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Generalized modeling of metal cutting mechanics Kaymakci, Mustafa 2009

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 GENERALIZED MODELING OF METAL CUTTING MECHANICS   by   MUSTAFA KAYMAKCI M.Sc. Koc University, Turkey, 2007     A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  in  The Faculty of Graduate Studies (Mechanical Engineering)      THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  August 2009 Β© Mustafa Kaymakci, 2009  ii  Abstract  Metal cutting is the most commonly used manufacturing process for producing parts with final dimensions. The aim of engineering science is to model the physics of the process which allows the simulation of part machining operations ahead of costly trials. There is a need to develop generalized models of cutting process which is applicable to various tool geometries and cutting processes in order to simulate machining of industri- al parts in virtual environment. This thesis presents a generalized mathematical model which can be used to predict turning, drilling, boring and milling processes. The tool geometry is adopted from ISO 13399 standards.  The rake face of the tool is mathematically modeled from ISO13399 model by considering tool geometry, en- gagement with the workpiece, feed and speed directions of cutting motion. Various geometric features of the tool, such as chamfer, nose radius, and cutting edge angles, are considered in developing coordinate transformation models between the machine motion and tool coordinate systems. The cutting forces on the rake face are defined in the direction of chip flow and per- pendicular to the rake face. The cutting force coefficients in the two directions are either identified mechanistically by conducting experiments specific to the tool geometry, or using orthogonal to oblique transformation of shear angle, average friction angle and shear stress. The friction and normal forces on the rake face are transformed to both stationary and rotating tool coordinate systems defined on the machine tool.  ii  Table of Contents  Abstract .............................................................................................................................. ii Table of Contents ............................................................................................................... ii List of Tables ..................................................................................................................... v List of Figures ................................................................................................................... vi Acknowledgements ......................................................................................................... xiii Nomenclature ..................................................................................................................... x 1. Introduction .................................................................................................................. 1 2. Literature Survey ......................................................................................................... 4 2.1 Overview ............................................................................................................ 4 2.2 Mechanics of Metal Cutting ............................................................................... 4 2.3 Cutting Force Models ......................................................................................... 6 2.3.1 Orthogonal Cutting ................................................................................ 6 2.3.2 Forces in Turning ................................................................................... 7 2.3.3 Forces in Milling .................................................................................. 10 2.4 Inserted Cutters ................................................................................................ 12 2.5 Generalized Mechanics of Machining ............................................................. 13 3. ISO Cutting Tool Geometry… .................................................................................. 14 3.1 Overview .......................................................................................................... 14 3.2 Planes ............................................................................................................... 14 3.3 Points and Angles ............................................................................................. 16 3.4 Dimensional Quantities .................................................................................... 20 3.5 Insert Shapes .................................................................................................... 21 3.5.1 Equilateral Equiangular Insert .............................................................. 21 3.5.2 Equilateral Nonequiangular Insert ....................................................... 22  iii  3.5.3 Nonequilateral Equiangular Insert ....................................................... 22 3.5.4 Nonequilateral Nonequiangular Insert ................................................. 22 3.5.5 Round Insert ......................................................................................... 23 3.6 Tool Coordinate Frames and Transformations ................................................ 24 3.6.1 Turning Operations .............................................................................. 25 3.6.2 Transformation of Rotary Tools........................................................... 31 3.7 Summary .......................................................................................................... 38 4. Generalized Geometric Model of Inserted Cutters .................................................... 39 4.1 Overview .......................................................................................................... 39 4.2 Mathematical Modeling of an Insert ................................................................ 40 4.3 Tool Coordinate Frames and Transformations ................................................ 42 4.3.1 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 43 4.3.2 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 44 4.3.3 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 45 4.4 Mathematical Relationships between Angles .................................................. 46 4.4.1 Normal Rake Angle ............................................................................. 46 4.4.2 True Cutting Edge Angle ..................................................................... 47 4.4.3 Inclination (Helix) Angle ..................................................................... 48 4.5 Examples .......................................................................................................... 51 4.5.1 Rectangular Insert with Corner Radius ................................................ 51 4.5.2 Rectangular Insert with Corner Chamfer ............................................. 59 4.5.3 Rhombic Turning Insert ....................................................................... 63 4.6 Summary .......................................................................................................... 67 5. Generalized Mechanics of Metal Cutting .................................................................. 68 5.1 Overview .......................................................................................................... 68 5.2 Rake Face Based General Force Model ........................................................... 68 5.3 Identification of Specific Cutting Coefficients ................................................ 72  iv  5.3.1 Orthogonal to Oblique Transformation ................................................ 72 5.3.2 Mechanistic Identification .................................................................... 79 5.4 Cutting Force Simulations and Validations ..................................................... 91 5.4.1 Turning Process Simulations................................................................ 91 Rhombic (ISO Style C) Insert: ........................................................................................ 95 5.4.2 Milling Process Simulations .............................................................. 102 5.5 Summary ........................................................................................................ 115 6. Conclusions .............................................................................................................. 116 Bibliography .................................................................................................................. 118   v  List of Tables   Table 3-I: Summary of Angles for Definition of Orientation of Cutting Edge and Rake Face. ................................................................................................................................. 20 Table 3-II: Insert Shapes Defined in ISO 13399. ............................................................ 23 Table 4-I: Inputs Used in the Model. ............................................................................... 51 Table 4-II: Inputs Used in the Analysis of the Insert with Chamfer. .............................. 59 Table 4-III: Inputs for the Rhombic Turning Insert. ........................................................ 63 Table 5-I: Cutting Coefficients and Tool Geometry Used in the Analysis. .................... 86 Table 5-II: Comparison of the Methods for the Solution of Oblique Shear Parameters. 88 Table 5-III: Orthogonal Turning Validation Experiments Cutting Conditions. .............. 93 Table 5-IV: Tool Geometry and Cutting Conditions for Rhombic Insert. ...................... 95 Table 5-V: Tool Geometry and Cutting Conditions for Square Insert. ......................... 100  vi  List of Figures  Figure 2-1: Orthogonal Cutting Process [5]. ..................................................................... 4 Figure 2-2: Illustration of Cutting Force Components [12][5]. ......................................... 5 Figure 2-3: Turning Operation Using a Tool with a Nose Radius and Colwell’s Approach. ........................................................................................................................... 8 Figure 2-4: Different Approaches for Predicting Chip Flow Direction: (a) Colwell, (b) Okushima and Minato, (c) Young et al. ............................................................................ 9 Figure 3-1: Planes Defined in ISO 3002 and ISO 13399. ............................................... 16 Figure 3-2: Illustration of the Cutting Reference Point (CRP). ....................................... 17 Figure 3-3: Demonstration of Tool Angles in ISO Standards for a Turning Tool. ......... 19 Figure 3-4: Dimensional Quantities Defined in the ISO Standards. ............................... 21 Figure 3-5: Mechanics of Oblique Cutting Process. ........................................................ 25 Figure 3-6: RTA and the Machine Coordinate Systems on a Lathe. ............................... 27 Figure 3-7: Cutting Edge Coordinate System along with the RTA Coordinate System. 28 Figure 3-8: Rake Face Coordinate System (System 3): Rotation around the 𝒀𝒀𝒀𝒀𝒀𝒀 Axis by the Amount of Rake Angle 𝜸𝜸𝜸𝜸. ....................................................................................... 29 Figure 3-9: Chip Flow Coordinate System (System 4): Rotation around the 𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀 Axis by the Amount of Chip Flow Angle 𝜼𝜼. ................................................................................. 30 Figure 3-10: Illustration of System * on a Rotary Tool with Four Flutes. ...................... 32 Figure 3-11: Transformation between System 1 and System *. ...................................... 34 Figure 3-12: Transformation between System 1 and System 2 for Rotary Tools. .......... 35 Figure 3-13: Transformation between Rake Face Coordinate System (System 3) and Cutting Edge Coordinate System (System 2). ................................................................. 36 Figure 3-14: Representation of the Chip Flow Coordinate System on a Milling Tool. .. 37 Figure 4-1: Local Coordinate Systems and Cutting Reference Points on Different Inserts.  ......................................................................................................................................... 40 Figure 4-2: Control Points on a Parallelogram Insert. ..................................................... 41 Figure 4-3: Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. ................................................................. 43 Figure 4-4: Frame  𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. ................................................................ 44  vii  Figure 4-5: Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. ................................................................. 45 Figure 4-6: Derivation of Normal Rake Angle. ............................................................... 47 Figure 4-7: Derivation of the True Cutting Edge Angle. ................................................. 48 Figure 4-8: Definition of Frame 𝑭𝑭𝑭𝑭 and Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. .................... 49 Figure 4-9: Definition of the Inclination (Helix) Angle. ................................................. 50 Figure 4-10: Illustrations of the Insert and the Cutter Body [55]. ................................... 51 Figure 4-11: Analytical Model of an Insert with a Corner Radius and Wiper Edge. ...... 52 Figure 4-12: Control Points of the Insert before the Rotations. ...................................... 54 Figure 4-13 : Control Points of the Insert after the Rotations. ........................................ 55 Figure 4-14: CAD Model of the Insert. ........................................................................... 56 Figure 4-15: Change of Local Radius along the Tool Axis. ............................................ 57 Figure 4-16: Change in Normal Rake Angle along the Tool Axis. ................................. 58 Figure 4-17: Change in Helix Angle along the Tool Axis. .............................................. 58 Figure 4-18: Catalogue Figure of the Insert Taken from Sandvik Coromant [55]. ......... 59 Figure 4-19: Illustration of Control Points and Dimensions for an Insert with Chamfer. 60 Figure 4-20: Change in the Local Radius along the Tool Axis for a Chamfered Insert. . 61 Figure 4-21: Change in Normal Rake Angle along the Tool Axis. ................................. 62 Figure 4-22: Change in Helix Angle along the Tool Axis. .............................................. 62 Figure 4-23: Catalogue Figures of the Turning Insert [55]. ............................................ 63 Figure 4-24: Control Points of the Rhombic Turning Insert. .......................................... 64 Figure 4-25: Position of the Cutting Edge for a Rhombic Turning Insert. ...................... 65 Figure 4-26: Change in the Normal Rake Angle along the Cutting Edge. ...................... 66 Figure 4-27: Change in the Helix Angle along the Cutting Edge. ................................... 66 Figure 5-1: Mechanics of Oblique Cutting [11]. ............................................................. 69 Figure 5-2: Summary of the Proposed Mechanistic Approach. ...................................... 71 Figure 5-3: Shear Plane Area Comparison: Orthogonal Cutting (Left) and Oblique Cutting (Right). ................................................................................................................ 73 Figure 5-4: Cutting Forces, Velocities, and Angles in Oblique Cutting [11]. ................. 74 Figure 5-5: Illustration of angle 𝜽𝜽𝜸𝜸. (𝑭𝑭𝑭𝑭,𝑭𝑭𝑭𝑭𝑭𝑭, and 𝑭𝑭𝑭𝑭𝑭𝑭 Are Projections of Forces onto Normal Plane). ................................................................................................................. 75 Figure 5-6: Summary of the Classical Approach in Cutting Coefficient Identification. . 78  viii  Figure 5-7: Block Diagram of the Iterative Method to Enhance the Theoretical Models.  ......................................................................................................................................... 84 Figure 5-8: Results of Helix Angle Predictions for Ti6Al4V. ......................................... 86 Figure 5-9: Results of Helix Angle Predictions for Al7075. ........................................... 87 Figure 5-10: Theoretical Identification Procedure. ......................................................... 89 Figure 5-11: Effect of Chip Thickness on the Cutting Coefficients. ............................... 90 Figure 5-12: Effect of Chip Thickness on the Chip Flow Angle. .................................... 90 Figure 5-13: Workpiece and 3- Component Dynamometer Fixed to Turret for Cutting Tests. ................................................................................................................................ 92 Figure 5-14: Sample Output Screen in CutPro 8.0 for Cutting Forces. ........................... 92 Figure 5-15: Forces simulated and measured for an orthogonal tool with 75Β° cutting edge angle for V=100 m/min. .................................................................................................. 94 Figure 5-16: Forces simulated and measured for an orthogonal tool with 75Β° cutting edge angle for V=150 m/min. .................................................................................................. 94 Figure 5-17: Illustration of Insert and the Holder Body [55]. ......................................... 95 Figure 5-18: Geometric Control Points of a Rhombic Turning Insert. ............................ 97 Figure 5-19: Comparison of Measured and Calculated Forces for the Rhombic Tool while the Width of Cut is 2 mm. ...................................................................................... 99 Figure 5-20: Comparison of Measured and Calculated Forces for the Rhombic Tool while the Width of Cut is 4 mm. ...................................................................................... 99 Figure 5-21: Illustration of Insert and the Holder Body [55]. ....................................... 101 Figure 5-22: Comparison of Measured and Calculated Forces for the Square Insert while the Width of Cut is 2 mm. ............................................................................................. 101 Figure 5-23: Comparison of Measured and Calculated Forces for the Square Insert while the Width of Cut is 4 mm. ............................................................................................. 102 Figure 5-24: Workpiece and 3-Component Dynamometer Fixed to Machine Table for Cutting Tests. ................................................................................................................. 103 Figure 5-25: Illustration of the Experimental Setup for Measurement of Cutting Forces.  ....................................................................................................................................... 104 Figure 5-26: 25 mm Diameter Shoulder End Mill from Sandvik Coromant [55]. ........ 104 Figure 5-27: Illustration of Control Points for the Shoulder Milling Insert. ................. 106  ix  Figure 5-28: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. 107 Figure 5-29: Measured and Predicted Forces for Al7050 Slot Milling. ........................ 108 Figure 5-30: 20 mm Diameter Bull-Nose Mill from Sandvik Coromant [55]. ............. 109 Figure 5-31: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. 110 Figure 5-32: Measured and Predicted Forces for Al7050 Slot Milling. ........................ 111 Figure 5-33: 20 mm Diameter Ball-End Mill from Sandvik Coromant [55]. ............... 112 Figure 5-34: Control Points of a Round Milling Insert ................................................. 113 Figure 5-35: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. 114 Figure 5-36: Measured and Predicted Forces for Al7050 Slot Milling. ........................ 