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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  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 industrial 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, engagement 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 perpendicular 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 ii  3.6  3.7  3.5.3  Nonequilateral Equiangular Insert ....................................................... 22  3.5.4  Nonequilateral Nonequiangular Insert ................................................. 22  3.5.5  Round Insert ......................................................................................... 23  Tool Coordinate Frames and Transformations ................................................ 24 3.6.1  Turning Operations .............................................................................. 25  3.6.2  Transformation of Rotary Tools........................................................... 31  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 4.3.2 4.3.3  4.4  4.5  4.6  Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 43 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 44 Frame 𝑭𝑭𝑭𝑭 to Frame 𝑭𝑭𝑭𝑭 Rotation.......................................................... 45  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  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  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 iii  5.4  5.3.1  Orthogonal to Oblique Transformation ................................................ 72  5.3.2  Mechanistic Identification .................................................................... 79  Cutting Force Simulations and Validations ..................................................... 91 5.4.1  Turning Process Simulations................................................................ 91  Rhombic (ISO Style C) Insert: ........................................................................................ 95 5.4.2 5.5  Milling Process Simulations .............................................................. 102  Summary ........................................................................................................ 115  6. Conclusions.............................................................................................................. 116 Bibliography .................................................................................................................. 118  iv  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  v  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 𝑿𝑿𝑿𝑿𝑿𝑿𝑿𝑿 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 vi  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 𝜽𝜽𝜽𝜽. (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 vii  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 viii  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  ix  Nomenclature πœ™πœ™π‘π‘  : Orthogonal shear angle  π›½π›½π‘Žπ‘Ž  : Friction angle  𝐴𝐴𝑠𝑠  : Shear plane area  𝐹𝐹  : Resultant force  𝐹𝐹𝑓𝑓  : Feed force  𝐹𝐹π‘₯π‘₯  : Force in 𝑋𝑋 direction  𝐹𝐹𝑧𝑧  : Force in 𝑍𝑍 direction  πœ™πœ™π‘›π‘›  : Normal shear angle  πœπœπ‘ π‘   : Shear Stress  πœ™πœ™π‘–π‘– , πœƒπœƒπ‘›π‘› , πœƒπœƒπ‘–π‘–  : Oblique angles  𝐹𝐹𝑑𝑑  : Tangential force  πΉπΉπ‘Ÿπ‘Ÿ  : Radial force  𝐹𝐹𝑦𝑦  : Force in π‘Œπ‘Œ direction  𝐹𝐹𝑒𝑒  : Friction force on the rake face  𝐹𝐹𝑣𝑣  : Normal force on the rake face  𝐹𝐹𝑛𝑛  : Normal force on the shear plane  πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ  : Radial cutting coefficient  𝐾𝐾𝑑𝑑𝑑𝑑  : Tangential edge coefficient  πΎπΎπ‘Žπ‘Žπ‘Žπ‘Ž  : Axial edge coefficient  𝐾𝐾𝑣𝑣𝑣𝑣  : Normal cutting coefficient  𝐾𝐾𝑣𝑣𝑣𝑣  : Normal edge coefficient  𝐾𝐾𝑓𝑓  : Feed cutting pressure  𝐹𝐹𝑠𝑠  : Shear force on the shear plane  𝐾𝐾𝑑𝑑𝑑𝑑  : Tangential cutting coefficient  πΎπΎπ‘Žπ‘Žπ‘Žπ‘Ž  : Axial cutting coefficient  πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ  : Radial edge coefficient  𝐾𝐾𝑒𝑒𝑒𝑒  : Friction cutting coefficient  𝐾𝐾𝑒𝑒𝑒𝑒  : Friction edge coefficient  𝐾𝐾𝑑𝑑  : Tangential cutting pressure  x  πΎπΎπ‘Ÿπ‘Ÿ  : Radial cutting pressure  β„Ž  : Uncut chip thickness  𝑏𝑏  : Width of cut  𝑐𝑐  : Feed rate  𝑃𝑃𝑑𝑑  : Cutting power  π‘šπ‘š1 & π‘šπ‘š2 : Cutting constants π‘Žπ‘Ž  𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒 𝑉𝑉𝑐𝑐  : Depth of cut  : Effective width of cut  : Cutting speed  𝑑𝑑𝐴𝐴𝑐𝑐  : Differential chip load  𝑑𝑑𝑑𝑑  : Differential cutting edge length  𝑁𝑁𝑓𝑓  : Number of cutting edges  πœ™πœ™  : Immersion angle  𝐴𝐴̅𝑐𝑐  : Average chip load  𝑑𝑑𝑑𝑑  : Differential cutting edge area  𝐾𝐾  : Total number of discrete points along the cutting edge  πœƒπœƒπ‘ π‘ π‘ π‘   : Tool entry angle  πœ…πœ…π‘Ÿπ‘Ÿ  : Cutting edge angle  πœ“πœ“π‘Ÿπ‘Ÿ  : Approach angle  Κ πœ€πœ€  : Corner chamfer angle  𝛾𝛾𝑓𝑓  : Radial rake angle  πœ‚πœ‚  : Chip flow angle  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  : Tool reference plane  πœƒπœƒπ‘’π‘’π‘’π‘’  : Tool exit angle  πœ€πœ€π‘Ÿπ‘Ÿ  : Tool included angle  πœ†πœ†π‘ π‘   : Inclination (oblique) angle  𝛾𝛾𝑛𝑛  : Normal rake angle  𝛾𝛾𝑝𝑝  : Axial rake angle  πœ‚πœ‚Μ…  : Average chip flow angle  𝑃𝑃𝑓𝑓  : Assumed working plane xi  𝑃𝑃𝑝𝑝  : Tool back plane  𝑃𝑃𝑛𝑛  : Cutting edge normal plane  𝐴𝐴𝛾𝛾  : Rake face  𝐷𝐷𝑐𝑐  : Cutting Diameter  𝑏𝑏𝑠𝑠  : Wiper edge length  𝑃𝑃𝑠𝑠  : Tool cutting edge plane  π‘ƒπ‘ƒπ‘šπ‘š  : Wiper edge normal plane  𝐢𝐢𝐢𝐢𝐢𝐢  : Cutting reference point  𝐿𝐿  : Insert length  𝑖𝑖𝑖𝑖  : Insert width  π‘Ÿπ‘Ÿπœ€πœ€  : Corner radius  𝑒𝑒 οΏ½βƒ—  : Unit vector from center of the arc to circumference  𝐢𝐢  : Center of the arc  π‘π‘π‘π‘β„Ž  : Corner chamfer length  οΏ½οΏ½βƒ— 𝑃𝑃𝑖𝑖  : Control Points  𝑛𝑛�⃗  : Unit vector perpendicular to arc  οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½οΏ½βƒ— 𝑍𝑍1.2  : Point on the tool axis  xii  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 cosupervised 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 Manufacturing 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 Kaymakci, 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.  