk450.3 0.4 227.5 ;arctan 0.4 0.0005 0.6 ;π26.8 0.0313 11.77 ;180limits : 0.08, 0.32 -mm; 50, 150 -m/min; 0, 0 -rad.vvvvhhhhτφβγ= + ⋅ + ⋅= + ⋅ + ⋅= − ⋅ + ⋅ ⋅∈ ∈ ∈ (4.22) Refs. [35,36] derive the total force acting on the rake face based on the orthogonal cutting principal of Merchant [24]. The total shearing force is projected onto the friction u and normal v directions using the friction a, jkiβ  angle:   c c c cu,jki jki v,jki jka, jk a,ii jkisin ; cosF F F Fβ β= =⋅ ⋅ . (4.23) The projected cutting forces are expressed in terms of the chip area c,jkiA  of the element Sjk at time ti and the cutting coefficient:   cc c c cu,jki u,jki v,jki v,j,jki c,jki ki;F K F KA A= =⋅ ⋅ . (4.24) 95  Equation (4.24) represents the shear component of the force expressed in Equation (4.21). The chip area ( c,jkiA ) is given in Equation (4.17). Using the orthogonal to oblique cutting transformation [6,35,36], the cutting coefficients of Equation (4.24) are obtained for the cutting edge element Sjk at time ti, as:   ( )( )2 2s, jki n,2 2 2s, jk n, n, n, n, n,2 2s, jki n,2 2jk jkijki jki jki jk jk jkijk jkijki jki jki jk jk j2s, jk n, n, n, n, n, kicu,jki jkicv,jkai,1 tan sinsincos sin cos tan sin1 tan sincoscos sin cos tan sin( d );k zKKτ η βλ φ φ β γ η βτ η βλ φ φ β γ η ββ−=+ − +−=+ − +kia, j .β (4.25) The projection of the friction angle on the cutting edge normal plane (Pn) is:  ( )jkn, jki a, jkiarctan tan cosβ β η= ⋅ . (4.26) Stabler’s rule [125] is assumed for the chip flow angle: jk s, jkη λ≅ . This is a fair assumption for large depth of cut where the chip flow angle approaches the cutting edge inclination angle [124,126]. 4.3 Force Transformations for Arbitrary Tools Forces predicted on the rake face are transformed to the machine coordinates with the following order: • From the rake face UV frame to the cutting edge RTA frame; • From the cutting edge RTA frame to the edge reference xRyRzR frame; • From the edge reference xRyRzR frame to the tool reference x0y0z0 frame; • From the tool reference x0y0z0 frame to the machine reference xyz frame. 96   Figure 4.5 Transformation of rake face forces (Fu and Fv) into the forces along the R,T and A axes of the cutting edge plane Ps. (Adapted from [1,127].) 4.3.1 Rake face UV (friction-normal) to Cutting Edge RTA (radial-tangential-axial) Starting from the rake face force vector, the transformations are sequenced in accordance with Figure 4.5:  • Negative chip flow angle ( jkη− ) rotation around V-axis; • Negative normal rake angle rotation (- kn, jγ ) around the new y-axis (YIII); • Negative cutting inclination angle (- s, jkλ ) around the new z axis (ZII); • RTA frame (xIyIzI frame) is obtained.  Resulting forces in RTA frame is represented by the rake face forces in UV frame, as: 97   c crta,jki IU, jk uv,jki3x23x1 2x1    = ⋅    F T F , (4.27) with Tc c c crta,jki r,jki t,jki a,jkiF F F =  F  and Tc c cuv,jki u,jki v,jkiF F =  F . Similarly, for the cutting coefficients, Equation (4.27) can be rewritten using Equation (4.24), as:  c c c crta,jki rta,jki c,jk uv,jki uv,jki c,jkc crta,jki IU, jk uv,jki3x23x1 2x1; ;.A A= ⋅ = ⋅    = ⋅    F K F KK T K (4.28) with Tc c c crta,jki r,jki t,jki a,jkiK K K =  K  and Tc c cuv,jki u,jki v,jkiK K =  K . The transformation for the element Sjk is,  n, jk jk n, jkIU, jk s, jk jk s, jk n, jk jk s, jk n, jks, jk jk s, jk n, jk jk s, jk n, jkcos cos sinsin sin cos sin cos cos coscos sin sin sin cos sin cosγ η γλ η λ γ η λ γλ η λ γ η λ γ − = +  − +  T . (4.29) If the cutting coefficients in RTA frame are identified mechanistically from cutting tests, they would be projected onto the UV frame (on rake face), as is similar to Equation (4.27):  ( ) 1c cuv IU, jk rta specificgeometry− = ⋅  K T K . (4.30) The edge forces or ploughing forces are defined in the RTA frame. The occurrence of ploughing forces is due to the indentation of the cutting edge into the workpiece, thus the creation of a normal force in R direction. A corresponding friction force acts in T direction, which is the same as the cutting velocity direction. Theoretically, the direction A has zero ploughing force as no indentation movement occurs towards the sides of the cutting edge. The edge forces are written from the Equations (4.15) and (4.19) as:  es erta,jk rta, jk *3x1 3x1r,jkdsinzκ   = ⋅   F K , (4.31) 98  where, Tes es es esrta,jk r,jk t,jk a,jkF F F =  F . The edge coefficients are Te e e erta,jki r,jk t,jk a,jkK K K =  K  and ea,jki 0K ≅ . The edge coefficients can also be identified from the generic database, which is obtained by conducting orthogonal cutting tests [34]. Similar to fitting shear parameters, edge coefficient equations (in N/mm) for AISI 1045 steel, for example, are as follows [118]:  ( ) ( )( ) ( )[ ] [ ]3 2et,jk c,jk c,jk c,jk3 2er,jk c,jk c,jk c,jkeastjk,jkc,jk n, ki j9.0688e-6 1.0382e-2 1.9856 55.431;2.9929e-4 9.8351e-2 10.299 294.96;0 (as expected);limits : 0.08, 0.32 -mm; 50, 150 -m/min; hK v v vK v v vKv γ= ⋅ − ⋅ + ⋅ −= ⋅ − ⋅ + ⋅ −=∈ ∈ [ ]0, 0 -deg.∈  (4.32) With no vibrations in the system, the total cutting force trta,jkF  acting on the element Sjk of the tool at time ti is found in RTA frame from Equation (4.28) and Equation (4.31):  t c esrta,jki rta,jki rta,jkt c erta,jki rta,jki rta,jkt c erta,jki rta,jki c,jk c,jkjjki rta,jk* *r,jk r,jkki;1d d .s;in sinh l lhz zκ κ= += ⋅ + ⋅= ⋅ + ⋅⋅F F FF K KF K K  (4.33) 4.3.2 RTA to Edge Reference Frame (xRyRzR), and to Process Frame (xyzθ) The tool coordinate frame (x0y0z0) is same as the process coordinate frame (xyzθ) for drilling, turning, boring and three axis milling operations.  For 5-axis milling, there would be additional lead and tilt angle rotation transformations [49]. Torsional (θ) degree of freedom of the tool (for milling, drilling, boring) or workpiece (for turning, boring, drilling) is added as the 4th degree of freedom. The corresponding torque is calculated using the force in T direction, which is in the velocity direction and is perpendicular to the radius vector (Figure 4.5).  99  Forces in Radial-Tangential-Axial (RTA or xIyIzI) coordinate system is transformed into process reference coordinates in the xyzθ frame. The intermediate step is the cutting edge reference coordinates in the xRyRzR frame. RTA (xIyIzI) is a right-hand coordinate system, and the cutting edge forces of all the machining operations are represented within this frame. Figure 4.6 shows the schematics of the turning, boring, drilling and milling operations. Since the cutting edge rake face (Figure 4.5) is standard for all cutters, the orientation of RTA (xIyIzI) along the cutting edge is the same for all the operations. Force transformation has the same form for all the operations:  0R, jki RI, jkc cjki rta,jkit jk0 0,R⋅ = ⋅   T TF F , (4.34) where the forces in tool reference frame include the forces in 3 axes ( cx,jkiF ,cy,jkiF ,cz,jkiF ) and the torque cjkiT  on the element Sjk at time ti:  Tc c c c cjk x,jk y,jk z,jk jki i i i iF F F T =  F . (4.35) t jk,R  in Equation (4.34) is the moment arm (radius) of the cutting edge element Sjk. Components of force vector are the feed ( cx,jkiF ), normal (cy,jkiF ) and axial (cz,jkiF ) forces and torque ( cjkiT ) on element Sjk at time ti. RI, jkT  is the transformation matrix from RTA to cutting edge reference xRyRzR, and 0R, jkiT  projects the forces onto the tool reference frame x0y0z0. The transformation for each operation can be derived from the given orientations shown in Figure 4.6.   100  Transformations for the rotating right hand boring and right hand drilling tools: Following from Figure 4.6d and Figure 4.6e, the transformations for the rotating boring and drilling operations are:  * *r, jk r, jkRI, jk* *r, jk r, jkcos 0 sin0 1 0sin 0 cosκ κκ κ − −  = −  T . (4.36) 0R, jkiT transforms force and deformation from cutting edge reference to tool coordinates:   jki jkijki j0R, kijkisin cos 0cos sin 00 0 1φ φφ φ−  =    T , (4.37) where jkiφ is the instantaneous angular position of the element Sjk at time ti:  ijki0jk ( )dtt t ψφ ω= +∫ . (4.38) ( )tω  is the angular speed of the cutter (in rad/s). jkψ  is the angular location of element Sjk and is relative to the +y0 axis (measured in CW direction).   Transformations for the nonrotating right-hand boring cutter: Traditional right-hand boring (Figure 4.6b) is a special case of the rotating boring operation. The equations are kept the same as in Equations (4.36) and (4.37), but the rotation angle term is removed:  jjki kφ ψ= . (4.39) The nominal value for the instantaneous angular position is 0-deg.   101  Transformations for the left-hand turning cutter: Left hand turning (Figure 4.6a) is the external scheme of the right hand boring operation. The model is the same as in Equations (4.36) and (4.37) for a rotating boring-cutter, but the rotation angle is removed from the instantaneous location angle term of Equation (4.38):  jjki kφ ψ= . (4.40) The nominal value for the angular position is -90 degrees.  Transformations for the right-hand milling cutter: Right hand milling (Figure 4.6c) has a similar transformation matrix RI, jkT  as the right hand turning operation:  * *r, jk r, jkRI, jk* *r, jk r, jksin 0 cos0 1 0cos 0 sinκ κκ κ − −  = −  T . (4.41) The 0R, jkiT  transformation is as the same as the drilling operation:  jki jkijki j0R, kijkisin cos 0cos sin 00 0 1φ φφ φ−  =    T , (4.42) with the same instantaneous location angle definition:  ijki0jk ( )dtt t ψφ ω= +∫ . (4.43) The rotation angle term in Equation (4.43) is written in general form to include spindle speed variation. For the constant rotational speed, 2π( )60tω =Ω :   i jkjki2π60t ψφ  Ω ⋅ += . (4.44) 102   Figure 4.6 Transformations from RTA to xRyRzR, and from xRyRzR to x0y0z0 for basic machining operations. (Images adapted from Sandvik and Ref.[3].) 103  Transformations for the right-hand turning cutter: Right hand turning cutter is shown in Figure 4.7a. The RI, jkT  transformation is written as:  * *r, jk r, jkRI, jk* *r, jk r, jkcos 0 sin0 1 0sin 0 cosκ κκ κ −  =   T . (4.45) The 0R, jkiT  transformation is as same as the traditional right-hand boring operation.   jki jkijki j0R, kijkisin cos 0cos sin 00 0 1φ φφ φ−  =    T   (4.46) With the constant angular location:  jjki kφ ψ=   (4.47) The nominal value for the angular position is 0-deg (Figure 4.7a), as similar to the traditional right-hand boring operation.    Figure 4.7 Right hand turning and parallel turning schematics. (Images adapted from Sandvik.) 104  A note for the parallel turning operation: The nominal angular positions of tool #1 and tool #2 of the parallel turning operation (Figure 4.7b) are 90 and 270 degrees, respectively.   Transformations for the left-hand milling cutter: Left-hand milling cutter is used for the application of proposed general mathematical model on a double milling application. As shown in Figure 4.8, left- and right-side indexable cutters are labeled according to left-hand and right-hand milling action, respectively. The transformations for the left-hand cutter (Figure 4.9a) are given as [4]:   jki,L jki,Ljki,* *r, jk r, jkRI, jk L jki,L 0R, jki L* *r, jk r, kLj,sin 0 cos sin cos 00 1 0 ; cos sin 00 0 1cos 0 sin,φ φφ φκ κκ κ − −     = =     −    T T , (4.48) and for the right-hand cutter (Figure 4.9b):  * *r, jk r, jkRI, jk,R 0R, jki R* *r, jkjki,R jki,Rjki,R jki,Rr, jksin 0 cos sin cos 00 1 0 ; cos sin 00 0 1cos 0 sin,φ φφκ κκφκ − − −     = =     −    T T . (4.49) The instantaneous angular positions for left- and right-side cutters are defined as:   i jk,xjki,x2π60t ψφ  Ω ⋅ +=   (4.50) where, for the left-side cutter: x=L, and for the right-side cutter: x=R.  105   Figure 4.8 Double-sided milling schematics. Figure 4.9 Double-sided milling tool geometry [4]: (a) Left-side cutter; (b) right-side cutter. 4.4 Transformation of the Vibration Vector to the Dynamic Chip Thickness The relative vibrations between the tool and workpiece are needed to calculate the dynamic cutting forces and chatter stability. A vibration vector defined in machine xyzθ frame at k-th segment at time ti is:  [ ]Tki ki ki ki kix y z θ=q . (4.51) The dynamic chip load jki∆q  of j-th tooth at segment-k is dependent on the difference of the surface left at the previous tooth pass:  jki ki k i jki( )t T∆ = − −q q q , (4.52) 106  where the delay period jkiT  for the cutting edge element Sjk is both geometry- (at tooth-j and segment-k) and time (ti) -dependent.  The vibrations in the tool reference frame are projected in the chip thickness direction. The positive chip thickness direction is in the negative-radial (R) direction (-xI) of Frame-I (xIyIzI frame) as shown in Figure 4.5. Thus. the dynamic chip thickness is:  d Ijki jkih x= −∆ . (4.53) The effect of the regenerative vibrations of the tool-workpiece system is projected onto the negative-R direction. The feed (i.e. rigid body motion) may contribute to the regenerative vibration in the torsional direction ( kiθ∆ ) in some applications, such as drilling and plunge milling [128], but this effect is negligible compared to the structural vibrations of the system. However, the torsional-axial dynamic coupling for most of the drilling tools (e.g. twist drills) is strong, with the torque consequently causing an axial deflection [129]. At time ti, the following transformation orients the regenerative vibration vector jki∆q  at k-th segment to the Ijkix−∆  vector of the element Sjk:  I djki jki jkix−∆ = ⋅∆e q , (4.54) where the general transformation vector from tool reference coordinates to the negative-R is [3]:  [ ] ( )T0R, jki RI, jkdjki 0M1 0 0= ⋅ ⋅−e T T T   (4.55) Since the translational axes of process (xyzθ) and tool (x0y0z0) coordinate systems are taken to be the same (in this thesis), the transformation is,  0M1 0 0 00 1 0 00 0 1 0  =    T . (4.56) 107  The last column is zero since the effect of torsional-axial coupling due to the feed (rigid body motion) is neglected. The dynamic chip thickness is represented by the relative vibrations between the tool and workpiece structure, which can be estimated from Equations (4.53), (4.54) and (4.55) as:  [ ] ( )T0R, jki RI, jkdjki 0M jki1 0 0h ⋅ ⋅= − ⋅∆T T T q . (4.57) The total chip thickness is represented in the general form from Equation (4.16):  st djki jki jki jkih h= + ⋅∆e q . (4.58)  4.5 Representation of General Force Vector The force vector is composed of shearing, ploughing and process damping components [2]. The shearing force is needed to plastically deform the material and form the chip, hence it is the major force with static and dynamic parts, as is expressed in Equations (4.18) and (4.19). The shearing force is written in general form by substituting Equation (4.58) into Equation (4.33), and into Equation (4.34) successively:  ( )st djki jki jki0R, jki RI, jkc cjki rta,jki *t jk r,jkdsin0 0,hzR κ+ ⋅∆⋅ = ⋅ ⋅   e qT TF K . (4.59) The shearing force can be represented by the static cs and dynamic cd shear components as:   c cs cdjki jki jkist0R, jki RI, jk jkics cjki rta,jki *t jk r,jkd0R, jki RI, jk jki jkicd cjki rta,jki *t jk r,jkd ;sind .sin;0 00 0,,hRRzzκκ= +⋅ = ⋅ ⋅   ⋅  ⋅ ∆= ⋅ ⋅   eF F FT TF KT T qF K (4.60) The dynamic component is written in a separated force coefficient and displacement terms:  108   cd djki jki jki= ⋅∆F F q , (4.61) where the dynamic force component is:  d0R, jki RI, jk jkid cjki rta,jki *t jk r,jk0 sin0d,Rzκ⋅ = ⋅ ⋅   eT TF K . (4.62) The ploughing force esjkiF  is the constant term of Equation (4.33) and does not depend on chip thickness, It is expressed from Equations (4.33) and (4.34) as:  0R, jki RI, jkes ejki rta,jki *t jk r,jkds10 in0,Rzκ⋅ = ⋅ ⋅   T TF K . (4.63) Process damping terms are caused by the plastic deformation of the edge of the tool pressing into the workpiece material [119]. The average force is modeled as the friction and normal terms in T and R directions, respectively [120]. As shown in Refs. [2,3], the process damping is generalized similar to the dynamic shearing force term ( cdjkiF  of Equation (4.60)) as:  2 d0R, jki RI, jk sp w jki kiedjki 0 *t jk c,jki r,jkdsi0 0 4 n,K LR vzκ⋅  ⋅= ⋅ ⋅ ⋅ ⋅  eT T qF P  (4.64) where { }T1 0µP =  and µ is the Coulomb friction coefficient. spK  is the material-dependent specific contact force; wL  is flank wear land length of the tool. Although it is not within the scope of this thesis, Equation (4.64) could be extended to more advanced models, such as the models proposed in Refs. [121,122]. Similar to the dynamic shear force, the process damping component is written in a separate force coefficient and velocity terms (see Equation (4.61)) as:   pedjki kijki= ⋅F F q , (4.65) where the average force component is: 109   2 d0R, jki RI, jk sp w jkipjki *t jk c, jki r,jkd0 s0 in4,KR vzLκ⋅ = ⋅ ⋅ ⋅ ⋅  eT TF P . (4.66) Then, the total force in machine coordinate frame (xyzθ) acting at the tool-workpiece contact is represented in the general form:    c es edjki jki jki jkishearing static ploughing process damping= + +F F F F . (4.67) Equation (4.67) represents the force acting on the tool:  tjki jki=F F . (4.68) Hence, the force acting on the workpiece is in the opposite direction:  wjki jki= −F F . (4.69) For the general case, the total cutting force is given in Equation (4.67). There are two special cases: Rigid system and stable cutting. If the tool-workpiece system was rigid, the vibration related displacement and velocity terms are set as zero, as shown in Equation (4.67). Thus, the static forces are:  st cs esjki jki jki= +F F F . (4.70) If the tool-workpiece is flexible and the process is stable, then the delay terms are set to zero as shown in Equation (4.67), giving the steady state forces:  ss cs es edjki jki jki jki= + +F F F F . (4.71) The differential force jkiF is applied at the cutting edge element Sjk of the arbitrary cutter. Although the differential component is general, forming the general cutting force vector iFdepends on the geometry and dynamics of the cutting operation.  110  4.5.1 Flexible system with lumped force and dynamics at one tool-workpiece contact point This is the most common occurrence when the dynamics is assumed constant along the tool-workpiece contact. This assumption is valid when the following conditions are true:  • The depth of cut is small; • The workpiece rigidity is high compared to tool rigidity; • Small variation of dynamics along the tool-workpiece contact. The cutting force vector at time ti is simply the summation of all the cutting edge elements at the point representing the average dynamics:  ,i 1,jk 2,jki 3,jki jki, 4x1N qj kg g g =    ∑F F , (4.72) where q is the number of levels for the depth of cut ap:  pdaqz= . (4.73) Figure 4.10a shows the lumping of cutting forces for the cylindrical end mill. The forces are predicted at each axial segment-k and all of the forces are lumped at the single point with the averaged dynamics. Since the variation of dynamics along the tool-workpiece is assumed to be low, the forces can be lumped at any point, preferably at the point of the dynamic identification (e.g., hammer test). The same principle is applied to the turning, boring and drilling operations. The forces generated at all the radial segments are lumped at the point where the dynamics are identified.  111   Figure 4.10 Modelling of cutting forces: (a) Lumped force when low flexilibility variation; (b) axially distributed force when high flexibility variation (adapted from Ref.[2]); (c) distributed force when high flexibility variation along tool profile. 112  4.5.2 Flexible system with distributed force along tool-workpiece contact If the flexibility of the system is varying over the tool-workpiece contact (as in Figure 4.10b-c), lumping the forces at a single point may not fully represent the system dynamics, especially when finding machining surface errors.   Axially distributed force: In the case of a long end mill with flexible tool-workpiece dynamics along the contact zone, the forces are axially distributed along the contact length (Figure 4.10b). This is common for finishing operations with large depth of cut, ap. The force vector is formed by keeping the forces on each differential segment-k:  ( )1,j1 2,j1i 3,j1i j1i1,j2 2,j2i 3,j2i j2ii1,jk 2,jki 3,jki jki1,jq 2,jqi 3,jqi jqi 4 x1Njqg g gg g gg g gg g g     =     ∑FFFFF. (4.74)  Radially distributed force: In the case of face milling operation, the axial depth of cut is small and the cutter is assumed to have constant dynamics along the tool-workpiece contact. The variation of dynamics comes from the flexible workpiece, and each differential dynamic section is radially positioned as shown in Figure 4.10c. The tool-workpiece is assumed to have constant flexibility along axial direction. As shown in Figure 4.11a, at each instance of cutting the force is lumped along axial direction (shown by Fji). At the current position of the cutter on the workpiece, if the flexible tool-workpiece contact is divided into s number of radial sections, the force vector is defined as: 113   Figure 4.11 Radial force distribution on the flexible tool-workpiece zone: (a) Instantaneous axially lumped forces; (b) limit angles for angular distribution.  [ ]( )( )1,jk 2,jki 3,jki 4,jki,1 jki1,jk 2,jki 3,jki 4,jki,2 jki,i 4 x1, 1,jk 2,jki 3,jki 4,jki,r jki1,jk 2,jki 3,jki 4,jki,s jki 4 x1N qsj ksg g g gg g g gg g g gg g g g     =     ∑FFFFF, (4.75) where the new function at point-r in the workpiece contact zone is defined as:  jk i4,jki,rjk iElement  falls in the -th radial zone at time 1 if 0 if Element  does not fall in the -th radial zone at time S r tgS r t= . (4.76) The r-th zone is defined by the angles bounding the radial sections in the tool-workpiece contact (Figure 4.11b). The entry and exit angles ( st,rφ  and ex,rφ ) for each zone-r can be assigned using unequal or equal step angles depending on the measured dynamics of the structure.  4.5.3 Multiple-Level Tool with Lumped Force at Each Station/Level If the tool has more than one dynamic levels (or stations), then the forces are lumped locally at each dynamic level [5].   114  Multifunctional tool: Multifunctional drilling/boring cutters have D levels (minimum D is 2) in contact with the workpiece. In this case, the dynamics is assumed to be constant within each dynamic level. The force vector is formed by lumping the force terms at each level accordingly. Forces at each level are lumped according to the limit radial level value (de with e=1..D), as:  [ ]121ee-1DD,1,jk 2,jki 3,jki jki,k 1,1,jk 2,jki 3,jki jki,k 1,i1,jk 2,jki 3,jki jki,k 1,1,jk 2,jki 3,jki jki,k 1 4 x1N djN dj dN dj dN dj d Dg g gg g gg g gg g g== += += + ⋅         =    •  •     ∑∑•∑∑FFFFF. (4.77) The dynamic level limit vector ( d ) is used for storing the last radial segment index of each level:   [ ]1 2 e Dd d d dd =   , (4.78) with de being the last radial segment index of the e-th dynamic level, e.g., radial segments from “d1+1” to “d2” is lumped onto dynamic level-2 (e=2). If the number of dynamic levels is one (i.e., D=1), then the total number of radial segments in contact (q) is assigned to d1=q and the Equation (4.77) reduces to the form of Equation (4.72). 115   Figure 4.12 Sandvik TM880 multifunctional tool with D=2 dynamic levels.  For example Sandvik multifunctional tool has 2 dynamic levels, therefore D=2. As shown in Figure 4.12, for Level-1 the forces from radial segment-1 to radial segment-d1 are lumped on Level-1 dynamics; forces from d1+1 to d2=K are lumped onto Level-2 dynamics.   Double-sided milling cutter: There are two sides in the double-sided milling cutter (Figure 4.13). Each side of the cutter has s number of sections (Figure 4.11) to approximate the dynamics along the tool-workpiece contact zone. The force vector is formed by writing the similar force vectors for both left- and right-side cutters, and joining them, as follows: 116   Figure 4.13 The double-sided milling cutter with forces applied at the opposite sides of the workpiece.  ( )( )Lii Ri 2 4 x4x x x x x1,jk 2,jki 3,jki 4,jki,1 jkix x x x x1,jk 2,jki 3,jki 4,jki,2 jki,xi x x x x x, 1,jk 2,jki 3,jki 4,jki,r jkix x x x x1,jk 2,jki 3,jki 4,jki,s jki 4 x1;sN qj ksg g g gg g g gg g g gg g g g⋅  =       =        FFFFFFFF;L if left-side cutterx .R if right-side cutter= ∑  (4.79)  Multiple-tool turning system (Parallel turning): Parallel turning application (Figure 4.14) is very similar to multifunctional cutter, except that it has two tools with individual dynamics for the tool #1 (j=1) and tool #2 (j=2): 117   Figure 4.14 Cutting forces of parallel turning operation.   [ ]1,1k 2,1ki 3,1ki 1kii1,2k 2,2ki 3,2ki 2ki4 2 x1qkqkg g gg g g⋅   =     ∑∑FFF. (4.80) Each tool experiences a lumped force, therefore acts like a separate dynamic level, as is shown in Figure 4.12. Following the principles of Figure 4.10a, forces on the radial segments of each tool are summed up at their own station.  4.6 Summary Chip geometry and tool angles at each edge element Sjk is used for computing the differential machining force. Shearing forces are transformed from rake face to process coordinate system. Ploughing forces are transformed from the RTA frame to process coordinate system. The forces are integrated along the cutting tool edges which are engaged with the workpiece. The flexibilities of the tool-workpiece are included in the model.   118  Chapter 5: Generalized Modelling of Cutting Dynamics  The previous chapters have focused on the modelling of tool and chip geometry, kinematics of chip generation and corresponding cutting forces with and without vibrations; these are the foundations of cutting dynamics model. Cutting forces excite the structural dynamics of the tool and part. This chapter models the interaction between the generalized metal cutting process and the vibrating structure. The resulting delayed differential equations are solved in semi-discrete time domain to simulate the dynamic cutting forces, surface form errors and vibration amplitudes.  5.1 General Equation of Motion The classical form for relating the force ( ( )sF ) and deformation ( ( )sQ ) in the Laplace (s)-domain can be expressed by:  [4 1] [4 4 ] [4 1]( ) ( ) ( )q q q qs s s× × ×=Q Φ F . (5.1) The transfer function matrix ( )sΦ  of the structure is represented by the identified modal parameters of the system [64]:  2 2 1 T[4 4 ] n n( ) ( 2 )q qs s s−× = + +Φ U I ζω ω U , (5.2) where [ ]m m×I  is the identity matrix, [ ]m m×ζ  is the diagonal damping ratio matrix, n [ ]m m×ω  is the diagonal natural frequency matrix, and [4 × ]q mU is the mass normalized mode shape matrix for the m number of identified tool and workpiece vibration modes. The transformation from physical to modal domain is carried out by:  ( ) ( )s s= ⋅Q U Γ , (5.3) 119  where [ 1]m×Γ is the modal displacement vector expressed explicitly using Equations (5.1) and (5.2):  2 2 1 Tn n( ) ( 2 ) ( )s s s s−= + +I ζω ω U FΓ . (5.4) The reduced order modal model equation (Equation (5.4)) is converted into the time domain as:   2 Tn n( ) 2 ( ) ( ) ( )t t t t+ + =ζω ω U F Γ Γ Γ . (5.5) The physical displacement vector ( )tQ  is obtained using Equation (5.3):  ( ) ( )t t= ⋅Q U Γ . (5.6) If the machine tool’s modeled mass, stiffness and damping matrices are available, the equation of motion of the full model is:  f f f f f f f( ) ( ) ( ) ( )t t t t+ + =M Q C Q K Q F  , (5.7) where subscript-f represents the parameters related to full order model of the machine tool. By applying the modal transformation:   f f( ) ( )t t= ⋅Q U Γ . (5.8) The modal displacement vector ( )tΓ  of the full order model is assumed to be equal to the one of the reduced order model presented in Equation (5.3). The Equation (5.7) is then reduced to the same equation as Equation (5.5):  2 Tn n f f( ) 2 ( ) ( ) ( )t t t t+ + =ζω ω U F Γ Γ Γ . (5.9) As long as the represented vibration modes are the same in reduced and full order models, their final modal equation would be the same (see Section 4.3 of Ref. [130]).  120  5.1.1 Reduced Dynamic Equation at the Non-cutting Zone The analysis of interaction between the tool and workpiece structure maybe required both at the cutting and non-cutting zones. The extended mode shape matrix eU  is defined to cover the non-cutting and cutting zone points:  enc =   UUU. (5.10) Similarly, the full order force vector f ( )tF  is renamed as extended force vector e ( )tF  and defined as follows:   enc( )( )( )ttt =   FFF. (5.11) If there are no external cutting forces acting in the non-cutting zone, the related force vector is set to zero, nc ( )t =F 0 , which yields to the extended form of Equation (5.9) as:  2 Tn n e( )( ) 2 ( ) ( )tt t t + + =   Fζω ω U0 Γ Γ Γ . (5.12) The mode shape matrix eU  is extended by including the contribution of the non-cutting zone to each existing mode. Moreover, more modes can be added to include the modes that do not contribute to the dynamic cutting force, which can then be used to predict the vibrations of the machine tool structure at the critical locations. The physical displacement vector then covers all cutting and noncutting zone points as:  e e( ) ( )t t= ⋅Q U Γ . (5.13) 5.1.2 Dynamic Interaction Between the Tool and Workpiece Structures  The cutting forces act on both tool and workpiece structures at the contact zone (Equation (5.5)): 121   2 Tt t n,t t n,t t t2 Tw w n,w w n,w w w( ) 2 ( ) ( ) ( );( ) 2 ( ) ( ) ( ),t t t tt t t t+ + =+ + = −ζ ω ω U Fζ ω ω U F  Γ Γ ΓΓ Γ Γ  (5.14) where the cutting force ( ( )tF ) acts both on the tool and workpiece with the same magnitude, but in opposite directions.. If q contacting points at the tool workpiece interface, with 4 flexibility directions at each point, are considered, the mass normalized mode shape matrices for the tool (tU ) and workiece ( wU ) are:   t tt tt ttw ww ww w1,1,t 1,n ,t 1,m ,tk,1,t k,n ,t k,m ,ttq,1,t q,n ,t q,m ,t 4q×m1,1,w 1,n ,w 1,m ,wk,1,w k,n ,w k,m ,wwq,1,w q,n ,w q,m ,w;    =          =      u u uu u uUu u uu u uu u uUu u u                      w4q×m. (5.15) The mode shape vector contributions at axial/radial segment-k on tn -th (tool) and wn -th (workpiece) modes are:   { }{ }t t t t tw w w w wTk,n ,t x,k,n ,t y,k,n ,t z,k,n ,t θ,k,n ,tTk,n ,w x,k,n ,w y,k,n ,w z,k,n ,w θ,k,n ,w;.u u u uu u u u=u =u (5.16) where x,y and z are the translation and θ is the torsional directions in the process coordinate system.  122  Natural frequency and modal damping ratio matrices are defined for each tool and workpiece for the corresponding number of vibration modes mt and mw (total vibration modes m=mt+mw):  t tt tt t t tw ww ww w w wn,1,t 1,tn,n ,t n ,tn,t tn,m ,t m ,tm ×m m ×mn,1,w 1,wn,n ,w n ,wn,w wn,m ,w m ,wm ×m m ×m0 0; ;0 00 0;0 0.ω ζω ζω ζω ζω ζω ζ= == =                                                ω ζω ζ      (5.17) Reference coordinate directions of tool and workpiece are set to be the same. The equation of motion (Equation (5.14)) is combined in one matrix form as:  2 Tt n,t n,t t tt t2 Tw n,w ww w n,w w2 ( )( ) ( )( )2 ( )( ) ( )tt tttt t            + + =           −       ζ ω 0 ω 0 UF0 ζ ω 0 ω U  ΓΓ ΓΓΓ Γ. (5.18) The tool and workpiece coordinates are combined in short but general form as:   2 Tc,i c n,c c,i n,c c,i i2+ + =ζ ω ω U F Γ Γ Γ , (5.19) where i i( )t=F F  is the force vector at time ti , and the combined tool and workpiece modal coordinate vectors are:   t i t i t ic,i c,i c,iw i w i w i( ) ( ) ( ); ;( ) ( ) ( )t t tt t t    = = =            Γ Γ ΓΓ Γ ΓΓ Γ Γ, (5.20) with the corresponding combined modal parameter matrices:  n,ttc n,cn,ww x x;m m m m  = =      ω 0ζ 0ζ ω0 ω0 ζ. (5.21) 123  The combined mass normalized mode shape matrix is defined as:  [ ]t w= −U U U . (5.22) 5.1.3 Cutting Force Representation The structural response can be analyzed by considering the static csif , dynamic cdif , delayed djkif , ploughing esif  and process damping edif  force vector components at time ti individually, as is shown in Equation (5.19):  ( )[4( 1) x ],cs cd d es edi i i c,i jki[4x ] c i jki i i c,i,[4( ) x ]k mN qmj kq k mt T−−      + − ⋅ − + +       ∑0F f f f f f0Γ Γ Γ=   (5.23) The static component csjkiF  of the shearing force at k-th axial/radial segment can be localized from Equation (4.60) as:  [ ][4( 1) x 4],cs csi 1,ki 2, jki 3, jki [4x 4] jki4 x1,[4( ) x 4]kN qqj kq kg g g−−  =     ∑0f I F0  (5.24) For the most general case, the tooth passing delay Tjki is a function of both geometry and time. From Equations (4.61) and (4.62), the local delayed force term at the element Sjk and time ti can be expressed as:  [4( 1) x ],cd di jki[4x ],[4( ) x ]k mN qmj kq k m−−   =   ∑0f f0; (5.25) [ ][4( 1) x ] [4( 1) x 4]d djki[4x ] 1,ki 2, jki 3, jki [4x 4] jki [4x 4( 1)] [4x 4] [4x 4( ))] t w[4( ) x 4][4( ) x ]k m km k q kq kq k mg g g− −− −−−         = −           0 0f I F 0 I 0 U U00.(5.26) 124  The ploughing term is derived from Equation (4.63):  [ ][4( 1) x 4],es esi 1,ki 2, jki 3, jki [4x 4] jki4 x1,[4( ) x 4]kN qqj kq kg g g−−  =     ∑0f I F0. (5.27) The process damping component is obtained from Equation (4.66):  [ ][4( 1) x 4],pedi [4x 4] [4x 4( 1)] [4x 4] [4x 4( ))] t wjki,[4( ) x 4]kN qk q kj kq k−− −−   = −      ∑0f I F 0 I 0 U U0. (5.28) The force components are for a general structure with axially and radially distributed delay and dynamics. The forces are later projected to the desired locations in special machining applications with multifunctional cutter, double milling cutter and parallel turning tools.  The generalized linear dynamic equation of Equation (5.19) is solved in time-domain using a variation of the discrete time method developed by Stepan et al.[81,131]. The stability of cutting operations, cutting forces, torque and power, structural vibrations, and surface deflection errors are predicted in discrete time intervals. Only the delayed states are approximated, while the other time terms are kept continuous [81].  5.2 Solution of the Equation of Motion in Modal Domain The general equation of motion represented in Equation (5.19) is converted into state space form to apply the time-marching solution method. An unconditionally A-stable 4th order method is used for solving the time history [132]. Delayed state vector is approximated using 4th order Lagrange polynomials. The convergence rate of the solution method is checked by testing the increase in eigenvalue accuracy versus the decrease in sampling time size.  125  5.2.1 State space representation The second-order dynamic equation of motion Equation (5.19) is reduced to first-order form by using two states. Here, displacement c ( )tΓ   and velocity c ( )tΓ  states are used. The modal state vector of the tool-workpiece system at a time t is:  t wcc 2( ) 1( )( )( )m mttt + × =   ΘΓΓ. (5.29) Equation (5.19) is then written in state-space form in continuous time domain as:  { },0 i 3 1 2,jk jk,( ) ( ) ( ) ( ) ( ) ( ( )) ( )N qj kt t t t t t T t t−= + − − +∑Θ A Θ B Θ B Θ S , (5.30) where the 0A  is constant matrix, 1( )tB  and 2, jk ( )tB  are the time dependent current and delayed state matrices, and ( )tS  is static force vector with process damping. The state matrices are derived from Equation (5.19) as,  t wt wt w2( )t w2( )t wTt0 cs es22( )n n T2( ) w 2( ) 1T Tt t1 cd edT Tw w2, jk( ) ;( ( ) ( ))d2( ) ;( )d ( )d( )m mm mm mm mm mtt t ztt z t zt+ ×++ ×++ ×     = =      +− −      −        =         − −     =00 IUA ; Sf fω ζωU0 0U UBf fU U0 0B 2( )t w2( )t w[4( 1) x ]Tt djki[4x ]Tw[4( ) x ].m mm mk mmq k m+ ×+−−           ⋅    −       0Uf 0U0  (5.31) 126  5.2.2 General Solution Method in Semi Discrete-Time Domain Following the method presented by Aguilar [132], the modal state vector ( )tΘ  is solved in t∆  discrete time intervals in three intermediate steps within time frame of 3i it t t− ≤ ≤  for s4,7,..., 1i n= + . The solution is carried out until the spindle period index sn  reached for the general cases, but shorter (regular pitch milling tools) or longer (multi-functional drills) periods are used in special cases. The start time of each interval is 3 1 ( 4)it t i t− = + − ⋅∆  with initial time 1t , and the final time of the simulation for one spindle period is sn 1 1 s( )t t n t+ = + ⋅∆ .The solution in each time interval is [81]:  ( ) { }00 33,( )3 1 2, jk jk,( ) e ( ) e ( ) ( ) ( ) ( ( )) ( ) diit N qtt tij ktt t Tτ τ τ τ τ τ τ τ−−−−− = + − − +   ∑∫AAΘ Θ B Θ B Θ S   (5.32) Sampling time t∆  is selected with respect to the vibration wave with the highest natural frequency of the combined tool-workpiece structure. The sampling time is set to:  ( )n,cπ3 maxt∆ =⋅ ω. (5.33) The integral part of Equation (5.32) is approximated by using two intermediate states, i.e. at 1i −  and 2i −  [132]. Since 0 3( )e it t −⋅ −A  part can be computed analytically, Equation (5.32) is put into following form:  ( ){ }00 33( )3,1 2, jk jk,( ) e ( ) e ( )d ;( ) ( ) ( ) ( ) ( ( )) ( ).iittt titN qj kt tTτ τ ττ τ τ τ τ τ τ−−−⋅ −−− == − − +∫∑AAΘ Θ hh B Θ B Θ S  (5.34) The following three equations are obtained by applying unconditionally A-stable, fourth-order integration method based on Simpson’s rule [132]: 127   ( )( ) ( ) ( )( )10 i 1030 i 1 i 3 0 i 1 i 2 0 i 1 i 10 i022i 1 i 3i 3 i 2 i 12i i 2( ) e ( ) e ( )d                                  e ( ) 4e ( ) e ( ) ;3( ) e ( ) e ( )d       iiiittttt t t t t tttttt ttt t tt tτττ ττ τ−−−− − − − − −−−⋅ ∆− −− − −− − −−⋅ ∆−− =∆  ≅ + +  − =∫∫AAA A AAAΘ Θ hh h hΘ Θ h( ) ( ) ( )( )( ) ( )0 i i 2 0 i i 1 0 i i0 i030 i i 3 0 i i 2i 2 i 1 i3i i 3i 3 i 2                        e ( ) 4e ( ) e ( ) ;3( ) e ( ) e ( )de ( ) 3e ( ) .3                               8iit t t t t tttttt t t ttt t tt tt ttτ τ τ− −−− −− − −− −−⋅ ∆−− −− −∆  ≅ + +  − =+ +∆≅∫A A AAAA Ah h hΘ Θ hh h( ) ( )0 i i 1 0 i ii 1 i...... 3e ( ) e ( )t t t tt t−− −−   + + A Ah h  (5.35) By substituting i 1 i 2( ), ( )t t− −h h  and i 3( )t −h  from Equation (5.34) at discrete-time intervals, the states become: ( )0 000 0jk,i-3 jk,i-2 jk,i-10021,i-3 i-3 1,i-2 i-2 1,i-1 i-12i-1 i-3 22,i-3 i-3,T 2,i-2 i-2,T 2,i-1 i-1,T2i-3i-2 i-1e 4e-e ...3 e 4ee... ;3 4 et ttt ttttt⋅ ∆ ⋅∆⋅ ∆⋅ ∆ ⋅∆⋅ ∆⋅∆ + + −∆  = + +   +∆  + + ⋅ + A AAA AAAB Θ B Θ B ΘΘ ΘB Θ B Θ +B ΘSS SΘ{ }0 000 0jk,i-2 jk,i-1 jk,i0021, 2 2 1, 1 1 1,2,i i-2 22,i-2 i-2,T 2,i-1 i-1,T 2,i i,T,2i-2i-1 ie 4e-e ...3 e 4ee... ;3 4 et ti i i i i itN qt tj ktttt⋅ ∆ ⋅∆− − − −⋅ ∆⋅ ∆ ⋅∆⋅ ∆⋅∆ + + − ∆ = +  + +      +∆  + ⋅ + ∑A AAA AAAB Θ B Θ B ΘΘB Θ B Θ B ΘSS S{ }00 0 00 0 0jk,i-3 jk,i-2 jk,i-1 jk,i3i i-33 21,i-3 i-3 1,i-2 i-2 1,i-1 i-1 1,i i,3 22,i-3 i-3,T 2,i-2 i-2,T 2,i-1 i-1,T 2,i i,T,-ee 3e 3e +38 e 3e 3e  tt t tN qt t tj kt⋅ ∆⋅ ∆ ⋅ ∆ ⋅∆∆ ∆ ∆= + + − ∆  +  + +     ∑AA A AA A AΘ ΘB Θ B Θ B Θ B ΘB Θ B Θ B Θ +B Θ0 003 2i-3 i-2i-1e 3 e3                                ...+ ,8 3 et ttit⋅ ∆ ⋅ ∆⋅∆ + ⋅ +∆   ⋅ + A AAS SS S (5.36) 128  where the notation is simplified for current terms, e.g., 1,i 1 1 i 1( )t− −=B B , 2,jk,i-3 2,jk i 3( )t −=B B , and are represented as follows for the delayed terms:  jkijk,i-1jk,i-2jk,i-3,2,i i,T 2,jk i i jk i,,2,i-1 i-1,T 2,jk i-1 i-1 jk i-1,,2,i-2 i-2,T 2,jk i-2 i-2 jk i-2,2,i-3 i-3,T 2,jk i-3 i-3 jk i-3( ) ( ( ));( ) ( ( ));( ) ( ( ));( ) ( ( ))N qj kN qj kN qj kjt t T tt t T tt t T tt t T t= −= −= −= −∑∑∑B Θ B ΘB Θ B ΘB Θ B ΘB Θ B Θ,,.N qk∑  (5.37) The number of discrete states (discretizations) is an integer multiple of 3. If motion in one spindle period (one rotation of cutter or workpiece) is simulated, the number of discretizations (ns) is:  ss 3 int 3Tnt = ⋅  ∆ , (5.38) where sT  is the period of one spindle rotation. Since the division inside int operator may not be exact, sampling time is updated as: s s/t T n∆ → . For a common (called general tool system here) application, the maximum delay occurring in the system is expected to be periodic at every spindle rotation (i.e., turning), which would be equal to the maximum delay period as:  ( )max jk smax ( ); 1 ; 1 ; 0T T t j N k q t T= = = =   , (5.39) which gives the spindle period delay index as:  ss ceilTnt =  ∆ . (5.40) Depending on the application, the maximum delay period may vary from the tooth passing period (i.e., regular pitch milling) to an infinite number. However, assuming that all parameters 129  of the operation are periodic at each spindle rotation, the maximum delay angle is set to 2π . Each of the delayed state vectors, jk,i-3i-3,TΘ , jk,i-2i-2,TΘ , jk,i-1i-1,TΘ and jkii,TΘ  are approximated using five neighbouring points, as is shown in Figure 5.1. The delay state index mjki of element Sjk at time ti corresponding to each delay period is set as:  jkijki ceilTmt =  ∆ , (5.41) where the input of ceil function is rounded off to the closest larger integer value. The maximum value (n) of the delay state index is:  ( )jki smax ; 1 ; 1 ; 1 +1n m j N k q i n= = = =   . (5.42) The states of current period are linked to states of previous tooth period through a state transition equation. The intermediate states ( s4,7,..., 1i n= + ) is expressed from Equation (5.36) in general form:  i-3 jk i-3i , i-2 jk i-21,i i-1 2,i i-3 3,jki 4,i, i-1 jk i-1i-2i jk i( ( ))( ( ))( ( ))( ( ))N qj kt T tt T tt T tt T t−    −  ⋅ = ⋅ + ⋅ +   −     − ∑ΘΘ ΘC Θ C Θ C CΘΘΘ, (5.43) where time- and geometry-dependent matrices are:   [ ][ ]0 1,i-2 1,i-1 2 x21, 0 1,i-2 0 1,i-1 1,i0 1,i-2 0 1,i-1 1,i3 2 x3 24exp( ) I-3 34exp(2 ) I exp( ) I ;3 3 39 9 3exp(2 ) exp( ) I8 8 8m mim mt ttt t tt tt t tt t⋅ ⋅∆ ∆ − ∆  ∆ ∆ ∆  = − ∆ + − ∆ −    ∆ ∆ ∆ − ∆ − ∆ −  A B B 0C A B A B BA B A B B  (5.44) 130   Figure 5.1 Interpolation of the delay state using five neighboring states.  [ ]0 1,i-32,i0 1,i-33 2 x2exp(2 ) I30 ;3exp(3 ) I8 m mtttt⋅ ∆ ∆ +     = ∆  ∆ +    A BCA B  (5.45) [ ][ ]0 00 00 0 022,jk,i-3 2,jk,i-2 2,jk,i-1 2 x222,jk,i-2 2,jk,i-1 2,jki2 x23 2 22,jk,i-3 2,jk,i-2 2,jk,i-1 2,jki3,jki4- e - e -3 3 34- e - e -3 3 33 9 9 3- e - e - e -8 8 8 8t tm mt tm mt t tt t tt t tt t t t∆ ∆∆ ∆∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆∆ ∆ ∆ ∆ =  A AA AA A AB B B 00 B B BB B B BC[ ]3 2 x 4 2m m⋅ ⋅; (5.46)  [ ][ ][ ][ ]0 i-3 0 i-2 i-14,i 0 i-2 0 i-1 i0 i-3 0 i-2 0 i-1 i3 2 x 1exp(2 ) 4 exp( )3exp(2 ) 4 exp( ) .33exp(3 ) 3 exp(2 ) 3 exp( )8 mtt ttt ttt t t⋅∆ ∆ + ⋅ ∆ +  ∆ = ∆ + ⋅ ∆ +  ∆ ∆ + ⋅ ∆ + ⋅ ∆ +  A S A S SC A S A S SA S A S A S S  (5.47) All the state transitions within one spindle period Ts is merged in one matrix equation as:   ,i 1,i-3 2,jk,i-3 i-3 i-3,N qj k = + ⋅ +   ∑D D Eϒ ϒ , (5.48) 131  where the state vector iϒ , previous state vector i-3ϒ and static force vector i-3E  at previous state are defined with an inclusion of all the states within the maximum delay period (the spindle period for the general case): jkijkijkijkijkii-n-4i-n-3i-m -5i-n-11i-n i-m -4 1,ii i-3 i-3i-3i-m +2i-2i-m +3i-1i-m +4i [2 ( 2)] x 1i-4i-3 [2 ( 2)] x 1; ;m nm n−++                    = = =                       ΘΘΘΘΘ Θ CEΘ ΘΘΘΘΘΘΘΘϒ ϒ( )4,i[2 3] x 1[2 ( 1)] x 1[2 ( 2)] x 1mm nm n⋅−+           C0. (5.49) Maximum delay index (n) may be different than the spindle period index ns in some cases. However, for the general case, the maximum delay is set to spindle period: sn n= . In Equation (5.48) the constant matrix is:  ( )1[2 3]x 1,i 2,i [2 3]x[2 ( 1)] [2 1]1,i-3[2 ( 1)x [2 ( 1)]x[2 ( 1)]2 3][2 ( 2)]x[2 ( 2)]m mm n mm n m n m nmm n m n−⋅ ⋅⋅ + ⋅− − −⋅+ +      =      0 C CD0 I. (5.50) 132  Before writing the matrix 2,jk,i-3D  related to the delay states, the delay states vector has to be expressed in terms of the previous state vector ( i-3ϒ ), and by means of using an approximated weight function matrix jkiW :  [2 4x 2 ( 2)] [2 ( 2)] x 1i-3 jk i-3i-2 jk i-2jki i-3i-1 jk i-1i jk i [2 4] x 1( ( ))( ( ))( ( ))( ( ))m m n m nmt T tt T tt T tt T t⋅ + +⋅−  − = ⋅ −  − ΘΘWΘΘϒ . (5.51) The matrix related to delay states is then expressed as follows:   ( )11,i 3,jki jki [2 3]x[2 ( 2)]2,jk,i-3[2 ( 1)x 2 ( 2)][2 ( 2)]x[2 ( 2)]mm nm n m nm n m n−⋅⋅ +− ++ +     =       C C WD0. (5.52)  Fourth-order Lagrange interpolation for the delay state: The interpolation of delay states in Equation (5.51) is carried out by a 5-point (4-th order) Lagrange interpolation scheme (Figure 5.2a). All of the delay states ( i jk i( ( )),t T t−Θ  i-1 jk i-1( ( )),t T t−Θ  i-2 jk i-2( ( ))t T t−Θ  and i-3 jk i-3( ( ))t T t−Θ ) are approximated similarly. A special case is shown in Figure 5.2b when the delay corresponds to the maximum delay of the operation. There are five points needed around the delay state. Thus, to account for this special case, (i-n-4)-th state is added to the calculations.  133   Figure 5.2 The neighboring states and the Lagrange- 4-th order interpolation: (a) xr values from reference state (i-mjki-1); (b) a special case for maximum delay term. As illustrated in Figure 5.1 and Figure 5.2a, the delay state i jk i( ( ))t T t−Θ  is approximated using the five neighbouring discrete states from jkii m 1− −Θ  to jkii m 3− +Θ  using the weighting function matrix jkiw  at time ti:  jkijkijkijkijki2 x 2 2 x 2 2 x 2 2 x 2 2 x 2i m 1i mi m 1i jk i jki 2 x(2 5)i m 2i m 3jki 1,jki 2,jki 3,jki 4,jki 5,jki( ( ))m m m m m m m m m mm mt T t− −−− +⋅− +− +     − = ⋅          =  ΘΘΘΘ wΘΘw w w w w w. (5.53) Each delay state vector is assumed to have the same weight in modal space, as: 134   [ ]r,jki r,jki 2 x2m mw= ⋅w I , (5.54) where r=1,2…5, and [ ]2 x2m mI is identity matrix with size 2m. For each modal state of i jk i( ( ))t T t−Θ , the weight terms r,jkiw  at time ti for the element Sjk are set as:  5L4 jki r,jki r1( ) ( )ry x w f x=≅ ∑ , (5.55) where the function r( )f x  takes on the values at the neighboring delay states (from jki 1i m− −  to jki 3i m− + ). The approximate value L4 jki( )y x  of the delay state corresponds to each modal state of i jk i( ( ))t T t−Θ . In Figure 5.2a, when the approximation is shown by taking the (i-mjki-1)-th state as a reference, the Equation (5.55) is expressed using the Lagrange 4-th order interpolation as:  55 jki sL4 jki r1 1 r s5 r,jkiL4 jki r1 r( ) ( ) ;( ) ( ).r ss rrx xy x f xx xcy x f xd= =≠=−≅−≅∑ ∏∑  (5.56) The weights are r,jki r,jki rw c d=  , where:  5 5r,jki jki s r r s1 1;s ss r s rc x x d x x= =≠ ≠= − = −∏ ∏ . (5.57) jkiT  is the instantaneous delay period for the k-th segment of j-th cutting edge. Because of the the effects of runout and spindle speed modulation, the delay period may be time-dependent. Also, because of the variable geometry of the cutter, the delay may also depend on the cutting edge and segment. For the reference State-1: jkiref i m 1t t − −= , the corresponding relative times ( rx  135  values in Figure 5.2a) of Equation (5.57) are found by subtracting the reference state from the actual values:  jki1 i m 1 ref 12 1 23 1 34 1 45 1 5; 0;; ;2 ; 2 ;3 ; 3 ;4 ; 4 .x t t xx x t x tx x t x tx x t x tx x t x t− −= − == + ∆ = ∆= + ∆ = ∆= + ∆ = ∆= + ∆ = ∆  (5.58) Since the time interval is fixed, the time delay values are also fixed. From Equation (5.57):  ( )( )( )( )( )( )( )( )( )( )541 1 s 1 2 1 3 1 4 1 5132 41542 2 s 2 1 2 3 2 4 2 512 3253 3 s 3 1 3 2 312324 ;6 ;st tt tsst tt tsst tsd x x x x x x x x x x td x x x x x x x x x x td x x x x x x x=−∆ − ∆− ∆ − ∆≠=∆ − ∆−∆ − ∆≠=∆ ∆≠= − = − − − − = ∆= − = − − − − = − ∆= − = − −∏∏∏(( (((( (((( (((( ((((((( )( )( )( )( )( )( )( )( )( )44 3 52544 4 s 4 1 4 2 4 3 4 513 24545 5 s 5 1 5 2 5 3 5 414 3 254 ;6 ;24 .t tst t t tsst t t tsx x x td x x x x x x x x x x td x x x x x x x x x x t−∆ − ∆=∆ ∆ ∆ −∆≠=∆ ∆ ∆ ∆≠− − = ∆= − = − − − − = − ∆= − = − − − − = ∆∏∏((((((((((((((((((((  (5.59) Neighboring states are chosen such that delay state falls between 2-nd and 3-rd states (Figure 5.2a). Relative time of delay state is:  jki jki jki( 1)x m t T= + ∆ − . (5.60) From Equation (5.57) the coefficients of weight function for cutting edge element jkS  at time ti are:  5r,jki jki s1ss rc x x=≠= −∏ , (5.61) 136  where differences with time reference are:   jki 1 jki jki jki 1 jki jkijki 2 jki jki jki 2 jki jkijki 3 jki jki jki 3 jki jkijki 4 jki jkijki 5 jki( 1) 0; ( 1) ;( 1) ; ;( 1) 2 ; ( 1) ;( 2) ;( 3x x m t T x x m t Tx x m t T t x x m t Tx x m t T t x x m t Tx x m t Tx x m − = + ∆ − − − = + ∆ −  − = + ∆ − − ∆ − = ∆ −  − = + ∆ − − ∆ − = − ∆ − − = − ∆ −− = − jki) .t T∆ −  (5.62) Putting Equations (5.59), (5.61) and (5.62) into Equation (5.56) gives the approximate value for the modal states p ( )tσ  (for p=1,2…2m) of the delay state vector ( )tΘ  in Equation (5.53):  ( ) ( )jki jki jki jki jki5p i jki r,jki p r1r,jki r,jki 1 i m 1 2 i m 3 i m 1 4 i m 2 5 i m 31p2m;/ ; ; ; ;  ; ;( )( )( ) ,( )rrt T w xw c d x t x t x t x t x tttttσ σσσσ=− − − − + − + − +− ≅ ⋅= = = = = =    =    ∑Θ  (5.63) which results in Equation (5.53):  ( ) ( )5i jki r,jki r1rt T x=− ≅ ⋅∑Θ w Θ . (5.64) Similarly, the weighting functions at other time instants are found and other delay state vectors are formed as in Equation (5.53):  ( ) ( )( ) ( )( ) ( )5i 1 jki r,jk,i-1 r15i 2 jki r,jk,i-2 r15i 3 jki r,jk,i-3 r1;2 ;3 .rrrt T x tt T x tt T x t−=−=−=− ≅ − ∆− ≅ − ∆− ≅ − ∆∑∑∑Θ w ΘΘ w ΘΘ w Θ  (5.65) The following expressions are written similar to Equation (5.53): 137   jk,i-1jk,i-1jk,i-1jk,i-1jk,i-1jk,i-2jk,i-2jk,i-2jk,i m 2i m 1i mi-1 jk i-1 jk,i-1 2 x(2 5)i m 1i m 2i m 3i m 2i m 1i-2 jk i-2 jk,i-2 2 x(2 5)i m( ( )) ;( ( ))m mm mt T tt T t− −− −−⋅− +− +− −− −− −⋅−     − = ⋅          − = ⋅ ΘΘΘΘ wΘΘΘΘΘΘ wΘi-2jk,i-2jk,i-3jk,i-3jk,i-3jk,i-3jk,i-3i m 1i m 4i m 3i m 2i-3 jk i-3 jk,i-3 2 x(2 5)i m 1i mjk,i-1 1,jk,i-1 2,jk,i-1 3,jk,i-1 4,jk,i-1 5;( ( )) ;m mt T t− +− −− −− −⋅− −−               − = ⋅         =ΘΘΘΘΘ wΘΘw w w w w w ,jk,i-1jk,i-2 1,jk,i-2 2,jk,i-2 3,jk,i-2 4,jk,i-2 5,jk,i-2jk,i-3 1,jk,i-3 2,jk,i-3 3,jk,i-3 4,jk,i-3 5,jk,i-3;;.   =   =  w w w w w ww w w w w w  (5.66) The delay state approximation is put into the general form as in Equation (5.51): By deriving the index of each delay state vector in the previous state vector ( i-3ϒ ). The first index of previous state vector is “i-n-4” and the final state is “i-3”. Thus, for example, for the vector in Equation (5.53), the start index is “i-mjki-1” and the final is “i-mjki+3”. Then, the first level of weight matrix jkiW  has  { }( )jkijki( 1) (i n 4) 2 ;3 2i m mn m m− − − − − ⋅− + ⋅  (5.67) preceding columns of zeros, and 138   { }( )jkijki(i 3) ( 3) 2 ;6 2i m mm m− − − + ⋅− ⋅  (5.68) succeeding columns of zeros after the delay approximation matrix jkiw . Similarly, the second level of weight matrix jkiW  has dimensions ( )jk,i-1 2 2n m m− + ⋅  and ( )jk,i-1 5 2m m− ⋅ . The third level has dimensions ( )jk,i-2 1 2n m m− + ⋅ , and the fourth level has dimenstions ( )jk,i-3 2n m m− ⋅  and ( )jk,i-3 3 2m m− ⋅  with preceding and succeeding columns of zeros before their corresponding delay approximation matrices, jk,i-1w , jk,i-2w  and jk,i-3w . The weight matrix jkiW  of the general form in Equation (5.51) is thus written as: ( ) ( )( ) ( )( ) ( )( ) ( )jk,i-3 jk,i-3jk,i-2 jk,i-2jk,i-1 jk,i-1jki jkijki2 x 2 2 x 3 2jk,i-12 x 1 2 2 x 4 2jkijk,i-22 x 2 2 2 x 5 2jk,i-32 x 3 2 2 x 6 2m n m m m m mm n m m m m mm n m m m m mm n m m m m m   − ⋅ − ⋅      − + ⋅ − ⋅      − + ⋅ − ⋅      − + ⋅ − ⋅   =0 w 00 w 0W0 w 00 w 0           . (5.69)  Time-domain stability simulation: The solution to the equation of motion of the tool-workpiece system with the defined cutting forces is:  i i-3 i-3 i-3= ⋅ +D Eϒ ϒ , (5.70) with  ,i-3 1,i-3 2,jk,i-3,N qj k= + ∑D D D . (5.71) The system is assumed to vibrate around its equilibrium state if the quasi-static 3i−E  terms in Equation (5.70) is dropped. Based on Floquet’s theory on periodically changing systems, 139  Insperger and Stepan [131] proposed a fast stability prediction for the equation of motion of machining operation. Depending on the periodicity of the system (tooth passing period for regular pitch cutters, spindle period for general tools), stability is predicted by checking eigenvalues of the state transition matrix of the system within its periodicity. To be consistent with general cutters, the periodicity is set as the spindle rotation period: s sT n t= ∆ . Thus, sn  time points are simulated in total:   s s s s si i-3 i-3i 3 i i i i-3 i-3i 6 i 3 i+3 i 3 i ii n i n 3 i n 3 i n 3 i n 6 i 3 i i .++ + ++ + − + − + − + − += ⋅= ⋅ = ⋅ ⋅= ⋅ = ⋅ ⋅••= ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅DD D DD D DD D D D Dϒ ϒϒ ϒ ϒϒ ϒ ϒϒ ϒ ϒ  (5.72) Since the geometry of the cutter is assumed general, and the spindle speed variation is assumed to have periodicity of spindle, the motion involves periodically changing parameters. Thus, the state transition matrix repeats itself after each spindle period: si i n+=D D . Floquet theory can be applied by evaluating the eigenvalues of period-transition matrix ( pΦ ) of the discrete system as:  s sp i n 3 i n 6 i 3 i+ − + − +Φ = ⋅⋅ ⋅ ⋅ ⋅D D D D . (5.73) If the dominant eigenvalue (with largest absolute value) is outside the unit circle, the operation is unstable. The operation is stable otherwise, i.e., if the eigenvalues are less than unity:  ( )( )p(unstable)1max eig 1 (stable)1 (marginally stable)>Φ = <=. (5.74) 140  The stability lobes are constructed by repeating the eigenvalue evaluation at the interested range of spindle speeds and dephts of cut.  Force/vibration simulation: The same scheme in Equation (5.72) is used for full time domain simulation of the system, which leads to the prediction of vibrations, forces, power, torque and surface errors of the machining operation. State transition matrices are minimized to the lowest possible size and is decided by the maximum delay index in the system, n  of Equation (5.42). For the general case, if periodicity of the system is the spindle rotation period, all the state transition matrices in one spindle/cutter/workpiece rotation (s si+n -3 i+n -6 i+3 i, , ⋅ ⋅ ⋅ ⋅D D D D  and s si+n -3 i+n -6 i+3 i, , ⋅ ⋅ ⋅ ⋅E E E E ) must be computed. Since these transition matrices repeat themselves after each spindle period, computing them for one rotation is sufficient. For the general case, the size of the transition matrix is set by the maximum delay state (n), and it is smaller than the spindle period index (ns). Similar to Equation (5.72), if the quasi-static component i 3−E  is included, the transition until the delay amount is as below:   ( ) ( )( )( )( )i i 3 i 3 i 3i 3 i i ii 6 i+3 i+3 i+3 i+3 i i i i+3 i+3 i i i+3 i i+3i 9 i+6 i+6 i+6 i+6 i+3 i i i+3 i i+3 i+6i+6 i+3 i i i+6 i+3 i i+6 i+3 i+6i+n*;;;;− − −+++= ⋅ += ⋅ += ⋅ + = ⋅ ⋅ + + = ⋅ ⋅ + ⋅ += ⋅ + = ⋅ ⋅ ⋅ + ⋅ + += ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ +=D ED ED E D D E E D D D E ED E D D D D E E ED D D D D E D E Eϒ ϒϒ ϒϒ ϒ ϒ ϒϒ ϒ ϒϒϒ( )i+n*-3 i+n*-3 i+n*-3 i+n*-6 i+3 i ii+n*-3 i+n*-6 i+3 i i+n*-3 i+n*-6 i+6 i+3 i+n*-3 i+n* i+n*-3...... .⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + +D D D D DD D D E D D D E D E Eϒ ϒ (5.75) Here, the spindle index ns term may not be an integer multiple of the delay index n term. Thus, as in Equation (5.75), modified delay “n*” term assures the integer number of transitions, by setting n* closest 3-multiplier of n while less than or equal to it: 141   * mod( ,3); *n n n n n= − ≤ . (5.76) The first part of the state equation Equation (5.75) is the state transition matrix, which is similar to Equations (5.72) and (5.73). The second part of the equation is the quasi-static component, and it is important for determining the resulting vibrations of the tool-workpiece system. Equation (5.75) represents the state transition with the maximum delay amount. Continuing the state transitions for the next cycles until the spindle period gives:  ( )i+n* i+n*-3 i+n*-3 i+n*-3 i+n-6 i+3 i ii+n*-3 i+n*-6 i+3 i i+n*-3 i+n*-6 i+6 i+3 i+n*-3 i+n* i+n*-3i+2n* i+2n*-3 i+2n*-6 i+n*+3 i+n* i+n*i+2n*-3 i+2n*-6...... ;......= ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + += ⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅D D D D DD D D E D D D E D E ED D D DD Dϒ ϒ ϒϒ ϒ( )i+3n*-3 i+3n*-6 i+2n*+3 i+2n* i+2n*i+3n*-3 i+3n*-6 i+2n*+3 i i+3n*-3 i+3n*-6 i+2n*+6 i+2n*+3 ii+n*+3 i i+2n*-3 i+2n*-6 i+n*+6 i+n*+3 i+2n*-3 i+2n* i+2n*-3i+3n* ......;⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ +⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + += D D D DD D D E D D D E DD E D D D E D E Eϒϒ( )+3n*-3 i+3n* i+3n*-3i+bn* i+bn*-3 i+bn*-6 i+(b-1)n*+3 i+(b-1)n* i+(b-1)n*i+bn*-3 i+bn*-6 i+(b-1)n*+3 i i+bn*-3 i+bn*-6 i+(b-1)n*+6 i+(b-1)n*+3i+bn*-3 i+bn* i+bn*-3;........+= ⋅ ⋅ ⋅ ⋅ ⋅ +⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ +++E ED D D DD D D E D D D ED E Eϒ ϒ( )s s ss s s s s s s s s ss i+n -3 i+n -6 i+n -bn*+3 i+bn* i+bn*i+n -3 i+n -6 i+n -n*+3 i+n -3 i+n -6 i+n -bn*+6 i+n -bn*+3 i+n -3 i+n i+n -3i+n ......;,i⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + +=D D D DD D D E D D D E D E Eϒϒ (5.77) where term b is for the last portion of the spindle rotation period. It is the largest multiple of modified delay term *n  while less than spindle period term sn :  sfloor( *)b n n= . (5.78) The floor function rounds the input down to the closest integer.   Algorithm for force/vibration simulation: The current state vector iϒ  of Equation (5.49) starts with the initial vector form for i=4. Initial condition vector for i-3ϒ  at i=4 is set to zero: 142   n1 n1 101 [2 ( 2)] x 1[2 ( 2)] x 1;m nm n−−++            = =           Θ 0Θ 0Θ 0Θ 0 ϒ ϒ . (5.79) Additionally, 4ϒ  is taken as zero initial condition, and all the states in 4ϒ  are set to zero:  3 n4 n4 434 [2 ( 2)] x 1[2 ( 2)] x 1;m nm n−−++            = =           Θ 0Θ 0Θ 0Θ 0 ϒ ϒ . (5.80) Assuming that the n* is not equal to ns, the first set of 3*n  transitions from 4ϒ  to 4+n*ϒ  (setting i=4 in Equation (5.77)) are set as:   st4 4+n*transition3 n 3 n+n*amount*4 n 4 n+n*33 3+n*4 4+n*1n− −− −            →            Θ ΘΘ ΘΘ ΘΘ Θ (( ((ϒ ϒ. (5.81) If the delay term n and modified delay term n* are the same, only the last two state vectors 4ϒ  ( 3Θ  and 4Θ ) are carried forward to be the first two state vectors of the 4+n*ϒ . In general,   4 (3 *) 1;* 2n nn n− − + +− +  (5.82) number of state vectors (at least 2, for *n n= ) are carried forward. This is important information when the state vectors are stitched to each other after each transformation. The rest of the transformations continue until the last one, as: 143   thst nd4 4+n* 4+2n*2transition transition 3 n+3 n 3 n+n* 3 n+2n*amount amount* *4 n 4 n+n* 4 n+2n*3 33 3+n* 3+2n*4 4+n* 4+2n*1n nb−− − −− − −                    → → → →                    ΘΘ Θ ΘΘ Θ ΘΘ Θ ΘΘ Θ Θ  (( (( ((ϒ ϒ ϒ 4+bn*bn*4 n+bn*3+bn*4+bn*−        ΘΘΘ((ϒ. (5.83) In total, * 3b n⋅  transformations are conducted from 4ϒ  to 4+bn*ϒ . The last transformation may be needed to complete the spindle period:  ( )sssssth4+bn*4+ntransition 3 n+n3 n+bn* amount* 4 n+n4 n+bn* 33+bn* 3+n4+bn* 4+n( +1)sn bnb−−− −−            →              ΘΘΘΘΘ ΘΘ Θ ((((ϒ ϒ. (5.84) For the final (b+1)-th transformation, last  ( ) ( )ss4 * 3 1;* 2bn n nbn n n+ − − + ++ − +  (5.85) state vectors of 4+bn*ϒ  are carried forward to the first state vectors of s4+nϒ . There is a special condition, for example, when maximum delay period is the same as the spindle period, or when the tool has single tooth: b=1 and n=n*=ns. In this case, the transformation step in Equation (5.84) is not needed, and only the 1st step in Equation (5.83) is conducted.  Finally, after stitching the consecutive state vectors ( 4ϒ , 4+n*ϒ ,.. 4+bn*ϒ , s4+nϒ ) to each other, overall modal displacement and velocity history in one spindle period is obtained: 144   ss34sp3 n4 n++    =    ΘΘΘΘΘ . (5.86) The rest of the simulation (after one spindle period) is carried out in a similar fashion by assignings3 n+Θ  and s4 n+Θ  as the initial condition of the next period. If P number of spindle rotations are simulated, the full history of the modal states are:   ss34fh3 Pn4 Pn++    =    ΘΘΘΘΘ . (5.87) From Equation (5.29), at each simulated time point ti from 3t  to s4 Pnt + , the relative deformations ( )itΨ  and velocities ( )itΨ  of the tool-workpiece system are found by back modal transformation of Equation (5.6), as:   [ ][ ]t wc iic i 2( ) 1t w c it w c i( )( ) ;( )( ) ( ) ;( ) ( ) ,m miitttt tt t+ × =   = − ⋅= − ⋅ΘU UU U ΓΓΨ ΓΨ Γ  (5.88) where the vector is composed of the vibration of each axial/radial segment: 145   [ ]1i1i1i1ikikii kikiqiqiqiqi 4  x 1( )qxyzxyt zxyzθθθ                     =                        Ψ . (5.89) The individual tool and workpiece deformation and velocity of the vibration are:   t t c i w w c it t c i w w c i( ) ( ); ( ) ( );( ) ( ); ( ) ( ).i ii it t t tt t t t⋅ − ⋅⋅ − ⋅Q = U Q = UQ = U Q = U  Γ ΓΓ Γ (5.90) Cutting forces (including the torque) along the contact zone is found at each time step it  from Equation (5.23), as:  ( )[4( 1) x ],cs cd d es edi i i c,i jki[4x ] c i jki i i c,i,[4( ) x ]( )k mN qmj kq k mt t T−−      + − ⋅ − + +       ∑0F f f f f f0Γ Γ Γ= . (5.91) At time ti, the approximation of delay term ( )c i jkit T−Γ  in Equation (5.91) is determined similarly as in Equation (5.53): 146   ( )[ ]jkijkijkijkijkix x x x xc,i m 1c,i mc,i m 1c i jki h,jki x( 5)c,i m 2c,i m 3h,jki h1,jki h2,jki h3,jki h4,jki h5,jkihr,jki r,jki x;;,m m m m m m m m m mm mm mt Tw− −−− +⋅− +− +     − = ⋅          =  = ⋅ww w w w w ww IΓΓΓΓΓΓ (5.92) where the weight matrices ( h,jkiw  and hr,jkiw ) are half the size of the ones given in Equations (5.53) and (5.54). r=1,2…5 and [ ]xm mI is the identity matrix with size m. The weight terms r,jkiw  at time ti for the element Sjk are derived in Equation (5.63).   Surface location error simulation: The surface location error (SLE) is the difference between the desired surface left by the rigid motion of the tool and the actual surface left by the flexible motion of the tool [133]. It is meaningful to define it only for stable operations, since unstable cutting will result in exponentially increased surface roughness and the process would be unacceptable in production.  Figure 5.3 shows the SLE for the undercutting and overcutting cases of milling operation: Part surface is undercut (Figure 5.3a) if the intended amount of material left is larger than the actual amount of machined material; it is overcut (Figure 5.3b) if the intended removed material is less than the actual material removed. If the feed travel motion of the cutter is ignored, then the SLE depends on the radius and the normal (y0) component of the relative vibration ( )itΨ  of tool-workpiece system (Equation (5.88)). 147   Figure 5.3 Surface location error for the milling operation. (a) Undercut surface; (b) Overcut surface. (c) significant transient vibrations (data from [96]).  SLE is calculated in the steady state region. SLE is defined only at the tool workpiece contact at the start and exit of the operation. Then, the maximum y0 location of the cutting edge element Sjk is the distance from the center to the machined surface. If only the rigid body motion is considered, the desired surface distance d,ky  (at axial segment-k) from the center would be the maximum of all the teeth cutting in one rotation period in the steady state region:   ( )( )jk jki,d,kjk jki,up milling: max cos( )down milling: max cos( )j ij iRyRφφ= −, (5.93) where the instantaneous angular position jkiφ  is measured from the +y0 axis in CW direction; the radius (Rjk) of the element Sjk includes the radial runout ( r,jkε ). If the vibrations in y0-direction 148  are added, then the actual surface distance a,ky  from the center would be the maximum of all teeth during one rotation:  ( )( )( )jk jki ki,a,kjk jki ki,up milling: max cos( ) down milling: max cos( )j ij iR yyR yφφ += − +. (5.94) Then the surface location error ek at axial segment-k is defined as the difference between the desired and actual surface [104]:   k d,k a,ke y y= − . (5.95) A positive ek implies that the workpiece is undercut (Figure 5.3a), whereas a negative value shows that the workpiece is overcut (Figure 5.3b). As shown in example case Figure 5.3c, the transient vibration period when the cutter is entering the cut may introduce more error on the workpiece surface. This effect is more pronounced for high feed and high speed cutting operations, because the cutter may travel a long distance in the transient region. To be consistent, SLE is defined at the steady state vibration period of the cutting operation. .  