115   x  Nomenclature πœ™πœ™π‘π‘  : Orthogonal shear angle πœ™πœ™π‘›π‘›  : Normal shear angle π›½π›½π‘Žπ‘Ž  : Friction angle πœπœπ‘ π‘  : Shear Stress 𝐴𝐴𝑠𝑠 : Shear plane area πœ™πœ™π‘–π‘– ,πœƒπœƒπ‘›π‘› ,πœƒπœƒπ‘–π‘–  : Oblique angles 𝐹𝐹 : Resultant force 𝐹𝐹𝑑𝑑  : Tangential force 𝐹𝐹𝑓𝑓  : Feed force πΉπΉπ‘Ÿπ‘Ÿ  : Radial force 𝐹𝐹π‘₯π‘₯  : Force in 𝑋𝑋 direction 𝐹𝐹𝑦𝑦  : Force in π‘Œπ‘Œ direction 𝐹𝐹𝑧𝑧  : Force in 𝑍𝑍 direction 𝐹𝐹𝑒𝑒  : Friction force on the rake face 𝐹𝐹𝑣𝑣 : Normal force on the rake face 𝐹𝐹𝑠𝑠 : Shear force on the shear plane 𝐹𝐹𝑛𝑛  : Normal force on the shear plane 𝐾𝐾𝑑𝑑𝑐𝑐  : Tangential cutting coefficient πΎπΎπ‘Ÿπ‘Ÿπ‘π‘  : Radial cutting coefficient πΎπΎπ‘Žπ‘Žπ‘π‘  : Axial cutting coefficient 𝐾𝐾𝑑𝑑𝑑𝑑  : Tangential edge coefficient πΎπΎπ‘Ÿπ‘Ÿπ‘‘π‘‘  : Radial edge coefficient πΎπΎπ‘Žπ‘Žπ‘‘π‘‘  : Axial edge coefficient 𝐾𝐾𝑒𝑒𝑐𝑐  : Friction cutting coefficient 𝐾𝐾𝑣𝑣𝑐𝑐  : Normal cutting coefficient 𝐾𝐾𝑒𝑒𝑑𝑑  : Friction edge coefficient 𝐾𝐾𝑣𝑣𝑑𝑑  : Normal edge coefficient 𝐾𝐾𝑑𝑑  : Tangential cutting pressure 𝐾𝐾𝑓𝑓  :  Feed cutting pressure  xi  πΎπΎπ‘Ÿπ‘Ÿ  : Radial cutting pressure π‘šπ‘š1 & π‘šπ‘š2 : Cutting constants β„Ž : Uncut chip thickness π‘Žπ‘Ž : Depth of cut 𝑏𝑏 : Width of cut 𝑏𝑏𝑑𝑑𝑓𝑓𝑓𝑓  : Effective width of cut 𝑐𝑐 : Feed rate 𝑉𝑉𝑐𝑐  : Cutting speed 𝑃𝑃𝑑𝑑  : Cutting power 𝑑𝑑𝐴𝐴𝑐𝑐  : Differential chip load ?Μ…?𝐴𝑐𝑐  : Average chip load 𝑑𝑑𝑑𝑑 : Differential cutting edge length 𝑑𝑑𝐴𝐴 : Differential cutting edge area 𝑁𝑁𝑓𝑓  : Number of cutting edges 𝐾𝐾 : Total number of discrete points along the cutting edge πœ™πœ™ : Immersion angle πœƒπœƒπ‘ π‘ π‘‘π‘‘  : Tool entry angle πœƒπœƒπ‘‘π‘‘π‘₯π‘₯  : Tool exit angle πœ…πœ…π‘Ÿπ‘Ÿ  :  Cutting edge angle πœ€πœ€π‘Ÿπ‘Ÿ  : Tool included angle πœ“πœ“π‘Ÿπ‘Ÿ  :  Approach angle πœ†πœ†π‘ π‘  : Inclination (oblique) angle Ξšπœ€πœ€  : Corner chamfer angle 𝛾𝛾𝑛𝑛  : Normal rake angle 𝛾𝛾𝑓𝑓  : Radial rake angle 𝛾𝛾𝑝𝑝  : Axial rake angle πœ‚πœ‚ : Chip flow angle ?Μ…?πœ‚ : Average chip flow angle π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  : Tool reference plane 𝑃𝑃𝑓𝑓  : Assumed working plane  xii  𝑃𝑃𝑝𝑝  : Tool back plane 𝑃𝑃𝑠𝑠 : Tool cutting edge plane 𝑃𝑃𝑛𝑛  : Cutting edge normal plane π‘ƒπ‘ƒπ‘šπ‘š  : Wiper edge normal plane 𝐴𝐴𝛾𝛾  : Rake face 𝐢𝐢𝐢𝐢𝑃𝑃 : Cutting reference point 𝐷𝐷𝑐𝑐  : Cutting Diameter 𝐿𝐿 : Insert length 𝑏𝑏𝑠𝑠 : Wiper edge length 𝑖𝑖𝑖𝑖 : Insert width π‘π‘π‘π‘β„Ž : Corner chamfer length π‘Ÿπ‘Ÿπœ€πœ€  : Corner radius 𝑃𝑃𝑖𝑖��⃗  : Control Points 𝑒𝑒�⃗  : Unit vector from center of the arc to circumference 𝑛𝑛�⃗  : Unit vector perpendicular to arc 𝐢𝐢 : Center of the arc 𝑍𝑍1.2οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½βƒ—  : Point on the tool axis   xiii  Acknowledgements  I am indebted to my supervisor, Prof. Yusuf Altintas, for the considerable guidance, support, and encouragement he has provided me throughout the duration of my studies at the University of British Columbia. I also wish to thank Dr. Doruk Merdol, who co- supervised much of this research, for his time and for teaching me many things. I also would like to thank Dr. Farrokh Sassani and Dr. Steve Feng for reading this thesis and involving in my thesis committee. It is an honor to work in an exceptionally professional environment in the Manufac- turing Automation Laboratory. I have enjoyed every moment I spent in and out of the lab with my colleagues. They are not just colleagues, but also good friends. I have lots of good memories I will remember for the rest of my life. I also would like to thank my friends in Vancouver for their support and friendship. Finally, I would like to thank my brother Orkun Kaymakci, my father Emin Kay- makci, and especially my mother Hamidiye Ozberk for their life-long love, for the kind of encouragement and support they have provided me throughout my entire education life, and for their absolute confidence in me. This thesis as well as all my previous success is dedicated to my family.  Chapter 1. Introduction  1  1. Introduction   In terms of operations that take place in industry, metal cutting is the most common manufacturing method in producing final shapes of mechanical parts with tight toler- ances and complex shapes. The main metal cutting processes can be listed as turning, milling, boring, and drilling. Metal cutting operations can be conducted manually or automatically by the help of Computer Numerical Control (CNC) tools. The motion of the machine tool on a CNC is controlled by the NC commands which are generated on computer – aided design/ computer – aided manufacturing (CAD/CAM) systems. The productivity and the output quality depends on the preparation of  NC programs, process planner, cutting conditions, workpiece material, cutter geometry, cutter material, machine tool rigidity and the performance of the CNC system. Economical and efficient manufacturing in metal cutting operations are vital in order to produce parts with desired accuracy and low cost. Understanding the mechanics of the metal cutting process assists in avoiding damages to the tool, machine and workpiece while improving productivity and accuracy. Cutting tools with replaceable inserts are widely used in machining industry. Insert geometries can vary depending on the cutting operation and workpiece material. While all turning tools are inserts, milling cutters use either inserts distributed on the cutter body or they are ground from solid carbides. Due to the geometric complexities of inserts with poorly defined rake face geometry, it is difficult to accurately model me- chanics of the cutting operation. As a result, an accurate cutting force model which can include various has been a challenge. Current literature focuses to determine separate mechanics models for each cutting process and even for each different cutter. In other words, researchers aimed to develop dedicated cutting models for turning, boring, drilling, and milling operations having different tool geometries. However, by understanding the fundamentals of metal cutting mechanics, it must be possible to develop a unified-generalized process model that can Chapter 1. Introduction  2  predict the cutting forces for a wide spectrum of machining operations practiced in industry. This thesis focuses on developing a generalized cutting mechanics model that can be applicable to most cutting processes. Cutting forces are first modeled on the rake face of the tool as friction and normal loads. The friction and normal forces are transformed to tool coordinate system. An improved and enhanced inserted cutter model has been developed based on the previous studies of Engin and Altintas [1]. Analytical model for all insert geometries defined in ISO 13399 standards have been derived. The forces are further transformed from the tool to cutting operation coordinate system to predict cutting forces, torque and power for various machining operations and cutting condi- tions. The thesis is organized as follows; Chapter 2 presents necessary background and literature review on metal cutting re- search. Cutting mechanics, previous models for chip thickness, chip flow models, as well as prediction of cutting forces for different operations are discussed. Previous geometric models of inserted cutters are summarized and generalization of metal cutting mechanics is reviewed. The geometric parameters of the cutting tools are defined according to ISO 13399 standards in Chapter 3. The reference planes and points required to describe the cutting angles are presented. Angular and dimensional quantities are considered to model an extensive variety of cutting tool geometries. The coordinate transformations needed to describe the cutting forces on rake face of the tool are presented. The geometric modeling of inserted cutters is presented in Chapter 4. Using the geometric identities described in Chapter 3, geometric control points are calculated analytically at the local coordinate system of the insert, which is placed on the cutter body using the orientation angles. Physical angles of the cutter required for cutting mechanics models are calculated with sample illustrations. The generalized modeling of cutting forces using friction and normal forces on the rake face of the cutter is presented in Chapter 5. The proposed mechanics model is Chapter 1. Introduction  3  compared with the current force models found in the literature. In addition to cutting forces, material model and calibration methods are also discussed and the transforma- tions between the models are summarized. Using the geometric modeling of inserted tools which is described in Chapter 4, experimental validations of the model are pre- sented. Simulations and measurements are presented for turning and milling operations for various types of inserted cutters and cutting conditions. The thesis is concluded in Chapter 6 with the summary of contributions and future research. Chapter 2. Literature Survey 4  2. Literature Survey 2.1 Overview A literature review of past research on modeling of cutting tool geometry and me- chanics are presented. The distribution of chip along the cutting edge is discussed, and the corresponding approaches in predicting the cutting forces are presented. 2.2 Mechanics of Metal Cutting Mechanics of metal cutting has been a subject of research for the last 60 years [2][3] [4] [5]. The process of mechanics are affected by parameters such as feed rate, depth of cut, cutting speed, cutting edge angle, rake angle, helix angle, and workpiece material [6]. Researchers have been trying to establish a relationship between these parameters and process mechanics. The early work done by Merchant [7][8] has been a foundation used in the modeling of cutting forces [9][10]. Merchant’s model is based on the concept of a steady process in which a chip is produced by shearing a strip of uncut metal continuously and uniformly, and the deformation of the chip takes place along a shear plane.  Figure 2-1: Orthogonal Cutting Process [5]. Chapter 2. Literature Survey 5  As shown in Figure 2-1, the uncut material approaches to the tool, sheared, and leaves parallel to the rake face of the tool with a new chip thickness. The width of the chip is assumed to be constant throughout the process. When the face of the tool is perpendicular to the plane of cutting (Figure 2-1), it is called orthogonal cutting, other- wise the process is considered to be oblique. Cutting forces occur in three directions in oblique cutting as shown in Figure 2-2 [11]. The component of the force acting on the rake face of the tool, normal to the cutting edge, is the tangential cutting force. The force component, acting in the radial direction, tending the push the tool away from the workpiece, is called the radial force. The third component is acting on the tool in the horizontal direction, parallel to the direction of feed, is referred as the feed force [12][13][5].  Figure 2-2: Illustration of Cutting Force Components [12][5]. Researchers [14] attempted to improve the models developed by Merchant. They in- cluded sophisticated mathematical formulations of frictional behavior on the tool rake face, high strain rate, work hardening of the workpiece material, and high temperature. Endres et al. [15] developed a cutting force model incorporating parameters of tool Chapter 2. Literature Survey 6  geometry. Lee and Shaffer [16] developed a more sophisticated model by introducing plasticity of the workpiece material into the solution. 2.3 Cutting Force Models Cutting force models using the tool geometry have been developed starting with simple orthogonal geometries and extended to the general turning and to milling processes. Following sections present a background on various cutting force models. 2.3.1 Orthogonal Cutting Orthogonal cutting provides the simplest geometry to study metal cutting mechan- ics, and models of the process have been advanced in the form of shear angle solutions. The best known approach is presented by Merchant [7] who defined the rake face contact as elastic with a constant coefficient of friction. The shear angle solution of this model is derived from minimum energy principle as:  πœ™πœ™π‘€π‘€π‘‘π‘‘π‘Ÿπ‘Ÿπ‘π‘ β„Žπ‘Žπ‘Žπ‘›π‘›π‘‘π‘‘ = πœ‹πœ‹4 + 𝛾𝛾𝑛𝑛2 βˆ’ π›½π›½π‘Žπ‘Ž2  (2.1) where 𝛾𝛾𝑛𝑛  and π›½π›½π‘Žπ‘Ž  are the normal rake angle and friction angle respectively. Lee and Shaffer [16] proposed the following shear angle relationship using a slip-line field approach:  πœ™πœ™πΏπΏπ‘‘π‘‘π‘‘π‘‘ = πœ‹πœ‹4 + 𝛾𝛾𝑛𝑛 βˆ’ π›½π›½π‘Žπ‘Ž  (2.2) The validity of the shear angle relationships have been evaluated [17], and improved models have been presented [18][19][20]. Assuming there is only sliding friction on the rake face and there is a thin primary shear deformation area, the above models lead to the following expressions for the magnitudes of the tangential force and feed force:  𝐹𝐹𝑑𝑑 = β„Ž 𝑏𝑏 οΏ½πœπœπ‘ π‘  cos(π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛)sinπœ™πœ™π‘π‘ cos(πœ™πœ™π‘π‘ + π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛)οΏ½ 𝐹𝐹𝑓𝑓 = β„Ž 𝑏𝑏 οΏ½πœπœπ‘ π‘  sin(π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛)sinπœ™πœ™π‘π‘ cos(πœ™πœ™π‘π‘ + π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛)οΏ½ (2.3) Chapter 2. Literature Survey 7  where β„Ž is the uncut chip thickness, 𝑏𝑏 is the width of cut and πœπœπ‘ π‘  is the shear yield stress of the workpiece material. Since it is difficult to predict the shear and the friction angles, a simplified mechanistic model of cutting forces has been developed. The normal approach in practice is to combine the effects of shear angle, rake angle, and friction angle under a parameter called as the specific cutting pressure:  𝐹𝐹𝑑𝑑 = β„Žπ‘π‘πΎπΎπ‘‘π‘‘ 𝐹𝐹𝑓𝑓 = β„Žπ‘π‘πΎπΎπ‘“π‘“  (2.4) where 𝐾𝐾𝑖𝑖  is the specific cutting pressure of direction 𝑖𝑖. Another approach to calcu- late the forces is using an exponential force model in which specific cutting force pressures have been expressed as an exponential function of the chip thickness. Sabber- wal and Koenigsberger [21][22] used this approach and obtained specific cutting coefficients experimentally. Their cutting force equation has been stated by:  𝐹𝐹𝑑𝑑 = πΎπΎπ‘‘π‘‘π‘π‘β„Žπ‘šπ‘š1 πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿπ‘π‘β„Žπ‘šπ‘š2  (2.5) where 𝐾𝐾𝑑𝑑 ,πΎπΎπ‘Ÿπ‘Ÿ ,π‘šπ‘š1, and π‘šπ‘š2 are experimentally calibrated empirical constants. It is possible to account for the edge forces by linearizing the cutting force expression. Linearization leads to a formulation of total cutting forces that are proportional to the undeformed chip cross sectional area and ploughing forces that are proportional to the length of the active cutting edge:  𝐹𝐹𝑑𝑑 = πΎπΎπ‘‘π‘‘π‘π‘π‘π‘β„Ž + 𝐾𝐾𝑑𝑑𝑑𝑑𝑏𝑏 πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿπ‘π‘π‘π‘β„Ž + πΎπΎπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘π‘ (2.6) In this thesis, a linear cutting force model with edge coefficients is used. The advan- tage of linear cutting coefficient model is that it is more compatible with other process models, i.e. stability calculations require constant cutting coefficient in order to solve the differential equations. With the nonlinear coefficient model, the differential equations in stability calculation will be nonlinear. Chapter 2. Literature Survey 8  2.3.2 Forces in Turning Most turning tools have oblique geometry with a nose radius (π‘Ÿπ‘Ÿπœ€πœ€), a cutting edge an- gle (πœ…πœ…π‘Ÿπ‘Ÿ ), and an inclination (oblique) angle (πœ†πœ†π‘ π‘ ). It is possible to extend the orthogonal cutting model by introducing the concept of equivalent chip thickness [23]. The equiva- lent chip thickness combines the effects of nose radius and cutting edge angle on the cutting forces. At constant velocity, it has been found that the cutting forces can be expressed as a function of equivalent chip thickness [23][24]. In this model, the direction of the in-plane feed force is described by the chip flow angle πœ‚πœ‚, using the following equations (Figure 2-3):  𝐹𝐹𝑦𝑦 = 𝐹𝐹𝑓𝑓 sin πœ‚πœ‚ 𝐹𝐹𝑧𝑧 = 𝐹𝐹𝑓𝑓 cos πœ‚πœ‚ (2.7)  Figure 2-3: Turning Operation Using a Tool with a Nose Radius and Colwell’s Approach. The chip flow angle (πœ‚πœ‚) has been modeled mostly empirically without considering the mechanics of the process. Nevertheless, these models are fairly successful in predict- ing chip flow angle, provided they are used at certain cutting conditions. Colwell [25] suggested that, without obliquity, the feed force is perpendicular to the line connecting the two end points of the active cutting edge. Okushima and Minato [26] proposed that the average chip flow (?Μ…?πœ‚) is the summation of elemental flow angles over the entire length of cutting edge: Chapter 2. Literature Survey 9   ?Μ…?πœ‚ = βˆ«πœ‚πœ‚(𝑠𝑠)𝑑𝑑𝑠𝑠 βˆ«π‘‘π‘‘π‘ π‘   (2.8) where πœ‚πœ‚(𝑠𝑠) is the direction of each unit’s elemental surface normal and 𝑠𝑠 is the arc length along the cutting edge. For the case of a straight oblique cutting edge, Stabler [27] stated that the chip flow angle is equal to the inclination angle. Young et al. [28] pub- lished a combined approach which assumed Stabler’s chip flow rule was valid for infinitesimal chip widths and summed the directions of the elemental friction forces in order to obtain the direction of chip flow. Dividing the tool – chip interface into small elements and calculating the force contribution of each element in 𝑋𝑋 and π‘Œπ‘Œ directions, they estimated the direction of chip flow as (Figure 2-4):  ?Μ…?πœ‚ = tanβˆ’1 �∫ sin πœ‚πœ‚(𝑠𝑠)𝑑𝑑𝐴𝐴 ∫ cos πœ‚πœ‚(𝑠𝑠)𝑑𝑑𝐴𝐴� (2.9)  Figure 2-4: Different Approaches for Predicting Chip Flow Direction: (a) Colwell, (b) Okushima and Minato, (c) Young et al. Wang [29] improved Young et al.’s method [28] by incorporating tool inclination angle and normal rake angle. In Wang’s model, cutting region is also divided into many local cutting elements and he assumed that the local chip flow for each element is collinear with that element’s friction force, i.e. , Stabler’s chip flow rule is applied for each element. The above methods for obtaining the direction of chip flow and feed force are empirical methods and cannot be applicable to all cutting operations. Usui et al. [30][31] have proposed and upper bound model for oblique cutting with a non-straight cutting edge. Chapter 2. Literature Survey 10  The first step in this thesis is to develop a simplified but complete cutting force model for oblique non-straight cutting edges. The model uses the mechanics on the rake face of the tool to account for friction and shear forces. This model is described in Chapter 5. 2.3.3 Forces in Milling The milling process differs from the turning process, because the generated chips, hence the forces are discontinuous, and periodic. Cutting forces occurred during milling is one of the most important parameters in order to improve the productivity and part quality, because deflection, tool breakage, surface quality, and form errors are mainly influenced by cutting forces. Determination of chip formation is the first step in mechanistic modeling of cutting forces. Early study of Martelotti [32][33] showed that the path of the tool is trochoidal, rather than circular because of the combined rotation and translation of the tool towards the workpiece. Martelotti also claimed that when the feed per tooth is much smaller than the tool radius, circular tool path assumption is valid and the error is negligible:  β„Ž(πœ™πœ™) = 𝑐𝑐 sinπœ™πœ™ (2.10) where β„Ž is the instantaneous chip thickness, 𝑐𝑐 is the feed rate, and πœ™πœ™ is the immer- sion angle of a tooth. Milling force models in the literature can be classified into two categories. In mechanistic models, the focus is to derive a relationship between cutting forces and process parameters such as tool geometry, workpiece material, and cutting conditions. Early work by Koeingsberger and Sabberwal [22] used experimentally determined cutting coefficients and related chip load to calculate cutting forces:  𝐹𝐹 = πΎπΎπ‘ π‘ π‘Žπ‘Žβ„Ž 𝐾𝐾𝑠𝑠 = πΆπΆβ„Žπ‘₯π‘₯  (2.11) where 𝐾𝐾𝑠𝑠 is the cutting pressure, π‘Žπ‘Ž is the radial depth of cut, β„Ž is the instantaneous chip thickness, 𝐢𝐢 and π‘₯π‘₯ are empirical constants. This model is considered as the first complete model in this category. However, calibration method used in this study is far from the physics of the process, because an empirical curve fitting technique was used Chapter 2. Literature Survey 11  instead of employing cutting laws. Tlusty and McNeil [34], Kline et al. [35], Sutherland and DeVor [36], and Altintas and Spence [37] have improved and adopted the empirical method in their models. Armarego and Deshpande [38] proposed linear cutting force model by introducing edge force components:  𝐹𝐹𝑑𝑑 = πΎπΎπ‘‘π‘‘π‘π‘π‘Žπ‘Žβ„Ž + πΎπΎπ‘‘π‘‘π‘‘π‘‘π‘Žπ‘Ž πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿπ‘π‘π‘Žπ‘Žβ„Ž + πΎπΎπ‘Ÿπ‘Ÿπ‘‘π‘‘π‘Žπ‘Ž πΉπΉπ‘Žπ‘Ž = πΎπΎπ‘Žπ‘Žπ‘π‘π‘Žπ‘Žβ„Ž + πΎπΎπ‘Žπ‘Žπ‘‘π‘‘π‘Žπ‘Ž (2.12) In Eq. (2.12), indices 𝑑𝑑 and 𝑐𝑐 represent the edge and cutting force components, re- spectively. Specific cutting coefficients and specific edge coefficients can be determined by applying linear regression to average cutting forces measured at different feed rates and this method is widely used in literature [39]. Mechanistic approach has limitations on milling with complex tools which have va- riable geometry along the axis of tool. Therefore, specific cutting coefficients are calculated as functions of shear stress, shear angle and friction angle [40] [41].  