xiii  Chapter 1. Introduction 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 tolerances 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 mechanics 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 1  Chapter 1. Introduction 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 conditions. The thesis is organized as follows; Chapter 2 presents necessary background and literature review on metal cutting research. 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 2  Chapter 1. Introduction 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 transformations between the models are summarized. Using the geometric modeling of inserted tools which is described in Chapter 4, experimental validations of the model are presented. 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.  3  Chapter 2. Literature Survey 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].  4  Chapter 2. Literature Survey 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, otherwise 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 included 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 5  Chapter 2. Literature Survey 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 mechanics, 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‘€π‘€π‘€π‘€ β„Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Žπ‘Ž =  πœ‹πœ‹ 𝛾𝛾𝑛𝑛 π›½π›½π‘Žπ‘Ž + βˆ’ 2 4 2  (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ΏπΏπΏπΏ =  πœ‹πœ‹ + 𝛾𝛾𝑛𝑛 βˆ’ π›½π›½π‘Žπ‘Ž 4  (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 ) οΏ½ sin πœ™πœ™π‘π‘ cos(πœ™πœ™π‘π‘ + π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛 )  sin(π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛 ) οΏ½ 𝐹𝐹𝑓𝑓 = β„Ž 𝑏𝑏 οΏ½πœπœπ‘ π‘  sin πœ™πœ™π‘π‘ cos(πœ™πœ™π‘π‘ + π›½π›½π‘Žπ‘Ž βˆ’ 𝛾𝛾𝑛𝑛 ) 6  (2.3)  Chapter 2. Literature Survey 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. Sabberwal and Koenigsberger [21][22] used this approach and obtained specific cutting coefficients experimentally. Their cutting force equation has been stated by: 𝐹𝐹𝑑𝑑 = 𝐾𝐾𝑑𝑑 π‘π‘β„Žπ‘šπ‘š 1  πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿ π‘π‘β„Žπ‘šπ‘š 2  (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 = 𝐾𝐾𝑑𝑑𝑑𝑑 π‘π‘β„Ž + 𝐾𝐾𝑑𝑑𝑑𝑑 𝑏𝑏  πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ π‘π‘β„Ž + πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ 𝑏𝑏  (2.6)  In this thesis, a linear cutting force model with edge coefficients is used. The advantage 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.  7  Chapter 2. Literature Survey 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 equivalent 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 predicting 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 (πœ‚πœ‚Μ… ) is the summation of elemental flow angles over the entire length of cutting edge: 8  Chapter 2. Literature Survey  πœ‚πœ‚Μ… =  ∫ πœ‚πœ‚(𝑠𝑠)𝑑𝑑𝑑𝑑 ∫ 𝑑𝑑𝑑𝑑  (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] published 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): πœ‚πœ‚Μ… = tanβˆ’1 οΏ½  ∫ sin πœ‚πœ‚(𝑠𝑠) 𝑑𝑑𝑑𝑑 οΏ½ ∫ cos πœ‚πœ‚(𝑠𝑠) 𝑑𝑑𝑑𝑑  (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.  9  Chapter 2. Literature Survey 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 sin π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 = 𝐾𝐾𝑠𝑠 π‘Žπ‘Žβ„Ž 𝐾𝐾𝑠𝑠 = πΆπΆβ„Ž π‘₯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 10  Chapter 2. Literature Survey 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 = 𝐾𝐾𝑑𝑑𝑑𝑑 π‘Žπ‘Žβ„Ž + 𝐾𝐾𝑑𝑑𝑑𝑑 π‘Žπ‘Ž  πΉπΉπ‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ π‘Žπ‘Žβ„Ž + πΎπΎπ‘Ÿπ‘Ÿπ‘Ÿπ‘Ÿ π‘Žπ‘Ž  (2.12)  πΉπΉπ‘Žπ‘Ž = πΎπΎπ‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Žβ„Ž + πΎπΎπ‘Žπ‘Žπ‘Žπ‘Ž π‘Žπ‘Ž  In Eq. (2.12), indices 𝑒𝑒 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 capable 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.  11  Chapter 2. Literature Survey 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 optimization 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 nonuniform 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 coefficients 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 12  Chapter 2. Literature Survey 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 resembling 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.  13  Chapter 3. ISO Cutting Tool Geometry 3.  3.1  ISO Cutting Tool Geometry  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 presented. 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.  14  Chapter 3. ISO Cutting Tool Geometry 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.  15  Chapter 3. ISO Cutting Tool Geometry  Figure 3-1: Planes Defined in ISO 3002 and ISO 13399.  3.3  Points and Angles Cutting Reference Point (𝐢𝐢𝐢𝐢𝐢𝐢): 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 only axial rake. The point is the intersection of: a plane perpendicular to the primary feed  16  Chapter 3. ISO Cutting Tool Geometry 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 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.  Figure 3-2: Illustration of the Cutting Reference Point (CRP).  17  Chapter 3. ISO Cutting Tool Geometry 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 (Κ π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 definitions of these angles are summarized in Table 3-I.  18  Chapter 3. ISO Cutting Tool Geometry  Figure 3-3: Demonstration of Tool Angles in ISO Standards for a Turning Tool.  19  Chapter 3. ISO Cutting Tool Geometry Table 3-I: Summary of Angles for Definition of Orientation of Cutting Edge and Rake Face.  