Steady state vibrations: The instantaneous location angle information for each time step ti for the milling operation is stored during each revolution. For stable cutting, when sufficient amount of rotations are covered in the simulations, the vibrations reach steady state and will continue repeating the same motion profile.  Ding et al. [107] employed an analytical method to predict the steady state vibrations using the state transition equations. Assuming that the maximum delay is the same as the spindle period (n=n*=ns), Equation (5.75) is rewritten to cover all spindle period: 149   ( ) ( )( )( )( )si i 3 i 3 i 3i 3 i i ii 6 i+3 i+3 i+3 i+3 i i i i+3 i+3 i i i+3 i i+3i 9 i+6 i+6 i+6 i+6 i+3 i i i+3 i i+3 i+6i+6 i+3 i i i+6 i+3 i i+6 i+3 i+6i+n;;;;− − −+++= ⋅ += ⋅ += ⋅ + = ⋅ ⋅ + + = ⋅ ⋅ + ⋅ += ⋅ + = ⋅ ⋅ ⋅ + ⋅ + += ⋅ ⋅ ⋅ + ⋅ ⋅ + ⋅ +=D ED ED E D D E E D D D E ED E D D D D E E ED D D D D E D E Eϒ ϒϒ ϒϒ ϒ ϒ ϒϒ ϒ ϒϒϒ( )s s s ss s s s s s si+n -3 i+n -3 i+n -3 i+n -6 i+3 i ii+n -3 i+n -6 i+3 i i+n -3 i+n -6 i+6 i+3 i+n -3 i+n i+n -3...... .⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ++ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + +D D D D DD D D E D D D E D E Eϒ ϒ (5.96) Similar to Equation (5.73), the state transition over a spindle period is expressed by including the quasi-static terms:  ss ss s s s s s si+n p i sp i n 3 i n 6 i 3 is i+n -3 i+n -6 i+3 i i+n -3 i+n -6 i+6 i+3 i+n -3 i+n i+n -3;;.+ − + − += Φ ⋅ + ΦΦ = ⋅⋅ ⋅ ⋅ ⋅Φ ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ + +D D D D= D D D E D D D E D E Eϒ ϒ  (5.97) Setting the state vectors equal at the steady state response,  si+n i s= =ϒ ϒ ϒ , (5.98) simplifies Equation (5.97) and the steady state vector sϒ  can be written as:  ( ) 1s p s−= −Φ ⋅ΦIϒ   (5.99) The limitation of using Equation (5.75) to predict the surface error is the assumption that the number of delay terms and spindle period terms are equal. If the delay terms are less than the spindle period, Equation (5.77) could be used for obtaining a similar expression to Equation (5.97).  5.3 Simplified Solution Methods for Special Cases  Although the presented general solution covers all possible machining cases, the solution can be simplified for special cases such as milling with low radial immersion case, where the 150  cutter out-of-cut time is considerably large. The solution can also be simplified for milling with regular (uniform) pitch cutters.  5.3.1 Low Radial Immersion with Free Vibration Condition (e.g. Double-Sided Milling)  The cutter is not in-cut during the full period of the spindle rotation The formulation in this section is based on the methods presented by Bayly et al. [106] and Ding et al. [134].  At a time instant ti, if both g1,jk (element-available-to-cut) and g2,jki (edge-in-cut) are zero among the cutting edges Sjk, then the cutter is practically out-of-cut and experiences free vibrations. All the force terms in Equation (5.30) are dropped, leaving only the system dynamics related component:  0 i 3( ) ( )t t −=Θ A Θ , (5.100) which is the free vibration equation, and its solution is analytically obtained from Equation (5.32) as:  0 3( ) 3( ) e ( )it tit t−−−=AΘ Θ . (5.101) In order to have efficient simulation, the cutter location angle is adjusted such that during each delay period of the cutter, the cutter enters cut once and exits cut once.   Adjusting the cutter location angle: The cutter location angle is adjusted such that the first state (state-1) becomes the start point of the in-cut region. This first state is called id1. Then, the second state id2 is selected as the last state in the in-cut region. The third state id3 is the last out-of-cut state.  If the cutting during each tooth period is analyzed, the cutter is adjusted to have an in-cut region from id1 to id2-4, and an out-of-cut region from id2-3 to id3. If there is no id2 assigned, the 151  cutter is always in cut. There may be more than one id2 assigned, thus implying that the cutter goes into in-cut and out-of-cut more than once during one spindle rotation.   Step-1: State transition in the in-cut region: The in-cut region is defined from the state id1 to state id2-3. If a nonzero force is always acting on the tool-workpiece system, then the solution presented in Equation (5.70) is valid in this region (for i=id1,id1+3,…id2-3). From Equation (5.72) the state transition for the dynamic component is written as:  1 11 1 1 1 1 12 2 2 2 2 1 1 1id id -3 i-3id 3 id id id id -3 id -3id 3 id 6 id 6 id 6 id 9 id 3 id id;;.+− − − − − += ⋅= ⋅ = ⋅ ⋅••= ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ ⋅DD D DD D D D Dϒ ϒϒ ϒ ϒϒ ϒ ϒ  (5.102) The first step presented in Equation (5.102) is the same as the original method and no computational advantage exists. Rewriting the state transition gives a shorter form:  2 12 2 1 1id 3 F idF id 6 id 9 id 3 id;,−− − += ⋅= ⋅ ⋅ ⋅ ⋅ ⋅DD D D D Dϒ ϒ  (5.103) where FD  is the state transition matrix for the forced vibration period. In the second step, the  state id2-3 is linked to the state id3 in one single equation, thus reducing the computation time.   Step-2: State transition in the free vibration region: The first step involves the state transition (from 1idϒ  to 2id 3−ϒ ) in the forced vibration region and is the same as the original procedure. Second step employs Equation (5.101) to link the state vectors of the free vibration region (from 2id 3−ϒ  to 3idϒ ): 152   3 2id f id 3−= Dϒ ϒ , (5.104) where fD  is the transition matrix in the noncutting duration of time. The state vectors are from Equation (5.49):   3 22332323 22 32 33 23 2id n 1 id n 4id n 3id nid n 2id 4id n 2id 3id id 3id 2 id n 1id 1 id nid 1 id 4id id 3− − − −− −−− −−− −−−− − −− −− −−                           = =                           Θ ΘΘΘΘΘΘΘΘ ΘΘ ΘΘ ΘΘ Θ ϒ ; ϒ . (5.105) In Equation (5.105), the states above the dashed line of 3idϒ  are same as the states below the dashed line of 2id 3−ϒ . These states are related by using identity matrix:   3 33 32 22 2id n 1 id n 1[2 2 ]id n id n[2 2 ][2 2 ]id 4 id 4[2 2 ]id 3 id 3mx mmx mmx mmx m− − − −− −− −− −                =                    Θ ΘIΘ ΘIIΘ ΘIΘ Θ , (5.106) where each [2 2 ]mx mI  is a 2m-by-2m identity matrix. Free vibration states starts from id2-3 until the state id3. Initial state vector 2id 3−Θ  is transferred to other states in the free vibration region by the matrix:  153   ( )( )( )020220 3 230 3 23id 22id 1id 3id id 2id 1id id 3ideeeetttt∆−∆−−− + ∆−− + ∆         = ⋅            AAAAΘΘΘΘΘ . (5.107) Thus, combining the matrices of Equations (5.106) and (5.107) gives the transition matrix of Equation (5.104) in the free vibration region:  332202233id n 1id nid 4id 3id 2id 1id 1id         eet− −−−−∆−−−                                                  =                                     AΘ  I   IΘ0ΘIΘΘΘ 0ΘΘ( )23303020 3 22id n 4id n 2id n 12id n3id 4id id 3id 3eettt− −− −− −∆−∆−− + ∆ −                        ⋅                                                        AAAΘΘΘΘΘΘ. (5.108)  Combining the forced and free vibration regions: After finding the forced vibration and free vibration state transitions, the total expression in one in-cut and out-of-cut period is found by putting Equation (5.103) into Equation (5.104):  3 1id f F id= ⋅ ⋅D Dϒ ϒ , (5.109) which gives the same result with Equation (5.73) for single in-cut and out-of-cut region pair. For single tooth cutting with low radial immersion, the period-state transition matrix would be:  p f FΦ = ⋅D D . (5.110) 154  The stability of the operation is described similar to Equation (5.74). If multiple in-cut and out-of-cut regions are involved during one rotation period, then the solution needs to be modified in order to continue the state transition from one region to another. The final state transition would look like:   ( ) ( ) ( ) ( )multiplep f,J F,J f,J-1 F,J-1 f,1 F,1regionsΦ = ⋅ ⋅ ⋅ ⋅D D D D D D , (5.111) where J is the total number of paired free-forced vibration periods. Equation (5.111) gives same result with Equation (5.110) for single region, J=1. 5.4 Summary This chapter presents the derivation of the equation of motion and its solution for general cutting operations. The presented model covers turning, boring, drilling and milling operations with arbitrary tool geometry and multiple tools. The chatter stability of general cutting operation, and the cutting states such as forces, vibrations and surface location errors can be predicted. The application of the generalized model on various practical machining operations is presented in the next chapter. 155  Chapter 6: Applications  6.1 Overview The proposed new, generalized mathematical model of machining processes can be used to model and predict the performance of various turning, boring, drilling and milling operations. In this chapter, the model is applied to several industrial machining operations. The validity of the proposed model is experimentally verified in milling with solid end mills, single and multiple indexable milling operations, drilling, boring, and in parallel machining operations. The validations are carried out by comparing the simulated and experimentally observed cutting forces, surface form errors, and chatter stability diagrams.  6.2 Application-1: Slender, Cylindrical Helical End Mill with Regular Pitch Angles [2] The parameters of the end mill is given in Table 6.1. The end mill has a long fluted section with varying flexibility along the tool-workpiece contact [64]. The runouts and the FRFs are measured along the tool axis. Radial runout is measured when the tool is on the spindle, by using a dial gage. In Table 6.2, minimum value of the measured runout values is normalized to zero. Runout of each element Sjk along the cutting edge is interpolated from the measured runout, and is assigned to measr,jkε . The generalized geometric, mechanics and dynamics modeling of the end mill is illustrated. The geometry is modeled by 15 parameters that are described in Section 3.1. The mode shapes along the tool axis are used to identify the distributed vibration amplitudes at cutting edge segments. The corresponding cutting forces, surface location errors and chatter stability conditions are predicted by the model with experimental validations.   156  Table 6.1 Information of the cylindrical end mill Type of end mill Regular pitch cylindrical end mill Manufacturer and model SGS Series 1EL-3/4 end mill Diameter (nominal) 19.05-mm (3/4-in) Length of fluted section 76.2-mm (3-in) Stick out from the shrink fit tool holder 110-mm Number of flutes N=4 Rake angle Not given by manufacturer Helix angle 30-deg (given by manufacturer) Cutting edge angle 90-deg Radial runout Only manufacturing error (measured data in Table 6.2, interpolated data in Figure 6.1c) Axial runout Negligible  Table 6.2 Radial runout measurement at five axial locations using a dial gage with 5-micrometer tolerance.  Axial distance from the tool tip (mm) 0 (tool tip) 9 19 29 39 Measured radial runout (micrometer) for all teeth (j=1,..4) j=1 5 0 10 20 17 j=2 19 10 7 0 4 j=3 33 28 21 15 3 j=4 22 25 29 22 13  Geometry: The end mill is axially discretized with dz=0.2-mm segments. Only 38 mm depth of the fluted section would be in cut. Therefore, the total number of axial segments yields to K=190:  38 38190d 0.2Kz= = = . (6.1) The end mill has four flutes (N=4). The 15 geometric parameters are defined at K locations of each flute. Each geometric parameter is represented by a column, and each level is represented 157  by a row. Thus, the whole geometry is stored in K-by-15N matrix (i.e., 190 x 15N ). Since all the cutting edges are available to cut, then g1 parameter is unity for all cutting edge elements:   1, jk 1 for all 1.. and 1..g j N k K= = = . (6.2) The axial location of each edge segment 0kz  (0k kz Z=  in Equation (3.2) is measured from the tooltip, and defined at the midpoint of the axial segment-k of the tool:   ( )k 0.5 dZ k z= − . (6.3) The angular location jkψ  (Figure 6.1a) of each cutting edge element Sjk is defined by the helix angle jβ , the axial location kZ  and the designed pitch angle pd,jφ :   ( ) kjk pd,j j1 tanπZjψ φ β= − − , (6.4) where the helix and pitch angles are uniform in this application:  j pd,j2π π30deg ; for all 1..2j NNβ φ= = = = . (6.5) The relative location angle reljkψ  (Figure 6.1b) is measured from the first S11 element in CW direction, as:   reljk jk 11ψ ψ ψ= − . (6.6) The nominal radius (R0) of all cutting edges is the same for the cylindrical end mill:   0 9.525 mm for all 1.. and 1..R j N k K= = = . (6.7) Radial runout along the cutting edge is found by linear approximation between the measurement locations and stored in measr,jkε  parameter. The highest measured radial runout value is 33-μm  at the 3rd tooth at the tool tip (Table 6.2). Assuming that the maximum radius is the 158  nominal radius (R0), the radius (Rjk) of each cutting edge element is found by normalizing the measured runout measr,jkε  to their maximum value. The radius Rjk (mm) at element Sjk is obtained as shown in Figure 6.1d:  ( )meas 3jk 0 r,jk33 10R R ε −= − − ⋅ . (6.8) The measured ( measr,kε ) and normalized ( r,kε ) radial runout vectors at axial segment-k are defined as:  meas meas meas measr,k r,1k r,jk r,Nkr,k r,1k r,jk r,Nk;.ε ε εε ε ε =   =    εε (6.9)     Figure 6.1 Geometry along cutting edge of cylindrical end mill: (a) Angular location; (b) relative angular location, (c) radial runout; (d) radius.  159  The normalized radial runout vector at each axial segment-k is set to have minimum value of zero, as:  meas measr,k r,k r,kmin( )j= −ε ε ε , (6.10) where the min( )j⋅  function takes the minimum of all teeth (j=1..N) values at axial segment-k (k=1..K). Figure 6.1c shows the radial runout ( r,jkε ) values after normalizing minimum value of each axial segment-k to zero. The locations ( 0jkx ,0jky ) of the cutting edge element Sjk is derived from the angular location jkψ  and the radius Rjk:   0 0jk jk jk jk jk jksin ; cosx R y Rψ ψ= ⋅ = ⋅ . (6.11) Each location for element Sjk is represented by point (0jkx ,0jky ,0jkz ) in the tool reference coordinate frame (Frame-0), as is shown in Figure 6.2.   Figure 6.2 Modelled 3D locations in the Frame-0 of cylindrical end mill. 160  The helical cylindrical end mill has pre-set true cutting edge angle ( *r,jkκ ), cutting edge inclination angle ( s,jkλ ) and normal rake angle ( n,jkγ ) for each element Sjk:  s,jk j s,jkn,jk*r,jk jk;  30deg;0deg;90deg;  for each element .Sλ β λγκ= === (6.12) The geometry matrix is formed according to the listed variables in Section 3.1. The cutting force algorithm directly uses this geometry matrix, along with two more instantaneous condition parameters, which are the edge-in-cut (g2,jki) condition from Equation (3.4) and the missed-cut (g3,jki) condition as explained in Section 3.1. If the cutter rotates at constant speed Ω  (rev/min), then the instantaneous angular position measured from +y0 axis is:   jki jk i2π60tφ ψ  = + Ω  . (6.13) Since the chip thickness is changing between a minimum and maximum value during the tool rotation, the effect of runout at each time instant (ti) is different. As a result, both the chip thickness and delay period become time-dependent. The cutting force is evaluated after calculating the instantaneous chip thickness using the feed per revolution and radial engagement of the tool.   Chip engagement calculation: For the 5% down milling case, the cutting edge element Sjk is supposed to cut if its instantaneous angular position ( jkiφ ) falls between the range of the entry and exit angles (Equation (3.4)). Engagement condition of element Sjk at axial segment-k is shown in Figure 6.3 for milling operation. Figure 6.3a shows the radial entry (ast,k) and exit 161  (aex,jk) distances from the cutter center in y0 direction. The engagement location depends on the maximum radius (Rmax,k), which is the maximum of the teeth radii (j=1..N) at segment-k:  ( )max,k jkmaxjR R= . (6.14) Effect of vibration in y0 is assumed negligible. The regular no runout case is shown in Figure 6.3a. The formulation is done for more general case with runout (Figure 6.3b). A 5% down milling geometry is shown in Figure 6.4. The radial runout also affects the cutter entry and exit angles due to the missed-cut (g3,jki) condition parameter (shown in Figure 6.4b).   Figure 6.3 Tool entry and exit parameters for milling operation: (a) No runout case; (b) Effect of runout.  Figure 6.4 (a) Schematics of 5% down milling. (b) In-cut and out-of-cut regions.  162  The entry and exit angles for element Sjk are evaluated at segment-k from Figure 6.3b and Figure 6.4a as:   st,kst,jkjkex,jkex,jkjkarccos ;arccos .aRaRφφ =     = −    (6.15) where, radial entry and exit distances (ast,k and aex,k) are the relative positions of the workpiece edges measured from the cutter center in y0 axis. For 5% down milling, the radial entry distance is 90% of the Rmax,k stated by tool envelope:   ( ) ( )st,k max,k max,k st,k max,k2 5% ; 0.9a R R a R = − − ⋅ = −  . (6.16) The entry distance ( st,ka ) is negative due to the definition in Figure 6.3a. The maximum radius at k-th segment is larger than or equal to the actual radius (Rjk) of element Sjk. Then, the radial exit distance for each element Sjk is equal to the actual radius:  ex,jk jka R= . (6.17)   Figure 6.5 Effects of radial tunout for 5% down milling of cylindrical helical end mill, on (a) tool entry angle; (b) Maximum effective chip load along the cutting edge.  163  The entry and exit angles are evaluated from Equation (6.15), as:  max,kst,jkjkjkex,jk ex,jkjkarccos 0.9 ;arccos ; 180deg.RRRRφφ φ = −    = − =    (6.18) The predicted cutter entry angle variation for 5% down milling is shown in Figure 6.5a. Cutter exit angle is 180-deg at all points along the cutting edge. The effect of the runout on the chip load can be seen from Figure 6.5b during one spindle rotation. Maximum effective chip load along the cutting edge (k=1..190) of each tooth (j=1..4) with dz=0.2-mm. In Figure 6.5b, for feed fr=1.0-mm, total effective chip load at each segment-k almost keeps constant (~0.42-mm). The algorithm presented in Section 4.1 is applied for the actual static chip thickness simulation. The edges j at each segment-k have different runout values. Therefore, each axial segment-k requires a separate analysis.  Analyses of edge-in-cut and missed-cut conditions are also combined in the simulation. For example, at 1.2-mm depth (k=6; kdz=1.2-mm), Figure 6.1a shows that the tooth j=3 has the highest runout, and tooth j=1 has the least runout. Hence, tooth j=3 cuts the most amount of chip whereas tooth j=1 cuts the smallest amount of chip. The flutes remove chips with the sequence of 4-3-2-1. In Figure 6.6, effect of runout on in-cut condition is simulated result for all teeth at the 1.2-mm depth (k=6). As tooth j=3 (Figure 6.6c) cuts during all instants in the engagement zone, j=4 (Figure 6.6d) has the second most cut amount, and teeth j=2 (Figure 6.6b) and j=1 (Figure 6.6a) cuts the least.  164      Figure 6.6 Zoomed tool-workpiece contact zone (from Figure 6.4b) at 1.2-mm axial depth, for tooth/edge numbers: (a) #1; (b) #2; (c) #3; (d) #4.  The effective chip load ( pa jkijkic γ− ) and the actual pitch angle p,jkiφ  for each tooth at 1.2-mm depth and fr=1.0-mm is presented for one spindle period in Figure 6.7a and Figure 6.7b, resepectively. As seen in Figure 6.7b, Tooth-4 has 270-deg pitch angle when exiting the cut. That is due to the material left from the previous teeth (j=1 and j=2). Tooth-3 cuts the most material. Tooth-3 also cuts the left portion by tooth-4 (shown in Figure 6.6d) and it gives a 360-deg pitch angle instantaneously.    165    Figure 6.7 Effect of runout on: (a) effective chip load; (b) actual instantaneous pitch angle.   