This method is called mechanics of milling model. In mechanics of milling model, oblique cutting force model is implemented to calculate the cutting coefficients [42][43]. This model is useful for general application to different cutters and it is applicable to cutters with variable geometry. Although orthogonal cutting database preparation is a time consuming process, it is very effective and accurate in calculating the cutting force coefficients [44]. In this thesis, since the aim is to obtain a generalized cutting force model that is ca- pable of covering different cutting processes and cutter geometries, mechanics of milling approach is used to calculate the cutting coefficients. Adaptation of this approach into the proposed cutting force model is demonstrated in Chapter 5. It is shown that the model can accurately predict the cutting forces for different cutter geometries and processes.   Chapter 2. Literature Survey 12  2.4 Inserted Cutters Inserted (indexable) cutters are widely used in industry. In turning operations, most of the tools use inserts with wide selection of shapes and geometries. In milling, inserted mills with large diameters are widely used for machining operations such as rough and finish machining. Compared to solid type cutters, inserted end mills have the following advantages: β€’ Higher material removal rate β€’ More stable machining without chatter β€’ Large cutter diameter availability β€’ Better chip extraction performance β€’ Longer tool life β€’ Lower setup costs There have been only few studies available in modeling indexed cutters. Fu et al. [45] presented a model for inserted face milling cutters. They included corner radius, calculated equivalent axial and radial rake angles, and used experimentally identified cutting coefficients in their force model, which is improved in [46]. Various optimiza- tion methods with feed rate scheduling and surface error models have been studied by implementing mechanistic models in milling. Kim et al. [47] presented a feed rate scheduling algorithm for indexable end mills. Gu et al. [48] presented a model to predict the surface flatness in face milling. Ko and Altintas [49] developed a model for plunge milling operation using inserted cutters. Choudhury and Mathew [50] adopted non- uniform pitch angles to their face milling model. Most of the work done on inserted cutters is for certain types of cutters and very few researchers attempted to generalize a cutting model which can be applied to all cutters used in industry. Engin and Altintas [1] presented a cutting force model for indexable end milling. They generalized the envelope of the indexable end mill, and predicted cutting forces to evaluate surface roughness and chatter stability. They considered the insert edge on the cutter envelope to be straight line and used average cutting coeffi- cients varying along the axial disk elements. Also, the uncut chip thickness was calculated from a geometric model. This is the first approach to generalized inserted Chapter 2. Literature Survey 13  cutter geometries. However, this model cannot account for corner radius, chamfer edge, and wiper edge. In addition, insert shapes used in this model are not capable of resem- bling all insert shapes currently being used in industry. 2.5 Generalized Mechanics of Machining As described throughout this chapter, there are various successful cutting force models to determine the cutting mechanics for different operations. However, these models are mostly limited by specific machining process, cutter geometry and cutting conditions. There is a lack of a fundamental approach that can combine different models by using the mechanics of metal cutting. Armarego [51][52] proposed a unified cutting modeling approach by combining oblique cutting operations and defining the cutting forces in shear and friction directions. These unique studies to generalize the cutting models are the first and only approach so far, but he concluded his work by defining individual empirical equations for different cutting processes to predict the machining parameters as functions of tool geometry and cutting conditions. The aim of this thesis is to determine the cutting forces by using fundamentals of metal cutting mechanics. The cutting forces are expressed as friction and normal forces on the rake face, and by using the cutter geometry, forces are transformed to machine coordinates. This model can be applied to different metal cutting processes, such as, turning, milling, boring, and other cutting operations.  Chapter 3. ISO Cutting Tool Geometry 14  3.  ISO Cutting Tool Geometry  3.1 Overview In this chapter, definitions of the cutting tool geometry elements stated in ISO 3002 [53]  and ISO 13399 [54] standards are summarized. These standards are published by the Organization for Standardization (ISO) and include the definitions of reference planes, major and minor cutting edge geometries, tool holders, connection elements, etc. Although the illustrations and figures in this chapter are mostly for turning tools and inserted cutters, they are applicable to the geometry of different individual tools, such as single-point tools, drills, and milling cutters. In the following sections, standard definitions are summarized and outlined in four different parts. In the first part, reference planes which are used to define the cutting angles are defined. The planes used in standards are defined for a selected point on the cutting edge. Since the most tool angles are designated by planes, so that the angles are defined for the same selected point on the cutting edge. In the second part, tool angles and other angular quantities are defined to describe the orientation of the cutting edge. In the third part, dimensional quantities necessary to describe the cutting edge are defined and finally, definitions of insert shapes, which are highly used in this thesis, are pre- sented. For the sake of simplicity, definitions of angular and dimensional quantities related only to major cutting edge are given. 3.2 Planes Tool Reference Plane (π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ): A plane through the selected point on the cutting edge, so chosen as to be either parallel or perpendicular to a plane or axis of the. It is generally oriented perpendicular to the assumed direction of primary motion. For an ordinary turning system, it is a plane parallel to the base of the tool. For a milling cutters and drills, it is a plane containing the tool axis. Chapter 3. ISO Cutting Tool Geometry 15  Assumed Working Plane (𝑃𝑃𝑓𝑓): A plane through the selected point on the cutting edge and perpendicular to the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  and so chosen as to be either parallel or perpendicular to a plane or an axis of the. It is generally oriented parallel to the assumed direction of feed motion. For ordinary lathe tools it is a plane perpendicular to the tool axis. For drills, facing tools, and parting-off tools, it is plane parallel to the tool axis. For milling cutters, it is a plane perpendicular to the tool axis. Tool Back Plane (𝑃𝑃𝑝𝑝): Tool back plane is a plane through a selected point on the cutting edge and perpendicular both to the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  and to the assumed working plane 𝑃𝑃𝑓𝑓 . Tool Cutting Edge Plane (𝑃𝑃𝑠𝑠): Tool cutting edge plane is a plane tangential to the cutting edge at the selected point and perpendicular to the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ . Cutting Edge Normal Plane (𝑃𝑃𝑛𝑛 ): Cutting edge normal plane is perpendicular to the cutting edge at the selected point on the cutting edge. Wiper Edge Normal Plane (π‘ƒπ‘ƒπ‘šπ‘š ): Wiper edge normal plane is a plane through the intersection of the reference planes 𝑃𝑃𝑝𝑝  and π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  perpendicular to the wiper edge. All the planes defined above are illustrated in Figure 3-1. Chapter 3. ISO Cutting Tool Geometry 16   Figure 3-1: Planes Defined in ISO 3002 and ISO 13399. 3.3 Points and Angles Cutting Reference Point (𝐢𝐢𝐢𝐢𝑃𝑃): Cutting reference point is the theoretical point of the tool from which the major functional dimensions are taken. For the calculation of this point the following cases are applied. Figure 3-2 shows the cutting reference point for different types of cutting edges: Case 1: Cutting Edge Angle (πœ…πœ…π‘Ÿπ‘Ÿ) ≀ 90Β° - the point is the intersection of:  the tool cutting edge plane 𝑃𝑃𝑠𝑠, the assumed working plane 𝑃𝑃𝑓𝑓 , and the tool rake plane. Case 2: Cutting Edge Angle (πœ…πœ…π‘Ÿπ‘Ÿ) β‰₯ 90Β°- the point is the intersection of: the as- sumed working plane 𝑃𝑃𝑓𝑓 , a plane perpendicular to assumed working plane and tangential to the cutting corner, and the tool rake plane. Case 3: ISO tool styles D and V (55Β° and 35Β° rhombic inserts respectively) with on- ly axial rake.  The point is the intersection of: a plane perpendicular to the primary feed Chapter 3. ISO Cutting Tool Geometry 17  direction and tangential to the cutting edge (tangential point), a plane parallel to the feed direction through the tangential point, and the tool rake plane. Case 4: Round inserts - a) feed direction parallel to the tool axis, primary used for turning tools.  The point is the intersection of: a plane perpendicular to the primary feed direction and tangential to the cutting edge, a plane parallel to the feed direction through the tangential point, and the tool rake plane;   b) feed direction perpendicular to the tool axis, primarily used for milling tools.  The point is the intersection of: a plane perpendi- cular to the primary feed direction and tangential to the cutting edge, a plane parallel to the feed direction through the tangential point, and the tool rake plane.  Figure 3-2: Illustration of the Cutting Reference Point (CRP).    Chapter 3. ISO Cutting Tool Geometry 18  Tool Radial Rake Angle (𝛾𝛾𝑓𝑓): Tool rake angle is the angle between rake face and tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  measured in assumed plane 𝑃𝑃𝑓𝑓 . It is also called side rake angle. Tool Axial Rake Angle (𝛾𝛾𝑝𝑝): Tool axial rake angle is the angle between rake face and tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  measured in the tool back plane 𝑃𝑃𝑝𝑝 . It is also called back rake angle. Tool Normal Rake Angle (𝛾𝛾𝑛𝑛 ): Tool normal rake is the angle for major cutting edge between the rake face and the reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  measured in plane 𝑃𝑃𝑛𝑛 . Tool Included Angle (πœ€πœ€π‘Ÿπ‘Ÿ ): It is the angle between the tool cutting edge plane 𝑃𝑃𝑠𝑠, and tool minor cutting edge plane. Simply, it is the angle between the major and minor cutting edges of a cutting item. Tool Cutting Edge Angle (πœ…πœ…π‘Ÿπ‘Ÿ ): Tool cutting edge angle is the angle between 𝑃𝑃𝑠𝑠 and plane 𝑃𝑃𝑓𝑓  measured in the reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ . In other words, it is the angle between major cutting edge and the direction of major feed. Tool Approach Angle (πœ“πœ“π‘Ÿπ‘Ÿ ): The angle between the tool cutting edge plane 𝑃𝑃𝑠𝑠 and the tool back plane 𝑃𝑃𝑝𝑝  measured in the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ . Tool approach angle is only defined for the major cutting edge. Thus at any selected point on the major cutting edge, the following equation is valid:  πœ…πœ…π‘Ÿπ‘Ÿ + πœ“πœ“π‘Ÿπ‘Ÿ = 90Β° (3.1) Tool Cutting Edge Inclination Angle (πœ†πœ†π‘ π‘ ): It is the angle between the cutting edge and the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  measured in the tool cutting edge plane 𝑃𝑃𝑠𝑠. Tool cutting edge inclination angle is also referred as oblique angle or helix angle in milling tools. Corner Chamfer Angle (Ξšπœ€πœ€): Corner chamfer angle is the angle of a chamfer on a corner measured from the major cutting edge. Angles defined in this section are shown in Figure 3-3 and Figure 3-4 and the defi- nitions of these angles are summarized in Table 3-I. Chapter 3. ISO Cutting Tool Geometry 19   Figure 3-3: Demonstration of Tool Angles in ISO Standards for a Turning Tool.      Chapter 3. ISO Cutting Tool Geometry 20  Table 3-I: Summary of Angles for Definition of Orientation of Cutting Edge and Rake Face. ANGLE Definition Angle Between Measured In Plane Orientation Of the Cutting Edge πœΏπœΏπ’“π’“  Tool Cutting Edge Angle 𝑃𝑃𝑠𝑠 𝑃𝑃𝑓𝑓  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ 𝝍𝝍𝒓𝒓 Tool Approach Angle 𝑃𝑃𝑠𝑠 𝑃𝑃𝑝𝑝  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ 𝝀𝝀𝑭𝑭 Tool Cutting Edge Inclination 𝑃𝑃𝑠𝑠 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑠𝑠 πœΊπœΊπ’“π’“ Tool Included Angle 𝑃𝑃𝑠𝑠 𝑃𝑃𝑠𝑠 π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ Orientation of the Rake Face (π‘¨π‘¨πœΈπœΈ) 𝜸𝜸𝜸𝜸 Tool Normal Rake 𝐴𝐴𝛾𝛾  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑛𝑛 πœΈπœΈπ’‡π’‡ Tool Side Rake 𝐴𝐴𝛾𝛾  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑓𝑓 πœΈπœΈπ’‘π’‘ Tool Back Rake 𝐴𝐴𝛾𝛾  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑝𝑝 3.4 Dimensional Quantities Dimensional quantities are illustrated in Figure 3-4. Cutting Diameter (𝐷𝐷𝑐𝑐): Cutting diameter is the diameter of a circle created by a cutting reference point (𝐢𝐢𝐢𝐢𝑃𝑃) revolving around the tool axis of a rotating tool item. Insert Length (𝐿𝐿): Insert length is the theoretical length of the cutting edge of a cut- ting item over sharp corners. Wiper Edge Length (𝑏𝑏𝑠𝑠): Wiper edge length is the measure of the length of a wiper edge of a cutting item. Insert Width (𝑖𝑖𝑖𝑖): Insert width is the distance between two sides of an insert when the inscribed circle cannot be used because of the shape of the insert. Corner Chamfer Length (π‘π‘π‘π‘β„Ž): Corner chamfer length is the nominal length of a chamfered corner measured in the π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  plane. Corner Radius (π‘Ÿπ‘Ÿπœ€πœ€): Corner radius is the nominal radius of a rounded corner meas- ured in the π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  plane. Chapter 3. ISO Cutting Tool Geometry 21   Figure 3-4: Dimensional Quantities Defined in the ISO Standards. 3.5 Insert Shapes In ISO 13399 standard, insert shapes are categorized in five main different shapes [54]. These categories are equilateral equiangular, equilateral nonequiangular, nonequi- lateral equiangular, nonequilateral nonequiangular and round inserts.  For each main category, there are different shapes of inserts are defined. Table 3-II summarizes the definitions stated below with sample figures. 3.5.1 Equilateral Equiangular Insert Equilateral equiangular insert is a type of cutting item of regular geometric shape with sides of equal length and equal tool included angles. This category contains inserts with the ISO shape codes: T, S, P, O, and H. 1. Triangular Insert (T): Insert with three equal sides and three equal in- ternal angles with the included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 60Β°. 2. Square Insert (S): Insert with four equal sides and four equal internal angles with the tool included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 90Β°. 3. Pentagonal Insert (P): Insert with five equal sides and five equal internal angles with the included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 108Β°. Chapter 3. ISO Cutting Tool Geometry 22  4. Hexagonal Insert (H): Insert with six equal sides and six equal internal angles with the tool included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 120Β°. 5. Octagonal Insert (O): Insert with eight equal sides and eight equal inter- nal angles with the tool included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 135Β°. 3.5.2 Equilateral Nonequiangular Insert It is the type of cutting item of regular geometric shape with sides of equal length and non-equal tool included angles. This category contains inserts with the following ISO shape codes: C, D, E, M, and V for rhombic (diamond) inserts and W for trigon inserts. 1. Rhombic (Diamond) Insert (C, D, E, M, V): Insert with two cutting corners, four sides of equal length and four internal angles none of which are equal to 90Β°. 2. Trigon Insert (W): Insert with a generally triangular shape with enlarged tool included angles. The edges between the corners may be curved or straight. 3.5.3 Nonequilateral Equiangular Insert It is the type of cutting item of regular geometric shape with sides of non-equal length and equal tool included angles. This category contains inserts with the ISO shape code L. Rectangular Insert (L): Insert with four sides and four equal internal an- gles with the tool included angle (πœ€πœ€π‘Ÿπ‘Ÿ ) of 90Β°. Opposing sides are in equal length but adjacent sides are not equal in length. 3.5.4 Nonequilateral Nonequiangular Insert It is the type of cutting item of regular geometric shape with sides of non-equal lengths and non-equal tool included angles. This category contains inserts with the following ISO shape codes: A, B, and K. Chapter 3. ISO Cutting Tool Geometry 23  1. Parallelogram Insert (A, B, K): Insert with four sides and four internal angles none of which are equal to ninety degrees. Opposing sides are parallel and equal in length. 3.5.5 Round Insert It is the type of cutting item with circular edges. This category contains inserts with the ISO shape code R. Table 3-II: Insert Shapes Defined in ISO 13399. Symbol Shape Insert Shape Nose Angle S  Square 90 T  Triangular 60 C  Rhombic (Diamond) 80 D 55 E 75 F 50 M 86 V 35 W  Trigon 80 H  Hexagonal 120 O  Octagonal 135 P  Pentagonal 108 L  Rectangular 90 A  Parallelogram 85 B 82 N/K 55 R  Round -   Chapter 3. ISO Cutting Tool Geometry 24  3.6 Tool Coordinate Frames and Transformations Coordinate transformations play an important role in this study, hence they are re- quired to define and transform the cutting mechanics from machine coordinate system to the rake face of the cutting tool. Machine coordinate system is defined by axes of the machine tool or the directions of the primary feed motion and primary cutting direction. On the other hand, chip flow coordinate system is defined in two axes; one is on the rake face and aligned in the direction of chip flow, and the second one is normal to the rake face and perpendicular to the chip flow axis. By using these transformations, it is possi- ble to remove the effects of tool geometry from the mechanics of cutting, and eliminate the complexity of different cutting operations such as, turning, milling etc. Two different coordinate frame definitions and rotations are considered in this the- sis. The following procedure is designed to transform the cutting forces or specific cutting force coefficients from machine coordinate system to the rake face. The mechan- ics of oblique cutting in turning operations is illustrated in Figure 3-5. In order to perform the transformations, chip flow angle (πœ‚πœ‚), which is not defined in ISO standards, is used. Chapter 3. ISO Cutting Tool Geometry 25   Figure 3-5: Mechanics of Oblique Cutting Process. 