UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Virtual five-axis flank milling of jet engine impellers Ferry, William Benjamin Stewart 2008

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2008_fall_ferry_william.pdf [ 5.01MB ]
Metadata
JSON: 24-1.0066463.json
JSON-LD: 24-1.0066463-ld.json
RDF/XML (Pretty): 24-1.0066463-rdf.xml
RDF/JSON: 24-1.0066463-rdf.json
Turtle: 24-1.0066463-turtle.txt
N-Triples: 24-1.0066463-rdf-ntriples.txt
Original Record: 24-1.0066463-source.json
Full Text
24-1.0066463-fulltext.txt
Citation
24-1.0066463.ris

Full Text

 VIRTUAL FIVE-AXIS FLANK MILLING OF JET ENGINE IMPELLERS   by  WILLIAM BENJAMIN STEWART FERRY   B.A.Sc., Queen’s University, 2000 M.A.Sc., University of Windsor, 2002     A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY   in   THE FACULTY OF GRADUATE STUDIES   (Mechanical Engineering)       THE UNIVERSITY OF BRITISH COLUMBIA  (Vancouver)     July 2008    © William Benjamin Stewart Ferry, 2008  ii Abstract This thesis presents models and algorithms necessary to simulate the five-axis flank milling of jet-engine impellers in a virtual environment. The impellers are used in the compression stage of the engine and are costly, difficult to machine, and time- consuming to manufacture. To improve the productivity of the flank milling operations, a procedure to predict and optimize the cutting process is proposed. The contributions of the thesis include a novel cutter-workpiece engagement calculation algorithm, a five-axis flank milling cutting mechanics model, two methods of optimizing feed rates for impeller machining tool paths and a new five-axis chatter stability algorithm. A semi-discrete, solid-modeling-based method of obtaining cutter-workpiece engagement (CWE) maps for five-axis flank milling with tapered ball-end mills is developed. It is compared against a benchmark z-buffer CWE calculation method, and is found to generate more accurate maps. A cutting force prediction model for five-axis flank milling is developed. This model is able to incorporate five-axis motion, serrated, variable-pitch, tapered, helical ball-end mills and irregular cutter-workpiece engagement maps. Simulated cutting forces are compared against experimental data collected with a rotating dynamometer. Predicted X and Y forces and cutting torque are found to have a reasonable agreement with the measured values. Two offline methods of optimizing the linear and angular feeds for the five-axis flank milling of impellers are developed. Both offer a systematic means of finding the highest feed possible, while respecting multiple constraints on the process outputs. In the  iii thesis, application of these algorithms is shown to reduce the machining time for an impeller roughing tool path. Finally, a chatter stability algorithm is introduced that can be used to predict the stability of five-axis flank milling operations with general cutter geometry and irregular cutter-workpiece engagement maps. Currently, the new algorithm gives chatter stability predictions suitable for high speed five-axis flank milling. However, for low-speed impeller machining, these predictions are not accurate, due to the process damping that occurs in the physical system. At the time, this effect is difficult to model and is beyond the scope of the thesis.              iv Table of Contents Abstract ............................................................................................................................. ii Table of Contents ........................................................................................................... iv List of Tables ................................................................................................................. viii List of Figures.................................................................................................................. ix Acknowledgements .................................................................................................... xxiii Chapter 1: Introduction ............................................................................................... 1 Chapter 2: Literature Review .................................................................................. 10 2.1 Overview........................................................................................................... 10 2.2 Cutter-Workpiece Engagement Calculations.................................................... 11 2.3 Milling Mechanics / Process Simulation .......................................................... 18 2.4 Feed Rate Optimization Techniques ................................................................. 22 2.5 Chatter Stability Prediction............................................................................... 25 Chapter 3: Calculation of Cutter-Workpiece Engagement Maps for Five- Axis Flank Milling by the Parallel Slicing Method........................................ 32 3.1 Overview........................................................................................................... 32 3.2 Cutter-Workpiece Engagements by the Parallel Slicing Method (PSM).......... 33 3.2.1 Generation of Removal Volume and Updated Workpiece ....................... 35 3.2.2 Division of Removal Volume into Parallel Planes ................................... 41 3.2.3 Calculation of Tool-Slice Intersection Curves.......................................... 45 3.2.4 Calculation of Tool Swept Area ............................................................... 48 3.2.5 Subtraction of Tool Swept Area and Generation of Engagement Arcs .... 54 3.2.6 Conversion of Engagement Arcs to Tool Coordinate System and Creation of Engagement Polygon............................................................................ 56 3.2.7 Creation of Cutter-Workpiece Engagement Maps.................................... 59 3.3 Comparison of Parallel Slicing Method with Z-Buffer Scheme....................... 61 3.4 Summary........................................................................................................... 72 Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry....................................................................................................................... 74 4.1 Overview........................................................................................................... 74  v 4.2 Geometric Modeling and Kinematics of Five-Axis Flank Milling with Serrated, Variable-Pitch, Helical, Tapered Ball-End Mills.............................................. 76 4.2.1 Feed Calculations for Five-Axis Flank Milling ........................................ 77 4.2.2 Distribution of Feed-Per-Tooth................................................................. 81 4.2.3 Distribution of Chip Thickness................................................................. 84 4.2.4 Effects of Feed Variation Along the Cutter Axis on Chip Thickness....... 88 4.2.5 Effects of Cutting Edge Serrations on Chip Thickness ............................ 91 4.2.6 Static Chip Thickness Expression for Five-Axis Milling with Serrated, Variable-Pitch, Helical General End Mills ............................................... 94 4.3 Prediction of Cutting Forces ............................................................................. 96 4.4 Comparison of Simulations with Experiments ............................................... 104 4.5 Summary......................................................................................................... 111 Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths According to Multiple Feed-Dependent Constraints .................................. 113 5.1 Overview......................................................................................................... 113 5.2 Modeling of Peak Feed-Dependent Outputs and Constraints......................... 114 5.2.1 Cutting Torque ........................................................................................ 116 5.2.2 Chip Thickness........................................................................................ 116 5.2.3 Tool Deflection ....................................................................................... 118 5.2.4 Tool Stress .............................................................................................. 119 5.3 Feed Rate Optimization By Multi-Constraint Virtual Adaptive Control........ 123 5.3.1 Modeling of the Cutting Process and Estimation of Parameters for Multi- Constraint Virtual Adaptive Feed Control .............................................. 124 5.4 Feed Rate Optimization By Non-Linear Root Finding................................... 131 5.5 Scaling of Angular Feeds................................................................................ 134 5.6 Filtering of Optimized Feeds .......................................................................... 136 5.7 Results............................................................................................................. 139 5.8 Summary......................................................................................................... 144 Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling ............................................................................................................ 147 6.1 Overview......................................................................................................... 147  vi 6.2 Chatter Stability Formulation For Five-Axis Flank Milling with General Milling Tools .................................................................................................. 150 6.2.1 Dynamic Chip Thickness Formulation ................................................... 151 6.2.2 Dynamic Cutting Forces ......................................................................... 156 6.2.3 Zero Order Expansion of Directional Coefficients................................. 164 6.2.4 Dynamic Displacements of the Tool and Workpiece ............................. 169 6.3  Solution of Chatter Stability Formulation Using Nyquist Theory.................. 174 6.4 Transfer Functions of the Tool and Blade ...................................................... 181 6.4.1 Transfer Functions of Blade.................................................................... 181 6.4.2 Transfer Function of Tool....................................................................... 190 6.5 Implementation of the Five-Axis Chatter Stability Theory ............................ 191 6.5.1 Comparison of New Stability Theory with Classical Stability Lobes for a 2D Cylindrical End Mill ......................................................................... 192 6.5.2 Comparison of New Stability Theory with Time Domain Process Simulation (CUTPRO) for a Serrated, Variable-Pitch, Cylindrical-End Mill in 2D Milling....................................................................................195 6.5.3 Stability Prediction of a Five-Axis Finishing Operation ........................ 199 6.6 Summary......................................................................................................... 205 Chapter 7: Conclusions............................................................................................. 207 7.1 Conclusions..................................................................................................... 207 7.2 Recommended Areas for Future Work ........................................................... 213 Bibliography ................................................................................................................. 216 Appendix A: Analytical Geometry for the Parallel Slicing Method......... 226    A1    Analytical Tapered Ball-End Mill–Plane Intersection Curves ........................ 226    A2    Conic Coefficients for Various Conic Forms .................................................. 235    A3    Calculation of Conic Parameters from Normalized Conic Coefficients.......... 239    A4    Conic-Conic Tangent Line Solution ................................................................ 240    A4.1             Removal of Cross Tangents .................................................................... 248    A5    Knot-Conic Tangent Line Solution.................................................................. 249 Appendix B: Cantilever Model of Tool For Static Deflection and Transverse Stress Calculations ........................................................................... 252  vii    B1   Overview of Cantilever Model of the Tool....................................................... 252    B2   Stiffness, Force and Displacement Equations for a Timoshenko Beam Element....... ...................................................................................................... 253    B3   Assembly of the Nodal Force Vector................................................................ 256    B4   Assembly of the Global Solution...................................................................... 258                   viii List of Tables Table 6-1 – Comparison of Natural Frequencies of the IBR From FE Analysis and Experiments................................................................................................................... 186 Table 6-2 – Experimentally Measured Damping Ratios for Modes 1-4 of the IBR..................................................................................................................................187 Table 6-3 – Summary of Cutting Conditions Used to Compare New Stability Algorithm Against CUTPRO (Analytical Zero-Order Solution) ............................. 192 Table 6-4 – Variable-Pitch Flute Spacing of Cylindrical Cutter Given in Table 6-3 ......................................................................................................................................... 194 Table 6-5 - Summary of Cutting Geometry and Modal Parameters Used to Compare New Stability Algorithm Against CUTPRO (Time Domain Process Simulation) .. 197   ix List of Figures Figure 1-1 – Diagram of a typical gas turbine jet engine [35] ...................................... 1 Figure 1-2 – Point milling of impeller blades (left). Flank milling of impeller blades (right). ................................................................................................................................ 2 Figure 1-3 – (a) Illustration of typical roughing and finishing cutters used to flank- mill impellers. (b) Bottom view of a four-fluted variable-pitch cutter showing the unequal spacing of the flutes............................................................................................ 4 Figure 1-4 – Flowchart of the thesis project – Virtual Five-Axis Flank Milling of Jet Engine Impellers. .............................................................................................................. 7 Figure 2-1 – Illustration of a cutter-workpiece engagement map (right) for an impeller roughing operation. ......................................................................................... 12 Figure 2-2 – Illustration of a Z-buffer method............................................................. 14 Figure 2-3 – (a) Spence and Altintas’s [112] solid modeling CSG tree approach to obtaining cutter-workpiece engagements. (b) Yip-Hoi and Huang’s [134] cutter- workpiece engagement feature extraction method. ..................................................... 16 Figure 2-4 – (a) Slot, cylindrical and ball-end milling cutters [130]. (b) Simple model of a down-milling operation. Flutes intermittently enter and exit the workpiece. ... 19 Figure 2-5 – Cutting force variation for an adaptively controlled five-axis rough- milling operation on a titanium impeller [25]. ............................................................. 24 Figure 2-6 – (a) Total chip thickness depends on the static chip thickness, present vibrations and vibrations from a previous revolution. (b) The orthogonal cutting process formulated as a control feedback system. (c) The phase shift between vibration waves can lead to chatter (dynamically unstable)....................................... 25 Figure 2-7 – A typical stability lobe chart .................................................................... 26 Figure 2-8 – Formulation of the dynamic milling problem [114]............................... 27 Figure 3-1 – Illustration of the Parallel Slicing Method (PSM) [56], [57]. ................ 34 Figure 3-2 – The seven parameter tool model used in Engin and Altintas [44] (upper left) and illustrations of helical, cylindrical-end, ball-end and tapered ball-end mills. ........................................................................................................................................... 36 Figure 3-3 – Illustration of a tool path segment, defined by a series of GOTO commands [52], [53]........................................................................................................ 38  x Figure 3-4 – Swept areas may not update the workpiece accurately if the slice axis is chosen improperly [56], [57]. ......................................................................................... 41 Figure 3-5 – (a) A practical method of determining the slice axis, { }S .  (b) Removal volume sliced along, { }S . [56], [57]. .............................................................................. 43 Figure 3-6 – Illustrations showing various tapered ball-end mill / plane intersection shapes [56], [57]. (a) Single conic (b) Composite conic (c) Truncated conic (d) Composite truncated conic............................................................................................. 47 Figure 3-7 – Calculation of the approximate tool swept area between two ellipses [56], [57]. .......................................................................................................................... 49 Figure 3-8 – Flow chart of the algorithm used to calculate the swept area between complex intersection shapes [56], [57]........................................................................... 52 Figure 3-9 – Graphical examples of the various steps of the algorithm used to calculate swept areas between complex intersection shapes [56], [57]....................... 53 Figure 3-10 – (a) An example of a self-intersecting tool path. (b) A more common example of self-intersection that can occur in five-axis flank milling. [56], [57]....... 55 Figure 3-11 – An engagement arc formed by the intersection of a tool-plane intersection curve and the updated removal volume slice. ......................................... 56 Figure 3-12 – (a) The engagement arcs are converted from the global coordinate system to the tool coordinate system. (b) The engagement polygon is created by moving around the perimeter of the arcs and joining their end points with linear segments. [56], [57].......................................................................................................... 57 Figure 3-13 – Close-up of a typical cutter-workpiece engagement map [56], [57] ... 60 Figure 3-14 – Figure showing how the MAL-VMI system obtains cutter-workpiece engagement maps. ........................................................................................................... 62 Figure 3-15 – The impeller roughing tool path used to compare the Parallel Slicing Method and MAL-VMI cutter-workpiece engagement calculation schemes [52], [53]. ................................................................................................................................... 63 Figure 3-16 – Calculation of the removal volume and finished workpiece for a five- axis flank milling operation on an integrally bladed rotor (IBR) [56], [57]. ............. 64 Figure 3-17 – (a)-(b) The removal volume sliced into 30 planes along the average tool axis orientation. (c)-(f) The removal volume at various stages of the flank- milling operation. [56], [57]............................................................................................ 65 Figure 3-18 – Table of engagement geometry for the IBR roughing tool path......... 67  xi Figure 3-19 – Illustration of cutter-workpiece engagement maps from a removal volume (a) 30 slices (b) 60 slices. [56], [57]. .................................................................. 68 Figure 3-20 – Comparison of engagement maps from the PSM (dashed boundaries) and the MAL-VMI (shaded boundaries) [56], [57]...................................................... 69 Figure 3-21 – (a) Illustration of an intersection grid returned by the API in Vericut. (b) Since the grid only stores a Boolean (true/false) intersection value, this may cause a truncation of the cutter-workpiece engagement zone in five-axis flank milling............................................................................................................................... 71 Figure 4-1 – Picture of an impeller being flank milled on a physical machine. ........ 74 Figure 4-2 – A five-axis sculptured surface machining tool path segment [52], [53] . ........................................................................................................................................... 77 Figure 4-3 – Illustration of cutter motion in five-axis flank milling [52], [53]. ......... 79 Figure 4-4 – Illustration showing how the total local feed, { }TF , can be split into a horizontal feed, { }X Tf X , and a vertical feed, { }Z Tf Z ................................................ 82 Figure 4-5 – Chip thickness distribution due to horizontal feed (feed along the { }TX direction) [52], [53].......................................................................................................... 84 Figure 4-6 – Chip thickness distribution due to vertical feed (feed parallel to the tool axis, { }TZ ) [52], [53]........................................................................................................ 86 Figure 4-7 – Chip thickness distribution due to a combination of horizontal and vertical feeds [52], [53].................................................................................................... 87 Figure 4-8 – (a) Feed variation along the cutting edge from combined translational and angular motion. (b) Total feed vector varies at each element. (c) Feed coordinate system at each element is shifted by an angle, ( )s zθ , relative to the feed coordinate system at the tool tip. [52], [53]................................................................... 89 Figure 4-9 – (a) Profile of a serrated cylindrical cutter. The radii of the flutes are different at each cross section. (b) Approximate path of each flute (c) and (d). [52], [53]. ................................................................................................................................... 93 Figure 4-10 – Some cutter-workpiece engagement maps from the IBR roughing tool path [52], [53]. ................................................................................................................. 98 Figure 4-11 – Close-up of the engagement map for tool path segment 67 [52], [53]. Each block is defined by four parameters: bz , ba , stφ  and exφ . .................................. 99  xii Figure 4-12 – Workpiece and tool path used to compare predicted and simulated cutting forces for a roughing operation on an integrally bladed rotor (IBR) [52], [53] ......................................................................................................................................... 105 Figure 4-13 – Comparisons of raw and filtered measured cutting forces and torques [52], [53]. (a)-(d) Raw and filtered data vs. time for the entire tool path. (e)-(f) Fourier spectrums of raw and filtered X and Z forces. (g)-(h) Close-ups of X and Z forces.. ............................................................................................................................ 107 Figure 4-14 – (a)-(d) Predicted and filtered experimental data vs. time for the entire tool path. (e)-(h) Close-ups of predicted and measured X-forces vs. time at various parts of the tool path. [52], [53]. .................................................................................. 109 Figure 5-1 – Finite element model of the tool with point-load cutting forces at the middle of each cutting force element [54], [55]. ......................................................... 115 Figure 5-2 – Cross section of a typical impeller milling tool showing rake and clearance angles............................................................................................................. 117 Figure 5-3 – Block diagram of the virtual cutting process [54], [55]. ...................... 125 Figure 5-4 – The z-domain block diagram of a closed loop, single-input single-output (SISO), virtual adaptive feed control system for a general, constraint normalized output (CNO) [54], [55]. ............................................................................................... 128 Figure 5-5 – Block diagram of the closed-loop multi-constraint virtual adaptive feed control system [54], [55]. .............................................................................................. 129 Figure 5-6 – Virtual cutting process modeled as a non-linear function of tool tip feed [54], [55]. ........................................................................................................................ 131 Figure 5-7 – Lines represent different CNOs as a function of feed [54], [55]. ........ 133 Figure 5-8 – Illustration showing how angular feeds must be scaled with tool tip feeds to preserve tool path geometry [54], [55]. ......................................................... 135 Figure 5-9 – Optimized tool tip feed vs. filtered optimized tool tip feed. (a) Filtering used to blend a sharp transition in feed where the tool enters the workpiece. (b) Filtering used to reduce fluctuations in the optimized feed profile. (c) Filtering used to blend a sharp transition in feed where the tool exits the workpiece. ................... 138 Figure 5-10 – Upper plots show tool stress and tool tip feed for the entire tool path. Middle and lower plots show close-ups of the tool stress limiting the output. [54], [55]. ................................................................................................................................. 141 Figure 5-11 – Multi-constraint virtual adaptive feed control optimization results for the IBR roughing tool path [54], [55]. ......................................................................... 142  xiii Figure 5-12 – Comparison of the multi-constraint virtual adaptive feed control and non-linear root finding algorithms [54], [55].............................................................. 143 Figure 6-1 – Illustration of the tool cutting the workpiece with vibrations in 2D milling............................................................................................................................. 148 Figure 6-2 – (left) Irregular cutter-workpiece engagement map, typical of five-axis flank milling. (right) The five-axis stability solution breaks this map into a series of discrete elements. .......................................................................................................... 150 Figure 6-3 – Figure showing the transformation of displacements at the cutting edge ( , ,j j ju v w ) to those in the feed coordinate system at the tool tip ( , ,j j jx y z ) ............ 151 Figure 6-4 – Illustration showing how the total chip thickness is a summation of the static and dynamic vibrations from the present and previous tooth periods [43]. . 155 Figure 6-5 – Square wave switching function [94]..................................................... 156 Figure 6-6 – Illustration showing a loss of contact when relative flute lengths are small in relation to the feed-per-tooth, jc .................................................................. 172 Figure 6-7 – Illustration showing typical Nyquist plots ............................................ 177 Figure 6-8 – Increasing the depth of cut in five-axis impeller machining can cause unpredictable changes in the cutter-workpiece engagements maps. ....................... 180 Figure 6-9 – General trend of the orthogonal-to-oblique cutting coefficients with chip thickness for 6 4Ti Al V− − alloy........................................................................... 182 Figure 6-10 – CAD model of an integrally bladed rotor (IBR)................................. 183 Figure 6-11 – Illustration of the finished integrally bladed rotor (IBR) used for test purposes ......................................................................................................................... 185 Figure 6-12 – (top) 256 nodes of the finite element blade structure were selected for FRF measurements by dividing the back area of the blade into a 16x16 grid. (bottom) The 3x3 FRF matrix at each node is obtained by measuring the displacements in x,y and z directions from an applied unit force. ........................... 187 Figure 6-13 – Comparison of stable, unstable and critically stable cutting conditions given by CUTPRO and the new chatter stability solution ........................................ 193 Figure 6-14 – Nyquist plots showing increased stability from use of a variable-pitch cutter. ............................................................................................................................. 195 Figure 6-15 – (a) A five-fluted serrated, variable-pitch, cylindrical end mill and (b) its corresponding serration wave profile used to compare the new, frequency domain stability algorithm vs. CUTPRO’s time domain process simulation.......... 196  xiv Figure 6-16 – (a) Stability comparison (CUTPRO vs. new stability algorithm) for the cutter defined in Figure 6-15 and cutting conditions given in Table 6-5. (b) Resultant force vs. time and FFT of resultant force for stable and unstable conditions......... 198 Figure 6-17 – (a) Tool position for the five-axis chatter stability analysis. Illustration shows the closest nodes to the engaged area in the 256 node selection set. (b) The cutter-workpiece engagement map for this tool move............................................... 200 Figure 6-18 – Nyquist plot of det[ ( )]I jω− Λ  for the five-axis finishing operation on the integrally bladed rotor ........................................................................................... 202 Figure 6-19 – Typical stability chart showing a process damping region at low cutting speeds ................................................................................................................ 203  Figure A-1 – A tapered ball-end mill. ......................................................................... 227 Figure A-2 – (left) The seven parameter tool model used in Engin and Altintas [44]. (right) Tapered ball-end mills have dimensions of the form shown......................... 230 Figure A-3 – Illustrations showing various tapered ball-end mill / plane intersection shapes [56], [57]. (a) Single conic (b) Composite conic (c) Truncated conic (d) Composite truncated conic........................................................................................... 232 Figure A-4 – Parameters of a rotated ellipse.............................................................. 235 Figure A-5 – Parameters of a rotated hyperbola. ...................................................... 236 Figure A-6 – Parameters of a rotated parabola. ........................................................ 237 Figure A-7 – Parameters of a circle............................................................................. 238 Figure A-8 – Examples of possible conic-line intersections with a line and an ellipse. ......................................................................................................................................... 242 Figure A-9 – Possible combinations of tangent lines between two ellipses.............. 247 Figure A-10 – Illustration of the four tangent lines that can be drawn between the two ellipses shown [54], [55]. ........................................................................................ 248  Figure B-1 – Illustration of the cantilever finite element model of the tool used to calculate tool deflection and transverse stress at each node [54], [55]. .................... 252 Figure B-2 – Nodal displacements and forces for the Timoshenko beam element in [30], [54], [55]................................................................................................................. 254 Figure B-3 – Free body diagrams of x,y and z forces on each beam element [54], [55]. ................................................................................................................................. 256  xv Figure B-4 – Illustration showing the finite element model of the cutting tool and the assembly of the global stiffness matrix and global nodal force vector [54], [55]. ... 259                         xvi Nomenclature qa  : Engagement block height ( )q b≡ ; Depth of cut ( )q c≡ ; Pole of   first-order virtual cutting process transfer function ( )q g≡ . A  : Area. { }qA  : Vector of displacement-independent cutting forces for all flutes at   an axial element ( )q a≡ ; Vector of displacement-independent   cutting forces for all flutes at an infinitesimal element ( )q dz≡ ;   Vector of displacement-independent cutting forces for one flute   ( )q j≡ ; Vector of displacement-independent cutting forces for all   flutes and all elements ( )q TA≡ . qA    : Coordinate transformation 1 2( )q q q≡  – gives the vectors of   coordinate system 2q  in the coordinate system of  1q .   1 2( )q or q G=  is the global coordinate system; 1 2( )q or q W=  is   the workpiece coordinate system; 1 2( )q or q T=  is the tool   coordinate system; 1 2( )q or q H=  is the coordinate system of the   hammer test. { }B  : Bounding box center coordinates. qc  : Horizontal feed-per-tooth for flute, j  ( )q Xj≡ ; Vertical feed-per-   tooth for flute, j  ( )q Zj≡ . qC  : Peak process constraints: tool stress ( )q TS≡ , tool deflection   ( )q TD≡ , cutting torque ( )q TQ≡  and chip thickness ( )q CL≡ . { }qC  : Center coordinates of slice plane, s , ( )q s≡ . qC    : Matrix of directional coefficients for all flutes at an axial element   ( )q a≡ ; Matrix of directional coefficients for all flutes at an   infinitesimal element ( )q dz≡ ; Matrix of directional coefficients   for one flute ( )q j≡ ; Matrix of directional coefficients for all   flutes  and all elements ( )q TA≡ ; Matrix of directional  xvii   coefficients for all flutes and all elements with zero order Fourier   expansion ( 0)q T≡ . db  : Projected length of an infinitesimal cutting flute in the direction   along the cutting velocity. dS  : Infinitesimal length of a cutting edge segment. dz  : Cutting force element height. qd∆  : Distance travelled in tool path segment, i  ( )q k≡ . qf  : Magnitude of angular feed in tool path segment, i , ( )q Ai≡ ;   Magnitude of tool tip feed (linear) in tool path segment, i ,   ( )q Li≡ ; Magnitude of total feed ( )q T≡ ; Magnitude of   horizontal feed ( )q X≡ ; Magnitude of vertical feed ( )q Z≡ . qF  : Cutting forces: tangential ( )q t≡ , radial ( )r t≡ , axial ( )q a≡ ,   x ( )q x≡ , y ( )q y≡ , z ( )q z≡ . { }qF  : Angular feed ( )q Ai≡  for tool path segment, i ; Tool tip feed   ( )q Li≡  for tool path segment, i ; Total feed vector ( )q T≡ ;   Vector of cutting forces for one flute ( )q j≡ ; Vector of cutting   forces for all flutes at an infinitesimal element ( )q dz≡ ; Vector of   cutting forces for or an axial element ( )q a≡ ; Vector of cutting   forces for all flutes and all axial elements ( )q TA≡ . { }qF  : Nodal force vector for Timoshenko beam element, e , ( )q e≡ . G  : Transfer function of PI controller. qh  : Intended chip thickness (no subscript); Chip thickness ( )q j≡  for   flute, j ; Chip thickness component due to horizontal feed for   flute, j  ( )q Xj≡ ; Chip thickness component due to vertical feed   for flute, j ( )q Zj≡ ; Static chip thickness for flute, j ( 0 )q j≡ ;   Maximum allowable chip thickness ( )q MAXALLOW≡ . qH  : Transfer function of cutting process (no subscript); Distance from   tool tip to top of cutting flutes ( )q f≡ . i  : Tool path segment number; Helix angle.  xviii qI  : Area moments of inertia ( )q xx≡  and ( )q yy≡ ; Area product of   inertia ( )q xy≡ . j  : Complex number = 1− ; Cutting flute number. J  : Normalized first-order virtual cutting process transfer function. qk  : Stiffness for mode of tool ( )q t≡  and ( )q w≡ . { }qk  : Vector of tool axis rotation for tool path segment, i  ( )q i≡ . [ ]qK  : Stiffness matrix for Timoshenko beam element, ( )q e≡ . qK  : Cutting coefficients: tangential, ( )q tc≡ , radial ( )r tc≡ , axial   ( )q ac≡ ; Edge force coefficients: tangential ( )q te≡ , radial   ( )r re≡ , axial ( )q ae≡ ; Gain of first-order virtual cutting process   transfer function ( )q g≡ ; Integral ( )q i≡  and proportional   ( )q p≡  gains of PI controller. qL  : Distance from the tool tip to the fixed support. wm  : Weibull exponent for tool stress model. qM  : Moments around x ( )q x≡  and y axes ( )q y≡ . { }qM  : Moving direction at tool tip between tool moves, i  and   i +1 ( )q i≡ . n  : Spindle speed (rev/min); Node number index. qN  : Number of cutting force elements ( )q a≡ ; Number of   Timoshenko beam elements ( )q e≡ ; Number of tool moves   ( )q i≡ ; Number of cutting flutes ( )q j≡ ; Number of   interpolation steps ( int)q ≡ ; Number of poles ( )q p≡ ; Number   of slice planes ( )q s≡ ; Number of zeros ( )q z≡ ; Number of   clockwise encirclements of the origin ( )q cwe≡ ; Number of   poles ( , )q p rhcp≡  and zeroes ( , )q z rhcp≡  in the right half of   the complex plane. qP  : Constraint normalized outputs (CNOs): tool stress ( )q TS≡ , tool  xix   deflection ( )q TD≡ , cutting torque ( )q TQ≡  and chip thickness   ( )q CL≡ ; Reference constraint normalized output ( )q r≡ . { }qP  : Tool tip position coordinates (no subscript); Tool tip position   coordinates at command, i  ( )q i≡ . r  : Displacement vector as a function of frequency. qR  : Radius (no subcript); Effective radius ( )q eff≡ ; Shank radius   ( )q s≡ ; Radius of flute, j , ( )q j≡ . qR    : Tool axis rotation matrix between tool move i  and tool move   1i + ( )q i≡ . s  : Laplace domain variable; Slice number. S  : Number of engagement pairs; Probability of survival for a   serrated, tapered ball-end mill. { }S  : Slice axis vector. qt  : Time (no subscript); Time interval for force prediction calculation   ( )q s≡ . t∆  : Time elapsed between start and end of tool path segment. qT  : Time period / control interval (no subscript); Cutting torque   ( )q c≡ ; Maximum cutting torque of the machine ( )q m≡ ; Time   delay for flute, j  ( )q j≡ . { }qT  : Tool axis vector (no subscript); Tool axis vector at tool move, i ,   ( )q i≡ . qu  : Parametric variable in the range of 0-1 (no subscript); Static   deflection in x ( )q x≡ , y ( )q y≡  and z ( )q z≡  directions; Total   static deflection ( )q total≡ ; Maximum allowable static   deflection ( limit)q ≡  ; Displacement in radial direction for   flute, j ( )q j≡ . Displacement in the chip thickness direction for   tool ( , )q t j≡ and workpiece ( , )q w j≡  for flute, j .   xx { }qu  : Vector of displacements for Timoshenko beam element,   e ( )q e≡ . qv  : Displacement in tangential direction for flute, j ( )q j≡ . { }qv  : Angular feed direction vector for tool path segment, i  ( )q Ai≡ ;   Tool tip feed direction for tool path segment, i  ( )q Li≡ ; Total   feed direction ( )q T≡ . qw  : Displacement in axial direction for flute, j ( )q j≡ . { }w  : Vector around which engagement arcs are rotated to convert from   global system to tool coordinate system. qx  : Displacement in x-direction for flute, j  ( )q j≡ . { }qX  : Vector of horizontal feed at the tool tip ( )q TT≡ ; Local vector of   horizontal feed ( )q T≡ . qy  : Peak process outputs: tool stress ( )q TS≡ , tool deflection   ( )q TD≡ , cutting torque ( )q TQ≡  and chip thickness ( )q CL≡ ;   Displacement in y-direction for flute, j  ( )q j≡ . { }qY  : Y-axis vector at tool tip ( )q TT≡ ; Local Y-axis vector ( )q T≡ . qz  : Distance from tool tip to bottom of engagement block ( )q b≡ ;   Distance from tool tip to middle of Timoshenko beam element   ( )q e≡ ; Lower ( )q l≡  and upper ( )q h≡  boundaries of a cutting   force element; Start ( serrmin)q ≡  and ending ( serrmax)q ≡   locations of serration waves along tool axis; Displacement in   z-direction for flute, j ( )q j≡ . { }qZ  : Z-axis vector at tool tip ( )q TT≡ ; Local Z-axis vector ( )q T≡ . r α  : Rake angle ( )q r≡ ; Normal rake angle ( )q n≡ . qβ  : Half-taper angle (no subscript); Friction angle ( )q s≡ ; Normal   friction angle ( )q n≡ . γ  : Angle of total feed from the { }TX (or forward) direction.  xxi ε  : Cutting force on/off function based on cutter-workpiece   engagement conditions. qζ  : Damping ratio (no subscript); Damping ratio for mode of tool   ( )q t≡  and workpiece ( )q w≡ . η  : Chip flow angle. qθ   Immersion coordinate shift angle ( )q s≡ . θ∆ , qθ∆ , : Half-angle of rotation between first and last tool move; Total   angle of tool rotation between tool move i  and tool move   1i + ( )q i≡ . qκ  : Axial immersion angle (no subscript); Axial immersion angle for   flute, j ( )q j≡ . Λ  : Total stability matrix for the system (includes transfer functions,   delay terms and zero order expanded directional coefficients for   all flutes and elements). qσ  : Effective max transverse stress on tool including stress   concentrations ( ZmaxEFF)q ≡ ; Max transverse stress on tool   ( Zmax)q ≡ ; Maximum failure stress for a carbide cutter in   service ( max)q cs= ; Maximum failure stress for a carbide blank   ( max)q blank= ; Max allowable transverse stress on tool   ( max )q Z ALLOW≡ . qτ  : Shear stress ( )q s≡ . qφ  : Immersion angle for flute, j , ( )q j≡ ; Immersion angle for   engagement arc endpoint ( )q ep≡ ; Start immersion angle   ( )q st≡ ; Exit immersion angle ( )q ex≡ ; Pitch angle for flute, j   ( , )q p j≡ ; Immersion angle of reference cutting edge ( )q r≡ ;   Shear angle ( )q c≡ ; Normal shear angle ( )q n≡ . qΦ  : Frequency response function for tool ( )q t≡  and workpiece   ( )q w≡ .  xxii ψ∆  : Rotation angle around vector, { }w , to convert engagement arcs   from global system to tool coordinate system. qψ  : Immersion lag angle due to helix on cutting flute, j ( )q j≡ . qΨ  : Total transfer function matrix for the system including delay   terms ( )q TA≡ . qω  : Angular velocity (no subscript); Angular velocity of tool axis in   tool path segment, i ( )q i≡ ; Natural frequency ( )q n≡ ; Natural   frequency for mode of tool ( )q nt≡ ; Natural frequency for mode   of workpiece ( )q nw≡ ; Highest flexible mode of the system   ( n,max)q ≡ . Ω  : Spindle speed (rad/sec).   xxiii Acknowledgements I would like to thank Dr. Yusuf Altintas for his assistance, guidance and instruction, and for giving me the opportunity to study in a world-class research facility like the Manufacturing Automation Laboratory (MAL). I would also like to express my appreciation to Dr. Derek Yip-Hoi for his supervision and expertise in the field of Computer Aided Design and Computer Aided Manufacturing (CAD/CAM), which was instrumental in my research. Many thanks are due to Mr. Don McIntosh, Dr. Serafettin Engin and Mr. Fuat Atabey at Pratt & Whitney Canada for allowing me to work closely with industry. Thanks are also due to members of the MAL for their help with my work and their friendship during my four-and-a-half year stay. Finally, I am very grateful to my wife, Robin, and my family for their love, support and patience during my long ascent to higher learning. 1 Chapter 1 Introduction  Modern jet-powered aircraft engines employ impellers to compress the incoming air (see Figure 1-1). This is required to increase its oxygen content, which is required for proper combustion of fuel and extraction of usable thrust. Small jet engines tend to use a small number of centrifugal impellers for compression, while larger jet engines use a higher number of axial impellers. Both types of impellers are composed of a series of thin, twisted blades attached to a central hub. Figure 1-1 – Diagram of a typical gas turbine jet engine [35]. Air is compressed by the impeller blades as it enters the engine, which is mixed and burned with fuel in the combustion section. The hot exhaust gases provide forward thrust and turn the turbines which drive the compressor stage [35].  Because these impellers experience high temperatures and forces and are subjected to air from the outside environment, they must be machined from strong, durable and Chapter 1: Introduction  2 corrosion-resistant materials such as the titanium alloy, 6 4Ti Al V− − , or the nickel alloy Inconel. As a consequence, these are some of the most costly, complicated and time- consuming parts to fabricate. They are milled directly from axisymmetric blanks, using five-axis machine tools. The extra two rotational degrees of freedom that the five-axis machine provides allow the cutter to rotate with respect to the workpiece in order to cut the complex twisted surfaces. In industry, there are two main methods of machining these parts (see Figure 1-2). Point Milling Flank Milling Start End X YZ  Figure 1-2 – Point milling of impeller blades (left). Flank milling of impeller blades (right).  The first is called point milling, which involves taking a high number of light, shallow cuts with a ball-end or tapered-ball end mill. In this case, usually only the ball-end of the tool is used to cut the workpiece. The second method is called flank-milling. Flank milling is a less common machining practice, but it is generally considered more efficient. In this process, the entire side (or flank) of the tapered-ball end mill is used to cut the ruled surfaces of the part. This greatly reduces the number of passes required to Chapter 1: Introduction  3 machine the blades, decreasing the machining time. Flank milling also eliminates scallops resulting from multiple tool passes on the same surface. The drawback to this method is that not all surfaces are flank-millable -- the desired surface must closely approximate a ruled surface [131]. Since flank milling is an aggressive form of cutting, the life of the tool is generally shorter and the process requires a much stiffer, more powerful machine tool to handle the high cutting forces and torques generated. Finally, due to the high depths of cut involved, the process requires tools to be more carefully designed in order to avoid self-excited vibrations. The mechanical properties and low thermal conductivities of titanium and Inconel present many difficulties during the machining process. As the heat generated during the cutting process does not dissipate through the part, it intensifies at the cutting area, which can lead to rapid tool damage. The elasticity of titanium creates additional manufacturing challenges. This property can cause the part to spring away from the tool during engagement, which makes the cutting edge rub the workpiece rather than cut it. The rubbing action causes increased friction and higher temperatures at the cutting area. Inconel is also difficult to machine using traditional machining techniques, due to rapid work hardening. Although each material has slightly different mechanical properties, an efficient method of flank-milling impellers from Titanium alloy or Inconel is to use large, solid- carbide, tapered helical ball-end mills with high depths of cut at low spindle speeds and moderate to low feeds. During roughing operations, serrated-cutting tools and cuts with large radial immersions are employed to remove high amounts of material while avoiding chatter (Figure 1-3(a)). Chapter 1: Introduction  4 φp,1φp,2 φp,3 φp,4 Serration  Waves Variable-Pitch Flutes Tapered Ball-End  Mill (Roughing) Tapered Ball-End  Mill (Finishing) (a) (b)  Figure 1-3 – (a) Illustration of typical roughing and finishing cutters used to flank mill impellers. (b) Bottom view of a four-fluted variable-pitch cutter showing the unequal spacing of the flutes.  