Cutting force prediction with runout: Cutting coefficients in RTA-frame are identified mechanistically for this application [6] and listed in Table 6.3. All the true cutting edge angles *r, jkκ  along each cutting edge element Sjk are 90-deg (see Equation (6.12)). Then, transformation from the RTA (cutting edge coordinate frame) to Frame-R (insert coordinate frame) is constant, as from Equation (4.41):  RI1 0 00 1 00 0 1−  =   − T . (6.19) The transformation from Frame-R to tool reference coordinate frame (Frame-0) is written from Equation (4.42), as:  jki jkijki j0R, kijkisin cos 0cos sin 00 0 1φ φφ φ−  =    T , (6.20) where the instantaneous angular position is expressed (Equation (6.13)) for a constant rotation. Due to vibration free nature of static condition, the static cutting force (Equation (4.70)) is 166  assumed not to be affected by the vibrations of the tool-workpiece structure. The static differential force at each element Sjk along the depth of cut is:  st cs esjki jki jki= +F F F , (6.21) where the static component of shearing force can be written from Equation (4.60) as:  st0R, jki RI, jk jkics cjki rta,jki *t jk r,jk0 0dsin,hRzκ⋅ = ⋅ ⋅   T TF K . (6.22) The transformation RI, jkT  from RTA to Frame-R is constant at all points. The moment arm t jk,R  is the radius (Rjk of Figure 6.1d). The true cutting edge angle is *r,jk 90degκ =  for all points. The cutting coefficient vector ( crta,jkiK ) is constant at all points and time instants (given in Table 6.3). The axial segment thickness dz=0.2-mm and the actual chip thickness ( stjkih ) is found from Equation (4.2) for milling. Equation (6.22) is rewritten as:   0R, jki RIcs c stjki rta jkijk0 0dhRz⋅ = ⋅ ⋅ ⋅   T TF K . (6.23) The static ploughing component of Equation (6.21) in simplified form is evaluated from Equation (4.63):   0R, jki RIes ejki rtajkd0 0zR⋅ = ⋅ ⋅   T TF K , (6.24) where the edge force coefficient vector ertaK  is constant at all points and time instants (given in Table 6.3). Integration of the total cutting force acting at the tool-workpiece contact gives the total lumped cutting force iF  at each discrete time instant (ti) from Equations (4.72), (6.21), (6.23) and (6.24), as: 167  Table 6.3 Mechanistic identification of the milling operation. Tool information in Table 6.1.  Operation scheme 5% down milling Workpiece and cutter material Aluminum 7050 and uncoated carbide Spindle speed 12,700-rev/min Axial depth of cut 25-mm Number of sample constant feed tests  4 Feed per revolutions of each sample test 0.840, 0.920, 1.00 and 1.02-mm/rev Identified cutting coefficients  Krc=106, Ktc=696, Kac=344-MPa Identified edge force coefficients  Kre=13.4, Kte=12.4, Kae=0-N/mm Curve fitting accuracy R2=99%   , 0R, jki RI 0R, jki RIc st ei 1,jk 2,jki 3,jki rta jki rtajk jk,d0 0 0 0N qj kzg g g hR R  ⋅ ⋅     = ⋅ ⋅ + ⋅ ⋅              ∑T T T TF K K , (6.25) with the total x ( x,iF ),y ( y,iF ),z ( z,iF ) and torque (ciT ) components:   Tci x,i y,i z,i iF F F T =  F. (6.26) The total cutting force is simulated for 3 spindle rotation periods for the given conditions in Figure 6.9 at depth of cut 25-mm (ap=25mm, dz=0.2mm). Then, q=125 from Equation (4.73), spindle speed Ω=12700-rev/min, and feed per revolution fr=1.0-mm/rev. The sampling time is4.1442e-5[s]t∆ = . The spindle rotation period is:  3s60 604.7244 10  [s]12700T −= = = ×Ω. (6.27) Each spindle rotation has the following number of simulation points (ns), as:   ss s; 114Tn nt= =∆. (6.28) The time instant (ti) is defined for 3 spindle periods as:  i s s, 1, 2...,3 ,3 1t i t i n n= ⋅∆ = + . (6.29) 168      Figure 6.8 Dynamometer direct force transfer function:Magnitude plot; (b) Phase plot; (c) Magnitude response in the vicinity of spindle rotation frequency; (d) in the vicinity of tooth passing frequency.  The direct magnitude (Figure 6.8a) and phase (Figure 6.8b) response of Kistler 9257 table type force dynamometer in each x, y and z directions are measured from 0 to 1500-Hz (modal hammer sensitivity 2.38 mV/N). The force measurement is acceptable for the cutting force input under 600-Hz, since the magnitude response plot is close to unity, i.e., the distortion is negligible in this range (see Figure 6.8c). The cutting forces in machine coordinates are compared with the experimentally measured forces in Figure 6.9, for fr=1.0-mm; depth of cut=25-mm; 12700-rev/min spindle speed; 5% down milling; tool information in Table 6.1; Radial runout in Table 6.2; Cutting coefficients in 169  Table 6.3. Kalman filter of our laboratory’s software is used for removing dynamometer distortion from x and y forces [135].    Figure 6.9 Experiment vs. predicted forces with runout effect.    170  Spindle rotation frequency is 12,700/60=211-Hz and tooth passing frequency is 211*4=844-Hz, which is greater than the dynamometer’s 600 Hz bandwidth. As seen in x force plot of Figure 6.9, the collected data has distortion due to the 844-Hz tooth passing frequency. Thus, the fundamental harmonics related to the spindle rotation frequency (211-Hz) is captured accurately. However, as seen in Figure 6.8d, there is a heavy distortion mode around the tooth passing frequency (844-Hz) due to the response peak in x direction. In Figure 6.9, Kalman filter is used to remove the distortion modes from the force in x and y directions [135]. Although the filter in x direction gives a clear output, the y direction filter does not change the results as much. The reason is that the distortion in y-direction is around 1055-Hz (see Figure 6.8a).  Dynamics of the tool workpiece system: The relative vibrations between the slender end mill and rigid workpiece occur due to the periodic cutting forces distributed at the contact zone. Figure 6.10 shows the Frequency Response Functions (FRF) of the tool. Machine tool is Mori Seiki NMV5000DCG; modal hammer sensitivity is 2.38-mV/N; Polytec laser Doppler velocity sensor sensitivity is 10-V/mm. Test locations on the tool are shown in Figure 6.10a: The impact location is point-3; output is measured at points 1 to 4. Figure 6.10b and Figure 6.10c give the magnitude plots of FRF measurements in feed (x0) and normal (y0) directions, respectively.  Modal parameters are identified using Cutpro [118]. Only tool related vibration modes (with subscript t) are considered. Modal damping ratio ( tζ ) and natural frequency ( n,tω ) matrices of Equation (5.17) are formed using the identified modal parameters in Table 6.4. Mode shape Ut matrix is formed by linear interpolation between the measurement locations. As shown in Figure 6.11, mode shape values at the measurement locations (Table 6.4) are interpolated at the mid-locations of each axial segment-k (1..K). 171   Figure 6.10 FRF measurements along the tool length: (a) Test locations; (b) Feed direction magnitude plot; (c) Normal direction magnitude plot.   Figure 6.11 Sketch of Figure 6.10a.  Table 6.4 Identified modal parameters in x0 and y0 directions.   Feed direction  Normal direction Number of vibration modes 8 8 Natural frequencies [Hz] 76,139,261,405,570,780,825,1143 190,265,424,566,794,830,1144,1508 Damping ratios [%] 3.50,4.89,4.55,6.45,3.17,3.18,1.10,3.34 4.90,3.12,1.88,3.13,2.53,0.92,2.98,5.36 Mass normalized mode shape vectors for each vibration mode [ 1/ kg ] in x and y directions.  Mode #1: 0.0574,0.0540,0.0566,0.0532 Mode #2: 0.0613,0.0577,0.0629,0.0684 Mode #3: 0.144,0.120,0.123,0.107 Mode #4: 0.272,0.170,0.213,0.163 Mode #5: 0.501,0.369,0.342,0.242 Mode #6: 1.61,1.05,0.939,0.577 Mode #7: 1.26,1.14,0.784,0.556 Mode #8: 2.01,1.56,0.800,0.238 Mode #9: 0.0781,0.0846,0.0947,0.104 Mode #10: 0.110,0.088,0.0904,0.0882 Mode #11: 0.154,0.127,0.138,0.103 Mode #12: 0.470,0.362,0.333,0.226 Mode #13: 1.71,1.18,0.910,0.589 Mode #14: 1.06,1.01,0.733,0.448 Mode #15: 2.04,1.41,0.753,0.257 Mode #16: 1.22,0.812,0.226,0.0153 172  Axial (z) and torsional (θ) directions are rigid, hence their mode shape values are set to zero in Equation (5.16), as:  t tz,k,n ,t θ,k,n ,t0; 0u u= = . (6.30) Mode shape values in x and y directions (tx,k,n ,tu  and ty,k,n ,tu ) at each axial segment-k are interpolated from Table 6.4; the hammer test locations are shown in Figure 6.10a. The mode shape vector tk,n ,tu  is formed at each level-k (k=1..K) and for each mode nt (1..mt) with the total number of tool modes mt=16 (8 modes each direction). Hence, the tool mode shape matrix Ut (4q x mt) for the operational depth of cut (ap=qdz) is obtained. For example, if ap=25-mm depth of cut and dz=0.2-mm segment thickness, then the number of contact elements is q=125, and the mode shape matrix would be 500 x 16. Since the workpiece is rigid, all the modal parameters of the tool represents the combined system: m=mt; U=Ut; n,c n,t=ω ω ; c t=ζ ζ .  The cutting force is axially distributed along the tool workpiece contact following from Equation (5.23). Process damping is neglected; the force at each cutting edge element Sjk is formed and the force vector at each axial level is summed up to obtain the distributed force vector (Equation (4.74)), as:  ( )1,j1 2,j1i 3,j1i j1i1,j2 2,j2i 3,j2i j2ii1,jk 2,jki 3,jki jki1,jq 2,jqi 3,jqi jqi 4 x1Njqg g gg g gg g gg g g     =     ∑FFFFF. (6.31) Rewriting the force in general form (Equation (5.23)) without the process damping leads to:  173   ( )[4( 1) x ],cs cd d esi i i c,i jki[4 x ] c i jki i,[4( ) x ]k mN qmj kq k mt T−−      + − ⋅ − +       ∑0F f f f f0= Γ Γ . (6.32) The static component of the shearing force vector ( csif ) at time ti is formed from the general representation of Equation (5.24), where the force csjkiF  at each segment-k is derived from Equation (4.60):  ( )cs1,j1 2,j1i 3,j1i j1ics1,j2 2,j2i 3,j2i j2icsi cs1,jk 2,jki 3,jki jkics1,jq 2,jqi 3,jqi jqi 4 x1Njqg g gg g gg g gg g g     =        ∑FFfFF. (6.33) Similarly, the static ploughing force vector ( esif ) at time ti is obtained from the general formula Equation (5.27), where the force esjkiF  at each segment-k is obtained from Equation (4.63):  ( )es1,j1 2,j1i 3,j1i j1ies1,j2 2,j2i 3,j2i j2iesi es1,jk 2,jki 3,jki jkies1,jq 2,jqi 3,jqi jqi 4 x1Njqg g gg g gg g gg g g     =        ∑FFfFF. (6.34) Dynamic component ( cdjkiF ) of the shearing force of element Sjk at time ti has two components (as seen in Equation (4.61)): One related to the current vibrations, and the other one 174  related to the delay vibrations. Rewriting the Equation (4.61) using the relative vibration vector of Equation (5.89) leads to:  ( )cd djki jki ki k i jki( )t T= − −F F Ψ Ψ , (6.35) where the relative displacement kiΨ  vector and the delay vector k i jki( )t T−Ψ  at segment k are:  [ ] [ ]k i jkikik i jkikiki k i jkiki k i jkiki 4 x 1 k i jki 4 x 1( )( ); ( )( )( )x t Txy t Tyt Tz z t Tt Tθ θ−      −  = − =   −      −   Ψ Ψ . (6.36) The overall displacement iΨ  vector becomes:  ( )1kiiqi 4 x1q    =      ΨΨΨΨ. (6.37) The overall delay vector d,jkiΨ  for segment-k is defined by only keeping the element at segment-k and setting the rest to zero, as:  ( )[4( 1) x1]d,jki k i jki[4( ) x1] 4 x1( )kq k qt T−−   = −    00Ψ Ψ . (6.38) Equations (6.37) and (6.38) are used to rewrite Equation (6.35), as:  ( )cd djki jki [4 x 4( 1)] [4 x 4] [4 x 4( ))] i d,jkik q k− − = − F F 0 I 0 Ψ Ψ , (6.39) which is reorganized with the modal transformation (Equation (5.88)), as:  ( )cd djki jki [4 x 4( 1)] [4 x 4] [4 x 4( ))] t w c i c i jki( ) ( )k q k t t T− − = − ⋅ − −   F F 0 I 0 U U Γ Γ . (6.40) 175  Since the only non-zero elements in d,jkiΨ  belong to the segment-k, the delay modal vector c i jki( )t T−Γ  is written using delay period jkiT  of tooth-j  in Equation (6.40) at the segment-k. The first component cdif  of the dynamic shear force vector is the multiplier of the current modal displacement vector c,iΓ  in Equation (6.40):  [ ]d1,j1 2,j1i 3,j1i j1i [4x 4] [4x 4( 1))] [4 x ]d1,j2 2,j2i 3,j2i j2i [4x 4] [4x 4] [4x 4( 2))] [4 x ]cdi 4 x d1,jk 2,jki 3,jki jki [4x 4( 1)] [4x 4] [4x 4( ))] [4 x ]1,jq 2,jqi 3,jqi jq q mq q mq mk q k q mg g gg g gg g gg g g−−− −    =  F I 0 UF 0 I 0 UfF 0 I 0 UF[ ]dqi [4x 4( 1)] [4x 4] [4 x ] 4 xNjq q m q m−              ∑0 I U, (6.41) or simply:  [ ]d1,j1 2,j1i 3,j1i j1i [4x 4] [4x 4( 1))]d1,j2 2,j2i 3,j2i j2i [4x 4] [4x 4] [4x 4( 2))]cdi 4 x d1,jk 2,jki 3,jki jki [4x 4( 1)] [4x 4] [4x 4( ))]d1,jq 2,jqi 3,jqi jqi [4x 4( 1)] [4x 4]qqq mk q kqg g gg g gg g gg g g−−− −−    =  F I 0F 0 I 0fF 0 I 0F 0 I[ ][4 x ]4 x 4Nq mjq q       ⋅       ∑ U . (6.42) The second component of the dynamic shear force vector includes the delay modal displacement vector c i jki( )t T−Γ : 176   ( )( )[4( 1) x ],djki[4x ] c i jki,[4( ) x ][4( 1) x ]d1,jk 2,jki 3,jki jki [4x 4( 1)] [4x 4] [4x 4( ))] c i jki [ x1][4( ) x ]......k mN qmj kq k mk mk q k mq k mt Tg g g t T−−−− −−       ⋅ − =               ⋅ ⋅ −      ∑0f00F 0 I 0 U0ΓΓ,,.N qj k ∑ (6.43) Thus, all the components of Equation (6.32) are obtained. The rest of the formulation is the same as the solution of Equation (5.19).  Prediction of stability chart: The chatter stability of the cylindrical end mill with the parameters given in Table 6.3 is verified. Only the lateral (x,y) directions of the cutter are flexible, and the size of the modal matrix U is halved by removing the terms related to the rigid z and torsional directions.  The sampling time is selected by considering the highest natural frequency and the desired number of points in the engagement zone.  The sampling time is initially chosen to be 6 times faster than the maximum natural frequency of the tool workpiece system, i.e., ( )n,cmax 2π 1508= ⋅ω  rad/s (see Table 6.4), hence ( )1 1.1052e-4t∆ = s. It is also desired to have at least 8 sample points in the tool workpiece engagement zone. For 5% down milling, the nominal entry and exit angles are 154.1-deg and 180-deg respectively. Then, the ratio of cutting time in one rotation period is360 (180 154.1) 13.9− = . If 8 sample points are taken in the cutting time, then in one rotation period, there are s 13.9 8 112n = ⋅ ≅  (integer value) sample points. For example, for 8000 rev/min spindle speed, one cutter rotation period is s 60 8000 7.5e-3T = = s. The sampling time is 177  ( ) s s2 0.696e-4t T n∆ = =  [s], which is almost half the value of the sampling time from the first criterion: ( )1 1.1052e-4t∆ = . The number of discretization is chosen as: ns=112. From Equation (5.38), ns has to be an integer multiple of 3, therefore ns is updated to the nearest higher integer multiple of 3: s 114n = . Updating the number of discrete states in one period also updates the sampling time: s s 0.678e-4t T n∆ = =  [s]. The procedure is applied to all cases.  The thickness of axial segments is set to dz=1-mm, which leads to K=38 segments for a 38-mm maximum simulated depth of cut.  The stability is predicted between the speed range of 8000-rev/min to 16000-rev/min at 100-rev/min increments. Scanned axial depth of cut are from 10-mm to 38-mm with 1-mm increments. The critically stable depth of cut is detected at one speed when an unstable eigenvalue (larger than 1) is reached. The cutting forces and surface finish are measured to assess the stability during experiments, as is shown in Figure 6.12. Stability chart is predicted separately when the effect of runout is included in one case, and neglected in the other case (Figure 6.12a). The simulation without runout is compared to the frequency domain solution [136]. Radial runout increases the stability limit due to the change in delay periods along the cutting edge. The resulting disturbance on the chip regeneration increases the stable depth of cut limit. Wan et al. [115] reported similar results on the effect of runout on stability.  While the chatter frequency is predicted to occur at 1152-Hz, it occurred at 1245-Hz during the milling experiments. The difference may be due to slight measurement error during impact testing, and also due to variations in the contact stiffness of spindle bearings and tool-holder-spindle interface joints during machining. For example, the presence of split modes at 790 and 1144 Hz leads to identification errors during impact testing.  178   Figure 6.12 Results of discrete-time simulation: (a) Stability predictions; (b) SLE measurement; (c) Predicted and measured SLE; (d) Example displacement simulation. The surface location error (SLE) is predicted from a window of 50 spindle rotation periods during stable machining after the transient vibrations diminish in seven spindle periods. (P=50 in Equation (5.87)), and is shown in Figure 6.13 for unstable (A) and stable (B) cases. The chatter frequency is predicted as 1152-Hz in Figure 6.13 whereas the measurement (Figure 6.12a) gives 1245-Hz.  179     Figure 6.13 Discrete time force simulation for the experiments (A) and (B) of Figure 6.12aExperimental validation of SLE has been carried out by machining a reference surface just before the actual machined area. The reference surface with 30-mm axial depth and 5 mm radial width has been pre-machined in multiple passes with very small forces to create smooth surface which is not touched by the tool during test cuts Also, another control surface with 5-mm depth is formed just below the machined area. The milled surface for SLE verification is measured by a dial-gage instrument at 3 different locations (Figure 6.12b) in the helix angle direction. Three measurement sets are presented in Figure 6.12c; zero SLE of this graph corresponds to the reference surface after compensating for the radial offset.  The predicted static flexibilities using the modal model at the 3rd and 4th points (Figure 6.10a) on the cutter are 1.60e-7 and 2.86e-7 m/N, respectively. However, the measured static 180  flexibilities at the same points are found to be 1.90e-7 and 3.40e-7 m/N, respectively. The static stiffness errors in the direction normal to the surface are compensated by creating a highly damped (50%) fictitious mode at a low frequency (10-Hz), which shifted the SLE upwards.  SLE is predicted by the steady-state section of discrete-time displacement simulation, an example of which is shown in Figure 6.12d, at 13-th axial segment along the tool axis. Predicted SLE is compared to the measurement in Figure 6.12c. Minima of the predicted and measured SLE are 28 and 56-micrometer respectively. The maxima of predicted and measured SLE are 75- and 95-micrometer respectively. It seems that the flexibility of the actual system during the machining at this spindle speed has less damping and stiffness than the measured system.  6.3 Application-2: Regular Pitch Serrated Cylindrical Helical End Mill [21] Serrated end mill, or the end mill with undulated cutting edge [137], is designed to increase the chatter stability of roughing operations. A three fluted serrated end mill is modelled. The model is validated with the experiments presented by Merdol and Altintas [21] and with the exact numerical  simulation given by Cutpro [118].  Geometry: As presented in Section 3.2.2, the geometry of the serrated flutes is measured as a radial distance from the tool axis, as is shown in Figure 3.35a. Measurement along the flute is then projected to the tool axis (Figure 3.35b). The nominal geometry of the end mill is given in Table 6.5. Cutting edge angle ( serr,s serrr,jr,jk ( d )k sκ κ= ) at midpoint of each axial segment-k is approximated numerically between the two neighboring measured points (from Figure 3.35a): 181   serr,sr,jk s sj,k+1 j,k-12darctansR Rκ  ≅ − , (6.44) where serrd d coss z β=  from Equation (3.53). The approximate cutting edge angle along the flute is given in Figure 3.36a. Cutting edge angle information is projected onto tool axis in Figure 3.36b and Figure 6.14b. Radial runout r,jkε  (Figure 6.14a) is obtained by normalizing each axial segment-k to its minimum (similar to Equation (6.10)) using the local radius (Figure 3.35b).  Angular location and radius of each element Sjk is defined as the location of each element Sjk relative to the origin of the tool reference frame. In Equation (3.55), the position vector 0 Dr of the coordinate system of each cutting edge element Sjk is obtained from the transformations of Figure 3.34a-d. The axial segment thickness is set to dz=0.1-mm; the geometry is interpolated at the midpoint of each segment. The maximum depth of cut is set to 39.2-mm, which makes K=39.2/0.1=392 axial levels in total. The angular location ( jkψ ) (Figure 6.15a) is found using the x0y0 coordinates (in tool reference frame) of the element Sjk: Table 6.5 Information of the serrated end mill Type of end mill Regular pitch cylindrical serrated end mill Manufacturer and model Sandvik R216.33-20040-AC38U H10F Diameter (nominal) 20-mm Length of fluted section 38-mm Number of flutes N=3 Designed pitch angle 120-deg Rake angle / Helix angle serrγ =10-deg / serrβ =40-deg Cutting edge angle, serr serrr,jk r,j ( d )k zκ κ=  Varying due to serrated profile (Figure 6.14b)  Radial runout Figure 6.14a.   Axial runout Negligible   182   ( )0 D 0 Djk y xπ atan2 ,2 r rψ = − . (6.45) Relative angular location is defined relative to the element S11 in CW direction. The modelled 3D geometry of cutting edge in Frame-0 (tool reference frame) is shown in Figure 6.16a. Close-up 3D view is given in Figure 6.16b with tooth numbers (j=1,2,3) shown. The remaining rotational transformations (Figure 3.34e-h) are done to obtain the cutting edge tangent ( )D jki , rake face normal ( )D jkj , and the rake face ( )D jkk  vectors (Equation (3.56)). Tool angles are computed from the formulation given in Section 3.3. The true cutting edge angle, cutting edge inclination angle and normal rake angle are presented in Figure 6.17.   Figure 6.14 Serrated end mill flute geometry: (a) Normalized radial runout; (b) cutting edge angle.   Figure 6.15 Serrated end mill: (a) angular location; (b) relative angular location.  183    Figure 6.16 Serrated end mill: (a) Modelled 3D geometry; (b) close-up 3D view.    Figure 6.17 Computed tool geometry of serrated end mill: (a) True cutting edge angle ; (b) Cutting edge inclination angle; (c) Normal rake angle . Computation of the tool angles along the cutting edge leads to the prediction of forces and stability for the given conditions.  