3.6.1 Turning Operations Machine Coordinate System (𝑿𝑿𝑭𝑭𝒀𝒀𝑭𝑭𝒁𝒁𝑭𝑭) Machine coordinate system (System 0) is given by the convention of the CNC sys- tem. The orientation of the machine coordinate system depends on the specific CNC tool, or the orientation of the dynamometer used to measure the cutting forces. In the most common approach, the 𝑍𝑍0 axis is collinear with the main spindle axis and the  𝑋𝑋0 and π‘Œπ‘Œ0 axes are perpendicular to the 𝑍𝑍0 axis. On a lathe, the 𝑋𝑋0 axis is the one on which the tool post moves to and away from the workpiece’s axis of rotation. The sign conven- tion for this coordinate system is determined such that the movement of the tool in a positive direction leads to a growing measurement of the workpiece. Figure 3-6 shows the machine coordinate system on a turning tool. Chapter 3. ISO Cutting Tool Geometry 26  Radial – Tangential – Axial Coordinate System (𝑹𝑹𝑹𝑹𝑨𝑨) (𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀𝒁𝒁𝒀𝒀) 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system is defined according to the directions of the cutting forces in orthogonal cutting. The forces in orthogonal turning are in the radial and the tangential directions, whereas the tangential direction is parallel to the direction of primary motion and the radial direction is perpendicular to the primary cutting edge and tool cutting edge plane 𝑃𝑃𝑠𝑠. Both tangential and radial directions point towards the tool and the coordinate frame’s origin is on the cutting edge. The axial direction is perpendicular to both of these directions, pointing away from the tool shaft. In turning, geometrical transformation from the machine coordinate system (𝑋𝑋0π‘Œπ‘Œ0𝑍𝑍0) to 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (π‘‹π‘‹πΌπΌπ‘Œπ‘ŒπΌπΌπ‘π‘πΌπΌ) can be performed by rotating the machine coordinate system around π‘Œπ‘Œ0 axis by amount of cutting edge angle (πœ…πœ…π‘Ÿπ‘Ÿ ) in negative direction. Figure 3-6 illustrates the machine coordinate system and the 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system on a turning tool. The transformation to the machine coordinate system can be expressed by a linear rotation matrix 𝐢𝐢01 , so that:  π‘Žπ‘Ž0 = 𝐢𝐢01 π‘Žπ‘Ž1 (3.2) where vector π‘Žπ‘Ž0 is defined in System 0 and vector π‘Žπ‘Ž1 is defined in System 1. Rota- tion matrix 𝐢𝐢01  can be expressed as follows:  𝐢𝐢01 = οΏ½cos πœ…πœ…π‘Ÿπ‘Ÿ 0 βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ0 1 0sinπœ…πœ…π‘Ÿπ‘Ÿ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ = ⎣ ⎒ ⎒ ⎒ ⎑cos οΏ½πœ‹πœ‹2 βˆ’ πœ“πœ“π‘Ÿπ‘ŸοΏ½ 0 βˆ’ sin οΏ½πœ‹πœ‹2 βˆ’ πœ“πœ“π‘Ÿπ‘ŸοΏ½0 1 0sin οΏ½πœ‹πœ‹2 βˆ’ πœ“πœ“π‘Ÿπ‘ŸοΏ½ 0 cos οΏ½πœ‹πœ‹2 βˆ’ πœ“πœ“π‘Ÿπ‘ŸοΏ½ ⎦βŽ₯βŽ₯βŽ₯ ⎀  𝐢𝐢01 = οΏ½sinπœ“πœ“π‘Ÿπ‘Ÿ 0 βˆ’ cosπœ“πœ“π‘Ÿπ‘Ÿ0 1 0cosπœ“πœ“π‘Ÿπ‘Ÿ 0 sinπœ“πœ“π‘Ÿπ‘Ÿ οΏ½ (3.3) Chapter 3. ISO Cutting Tool Geometry 27   Figure 3-6: RTA and the Machine Coordinate Systems on a Lathe. Similarly, the inverse transformation from System 0 to System 1 can be determined by calculating the inverse of 𝐢𝐢01:  𝐢𝐢10 = 𝐢𝐢01βˆ’1 = 𝐢𝐢01𝑅𝑅 𝐢𝐢10 = οΏ½ cos πœ…πœ…π‘Ÿπ‘Ÿ 0 sinπœ…πœ…π‘Ÿπ‘Ÿ0 1 0 βˆ’sinπœ…πœ…π‘Ÿπ‘Ÿ 0 cos πœ…πœ…π‘Ÿπ‘ŸοΏ½ = οΏ½ sinπœ“πœ“π‘Ÿπ‘Ÿ 0 cosπœ“πœ“π‘Ÿπ‘Ÿ0 1 0βˆ’cosπœ“πœ“π‘Ÿπ‘Ÿ 0 sinπœ“πœ“π‘Ÿπ‘Ÿ οΏ½ (3.4) Cutting Edge Coordinate System (𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒁𝒁𝒀𝒀𝒀𝒀) Figure 3-7 illustrates the cutting edge coordinate system (System 2, (𝑋𝑋𝐼𝐼𝐼𝐼 ,π‘Œπ‘ŒπΌπΌπΌπΌ ,𝑍𝑍𝐼𝐼𝐼𝐼)). The π‘Œπ‘ŒπΌπΌπΌπΌ  axis of the cutting edge coordinate system lies along the cutting edge of the tool, 𝑍𝑍𝐼𝐼𝐼𝐼  axis is collinear with 𝐢𝐢 axis, and 𝑋𝑋𝐼𝐼𝐼𝐼  axis is defined perpendicular to π‘Œπ‘ŒπΌπΌπΌπΌ  and 𝑍𝑍𝐼𝐼𝐼𝐼 axes. This coordinate frame is also used for the definition of the oblique cutting mechan- ics in Altintas [5]. In order to transform the 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (System 1) to cutting edge coordinate system (System 2), 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system needs to be rotated about 𝐢𝐢 axis by the amount of inclination (helix) angle (πœ†πœ†π‘ π‘ ) and additionally, a 90Β° rotation around the 𝐴𝐴 axis and 90Β° rotation around the 𝐢𝐢 axis need to be performed, in order to get the axes pointing the proper directions. Chapter 3. ISO Cutting Tool Geometry 28    Figure 3-7: Cutting Edge Coordinate System along with the RTA Coordinate System. Therefore, the transformation matrix is:  𝐢𝐢21 = οΏ½οΏ½0 0 11 0 00 1 0οΏ½ οΏ½cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  0sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘  00 0 1οΏ½οΏ½ βˆ’1 = οΏ½0 cos πœ†πœ†π‘ π‘  sin πœ†πœ†π‘ π‘ 0 βˆ’ sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘ 1 0 0 οΏ½ (3.5) And the resultant transformation from machine coordinate system can be accom- plished by Eq. (3.6).  Chapter 3. ISO Cutting Tool Geometry 29   𝐢𝐢20 = 𝐢𝐢21 𝐢𝐢10 𝐢𝐢20 = οΏ½βˆ’ cosπœ“πœ“π‘Ÿπ‘Ÿ sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘  sinπœ“πœ“π‘Ÿπ‘Ÿ sin πœ†πœ†π‘ π‘ βˆ’ cosπœ“πœ“π‘Ÿπ‘Ÿ cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  sinπœ“πœ“π‘Ÿπ‘Ÿ cos πœ†πœ†π‘ π‘ sinπœ“πœ“π‘Ÿπ‘Ÿ 0 cosπœ“πœ“π‘Ÿπ‘Ÿ οΏ½ (3.6) Rake Face Coordinate System (𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒁𝒁𝒀𝒀𝒀𝒀𝒀𝒀) Rake face coordinate system (System 3, (π‘‹π‘‹πΌπΌπΌπΌπΌπΌπ‘Œπ‘ŒπΌπΌπΌπΌπΌπΌπ‘π‘πΌπΌπΌπΌπΌπΌ)) is a transitionary coordinate system used in this study. The π‘Œπ‘ŒπΌπΌπΌπΌπΌπΌ  axis of the rake face coordinate system is defined on the cutting edge and 𝑍𝑍𝐼𝐼𝐼𝐼𝐼𝐼  axis is defined on the rake face of the cutting tool. The trans- formation from cutting edge coordinate system (System 2) to the rake face coordinate system (System 3) can be accomplished by rotating the cutting edge coordinate system around π‘Œπ‘ŒπΌπΌπΌπΌ  axis by the amount of normal rake angle (𝛾𝛾𝑛𝑛 ). Figure 3-8 shows the orienta- tion of the cutting edge coordinate system and rake face coordinate system, and the transformation matrix is shown in Eq. (3.7).  Figure 3-8: Rake Face Coordinate System (System 3): Rotation around the 𝒀𝒀𝒀𝒀𝒀𝒀 Axis by the Amount of Rake Angle 𝜸𝜸𝜸𝜸. Chapter 3. ISO Cutting Tool Geometry 30   𝐢𝐢32 = οΏ½cos 𝛾𝛾𝑛𝑛 0 βˆ’ sin 𝛾𝛾𝑛𝑛0 1 0sin 𝛾𝛾𝑛𝑛 0 cos 𝛾𝛾𝑛𝑛 οΏ½ (3.7) Chip Flow Coordinate System (𝑼𝑼𝑼𝑼) Chip Flow Coordinate System (System 4, (π‘‹π‘‹πΌπΌπ‘‰π‘‰π‘Œπ‘ŒπΌπΌπ‘‰π‘‰π‘π‘πΌπΌπ‘‰π‘‰)) can be obtained by rotating the rake face coordinate system (System 3) by its 𝑋𝑋𝐼𝐼𝐼𝐼𝐼𝐼  axis by the amount of chip flow angle (πœ‚πœ‚). With this transformation operation, π‘ˆπ‘ˆ(𝑋𝑋𝐼𝐼𝑉𝑉) axis of the chip flow coordinate system becomes parallel to the friction force (𝐹𝐹𝑒𝑒 ) and the 𝑉𝑉 (π‘Œπ‘ŒπΌπΌπ‘‰π‘‰) axis becomes collinear with the normal force (𝐹𝐹𝑣𝑣) acting on the rake face. 𝑖𝑖 or 𝑍𝑍𝐼𝐼𝑉𝑉  axis has no physical mean- ing and will be omitted in this study. Figure 3-9 shows the relationship between the rake face coordinate system and the chip flow coordinate system. The transformation between rake face coordinate system and the chip flow coordinate system can be described by the following equation:  οΏ½π‘ˆπ‘ˆ 𝑉𝑉 οΏ½ = οΏ½0 βˆ’ sin πœ‚πœ‚ cos πœ‚πœ‚1 0 0 οΏ½  οΏ½π‘‹π‘‹πΌπΌπΌπΌπΌπΌπ‘Œπ‘ŒπΌπΌπΌπΌπΌπΌ 𝑍𝑍𝐼𝐼𝐼𝐼𝐼𝐼 οΏ½ (3.8)  Figure 3-9: Chip Flow Coordinate System (System 4): Rotation around the 𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀 Axis by the Amount of Chip Flow Angle 𝜼𝜼. Chapter 3. ISO Cutting Tool Geometry 31   Furthermore:  𝐢𝐢41 = 𝐢𝐢43 𝐢𝐢32 𝐢𝐢21 𝐢𝐢41 = οΏ½ cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ βˆ’ sin 𝛾𝛾𝑛𝑛sin πœ†πœ†π‘ π‘  sin πœ‚πœ‚ + cos πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ cos πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 βˆ’ cos πœ†πœ†π‘ π‘  sin πœ‚πœ‚ + sin πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ sin πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 οΏ½ βˆ’1 = 𝐢𝐢41𝑅𝑅  (3.9) Finally, general transformation can be calculated by multiplying the transformation matrices from machine coordinate system to chip flow coordinate system. The resulting transformation is:  οΏ½π‘ˆπ‘ˆ 𝑉𝑉 οΏ½ = 𝐴𝐴 Γ— οΏ½π‘‹π‘‹π‘Œπ‘Œ 𝑍𝑍 οΏ½ (3.10) 𝐴𝐴 = οΏ½0 βˆ’sin πœ‚πœ‚ cos πœ‚πœ‚1 0 0 οΏ½ οΏ½cos 𝛾𝛾𝑛𝑛 0 βˆ’sin 𝛾𝛾𝑛𝑛0 1 0sin 𝛾𝛾𝑛𝑛 0 cos 𝛾𝛾𝑛𝑛 οΏ½ οΏ½0 cos πœ†πœ†π‘ π‘  sin πœ†πœ†π‘ π‘ 0 βˆ’ sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘ 1 0 0 οΏ½ οΏ½sinπœ“πœ“π‘Ÿπ‘Ÿ 0 βˆ’cosπœ“πœ“π‘Ÿπ‘Ÿ0 1 0cosπœ“πœ“π‘Ÿπ‘Ÿ 0 sinπœ“πœ“π‘Ÿπ‘Ÿ οΏ½ 3.6.2 Transformation of Rotary Tools As in milling and drilling, instead of workpiece, cutting tool rotates and rotary tools may have more than one cutting edge. Therefore, for each flute, following transforma- tions should be defined individually. Machine Coordinate System (𝑿𝑿𝒀𝒀𝒁𝒁) Similar to a turning machine, the 𝑍𝑍 axis of the machine coordinate system (System 0, (π‘‹π‘‹π‘Œπ‘Œπ‘π‘)) is parallel to the axis of the spindle. In milling machines, the 𝑋𝑋 axis is the main axis parallel to the working surface, and the π‘Œπ‘Œ axis can be found accordingly. Radial – Tangential – Axial Coordinate System (𝑹𝑹𝑹𝑹𝑨𝑨) 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (System 1, (π‘‹π‘‹πΌπΌπ‘Œπ‘ŒπΌπΌπ‘π‘πΌπΌ)) is defined similar to a turning tool. Since the cutting edge on a rotary tool is positioned differently relative to the machine coordinate system (System 0), there is a slight difference between transformations in turning and other operations. Chapter 3. ISO Cutting Tool Geometry 32  In order to perform the transformations from machine coordinate system to RTA coordinate system, an additional coordinate system (System *) is introduced for each flute. System * has its origin located at the selected point on the cutting edge and rotates with the tool. The angle between the machine coordinate system and this coordinate frame is the immersion angle (πœ™πœ™). Figure 3-10 illustrates the introduced coordinate system. Immersion angle πœ™πœ™ is measured clockwise between the positive π‘Œπ‘Œ axis and the flutes of the cutting tool. The axes of the System * are aligned so that they are parallel to π‘‹π‘‹π‘Œπ‘Œπ‘π‘ for a cutting flute positioned on the 𝑋𝑋 axis, in which the immersion angle is equal to 90Β° (Flute 2 in Figure 3-10).  For instance, when πœ™πœ™ is equal to 0Β° (Flute 1 in Figure 3-10) the transformation between System 0 and System * is a rotation around the 𝑍𝑍 axis by an amount of 90Β°.  Figure 3-10: Illustration of System * on a Rotary Tool with Four Flutes. Chapter 3. ISO Cutting Tool Geometry 33  Consequently, the transformation from tool coordinate system (System *) to ma- chine coordinate system (System 0) is a combination of two basic rotations:  πΆπΆβˆ—0 = οΏ½cosπœ™πœ™ βˆ’ sinπœ™πœ™ 0sinπœ™πœ™ cosπœ™πœ™ 00 0 1οΏ½  οΏ½ cos 90Β° sin 90Β° 0βˆ’ sin 90Β° cos 90Β° 00 0 1οΏ½ πΆπΆβˆ—0 = οΏ½ sinπœ™πœ™ cosπœ™πœ™ 0βˆ’cosπœ™πœ™ sinπœ™πœ™ 00 0 1οΏ½ (3.11) In 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system, 𝑅𝑅 axis is defined parallel to the primary rotational mo- tion of the tool, 𝐢𝐢 axis is aligned towards the axis of the tool while being normal to the cutting edge, and 𝐴𝐴 axis is defined on the cutting edge. The direction of 𝐢𝐢 axis is represented with cutting element position angle (180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ). So, the transformation from 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (System 1) to System * is a positive rotation around the π‘Œπ‘Œβˆ— axis by an amount of  πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°. Figure 3-11 shows the transformation and the resulting transformation matrix is:  𝐢𝐢1βˆ— = οΏ½ cos(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°) 0 sin(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°)0 1 0 βˆ’sin(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°) 0 cos(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°)οΏ½ = οΏ½βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ0 1 0cos πœ…πœ…π‘Ÿπ‘Ÿ 0 βˆ’ sin πœ…πœ…π‘Ÿπ‘ŸοΏ½ 𝐢𝐢1βˆ— = οΏ½ cos(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ) 0 sin(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ)0 1 0 βˆ’sin(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ) 0 cos(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ)οΏ½ = οΏ½βˆ’ cosπœ“πœ“π‘Ÿπ‘Ÿ 0 sinπœ“πœ“π‘Ÿπ‘Ÿ0 1 0sinπœ“πœ“π‘Ÿπ‘Ÿ 0 βˆ’ cosπœ“πœ“π‘Ÿπ‘ŸοΏ½ (3.12) Chapter 3. ISO Cutting Tool Geometry 34   Figure 3-11: Transformation between System 1 and System *. Thus, the resultant transformation from machine coordinate system to 𝐢𝐢𝑅𝑅𝐴𝐴 coordi- nate system can be found as the following:  𝐢𝐢10 = 𝐢𝐢1βˆ— πΆπΆβˆ—0 = οΏ½βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ0 1 0cos πœ…πœ…π‘Ÿπ‘Ÿ 0 βˆ’ sin πœ…πœ…π‘Ÿπ‘ŸοΏ½ οΏ½ sinπœ™πœ™ cosπœ™πœ™ 0βˆ’cosπœ™πœ™ sinπœ™πœ™ 00 0 1οΏ½ 𝐢𝐢10 = οΏ½βˆ’sinπœ™πœ™ sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’cosπœ™πœ™ sinπœ…πœ…π‘Ÿπ‘Ÿ cos πœ…πœ…π‘Ÿπ‘Ÿβˆ’cosπœ™πœ™ sinπœ™πœ™ 0 βˆ’sinπœ™πœ™ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’cosπœ™πœ™ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin πœ…πœ…π‘Ÿπ‘ŸοΏ½ (3.13) Cutting Edge Coordinate System (𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒁𝒁𝒀𝒀𝒀𝒀) For rotary tools, cutting edge coordinate system (System 2, (π‘‹π‘‹πΌπΌπΌπΌπ‘Œπ‘ŒπΌπΌπΌπΌπ‘π‘πΌπΌπΌπΌ)) is defined the same as turning operations. The transformation from cutting edge coordinate system (System 2) and the 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (System 1) is stated in Eq. (3.14). The coordinate system is demonstrated in Figure 3-12. Chapter 3. ISO Cutting Tool Geometry 35   𝐢𝐢21 = οΏ½0 cos πœ†πœ†π‘ π‘  sin πœ†πœ†π‘ π‘ 0 βˆ’ sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘ 1 0 0 οΏ½ (3.14)   Figure 3-12: Transformation between System 1 and System 2 for Rotary Tools. Rake Face Coordinate System (𝑿𝑿𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒀𝒁𝒁𝒀𝒀𝒀𝒀𝒀𝒀) As it can be seen from Figure 3-13, rake face coordinate system (System 3) is de- fined same as the turning operations, which is described in Section 0. As a result,  𝐢𝐢32 = οΏ½cos 𝛾𝛾𝑛𝑛 0 βˆ’ sin 𝛾𝛾𝑛𝑛0 1 0sin 𝛾𝛾𝑛𝑛 0 cos 𝛾𝛾𝑛𝑛 οΏ½ (3.15) Chapter 3. ISO Cutting Tool Geometry 36    Figure 3-13: Transformation between Rake Face Coordinate System (System 3) and Cutting Edge Coordinate System (System 2). Chip Flow Coordinate System (𝑼𝑼𝑼𝑼) Chip flow coordinate system (System 4, (π‘‹π‘‹πΌπΌπ‘‰π‘‰π‘Œπ‘ŒπΌπΌπ‘‰π‘‰π‘π‘πΌπΌπ‘‰π‘‰)) for rotary tools is defined same as turning. It has only two axes: π‘ˆπ‘ˆ axis is on the rake face, aligned with the direc- tion of chip flow, and 𝑉𝑉 axis is normal to the rake face. Figure 3-14 shows the representation of chip flow coordinate system on a rotary tool. In this figure, 𝑖𝑖 axis has no physical meaning and is shown only to complete the coordinate system. Chapter 3. ISO Cutting Tool Geometry 37   Figure 3-14: Representation of the Chip Flow Coordinate System on a Milling Tool. The transformation between rake face coordinate system (System 3) and chip flow coordinate system (System 4) is defined by the following matrix:  𝐢𝐢43 = οΏ½0 βˆ’ sin πœ‚πœ‚ cos πœ‚πœ‚1 0 0 οΏ½ (3.16)    Chapter 3. ISO Cutting Tool Geometry 38  3.7 Summary If the transformation operations in turning and rotating tools are compared, it can be observed that the transformation matrices from 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system (System 1) to π‘ˆπ‘ˆπ‘‰π‘‰ coordinate system (System 4) are identical. The only difference between these opera- tions comes from the rotation of multiple flutes in rotary tools. This difference is explained in Section 0. By performing these transformations, it is possible to describe the kinematics of the cutting process and cutting forces on the rake face of a cutter regardless of the operation. Thus, cutting forces measurable in machine coordinate system can be described as the friction and normal forces which are in the chip flow coordinate system. Chapter 4. Generalized Geometric Model of Inserted Cutters      39 4. Generalized Geometric Model of Inserted Cutters  4.1 Overview In this chapter, a generalized model for inserted cutters is presented. Inserted cutters are widely used in turning and milling processes, both in roughing and finishing opera- tions because of their low cost advantage on solid cutters. When tool wear or breakage occurs, instead of replacing the cutter body, replacing the insert is sufficient. Due to their popularity and availability in different types of applications, many different inserts are available in terms of shapes and dimensions. As described in Section 3.5, 17 different insert shapes are defined in ISO 13399 standard. Details of these shapes can be seen in Table 3-II of Chapter 3. Since this thesis aims to develop a generalized model, it covers not only insert shapes defined in ISO 13399, but also any arbitrary insert geometry defined by control points. Modeling of inserted cutters is more complicated than solid body cutters, because number of parameters used to define the insert geometry as well as to place on the cutter body is significantly higher. The output of this model is the axial locations, radii, helix / inclination angles, normal rake angles, and cutting edge angles of the cutting edge(s) at each specified point along the cutter axis. By using this data, it is possible to determine the cutting forces, tool vibrations, static deflections, and stability lobes for almost any kind of machining operations, such as, turning, boring, milling, etc. In the model, first of all, geometry of the insert is defined insert’s local coordinate system analytically and it is placed on the cutter body using the orientation angles, i.e. cutting edge angle πœ…πœ…π‘Ÿπ‘Ÿ , axial rake angle 𝛾𝛾𝑝𝑝 , and radial rake angle 𝛾𝛾𝑓𝑓 . By using these angles, cutting edge positions are transformed to global coordinate system which is located at the cutter tip. After the transformation, for cutting mechanics which is described in Chapter 5.2, normal rake angle 𝛾𝛾𝑛𝑛  and helix angle πœ†πœ†π‘ π‘  are calculated using the position of the cutting edge(s). Sample examples are presented at the end of this chapter. Chapter 4. Generalized Geometric Model of Inserted Cutters      40 Inputs required to define the cutting edge on an insert are defined in Chapter 3 with illustrative figures, along with the coordinate axes used in this model. 4.2 Mathematical Modeling of an Insert In mathematical modeling, a similar approach to Engin and Altintas’ study [1] has been used with certain modifications and improvements. Firstly, using the inputs, mathematical model of one insert was developed on the local coordinate system (𝑋𝑋0π‘Œπ‘Œ0𝑍𝑍0) positioned at the cutting reference point (𝐢𝐢𝐢𝐢𝑃𝑃) of the insert. Figure 4-1 illustrates the local coordinate system and the cutting reference point on two different inserts. The aim of this model is to calculate the control points that are sufficient to define the features (nose radius, corner chamfer, wiper edge etc.) on an insert.  Figure 4-1: Local Coordinate Systems and Cutting Reference Points on Different Inserts.    Chapter 4. Generalized Geometric Model of Inserted Cutters      41 In ISO 13399 standard, there are various shapes defined for inserts. Therefore, for this study, for each insert shape, control points were formulated for both corner chamfer and nose radius cases.  Control points include the positions of the start and the end points of a feature on the cutting edge. For ISO type inserts, these control points are all in π‘₯π‘₯0𝑧𝑧0 plane, therefore they were all assumed to be flat inserts. For instance, Figure 4-2 shows the control points and dimensions of a parallelogram insert with a wiper edge and a corner radius. In this Figure, points 𝐴𝐴,𝐡𝐡,𝐷𝐷, and 𝐸𝐸 are the control points on the cutting edge, 𝐢𝐢 and 𝐼𝐼 used to locate the insert center and the center of corner radius, 𝐢𝐢𝐢𝐢𝑃𝑃 is the cutting reference point which is the origin for specific dimensions and rotations, point 𝐹𝐹 is the theoretical sharp point of the insert.  Figure 4-2: Control Points on a Parallelogram Insert.   Chapter 4. Generalized Geometric Model of Inserted Cutters      42 Cutting reference point was selected as the origin of the local coordinate system. Locations of each control point as well as insert and radius centers were calculated analytically. These control points are rotated according to the given axial and radial rake angles, and transformed to the global coordinate system which is located at the tip of the cutter body. Finally, for modeling of mechanics, normal rake angle, helix angle, and true cutting edge angle were calculated for each point on the cutting edge. In the following sections, transformations and calculations of angles are described and the whole procedure is applied as examples for two milling inserts with different corner modifications and a turning insert. 4.3 Tool Coordinate Frames and Transformations Since the cutting reference point (𝐢𝐢𝐢𝐢𝑃𝑃) is used to define all the angular and dimen- sional values to define an insert, all of the coordinate axes’ origins used in this model to orient and place the insert on the cutter body are located at the 𝐢𝐢𝐢𝐢𝑃𝑃. Four different coordinate systems were defined to accomplish the required rotations for the orientation of an insert on the cutter body. Initial coordinate frame 𝐹𝐹0 is located with its origin at 𝐢𝐢𝐢𝐢𝑃𝑃 and the 𝑋𝑋 axis (𝑋𝑋0) along the primary feed direction as seen in Figure 4-3. 𝑍𝑍 axis (𝑍𝑍0) of the 𝐹𝐹0 frame is parallel to the cutter body rotation axis and positive 𝑍𝑍0 is pointing towards the cutter body. In 𝐹𝐹0 frame, 𝑋𝑋0𝑍𝑍0 plane corresponds to tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ , and 𝑋𝑋0π‘Œπ‘Œ0 plane corresponds to assumed working plane 𝑃𝑃𝑓𝑓 . Final coordinate frame 𝐹𝐹3 has its 𝑋𝑋 axis (𝑋𝑋3) directed along the cutting edge and the 𝑍𝑍 axis (𝑍𝑍3) on the rake face of the insert. In order to transform the 𝐹𝐹0 frame to 𝐹𝐹3 frame, two intermediate coordinate frames were defined. Chapter 4. Generalized Geometric Model of Inserted Cutters      43  Figure 4-3: Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. 