Definition  ANGLE  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  𝝀𝝀𝒔𝒔 Tool Cutting Edge Inclination  𝑃𝑃𝑠𝑠  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑠𝑠  𝝍𝝍𝒓𝒓 Tool Approach Angle  𝑃𝑃𝑠𝑠  πœΊπœΊπ’“π’“ Tool Included Angle  𝑃𝑃𝑠𝑠  𝑃𝑃𝑝𝑝 𝑃𝑃𝑠𝑠  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  Orientation of the Rake Face (π‘¨π‘¨πœΈπœΈ )  πœΈπœΈπ’π’ Tool Normal Rake  𝐴𝐴𝛾𝛾  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑛𝑛  πœΈπœΈπ’‘π’‘ Tool Back Rake  𝐴𝐴𝛾𝛾  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑝𝑝  πœΈπœΈπ’‡π’‡ Tool Side Rake  3.4  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝐴𝐴𝛾𝛾  Dimensional Quantities  π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ  𝑃𝑃𝑓𝑓  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 (𝐢𝐢𝐢𝐢𝐢𝐢) 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 cutting 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.  20  Chapter 3. ISO Cutting Tool Geometry  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, nonequilateral 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 internal 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°. 21  Chapter 3. ISO Cutting Tool Geometry 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 angles 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.  22  Chapter 3. ISO Cutting Tool Geometry 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 S  T C  D E F  M V  W H O P L  A B  N/K R  Insert  Shape  Square  Triangular Rhombic (Diamond) Trigon  Hexagonal Octagonal  Pentagonal  Rectangular Parallelogram Round  23  Nose Angle 90 60 80 55 75 50 86 35 80  120 135 108 90 85 82 55 -  Chapter 3. ISO Cutting Tool Geometry 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 possible 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 thesis. The following procedure is designed to transform the cutting forces or specific cutting force coefficients from machine coordinate system to the rake face. The mechanics 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.  24  Chapter 3. ISO Cutting Tool Geometry  Figure 3-5: Mechanics of Oblique Cutting Process.  3.6.1  Turning Operations Machine Coordinate System (π‘Ώπ‘ΏπŸŽπŸŽ π’€π’€πŸŽπŸŽ π’π’πŸŽπŸŽ )  Machine coordinate system (System 0) is given by the convention of the CNC system. 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 𝑍𝑍 0 axis is collinear with the main spindle axis and the 𝑋𝑋 0  and π‘Œπ‘Œ 0 axes are perpendicular to the 𝑍𝑍 0 axis. On a lathe, the 𝑋𝑋 0 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. 25  Chapter 3. ISO Cutting Tool Geometry Radial – Tangential – Axial Coordinate System (𝑹𝑹𝑹𝑹𝑹𝑹) (𝑿𝑿𝑰𝑰 𝒀𝒀𝑰𝑰 𝒁𝒁𝑰𝑰 )  𝑅𝑅𝑅𝑅𝑅𝑅 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 (𝑋𝑋 0 π‘Œπ‘Œ 0 𝑍𝑍 0 ) to 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (𝑋𝑋 𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼 𝑍𝑍 𝐼𝐼 ) can be performed by rotating the machine  coordinate system around π‘Œπ‘Œ 0 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 𝑅𝑅𝑅𝑅𝑅𝑅 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: cos πœ…πœ…π‘Ÿπ‘Ÿ 𝐢𝐢01 = οΏ½ 0 sin πœ…πœ…π‘Ÿπ‘Ÿ  πœ‹πœ‹ βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ οΏ½ 0 cos οΏ½ ⎑ 0 βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ 2 ⎒ 1 0 οΏ½=⎒ 0 1 0 cos πœ…πœ…π‘Ÿπ‘Ÿ ⎒ sin οΏ½πœ‹πœ‹ βˆ’ πœ“πœ“ οΏ½ 0 π‘Ÿπ‘Ÿ ⎣ 2 sin πœ“πœ“π‘Ÿπ‘Ÿ 0 βˆ’ cos πœ“πœ“π‘Ÿπ‘Ÿ 1 0 𝐢𝐢01 = οΏ½ 0 οΏ½ cos πœ“πœ“π‘Ÿπ‘Ÿ 0 sin πœ“πœ“π‘Ÿπ‘Ÿ  26  πœ‹πœ‹ βˆ’ sin οΏ½ βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ �⎀ 2 βŽ₯ 0 βŽ₯ πœ‹πœ‹ cos οΏ½ βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ οΏ½ βŽ₯⎦ 2  (3.3)  Chapter 3. ISO Cutting Tool Geometry  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  cos πœ…πœ…π‘Ÿπ‘Ÿ =οΏ½ 0 βˆ’sin πœ…πœ…π‘Ÿπ‘Ÿ  βˆ’1 𝑇𝑇 𝐢𝐢10 = 𝐢𝐢01 = 𝐢𝐢01  0 1 0  sin πœ…πœ…π‘Ÿπ‘Ÿ sin πœ“πœ“π‘Ÿπ‘Ÿ 0 οΏ½=οΏ½ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’cos πœ“πœ“π‘Ÿπ‘Ÿ  Cutting Edge Coordinate System (𝑿𝑿𝑰𝑰𝑰𝑰 𝒀𝒀𝑰𝑰𝑰𝑰 𝒁𝒁𝑰𝑰𝑰𝑰 )  0 1 0  cos πœ“πœ“π‘Ÿπ‘Ÿ 0 οΏ½ sin πœ“πœ“π‘Ÿπ‘Ÿ  (3.4)  Figure 3-7 illustrates the cutting edge coordinate system (System 2, (𝑋𝑋 𝐼𝐼𝐼𝐼 , π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 , 𝑍𝑍 𝐼𝐼𝐼𝐼 )).  The π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 axis of the cutting edge coordinate system lies along the cutting edge of the tool, 𝑍𝑍 𝐼𝐼𝐼𝐼 axis is collinear with 𝑅𝑅 axis, and 𝑋𝑋 𝐼𝐼𝐼𝐼 axis is defined perpendicular to π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 and 𝑍𝑍 𝐼𝐼𝐼𝐼  axes. This coordinate frame is also used for the definition of the oblique cutting mechanics in Altintas [5]. In order to transform the 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (System 1) to cutting edge coordinate system (System 2), 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system needs to be rotated about 𝑅𝑅  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 𝑅𝑅 axis need to be performed, in order to get the axes pointing the proper directions.  27  Chapter 3. ISO Cutting Tool Geometry  Figure 3-7: Cutting Edge Coordinate System along with the RTA Coordinate System.  Therefore, the transformation matrix is: 𝐢𝐢21  0 = οΏ½οΏ½1 0  0 0 1  1 cos πœ†πœ†π‘ π‘  0οΏ½ οΏ½ sin πœ†πœ†π‘ π‘  0 0  βˆ’ sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘  0  0 βˆ’1 0 0οΏ½οΏ½ = οΏ½0 1 1  cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  0  sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘  οΏ½ 0  (3.5)  And the resultant transformation from machine coordinate system can be accomplished by Eq. (3.6).  