Force prediction: The actual chip load distribution is found by considering the serrations as radial runouts. An example of radial runout for the first 8-mm depth along the cutting edge is shown in Figure 6.18a. Maximum effective chip load simulations in Figure 6.18b-f show that the chip load reaches the feedrate at most of the cutting section due to the serrated profile. Only one tooth is in-cut (while the other two miss the cut) at the sections where chip load reaches the 184  feedrate (fr) value. Cutting engagement, depth of cut and feedrate of five simulations are given in Figure 6.18.        Figure 6.18 (a) Radial runout for the first 8-mm depth; (b-f) Maximum effective chip load along the cutting edge for five cases: (b) 50% down m., ap=8-mm, fr=0.24-mm/rev; (c) 50% down m., ap=8-mm, fr=0.30-mm/rev; (d) 50% up m., ap=4-mm, fr=0.24-mm/rev; (e) 50% up m., ap=6-mm, fr=0.24-mm/rev; (f) Slot milling, ap=14-mm, fr=0.12-mm/rev.   185  Cutting forces are distributed along the cutting edge and lumped at the tool tip (Figure 4.2). Equation (6.25) is used for predicting the static cutting forces with the application of transformation matrix ( RI, jkT ) and the cutting (crta,jkiK ) and edge (erta,jkK ) coefficient vectors The matrix ( RI, jkT ) is used to transform the forces from the geometry dependent cutting edge frame (RTA) to the edge reference coordinate frame (R) (Equation  (4.41)) :  * *r, jk r, jkRI, jk* *r, jk r, jksin 0 cos0 1 0cos 0 sinκ κκ κ − −  = −  T . (6.46) where the variation of true cutting edge *r, jkκ  angle along the tool axis is shown in Figure 6.17a. Cutting coefficients cu,jkiK  and cv,jkiK  on the rake face (UV-Frame) are found from Equation (4.25) as:  ( )( )2 2s, jki n,2 2 2s, jk n, n, n, n, n,2 2s, jki n,2 2jk jkijki jki jki jk jk jkijk jkijki jki jki jk jk j2s, jk n, n, n, n, n, kicu,jki jkicv,jkai,1 tan sinsincos sin cos tan sin1 tan sincoscos sin cos tan sin( d );k zKKτ η βλ φ φ β γ η βτ η βλ φ φ β γ η ββ−=+ − +−=+ − +kia, j ,β (6.47) where the shear parameters are approximated using the orthogonal cutting tests. The workpiece material is Aluminum 7050-T7451. Low speed Al7050-T7451 orthogonal cutting database of Cutpro [118] is used with the following shear variables:   ( )( )[ ]s,jki c,jk n,jkn,jki c,jk n,jka,jki c,jk nstjkistjkist,jkjkistj c,jki k180266 0.0437 174 ;ππ19.4 42.0 0.384 ;180π25.8 1.28 0.0075 0.181 ;180limits : 0.04, 0.20 -mm; 14.3, 20.8960.02hhhvvhvvτ γφ γβ γ= + ⋅ + ⋅ ⋅= + ⋅ ⋅ ⋅ += − ⋅ ⋅ ⋅ +∈ ∈+ ⋅+ ⋅− ⋅[ ] [ ]n,jk00 -m/min; 0, 0.5235 -rad .γ ∈ (6.48) 186  Shear stress s,jkiτ  (MPa), shear angle n,jkiφ  (rad) and average friction angle a,jkiβ  (rad) of Equation (6.48) are stored as functions of static chip thickness stjkih (mm), cutting velocity c,jkv  (m/min) and rake angle n,jkγ  (rad). The cutting coefficients are transferred to RTA frame using Equation (4.28), where the IU, jkT  matrix transforming rake face (UV-frame) to RTA frame is from Equation (4.29). The edge force coefficients are directly found in RTA frame using the orthogonal cutting force database. The edge force coefficients defined in Al7050-T7451 database of Cutpro [118] are given as:     Figure 6.19 Force simulations compared to Cutpro [118]: (a) 50% down m., ap=8-mm, fr=0.24-mm/rev; (b) 50% down m., ap=8-mm, fr=0.30-mm/rev; (c) 50% up m., ap=4-mm, fr=0.24-mm/rev; (d) 50% up m., ap=6-mm, fr=0.24-mm/rev.   187   [ ] [ ] [ ]et,jk c,jk n,jker,jk c,jk n,jkea,jkc,jk nsj jkk ,ti18020.9 0.0152 0.358 ;π18033.6 0.00598 0.151 ;π0 (as expected for orthogonal cutting);limits : 0.04, 0.20 -mm; 14.3, 200 -m/min; 0, 30 -deg .K vK vhKvγγγ= − ⋅ − ⋅ ⋅= − ⋅ − ⋅ ⋅=∈ ∈ ∈ (6.49) Cutting forces are predicted at dz=0.1-mm increments with Δt=1.25e-4 [s] sampling time; spindle speed: Ω=1600-rev/min; workpiece material: Al7050-T7451. Cutting engagement, depth of cut and feedrate of four simulations are given in Figure 6.19.   Figure 6.20 Force simulation compared to the experiment plots of [21]  188  Predicted forces are consistent with the numerical, time domain simulation results obtained from Cutpro [118] in Figure 6.19, and with the experimental cutting forces [21] in Figure 6.20. The phase difference between predicted forces and measured forces (Figure 6.20) can be neglected due to no information on dynamometer and measurement characteristics. Cutting engagement, depth of cut and feedrate of four simulations of Figure 6.20 are same as in Figure 6.19.  Stability chart: Tool is assumed to have lumped dynamics at its tip where the total forces are applied. The total cutting forces are expressed from Equation (6.32), as:  ( ){ },cs,lumped cd,lumped es,lumpedlumped di c,i jki[4 x ] c i jkii i i,N qmj kt T+ − ⋅ − +∑F f f f f= Γ Γ , (6.50) where the total force vector is ( )lumpedi [4 1]×F . The static component of the shearing force ( csif ) is modified to combine all the axial segments (from k=1..q) along the tool workpiece contact:  ( )[ ] ( ),cs,lumped cs1,jk 2,jki 3,jki jkii 4x14x1 ,N qj kg g g= ∑f F . (6.51) The ploughing force vector is also combined using the Equation (6.34):  ( )[ ] ( ),es,lumped es1,jk 2,jki 3,jki jkii 4x14x1 ,N qj kg g g= ∑f F . (6.52) The component related to current state of the dynamic shear force vector is written from Equation (6.41). Since the modal identification is done at the tool tip, the size of the mode shape is [4 x ]mU  . The Equation (6.41) is modified by multiplying each segment by the lumped dynamics mode shape matrix, as:  189   [ ][ ]d1,j1 2,j1i 3,j1i j1i [4x ]d1,j2 2,j2i 3,j2i j2i [4x ]cdi 4 x d1,jk 2,jki 3,jki jki [4x ]d1,jq 2,jqi 3,jqi jqi [4x ] 4 xmmNq mj mm q mg g gg g gg g gg g g     =        ∑F UF UfF UF U. (6.53) The lumped force vector is obtained by summing the contributions of all segments:   ( )[ ] ( ),cd,lumped d1,jk 2,jki 3,jki jki [4x ]i 4x44x ,N qmm j kg g g =    ∑f F U . (6.54) Following the Equation (6.54), the component related to the delay state of the dynamic shear force vector is obtained by combining the contribution of all segments, as shown in Equation (6.43):  ( ){ } ( ){ }, ,d djki[4x ] c i jki 1,jk 2,jki 3,jki jki [4 x ] c i jki [ x1], ,N q N qm q m mj k j kt T g g g t T⋅ − = ⋅ ⋅ −∑ ∑f F UΓ Γ . (6.55) The rest of the formulation is the same as in Section 5.2. The hammer test is conducted at the tool tip, and tool information is found in Table 6.5. Modal parameters of the tool dynamics are taken from [21], and listed in Table 6.6. Since the two x and y directions do not have coupled dynamics, their mass normalized mode shape values are found as (Equation (5.16)): Table 6.6 Identified modal parameters in x and y directions (taken from Table 2 of Ref. [21]).   Feed direction Normal direction Number of vibration modes 7 7 Natural frequencies [Hz] 480,530,893,1216,1433,2085,2691 470,533,884,1155,1270,1946,2670 Damping ratios [%] 1.52,2.80,3.03,3.33,4.11,2.94,2.65 2.81,2.67,2.41,1.61,2.80,2.44,1.89 Normalized mode shape values [1/ kg ] Mode #1: 0.2740; Mode #2: 0.1920; Mode #3: 0.5080; Mode #4: 1.5100; Mode #5: 0.9250; Mode #6: 1.7480; Mode #7: 1.7160. Mode #8: 0.1020; Mode #9: 0.2820; Mode #10: 0.5600; Mode #11: 0.6320; Mode #12: 1.0400; Mode #13: 1.9600; Mode #14: 1.6070.   190   t tt tt tn,n ,t n,n ,tt tx,n ,t y,n ,tx,k,n ,t y,k,n ,t, 1, 2...7; , 8, 2...14,n nk ku uω ω= = = =  (6.56) where the 1st mode to 7th mode of Table 6.6 are listed for x-direction, and the 8th mode to 14th mode are listed for y-direction. tx,n ,tk  and ty,n ,tk  are the modal stiffnesses of x and y directions from Table 2. The vibration modes with the highest natural frequency have little effect on the overall flexibility, therefore reduced dynamics is formed by neglecting modes 6, 7, 13 and 14, as is found in Table 6.6.  The stability of slot milling of Aluminum 7050 is simulated for the feedrate fr=0.12-mm/rev, and is compared with the experimental and numerical results given by Merdol and Altintas [21]. Spindle speed is scanned from 4000 to 5500-rev/min with 100-rev/min increments. The depth of cut is scanned from 7 to 11-mm with 1-mm increments at each speed. The sampling frequency is set to 3-times the maximum natural frequency, and the axial segment thickness is set to dz=0.1-mm.   Figure 6.21 Comparison of the stability prediction to the results presented in [21]. 191  Simulation results are shown in Figure 6.21. Full dynamics (Table 6.6) and reduced dynamics (Table 6.6 when 6th,7th,13th and 14th modes removed) models give similar results. As seen in Figure 6.21, the predicted results are slightly lower than the results presented in the original study reported in “Fig.13 of Ref. [21]”, the latter employing numerical, time domain simulation that includes nonlinear effects such as vibrations affecting the chip thickness. In addition, the numerical simulation assumes instability if the vibrations exceed more than 20% of the forced vibrations [21]. However, the stability decision method presented in this thesis is analytical and based on the eigenvalues.   6.4 Application-3: Two-Level Multifunctional Drilling/Boring Cutter [5] A multifunctional (boring and drilling combined) tool case presented by Wan et al. [5] is used to illustrate the application of the proposed general formulation on a cutter with complex geometry and multiple functionalities. The cutter is a custom made multifunctional drill with the Sandvik TM880 type body. It has two drilling inserts at the bottom level (Level-2) and two step-boring inserts at the top level (Level-1) with different dynamics. Only the lateral flexibility is critical for this cutter; it is rigid in the torsional-axial directions. The toolholder is Sandvik Coromant HydroGrip (HSK-63A interface) 392.410CGA-63 20 088B.  Geometry: 3D geometry of the cutting edge on the cutter body is shown in Figure 6.22a. Level-1 inserts are #1 and #2, and Level-2 inserts are #3 and #4. As shown in in Figure 6.22b, radial depth vs. radius of the midpoints is a linear plot. 9.9-mm radial depth corresponds to cutter center segment, and 0.1-mm corresponds to the perimeter segment of the cutter.  192  (a)   Figure 6.22 Sandvik multifunctional tool: (a) 3D geometry; (b) Radial depth of the midpoints of the segments along the cutter body.  The geometry and locations of the two inserts at the bottom side (Level-2) of the cutter are identified from tool microscope measurements and is given in Figure 6.22a. The radial depth is discretized at dz=0.2-mm segment increments. Thus, the 10-mm radius is divided into K=10/0.2=50 elements.  Figure 6.23a is the angular location of element Sjk measured from +y0 axis of the cutter reference frame, and Figure 6.23b is the relative angular location is measured from S11 element in CW direction. Axial location (Ajk) of each element is measured from the tool tip. As shown in Figure 6.24, the two levels are given separately for easier visualization. It is assumed that there is no axial difference (Figure 6.24a) between the 1st and 2nd inserts at Level-1. Thus, there is no runout between Level-1 inserts (Figure 6.25a). The axial height difference between the tips of 3rd and 4th inserts of Level-2 are measured after installing the cutter on the spindle. The chip engagement is shared in the overlapping zone (Figure 6.24b) of the Level-2 inserts. Figure 6.25b shows the axial runout as a result of the axial height difference in the overlapping zone (see Figure 6.24b) of Level-2 inserts.   193    Figure 6.23 Angular location (a) and relative angular location (b) of element Sjk.   Figure 6.24 Axial location: (a) Level-1; (b) Level-2.    Figure 6.25 (a) Axial runout: (a) Level-1; (b) Level-2.  Mechanics of cutting: The peripheral radius section starts at k=1 element, and k=50 element corresponds to the center segment in Figure 6.22b. The number of dynamic levels D in Figure 194  4.12 depends on the depth of cut, qdz. The depth of cut qdz is defined as the difference in the outer radius of cutter and the final radius of the drilled/bored hole; the number of contact elements is q. If the cutter is fully in cut with both levels, then D=2. If only the bottom or top level inserts are in cut, then D=1.  The following three cases are considered: • When the cutter starts the engagement only at the bottom side, Level-2 inserts are in cut. The cutter acts as a regular drilling cutter. The cutter radius for the bottom section is 7-mm and the number of elements is redefined as K=7/dz=35 for dz=0.2-mm segment thickness. Number of dynamic levels is D=1. Depending on the number of contact elements q for the depth of cut qdz, the limit for the level is d1=q. • After dilling 14-mm depth hole, the topside Level-1 inserts engage with the workpiece. The cutter radius is 10-mm and the number of elements is the same as the original value (K=50). The number of dynamic levels is D=2. The depth of cut is qdz. The limit for the second level is d2=q. The limit for the first level is constant since Level-1 is always fully engaged if bottom inserts are in cut. For 3-mm radius area, the limit for first level is d1=3/dz=15. • If the predrilled hole is larger than 7-mm, then only Level-1 inserts are in cut. The cutter acts as a regular boring cutter. The cutter radius for the top section is 10-mm and the number of elements is redefined as K=3/dz=15. The number of dynamic levels is D=1. Depending on the number of contact elements q for the depth of cut qdz, the limit for the level is d1=q. The geometric parameters are evaluated for all the elements (Sjk) of the cutter at both levels. The axial runout of the Level-1 inserts are zero (Figure 6.25a), and the runout for the Level-2 inserts are given in Figure 6.25b. The designed pitch angle (Figure 6.26a) is derived when the feedrate effect is neglected, i.e., when assuming that there is no overlap or runout between the 195  inserts. It is found by using the angular location of Figure 6.23. The actual pitch angle due to runout is presented in Figure 6.26b; it is determined when the overlapping zone is considered for fr=0.1-mm commanded feedrate.  The cutting edge tangent, rake face normal and rake face vectors are identified from the measured geometry. The effective tool geometry is identified from the theory presented in Section 3.3 and is shown in Figure 6.27. Cutting coefficients in RTA directions are presented in Figure 6.28a-c, for machining AISI 1045 steel at 1000-rev/min cutting speed and 0.1-mm/rev feedrate; edge coefficients in RTA arepresented in Figure 6.28d-f.   Figure 6.26 Pitch angle: (a) Feedrate neglected (fr≈0); (b) fr=0.1-mm.     Figure 6.27 Effective tool geometry calculated from the rake face orientation: (a) True cutting edge angle; (b) Cutting edge inclination angle; (c) normal rake angle at each element Sjk. 196        Figure 6.28 Cutting force coefficients (a,b,c) and edge coefficients (d,e,f) in RTA directions. Total static cutting forces acting at the tool-workpiece contact is found by integrating the forces as derived in Equations (4.70) and (4.72). Process damping ( edjkiF ) forces are negligible. Force vector at each segment consists of static shear Equation (4.60) and static ploughing forces Equation (4.63). The moment arm is the radius at each element: t,jk jkR R= . The static chip 197  thickness of drilling operation is written from Equation (4.2), and the total static cutting force at element Sjk is expressed as:   ( )0R, jki RI, jk 0R, jki RI, jkst c a ejki rta,jk jk jk rta,jk *jk jk r,jk10 0 0 0dsinzcR Rγκ ⋅ ⋅    = ⋅ − ⋅ ⋅           + T T T TF K K . (6.57) Transformation matrices at each element Sjk are given in Equations (4.36) and (4.37). For constant rotation speed Ω (rev/min), the instantaneous angular position is found from Equation (4.38):  kjki i j2π60t ψφ  = Ω +  . (6.58) The edge-in-cut (g2,jki) parameter takes the value 1 due to the continuous cutting action of drilling:  2, jki s1 for all 1.. , 1.. and 1.. +1 (one rotation period)g j N k K i n= = = = . (6.59)  Since the chip engagement is time invariant for drilling operation (Equation (4.2)), the missed cut condition (g3,jki) parameter becomes time invariant as well: g3,jki=g3,jk. Figure 6.29 shows that for Level-1 inserts (#1 and #2), g1 and g3 are the same due to zero axial runout (see Figure 6.25a). However, g1 and g3 parameters for Level-2 inserts (#3 and #4) are not the same because of the nonzero axial runout (see Figure 6.25b-e). Missed-cut conditions for the Level-2 inserts depend on feedrate (Figure 6.29d-e shows for fr=0.1-mm/rev). Substituting the force terms from Equation (6.57) into Equation (4.72) gives:   ,st sti 1,jk 3,jk jki, 4x1N qj kg g =    ∑F F . (6.60) 198        Figure 6.29 Element-available-to-cut (a) and missed-cut (b,c,d,e) conditions for Sandvik multifunctional tool.  The prediction of cutting forces is validated experimentally. Number of data points per spindle period is kept constant for each simulation, ns=360. The simulation is carried out for one spindle period: i s s, 1, 2..., , 1t i t i n n= ⋅∆ = + . The work materials were AISI 1045 steel (Equation (4.22)) and Aluminum 7050-T7451 Equation (6.48), and the cutting coefficients were obtained 199  from the Cutpro database [118]. The predicted forces (in tool reference frame) for drilling AISI 1045 steel are compared against experiments, as shown in Figure 6.30. For the test of Figure 6.30a1-a3: Only Level-2 of Sandvik TM880 cutter is in cut; it is full hole drilling; 3000-rev/min spindle speed; no pre-drilled hole with 7-mm radial depth (dz=0.2-mm, q=35), fr=0.1-mm/rev. For the test of Figure 6.30b1-b3: Both levels (all inserts) of Sandvik TM880 cutter are in cut; it is combined full hole drilling / stepboring operation; 1000-rev/min spindle speed; 10-mm radial depth (dz=0.2-mm, q=50); fr=0.1-mm/rev.       Figure 6.30 Predicted cutting forces for Sandvik multifunctional tool: (a) Only Level-2 in cut with full hole drilling; (b) both levels in cut with combined drilling / stepboring.   200  Dynamics: The formulation is shown for the second case, where both levels of the cutter are in cut. Level-1 inserts cover the outer 3-mm radial section, thus from k=1 to k=15, and d1=15, as shown in Figure 4.12. Level-2 inserts cover the inner 7-mm radial section; starting from k=d1+1=16 to q. The parameter d2 in Figure 4.12 is dD=q. The force vector along the tool workpiece contact is expressed from Equation (4.77):  [ ]151,jk 3,jk jkik 1i1,jk 3,jk jkik 16 4 2 x1Nqjg gg g== ⋅   =     ∑∑∑FFF. (6.61) The rest of the formulation is similar to the formulation given in Section 5.2. The flank wear land is assumed to be Lw=20-micrometer in the process damping force model. Specific indentation force for Aluminum and Steel is taken as Ksp=1.5e5 N/mm3 [120] and 4e5 N/mm3 [119] respectively. Coulomb friction coefficient is assumed as µ =0.3.  Table 6.7 Identified modal parameters for Sandvik and Kennametal multifunctional tools. Cutter type Vibration mode Natural frequency (Hz) Damping ratio (%) Mass normalized mode shape matrix Sandvik TM880 1 ( x direction) 936.66 2.51 tx,1,1,t x,1,2,ty,1,1,t y,1,2,tz,1,1,t z,1,2,tθ,1,1,t θ,1,2,tx,2,1,t x,2,2,ty,2,1,t y,2,2,tz,2,1,t z,2,2,tθ,2,1,t θ,2,2,t0.801 00 0.8140 00 00.838 00 0.8810 00 0u uu uu uu uu uu uu uu u= =            =         U U 1 kg          2 ( y direction) 928.57 3.93 Kennametal HTS-R 1 (torsional-axial, zθ) 3677 0.51 x,1,1,ty,1,1,ttz,1,1,t2θ,1,1,t000.1761 1 kg188.4 1 kg muuuu           = = = −           ⋅     U U  201  Instead of lumping the forces at each segment as in Equation (6.31), there are two dynamic levels. The corresponding segments are summed within each dynamic level. The related mode shape matrix includes only two dynamic levels (1 and 2) instead of every radial segment (1,2...q). Only the principal lateral directions are measured since the torsional axial flexibilities are comparatively more rigid. The principal directions are assumed to be fixed and nonrotating. The identified modal parameters are listed in Table 6.7. There are two mode shape te,n ,tu  vectors for each dynamic level (e=1,2) and for each mode (nt=1,2) of the cutter. Only the lateral related tx,e,n ,tu  and ty,e,n ,tu  are nonzero, and the torsional axial components are zero: t tz,e,n ,t θ,e,n ,t0; 0u u= = . Prediction of stability chart is validated experimentally for each case. When only Level-2 (bottom side) is in cut (first case), the process is stable at all speeds for full hole drilling‒see Figure 6.31a5. When only the top inserts are in cut, the predictions are highly sensitive to the input geometry. There are two lateral stability predictions: The first prediction is when the measured geometry is used (Figure 6.31a1 and Figure 6.31a2), and the second is when the top side inserts are assumed regular-pitch 2-tooth boring cutter (Figure 6.31a3 and Figure 6.31a4). When only Level-1 inserts (#1 and #2) are in cut, Figure 6.31a3 gives a more accurate estimation than Figure 6.31a1. This shows that the measured geometry is inaccurate and the actual dimensions are closer to that of a regular 2-tooth boring cutter. For the case when all inserts in cut both simulations (Figure 6.31a4 and Figure 6.31a2) give inaccurate results. Cutting tests are all stable in the measured range most probably due to the stiffening effect from the hinge or pinned contact at the cutting zone of the drill cutter. When only the bottom-side (Level-2) inserts are in cut, the simulation and prediction are matching as seen in Figure 6.31a5.  