Cutting forces in roughing operations are typically high – usually in the range of 1- 10 kilonewtons. For semi-finishing and finishing operations, non-serrated tapered ball- end mills with a higher number of cutting flutes and low radial immersions are used to obtain a smooth surface finish (Figure 1-3(a)). Cutting forces in these tool paths are generally lower. However the low radial immersion cutter-workpiece engagement area, small chip thicknesses and high flexibility of the part can cause large unwanted vibrations. Both roughing and finishing cutters employ carefully-designed variable-pitch flutes to prevent the onset of chatter (Figure 1-3(b)). Since the blades of the impellers must be machined within tight tolerances, tool damage and deflections of the tool and flexible workpiece can jeopardize the precision of the process. Chatter vibrations can destroy the surface finish of the part and cause Chapter 1: Introduction  5 accelerated tool wear or even damage the machine tool itself. Substantial damage to the impeller during the machining process means the whole impeller must be scrapped, at a cost of $15,000 to $50,000 per unit, depending on its size. The impellers are often large in size and can take a few days to produce. Also, each engine may require multiple impellers to adequately compress the incoming air. Consequently, any reductions in cycle time can result in increased productivity and substantial cost savings. In the quest for increased efficiency, the industry goal is to model and predict the physics of the cutting process and then select optimum cutting conditions based upon the simulated outputs. Literature on process simulation of the five- axis flank milling process, however, is relatively sparse. Currently, several commercial software packages [32], [38], [67], [91] can simulate the five-axis flank milling of an impeller in a virtual environment. However, these programs are used mostly for verification of numerical control (NC) code and for collision detection. They do not take in account the mechanics and dynamics of the cutting process and they do not allow the user to calculate useful quantities such as cutting forces, tool stress and deflection, chip thickness and cutting torque / power. Also, these programs do not predict when damaging, high-amplitude, self-excited vibrations will occur. The current research aims to fill this gap by presenting modeling techniques and algorithms required to simulate and optimize five-axis flank milling operations for jet engine impellers. The aim is to machine the part in a virtual environment and have optimum cutting conditions ready for the process planner without the need for costly Chapter 1: Introduction  6 physical testing. A flowchart of the overall scheme described in the thesis is shown in Figure 1-4. The areas studied in the thesis are shown by the frames with dashed lines. To predict the cutting forces and chatter stability of the five-axis flank milling process, the engaged area of the tool with the workpiece must be obtained. This area varies continuously along the tool path, due to the changing position and orientation of the tool and the complex boundaries of the workpiece. The contact zone is called a cutter- workpiece engagement (CWE), and usually takes the form of a map showing the entry and exit locations of the cutting flutes as a function of height along the tool axis. Calculating this area can be a difficult problem, particularly in five-axis flank milling, because the boundaries of the workpiece are constantly being changed as the machining operation progresses. Few methods of calculating CWEs for five-axis machining appear in the available literature -- especially those employing solid modeling methods, which are generally considered more accurate than the more common, discrete, z-buffer methods. A new, solid modeling method which updates the workpiece and obtains CWE maps has been developed in the thesis. It is able to handle five-axis motion, tapered ball- end mills and complicated workpiece geometry. In industry, five-axis flank milling roughing operations employ tapered, helical, ball-end mills with features such as serrated cutting edges and variable-pitch flutes. Cutting force models for ball-end milling, general end milling, inserted and serrated cutters have been developed for 2D machining cases, as well as some for five-axis ball- end milling. However, at this time, a model that incorporates five-axis motion, general end mill geometry, serrated cutting edges and variable-pitch teeth, complex cutter- workpiece engagement maps, and non-linear cutting coefficients that vary over the axis Chapter 1: Introduction  7  Time(Sec) F o rc e  ( N ) Optimized NC Code  - Max. material    removal rate -  Part tolerances    not violated - Process is   chatter free Final Part Spindle SpeedD e p th  o f C u t z φ XT n Zc YT φst φex φ 300 200 100 200 400 600 800 S tr e s s  [ M P a ] 0 Time [s] 0 400 500 Mechanics Model Chatter Stability Prediction Tool / Workpiece CAD models Cutter-Workpiece Engagements Feed Rate Optimization Start End Five-Axis Cutting Virtual Five-Axis Flank Milling of an Impeller  Figure 1-4 – Flowchart of the thesis project – Virtual Five-Axis Flank Milling of Jet Engine Impellers. Chapter 1: Introduction  8 of the cutter does not appear to exist, based upon a review of the available literature. As a consequence, industrial impeller flank milling operations cannot be accurately simulated. In this thesis, a cutting mechanics model that can integrate these complexities is developed. The cutting force model, along with a cantilever finite-element model of the tool, is used to predict other useful outputs such as tool stress, tool deflection, cutting torque and chip thickness. To maximize productivity, a method of finding the optimum feed that does not violate any of these outputs is desired. In the open literature, several works have presented offline feed rate scheduling strategies for 2D, 3D and multi-axis milling based on a single constraint. However, there do not appear to be any articles presenting feed rate optimization techniques for five-axis flank milling operations according to multiple process constraints. Two such methods are presented in the thesis, which can reduce the machining time of jet-engine impellers. Unstable, self-excited vibrations – also known as chatter – can occur in five-axis flank milling if cutting conditions are not properly selected. Chatter destroys the surface of the part and may cause damage to the tool and workpiece. Predicting the chatter stability of 2D milling operations has been widely studied. However, no such solution has been developed for five-axis flank milling. In the thesis, a new method of determining the stability of this process is developed. It uses the Nyquist Stability Criterion to check whether a given set of conditions is stable or unstable and allows integration of general cutting edge geometry, irregular cutter-workpiece engagements and tool and workpiece dynamics that vary over the axis of the cutter. Chapter 1: Introduction  9 Although still in the research stage of development, the four modules together form a process simulation and optimization software package for five-axis flank milling. This allows the user to machine the part in a virtual environment and optimize the process with a reduced number of physical tests. Henceforth, this thesis is organized as follows: The review of related literature is presented in Chapter 2. Chapter 3 details a new method of calculating cutter-workpiece engagement maps for five-axis flank milling, which are a requirement for chatter stability predictions and cutting force simulations. Chapter 4 presents a novel five-axis mechanics model applicable to sculptured-surface machining with general, serrated, variable-pitch, helical ball-end mills. In Chapter 5, two methods of optimizing five-axis tool paths according to multiple feed-dependent constraints are discussed and demonstrated. Chapter 6 presents a chatter stability algorithm for five-axis flank milling. Conclusions and future research are given in Chapter 7 and specifics of the mathematics and algorithms used to implement these theories are given in the Appendices. 10 Chapter 2 Literature Review  2.1 Overview The main goal of the thesis project is to present models and algorithms required to flank mill jet-engine impellers in a virtual environment. The information from these simulations can be used to increase the productivity of the physical process and reduce manufacturing costs. In this chapter, previous research related to the intended contributions of the thesis is reviewed, and state-of-the-art theories and modeling techniques that may be relevant to the current research project are discussed.  Section 2.2 gives a background of methods of obtaining cutter-workpiece engagements (CWEs) for milling operations. CWEs, or entry/exit locations of the cutting flutes as a function of height, are a requirement for cutting force and chatter stability predictions. Calculating this area can be a difficult problem, particularly in five-axis milling, as the boundaries of the workpiece are constantly being changed as the machining operation progresses. Section 2.3 details past literature relating to the cutting mechanics of milling. Cutting mechanics is the area of study that attempts to describe the interaction of the cutting edge with the workpiece, which is required to predict cutting forces. The cutting forces can be used to calculate other useful quantities such as tool stress, tool deflection, cutting torque and power. Chapter 2: Literature Review  11   Section 2.4 examines previous feed rate optimization techniques for various milling processes. The feed, or velocity, of the tool as it moves through the workpiece directly governs the cycle time of the machining operation. A higher feed rate results in a shorter machining time, but also increases the stress on the tool and part and the required cutting torque of the machine. Feed rate optimization involves a systematic method of finding the highest feed possible according to a given set of process constraints (such as cutting force, torque, tool deflection, chip thickness and bending moment).  Section 2.5 presents literature related to chatter stability analysis. Chatter is a complex process that results from the tool recutting the wavy surface of the workpiece left by vibrations from the previous tooth or tool pass. Depending on the phase shift between waves, the vibrations may grow exponentially until the tool jumps out of cut or is damaged. Although vibrations of the tool and workpiece during machining cannot be avoided, damaging chatter vibrations can be eliminated if adequate cutting conditions are selected.  2.2 Cutter-Workpiece Engagement Calculations Cutter-workpiece engagement maps, or entry/exit locations of the cutting flutes as a function of height (see Figure 2-1), are a requirement for cutting-force and chatter stability predictions in milling. Chapter 2: Literature Review  12 Cutter-Workpiece Engagement Map Immersion Angle D is ta n ce  F ro m  T o o l T ip  ( m m ) Tool Workpiece Cutter-Workpiece Engagement Area  Figure 2-1 – Illustration of a cutter-workpiece engagement map (right) for an impeller roughing operation.  Although dynamic displacements of the tool and part can cause the engaged area to change during machining, process engineers can minimize these effects by eliminating chatter with variable-pitch cutters [26], [27] and by selecting suitable spindle speeds in order to avoid large, forced vibrations. If deflections of the tool and workpiece are considered small, calculation of the cutter-workpiece engagement (CWE) area becomes a purely geometric problem. Unfortunately, this problem can be a difficult one, particularly when five-axis milling a part with complex boundaries -- boundaries that are constantly being altered as the tool removes more material. This means that entry/exit locations of the flutes at the present location may depend upon surfaces generated by one or more previous tool paths. Calculation of CWEs for machining processes falls into the field of Computer-Aided Design (CAD) and Computer-Aided Manufacturing (CAM), where analytical and numerical geometric algorithms are applied to update the in-process workpiece and extract the required engagement zone at each discrete position of the tool. However, it is only in the past two decades that computers have become fast enough to Chapter 2: Literature Review  13 solve many CWE problems in a practical time frame. As a result, calculation of CWEs for machining processes is a relatively new, evolving area of research.  As early as 1981, Voelcker and Hunt [127] used the PADL Constructive Solid Geometry solid-modeling system for simulation of NC programs. The simulation involved performing Boolean subtraction of the tool movement volume from the workpiece. However, the cost of the simulation was reported to be high -- on the order of O(N4), where N represents the number of tool movements. In 1986, Wang and Wang [130] used a view-based method to perform NC simulation and verification of parts. Drawbacks included the fact that small errors occurring outside the chosen field of view could not be detected. Also, changing to a different viewpoint required re-running the simulation.  Today, the most common method used to model the removal of material in machining processes is the Z-buffer approach. This method was applied as early as the 1980s by researchers such as Chappel [31], Van Hook [125] and Jerard [79]. In this method, the CAD model of the workpiece is divided into an x-y grid. The surface is then stored as a series of z-vectors – one for each x-y point (see Figure 2-2). As the cutter moves through the workpiece, these vectors are reduced in length – similar to cutting blades of grass [76]. This leads to short simulation times and a reduction in the complexity of the intersection calculations, meaning that these methods are both fast and robust. Kim et al. [81], [82] used a z-buffer algorithm to obtain cutter-workpiece engagements for ball-end milling of inclined surfaces and varying cutter inclination angles. Fussell et al [59] used an extended z-buffer algorithm to calculate cutter- Chapter 2: Literature Review  14 X Y Z Workpiece Surface Tool Swept Volume Z Vector Intersection Points Z Vectors are reduced in height  Figure 2-2 – Z-buffer methods store the workpiece as a series of vectors. As the tool moves through vectors, it reduces them in size – similar to cutting blades of grass [76].  workpiece engagements for five-axis machining of a centrifugal impeller with a ball-end mill. This algorithm allowed overhangs, pockets and voids in the geometry to be incorporated. Roth et. al used an "adaptive" depth buffer to calculate cutter-workpiece engagements and then predict cutting forces in five-axis [108], and three-axis [109] cases. Their adaptive z-buffer system matched the depth z-buffer orientation to the tool axis orientation and sized the depth-buffer relative to the tool rather than the workpiece, which reduced the data storage and increased the accuracy of computations. All of the above mentioned authors used the engagement zones as an input to cutting force prediction algorithms. Other research works employing z-buffer techniques to machining problems are given in [40], [65], [118] and [137]. Z-buffer techniques are renowned for their high computational speed and are often used in commerical CAD/CAM software packages such as CGTech’s Vericut [126]. Chapter 2: Literature Review  15 In contrast to Z-buffer schemes, which employ computer-aided design (CAD) geometry in a discretized format, solid modeling methods store the CAD geometry as a series of simpler entities (i.e. faces, curves, lines) or features which are connected together in a hierarchical structure. These techniques are generally considered more accurate than z-buffer methods, but more computationally expensive. Spence and Altintas [13], [114] used a solid-modeling-based approach to extract the immersion angles and calculate the cutting forces during simulation of 2D milling with a cylindrical tool and a rectangular workpiece. They used a constructive solid geometry (CSG) tree approach that divided the cutter engagement arc into several segments ( , , ,st i ex iφ φ   ). The CSG solid modeler determined the cutter arc interaction with each primitive and then combined the results (see Figure 2-3(a)). Imani et. al [68] used the ACIS (Andy, Charles, Ian’s System) boundary representation (B-rep) modeler to update the workpiece and calculate the uncut chip geometry for 2 and 3-axis milling operations. Non-Uniform Rational B-Spline (NURBS) edge geometry was used to model the cutting edge. Yip-Hoi and Huang [135] used the ACIS solid modeling package, to extract and decompose cutter engagement features during 2D milling (see Figure 2-3(b)). This information was used to generate cutter-workpiece engagement maps with respect to immersion-angle along the height of the tool. These maps were used in conjunction with a variant of the cutting force model mentioned in Altintas and Spence's work [13], [114]. Predicted cutting forces were compared to those obtained from experimental testing, and were found to match well – usually to within 10%. Limitations of this method were the need of the sub-sections on the cutter engagement features to be rectangular and for the machining to be orthogonal to the workpiece. Chapter 2: Literature Review  16 Ri Pi,0 Pi,1 Pi,j Pi,ni Ri+1 ei,0 ei,1 ei,j di,0 di,1 |di,j+1 – di,j| PCL ei,j+1 Pi,j+1 φ i,j=φ i,j+1 di,j di,j+1 φ i,j+2 |di,1 – di,0| Cutter Workpiece R1 R2 n ar n ar ei,ni-1 (b) st, 1 ex, 1 + * - *φ φ st, 2φ ex, 2φ st, 3φ ex, 3φ (a)  Figure 2-3 – (a) Spence and Altintas’s [114] solid modeling CSG tree approach to obtaining cutter-workpiece engagements. (b) Yip-Hoi and Huang’s [135] cutter- workpiece engagement feature extraction method.  Other methods described in the literature that have been used to extract cutter- workpiece intersections in machining processes include voxels [72], faceted-models [133] and closed-form analytic solutions [63]. Voxels and faceted-models can be fast and efficient, but often require substantial programming effort to achieve these benefits. Closed-form analytic solutions provide the fastest methods for obtaining CWE zones, but are usually limited to simple cases and are not applicable to five-axis tool paths with complex workpiece geometry. Chapter 2: Literature Review  17 So far, most of the techniques used to extract cutter-workpiece engagements in milling have been developed for 2D and 3D cases only. For five-axis machining, nearly all of the tool-workpiece intersection calculation schemes rely on z-buffer methods. These algorithms are fast and robust, but are generally not as accurate as their solid model counterparts. A new approach called the Parallel Slicing Method (PSM) is explained in Chapter 3 of the thesis. It is able to handle complex, multi-axis milling and employs B-rep solid models to increase accuracy. Unlike conventional 3D solid modeling techniques, the method reduces the data storage required and computational complexity. It does this by reducing 3D cutter-workpiece intersection problems to sets of simpler, 2D planar problem and by using analytical planar geometry to decrease the number of Boolean operations required. This approach makes updating the workpiece easier and could facilitate a parallel processing approach for each 2D planar problem in the future [115]. Also, in the PSM, all Boolean intersection and subtraction operations are performed on the removal volume, rather than the workpiece. This simplifies calculations as the geometry is gradually deleted, and reduced in size, as the simulations take place. When compared with a benchmark z-buffer method, the PSM is shown to produce smoother engagement boundaries and capture a greater range of entry/exit angle pairs along the axis of the cutter. The drawback is that, in its present form, it requires more time to calculate the cutter-workpiece engagement maps. The results of this work are given in Chapter 3 of the thesis and reviewed articles [56], [57].  Chapter 2: Literature Review  18 2.3 Milling Mechanics / Process Simulation Milling is a machining operation in which a rotating cutter with one or more flutes (Figure 2-4(a)) is fed into the workpiece. The cutting teeth shear the material from the part, which is evacuated through the flute cavities. Unlike a single-point turning operation that has a constant chip thickness, the chip thickness for each flute varies over the spindle period in milling. Also, the chip thickness and cutting forces for each flute are discontinuous over time, as each tooth enters and exits the workpiece intermittently (see Figure 2-4(b)). In milling, the cutting forces, vibrations and chatter depend on chip thickness, so accurate chip thickness estimation is important for predicting these quantities. Generally speaking, the mechanics of milling refers to the study of the interaction of the cutting edge as it shears material away from the workpiece. The physics of the process, along with mathematical models, are used to predict cutting coefficients, chip thickness and cutting forces acting on the tool and workpiece [5], [15], [16], [93], [110]. These quantities can be used to predict other useful outputs such as stress on the tool, deflection of the tool tip, required cutting torque or power and the stability of the process. The mechanics of ball end milling [49], [50], [81], [87], [136], tapered ball-end milling [107], face milling [58], general helical end-milling [45] and milling with inserted [46] and serrated, helical [94] cutters for two dimensional and three dimensional milling have been studied in detail. However, most of the research in literature for five-axis milling has focused on tool path generation, collision detection and geometric verification of the tool path. Current CAM software systems can simulate the five-axis milling of a Chapter 2: Literature Review  19 (a) (b) Y X φex φst φ Workpiece Chip n Tool Feed  Figure 2-4 – (a) Slot, cylindrical and ball-end milling cutters [92]. (b) Simple model of a down-milling operation. Flutes intermittently enter and exit the workpiece.  part geometrically in a virtual environment [32], [38], [67], [91], but they do not consider the physics or mechanics of the cutting process. As a result, the performance and faults of the process cannot be predicted before an actual machining operation begins. Although few articles exist in the available literature, there has been some notable research in cutting force predictions for five-axis machining. Chapter 2: Literature Review  20 Zhu et al. [137] developed a process simulation package for five-axis ball end milling, using a Z-buffer algorithm to calculate cutter-workpiece engagements. Employing experimentally calibrated cutting coefficients, accurate cutting force predictions were demonstrated for roughing, semi-finishing and finishing operations on a 6 4Ti Al V− −  workpiece. The model was able to incorporate process faults such as runout, flute chipping and flute deviation in the cutting force predictions. Hemmett [65] developed an integrated software package consisting of three models to predict cutting forces and automatically determine the best feed rate for 3-axis and 5- axis ball-end and cylindrical-end milling operations. The package consisted of a discrete mechanistic force model, which was used estimate cutting forces, a discrete geometric model to keep track of the changing workpiece geometry and a CNC machine model to calculate the cutter-stock relative velocity based on feed inputs, machine kinematics and controller behavior. Fussell et al. [61] presented an algorithm to calculate cutter-workpiece engagements using an extended Z-buffer approach. The authors used this information, along with a discrete force-prediction model and average cutting coefficients, to predict the cutting forces when milling a centrifugal impeller with a ball-end mill. Bailey et al. [17], [18] presented a two-part paper on the simulation of cutting forces and optimization of feed rates for five-axis sculptured surface machining. A Non- Uniform Rational B-Spline (NURBS) curve was used to model the cutting edge. Force predictions and feed optimizations were demonstrated on the machining of a complex surface and an airfoil-like shape. Mechanistic cutting force coefficients were used along with a discrete force model for cutting force simulations. Chapter 2: Literature Review  21 Finally, Becze et al. [21] developed analytical equations for chip area and used neural-networks to obtain mechanistic cutting coefficients. The authors used these quantities to predict cutting forces for the five-axis ball-end milling of a hardened die steel. In Chapter 4, a generalized, five-axis cutting force model is presented, with experimental validation from an actual impeller roughing operation. The proposed cutting force model presented in this paper is based on the discrete 2D milling force prediction models of Engin and Altintas [45] and Merdol and Altintas [94], but has been extended to include five-axis motion. It allows simulation of cutting forces using general end mills with serrated cutting edges and variable-pitch flutes -- tools which are used by the aircraft engine industry to machine jet engine impellers [25]. Since feeds are generally quite low when machining titanium impellers, an approximate model is used for chip thickness calculations rather than the true kinematics of milling [99]. This eliminates the need for storing and updating the workpiece surface, which reduces computation time. The Orthogonal-to-Oblique Cutting Mechanics Transformation [5], [15], along with an orthogonal cutting database, are used to obtain cutting force coefficients that consider the effects of flute geometry and chip thickness variation along the axis of the cutter. As shown in Chapter 4, for an impeller roughing operation, the predicted X and Y cutting forces and cutting torque are in agreement with the experimental values – usually to within 20%. The five-axis mechanics model is presented in Chapter 4 of the thesis and journal articles [52], [53].  Chapter 2: Literature Review  22 2.4 Feed Rate Optimization Techniques The impeller blades are modeled as flank-millable ruled surfaces in a dedicated CAD /CAM environment. When machining impellers from Titanium and Nickel alloys, cutting speeds are maintained at a constant level in order to avoid accelerated tool wear. Also, the spindle speed and spacing between flutes is carefully designed around the part and spindle dynamics in order to minimize forced and chatter vibrations [26], [27]. One variable that can be adjusted is the feed along the tool path. Increasing the feed of the tool reduces the machining time of the part. However, it also increases the chip thickness and cutting forces which, if excessive, can lead to tool breakage, stalling of the machine or large, unwanted tool deflections. Since aircraft impellers are costly and have long machining times, finding the highest feed rate possible without violating any constraints on the process has the potential to greatly improve productivity, while avoiding damage to the tool, part and machine. Feed rate optimization of machining processes has been implemented in various forms in literature. Feng and Su [51] optimized the orientation angle and feed rate of a ball-end mill simultaneously while milling an inclined surface. Tounsi and Elbestawi [124] optimized both the tool path and feed rate in 3-axis ball-end milling. The authors used cutting forces, contour errors, feed rate boundaries and feed drive acceleration/deceleration profiles for constraints. They also took into account the dynamics of the machine tool drives. Zhu et al. [137] scheduled feed rates according to a maximum chip thickness value during five-axis ball-end milling of a 6 4Ti Al V− −  alloy workpiece. Jerard et al. [77], Fussell et al. [59] and Hemmett [65] performed feed rate scheduling for 3- and 5-axis ball-end milling to maintain cutting forces at a constant Chapter 2: Literature Review  23 level. All of the above mentioned authors used iterative solutions to obtain optimal conditions. Chu et. al [33] developed empirical equations relating optimal feed rate selection to local shape features of the tool path. Local features included concave corners, contour cutting and upward and downward cutting. Experiments were performed with a ball-end mill and a zinc alloy workpiece. Yazar et. al [134] presented a feed rate optimization strategy based on cutting force calculations in 3-axis milling of dies and molds with ball- end and flat-end mills. The algorithm scaled the maximum resultant cutting force according to a desired value, assuming a linear relation between feed and force when outside a specified tolerance band. More recently, Erdim et al. [47] and Guzel and Lazoglou [64] used a similar feed rate scheduling strategy based on maximum resultant force and metal removal rate (MRR) when ball-end milling sculptured surfaces. Jerard et al. [78] optimized feed rates in 3-axis ball-end milling according to maximum force, maximum chip thickness and surface finish limits simultaneously through an iterative method. Jerard et al. [78] made the important observation that different constraints may be important at various parts of the tool path. Finally, Budak [25] implemented Altintas and Erol’s [10] online, single-constraint based adaptive controller on a physical 5-axis machining centre during machining of a titanium alloy impeller. A significant reduction in the machining time of the impeller was realized, while respecting a maximum peak force threshold (see Figure 2-5). So far in literature, most of the past research has focused on feed rate optimizations for cylindrical-end or ball-end milling of sculptured surfaces according to a single constraint. Chapter 5 presents two methods of off-line scheduling of feed rates according Chapter 2: Literature Review  24 to multiple feed-dependent constraints for five-axis flank milling with serrated, tapered, helical, ball-end milling cutters with variable-pitch flutes. The process is non-linear, and requires accurate simulation of the physics while adjusting the feed rate to satisfy multiple constraints during the flank milling operation. The feed rate optimization schemes and their application to an impeller roughing operation are given in Chapter 5 and journal articles [54], [55].   Figure 2-5 – Cutting force variation for an adaptively controlled five-axis rough- milling operation on a titanium impeller [25]. Budak [25] implemented Altintas and Erol’s adaptive controller [10] in real-time. Note the significant reduction in cycle time shown on the top graph.  Chapter 2: Literature Review  25 2.5 Chatter Stability Prediction Chatter in machining operations is a complex phenomenon that results from the tool recutting the wavy surface of the workpiece left by vibrations from the previous tool or tooth pass (see Figure 2-6(a)). In the single cutting point (or orthogonal cutting) case, as in a typical turning operation, the chatter process can be formulated in a control loop block diagram. Vibrations from the present revolution are modeled as a negative feedback, and vibrations from the previous revolution as a positive feedback (see Figure 2-6(b)). Depending on the phase shift between the present and previous vibrations (see Figure 2-6(c)), the chip thickness and cutting forces may grow exponentially until the tool jumps out of cut or is damaged. h ε Dynamically Stable Dynamically Unstable h Static (c) (a) h y0(s) y(s) y(s) Static chip thicknessh(t) y(t-T) y(t) (b)  Figure 2-6 – (a) Total chip thickness depends on the static chip thickness, present vibrations and vibrations from a previous revolution. (b) The orthogonal cutting process formulated as a control feedback system. (c) The phase shift between vibration waves can lead to chatter (dynamically unstable).  Chapter 2: Literature Review  26 Chatter was identified as early as 1907 by F.W. Taylor [119]. But Tobias and Fishwick [123] and Tlusty and Polacek [121] were the first to develop chatter stability models for one-dimensional orthogonal cutting, and introduced the classic equation: [ ]lim min 1 2 Res a K G = −  (2.1)  Where lima is the critical width of cut for stable cutting, sK , is the specific cutting coefficient and Re[ ] min G  is the minimum real part of the machine-tool structure’s frequency response function (FRF). Tobias and Fishwick [123] also introduced the notion of stability lobes (see Figure 2-7) – sawtooth-shaped graphs that show the boundaries of stability according to depth of cut and spindle speed.  3000 3400 3600 3800 40003200 0 1 2 3 4 5 6 Spindle Speed (rpm) D e p th  o f C u t (m m ) X X  Figure 2-7 – A typical stability lobe chart. A point above the lobes indicates unstable cutting conditions. A point below the lobes indicates stable conditions. Chapter 2: Literature Review  27 Merritt [97] formulated the chatter-stability problem for the orthogonal case, using feedback control theory (see Figure 2-6(b)) and solved for the critical width of cut using the Nyquist Stability Criterion. For milling, Tlusty [84] introduced theories involving orientation of cutting forces, average number of teeth in cut and flexibilities in the chip direction. Sridhar et al. [116] formulated the dynamics of milling for a straight tooth-cutter (Figure 2-8) and used a numerical algorithm to analyze the stability of the process. Opitz [103] solved the dynamic milling stability problem by orienting all cutting forces and vibrations in an average direction, then using Tobias and Fishwick’s [123] and Tlusty and Polacek’s [121] orthogonal cutting solution (Equation (2.1)) to obtain the critical depth of cut.  kx cx kycy y x Frj Ftj uj vj tooth (j) tooth (j-1) tooth (j-2) vibration marks left by tooth (j-1) j φ  Figure 2-8 – Formulation of the dynamic milling problem [116]. The flutes on the cutter sweep through modes in both x and y directions. Chapter 2: Literature Review  28 During the 1980s, Tlusty and Ismail, [120], [122] used time-domain simulations to calculate the stability of 2D milling through numerical process simulations. They included the effects of the tool jumping out of cut and time-varying force directions. Minus and Yanushevsky [98] solved the coupled dynamics of milling, using the Floquet theory [88]. They formulated an analytical solution, and used an iterative technique to obtain results. In 1995, Altintas and Budak developed an analytical, frequency-domain solution for 2.5D milling [5], [8], [23], [24]. The authors formulated the dynamic equations of the milling process, where the cutting flutes sweep through modes of vibration in the x and y directions. The matrix of directional coefficients was made time-invariant by taking a zero-order Fourier series expansion and an eigenvalue solution was used to solve for the critical depth of cut. The chatter-stability solution presented by Altintas and Budak [23] was considered ground-breaking, since it was able to predict stability lobes that are very close to those predicted by lengthy time-domain simulations at a fraction of the computational expense. Since then, Altintas and Budak’s analytical chatter solution has been extended to ball-end mills [12], variable-pitch cutters [9], [26], [27] and face milling cutters with inclination angle [74], [75]. The theory has also been used to solve three-dimensional milling [6] for such cases as face milling with circular inserts. In this case, the inserts excite modes in x,y and z directions. Articles presented by Davies et. al [36], Endres et. al [43] and Bayley et. al [19] proved that classical chatter stability solutions, which neglect the intermittency of the process, may not accurately predict the stability lobes at low radial immersion milling Chapter 2: Literature Review  29 operations. Merdol and Altintas [95] proved that Altintas and Budak’s chatter theory [8] can predict the chatter stability in low radial immersion milling (i.e. finish milling of thin- walled structures) by including the harmonics of the tooth-passing frequency in the eigenvalue solution. They showed that when the harmonics of the tooth passing frequency shift the transfer function to the region of the natural modes, additional narrow stability pockets are created which differ from the lobes predicted by the single frequency or classical solutions. Although chatter stability predictions and experiments agree well for milling, turning and drilling at high cutting speeds (i.e. spindle speeds), experiments show a large increase in stability at low cutting speeds whereas predictions show the opposite effect. The increase in observed stability is known as process damping and has been identified at least as early as 1958 [123]. Several theories have been put forth to explain this phenomenon – the main one being a damping effect caused by contact interference of the rake face of the tool with the wavy surface of the workpiece [80], [112]. Since many flank milling operations for impellers must be conducted at low cutting speeds due to the thermal properties of the workpiece material, process damping can be employed to achieve high, chatter-free depths of cut in these situations. Although rules of thumb have been developed to predict when it will occur and the severity of its effects, it is a complex process. Attempts of modeling and predicting it continue today [11] -- albeit with a somewhat limited success. The stability of turning and milling processes has also been modeled in the time domain by experts in delayed differential equations (DDEs), Stepan and Insperger [69], [70], [71]. The authors used a technique called the semi-discretization method to predict Chapter 2: Literature Review  30 the stability of the process. In the semi-discretization method, the delay term of a DDE is approximated by a series of piece-wise autonomous ordinary differential equations (ODEs). This approximation allows the stability of the system to be assessed using linear solution methods. In the available literature, a chatter stability solution for five-axis milling which is applicable to flank milling operations for jet-engine impellers, has not yet been developed. Some of the challenges involved in development of a solution are as follows: • The depth of cut may vary with immersion angle • The entry and exit immersion angles of the cutting flutes with the workpiece can vary along the axis of the tool • Tools may have serration waves on the cutting edges and employ variable- pitch flutes (Figure 1-3) • Both tool and workpiece transfer functions can vary along the axis of the cutter • Cutting coefficients may vary along the axis of the cutter. The frequency domain five-axis chatter stability solution detailed in Chapter 6 allows these complexities to be modeled. It uses the Nyquist Stability Criterion to determine whether a given set of cutter-workpiece engagement conditions at a given spindle speed is stable or unstable. Although the process damping effect causes the physical system to be more stable than the simulated one at low cutting / spindle speeds [11], [80], [112] (conditions at which impellers are often flank milled), it is difficult to model and beyond the scope of this work. The objective of the algorithm discussed in the Chapter 2: Literature Review  31 thesis is to present the basic five-axis chatter stability model which should give accurate stability predictions for spindle speeds above the process damping range. If a suitable process damping model becomes available, it could be added to the current stability algorithm at a later date. 32 Chapter 3 Calculation of Cutter-Workpiece Engagement Maps for Five-Axis Flank Milling by the Parallel Slicing Method  3.1 Overview Cutter-workpiece engagement (CWE) maps, or cutting flute entry/exit locations as a function of height, are a requirement for cutting-force and chatter stability predictions of five-axis machining operations. Dynamic displacements of the tool and part can cause the engaged area to change during machining. However process engineers in the jet- engine manufacturing industry minimize these effects by eliminating chatter with variable-pitch cutters [26], [27] and by selecting suitable spindle speeds in order to avoid large, forced vibrations. If static deflections of the tool and workpiece are small, calculation of the CWE area becomes a purely geometric problem. For 2D slot milling cases on a simple block, this area is rectangular and straightforward to calculate. However, when five-axis milling a complicated workpiece with irregular boundaries, the engagement zone can have a complex shape and is much more difficult to determine. Described in this chapter is a new method of calculating cutter-workpiece engagement maps for five-axis milling -- The Parallel Slicing Method (PSM). The algorithms are implemented in C++ using the Andy, Charles, Ian’s System (ACIS) boundary representation (B-rep) solid modeling kernel. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  33 Section 3.2 outlines the PSM and explains each of the required steps in detail. Appendix A presents the mathematics behind the analytical geometry and in Section 3.3, the PSM is tested on a CAD model of an experimental integrally bladed rotor (IBR). The CAD model of the workpiece, tool geometry and tool path were provided by Pratt & Whitney Canada1. Cutter-workpiece engagement maps from the PSM are shown at various stages of the flank milling process, and are compared against those given by a conventional z-buffer method. Finally, a summary of the chapter is given in Section 3.4.  3.2 Cutter-Workpiece Engagements by the Parallel Slicing Method (PSM) The Parallel Slicing Method (PSM) used to obtain cutter-workpiece engagement maps for multi-axis machining is illustrated in Figure 3-1. It uses a discretized 2D form of the 3D geometry found in a B-rep solid model in an attempt to reduce data storage, increase computation speed and make calculation of removed material more straightforward. Also, using this scheme, parallel processing could be employed to solve each planar problem independently, significantly decreasing calculation time [115]. As shown in Figure 3-1, there are seven main steps. Steps 3 to 7 are performed in a loop, for each tool move, until all moves have been processed. Each of the steps is explained in detail in the following sections.   1The research partner in the project, Pratt  & Whitney Canada (PW&C) is a large commercial aircraft engine manufacturer based in Longueuil, Quebec that uses the IBR for testing purposes. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  34  Swept area Calculate tool-plane intersection curves. Calculate swept area between intersection curves. Subtract swept area from slice plane. Intersect curve with remaining planar material. Convert engagement "arcs" to tool coordinate system. Join endpoints of arcs with lines to form engagement polygon. 3. 4. 5. Generate removal volume and updated workpiece. 1. T6. Generate cutter- workpiece engagement maps from polygon. 7. Slice removal volume into planes along a common axis. 2. 1 2 1 2 1 2 z φ T Removal volume Updated workpiece S  Figure 3-1 – Illustration of the Parallel Slicing Method (PSM) [56], [57].   Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  35 3.2.1 Generation of Removal Volume and Updated Workpiece The first step in the PSM is to generate the solid volume removed by the tool cutting the workpiece. This is called the removal volume and can be thought of as the Boolean intersection of the tool swept envelope with the workpiece. Once the removal volume has been obtained, it is used for cutter-workpiece engagement calculations along the tool path. The swept volume of the tool path is also used to update the part for the next milling operation. Illustrations of the removal volume and updated workpiece are shown in Step 1 in Figure 3-1. In this research, the removal volume is used for cutter-workpiece engagement calculations rather than the entire workpiece. This allows Boolean intersections and subtractions to be performed on a smaller, simpler data structure. It also ensures that the geometry of the workpiece is gradually deleted as the simulation takes place, rather than becoming larger and more complex. This reduces the number of edges that the solid modeller must search through when querying and modifying the workpiece geometry. Generation of swept volumes for 2-axis and 3-axis tool paths is relatively easy to implement. However, generation of accurate swept volumes for multi-axis tool paths with general milling cutters is a challenging task. A few authors addressed this area [1], [41], [111] but the solutions presented are fairly involved. Since the focus of this work is not on swept volume generation, but on cutter-workpiece engagement calculations, the swept volume of the tool path was obtained by uniting solid models of the cutter together at various points along the tool path. A text parser was coded to read five-axis Automatically Programmed Tools - Cutter Location (APT-CL) data files. The APT-CL file contains the dimensions of the tool and Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  36 the locations and orientations of the tool along the path. The dimensions of the tool are given after the CUTTER keyword and follow the 7-parameter general end mill given in Engin and Altintas [45] as shown in the upper left of Figure 3-2.  M u N M D O R R C R L M N h ST r z r z r z N α β z r D≠0 , R=R ≠0 , R =0 α=0 , β≠0 , h≠0 z r N M General End Mill Model Tapered Ball-End MillBall-End Mill D≠0 , R=R =D/2 , R =0 α=β=0 , h≠0 z r D≠0 , R=0 , R =D/2 R =0 , α=β=0 , h≠0 r z Cylindrical-End Mill M N N  Figure 3-2 – The seven parameter tool model used in Engin and Altintas [45] (upper left) and illustrations of helical, cylindrical-end, ball-end and tapered ball-end mills.   Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  37 At this time, tapered ball-end mills, ball-end mills and cylindrical-end mills are the only types of tools supported by the PSM (Figure 3-2). As explained in Section 3.2.3, additional mathematics may be required for other cutters and is an area for future study. This means that all tools supported by the PSM have dimensions of the following forms, 0, 0, 0, 0, 0, 0z rD R R R hα β≠ = ≠ = = ≠ ≠    for tapered ball-end mills  0, / 2, 0, 0, 0z rD R R D R hα β≠ = = = = = ≠      for ball-end mills  0, 0, / 2, 0, 0, 0 r zD R R D R hα β≠ = = = = = ≠  for cylindrical end mills   (3.1)  where , , , , , , r zD R R R hα β  are the seven given parameters. , , ,r z r zM M N N  are the radius and height at points, M  and N , respectively (see Figure 3-2). , , , r z r zM M N N  are calculated from algorithms given in [45]. The location and orientation of the tool at each move are given by GOTO commands in an Automatically Programmed Tools-Cutter Location file (APT-CL). An example is shown below:           GOTO  /3.85223,-5.59226, 1.83353,  .098484, -.070405,  .992645           GOTO  /3.86926,-5.57784, 1.85518,  .097217, -.070637,  .992753           GOTO  /3.88629,-5.56310, 1.87674,  .095971, -.070943,  .992853  The transition between GOTO commands is called a tool path segment, which is illustrated in Figure 3-3. The first three columns give the x,y and z coordinates of the tool tip, { }iP , and the last three columns give the unit vector orientation of the tool axis vector, { }iT . The subscript, i , denotes tool path command, i . At each discrete position, a solid model of the tool was generated by revolving a wire-body of its profile around the Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  38 CL Command (i) {Pi+1} {Pi} CL Command (i+1) ∆θi u {ki} Tool Path Segment i {Ti+1} {Ti} u=0 u=1  Figure 3-3 – Illustration of a tool path segment, defined by a series of GOTO commands [52], [53].  tool axis. Then, each revolved profile was rotated and translated to the position and tool axis orientation given in the APT tool path. All solid models of the tool were united to form the swept volume with the Boolean api_unite() application programming interface (API) in ACIS. If the tool moves are spaced far apart, or contain large changes in tool axis orientation, the solid models of the tool may not intersect. In this case, additional tool moves can be inserted into the tool path to obtain a smooth, continuous shape. As an approximation, the tool tip position can be linearly interpolated and the tool axis spherically interpolated. This is given by, { } { } { }( ) { } { } [ ]{ }1i i i iP P P u P T R T+= − + =  (3.2)  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  39 where { }iP  and { }iT  are the tool tip position and unit vector tool axis orientation at tool path command, i , respectively. 1 ii N= …  is the tool move number, where iN  is the number of tool moves. The spherical interpolation matrix, [ ]R , is given by, [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 2 2 2 1 1 1 1 1 1 1 1 1 x x y z x z y x y z y y z x x z y y z x z k h h k k h k g k k h k g R k k h k g k h h k k h k g k k h k g k k h k g k h h  − + − − − +   = − + − + − −    − − − + − +    (3.3)  where, { } { } { }{ } { } { } { } { } { } 11 11 sin( ) cos( ) arctan2 i i x i ii i i y i i ii i z g u h u k T TT T k k T TT T k θ θ θ ++ ++ = ∆ = ∆    ××  = = ∆ =    •×           (3.4)  The unit vector axis of rotation is given by { }ik  and the total angle of rotation between command, i  and 1i +  (tool path segment i ) is given by iθ∆ . The arctan2 function in Equation (3.4) is the four quadrant arctangent function, which allows calculation of angles between –pi  and pi . The parametric variable, u , varies between 0 and 1 and represents the position and orientation of the tool between command i  and 1i + . When u is 0, the tool tip is at position, { }iP , tool axis orientation, { }iT . When u  is 1, the tool tip is at position, { }1iP+ , tool axis orientation, { }iT . To divide the interval of 0...1u =  into intN  additional, equally-spaced points, the equation, ( )int 1m m u N = +   (3.5) Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  40 is used. Where int1, 2,3m N= …  is the number of the interpolated tool move. Note that the above method for interpolating between tool moves is based on an approximate model. A more accurate approach would be to fit the tool path with splines and calculate intermediate tool moves from the spline equations. An even more exact approach would be to model the kinematics of the machine tool and use its joint space to interpolate tool moves. Once the swept volume has been generated, it is intersected with the solid model of the workpiece to obtain the removal volume. The swept volume is also subtracted from the part to obtain the finished workpiece. Both Boolean intersection and subtraction operations are performed using application programming interfaces (APIs) in ACIS. An example of the removal volume and finished workpiece for a simple case are shown in Step 1 in Figure 3-1. Uniting successive tool moves to generate swept volumes is recognized as a computationally inefficient approach, even though it is simple to implement, as it produces an entity with many small edges and surfaces. When this swept volume is intersected with the workpiece, these edges and surfaces will appear in the removal volume. This will increase the number of entities that the solid modeler must search through in order to correctly query and modify the workpiece. Also, when the swept volume is subtracted from the workpiece, it will imprint many of these tiny edges and surfaces on the finished part. As more and more machining operations are performed, the number of edges and surfaces in the finished workpiece and removal volumes will grow, leading to an increase in computation time. In these situations, an improvement in calculation speed could be realized by use of an approximated or analytically generated Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  41 smooth swept volume with non-uniform rational B-spline (NURBS) surfaces. This is left as an area for further study. It is also possible that the removal volume for each machining operation could be obtained from the CAD designer during the design stage of the part.  3.2.2 Division of Removal Volume into Parallel Planes After the removal volume is generated, it is divided into a number of planes along an axis, { }S  (see Figure 3-1 Step 2). The slice normal can be in any direction, but it should be chosen so that it is as orthogonal as possible to the direction of motion. This reason for this that during simulations, the workpiece is updated using 2D swept areas. When the tool movement is parallel to the slice normal, the swept areas may not capture all of the material cut between the tool moves (Figure 3-4). s=1 s=2 s=3 s=1 s=2 s=3 Slice Normal Slice Normal {Ti} {Ti+1} {Ti} {Ti+1} Moving Direction Moving Direction Swept Areas No swept area possible!  