28  Chapter 3. ISO Cutting Tool Geometry  𝐢𝐢20  𝐢𝐢20 = 𝐢𝐢21 𝐢𝐢10  βˆ’ cos πœ“πœ“π‘Ÿπ‘Ÿ sin πœ†πœ†π‘ π‘  = οΏ½βˆ’ cos πœ“πœ“π‘Ÿπ‘Ÿ cos πœ†πœ†π‘ π‘  sin πœ“πœ“π‘Ÿπ‘Ÿ  cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  0  Rake Face Coordinate System (𝑿𝑿𝑰𝑰𝑰𝑰𝑰𝑰 𝒀𝒀𝑰𝑰𝑰𝑰𝑰𝑰 𝒁𝒁𝑰𝑰𝑰𝑰𝑰𝑰 )  sin πœ“πœ“π‘Ÿπ‘Ÿ sin πœ†πœ†π‘ π‘  sin πœ“πœ“π‘Ÿπ‘Ÿ cos πœ†πœ†π‘ π‘  οΏ½ cos πœ“πœ“π‘Ÿπ‘Ÿ  (3.6)  Rake face coordinate system (System 3, (𝑋𝑋 𝐼𝐼𝐼𝐼𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼𝐼𝐼𝐼𝐼 𝑍𝑍 𝐼𝐼𝐼𝐼𝐼𝐼 )) is a transitionary coordinate  system used in this study. The π‘Œπ‘Œ 𝐼𝐼𝐼𝐼𝐼𝐼 axis of the rake face coordinate system is defined on  the cutting edge and 𝑍𝑍 𝐼𝐼𝐼𝐼𝐼𝐼 axis is defined on the rake face of the cutting tool. The transformation 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‘Œ 𝐼𝐼𝐼𝐼 axis by the amount of normal rake angle (𝛾𝛾𝑛𝑛 ). Figure 3-8 shows the orientation 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 𝒀𝒀𝑰𝑰𝑰𝑰 Axis by the Amount of Rake Angle πœΈπœΈπ’π’ .  29  Chapter 3. ISO Cutting Tool Geometry  𝐢𝐢32  cos 𝛾𝛾𝑛𝑛 =οΏ½ 0 sin 𝛾𝛾𝑛𝑛  0 1 0  Chip Flow Coordinate System (𝑼𝑼𝑼𝑼)  βˆ’ sin 𝛾𝛾𝑛𝑛 0 οΏ½ cos 𝛾𝛾𝑛𝑛  (3.7)  Chip Flow Coordinate System (System 4, (𝑋𝑋 𝐼𝐼𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 𝑍𝑍 𝐼𝐼𝐼𝐼 )) can be obtained by rotating  the rake face coordinate system (System 3) by its 𝑋𝑋 𝐼𝐼𝐼𝐼𝐼𝐼 axis by the amount of chip flow  angle (πœ‚πœ‚). With this transformation operation, π‘ˆπ‘ˆ(𝑋𝑋 𝐼𝐼𝐼𝐼 ) axis of the chip flow coordinate system becomes parallel to the friction force (𝐹𝐹𝑒𝑒 ) and the 𝑉𝑉 (π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 ) axis becomes collinear  with the normal force (𝐹𝐹𝑣𝑣 ) acting on the rake face. π‘Šπ‘Š or 𝑍𝑍 𝐼𝐼𝐼𝐼 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: 0 π‘ˆπ‘ˆ οΏ½ οΏ½=οΏ½ 𝑉𝑉 1  βˆ’ sin πœ‚πœ‚ 0  𝐼𝐼𝐼𝐼𝐼𝐼 cos πœ‚πœ‚ 𝑋𝑋𝐼𝐼𝐼𝐼𝐼𝐼 οΏ½ οΏ½π‘Œπ‘Œ οΏ½ 0 𝑍𝑍 𝐼𝐼𝐼𝐼𝐼𝐼  (3.8)  Figure 3-9: Chip Flow Coordinate System (System 4): Rotation around the 𝑿𝑿𝑰𝑰𝑰𝑰𝑰𝑰 Axis by the Amount of Chip Flow Angle 𝜼𝜼.  30  Chapter 3. ISO Cutting Tool Geometry  Furthermore:  𝐢𝐢41  𝐢𝐢41 = 𝐢𝐢43 𝐢𝐢32 𝐢𝐢21  cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ = οΏ½ sin 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‚ 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‚  βˆ’ sin 𝛾𝛾𝑛𝑛 βˆ’1 𝑇𝑇 cos πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 οΏ½ = 𝐢𝐢41 sin πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛  (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:  𝐴𝐴 = οΏ½  3.6.2  0 1  βˆ’sin πœ‚πœ‚ 0  cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ οΏ½οΏ½ 0 0 sin 𝛾𝛾 𝑛𝑛  0 1 0  𝑋𝑋 π‘ˆπ‘ˆ οΏ½ οΏ½ = 𝐴𝐴 Γ— οΏ½π‘Œπ‘Œ οΏ½ 𝑉𝑉 𝑍𝑍  βˆ’sin 𝛾𝛾𝑛𝑛 0 0 οΏ½ οΏ½0 cos 𝛾𝛾𝑛𝑛 1  Transformation of Rotary Tools  cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  0  (3.10) sin πœ†πœ†π‘ π‘  sin πœ“πœ“π‘Ÿπ‘Ÿ cos πœ†πœ†π‘ π‘  οΏ½ οΏ½ 0 cos πœ“πœ“π‘Ÿπ‘Ÿ 0  0 1 0  βˆ’cos πœ“πœ“π‘Ÿπ‘Ÿ 0 οΏ½ sin πœ“πœ“π‘Ÿπ‘Ÿ  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 transformations should be defined individually. Machine Coordinate System (𝑿𝑿𝑿𝑿𝑿𝑿) Similar to a turning machine, the 𝑍𝑍 axis of the machine coordinate system (System  0, (𝑋𝑋𝑋𝑋𝑋𝑋)) 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 (𝑹𝑹𝑹𝑹𝑹𝑹)  𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (System 1, (𝑋𝑋 𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼 𝑍𝑍 𝐼𝐼 )) 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.  31  Chapter 3. ISO Cutting Tool Geometry 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 𝑋𝑋𝑋𝑋𝑋𝑋 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.  32  Chapter 3. ISO Cutting Tool Geometry Consequently, the transformation from tool coordinate system (System *) to machine coordinate system (System 0) is a combination of two basic rotations: cos πœ™πœ™ πΆπΆβˆ—0 = οΏ½ sin πœ™πœ™ 0  βˆ’ sin πœ™πœ™ 0 cos 90Β° sin 90Β° cos πœ™πœ™ 0οΏ½ οΏ½βˆ’ sin 90Β° cos 90Β° 0 0 0 1 sin πœ™πœ™ cos πœ™πœ™ 0 πΆπΆβˆ—0 = οΏ½βˆ’cos πœ™πœ™ sin πœ™πœ™ 0οΏ½ 0 0 1  0 0οΏ½ 1  (3.11)  In 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system, 𝑇𝑇 axis is defined parallel to the primary rotational mo-  tion of the tool, 𝑅𝑅 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 𝑅𝑅 axis is  represented with cutting element position angle (180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ ). So, the transformation from 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (System 1) to System * is a positive rotation around the π‘Œπ‘Œ βˆ—  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:  cos(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°) 𝐢𝐢1βˆ— = οΏ½ 0 βˆ’sin(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°)  cos(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ ) 𝐢𝐢1βˆ— = οΏ½ 0 βˆ’sin(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ )  0 1 0  0 1 0  βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ sin(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°) 0 οΏ½=οΏ½ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ cos(πœ…πœ…π‘Ÿπ‘Ÿ + 90Β°)  βˆ’ cos πœ“πœ“π‘Ÿπ‘Ÿ sin(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ ) 0 0 οΏ½=οΏ½ sin πœ“πœ“π‘Ÿπ‘Ÿ cos(180Β° βˆ’ πœ“πœ“π‘Ÿπ‘Ÿ )  33  0 1 0  0 1 0  cos πœ…πœ…π‘Ÿπ‘Ÿ 0 οΏ½ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ  sin πœ“πœ“π‘Ÿπ‘Ÿ 0 οΏ½ βˆ’ cos πœ“πœ“π‘Ÿπ‘Ÿ  (3.12)  Chapter 3. ISO Cutting Tool Geometry  Figure 3-11: Transformation between System 1 and System *.  