202       Figure 6.31 Sandvik multifunctional tool stability predictions compared to experiments: (a1,a2) With the measured geometry of Level-1 inserts; (a3,a4) with the Level-1 inserts assumed regular 2-tooth boring geometry; (a5) when only Level-2 inserts are in cut; (b) results of a sample unstable cutting test [5]. During measurements, the cutting force and sound data are collected and the resulting surface is checked to decide if chatter occured. Measured force, sound and surface image of a 203  sample unstable cutting test (from Ref. [5]) is presented in Figure 6.31b. The sample data is for the case in Figure 6.31a1 and Figure 6.31a3, at 1,200-rev/min unstable cutting test.  When all the inserts are in cut, the stability is predicted with better accuracy if the measured geometry is used. Therefore, following are concluded: • The stability prediction depends directly on the geometric identification. Accuracy of geometry directly affects the delay period estimation which influences the stability predictions.  • One other reason may be the stiffening effect of the drilling cutter during the operation, as is argued by Tekinalp and Ulsoy [138].   6.5 Application-4: Drilling with a Multifunctional, Indexable Cutter A drill with four inserts [5] is used to validate the generalized model for the prediction of drilling operations‒see Figure 6.32. The geometry of each insert is presented in Figure 3.31. Locations of inserts are measured with an optical tool microscope and the geometry is constructed as shown in Figure 6.32. The diameter of the cutter body is 46-mm with a 10-mm diameter twist drill with its center having a 118-deg point angle and a 30-deg helix angle (Kennametal catalog #B513S10000; order #1940741-HSS coated; grade AS3).  The cutting force and chatter stability models are verified within 18 mm radial depth of cut by excluding the pilot drilling operation. Torsional-axial flexibility of the drill given in Table 6.7 is considered as the dominant source of regeneration [5]. The model is validated in drilling steel and aluminum alloys. 204    Figure 6.32 3D locations of the segments on the Kennametal cutter body. Insert numbers are labeled. Geometry of the tool is defined by the angular location (Figure 6.33a, measured from +y0 axis of the tool reference frame) and axial location (Figure 6.34a) of each element Sjk. The relative angular location (Figure 6.33b) is measured from the reference element S11 in CW direction. The axial runout (Figure 6.34b) is derived by shifting the minimum axial location (Figure 6.34a) value to zero at each radial segment-k. The designed pitch angle (Figure 6.35a) depends on the angular location (Figure 6.33b) of the elements, and it is only meaningful for the static condition (when feedrate is zero). It is calculated by assuming that there is no overlap or runout between the inserts. The actual pitch angle additionally depends on the axial runout and feedrate, and is plotted in Figure 6.35b (for 0.1-mm/rev feedrate). The resulting effective tool geometry is given in Figure 6.36.           ZX(a)Bottom viewBottom view132 4Bottom level    205    Figure 6.33 Angular location (a) and relative angular location (b) of element Sjk.   Figure 6.34 (a) Axial location (height), and (b) corresponding axial runout.    Figure 6.35 Pitch angle: (a) Norunout/zero feedrate; (b) fr=0.1-mm/rev  A pilot hole with a 10-mm diameter is predrilled before expanding the hole to a 46-mm diameter with indexed region of the cutter. However, since the 10-mm diameter twist drill is not removed prior to cutting experiment, it adds rubbing force to the measurement.  206     Figure 6.36 Effective tool geometry: (a) True cutting edge angle; (b) Cutting edge inclination angle; (c) normal rake angle. As seen in Figure 6.37, z0-force does not show this effect, but it is clearly seen in the x0 and y0-force data. Test conditions (Figure 6.37 and Figure 6.38) for drilling AISI 1045 steel are: Ω=900-rev/min; 10-mm pilot hole diameter (18-mm depth of cut); fr=0.1-mm/rev; total depth of the drilled-hole is 2-mm (~1.3-s). Simulation conditions (Figure 6.37 and Figure 6.38): dz=0.2-mm (q=90), ns=360.  From the measured x0 and y0 forces, the rubbing effect is clearly seen in the noncutting period-A, where the rubbing force amplitude is 100-N. In the cutting period-B, the combined rubbing and shearing action gives 200-N amplitude force. The estimated x0 and y0 cutting forces in Figure 6.38 have 73-N amplitude. z0-Force has no rubbing effect with zero force in period-A, therefore the estimated z0-Force in Figure 6.38 is more accurate. When the pilot hole diameter is set larger than the pilot drill diameter (10-mm), the rubbing effect is avoided (see Figure 6.39). Therefore, the force predictions become more accurate (see Figure 6.40) with some error left, which may be attributed to the ploughing of the peripheral inserts.  Test conditions (Figure 6.39 and Figure 6.40) for drilling Aluminum 7050-T7451 are: Ω=300-rev/min; 17-mm pilot hole diameter (14.5-mm depth of cut); fr=0.1-mm/rev; total depth 207  of the drilled-hole is 2-mm (~1.3-s). Simulation conditions (Figure 6.39 and Figure 6.40): dz=0.05-mm (q=290), ns=360.    Figure 6.37 Effect of the rubbing of the pilot drill on the cutting forces of Kennametal drill.     Figure 6.38 Predicted and measured cutting forces at period-B of Figure 6.37.   208     Figure 6.39 Cutting forces without rubbing effect. Pilot hole diameter is larger than 10-mm.    Figure 6.40 Predicted and measured cutting forces (close-up of Figure 6.39). For the stability simulation, only the z0 and torsion directions are solved. In this case, the related state transition matrix in Equation (5.70) becomes time invariant. Therefore, finding the eigenvalues for the constant state transition matrix is sufficient and thus saves significant computation time. Process damping parameters are taken to be the same as the Sandvik TM880 cutter of Section 6.4: Lw=20-micrometer; specific indentation force for aluminum, Ksp=1.5e5 N/mm3; Coulomb friction coefficient, µ =0.3. Simulated torsional-axial stability chart for drilling Aluminum 7050-T7451 is compared to experiments in Figure 6.41a. The scanned depths of cut are from 8 to 18-mm with 0.4-mm increments, and the scanned spindle speeds are from 500 to 5300-rev/min with 15-rev/min increments; a total of 321x26 grid of 8346 points is solved.  209    Figure 6.41 Stability test with Kennametal drill: (a) Stability chart compared to experiments; (b) Forces with the resulting surface finish photo of the selected unstable cutting test [5]. Cutting forces and sound are measured (total 4 channels) during the tests. Sample (taken from Ref. [5]) force and sound measurement data, and resulting surface finish for an unstable operation, are shown in Figure 6.41b. Forces in y0 and z0 directions are shown for Ω=1000-rev/min and ap=17-mm. The resulting sutface finish has sunray pattern which is a sign for torsional-axial chatter.  210  6.6 Application-5: Double-Sided Parallel Milling Double-sided milling application is used in the industry for machining flexible, cantilevered plates (e.g., elevator guide) from both sides. The proposed model is verified in modeling the geometry of the double cutter and predicting the simultaneous milling of a cantilevered plate from both sides  Geometry: The inserts are either brazed permanently or clamped to the insert pockets. Although detachable inserts are more practical, they have less location accuracy in the pocket (with up 40 µm). The runout of inserts are measured with an optical microscope (KeeJaan KJ-340) with a 10 µm measurement accuracy, as is shown in Figure 6.42a. The optical camera with a high resolution lens is aimed at the cutting edge of the cutter which is mounted on the spindle of the instrument. The cutter is free to rotate and can translate in the axis towards the column. The lens can move along the cutter axis. The radial and axial runout references are set by moving the axis and rotating the cutter to match the red broken line on the screen. Figure 6.42b shows the sample output screen view for measuring the cutting edge for insert#1 of right-side cutter. The cutting tests have been conducted on a highly rigid DIXI DJC-350 CNC machining center at ITRI-IMTTC (Taiwan) [4].  (a)  (b)  Figure 6.42 Measurement of radial runouts on the double-sided milling tool: (a) setup; (b) screen view. 211     (a) (b) (c) Figure 6.43 Double-sided milling tool: (a) In operation; (b) with brazed inserts; (c) with detachable inserts.   Milling experiments with an eight-tooth brazed cutter: KOVA company (Taiwan) custom produced the double-sided tool with brazed inserts shown in Figure 6.43b. The cutters are used in parallel milling of a plate shown in Figure 6.43a and Figure 6.44a. Engagement of the down milling operation is approximately 20% (Figure 6.44b). The left-side (KOVA D125-20L-D109.7-8T) and right-side (KOVA D125-20R-D109.7-8T) cutters have non-detachable, brazed inserts. The brazed inserts are ground on the cutter body to minimize the radial and axial runouts. The cutters are installed on keyways of the shank of the custom tool holder, with BT-50 taper at spindle interface. The keyway is the supporting edge when the cutters are mounted on the body. The supporting edges are aligned at the tool holder reference line (Figure 4.9) for angular location measurement. Following measured geometry of the left and right-side cutters are listed in Table 6.8: Measured angular location (with respect to tool holder reference line in Figure 6.46), corresponding designed pitch angle (Figure 4.9) and radial runout (using setup of Figure 6.42).   212  Table 6.8 Measured angular location, pitch angle and radial runout of the left- and right-side cutters. Insert number Angular location from toolholder reference (deg) Pitch angle (deg) Radial runout (micrometer) Left cutter j,Lψ  Right cutter j,Rψ  Right cutter p,j,Rφ  Left cutter p,j,Lφ  Left cutter Lr,jε  Right cutter Rr,jε  1 -8 -9 45 44.5 0 18 2 36.5 36 45 45.5 6 18 3 82 81 45 45.5 6 9 4 127.5 126 44.5 45 22 0 5 172.5 170.5 45.5 44.5 21 4 6 217 216 45.5 45 17 4 7 262 261.5 45 45 17 14 8 307 306.5 44.5 45 0 16   Figure 6.44 Double-sided milling tool: (a) Components‒see Figure 4.8; (b) operation is ~20% down milling.   213   (a)  (b)  Figure 6.45 Supporting edge line (tool holder reference line) shown on: (a) Left-side and (b) right-side cutters with labeled insert numbers.  The inserts have true cutting edge angle of *r 15-degκ = , 0-deg normal rake angle, and 0-deg inclination angle. The cutting forces are measured by Kistler 9257BA dynamometer during parallel milling of the AISI 1045 steel rigid workpiece (170-mm length). Cross section of the workpiece/part is given in Figure 6.45a; it is mounted on the dynamometer using an adapter plate with 6 clamps, as is shown in Figure 6.45b. The clamps are fixed with M8 screws which are tightened at controlled 400-kgf.cm torque using a torque wrench. (a)  (b)  Figure 6.46 Double-sided milling experiment setup: (a) Part cross section; (b) 3D view.   214   Figure 6.47 Predicted (solid line) and experimental (broken line) cutting forces: (a) Only left-side milling; (b) only right-side milling; (c) double-sided milling.   215  Table 6.9 Average forces derived from experiment and theoretical calculations.   Average theoretical force (N) Average experiment force (N) % Prediction error ((Theory-Exp)/Exp.. ..*100) Figure 6.47a Left-side x0-force 105 118 -11 % Left-side y0-force 138 147 -6.1 % Left-side z0-force -133 -132 -0.75 % Figure 6.47b Right-side x0-force 105 116 -9.4 % Right-side y0-force 138 153 -9.8 % Right-side z0-force 133 141 -5.6 % Figure 6.47c Double-sided x0-force 211 240 -12 % Double-sided y0-force 276 305 -9.5 % Double-sided z0-force 0.11 6.1 -98 %  Three cutting tests have been conducted: Only left-side milling, only right-side milling and double-sided milling. The predictions and experiment results for all tests are given in Figure 6.47a-c and the average forces are compared in Table 6.9. Test conditions of milling AISI 1045 steel part: N=8-tooth; Ω=250-rev/min; fr=0.16-mm/rev; dry cutting (with no cutting fluid); orthogonal cutting edge geometry (0-deg normal rake and 0-deg cutting edge inclination angle) with *r 15-degκ = cutting edge angle; ap=0.5-mm depth of cut from each side. The geometry is discretized by dz=0.1-mm axial segment height (q=5-segments in tool-workpiece contact zone). The time step of the simulation is t∆ =0.6-ms. The orthogonal cutting database for steel is from Equation (4.22). The angular location, pitch angle and radial runout is found in Table 6.8. The average forces (Table 6.9) of the single sided cutting are predicted accurately, within 11% error. Prediction accuracy of average forces in z0-direction of double-sided milling tool is low due to the geometrical differences of left- and right-side cutters. As seen in Figure 6.47, the peak amplitudes do not match precisely. The orthogonal cutting edge of the cutter creates an impact force during the entry of the cutter into the plate, thus exciting the dynamometer to produce peaks. A non-zero cutting edge inclination angle could reduce the effect of the impacts.  216    Figure 6.48 Experiment setup with flexible workpiece: (left) CAD schematics; (right) photo.   Figure 6.49 Pre-set spacer width of the double-sided milling tool.   Figure 6.50 Engagement condition of double-sided cutting tests.  217  Milling experiments with parallel cutters having single tooth: A single detachable insert (KOVA XNEX080608), which avoids the presence of runout, is used at each side of the double-sided cutter, as is shown in Figure 6.43c. A flexible workpiece set-up is used to verify the prediction of the stability. Constant 400-kgf.cm torque is applied on M8 screws to clamp the workpiece (Figure 6.48); this ensures repeatable dynamics of the workpiece when it is demounted and mounted. A spacer is put in between the cutters to fix the machined width of the workpiece at 23mm, see Figure 6.49. The initial width of the workpiece is 24-mm (Figure 6.48a), thus the spacer between the cutters pre-sets the axial depth of cut to 0.5-mm from each side.The positioning of the cutter reference, with respect to workpiece, is set by removing the left-side cutter and measuring the position of the insert tip at the right-side cutter. The radial depth of cut of the operation is set by positioning the insert tip 2-mm below the machined zone of the workpiece. As shown in Figure 6.50, the periphery circle of the tip of the insert (zero depth of cut location) is set 2-mm below the machined zone on the workpiece. The entrance into the cut happens 25-mm above that point for the tip of the insert. According to the engagement condition in Figure 6.50, tool entry and exit angles change along the cutting edge, as shown in Figure 6.51a-b.    Figure 6.51 Tool entry (a) and exit (b) angles for the indexable double-sided milling tool.  218  The cutter center to insert tip distance is 109.7-mm, then the cutter entry and exit angles along the cutting edge are found as in Equation (6.16). For 25-mm machined zone thickness, the radial entry and exit distances are identified as:  ( )ex st ex109.72 52.8 mm; 25 27.8 mm,2a a a= − = = − − =  (6.62) which leads to the entry (Figure 6.51a) and exit (Figure 6.51b) angles at each axial segment-k, from Equation (6.15), as:  st exst,jk ex,jkjk jkarccos ; arccosa aR Rφ φ   = = −         . (6.63) The finite element simulation of the cantilevered plate gives dominant bending (762 Hz) and torsional (2102 Hz) vibration modes, as shown in Figure 6.52. Standard impact hammer test is done using accelerometer, at the location shown in Figure 6.53 (A=48-mm; B=4-mm). Identified  modal parameters (from Cutpro [118]) of first mode: n,1,wω =739-Hz, 1,wζ =1.2%, z,k,1,w 0.99-[1 kg]u = ; of second mode: n,2,wω =2089-Hz, 2,wζ =0.37[%], z,k,2,w 2.13-[1 kg]u = . The difference in identified and predicted natural frequencies can be attributed to the assumption that the clamps are fixed rigidly. Fitted FRF is constructed by identified modal parameters. Figure 6.54a-b show the measured FRF and fitted FRF around the frequencies of first two vibration modes. Identified modal parameters are used during the dynamics analysis.At each time instance, the workpiece is assumed to have 6 dynamic levels, i.e., s=3 in Equation (4.79) for each cutter. Dynamic level definition is similar to the multifunctional tool case; the difference is that the forces are lumped onto the dynamic levels on the workpiece at each radial location r=1,2,3 (in general r=1..s) on the workpiece. Force functions LjkiF  and RjkiF  for left and right side cutters are 219  found from Equation (4.67) using the transformation matrices given in Equation (4.48) and Equation (4.49).  (a)  (b)  Figure 6.52 First two vibration mode shapes from finite element simulation: (a) Bending mode shape at 762-Hz; (b) Torsional mode shape at 2102-Hz.  Figure 6.53 Measurement at selected  point (at A and B distances) on the steel part.  (a) Around first vibration mode (b) Around second vibration mode   Figure 6.54 Real and imaginary parts of measured and fitted FRF.   220   Figure 6.55 Predicted limit depth of cut per side for stable double sided milling.  The predicted stability is given in Figure 6.55 for the depth of cut per side. AISI 1045 steel part is machined with the radial depth condition given in Figure 6.50. Average FRF (Figure 6.54) is assumed along the tool-workpiece contact zone. AISI 1045 steel database given in Equation (4.22). KOVA XNEX080608 inserts (Figure 3.11) used with double milling cutter (left-hand side Figure 3.19c; right-hand side Figure 3.19b).   (a)  (b)  Figure 6.56 Identified cutting edge of the two-insert end mill: (a) View in Frame-i; (b) close up view.  221   Figure 6.57 Predicted forces compared to the experimental results reported by Eynian [75].  6.7 Application-6: Prediction of Cutting Forces for a Two-Insert Indexable End Mill The generalized model is applied on an indexable end mill produced (Sandvik R390-020A20L-11L) with two inserts having 0.2 mm nose radius (R390-11 T302E-PM-4240). The experimental results were reported by Eynian [75] for this tool, but only the CAD model of a similar inserts (R390-11T3 04M-PM 1025) with 0.4 mm nose radius is available. The placement of inserts on the tool body is given in Figure 3.26, and the modelled cutting edge is shown in insert reference frame (Frame-i) in Figure 6.56a. The cutting edge geometry is modified to have a nose radius of 0.2 mm as shown in Figure 6.56b. The runout values were not reported in Ref. [75]. The predicted cutting forces reasonably agree with the experimental measurements reported by Eynian [75], see Figure 6.57. Simulated spindle speed is Ω=4125-rev/min; number of inserts is N=2-tooth; feedrate is fr=0.2-mm/rev; number of tool workpiece contact segments is q=ap/dz=0.25/0.001=250. 6.8 Application-7: Three Holders with the Same General Purpose Insert KOVA insert XNEX080608 (Figure 3.11) has been used on three different tool holders to illustrate the adaptability of the proposed model on face milling (Figure 3.19b), drilling (Figure 3.23) and double-sided milling (Figure 3.19a). The experimental validation of double-sided 222  milling is presented in Section 6.6 and the applications for face milling and drilling are given below.   Face Milling Cutter: The geometry of the face mill is shown in Figure 6.58. The radial runout of inserts are measured when the cutter was mounted on the spindle. Forces are simulated with ns=360 (number of points in one spindle period), and they are given in Figure 6.59. Slot milling operation is simulated: Spindle speed is 1000-rev/min; axial depth of cut is 2-mm; feedrate is 1.0-mm/rev for machining AISI 1045 steel (Figure 6.59a-b), and it is 0.5-mm/rev for milling Aluminum 7050-T7451 (Figure 6.59c-d).    Figure 6.58 KOVA U140 face mill geometry. (Courtesy of KOVA.)  223      Figure 6.59 Face milling operation cutting force prediction.  Drilling Cutter: The drilling tool geometry with two inserts is modeled as shown in Figure 6.60. Axial runout is measured after the cutter is installed on the spindle. Force is simulated with ns=360 (number of points in one spindle period) and is presented in Figure 6.61.  Simulated spindle speed is 1000-rev/min, and feedrate is 0.1-mm/rev. There exists a 0.5-mm diameter pre-drilled pilot hole. Workpiece materials of simulations in Figure 6.61a-b is AISI 1045 steel, and in Figure 6.61c-d is Aluminum Al7050-T7451.  224   Figure 6.60 KOVA U502 drill geometry. (Courtesy of KOVA.)     Figure 6.61 Drilling operation cutting force predictions. 