4.3.1 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation The first step is to align the 𝑋𝑋1 axis with the wiper edge or parallel land in a coordi- nate frame 𝐹𝐹1 by rotating the frame 𝐹𝐹0 around 𝑍𝑍0 axis by an amount of radial rake angle in the reverse direction (βˆ’π›Ύπ›Ύπ‘“π‘“). As a result, the rotation matrix 𝐢𝐢01 from frame 𝐹𝐹0 to frame 𝐹𝐹1 becomes:  πœƒπœƒπ‘§π‘§01 = βˆ’π›Ύπ›Ύπ‘“π‘“ 𝐢𝐢01 = οΏ½cos(βˆ’π›Ύπ›Ύπ‘“π‘“) βˆ’ sin(βˆ’π›Ύπ›Ύπ‘“π‘“) 0sin(βˆ’π›Ύπ›Ύπ‘“π‘“) cos(βˆ’π›Ύπ›Ύπ‘“π‘“) 00 0 1οΏ½ (4.1)      Chapter 4. Generalized Geometric Model of Inserted Cutters      44 4.3.2 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation The second step is to align the 𝑍𝑍2 axis of the frame 𝐹𝐹2 with the rake face of the in- sert by rotating the frame 𝐹𝐹1 around 𝑋𝑋1 axis by an amount of axial rake angle also in reverse direction (βˆ’π›Ύπ›Ύπ‘π‘ ). Consequently, the rotation matrix 𝐢𝐢12  which defines the rotation from frame 𝐹𝐹1 to frame 𝐹𝐹2 becomes:  πœƒπœƒπ‘₯π‘₯12 = βˆ’π›Ύπ›Ύπ‘π‘ 𝐢𝐢12 =  οΏ½1 0 00 cos(βˆ’π›Ύπ›Ύπ‘π‘) βˆ’ sin(βˆ’π›Ύπ›Ύπ‘π‘)0 sin(βˆ’π›Ύπ›Ύπ‘π‘) cos(βˆ’π›Ύπ›Ύπ‘π‘) οΏ½ (4.2)  Figure 4-4: Frame  𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.    Chapter 4. Generalized Geometric Model of Inserted Cutters      45 4.3.3 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation The final step is to align the 𝑋𝑋3 axis of the frame 𝐹𝐹3 with the cutting edge of the in- sert by rotating the frame 𝐹𝐹2 around the π‘Œπ‘Œ2 axis by an amount of cutting edge angle in the reverse direction (βˆ’πœ…πœ…π‘Ÿπ‘Ÿ). So that the rotation matrix 𝐢𝐢23 defining the rotation from frame 𝐹𝐹2 to frame 𝐹𝐹3 is:  πœƒπœƒπ‘¦π‘¦23 = βˆ’πœ…πœ…π‘Ÿπ‘Ÿ 𝐢𝐢23 =  οΏ½ cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ) 0 sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ)0 1 0 βˆ’ sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ) 0 cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ)οΏ½ (4.3)  Figure 4-5: Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.      Chapter 4. Generalized Geometric Model of Inserted Cutters      46 Finally, in order to complete the rotation matrix between frames 𝐹𝐹0 and 𝐹𝐹3 can be calculated as the following:  𝐢𝐢03 = 𝐢𝐢01 𝐢𝐢12 𝐢𝐢23 (4.4) where;  𝐢𝐢03 = οΏ½cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ sin 𝛾𝛾𝑓𝑓 cos 𝛾𝛾𝑝𝑝 βˆ’ cos 𝛾𝛾𝑓𝑓 sin πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿsin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑓𝑓 cos 𝛾𝛾𝑝𝑝 sin 𝛾𝛾𝑓𝑓 sin πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿcos 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin 𝛾𝛾𝑝𝑝 cos 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ (4.5) 4.4 Mathematical Relationships between Angles Next task in mathematical modeling of insert geometry is to derive the angles used in mechanics of metal cutting which are normal rake angle 𝛾𝛾𝑛𝑛 , helix / inclination gle πœ†πœ†π‘ π‘ , and cutting edge angle πœ…πœ…π‘Ÿπ‘Ÿ . These angles were derived according to the definitions summarized in Chapter 3. 4.4.1 Normal Rake Angle The normal rake angle 𝛾𝛾𝑛𝑛  can be described as the angle between the direction of primary motion vector (βˆ’π‘Œπ‘Œ0) and the normal of the rake face (π‘Œπ‘Œ3) measured in the cutting edge normal plane 𝑃𝑃𝑛𝑛  (π‘Œπ‘Œ3𝑍𝑍3 plane). In order to determine the normal rake angle using the tool design angles (𝛾𝛾𝑓𝑓  and 𝛾𝛾𝑝𝑝), a vector (𝑣𝑣0) directed along the direction of primary motion was defined in frame 𝐹𝐹0. Vector 𝑣𝑣0 later can be described in frame 𝐹𝐹3 with the inverse of the rotation matrix 𝐢𝐢03. Therefore:  𝑣𝑣0 = οΏ½ 0βˆ’10 οΏ½  β‡’  𝑣𝑣3 =  𝐢𝐢03βˆ’1 𝑣𝑣0 = 𝐢𝐢03𝑅𝑅  𝑣𝑣0 𝑣𝑣3 = οΏ½ cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿβˆ’ cos 𝛾𝛾𝑓𝑓 cos 𝛾𝛾𝑝𝑝 βˆ’ sin 𝛾𝛾𝑓𝑓 sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘ŸοΏ½ = οΏ½ 𝑣𝑣3π‘₯π‘₯ 𝑣𝑣3𝑦𝑦 𝑣𝑣3𝑧𝑧 οΏ½ (4.6)   Chapter 4. Generalized Geometric Model of Inserted Cutters      47 In order to measure the normal rake angle 𝛾𝛾𝑛𝑛  in 𝑃𝑃𝑛𝑛  plane, π‘₯π‘₯ component of the 𝑣𝑣3 vector (𝑣𝑣3π‘₯π‘₯ ) was set to zero. From Figure 4-6, it can be observed that:  𝛾𝛾𝑛𝑛 = atan(βˆ’π‘£π‘£3π‘§π‘§βˆ’π‘£π‘£3𝑦𝑦) = atan(𝑣𝑣3𝑧𝑧𝑣𝑣3𝑦𝑦) 𝛾𝛾𝑛𝑛 = atan(sin 𝛾𝛾𝑓𝑓 sin πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿcos 𝛾𝛾𝑓𝑓 cos 𝛾𝛾𝑝𝑝 ) (4.7)  Figure 4-6: Derivation of Normal Rake Angle. 4.4.2 True Cutting Edge Angle True cutting edge πœ…πœ…π‘Ÿπ‘Ÿβˆ— is defined as the angle between the cutting edge (𝑋𝑋3) and the assumed working plane 𝑃𝑃𝑓𝑓  (𝑋𝑋0π‘Œπ‘Œ0 plane) measured in the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  (𝑋𝑋0𝑍𝑍0 plane). The mathematical relationship between the true cutting edge angle πœ…πœ…π‘Ÿπ‘Ÿβˆ— and the tool design angles (𝛾𝛾𝑓𝑓 ,  𝛾𝛾𝑝𝑝  and πœ…πœ…π‘Ÿπ‘Ÿ ) can be calculated by defining a vector, 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒 𝑑𝑑3 , along the cutting edge in frame 𝐹𝐹3 and using the rotation matrix 𝐢𝐢03 to describe the vector in 𝐹𝐹0 coordinate frame. In order to measure the angle in tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ , 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑0𝑦𝑦 component of the vector 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑3  was set to zero. As a result, true cutting edge angle πœ…πœ…π‘Ÿπ‘Ÿβˆ— can be determined as the angle between the cutting edge vector 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑0  and the 𝑋𝑋0 axis.  Chapter 4. Generalized Geometric Model of Inserted Cutters      48  𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 3 = οΏ½100οΏ½ , 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0 = 𝐢𝐢03 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 3 = �𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0π‘₯π‘₯𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0𝑦𝑦𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0𝑧𝑧 οΏ½ 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0 = οΏ½ cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿβˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿcos 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿ οΏ½ (4.8)  πœ…πœ…π‘Ÿπ‘Ÿ βˆ—  = atan�𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0𝑧𝑧 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 0π‘₯π‘₯οΏ½ πœ…πœ…π‘Ÿπ‘Ÿ βˆ—  = atanοΏ½ cos 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿcos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘ŸοΏ½ (4.9)  Figure 4-7: Derivation of the True Cutting Edge Angle. 4.4.3 Inclination (Helix) Angle The inclination angle πœ†πœ†π‘ π‘  is defined as the angle between the cutting edge (𝑋𝑋3) and the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  (𝑋𝑋0𝑍𝑍0 plane) measured in tool cutting edge plane 𝑃𝑃𝑠𝑠. In order to relate the tool design angles, a new coordinate frame must be defined such that the frame 𝐹𝐹𝑠𝑠 has its 𝑋𝑋𝑠𝑠 axis aligned with the tool cutting edge plane 𝑃𝑃𝑠𝑠, and its π‘Œπ‘Œπ‘ π‘  axis aligned with π‘Œπ‘Œ0 axis of the coordinate frame 𝐹𝐹0. Figure 4-8 illustrates the frame 𝐹𝐹𝑠𝑠. Since a rotation of – οΏ½πœ‹πœ‹2 βˆ’ πœ…πœ…π‘Ÿπ‘ŸοΏ½ around the π‘Œπ‘Œ0 axis is needed to align the 𝑋𝑋𝑠𝑠 axis with the plane 𝑃𝑃𝑠𝑠, the rotation matrix between frames 𝐹𝐹0 and 𝐹𝐹𝑠𝑠 becomes: Chapter 4. Generalized Geometric Model of Inserted Cutters      49  𝐢𝐢0𝑠𝑠 = οΏ½ cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ) 0 sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ)0 1 0 βˆ’ sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ) 0 cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ)οΏ½ = οΏ½cos(πœ…πœ…π‘Ÿπ‘Ÿ) 0 βˆ’ sin(πœ…πœ…π‘Ÿπ‘Ÿ)0 1 0sin(πœ…πœ…π‘Ÿπ‘Ÿ) 0 cos(πœ…πœ…π‘Ÿπ‘Ÿ) οΏ½ (4.10)  Figure 4-8: Definition of Frame 𝑭𝑭𝑭𝑭 and Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation. As a result, vector 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑠𝑠  can be calculated as:  𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑠𝑠 = 𝐢𝐢0𝑠𝑠𝑅𝑅  𝐢𝐢03 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 3 = �𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒 𝑑𝑑𝑠𝑠π‘₯π‘₯𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒 𝑑𝑑𝑠𝑠𝑦𝑦 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒 𝑑𝑑𝑠𝑠𝑧𝑧 οΏ½ 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑠𝑠 = οΏ½ cos2 πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑓𝑓 + cos πœ…πœ…π‘Ÿπ‘Ÿ sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ + sin2 πœ…πœ…π‘Ÿπ‘Ÿ cos π›Ύπ›Ύπ‘π‘βˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin2 πœ…πœ…π‘Ÿπ‘Ÿ sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 + cos πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘ŸοΏ½ (4.11)  Following that, the inclination angle πœ†πœ†π‘ π‘  can be defined as the angle between the 𝑋𝑋𝑠𝑠 axis in π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  plane and the edge vector 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑠𝑠  with 𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑𝑠𝑠𝑧𝑧  component equal to zero.  Chapter 4. Generalized Geometric Model of Inserted Cutters      50  πœ†πœ†π‘ π‘  = atan�𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 _𝑠𝑠𝑦𝑦𝑣𝑣𝑑𝑑𝑑𝑑𝑒𝑒𝑑𝑑 _𝑠𝑠π‘₯π‘₯οΏ½ πœ†πœ†π‘ π‘  = atanοΏ½ βˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿcos2 πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑓𝑓 + cos πœ…πœ…π‘Ÿπ‘Ÿ sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sinπœ…πœ…π‘Ÿπ‘Ÿ + sin2 πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑝𝑝� (4.12)  Figure 4-9: Definition of the Inclination (Helix) Angle. Calculation of these angles is important to determine cutting forces along the cutting edge. In most cases, radial and axial rakes are constant along the cutting edge of the insert, and cutting edge angle changes with the edge modification; it becomes a different constant value along the chamfered corner, and quadratically changes along the round edges. Using the vectors and formulations derived in this chapter, it is possible to calculate the angles at each selected point on the cutting edge. In the next section, as case examples, several different inserts were selected and modeled using the proposed model.   Chapter 4. Generalized Geometric Model of Inserted Cutters      51 4.5 Examples 4.5.1 Rectangular Insert with Corner Radius The first selected sample case is a rectangular milling insert with a corner radius. In- sert and cutter body geometry were taken from Sandvik Coromant [55]. These dimensions are listed in Table 4-I and illustrated in Figure 4-10. Table 4-I: Inputs Used in the Model. Inputs 𝑳𝑳 (Insert Length) 11 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 90Β° π’Šπ’Šπ’Šπ’Š (Insert Width) 6.8 mm πœΈπœΈπ’‡π’‡ (Radial Rake Angle) 9Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) 0.5 mm πœΈπœΈπ’‘π’‘ (Axial Rake Angle) 10Β° π’“π’“πœΊπœΊ (Corner Radius) 0.4 mm πœΊπœΊπ’“π’“ (Tool Included Angle) 90Β° 𝑫𝑫𝒄𝒄 (Cutting Diameter) 20 mm   Figure 4-10: Illustrations of the Insert and the Cutter Body [55]. After obtaining the dimensions and the angles required for the placement of the in- sert on the cutter body, insert center is calculated in (𝑋𝑋0𝑍𝑍0) plane. In Figure 4-11, insert center is defined by point 𝐼𝐼(π‘₯π‘₯, 𝑧𝑧): Chapter 4. Generalized Geometric Model of Inserted Cutters      52  𝐼𝐼π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 + π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ (1 βˆ’ cos πœ…πœ…π‘Ÿπ‘Ÿ) βˆ’ 𝑖𝑖𝑖𝑖 sin πœ…πœ…π‘Ÿπ‘Ÿ2 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sinπœ…πœ…π‘Ÿπ‘Ÿ + 𝐿𝐿 cos πœ…πœ…π‘Ÿπ‘Ÿ2+ π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝑏𝑏𝑠𝑠 cos2 πœ…πœ…π‘Ÿπ‘Ÿ 𝐼𝐼𝑧𝑧 = βˆ’πΏπΏ sin πœ…πœ…π‘Ÿπ‘Ÿ2 + π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝑖𝑖𝑖𝑖 cos πœ…πœ…π‘Ÿπ‘Ÿ2 + 𝑏𝑏𝑠𝑠 sin πœ…πœ…π‘Ÿπ‘Ÿ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cos2 πœ…πœ…π‘Ÿπ‘Ÿ (4.13)   Figure 4-11: Analytical Model of an Insert with a Corner Radius and Wiper Edge.       Chapter 4. Generalized Geometric Model of Inserted Cutters      53 The second step is to calculate locations of control points which define the cutting edge. In this case, these points are 𝐴𝐴, 𝐡𝐡, 𝐷𝐷, and 𝐸𝐸. Wiper edge is on the feed plane and defined by a between points 𝐴𝐴 and 𝐡𝐡. The locations of these points can be calculated as:  𝐴𝐴π‘₯π‘₯ = βˆ’π‘π‘π‘ π‘  + 𝐷𝐷𝑐𝑐2 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sinπœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ    ,    𝐴𝐴𝑧𝑧 = 0 (4.14)  𝐡𝐡π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sin πœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ    ,    𝐡𝐡𝑧𝑧 = 0 (4.15) Corner edge is also defined by two points; points 𝐡𝐡 and 𝐷𝐷. Similarly, location of point 𝐷𝐷 can be calculated by the following equations:  𝐷𝐷π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 + π‘Ÿπ‘Ÿπœ€πœ€ sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sin πœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿsin πœ…πœ…π‘Ÿπ‘Ÿ    ,   𝐷𝐷𝑧𝑧 = |π‘Ÿπ‘Ÿπœ€πœ€(βˆ’1 + cos πœ…πœ…π‘Ÿπ‘Ÿ)| (4.16) Since there are only two symmetrical edges on a rectangular insert (Figure 4-10), corner modification is present only at two corners, thus main cutting edge is defined by points 𝐷𝐷 and 𝐸𝐸. The position of point 𝐸𝐸 can be calculated as:  𝐸𝐸π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 + π‘Ÿπ‘Ÿπœ€πœ€ sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sinπœ…πœ…π‘Ÿπ‘Ÿ + (βˆ’π‘π‘π‘ π‘  + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘Ÿ) cos2 πœ…πœ…π‘Ÿπ‘Ÿ+ �𝐿𝐿 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ 𝑏𝑏𝑠𝑠 sinπœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘ŸοΏ½ cos πœ…πœ…π‘Ÿπ‘Ÿ 𝐸𝐸𝑧𝑧 = |βˆ’πΏπΏ sinπœ…πœ…π‘Ÿπ‘Ÿ + 𝑏𝑏𝑠𝑠 cot πœ€πœ€π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘Ÿ + (βˆ’π‘Ÿπ‘Ÿπœ€πœ€ βˆ’ 𝑏𝑏𝑠𝑠 cot πœ€πœ€π‘Ÿπ‘Ÿ) cos2 πœ…πœ…π‘Ÿπ‘Ÿ+ (π‘Ÿπ‘Ÿπœ€πœ€ + 𝑏𝑏𝑠𝑠 sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sinπœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘Ÿ) cos πœ…πœ…π‘Ÿπ‘Ÿ | (4.17) After the calculation of control points and the cutting edges between these points, cutting edge is defined in (𝑋𝑋0𝑍𝑍0) plane in two dimensions. Rotation about π‘Œπ‘Œ0 axis by πœ…πœ…π‘Ÿπ‘Ÿ has been already implemented in these equations in order to make sure that the wiper edge is parallel to (π‘‹π‘‹π‘Œπ‘Œ) plane. Moreover, these locations were defined in global π‘‹π‘‹π‘Œπ‘Œπ‘π‘ coordinate frame with the tool tip as origin, therefore, these points must be translated to cutting reference point then the rotations presented in Section 4.3 must be performed.     Chapter 4. Generalized Geometric Model of Inserted Cutters      54 Transformations from a fixed frame is not commonly used in robotics, however, placement of the insert on the cutter body is a fixed frame transformation. Fixed frame transformation matrix can be calculated by taking the inverse of the current frame transformation using the same rotations [56]. After the rotations, all control points were transformed back to global π‘‹π‘‹π‘Œπ‘Œπ‘π‘ coordinate frame. Following figures show the locations of the control points before and after the transformations in global coordinate system.   Figure 4-12: Control Points of the Insert before the Rotations.  Chapter 4. Generalized Geometric Model of Inserted Cutters      55  Figure 4-13 : Control Points of the Insert after the Rotations. Following the rotations of the control points, lines and arcs between the consecutive points were defined in three dimensions. Since the performed rotations were solid body rotations, the relationship between any consecutive points would be same. However, these lines and arcs were no longer in (𝑋𝑋0𝑍𝑍0) plane; therefore, parametric equations were defined to obtain the location of each selected point on the cutting edge. For linear cutting edges:  𝑃𝑃 = π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1 + �𝑃𝑃�⃗𝑖𝑖 βˆ’ π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1οΏ½ 𝑑𝑑   ,   0 ≀ 𝑑𝑑 ≀ 1 (4.18) For arcs:  𝑃𝑃 = 𝐢𝐢 cos 𝑑𝑑 𝑒𝑒�⃗ + 𝐢𝐢 sin 𝑑𝑑  𝑛𝑛�⃗ Γ— 𝑒𝑒�⃗ + 𝐢𝐢,   0 ≀ 𝑑𝑑 ≀ πœ‹πœ‹2 (4.19)   Chapter 4. Generalized Geometric Model of Inserted Cutters      56 where 𝑃𝑃 is any point between 𝑃𝑃�⃗𝑖𝑖  and π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1 which are two consecutive control points, 𝐢𝐢 is the radius of the arc, 𝑒𝑒�⃗  is a unit vector from the center of the arc to any point on the circumference, 𝑛𝑛�⃗  is a unit vector perpendicular to the plane of the arc, and 𝐢𝐢 is the center vector of the arc. Note that these equations are parametric equations, in order to calculate the positions along the cutter axis, the relationship between any axial point and corres- ponding parameter value should be calculated. During the calculation of geometric control points, cutting edge shape between any two consecutive points are stored and during the general model, corresponding parametric equation is called automatically. The radius of any point on the cutting edge can be calculated by a point – line distance equation in space. Hence:  π‘Ÿπ‘Ÿ = οΏ½(𝑍𝑍2 βˆ’ 𝑍𝑍1) Γ— (𝑍𝑍1 βˆ’ 𝑃𝑃�⃗ )οΏ½ �𝑍𝑍2 βˆ’ 𝑍𝑍1οΏ½  (4.20) where π‘Ÿπ‘Ÿ is the local radius, 𝑍𝑍1 and 𝑍𝑍2 are any two points along the cutter axis. Using these formulations and rotation matrix developed in section 4.3, it is possible to calculate the local radius, normal rake angle, and helix angle of any point on the cutting edge of the insert.  Figure 4-14: CAD Model of the Insert. Chapter 4. Generalized Geometric Model of Inserted Cutters      57 Figure 4-15 shows the local radii of the insert along the cutter axis with comparison of the local radii extracted from the CAD part of the insert which is shown in Figure 4-14. As stated before, this model is valid for flat inserts; therefore, due to the complexi- ty of the insert geometry, mathematical model and CAD data does not match perfectly. However, the error is acceptable; since the maximum error between the model and the insert is around 50 microns. Figure 4-16 and Figure 4-17 show the change in the normal rake angle and helix angle along the cutter axis respectively.  Figure 4-15: Change of Local Radius along the Tool Axis. Chapter 4. Generalized Geometric Model of Inserted Cutters      58  Figure 4-16: Change in Normal Rake Angle along the Tool Axis.  Figure 4-17: Change in Helix Angle along the Tool Axis. Chapter 4. Generalized Geometric Model of Inserted Cutters      59 4.5.2 Rectangular Insert with Corner Chamfer Similar procedure was applied to an insert with a chamfered corner. Table 4-II and Figure 4-18 show the angles and dimensions of the insert. Table 4-II: Inputs Used in the Analysis of the Insert with Chamfer. Inputs 𝑳𝑳 (Insert Length) 11 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 90Β° π’Šπ’Šπ’Šπ’Š (Insert Width) 11.5 mm πœΈπœΈπ’‡π’‡ (Radial Rake Angle) 5Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) 1.5 mm πœΈπœΈπ’‘π’‘ (Axial Rake Angle) 5Β° π‘²π‘²πœΊπœΊ (Corner Chamfer) 1 mm x45Β° πœΊπœΊπ’“π’“ (Tool Included Angle) 90Β° 𝑫𝑫𝒄𝒄 (Cutting Diameter) 63 mm  Figure 4-18: Catalogue Figure of the Insert Taken from Sandvik Coromant [55].     Chapter 4. Generalized Geometric Model of Inserted Cutters      60 Analytical model of the insert can be seen in Figure 4-19. Insert center is defined by point 𝐼𝐼. Coordinates of this point in (𝑋𝑋0𝑍𝑍0) plane is:  𝐼𝐼π‘₯π‘₯ = 12 |𝐷𝐷𝑐𝑐 + 𝑖𝑖𝑖𝑖 cos(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) csc πœ€πœ€π‘Ÿπ‘Ÿ+ cos πœ…πœ…π‘Ÿπ‘Ÿ (𝐿𝐿 βˆ’ 2𝑏𝑏𝑠𝑠 csc πœ€πœ€π‘Ÿπ‘Ÿ sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ)) βˆ’ 2π‘π‘π‘π‘β„Ž cot πœ…πœ…π‘Ÿπ‘Ÿ csc πœ€πœ€π‘Ÿπ‘Ÿ sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) sin(πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ πΎπΎπœ€πœ€)| 𝐼𝐼𝑧𝑧 = 12 οΏ½csc πœ€πœ€π‘Ÿπ‘Ÿ �𝐿𝐿 sin πœ€πœ€π‘Ÿπ‘Ÿ sin πœ…πœ…π‘Ÿπ‘Ÿ+ sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) (𝑖𝑖𝑖𝑖 βˆ’ 2𝑏𝑏𝑠𝑠 sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 2π‘π‘π‘π‘β„Ž sin(πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ πΎπΎπœ€πœ€))οΏ½οΏ½ (4.21)  Figure 4-19: Illustration of Control Points and Dimensions for an Insert with Chamfer. For this type of insert, control points are points 𝐴𝐴,𝐡𝐡,𝐢𝐢, and 𝐷𝐷. Locations of these points can be calculated as the following:  Chapter 4. Generalized Geometric Model of Inserted Cutters      61  𝐴𝐴π‘₯π‘₯ = βˆ’π‘π‘π‘ π‘  + 𝐷𝐷𝑐𝑐2 + 𝐡𝐡𝐢𝐢𝐡𝐡 sinπΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐡𝐡𝐢𝐢𝐡𝐡 cosπΎπΎπœ€πœ€    ,   𝐴𝐴𝑧𝑧 = 0   (4.22)  𝐡𝐡π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 + 𝐡𝐡𝐢𝐢𝐡𝐡 sinπΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐡𝐡𝐢𝐢𝐡𝐡 cosπΎπΎπœ€πœ€    ,   𝐡𝐡𝑧𝑧 = 0 (4.23)  𝐢𝐢π‘₯π‘₯ = 𝐷𝐷𝑐𝑐2 + 𝐡𝐡𝐢𝐢𝐡𝐡 sinπΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ    ,   𝐢𝐢𝑧𝑧 = 𝐡𝐡𝐢𝐢𝐡𝐡 sinπΎπΎπœ€πœ€  (4.24)  𝐷𝐷π‘₯π‘₯ = �𝑏𝑏𝑠𝑠 βˆ’ 𝐷𝐷𝑐𝑐2 βˆ’ 𝐿𝐿 cos πœ…πœ…π‘Ÿπ‘Ÿ + 𝑏𝑏𝑠𝑠 cos(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) csc πœ€πœ€π‘Ÿπ‘Ÿ sinπœ…πœ…π‘Ÿπ‘Ÿ+ π‘π‘π‘π‘β„Ž cot πœ…πœ…π‘Ÿπ‘Ÿ csc πœ€πœ€π‘Ÿπ‘Ÿ sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) sin(πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ πΎπΎπœ€πœ€)οΏ½ 𝐷𝐷𝑧𝑧 = |𝐿𝐿 sinπœ…πœ…π‘Ÿπ‘Ÿ βˆ’ csc πœ€πœ€π‘Ÿπ‘Ÿ sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ) (𝑏𝑏𝑠𝑠 sin πœ…πœ…π‘Ÿπ‘Ÿ + π‘π‘π‘π‘β„Ž sin(πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ πΎπΎπœ€πœ€))| (4.25) After the calculations of the locations of control points in (𝑋𝑋0𝑍𝑍0) plane, same pro- cedure explained in the previous example is applied. As a result, tool angles and local radii were calculated along the cutting edge of this insert. Following figures illustrate these results.  