Thus, the resultant transformation from machine coordinate system to 𝑅𝑅𝑅𝑅𝑅𝑅 coordi-  nate system can be found as the following: 𝐢𝐢10 = 𝐢𝐢1βˆ— πΆπΆβˆ—0 𝐢𝐢10  βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ =οΏ½ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ  βˆ’sin πœ™πœ™ sin πœ…πœ…π‘Ÿπ‘Ÿ = οΏ½ βˆ’cos πœ™πœ™ βˆ’sin πœ™πœ™ cos πœ…πœ…π‘Ÿπ‘Ÿ  0 1 0  cos πœ…πœ…π‘Ÿπ‘Ÿ sin πœ™πœ™ 0 οΏ½ οΏ½βˆ’cos πœ™πœ™ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ 0  βˆ’cos πœ™πœ™ sin πœ…πœ…π‘Ÿπ‘Ÿ sin πœ™πœ™ βˆ’cos πœ™πœ™ cos πœ…πœ…π‘Ÿπ‘Ÿ  Cutting Edge Coordinate System (𝑿𝑿𝑰𝑰𝑰𝑰 𝒀𝒀𝑰𝑰𝑰𝑰 𝒁𝒁𝑰𝑰𝑰𝑰 )  cos πœ™πœ™ sin πœ™πœ™ 0 cos πœ…πœ…π‘Ÿπ‘Ÿ 0 οΏ½ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ  0 0οΏ½ 1  (3.13)  For rotary tools, cutting edge coordinate system (System 2, (𝑋𝑋 𝐼𝐼𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 𝑍𝑍 𝐼𝐼𝐼𝐼 )) is defined  the same as turning operations. The transformation from cutting edge coordinate system (System 2) and the 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (System 1) is stated in Eq. (3.14). The coordinate system is demonstrated in Figure 3-12. 34  Chapter 3. ISO Cutting Tool Geometry  𝐢𝐢21  0 = οΏ½0 1  cos πœ†πœ†π‘ π‘  βˆ’ sin πœ†πœ†π‘ π‘  0  sin πœ†πœ†π‘ π‘  cos πœ†πœ†π‘ π‘  οΏ½ 0  (3.14)  Figure 3-12: Transformation between System 1 and System 2 for Rotary Tools.  Rake Face Coordinate System (𝑿𝑿𝑰𝑰𝑰𝑰𝑰𝑰 𝒀𝒀𝑰𝑰𝑰𝑰𝑰𝑰 𝒁𝒁𝑰𝑰𝑰𝑰𝑰𝑰 )  As it can be seen from Figure 3-13, rake face coordinate system (System 3) is defined same as the turning operations, which is described in Section 0. As a result, 𝐢𝐢32  cos 𝛾𝛾𝑛𝑛 =οΏ½ 0 sin 𝛾𝛾𝑛𝑛  35  0 1 0  βˆ’ sin 𝛾𝛾𝑛𝑛 0 οΏ½ cos 𝛾𝛾𝑛𝑛  (3.15)  Chapter 3. ISO Cutting Tool Geometry  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, (𝑋𝑋 𝐼𝐼𝐼𝐼 π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 𝑍𝑍 𝐼𝐼𝐼𝐼 )) 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 direction 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, π‘Šπ‘Š axis has no physical meaning and is shown only to complete the coordinate system. 36  Chapter 3. ISO Cutting Tool Geometry  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 = οΏ½  0 1  βˆ’ sin πœ‚πœ‚ 0  37  cos πœ‚πœ‚ οΏ½ 0  (3.16)  Chapter 3. ISO Cutting Tool Geometry 3.7  Summary If the transformation operations in turning and rotating tools are compared, it can be  observed that the transformation matrices from 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system (System 1) to π‘ˆπ‘ˆπ‘ˆπ‘ˆ  coordinate system (System 4) are identical. The only difference between these operations 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.  38  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 operations 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.  39  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 (𝑋𝑋 0 π‘Œπ‘Œ 0 𝑍𝑍 0 ) positioned at the cutting reference point (𝐢𝐢𝐢𝐢𝐢𝐢) 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.  40  Chapter 4. Generalized Geometric Model of Inserted Cutters 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‘₯ 0 𝑧𝑧 0  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, 𝐢𝐢𝐢𝐢𝐢𝐢 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.  41  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 (𝐢𝐢𝐢𝐢𝐢𝐢) 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 𝐢𝐢𝐢𝐢𝐢𝐢. Four different coordinate systems were defined to accomplish the required rotations for the orientation of an insert on the cutter body. Initial coordinate frame 𝐹𝐹 0 is located with its origin at 𝐢𝐢𝐢𝐢𝐢𝐢 and the 𝑋𝑋 axis (𝑋𝑋 0 )  along the primary feed direction as seen in Figure 4-3. 𝑍𝑍 axis (𝑍𝑍 0 ) of the 𝐹𝐹 0 frame is  parallel to the cutter body rotation axis and positive 𝑍𝑍 0 is pointing towards the cutter body. In 𝐹𝐹 0 frame, 𝑋𝑋 0 𝑍𝑍 0 plane corresponds to tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ , and 𝑋𝑋 0 π‘Œπ‘Œ 0 plane corresponds to assumed working plane 𝑃𝑃𝑓𝑓 .  Final coordinate frame 𝐹𝐹 3 has its 𝑋𝑋 axis (𝑋𝑋 3 ) directed along the cutting edge and the  𝑍𝑍 axis (𝑍𝑍 3 ) on the rake face of the insert. In order to transform the 𝐹𝐹 0 frame to 𝐹𝐹 3 frame, two intermediate coordinate frames were defined.  42  Chapter 4. Generalized Geometric Model of Inserted Cutters  4.3.1  Figure 4-3: Frame π‘­π‘­πŸŽπŸŽ to Frame π‘­π‘­πŸπŸ Rotation.  Frame π‘­π‘­πŸŽπŸŽ to Frame π‘­π‘­πŸπŸ Rotation  The first step is to align the 𝑋𝑋 1 axis with the wiper edge or parallel land in a coordi-  nate frame 𝐹𝐹1 by rotating the frame 𝐹𝐹 0 around 𝑍𝑍 0 axis by an amount of radial rake angle  in the reverse direction (βˆ’π›Ύπ›Ύπ‘“π‘“ ). As a result, the rotation matrix 𝑅𝑅01 from frame 𝐹𝐹 0 to frame 𝐹𝐹1 becomes:  𝑅𝑅01  πœƒπœƒπ‘§π‘§01 = βˆ’π›Ύπ›Ύπ‘“π‘“  cos(βˆ’π›Ύπ›Ύπ‘“π‘“ ) = οΏ½ sin(βˆ’π›Ύπ›Ύπ‘“π‘“ ) 0  43  βˆ’ sin(βˆ’π›Ύπ›Ύπ‘“π‘“ ) cos(βˆ’π›Ύπ›Ύπ‘“π‘“ ) 0  0 0οΏ½ 1  (4.1)  Chapter 4. Generalized Geometric Model of Inserted Cutters 4.3.2  Frame π‘­π‘­πŸπŸ to Frame π‘­π‘­πŸπŸ Rotation  The second step is to align the 𝑍𝑍 2 axis of the frame 𝐹𝐹 2 with the rake face of the in-  sert by rotating the frame 𝐹𝐹1 around 𝑋𝑋 1 axis by an amount of axial rake angle also in  reverse direction (βˆ’π›Ύπ›Ύπ‘π‘ ). Consequently, the rotation matrix 𝑅𝑅12 which defines the rotation from frame 𝐹𝐹1 to frame 𝐹𝐹 2 becomes: 𝑅𝑅12  1 = οΏ½0 0  πœƒπœƒπ‘₯π‘₯12 = βˆ’π›Ύπ›Ύπ‘π‘  0 cos(βˆ’π›Ύπ›Ύπ‘π‘ ) sin(βˆ’π›Ύπ›Ύπ‘π‘ )  0 βˆ’ sin(βˆ’π›Ύπ›Ύπ‘π‘ )οΏ½ cos(βˆ’π›Ύπ›Ύπ‘π‘ )  Figure 4-4: Frame π‘­π‘­πŸπŸ to Frame π‘­π‘­πŸπŸ Rotation.  44  (4.2)  Chapter 4. Generalized Geometric Model of Inserted Cutters 4.3.3  Frame π‘­π‘­πŸπŸ to Frame π‘­π‘­πŸ‘πŸ‘ Rotation  The final step is to align the 𝑋𝑋 3 axis of the frame 𝐹𝐹 3 with the cutting edge of the in-  sert by rotating the frame 𝐹𝐹 2 around the π‘Œπ‘Œ 2 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 𝑅𝑅23 defining the rotation from frame 𝐹𝐹 2 to frame 𝐹𝐹 3 is:  𝑅𝑅23  πœƒπœƒπ‘¦π‘¦23 = βˆ’πœ…πœ…π‘Ÿπ‘Ÿ  cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ ) 0 = οΏ½ βˆ’ sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ )  0 1 0  sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ ) 0 οΏ½ cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ )  Figure 4-5: Frame π‘­π‘­πŸπŸ to Frame π‘­π‘­πŸ‘πŸ‘ Rotation.  