Figure 3-4 – Swept areas may not update the workpiece accurately if the slice axis is chosen improperly [56], [57].  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  42 Another objective for the slice plane orientation is to attempt to orient the planes so that they cut the tool at locations where the intersection graphs will be composed of only one conic. This reduces the complexity of the swept area calculations (Section 3.2.4). For the impeller machining tool path on which the PSM is demonstrated in Section 3.3, a practical method of determining an optimum slice plane-orientation is to take the half angle tool-axis orientation between the first and last tool moves. This method is acceptable, since the total angle swept out by the tool over the tool path is relatively small. This technique is shown in Figure 3-5(a) and given by, { } [ ]{ }1S R T=  (3.6)  with [ ]R  calculated from Equations (3.3) and (3.4) but using, { } { } { }{ } { } { } { } { } { } 11 11 1 arctan2 2 ii ii x NN y NN z k T TT T k k T TT Tk θ    ××   = = ∆ =   •×       (3.7)  instead of { }ik  and iθ∆ . The unit-vector slice normal (slice axis) is given by { }S . The tool-axis rotation vector is denoted by { }k  and the half-angle of the total rotation between the first and last tool moves is given by θ∆ . The orientation of the slice normal, according to Equations (3.3), (3.6) and (3.7), can be considered as an average tool-axis orientation over the entire tool path. For tool paths where the angular motion is large, this may not be adequate. Tool paths should be checked at run time for large changes in angular motion, and then split into smaller Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  43 2∆θ ∆θ{T1} {TNi} {k} S Plane 1 Plane Ns Removal volume sliced along {S} dmax dmin {B} {C1} {CNs} Extents of workpiece touching Plane 1 Bounding Box Extents of workpiece touching Plane Ns (a) (b) {S} Removal volume  Figure 3-5 – (a) A practical method of determining the slice axis, { }S .  (b) Removal volume sliced along, { }S . [56], [57].  sections according to a predefined angular limit. Each of these sections will have its own removal volume, which will be sliced along a different direction. Once the slice axis has been determined, the removal volume is divided into sN planar slices along { }S . The center coordinates of the first and last plane are determined as shown in Figure 3-5(b). The bounding box of the removal volume is obtained by calling an API in ACIS and a large plane is created with normal, { }S , and center coordinates, { } { } { }1C B d S= −  (3.8)  { }1C  is the center coordinates of the first slice plane, { }B  is the bounding box center coordinates and d  is half diagonal of the bounding box. To maximize the resolution of the method, the first and last slice planes should just touch the extents of the Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  44 removal volume. However, the removal volume is usually within the full-extents of the bounding box and so a plane with this center may not touch the object. After the plane is generated, it is intersected with the removal volume. If the result of the operation is a null body (no intersection), then the plane is moved inwards along { }S  in small steps (by decreasing d ) until an intersection is found. When one is detected, the center coordinates of the first plane are calculated from Equation (3.8). This procedure is repeated to find the center coordinates of the last plane, { }NsC : { } { } { }NsC B d S= +  (3.9)  Once { }1C  and { }NsC  have been determined, the removal volume is sliced into sN  slices using, { } { } { }( ) ( )( ) { }1 111Nss s s C C C C N − = − + −   (3.10)  which is shown in Figure 3-5(b). { }sC  denotes the center coordinates for plane, 1,2 ss N= … . The spacing between each plane will be, { } { }1 1 N s s C C s N − ∆ = −   (3.11)  Rearranging Equation (3.11) for sN  gives, Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  45 { } { }1 1 N s s C C N s − = + ∆   (3.12)  which expresses the number of planes, sN , as a function of their spacing. The spacing is equivalent to the resolution of the cutter-workpiece engagement maps and should be based on their desired accuracy. A higher number of planes will give more accurate maps, but at a greater computational expense.  3.2.3 Calculation of Tool-Slice Intersection Curves After the removal volume has been divided into slices, the intersection curves between the tool and each slice are determined (see Figure 3-1 Step 3). The equations for the tool-slice intersection curves are used to determine the 2D swept area cut away between tool moves. Also, the intersection of the tool-slice curves with the updated planes of the removal volume (Section 3.2.5) forms the boundaries of the engagement zone. The curves are calculated by evaluating the intersection of the solid cutter with the equation for the removal volume plane analytically. Calculating the equations of the curves in this fashion and creating them with ACIS helps reduce the number of costly Boolean intersection operations required. The mathematics of the procedure is explained in detail in Appendix A1-A3. The intersection of a tapered ball-end mill with a plane can produce a conic, composite conic, truncated conic or a truncated composite conic. Ball-end mills and Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  46 cylindrical-end mills can be considered as subsets of tapered ball-end mills. This means they will have tool-plane intersection shapes with similar conic forms. Examples of tool- plane intersection shapes with a tapered ball end mill at different orientations are shown in Figure 3-6. The simplest shape, the conic, occurs when the plane cuts the tool along only one solid body (see Figure 3-6(a)). Composite conics mean that the intersection curve is composed of more than one conic entity. These shapes arise from the fact that a tapered-ball end mill is a combination of two solids -- a sphere and an inverted conical frustum. Intersecting the tool with a plane where the solids are joined will produce an intersection shape that is composed of two conics joined at "knot points", or points of discontinuity (see Figure 3-6(b)). Truncated shapes can appear due to the cutter’s finite length (Figure 3-6(c)). For example, the top of the tool will create discontinuities in the intersection shapes when the plane cuts it at this point. In this case the conic will not be complete; the knots in the intersection graph will be joined by a line. Composite truncated conics are a combination of the two, previously mentioned shapes (Figure 3-6(d)). For tapered-ball end mills, the intersection shapes will be composed of one or more conics. However, for other cutters, such as bull-nose end mills, the tool-plane intersection shapes may include higher-order curves such as toric sections. Development of additional mathematics for these intersection curves and swept areas is an area for future study.  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  47 X Y Z X Z Y X Y Z XY Z Ellipse Composite Conic - Ellipse-Circle Truncated Conic - Ellipse-Line Composite Truncated Conic - Ellipse-Circle-Line (a) (b) (c) (d) Ellipse Circle Ellipse Line Circle Ellipse Line Ellipse Knot 1 Knot 2 Knot 1 Knot 2 Knot 3 Knot 4 X Y Z X Y Z X Y Z X Y Z  Figure 3-6 – Illustrations showing various tapered ball-end mill / plane intersection shapes [56], [57]. (a) Single conic (b) Composite conic (c) Truncated conic (d) Composite truncated conic   Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  48 3.2.4 Calculation of Tool Swept Area Tool Swept Area Between Conic Curves If the angular motion between tool moves is small, then the curved boundaries of the tool swept envelope on each removal volume slice can be approximated by a set of two bounded lines. For tapered ball-end mills, ball-end mills and cylindrical-end mills these lines can be approximated by the common tangents of the intersection shapes - See Figure 3-1 Step 4. The end points of these tangent lines (tangent points) are joined by two additional lines to create a quadrilateral area that approximates the swept area of the tool. Common conic-conic tangent line solutions are determined analytically to reduce the number of API calls to the solid modeler (ACIS), which aids computational efficiency. Determining the common tangents between two conics requires solving a fourth-order polynomial equation, which is given in Appendix A4. Polynomial equations less than fifth order can be solved with a closed-form analytical solution [2]. An alternative is to use a numerical polynomial solver such as the Jenkins-Traub root finding algorithm [73]. The C++ code for the Jenkins-Traub algorithm is widely available in the open literature. It is known to provide extremely accurate solutions rapidly. Figure 3-7 shows an example of the swept area quadrilateral solution between two rotated ellipses given by tangent lines 1 and 2 and the dotted lines connecting the tangent points. The number of real solutions to the fourth order equation indicates how many tangent lines exist. Solutions occur in pairs -- zero, two and four lines are possible. The tangent lines can either be classified as “straight” or “cross”. Straight tangent lines have outward normals at the tangent points that point in the same direction. Cross tangent lines Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  49 have outward normals that point in opposite directions. An example of this is shown in Figure 3-7. Lines 3 and 4 are cross tangents and should be eliminated from the solutions, as they will not be part of the swept area. A2 = 84.00 B2 = 52.00 C2 = -10894.15 D2 = 7451.28 E2 = 55.43 F2 = 436407.66 A1 = 117.00 B1 = 63.00 C1 = -4210.61 D1 =-3455.30 E1 = 93.53 F1 = 50422.15 Conic 2 (Ellipse) ae2 = 10.00 be2 = 6.00 φe2 = -60 deg xe2 = 50.00 ye2 = 45.00 Conic 1 (Ellipse) ae1 = 12.00 be1 = 6.00 φe1 = -60 deg xe1 = 10.00 ye1 = 20.00 Tangent Lines p qNo. Type Straight Straight Cross Cross -5.00546e-1 7.30132e-1 2.01801e-2 -3.55589e-2 -9.71235e-2 6.19068e-2 3.47211e-3 -3.30663e-2 1 2 3 4 (px+qy=1) 0 20 40 60 0 20 40 60 x-coordinate y -c o o rd in a te 1 2 3 4 aebe xe ye φe x y Outward Normals at Tangent Points  Figure 3-7 – Calculation of the approximate tool swept area between two ellipses [56], [57]. An analytical tangent line solution was used to generate the swept area.  As mentioned previously, for tapered-ball end mills the intersection curves for each tool move may be a conic, composite conic, truncated conic or truncated composite conic. The swept area solutions for truncated and composite (or truncated composite) conic shapes are more complicated. In these cases, the boundaries of the swept area are a subset of the common tangents, knot-tangents and the knot-knot lines. Currently, a sorting algorithm is used to determine the correct lines for the swept area calculations. Calculation of the swept area for these shapes increases the computational load, so it is preferable to try and avoid as many of these shapes as possible. Minimization of these Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  50 shapes is usually achieveable by orienting the slice axis as closely as possible to the tool axis.  Tool Swept Area for Truncated and Composite Conics When cutters composed of two or more primitive solids (e.g. tapered ball end mills and ball-end mills) are intersected with a plane, it is possible that the resulting tool-plane intersection curve will be composed of two or more conics -- see Figure 3-6(b) and Figure 3-6(d). This can occur when the slice plane intersects a location where the solids are joined. It is also possible for the conics to be truncated -- for example, a partial conic with two knots joined by a line -- see Figure 3-6(c) and Figure 3-6(d). Truncated shapes are due to the cutter being of finite length. Determining the swept area between two intersection curves that are composed of these shapes is more involved. An algorithm has been developed to perform this task. Like conic-conic intersection swept areas, the lines that bound the swept area between composite and truncated conics can be obtained with the use of common tangents. However, as there can be multiple entities and discontinuities in the graphs at the knot points, the solution is not as straightforward. For instance, when computing common tangents, it is possible that some of the calculated lines will not be on the bounded portion of the intersection graphs. Lines such as these should be removed. The lines that bound the swept area between complex intersection curves are a subset of all conic-conic tangent lines, knot-conic tangent lines and knot-knot lines between all curves on each body. To calculate this area, all possible combinations of lines Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  51 are computed and those which do not touch the same side of each intersection graph only once are removed. This procedure is accomplished with a sorting algorithm that has the following set of rules: 1. The tangent points of candidate lines must be on the bounded part of the intersection graph. 2. The perpendicular components of the outward normals where the lines touch each intersection shape must point in the same direction. 3. The lines of the swept area must not cross through the interior of an intersection graph. The algorithm is explained in the form of a flow chart – see Figure 3-8. Examples of the steps of the flow chart in Figure 3-8 are shown in Figure 3-9. Step 1 in Figure 3-9 shows identification of all entities on the ellipse-circle intersection graphs. Figure 3-9 shows all possible combinations of lines between the two composite conic intersection curves. In Figure 3-9, the Step 3 example shows conic-conic tangent lines (dashed) whose endpoints are not on the bounded portion of the intersection graphs – these should be removed. The Step 4 example shows a candidate knot-conic line and how perpendicular components of the outward normals can point in opposite directions. This line should be excluded. The illustration in Step 5 shows a knot-conic line passing through the interior of the first complex conic. It has 4 intersection points. Knot-conic lines that are part of the final solution will have only 3 intersection points – one at the tangency point and two at the knot. Since this line has four, it will be discarded. The lines remaining after Steps 1-5 give the swept area solution, which is shown in Figure 3-9. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  52 5(a) Remove lines with more than two intersection points (one on each curve) START - 1 Identify all boundary lines, knots and conic shapes on both intersection curves 2(a) Calculate conic- conic tangent lines 2(b) Calculate knot- conic tangent lines 2(c) Calculate knot-knot lines 3 Remove lines where tangent points are not on intersection graphs 4 Remove lines where perp. components of outward normals at endpoints point in opposite directions 5(b) Remove lines with more than three intersection points (one on curve, two at knot) 5(c) Remove lines with more than four intersection points (two at each knot) END - Two lines remain. Join end points on each body with lines to form quadrilateral area CC KC KK KK KCCC CC KC CC KC KK CC = Conic-conic tangent lines KC = Knot-conic tangent lines KK = Knot-knot lines (a)  Figure 3-8 – Flow chart of the algorithm used to calculate the swept area between complex intersection shapes [56], [57].   Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  53 Knot-Knot Lines END - Final Swept Area Knot 1 Knot 2 Circle Center Ellipse Center Circle Center Ellipse Center Knot 1 Knot 2 Composite Conic 2 Composite Conic 1 Knot-Conic Lines Line 2 Line 1 Example - Step 4 Example - Step 5 Step 1 Step 2 (b) Example - Step 3 Conic-Conic Lines  Figure 3-9 – Graphical examples of the various steps of the algorithm used to calculate swept areas between complex intersection shapes [56], [57]. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  54 From the steps above, it is obvious that calculation of the swept area between composite and truncated intersection curves is more involved and, as a result, requires additional computational effort. These shapes appear more frequently when the slice plane’s normal is almost orthogonal to the tool axis vector. Ideally, simple conic-conic intersections curves are desired. This means the slice axis should be carefully chosen in a manner that minimizes complex shapes.  3.2.5 Subtraction of Tool Swept Area and Generation of Engagement Arcs Once all swept areas for a tool move are calculated, they are subtracted from the removal volume slices using 2D Boolean subtraction. Although the intersection of the tool-plane intersection curve with the sides of the removal volume will usually give correct engagement conditions (i.e. no removal of the swept area), this is often not the case when self-intersection arises. An example of this is shown in Figure 3-10(a). Figure 3-10(a) shows a top view of the planar removal volume slice from a self-intersecting tool path. If the swept area is not removed between tool moves 1-2, then for tool move, i , the engagement will appear to be full rather than zero. A more common type of self intersection that can appear in five-axis flank milling is when the tool moves in the manner shown at the bottom of Figure 3-10(b). Note that the movement here is exaggerated for illustration purposes. In this figure, the tool changes from orientation 1 to 2 and then from 2 to 3. On slice plane, s , if the material  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  55 1 If swept area is not removed, then at tool move, i,  engagement will be full rather than zero 1 2 3 1 2 3 1-2 swept area not removed and using sides of removal volume - full engagement at 3 1-2 2-3 12 3 (a) (b) Slice plane, s 2 ii-1 1-2 2-3 Side view Top view Top viewTop view Sides of removal volume Slice plane, s Slice plane, s 1-2 swept area removed - no engagement at 3  Figure 3-10 – (a) An example of a self-intersecting tool path. (b) A more common example of self-intersection that can occur in five-axis flank milling. [56], [57].  between 1-2 is not removed, then the intersection curve will incorrectly show engagement at 3. After all swept areas have been subtracted from the slices of the removal volume at a given tool move, the engagement arcs are formed. This is performed by modifying the boundaries of the tool-plane intersection curves with the boundaries of the updated removal volume slice – a task performed using a Boolean intersection operation in ACIS. The resulting arcs give the entry and exit points of the cutting flute with the workpiece - see Figure 3-11 (also Figure 3-1 Step 5). Together, the boundaries of all arcs will give the total entry and exit points of the cutter with the removal volume. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  56 X Y Swept Area Feed direction φst φex Tool-plane Intersection curve i+1 Tool-plane Intersection curve i Engagement arc i+1 Slice plane, s  Figure 3-11 – An engagement arc formed by the intersection of a tool-plane intersection curve and the updated removal volume slice.  3.2.6 Conversion of Engagement Arcs to Tool Coordinate System and Creation of Engagement Polygon When all engagement arcs for a particular tool move are created, they are transformed from the global coordinate system to the tool coordinate system. The origin of the tool coordinate system is at the tool tip – see Figure 3-12(a). This step is performed as the force prediction model in Chapter 4, and the chatter stability solution in Chapter 6, requires the cutter-workpiece engagements to be referenced to this coordinate system. The tool coordinate system is composed of the tool axis, { }T  or { }TTZ , the direction of forward motion perpendicular to tool axis, { }TTX , and the axis of zero immersion angle, { }TTY . The immersion angle is the angle of the cutting edge from the { }TTY  axis, measured in the clockwise direction. The subscript,  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  57  (a) (b) Tool Coordinate SystemGlobal Coordinate System {Pi} = (0,0,0) [0,1,0] [0,0,1] Tool Axis {Xg} {Yg} {Zg} {YTT} {ZTT} (0,0,0) Moving Direction Perp. to Tool Axis Normal to     and {XTT} [1,0,0] Global Origin Start End Joining End Points of Arcs with Lines All Points Joined Closed Polgon of Lines [0,0,1] [0,1,0] [1,0,0] Engagement Arcs {XTT} {YTT} {XTT} {ZTT} {ZTT} {ZTT} {ZTT} {XTT} {ZTT} {YTT} {Pi} {ZTT}  Figure 3-12 – (a) The engagement arcs are converted from the global coordinate system to the tool coordinate system. (b) The engagement polygon is created by moving around the perimeter of the arcs and joining their end points with linear segments. [56], [57].   Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  58 TT , indicates that the tool coordinate system is measured at the tool tip. Relative to the global system, the vectors are given by, { } { } { }{ } { } { } { } { } { } { } { } { } TT TT TT TT TT Y T T M X Y Z T Y T T M × × = = = × ×   (3.13)  For sculptured-surface machining tool paths, where the tool path segments are small, { } { } { }{ } { } 1 1 i i i i P P M P P + + − = −   (3.14)  where { }M  is the direction of tool tip motion and { }iP  is the tool tip coordinates at tool move, i . The vectors given by Equation (3.13) can be put into a matrix that defines the coordinates of the tool coordinate system in those of the global coordinate system: [ ] { } { } { } 11 12 13 21 22 23 31 32 33 GT TT TT TT a a a A X Y Z a a a a a a     = =          (3.15)  To transform the arcs at each tool path command, i , to the tool coordinate system, each arc is first translated by { }iP− , and rotated by an angle ψ∆  around an axis, { }w . ψ∆  and { }w  are given by, [ ] { } ( ) 32 23 1 13 31 21 12 1 1 cos 2 2sin GT a a Tr A w a a a a ψ ψ − −   −  ∆ = = −   ∆    −    (3.16)  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  59 The end points of the arcs are converted into immersion angle coordinates by, arctan2 2 ep ep ep ep ep y z z x piφ  = − =       (3.17)  where epφ  is the immersion angle for end point, ep , measured from the { }TTY  axis. epx , epy  and epz  are the coordinates of point, ep , in the tool coordinate system. The arctan2 function in Equation (3.17) is the four-quadrant arctangent function. The end points of the arcs are ordered and connected with linear segments to create the boundary of the cutter-workpiece engagement map. The algorithm that performs the joining process starts from the lowest arc and smallest engagement angle and moves around the perimeter of the engagement zone, until it returns to the starting point – see Figure 3-12(b). The algorithm that joins the endpoints of the arcs was written in C++. It automatically performs the joining process, after the arcs are converted to the tool- coordinate system.  3.2.7 Creation of Cutter-Workpiece Engagement Maps The engagement zone is now a closed polygon of LN  contiguous lines. At this point, the cutter-workpiece engagement maps can be generated. These maps give the entry and exit immersion angle pairs of the cutting flutes with the workpiece as a function of distance from the tool tip. Although there are many possible methods of storing the engagement maps, a method commonly used by force prediction algorithms is by a series of rectangular blocks – see Figure 3-13. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  60  1.8 2.0 2.2 2.4 2.6 2.8 Close-up of a Typical Cutter- Workpiece Engagement Map Immersion Angle [rad] 144 146 150 152 154 148 D is ta n c e  F ro m  T o o l T ip  [ m m ] zb ab φ st φ ex z = 0  Figure 3-13 – Close-up of a typical cutter-workpiece engagement map [56], [57]. The maps are defined by a series of blocks.  Each block represents an engagement angle pair at a given height and is defined by four parameters - bz , ba , stφ  and exφ  (see right of Figure 3-13). The distance from the tool tip to the bottom of the engagement block is bz  and the height of the block is ba . The entry and exit immersion angles of the block are defined by stφ  and exφ , respectively. To generate the engagement maps, the engagement polygon is divided along its height into intervals of ba , according to a user-specified resolution. At each interval, the z  coordinate at the middle of each block is checked for intersection with all lines of the engagement polygon using, , , , s k k e k s k z z u z z − = −   (3.18)  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  61 where ,s kz  and ,e kz  are the starting and ending z-coordinates of line number, k . The parametric variable, ku , represents the position of the given z  coordinate along line, k . If 0 1ku≤ ≤  on any of the lines, then there is an intersection at that height. If 0ku <  or 1ku > , then no intersection exists. If an intersection does exist, the parametric variable, ku , can be substituted into the equation, ( ), , ,st ex e k s k k s kor uφ φ φ φ φ= − +  (3.19)  to find the corresponding entry/exit angle, stφ  and exφ , at the height in question. Since the boundary area is a closed polygon of continuous lines, the entry exit angles will occur in pairs.  Each pair ( stφ , exφ ) forms the boundaries of one block and there may be more than one block at each height. The bottom of each block, bz , is calculated from, 2 b b a z z= − (3.20)  After all blocks have been calculated, they are written to a text file which can be used by force-prediction and chatter stability algorithms.  3.3 Comparison of Parallel Slicing Method with Z-Buffer Scheme An impeller roughing tool path for a prototype integrated bladed rotor (IBR) was used as a case study to obtain cutter-workpiece engagements (CWE) maps using the Parallel Slicing Method (PSM). These maps are compared against those obtained using the Manufacturing Automation Laboratory’s Virtual Machining Interface (MAL-VMI). Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  62 Cutting force predictions using CWE maps from the MAL-VMI method have been found to agree well with experiments in 2,3 and 5-axis milling – usually to within 20%. As a result, the MAL-VMI method can be considered a good benchmark. The Virtual Machining Interface uses an application programming interface (API) in the commercial numerical control (NC) verification software package, Vericut [126], to obtain cutter-part intersections. The API in Vericut employs a z-buffer scheme that calculates the intesection of the tool with the workpiece and then returns a grid of intersection points. As shown in Figure 3-14, the intersection of rays cast from these points (along the moving direction) with the solid model of the tool form the cutter-workpiece intersection maps. Grid Rays Tool Cutter-workpiece engagement map Moving Direction  Figure 3-14 – Figure showing how the MAL-VMI system obtains cutter-workpiece engagement maps. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  63 The impeller roughing tool path used to compare the two methods is composed of approximately 300 tool moves and is illustrated in Figure 3-15. The cutter is brought to the top of the impeller blank with the tool axis oriented roughly parallel to the workpiece. It is gradually lowered into the workpiece and, when fully immersed, plunges through the blank with a curved motion until it exits through the bottom of the part. The tool used for this tool path is a tapered ball-end mill, which has a shank diameter of 38.1mm and a length of 203mm long from tip to tool holder. The distance from the tip of the cutter to the top of the tapered section (where the cutting flutes end) is 170mm. Start End X YZ  Figure 3-15 – The impeller roughing tool path used to compare the Parallel Slicing Method and MAL-VMI cutter-workpiece engagement calculation schemes  [52], [53]. Illustration was generated with the help of CGTech’s Vericut (TM) [32].  The swept volume, removal volume and updated workpiece are shown in Figure 3-16. The swept volume of the operation is calculated by uniting all tool moves in the tool path (Section 3.2.1) together and is used to create the removal volume and finished workpiece. Due to the tight spacing between tool moves in the tool path, cusps near the sides of the removal volume were negligible. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  64 Initial Workpiece (W) Toolpath Swept Volume (SV) Initial Workpiece with Tool Swept  Volume (W )     (SV) U (W )     (SV)- Removal Volume Finished Workpiece Start End  Figure 3-16 – Calculation of the removal volume and finished workpiece for a five- axis flank milling operation on an integrally bladed rotor (IBR) [56], [57].  Figure 3-17(a)-(b) shows how the removal volume is sliced into 30 planes along the half-angle of the start and end tool axis orientations (Section 3.2.2). Figure 3-17(c)-(f) illustrates how the sliced removal volume is gradually machined away during simulation. Performing Boolean operations on the removal volume rather than the entire workpiece has the benefit that the structure of the CAD model is simpler and smaller. Also, the file size is gradually reduced in size as the machining operation progresses, as edges and surfaces are deleted from the model during simulation. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  65  Initial Sliced Removal Volume (30 Planes) Sliced Removal Volume       - Tool Move 130 Sliced Removal Volume      - Tool Move 260 Removal Volume Sliced Removal Volume       - Tool Move 280 Sliced Removal Volume       - Tool Move 180 (a) (b) (c) (d) (e) (f )   Figure 3-17 – (a)-(b) The removal volume sliced into 30 planes along the average tool axis orientation. (c)-(f) The removal volume at various stages of the flank- milling operation. [56], [57].    Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  66 Figure 3-18 shows a table of the engagement geometry at various points in the tool path. The second column from the left shows the engagement arcs left by the intersections of the tool-plane intersection shapes with the 2D removal volume slices. The third column shows how these arcs appear on the surface of the cutter. The fourth column shows the engagement polygon, which gives the boundaries of the engagement zones. The last column shows the engagement maps created from the polygon. Figure 3-18 illustrates that the Parallel Slicing Method can obtain complex engagements for multi-axis machining with a relatively high degree of accuracy. Figure 3-19 shows the difference in map resolution realized by a greater number of removal volume slices. Figure 3-19(a) shows an engagement map produced using a removal volume sliced into 30 planes, whereas Figure 3-19(b) shows the map given when the removal volume sliced into 60 planes. For this case, the distance between the slices drops from 5.33mm to 2.667mm. The higher number of planes more accurately captures the curved hole evident in the engagement maps, as the hole appears to be deeper in Figure 3-19(b) than in Figure 3-19(a). This hole is due to the tool exiting the workpiece near the end of the tool path. The engagement map with the higher number of slices is more continuous and less "blocky" than with the lower resolution. However, both cases show that the differences in the CWE maps are relatively minor even though the 60 slice removal volume requires significantly more computational effort.    Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  67  -1.0 3.02.01.00.0 -1.0 3.02.01.00.0 -1.0 3.02.01.00.0 -1.0 3.02.01.00.0 40 80 120 160 0 40 80 120 160 0 40 80 120 160 0 40 80 120 160 0 Immersion angle (rad) Immersion angle (rad) Immersion angle (rad) Immersion angle (rad) z (m m ) Cutter-Workpiece Engagement Maps  Arc BoundariesEngagement         Arcs      Arcs on Cutting Tool Tool Move   Number 130 180 260 280 X Z Y X Y Z z (m m ) z (m m ) z (m m ) X Z Y X Z Y X Z Y X Z Y X Z Y X Z Y X Z Y X Z Y X Y Z X Y Z  Figure 3-18 – Table of engagement geometry for the IBR roughing tool path. Shown in the figure are the engagement arcs, the engagement arcs on the tool, the engagement polygon and the cutter-workpiece engagement maps for various tool moves.  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  68 -1.0 0.0 1.0 3.02.0 -1.0 0.0 1.0 3.02.0 0 40 80 120 160 0 40 80 120 160 Immersion Angle        (radians) Immersion Angle         (radians) z (m m ) Tool Move 268 - 30 Slices Tool Move 268 - 60 Slices Engagements Arcs Engagement Map Engagements Arcs Engagement Map (a) (b) z (m m )  Figure 3-19 – Illustration of cutter-workpiece engagement maps from a removal volume (a) 30 slices (b) 60 slices. [56], [57].  Figure 3-20 compares the engagement maps from the Parallel Slicing Method (PSM) with those obtained from the MAL-VMI cutter-workpiece engagement (CWE) method. Although the maps match quite well, the Parallel Slicing Method captures the boundaries of the engagement zone more accurately. The MAL-VMI scheme shows serrated boundaries on the borders of the engagement maps, which result from the projection of the intersection grid into cylindrical coordinates. In contrast, the PSM shows smooth borders, which is a closer approximation of the true cutter-workpiece engagements. The PSM also captures a wider range of engagement angles. Figure 3-20 shows that the maps produced by the PSM include engagement angles less than zero degrees, whereas the MAL-VMI scheme does not. In five-axis flank milling, the MAL- VMI method often loses engagement information due to its method of CWE calculation. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  69 Toolpath Command 130 -1.0 0.0 1.0 2.0 3.0 Immersion Angle (radians) 40 80 120 160 0 Toolpath Command 180 Toolpath Command 260 Toolpath Command 280 -1.0 0.0 1.0 2.0 3.0 -1.0 0.0 1.0 2.0 3.0 -1.0 0.0 1.0 2.0 3.0 Immersion Angle (radians) Immersion Angle (radians) Immersion Angle (radians) 40 80 120 160 0 40 80 120 160 0 40 80 120 160 0 D is ta n c e  F ro m  T o o l T ip  ( m m ) D is ta n c e  F ro m  T o o l T ip  ( m m ) D is ta n c e  F ro m  T o o l T ip  ( m m ) D is ta n c e  F ro m  T o o l T ip  ( m m ) PSM MAL-VMI  Figure 3-20 – Comparison of engagement maps from the PSM (dashed boundaries) and the MAL-VMI (shaded boundaries) [56], [57].  The API in Vericut returns a grid of points, with the grid normal aligned with the direction of motion at the tool tip (Figure 3-21(a)). This grid is flat and gives a series of Boolean values – either a zero (no engagement) or a one (engagement). The intersection Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  70 of rays projected from these points with the solid model of the tool form the boundaries of the maps. This is sufficient to capture 2D and 3D motion, where the direction of motion is constant along the tool axis. However, when five-axis motion is involved, the direction of motion is different at each point along the tool axis, which may lead to CWEs at the side or back of the cutter. In reality, these engagement conditions will have multiple intersections of the ray with the tool at each height, relative to the grid as shown in Figure 3-21(b). Unfortuntately, since the grid only stores a Boolean value (0 or 1), multiple intersections of the same ray are lost (Figure 3-21(b)). Consequently, the engagement maps become truncated or clipped at the borders (Figure 3-20). Where the MAL-VMI scheme is superior to the PSM is in computation time. On a PC system with an AMD Athlon 64 3000+ CPU and 1.0GB of ram, the MAL-VMI method took about 2 minutes to calculate engagement maps for just over 300 tool moves. By comparison, using a removal volume of 60 slices, the PSM took 38 minutes. When the number of slices was reduced to 30, the computation time was reduced to 19 minutes. A further reduction in calculation time could be realized by optimization of the C++ subroutines, use of a smooth tool swept volume / removal volume, and implementation of parallel processing [115] to compute the required Boolean intersections, subtractions and analytical geometry on each slice.  Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  71 Forward motion       at tool tip This point is not captured Ray This point is captured Top view of tool at height, z Grid 1 (b) Grid returns 0 (no int.) or 1 (int.) Grid normal (a) True engagement zone This zone is lost Grid normal Normal to tool tip motion Tool tip motion Normal to tool tip motion z Rays Grid    Forward motion at z Forward motion       at tool tip    Forward motion at z  Figure 3-21 – (a) Illustration of an intersection grid returned by the API in Vericut. (b) Since the grid only stores a Boolean (true/false) intersection value, this may cause a truncation of the cutter-workpiece engagement zone in five-axis flank milling. Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  72 Use of a tool-united swept volume to compute the removal volume and updated workpiece is recognized as a computationally inefficient approach. The number of small edges and surfaces will grow and slow down calculations as more machining operations are performed on the part. Use of smooth or analytically generated tool-swept volumes when generating the removal volume and updated workpiece would help to increase calculation speed – especially over multiple tool paths. This is an area for future work.  3.4 Summary A new method of calculating cutter-workpiece engagement maps for five-axis flank milling of jet engine impellers has been presented. These maps, along with tool geometry and cutting mechanics parameters, are required when predicting the cutting forces and chatter stability of the five-axis flank milling process. The algorithm, called the Parallel Slicing Method (PSM), is a semi-discrete, solid modelling-based, cutter-workpiece engagement calculation scheme. ACIS, a commercial boundary representation (B-rep) solid modeling environment, and was used to store and update the workpiece and perform Boolean operations. The Parallel Slicing Method was demonstrated on a prototype integrated bladed rotor (IBR) and cutter-workpiece engagement maps were compared with those obtained from the Manufacturing Automation Laboratory’s Virtual Machining Interface (MAL-VMI). The MAL-VMI method uses an API in Vericut [126] to obtain cutter-workpiece intersections through a z- buffer type method. The maps obtained from the two methods were similar, although the PSM more accurately captured the boundaries of the engagement maps -- it showed Chapter 3: Cutter-Workpiece Engagements By The Parallel Slicing Method  73 smoother borders and captured a greater range of immersion angles. Currently, the main drawback to the PSM is computation time - it is significantly slower than the MAL-VMI scheme. On the same computer, the MAL-VMI took 2 minutes to calculate approximately 300 tool moves, whereas the parallel slicing method took 38 minutes with 60 slices and 19 minutes with 30 slices. In summary the Parallel Slicing Method offers a degree of flexibility, not common to solid-modelers. Although it is slower than the MAL-VMI (based on a z-buffer method), it more accurately captures the cutter-workpiece engagement area encountered in five-axis flank milling. Further increases in computation speed could be obtained by optimization of C++ functions, use of a smooth tool swept volume (removal volume) and implementation of parallel processing. These are currently areas for future improvement. At this time, the analytical mathematics for the PSM has been developed for tapered-ball end mills, ball-end mills and cylindrical-end mills only. The development of mathematics for other tools is an area for further study. 74 Chapter 4 Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry 4.1 Overview Jet engine impellers are flank milled on five-axis computer numeric control (CNC) machining centers as shown in Figure 4-1.  Figure 4-1 – Picture of an impeller being flank milled on a physical machine. Picture was taken from [48].  In order to maintain tangential contact between the ruled surface of the blade and the tapered, helical, ball-end periphery of the cutter, the flank milling process requires three translational and two rotational degrees of freedom. The impellers are made from Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  75 titanium or nickel alloys due to their high mechanical and thermal strength. The thin webs of the impeller, the strength and low thermal conductivity of the workpiece material present difficulties during flank milling of the part. To minimize heat generation and tool wear, low cutting speeds and feeds are used. However, to maximize the metal removal rate during roughing, large radial and axial depths of cuts are employed. As the tool is generally quite long for clearance purposes, the high cutting forces can lead to large stresses on the tool. If the cutting forces become too high, the tool may fracture completely, causing the costly workpiece to be scrapped. The five-axis flank milling process is more complex than that of two-dimensional and three-dimensional milling due to the addition of angular motion from the rotary joints of the machine tool. This motion causes feeds to vary in magnitude and direction along the axis of the cutter, which leads to varying chip thickness distribution along the cutting edge. Also, the tilting of the tool with respect to the workpiece permits irregular cutter- workpiece engagement conditions to exist. In this chapter, a mathematical model for predicting static cutting forces during the five-axis flank milling of impellers is presented. The model is based on the discrete 2D milling force prediction models of Engin and Altintas [45] and Merdol and Altintas [94], but has been extended to include five-axis motion and irregular cutter-workpiece engagement maps. Since feeds are low when machining titanium impellers, the approximate chip thickness model is used for cutting force calculations rather than the true kinematics of milling [99]. This eliminates the need for storing and updating the surface of the workpiece, which reduces computation time. Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  76 This chapter comprises the following sections. Section 4.2 presents feed and chip- thickness calculations for five-axis flank milling with tapered, helical ball-end mills with serrated cutting edges and variable-pitch flutes (Figure 1-3). These tools are used by the aircraft engine industry to rough-machine jet engine impellers [25], [94]. Although the model has been developed for use with these complex tools, it can also be applied to standard tapered ball-end mills, such as those used in semi-finishing and finishing operations (Figure 1-3). Section 4.3 describes how the total cutting forces on the tool and workpiece can be obtained by digitally summing forces at discrete axial elements. This section also introduces non-linear cutting coefficients that consider the effects of flute geometry and chip thickness variation along the axis of the cutter. In Section 4.4, predicted cutting forces are compared with those collected during a roughing operation on a prototype integrally bladed rotor (IBR). Finally, Section 4.5 gives a summary of the chapter.  4.2 Geometric Modeling and Kinematics of Five-Axis Flank Milling with Serrated, Variable-Pitch, Helical, Tapered Ball-End Mills A five-axis tool path for sculptured surface machining is composed of a number of small segments connected together in series as shown in Figure 4-2. In each tool path segment, the translational and angular velocities can be assumed to be constant, with changes occurring at the segment connection nodes or at the end of each numerical control (NC) block. To calculate the cutting forces along each segment, the depth of cut is divided into a number of differential elements along the axis of the cutter. If the chip Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  77 thickness, edge geometry and cutting coefficients for each element are known, the elemental cutting forces can be calculated and then summed to obtain the total forces cutting acting on the tool and workpiece. The chip thickness at each element depends on the local feed rate, tool geometry and cutter-workpiece engagement conditions. CL Command i {Pi+1} {Pi} CL Command i+1 ∆θi ∆di {vLi} {ki} Tool Path Segment i {Ti+1} {Ti} ωi fLi (u=0) (u=1)  Figure 4-2 – A five-axis sculptured surface machining tool path segment [52], [53].  4.2.1 Feed Calculations for Five-Axis Flank Milling A typical tool path segment is given by two "GOTO" commands in an Automatically Programmed Tools - Cutter Location (APT-CL) File. An example of a sequence of commands is shown below:  GOTO /-1.634070, -0.985070, 2.632550, -0.000050, -0.241677, 0.970357, 90.70, 0.1490 GOTO /-1.626410, -0.971700, 2.636810, -0.000414, -0.241885, 0.970305, 90.20, 0.1597 GOTO /-1.618720, -0.958300, 2.641010, -0.000783, -0.242102, 0.970250, 89.80, 0.1709  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  78 The first three columns of each line give the x,y and z coordinates of the tool tip. For tool path command, i , these coordinates are denoted by the vector, { }iP , in Figure 4-2. The next columns three give the unit-vector orientation of the tool axis in the global system. For tool path command, i , the tool axis vector is given by { }iT  in Figure 4-2. The seventh column gives the feed rate between commands in inverse time feed (G93 command in NC program standards) and the eighth column gives the running, estimated elapsed time. If the tool path segment, i , is small, the tool tip can be assumed to move a distance id∆ , (from coordinates { }iP  to { }1iP+ ) along a vector, { }Liv , at a constant linear velocity, Lif  (see Figure 4-2). During the same time, the tool axis rotates from orientation { }iT  to { }1iT +  at a constant angular velocity, iω  (Figure 4-2). The distance travelled by the tool tip and magnitude and direction of the tool tip feed are given by, { } { } { } { } { }{ } { } 1 1 ( ) 1 i ii i i i Li i if i Li i i i P Pdd P P f d t v t P P + + + −∆∆ = − = = ∆ ∆ = ∆ −   (4.1)  where it∆  and ( )if it∆  are the time taken and inverse time taken respectively, for the tool to travel through tool path segment, i . In each tool path segment, the tool axis also rotates from { }iT  to { }1iT +  around a fixed axis, { }ik , which is assumed to be centered at the tool tip. The tool axis rotates a total angle, iθ∆ , at a constant angular velocity, iω . The axis, { }ik , and angle, iθ∆ , can be evaluated from Equation (3.4). The angular velocity is calculated using, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  79 ( ) i i i if i i t t θ ω θ∆= = ∆ ∆ ∆  (4.2)   Since the motion of the tool is a combination of linear (left of Figure 4-3) and angular velocities (right of Figure 4-3) the sum of these vectors, or total feed velocity, will vary along the tool axis. The total feed vector at a height, z , along the axis of the cutter is: { } { } { } { } { } { } { } { } { } ( , ) ( , ) ( , ) ( , ) ( , ) ( )T T T Li Ai Li Li Ai Ai Li Li i i F z T f z T v z T F F z T f v f z T v T f v k z Tω = = + = + = + ×   (4.3)   where, { } { } { } { }( , ) ( ) iAi i Ai ik Tf z T zQ v T Q k TQω × = = = ×  (4.4)   Xg Zg Yg Xg Zg Yg z {FLi}5 = fLi{vLi} 1 2 3 4 5 {FAi}5 = (ωiz5Q){vA(T)} {ki} {vLi} {vA(T)} = {ki}x{T} Side View Front View {T} ω i fLi {FAi}4 = (ωiz4Q){vA(T)} {FAi}3 = (ωiz3Q){vA(T)} {FAi}2 = (ωiz2Q){vA(T)} {FAi}1 = (ωiz1Q){vA(T)} {T} z 1 2 3 4 5 {ki} {FLi}4 = fLi{vLi} {FLi}3 = fLi{vLi} {FLi}2 = fLi{vLi} {FLi}1 = fLi{vLi} Q =   {ki}x{T} Q ω i  Figure 4-3 – Illustration of cutter motion in five-axis flank milling [52], [53]. Tool is translating along vector, { }Liv  (left), while rotating around, { }ik  (right). At each point along its axis, the total feed varies due to linear and angular motion.  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  80 { }( , )TF z T  is the total feed vector at a height, z , in tool path segment, i . { }( , )TF z T  has a magnitude of ( , )Tf z T  and a direction given by { }( , )Tv z T . It can be further decomposed into the sum of two vectors. The first is a linear feed vector at the tool tip, { }LiF , which is constant along the tool axis and the tool path segment. { }LiF  has magnitude, Lif , and direction, { }Liv , which are given by Equation (4.1). The second component is the angular feed vector, { }( , )AiF z T , which is a function of axial height and tool axis orientation. It has a magnitude of ( , )Aif z T , which varies with height and tool axis orientation. { }( , )AiF z T  has direction, { }( )Aiv T , which is a function of tool axis orientation. At an intermediate point along the tool path segment, the tool axis, { }T , can be spherically interpolated between, { }iT  and { }1iT +  by, { } [ ]{ }iT R T=  (4.5)  where [ ]R  is given by Equations (3.3) and (3.4). In the equation for [ ]R , u  is a parametric variable that varies between 0 and 1 and represents the position of the tool in the tool path segment. When u  is 0, the tool is at location { }iP  with orientation, { }iT . When u  is 1, the tool is at { }1iP+  with orientation, { }1iT + . Since the tool tip is assumed to translate linearly, it is convenient to call u  the fraction of the tool path segment along which the tool tip has travelled. This means: Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  81 { } { } { } { }1 i i i P P u P P+ − = −   (4.6)  Where { }P  is the current position of the tool tip. In simulations, for a given calculation sampling time interval, st , the tool tip position is updated at each time step as: { } { } { }Li s LiP P f t v= +  (4.7)  When { }P  has been updated, u  can be calculated from Equation (4.6) and used to interpolate the tool axis position with Equation (4.5). If u  becomes greater than unity, it means that the tool has entered into the next tool path segment.  4.2.2 Distribution of Feed-Per-Tooth In five-axis machining, the tool can tilt as well as translate, which leads to feeds in multiple directions that vary along the axis of the cutter. At a given height, the total feed can be decomposed into two principal directions - a horizontal feed and a vertical feed. These two feeds contribute to the total chip thickness. At tool axis orientation, { }T , the total feed vector, { }( , )TF z T , varies only with axial position and can be written simply as { }( )TF z . { }( )TF z  can be split into a feed perpendicular to the tool axis, { }X Tf X , and a feed parallel to the tool axis, { }Z Tf Z , as shown in Figure 4-4. The directions of horizontal and vertical feeds, { }( )TX z  and { }( )TZ z , at a height along the tool axis, z , are determined from cross products as follows: Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  82 {Y T } {X T } f Z f X γ Small Cutting Force Element {T}, {Z T } {F T }  Figure 4-4 – Illustration showing how the total local feed, { }TF , can be split into a horizontal feed, { }X Tf X , and a vertical feed, { }Z Tf Z .  { } { } { }{ } { } { } { } { } { } { } { } { } ( ) ( )( ) ( ) ( )( ) ( ) T T T T T T T Y z T T F z X z Y z Z z T Y z T T F z × × = = = × ×   (4.8)  The vectors in Equation (4.8), are also the x,y and z axes of the feed coordinate system at a height, z . The angle between { }( )TF z  and { }( )TX z  is given by, { } { } { } { } ( ) ( )( ) arctan2 ( ) ( ) T T T T X z F z z X z F z γ  × =   •     (4.9)  where the arctan2 function is the four-quadrant arctangent function. The magnitude of feed in the { }( )TX z  direction at a height, z , is given by, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  83 { } ( )( ) ( ) cos ( )X Tf z F z zγ=  (4.10)  and the correct magnitude of the vertical feed ({ }( )TZ z direction) can be evaluated by, ( )Z mvf z f=  if { } { } { }( ) ( ) ( ) ( )T X T T mv F z f z X z Z zf − = ( )Z mvf z f= −     if { } { } { }( ) ( ) ( ) ( )T X T T mv F z f z X z Z zf − = −   (4.11)  where,  { } { }( ) ( ) ( )mv T X Tf F z f z X z= −  (4.12)  The direction of vertical feed is assumed to coincide with the tool axis. ( )Zf z  holds the sign of the vertical feed. When ( )Zf z  is positive, the vertical feed is along the direction of the tool axis, or upwards. When ( )Zf z  is negative, the vertical feed is in the opposite direction of the tool axis, or downwards. Since the total feed has been decomposed into two directions, feed-per-tooth calculations for the horizontal and vertical directions are given as follows, , , ( ) ( )( ) ( ) 2 2 X p j Z p j Xj Zj f z f z c z c z n n φ φ pi pi = =  (4.13)   where ( )Xjc z  and ( )Zjc z  are the feed-per-flutes in the horizontal and vertical directions at height, z  for flute, j . The pitch angle for flute, j , is given by ,p jφ  and the spindle speed Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  84 in revs/sec is denoted by n . When constant pitch cutters are used, the pitch angle is , 2 p j jN piφ =  for every flute, where jN  is the number of flutes on the cutter. If variable- pitch cutters are used, the pitch angle, ,p jφ , will vary on each tooth.  4.2.3 Distribution of Chip Thickness Figure 4-5 shows the approximate chip thickness distribution along the cutting edge given by the 2D general end mill model [45]. z hXj = h1sin(κ) Feed Direction{YT} φex h1 = cXjsin(φj) Feed Direction Static Chip Thickness For Pure Horizontal Feed (Feed Along {X T } - Dir) {ZT} κ(z) cXj =  fXφp,j κ(z) φst φj(z) Chip Thickness Due to Horizontal Feed Workpiece Top View of Cutter Side view of Cutter n h1 = cXjsin(φj) {XT} {XT} 2pin {ZT} hXj = ε(φj)[cXjsin(φj)sin(κ)]  Figure 4-5 – Chip thickness distribution due to horizontal feed (feed along the { }TX direction) [52], [53].  Chip thickness is measured perpendicular to the cutting edge and given as, ( ) ( ) ( ) ( ), sin ( ) sin ( )Xj j j Xj jh z c z zφ ε φ φ κ =    (4.14)  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  85 where, ( ) 1jε φ =   if             ( )st j exzφ φ φ≤ ≤             ( ) 0jε φ =                    if                   ( )j stzφ φ<       or    ( )j exzφ φ>  (4.15)  and, ( , ) 0j jh zφ =              if               ( , ) 0j jh zφ <  (4.16)  The feed-per-tooth along the feed direction for flute, j , is denoted by Xjc  and the immersion angle of flute, j , is given by ( )j zφ . The immersion angle is defined as the angle of the cutting edge from the y-axis of the feed coordinate system at the tool tip. The entry and exit angles of the cutter with the workpiece are stφ  and exφ , respectively and ( )jε φ  is an on/off switching function that makes the chip thickness zero when the flute is outside the entry/exit immersion angle zone. In Engin and Altintas’s [45] and Merdol and Altintas’s models [94], the start and exit immersion angles are considered constant along the axis of the cutter. The axial immersion angle, ( )zκ , is the angle between the tool axis and the outward normal of the cutter’s surface. Evaluation of ( )zκ for general end mills and serrated cutters is given in [45] and [94], and is not repeated here. The axial immersion angle has the effect of reducing the chip thickness perpendicular to the cutting edge. In Engin and Altintas’s [45] and Merdol and Altintas’s [94] models, feeds have constant magnitude and direction at each location along the tool axis. In addition, these feeds are always perpendicular to the tool axis. In five-axis machining, feeds can vary in Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  86 magnitude and direction over the axis of the cutter. Furthermore, the feeds at each axial position can have tool-axis perpendicular and parallel components. Both of these feeds contribute to the total chip thickness. The effects of vertical feed on the chip thickness are shown in Figure 4-6, which describes two discrete positions and the chip cut during movement from one to the other.  z κ(z) hZj = cZjcos(κ) Feed Direction n Zc Side view of Cutter Top View of Cutter Feed Direction (into page) κ(z) hZj =  ε(φj)[cZjcos(κ)] cZj =  fZφp,j 2pin Static Chip Thickness for Pure Vertical Feed (Feed Along {Z T  } - Dir) Chip Thickness Due to Vertical Feed {XT} {ZT} {XT} {YT} φj(z) Edge can cut over full 2pi immersion angle range Projection of   hZj on the {XT}-{YT} plane  Figure 4-6 – Chip thickness distribution due to vertical feed (feed parallel to the tool axis, { }TZ ) [52], [53].  The illustration on the right shows that the chip thickness does not vary with instantaneous immersion angle. Also, the tool can cut during the entire 0 to 2pi immersion range. The chip thickness contribution for flute, j , due to the vertical feed component is, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  87 ( ) ( ) ( )cos ( )Zj j Zjh z c zε φ κ =    (4.17)  where Zjc  is the feed-per-tooth along the vertical direction for flute, j . When milling with a tool that has zero radius at the tool tip (e.g. a ball-end mill), the cutting speed at the tip will be zero. The cutting mechanism at this point will be indentation or spreading of the workpiece material, which is a complex process. In practice, the predicted cutting forces at the bottom of the ball are experimentally calibrated. In simulations, to avoid this singularity, the midpoints of the cutting edge elements are used as cutting force calculation locations. When horizontal and vertical feeds are combined, they interact as shown in Figure 4-7. κ(z) cZj cXjsin(φj)sin(κ) cZjcos(κ) Static Chip Thickness for Combined Horizontal and Vertical Feeds hj = ε(φj)[cXjsin(φj)sin(κ) - cZjcos(κ)] Chip Thickness Due to Horizontal and Vertical Feeds   cXjsin(φj) Side view of Cutter κ(z) = + = - {XT} {ZT} cZj cZj  Figure 4-7 – Chip thickness distribution due to a combination of horizontal and vertical feeds [52], [53]. Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  88 The chip thickness contributions from combined horizontal and vertical motions can be summed to obtain the total chip thickness. In other words: ( ) ( ) ( ) ( )( , ) sin ( ) sin ( ) cos ( )j j j Xj j Zjh z c z z c zφ ε φ φ κ κ = −   (4.18)  The negative sign in Equation (4.18) indicates that downward feeds will increase the chip thickness, while upwards feeds will decrease the chip thickness. If the feed-per-tooth varies along the tool axis, the chip thickness equation can be written as: ( ) ( ) ( ) ( )( , ) ( ) sin ( ) sin ( ) ( ) cos ( )j j j Xj j Zjh z c z z z c z zφ ε φ φ κ κ = −   (4.19)  4.2.4 Effects of Feed Variation Along the Cutter Axis on Chip Thickness  In five-axis machining, the combined translational and angular feed components cause the total feed to vary along the axis of the cutter. This, in turn, causes the feed direction to change along the axis of the cutter as illustrated in Figure 4-8. As an example, the cutter is divided into four elements. As shown in Figure 4-8(a), the tool is translating along vector, { }Liv , and rotating around vector, { }ik , at angular velocity, iω . Although in practice, { }Liv  and { }ik  can be in different directions, for illustration purposes in this example, { }Liv  and { }ik  have the same direction. Due to the angular Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  89 θs,2 θs,3 θs,4 Top View of Cutter - Summation of Feed Vectors 3D View of Cutter - Linear and Angular Feeds φj φj-θs,1 Top View of Cutter - Coordinate Systems and Shift Angles ωi {FAi}4 {FAi}3 {FAi}2 {FAi}1 {FLi}4 {FLi}3 {FLi}2 {FLi}1 {T} {vLi},{ki} {FLi}1-4 {YTT} {YT}4 {YT}3 {YT}2 {YT}1 {XT}1{XT}2 {XT}3 {XT}4 {XTT} θs,1 {FT}1{FT}2 {FT}4{FT}3 {FAi}1 {FAi}2 {FAi}3 {FAi}4 {FT}1 {FT}2 {FT}3 {FT}4 {XT}1-4 {XTT} (a) (b) (c)  Figure 4-8 – (a) Feed variation along the cutting edge from combined translational and angular motion. (b) Total feed vector varies at each element. (c) Feed coordinate system at each element is shifted by an angle, ( )s zθ , relative to the feed coordinate system at the tool tip. [52], [53].  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  90 motion of the cutter, the angular feed vectors, { } { }1 4Ai AiF F… , vary in magnitude at each element. Looking from the top of the cutter, Figure 4-8(b) illustrates that the translational feed vectors, { } { }1 4Li LiF F… , and angular feed vectors sum to total feed vectors with different directions, { } { }1 4T TF F… . In this case, the direction of the horizontal feed vector at each height, { } { }1 4T TX X… , has the same direction as the total feed vectors at each element. Figure 4-8(c) shows that, if the coordinate system at the bottom of the cutter ({ } { } { }, ,TT TT TTX Y Z ) is taken as a reference, the horizontal feed direction for each element will be shifted by an angle, ,1 ,4s sθ θ… .  The horizontal feed direction vectors determine the immersion angle at which the maximum chip thickness occurs. The shift angle changes the effective immersion angle in the chip thickness calculations at each axial element. For example, in Figure 4-8(c), the cutting edge at Element 1 is at immersion angle, jφ , measured from the y-axis at the bottom of the cutter, { }TTY . However, in the feed coordinate system at Element 1, the cutting edge is at immersion angle, ,1j sφ θ− , measured from { }1TY . In other words, the maximum chip thickness at Element 1 occurs at a greater immersion angle than at the tool tip due to the effects of angular feed. In general, the shift angle along the axis of the cutter relative to the tool tip is a function of height and given by, ( )( ) arctan2 ( ) XT s XT y z z x z θ  = −       (4.20)   Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  91 where ( )XTx z  and ( )XTy z  are the x- and y-coordinates of the feed vector, { }( )TX z , measured from the feed direction at the tool tip, { }TTX . The negative sign in Equation (4.20) arises because angles returned by the arctan2 (four-quadrant arctangent) function are considered positive in the counterclockwise direction, whereas in the feed-coordinate system, angles in the clockwise direction are considered positive. To calculate chip thickness at a height, z , the approximate chip thickness equation becomes: ( ) ( ) ( ) ( )( , ) ( )sin ( ) ( ) sin ( ) ( ) cos ( )j j j Xj j s Zjh z c z z z z c z zφ ε φ φ θ κ κ = − −   (4.21)  4.2.5 Effects of Cutting Edge Serrations on Chip Thickness For roughing operations in impeller machining, serrations are ground into the cutting edges in order to reduce the cutting forces and self-excited vibrations (Figure 1-3). On each flute, serrated, tapered, ball-end roughing cutters employ a unique vertical offset, or phase shift, of the serration wave profile from the top of the ball. If a cross- section of the cutter is taken at a particular height, each flute will have a different radius. The unequal cutter radii at the cross-section induce a runout-type effect. The effects of radial runout on chip thickness have been investigated for the case of 2D milling by Kline and Devor [83] and Wang and Liang [129]. For a cylindrical end mill with constant pitch flutes, the expression for the thickness of a chip cut by the j th flute with uneven radii can be written as: ( ) { },min( ) ( ) sin( )1j j j j m j X j j j mjh h mc R Rm Nφ ε φ φ φ −   = = + −  = …   (4.22)  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  92 Where Xc  is the feed-per-tooth and ( )j jh φ  is the chip thickness for cutting flute, j . The number of flutes on the cutter is given by jN  and 1 jm N= …  is the index of previous cutting flutes. jR  and j mR −  are the radius for tooth j  and tooth j m− , respectively. If the cutting teeth are unequally spaced, the feed-per-tooth for each cutting edge will vary. If the direction of rotation is assumed to be clockwise and the teeth are numbered so that the tooth index increases in a counter-clockwise direction (Figure 4-9(a)) the expression changes to, ( ) , ( 1) 1 min( ) ( ) sin( ) 1 m j j j j m j X j p j j j m pj h h c R R m N φ ε φ φ φ − + − =      = = + −    =       ∑ …   (4.23)  where  1p m= …  is the feed-per-tooth summation index. Note that, if the indices of the horizontal feed-per-tooth, 1j p− +  or j m− , become less than 1, then jN  is added to them. If the indices become greater than jN , then jN  is subtracted from them. For clarity, Figure 4-9(a) shows a serrated cylindrical cutter with variable-pitch teeth cutting along the feed direction. The figure shows that the cutting flutes will have unequal radii at a particular cross-section due to the phase-shifted serration waves. Figure 4-9(b) shows that the unequal radii, coupled with varying feed-per-tooth values will cause the chip thickness profile to vary. Figure 4-9(c)-(d) shows how Equation (4.23) searches backwards through the previous feed-per-tooth values and differences in cutting flute radii to calculate the minimum chip thickness. Calculations for one and two teeth  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  93  Rj(z) = R(z) - Rsj(z) Xc Yc n Rs1(z) Rs4(z) Rs2(z)Rs3(z) R(z)R1(z) R4(z) R3(z) R2(z) R1(z) = R2(z) = R3(z) = R4(z) 14 3 2 1 cX2 cX3 cX4 cX1 1 Flute Back R1 R4 cX1 cX1sin(φ1) (R1 - R4) φ1 h1 = h1,1 h1 = min(h1,1 , h1,2) = h1,1 Profile of Serrated Cutter Cross-Section of Serrated Cutter Approximate Path of Each Flute Feed Direction  = Radius of flutes 1...N at height, z.  = Radius of base cylinder at height, z.  = Distance of serration spline from     base circle at height, z, for flutes 1...N. Phase Shift Between Waves Rs1...sN(z) R(z) R1...N(z) R1 R4 R3 h1,1 = cX1sin(φ1) + (R1 - R4) 2 Flutes Back h1,2 = cX1sin(φ1) + cX4sin(φ1) + (R1 - R4) + (R4- R3) = (cX1+ cX4)sin(φ1) + (R1 - R3) (R1 - R4) cX1sin(φ1) (R4 - R3) cX4sin(φ1)cX1cX4 YT XT YT XT (a) h1,2 > h1,1 φ φ1 (b) (c) (d) Zc  Figure 4-9 – (a) Profile of a serrated cylindrical cutter. The radii of the flutes are different at each cross section. (b) Approximate path of each flute (c) and (d) Illustrations showing how Equation (4.23) searches back over previous cuts to find the minimum chip thickness. [52], [53].  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  94 back are shown. The minimum chip thickness between the present and all previous teeth, up until the last pass of the present tooth, will be the actual approximate chip removed. If the radius of the present tooth is larger than the previous tooth, it will cut more than the feed-per-tooth. If the radius of the present tooth is smaller than the previous tooth, it will cut less. If the radius of the present tooth is sufficiently small in comparison to the previous one, it will not cut at all. The effects of runout due to unequal flute lengths are most apparent when differences in tooth radii are large in comparison to the feed-per- tooth For a general, serrated end mill with variable-pitch teeth in a 2D slotting operation, the chip thickness for tooth, j , at a height, z , becomes: ( ) ( ) ( ) , ( 1) 1 min( , ) ( ) sin ( ) ( ) ( ) sin ( ) 1 m j j j j m j X j p j j j m j pj h z h c z R z R z z m N φ ε φ φ φ κ − + − =      = = + −    =       ∑ …   (4.24)  In Equation (4.24), the subscript, j , on ( )j zκ  denotes the axial immersion angle for flute, j . This takes allows a unique axial immersion angle on each flute (at height, z ) to be incorporated.  4.2.6 Static Chip Thickness Expression for Five-Axis Milling with Serrated, Variable-Pitch, Helical General End Mills Combining the effects of the varying horizontal and vertical feeds (Section 4.2.3), changing feed coordinate system (Section 4.2.4) and varying radius due to the serration Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  95 profile (Section 4.2.5), the approximate chip thickness expression for five-axis machining with a general, serrated, variable-pitch end mill is given by, ( ) ( ) ( ) ( ) , ( 1) 1 min( , ) , ( , ) ( ) sin ( ) ( ) 1 ( ) ( ) sin ( ) ( ) cos ( ) m j j j j m j X j p j s pj j j m j Zj j h z z h z c z z z m N R z R z z c z z φ ε φ φ φ θ κ κ − + = −     = = −    =     + − −    ∑ …    (4.25)  and ( , ) 0j jh zφ =              if               ( , ) 0j jh zφ <  (4.26)  In Equation (4.25) the ε  function has been expanded to include irregular cutter- workpiece engagements. It is given as follows,      ( , ) 1j zε φ =      if            , ,( ) ( ) ( )st q j ex qz z zφ φ φ≤ ≤             for any   1 ( )q S z= …     ( , ) 0j zε φ =       if  ,( ) ( )j st qz zφ φ<  or ,( ) ( )j ex qz zφ φ>    for  all 1 ( )q S z= …  (4.27)  The added dependency of ε  on z  shows that the engagement conditions can vary over the axis of the cutter. Also, the ε  function in Equation (4.27) can include multiple engagement conditions at a given height. The start and exit immersion angle pairs of engagement zone, q , at height, z  are denoted by , ( )st q zφ  and , ( )ex q zφ , and ( )S z  is the number of pairs at the element’s axial position.  Values for , ( )st q zφ , , ( )ex q zφ  and ( )S z  are obtained from the cutter-workpiece engagment maps (see Chapter 3) as discussed in Section 4.3.  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  96 4.3 Prediction of Cutting Forces The depth of cut is divided into a number of small elements along the axis of the cutter and the local chip thickness at each element is evaluated from Equation (4.25). The total cutting forces acting on the tool are determined by summing the contributions of all cutting edge elements that are in cut. The immersion angle of the elemental cutting edge is evaluated as proposed in [45], 1 , 1 ( ) ( ) j j r p n j n z zφ φ φ ψ − = = − −∑  (4.28)   where ( )j zψ  is the lag angle due to the cutter’s helix on flute, j , where 1 jj N= … . In the above expression, the numbering of the flutes increases in a counterclockwise direction (negative sign before summation of ,p nφ ), as opposed to [45] where the flute number increases in the clockwise direction (positive sign before summation of ,p nφ ). This numbering format is maintained throughout the thesis. Calculation of the lag angle can be found in [5]. The reference immersion angle, r φ , is the radial position of the first flute at the tool tip measured from { }TTY . The reference immersion angle is advanced during the simulation using the equation, 2 60 s r r ntpiφ φ= ±  (4.29)  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  97 where n  is the spindle speed in rev/min and st  is the cutting force calculation time interval. For clockwise rotation the reference angle increases and for counter-clockwise rotation the reference angle decreases. Once the radial position of the cutting edge is calculated using Equation (4.28), it is checked whether it is within cut using, ( )j zφ    is in cut if      , ,( ) ( ) ( )st q j ex qz z zφ φ φ≤ ≤       for any     1 ( )q S z= …  (4.30)  As explained in Section 4.2.6, , ( )st q zφ  and , ( )ex q zφ  are the start and exit immersion angle pairs of engagement zone, q , at height, z . ( )S z  gives the number of entry/exit angle pairs at height, z . Values of , ( )st q zφ , , ( )ex q zφ  and ( )S z  are obtained from the cutter-workpiece engagment maps. If the system is free of chatter and forced vibrations, and deflections of the tool and workpiece are small, calculations of the cutter-workpiece engagements (CWEs) can be considered as a purely geometric problem (see Chapter 3). CWEs depend upon the geometry of the tool and workpiece and the tool’s position relative to the workpiece. They take the form of maps with immersion angle on the x-axis and distance from the tool tip on the y-axis. Examples of these maps for the tool path demonstrated later in the chapter are shown in Figure 4-10. The CWEs can be calculated using the Parallel Slicing Method (Chapter 3) or a z- buffer method such as the one available in the Manufacturing Automation Laboratory’s Virtual Machining Interface (MAL-VMI) [104] (see Chapter 3). Since the Parallel Slicing Method is slower and is still in the research stage of development, the MAL-VMI method Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  98 0 40 80 120 160 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 1.0 1.5 2.0 2.5 3.0 Cutter-Workpiece Engagement Maps for the IBR Roughing Tool Path Tool Path Segment 19 Tool Path Segment 67 Tool Path Segment 280 D is ta n c e  F ro m  T o o l T ip  [ m m ] Immersion Angle Immersion Angle Immersion Angle [radians] [radians] [radians] 0 40 80 120 160 0 40 80 120 160  Figure 4-10 – Some cutter-workpiece engagement maps from the IBR roughing tool path [52], [53]. These maps were calculated with the Manufacturing Automation Laboratory’s Virtual Machining Interface (MAL-VMI).  [104] was used to obtain cutter-workpiece engagement maps in this chapter. As explained in Chapter 3, the Virtual Machining System uses an application programming interface (API) from the "OptiPath" library [126] in the commercial NC verification software, Vericut. This API returns a grid of points that corresponds to the intersection of the tool with the workpiece. The intersection of rays cast from these points along the feed direction with the solid model of the tool forms the cutter-workpiece engagement maps. In the CWE maps, zero degrees immersion angle is measured from the Y-axis in the feed (or tool) coordinate system at the tool tip, { }TTY . Figure 4-11 shows that each map is composed of a series of rectangular blocks that each give a pair of entry and exit angles of the cutting edge with the workpiece at a particular height. For each block, the location of the bottom of the block, bz , along with the block height, ba , and the entry and exit immersion angles, stφ  and exφ , are given. To test whether a cutting edge is engaged with the workpiece, all engagement blocks at the Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  99 1.8 2.0 2.2 2.4 2.6 2.8 Tool Path Segment 67 Immersion Angle [radians] 144 146 150 152 154 148 D is ta n c e  F ro m  T o o l T ip  [ m m ] zb ab φ st φ ex z = 0  Figure 4-11 – Close-up of the engagement map for tool path segment 67 [52], [53]. Each block is defined by four parameters: bz , ba , stφ  and exφ .  element’s height are retrieved. If any are found, the position of the cutting edge is tested to see whether it lies within any using Equation (4.30). If the edge does not lie within any blocks, the cutting forces are set to zero. Cutter-workpiece engagements are calculated off-line before the force prediction algorithm starts. Since the tool path segments are small when flank milling impellers, the engagements are changed at the beginning of each tool path segment during cutting force simulations. Once changed, they are kept constant for the duration of the tool path segment. For cases where the boundaries of the workpiece change more abruptly, the engagement maps may need to be updated several times along each tool path segment. If the edge element is engaged with the workpiece, (the condition of Equation (4.30) is satisfied) the differential radial, tangential and axial cutting forces acting on an infinitesimal cutting edge segment can be determined from, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  100 ( ) ( , ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) r rc j j re t tc j j te a ac j j ae dF K z h z db z K z dS z dF K z h z db z K z dS z dF K z h z db z K z dS z φ φ φ = + = + = +   (4.31)  otherwise they are set to zero. rcK , tcK  and acK  are the radial, tangential and axial cutting force coefficients and reK ,  teK  and aeK  are the edge force coefficients. ( , )j jh zφ is the elemental chip thickness calculated from Equation (4.25) and ( )csc ( )db dz zκ= , which is the projected length of an infinitesimal cutting flute in the direction along the cutting velocity. dz  is the height of the cutting force element and dS  is the infinitesimal length of cutting edge segment and is evaluated as, 2 2 1 h l z j j j z dR d dS R dz dz dz ψ      = + +         ∑  (4.32)   where 2lz z dz= −  and 2hz z dz= +  are the upper and lower boundaries of the cutting force element. j dR dz  and j d dz ψ  are the derivatives of the radius and lag angle of flute, j , with respect to z  and can be found in [45] and [94]. The integral in Equation (4.32) can be evaluated using Gaussian quadrature [3]. Cutting forces, as well as the cutting force and edge force coefficients, are all measured perpendicular to the cutting edge. The cutting force coefficients at each edge element vary and are obtained from the Orthogonal-to-Oblique Cutting Mechanics Transformation [5], [15] for titanium 6 4Ti Al V− −  alloy as given by Budak et. al [28]. In [28] the following expressions for shear stress, friction angle and shear angle on the orthogonal cutting plane were determined from experiments with sharp tools, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  101 ( ) ( ) 1 1 1 01 0 0 0 1 613 19.1 0.29 (deg) cos tan 1 sin 1.755 0.028 (deg) 0.331 0.0082 (deg) s s r C r c C r C c r r MPa C h C h r C h C C τ β α αφ α α α − = = +   =   −  = = − = −      (4.33)  where r α  is the rake angle of the tool, sτ  is the shear stress, and sβ  is the friction angle. cφ  is the shear angle, cr  is the chip compression ratio and h  is the uncut chip thickness. From the orthogonal data above, the tangential, radial and axial cutting coefficients on an oblique plane can be calculated using the following equations [5], [15], ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 cos tan tan sin sin cos tan sin sin sin cos cos tan sin cos tan tan sin sin cos tan sin n n ns tc n n n n n n ns rc n n n n n n n ns ac n n n n n i K K i i K β α η βτ φ φ β α η β β ατ φ φ β α η β β α η βτ φ φ β α η β − + = + − + − = + − + − − = + − +      (4.34)  where nβ , nφ , nα  are the normal friction, normal shear and normal rake angles respectively. η  is the chip flow angle and i  is the oblique angle. In milling, the oblique angle is equal to the helix angle of the cutting edge. nβ  is related to sβ  and nφ  is related to cφ  by, Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  102 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 1 1 tan tan cos cos cos cos tan cos 1 sin cos n s c n n c n r i r i β β η η α φ η α − − =             =      −           (4.35)  However, employing Stabler’s chip flow rule [117] and making the following practical assumptions [5] for Equation (4.34), n c n r n s iη φ φ α α β β = = = =    (4.36)  the oblique cutting coefficients for the 6 4Ti Al V− −  alloy become, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 cos tan sin( , , ) sin cos tan sin sin( , , ) sin cos cos tan sin cos tan tan sin( , , ) sin cos tan sin s r ss tc r c c s r s s rs rc r c n n n s s r ss ac r c c s r s i K h i i K h i i i i i K h i i β α βτ α φ φ β α β β ατ α φ φ β α β β α βτ α φ φ β α β − + = + − + − = + − + − − = + − +      (4.37)  which shows that the elemental cutting coefficients for 6 4Ti Al V− −  are a function of chip thickness, h  (which affects the shear angle), helix angle and rake angle (which affects both the friction angle and shear angle). For a particular cutting tool where the rake angle is constant, the cutting coefficients are a function of chip thickness and helix angle only. To obtain the cutting coefficients for each element using this material model, these quantities must first be evaluated (Equation (4.25), [45]). Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  103 The tangential, radial and axial edge forces for the 6 4Ti Al V− −  alloy in oblique cutting are approximately constant [28] and given as, 24.00 43.00 3.00 te re ae K K K = = = −   (4.38)  Equation (4.31) calculates cutting forces perpendicular to the cutting edge. These forces can be transformed into feed, global or rotating dynamometer coordinates through the following relation, [ ][ ] x r y t z a F dF F G H dF F dF         =              (4.39)  where [ ]H is given in [45], [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) cos ( ) sin ( ) sin ( ) cos ( ) cos ( ) cos ( ) 0 sin ( ) j j j j j j j j j j j j z z z z z H z z z z z z z φ κ φ φ κ φ κ φ φ κ κ κ   − − −    = − −     −     (4.40)  and [ ]G  is given by: [ ] 1 0 0 0 1 0 0 0 1 G     =                                for feed coordinates  (4.41)    [ ] { } { } { }TT TT TTG X Y Z=                         for global coordinates  (4.42)  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  104 [ ] ( ) ( ) ( ) ( ) cos sin 0 sin cos 0 0 0 1 r r r rG φ φ φ φ −    =             for rotating dynamometer coordinates   (4.43)   The cutting forces for each element and each flute in cut are summed to obtain the total forces acting on the tool at a particular reference immersion angle. This procedure is repeated for each reference immersion angle increment over a given number of rotations per tool path segment.  4.4 Comparison of Simulations with Experiments Pratt & Whitney Canada2 performed five-axis flank milling of a prototype integrally bladed rotor (IBR). They measured cutting forces and torques using a rotating dynamometer attached to the spindle of the machine. Although the cutting forces for several tool paths were measured, in this chapter simulations are compared against experiments for one tool path only. An illustration of the tool path is shown in Figure 4-12. The cutter is brought to the top of the blank with the tool axis oriented roughly parallel to the workpiece. It is gradually lowered into the workpiece and, when fully immersed, plunges through the blank in curved motion until it exits through the bottom of the part.  2  The research partner in the project.  Pratt and Whitney Canada (PW&C) is a large commercial aircraft engine manufacturer based in Longueuil, Quebec that uses the IBR for testing purposes.  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  105 Start End X YZ  Figure 4-12 – Workpiece and tool path used to compare predicted and simulated cutting forces for a roughing operation on an integrally bladed rotor (IBR) [52], [53]. Illustration was generated using CGTech’s Vericut (TM) [32].  The workpiece material for the IBR was 6 4Ti Al V− − alloy and the spindle speed for this tool path was a constant 600 [rev/min]. The tool tip feed rate, while in cut, varied from 12 to 33 mm/min (0.20 to 0.55 mm/sec) and angular feeds varied from 0.03 to 0.12 rad/min (0.0005 to 0.002 rad/sec). The tool used for this operation was a long, four fluted, tapered, serrated, variable-pitch, helical ball-end mill with the end portion of the ball ground off for clearance. A similar tool is shown at the top of Figure 4-14. The cutter was approximately 200 mm long, measured from tip to tool-holder and the shank diameter was 37.5 mm. The cutter was ground from cemented Tungsten Carbide. The tool path was given in APT-CL data format. It was composed of just over 300 tool path segments. Depths of cut range from under 25 mm to over 150 mm. Figure 4-10 shows some cutter-workpiece engagements maps along the tool path. Figure 4-13 illustrates the measured X, Y and Z forces along with the cutting torque, or MZ moment. In Figure 4-13(g)-(h) one can note that the raw data contains a Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  106 significant amount of higher frequency noise due to the dynamics of the system and self- excited vibrations (chatter). Figure 4-13(e)-(f) show frequency spectrums of the X and Z cutting forces and that the dominant chatter frequency is centered at 410Hz. This is due to a vibration mode of the spindle. In the jet engine impeller manufacturing industry, the impellers are cut very close to the stability borders of the system and small variations in tool and workpiece geometry can cause occasional light chatter. The strong chatter frequencies observed in the raw data are the result of the rotating dynamometer reducing the stiffness of the spindle-tool holder assembly due to increased stick-out and added mass. Severe chatter does not occur when machining the production part as reported in [25]. The model presented in the current research project predicts only static cutting forces and so the higher frequency signals were filtered out in order to make a fair comparison. The experimental data was forward and reverse low-pass filtered using the Matlab command, “filtfilt”, with a 10th-order Butterworth filter with a cut-off frequency of 80Hz, which is twice the tooth passing frequency. Forward and reverse filtering ensures that the magnitude and phase of the low-frequency components are not altered significantly so as to minimize distortion of the data. Time history plots of the filtered X and Z forces are shown in Figure 4-13(g)-(h). Figure 4-13(a)-(d) show the differences between the raw and filtered data over the entire tool path. The effects of removing the chatter signals in the raw data are most noticeable in the cutting torque measurements and in the Z forces (axial direction). In parts, the peak dynamic torque is approximately twice the static. Also, in certain sections, the magnitude of the filtered Z-forces is one-third of those in the raw data. Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  107 2.0 4.0 6.0 0 -2.0 -4.0 -6.0 441.00 441.05 F o rc e  [ k N ] Closeup of X Forces vs. Time - Frequency [Hz] Time [s] F o rc e  [ N ] 441.10 441.15 441.00 441.05 441.10 441.15 0 FFT of X Forces - Magnitude Frequency [Hz] Time [s] F o rc e  [ k N ] F o rc e  [ N ] 0 Closeup of Z Forces vs. Time Time [s] F o rc e  [ k N ] X Forces vs. Time Time [s] 0.5 1.0 1.5 -2.0 0 Time [s] T o rq u e  [ N m ] 60 80 100 120 40 20 0 -20 2.0 4.0 6.0 -2.0 -4.0 -6.0 0 -0.5 -1.0 -1.5 8.0 -8.0 Y Forces vs. Time Z Forces vs. Time Cutting Torque vs. Time F o rc e  [ k N ] F o rc e  [ k N ] Raw Data Filtered FFT of Z Forces - Magnitude 200 400 600 8000 200 400 600 800 50 0 10 20 30 40 0 200 400 600 800 1.0 1.5 0.5 -1.5 -0.5 -1.0 Cutting forces measured with Kistler 4-Component Rotating Dynamometer - Type 9124A X and Y axes spin with tool (rotating coordinates) Sampling frequency =  4000Hz 100 150 200 250 50 60 0 Time [s] 200 400 600 8000 2.0 4.0 6.0 -2.0 -4.0 -6.0 0 8.0 -8.0 200 400 600 8000 200 400 600 8000 70 (a) (b) (c) (d) (e) (f ) (g) (h)  Figure 4-13 – Comparisons of raw and filtered measured cutting forces and torques [52], [53]. (a)-(d) Raw and filtered data vs. time for the entire tool path. (e)-(f) Fourier spectrums of raw and filtered X and Z forces. (g)-(h) Close-ups of X and Z forces.  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  108 Cutting forces simulations were run for the impeller roughing tool path using a cutting force element size of 0.1mm. This high resolution was necessary to accurately capture the serration wave geometry on the cutting edges. The simulations were set to produce measurements for every two degrees of rotation, or a calculation sampling time interval of 45.555 10st − = ×  seconds. To speed up computations, only two revolutions per tool path segment were simulated. However, on PC with a 3.4GHz Pentium central processing unit (CPU) and 2 gigabytes of RAM, the simulation took approximately 2 days. The long simulation time results from the small element size necessary to capture the serration waves, in combination with the large depth of cuts that occur in this flank- milling operation. Efficient calculation of cutting forces for serrated-cutters with large axial depths of cut is an area for future study. Figure 4-14(a)-(b) illustrates that the overall cutting force predictions compare quite well with the filtered, experimental measurements for the X and Y forces. The shape of both is almost identical. However, cutting force predictions for the X and Y forces tend to underpredict the filtered, experimental forces by about 20%. The simulated and filtered experimental Z forces (Figure 4-14(c)) do not match as well, but the overall trend of the predicted Z forces appears to be similar to the experimental, but with smaller amplitude.  Close-ups of cutting forces in the X direction at various parts of the tool path are shown in Figure 4-14(e)-(h). The shape and amplitude of the predicted X (Y is almost identical to X) forces matches quite well with those of the experimental throughout the duration of the tool path. The main differences occur at the peaks of the cutting force waves. There is some minor discrepancy in the shape of the cutting force waves towards Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  109 2.0 4.0 6.0 0 -2.0 -4.0 -6.0 200 400 600 800 X Force vs. Time (Rotating Coordinates) 0 200 400 600 8000 0 0.2 0.4 0.6 0 -0.2 40 60 20 0 -10 Time [s] T o rq u e  [ N m ] Y Force vs. Time  (Rotating Coordinates) Z Force vs. Time Cutting Torque vs. Time -0.4 -0.6 -0.8 F o rc e  [ k N ] Exp. - Filtered Predicted Predicted and Filtered Experimental Cutting X Forces vs. Time (Rotating Coordinates) 40.4040.15 121.0120.75 379.40 4.0 0 0.6 0.4 0 0.2 379.6 706.8 707.0 -0.6 -0.4 -0.2 2.0 -4.0 -2.0 4.0 2.0 -4.0 -2.0 0 0.2 0 -0.2 -0.4 Predicted and Filtered Experimental Cutting Forces and Torques vs. Time F o rc e  [ k N ] F o rc e  [ k N ] Time [s] Time [s] Time [s] Time [s] Time [s] Time [s] Time [s] F o rc e  [ k N ] F o rc e  [ k N ] F o rc e  [ k N ] F o rc e  [ k N ] Cutting force element size = 0.1mm Spindle speed = 600 RPM ~300 tool path segments Sampling (calculation) frequency = 1800Hz 2 tool rotations simulated per tool path segment Cutter = 4-fluted tapered, serrated, variable-pitch ball-end mill (200 mm long) 2.0 4.0 6.0 -2.0 -4.0 -6.0 200 400 600 8000 200 400 600 8000 (a) (b) (c) (d) (e) (f ) (g) (h)  Figure 4-14 – (a)-(d) Predicted and filtered experimental data vs. time for the entire tool path. (e)-(h) Close-ups of predicted and measured X-forces vs. time at various parts of the tool path. [52], [53].  Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  110 the end of the tool path, but this is most likely due to the lower resolution of the engagement maps at smaller radial immersions. The main differences in cutting force predictions (X, Y and Z directions) with experiments are thought to be caused by the honing (or rounding) of the physical cutting edge in order to prevent edge chipping. The orthogonal cutting tests, which were used to obtain cutting coefficients for the titanium alloy, were performed with sharp tools. In the physical process, the cutting edge of the tool maybe rubbing and spreading the workpiece material, rather than shearing it cleanly. This effect may cause the orthogonal cutting database to be less accurate. Tool wear could also be responsible for some of the differences, as the predicted cutting forces appear to be in closer agreement at the beginning of the tool path. Xu et. al [132] and Desfosses et al. [39] have shown that the tangential cutting coefficent, tcK , and edge force coefficient, teK , begin changing almost immediately as the tool wears in ball-end milling. The authors’ [39], [132] experiments showed that teK  increased almost linearly with time according to flank wear of the tool, while tcK  varied less but rose sharply near the end of the tool’s life as the cutting edge began to chip. Mechanistic calibration of the cutting coefficients with honed and worn cutting edges may improve the accuracy of predictions. Also, the chatter signals in the raw data (particularly in the Z forces) were quite severe and they may have affected low frequency static cutting force measurements. Filtering may not have been able to remove the distortion. Finally, the MAL-VMI method may have introduced some loss of accuracy in the cutter-workpiece engagements maps due to truncation or clipping of the borders (see Section 3.3). Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  111 In spite of these differences, for the tool path demonstrated, the predicted cutting torque and X-Y cutting forces are usually within 20% of the experimentally measured results. These components are responsible for the majority of the shearing action in the cutting process. The agreement between the simulated and measured results provides a solid verification of the cutting force model.  4.5 Summary In this chapter, a virtual model of the impeller machining process is presented. The model is able to predict the performance of helical, tapered, serrated, variable-pitch, ball- end milling cutters when five-axis flank milling a jet engine impeller. Unlike 2D milling operations, the five-axis motion of the cutter, combined with complex tool and workpiece geometry, creates irregular cutter-workpiece engagements, uneven chip thickness distributions and varying feed vectors along the periphery of the cutter. The mechanics of cutting requires the evaluation of the chip thickness along the feed motion, which varies along the tool path and the cutter axis in five-axis flank milling. This chapter presents a mathematical model of feed decomposition and chip thickness distribution along the flutes engaged with the impeller blade. The simulated cutting forces agree reasonably well with measurements collected during the actual impeller machining operation. However, the extension of the tool holder due to the rotating dynamometer led to a flexible measurement system which produced chatter vibrations during the test. In addition, the cutting force coefficients will vary along the serrated cutting edges, which have different cutting speeds, chip thickness Chapter 4: Mechanics of Five-Axis Flank Milling with Complex Cutter Geometry  112 distributions, normal rake and oblique angles, as well as honed cutting edges. The model used the Orthogonal-to-Oblique Cutting Mechanics Transformation [5], [15] which considered the speed, chip thickness, rake and oblique angles but not the wear and honing of the edge. Finally, the MAL-VMI method may have introduced some loss of accuracy in the cutter-workpiece engagements maps due to truncation or clipping of the borders (see Section 3.3). Despite these measurement difficulties and approximations for a five- axis impeller machining tool path, the model predicted the X-Y cutting forces and cutting torque reasonably well – usually to within 20% of the experimentally measured values. Prediction and optimization of these parts in a virtual environment has the potential to reduce the machining time of these costly impellers as illustrated in Chapter 5 of the thesis. 113 Chapter 5 Feed Rate Optimization of Five-Axis Flank Milling Tool Paths According to Multiple Feed-Dependent Constraints  5.1 Overview The blades of jet-engine impellers can be modeled as flank-millable ruled surfaces in a dedicated CAD /CAM environment. When machining impellers from titanium and nickel alloys, cutting speeds are maintained at a constant level in order to avoid accelerated tool wear. Also, the spacing between the flutes is designed around the part and spindle dynamics, and is chosen to reduce chatter vibrations at a fixed spindle speed [26], [27]. For these reasons, changing the spindle speed of the process is undesirable. One variable that can be adjusted easily is the feed along the tool path. Increasing the feed of the tool reduces the machining time of the part. However, it also increases the chip thickness and cutting forces which, if excessive, can lead to tool breakage, stalling of the machine or large, unwanted tool deflections. Since aircraft impellers are costly with long machining times, optimizing the feed rate with respect to several constraints can improve productivity, while avoiding damage to the part, tool and machine. This chapter presents offline process optimization schemes for the five-axis flank milling of jet engine impellers based on the mechanics model in Chapter 4. The process is optimized by varying the feed automatically as the tool-workpiece engagements, i.e. the process, varies along the tool path. The feed is adjusted by limiting feed-dependent peak outputs to a set of user-defined constraints. The constraints are tool shank bending stress, Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  114 tool deflection, maximum chip thickness (to avoid edge chipping) and the torque limit of the machine. The linear and angular feeds of the machine are optimized by two different methods -- a multi-constraint-based virtual adaptive control of the process and a non- linear root finding algorithm. The chapter is organized as follows. Predictions of maximum tool stress, tool deflection, chip thickness and cutting torque for serrated, tapered, variable-pitch, helical ball-end mills are discussed in Section 5.2. These are predictions based on the mechanics model given in Chapter 4 and a finite-element, cantilever model of the tool given in Appendix B. The multi-constraint virtual adaptive feed control system is presented in Section 5.3. This control system adaptively adjusts the feed at each tool path segment while respecting all constraints. Section 5.4 details the non-linear root finding feed optimization algorithm, which iteratively solves for the greatest feed rate based on the worst process output using Brent’s method [106]. Section 5.5 discusses how angular feeds must be scaled with tool tip feeds in order to preserve tool path geometry, while Section 5.6 discusses filtering of the optimized feed rate profiles. The application of both algorithms, with a significant machining time reduction, is presented in Section 5.7 and a summary of the chapter is given in Section 5.8.  5.2 Modeling of Peak Feed-Dependent Outputs and Constraints The cutting process outputs, namely torque, maximum deflection at the tool tip, maximum chip thickness over the cutting flutes and bending stress along the axis of the tool are evaluated for each angular increment of reference cutting edge. Feeds are then Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  115 scheduled to satisfy peak values of these outputs, as the cutter-part engagement conditions vary along the five-axis tool path. As explained in Chapter 4, the depth of cut of the variable-pitch, serrated, tapered, helical, ball-end mill is divided into a number of differential cutting force elements. The cutter-workpiece engagements of the elements are calculated from a solid model-based CAM system. The local rake and helix angles and radius of the cutter are also evaluated, and the corresponding elemental cutting forces, torques and maximum chip thickness are determined. Using a simple finite element analysis with a number of Timoshenko beam elements, tool stress and tool deflection are evaluated using the model shown in Figure 5-1. For a more detailed explanation on the finite element calculations, please refer to Appendix B. Hf ac Lfixed Cutting Forces Fixed Support (Tool Holder) Z Cutting Flute B e a m  E le m e n t Cutting Force Element Cutting Forces Workpiece Chip  Figure 5-1 – Finite element model of the tool with point-load cutting forces at the middle of each cutting force element [54], [55].  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  116 5.2.1 Cutting Torque The total cutting torque, cT , at a fixed angular position of the cutter is evaluated by summing the elemental cutting torques, , , 1 1 ja NN c ta j a j a j T F R = =   =      ∑ ∑ (5.1)  where ,ta jF  is the tangential cutting force for cutting force element, a , flute j . The local radius of the cutter at element a , flute j , is denoted by ,a jR . aN  is the number of cutting force elements and jN  is the number of flutes on the cutter. Jet engine impellers are made from titanium or nickel alloys -- materials which have high thermal resistance and strength. In order to machine these parts effectively, high torque at low cutting speeds is required. The maximum cutting torque must not exceed the torque capacity of the machine tool spindle, m T , at the given operating speed. In other words: c mT T≤  (5.2)  5.2.2 Chip Thickness  In titanium or nickel alloy machining operations, the rake angle of the tool is typically +5 to +10 degrees and the clearance angle of the cutting edge is usually greater than 5 degrees (Figure 5-2).  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  117 Rake Angle (αn) (Normal Rake) ~ (5-10 deg) Clearance Angle (γ CL ) ~ (> 5 deg) Cross Section of Cutter  Figure 5-2 – Cross section of a typical impeller milling tool showing rake and clearance angles.  As a result, the serrated cutting edge becomes weak and can fracture easily if the chip thickness exceeds a certain limit. At a particular angular position of the cutter following relation must be satisfied, ( )maxmax ( )11 j a cMAXALLOWja h z hj Na N     ≤    ==    ……   (5.3)  where az  is the position along the tool axis at the middle of cutting force element, a . ( )j ah z  is the chip thickness for flute, j , axial position, az , and cMAXALLOWh  is the chip thickness limit. The maximum chip thickness that will not damage the cutting edge is determined through finite element models of the metal cutting process calibrated with experimental tests. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  118 When using serrated cutters, the thickness of the chip can be higher than the feed- per-tooth due to the unequal flute lengths at the cross section.  