Figure 4-20: Change in the Local Radius along the Tool Axis for a Chamfered Insert. Chapter 4. Generalized Geometric Model of Inserted Cutters      62  Figure 4-21: Change in Normal Rake Angle along the Tool Axis.  Figure 4-22: Change in Helix Angle along the Tool Axis. Chapter 4. Generalized Geometric Model of Inserted Cutters      63 4.5.3 Rhombic Turning Insert As stated before, geometric model of inserted cutters can be applied not only for milling inserts, but also turning inserts. For that purpose, a rhombic (diamond) shaped insert was selected for analysis. Table 4-III and Figure 4-23 show the insert geometry. Table 4-III: Inputs for the Rhombic Turning Insert. Inputs 𝑳𝑳 (Insert Length) 15 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 93Β° π’Šπ’Šπ’Šπ’Š (Insert Width) - πœΈπœΈπ’‡π’‡ (Radial Rake Angle) -8Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) - πœΈπœΈπ’‘π’‘ (Axial Rake Angle) -10Β° π’“π’“πœΊπœΊ (Corner Radius) 1.6 mm  πœΊπœΊπ’“π’“ (Tool Included Angle) 55Β°  Figure 4-23: Catalogue Figures of the Turning Insert [55]. In the analysis of turning tools, similar procedure to previous examples was applied with some certain modifications. Since turning tools do not rotate, 𝐷𝐷𝑐𝑐  (Cutting Diameter) is not defined. Moreover, wiper edge was also not considered for turning tools. As a result, instead of tool axis tip, intersection between cutting edge and feed plane (Point 𝐴𝐴 in Figure 4-24) was selected as the origin of the global coordinate system. Figure 4-24 shows the variable and control points used in the analysis of this type of insert. Chapter 4. Generalized Geometric Model of Inserted Cutters      64  Figure 4-24: Control Points of the Rhombic Turning Insert. Location of insert center with respect to the point 𝐴𝐴 can be calculated as:  𝐼𝐼π‘₯π‘₯ = (βˆ’πΏπΏ + 2π‘Ÿπ‘Ÿπœ€πœ€ sin πœ€πœ€π‘Ÿπ‘Ÿ + 𝐿𝐿 cos2 πœ€πœ€π‘Ÿπ‘Ÿ) sin πœ…πœ…π‘Ÿπ‘Ÿ2 sin πœ€πœ€π‘Ÿπ‘Ÿ+ (βˆ’2π‘Ÿπ‘Ÿπœ€πœ€ + 𝐿𝐿 sin πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ 2π‘Ÿπ‘Ÿπœ€πœ€ cos πœ€πœ€π‘Ÿπ‘Ÿ + 𝐿𝐿 sin πœ€πœ€π‘Ÿπ‘Ÿ cos πœ€πœ€π‘Ÿπ‘Ÿ) cos πœ…πœ…π‘Ÿπ‘Ÿ2 sin πœ€πœ€π‘Ÿπ‘Ÿ 𝐼𝐼𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€ + 𝐿𝐿 sinπœ…πœ…π‘Ÿπ‘Ÿ2 (1 + cos πœ€πœ€π‘Ÿπ‘Ÿ) βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿsin πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿ+ 𝐿𝐿 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐿𝐿 cos2 πœ€πœ€π‘Ÿπ‘Ÿ cos πœ…πœ…π‘Ÿπ‘Ÿ2 sin πœ€πœ€π‘Ÿπ‘Ÿ (4.26)   Chapter 4. Generalized Geometric Model of Inserted Cutters      65 The locations of control points 𝐴𝐴,𝐡𝐡, and 𝐷𝐷 can be described as the following equa- tions. The results of the model are shown in the following figures.  𝐴𝐴π‘₯π‘₯ = 0   ,   𝐴𝐴𝑧𝑧 = 0 (4.27)  𝐡𝐡π‘₯π‘₯ = π‘Ÿπ‘Ÿπœ€πœ€ sinπœ…πœ…π‘Ÿπ‘Ÿ    ,   𝐡𝐡𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€(1 βˆ’ cos πœ…πœ…π‘Ÿπ‘Ÿ) (4.28)  𝐷𝐷π‘₯π‘₯ = π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ + �𝐿𝐿 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€sin πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘ŸοΏ½ cos πœ…πœ…π‘Ÿπ‘Ÿ 𝐷𝐷𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€ + 𝐿𝐿 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿsin πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sinπœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿ  (4.29)  Figure 4-25: Position of the Cutting Edge for a Rhombic Turning Insert. Chapter 4. Generalized Geometric Model of Inserted Cutters      66  Figure 4-26: Change in the Normal Rake Angle along the Cutting Edge.  Figure 4-27: Change in the Helix Angle along the Cutting Edge. Chapter 4. Generalized Geometric Model of Inserted Cutters      67 4.6 Summary In this chapter, the analytical calculations needed to define and orient the cutting edge on a cutter or tool holder is presented. Variables used in the equations have been obtained from ISO standards ISO 3002 and ISO 13399. It is shown that for any type of insert geometry, it is possible to define the cutting edge on the tool axis and to convert from design angles to physical angles. From the definitions and transformation equations, it can be observed that normal rake angle 𝛾𝛾𝑛𝑛  and helix (inclination) angle πœ†πœ†π‘ π‘  are functions of the cutting edge angle πœ…πœ…π‘Ÿπ‘Ÿ , radial rake angle 𝛾𝛾𝑓𝑓 , and axial rake angle 𝛾𝛾𝑝𝑝 . As a result, any change in one of these angles along the cutting edge alters others. However, in this study, axial and radial rake angles are kept constant; therefore, changes observed in helix and normal rake angles in the presented figures are due to the cutting angle changes along the corner modifications. For instance, for an insert with -10Β° axial rake and -8Β° radial rake, as the cutting edge angle changes from 0Β° to 93Β°, normal rake angle changes from -13Β° to -9Β° and the helix angle changes from +6Β° to -10Β° along 1.6 mm corner radius. In the next chapter, generalized mechanics of metal cutting is presented. A new cut- ting force model which is applicable to different cutting processes is introduced and simulations are presented with experimental verifications. Chapter 5. Generalized Mechanics of Metal Cutting      68 5. Generalized Mechanics of Metal Cutting  5.1 Overview In this chapter, generalized modeling of cutting forces using the mechanics on the rake face, the friction force 𝐹𝐹𝑒𝑒  and the normal force 𝐹𝐹𝑣𝑣, is presented. The generalized model is compared with the current cutting force models in the literature, and the rela- tionships between the models are presented. In addition to cutting force models, calibration and material models are summarized and their transformations to the proposed model are presented. With this information, it is possible to convert most of the models currently being used in the literature to the proposed model. 5.2 Rake Face Based General Force Model The normal and friction forces on the rake face are used as the base in predicting the cutting forces in the machining system. Figure 5-1 shows a sample oblique cutting process with all the variables used in this study. In Figure 5-1, ?⃑?𝐹 is the total resultant force, 𝐹𝐹𝑒𝑒���⃑  is the friction force on the rake face, 𝐹𝐹𝑣𝑣���⃑  is the normal force on the rake face, 𝐹𝐹𝑠𝑠���⃑ and 𝐹𝐹𝑛𝑛���⃑  are the shearing and normal forces acting on the shear plane respectively, 𝛾𝛾𝑛𝑛  is the rake angle, πœ†πœ†π‘ π‘  is the inclination / helix angle, πœ‚πœ‚ is the chip flow angle, πœ™πœ™π‘›π‘›  is the normal shear angle, π›½π›½π‘Žπ‘Ž  is the friction angle, and finally, 𝑏𝑏 and β„Ž are the width of cut and uncut chip thickness respectively. Chapter 5. Generalized Mechanics of Metal Cutting      69  Figure 5-1: Mechanics of Oblique Cutting [11]. For a differential chip load (𝑑𝑑𝐴𝐴𝑐𝑐) in an engagement with a selected point on the cut- ting edge of the tool, the differential friction (𝑑𝑑𝐹𝐹𝑒𝑒 ) and the differential normal force (𝑑𝑑𝐹𝐹𝑣𝑣) acting on the rake face can be expressed as,  𝑑𝑑𝐹𝐹𝑒𝑒 = 𝐾𝐾𝑒𝑒𝑐𝑐  𝑑𝑑𝐴𝐴𝑐𝑐 + 𝐾𝐾𝑒𝑒𝑑𝑑  𝑑𝑑𝑑𝑑 𝑑𝑑𝐹𝐹𝑣𝑣 = 𝐾𝐾𝑣𝑣𝑐𝑐  𝑑𝑑𝐴𝐴𝑐𝑐 + 𝐾𝐾𝑣𝑣𝑑𝑑  𝑑𝑑𝑑𝑑 (5.1) where 𝐾𝐾𝑒𝑒𝑐𝑐  and 𝐾𝐾𝑣𝑣𝑐𝑐  are the friction and normal cutting coefficients and 𝐾𝐾𝑒𝑒𝑑𝑑  and 𝐾𝐾𝑣𝑣𝑑𝑑 are the related edge coefficients. These specific cutting coefficients depend on the tool – workpiece combination. For a specific workpiece, these cutting coefficients can be described as a function of chip thickness, rake angle, and the cutting speed (𝑉𝑉𝑐𝑐), they are called size, geometry and speed effects, respectively.  The identification of the cutting Chapter 5. Generalized Mechanics of Metal Cutting      70 force coefficients are explained in the following sections. In Eq. (5.1), 𝑑𝑑𝐴𝐴𝑐𝑐  and 𝑑𝑑𝑑𝑑 are differential chip area and differential cutting edge length can be calculated with the following;  𝑑𝑑𝐴𝐴𝑐𝑐 = β„Ž 𝑏𝑏 𝑑𝑑𝑑𝑑 = 𝑏𝑏𝑑𝑑𝑓𝑓𝑓𝑓 = 𝑏𝑏sin πœ…πœ…π‘Ÿπ‘Ÿ  (5.2) Once the differential friction (𝑑𝑑𝐹𝐹𝑒𝑒 ) and differential normal (𝑑𝑑𝐹𝐹𝑣𝑣) cutting forces are evaluated through the use of Eq. (5.1), they can be transformed into machine coordinate system (π‘‹π‘‹π‘Œπ‘Œπ‘π‘) with procedure described in Chapter 4.3. Since the tool geometry (rake, helix, cutting edge angles) may change along the cutting edge of the tool, these trans- formations must be repeated for each differential part of the cutting edge(s). After the transformation process, the differential forces are summed to determine the total cutting forces acting on the machine coordinates (𝐹𝐹π‘₯π‘₯ , 𝐹𝐹𝑦𝑦 , and 𝐹𝐹𝑧𝑧), as:  οΏ½ 𝐹𝐹π‘₯π‘₯ 𝐹𝐹𝑦𝑦 𝐹𝐹𝑧𝑧 οΏ½ = ���𝑑𝑑𝐹𝐹π‘₯π‘₯𝑑𝑑𝐹𝐹𝑦𝑦 𝑑𝑑𝐹𝐹𝑧𝑧 οΏ½ 𝐾𝐾 π‘˜π‘˜=1 𝑁𝑁𝑓𝑓 𝑛𝑛=1 οΏ½ π‘˜π‘˜ ,𝑛𝑛  (5.3) where 𝑁𝑁𝑓𝑓  is the total number of cutting edges and 𝐾𝐾 represents the total number of discrete points along the cutting edge 𝑛𝑛.  Proposed cutting force model is summarized in Figure 5-2. Chapter 5. Generalized Mechanics of Metal Cutting      71  Figure 5-2: Summary of the Proposed Mechanistic Approach. Chapter 5. Generalized Mechanics of Metal Cutting      72 5.3 Identification of Specific Cutting Coefficients Accurate determination of cutting force coefficients is critical to cutting force pre- diction. There are several parameters influencing the cutting force coefficients and they can be estimated either mechanistically or using classical orthogonal to oblique trans- formation method.  The cutting force coefficients depend on: - Cutting Method : constant or varying chip volume removal (Turning vs. Milling) - Cutting Conditions: feed rate, depth of cut, cutting speed, use of coolant; - Workpiece Material: chemical composition; - Tool: tool material, chip breaker, tool wear. Since it is not possible to account for all of these variables simultaneously, only the effects of chip thickness, cutting speed, and normal rake angle have been considered which are most dominant. In the following sections, both mechanistic approach and classical approach are explained in detail to determine the specific cutting coefficients in friction (𝐾𝐾𝑒𝑒 ) and normal (𝐾𝐾𝑣𝑣) directions. 5.3.1 Orthogonal to Oblique Transformation Orthogonal cutting tests can be used to calculate the shear stress (πœπœπ‘ π‘ ), shear angle (πœ™πœ™π‘π‘), and the friction angle (π›½π›½π‘Žπ‘Ž ) as a function of chip thickness, rake angle, and cutting speed. However, cutting edge(s) are usually not orthogonal to the cutting velocity, but inclined with helix (πœ†πœ†π‘ π‘ ) or inclination (𝑖𝑖) angle. Thus, an orthogonal to oblique transfor- mation needs to be applied. The same concept has been implemented stated in Shamoto and Altintas [11], except the cutting coefficients are calculated in normal and friction directions instead of radial, tangential, and axial directions. In orthogonal cutting, the rectangular shear plane area can be calculated as (Figure 5-3):  𝐴𝐴𝑠𝑠,π‘œπ‘œπ‘Ÿπ‘Ÿπ‘‘π‘‘ β„Žπ‘œπ‘œπ‘’π‘’π‘œπ‘œπ‘›π‘›π‘Žπ‘Žπ‘œπ‘œ = 𝑏𝑏 β„Žsinπœ™πœ™π‘π‘  (5.4) However, due to the helix angle (πœ†πœ†π‘ π‘ ) in oblique cutting, the shear plane area is a pa- rallelogram and its area can be calculated by: Chapter 5. Generalized Mechanics of Metal Cutting      73  𝐴𝐴𝑠𝑠,π‘œπ‘œπ‘π‘π‘œπ‘œπ‘–π‘–π‘œπ‘œπ‘’π‘’π‘‘π‘‘ = 𝑏𝑏 β„Žcos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘›  (5.5) Where πœ™πœ™π‘›π‘›  is the normal shear angle, i.e. shear angle measured in the normal plane 𝑃𝑃𝑛𝑛 . In this study, using the same assumptions stated in Altintas [5], normal shear angle (πœ™πœ™π‘›π‘› ) has been assumed equal to the shear angle in orthogonal cutting (πœ™πœ™π‘π‘).  Figure 5-3: Shear Plane Area Comparison: Orthogonal Cutting (Left) and Oblique Cutting (Right). In oblique cutting, shear force is equal to:  𝐹𝐹𝑠𝑠 = πœπœπ‘ π‘ π΄π΄π‘ π‘  = πœπœπ‘ π‘  𝑏𝑏 β„Žcos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘›  (5.6) Detailed illustration of the oblique cutting geometry is shown in Figure 5-4 [11]. When the shearing force is transformed to Cartesian coordinates, following equations can be obtained:  𝐹𝐹𝑋𝑋𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠 cosπœ™πœ™π‘–π‘– cos πœƒπœƒπ‘›π‘›cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘›) (5.7)  πΉπΉπ‘Œπ‘ŒπΌπΌπΌπΌ = βˆ’πΉπΉπ‘ π‘  sinπœ™πœ™π‘–π‘–  (5.8)  𝐹𝐹𝑍𝑍𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠 cosπœ™πœ™π‘–π‘– sinπœƒπœƒπ‘›π‘›cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘›) (5.9) Chapter 5. Generalized Mechanics of Metal Cutting      74  Figure 5-4: Cutting Forces, Velocities, and Angles in Oblique Cutting [11]. In order to use Eq. (5.7), Eq. (5.8), and Eq. (5.9), the unknown angles πœƒπœƒπ‘›π‘›  and πœ™πœ™π‘–π‘– must be expressed with known angles. πœƒπœƒπ‘›π‘›  is the angle between cut surface and the resultant cutting force 𝐹𝐹. Due to the simplified geometry in Figure 5-5:    πœƒπœƒπ‘›π‘› = 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛  (5.10)   Chapter 5. Generalized Mechanics of Metal Cutting      75 Similarly, the tangent of πœ™πœ™π‘–π‘–  is equal to the following:  tanπœ™πœ™π‘–π‘– = 𝐹𝐹𝑦𝑦𝐼𝐼𝐼𝐼𝐹𝐹𝑠𝑠𝑁𝑁  (5.11) where 𝐹𝐹𝑠𝑠𝑁𝑁  is the projection of shear force 𝐹𝐹𝑠𝑠 onto normal plane 𝑃𝑃𝑛𝑛 .  Figure 5-5: Illustration of angle 𝜽𝜽𝜸𝜸. (𝑭𝑭𝑭𝑭,𝑭𝑭𝑭𝑭𝑭𝑭, and 𝑭𝑭𝑭𝑭𝑭𝑭 Are Projections of Forces onto Normal Plane).  In Figure 5-4, 𝐹𝐹𝑁𝑁  can be expressed as,  𝐹𝐹𝑁𝑁 = 𝐹𝐹𝑠𝑠 cosπœ™πœ™π‘–π‘–cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘›) = 𝐹𝐹𝑒𝑒 cos πœ‚πœ‚sin𝛽𝛽𝑛𝑛 𝐹𝐹𝑒𝑒 = 𝐹𝐹𝑠𝑠 sinπœ™πœ™π‘–π‘–sin πœ‚πœ‚  (5.12) Combining equations (5.10) and (5.12) gives the following relationship:  tanπœ™πœ™π‘–π‘– = tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛cos(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) (5.13)    Chapter 5. Generalized Mechanics of Metal Cutting      76 And furthermore:  cosπœ™πœ™π‘–π‘– = cos(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  sinπœ™πœ™π‘–π‘– = tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛 οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  (5.14) Combining equations (5.7), (5.8), (5.9), and (5.10) with (5.14) results in:  𝐹𝐹𝑋𝑋𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠 cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  = π‘π‘β„Ž οΏ½ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛)οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� 𝐹𝐹𝑍𝑍𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  = π‘π‘β„Ž οΏ½ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛)οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� (5.15) Similarly, by substituting equations (5.6) and (5.16) into Equation (5.8):  πΉπΉπ‘Œπ‘ŒπΌπΌπΌπΌ = βˆ’πΉπΉπ‘ π‘   tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛 οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  = π‘π‘β„Ž οΏ½βˆ’ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛�cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� (5.16) Corresponding specific cutting coefficients in 𝑋𝑋𝐼𝐼𝐼𝐼 ,π‘Œπ‘ŒπΌπΌπΌπΌ , and 𝑍𝑍𝐼𝐼𝐼𝐼  directions can be eva- luated as:  𝐾𝐾𝑋𝑋𝐼𝐼𝐼𝐼 = οΏ½ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛)οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� πΎπΎπ‘Œπ‘ŒπΌπΌπΌπΌ = οΏ½βˆ’ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛�cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� 𝐾𝐾𝑍𝑍𝐼𝐼𝐼𝐼 = οΏ½ πœπœπ‘ π‘ cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛)οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛� (5.17)  Chapter 5. Generalized Mechanics of Metal Cutting      77 The cutting coefficients are transformed from the cutting edge coordinate system (System 2) to chip flow coordinate system (System 4) to calculate the cutting coeffi- cients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣. It is possible to accomplish this task in two different ways. Firstly, these coefficients in the cutting edge coordinate system can be geometrically trans- formed into the chip flow coordinate system as described in Chapter 4.3. The transformation matrix required for this task 𝐢𝐢42  can be obtained by:  𝐢𝐢42 = (𝐢𝐢23 𝐢𝐢34)𝑅𝑅  (5.18) Hence:  οΏ½ 𝐾𝐾𝑒𝑒 𝐾𝐾𝑣𝑣 οΏ½ = 𝐢𝐢42  οΏ½πΎπΎπ‘‹π‘‹πΌπΌπΌπΌπΎπΎπ‘Œπ‘ŒπΌπΌπΌπΌ 𝐾𝐾𝑍𝑍𝐼𝐼𝐼𝐼 οΏ½ οΏ½ 𝐾𝐾𝑒𝑒 𝐾𝐾𝑣𝑣 οΏ½ = οΏ½sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ βˆ’ sin πœ‚πœ‚ cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚cos 𝛾𝛾𝑛𝑛 0 βˆ’ sin 𝛾𝛾𝑛𝑛 οΏ½  οΏ½πΎπΎπ‘‹π‘‹πΌπΌπΌπΌπΎπΎπ‘Œπ‘ŒπΌπΌπΌπΌπΎπΎπ‘π‘πΌπΌπΌπΌ οΏ½ (5.19) As a result:  𝐾𝐾𝑒𝑒 = sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚  𝐾𝐾𝑋𝑋𝐼𝐼𝐼𝐼 βˆ’ sin πœ‚πœ‚  πΎπΎπ‘Œπ‘ŒπΌπΌπΌπΌ + cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚  𝐾𝐾𝑍𝑍𝐼𝐼𝐼𝐼 𝐾𝐾𝑣𝑣 = cos 𝛾𝛾𝑛𝑛  𝐾𝐾𝑋𝑋𝐼𝐼𝐼𝐼 βˆ’ sin 𝛾𝛾𝑛𝑛  𝐾𝐾𝑍𝑍𝐼𝐼𝐼𝐼  (5.20) Alternatively, specific cutting coefficients in chip flow coordinate system can be calculated by evaluating the resultant force:  𝐹𝐹 = �𝐹𝐹𝑋𝑋𝐼𝐼𝐼𝐼2 + πΉπΉπ‘Œπ‘ŒπΌπΌπΌπΌ2 + 𝐹𝐹𝑍𝑍𝐼𝐼𝐼𝐼2  (5.21) By assuming that the average friction angle (π›½π›½π‘Žπ‘Ž ) is equal to the friction angle (𝛽𝛽) in orthogonal cutting [5].  𝐾𝐾𝑒𝑒 = πΉπΉπ‘π‘β„Ž sinπ›½π›½π‘Žπ‘Ž 𝐾𝐾𝑣𝑣 = πΉπΉπ‘π‘β„Ž cosπ›½π›½π‘Žπ‘Ž  (5.22) Chapter 5. Generalized Mechanics of Metal Cutting      78 Figure 5-6 illustrates and summarizes the procedure outlined above. As result, spe- cific cutting coefficients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣 on friction and normal directions can be evaluated with the following equations:  𝐾𝐾𝑒𝑒 = πœπœπ‘ π‘ οΏ½1 βˆ’ tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 sinπ›½π›½π‘Žπ‘Ž 𝐾𝐾𝑣𝑣 = πœπœπ‘ π‘ οΏ½1 βˆ’ tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 π‘π‘π‘œπ‘œπ‘ π‘  π›½π›½π‘Žπ‘Ž (5.23) Or applying the geometrical transformation in Equation (5.20):  𝐾𝐾𝑒𝑒 = πœπœπ‘ π‘ π‘π‘π‘œπ‘œπ‘ π‘ πœ‚πœ‚(sin 𝛾𝛾𝑛𝑛 cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin𝛽𝛽𝑛𝑛 + cos 𝛾𝛾𝑛𝑛 sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛))cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 𝐾𝐾𝑣𝑣 = πœπœπ‘ π‘  cos 𝛾𝛾𝑛𝑛 cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) (1 βˆ’ tan 𝛾𝛾𝑛𝑛 tan(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛))cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘› οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  (5.24)  Figure 5-6: Summary of the Classical Approach in Cutting Coefficient Identification. In summary, cutting coefficients in friction and normal directions can be evaluated using orthogonal shear parameters by Eq. (5.24). After obtaining the cutting coefficients, cutting force model explained in Section 5.2 can be applied. The edge coefficients 𝐾𝐾𝑒𝑒𝑑𝑑 and 𝐾𝐾𝑣𝑣𝑑𝑑  can be calculated by transforming the edge coefficients from 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system to π‘ˆπ‘ˆπ‘‰π‘‰ coordinate system. Alternatively, they can be added when the cutting forces are calculated and transformed back to the 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system or machine coordinate system. In the simulations and validations shown in the following sections, second method has been used. Shear Parameters From Orthogonal Cutting Data πœ™πœ™π‘π‘ , πœπœπ‘ π‘  ,π›½π›½π‘Žπ‘Ž 𝐹𝐹𝑋𝑋 𝐼𝐼𝐼𝐼 πΉπΉπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 𝐹𝐹𝑍𝑍 𝐼𝐼𝐼𝐼  𝐾𝐾𝑒𝑒 𝐾𝐾𝑣𝑣  (5.15) (5.16) (5.19) 𝐹𝐹𝑠𝑠 (5.6) πœ†πœ†π‘ π‘  , 𝛾𝛾𝑛𝑛 , πœ‚πœ‚ 𝑏𝑏,β„Ž Tool Geometry And Assumed Chip Flow Direction Cutting Conditions Chapter 5. Generalized Mechanics of Metal Cutting      79 5.3.2 Mechanistic Identification Currently, the most commonly used approach to determine the cutting coefficients is the mechanistic identification. Although this approach cannot provide the detailed microscopic effects of the machining process, such as shearing and chip flow, it allows predicting the cutting forces without extensive turning tests. However, in mechanistic modeling, accurate determination of these cutting coefficients over a wide range of cutting conditions is time consuming, since it requires a large number of experiments with many parameters such as cutting tool – workpiece material combination, cutting tool geometry, cutting speed, and depth of cut. In the following sections, two different cases to identify the cutting coefficients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣 are discussed. Empirical Approach In the first case, calculations that are needed to identify the cutting coefficients from experiments are presented. By using this method, it is also possible to calculate the true chip flow angle (πœ‚πœ‚). Since edge forces have no effect on shearing, they must be calculated and subtracted from total cutting forces in radial, tangential, and axial directions [5]. Thus, it is possible to reduce the cutting force equations as the following:  πΉπΉπ‘Ÿπ‘Ÿπ‘π‘ = πΎπΎπ‘Ÿπ‘Ÿ  𝐴𝐴𝑐𝑐 𝐹𝐹𝑑𝑑𝑐𝑐 = 𝐾𝐾𝑑𝑑  𝐴𝐴𝑐𝑐 πΉπΉπ‘Žπ‘Žπ‘π‘ = πΎπΎπ‘Žπ‘Ž  𝐴𝐴𝑐𝑐  (5.25) where 𝐹𝐹𝑖𝑖𝑐𝑐  is the cutting force in 𝑖𝑖 ∈ (π‘Ÿπ‘Ÿ, 𝑑𝑑,π‘Žπ‘Ž) direction, 𝐾𝐾𝑖𝑖  is the specific cutting coefficient in 𝑖𝑖 direction, and 𝐴𝐴𝑐𝑐  is the chip load. In order to evaluate the coefficients in normal and friction directions, the transformation equations derived in Chapter 4.3 are applied.  