45  (4.3)  Chapter 4. Generalized Geometric Model of Inserted Cutters Finally, in order to complete the rotation matrix between frames 𝐹𝐹 0 and 𝐹𝐹 3 can be  calculated as the following:  𝑅𝑅03 = 𝑅𝑅01 𝑅𝑅12 𝑅𝑅23  where;  cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ  𝑅𝑅03 = οΏ½sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ  4.4  (4.4)  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‘Ÿ  βˆ’ sin 𝛾𝛾𝑝𝑝  cos 𝛾𝛾𝑝𝑝 cos 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‘Ÿ οΏ½  (4.5)  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‘Œ 0 ) and the normal of the rake face (π‘Œπ‘Œ 3 ) measured in the cutting edge normal plane 𝑃𝑃𝑛𝑛 (π‘Œπ‘Œ 3 𝑍𝑍 3 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 𝐹𝐹 0 . Vector 𝑣𝑣0 later can be described in frame 𝐹𝐹 3 with the inverse of the rotation matrix 𝑅𝑅03 . Therefore:  𝑣𝑣3 = οΏ½  0 βˆ’1 𝑇𝑇 𝑣𝑣0 = οΏ½βˆ’1οΏ½ β‡’ 𝑣𝑣3 = 𝑅𝑅03 𝑣𝑣0 = 𝑅𝑅03 𝑣𝑣0 0 cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ cos 𝛾𝛾𝑓𝑓 cos 𝛾𝛾𝑝𝑝  βˆ’ sin 𝛾𝛾𝑓𝑓 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿ  46  𝑣𝑣3π‘₯π‘₯  οΏ½ = �𝑣𝑣3𝑦𝑦 οΏ½ 𝑣𝑣3𝑧𝑧  (4.6)  Chapter 4. Generalized Geometric Model of Inserted Cutters 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: 𝑣𝑣3𝑧𝑧 βˆ’π‘£π‘£3𝑧𝑧 ) = atan( ) 𝛾𝛾𝑛𝑛 = atan( 𝑣𝑣3𝑦𝑦 βˆ’π‘£π‘£3𝑦𝑦  sin 𝛾𝛾𝑓𝑓 sin πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 cos πœ…πœ…π‘Ÿπ‘Ÿ ) 𝛾𝛾𝑛𝑛 = atan( cos 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 (𝑋𝑋 3 ) and the  assumed working plane 𝑃𝑃𝑓𝑓 (𝑋𝑋 0 π‘Œπ‘Œ 0 plane) measured in the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ (𝑋𝑋 0 𝑍𝑍 0  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, 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒3 , along  the cutting edge in frame 𝐹𝐹 3 and using the rotation matrix 𝑅𝑅03 to describe the vector in 𝐹𝐹 0 coordinate frame. In order to measure the angle in tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ , 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒0𝑦𝑦  component of the vector 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒3 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 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒0 and the 𝑋𝑋 0 axis.  47  Chapter 4. Generalized Geometric Model of Inserted Cutters 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0π‘₯π‘₯ 1 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 3 = οΏ½0οΏ½ , 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0 = 𝑅𝑅03 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 3 = �𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0𝑦𝑦 οΏ½ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0𝑧𝑧 0 cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0 = οΏ½βˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ cos 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ πœ…πœ…π‘Ÿπ‘Ÿβˆ— = atan οΏ½  𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0𝑧𝑧 οΏ½ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 0π‘₯π‘₯  cos 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ πœ…πœ…π‘Ÿπ‘Ÿβˆ— = atan οΏ½ cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + sin 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ  (4.8)  (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 (𝑋𝑋 3 ) and  the tool reference plane π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ (𝑋𝑋 0 𝑍𝑍 0 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 𝐹𝐹 𝑠𝑠 has its 𝑋𝑋 𝑠𝑠 axis aligned with the tool cutting edge plane 𝑃𝑃𝑠𝑠 , and its π‘Œπ‘Œ 𝑠𝑠 axis aligned with π‘Œπ‘Œ 0 axis of the coordinate frame 𝐹𝐹 0 . Figure 4-8 illustrates the frame 𝐹𝐹 𝑠𝑠 . πœ‹πœ‹  Since a rotation of – οΏ½ βˆ’ πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ around the π‘Œπ‘Œ 0 axis is needed to align the 𝑋𝑋 𝑠𝑠 axis with the 2  plane 𝑃𝑃𝑠𝑠 , the rotation matrix between frames 𝐹𝐹0 and 𝐹𝐹𝑠𝑠 becomes: 48  Chapter 4. Generalized Geometric Model of Inserted Cutters  𝑅𝑅0𝑠𝑠  cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ ) 0 =οΏ½ βˆ’ sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ )  0 1 0  sin(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ ) cos(πœ…πœ…π‘Ÿπ‘Ÿ ) 0 οΏ½=οΏ½ 0 cos(βˆ’πœ…πœ…π‘Ÿπ‘Ÿ ) sin(πœ…πœ…π‘Ÿπ‘Ÿ )  0 1 0  βˆ’ sin(πœ…πœ…π‘Ÿπ‘Ÿ ) 0 οΏ½ cos(πœ…πœ…π‘Ÿπ‘Ÿ )  (4.10)  Figure 4-8: Definition of Frame 𝑭𝑭𝒔𝒔 and Frame π‘­π‘­πŸŽπŸŽ to Frame 𝑭𝑭𝒔𝒔 Rotation.  As a result, vector 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠 can be calculated as: 𝑇𝑇 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠 = 𝑅𝑅0𝑠𝑠 𝑅𝑅03 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 3  𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠  𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠𝑠𝑠 = �𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠𝑠𝑠 οΏ½ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠𝑠𝑠  cos 2 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 𝛾𝛾𝑝𝑝 βˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ =οΏ½ οΏ½ 2 βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ cos 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin 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‘Ÿ  (4.11)  Following that, the inclination angle πœ†πœ†π‘ π‘  can be defined as the angle between the 𝑋𝑋 𝑠𝑠  axis in π‘ƒπ‘ƒπ‘Ÿπ‘Ÿ plane and the edge vector 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠 with 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒 𝑒𝑒𝑠𝑠𝑠𝑠 component equal to zero.  49  Chapter 4. Generalized Geometric Model of Inserted Cutters  πœ†πœ†π‘ π‘  = atan οΏ½  𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 _𝑠𝑠𝑠𝑠 οΏ½ 𝑣𝑣𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 _𝑠𝑠𝑠𝑠  βˆ’ sin 𝛾𝛾𝑓𝑓 cos πœ…πœ…π‘Ÿπ‘Ÿ + cos 𝛾𝛾𝑓𝑓 sin 𝛾𝛾𝑝𝑝 sin πœ…πœ…π‘Ÿπ‘Ÿ οΏ½ πœ†πœ†π‘ π‘  = atan οΏ½ 2 cos 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 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.  