225  6.9 Application-8: Parallel Turning with Two Tools A cylindrical part is turned from two opposite sides simultaneously in parallel turning application, as reported in Ref. [139]. The regenerative delay between the two tools is usually equal to half of the spindle period, but a more general form is obtained here with the application of Equation (4.52). Assuming constant dynamics along the tool-workpiece contact for each tool, the regenerative, dynamic chip thickness of the two tools can be expressed as:   1ki 1i 2 i 2ki2ki 2i 1 i 1ki( );( ),t Tt T∆ = − −∆ = − −q q qq q q (6.64) where the dynamic chip thickness 1i∆q  for tool #1 is related to the delay displacement 2 i 2ki( )t T−q  of tool #2, and the dynamic chip thickness 2i∆q  for tool #2 is related to the delay displacement 1 i 1ki( )t T−q  of tool #1. The displacement vector for each tool is expressed from Equation (4.51), but together with the constant dynamics along tool workpiece contact (k=1..q) for each tool:   [ ][ ]T1i 1i 1i 1i 1iT2i 2i 2i 2i 2i;.x y zx y zθθ==qq (6.65) The dynamic shear force vector is rewritten by substituting Equation (6.64) in Equation (4.61):  ( )( )cd d1ki 1ki 1i 2 i 2kicd d2ki 2ki 2i 1 i 1ki( ) ;( ) ,t Tt T= − −= − −F F q qF F q q (6.66) where the force component d1kiF  and d2kiF  vectors for tool #1 and tool #2 are given in Equation (4.62). If the flexible workpiece is added to the formula, Equation (6.64) has to be modified since the delay vibrations of the workpiece affects both tools [63]. For a rigid workpiece, the mode shape matrix is composed of flexible tool #1 and tool #2, as: 226   [ ] [ ]t1 t2 t1 t28x( )m m+U = U U , (6.67) where tool #1 and tool #2 mode shape matrices are t1U  and t2U  respectively; mt1 and mt2 are the number of vibration modes of each tool. Assuming a continuous turning operation, the in cut parameters for both tools are set as 2,1ki 2,2ki 1g g= = , as in Equation (4.80):  ( )[ ]1,1k 3,1ki 1kii 8x11,2k 3,2ki 2kiqkqkg gg g   =     ∑∑FFF. (6.68) The first 4 rows of the force ( iF ) vector are related to tool #1, and the last 4 rows are related to tool #2. The forces generated on each tool are coupled due to the delay effect in Equation (6.66), which affect  the forces expressed in Equation (5.23) as follows: Static shear force vector from Equation (5.24):  [ ]cs1,ki 3,1ki 1kicsi 8x1 cs1,ki 3,2ki 2kiqkg gg g =   ∑FfF. (6.69) Ploughing force vector from Equation (5.27):  [ ]es1,ki 3,1ki 1kiesi 8x1 es1,ki 3,2ki 2kiqkg gg g =   ∑FfF. (6.70) Dynamic shear force component from Equation (5.25):  [ ]d1,ki 3,1ki 1kicdi 8x1 d1,ki 3,2ki 2kiqkg gg g  = ⋅     ∑F 0f U0 F. (6.71) Process damping force vector from Equation (5.28):  [ ]p1,ki 3,1ki 1kiedi 8x1 p1,ki 3,2ki 2kiqkg gg g  = ⋅     ∑F 0f U0 F. (6.72) 227  Delay related force vector has an updated definition from Equation (6.66) as:  ( ) ( )d1,ki 3,2ki 2kic i 2ki c i 1kid1,ki 3,1ki 1kiqkg gt T t Tg g      − ⋅ ⋅ − + ⋅ ⋅ −            ∑0 00 FU UF 00 0Γ Γ , (6.73) where the combined modal displacement vector is composed only of tool vibration modes:  ( ) ( )( )t1 ic it2 ittt =    ΓΓΓ. (6.74) The force vector is constructed from Equation (5.23): ( ) ( )dcs cd 1,ki 3,2ki 2kii i i c,i c i 2ki c i 1kid1,ki 3,1ki 1kies edi i c,i... .qkg gt T t Tg g      + − ⋅ ⋅ − + ⋅ ⋅ −            + +∑0 00 FF f f U UF 00 0f f Γ Γ ΓΓ= (6.75) The remaining part of the formulation is the same as the general model given in Section 5.2.   Geometry: Figure 4.14 shows the two dimensional geometry when both tools have identical nose radii. The transformation matrices for the right hand turning operation of both tools are found using Equations (4.45) and (4.46). The angular locations for the tools are taken as 1ki 90φ =deg for tool #1 (j=1), and 2ki 702φ = deg for tool #2 (j=2). The angular location of each cutting edge element Sjk along each tool is assumed to be constant.   Stability calculation and comparison to the literature: In Figure 4.7, the cutting edge elements S1k on tool #1 and S2k on tool #2 are at the same radial segment-k along the tool workpiece contact. There are three possible delay conditions at the radial segment-k: 228  • If both tools are in cut (g1,1k=g1,2k=1 and g3,1ki=g3,2ki=1), then the delay period of the tools is equal to half of the spindle rotation period at the radial segment-k: s s1ki 2ki;2 2T TT T= = ; • If only tool #1 is in cut (g1,1k=g3,1ki=1 and g1,2k=g3,2ki=0), then the delay period for element S1k is equal to the full spindle period: 1ki sT T= . • If only element S2k of tool #2 is in cut (g1,2k=g3,2ki=1 and g1,1k=g3,1ki=0), then the full spindle rotation period is the delay period for element S2k: 2ki sT T= . Budak and Ozturk [59] reported parallel turning experiments (Figure 6.62), which were used to validate various stability prediction methods reported in the literature [61]. They fixed the milling spindle to install the tool #1 and used the turret chuck to fix the tool #2. The tools are moved in z0 by the two individual feed drives. If the two drives are not synchronized to be located at same axial coordinates, there would be an axial offset between the tools. The effect of this offset on the chip thickness is the same as axial runout, as is shown in Figure 4.4. Delay period of the tools would be affected as well.   Figure 6.62 Parallel turning setup. (Taken from Figure 1 and Figure 2 of Ref. [59].)   229  The predicted stability chart proposed by the generalized model is compared against the experimental and simulation results reported in Refs. [59,61], see Figure 6.63. Modal parameters are given below each chart (Figure 6.63a-b). The following assumptions are adopted from the reported tests: Sharp cutter is assumed for the inserts, i.e., the nose radii are zero and the true cutting edge angles are *r 90κ = deg for each element Sjk. Zero axial offset is assumed between tools; both tools are only flexible in the z0 direction, only the radial cutting force acting in z0 direction affects the dynamics. As a result, only the radial cutting coefficient Krc affects the stability.    Figure 6.63 Comparison of the predicted stability chart (modelled) to the literature.   230  Chapter 7: Conclusions and Future Research Directions  7.1 Conclusions Metal cutting tools have edges which may have various shapes such as slanted edge with a nose radius in turning, helical edge in milling or curved lip in drilling operations. Traditional metal cutting studies discretize the curved cutting edge of the tool into small, differential linear segments. The cutting force generated by each differential segment is predicted as a function of chip thickness and cutting force coefficients which depend on the tool-workpiece material pair. The chip thickness is modeled as a function of rigid body kinematics of individual machining operation, relative vibrations between the tool and workpiece and geometry of each cutting tool. The total force acting on the tool body is predicted by integrating the distributed differential forces along the entire cutting edge which is engaged with the workpiece. The traditional approach requires dedicated mathematical models for each tool geometry and machining operation. This thesis introduces a novel approach in unifying all cutting operations with arbitrary tool geometries in one generalized mathematical model. One parametric, mathematical model simulates all machining operations, as opposed to developing dedicated models for each tool geometry and operation; it simulates cutting forces, torque, and power of the machining process; surface form errors on the workpiece; relative tool-workpiece vibrations and chatter stability lobes for all turning, boring, drilling and milling operations. The generalized mathematical model is achieved with the following contributions to the literature: The tool’s cutting edges are either discretized as axial disk segments for milling, or as radial ring segments for turning, boring and drilling operations. Each differential cutting edge element 231  is defined by 15 geometric parameters which define the edge geometry, rake face of the tool, location of the edge in Cartesian coordinate system, axial and radial runouts and the engagement conditions of the edge with the workpiece. The geometric parameters of the cutting edge is transformed to tool body by orienting the edge in accordance with the kinematic model of the operation and tool geometry. The general transformation matrices are configured parametrically to define any tool for any metal cutting configuration. It is shown that solid tools as well as indexable cutter bodies can be modeled by the proposed approach, which is the foundation in generalizing the metal cutting operations. The model is demonstrated for milling and drilling operations, multifunctional drilling/boring tools, and for solid tool with arbitrary (i.e., serrated/undulated) edge of end mills and twist drills. The chip geometry that is removed by each differential cutting edge element is modeled by considering both rigid body motion and structural vibrations of tool-workpiece system. The runout of the edge is considered. The time delay contributed by the regenerative vibrations is modeled as a function of machining operation and tool geometry. The differential cutting force for the edge segment is modeled by considering both chip shearing and ploughing forces. The differential forces are integrated in tool body coordinates to evaluate the total force acting on the tool and workpiece. It is shown that the proposed model can predict cutting forces for turning, boring, drilling, milling and variety of multifunctional tool geometries. The relative vibrations between the tool and workpiece can either be modeled at a lumped point (e.g., tip of the end mill) or distributed along the tool-workpiece contact zone (e.g., flexible end mill). The cutting forces can be distributed by considering the mode shapes of the structure. The equation of motion between the cutting process and machine tool/part is derived by 232  considering single or multiple delays depending on the tool geometry. The set of delay-differential equations are reduced to a set of first order differential equations, which are solved in semi-discrete time domain. Both stability as well as cutting forces, surface location errors, vibration amplitudes and stability diagrams are predicted for arbitrary tool and machining operations by setting the process parameters.  The proposed method is experimentally proven in turning, boring, drilling and milling. In addition to predicting the classical tools and operations, the thesis includes the applications on a multifunctional drill with multiple cutting edges and parallel face milling cutters in machining thin workpieces with position dependent dynamics.  The presented genaralized mathematical modeling of tools, operations, cutting mechanics, cutting dynamics and chatter stability is the first research in the literature. The generalized model allows the virtual simulation and optimization of various machining operations in an integrated manner in Compuer Aided Manufacturing (CAM) platforms.   7.2 Future Research Directions Both the concept and developed mathematical model in generalizing the process mechanics and dynamics are general, and can be extended to other machining operations as follows: • Current model is for 2-axis machining operations in xy direction of the machine-tool. The mechanics and dynamics model can be extended to include the additional feed direction (i.e., 3-axis operations), and to include varying feed along the tool axis (i.e., 5-axis operations). The fundamental mathematical models presented in this thesis required only coordinate transformations from the feed-cutting velocity plane to machine tool frame, which requires kinematic model of the machine. 233  • The generalized model can be adopted to simulate broaching operations by modeling the broach edge geometry and spacing of the broaching teeth only.  • As illustrated in the thesis, multifunctional tools can easily be modeled by the proposed general model. Complex operations such as turn-milling can also be simulated by the proposed generalized model by incorporating the forward and inverse kinematics of these machines.   • The cutting mechanics model can be improved by considering fklank wear, back cutting and tool indentation mechanics as suggested by Tuysuz et al. [48] for balll end milling operations. • The solution can be extended to include nonlinear chip thickness and velocity effects, based on the works of Montgomery and Altintas [19], Campomanes and Altintas [101] and Ratchev et al. [140]. Estimating the vibration amplitude is important for different tool and edge geometry, and it can help to predict tool life, force and vibration during the design.  • Variable delay along the cutting edge (the most general case) is presented in this thesis. 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G., 1989, “Modeling and Finite Element Analysis of Drill Bit Vibrations,” J. Vib. Acoust. Stress Reliab. Des., 111(2), p. 148. [139] Ozturk, E., and Budak, E., 2010, “Dynamics and Stability of Five-Axis Ball-End Milling,” J. Manuf. Sci. Eng., 132(2), p. 021003. [140] Ratchev, S., Liu, S., Huang, W., and Becker, A. A., 2004, “A flexible force model for end milling of low-rigidity parts,” J. Mater. Process. Technol., 153-154, pp. 134–138. [141] Seguy, S., Insperger, T., Arnaud, L., Dessein, G., and Peigné, G., 2009, “On the stability of high-speed milling with spindle speed variation,” Int. J. Adv. Manuf. Technol., 48(9-12), pp. 883–895.     245  Appendix A  Transforming the basis from Frame-1 to Frame-2  The Frame-2 origin and vectors can be represented in Frame-1 coordinates using translational and rotational transformations.  A.1 Pure translation If the relative Frame-1 position of the origin of the Frame-2 is only translational, then the following transformation is used:  [ ]T T1 1 1 12 2 2 2 2 2 2, , ,1 , , ,1 .x y z x y z  =  T (A.1) ( )1 1 12 2 2, ,x y z  is the Frame-1 representation of the ( )2 2 2, ,x y z  vector that is represented in Frame-2. The homogeneous transformation matrix 12T  only translates the pre-multiplied frame (Frame-2) to the new frame (Frame-1)  ( )1 1 2 1 2 1 22 x y zTrans , ,r r r=T , (A.2) where the 1 2 1 2 1 2x y z, ,r r r  are the relative x,y,z components in the Frame-1 of the origin of the Frame-2. The translation matrix equation of Equation (A.2) is:  ( )1 0 00 1 0Trans , ,0 0 10 0 0 1aba b cc   =   . (A.3) The a,b,c components are the x,y,z axis translations of the pre-multiplied frame (Frame-2) to transform it to the new frame (Frame-1). For example, if the origin 2r  of the Frame-2 represented in Frame-2 is:  [ ]T2 0 0 0 1=r , (A.4) then, its relative location in Frame-1 is found by Equation (A.1): 246   1 21 21 2 2 21 21 0 0 000 1 0;00 0 110 0 0 1xyzrrr         = ⋅ =           r r r . (A.5) The resulting vector is as expected:  T1 2 1 2 1 2 1 2 1x x zr r r =  r. (A.6) A.2 Pure rotations The transformation of the Equation (A.1) can be used for rotations around one of the x,y,z axes of the Frame-1. The functions for the rotation around x,y, and z axes are named Rotx,Roty and Rotz. If Frame-2 is obtained by rotating the Frame-1 around x-axis by α angle, the Rotx transformation is applied:  ( )1 0 0 00 cos sin 0Rotx0 sin cos 00 0 0 1α ααα α  − =   . (A.7) The last column of Rotx matrix is [ ]T0 0 0 1  because of the pure rotation and no translation. If Frame-2 is obtained by rotating the Frame-1 around y-axis by φ  angle, the Roty transformation is applied:  ( )cos 0 sin 00 1 0 0Rotysin 0 cos 00 0 0 1φ φφφ φ   = −  . (A.8) If Frame-2 is obtained by rotating the Frame-1 around z-axis by ψ  angle, the Rotz transformation is applied: 247   ( )cos sin 0 0sin cos 0 0Rotz0 0 1 00 0 0 1ψ ψψ ψψ−   =   . (A.9) , ,α φ ψ  angles are also called Euler angles [114]. The order of rotations is important because each transformation is done with respect to the previous frame. For large angle rotations, the difference is more pronounced than the small angle rotations. When the rotations are done in infinitesimal way, the accuracy is increased. The three procedures are compared in the following sections.   A.3 Procedure A: Large angle rotations The following two rotations are considered: 1. Rotating first rotate 30 [deg] around x-axis of Frame-1, and then rotate 60 [deg] around the y-axis of the intermediate frame, Frame-i:  ( )( )12A112A1=Rotx(30[deg]) Roty(60[deg]);0.500     0        0.866    00.433   0.866   -0.250   0;-0.750  0.500   0.433    0 0           0           0       1⋅   =   TT (A.10) 2. Rotating first rotate 60 [deg] around x-axis of Frame-1, and then rotate 30 [deg] around the y-axis of the intermediate frame, Frame-i*:  ( )( )12A212A2=Roty(60[deg]) Rotx(30[deg]);0.5000    0.4330    0.7500      0     0       0.8660    -0.5000     0.-0.8660   0.2500    0.4330      0    0         0              0             1⋅   =   TT (A.11) A test vector 2testr  is chosen in Frame-2 to check the accuracy of the transformation matrices: 248   2 Ttest [2 3 4 0]=r . (A.12) From two different order transformations of Equations (A.10) and (A.11) the test vector can be represented in Frame-1,  ( ) ( ) ( ) ( )( ) ( )1 2 1 2 1 2 1 2test 2 test test 2 testA1 A1 A2 A21 2 1 2test testA1 A2; ;4.464 5.2992.464 0.5981; .1.732 0.7500   0     0= =         = =         r T r r T rr r (A.13) The two orders give quite different vectors. The off-diagonal terms of the two transformations of Equations (A.10) and (A.11) are not equal, and they have a large difference. The difference is expected to decrease when the angles get smaller.  A.4 Procedure B: Small angle rotations  For small angles, the following two ways would give almost the same transformations: 1. Rotating first rotate 10 [deg] around x-axis of Frame-1, and then rotate 20 [deg] around the y-axis of the intermediate frame, Frame-i; 2. Rotating first rotate 20 [deg] around x-axis of Frame-1, and then rotate 10 [deg] around the y-axis of the intermediate frame, Frame-i*. Resulting  transformations are:  ( ) ( )1 12 2B1 B20.9397      0          0.3420       0 0.9397    0.0594    0.3368   0.0594    0.9848    -0.1632     0;-0.3368   0.1736    0.9254      0 0             0              0           1   = =   T T  0    0        0.9848    -0.1736    0-0.3420   0.1632   0.9254      0    0            0            0          1      . (A.14) The corresponding transformed test vectors are: 249   ( ) ( ) ( ) ( )( ) ( )1 2 1 2 1 2 1 2test 2 test test 2 testB1 B1 B2 B21 2 1 2test testB1 B2; ;3.247 3.4042.420 2.259; .3.549 3.507   0    0= =         = =         r T r r T rr r  (A.15) The vectors are closer to each other. To make it more accurate, the rotations can be done in multiple steps.  A.5 Procedure C: Rotating in multiple steps If the ns is the number of rotations for a rotation matrix, then the original input angles should be divided to ns and the transformation is done ns times. For example, to implement Procedure A in multiple steps, the following steps should be performed ns times: 1. Rotating first rotate 30/ns [deg] around x-axis of Frame-1, and then rotate 60/ns [deg] around the y-axis of the intermediate frame, Frame-i; 2. Rotating first rotate 60/ns [deg] around x-axis of Frame-1, and then rotate 30/ns [deg] around the y-axis of the intermediate frame, Frame-i*. For 10-steps of rotation, 10sn = :  ( ) ( )( ) ( )( )ss12 s sC112 s sC212C1= Rotx(30 [deg]) Roty(60 [deg]) ;= Roty(60 [deg]) Rotx(30 [deg]) ;0.5114    0.2224    0.8301   00.2655    0.8778   -0.3988   0-0.8173   0.4243   0.3898    0    0            0      nnn nn n⋅⋅=TTT ( )12C20.5114    0.2655    0.8173   00.2224    0.8778   -0.4243   0; .-0.8301   0.3988   0.3898    0      0        1      0           0            0        1         =         T  (A.16) Then, the transformation matrices become more comparable and the dependence on the order is weaker. Corresponding test vector representations in Frame-1 also become very close to each other: 250   ( ) ( ) ( ) ( )( ) ( )1 2 1 2 1 2 1 2test 2 test test 2 testC1 C1 C2 C21 2 1 2test testC1 C2; ;5.010 5.0881.569 1.380; .1.197 1.095   0    0= =         = =         r T r T rr rr (A.17) A.6 Finding the transformation matrix using the known vectors The 4-by-4 homogenous matrix transformation from Frame-2 to Frame-1 can be found if the representation of the four vectors of Frame-2 are known in Frame-1. The three axis vectors of Frame-2 are the easiest ones to use, as:  [ ] [ ] [ ]T T T2 2 21 0 0 0 ; 0 1 0 0 ; 0 0 1 0= = =i j k . (A.18) Their vector representations 0 0 0i i i, andi j k  in Frame-1 are assumed as known. The fourth vector would be position of the origin of Frame-2:  [ ]T2 0 0 0 1 .=r  (A.19) Its vector representation 1 2r  in Frame-1 is also assumed known. Then, the transformation matrix 12T  from Frame-2 to Frame-1 is found by joining the four vectors:  1 1 1 1 2 1 2 22 2 2 2 2 2 2 2 2 2 41 1 1 1 1 22 2 2 2: : : : : : ; : : : ;: : : .     = ⋅ =      =  i j k r T i j k r i j k r IT i j k r  (A.20) The form of homogeneous transformation matrix is:  1 1 21 221x3 1 =    R rT0. (A.21) Although the inverse of the rotation matrix is its transpose:  ( ) ( )1 T2 1 2 11 2 1 2;−= =R R R R , (A.22) 251  then, the inverse of the homogeneous transformation matrix is:  ( ) ( ) ( )T T1 1 1 2 2 2 1 212 1 2 22 2 1 11 2 1 11x31x3; ;11−    − − = = =        R R r R R rT T T T00. (A.23)  

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