πΎπΎπ‘Ÿπ‘Ÿ = (𝐾𝐾𝑒𝑒 cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ βˆ’ 𝐾𝐾𝑣𝑣 sin 𝛾𝛾𝑛𝑛) 𝐾𝐾𝑑𝑑 = (𝐾𝐾𝑒𝑒 sin Ξ»s sin πœ‚πœ‚ + 𝐾𝐾𝑒𝑒 cos πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ + 𝐾𝐾𝑣𝑣 cos πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛) πΎπΎπ‘Žπ‘Ž = (βˆ’πΎπΎπ‘’π‘’ cos πœ†πœ†π‘ π‘  sin πœ‚πœ‚ + 𝐾𝐾𝑒𝑒 sin πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ + 𝐾𝐾𝑣𝑣 sin πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛) (5.26) Chapter 5. Generalized Mechanics of Metal Cutting      80 In Equation (5.26), πΎπΎπ‘Ÿπ‘Ÿ , 𝐾𝐾𝑑𝑑 , and πΎπΎπ‘Žπ‘Ž  can be obtained from cutting tests, normal rake (𝛾𝛾𝑛𝑛 ) and oblique / helix angle (πœ†πœ†π‘ π‘ ) can be obtained from tool geometry, and 𝐾𝐾𝑒𝑒 , 𝐾𝐾𝑣𝑣, and chip flow angle (πœ‚πœ‚) are unknown. There are three equations and three unknowns, howev- er, chip flow angle (πœ‚πœ‚) is nonlinear. To overcome this problem, two new variables, 𝐾𝐾𝑒𝑒1 and 𝐾𝐾𝑒𝑒2 were introduced:  𝐾𝐾𝑒𝑒1 = 𝐾𝐾𝑒𝑒 cos πœ‚πœ‚ 𝐾𝐾𝑒𝑒2 = 𝐾𝐾𝑒𝑒 sin πœ‚πœ‚ (5.27) Substituting Equation (5.27) into Equation (5.26):  πΎπΎπ‘Ÿπ‘Ÿ = πΆπΆπ‘’π‘’π‘Ÿπ‘Ÿ 1𝐾𝐾𝑒𝑒1 + πΆπΆπ‘£π‘£π‘Ÿπ‘ŸπΎπΎπ‘£π‘£ 𝐾𝐾𝑑𝑑 = 𝐢𝐢𝑒𝑒𝑑𝑑 1𝐾𝐾𝑒𝑒1 + 𝐢𝐢𝑒𝑒𝑑𝑑 2𝐾𝐾𝑒𝑒2 + 𝐢𝐢𝑣𝑣𝑑𝑑𝐾𝐾𝑣𝑣 πΎπΎπ‘Žπ‘Ž = πΆπΆπ‘’π‘’π‘Žπ‘Ž 1𝐾𝐾𝑒𝑒1 + πΆπΆπ‘’π‘’π‘Žπ‘Ž 2𝐾𝐾𝑒𝑒2 + πΆπΆπ‘£π‘£π‘Žπ‘ŽπΎπΎπ‘£π‘£ (5.28) where πΆπΆπ‘’π‘’π‘Ÿπ‘Ÿ 1 = cos 𝛾𝛾𝑛𝑛 ,πΆπΆπ‘£π‘£π‘Ÿπ‘Ÿ = βˆ’sin 𝛾𝛾𝑛𝑛 ,𝐢𝐢𝑒𝑒𝑑𝑑 1 = cos πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 ,𝐢𝐢𝑒𝑒𝑑𝑑 2 = sin Ξ»s ,𝐢𝐢𝑣𝑣𝑑𝑑 = cos πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 πΆπΆπ‘’π‘’π‘Žπ‘Ž 1 = sin πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 ,πΆπΆπ‘’π‘’π‘Žπ‘Ž 2 = βˆ’cos πœ†πœ†π‘ π‘  ,πΆπΆπ‘£π‘£π‘Žπ‘Ž = sin πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 These coefficients are only functions of tool geometry, and they are constant for a specific experiment. As a result, there are 3 linear equations with three unknowns. In matrix form:  οΏ½ πΎπΎπ‘Ÿπ‘Ÿ 𝐾𝐾𝑑𝑑 πΎπΎπ‘Žπ‘Ž οΏ½ = οΏ½πΆπΆπ‘’π‘’π‘Ÿπ‘Ÿ 1 0 πΆπΆπ‘£π‘£π‘Ÿπ‘ŸπΆπΆπ‘’π‘’π‘‘π‘‘ 1 𝐢𝐢𝑒𝑒𝑑𝑑 2 𝐢𝐢𝑣𝑣𝑑𝑑 πΆπΆπ‘’π‘’π‘Žπ‘Ž 1 πΆπΆπ‘’π‘’π‘Žπ‘Ž 2 πΆπΆπ‘£π‘£π‘Žπ‘ŽοΏ½  οΏ½ 𝐾𝐾𝑒𝑒1 𝐾𝐾𝑒𝑒2 𝐾𝐾𝑣𝑣 οΏ½ (5.29) Solving these equations in Eq. (5.29) results in 𝐾𝐾𝑒𝑒1, 𝐾𝐾𝑒𝑒2, and 𝐾𝐾𝑣𝑣. It should be noted that:  𝐾𝐾𝑒𝑒 = �𝐾𝐾𝑒𝑒12 + 𝐾𝐾𝑒𝑒22 (5.30)  πœ‚πœ‚ = tanβˆ’1 𝐾𝐾𝑒𝑒2 𝐾𝐾𝑒𝑒1  (5.31) Consequently, a set of simple experiments with known cutting conditions and tool geometry are sufficient to calculate the cutting coefficients and chip flow angle. Since most of the chip flow models in the literature are based on empirical equations, and the Chapter 5. Generalized Mechanics of Metal Cutting      81 theoretical models are difficult to apply and generalize, this method provides a quick way to determine the chip flow angle. During milling, chip thickness varies continuously as tool motion is a trochoidal motion. In order to apply proposed mechanistic identification method, cutting force coefficients can be obtained in terms of the average chip thickness defined as [45]:  𝐴𝐴𝑐𝑐��� = 1πœƒπœƒπ‘‘π‘‘π‘₯π‘₯ βˆ’ πœƒπœƒπ‘ π‘ π‘‘π‘‘ οΏ½ 𝐴𝐴𝑐𝑐(πœƒπœƒ)π‘‘π‘‘πœƒπœƒπœƒπœƒπ‘‘π‘‘π‘₯π‘₯πœƒπœƒπ‘ π‘ π‘‘π‘‘  (5.32) where πœƒπœƒπ‘ π‘ π‘‘π‘‘  and πœƒπœƒπ‘‘π‘‘π‘₯π‘₯  are tool entry and exit angles, respectively. Theoretical Approach In the second case, it has been assumed that the cutting coefficients in radial, tan- gential, and axial directions as well as tool geometry are known, but the objective is to calculate the shear parameters (shear stress πœπœπ‘ π‘ , shear angle πœ™πœ™π‘π‘ , and friction angle π›½π›½π‘Žπ‘Ž ) only using this set of experiments. There have been two fundamental approaches for the solution of oblique cutting parameters, namely maximum shear stress principle and the minimum energy principle. Although these principles are iterative models, it is possible to obtain an approximate solution using Stabler’s chip flow assumption (πœ‚πœ‚ = πœ†πœ†π‘ π‘ ). Maximum Shear Stress Principle Maximum shear stress principle states that the resultant cutting force (𝐹𝐹) makes an angle amount of (πœ™πœ™π‘π‘ + π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛 ) with the shear plane, and the angle between the maxi- mum shear stress and the principal stress is 45Β° [57]. So:  𝐹𝐹𝑠𝑠 = 𝐹𝐹(cos πœƒπœƒπ‘–π‘– cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) cosπœ™πœ™π‘–π‘– + sinπœƒπœƒπ‘–π‘– sinπœ™πœ™π‘–π‘–) 𝐹𝐹𝑠𝑠 = 𝐹𝐹 cos 45Β° (5.33) Moreover, another statement of this principle dictates that the projection of the re- sultant force to the shear plane coincides with the shear direction. This results in:  𝐹𝐹(cos πœƒπœƒπ‘–π‘– cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) sinπœ™πœ™π‘–π‘– βˆ’ sinπœƒπœƒπ‘–π‘– cosπœ™πœ™π‘–π‘–) = 0 (5.34) Eq. (5.33) and Eq. (5.34) are used to derive the following relationships: Chapter 5. Generalized Mechanics of Metal Cutting      82  πœ™πœ™π‘–π‘– = sinβˆ’1�√2 sinπœƒπœƒπ‘–π‘–οΏ½ (5.35)  πœ™πœ™π‘›π‘› = cosβˆ’1 οΏ½tanπœƒπœƒπ‘–π‘–tanπœ™πœ™π‘–π‘–οΏ½ βˆ’ πœƒπœƒπ‘›π‘›  (5.36)  Figure 5-4 illustrates the oblique cutting mechanics. In order to avoid dividing by zero for orthogonal cases, Eq. (5.36) can be rewritten as:  πœ™πœ™π‘›π‘› = cosβˆ’1 οΏ½ 1 √2 cosπœ™πœ™π‘–π‘–cos πœƒπœƒπ‘–π‘– οΏ½ βˆ’ πœƒπœƒπ‘›π‘›  (5.37) Using Eq. (5.35), Eq. (5.37), and the tool geometry (𝛾𝛾𝑛𝑛 , πœ†πœ†π‘ π‘ ), new chip flow angle (πœ‚πœ‚π‘‘π‘‘) can be calculated using the velocity relation:  πœ‚πœ‚π‘‘π‘‘ = tanβˆ’1 οΏ½tan πœ†πœ†π‘ π‘  cos(πœ™πœ™π‘›π‘› βˆ’ 𝛾𝛾𝑛𝑛) βˆ’ cos 𝛾𝛾𝑛𝑛 tanπœ™πœ™π‘–π‘–sinπœ™πœ™π‘›π‘› οΏ½ (5.38) In order to calculate the true chip flow angle, following interpolation algorithm is applied by using new and previous chip flow angles:  πœ‚πœ‚(π‘˜π‘˜) = 𝜐𝜐 πœ‚πœ‚(π‘˜π‘˜ βˆ’ 1) + (1 βˆ’ 𝜐𝜐) πœ‚πœ‚π‘‘π‘‘  (5.39) Using the above equations, true chip flow angle can be calculated iteratively. In this equation, 𝜈𝜈 is the convergence parameter. When the iteration is completed, the shear stress can be calculated as the following:  𝐹𝐹 = �𝐹𝐹𝑒𝑒2 + 𝐹𝐹𝑣𝑣2 = π‘π‘β„ŽοΏ½πΎπΎπ‘’π‘’2 + 𝐾𝐾𝑣𝑣2 (5.40)  𝐹𝐹𝑠𝑠 = πœπœπ‘ π‘ π΄π΄π‘ π‘  = πœπœπ‘ π‘  οΏ½ 𝑏𝑏cos πœ†πœ†π‘ π‘ οΏ½ οΏ½ β„Žsinπœ™πœ™π‘›π‘›οΏ½ = πΉπΉπ‘π‘π‘œπ‘œπ‘ π‘  45Β° (5.41)  πœπœπ‘ π‘  = �𝐾𝐾𝑒𝑒2 + 𝐾𝐾𝑣𝑣22  cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘›  (5.42) Results and discussions about this approach are given in the following sections.   Chapter 5. Generalized Mechanics of Metal Cutting      83 Minimum Energy Principle In this approach work done by Shamoto and Altintas [11] is used. In their study, re- sultant cutting force is expressed as:  𝐹𝐹 = πœπœπ‘ π‘ π‘π‘β„Ž[cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) cos πœƒπœƒπ‘–π‘– cosπœ™πœ™π‘–π‘– + sinπœƒπœƒπ‘–π‘– sinπœ™πœ™π‘–π‘–] cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘›  (5.43) And the cutting power (𝑃𝑃𝑑𝑑) required during cutting can be expressed as:  𝑃𝑃𝑑𝑑 = 𝐹𝐹(cos πœƒπœƒπ‘–π‘– cos πœƒπœƒπ‘›π‘› cos πœ†πœ†π‘ π‘  + sinπœƒπœƒπ‘–π‘– sin πœ†πœ†π‘ π‘ ) 𝑉𝑉 (5.44) where 𝑉𝑉 is the cutting speed. Non-dimensional cutting power (𝑃𝑃𝑑𝑑′ ) can be derived by substituting Eq. (5.43) into Eq. (5.44).  𝑃𝑃𝑑𝑑 β€² = 𝑃𝑃𝑑𝑑 π‘‰π‘‰πœπœπ‘ π‘ π‘π‘β„Ž = cos πœƒπœƒπ‘›π‘› tanπœƒπœƒπ‘–π‘– tan πœ†πœ†π‘ π‘ [cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) cosπœ™πœ™π‘–π‘– + tanπœƒπœƒπ‘–π‘– sinπœ™πœ™π‘–π‘–] sinπœ™πœ™π‘›π‘›  (5.45) Minimum energy principle states that the cutting power drawn must be minimal for a unique shear angle solution [11]. Therefore:  πœ•πœ•π‘ƒπ‘ƒπ‘‘π‘‘β€² πœ•πœ•πœ™πœ™π‘›π‘› = 0 πœ•πœ•π‘ƒπ‘ƒπ‘‘π‘‘ β€² πœ•πœ•πœ™πœ™π‘–π‘– = 0 (5.46) Using Stabler’s chip flow angle rule and maximum shear stress equations for the initial guess, shear angles (πœ™πœ™π‘›π‘›  and πœ™πœ™π‘–π‘–) are iterated to minimize the non-dimensional cutting power 𝑃𝑃𝑑𝑑′  using the following iterative equations:  οΏ½ πœ™πœ™π‘›π‘›(π‘˜π‘˜) πœ™πœ™π‘–π‘–(π‘˜π‘˜)οΏ½ = οΏ½πœ™πœ™π‘›π‘›(π‘˜π‘˜ βˆ’ 1)πœ™πœ™π‘–π‘–(π‘˜π‘˜ βˆ’ 1)οΏ½ βˆ’ 𝜁𝜁 �Δ𝑃𝑃𝑑𝑑′ Ξ”πœ™πœ™π‘›π‘›β„Ξ”π‘ƒπ‘ƒπ‘‘π‘‘β€² Ξ”πœ™πœ™π‘–π‘–β„ οΏ½ (5.47) In this method, convergence depends on the step size 𝜁𝜁 and the marginal increase during the perturbation (Ξ”πœ™πœ™π‘›π‘› ,𝑖𝑖). After the iterations are completed, shear stress can be calculated by the following equations:  𝐹𝐹𝑠𝑠 = 𝐹𝐹[cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) cosπœ™πœ™π‘–π‘– cosπœƒπœƒπ‘–π‘– + sinπœƒπœƒπ‘–π‘– sinπœ™πœ™π‘–π‘–] πœπœπ‘ π‘  = �𝐾𝐾𝑒𝑒2 + 𝐾𝐾𝑣𝑣2 [cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘›) cosπœ™πœ™π‘–π‘– cos πœƒπœƒπ‘–π‘– + sinπœƒπœƒπ‘–π‘– sinπœ™πœ™π‘–π‘–] cos πœ†πœ†π‘ π‘  sinπœ™πœ™π‘›π‘›  (5.48) Chapter 5. Generalized Mechanics of Metal Cutting      84 Enhancement of the Theoretical Models Stabler’s rule of chip flow is used as the initial guess in most theoretical models. However, since it is possible to obtain true chip flow angle using maximum shear stress and minimum energy principles, an iterative method was adapted to improve the calcula- tion of chip flow angle and cutting coefficients. Block diagram of this enhancement is illustrated in Figure 5-7. Stabler’s rule (πœ‚πœ‚ = πœ†πœ†π‘ π‘ ) was used as the initial value and cutting coefficients in π‘ˆπ‘ˆπ‘‰π‘‰ coordinate system were calculated by using geometrical transforma- tions. Then using the minimum energy or maximum shear stress principle, new chip flow angle was calculated and the procedure was repeated until the difference between the new and previous chip flow angle is less than the specified tolerance value.  Figure 5-7: Block Diagram of the Iterative Method to Enhance the Theoretical Models. By using the proposed method, it is possible to determine the inclination angle πœ†πœ†π‘ π‘  of the tool used to obtain the mechanistically determined cutting, πΎπΎπ‘Ÿπ‘Ÿ , 𝐾𝐾𝑑𝑑 , and πΎπΎπ‘Žπ‘Ž . In order to accomplish this task, following procedure was applied: 1. Cutting coefficients in radial, tangential, and axial directions were trans- formed to friction and normal coefficients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣 using the geometrical transformation equations described in Chapter 4.3. In order to complete the transformations, an arbitrary value for πœ†πœ†π‘ π‘  was selected. Since πœ†πœ†π‘ π‘  is unknown, this procedure must be repeated for each value of πœ†πœ†π‘ π‘  in a predetermined range. 2. Using the minimum energy principle, shear parameters (πœπœπ‘ π‘ , πœ™πœ™π‘›π‘› , and π›½π›½π‘Žπ‘Ž ) as well as true chip flow angle πœ‚πœ‚ were calculated. πΎπΎπ‘Ÿπ‘Ÿ ,𝐾𝐾𝑑𝑑 ,πΎπΎπ‘Žπ‘Ž  Geometrical Transformation 𝐾𝐾𝑒𝑒 ,𝐾𝐾𝑣𝑣 Max. Shear Stress Min. Energy (5.39), (5.47) πœ‚πœ‚ 𝛾𝛾𝑛𝑛 , πœ†πœ†π‘ π‘  Chapter 5. Generalized Mechanics of Metal Cutting      85 3. Following that, πΎπΎπ‘Ÿπ‘Ÿ ,𝐾𝐾𝑑𝑑 , and πΎπΎπ‘Žπ‘Ž  were calculated back using the orthogonal to oblique transformation using the following equations [5]:  πΎπΎπ‘Ÿπ‘Ÿ = πœπœπ‘ π‘ sinπœ™πœ™π‘›π‘› cos πœ†πœ†π‘ π‘  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛)οΏ½cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 𝐾𝐾𝑑𝑑 = πœπœπ‘ π‘ sinπœ™πœ™π‘›π‘› cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan πœ†πœ†π‘ π‘  tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛�cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 πΎπΎπ‘Žπ‘Ž = πœπœπ‘ π‘ sinπœ™πœ™π‘›π‘› cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) tan πœ†πœ†π‘ π‘  βˆ’ tan πœ‚πœ‚ sin𝛽𝛽𝑛𝑛�cos2(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 (5.49) 4. Calculated and original values of the cutting coefficients were compared and the errors for each πœ†πœ†π‘ π‘  were calculated using Eq. (5.50).  % πΈπΈπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿπ‘œπ‘œπ‘Ÿπ‘Ÿ = οΏ½πΎπΎπ‘π‘π‘Žπ‘Žπ‘œπ‘œπ‘π‘π‘’π‘’π‘œπ‘œπ‘Žπ‘Žπ‘‘π‘‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πΎπΎπ‘œπ‘œπ‘Ÿπ‘Ÿπ‘–π‘–π‘’π‘’π‘–π‘–π‘›π‘›π‘Žπ‘Žπ‘œπ‘œ οΏ½ πΎπΎπ‘œπ‘œπ‘Ÿπ‘Ÿπ‘–π‘–π‘’π‘’π‘–π‘–π‘›π‘›π‘Žπ‘Žπ‘œπ‘œ Γ— 100 (5.50) It can be observed that for a unique oblique angle, percentage error for axial cutting coefficient πΎπΎπ‘Žπ‘Ž  is minimal. Mechanistically determined cutting coefficients for two different workpiece mate- rials are used as inputs. Although the tool geometry (rake and helix angles) was known, helix angle (πœ†πœ†π‘ π‘ ) has been assumed to be unknown to check the validity of the proposed method. The first analysis was made on Ti6Al4V workpiece. Results of Budak’s cutting coefficient identification [58] were used. Tool used for the mechanistic calibration has 15Β° rake angle (𝛾𝛾𝑛𝑛 = 15Β°) and 30Β° helix angle (πœ†πœ†π‘ π‘  = 30Β°). In order to apply the model, helix angle is assumed to be unknown and all angles between 0Β° and 40Β° were scanned in the analysis and errors were calculated using the procedure outlined above. Table 5-I presents the cutting coefficients and tool workpiece used in both cases. Chapter 5. Generalized Mechanics of Metal Cutting      86  Figure 5-8: Results of Helix Angle Predictions for Ti6Al4V. Table 5-I: Cutting Coefficients and Tool Geometry Used in the Analysis. Material πΎπΎπ‘Ÿπ‘Ÿ  [MPa] 𝐾𝐾𝑑𝑑  [MPa] πΎπΎπ‘Žπ‘Ž  [MPa] Helix πœ†πœ†π‘ π‘  [Β°] Rake 𝛾𝛾𝑛𝑛  [Β°] Ti6Al4V 340 1630 608 30 15 Al7075 788.83 1319.41 48.751 5 0  Second analysis was performed using the coefficients calculated in Engin’s study [44]. Workpiece material has been Al7075, and the tool used in mechanistic identifica- tion has been a bull-nose end mill with 5/4” diameter, 0Β° rake angle, and 5Β° helix angle. Cutting coefficients are listed in Table 5-I and error results are presented in Figure 5-9. Chapter 5. Generalized Mechanics of Metal Cutting      87  Figure 5-9: Results of Helix Angle Predictions for Al7075. It can be observed from Figure 5-8 and Figure 5-9 that πΎπΎπ‘Žπ‘Ž  is very sensitive to the changes in the helix angle and it is possible to accurately determine the helix angle by applying the proposed method. On the other hand, theoretical proof of this method is rather difficult, because chip flow angle also depends on the helix angle and the equations used in minimum energy principle is rather complex and they can be solved only using numerical methods.  Chapter 5. Generalized Mechanics of Metal Cutting      88 Results and Summary Block diagram of the procedures explained in this section is illustrated in Figure 5-10. As a case study, to validate the models discussed in this section, mechanistically determined average cutting coefficients in 𝐢𝐢𝑅𝑅𝐴𝐴 coordinate system were transformed to π‘ˆπ‘ˆπ‘‰π‘‰ coordinate system and shear parameters were calculated using both maximum shear stress principle and minimum energy principle. Table 5-II summarizes the results: Table 5-II: Comparison of the Methods for the Solution of Oblique Shear Parameters. Maximum Shear Stress Principle Cutting Tool Rake Angle 𝛾𝛾𝑛𝑛  [Β°] Oblique Angle  πœ†πœ†π‘ π‘  [Β°] Friction Angle π›½π›½π‘Žπ‘Ž  [Β°] Shear Angle πœ™πœ™π‘›π‘›  [Β°] Shear Stress πœπœπ‘ π‘ [MPa] Chip Flow Angle πœ‚πœ‚ [Β°] Comp. Time [ms] # of Iterations CTGP 5 0 26.9 23 508 0 6.6 ms 2 SCGP 3.5 3.5 32 16.5 459.2 3.3 6.4 ms 31 MTGN -5 -5 20.3 17.1 477.8 -5.4 7.4  ms 14 Orthogonal 5 0 28.1 31.2 549 0 - -  Minimum Energy Principle Cutting Tool Rake Angle 𝛾𝛾𝑛𝑛  [Β°] Oblique Angle  πœ†πœ†π‘ π‘  [Β°] Friction Angle π›½π›½π‘Žπ‘Ž  [Β°] Shear Angle πœ™πœ™π‘›π‘›  [Β°] Shear Stress πœπœπ‘ π‘ [MPa] Chip Flow Angle πœ‚πœ‚ [Β°] Comp. Time [ms] # of Iterations CTGP 5 0 26.9 33.1 573.8 0 13.2 140 SCGP 3.5 3.5 32 30.1 596 1.72 14.5 115 MTGN -5 -5 20.3 31.6 600 -3.1 14.4 124 Orthogonal 5 0 28.1 31.2 549 0 - -  Workpiece used in these experiments was AISI Steel 1045, and Kennametal tools were used. Cutting speed (𝑉𝑉) was 150 m/min and the width of cut (𝑏𝑏) was 3.05 mm. Highlighted row is the experimentally measured orthogonal reference data. From Table 5-II, it can be observed that the minimum energy principle gives more accurate results than the maximum shear stress principle, since maximum shear stress principle is more conservative approach. However, minimum energy principle needs more iteration to calculate the shear parameters than the maximum shear stress principle. Chapter 5. Generalized Mechanics of Metal Cutting      89  Figure 5-10: Theoretical Identification Procedure. In mechanistic modeling, it is possible to determine the cutting coefficients by using a single test. Therefore, it provides an opportunity to observe the effect of chip thickness on the cutting coefficients. On the other hand, theoretical methods use average cutting coefficients, in which one set of coefficients can be obtained by conducting tests with different chip thicknesses. Figure 5-11 demonstrates this argument with SCGP tool. Tool geometry and cutting conditions are listed above. In Figure 5-11, it can be observed that specific cutting coefficients increase exponentially as the chip thickness decreases. This phenomenon is called size effect. Figure 5-12 shows the comparison of chip flow angle of different methods. Stabler rule states that the chip flow angle is equal to the inclination angle πœ†πœ†π‘ π‘ . It can be seen that according to the mechanistic approach, there is also an exponential relationship between chip thickness and chip flow angle, similar to the size effect. π›½π›½π‘Žπ‘Ž 𝛾𝛾𝑛𝑛 πœ†πœ†π‘ π‘  Force Relation (5.35) & (5.36) Velocity Relation (5.38) πœ‚πœ‚ (5.39) πœƒπœƒπ‘›π‘› ,πœƒπœƒπ‘–π‘–  Max. Shear Stress  (5.35) & (5.37) Minimum Energy (5.45) & (5.46) πœ™πœ™π‘›π‘› ,πœ™πœ™π‘–π‘– Input Angles Chapter 5. Generalized Mechanics of Metal Cutting      90  Figure 5-11: Effect of Chip Thickness on the Cutting Coefficients.  Figure 5-12: Effect of Chip Thickness on the Chip Flow Angle.   Chapter 5. Generalized Mechanics of Metal Cutting      91 5.4 Cutting Force Simulations and Validations Proposed insert geometry and mechanics models discussed in previous chapter are combined to simulate cutting forces in turning and milling using different tools. Several different insert geometries are considered for each type of cutting process. 5.4.1 Turning Process Simulations To validate the model in turning, two different cases are selected. First case is or- thogonal cutting with a triangular insert, and the second case is oblique cutting with a square insert.  Experimental Setup Validation experiments for turning process have been performed on Hardinge Su- perslant lathe. The workpiece material was AISI Steel 1045 bar with 255HB hardness. The diameter of the workpiece was 38.1 mm. Kistler three component dynamometer (Model 9121) and charge amplifier were used to measure the cutting forces. The dyna- mometer was mounted on the turret as shown in Figure 5-13. Displaying and recording of the measured data were performed with data acquisi- tion software, MalDAQ module of CutPro 8.0. Figure 5-14 shows a sample output screen of MalDAQ. Chapter 5. Generalized Mechanics of Metal Cutting      92  Figure 5-13: Workpiece and 3- Component Dynamometer Fixed to Turret for Cutting Tests.  Figure 5-14: Sample Output Screen in CutPro 8.0 for Cutting Forces. Chapter 5. Generalized Mechanics of Metal Cutting      93 Orthogonal Cutting Tests A carbide triangular insert (TPG322) from Kennametal and a shank style tool holder (CTGP) with 5Β° normal rake angle (𝛾𝛾𝑛𝑛 ), 75Β° cutting edge angle (πœ…πœ…π‘Ÿπ‘Ÿ ) and 0Β° inclination angle (πœ†πœ†π‘ π‘ ) was used in orthogonal cutting tests. The influence of nose radius was avoided by using a tube with a wall thickness of 3.05 mm. Orthogonal cutting parameters were used to calculate cutting coefficients in friction and normal directions at each selected point on the cutting edge. The tests were performed at different cutting speeds and feed rates. Cutting conditions for orthogonal cutting validation experiments are summarized in Table 5-III. As the cutting speed changes along the cutting edge, cutting coefficients become different at each differential disk on the tool. Therefore, at each differential disk, cutting coefficients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣 are calculated by using the following equations:  πœ™πœ™π‘›π‘› = 𝑑𝑑(0.6123 β„Žβˆ’0.6988) π›½π›½π‘Žπ‘Ž = 𝑑𝑑(βˆ’0.1515 β„Žβˆ’0.5453) πœπœπ‘ π‘  = 𝑑𝑑0.0417 β„Žβˆ’0.0079 𝑉𝑉+6.2938  (5.51) The model is validated by comparing the predicted forces against new experimental results as shown in Simulated and experimental forces are plotted versus chip thickness in Figure 5-15 and Figure 5-16. Table 5-III: Orthogonal Turning Validation Experiments Cutting Conditions. Tool Geometry 𝑳𝑳 (Insert Length) 12 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 75Β° π’Šπ’Šπ’Šπ’Š (Insert Width) - πœΈπœΈπ’‡π’‡ (Radial Rake Angle ) 0Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) - πœΈπœΈπ’‘π’‘ (Axial Rake Angle) 0Β° π’“π’“πœΊπœΊ (Corner Radius) - πœΊπœΊπ’“π’“ (Tool Included Angle) 60Β° Cutting Conditions Width Of Cut 3.05 mm Feed Rate 0.1 – 0.3 mm/rev Cutting Speed 100 – 150 m/min Workpiece Material AISI 1045 Steel Chapter 5. Generalized Mechanics of Metal Cutting      94  Figure 5-15: Forces simulated and measured for an orthogonal tool with 75Β° cutting edge angle for V=100 m/min.  