50  Chapter 4. Generalized Geometric Model of Inserted Cutters 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. Insert 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œΈπœΈπ’‘π’‘ (Axial Rake Angle)  10Β°  πœΈπœΈπ’‡π’‡ (Radial Rake Angle)  π’Šπ’Šπ’Šπ’Š (Insert Width)  6.8 mm  π’“π’“πœΊπœΊ (Corner Radius)  0.4 mm πœΊπœΊπ’“π’“ (Tool Included Angle)  𝒃𝒃𝒔𝒔 (Wiper Edge Length) 0.5 mm  𝑫𝑫𝒄𝒄 (Cutting Diameter)  9Β°  90Β° 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 insert on the cutter body, insert center is calculated in (𝑋𝑋 0 𝑍𝑍 0 ) plane. In Figure 4-11, insert center is defined by point 𝐼𝐼(π‘₯π‘₯, 𝑧𝑧):  51  Chapter 4. Generalized Geometric Model of Inserted Cutters 𝐼𝐼π‘₯π‘₯ = 𝐼𝐼𝑧𝑧 =  𝐿𝐿 cos πœ…πœ…π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿπœ€πœ€ 𝑖𝑖𝑖𝑖 sin πœ…πœ…π‘Ÿπ‘Ÿ 𝐷𝐷𝑐𝑐 + βˆ’ + 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’ 2 sin πœ…πœ…π‘Ÿπ‘Ÿ 2 2 + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝑏𝑏𝑠𝑠 cos 2 πœ…πœ…π‘Ÿπ‘Ÿ  𝑖𝑖𝑖𝑖 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’πΏπΏ sin πœ…πœ…π‘Ÿπ‘Ÿ + 𝑏𝑏𝑠𝑠 sin πœ…πœ…π‘Ÿπ‘Ÿ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cos 2 πœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 2 2  Figure 4-11: Analytical Model of an Insert with a Corner Radius and Wiper Edge.  52  (4.13)  Chapter 4. Generalized Geometric Model of Inserted Cutters 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‘  + 𝐡𝐡π‘₯π‘₯ =  π‘Ÿπ‘Ÿπœ€πœ€ 𝐷𝐷𝑐𝑐 + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ , 𝐴𝐴𝑧𝑧 = 0 βˆ’ 2 sin πœ…πœ…π‘Ÿπ‘Ÿ  π‘Ÿπ‘Ÿπœ€πœ€ 𝐷𝐷𝑐𝑐 + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ , 𝐡𝐡𝑧𝑧 = 0 βˆ’ 2 sin πœ…πœ…π‘Ÿπ‘Ÿ  (4.14) (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‘₯ =  π‘Ÿπ‘Ÿπœ€πœ€ cos πœ…πœ…π‘Ÿπ‘Ÿ π‘Ÿπ‘Ÿπœ€πœ€ 𝐷𝐷𝑐𝑐 , 𝐷𝐷𝑧𝑧 = |π‘Ÿπ‘Ÿπœ€πœ€ (βˆ’1 + cos πœ…πœ…π‘Ÿπ‘Ÿ )| + + π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ sin πœ…πœ…π‘Ÿπ‘Ÿ 2  (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‘₯ =  π‘Ÿπ‘Ÿπœ€πœ€ 𝐷𝐷𝑐𝑐 + (βˆ’π‘π‘π‘ π‘  + π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘Ÿ ) cos 2 πœ…πœ…π‘Ÿπ‘Ÿ + π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin πœ…πœ…π‘Ÿπ‘Ÿ 2 π‘Ÿπ‘Ÿπœ€πœ€ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘Ÿ βˆ’ 𝑏𝑏𝑠𝑠 sin πœ…πœ…π‘Ÿπ‘Ÿ cot πœ€πœ€π‘Ÿπ‘Ÿ οΏ½ cos πœ…πœ…π‘Ÿπ‘Ÿ + �𝐿𝐿 βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ + sin πœ…πœ…π‘Ÿπ‘Ÿ  (4.17)  𝐸𝐸𝑧𝑧 = |βˆ’πΏπΏ sin π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 ) cos 2 πœ…πœ…π‘Ÿπ‘Ÿ + (π‘Ÿπ‘Ÿπœ€πœ€ + 𝑏𝑏𝑠𝑠 sin 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 ) cos πœ…πœ…π‘Ÿπ‘Ÿ |  After the calculation of control points and the cutting edges between these points, cutting edge is defined in (𝑋𝑋 0 𝑍𝑍 0 ) plane in two dimensions. Rotation about π‘Œπ‘Œ 0 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 (𝑋𝑋𝑋𝑋) plane. Moreover, these locations were defined in global 𝑋𝑋𝑋𝑋𝑋𝑋  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.  53  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 𝑋𝑋𝑋𝑋𝑋𝑋 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.  54  Chapter 4. Generalized Geometric Model of Inserted Cutters  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 (𝑋𝑋 0 𝑍𝑍 0 ) plane; therefore, parametric equations were defined to obtain the location of each selected point on the cutting edge. For linear cutting edges:  For arcs:  𝑃𝑃 = π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1 + �𝑃𝑃�⃗𝑖𝑖 βˆ’ π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1 οΏ½ 𝑑𝑑 , 0 ≀ 𝑑𝑑 ≀ 1 𝑃𝑃 = 𝑅𝑅 cos 𝑑𝑑 𝑒𝑒 οΏ½βƒ— + 𝑅𝑅 sin 𝑑𝑑 𝑛𝑛�⃗ Γ— 𝑒𝑒 οΏ½βƒ— + 𝐢𝐢, 0 ≀ 𝑑𝑑 ≀  55  (4.18)  πœ‹πœ‹ 2  (4.19)  Chapter 4. Generalized Geometric Model of Inserted Cutters where 𝑃𝑃 is any point between 𝑃𝑃�⃗𝑖𝑖 and π‘ƒπ‘ƒοΏ½βƒ—π‘–π‘–βˆ’1 which are two consecutive control points,  𝑅𝑅 is the radius of the arc, 𝑒𝑒 οΏ½βƒ— 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 corresponding 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(𝑍𝑍⃗2 βˆ’ 𝑍𝑍⃗1 ) Γ— (𝑍𝑍⃗1 βˆ’ 𝑃𝑃�⃗)οΏ½ �𝑍𝑍⃗2 βˆ’ 𝑍𝑍⃗1 οΏ½  (4.20)  where π‘Ÿπ‘Ÿ is the local radius, 𝑍𝑍⃗1 and 𝑍𝑍⃗2 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.  56  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 complexity 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.  57  Chapter 4. Generalized Geometric Model of Inserted Cutters  Figure 4-16: Change in Normal Rake Angle along the Tool Axis.  Figure 4-17: Change in Helix Angle along the Tool Axis.  58  Chapter 4. Generalized Geometric Model of Inserted Cutters 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’Šπ’Šπ’Šπ’Š (Insert Width)  π‘²π‘²πœΊπœΊ (Corner Chamfer)  90Β°  πœΈπœΈπ’‘π’‘ (Axial Rake Angle)  5Β°  πœΈπœΈπ’‡π’‡ (Radial Rake Angle)  5Β°  1 mm x45Β° πœΊπœΊπ’“π’“ (Tool Included Angle)  90Β°  11.5 mm  𝒃𝒃𝒔𝒔 (Wiper Edge Length)  πœΏπœΏπ’“π’“ (Cutting Edge Angle)  1.5 mm  𝑫𝑫𝒄𝒄 (Cutting Diameter)  63 mm  Figure 4-18: Catalogue Figure of the Insert Taken from Sandvik Coromant [55].  59  Chapter 4. Generalized Geometric Model of Inserted Cutters Analytical model of the insert can be seen in Figure 4-19. Insert center is defined by point 𝐼𝐼. Coordinates of this point in (𝑋𝑋 0 𝑍𝑍 0 ) plane is: 𝐼𝐼π‘₯π‘₯ =  𝐼𝐼𝑧𝑧 =  1 |𝐷𝐷 + 𝑖𝑖𝑖𝑖 cos(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ ) csc πœ€πœ€π‘Ÿπ‘Ÿ 2 𝑐𝑐 + 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€ )|  (4.21)  1 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 2  + sin(πœ€πœ€π‘Ÿπ‘Ÿ + πœ…πœ…π‘Ÿπ‘Ÿ ) (𝑖𝑖𝑖𝑖 βˆ’ 2𝑏𝑏𝑠𝑠 sin π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  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:  60  Chapter 4. Generalized Geometric Model of Inserted Cutters 𝐴𝐴π‘₯π‘₯ = βˆ’π‘π‘π‘ π‘  +  𝐷𝐷𝑐𝑐 + 𝐡𝐡𝐡𝐡𝐡𝐡 sin πΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐡𝐡𝐡𝐡𝐡𝐡 cos πΎπΎπœ€πœ€ , 𝐴𝐴𝑧𝑧 = 0 2  𝐷𝐷𝑐𝑐 + 𝐡𝐡𝐡𝐡𝐡𝐡 sin πΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐡𝐡𝐡𝐡𝐡𝐡 cos πΎπΎπœ€πœ€ , 𝐡𝐡𝑧𝑧 = 0 2 𝐷𝐷𝑐𝑐 + 𝐡𝐡𝐡𝐡𝐡𝐡 sin πΎπΎπœ€πœ€ cot πœ…πœ…π‘Ÿπ‘Ÿ , 𝐢𝐢𝑧𝑧 = 𝐡𝐡𝐡𝐡𝐡𝐡 sin πΎπΎπœ€πœ€ 𝐢𝐢π‘₯π‘₯ = 2 𝐷𝐷𝑐𝑐 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 sin πœ…πœ…π‘Ÿπ‘Ÿ 𝐷𝐷π‘₯π‘₯ = �𝑏𝑏𝑠𝑠 βˆ’ 2 𝐡𝐡π‘₯π‘₯ =  + 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  (4.22) (4.23) (4.24)  (4.25)  𝐷𝐷𝑧𝑧 = |𝐿𝐿 sin π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€ ))|  After the calculations of the locations of control points in (𝑋𝑋 0 𝑍𝑍 0 ) 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.  