5.2.3 Tool Deflection The deflection of the tool during milling is calculated through finite element analysis by considering differential tool segments as Timoshenko beam elements connected in series (Figure 5-1). If they are available, the clamping stiffness of the tool holder and spindle can be incorporated into the model as boundary conditions. Details of the finite element model used to model tapered, serrated, variable-pitch, helical, ball-end mills are summarized in the Appendix B. The total displacement at a position along the cutter’s axis, z , is the magnitude of the displacement in all three directions (x,y and z). In other words, 2 2 2( ) ( ) ( ) ( )total x y zu z u z u z u z= + +  (5.4)  In practice, cutting forces in the axial direction are usually significantly smaller than those in the transverse direction. Also, the stiffness of the beam in tension / compression is much higher than in bending. As a result, the axial component of deflection, ( )zu z , is often substantially less than the transverse deflections ( ( )xu z  and ( )yu z ) and can usually be neglected. The maximum deflection of the tool at a reference position of the cutter must be limited according to a deflection constraint, Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  119 [ ] limitmax ( )1 1 total ne u z un N   ≤  = + …   (5.5)  where n z  indicates the axial position of node, n , on the cantilever model given in Appendix B. The model has 1eN +  nodes, where eN  is the number of elements. limitu  is the maximum allowable deflection that will not cause violation of part tolerances and is imposed as a constraint in feed rate optimizations.  5.2.4 Tool Stress Popov [105] explains that the stress in the transverse direction for a general, three- dimensional beam at a particular cross-section is, ( )( ) ( ) ( )2 x xy y xx x yy y xy Z xx yy xy M I M I x M I M I y I I I σ − + + + = −   (5.6)  where xM  and yM  are the moments about the x and y axes of the centroid of the beam. xxI  and yyI  are the area moments of inertia and xyI  is the area product of inertia for the beam’s cross section. The total nodal displacements are solved for, as documented in the tool deflection section and Appendix B. The moments, xM  and yM , can be recovered from the element nodal force vector, { } e F , (Equation (B.1) – Appendix B), and by solving the equation, [ ] { } { } e ee K u F=  (5.7)  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  120 for each Timoshenko beam element, e . If the beam’s cross section is assumed to be circular, the area moments of inertia will be equal xx yyI I I= =  and the product of inertia term will disappear, 0xyI = . If tensile / compressive stresses from the axial forces are also included, Equation (5.6) becomes, y x z Z M x M y F I A σ − + = +  (5.8)  where ZF  is the total normal force at the cross section and A  is the cross-sectional area of the cutter. Taking the total resultant moment at the radius, R , and inserting 4 4 RI pi= and 2A Rpi=  into Equation (5.8) and rearranging gives, 2 2 max 3 4 x y z Z M M F R R σ pi + + =  (5.9)  As explained in Appendix B, a scaling factor is used on the radius in the cutting flute zone. The resulting maximum stress is, 2 2 max 3 4 x y z eff Z eff M M F R R σ pi + + =  when              0 e fz H≤ ≤  2 2 max 3 4 x y z Z M M F R R σ pi + + =              when               f e fixedH z L≤ ≤   (5.10)  where fH  is the distance from the tool tip to the top of the cutting flutes and fixedL  is the distance from the tool tip to the fixed support (tool holder). According to finite element simulations and analysis performed by Nemes et al. [101], the serrations in the cutting Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  121 flute zone induce stress concentrations in the tool. On the serration profiles tested in [101], they magnified the local stresses within the cutting flute zone by an average factor of approximately 2.1. This means,         max max2.1Z EFF Zσ σ=          when                     min maxserr e serrz z z≤ ≤        max maxZ EFF Zσ σ=               when        min0 e serrz z≤ <    or   maxserr e fixedz z L< ≤  (5.11)  where maxZ EFFσ  is the effective transverse stress including stress concentrations.  minserrz and maxserrz  are the locations along the tool axis at which the serrations on the cutting edge start and end, respectively. The maximum stress to which the cutter is subjected, maxZ EFFσ , is compared against the yield criteria for metallic cutters or, for carbide cutters, an equivalent tool breakage stress obtained from Weibull probability failure distribution. Nemes et al. [101] shows that failure stress for a tapered carbide cutter in service can be related to failure of a cylindrical carbide blank through, ( ) ( ) 1 2 max max 87.63 9.531.25 w w m m cs blank f sH R σ σ     =           (5.12)  where max( )csσ  is the maximum failure stress for a carbide cutter in service. fH  is the flute length measured from the tool tip (mm) and sR  is the shank radius (mm). The Weibull exponent, wm , and the transverse rupture strength of the carbide rod, max( )blankσ , are highly sensitive to the material composition of the carbide and to the diameter of the rod. Both values are determined through experimental testing. Nemes et. al [101] gives Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  122 Weibull exponents of 5.64 and 3.54 for the 25.4mm and 12.5mm carbide rods tested, respectively. Nemes et. al [101] also gives the transverse rupture strength of the 25.4mm blanks as 1262 MPa, with a standard deviation of 214 MPa, and for the 12.5mm blanks as 2131 MPa with a standard deviation of 503 MPa. It is important to emphasize the Weibull exponents, transverse rupture stengths and Equation (5.12) itself was derived from experimental data using Tungsten Carbide blanks of 12.5mm and 25mm in diameter. Nemes et. al [101] cautions against extrapolating Equation (5.12) to cutters outside this diameter range. Since carbide cutters have high variability in breakage limits from transverse stress, Nemes et. al [101] explains that a better limit is the stress that will give the cutter a certain probability of survival. Using Weibull failure distribution, the probability of survival for the carbide cutter is, max( ) mw z csS e σ σ   −    =   (5.13)  where S  is the probability of survival measured in percent and Zσ  is the applied stress. Solving Equation (5.13) for the applied stress gives: ( )max1ln ln ln max w cs w m S m Z ALLOW Z e σ σ σ     +               = =  (5.14)  Equation (5.14) gives the transverse stress that will give the cutter a desired probability of survival. The stress calculated from Equation (5.14) can be used as an estimate for the tool breakage limit for serrated tungsten carbide cutters. At a fixed Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  123 angular position of the cutter, the maximum stress over all nodes of the finite element model must be less than maxZ ALLOWσ .. In other words,     [ ]max maxmax ( )1 1 Z EFF n Z ALLOWe zn N σ σ≤= +…  (5.15)  5.3 Feed Rate Optimization By Multi-Constraint Virtual Adaptive Control The purpose of multi-constraint virtual adaptive feed control is to model the virtual cutting process and then to optimize it during a simulation by increasing / decreasing feeds so that peak process outputs are kept at or below user-specified values. Multi-constraint adaptive feed control is performed using two parallel systems. The first is the simulated model of the cutting process. The complete process physics are simulated at one degree cutter rotation increments for a given set of material properties, tool-workpiece engagements, cutter geometry and feeds, as presented in Chapter 4 of the thesis. The maximum values of torque, chip thickness, tool deflection and stress over one spindle revolution are selected from these calculations. The peak outputs are normalized with respect to their constraints and the most conservative of these are used to control the feed of the system. The process model is a virtual representation of the physical machining process. The outputs of the virtual cutting process correspond to measured values from sensors on the actual machine. The second process model is a first-order transfer function with a fixed time constant and time-varying gain. The transfer function is used as an approximate Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  124 representation of the process for virtual adaptive control purposes only. Similar to real time adaptive control, the input to the process is the feed and the output is the maximum normalized process output obtained from virtual simulation of the process physics. The process is simulated at very slow time intervals (i.e. discrete positions along the tool path), which is a few times longer than the time constant of the servo drives in order to avoid the influence of the CNC’s transfer function. The estimated process gain and fixed time constant are used to adaptively tune the proportional-integral (PI) controller, which manipulates the feed rate in order to satisfy the process output constraints. The details of the process control application are given as follows.  5.3.1 Modeling of the Cutting Process and Estimation of Parameters for Multi-Constraint Virtual Adaptive Feed Control The general block diagram of the five-axis cutting process in the z-domain is shown in Figure 5-3. ( )H z  is the transfer function of the cutting process where ( )Lf z  is the input (tool tip feed) and ( )TSy z , ( )TDy z , ( )TQy z  and ( )CLy z  are the peak tool stress, tool deflection, cutting torque and chip thickness outputs over one spindle period, respectively. Although not shown on Figure 5-3, the angular velocity, ( )zω , is also an input to the process. This value is scaled with tool tip feed in a fixed ratio according to the tool path trajectory as shown in Section 5.5. The outputs are normalized with respect to user-defined constraints as shown in Figure 5-3. Normalizing the outputs gives an indication of how far each output is from its corresponding constraint. It also shows which output is the closest to being violated. The outputs are normalized as follows, Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  125 H(z) fL(z) yTS(z) yTQ(z) 100 C TS yTD(z) yCL(z) 100 C TD 100 C TQ 100 C CL PTS(z) PTD(z) PTQ(z) PCL(z) Virtual Cutting Process Peak Outputs Tool Tip Feed Constraint Normalized Outputs (CNOs)  Figure 5-3 – Block diagram of the virtual cutting process [54], [55].  ( ) ( )( ) 100 ( ) 100 ( ) ( )( ) 100 ( ) 100 TS TD TS TD TS TD TQ CL TQ CL TQ CL y z y zP z P z C C y z y zP z P z C C     = =           = =           (5.16)  where TSC , TDC , TQC  and CLC  are the tool stress, tool deflection, cutting torque and max chip thickness constraints. These are constant threshold values given by the user. ( )TSy z , ( )TDy z , ( )TQy z  and ( )CLy z  are the peak tool stress, tool deflection, cutting torque and chip thickness outputs over one spindle period. ( )TSP z , ( )TDP z , ( )TQP z  and ( )CLP z  are called constraint normalized outputs (CNOs) and express the outputs as a percentage of their constraints. In the following section, the variable, ( )P z , will refer to a generalized CNO. The virtual five-axis flank milling process simulation contains a significant amount of non-linear algorithms, but the process varies very slowly at long time intervals. This is Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  126 due to the gradual change in cutter-workpiece engagements that occurs when flank milling jet engine impellers. With this assumption, the process can be approximated by a first order transfer function [4], [113] with feed as the input. In the z-domain,  ( )( ) ( ) g L g KP zJ z f z z a= = +  (5.17)  where ( )J z  is the transfer function of the virtual cutting process with respect to a CNO. The parameters of the plant's transfer function can be estimated during simulations by assuming that the pole, ga , is fixed and then solving for the time varying gain, gK . The pole is fixed as,  g T ga e τ   −      = −   (5.18)  where T  is the length of the control interval, which is equal to one spindle period. gτ  is the time constant of the virtual cutting process. Since the virtual cutting process has a fast response in comparison to the length of the control intervals, the time constant should be small. It can be selected as 4g T τ = . In discrete time, the gain, gK , can be calculated at each control interval from the past and present inputs and outputs,  ( ) ( 1) ( 1) g g L P k a P k K f k + − = −   (5.19)  where 1,2,3...k =  is the discrete time sample counter. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  127 Modeling the cutting process with a first-order transfer function [4], [113] and a fixed pole, ga , greatly simplifies estimation of the transfer function's parameters. If the system is modeled with a higher order transfer function, or if ga  is not fixed, the parameters must be estimated using recursive least-squares (RLS) parameter estimation [4], [62], [113] which is unnecessary here due to the long control intervals. In order to avoid frequent updates of the feed, which leaves marks on the surface finish, and to prevent the continuous acceleration mode of the machine, long time intervals are selected which typically correspond to 1-2 mm intervals along the tool path. A simple proportional-integral controller (PI) can be used to control the system effectively: ( )( ) ( ) 1 L P i f z TG z K K err z z   = = +   −    (5.20)  ( ) ( ) ( ) r err z P z P z= −  is the error between the CNO and the desired CNO reference value, ( ) 100 r P z = . pK  and iK  are the proportional and integral gains of the controller, ( )G z , respectively. The integral action of the controller assures elimination of steady- state error. Figure 5-4, shows the block diagram for an adaptively controlled virtual cutting process in the z-domain.   Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  128 J(z)G(z) err(z)Pr = 100 Error Feed Desired Output P(z) Normalized Output fL(z) Virtual Cutting Process Controller Estimate Parameters of J(z)  Figure 5-4 – The z-domain block diagram of a closed loop, single-input single-output (SISO), virtual adaptive feed control system for a general, constraint normalized output (CNO) [54], [55].  If the virtual cutting process is modeled with a first order plant and controlled with a PI controller, the transfer function between the desired output and the constraint normalized output is: ( ) 2 ( ) ( ) 1 p g g i P r p g g p g g i g K K z K K T KP z P z z K K a z K K K K T a    + −    =    + + − + − + −      (5.21)  The denominator of Equation (5.21) can be expressed in the form: 2 1 2CLdenTF z m z m= + +        (5.22)  In terms of natural frequency and damping ratio, 1m  and 2m  are equal to: ( ) 221 22 cos 1n nT Tnm e T m eζω ζωω ς− −= − − =        (5.23)  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  129 From Equations (5.21)-(5.23), the proportional and integral gains of the controller are calculated as, ( )2 21 2 cos 1n nT Tg n p g g p i g g a e T e K K a K K K K T ζω ζωω ζ− −− − − + + = =  (5.24)  It is desired to have an overdamped damped response to obtain a smooth convergence of feed to the optimized value. It is also desired to have the peak output reach the desired constraint within at least 4-6 spindle revolutions. A damping ratio of 0.80ζ =  and a natural frequency of 2 n Tω pi=  are selected. An algorithm was coded to perform multi-constraint virtual adaptive feed control of the modeled cutting process in discrete time. The closed-loop block diagram for the system is shown in Figure 5-5. J(z)G(z) err(z) Controller Cutting Process Pmax(z) MAX PTS(z) Pr = 100 Error Feed Estimate Parameters of J(z) Array of Previous CNO Values PTD(z) PCL(z) PTQ(z) Output Selector (Index of Max P) Pmax(z) z-1Pmax(z) z-1PTS(z) z-1PTD(z) z-1PCL(z) z-1PTQ(z) z-1 Delay Desired Output fL(z)  Figure 5-5 – Block diagram of the closed-loop multi-constraint virtual adaptive feed control system [54], [55]. The maximum P  value from the cutting process is used to switch to the ( )P z  and 1 ( )z P z−  values that will correctly limit the feed.  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  130 At each control interval, all ( )P z  values are calculated and the maximum of these, ( )max ( ) max ( ), ( ), ( ), ( )TS TD TQ CLP z P z P z P z P z= , is subtracted from the reference value of 100 to determine the error term for the controller. This ensures that the controller will drive the feed to the most conservative constraint, while keeping the system as single- input single-output (SISO). The index of the limiting CNO is stored and used to switch between outputs for calculation of gK , pK , iK  and the new feed command, ( )Lf k . An array holds previous ( )P z  values for each peak output. If the controller switches to a new limiting constraint (i.e. the active constraint changes), then the discrete control law switches to the corresponding past output, 1 ( )z P z− , to calculate the new controller gains and feed. For example, if  max ( ) ( )TSP z P z=  changes to max ( ) ( )CLP z P z=  then 1 1 max ( ) ( )TSz P z z P z− −=  will switch to 1 1max ( ) ( )CLz P z z P z− −= . To calculate the required feed at each control interval in discrete time, max max( ) ( ) ( ) ( 1) ( 1) ( 1) L p r p i p r p i L f k K P k K P k K T K P k K K T P k f k    = − + − − + − −    + −   (5.25)  During optimization simulations, the converged feed value at the end of each tool path segment is written to an APT-CL data file in inverse time feed. This file can be read by the postprocessor of the CNC machine.  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  131 5.4 Feed Rate Optimization By Non-Linear Root Finding An alternate method of obtaining optimum feeds is through non-linear root finding. Similar to the adaptive feed control optimization, the user specifies a number of constraints on the peak outputs that can not be violated. However, the peak outputs are normalized in such a way that they become zero when equal to their respective constraints. A non-linear root finding algorithm is used to iteratively solve for the zero of the worst output, which is the optimum feed. Convergence is achieved when the given feed is satisfied to within a given tolerance. As illustrated in Figure 5-6, the virtual cutting process can be considered as a non- linear "black box" type function with tool tip feed being the input and tool stress, tool deflection, cutting torque and max chip thickness being the outputs. It is known that each of the outputs will increase with an increase in feed, however the rate at which each increases can be non-linear. yTS yTQ 1 C TS yTD yCL C TD C TQ C CL PTS PTD PTQ PCL Virtual Cutting Process Function Peak Outputs Constraint Normalized Outputs (CNOs) Tool Tip Feed 1 1 1 C TS C TD C TQ C CL fL  Figure 5-6 – Virtual cutting process modeled as a non-linear function of tool tip feed [54], [55]. Peak outputs are normalized so they become zero when equal to their respective constraints.   Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  132  At each tool path segment, the cutter is rotated once and the peak outputs for tool stress, tool deflection, cutting torque and max chip thickness are collected. These peak values are normalized with respect to their constraints using the following equations: TS TS TD TD TS TD TS TD TQ TQ CL CL TQ CL TQ CL y C y CP P C C y C y CP P C C    − − = =         −   − = =            (5.26)  TSP , TDP , TQP  and CLP  are the constraint normalized outputs (CNOs) for the non- linear root finding feed optimization. Note that the CNOs in Equation (5.26) are different from those used in the virtual adaptive control algorithm given by Equation (5.16). Normalizing the peak outputs using Equation (5.26) causes the CNO values to equal zero when a peak output is equal to its constraint. The optimum feed, as illustrated in Figure 5-7, is now a matter of finding the feed at which one P  value is zero and all others are below zero -- in other words, finding the root of the limiting CNO. Calculation of the root is achieved using a non-linear root finding algorithm. The root-finding method used in the current research project is called the Van Wijngaarden- Dekker-Brent Method, usually called "Brent's Method" [106]. An algorithm was written to perform non-linear root finding feed optimizations on five-axis flank milling tool paths. The algorithm optimizes an entire tool path segment by finding the optimum feed for one cutter rotation at the beginning of each segment. This is deemed appropriate as in each tool path segment, the engagements are constant and the tool path segments are short. Also, the tool axis does not undergo significant changes in tool-axis orientation. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  133 fLA P(fLA) P(fLoptimum) = 0 P P(fLC) fLC fLoptimum fL Constraint Line 0  Figure 5-7 – Lines represent different CNOs as a function of feed [54], [55]. The optimum feed is the feed at which one CNO is zero and all others are below zero.  The algorithm begins by determining the upper feed bracket value, or the upper ordinate and abscissa pair, , ( ) C CL L f P f   , where only one of the outputs is above its constraint (Figure 5-7). This is accomplished in iterations by increasing the feed from the initial value given in the APT-CL data file, rotating the cutter once, and checking whether only one CNO is greater than zero. When only one CNO is above zero, this feed-CNO pair is chosen as the upper bracketing limit. If more than one CNO is already above zero with the feed given in the APT-CL data file, then the algorithm will decrease the feed until only one CNO is above zero. Next, the algorithm finds the lower bracketing pair, , ( ) A AL L f P f   , where the limiting CNO determined previously is below zero (Figure 5-7). This is accomplished iteratively by decreasing the feed from the initial value in the APT-CL data file, rotating the cutter once and observing whether the CNO chosen for the upper feed bracket is less Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  134 than zero. In most cases, the limiting CNO in the APT-CL data file is already below zero (no constraints are violated) and no iterations are necessary. Once the upper and lower bracketing pairs, , ( ) A AL L f P f    and , ( )C CL Lf P f    are determined, it is known that a root for the limiting CNO exists between these two limits. Brent’s method uses iterations to "home in" on the root until it is within a given tolerance. Again, it is important to mention that, even though the tool tip feed is used as the input to the function, the angular velocity is also changed according to Equation (5.30) shown in Section 5.5. When optimizing the feed rates for this tool path, Brent's method usually converged to less than 1% of the optimized feed within 3-5 iterations. The output of the algorithm is an optimized APT-CL data file with the new feeds written in inverse time feed.  5.5 Scaling of Angular Feeds When controlling feeds using the multi-constraint adaptive feed control and non- linear root finding methods, the tool tip feed rate is used as the control variable. However, when the tool tip feed rate is increased or decreased, the angular velocity must be scaled up / down as well in order for the trajectory of the cutter to remain fixed. The tool path is composed of a series of segments where the tool tip and angular velocities are assumed to remain constant. Figure 5-8 shows a typical five-axis sculptured surface machining tool path segment. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  135 CL Command (i) {Pi+1} {Pi} CL Command (i+1) Tool Path Segment i {Ti+1} {Ti} {Pi+1} {Pi} ∆di fLi ∆θi {ki} {Ti+1} {Ti} ωi ∆t Tool Tip Motion Angular Motion ∆t  Figure 5-8 – In tool path segment, i , the tool travels a distance,  id∆   [54], [55]. In the same time period, t∆ , it rotates a total angle, iθ∆ . Angular feeds must be scaled with tool tip feeds to preserve this tool path geometry.  The time taken for the tool tip to travel through segment, i , is,  i Li d t f ∆∆ =        (5.27)   where id∆  is the distance travelled by the tool tip in tool path segment, i . t∆  is the time elapsed between the beginning and end of the segment and Lif  is the tool tip feed. Figure 5-8 also shows that in the same time period the tool-axis rotates an angle, iθ∆ , at angular velocity iω . Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  136 i i t θ ω ∆∆ =     (5.28)   Setting Equation (5.27) equal to Equation (5.28) and rearranging gives:  i i Li if d ω θ∆ = ∆   (5.29)   In other words, for the tool path trajectory to remain fixed, the ratio between the angular velocity and the tool tip feed must remain constant. This relation can also be generalized to a fraction of the tool path segment. In other words, when the tool tip feed is changed the new angular velocity is,  L i i f d θ ω ∆ = ∆   (5.30)  5.6 Filtering of Optimized Feeds The results of the optimization process give the highest feed that will not violate the most conservative constraint in each tool path segment. However, due to the changing tool-tip and angular feeds, complex cutter-workpiece engagements and non-linearities of the process, the optimum feed profile generated is often noisy. If these feeds are run on a physical machine, the machine may be in a constant state of acceleration / deceleration, which can leave feed marks on the surface of the part. To overcome this, the optimum feed profile should be smoothed. When flank-milling impellers from alloys such as 6 4Ti Al V− −  and Inconel, changes in feed along the tool path are usually low in relation to the acceleration capabilities of the machine. In this case, a discrete averaging filter can Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  137 be used to reduce fluctuations. If the accleration of the machine cannot be neglected (i.e. changes in feeds are large) then feed rate filtering must be based the machine’s acceleration specifications [78]. The averaging filter was chosen as it is relatively simple to implement and works well to remove high frequency noise from low frequency signals. The equation for a one- dimensional discrete digital filter is given as, (1) ( ) (1) ( ) (2) ( 1) ( 1) ( ) (2) ( 1) ( 1) ( ) b b a a a y n b x n b x n b n x n n a y n a n y n n = + − + + + − − − − + − … …   (5.31)   where a  and b  are the filter coefficients for the output and input respectively and x  and y  are the input and output values at discrete time intervals. an  and bn  are the total number of coefficients for a  and b , respectively. For a simple averaging filter the coefficients are as follows, { }1 1 1 1 1 a b filtorder filtorder filtorder filtorder =   =     …   (5.32)  where b  is a vector having a dimension of 1x filtorder , where filtorder  is the number of points to be averaged. The higher the filter order, the more the feed fluctuations are reduced. The lower the order, the more closely the filtered signal follows the original data. The MATLAB forward-reverse filtering command, “filtfilt”, is used to filter the feeds. This command first filters the input data, reverses it and filters it again, which minimizes the phase distortion of the signal. The feeds are filtered by tool path segment Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  138 rather than time, as altering the feeds changes the duration of the tool path segments. An example is shown in Figure 5-9.  0 Tool Tip Feed vs. Time  F e e d  [ m m /s ]  F e e d  [ m m /s ] Optimized Tool Tip Feed Filtered Optimized Tool Tip Feed 100 300200 16 3224 28 1.2 2.0 0.4 1.0 2.0 3.0 0.8 1.6 20 Tool Path Segment Number 282 298290 294 0.6 1.0 0.2 0.4 0.8 286 Tool Path Segment Number Tool Path Segment Number Tool Path Segment Number 130 146138 142 0.5 0.54 134 150 0.58 0.62 (a) (b) (c)  Figure 5-9 – Optimized tool tip feed vs. filtered optimized tool tip feed. (a) Filtering used to blend a sharp transition in feed where the tool enters the workpiece. (b) Filtering used to reduce fluctuations in the optimized feed profile. (c) Filtering used to blend a sharp transition in feed where the tool exits the workpiece.  The data in Figure 5-9 was filtering using a 10th order averaging filter with the MATLAB “filtfilt” command. As can be seen in Figure 5-9(b), the filtered tool tip feed profile is much less noisy. The optimization algorithms do not adjust feeds in tool path segments with no-engagement (air-cutting). Filtering can be used to smooth the Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  139 transitions from the original and optimized feeds in these sections. Examples of this are shown in Figure 5-9(a) and Figure 5-9(c) where the tool enters and exits the workpiece, respectively. The averaging of the feed profile in these areas reduces the shock on the tool and part and mitigates transient vibrations. Care must be taken when filtering the feed profiles, as the filtering process may cause feeds to exceed those of the optimized results. This is evident in Figure 5-9(b). If the filtered feeds become greater than the optimized values, the peak outputs may violate their respective constraints. In practice, this is usually not an issue. If the order of the filter is chosen correctly, it should not change feeds significantly (usually under 5% - Figure 5-9(b)). Also the constraints on the process should be set slightly below maximum allowable values to allow some tolerance for filtering. Finally, if desired, the optimum profile can be manually adjusted so that, when filtered, the feeds are below the original optimized points at all parts of the tool path.  5.7 Results The roughing tool path for the integrally bladed rotor (IBR) described in Chapter 4 was optimized using the multi-constraint virtual adaptive feed control and non-linear feed rate optimization methods. The tool stress constraint was set to 480TSC MPa= , the deflection constraint was set to 0.70TDC mm= , the torque constraint was set at 60TQC Nm=  and the max chip thickness constraint was 0.090CLC mm= . The values for the constraints were based on realistic and tested values received from industry. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  140 A simulation was run with the adaptive feed rate control enabled. Figure 5-10 shows how the controller regulates the feeds according to constraints over the course of a simulation. The upper plots show the tool stress (the most dominant constraint) and feed rate vs. time for the entire tool path. The middle and lower plots show close-ups of tool path segments where the tool stress is the limiting constraint. When the engagements change at the beginning of the tool path segment, the virtual adaptive feed control algorithm automatically adjusts the gains of the controller to bring the tool stress to the constraint. Figure 5-10 shows the transient output vs. time and that after roughly 4-6 spindle revolutions, the feed and peak output reach a steady state value. In Figure 5-10, it observed that constraint violations exist when the cutter-workpiece engagements change at the beginning of each tool path segment. During the adaptive feed rate control simulation, the feeds at the end of six cutter revolutions are written to a new CL-data file. A force prediction simulation was run using the new feeds as input. Figure 5-11 shows the results of this simulation and compares the optimized feed profile and process outputs with those of the original tool path. Notice that no constraint violations exist. Figure 5-11 shows that, with the given constraints, the cycle time has been reduced by about 45%. The most dramatic changes in feed are observed at the beginning and end of the tool path. It is also interesting to note how, at various parts of the tool path, different constraints are active. For example, Figure 5-11 shows that the maximum chip thickness is the limiting constraint at the beginning and the end of the simulation, whereas the tool stress and cutting torque limit the feed in the middle of the simulation. Figure 5-11 also shows that the tool deflection constraint is not active for the duration of the simulation. Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  141 300 Time [s] Time [s] Closeup of Tool Stress vs. Time - Adaptive Feed Control Closeup of Tool Tip Feed vs. Time - Adaptive Feed Control Closeup of Tool Tip Feed vs. Time - Adaptive Feed Control  F e e d  [ m m /s ]  T o o l S tr e s s  [ M P a ] 302 304 306 Tool Path Segments Beginning of Segment Controller moves peak tool stress towards constraint Closeup of Tool Stress vs. Time - Adaptive Feed Control  F e e d  [ m m /s ] Engagements Change Here  T o o l S tr e s s  [ M P a ] Tool Stress vs. Time - Adaptive Feed Control Tool Tip Feed vs. Time - Adaptive Feed Control Time [s] 0 100 200 300 400  F e e d  [ m m /s ] 0.4 0.8 1.2 1.6 2.0 Time [s] 0 100 200 300 400 100 200 300 400 500 Time [s]  T o o l S tr e s s  [ M P a ] 300 302 304 306 360 400 440 480 500 0.47 0.49 0.51 0.53 360 400 440 480 500 0.47 0.49 0.51 0.53 Time [s] Tool Stress Tool Tip Feed Tool Stress Constraint Tool Stress Constraint Tool Stress Constraint 300.5 300.7 300.9 301.1 300.5 300.7 300.9 301.1  Figure 5-10 – Upper plots show tool stress and tool tip feed for the entire tool path [54], [55]. Middle and lower plots show close-ups of the tool stress limiting the output. When the engagements change, both tool stress and feeds are increased / decreased until the peak tool stress is equal to its constraint.  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  142 300 200 100 200 400 600 800 S tr e s s  [ M P a ] Time [s] 0 Time [s] D e fl e c ti o n  [ m m ] 0 0.1 0 Tool Stress vs. Time Tool Deflection vs. Time Cutting Torque vs. Time Max Chip Load vs. Time 0.3 0.5 0.7 60 40 20  T o rq u e  [ N m ] 0  C h ip  T h ic k n e s s  [ µ m ] 50 70 30 10 0 90 Original Adaptive Control Torque Constraint Original Adaptive Control Chip Load Constraint Original Adaptive Control Deflection Constraint Original Adaptive Control Stress Constraint 400 Tool Tip Feed vs. Time F e e d  [ m m /s ] F e e d  [ ra d /s ] Time [s] 0.5 1.0 1.5 0 2.0 2.5 2 0 4 6 8 x10-3 C TS = 480 MPa C TD = 0.7 mm C TQ = 60 Nm C CL = 90µm Constraints Angular Feed  vs. Time 200 400 600 8000 200 400 600 8000200 400 600 8000 200 400 600 8000200 400 600 8000 Time [s]Time [s] Original Feed Adaptive Control Original Feed Adaptive Control 500 Time [s]  Figure 5-11 – Multi-constraint virtual adaptive feed control optimization results for the IBR roughing tool path [54], [55]. With the given constraints, the cycle time has been reduced by about 45%. Note how different constraints are active during different parts of the tool path.  Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  143 Figure 5-12 compares the feeds from the non-linear root finding optimization with those from the virtual adaptive control algorithm. With the same constraints, a 45% reduction in cycle time is observed. Note that the two methods produce nearly identical feed profiles. 0 Time [s]Time [s] Tool Tip Feed vs. Time  F e e d  [ m m /s ]  F e e d  [ m m /s ]  F e e d  [ ra d /s ] Angular Feed vs. Time  F e e d  [ ra d /s ] x10-3 Time [s]Time [s] x10-3 Closeup of Tool Tip Feed vs. Time Adaptive Control Non-Linear Root Finding 100 300 400200 0 100 300 400200 80 11090 100 0.7 0.8 0.6 1.6 1.8 2.0 2.2 2 3 4 5 1 7 8 6 0.6 1.0 0.2 1.4 1.8 Closeup of Angular Feed vs. Time 2.4 80 11090 100 2.6  Figure 5-12 – Comparison of the multi-constraint virtual adaptive feed control and non-linear root finding algorithms [54], [55]. Feed profiles from both methods are almost identical. In both methods, a 45% reduction in cycle time was achieved with the given constraints.  Both approaches work rather well for the tool path mentioned in this project. However, the non-linear root-finding optimization approach is computationally more convenient and yields better convergence for small tool path segments. It is also more precise in that it uses a robust non-linear solver, which converges quickly on the limiting Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  144 peak output. Although not shown in the thesis due to confidentiality restrictions, the non- linear root-finding feed optimization method has been applied to another impeller roughing tool path at Pratt & Whitney Canada. The optimized tool path was verified on a production impeller and reduced the machining time by approximately 20%. The adaptive control algorithm also works well. However, using this method, it is difficult to set a minimum convergence error on the feed or CNO values. It is also difficult to predict the exact behavior of the controller because the transfer function of the virtual cutting process changes over time. The proportional-integral controller used in this project takes 4-6 tool rotations to adjust to changes in feed. For short tool path segments the controller may not be able to bring the feed to the required set point quickly enough. This could lead to feeds that are far from optimum or feeds that are too high, which may cause constraint violations. The adaptive control approach works much better when cutter/workpiece engagement maps change slowly or are constant over a long period of time. An advantage of the virtual adaptive control algorithm is that it allows the user to simulate the physical response of the machine to changes in feed. Also, for faster processes, the transfer function of the feed drive and CNC system can be included in the adaptive control system.  5.8 Summary In this chapter, optimization of feed rates for five-axis impeller flank milling operations with complex milling cutters is presented. The optimization schemes allow the feed to be limited according to multiple feed-dependent constraints. The process outputs, Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  145 such as peak values of tool bending stress, tool deflection, cutting torque and maximum chip thickness are predicted from the cutting mechanics model described in Chapter 4 of the thesis and a finite element model of the tool, described in Appendix B. The process model is optimized offline using two methods -- a virtual adaptive control system and a non-linear root finding feed rate optimization scheme. In the virtual adaptive control system, the virtual cutting process is modeled with a first order transfer function and a fixed time constant. This system has multiple outputs limited by multiple constraints but is reduced to a single input-single output (SISO) control system through normalization and selection of the worst output at each control interval. Due to the slow variation of cutter-workpiece engagements in impeller flank- milling, adaptation of a simple PI controller to the changing process gain is usually sufficient to control the system. The feed rate optimization demonstrated Section 5.7 met the set tool stress, tool deflection, cutting torque and maximum chip thickness constraints by adaptively varying the feed. In the non-linear root finding feed optimization algorithm, the peak outputs are normalized so that they equal zero when equal to their respective constraints. The zero of the worst normalized output is then solved for iteratively using Brent's Method [106]. Both the adaptive control and non-linear root finding algorithms were tested on the impeller tool path given in Chapter 4 of the thesis and gave almost identical optimum feed profiles -- profiles that realized a 45% reduction in cycle time. The non-linear root finding method is more robust as it allows a tolerance to be set on the optimimum feed and can better handle shorter tool path segments. This optimization scheme has been experimentally verified on a physical machine tool at Pratt & Whitney Canada, reducing Chapter 5: Feed Rate Optimization of Five-Axis Flank Milling Tool Paths  146 the cycle time of an impeller roughing tool path by 20%. The adaptive feed control optimization algorithm more closely mimics the response of the physical machine, but may not arrive at an optimum feed if the cutter-workpiece engagements change too quickly. Although this work has focused on impeller machining, the theories and algorithms explained in this work can be applied to general five-axis milling processes.  147 Chapter 6 Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  6.1 Overview During the flank milling of jet-engine impellers, vibrations of the tool and workpiece are unavoidable. However chatter vibrations, which can destroy the surface finish of the part and cause damage to the tool and workpiece, can be avoided if adequate cutting conditions are selected. Chatter is a complex process that results from the tool recutting the wavy surface of the workpiece left by vibrations from a previous tool pass (see Figure 6-1). Depending on the phase shift between the present and previous vibrations, the chip thickness and cutting forces may grow exponentially until the tool jumps out of cut or is damaged. The chatter process can be modeled mathematically using a delayed-differential equation (DDE). This DDE is difficult to solve analytically due to the infinite number of roots of the characteristic equation. Approximate solutions have been presented for several machining operations, but for five-axis flank milling with general tool geometry, no solutions have been observed in literature at this time. In this chapter, a new chatter stability solution for five-axis flank milling is presented. The solution permits identification of chatter conditions that may occur in five-axis milling with tapered, serrated, variable-pitch, helical ball-end mills. The method is robust in that it allows irregular cutter-workpiece engagements to be taken into  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  148 kx cx kycy y x FrjFtj flute (j) flute (j-1) flute (j-2) vibration marks left by tooth (j-1) No vibrations In phase waves Constant chip thickness Forced vibrations No chatter Out of phase waves Chip thickness grows exponentially NO VIBRATIONS FORCED VIBRATIONS CHATTER VIBRATIONS ε = 0 ε = 0 Ω φj  Figure 6-1 – Illustration of the tool cutting the workpiece with vibrations in 2D milling. Chatter vibrations may occur, depending on the phase shift between the present and previous vibrations of the tool.  account, as well as dynamic flexibilities of the tool and workpiece. These conditions are encountered in jet-engine impeller machining and play a significant role in the stability of the process.  The solution presented in this chapter is adopted from the formulation given by Altintas and Budak [8] for 2D milling with a cylindrical end mill. However, since the Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  149 depth of cut and entry/exit conditions of the flutes can vary along the axis of the cutter, the solution of Altintas and Budak [8] can not be used in its original form. Furthermore, in five-axis machining, increasing the depth of cut along the tool axis may change the shape of the cutter-workpiece engagement area in an unpredictable manner. In this case the depth of cut can not be used as a simple gain in order to determine a critical depth of cut. The method employed to solve for the chatter stability in this chapter involves selecting a spindle speed, imposing a cutter-workpiece engagement map and then formulating the dynamic stability equations with discrete elements along the depth of cut. The Nyquist stability criterion is used to check whether the system is unstable or not. This chapter is organized as follows. Section 6.2 presents formulation of the dynamic chip thickness and cutting force equations for the five-axis chatter stability solution. Section 6.3 discusses the stability analysis of the characteristic equation using the Nyquist Stability Criterion. Section 6.4 analyzes the dynamics of the tool and flexible impeller blades and demonstrates how they are integrated into the present solution. Section 6.5 compares stability predictions from the new method against those given by Altintas and Budak’s chatter stability theory [8] for a simple 2D milling case. It also attempts to correlate stability predictions for a five-axis flank milling finishing pass on an industrial jet-engine impeller with observations from experiments. Finally, a summary of the chapter is given in Section 6.6. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  150 6.2 Chatter Stability Formulation For Five-Axis Flank Milling with General Milling Tools The stability solution presented in this chapter calculates whether the process is stable or unstable for a given spindle speed and cutter-workpiece engagement map using Nyquist plots. The solution is derived by breaking the cutter-workpiece engagement maps into a series of small elements (see Figure 6-2), formulating expressions for dynamic chip thickness and cutting forces for each element, and then summing the differential forces to obtain the total forces in the x,y and z directions. -1.0 0.0 1.0 3.02.0 0 40 80 Immersion Angle        (radians) z (m m ) Engagement Map D e p th  o f cu t 0 40 80 -1.0 0.0 1.0 3.02.0 dz  Figure 6-2 – (left) Irregular cutter-workpiece engagement map, typical of five-axis flank milling. (right) The five-axis stability solution breaks this map into a series of discrete elements.  The solution allows the following to be incorporated if desired: • Cutter-workpiece engagement maps with varying radial immersions and depths of cut. • General end mill geometry [45]. • Serrated cutting edges (Figure 1-3), [94]. • Variable-pitch flute spacing (Figure 1-3), [26], [27]. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  151 • Tool and workpiece transfer functions that change along the axis of the cutter. • Cutting coefficients that vary along the axis of the cutter. • Loss of contact between successive cutting flutes.  6.2.1 Dynamic Chip Thickness Formulation The dynamic chip thickness equations consider vibrations at each element in x,y and z directions only. Torsional vibrations of the cutter are neglected, as the cutter is considered to be sufficiently stiff in these modes. Figure 6-3 shows the how displacements at a small cutting edge segment for flute, j , ( ju , jv , jw ) can be projected into x,y and z coordinates ( jx , jy , jz ) in the feed (or tool) coordinate system ( ), ,TT TT TTX Y Z . XTT YTT κ(z) z Y1 X1 n YTT φj(z) vj wj ZTT Z1 xj yj zj , Cutting Edge Cutter Envelope uj  Figure 6-3 – Figure showing the transformation of displacements at the cutting edge ( , ,j j ju v w ) to those in the feed coordinate system at the tool tip ( , ,j j jx y z ). This figure is based on one in [42] Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  152 Since the cutting edge can be inclined, due to the edge geometry and envelope of the cutter, and rotates around the axis of the tool, the total coordinate transformation is obtained by multiplying two matrices together. The first transformation matrix prescribes a rotation of angle, ( )j zκ , around axis, 1Y , which changes the displacement coordinates in the edge coordinate system ( , ,j j ju v w ) to those of the rotating coordinate system ( 1 1 1, ,X Y Z ). The axial immersion angle, ( )j zκ , is given in [45] for general end mills and in [94] for serrated end mills. The second transformation matrix is a rotation of angle, ( )j zφ , around the TTZ  axis, which projects the displacements in the rotating coordinate system ( 1 1 1, ,X Y Z ) to the feed coordinate system ( , ,TT TT TTX Y Z ). For flute j , the total coordinate transformation matrix is given as follows, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) cos ( ) sin ( ) sin ( ) cos ( ) cos ( ) cos ( ) 0 sin ( ) j j j j jj j j j j j j j j j jj j z z z z zx u y z z z z z v z wz z φ κ φ φ κ φ κ φ φ κ κ κ   − − −        = − −             −     (6.1)  where , ,j j ju v w  are the radial, tangential and axial displacements at the cutting edge, respectively. Since Equation (6.1) is an orthogonal matrix, the inverse relation is simply the transpose. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) cos ( ) sin ( ) 0 sin ( ) cos ( ) cos ( ) cos ( ) sin ( ) j j j j jj j j j j j j jj j j j j z z z z zu x v z z y w zz z z z z φ κ φ κ κ φ φ φ κ φ κ κ   − −        = −             − − −     (6.2) Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  153 The displacement along the chip thickness direction, ju , at axial position, z , in terms of displacements in jx , jy  and jz  is obtained by multiplying out the column for ju in Equation (6.2). ( ) ( ) ( ) ( ) ( )( , ) ( )sin ( ) sin ( ) ( )cos ( ) sin ( ) ( )cos ( )j j j j j j j j j ju z x z z z y z z z z z zφ φ κ φ κ κ= − − +  (6.3)  As the cutter rotates, the reference immersion angle changes with time according to [45], 1 , 1 ( , ) ( ) ( ) j j r p n j n z t t zφ φ φ ψ − = = − −∑         and    ( ) 30r n t t piφ =   (6.4)   Where ( ) r tφ  is the reference immersion angle at time, t , or the angle of the first flute from TTY . ,p jφ  is the pitch angle of flute, 1 jj N= … , for a cutter with jN  flutes and ( )j zψ  is the lag angle of flute, j , due to the cutter’s helix. As explained in Section 4.3, flutes are numbered so that their index increases in a counterclockwise direction. This is opposite to the convention given in [45]. Calculation of the lag angle is not discussed here but can be found in [45] for general end mills. The lag angle is the same in serrated cutters as it is for general end-mills. The spindle speed of the tool is given by n (rev/min). If Equation (6.4) is substituted into (6.3), Equation (6.3) becomes: ( ) ( ) ( ) ( ) ( )( , ) ( , )sin ( , ) sin ( ) ( , )cos ( , ) sin ( ) ( , )cos ( )j j j j j j j j ju t z x t z t z z y t z t z z z t z zφ κ φ κ κ= − − +  (6.5)  For a previous flute in the chip thickness direction of the present flute, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  154 ( ) ( ) ( ) ( ) ( ) 1 1 1 1 1 1 ( , ) ( , )sin ( , ) sin ( ) ( , )cos ( , ) sin ( ) ( , )cos ( ) j j j j j j j j j j j j j j u t z x t T z t T z z y t T z t T z z z t T z z φ κ φ κ κ − − − − − − = − − − − − − + −   (6.6)  Where jT  is the time delay between the present and previous tooth pass. , 30 p j jT n φ pi = . Since 1( , ) ( , )j j jt T z t zφ φ− − = , Equation (6.6) is equivalent to, ( ) ( ) ( ) ( ) ( ) 1 1 1 1 ( , ) ( , )sin ( , ) sin ( ) ( , )cos ( , ) sin ( ) ( , )cos ( ) j j j j j j j j j j j j u t z x t T z t z z y t T z t z z z t T z z φ κ φ κ κ − − − − = − − − − + −   (6.7)  As shown in Figure 6-4, the total chip thickness for flute, j , at a position along the tool axis, z , can be written as a summation of the static, dynamic and regenerative chip thickness contributions. If the dynamics of the workpiece and cutter-workpiece boundaries are included, the expression for the total dynamic chip thickness becomes, ( ) ( ) ( ) ( )0 , 1 , 1 , ,, , ( , ) ( , ) ( , ) ( , ) ( , )j j j t j j w j j t j w jh t z z h t z u t T z u t T z u t z u t zε φ − − = + − − − − −   (6.8)  where ( , )j zε φ  is the square wave function shown in Figure 6-5 and is discussed in Section 4.2.6. It changes between 1 and 0 depending upon whether cutting edge is engaged with the workpiece [96]. , ( , )t ju t z  and , ( , )w ju t z  are the present displacements of the tool and workpiece respectively in the direction of the chip thickness. , 1( , )t j ju t T z− − and , 1( , )w j ju t T z− −  are the radial displacements of the previous tooth (regeneration terms). 