Figure 5-16: Forces simulated and measured for an orthogonal tool with 75Β° cutting edge angle for V=150 m/min. Chapter 5. Generalized Mechanics of Metal Cutting      95 Oblique Cutting Tests After obtaining satisfactory validation results in orthogonal cutting validation tests, mechanics model has been tested for oblique tools with nose radius. Several tests were performed with different tools and the results are presented in the following sections. Rhombic (ISO Style C) Insert: A rhombic insert with a tool included angle of 80Β° with -8Β° rake angle and -8Β° inclination angle was simulated and tested. The tool holder was Sandvik DCKNL 2020K 12, and the insert was Sandvik CNMA 12 04 08 rhombic insert with 0.8 mm nose radius. Insert geometry and cutting conditions are listed in Table 5-IV. Table 5-IV: Tool Geometry and Cutting Conditions for Rhombic Insert. Tool Geometry 𝑳𝑳 (Insert Length) 12 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 75Β° π’Šπ’Šπ’Šπ’Š (Insert Width) - πœΈπœΈπ’‡π’‡ (Radial Rake Angle ) -8Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) - πœΈπœΈπ’‘π’‘ (Axial Rake Angle) -8Β° π’“π’“πœΊπœΊ (Corner Radius) 0.8 mm πœΊπœΊπ’“π’“ (Tool Included Angle) 80Β° Cutting Conditions Width Of Cut 2 – 4  mm Feed Rate 0.1 – 0.24 mm/rev Cutting Speed 150 m/min Workpiece Material AISI 1045 Steel   Figure 5-17: Illustration of Insert and the Holder Body [55].  Chapter 5. Generalized Mechanics of Metal Cutting      96 The equations for the control points of this type of insert are given in Section 4.5.3. Hence, the locations of the geometric control points are (Figure 5-18): Global coordinate system origin (Point A):  𝐴𝐴π‘₯π‘₯ = 0,     𝐴𝐴𝑧𝑧 = 0 (5.52) Insert center (Point I):  𝐼𝐼π‘₯π‘₯ = βˆ’3.359 ,     𝐼𝐼𝑧𝑧 = 8.076 (5.53) Insert cutting reference point (Point CRP):  𝐢𝐢𝐢𝐢𝑃𝑃π‘₯π‘₯ = 0.613 ,     𝐢𝐢𝐢𝐢𝑃𝑃𝑧𝑧 = 0  (5.54) Control Points (Points A, B, and D):  𝐴𝐴π‘₯π‘₯ = 0,     𝐴𝐴𝑧𝑧 = 0 𝐡𝐡π‘₯π‘₯ = 0.772,     𝐡𝐡𝑧𝑧 = 0.592 𝐷𝐷π‘₯π‘₯ = 3.651,     𝐷𝐷𝑧𝑧 = 11.336 (5.55) Using the orientation angles (πœ…πœ…π‘Ÿπ‘Ÿ = 75Β°, 𝛾𝛾𝑓𝑓 = 𝛾𝛾𝑝𝑝 = βˆ’8Β°), control points are trans- formed and described in the global coordinate system by implementing the procedure presented in Section 4.3. After the transformations, the locations of the control points become:  𝐴𝐴π‘₯π‘₯ = 0,     𝐴𝐴𝑦𝑦 = 0,   𝐴𝐴𝑧𝑧 = 0 𝐡𝐡π‘₯π‘₯ = 0.753,   𝐡𝐡𝑦𝑦 = βˆ’0.189,   𝐡𝐡𝑧𝑧 = 0.587 𝐷𝐷π‘₯π‘₯ = 3.396,   𝐷𝐷𝑦𝑦 = βˆ’2.070,    𝐷𝐷𝑧𝑧 = 11.226 (5.56) Chapter 5. Generalized Mechanics of Metal Cutting      97  Figure 5-18: Geometric Control Points of a Rhombic Turning Insert. For the arc between points A and B, following parametric arc equation is employed:  𝑃𝑃𝐴𝐴𝐡𝐡 = 𝐢𝐢 cos 𝑑𝑑 𝑒𝑒�⃗ + 𝐢𝐢 sin 𝑑𝑑  𝑛𝑛�⃗ Γ— 𝑒𝑒�⃗ + 𝐢𝐢,   0 ≀ 𝑑𝑑 ≀ πœ‹πœ‹2 𝐢𝐢 = 0.8, 𝑒𝑒�⃗ = οΏ½0.0190.138 βˆ’0.99οΏ½ ,   𝑛𝑛�⃗ = οΏ½βˆ’0.138βˆ’0.980βˆ’0.139οΏ½ ,   𝐢𝐢 = οΏ½βˆ’0.015βˆ’0.1100.792 οΏ½ 𝑃𝑃𝐴𝐴𝐡𝐡 = οΏ½0.015 cos 𝑑𝑑 + 0.792 sin 𝑑𝑑 βˆ’ 0.0150.110 cos 𝑑𝑑 βˆ’ 0.111 sin 𝑑𝑑 βˆ’ 0.1100.792 βˆ’ 0.792 cos 𝑑𝑑 οΏ½ (5.57)     Chapter 5. Generalized Mechanics of Metal Cutting      98  For the linear edge between points B and D, linear parametric equation was used:  𝑃𝑃𝐡𝐡𝐷𝐷 = 𝑃𝑃�⃗𝐡𝐡 + �𝑃𝑃�⃗𝐷𝐷 βˆ’ 𝑃𝑃�⃗𝐡𝐡� 𝑑𝑑   ,   0 ≀ 𝑑𝑑 ≀ 1 𝑃𝑃�⃗𝐡𝐡 = οΏ½ 0.753βˆ’0.1890.587 οΏ½ ,   𝑃𝑃�⃗𝐷𝐷 = οΏ½ 3.396βˆ’2.07011.226οΏ½ 𝑃𝑃𝐡𝐡𝐷𝐷 = οΏ½ 2.642 𝑑𝑑 + 0.753βˆ’1.881 𝑑𝑑 βˆ’ 0.18910.639 𝑑𝑑 + 0.587οΏ½ (5.58) By changing the parameter 𝑑𝑑 from 0 to 1 for each equation, the location of any point on the cutting edge can be calculated. Moreover, using the equations presented in Section 4.4, physical angles along the cutting edge were calculated. Finally, orthogonal parameters shown in (5.51) were used to calculate cutting coefficients 𝐾𝐾𝑒𝑒  and 𝐾𝐾𝑣𝑣. Cutting forces were calculated using the rake face based force model which is described in Section 5.2. Simulated and experimental cutting forces for rhombic insert are shown in Figure 5-19 and Figure 5-20. The simulated forces showed good agreement with the experimental forces.  Chapter 5. Generalized Mechanics of Metal Cutting      99  Figure 5-19: Comparison of Measured and Calculated Forces for the Rhombic Tool while the Width of Cut is 2 mm.  Figure 5-20: Comparison of Measured and Calculated Forces for the Rhombic Tool while the Width of Cut is 4 mm. Chapter 5. Generalized Mechanics of Metal Cutting      100 Square (ISO Style S) Insert: Next tool used to validate the cutting force model in turning was a Sandvik DSBNL 2020K 12 holder with 75Β° cutting edge angle, -6Β° rake and inclination angle and Sandvik SNMA 12 04 08 insert with 0.8 mm corner radius. Geometry inputs and experiment conditions are listed in Table 5-V. Tool geometry is illustrated in Figure 5-21. Similar to rhombic insert 2 sets of experiments were con- ducted with 2 mm and 4 mm widths of cut. Using the same procedure described in the previous example, position of each se- lected point is calculated with the associated physical angles. Orthogonal cutting parameters shown in Eq. (5.51) and same cutting force model used to simulate the cutting forces. Although the validations are not as accurate as orthogonal cutting, results are satisfactory. The main reason behind the error is that cutting coefficients were determined using a positive rake tool, however the tools used in oblique cutting have negative rake angles. Following figures can be seen to compare the simulations and measured forces. Table 5-V: Tool Geometry and Cutting Conditions for Square Insert. Tool Geometry 𝑳𝑳 (Insert Length) 12 mm πœΏπœΏπ’“π’“ (Cutting Edge Angle) 75Β° π’Šπ’Šπ’Šπ’Š (Insert Width) - πœΈπœΈπ’‡π’‡ (Radial Rake Angle ) -6Β° 𝒃𝒃𝑭𝑭 (Wiper Edge Length) - πœΈπœΈπ’‘π’‘ (Axial Rake Angle) -6Β° π’“π’“πœΊπœΊ (Corner Radius) 0.8 mm πœΊπœΊπ’“π’“ (Tool Included Angle) 90Β° Cutting Conditions Width Of Cut 2 – 4  mm Feed Rate 0.12 – 0.24 mm/rev Cutting Speed 150 m/min Workpiece Material AISI 1045 Steel Chapter 5. Generalized Mechanics of Metal Cutting      101  Figure 5-21: Illustration of Insert and the Holder Body [55].  Figure 5-22: Comparison of Measured and Calculated Forces for the Square Insert while the Width of Cut is 2 mm. Chapter 5. Generalized Mechanics of Metal Cutting      102  Figure 5-23: Comparison of Measured and Calculated Forces for the Square Insert while the Width of Cut is 4 mm. 5.4.2 Milling Process Simulations Experimental Setup The experiments for milling force validations have been performed on Mori Seiki NMV5000DCG 5 – axis machining center with 20000 rpm spindle. Various inserted milling tools were used from Sandvik Coromant. The workpiece material was aluminum blocks (Al7050-T451) with dimensions of 160x100x100 mm. Kistler three component dynamometer (Model 9257B) and a charge amplifier was used to measure the cutting forces. The dynamometer was mounted on the machine table using fixtures and the aluminum block was attached to the dynamometer as seen in Figure 5-24. Similarly, displaying and recording the measured data was performed with data ac- quisition software MalDAQ module of CutPro 9.0. The complete actual testing environment is illustrated in Figure 5-25. Chapter 5. Generalized Mechanics of Metal Cutting      103 Various tests were performed for validation of the proposed force model. Differen- tial cutting edge disk height and the differential rotation angle in simulations were selected to be 0.01 mm and 4Β° respectively which adequately resembled the actual cutting conditions for the mathematical cutting force model. The applied conditions and the results of the performed milling tests are described throughout this section. The cutting coefficients were described as functions of rake angle, cutting speed and chip thickness. As a result, for each differential cutting edge element, corresponding orthogonal parameter were calculated and converted to the cutting coefficients in normal and friction directions individually. The list of shear parameters used in the mathemati- cal model is given as:  πœ™πœ™π‘›π‘› = 19.4004 + 42.0174 β„Ž + 0.02 𝑉𝑉 + 0.3842 𝛾𝛾𝑛𝑛 πœπœπ‘ π‘  = 266.8047 + 174.1289 β„Ž βˆ’ 0.0437 𝑉𝑉 + 0.8961 𝛾𝛾𝑛𝑛 π›½π›½π‘Žπ‘Ž = 25.8772 βˆ’ 1.2837 β„Ž βˆ’ 0.0075 𝑉𝑉 + 0.1818 𝛾𝛾𝑛𝑛  (5.59)  Figure 5-24: Workpiece and 3-Component Dynamometer Fixed to Machine Table for Cutting Tests. Chapter 5. Generalized Mechanics of Metal Cutting      104  Figure 5-25: Illustration of the Experimental Setup for Measurement of Cutting Forces. Shoulder End Mill with Rectangular Insert A two fluted inserted cutter with built-in HSK holder and 25 mm diameter was se- lected. The cutter body is Sandvik R790-025HA06S2-16L and the inserts are R790- 160408PH-PL with 16 mm insert length, 1 mm wiper edge, and 0.8 mm corner radius. Cutter body has 10Β° axial rake angle, 8Β° radial rake angle, and 90Β° cutting edge angle. Details of the cutter body can be seen in Figure 5-26.  Tool 25 mm End Mill w/ 0.8 mm Nose Radius Axial Rake: 10Β° Radial Rake: 8Β° Insert Rake: 0Β° Insert Helix: 0Β°  Half Immersion Full Immersion Width of Cut: 12.5 mm 25 mm Depth Of Cut: 4 mm 4 mm Material: Al7050-T541 Al7050-T541 Spindle Speed: 1500 rpm 2000 rpm Feed Rate : 0.08 mm/tooth 0.1 mm/tooth Figure 5-26: 25 mm Diameter Shoulder End Mill from Sandvik Coromant [55]. Chapter 5. Generalized Mechanics of Metal Cutting      105 Control Points in global coordinate system (A, B, D, and E) (Figure 5-27):  𝐴𝐴π‘₯π‘₯ = 11.2,     𝐴𝐴𝑧𝑧 = 0 𝐡𝐡π‘₯π‘₯ = 11.7,     𝐡𝐡𝑧𝑧 = 0 𝐷𝐷π‘₯π‘₯ = 12.5,     𝐷𝐷𝑧𝑧 = 0.8 𝐸𝐸π‘₯π‘₯ = 12.5,     𝐸𝐸𝑧𝑧 = 16 (5.60) Using the orientation angles (πœ…πœ…π‘Ÿπ‘Ÿ = 90Β°, 𝛾𝛾𝑓𝑓 = 10Β°, 𝛾𝛾𝑝𝑝 = 8Β°), control points are trans- formed using Eq. (4.5). After the transformations, the locations of the control points become:  𝐴𝐴π‘₯π‘₯ = 11.212,     𝐴𝐴𝑦𝑦 = βˆ’0.180,   𝐴𝐴𝑧𝑧 = 0 𝐡𝐡π‘₯π‘₯ = 11.707,   𝐡𝐡𝑦𝑦 = βˆ’0.111,   𝐡𝐡𝑧𝑧 = 0 𝐷𝐷π‘₯π‘₯ = 12.480,   𝐷𝐷𝑦𝑦 = 0.137,    𝐷𝐷𝑧𝑧 = 0.787 𝐸𝐸π‘₯π‘₯ = 12.113,   𝐸𝐸𝑦𝑦 = 2.751,    𝐸𝐸𝑧𝑧 = 15.756 (5.61) In this example there are two linear edges (|𝐴𝐴𝐡𝐡| and |𝐷𝐷𝐸𝐸|) and one arc (|𝐡𝐡𝐷𝐷|). Therefore, two linear and one arc parametric equations are used:  𝑃𝑃𝐴𝐴𝐡𝐡 = 𝑃𝑃�⃗𝐴𝐴 + �𝑃𝑃�⃗𝐡𝐡 βˆ’ 𝑃𝑃�⃗𝐴𝐴� 𝑑𝑑   ,   0 ≀ 𝑑𝑑 ≀ 1 𝑃𝑃𝐷𝐷𝐸𝐸 = 𝑃𝑃�⃗𝐷𝐷 + �𝑃𝑃�⃗𝐸𝐸 βˆ’ 𝑃𝑃�⃗𝐷𝐷� 𝑑𝑑   ,   0 ≀ 𝑑𝑑 ≀ 1 𝑃𝑃𝐴𝐴𝐡𝐡 = οΏ½0.495 𝑑𝑑 + 11.2120.069 𝑑𝑑 βˆ’ 0.1800 οΏ½ 𝑃𝑃𝐷𝐷𝐸𝐸 = οΏ½βˆ’0.367 𝑑𝑑 + 12.4802.613 𝑑𝑑 + 0.13714.969 𝑑𝑑 + 0.787 οΏ½ (5.62) Chapter 5. Generalized Mechanics of Metal Cutting      106  Figure 5-27: Illustration of Control Points for the Shoulder Milling Insert.  𝑃𝑃𝐡𝐡𝐷𝐷 = 𝐢𝐢 cos 𝑑𝑑 𝑒𝑒�⃗ + 𝐢𝐢 sin 𝑑𝑑  𝑛𝑛�⃗ Γ— 𝑒𝑒�⃗ + 𝐢𝐢,   0 ≀ 𝑑𝑑 ≀ πœ‹πœ‹2 𝐢𝐢 = 0.8, 𝑒𝑒�⃗ = οΏ½ 0.024βˆ’0.172 βˆ’0.984οΏ½ ,   𝑛𝑛�⃗ = οΏ½ 0.137βˆ’0.9750.173 οΏ½ ,   𝐢𝐢 = οΏ½11.6880.0260.787 οΏ½ 𝑃𝑃𝐡𝐡𝐷𝐷 = οΏ½0.019 cos 𝑑𝑑 + 0.792 sin 𝑑𝑑 + 11.6880.111 cos 𝑑𝑑 βˆ’ 0.137 sin 𝑑𝑑 + 0.02620.787 βˆ’ 0.787 cos 𝑑𝑑 οΏ½ (5.63) Using 𝑃𝑃𝐴𝐴𝐡𝐡 , 𝑃𝑃𝐡𝐡𝐷𝐷 , and 𝑃𝑃𝐷𝐷𝐸𝐸 , local radii and physical angles of each selected point on the cutting edge were calculated and cutting forces were simulated with the same force model used in turning simulations. Validations of cutting forces were performed for half immersion up milling and full immersion slot cutting tests. In simulations, orthogonal parameters shown in Eq. (5.59) are used. The tests were conducted for different feed rates and spindle speeds. Cutting conditions for validation tests are summarized also in Figure 5-26. Simulated and experimental cutting forces are plotted versus time for two revolution of the tool in Figure 5-28 and Figure 5-29 for 4 mm axial depth of cut. Chapter 5. Generalized Mechanics of Metal Cutting      107 Although there was a forced vibration during cutting tests due to the vibration of the dynamometer, simulated cutting force patterns and their magnitudes showed very good agreement with the measurements. The tool was two fluted end mill, as a result two peaks can be observed in a single rotation of the tool. For consistency, cutting force of two rotations (720Β°) of the tool is presented for each width of cut.  Figure 5-28: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. Chapter 5. Generalized Mechanics of Metal Cutting      108  Figure 5-29: Measured and Predicted Forces for Al7050 Slot Milling.       Chapter 5. Generalized Mechanics of Metal Cutting      109 Bull-Nose End Mill with Parallelogram Insert In the second case, same workpiece (Al7050 – T451) was used to validate the cut- ting force model using a 20 mm diameter Sandvik R216-20B25-050 cutter with two flutes and two parallelogram inserts (Sandvik R216-20 T3 M-M 1025) with 10 mm nose radius. Cutter body has 10Β° axial rake angle and 5Β° radial rake angle. For the sake of simplicity, control points and parametric equations are not shown in this insert; hence cutting edges are very similar to the shoulder end mill. Only difference between two inserts is that bull nose insert has a bigger corner radius. One slot milling and one half immersion up-milling tests were performed with different spindle speeds, feed rates, and depth of cuts. Cutting conditions are listed in Figure 5-30. Comparison of the measured and calculated cutting forces can be seen Figure 5-31 and Figure 5-32. It can be observed from the figures that the simulations and the experimental values are in good agreement; this proves that the cutting force model proposed in this chapter is valid not only for straight but also for round cutting edges.  Tool 20 mm Bull Nose Mill w/ 10 mm Nose Radius Axial Rake: 12Β° Radial Rake: 5Β° Insert Rake: 0Β° Insert Helix: 0Β°  Half Immersion Full Immersion Width of Cut: 10 mm 20 mm Depth Of Cut: 2 mm 3 mm Material: Al7050-T541 Al7050-T541 Spindle Speed: 2000 rpm 1000 rpm Feed Rate : 0.1 mm/tooth 0.05 mm/tooth Figure 5-30: 20 mm Diameter Bull-Nose Mill from Sandvik Coromant [55]. Chapter 5. Generalized Mechanics of Metal Cutting      110  Figure 5-31: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. Chapter 5. Generalized Mechanics of Metal Cutting      111  Figure 5-32: Measured and Predicted Forces for Al7050 Slot Milling.      Chapter 5. Generalized Mechanics of Metal Cutting      112 Ball-End Mill with a Circular Insert In the final case of milling validation tests, a ball end mill was selected with a circu- lar (round) insert. Cutter body was Sandvik RF216F-20A20S-038, a 20 mm ball-end mill cutter. Unlike the other cutters presented in this section, this cutter has some unique properties. First of all, although it is a two fluted cutter, it has only one flat insert that cuts with both sides. Therefore, it does not have any axial or radial rake. The insert used with this cutter was Sandvik R216F-20 50E-L P10A circular insert with diameter of 20 mm.  Tool 20 mm Ball End Mill Axial Rake: 0Β° Radial Rake: 0Β° Insert Rake: 0Β° Insert Helix: 0Β°  Half Immersion Full Immersion Width of Cut: 10 mm 20 mm Depth Of Cut: 3 mm 3 mm Material: Al7050-T541 Al7050-T541 Spindle Speed: 5000 rpm 2000 rpm Feed Rate : 0.1 mm/tooth 0.08 mm/tooth Figure 5-33: 20 mm Diameter Ball-End Mill from Sandvik Coromant [55]. Control points in global coordinate system (A and B) (Figure 5-34):  𝐴𝐴π‘₯π‘₯ = 0,    𝐴𝐴𝑦𝑦 = 0,   𝐴𝐴𝑧𝑧 = 0 𝐡𝐡π‘₯π‘₯ = 10,    𝐡𝐡𝑦𝑦 = 0,    𝐡𝐡𝑧𝑧 = 10 (5.64) In this inserted cutter, since all orientation angles are 0Β°, the transformation matrices will be identity; therefore, the final positions of the control points do not change. There is only one parametric equation used in this type of insert:   Chapter 5. Generalized Mechanics of Metal Cutting      113  𝑃𝑃𝐴𝐴𝐡𝐡 = 𝐢𝐢 cos 𝑑𝑑 𝑒𝑒�⃗ + 𝐢𝐢 sin 𝑑𝑑  𝑛𝑛�⃗ Γ— 𝑒𝑒�⃗ + 𝐢𝐢,   0 ≀ 𝑑𝑑 ≀ πœ‹πœ‹2 𝐢𝐢 = 10, 𝑒𝑒�⃗ = οΏ½ 00 βˆ’1οΏ½ ,   𝑛𝑛�⃗ = οΏ½ 0βˆ’10 οΏ½ ,   𝐢𝐢 = οΏ½ 0010οΏ½ 𝑃𝑃𝐴𝐴𝐡𝐡 = οΏ½ 10 sin 𝑑𝑑010 βˆ’ 10 cos 𝑑𝑑� (5.65)  Figure 5-34: Control Points of a Round Milling Insert By changing parameter 𝑑𝑑 from 0 to 1, local radii and physical angles along the cut- ting edge can be determined. Similar to the previous milling examples, one half immersion and one full immersion slot milling tests were conducted with different spindle speeds and feed rates. Cutter geometry and tests conditions can be seen in Figure 5-33. Figure 5-35 and Figure 5-36 show the comparison of simulated values and experi- mental values. It can be noted that the measured and predicted cutting forces are found to be in good agreement. Chapter 5. Generalized Mechanics of Metal Cutting      114   Figure 5-35: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling. Chapter 5. Generalized Mechanics of Metal Cutting      115  Figure 5-36: Measured and Predicted Forces for Al7050 Slot Milling. 5.5 Summary In this chapter, generalized cutting mechanics and mathematical cutting force model for inserted cutters are presented. Mechanics model calculates the cutting forces on the rake face along the normal and friction directions. By doing so, it is possible to model any type of cutting process and tool geometry. This model also allows using any type of ISO insert as well as any insert with a custom shape. Proposed cutting force model is verified both in turning and milling operations using various tool geometries and cutting conditions.  Chapter 6. Conclusions      116 6. Conclusions  The aim of this thesis has been to develop a generalized mathematical modeling of metal cutting mechanics which allows prediction of cutting forces for a variety of machining operations. The cutting forces are used to analyze torque, power and stiffness requirements from a machine tool. They are also primary variables in simulating and optimizing machining operations in virtual environment. The proposed generalized model has three fundamental steps: 1) generalized geome- tric model of cutting tool based on ISO standards; 2) kinematic transformation of force vectors in machining systems; 3) and modeling of two principal cutting forces acting on the rake face of the tool. The two principal forces, namely the friction and normal forces on the rake face, are transformed to both cutting tool and machine tool coordinate systems. The generalized transformations allow the use of same material properties and rake face forces in predicting the loads in a variety of machining processes such as drilling, turning, boring, milling and other operations conducted with defined cutting edges. The main contributions of the thesis can be summarized as follows: β€’ Instead of developing dedicated and tool geometry and cutting operation specific cutting force models as reported in the literature, an integrated geo- metric-mechanic and kinematic model of the process is presented. The model can be used to predict forces in various cutting operations with gener- al tool geometry. β€’ A generalized geometric model of inserts and their placement on the tools have been developed using ISO standards for cutting tool geometry. The cut- ting edge coordinates, where the force is generated, are analytically evaluated from the common geometric model.  The model allows the use of multiple cutting edges mounted on the cutter body. The required normal and oblique angles, which are needed in mechanics models, are evaluated analyt- ically using general geometric model. Chapter 6. Conclusions      117 β€’ The normal and friction forces on the rake face are used as the principal cut- ting forces. They are predicted either from orthogonal parameters (shear stress, shear angle and average friction coefficient) or empirical cutting force coefficients calibrated from mechanistic experiments. The principal cutting forces are transformed to both stationary and rotating tool coordinates de- pending on machining operations. β€’ Generalized coordinate transformation models are developed for both statio- nary and rotating tools. The cutting forces acting on the rake face are transformed to feed, normal and axial directions of the machine tool motion. β€’ The proposed general mechanics and geometric models are experimentally validated in turning and milling experiments with inserts having complex geometries. The generalized mechanics model allow prediction of cutting forces, torque and power in a number of cutting operations conducted with tools having arbitrary geome- tries. The proposed model improves the computational efficiency and accuracy in simulating process physics and optimizing the operations in virtual machining of parts. The most important limitation to the generalized cutting force model is the accuracy of the material model. Since the physical angles and cutting conditions significantly change along the cutting edge, material model (shear parameters or mechanistically determined cutting coefficients) should cover these changes. Otherwise, cutting coeffi- cients will be extrapolated and simulated cutting forces will be inaccurate. This situation is already observed in the cutting force validations in 𝑍𝑍 direction. The proposed model can be extended to drilling, boring, broaching, reaming and gear shaping in the future research. 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