0 jh  is the static chip thickness due to five-axis motion, which was presented in Chapter 4 (Equation (4.25)). Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  155  Radial Cutting Force Tangential Cutting Force Axial Cutting Force YT XT φj(t,z) Static Chip  Thickness h0j(t,z) Vibration at present tooth period Vibration at previous tooth period Total chip thickness YT XT uj(t,z) uj-1(t-Tj,z) [-xj(t,z)sin(φj(t,z))sin(κj(z)) -yj(t,z)cos(φj(t,z))sin(κj(z)) +zj(t,z)cos(κj(z))] [-xj-1(t-Tj,z)sin(φj(t,z))sin(κj(z)) -yj-1(t-Tj,z)cos(φj(t,z))sin(κj(z)) +zj-1(t-Tj,z)cos(κj(z))] h(t,Tj,z) = h0j(t,z)+uj-1(t-Tj,z)-uj(t,z) φj(t,z)  Figure 6-4 – Illustration showing how the total chip thickness is a summation of the static and dynamic vibrations from the present and previous tooth periods [44].  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  156 φst(z) 1 0 φex(z) φj(t,z) ε(φj,z)  Figure 6-5 – Square wave switching function [96]. This function makes the cutting forces zero when the cutting edge is outside the engagement zone.  Inserting the expressions for , ( , )t ju t z , , ( , )w ju t z , , 1( , )t j ju t T z− −  and , 1( , )w j ju t T z− − into Equation (6.8) gives, ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) 0 , , 1 , , 1 , , 1 , , 1 , , 1 , ( , ) ( , ) ( , ) ( , ) ( , ) sin ( , ) sin ( ) , , , ( , ) ( , ) ( , ) ( , ) cos ( , ) sin ( ) ( , ) ( , ) ( , j t j t j j w j w j j j j j j j t j t j j w j w j j j j t j t j j w j h t z x t z x t T z x t z x t T z t z z h t T z z y t z y t T z y t z y t T z t z z z t z z t T z z t z φ κ ε φ φ κ − − − − −  + − − − − −   =  + − − − − −  − − − −( ) ( ), 1) ( , ) cos ( )w j j jz t T z zκ−                − −        (6.9)  which is the total chip thickness at height, z , including vibrations.  6.2.2 Dynamic Cutting Forces Using the coordinate transformation in Eq (6.1), the expression for infinitesimal x,y and z cutting forces at a height z , flute, j , in the feed coordinate system at the tool tip is, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  157 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin ( ) sin ( ) cos ( ) sin ( ) cos ( ) cos ( ) sin ( ) sin ( ) cos ( ) cos ( ) cos ( ) 0 sin ( ) j j j j j xj rj yj j j j j j tj zj ajj j z z z z zdF dF dF z z z z z dF dF dFz z φ κ φ φ κ φ κ φ φ κ κ κ   − − −          = − −             −     (6.10)  where rjdF , tjdF  and ajdF  are the tangential, radial and axial forces at the cutting edge. The linear chip thickness model for an infinitesimal piece of cutting edge segment (Chapter 4) is, ( ) ( ) ( ) , ( ) ( )( , , ) ( , , ) , ( ) ( ) ( , ) ( , , ) , ( ) ( ) rc j j re rj j tj j tc j j te j aj j ac j j ae K h z db z K dS zdF z h dF z h K h z db z K dS z z dF z h K h z db z K dS z φφ φ φ ε φ φ φ  +        = +          +      (6.11)  and,  ( )( ) sin ( )j dzdb z zκ =  (6.12)  If the element size is small, dS  becomes very close to db , even though the exact expression is given in Chapter 4 by Equation (4.33). If small cutting force elements are used, this approximation can be used to simplify Equation (6.11) to,  ( ) ( ) ( ) ,( , , ) ( , , ) , csc( ( )) ( , ) ( , , ) , rc j j re rj j j tj j j tc j j te j j aj j j ac j j ae K h z KdF z h dF z h K h z K dz z z dF z h K h z K φφ φ φ κ ε φ φ φ  +        = +          +      (6.13)  The immersion angle varies with time (Equation (6.4)) so Equation (6.13) becomes, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  158  ( , , ) ( , ) ( , , ) ( , ) csc( ( )) ( , ) ( , , ) ( , ) rj j rc j re tj j tc j te j j aj j ac j ae dF t z h K h t z K dF t z h K h t z K dz z z dF t z h K h t z K κ ε φ    +     = +       +      (6.14)  Substituting Equation (6.13) into Equation (6.10) gives, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) sin ( , ) sin ( ) cos ( , ) sin ( , ) cos ( )( , , ) ( , , ) cos ( , ) sin ( ) sin ( , ) cos ( , ) cos ( ) ( , , ) cos ( ) 0 sin ( ) , , j j j j j xj j yj j j j j j j zj j j j rc j re tc j te t z z t z t z zdF t z h dF t z h t z z t z t z z dF t z h z z K h t z K K h t z K K φ κ φ φ κ φ κ φ φ κ κ κ   − − −        = − −        −  + × + ( ) csc( ( )) ( , ) , j j ac j ae dz z z h t z K κ ε φ        +       (6.15)  Expanding this expression and putting it in a more compact form gives, ( ) ( ) ( ) ( )[ ] [ ]( ) ( ) ( )[ ] [ ]( ) ( ) ( )[ ] 11 12 13 11 12 13 21 22 23 21 22 23 31 32 33 31 32 33 , , , ( ) , , , , ( ) , , , , xj j j rc tc ac re te ae j yj j j rc tc ac re te ae j zj j j rc tc ac re te a dF t z h h t z a K a K a K a K a K a K dz z dF t z h h t z a K a K a K a K a K a K dz z dF t z h h t z a K a K a K a K a K a K ε φ ε φ   + + + + +     = + + + + +    + + + + +   [ ]( ) ( )( ) ,e jdz zε φ                 (6.16)  where, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 11 12 13 21 22 23 31 32 33 sin ( , ) cos ( , ) csc ( ) sin ( , ) cot ( ) cos ( , ) sin ( , ) csc ( ) cos ( , ) cot ( ) cot ( ) 0 1 j j j j j j j j j j j a t z a t z z a t z z a t z a t z z a t z z a z a a φ φ κ φ κ φ φ κ φ κ κ =− =− =− =− = =− = = =−    (6.17)  The chip thickness equation (Equation (6.9)) can be written more compactly as, ( ) ( )0 1 2 3, , ( , ) ( , , ) ( , , ) ( , , ) ,j j j j j j j j j jh t T z h t z b x t T z b y t T z b z t T z zε φ = + ∆ + ∆ + ∆   (6.18)  where, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  159 ( ) ( ) ( ) ( ) ( )1 2 3sin ( , ) sin ( ) cos ( , ) sin ( ) cos ( )j j j j jb t z z b t z z b zφ κ φ κ κ= = = −  (6.19)  and, ( ) ( ) ( ) ( ) ( ) ( ) , , 1 , , 1 , , 1 , , 1 , , 1 , , 1 ( , , ) ( , ) ( , ) ( , ) ( , ) ( , , ) ( , ) ( , ) ( , ) ( , ) ( , , ) ( , ) ( , ) ( , ) ( , ) j j t j t j j w j w j j j j t j t j j w j w j j j j t j t j j w j w j j x t T z x t z x t T z x t z x t T z y t T z y t z y t T z y t z y t T z z t T z z t z z t T z z t z z t T z − − − − − − ∆ = − − − − − ∆ = − − − − − ∆ = − − − − −    (6.20)  Inserting Equation (6.18) into (6.16), [ ] [ ] 0 1 2 3 11 12 13 0 1 2 3 21 22 23 0 ( , ) ( , , ) ( , , ) ( , , )( , , ) ( , , ) ( , ) ( , , ) ( , , ) ( , , ) ( , , ) ( , ) j j j j rc tc ac xj j yj j j j j j rc tc ac zj j j h t z x t T z b y t T z b z t T z b a K a K a KdF t T z dF t T z h t z x t T z b y t T z b z t T z b a K a K a K dF t T z h t z x  +∆ +∆ +∆ + +       = +∆ +∆ +∆ + +        +∆ [ ] [ ] [ ] [ ] ( ) ( ) 1 2 3 31 32 33 11 12 13 21 22 23 31 32 33 ( , , ) ( , , ) ( , , ) , j j j rc tc ac re te ae re te ae j re te ae t T z b y t T z b z t T z b a K a K a K a K a K a K a K a K a K dz z a K a K a K ε φ       +∆ +∆ + +  + +  + + +  + +       (6.21)  which can be written more simply as, { } ( , , ) ( , , ) ( , , ) ( , ) ( , ) ( , , ) ( , , ) ( , , ) xj j j j yj j j j j j zj j j j dF t T z x t T z dF t T z A t z C t z y t T z dF t T z z t T z    ∆      = + ∆        ∆       (6.22)   where { }( , )jA t z  is the vector of displacement-independent forces and ( , )jC t z    is the matrix of directional coefficients for flute, j . { }( , )jA t z  and ( , )jC t z    are given by,  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  160 { } [ ] [ ] [ ] , 0 11 12 13 11 12 13 , 0 21 22 23 21 22 23 , 0 31 32 33 31 32 33 ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) x j j rc tc ac re te ae j y j j rc tc ac re te ae z j j rc tc ac re te ae A t z h t z a K a K a K a K a K a K A t z A t z h t z a K a K a K a K a K a K A t z h t z a K a K a K a K a K a K    + + + + +    = = + + + + +       + + + + +   ( ) ( ),jdz zε φ     , , , , , , , , , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) xx j xy j xz j j yx j yy j yz j zx j zy j zz j C t z C t z C t z C t z C t z C t z C t z C t z C t z C t z       =             (6.23)  and the entries of ( , )jC t z    are given by, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , 2 2 , , 1 sin ( , ) sin ( ) sin 2( ( , )) 2( , ) ( , ) sin ( , ) cos ( ) 1 sin 2( ( , )) sin ( ) cos ( , ) 2( , ) ( , ) 1 sin 2( ( , )) cos ( ) 2 j j rc j tc xx j j j j ac j j rc j tc xy j j j j ac xz t z z K t z K C t z dz z t z z K t z z K t z K C t z dz z t z z K C φ κ φ ε φ φ κ φ κ φ ε φ φ κ   − −  =   −     − −  =     −    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , sin ( , ) cos ( ) cos ( , ) cot ( ) ( , ) ( , ) sin ( , ) cos ( ) cot ( ) 1 sin 2( ( , )) sin ( ) sin ( , ) 2( , ) ( , ) 1 sin 2( ( , )) cos ( ) 2 j j rc j j tc j j j j j ac j j rc j tc yx j j j j ac y t z z K t z z K t z dz z t z z z K t z z K t z K C t z dz z t z z K C φ κ φ κ ε φ φ κ κ φ κ φ ε φ φ κ  +  =  +    − +  =     −    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 , 2 , , 1 cos ( , ) sin ( ) sin 2( ( , )) 2( , ) ( , ) cos ( , ) cos ( ) cos ( , ) cos ( ) sin ( , ) cot ( ) ( , ) ( , ) cos ( , ) cos ( ) cot ( ) ( j j rc j tc y j j j j ac j j rc j j tc yz j j j j j ac zx j t z z K t z K t z dz z t z z K t z z K t z z K C t z dz z t z z z K C φ κ φ ε φ φ κ φ κ φ κ ε φ φ κ κ   − +  =   −     −  =  +  ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , , ) sin ( , ) cos ( ) sin ( , ) sin ( ) ( , ) ( , ) cos ( , ) cos ( ) cos ( , ) sin ( ) ( , ) ( , ) cos ( ) cot ( ) cos ( ) ( , ) j j rc j j ac j zy j j j rc j j ac j zz j j j rc j ac j t z t z z K t z z K dz z C t z t z z K t z z K dz z C t z z z K z K dz z φ κ φ κ ε φ φ κ φ κ ε φ κ κ κ ε φ  = −   = −   = − +                  (6.24)   Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  161 The total dynamic milling forces at a differential element are a summation of the cutting forces acting on each flute. In matrix form, this can be written as, { } { } [ ]{ }1 1( , , ) ( , ) ( , ) ( , , )dz j dz dz dz jF t T T z A t z C t z t T T z= + ∆… …  (6.25)   where, { } { }1( , , ) Tdz j x y zF t T T z dF dF dF=…  { } { } { }{ }1 1 1 1( , , ) j j j Tdz j N N Nt T T z x y z x y z∆ = ∆ ∆ ∆ ∆ ∆ ∆… …  { } , ,1 ,1 , ,1 , ( , ) j j j x Nx dz y y N z z N AA A t z A A A A          = + +              …  [ ] , , , ,1 ,1 ,1 ,1 ,1 ,1 , , , ,1 ,1 ,1 , , , ( , ) j j j j j j j j j xx N xy N xz Nxx xy xz dz yx yy yz yx N yy N yz N zx zy zz zx N zy N zz N C C CC C C C t z C C C C C C C C C C C C         =            …         (6.26)    { }1( , , )dz jF t T T z…  is the vector of infinitesimal cutting forces from all flutes at an axial position, z . { }1( , , )dz jt T T z∆ …  is the vector of displacements in x,y and z directions for each flute and { }( , )dzA t z  is the total vector of infinitesimal displacement-independent cutting forces. { }( , )dzF t z  and { }( , )dzA t z  have dimensions of 3x1 and { }1( , , )dz jt T T z∆ … has dimension (3 ) 1jN × , where jN  is the number of cutting flutes. [ ]( , )dzC t z  comprises the matrices for all directional coefficients of each flute and has dimension (3) (3 )jN× . Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  162 Equation (6.25) gives the equation for the dynamic forces over all teeth for an infinitesimal force element. If the total depth of cut is broken into a series of small slices, then for each slice, Equation (6.25) must be integrated over its lower and upper boundaries to obtain the expression for the total cutting forces. This is required as terms such as the immersion angle, ( , )j t zφ  (due to the lag angle of the helix), and axial immersion angle, ( )j zκ  vary with height. However, if the elements are small enough, the effects of these terms will be negligible [22], [44]. If this is the case, the integration of the entries of Equation (6.25) can be performed by multiplying all entries of { }( , )dzA t z  and [ ]( , )dzC t z  by the element height, dz . Equation (6.25) becomes, { } { } [ ]{ }1 1( , , ) ( , ) ( , ) ( , , )a j a a a jF t T T z A t z C t z t T T z= + ∆… …  (6.27)   All entries of Equation (6.26) will remain unchanged. The subscript, a , in each of the terms in (6.27) means that the vectors and matrices are identical to those in (6.26), but for an axial element, a . In 2D milling, the total forces due to the entire depth of cut would be formulated by simply setting cdz a= , where ca  is the depth of cut. However, in five-axis jet-engine impeller machining, the cutter-workpiece engagement (CWE) maps are usually non- rectangular (see Figure 6-2). In this case, the total cutting forces can be obtained by breaking the CWE map into small elements and summing the contributions from each element. This requires enlarging some of the matrices and vectors. For aN  cutting force elements, this is written as follows, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  163 { } { } [ ]{ }1 1( , ) ( ) ( ) ( , )TA j TA TA TA jF t T T A t C t t T T= + ∆… …  (6.28)   where, { } { }1( , ) TTA j x y zF t T T F F F=…  { } { } { }{ } { } (1,1) (1,1) (1,1) (1, ) (1, ) (1, ) 1 ( ,1) ( ,1) ( ,1) ( , ) ( , ) ( , ) ( , ) j j j j a a a a j a j a j T N N N TA N N N N N N N N N N x y z x y z t T T x y z x y z  ∆ ∆ ∆ ∆ ∆ ∆ ∆ =   ∆ ∆ ∆ ∆ ∆ ∆    … … … …   { } ,(1, ) ,( , ),( ,1) ,(1,1) ,(1,1) ,(1, ) ,( ,1) ,( , ) ,(1,1) ,( ,1),(1, ) ,( , ) ( ) j a ja j a a j aj a j x N x N Nx Nx TA y y N y N y N N z z Nz N z N N A AAA A t A A A A A AA A                          = + + + + + +                                    … … …                [ ] ,(1, ) ,(1, ) ,(1, ) ,(1,1) ,(1,1) ,(1,1) ,(1,1) ,(1,1) ,(1,1) ,(1, ) ,(1, ) ,(1, ) ,(1,1) ,(1,1) ,(1,1) ,(1, ) ,(1, ) ,(1, ) ( ) j j j j j j j j j xx N xy N xz N xx xy xz yx yy yz yx N yy N yz N zx zy zz zx N zy N zz N TA C C CC C C C C C C C C C C C C C C C t                 = … ,( , ) ,( , ) ,( , ),( ,1) ,( ,1) ,( ,1) ,( ,1) ,( ,1) ,( ,1) ,( , ) ,( , ) ,( , ) ,( ,1) ,( ,1) ,( ,1) ,( , ) ,( a j a j a ja a a a a a a j a j a j a a a a j xx N N xy N N xz N Nxx N xy N xz N yx N yy N yz N yx N N yy N N yz N N zx N zy N zz N zx N N zy N C C CC C C C C C C C C C C C C C                       … … , ) ,( , )a j a jN zz N NC                                                  (6.29)   { }1( , )TA jF t T T…  is the vector of total cutting forces from all flutes and elements, { }1( , )jTA Nt T T∆ …  is the vector of total displacements and { }( )TAA t  is the vector of the total displacement-independent cutting forces. { }1( , )TA jF t T T…  and { }( )TAA t  have dimensions of 3x1. { }1( , )jTA Nt T T∆ …  has dimensions of 1 (3 )a jN N× , where aN  is the Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  164 number of axial elements. [ ]( )TAC t  is the matrix of directional coefficients for all teeth and all cutting force elements. [ ]( )TAC t  has dimensions of (3) (3 )a jN N× . The vector of static cutting forces and edge forces can be dropped from the formulation as it will not contribute to the stability of the system [22], [44]. This means that the total dynamic cutting force equation for stability analysis becomes, { } [ ]{ }1 1( , ) ( ) ( , )TA j TA TA jF t T T C t t T T= ∆… …  (6.30)   6.2.3 Zero Order Expansion of Directional Coefficients From Equation (6.30) it is observed that [ ]( )TAC t , the total matrix of directional coefficients for all elements and flutes, varies with time. However, it is periodic with tooth passing frequency for constant-pitch cutters and periodic with spindle frequency for variable-pitch cutters. ( , ) ( , )a jC t z   , the directional coefficient matrix of [ ]( )TAC t  for a single flute, j , cutting force element, a , has entries ,( , ) ,( , )xx a j zz a jC C…  which are periodic harmonic functions (except ,( , )zz a jC  which is constant) multiplied by square wave functions, ( ),j zε φ . Since these terms are dependent upon time and periodic with spindle frequency, they can be expressed using Fourier series expansion. As suggested by Altintas and Budak [8], if the average or zero-order expansion -- is considered then ( , ) ( , )a jC t z   becomes, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  165 ( ) 2 1 0,( , ) , 1( ) ( , ) t a j a j t C z C t z dt T    =   ∫  (6.31)  The zero-order expansion eliminates the time dependency of the directional coefficients. It has been shown to provide accurate chatter stability predictions in nearly all cases with the exception of very low radial immersion milling [20], [36], [34], [95], where the directional coefficients are highly intermittent. This may occur in impeller machining during semi-finishing and finishing operations. In these situations, the harmonic terms of the directional coefficients and frequency response function matrices should be included in the solution. However, this increases the size of the matrices and the complexity of the problem. Since this has already been covered in previous research [22], [95] it is not repeated here. The time variable in Equation (6.31) can be changed to angular coordinates as, 1 1 , , 1 1 ( , ) ( ) ( ) j j j r p n j p n lag n n t z t z t T Tφ φ φ ψ τ − − = = = − − = Ω − Ω − Ω = Ω∑ ∑  (6.32)  where 2 60 npiΩ =  is the spindle speed in radians per second and, , , , 1 ( ) jNp j j p j lag p j lag j z T T t T T φ ψ τ = = = = − − Ω Ω ∑   (6.33)  Equation (6.31) becomes, 0,( , ) ( , ) 1( ) ( , ) 2 ex st a j a jC z C z d φ φ φ φ pi    =   ∫    (6.34) Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  166 If there are multiple engagement pairs for each tooth at the element, the integral can be broken into parts. 0,( , ) ( , ) ( , ) ( , ) , ( ),1 ,2 ,1 ,2 , ( ) 1( ) ( , ) ( , ) ...... ( , ) 2a j a j a j a j ex S zex ex st st st S z C z C z d C z d C z d φφ φ φ φ φ φ φ φ φ φ φ pi           = + + +           ∫ ∫ ∫  (6.35)  Where ,st qφ  and ,ex qφ  are the q th pair of entry and exit conditions at element, a . ( )S z  is the total number of engagement pairs at height, z . The integrated entries of 0,( , )a jC    for one engagement pair are, ( ) ( ) ( ) ( ) 2 ,( , ) ,( , ) 0, ,( , ) ,( , ) 2 ,( , ) 0, ,( , ) , ,1 1 1 sin( ( )) sin 2( ) sin 2 2 2 1 1 cos( ( )) sin 2( ) 2 2 1 1 1 sin ( )sin( ( )) sin 2( ) 2 2 2 j a rc a j tc a j xx a j j a ac a j j a rc a j tc xy a j st q ex q z K K C dz z K z K K C dz φ φ κ φ φ φ κ φ φ φ κ φ φ    − −     =    + −        − + − −    = ,( , ) 2 ,( , ) ,( , ) ,( , ) 0, ,( , ) ,( , ) , , , , 1 sin ( )cos( ( )) 2 cos( )cos( ( )) sin( )cot( ( )) cos( )cos( ( ))cot( ( )) a j j a ac a j j a rc a j j a tc a j xz a j j a j a ac a j ex q st q ex q st q z K z K z K C dz z z K φ φ φ φ φ κ φ κ φ κ φ κ κ         −   − +  =   −          (6.36) Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  167 ( ) ( ) 2 ,( , ) ,( , ) 0, ,( , ) 2 ,( , ) 2 ,( , ) ,( , ) 0, ,( , ) , , 1 1 1 sin ( )sin( ( )) sin 2( ) 2 2 2 1 sin ( )cos( ( )) 2 1 1 1 sin( ( )) sin 2( ) sin ( ) 2 2 2 j a rc a j tc a j yx a j j a ac a j j a rc a j tc a j yy a j ex q st q z K K C dz z K z K K C dz φ φ φ κ φ φ φ κ κ φ φ φ    − + − +     =   −     − − +    = + ( ) ,( , ) ,( , ) ,( , ) 0, ,( , ) ,( , ) , , , , 1 1 cos( ( )) sin 2( ) 2 2 sin( )cos( ( )) cos( )cot( ( )) sin( )cos( ( ))cot( ( )) j a ac a j j a rc a j j a tc a j yz a j j a j a ac a j ex q st q ex q st q z K z K z K C dz z z K φ φ φ φ κ φ φ φ κ φ κ φ κ κ          − −      +  =   +          (6.37)  0, ,( , ) ,( , ) ,( , ) 0, ,( , ) ,( , ) ,( , ) 0, ,( , ) , , , , cos( )cos( ( )) cos( )sin( ( )) sin( )cos( ( )) sin( )sin( ( )) cos( ( )) cot( ( ) zx a j j a rc a j j a ac a j zy a j j a rc a j j a ac a j zz a j j a j a ex q st q ex q st q C dz z K z K C dz z K z K C dz z z φ φ φ φ φ κ φ κ φ κ φ κ φ κ κ  = − +   = −  = − ,( , ) ,( , ) , , ) cos( ( )) rc a j j a ac a j ex q st q K z K φ φφ κ +      (6.38)  In Chapter 4 it was established that the cutting coefficients for 6 4Ti Al V− −  alloy are non-linear functions of chip-thickness (and helix angle). However, in the above formulations the chip thickness dependency has been neglected – the entries are now constants at an axial element, a , flute, j . If this dependency is included, it greatly increases the complexity of the formulation. However, the directional coefficient matrices do allow different coefficients to be used at each flute and each element. ( ,( , )rc a jK , ,( , )tc a jK and ,( , )ac a jK ). Since the average directional coefficients are used in the stability formulation, a reasonable approximation of the cutting force coefficients at cutting force element, a , Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  168 flute, j , can be obtained by calculating the average static cutting coefficients over an engagement pair, , , ( , ) ( ), ( )st q ex q a jz zφ φ   . This is performed with the equation, ( ) ( ) 0( ) ,( , ) ( ) ,( , ) 0( ) ,( , ) ( ) 0( ) , , , , , , ( ( ), ) 1 ( ( ), ) ( ( ), ) a a a a a a z rc j az rc a j z tc a j tc j az ex st ac a j z ac j az ex q st q ex q st q ex q st q K h z d K K K h z d K K h z d φ φ φ φ φ φ φ φ φ φφ φ φ φ             =    −            ∫ ∫ ∫     (6.39)  where rcK , tcK  and acK  are the tangential, radial and axial chip thickness-dependent, nonlinear cutting coefficients and az  is the axial position of the middle of element a . The integration in Equation (6.39) can be implemented numerically using Gaussian quadrature [106] and the cutting coefficients for the static chip thickness are given by the Orthogonal-to-Oblique Transformation for 6 4Ti Al V− −  alloy presented in Chapter 4 (Equation (4.37)). The values calculated from Equation (6.39) will give the cutting coefficents for each engagement pair (Equations (6.36)-(6.38)) of the directional coefficient matrix (Equation (6.35)). If the tooth is not cutting due to the unequal flute lengths (serrations), this can be modeled by making the cutting coefficients zero. The total dynamic cutting force equation (Equation (6.30)) with the zero-order approximation for the directional coefficients becomes, { } [ ]{ }1 0 1( , ) ( , )j jTA N T TA NF t T T C t T T= ∆… …  (6.40)  where [ ]0TC  is the matrix of all directional coefficients with the zero order expansion. [ ]0TC  has the same form as [ ]( )TC t  in Equation (6.29). Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  169  6.2.4 Dynamic Displacements of the Tool and Workpiece The dynamic displacements (Equation (6.20)) at element, a , flute, j , can be written, ( ) ( ) ( ) ( ) ( ) ( , ) ,( , ) ,( , 1) ,( , ) ,( , 1) ( , ) ,( , ) ,( , 1) ,( , ) ,( , 1) ( , ) ,( , ) ,( , 1) ,( , ) ,( , ( , ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) ( ) ( , ) ( ) ( ) ( ) a j j t a j t a j j w a j w a j j a j j t a j t a j j w a j w a j j a j j t a j t a j j w a j w a x t T x t x t T x x t T y t T y t y t T y t y t T z t T z t z t T z t z − − − − − ∆ = − − − − − ∆ = − − − − − ∆ = − − − −( )1) ( )j jt T− −    (6.41)  Describing the time varying displacements of Equation (6.41) in the Laplace domain using frequency response functions for the tool and workpiece, { } ( ) ( ) ( ) { } { } ( ) ( ) ( ) { } , , ,( , ) ,( , ) ( , ), , , , , , ,( , ) ,( , ) ( , ), , , , ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t a j t a j t a j a jt a j t a j w a j w a j w a j a jw a j w a j x s r s y s s F s z s x s r s y s s F s z s       = = Φ               = = − Φ              (6.42)  where ,( , ) ( )t a j s Φ   and ,( , ) ( )w a j s Φ   are the 3x3 frequency response functions of the tool and workpiece. The negative sign for { },( , ) ( )w a jr s  in Equation (6.42) is due to the fact that milling forces applied on the cutter and the workpiece are in opposite directions. ,( , ) ( )t a j s Φ   and ,( , ) ( )w a j s Φ   are given by, Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  170 , ,( , ) , ,( , ) , ,( , ) ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) t xx a j t xy a j t xz a j t a j t yx a j t yy a j t yz a j t zx a j t zy a j t zz a j w xx a j w xy a j w x w a j s s s s s s s s s s s s s  Φ Φ Φ    Φ = Φ Φ Φ    Φ Φ Φ  Φ Φ Φ  Φ =  ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) , ,( , ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) z a j w yx a j w yy a j w yz a j w zx a j w zy a j w zz a j s s s s s s s    Φ Φ Φ   Φ Φ Φ       (6.43)  and, ( ) ( ) ( )( ) ( ) m 2 ,( , , , ) ,( , , , ) , ,( , ) 221 ,( , , , ) ,( , , , ) ,( , , , ) ( )( ) ( ) 2 N nt pq a j m t pq a j mt t pq a j m t pq a j m nt pq a j m nt pq a j m kX s s F s s s ω ζ ω ω=     Φ = =   + +  ∑  ( ) ( ) ( )( ) ( ) m 2 ,( , , , ) ,( , , , ) , ,( , ) 221 ,( , , , ) ,( , , , ) ,( , , , ) ( )( ) ( ) 2 N nw pq a j m w pq a j mw w pq a j m w pq a j m nw pq a j m nw pq a j m kX s s F s s s ω ζ ω ω=     Φ = =   + +  ∑     (6.44)  ,( , , , )nt pq a j mω , ,( , , , )t pq a j mk , ,( , , , )t pq a j mζ  are the natural frequency, stiffness and damping ratio for mode, m , element, a , flute, j , displacement direction p , force direction, q , for the tool. p  and q  can be in directions x,y or z. The same nomenclature applies to the modal parameters of the workpiece except that the subscript, t , becomes a w . Equation (6.44) implies that the displacement due to a unit force is simply a summation of all contributions from each mode of the structure. In the Laplace domain, the equations for the displacements of the tool from the previous tooth period (for element a , flute j ) are, { } { } { } { } ,( , 1) ,( , ) ,( , 1) ,( , )( ) ( ) ( ) ( )j jsT sTt a j t a j w a j w a jr s e r s r s e r s− −− −= =  (6.45)  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  171 where jsTe−  in Equation (6.45) is the delay term. This term can cause the system to become unstable. Without it, the roots of the characteristic equation will always have negative real parts, which means that the system will always be stable [7]. jT  is the time period that has passed between the present and previous flute. For constant pitch cutters this is, 60 j j T N n =  (6.46)  and for variable-pitch cutters, jT  is given by, n T jpj 6 (deg),φ =  (6.47)  jp(deg),φ  is the pitch angle in degrees for tooth, j . It is important to mention that when cutters with unequal flute lengths at their cross-sections are used (serrated-cutters), it is possible that some of the flutes at an element may not cut. This is illustrated in Figure 6-6. Assuming small vibrations, if one of the flutes, j , at an element, a , is small enough in relation to the feed-per-tooth, it will not cut. The cutting coefficients in the directional coefficient matrix for this flute, 0,( , )a jC   , will be set to zero. Additionally, the delay term between 1j −  and 1j +  will increase (Figure 6-6) and at the cross-section, the tool will behave like a cutter with 1jN −  teeth. This illustrates how the stability of cutters with unequal flute lengths at the cross section (i.e. serrated cutters) can be dependent Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  172 Flute j is too small to cut j j-1 j+1 Tj Tj+1 Flute j+1 cuts cj Ω Flute j-1 cuts Since flute j is too small to cut, delay between flutes in cut ( j+1 and j-1) = Workpiece Tj + Tj+1 Tj+1 = φp,j+1 Ω Tj = φp,j Ω  Figure 6-6 – Illustration showing a loss of contact when relative flute lengths are small in relation to the feed-per-tooth, jc . This increases the delay term between flutes in cut.  upon feed rate. Lower feed rates may decrease the number of teeth in cut and increase the delay period, while higher feed rates may increase the number of teeth in cut and decrease the delay period. To approximate the true delay terms between teeth on a serrated cutter, the average static chip thicknesses of all teeth at the cross section is taken before forming the delay terms. If the chip thickness term for a flute is zero, then the delay term between j  and 1j +  must be added to j  and 1j −  for flute 1j − . This has been implemented in the proposed stability algorithm. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  173 In the Laplace domain, the displacements of Equation (6.41) can be written as ( , ) ( )a j s∆ . Substituting Equations (6.42) and (6.45) into the terms of { }, ( )a j s∆  gives, { } { } { } { } { } ( ){ } ( , ) ,( , ) ,( , 1) ,( , ) ,( , 1) ,( , ) ,( , ) ( , ) ( ) ( ) ( ) ( ) ( ) (1 ) ( ) ( ) ( )j a j t a j t a j w a j w a j sT t a j w a j a j s r s r s r s r s e s s F s − − −    ∆ = − − −       = − Φ + Φ       (6.48)  Generating an expression of Equation (6.48) for each flute and each element, then substituting into Equation (6.40) gives, { } [ ][ ]{ }0( ) ( ) ( )TA T TA TAF s C s F s= Ψ  (6.49)  where,  [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 1 , 1,1 , 1,1 , 1,1 , 1,1 , 1,1 , 1,1 , 1,1 , 1,1 , 1,1 , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, , 1, 1 ( ) 1 j j j Nj j j j j j j xx xy xz sT yx yy yz zx zy zz TA xx N xy N xz N sT yx N yy N yz N zx N zy N zz N e s e     Φ Φ Φ   − Φ Φ Φ    Φ Φ Φ    Ψ =    Φ Φ Φ      − Φ Φ Φ     Φ Φ Φ      ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) 1 , ,1 , ,1 , ,1 , ,1 , ,1 , ,1 , ,1 , ,1 , ,1 , , , , , , , , , , , , , , , , , , 1 1 a a a a a a a a a a j a j a j Nj a j a j a j a j a j a j xx N xy N xz N sT yx N yy N yz N zx N zy N zz N xx N N xy N N xz N N sT yx N N yy N N yz N N zx N N zy N N zz N N e e  Φ Φ Φ    − Φ Φ Φ   Φ Φ Φ        Φ Φ Φ    − Φ Φ Φ   Φ Φ Φ                                         (6.50 )  [ ]( )TA sΨ  is the frequency response function matrix of the system. [ ]( )TA sΨ consists of the delay terms for each flute and frequency response function terms for all elements and flutes. ( ) ,( , ) , ,( , ) , ,( , )pq a j t pq a j w pq a jΦ = Φ + Φ  where , ,( , )t pq a jΦ  and , ,( , )w pq a jΦ  are Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  174 the transfer functions of the tool and workpiece for displacement direction, p , force direction, q , element, a , flute, j . [ ]( )TA sΨ  has dimensions of ( ) ( )3 3a jN N × .  6.3 Solution of Chatter Stability Formulation Using Nyquist Theory Even though Equation (6.49) appears to be an eigenvalue problem, the varying delay terms in the [ ]( )TA sΨ  matrix cause the system to have an infinite number of eigenvalues. This makes assessment of the system’s stability difficult. However, the Nyquist Stability Criterion [102] can be used to determine the stability of the system as demonstrated below.  Equation (6.49) can be written in the following format, { } [ ]{ }( ) ( ) ( )TA TAF s s F s= Λ  (6.51)  where, [ ] [ ][ ]0( ) ( )T TAs C sΛ = Ψ  (6.52)  To check the stability of the system, a vector of external disturbance forces, { }( )dF s , can be added to the right hand side of Equation (6.51). { } [ ]{ } { }( ) ( ) ( ) ( )TA TA dF s s F s F s= Λ +  (6.53)  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  175 { }( )dF s  can be thought of as a vector of external, unmodeled forces that may be introduced into the system. The stability of the system can be defined by its response to these applied forces. Solving (6.53) for { }( )TAF s : { } [ ] [ ]( ) ( ){ }1( ) ( )TA dF s I s F s−= − Λ  (6.54)  Where [ ]I  is the 3x3 identity matrix. From Equation (6.54) it can be seen that [ ] [ ]( ) 1( )I s −− Λ  governs whether the cutting forces will grow or decay exponentially from an applied external force. The inverse operation on [ ] [ ]( )I s− Λ  can be expressed analytically [42], [ ] [ ]( ) [ ] [ ]( ) [ ] [ ]( ) 1 1( ) ( ) det ( )I s adj I sI s − − Λ = − Λ − Λ   (6.55)  where [ ] [ ]( )det ( )I s− Λ  is the determinant of [ ] [ ]( )I s− Λ  and [ ] [ ]( )( )adj I s− Λ  is the adjugate matrix, or classical adjoint matrix, of [ ] [ ]( )I s− Λ . The adjugate matrix can be defined for any square matrix without the need to perform divisions [42]. This fact means that the only divisions that result the inverse operation of [ ]( ) 1( )I s −− Λ  are due to the [ ] [ ]( ) 1 det ( )I s− Λ  term. The poles of the total dynamic force equation, [ ]( )sΛ , come solely from the transfer function matrices for each element and flute, ( , ) ( )a j s Φ   and are second order and stable. This means that the source of possible instability will be additional poles Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  176 introduced by the [ ] [ ]( ) 1 det ( )I s− Λ  term. An infinite number of poles are introduced to the system by the delay terms, jsTe , in the [ ]( )T sΨ  matrix. Expanding sTe−  into its Taylor series, 2 2 3 3 0 ( 1) ( 1)1 .... 2! 3! ! ! n n n n n n sT s T s T s T s Te sT n n ∞ − − − = − + − + + =∑  (6.56)  it is evident that the exponential term is in fact, an infinite summation polynomial. The division of these terms in [ ] [ ]( ) 1 det ( )I s− Λ  will create an infinite number of poles, some or which may or may not have positive real parts. Poles which have positive real parts will cause the system to have an unstable response (chatter). The determinant of [ ] [ ]( )I s− Λ can be written as, [ ] [ ]( ) 1 2 1 2 ( )( )....( )( )det ( ) ( ) ( )( ).....( ) z p N N s b s b s bZ sI s P s s a s a s a + + + − Λ = = + + +   (6.57)  where 1... pNa a  are the poles, and 1... zNb b  are the zeros, of the determinant. pN  is the total number of poles and zN  is the total number of zeros.  Since [ ] [ ]( ) 1 det ( )I s− Λ  is the characteristic equation of the system’s response, the zeroes of [ ] [ ]( )det ( )I s− Λ  govern the stability of the system. The zeros of [ ] [ ]( )det ( )I s− Λ  can be checked using the Nyquist Stability Criterion [102]. If s jω=  (the imaginary axis – the border of stability) is substituted into a Laplace Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  177 transfer function, and the real and imaginary parts of the function are plotted over the interval, [ ],jω = −∞ ∞ , a series of circles will be observed. The plot of [ ]0,jω = ∞  will make clockwise circles with increasing frequency, while the plot over the frequency range [ ]0,jω = −∞  will make counterclockwise circles and with decreasing frequency. The plot of [ ]0,jω = −∞  will be the mirror image of the curve formed by [ ]0,jω = ∞ , reflected across the real axis. To reduce redundancy, usually only the positive range is plotted. The plots themselves are called Nyquist plots, some examples of which are shown in Figure 6-7. -1 -2 -1 1 Real Im ag . 1 2 3-1 -2 -1 1 Real Im ag . 1 2 3 Stable Unstable s=jω=[0,   ]8 s=jω=[0,−   ]8 s=jω=[0,−   ]8 s=jω=[0,   ]8  Figure 6-7 – Illustration showing typical Nyquist plots. The Nyquist curve in the stable case does not encircle the origin, whereas in the unstable case it does.  The Nyquist Stability Criterion, which employs Cauchy’s Argument Principle [86], states that over the range of [ ]0,jω = ∞ , the number of clockwise encirclements of the origin must be the number of zeros of [ ] [ ]( )det ( )I s− Λ  in the right-half complex plane Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  178 minus the poles of [ ] [ ]( )det ( )I s− Λ  in the right-half complex plane. This can be formally stated as, , ,cwe z rhcp p rhcpN N N= −  (6.58)  where cweN  is the number of clockwise encirclements of the origin and ,z rhcpN  is the number of zeroes in the right-half of the complex plane. ,p rhcpN  is the number of poles in the right-half of the complex plane. The poles of [ ] [ ]( )det ( )I s− Λ  are all stable, which means that , 0p rhcpN = . Equation (6.58) becomes, ,cwe z rhcpN N=  (6.59)  which means that the number of unstable zeroes will equal the number of clockwise encirclements of the origin. An example is shown in Figure 6-7. Interpolating the value of the plot when it crosses the real axis, going from a negative to a positive imaginary part, will tell whether the system is stable ( Re 0> ), critically stable (Re 0)= , or unstable (Re 0)< . Linear interpolation can be used if the frequency interval is small enough. Obviously, scanning from [ ]0,∞  – an infinite frequency range – is impossible. However, since it is known that chatter always occurs near a mode of the system (usually the most flexible ones) and that the transfer function values for the tool and workpiece become small when far away from the modes of the system, a frequency range of ,max0, 2 nω   , where ,maxnω  is the highest flexible mode of the system, should be sufficient to identify all unstable zeros of the system. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  179 To calculate the stability of the entire system, the following procedure is employed: 1. Identify the most flexible modes of the system. 2. Choose the spindle speed of the operation. 3. Calculate the cutter-workpiece engagement map for the tool move. 4. Substitute s jω=  into [ ] [ ]( )det ( )I s− Λ  and plot [ ] [ ]( )det ( )I jω− Λ  over the range of ,max0 2 nω ω≤ ≤  in small frequency intervals. 5. Check whether the plot encircles the origin. If it does, the system is unstable, otherwise it is stable.  Currently, checking whether the system is unstable or not, is all that has been implemented in the current thesis project. Even still, this maybe useful to process engineers as it could allow them to identify parts of the tool path that could be unstable. Unfortunately, there is no simple expression for the critical depth of cut, unlike the 2D chatter stability theory of Altintas and Budak [8], because the depth of cut can vary with immersion angle. Also, the engagement conditions may change over the axis of the tool. As a consequence, the critical depth of cut can not be factored out of the eigenvalue problem as a constant value as given in Altintas and Budak [8]. As shown in Figure 6-8, due to the complex boundaries of the workpiece and the fact that the tool axis is usually not 90 degrees to the surface of the part, increasing the depth of cut may cause the cutter- workpiece engagement maps to change in an unpredictable manner. To develop a stability chart for five-axis machining in the future, the following scheme is proposed for future work, 1. Identify the most flexible modes of the system. 2. Choose a lower bound on spindle speed for the stability chart. 3. Calculate the cutter-workpiece engagement map for the tool move. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  180 4. Substitute s jω=  into [ ] [ ]( )det ( )I s− Λ  and plot [ ] [ ]( )det ( )I jω− Λ  over the range of ,max0 2 nω ω≤ ≤  in small frequency intervals. 6. Check whether the plot encircles the origin. If it does, the system is unstable, otherwise it is stable. 5. If the system is stable, increase the depth of cut along the tool axis and go to step 3. If the system is unstable, decrease the depth of cut and go to step 3. Repeat steps 3-5 until a given tolerance on the depth of cut is reached. 6. Increment the spindle speed and repeat steps 3-6. Stop when the upper bound of the spindle speed is reached.  Original Position Cutter-Workpiece Engagement Map Tool Tip Offset 2.54 mm (0.1 in) Along Tool Axis Cutter-Workpiece Engagement Map Tool Tip Offset 1.27 mm (0.05 in) Along Tool Axis Cutter-Workpiece Engagement Map Tool Tip Offset 0.254 mm (0.01 in) Along Tool Axis Cutter-Workpiece Engagement Map Tool Tip Offset -2.54 mm (-0.1 in) Along Tool Axis Cutter-Workpiece Engagement Map No Offset Along Tool Axis (Original Tool Tip Position) Immersion Angle Immersion AngleImmersion Angle Immersion AngleImmersion Angle D is ta n ce  A lo n g D is ta n ce  A lo n g To o l A x is To o l A x is D is ta n ce  A lo n g To o l A x is D is ta n ce  A lo n g To o l A x is Tool Axis  Figure 6-8 – Increasing the depth of cut in five-axis impeller machining can cause unpredictable changes in the cutter-workpiece engagements maps.  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  181 In practice, implementation of this scheme would require substantial programming effort, computational power and time – at least with current computational resources. Also, it is possible that increasing or decreasing the depth of cut along the tool axis could change the entire shape of the impeller blade, which could have disastrous effects on the operating efficiency of the part. In this case, the entire tool path might need to be recalculated and so the search algorithm would need to be coupled to a five-axis trajectory generation module. Because of these complexities, implementation of stability lobe calculation for five-axis machining is left as an area of future research.  6.4 Transfer Functions of the Tool and Blade In Section 6.5.3, the Nyquist Stability Criterion [102] is used to calculate the stability of a five-axis impeller finishing operation. In this case, the blade is thin and flexible and has frequency response functions (FRFs) that vary over its surface. The following section discusses how approximations of these FRFs can be obtained through finite element analysis. It also examines the necessary coordinate transformations required to project the FRFs of the tool and workpiece in the directions required for stability analysis.  6.4.1 Transfer Functions of Blade  In the five-axis flank milling jet-engine impellers, the dynamics of the workpiece change the most during roughing operations. However it is at the start of these operations Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  182 where the workpiece is stiffest, as the bulk of the material to be removed is still attached to the workpiece. Also, since the surface finish of the part is not important during these operations, serrated, variable-pitch cutters can be used to make the tool and workpiece resistant to chatter. During these operations chatter generally occurs due to flexibility of the spindle or machine tool structure. During semi-finishing and finishing operations, chatter can become an issue for several reasons. First is that the surface finish of the part is important and so serrated cutters can not be used. Also, the thickness of the chips removed during these operations is small, which leads to an increase in the cutting coefficients of the 6 4Ti Al V− −  alloy – See Figure 6-9, further reducing stability. 0.02 0.04 0.06 0.08 0.100 Chip Thickness (mm) C u tt in g  C o e ff ic e n ts  ( N /m m 2 ) 2000 4000 6000 Cutting Coefficents vs. Chip Thickness Ti-6Al-4V Alloy - 5 deg rake, 15 deg helix Ktc Krc Kac  Figure 6-9 – General trend of the orthogonal-to-oblique cutting coefficients with chip thickness for 6 4Ti Al V− − alloy. Cutting coefficients rise sharply at small chip thicknesses.  Finally, the thin-walled blades of the impeller have lowly-damped ( 0.01ζ < ), highly-flexible modes of vibration. During these operations, the dynamics of the Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  183 workpiece are critical to the stability of the process and cannot be neglected from chatter prediction calculations. A CAD model of an integrally bladed rotor (IBR) is shown in Figure 6-10.  Figure 6-10 – CAD model of an integrally bladed rotor (IBR)  From Figure 6-10, it is observed that the blades are tightly packed together, which is typical of these parts. Unfortunately, the tight spacing and thin-walled flexible structure makes obtaining transfer function measurements by impact testing a very difficult chore. The overlap of the blades blocks hammer access to all points except the outside edge, which is troublesome as the transfer functions of the structure vary continuously over its surface. The flexibility of the thin webs makes it very difficult to impart a half-sine wave impact to the structure, which can cause the frequency response function measurements to become distorted. Finally, obtaining the direct and cross transfer function measurements by experiments in all three directions is time consuming and impractical. To approximate the direct ( ), ,xx yy zzΦ Φ Φ  and cross ( ),, , , ,xy yx xz zx yz zyΦ Φ Φ Φ Φ Φ transfer functions of the impeller blade in all three directions, a finite-element approach Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  184 can be used with damping values borrowed from experiments. The transfer functions are calculated in the workpiece coordinate system and then transformed to tool coordinate system using similarity transforms. A CAD model of an integrally bladed rotor (IBR) was obtained from the research partner in the project -- Pratt & Whitney Canada. This was a model of the finished workpiece, as the experimental transfer function measurements were performed on the production part. Also, the initial workpiece for the finishing pass demonstrated in Section 6.5.3 is very close in structure to the finished part. One sector of the 19-bladed IBR was removed from CAD model of the finished part for FRF measurements. This was done in order to decrease the complexity of the meshing procedure and to increase the computational efficiency. The sector contained several small geometric features, as well as parts of overlapping blades, which were removed to simplify the meshing of the structure. This left the main part of the blade and a small part of the attachment to the central hub, which is shown in Figure 6-11. As seen in Figure 6-11, even after the simplifications, the blade is still a complex object having a thin, twisted surface that varies in width and thickness along its length. The structure was imported into ANSYS v10 and meshed with tetrahedrons using SOLID92 elements. These elements have quadratic displacement behavior and are well suited to modeling irregular meshes (such as those produced from various CAD/CAM systems) [14]. Displacements on the side surfaces of the hub were constrained on lines of symmetry. Impact analysis was performed by engineers at Pratt & Whitney Canada on the finished part to compare the frequencies of the meshed structure with the physical structure. Four modes were identified as being significant in the physical part. Table 6-1 Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  185 compares the experimentally measured natural frequencies with the finite element results. Table 6-1 shows some differences in the calculated and experimentally measured natural frequencies. These are most likely due the fact that simplifications were made to the structure of the model and that the impact testing was performed when the impeller was on a large cart, which may have added mass to the structure.  One Sector Removed Simplified Blade Full CAD model of IBR Finite Element Mesh  Figure 6-11 – Illustration of the finished integrally bladed rotor (IBR) used for test purposes. One sector of the IBR is removed, simplified and then meshed for frequency response function (FRF) analysis.   Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  186 Table 6-1 – Comparison of Natural Frequencies of the IBR From FE Analysis and Experiments Mode Number of IBR Natural Frequency – Impact Testing (Hz) Natural Frequency - ANSYS (Hz) Percent Difference (%) 1 165.80 195.44 17.9% 2 548.87 688.02 25.4% 3 933.36 995.82 6.7% 4 1248.38 1642.57 31.2%  As the finishing pass presented in Section 6.5.3 cuts the back surface of the blade only, 256 nodes of the back face were selected for FRF calculation as shown in Figure 6-12. These 256 nodes were selected by dividing the area into a 16 x 16 grid and then choosing nodes on the structure that were closest to these points. Harmonic analysis in ANSYS (batch mode) was used to apply a unit force in all three directions and record the frequency domain displacement responses (receptance) in all three directions (see Figure 6-12). Modal damping values were imposed on the structure using the DAMPRAT command in ANSYS. The experimentally measured damping ratios are shown in Table 6-2 and were obtained using linear least-squares curve fitting [29] of the physical data. From the table it can be seen that all modes have very low damping – well under 1%. The modal stiffness of the blade ( ,( , , , )w pq a j mk  in Equation (6.44)) , for each mode at each node, in all cross and direct transfer functions was obtained using linear least squares fitting [29] of the ANSYS data. The natural frequencies and damping values for each mode are the same at each point of the structure. All modal values for each node are stored in a text file and retrieved at the beginning of the chatter stability prediction  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  187 Y Z X z(ω) y(ω) x(ω) Fx(ω) z(ω) y(ω) Fy(ω)x(ω) Fz(ω) y(ω) x(ω) z(ω)  Figure 6-12 – (top) 256 nodes of the finite element blade structure were selected for FRF measurements by dividing the back area of the blade into a 16x16 grid. (bottom) The 3x3 FRF matrix at each node is obtained by measuring the displacements in x,y and z directions from an applied unit force.  Table 6-2 – Experimentally Measured Damping Ratios for Modes 1-4 of the IBR Mode Number of IBR Damping Ratio % 1 0.6740 2 0.4118 3 0.1895 4 0.2016  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  188 algorithm. Since the direct and cross transfer functions are referenced to coordinate system of the workpiece, they need to be transformed to the tool coordinate system for chatter stability predictions. The equation for a general FRF matrix in the workpiece coordinate system is, , , , , , , , , , w w xx w xy w xz xw w w yx w yy w yz yw w w zx w zy w zz zw x F y F z F  Φ Φ Φ         = Φ Φ Φ         Φ Φ Φ        (6.60)  The frequency response functions can be transformed from one coordinate system to the other by a similarity transform. For example, to transform from the workpiece coordinate system to a fixed frame of reference (the global system), the following relations can be used, [ ] [ ] w g xw xg T T w GW g yw GW yg w g zw zg x x F F y A y F A F z z F F               = =                          (6.61)  where, [ ]GWA  is the 3x3 rotation matrix that gives the unit vectors of the workpiece coordinate system in the global system. Inserting Equations (6.61) into (6.60) and multiplying the left side of both terms in Equation (6.60) by [ ]GWA , [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] xx xy xzw wwg xg T g GW yx yy yz GW ygw w w g zg zx zy zzw ww x F y A A F z F  Φ Φ Φ         = Φ Φ Φ             Φ Φ Φ     (6.62)  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  189 which transforms the directions of the frequency response functions from the workpiece coordinate system to the global coordinate system. To transform the workpiece FRFs to the tool coordinate system, another similarity transform can be used. [ ]GTA , the matrix of vectors of the tool coordinate system in the global coordinate system, can be used to relate the displacements and forces in the tool coordinate system to the global coordinate system. [ ] [ ] g t xg xt g GT t yg GT yt g t zg zt x x F F y A y F A F z z F F               = =                          (6.63)  Inserting these into Equation (6.62) and multiplying the left side of both terms by [ ]TGTA  gives, [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] xx xy xzw wwt xt T T t GT GW yx yy yz GW GT ytw w w t zt zx zy zzw ww x F y A A A A F z F  Φ Φ Φ          = Φ Φ Φ             Φ Φ Φ     (6.64)  which projects the FRFs in the workpiece coordinate system to the tool coordinate system. In the current research project, the workpiece and the global coordinate systems were identical [ ] [ ]GWA I= . Also, for a tool move,[ ]GTA  is formed by the vectors of the tool coordinate system. In other words, [ ] { } { } { }GT TT TT TTA X Y Z=     (6.65)  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  190 Note that [ ]GTA  varies continuously along the tool path. It can be calculated at each tool move using the vectors given by Equation (4.8) in Chapter 4.  6.4.2 Transfer Function of Tool In five-axis machining, the dynamics of the machine can change continuously throughout the machining operation. Although this can occur in three axis machining, it becomes more apparent due to the rotational degrees of freedom of tool and workpiece. However, the extent of the change depends upon the configuration of the machine. At Pratt & Whitney Canada, the machines used to flank mill impellers have two rotary axes on the table and three translational axes on the spindle. The curved surfaces of the blades are cut by moving the rotational and translational axes simultaneously. For maximum accuracy in chatter stability predictions, experimental modal testing could be performed at several positions and orientations. This information can be stored in a file and then retrieved when the simulated tool is at the correct position / orientation. Another method of predicting the change in dynamics of the machine would be to construct a full finite element model and obtain FRFs of the structure at each point along the tool path. However, both of these schemes could be extremely time consuming and unnecessary, particularly when the change in configuration of the machine during the flank milling is small. The translational joints on the machine used to cut the impellers at Pratt & Whitney Canada are stiff and have a relatively short travel distance. Also, the table is almost symmetrical. In this case, impact testing can be performed in the “home position” Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  191 and then transformed to the orientation of the tool axis. Again, this is accomplished using a double similarity transform, [ ] [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] xx xy xzh hht xt T T t GT GH yx yy yz GH GT yth h h t zt zx zy zzh hh x F y A A A A F z F  Φ Φ Φ          = Φ Φ Φ             Φ Φ Φ     (6.66)  where [ ]GHA  is the rotation matrix between the global, or fixed, coordinate system and the coordinate system of the hammer test. [ ]GTA  is the transformation between the global coordinate system and the tool coordinate system.  6.5 Implementation of the Five-Axis Chatter Stability Theory The five-axis chatter stability theory presented in this chapter is implemented in 2D and five-axis cases. The 2D case is used to verify that the Nyquist stability solution agrees with stability predictions from a commercial process simulation software package – CUTPRO [89].  Currently there appears to be no methods of calculating the stability of five-axis machining operations in the available literature and no commercial programs that can be used to test the theory. Furthermore, no five-axis machine tools at Pratt & Whitney Canada were available for chatter stability tests as this would require stoppage of the production part, which is costly for the company. The stability predictions for the five-axis machining case are presented as an example only.  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  192 6.5.1 Comparison of New Stability Theory with Classical Stability Lobes for a 2D Cylindrical End Mill  The stability chart for a four-fluted cylindrical cutter is shown in Figure 6-13. This chart is based on Altintas and Budak’s chatter stability theory [8] with the zero order expanded directional coefficients. The chart was generated using CUTPRO [89] – a process simulation package developed by our laboratory and used by large manufacturing companies such as Boeing, Caterpillar and General Motors [90]. The cutting conditions for the test are given in Table 6-3.  Table 6-3 – Summary of Cutting Conditions Used to Compare New Stability Algorithm Against CUTPRO (Analytical Zero-Order Solution) Cutter Cylindrical End Mill Diameter 19.05 mm Number Of Flutes 4 Pitch Angle of Flutes 90 deg Engagement Conditions stφ  = 0 deg        exφ  = 180 deg Number of Modes 2 orthogonal – 1 mode in X, 1 mode in Y  Natural Frequency n ω  (Hz) Stiffness k  (N / m) Damping Ratio ς Mode XX 560 1.0e8 0.05 Mode YY 800 3.03e8 0.05   Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  193 The five-axis chatter stability algorithm was used to generate Nyquist plots at various depths of cut for stable, unstable and critically stable cutting conditions (see Figure 6-13). As observed in Figure 6-13, the stability predictions from this algorithm are in exact agreement with those of CUTPRO. In the stable case, there is no encirclement of the origin. In the critically stable case, one circle of the Nyquist plot passes exactly through the origin. In the unstable case, there are two encirclements of the origin, which means the determinant has at least two zeroes with positive real parts.  4500 RPM DOC = 55.0 mm (unstable) DOC = 55.0 mm Origin Start Freq = 0 Hz End Freq = 1600.00 Hz Real Part Im a g in a ry  P a rt Nyquist Plot of Complex Determinant Function DOC = 45.24 mm CRITICALLY  STABLE Origin Start Freq = 0 Hz End Freq = 1600.00 Hz Real Part Im a g in a ry  P a rt Nyquist Plot of Complex Determinant Function UNSTABLE CASE Nyquist Plot of Complex Determinant Function DOC = 35.00 mm STABLE CASE Real Part Im a g in a ry  P a rt Start Freq = 0 Hz End Freq = 1600.00 Hz Chatter Freq = 567.29 Hz DOC = 45.24 mm (critically stable) DOC =35.00 mm (critically stable) 1152.8 2254.1 3355.4 4456.7 5558.0 6659.4 60.12 48.10 36.07 24.05 12.02 0.00 -12.02 D e p th  o f C u t (m m ) Spindle Speed (RPM) Stability Lobes (Analytical) Origin 0.0 0.0 1.0 2.0-1.0 -1.0 1.0 0.0 0.0 -1.0 1.0 1.0 2.0-1.0 0.0-1.0 1.0 2.0 0.0 -1.0 1.0  Figure 6-13 – Comparison of stable, unstable and critically stable cutting conditions given by CUTPRO and the five-axis chatter stability solution. Parameters for the cutting conditions are given in Table 6-3. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  194 The Nyquist plots also appear to give the user a measure of the relative stability of the process. This is realized by taking the minimum real value of all real-axis crossing points where the determinant’s imaginary part changes from negative to positive. The minimum real value of all crossings will indicate how stable the process is. The more negative the real part, the more unstable the process. The more positive the minimum real, part the more stable the process. When the curve crosses the real axis exactly at the origin, the process is critically stable. In this case, the frequency at which the Nyquist curve passes through the zero is the chatter frequency. The crossing frequency in Figure 6-13 is 567.288Hz, which exactly matches the chatter frequency given by CUTPRO. To illustrate the stability gains from variable-pitch flutes, each flute is given the spacing shown in Table 6-4. Figure 6-14 shows the revised stability charts. It can be seen from the new Nyquist plots that the unequal pitch spacing has increased the stability of the process. The minimum real part of all real axis crossings is now to the right of origin. Seen in this way, Nyquist plots can be a useful tool to the user when analyzing the relative changes in stability due to changes in tool geometry, cutting conditions and material properties.  Table 6-4 – Variable-Pitch Flute Spacing of Cylindrical Cutter Given in Table 6-3 Flutes Pitch (deg) 1-2 82.5 2-3 87.5 3-4 92.5 4-1 97.5  Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  195 DOC = 55.0 mm Origin Start Freq = 0 Hz End Freq = 1600.00 Hz Real Part Im a g in a ry  P a rt Nyquist Plot of Complex Determinant Function UNSTABLE DOC = 55.0 mm Nyquist Plot of Complex Determinant Function STABLE CONSTANT PITCH VARIABLE PITCH Real Part Im a g in a ry  P a rt 0.0 1.0 2.0-1.0 0.0 -1.0 1.0 1.0 0.0 -1.0 0.0 1.0 2.0-1.0 -2.0 Origin Start Freq = 0 Hz End Freq = 1600.00 Hz Spindle Speed = 4500 RPM Spindle Speed = 4500 RPM  Figure 6-14 – Nyquist plots showing increased stability from use of a variable-pitch cutter. With the variable-pitch tool, there are no encirclements of the zero at the same depth of cut.   6.5.2 Comparison of New Stability Theory with Time Domain Process Simulation (CUTPRO) for a Serrated, Variable-Pitch, Cylindrical-End Mill in 2D Milling To further verify the stability algorithm developed in this chapter, results from the new algorithm are compared against those given by a time domain process simulation in CUTPRO [89] for a serrated, variable-pitch, cylindrical-end mill. Since a frequency domain chatter stability solution for these types of cutters is not available in literature, CUTPRO determines the stability of complex tools by numerically simulating the dynamic chip thickness and cutting forces in the time domain. The dynamic equations of motion are integrated through time for specified number of rotations of the tool using a fourth-order Runge-Kutta method (RK4) [106] and a 2D grid is used to store the surface Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  196 of the workpiece during the simulation. If the peak-to-peak cutting forces have grown by a certain threshold (usually 20%) over the course of the simulation, the process is deemed to be unstable. A five-fluted serrated, variable-pitch, helical, cylindrical-end mill in a 2D slotting operation was chosen as a test case. An illustration of the cutter and the serrated edge geometry is given in Figure 6-15. The cutting conditions are listed in Table 6-5. D = 18.55mm h = 60 mm i = 15 deg 0 0.1 A m p lit u d e  ( m m ) Distance Along Cutting Edge (mm) 0.2 0.3 0.4 0.5 0.6 0.7 0.4 0.8 1.2 1.6 2.00 Tool Envelope Serration Wave Serration Wave Profile  Figure 6-15 – (a) A five-fluted serrated, variable-pitch, cylindrical end mill and (b) its corresponding serration wave profile used to compare the new, frequency domain stability algorithm vs. CUTPRO’s time domain process simulation.  The stability of serrated cutters is affected by feed rate as at low feeds, only a few teeth will be in cut at each axial element. The causes serrated cutters to have the stability characterisitics of a tool with a lower number of flutes (the critical depth of cut varies inversely with the number of flutes – Altintas and Budak [8]) even though the same volume of material is removed. At higher feeds, the runout effect from unequal flute lengths is eliminated as the feed-per-tooth becomes greater than the differences in the Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  197  Table 6-5 - Summary of Cutting Geometry and Modal Parameters Used to Compare New Stability Algorithm Against CUTPRO (Time Domain Process Simulation)  Cutter Serrated, Variable-Pitch, Helical, Cylindrical-End Mill Diameter (D) 18.55 mm Height (h) 60.00 mm Number Of Flutes [N] 5 Helix Angle (i) [deg] 15 Engagement Conditions stφ  = 0 deg        exφ  = 180 deg Number of Modes 2 orthogonal – 1 mode in X, 1 mode in Y Average Feed Per Tooth ( Xc ) [mm/flute] 0.050 Feed rate [mm/s] 60 X X c nNf =  Pitch Angles Flutes Pitch (deg) 1-2 66.15 2-3 69.07 3-4 72.00 4-5 74.93 5-1 77.85   Modal Parameters Mode Direction Natural Frequency n ω  (Hz) Stiffness k  (N / m) Damping Ratio ς Mode XX 560 5.0e7 0.03 Mode YY 800 8.03e7 0.03    Cutting Coefficients (N/mm^2) rcK  760.22 tcK  1742.34 acK  458.32 Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  198 flute lengths. This causes more teeth to bite into the material and reduces the stability of the cutter. When generating the stability chart for the new stability algorithm in this section, the feed was recalculated at each spindle speed according to the equation in Table 6-5. This was performed to ensure that the same average feed-per-tooth value was used for each spindle speed, giving the same number of flutes in cut at each axial element. Figure 6-16(a) shows how the new stability algorithm’s critical depth of cut at a given spindle speed compares to the results from CUTPRO’s time domain process simulation. The CUTPRO results were generated using 20 revolutions of the tool. Figure 6-16(b) shows the resultant X-Y cutting forces vs. time and fast-fourier transform (FFT) of the resultant X-Y cutting forces given by CUTPRO for the stable and unstable conditions given in Figure 6-16(a). 0.1 0.2 0.3 0.40.0 0.1 0.2 0.3 0.40.0 Time (Sec) Time (Sec) X -Y  R e su lt a n t F o rc e  ( N ) 1000 2000 3000 1000 2000 3000 4000 5000 6000 Frequency (Hz) Frequency (Hz)  M a g n it u d e  o f X -Y  R e su lt a n t F o rc e  ( N ) Chatter (662.3 Hz) 100 200 0 40 80 120 0 1000 2000 3000 4000 1000 2000 3000 400000 DEPTH OF CUT = 20mm DEPTH OF CUT = 22mm X-Y Resultant Force vs. Time X-Y Resultant Force vs. Time FFT (Mag) of X-Y Resultant        Force vs.  Frequency CUTPRO Time Domain Process Simulation FFT (Mag) of X-Y Resultant        Force vs.  Frequency 3000 RPM 3000 RPM STABLE UNSTABLE UNSTABLE STABLE 2500 3000 3500 4000 12 16 20 24 D e p th  o f C u t (m m ) Spindle Speed (RPM) Cutpro - Time Domain New Stability Algorithm  - Frequency Domain Stability Chart Comparison(a) (b) Unstable (3000 RPM, 22 mm) Stable (3000 RPM, 20 mm) Limit (3000 RPM, 21.35 mm)  Figure 6-16 – (a) Stability comparison (CUTPRO vs. new stability algorithm) for the cutter defined in Figure 6-15 and cutting conditions given in Table 6-5. (b) Resultant force vs. time and FFT of resultant force for stable and unstable conditions. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  199 Note that in Figure 6-16(a), the stability chart from the two methods compares quite well – usually to within 20% up to 3500 rpm. It appears that the lobe given by the proposed method is somewhat wider than CUTPRO’s. The differences in the stability charts may be due to the fact that the new algorithm uses a zero order expansion of the directional coefficients [8] whereas CUTPRO’s is able to consider the full solution. A multi-frequency solution, such as the one presented by Merdol and Altintas [95], may provide a closer match to CUTPRO’s predictions. Also, the vibrational motion of the tool may dynamically add or subtract to the feed per tooth, causing additional variation in the number of flutes engaged in the workpiece. CUTPRO can consider these effects. In contrast, the proposed algorithm calculates the number of teeth in cut at each cutting force element based on the static chip thickness.  6.5.3 Stability Prediction of a Five-Axis Finishing Operation  To demonstrate implementation of the five-axis chatter stability solution, a finishing operation for an integrally bladed rotor (IBR) was selected. During these operations, the blade is the most flexible and has the greatest limiting effect on the stability of the system. The workpiece material used was 6 4Ti Al V− −  alloy and one tool move from the tool path was selected for analysis. The spindle speed for the operation was 400 RPM. This low spindle speed is required to limit heat generation during machining as the 6 4Ti Al V− − alloy has a very low thermal conductivity. Also, low cutting speeds allow for an increase in stability due to process damping effects, which will be discussed later in the section. The tool used is a six-fluted, non-serrated, tapered Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  200 ball-end mill with variable-pitch flutes. The position of the tool in relation to the blade, and the corresponding cutter-workpiece engagement map are shown in Figure 6-17. 150 100 50 0 -1.0 0.0 1.0 2.0 3.0 Immersion Angle (φ) D is ta n ce  f ro m  t o o l t ip  ( m m ) Cutter-Workpiece Engagement Map      for Impeller Finishing Operation (b) (a) Tool Position on Workpiece Closest Nodes of Selection Set to Tool Corresponding Nodes on Finite Element Model  Figure 6-17 – (a) Tool position for the five-axis chatter stability analysis. Illustration shows the closest nodes to the engaged area in the 256 node selection set. (b) The cutter-workpiece engagement map for this tool move.  It can be seen in Figure 6-17(b) that the cutter-workpiece engagement map for this operation resembles a tall, narrow band with low radial immersion. The depth of cut is very high – 120 mm (~ 5 inches). Figure 6-17(a) shows the nodes of the blade closest to the engaged area of the tool on the 256 node selection set. Also shown is where these points are located on the finite element model. The frequency reponse functions for direct and cross orientations (at the selected nodes) were obtained using the procedure Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  201 documented in 6.4.1. The tool was assumed to be rigid as the blade is the most flexible, lowly damped structure and will have the greatest effects on limiting the stability of the system. An algorithm was written in C++ to implement the solution presented in Chapter 6.2 to assess the stability of the system. This algorithm was used to generate the Nyquist plots of the system’s complex determinant. The highest mode of the system was 1642 Hz (Table 6-1) and so frequencies in the range of [0,3300]jω =  were scanned in increments of 0.1Hz. A cutting force element size of 2mm was chosen. With these set of given conditions, the simulation took approximately 11 hours to calculate, which is slow compared with the analytical solution of Altintas and Budak [8]. The reason for the long computation time is the large series of nested loops in the code resulting from the number of discrete cutting elements, flutes and frequency increments. As computers continue to get faster, this may cease to be a problem. Figure 6-18 shows the Nyquist plots of the determinant function for this case, The plot in Figure 6-18 shows a large number of circles on the graph. As shown below, the Nyquist plots show that the system is extremely unstable with multiple encirclements of the zero, and large real values at the negative-to-positive real axis crossings. This is typical of highly unstable systems. However, the prediction is in disagreement with the stable finishing operation on the production part. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  202 0 0 20 40 60 0.0 0.0 Determinant of [I-Λ(jω)] No Zoom Zoom x 1 Zoom x 2 -20-40-60 -20 -40 20 40 4.0 8.0-4.0-8.0 4.0 -4.0 -8.0 8.0 0.0 1.0-1.0 0.0 -1.0 1.0 Origin Unit Circle  Figure 6-18 – Nyquist plot of det[ ( )]I jω− Λ  for the five-axis finishing operation on the integrally bladed rotor. Plots show a highly unstable system with multiple encirclements of the origin.  There are several reasons for the differences between the experiments and the computer simulation. The main reason is process damping. Process damping is the name given to the large increase in the stability limit at low cutting speeds observed in experiments, but not predicted by classical chatter theory. An example of this is shown in Figure 6-19. In fact, classical chatter theory predicts the opposite effect – that the stability of the process decreases to a constant depth of cut at low cutting speeds. Several theories Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  203 have been put forth to explain this phenomena – the chief one being a damping effect caused by contact interference of the rake face of the tool with the workpiece [80], [112]. Currently, no reliable models of this mechanism have implemented in literature and it continues to be an ongoing area of research, particularly in our laboratory [11]. It is beyond the current scope of this project. However, if a breakthrough in this area is achieved, it could be added to the current model for a more accurate prediction. 0.0 0.5 1.0 1.5 2.0 0 1000 2000 3000 4000 D e p th  O f C u t (m m )  Spindle speed (rev/min) Process Damping Zone Experiment - Stable Experiment - Chatter Prediction  Figure 6-19 – Typical stability chart showing a process damping region at low cutting speeds. Experiments show a large increase in stability whereas chatter predictions show the opposite effect [11].  A second reason for the disagreement between the predicted and observed stability of the process is the fact that the natural frequencies of the CAD model calculated by ANSYS were observed to differ somewhat from those of the physical blade. This indicates that FRF measurements are also likely to differ from those of the physical blade, which could alter the stability predictions. It is unknown whether these differences Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  204 are due to distortion of the physical natural frequencies by the cart on which the impeller was measured or if this was due to errors introduced by simplifying the CAD model. FRF testing of the impeller in a free-free condition would be a recommended area for future work. In the five-axis chatter stability algorithm, the zero-order Fourier expansion of the directional coefficients was taken. Although this solution has been known to provide accurate chatter predictions in most milling operations, it may not be acceptable in low- radial immersion milling [22], [95]. Unfortunately, the test case had low-radial immersion engagement maps, which may require expansion of the higher-order terms of the directional coefficients. As this has already been presented in [22] and [95] and involves a fairly lengthy proof, inclusion of the higher order harmonics is left as an area of future study. Finally, the thickness of the chips removed in this operation is small [< 0.020mm] as the tool cuts near the 0-10 degree immersion angle zone. The orthogonal cutting database for the 6 4Ti Al V− −  alloy was not calibrated at these low chip thicknesses, which may cause the calculated cutting coefficients to deviate from the true values. Also, at these small chip thicknesses, the tool does not cut very effectively – it tends to grind the material rather than shear it – meaning that the cutting coefficients will be higher, further reducing the stability of the system. When cutting thin chips, the cutting coefficients are much more sensitive to change (see Figure 6-9) and small errors in the approximate chip thickness model could have large effects on the cutting coefficients, which would cause a drop in stability. Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  205 In summary, the unstable predictions of the five-axis chatter stability solution presented in this section deviated substantially from the observed stability in the real machining process. The process damping problem must be solved before an accurate prediction can be achieved. However, the theory may be applicable to high-speed machining in cases where half to full immersion angle maps are used. Further experimental testing should be conducted to establish the accuracy of the theory in these cases, but at present time no five-axis machine tools were available for testing. This remains an area for future work.  6.6 Summary  In this chapter, a new chatter stability solution for five-axis machining is presented. The solution permits identification of chatter conditions that may occur in five-axis flank milling with general, serrated, variable-pitch cutters. The algorithm is robust in that it allows irregular cutter-workpiece engagements to be taken into account, as well as dynamic flexibilities of the tool and workpiece that vary along the axis of the tool. These are all conditions that occur in jet-engine impeller machining. The solution can also be used for simpler, 2D and 3D machining cases when required. In other words, the solution can be applied to general milling processes. Chapter 6.5 showed that the Nyquist plots produced by the stability theory compare exactly with those of CUTPRO for a simple 2D planar machining case. The ability of Nyquist plots to assess relative stability was demonstrated using variable-pitch flutes to turn unstable cutting conditions into stable cutting conditions. Also, a stability chart for a Chapter 6: Frequency Domain Chatter Stability Solution for Five-Axis Flank Milling  206 five-flute serrated, variable-pitch, cylindrical-end mill was generated with the new algorithm, which compared favorably with results from time domain process simulations in CUTPRO. Finally, the new chatter prediction algorithm was demonstrated on a five- axis impeller machining example. In this example, Nyquist plots were generated for a finishing operation on an integrally bladed rotor (IBR) with a thin, flexible blade. The predictions from the algorithm did not match well with observations of the physical process, as it simulated an extremely unstable response. In contrast, the actual machining operation is stable. Differences in predictions and experiments were attributed to the process damping effect, differences in the FRF measurements of the physical blade and approximated CAD model, zero-order expansion of directional coefficients and inaccuracy / sensitivity of the cutting coefficients at small chip thicknesses. 207 Chapter 7 Conclusions  7.1 Conclusions In this thesis, the required models and algorithms necessary to perform five-axis flank milling of jet-engine impellers in a virtual environment were presented. A flowchart of the overall scheme described in the thesis is shown in Figure 1-4 and the areas studied are highlighted by the frames with dashed lines. The five-axis flank milling research topics consist of cutter-workpiece engagment calculations, cutting force mechanics modelling, feed rate optimization schemes and chatter stability analysis. The contributions of this thesis can be summarized in the following main categories with the related results:  • A new method of calculating the cutter-workpiece engagement area in five-axis flank milling operations. A semi-discrete solid model-based Parallel Slicing Method (PSM), was developed using the ACIS solid modeling kernel. It was used to calculate cutter-workpiece engagement maps for a roughing operation on a prototype integrally bladed rotor (IBR). These maps were compared with those obtained from the Manufacturing Automation Laboratory’s Virtual Machining Interface (MAL-VMI). The MAL-VMI scheme uses an Application Programming Interface (API) in the commercial numerical control (NC) verification software package, Vericut, Chapter 7: Conclusions  208 to obtain cutter-workpiece intersections through a z-buffer type method. The engagement maps obtained from the two methods compared reasonably well, although the proposed PSM captured the boundaries more accurately. The main drawback to the PSM is computation time – currently it is significantly slower than the z-buffer based MAL-VMI scheme. On the same computer, the MAL-VMI method took 2 minutes to calculate approximately 300 tool moves, whereas the PSM took 38 minutes with 60 slices and 19 minutes with 30 slices. The Parallel Slicing Method is novel in that it discretizes 3D models of the tool and workpiece into a series of 2D slices. It then employs analytical geometry and planar intersection/subtraction operations to generate cutter-workpiece engagement maps for five-axis flank milling. The PSM is based on a solid modeling environment and is able to update the workpiece for sucessive machining operations. The scholarly contributions will be published in the ASME Journal of Manufacturing Science and Engineering [57] some time in 2008.  • An experimentally verified cutting mechanics model that allows cutting force predictions for five-axis flank milling operations. The model was used to predict cutting forces for a roughing operation on an integrally bladed rotor (IBR) with a serrated, tapered, variable-pitch, helical ball-end mill. Discrepancies between predicted and measured forces were attributed to three points. First, the orthogonal cutting database used to predict the cutting force coefficients was developed with sharp tools and not honed or Chapter 7: Conclusions  209 worn cutting edges, which appear on the physical tool. Second, the experimental cutting force measurements where somewhat distorted by chatter resulting from the extension of the tool holder due to the added length of the rotating dynamometer. Finally, the MAL-VMI method may have introduced some loss of accuracy in the cutter-workpiece engagements maps due to clipping of the borders (see Section 3.3). Despite the measurement difficulties and approximations, the model predicted the X and Y cutting forces and cutting torque reasonably well – usually to within 20% of the experimental values. The new cutting force model presented in the thesis permits calculation of cutting forces for flank milling tool paths with five-axis motion, general end mill geometry (including serrated cutting flutes and variable-pitch teeth), irregular cutter-workpiece engagement maps and non-linear cutting coefficients that vary over the axis of the cutter. At this time, a model that is able to incorporate these complexities does not appear in the available literature. The scholarly contributions have been published in the ASME Journal of Manufacturing Science and Engineering [52].  • Two offline feed rate optimization algorithms that are able to decrease the cycle time of five-axis flank milling tool paths according to multiple feed-dependent constraints. These include a virtual adaptive control system and a non-linear root finding feed rate optimization scheme. Both of these methods were tested on the tool path given in Chapter 4 of the thesis Chapter 7: Conclusions  210 and gave almost identical optimum feed profiles -- profiles that realized a 45% reduction in cycle time. The process outputs from the optimized tool path met the set tool stress, tool deflection, cutting torque and chip thickness constraints, which were limited by intelligently varying the feed at each tool path segment. The tool stress and tool deflection at each angular increment of the cutter were calculated from a cantilever finite element model of the tool, which is presented in Appendix B. The non-linear root finding method is more robust as it calculates the optimum feed to within a given tolerance and can also better handle shorter tool path segments. The adaptive feed control optimization algorithm more closely mimics the response of the physical machine, but may not arrive at an optimum feed if the cutter- workpiece engagements change too quickly. Both methods allow feed rate optimization of five-axis flank milling tool paths based on multiple process constraints, which does not exist in open literature. The non-linear root finding feed rate optimization has been implemented on a physical machine at Pratt & Whitney Canada, reducing the cycle time of an impeller roughing tool path by approximately 20%. The scholarly contributions have been published in the ASME Journal of Manufacturing Science and Engineering [54].  • A chatter stability algorithm for five-axis flank milling, which is based on the five-axis cutting force model described in Chapter 4 of the thesis. This algorithm uses the Nyquist Stability Criterion to test whether a given Chapter 7: Conclusions  211 set of conditions is stable or unstable. The algorithm permits identification of chatter conditions that occur in five-axis flank milling with general, serrated, variable-pitch cutters. The proposed novel formulation is robust in that it allows complex cutter-workpiece engagements to be taken into account, as well as dynamic flexibilities of the tool and workpiece that vary along the axis of the tool. It also incorporates modeling of the effects of feed rate on stability for serrated cutters by varying the number of flutes in cut based on a static chip thickness approximation. So far, no such method has been presented in literature. Unlike the classical stability formulation for milling presented by Altintas and Budak [8], the solution presented in this Chapter does not generate a stability chart for a set of spindle speeds and depths of cut. Rather, it checks whether the system is stable or unstable for a given spindle speed and cutter-workpiece engagement map using Nyquist plots. Since increasing the depth of cut along the tool axis may distort the shape of the blade and cause unpredictable changes in the cutter-workpiece engagement maps, the solution must be coupled with an iterative trajectory generation module and cutter-workpiece engagement calculation algorithm in order to provide optimal chatter free conditions. This is a complex problem and is left for an area of future study. Section 6.5 showed that Nyquist plots produced by the new stability algorithm compare exactly with those of an experimentally and numerically proven method [8], [89] for a simple 2D planar machining case. The new method was also used to produce a chatter stability chart for a five-fluted serrated, variable-pitch Chapter 7: Conclusions  212 cylindrical end mill. This chart compared well with that generated from a time domain process simulation in CUTPRO – a commercial machining process simulation package. The five-axis chatter stability algorithm was also demonstrated on a five-axis flank milling operation. In this case, Nyquist plots were generated for a low-speed finishing operation on an integrally bladed rotor (IBR) with a thin, flexible blade. The frequency response functions over the blade’s surface were approximated with the help of ANSYS – a commercial finite element simulation package. The theory did not match well with observations of the physical process as it predicted an extremely unstable response. In contrast, the actual machining operation is known to be stable. Differences in predictions and experiments were attributed to the process damping effect, differences in the FRF measurements of the physical blade and approximated CAD model, zero- order expansion of directional coefficients and inaccuracy / sensitivity in the cutting coefficients at small depths of cut. Of these sources of error, process damping is thought to be the main reason for the discrepancies. Modeling this effect is a difficult problem and beyond the scope of the thesis. Although the theory is unable to offer accurate predictions for low speed machining at this time, it could be useful in high-speed flank milling where process damping is not active. Unfortunately, in this work, no five-axis machine tools were available for chatter tests. Verification of the theory with experiments for high-speed five-axis flank milling tool paths is left as an area for future study. Chapter 7: Conclusions  213  7.2 Recommended Areas for Future Work Although the core modules necessary for a virtual five-axis flank milling process simulation system are presented in the thesis, there is some room for improvement. Investigation of these suggested areas of research should help to decrease simulation time and increase the fidelity of the models, which will allow for more accurate predictions and additional optimization of the process. Currently, the Parallel Slicing Method (PSM) discussed in Chapter 3 only supports tapered ball-end mills, ball-end mills and cylindrical end mills. As explained in Section 3.2.1, additional mathematics for other cutters and an area for future study. This could include an arbitrary revolved profile that could be used to model all types of milling cutters. The calculation speed of the PSM could also be improved by optimization of the C++ subroutines, use of a smooth, analytical tool swept volume / removal volume, and implementation of parallel processing to compute the required Boolean intersections, subtractions and analytical geometry on each removal volume slice. Simulated cutting forces agreed reasonably well with measurements collected during an actual impeller machining operation (Chapter 4). However, improvements in predictions could be made by using an orthogonal cutting database calibrated for honed and worn cutting edges. Further experimental cutting force measurements could be performed with a stiffer tool holder setup to remove chatter from the system. This would allow for a better comparison of static cutting forces. Currently, the force prediction algorithm is rather slow – especially for serrated cutters where the element size must be Chapter 7: Conclusions  214 small to capture the features of the wavy edge geometry. Implementing methods to reduce the number of cutting force elements without loss of accurarcy, and optimization of the C++ algorithms, would help decrease simulation times. The non-linear root-finding feed rate optimization method discussed in Chapter 5 has been verified on a physical machine at Pratt & Whitney Canada, reducing the machining time of an impeller roughing tool path by approximately 20%. The multi- constraint adaptive feed-control method yields almost identical feed rate profiles. However, the main drawback to both methods is the long simulation times (often several days). Although the adaptive feed control and non-linear root-finding algorithms converge in only a few iterations, each iteration requires that the maximum process outputs be determined over one revolution of the cutter. Currently this is implemented by simulation of the outputs in one to two degree increments, which is time consuming. If a fast, numerical technique could be employed to find the maximum process outputs over one cutter revolution quickly, the speed of the optimizations would be greatly increased. The five-axis flank milling chatter stability model matched exactly with the zero- order eigenvalue solution of Altintas and Budak [8] for 2D milling, but not with the five- axis finishing operation on the flexible impeller blade. To improve correlations between the simulated and physical systems, the multi-frequency solution could be applied to include the harmonics of the directional coefficients and frequency response functions. Also, a process damping model could be added to the stability algorithm when one becomes available. This is still a challenging area of research. Modal analysis of the physical impeller in free-free conditions would allow for more accurate FRF comparisons with the CAD model of the blade. The CAD model could then be fine-tuned to bring Chapter 7: Conclusions  215 predicted natural frequencies into closer correlation with those of the physical blade. As mentioned in Chapter 6, in five-axis flank milling, increasing the depth of cut along the tool axis may distort the shape of the blade and cause non-linear changes in the cutter- workpiece engagement maps. To find optimal chatter-free conditions the chatter stability solution could be coupled with an iterative trajectory generation module and cutter- workpiece engagement calculation algorithm. Finally, the semi-discretization technique presented in articles [69], [70], [71] for time-domain stability analysis could be applied to five-axis flank milling. This method may allow for conditions such as dependency of the cutting coefficients on chip thickness to be incorporated.       216 Bibliography [1] Abdel-Malik K., Yeh H.-J., 1997, "Geometric Representation of the Swept Volume using Jacobian Rank-Deficiency Conditions, Computer-Aided Design, Vol. 29, No. 6, pp. 457-468. [2] Abramowitz, M. and Stegun, I. A. (Eds.). 1972, “Solutions of Quartic Equations”, 3.8.3 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 17-18. [3] Acton, F. S. 1990, “Numerical Methods That Work, 2nd printing”. Washington, DC: Math. Assoc. Amer., p. 103. [4] Altintas Y., 1992, “Direct Adaptive Control of End Milling Process”, International Journal of Machine Tools and Manufacture, Vol. 34, No. 4, pp. 461-472. [5] Altintas Y., 2000, "Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations and CNC Design", and CNC Design, Cambridge University Press. [6] Altintas Y., 2001, “Analytical Prediction of Three Dimensional Chatter Stability in Milling”, Japan Society of Mechanical Engineers (JSME) International Journal, Series C, Vol 44, No. 3, pp. 717-723. [7] Altintas Y., 2003, “Proof That The Cutting System Can Not Be Unstable If The Regeneration Term Does Not Exist – Note to Bill Endres, Editor of ASME Journal of Manufacturing Science and Engineering”. [8] Altintas, Y., Budak, E., 1995, "Analytical Prediction of Stability Lobes in Milling", Annals of the CIRP, Vol. 44/1, pp. 357-362. [9] Altintas Y., Engin S., Budak E., 1999, “Analytical Stability Prediction and Design of Variable Pitch Cutters”, Transactions of the ASME Journal of Manufacturing Science and Engineering, vol. 121, pp. 586-592. [10] Altintas Y., Erol N. A., 1998, “Open Architecture Modular Tool Kit for Motion and Machining Process Control”, Annals of the CIRP Vol. 47, pp. 295-300. [11] Altintas Y., Eynian M., Onozuka H., 2008, “Identification of Dynamic Cutting Force Coefficients and Chatter Stability with Process Damping”, Annals of the CIRP, Vol. 56/1. [12] Altintas Y., Shamoto E., Lee P., Budak E., 1999, “Analytical Prediction of Stability Lobes in Ball End Milling”, Transactions of ASME Journal of Manufacturing Science and Engineering vol 121, pp 586-592. [13] Altintas Y., Spence A., 1991, “End Milling Force Algorithms for CAD Systems”, Annuals of the CIRP, Vol 40/1, pp. 31-34. [14] ANSYS Inc., 2006, “Release 10 Documentation for ANSYS v10”. [15] Armarego E. J. A., Brown R. H., 1969, "The Machining of Metals", Prentice-Hall Inc. Bibliography  217 [16] Armarego E. J. A., 1994, “Material Removal Processes – An Intermediate Course”, Manufacturing Science Group, Department of Mechanical and Manufacturing Engineering, The University of Melbourne. [17] Bailey T., Elbestawi M.A., El-Wardany T. I., Fitzpatrick P., Aug. 2002, "Generic Simulation Approach for Five-Axis Machining, Part I: Modeling Methodology", ASME Journal of Manufacturing Science and Engineering, Vol. 124, pp 624-633. [18] Bailey T., Elbestawi M.A., El-Wardany T. I., Fitzpatrick P., Aug. 2002, "Generic Simulation Approach for Five-Axis Machining, Part II: Model Calibration and Feed Rate Scheduling", ASME Journal of Manufacturing Science and Engineering, Vol. 124, pp 634-642. [19] Bayly, P. V., Mann, B. P., Schmitz, T. L., Peters, A. P., Stepan, G., and Insperger, T., 2002, ‘‘Effects of Radial Immersion and Cutting Direction on Chatter Instability in End-Milling,’’ Proceedings of IMECE, MED-39116. [20] Bayly P. V., Halley J. E., Mann B. P., Davies M. A., 2003, “Stability of Interrupted Cutting By Temporal Finite Element Analysis”, ASME Journal of Manufacturing Science and Engineering, Vol. 125, pp. 220-225. [21] Becze C.E., Clayton P., Chen L., El-Wardany T.I., Elbestawi M.A., 2000, "High- speed Five-Axis Milling of Hardened Tool Steel", International Journal of Machine Tools & Manufacture, Vol. 40, pp. 869-885. [22] Budak, E., 1994, “Mechanics and Dynamics of Milling Thin Walled Structures”, Ph.D. Thesis, The University of British Columbia. [23] Budak E., Altintas Y., 1998, “Analytical Prediction of Chatter Stability in Milling. Part I: Modeling”, Transactions of ASME, Journal of Dynamic Systems, Measurement and Control, Vol. 120, pp 22-30. [24] Budak E., Altintas Y., 1998, “Analytical Prediction of Chatter Stability in Milling. Part II: Applications”, Transactions of ASME, Journal of Dynamic Systems, Measurement and Control, Vol. 120, pp 31-36. [25] Budak, E., 2000, "Improving Productivity and Part Quality in Milling of Titanium Based Impellers by Chatter Suppression and Force Control", Annals of the CIRP, Vol. 49/1, pp. 31-36. [26] Budak E., Feb. 2003, "An Analytical Design Method for Milling Cutters With Nonconstant Pitch to Increase Stability, Part I: Theory", ASME Journal of Manufacturing Science and Engineering, Vol. 125, pp 29-34. [27] Budak E., Feb. 2003, "An Analytical Design Method for Milling Cutters With Nonconstant Pitch to Increase Stability, Part II: Application", ASME Journal of Manufacturing Science and Engineering, Vol. 125, pp 35-38. [28] Budak E., Altintas Y., Armarego E. J. A., May 1996, "Prediction of Milling Force Coefficients from Orthogonal Cutting Data", ASME Journal of Manufacturing Science and Engineering, Vol. 118, pp. 216-224. [29] Campomanes M. L., 1998, “Dynamics of Milling Flexible Structures”, M.A.Sc Thesis, Department of Mechanical Engineering. University of British Columbia. Bibliography  218 [30] Cao. Y, 2006, "Modeling of High-Speed Machine-Tool Spindle Systems", Doctor of Philosophy Thesis, The University of British Columbia. [31] Chappel I.T., 1983, “The use of vectors to simulate material removed by numerically control milling”, Computer Aided Design 15(3): pp. 156-158. [32] CGTech, 2004, “Vericut – Machine Simulation, Cutting Speed Optimization, Program Verification, Inspection and Analysis”, Information Sheet for Industry. [33] Chu C. N., Kim S. Y., Lee J. M., 1997, "Feed-Rate Optimization of Ball End Milling Considering Local Shape Features", Annuals of the CIRP, Vol. 46, No. 1, pp. 433-436. [34] Corpus, W.T., and Endres, W.J., 2000, “A high-order solution for the added stability lobes in intermittent machining,” MED-Vol. 11, Proceedings of the ASME Manufacturing Engineering Division, pp. 871-878. [35] Dahl J., 2008, “Jet engine.svg – Wikimedia Commons”, http:://commons.wikimedia.org/wiki/Image:Jet_engine.svg (current as of July 9, 2008), Licenseed under the Creative Commons Attribution ShareAlike Licence Versions 3.0 (http://creative commons.org/licenses/by-sa/3.0/), 2.5 (http://creative commons.org/licenses/by-sa/2.5/), 2.0 (http://creative commons.org/licenses/by- sa/2.0/), and 1.0 (http://creative commons.org/licenses/by-sa/1.0/). [36] Davies M. A., Pratt J. R., Dutter B., Burns T. J., 2002, “Stability Prediction for Low Radial Immersion Milling”, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol. 124, pp 217-225. [37] Davies R., "Documentation for NewMat10", "http://objcryst.sourceforge.net /ObjCryst/newmat.htm", current as of August 22, 2007. [38] Delmia Corp., 2004, “Virtual NC – Rapid Emulating, Validating and Optimizing the NC Machine Process”, Information Sheet for Industry. [39] Desfosses B., Jerard R. B., Fussell B. K., Xu M., 2008, “An Improved Power Threshold Method for Estimating Tool Wear During Milling”, Transactions of the North American Manufacturing Research Institution / Society of Manufacturing Engineers (NAMRI/SME), Vol. 36, pp. 541-548. [40] Drysdale R. L., Jerard R. B., Schaudt B., Hauck K., 1989, “Discrete simulation of NC machining”, Algorithmica, Springer New York, Vol. 4, No. 1, pp. 33-60 [41] Du. S., Surmann T., Webber O., Weinert K., 2005, "Formulating Swept Profiles for Five-Axis Tool Motions", International Journal of Machine Tools and Manufacture, Vol. 45, pp. 849-861. [42] Edwards C. H., Penney D. E., 1998, “Elementary Linear Algebra – Inverses and the Adjoint Matrix”, Prentice-Hall Inc., New Jersey, USA, pp. 103-105. [43] Endres, W. J., and Corpus, W. T., 2000, “A High-Order Solution for the Added Stability Lobes in Intermittent Machining,” Proceedings of the Symposium on Machining Processes, MED-11, pp. 871–878. Bibliography  219 [44] Engin, S., 1999, “Mechanics and Dynamics of Milling with Generalized Geometry”, Ph.D. Thesis, The University of British Columbia. [45] Engin S., Altintas Y., 2001, "Mechanics and Dynamics of General Milling Cutters. Part I: Helical End Mills", International Journal of Machine Tools & Manufacture, Vol 41, No. 15, pp 2195-2212. [46] Engin S., Altintas Y., 2001, "Mechanics and Dynamics of General Milling Cutters. Part II: Inserted Cutters", International Journal of Machine Tools & Manufacture, Vol 41, No. 15, pp 2213-2231. [47] Erdim H., Lazoglu I., Ozturk B., 2006, "Feedrate scheduling strategies for free- form surfaces", International Journal of Machine Tools and Manufacture, Vol. 46, Issues 7-8,  Pages 747-757. [48] Erkorkmaz K., 2003, “Optimal Trajectory Generation and Precision Tracking Control For Multi-Axis Machines”, Ph.D. thesis, The University of British Columbia, pp. 3. [49] Feng H-Y., Menq C-H., 1994, "Prediction of Cutting Forces in the Ball-End Milling Process 1. Model Formulation and Model Building Procedure", International Journal of Machine Tools & Manufacture, Vol. 34., No. 5., pp. 697- 710. [50] Feng H-Y., Menq C-H., 1994, "The Prediction of Cutting Forces in the Ball-End Milling Process. II. Cut Geometry Analysis and Model Verification", International Journal of Machine Tools & Manufacture, Vol. 34., No. 5., pp. 711-719. [51] Feng H. Y., Su N., 2000, "Integrated Tool Path and Feed Rate Optimization for the Finishing Machining of 3D Plane Surfaces", International Journal of Machine Tools and Manufacture, Vol. 40, pp. 1557- 1572. [52] Ferry W. B., Altintas Y., 2007, “Virtual Five-Axis Flank Milling of Jet Engine Impellers—Part I: Mechanics of Five-Axis Flank Milling”, Proceedings of IMECE2007 2007 ASME International Mechanical Engineering Congress and Exposition, November 11-15, 2007, Seattle, Washington, USA. [53] Ferry W. B., Altintas Y., Feb. 2008, “Virtual Five-Axis Flank Milling of Jet Engine Impellers—Part I: Mechanics of Five-Axis Flank Milling”, ASME Journal of Manufacturing Science and Engineering, Vol. 130, pp. 011005-1 to 011005-11. [54] Ferry W. B., Altintas Y., 2007, “Virtual Five-Axis Flank Milling of Jet Engine Impellers—Part II: Feed Rate Optimization of Five-Axis Flank Milling”, Proceedings of IMECE2007 2007 ASME International Mechanical Engineering Congress and Exposition, November 11-15, 2007, Seattle, Washington, USA. [55] Ferry W. B., Altintas Y., Feb. 2008, “Virtual Five-Axis Flank Milling of Jet Engine Impellers—Part II: Feed Rate Optimization of Five-Axis Flank Milling”, ASME Journal of Manufacturing Science and Engineering, Vol. 130, pp. 011013-1 to 011013-13. [56] Ferry W. B., Yip-Hoi D., 2007, “Cutter-Workpiece Engagement Calculations By Parallel Slicing For Five-Axis Flank Milling Of Jet Engine Impellers”, Proceedings Bibliography  220 of IMECE2007 2007 ASME International Mechanical Engineering Congress and Exposition, November 11-15, 2007, Seattle, Washington, USA. [57] Ferry W. B., Yip-Hoi D., 2007, “Cutter-Workpiece Engagement Calculations By Parallel Slicing for Five-Axis Flank Milling of Jet Engine Impellers, ASME Journal of Manufacturing Science and Engineering, Accepted for Publication November 2007. [58] Fu, H. J., Devor, R. E., and Kapoor, S. G., 1984, “A Mechanistic Model for the Prediction of the Force System in Face Milling Operation”, ASME Journal of Engineering for Industry, Vol. 106(1), pp 81-88. [59] Fussell B. K., Hemmett J. G., Jerard R. B., 1999, "Modeling of Five-Axis End Mill Cutting Using Axially Discretized Tool Moves", Transactions of NAMRI / SME, Vol. 27, pp. 81-86. [60] Fussell B. K., Jerard R. B., Hemmett J.G., 2001, "Robust Feedrate Selection for 3- Axis NC Machining Using Discrete Models", ASME Journal of Manufacturing Science and Technology, Vol. 123, pp.214-224. [61] Fussell B.K., Jerard R.B., Hemmett J.G., 2003, "Modeling of Cutting Geometry and Forces for 5-Axis Sculptured Surface Machining", Computer Aided Design, Vol. 35, pp. 333-346. [62] Goodwin C. G., Sin K. S., 1984, "Adaptive Filtering Prediction and Control", Prentice-Hall. [63] Gupta S. K., Saini S.K., Spranklin B.W., Yao Z., 2005, "Geometric Algorithms for Computing Cutter Engagement Functions in 2.5D Milling Operations", Computer- Aided Design, Vol. 37, pp. 1469-1480. [64] Guzel B. U., Lazoglu I., 2003, "Increasing Productivity in Sculptured Surface Machining via Off-line Piecewise Variable Feedrate Scheduling based on the Force System Model", International Journal of Machine Tools and Manufacture, Vol. 44, pp. 21-28. [65] Hemmett J. G., 2001, “Discrete Modeling of Sculptured Surface Machining for Robust Automatic Feedrate Selection”, Ph.D. Thesis, University of New Hampshire. [66] Hutchinson J.R., 2001, "Shear Coefficients for Timoshenko Beam Theory", Transactions of the ASME, Journal of Applied Mechanics, Vol. 68, pp 87-92. [67] ICAM Technologies Corporation, “CAM-POST V-15”, www.icam.com, current as of Jan 17, 2008 [68] Imani B.M., Sadeghi M.H., Elbestawi M.A., 1998, "An Improved Process Simulation System for Ball-End Milling of Sculptured Surfaces", International Journal of Machine Tools and Manufacture, Vol. 38, pp. 1089-1107. [69] Insperger T., Stepan G., 2002, “Semi-discretization method for delayed systems”, International Journal For Numerical Methods in Engineering, Vol. 55, pp. 503–518. Bibliography  221 [70] Insperger T., Stepan G., Bayly P. V., Mann B. P., 2003, “Multiple Chatter Frequencies In Milling Processes”, Journal of Sound and Vibration, Vol. 262, pp. 333-345. [71] Insperger T., Stepan G., 2004, “Stability Analysis of Turning With Periodic Spindle Speed Modulation Via Semidiscretization”, Journal of Vibration and Control, Vol. 10, pp. 1835-1855. [72] Jang D., Kim K., Jung J., 2000, "Voxel-Based Virtual Multi-Axis Machining", The International Journal of Advanced Manufacturing Technology", Vol. 16, pp. 709- 713. [73] Jenkins M.A., Traub J.F., December 1970, "A three-stage algorithm for real polynomials using quadratic iteration", SIAM Journal on Numerical Analysis, Vol. 7, No. 4, pp. 545-566. [74] Jensen S.A., Shin Y.C., 1999, “Stability Analysis in Face Milling Operations, Part 1: Theory of Stability Lobe Prediction”, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol. 121, No. 4, pp. 600-605. [75] Jensen S.A., Shin Y.C., 1999, “Stability Analysis in Face Milling Operations. II. Experimental Validation and Influencing Factors”, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol. 121, No. 4, pp. 606-614. [76] Jerard R.B., Drysdale R.L., Hauck K., Schaudt B., Ford J.M., January 1989, "Methods for Detecting Errors in Numerically Controlled Machining of Sculptured Surfaces", IEEE Computer Graphics & Applications, pp. 26-39. [77] Jerard R. B., Fussell, B. K., Hemmett  J. G., Ercan M. T., 2000, "Toolpath Feedrate Optimization: A Case Study", Proceedings of the 2000 NSF Design & Manufacturing Research Conference, Jan 3-6, Vancouver, British Columbia, Canada, pp. 1-6. [78] Jerard R. B., Fussell B. K., Xu M., Yalcin C., 2006, “Process Simulation and Feedrate Selection for Three-Axis Sculptured Surface Machining”, International Journal of Manufacturing Research, Vol. 1, No. 2, pp. 136-156. [79] Jerard R.B., S.Z. Hussani, R.L Drysdale and B. Schaudt, 1989, “Approximate Methods for Simulation and Verification of Numerically Controlled Machining Programs”, The Visual Computer, No. 5, pp. 329-348. [80] Kegg. R. L, November 1965, “Cutting Dynamics in Machine Tool Chatter – Contribution to Machine-Tool Chatter Research-3”, Transactions of the ASME – Journal of Engineering for Industry, pp. 464-470. [81] Kim G. M., Cho P.J., Chu C.N., 2000, "Cutting Force Prediction of Sculptured Surface Ball-End Milling Using Z-Map", International Journal of Machine Tools and Manufacture, Vol. 40, No. 2, pp. 277-291. [82] Kim G. M., Chu C. N., 2004, "Mean cutting forces prediction in ball-end milling using force map method", Journal of Materials Processing Technology, Vol. 146, pp. 303-310. Bibliography  222 [83] Kline W.A., DeVor R. E., 1983, “The Effect of Runout on Cutting Geometry and Forces in End Milling”, International Journal of Machine Tool Design & Research, Vol. 23, No. 2-3, pp. 123-140. [84] Koenigsberger F., Tlusty J, 1967, “Machine Tool Structures – Vol I: Stability Against Chatter”, Pergamon Press. [85] Kops L., Vo. D. T., 1990, "Determination of the Equivalent Diameter of an End Mill Based on its Compliance", Annuals of the CIRP, Vol. 39, No. 1. [86] Krantz, S. G., 1999, “Handbook of Complex Variables”, Birkhauser Boston, Massachusettes, USA. Ch. 5, pp. 69-78. [87] Lee P., Altintas Y., 1996, "Prediction of Ball-End Milling Forces From Orthogonal Cutting Data", International Journal of Machine Tools and Manufacturing, Vol. 36, No. 9, pp.1059-1072. [88] Magnus, W. and Winkler, 1979, S. "Floquet's Theorem." Chapter 1.2 in Hill’s Equation, New York: Dover, pp. 3-8. [89] MAL Inc., 2003, “CUTPRO Version 8”. Process Simulation Software Package. [90] MAL Inc., 2008, “MAL Inc. Website - Customers”,http://www.malinc.com/ customers.html, current as of Jan 8, 2008. [91] Manufacturing Information Systems (MIS), 2008, “Predator – Virtual CNC” http://www.mis-group.com/virtualcnc/virtual.php, current as of Jan 17, 2008. [92] McKechnie G., 2005, “Image:MillingCutterSlotEndMillBallnose.jpg”, http://en.wikipedia.org/wiki/Image:MillingCutterSlotEndMillBallnose.jpg (current as of  2008), File from Wikimedia Commons, Licensed under the under Creative Commons Attribution ShareAlike 2.0 License. [93] Merchant M. E., 1994, “Basic Mechanics of the Metal-Cutting Process”, Journal of Applied Mechanics, pp. A-168 to A-175. [94] Merdol S.D., Altintas Y., 2004, "Mechanics and Dynamics of Serrated Cylindrical and Tapered End Mills", ASME Journal of Manufacturing Science and Engineering, Vol. 126, pp. 317-326. [95] Merdol S. D., Altintas Y., 2004, “Multi Frequency Solution of Chatter Stability for Low Immersion Milling”, Transactions of the ASME, Journal of Manufacturing Science and Engineering, Vol. 126, pp. 459-466. [96] Merdol S. D., 2002, “Mechanics and Dynamics of Serrated End Mills”, Masters Thesis, University of British Columbia. [97] Merritt H.E., 1965, “Theory of Self-Excited Machine Tool Chatter”, Transactions of the ASME, Journal of Engineering for Industry, 87:447-454. [98] Minis I., Yanushevsky R., 1993, “A New Theoretical Approach to the Prediction of Machine Tool Chatter in Milling”, Transactions of the ASME, Journal of Engineering for Industry, Vol. 115, No. 1, pp. 1-8. Bibliography  223 [99] Montgomery D., Altintas Y., 1991, "Mechanism of Cutting Force and Surface Generation in Dynamic Milling", ASME Journal of Engineering for Industry, 113, pp 160-168. [100] Morrison, N., 1994, “Introduction to Fourier Analysis”, New York, Wiley- Interscience. [101] Nemes J.A., Asamoah-Attiah S., Budak E., 2001, "Cutting Load Capacity of End Mills with Complex Geometry", Annuals of the CIRP, Vol. 50, No. 1, pp.65-68. [102] Ogata K., 1970, “Modern Control Engineering”, Prentice-Hall, Englewood Cliffs New Jersey, USA. [103] Optiz H., 1969, “Investigation and Calculation of the Chatter Behavior of Lathes and Milling Machines”, Annuals of the CIRP, vol. 18, pp 335-342. [104] OuYang D., 2007, “Virtual Machining System – Instruction Manual”, Manufacturing Automation Laboratories (MAL) Inc. [105] Popov E., 1990, "Engineering Mechanics of Solids", Prentice-Hall Inc. [106] Press W. H., Vetterling W. T., Teukolsky S.A., Flannery B.P., 2002, "Numerical Recipes in C++ - The Art of Scientific Computing - Second Edition", Cambridge, University Press, 2nd edition. [107] Ramaraj T. C., Eleftheriou E., 1994, “Analysis of the Mechanics of Machining With Tapered End Milling Cutters”, Transactions of the ASME, Journal of Engineering for Industry, Vol. 116, pp. 398–404. [108] Roth D., Ismail F., Bedi S., 2003, "Mechanistic Modelling of 5-axis Milling using an Adaptive Depth Buffer", University of Waterloo. [109] Roth D., Ismail F., Bedi S., 2003, "Mechanistic Modelling of the Milling Process using an Adaptive Depth Buffer", Computer-Aided Design, Vol. 35, pp. 1287- 1303. [110] Shaw M. C., Cook N. H., Smith P.A., 1951, “The Mechanics of Three-Dimensional Cutting Operations”, Journal of Applied Mechanics, pp. 1055-1064. [111] Sheltami K., Bedi S., Ismail F., 1998, "Swept Volumes of Toroidal Cuttings Using Generating Curves", International Journal of Machine Tools and Manufacture, Vol. 38, pp. 855-870. [112] Sisson T. R., Kegg R. L., November 1969, “An Explanation of Low-Speed Chatter Effects”, Transactions of the ASME – Journal of Engineering for Industry, pp. 951- 958. [113] Spence A.D., Altintas Y., 1991, “CAD Assisted Adaptive Control for Milling”, Transactions of the ASME, Journal of Dynamics Systems, Measurement and Control, Vol. 113, pp. 444-450. [114] Spence A.D., 1994, “A Solid Modeller Based Milling Process Simulation and Planning System”, Transactions of ASME, Journal of Engineering for Industry”, Vol. 116, pp. 61-69. Bibliography  224 [115] Spence A.D., 2001, "Parallel Processing for 2-1/2D Machining Simulation", ACM Symposium on Solid and Physical Modeling - Proceedings of the sixth ACM symposium on Solid modeling and Applications. [116] Sridhar R., Hohn R.E., Long G.W., 1968, “A Stability Algorithm for the General Milling Process”, Transactions of the ASME Journal of Engineering for Industry, Vol. 90, pp. 330-334. [117] Stabler, G. V., 1964, “The Chip Flow Law and Its Consequences”, Advances in Machine Tool Design and Research, pp. 243-251. [118] Takata S., Tsai M.D., Inui M., 1989, "A Cutting Simulation System for Machinability Evaluation Using a Workpiece Model", Annals of the CIRP Vol. 38, pp. 417-420. [119] Taylor F. W., 1907, “On the Art of Cutting Metals”, Transactions of ASME, 28. [120] Tlusty, J. 1986, “Dynamics of High Speed Milling”, Transactions of the ASME, Journal of Engineering for Industry, Vol. 108, pp 59-67. [121] Tlusty J., Polacek M., 1957, “Besipiele der behandlung der Selbsterregten Schwingung der Werkzeugmaschinen”, FoKoMa, Hanser Verlag, Munchen. [122] Tlusty, J. and Ismail, F., 1981, “Basic Nonlinearity in Machining Chatter”, Annals of the CIRP, Vol. 30, pp. 21-25. [123] Tobias S.A., Fishwick W., 1958, “Theory of Regenerative Machine Tool Chatter”, Engineering London, Vol 258. [124] Tounsi N., Elbestawi M.A., 2001, "Enhancement of Productivity by Intelligent Programming of Feed Rate in 3-Axis Milling, Machining Science and Technology, Vol. 5(3), pp. 393-414. [125] Van Hook T., 1986, "Real-Time Shaded NC Milling Display", Computer Graphics, Vol. 20, No. 4, pp. 15-20. [126] Vericut, 2004, "Vericut Online Help Version 5.4 - OptiPath API", CGTech. [127] Voelcker H.B., Hunt W.A., 1981, “The role of solid modeling in machining process modeling and NC verification”, SAE technical paper, 810195. [128] Vossler D. L., Engin S., Altintas Y., 2001, "Exploring Analytic Geometry with Mathematica - 12.1 - Conic From Quadratic Equation”, Academic Press, pp. 175- 184. [129] Wang J.-J. Junz, Liang S. Y., February 1996, "Chip Load Kinematics in Milling with Radial Cutter Runout", Transactions of the ASME, Vol. 118, pp. 111-118. [130] Wang W.P., Wang K.K., 1986, “Real-Time Verification of Multiaxis NC Programs with RasterGraphics”, IEEE paper, CH2282-2/86/0000/0166S01.00. [131] Wu C. Y., 1995, “Arbitrary Surface Flank Milling of Fan, Compressor and Impeller Blades”, Transactions of the ASME -- Journal of Engineering for Gas Turbines and Power, Vol. 117, pp. 534-539. Bibliography  225 [132] Xu M., Jerard R. B., Fussell B. K., 2007, “Energy Based Cutting Force Model Calibration for Milling”, Computer-Aided Design & Applications, Vol. 4, No. 1-4, pp. 341-351. [133] Yao Z., 2005, "Finding Cutter Engagement For Ball End Milling of Tessellated Free-Form Surfaces", Proceedings of IDETC/CIE 2005 ASME 2005 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference September 24-28, Long Beach, California, USA. pp. 1-7. [134] Yazar Z., Koch. K., Merrick, T., Altan T., 1994, "Feed Rate Optimization Based on Cutting Force Calculations in 3-Axis Milling of Dies and Molds With Scuptured Surfaces", International Journal of Machine Tools and Manufacture, Vol. 34, No 3., pp. 365-377. [135] Yip-Hoi D., Huang X., 2006, “Cutter-Workpiece Engagement Feature Extraction from Solid Models for End Milling”, Journal of Manufacturing Science and Engineering, Vol. 128, pp. 249-260. [136] Yucesan G., Altintas Y., 1996, "Prediction of Ball End Milling Forces", ASME Journal of Engineering for Industry, Vol. 118, pp. 95-103. [137] Zhu R., Kapoor S.G., DeVor R.E., Aug. 2001, "Mechanistic Modeling of the Ball End Milling Process for Multi-Axis Machining of Free-Form Surfaces", ASME Journal of Manufacturing Science and Engineering, Vol. 123, pp. 369-379.   226 Appendix A Analytical Geometry for the Parallel Slicing Method    A1  Analytical Tapered Ball-End Mill–Plane Intersection Curves   The following section presents an analytical solution for determining the intersection shapes between a tapered ball-end mill and a plane. This is required in Sections 3.2.3 and 3.2.4 for determining the swept area used to update the in-process removal volume. These shapes are uses to generate the engagement arcs that form the engagement polygon. Although the intersection curves can be obtained using the ACIS solid modeling environment, ACIS uses numerical methods to calculate Boolean intersection operations, which are computationally expensive. An analytical solution is preferable as it calculates the curve directly. Since ball-end mills and cylindrical-end mills can be considered as subsets of tapered ball-end mills, the mathematics discussed in this appendix can also be applied to these types of cutters. A ball-end mill can be thought of as a tapered ball-end mill with a 0 degree taper angle and a cylindrical end mill is a tapered ball-end mill with no ball and a 0 degree taper angle. Examples of these cutters and their dimensional forms are shown in Figure 3-2. A tapered ball-end mill is composed of two solid objects – an inverted conical frustum and a sphere (Figure A-1). The curve formed by the intersection of the tapered section with a plane will be a conic, since the conical frustum is actually a truncated cone. A conic, by definition, is the family of curves formed from the intersection of a plane Appendix A: Analytical Geometry For The Parallel Slicing Method  227 with a cone. The intersection curves for the ball with a plane will be a circle, since the intersection of a sphere with a plane is always a circle. The intersection of the solids with a plane can also produce partial conics. These arise from the discontinuities in the solids, such as at the section where the ball and taper are joined and at the top of the cutter, where the frustum is bounded. Ball (Sphere) Taper (Inverted Conical Frustum) Tapered Ball-End Mill  Figure A-1 – A tapered ball-end mill.   As explained in Section 3.2.1, the coordinates of the cutter at each tool move are given as a series of GOTO commands. The first three columns specify the x,y and z coordinates of the tool tip in the global system, { }P , and the next three columns specify the coordinates of the unit vector tool-axis, { }TTZ  or { }T . To simplify calculations, the location and orientation of the tool in the global coordinate system is transformed to the slice plane coordinate system. This is performed through homogenous transformations. Appendix A: Analytical Geometry For The Parallel Slicing Method  228 The homogenous transformation matrices from the global system to the tool coordinate system, [ ]GTH , and global system to the slice plane, [ ]GS sH , are evaluated as, [ ] { } { } { } { } [ ] , 1 2 1 , 1 2 1 1 , 2 0 0 0 1 0 0 0 0 1 yx z x x j y z x TT TT TT y y j GT GS s z z j ss s s c C C C s s sX Y Z P s c H H C C C C s c C  −       − −     = =                     (A.1)  where, 2 2 2 2 2 1 2x y x y zC s s C s s s= + = + + (A.2)  Here { }Tx y zS s s s=  is the slice normal and { }, , , Ts x s y s z sC c c c=   is the center coordinates of the slice plane for slice, s . { }TTX , { }TTY  and { }TTZ  are the axes of the tool coordinate system at the tool tip given in Sections 3.2.6 and 4.3. Eq (A.1) is valid for any normal of the slice plane except when it is parallel with the Z-axis of the global system. This orientation will cause singularities in some of the entries and is handled separately. Using the above homogenous transformation matrices, the homogenous transformation matrix between the slice plane and the position / orientation of the tool, [ ]ST sH , can be calculated through: Appendix A: Analytical Geometry For The Parallel Slicing Method  229 [ ] [ ] [ ] 11 12 13 14 1 21 22 23 24 31 32 33 34 0 0 0 1 ST GS GTs s a a a a a a a a H H H a a a a −      = =          (A.3)  To convert vectors and coordinates from slice plane coordinates back to global coordinates the following equation can be used, { } [ ] { }GS sg H s=  (A.4)  where { }s  is a homogenous vector (4x1) in the slice coordinate system and { }g  is the vector in global coordinates. A tapered ball-end mill is composed of an inverted, truncated cone (conical frustum) and a partial sphere. The parametric equations for a revolved profile at an arbitrary position and orientation in the slice coordinate system are, 11 12 13 14 21 22 23 24 31 32 33 34 ( ) cos( ) ( )sin( ) ( ) cos( ) ( )sin( ) ( ) cos( ) ( )sin( ) x a F v u a F v u a v a y a F v u a F v u a v a z a F v u a F v u a v a = + + + = + + + = + + +   (A.5)  where, ( ) ( ) tan( ) r zF v N v N β= + −         for the taper  2 2( ) ( )F v R v R= − −               for the ball  (A.6)  r N  and zN  are the radius and height of the bottom of the frustum (top of the ball) measured from the tool tip. β  is the half-taper angle and R  is the radius of the ball. Appendix A: Analytical Geometry For The Parallel Slicing Method  230 These quantities are given by the user and are based on the seven-parameter tool model in Engin and Altintas [45] (see Figure A-2). The radial parametric variable, u , varies from 0 2pi…  for both ball and taper. The axial parametric variable, v , varies from 0 zv N= … for the ball and zv N h= …  for the taper. h  is the height of the taper measured from the tool tip. M u N M D O R R C R L M N h ST r z r z r z N α β z r D≠0 , R=R ≠0 , R =0 α=0 , β≠0 , h≠0 z r General End Mill Model Tapered Ball-End Mill N M  Figure A-2 – (left) The seven parameter tool model used in Engin and Altintas [45]. (right) Tapered ball-end mills have dimensions of the form shown.  The equation of the slice plane in the slice coordinate system is given by 0z = . Substituting the parametric form of each solid into 0z =  and rearranging allows isolation of one parametric variable. For the taper, v , can be solved for whereas for the ball, u  is easier. The isolated parametric variables for the ball and taper are given by, ( ) ( ) 2 2 1 31 31 32 34 32 2 2 1 31 31 32 33 32 tan( ) sin tan tan sin tan z r a a a N N u a a v a a a u a a β β − −    + − + −      =    + + +              for the taper   (A.7) Appendix A: Analytical Geometry For The Parallel Slicing Method  231  ( ) ( )( ) 1 134 33 31 22 2 2 32 31 32 sin tana a v au aa a R v R − −     − − = −     + − −             for the ball  (A.8)  Substituting the expressions for the isolated parametric variables, (Equations (A.7) and (A.8)), into Equations (A.5) and (A.6) gives the equation of the planar intersection curve as a function of one parametric variable only. Since a tapered-ball end mill is a composite three-dimensional solid with finite boundaries, the intersection of the tool with a plane will produce one of the following (see Section 3.2.3): • A single conic. • Multiple conics (i.e. a composite conic). • A truncated conic. • A truncated composite conic Examples of these intersection conditions are shown in Figure A-3. To predict the type of intersection graph, the number of knot points – or points of discontinuity – must be calculated. The number of knot points in the tool-plane intersection curves can be determined by setting the z-coordinate of the taper’s parametric equation to zero and then solving for u .   Appendix A: Analytical Geometry For The Parallel Slicing Method  232 X Y Z X Z Y X Y Z XY Z Ellipse Composite Conic - Ellipse-Circle Truncated Conic - Ellipse-Line Composite Truncated Conic - Ellipse-Circle-Line (a) (b) (c) (d) Ellipse Circle Ellipse Line Circle Ellipse Line Ellipse Knot 1 Knot 2 Knot 1 Knot 2 Knot 3 Knot 4 X Y Z X Y Z X Y Z X Y Z  Figure A-3 – Illustrations showing various tapered ball-end mill / plane intersection shapes [56], [57]. (a) Single conic (b) Composite conic (c) Truncated conic (d) Composite truncated conic   Appendix A: Analytical Geometry For The Parallel Slicing Method  233 1 131 31 32 32 ( ) tan , ( ) tana au G v G v a a pi− −        = − − −                (A.9)  Where, ( ) ( )( ) 1 34 33 2 2 31 32 ( ) sin ( ) tanr z a a vG v a c N v N β −   − −  =  + + −     (A.10)  Substituting the limits of v  ( zv N=  and v h=  ) into Equations (A.9) and (A.10) and calculating the number of real solutions for u  will give the number of knots on the graph. The real solutions are governed by the expression inside the 1sin−  of ( )G v , and will always occur in pairs. No real solutions for u  indicates either a full conic or no- intersection. Two or four real solutions imply that the graph is a truncated, composite or truncated composite conic. Two real solutions from zv N=  indicate that the discontinuities on the graph are due to the taper-ball connection, whereas two real solutions from v h=  show that the discontinuities are due to the bounded top of the frustum. The equations of the intersection curves resulting from the substitution of Equations (A.7) and (A.8) into (A.5) and (A.6) are complicated, so it is easier to identify the conics and reparameterize them into a simpler form. This form makes it easier to calculate common tangents between the intersection curves. The general equation of a conic is given by, 2 2 0Ax By Cx Dy Exy F+ + + + + =  (A.11)  Appendix A: Analytical Geometry For The Parallel Slicing Method  234 If F is not zero, Equation (A.11) can be written as, 2 2 1xx yy x y xym x m y m x m y m xy+ + + + = −  (A.12)  where A F… are the generalized conic coefficients and, xx yy x y xy xy A B C D E A m m m m m m F F F F F F = = = = = = (A.13)  If five discrete points on the intersection curve, ( ) ( )1 1 5 5, ,x y x y… , are calculated using Equations (A.5)-(A.8), the m-coefficients of the conic can be determined by solving a linear system of five equations. In other words, 12 2 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 2 2 4 4 4 4 4 4 2 2 5 5 5 5 5 5 1 1 1 1 1 xx yy x y xy m x y x y x y m x y x y x y m x y x y x y m x y x y x y m x y x y x y −   −         −        = −         −       −            (A.14)  Assuming that the conic is non-degenerate, the equation 2 4conic xy xx yyC m m m= −  (A.15)  can be used to determine the type of conic [128]. If 0conicC < , the conic is an ellipse and if 0conicC > , the conic is a hyperbola.. If 0conicC =  the conic is a parabola and if 0conicC < and xx yym m= , the conic is a circle. Knowing the type of conic and its normalized coefficients allows its parameters to be determined. This is explained in Section A2. Once the parameters of all conics in the Appendix A: Analytical Geometry For The Parallel Slicing Method  235 intersection curve are evaluated, the knots (if any) are reparameterized in terms of the new 2D parametric variable. Since the number and type of conics on the intersection graph are known, along with their defining parameters and knot points, the intersection curve can be created easily in a computer CAD software package. Knowledge of these properties also permits the swept area between successive intersection curves to be evaluated using analytical geometry. For tapered ball-end mills, the intersection shapes will be composed of conics. However, for other cutters, such as bull-nose end mills, the tool-plane intersection shapes may include higher-order curves such as toric sections. Toric sections are fourth order curves. Development of additional mathematics is necessary for these intersection curves and swept areas.  A2 Conic Coefficients for Various Conic Forms Rotated Ellipse aebe xe ye φe x y Rotated Ellipse  Figure A-4 – Parameters of a rotated ellipse. Appendix A: Analytical Geometry For The Parallel Slicing Method  236 The conic coefficients for a rotated ellipse (see Figure A-4) with center coordinates, ( ),e ex y , semi-major and semi-minor axes lengths, ea  and eb , rotated at an angle, eφ , are given as follows, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 cos sin sin cos 2 2 2cos sin ( ) e e e e e e e e e e e e e e e e e e e e e e A b a B b a C Ax Ey D By Ex E b a F Ax By Ex y a b φ φ φ φ φ φ = + = + = − + = − + = − = + + −   (A.16)  Rotated Hyperbola ah xh yh φh x y bh vertex STA SC A HyperbolaRotated  Figure A-5 – Parameters of a rotated hyperbola.   The conic coefficients for a rotated hyperbola (Figure A-5) with asymptote center coordinates, ( ),h hx y , semi-transverse and semi-conjugate lengths, ha  and hb , rotated at an angle, hφ  are as follows, Appendix A: Analytical Geometry For The Parallel Slicing Method  237 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 cos sin sin cos 2 2 2cos sin h h h h h h h h h h h h h h h h h h h h h h A b a B b a C Ax Ey D By Ex E b a F Ax By Ex y a b φ φ φ φ φ φ = − = − = − + = − + = + = + + −   (A.17)  In Figure A-5, STA is the semi-transverse axis and SCA is the semi-conjugate axis. The semi-transverse length of the hyperbola, ha , is the distance from the asymptote center to the vertex along the semi-transverse axis. The slope of the asymptotes is, h h b a ± Rotated Parabola xp yp φp x y vertex p Parabola focus Rotated  Figure A-6 – Parameters of a rotated parabola.   The conic coefficients for a rotated parabola (Figure A-6) opening along the positive x-axis with vertex coordinates, ( ),p px y  rotated at an angle, pφ , and distance from the vertex to the focus, p  are, Appendix A: Analytical Geometry For The Parallel Slicing Method  238 ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) 2 2 2 2 sin cos 2 4 cos 2 4 sin sin 2 4 cos sin p p p p p p p p p p p p p p p p p A B C Ax Ey p D Ex By p E F Ax By Ex y p x y φ φ φ φ φ φ φ = = = − − − = − − − = − = + + + +      (A.18)  Circle xc yc x y r Circle  Figure A-7 – Parameters of a circle.  The conic coefficients for a circle (Figure A-7) with center coordinates, ( ),c cx y and radius, cr , are, 2 2 2 1 1 2 2 0 c c c c c A B C x D y E F x y r = = = − = − = = + −   (A.19)  Appendix A: Analytical Geometry For The Parallel Slicing Method  239 A3 Calculation of Conic Parameters from Normalized Conic Coefficients The following equations give the characteristic parameters (see Section A2) for the various conic forms, in terms of normalized conic coefficients ( , , , ,xx yy x y xym m m m m ). Rotated ellipse (Figure A-4) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 4 4 1 arctan2 2 4 1 4 2 2 y xy x yy x xy y xx e e xx yy xy xx yy xy xy e yy xx xx e yy e xy e e xx yy xy e xx yy xx yy xy e xx yy xx yy xy m m m m m m m m x y m m m m m m m m m m x m y m x y F m m m F a m m m m m Fb m m m m m φ − − = = − −   = −   −   + + − = −   = + + − +     = + − − +           (A.20)  Rotated Hyperbola (Figure A-5) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 4 4 1 arctan2 2 4 1 4 2 2 y xy x yy x xy y xx h h xx yy xy xx yy xy xy h yy xx xx h yy h xy h h xx yy xy h xx yy xx yy xy h xx yy xx yy xy m m m m m m m m x y m m m m m m m m m m x m y m x y F m m m F a m m m m m Fb m m m m m φ − − = = − −   = −   −   − − − = −   = − + + − +     = + − − +          (A.21)  Appendix A: Analytical Geometry For The Parallel Slicing Method  240 Rotated Parabola (Figure A-6) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 1 arctan 2 2 sin cos sin 2 cos sin 4 sin cos cos sin cos sin 4 2 sin cos 4 2 sin cos 2 sin xy e yy xx p p p xx yy xy x p y p xx yy xy p p p p p p p p m m m F m m m m m p m m m JI x F H G H JIy F H G H EG D C H A B I D φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ φ   = −   −   = = = + = − + −   = − −     = − +   = + = + + = ( ) ( ) ( ) ( )cos sin cosp p p pC J C Dφ φ φ φ− = −           (A.22)  Circle (Figure A-7) 2 2 1 1 2 2 xx yy yx c c c c F m m m Fm F x y r x y F = = = − = − = + −     (A.23)  A4 Conic-Conic Tangent Line Solution As shown in Section 3.2.4 (Figure 3-7), if the distance between tool moves is small, a good approximation of the area cut between tool moves is given by the quadrilateral area formed from the tangent lines of successive intersection curves. These lines can be calculated through analytical planar 2D geometry. The general equation of a conic is as follows: Appendix A: Analytical Geometry For The Parallel Slicing Method  241 2 2 0Ax By Cx Dy Exy F+ + + + + =  (A.24)  The tangent lines can be expressed in the form, 1 0px qy+ + =  or 1 1p pxy x q q q  + = − − = −      (A.25)  Substituting the above expression for y  into the conic equation gives, 2 2 1 1 1 0px px pxAx B Cx D Ex F q q q      + + + + + − − + =              (A.26)  Expanding Equation (A.26) and collecting x terms yields, 2 2 2 2 2 2 0Bp Ep Bp Dp E B DA x C x F q q q q q q q       + − + + − − + + − =             (A.27)  which is the same as, ( ) ( ) ( )2 2 2 2 22 0Aq Bp Epq x Cq Bp Dpq Eq x Fq B Dq+ − + + − − + + − =  (A.28)  Using the quadratic equation to solve for x  gives: J K x L − ± = (A.29)  where, Appendix A: Analytical Geometry For The Parallel Slicing Method  242 ( ) ( ) ( ) ( ) ( ) 2 22 2 2 2 2 2 2 2 4 2 J Cq Bp Dpq Eq K Cq Bp Dpq Eq Aq Bp Epq Fq B Dq L Aq Bp Epq = + − − = + − − − + − + − = + −    (A.30)  From Equation (A.29) it is evident that there are three possible outcomes. Examples of these outcomes with a line and an ellipse are shown in Figure A-8. Ellipse-Line Intersections No Real Solutions Two Real Solutions One Real Solution (Tangency)  Figure A-8 – Examples of possible conic-line intersections with a line and an ellipse.   1. 0 real solutions:  ( ) ( )( )22 2 2 22 4Cq Bp Dpq Eq Aq Bp Epq Fq B Dq+ − − < + − + −  (A.31)  This situation occurs when a line does not intersect the conic at any point. 2. 2 real solutions: ( ) ( )( )22 2 2 22 4Cq Bp Dpq Eq Aq Bp Epq Fq B Dq+ − − > + − + −  (A.32)  Appendix A: Analytical Geometry For The Parallel Slicing Method  243 This occurs when a line passes in and out of conic. 3. 1 real solution (2 repeated solutions): ( ) ( )( )22 2 2 22 4Cq Bp Dpq Eq Aq Bp Epq Fq B Dq+ − − = + − + −  (A.33)  When only one real solution is possible, the line touches the conic at one point only. This is equivalent to a tangency. For tangency, the term under the square root must be zero. This means: ( ) ( ) ( )22 2 2 22 4 0Cq Bp Dpq Eq Aq Bp Epq Fq B Dq+ − − − + − + − =  (A.34)  Expanding (A.34), collecting q  terms and factoring out the 2q  term gives, ( ) ( ) ( ) 2 2 2 2 2 2 4 4 2 2 4 4 2 4 4 0 C AF q AD CE CDp EFp q E AB EDp CBp D p BFp − + − − + + − − + + − =   (A.35)  which gives an expression for q  in terms of the conic coefficents, A F… , and p at tangency. To solve for the common tangents between two conics, two of these expressions must be developed. For the first conic (A.35) is given by, ( ) ( ) ( ) 2 2 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 1 1 1 1 1 1 1 1 1 1 4 4 2 2 4 4 2 4 4 0 C A F q A D C E C D p E F p q E A B E D p C B p D p B F p − + − − + + − − + + − =   (A.36)  and for the second, ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 2 2 4 4 2 4 4 0 C A F q A D C E C D p E F p q E A B E D p C B p D p B F p − + − − + + − − + + − =   (A.37) Appendix A: Analytical Geometry For The Parallel Slicing Method  244 There are now two equations -- (A.36) and (A.37) – and two unknowns, p and q . To solve for p , each equation is divided by the leading coefficient of its 2q  term. [ ] [ ]( ) ( ) [ ]( ) ( ) 1 1 1 1 1 1 1 12 2 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 2 1 1 1 4 2 4 2 4 4 4 2 4 0 4 E F C D p A D C E q q C A F D B F p C B E D p E A B C A F − + − + −    − + − + −    + = −   and  [ ] [ ]( ) ( ) [ ]( ) ( ) 2 2 2 2 2 2 2 22 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 2 4 2 4 4 4 2 4 0 4 E F C D p A D C E q q C A F D B F p C B E D p E A B C A F − + − + −    − + − + −    + = −           (A.38)  As the equations get large, a substitution of variables simplifies the notation, ( ) ( )21 1 11 12 1 1 0 Q p R p SN p P q q M M + ++ + + =  and  ( ) ( )22 2 22 22 2 2 0 Q p R p SN p P q q M M + ++ + + =     (A.39)  where, Appendix A: Analytical Geometry For The Parallel Slicing Method  245 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 1 1 1 1 1 2 2 2 2 2 2 2 1 1 1 1 2 2 2 2 4 4 4 2 4 2 4 2 4 2 4 4 4 2 4 2 4 4 M C A F M C A F N E F C D N E F C D P A D C E P A D C E Q D B F Q D B F R C B E D R C B E D S E A B S E A B = − = − = − = − = − = − = − = − = − = − = − = −     (A.40)  Equation 2 in Equation (A.39) is subtracted from equation 1 to eliminate the 2q term. q  is obtained as, ( ) ( ) ( ) 2 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 ( ) ( ) M Q M Q p M R M R p M S M S q M N M N p M P M P − + − + − = − + −   (A.41)  Another substitution is used to reduce the of Equation (A.41) form to, 2Tp Up Vq Wp X + + = +   (A.42)  Where, 1 2 2 1 1 2 2 1 1 2 2 1 2 1 1 2 2 1 1 2 T M Q M Q U M R M R V M S M S W M N M N X M P M P = − = − = − = − = −     (A.43)  The expression for q  in Equation (A.42) can be substituted into either of the tangent-conic equations in Equation (A.39) to isolate an expression for p . Proceeding by substituting Equation (A.42) into the first equation of Equation (A.39), dividing out the Appendix A: Analytical Geometry For The Parallel Slicing Method  246 denominator and collecting terms, an expression is obtained for p  in terms of conic coefficients only. 4 3 2 1 2 3 4 5 0Z p Z p Z p Z p Z+ + + + =  (A.44)  Where, 2 2 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 3 1 1 1 1 1 1 1 1 1 2 4 1 1 1 1 1 1 2 2 5 1 1 1 2 2 2 2 2 2 Z QW M T N TW Z PTW N TX QWX M TU RW N UW Z M TV S W N VW Q X M U N UX PUW PTX RWX Z R X N VX PUX M UV PVW S WX Z M V PVX S X = + + = + + + + + = + + + + + + + + = + + + + + = + +    (A.45)  Equation (A.44) is a quartic polynomial and its roots can be determined analytically [2]. An alternative is to use the Jenkins-Traub numerical polynomial root-finding method [73] to calculate a fast, accurate solution. By solving Equation (A.44), four values for p can be calculated. Three outcomes are possible: • Four real solutions – four tangent lines (e.g. two rotated ellipses – two straight tangents, two cross tangents) • Two real solutions – two tangent lines. (e.g. two rotated ellipses – one ellipse is partially inside the other – two straight tangents) • No real solutions – no tangent lines. (e.g. two rotated ellipses – one ellipse is completely inside the other) The cases for two rotated ellipses in various positions and orientations are shown in Figure A-9. When p  is obtained, q  can be calculated from Equation (A.42). As Appendix A: Analytical Geometry For The Parallel Slicing Method  247 Conic-Conic Tangent Line 4 Tangent Rot. Ellipse - Rot. Ellipse Example 2 Tangent 0 Tangent Cross Tangent Straight Tangent Straight Tangent Lines Possible  Lines PossibleLines Possible  Figure A-9 – Possible combinations of tangent lines between two ellipses. The number of tangent lines is related to the position and orientation of each conic shape.  mentioned previously, there can be up to four pairs of ( ),p q  values. With these known, the y  and b  (slope and y-intercept) of the tangent lines can be determined through, 1p m b q q = − = −  (A.46)  Once the equations of the tangent lines are known, the points of tangency can be calculated. As stated earlier, the x -coordinate for a line-conic intersection is given by Eqs (A.29) and (A.30). When the line is tangent to the conic, the term under the square root vanishes. This reduces Equations (A.29) and (A.30) to, ( ) ( ) 2 2 2 2 2t Dpq Eq Bp Cq x Bp Epq Aq + − − = − +   (A.47)  Appendix A: Analytical Geometry For The Parallel Slicing Method  248 Substituting the tx  coordinate back into the expression for ty  in (A.25) and repeating for each conic and each set of lines gives all ( ),t tx y  points of tangency.  A4.1     Removal of Cross Tangents If the intersection curves are spaced relatively far apart (that is one is not partially or fully inside the other), four tangent line solutions will be calculated. Two of these are “cross tangents” and are not needed for swept area calculations. These lines must be identified and eliminated. Examples of cross tangents are shown in Figure A-10 (Lines 3 and 4). 0 20 40 60 0 20 40 60 x-coordinate y -c o o rd in a te 1 2 3 4 Outward Normals at Tangent Points 1 = straight tangents2 3 4 = cross tangents , ,  Figure A-10 – Illustration of the four tangent lines that can be drawn between the two ellipses shown [54], [55]. Lines 3 and 4 are cross tangents and must be eliminated from the swept area solution.  Appendix A: Analytical Geometry For The Parallel Slicing Method  249 Cross tangents have the property that the outward normals at tangent locations point in opposite directions.  Straight tangent lines have the same outward normal at each tangent point. In the x-y plane, the outward normal vector at the tangent points is: { } ( ) ( ) ( ) ( ) 2 2 2 2 2 2 2 2 2 2 x y Ax C Ey Ax C Ey Bx D Exn N n Bx D Ex Ax C Ey Bx D Ex + +    + + + + +    = =    + +     + + + + +     (A.48)  To check for cross tangents, the normal vectors at each tangent point (at each conic), 1N  and 2N , are calculated. These vectors are subtracted from each other. If the magnitude is zero, this line is a straight tangent (as both vectors are the same). If the result is 2 then the line is a cross tangent (as the vectors are in opposite directions).  A5 Knot-Conic Tangent Line Solution The following solution is used to calculate knot-conic lines for the parallel slicing method. Knot-conic lines are used to solve for the tangent lines between composite conics. The equation of a line through a point, ( 1x , 1y ) can be written as, 1 1 1 0px qy+ + =  (A.49)  Solving this expression for q  gives, Appendix A: Analytical Geometry For The Parallel Slicing Method  250 1 1 1 pxq y − − =  (A.50)  Substituting this expression into Equation (A.35) – the expression for line-conic tangency gives: ( ) ( ) ( ) 2 2 1 1 1 1 2 2 2 2 1 14 4 2 2 4 4 2 4 4 0 px pxC AF AD CE CDp EFp y y E AB EDp CBp D p BFp     − − − − − + − − +        + − − + + − =    (A.51)  Expanding, collecting terms, dividing out the 1y  term and collecting terms for p gives: 2 0Xp Yp Z+ + =  (A.52)  where, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 1 1 1 2 2 1 1 1 1 1 2 2 2 1 1 4 4 2 4 2 8 2 4 2 4 4 2 4 2 4 4 X C AF x D BF y CD EF x y Y C AF x CD EF y CE AD x y CB ED y Z E AB y CE AD y C AF = − + − + − = − + − + − + − = − + − + −    (A.53)  Solving Equation (A.52) with the quadratic equation gives: 2 1,2 4 2 Y Y XZp X − ± − =  (A.54)  Appendix A: Analytical Geometry For The Parallel Slicing Method  251 Note that p  becomes indefinite when 0X =  -- for instance when 1 0x =  and 1 0y = . In this case, the entire solution can be rotated and translated to a new point, then solved and rotated back to the original orientation. Once both values for p  have been determined, the q  values can be solved for using Equation (A.50). The slopes and y intercepts of the lines can be calculated as follows: 1p m b q q = − = −  (A.55)  The points of tangency can be determined using the line-conic tangency expression for x  (Equation (A.29)), and setting the expression under the square-root term to zero, ( ) 2 2 2 2 2t Dpq Eq Bp Cq x Bp Epq Aq + − − = − +   (A.56)  The y  coordinate of the tangent point is determined through, t ty mx b= +  (A.57)  252 Appendix B Cantilever Model of Tool for Static Deflection and Transverse Stress Calculations  B1 Overview of Cantilever Model of the Tool  Hf ac Lfixed E1 E2 E3 ENe-1 Cutting Forces Fixed Support (Tool Holder) E... E... Z ENe E4 Cutting Flute B e a m  E le m e n t Cutting Force Element Cutting Forces Workpiece Chip  Figure B-1 – Illustration of the cantilever finite element model of the tool used to calculate deflection and transverse stress at each node [54], [55].  To calculate the static deflection and transverse bending stress of the tool at each angular increment of reference cutting edge, a simple finite element model of the tool is employed. The model uses Timoshenko beam elements connected together in series with fixed boundary conditions at the tool holder (displacements in all directions equal zero at Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  253 the first node of the first element). This is essentially a cantilever bending problem. Using a series of elements allows local shape features to be taken into account, such as a reduced cross section and area moment of inertia in the cutting flute zone. By assembling the element stiffness matrices into a global stiffness matrix and the element force vectors into a global force vector, the nodal displacements can be solved for using a simple matrix inversion and multiplication. The transverse bending stress at each node can be calculated from the nodal displacements. An illustration of the cantilever model is shown in Figure B-1.  B2 Stiffness, Force and Displacement Equations for a Timoshenko Beam Element The elemental equations for static deflection of a 3D Timoshenko beam (see Figure B-2), neglecting torsion, are given in [30] as,  [ ] { } { } e ee K u F=  (B.1)  where, [ ] ( ) 1 1 2 3 2 3 3 1 1 3 2 3 2 0 0 0 0 0 0 0 0 0 12 0 0 6 0 12 0 0 6 0 0 12 6 0 0 0 12 6 0 0 0 6 0 0 0 6 0 0 6 0 0 0 6 0 0 0 0 0 0 0 0 0 01 0 12 0 0 6 0 12 0 0 6 0 0 12 6 0 0 0 12 6 0 0 0 6 0 0 0 6 0 0 6 0 0 0 6 0 0 e k k L L L L L k L k L k L kEIK k kL L L L L L k L k L k L k −    −   − − −   −    − =   −+ Φ     − − −   −    −    −         (B.2) Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  254  y x z Fx2 O θy θx 2 1 Fz2 Mx2 My2 Mx1 My1 L Fy2 Fx1 Fy1 Fz1  Figure B-2 – Nodal displacements and forces for the Timoshenko beam element in [30]. [54], [55].  and, { } { } 1 1 1 1 1 2 2 2 2 2 1 1 1 1 1 2 2 2 2 2 z y x y x z y x y xe z y x y x z y x y xe F F F F M M F F F M M u u u u u u uθ θ θ θ  =    =      (B.3)  [ ] e K  is the element stiffness matrix, { } e u  is the element deflection vector and { } e F  is the element force vector. The stiffness parameters in Equation (B.4) are given in [30]: Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  255 ( ) 2 2 2 1 2 3 2 2 2 4 2 (1 ) (4 ) (2 ) 12 6 12 6 4 12 7 2 1 4 e s s e e e e e A Lk k L k L I EI Ek G k A GL RI A R ν ν ν ν ν pi pi + Φ = = + Φ = − Φ + +Φ = = = + + + = =     (B.4)  sk  is the transverse shear form factor for a Timoshenko beam with a circular cross- section [66], υ  is the Poisson’s ratio of the tool’s material, E  is the Young’s modulus and G  is the shear modulus of the material. eI  is the element’s moment of inertia for a circular cross section, eA  is the element’s cross sectional area and L  is the length of the element. Both eI  and eA  are calculated at the middle of the element. The stiffness matrices for all eN  elements are constructed using Equation (B.2), where each element has a length of fixed e L L N =  length. The effective radius of each element, eR , is taken as the average radius at the middle of each element. Details of the tapered cutter geometry are given in [45] and the serrated edge parameters are given in [94]. The radius is scaled when calculating the area moment of inertia for the cross section, eI , and the cross-sectional area, eA , in the cutting flute zone. The scaling is to account for missing material at the flute cavities. In reality, the cross section over this zone has a complex, non-circular shape, which may or may not be symmetrical. Nemes et al. [101] developed equations for the cross-sectional moment of inertia and area of three and four fluted cutters and Hutchinson [66] gives the Timoshenko shear factor equation for a beam of arbitrary cross section. However, finite element models and experiments Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  256 [85] indicate that an effective radius of 0.8eff eR R=  in the cutting flute zone has been shown to provide a reasonable estimate of deflection. In other words:      0.8eff eR R=                      in the flute zone                       0 e fz H≤ ≤           eff eR R=                      in sold shank zone                   f eH z L< ≤   (B.5)  Where fixedL  is the distance from the tool tip to the fixed support (usually the tool holder) and ez  is the distance along the tool axis to the center of Timoshenko beam element, e , measured from the tool tip.  B3 Assembly of the Nodal Force Vector Once the element stiffness matrices have been formed, the nodal force vectors are assembled. This is performed by replacing the point load cutting forces (from the force prediction model) with equivalent forces at each node according to Figure B-3. X FEQX1 FEQX2 FX1 FX2 FX(Nb-1) FX(Nb) FX... Point Cutting Forces in X-Direction for element , Ee Equivalent Nodal Forces in X-Direction Node 1 Node 2 Y FEQY1 FEQY2 FY1 FY2 FY... Equivalent Nodal Forces in Y-Direction Point Cutting Forces in Y-Direction for element , Ee FY(Nb-1) FY(Nb) Node 2Node 1 FEQZ1 FEQZ2 FZ1 FZ2 FZ...Z Equivalent Nodal Forces in Z-Direction Point Cutting Forces in Z-Direction for element , Ee FZ(Nb-1) FZ(Nb) Node 1 Node 2  Figure B-3 – Free body diagrams of x,y and z forces on each beam element [54], [55].   Using free body diagrams and force and moment balances, Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  257 2 1 1 2 1 2 1 1 2 1 b b b b N N Xb b Xb EQX b b EQX EQX N N Yb b Yb EQY b b EQY EQY F z F F F F L L F z F F F F L L = = = = − = = − = = ∑ ∑ ∑ ∑     (B.6)  where 1EQXF , 2EQXF , 1EQYF  and 2EQYF  are the equivalent nodal forces in the x and y directions at nodes 1 and 2. 1 bb N= …  is the point load cutting force index and is a subset of the total cutting force index, 1 aa N= … . bN  is the number of point load cutting forces that are located along the length of the beam element. XbF  and YbF  are x and y point-load cutting forces at location along the tool axis, bz . bz  is measured in the element coordinate system with 0bz =  at node 1 and bz L=  at node 2. Equations (B.6) calculate the nodal forces by balancing the combined point-load forces and moments for that element. In the axial direction, the forces do not induce moments on the beam and so a force / moment balance is not necessary - the forces can simply be lumped at the nodes. A reasonable method is to lump all point-load cutting forces that are closer to node 1 at node 1. The same logic can be applied at node 2. This means,          1EQZ ZbF F=∑      for all point forces where            2b L z ≤           2EQZ ZbF F=∑     for all point forces where            2b L z >   (B.7)  Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  258 where 1EQZF  and 2EQZF  are the equivalent nodal forces in the z-direction at nodes 1 and 2. In other words, 1EQZF , the equivalent nodal z-force at node 1, is the sum of all axial cutting forces that are located to the left of the element’s mid point. Similarly, 2EQZF  is the sum of all axial cutting forces that are located to the right of the element’s mid point.  B4 Assembly of the Global Solution Once the element stiffness matrices and force vectors are formed, the global (or total) stiffness matrix, [ ]GK , and the global nodal force vector, { }GF  are assembled. The illustration on the left of Figure B-4 shows a finite element model of a tapered cutting tool. ,e fN  is node number f  for element e . eE  indicates Timoshenko beam element number e . The illustrations in the upper right of Figure B-4 show how the entries of the elemental stiffness matrices and nodal force vectors are assembled. Each element stiffness matrix, [ ] e K , which is 10 x 10, can be partitioned into four quadrants, [ ] ,11eK , [ ] ,12eK , [ ] ,21eK  and [ ] ,22eK , each of which are 5 x 5. [ ] ,e fgK  is a stiffness submatrix for element, e , that relates forces and moments at node, f , to displacements and slopes at node, g . Each element nodal force vector, which is 10 x 1, can be partitioned into two parts, { } ,1eF  and { } ,2eF , each of which are 5 x 1. { } ,e fF  is the nodal force subvector at node, f , for element, e . The total stiffness matrix is assembled by partially overlapping [ ] e K  with [ ] 1eK +  and adding the entries of [ ] ,22eK  with [ ] 1,11eK +  for all elements. The Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  259  Global Stiffness Matrix Global Nodal Force Vector E1 E2 E... E... ENe ENe-1 N 1,1 N Ne,2 + 5DOF 5DOF [K] e,11 [K]e,12 [K] e,21 [K] e+1,22 [K] e+1,21 [K] e+1,12 [K] e+1,11 [K] e,22 Addition of Element Stiffness Matrices Addition of Element Nodal Force Vectors 5DOF 5DOF+ {F} e,1 {F} e+1,2 {F} e,2 {F} e+1,2 [K] e [K] e+1 5DOF 5DOF Timoshenko Beam Element Model of Cutter {F} e {F} e+1 5DOF [K] 1 [K] 2 [K]Ne-1 [K]Ne {F} 1 {F} 2 {F} ... {F}Ne-1 {F}Ne {F} ... [K]... [K]... N 1,2 N 2,1 N 2,2 N ...,1 N Ne,1 N Ne-1,2 N Ne-1,1 N ...,2 5(Ne+1) 5 (N e + 1 ) 5 (N e + 1 ) + + + + + + + + + + + + Tool Holder  Figure B-4 – Illustration showing the finite element model of the cutting tool and the assembly of the global stiffness matrix and global nodal force vector [54], [55].   Appendix B: Cantilever Model of Tool for Deflection and Stress Calculations  260 total nodal force vector is assembled in a similar fashion -- the entries of { } ,2eF  are overlapped and summed with { } 1,1eF +  for each element. The illustration on the bottom right of Figure B-4, shows the assembled, global stiffness and global nodal force vectors. In the end, the total stiffness matrix will have dimensions of 5( 1)eN +  x 5( 1)eN +  and the total nodal force vector will be a 5( 1)eN +  x 1 vector. Boundary conditions are applied when solving for displacements. Since the tool is modeled as a cantilever beam, the displacements and slopes at the fixed support (node 1 of first element) will be zero. The first five rows and first five columns of the total stiffness matrix are "crossed-out" or removed when solving for the displacements.  The unknown displacements along the tool are obtained by solving the equation,     [ ] { } { }1G G GK F u− =       (B.8)  where [ ]GK  is the global stiffness matrix of the system, { }GF  is the global nodal force vector and { }Gu  is the vector of system displacements. In Equation (B.8), inverting the stiffness matrix is a computationally expensive operation, especially if the number of elements is large. Since the entries of the stiffness matrix do not change during simulations, the inversion [37] is performed only once at the beginning of the program and stored for later use during optimizations. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
https://iiif.library.ubc.ca/presentation/dsp.24.1-0066463/manifest

Comment

Related Items