61  Chapter 4. Generalized Geometric Model of Inserted Cutters  Figure 4-21: Change in Normal Rake Angle along the Tool Axis.  Figure 4-22: Change in Helix Angle along the Tool Axis.  62  Chapter 4. Generalized Geometric Model of Inserted Cutters 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  𝒃𝒃𝒔𝒔 (Wiper Edge Length)  -  π’Šπ’Šπ’Šπ’Š (Insert Width)  π’“π’“πœΊπœΊ (Corner Radius)  πœΏπœΏπ’“π’“ (Cutting Edge Angle)  93Β°  πœΈπœΈπ’‘π’‘ (Axial Rake Angle)  -10Β°  πœΈπœΈπ’‡π’‡ (Radial Rake Angle)  -  1.6 mm πœΊπœΊπ’“π’“ (Tool Included Angle)  -8Β°  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.  63  Chapter 4. Generalized Geometric Model of Inserted Cutters  Figure 4-24: Control Points of the Rhombic Turning Insert.  Location of insert center with respect to the point 𝐴𝐴 can be calculated as: (βˆ’πΏπΏ + 2π‘Ÿπ‘Ÿπœ€πœ€ sin πœ€πœ€π‘Ÿπ‘Ÿ + 𝐿𝐿 cos 2 πœ€πœ€π‘Ÿπ‘Ÿ ) sin πœ…πœ…π‘Ÿπ‘Ÿ 𝐼𝐼π‘₯π‘₯ = 2 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‘Ÿ 2 sin πœ€πœ€π‘Ÿπ‘Ÿ  π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ 𝐿𝐿 sin πœ…πœ…π‘Ÿπ‘Ÿ (1 + cos πœ€πœ€π‘Ÿπ‘Ÿ ) βˆ’ 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 𝐼𝐼𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€ + sin πœ€πœ€π‘Ÿπ‘Ÿ 2 𝐿𝐿 cos πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ 𝐿𝐿 cos 2 πœ€πœ€π‘Ÿπ‘Ÿ cos πœ…πœ…π‘Ÿπ‘Ÿ + 2 sin πœ€πœ€π‘Ÿπ‘Ÿ  64  (4.26)  Chapter 4. Generalized Geometric Model of Inserted Cutters 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  𝐡𝐡π‘₯π‘₯ = π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ , 𝐡𝐡𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€ (1 βˆ’ cos πœ…πœ…π‘Ÿπ‘Ÿ )  𝐷𝐷π‘₯π‘₯ = π‘Ÿπ‘Ÿπœ€πœ€ sin πœ…πœ…π‘Ÿπ‘Ÿ + �𝐿𝐿 βˆ’  π‘Ÿπ‘Ÿπœ€πœ€ βˆ’ π‘Ÿπ‘Ÿπœ€πœ€ cot πœ€πœ€π‘Ÿπ‘Ÿ οΏ½ cos πœ…πœ…π‘Ÿπ‘Ÿ 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€ 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 𝐷𝐷𝑧𝑧 = π‘Ÿπ‘Ÿπœ€πœ€ + 𝐿𝐿 sin πœ…πœ…π‘Ÿπ‘Ÿ βˆ’ sin πœ€πœ€π‘Ÿπ‘Ÿ  Figure 4-25: Position of the Cutting Edge for a Rhombic Turning Insert.  65  (4.27) (4.28)  (4.29)  Chapter 4. Generalized Geometric Model of Inserted Cutters  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.  66  Chapter 4. Generalized Geometric Model of Inserted Cutters 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 cutting force model which is applicable to different cutting processes is introduced and simulations are presented with experimental verifications.  67  Chapter 5. Generalized Mechanics of Metal Cutting 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 relationships 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, 𝐹𝐹⃑ 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.  68  Chapter 5. Generalized Mechanics of Metal Cutting  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 = 𝐾𝐾𝑒𝑒𝑒𝑒 𝑑𝑑𝐴𝐴𝑐𝑐 + 𝐾𝐾𝑒𝑒𝑒𝑒 𝑑𝑑𝑑𝑑 𝑑𝑑𝐹𝐹𝑣𝑣 = 𝐾𝐾𝑣𝑣𝑣𝑣 𝑑𝑑𝐴𝐴𝑐𝑐 + 𝐾𝐾𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑  (5.1)  where 𝐾𝐾𝑒𝑒𝑒𝑒 and 𝐾𝐾𝑣𝑣𝑣𝑣 are the friction and normal cutting coefficients and 𝐾𝐾𝑒𝑒𝑒𝑒 and 𝐾𝐾𝑣𝑣𝑣𝑣  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  69  Chapter 5. Generalized Mechanics of Metal Cutting 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 = 𝑏𝑏𝑒𝑒𝑒𝑒𝑒𝑒 =  𝑏𝑏 sin 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 (𝑋𝑋𝑋𝑋𝑋𝑋) 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 transformations 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: 𝑁𝑁𝑓𝑓 𝐾𝐾 𝑑𝑑𝑑𝑑π‘₯π‘₯ 𝐹𝐹π‘₯π‘₯ �𝐹𝐹𝑦𝑦 οΏ½ = οΏ½ οΏ½ �𝑑𝑑𝑑𝑑𝑦𝑦 οΏ½οΏ½ 𝐹𝐹𝑧𝑧 𝑛𝑛=1 π‘˜π‘˜=1 𝑑𝑑𝑑𝑑𝑧𝑧 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.  70  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-2: Summary of the Proposed Mechanistic Approach.  71  Chapter 5. Generalized Mechanics of Metal Cutting 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 transformation 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 transformation 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‘œπ‘œπ‘œπ‘œ β„Žπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œ =  𝑏𝑏 β„Ž sin π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:  72  Chapter 5. Generalized Mechanics of Metal Cutting  𝐴𝐴𝑠𝑠,π‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œ =  𝑏𝑏 β„Ž cos πœ†πœ†π‘ π‘  sin π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‘  𝐴𝐴𝑠𝑠 = πœπœπ‘ π‘   𝑏𝑏 β„Ž cos πœ†πœ†π‘ π‘  sin π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: 𝐹𝐹𝑋𝑋 𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠  cos πœ™πœ™π‘–π‘– cos πœƒπœƒπ‘›π‘› cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘› )  𝐹𝐹𝑍𝑍 𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠  cos πœ™πœ™π‘–π‘– sin πœƒπœƒπ‘›π‘› cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘› )  πΉπΉπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 = βˆ’πΉπΉπ‘ π‘  sin πœ™πœ™π‘–π‘–  73  (5.7) (5.8) (5.9)  Chapter 5. Generalized Mechanics of Metal Cutting  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  74  (5.10)  Chapter 5. Generalized Mechanics of Metal Cutting Similarly, the tangent of πœ™πœ™π‘–π‘– is equal to the following: 𝐹𝐹𝑦𝑦𝐼𝐼𝐼𝐼 tan πœ™πœ™π‘–π‘– = 𝐹𝐹𝑠𝑠𝑠𝑠  (5.11)  where 𝐹𝐹𝑠𝑠𝑠𝑠 is the projection of shear force 𝐹𝐹𝑠𝑠 onto normal plane 𝑃𝑃𝑛𝑛 .  Figure 5-5: Illustration of angle πœ½πœ½π’π’ . (𝑭𝑭𝑡𝑡 , 𝑭𝑭𝒖𝒖𝒖𝒖 , and 𝑭𝑭𝒗𝒗𝒗𝒗 Are Projections of Forces onto Normal Plane).  In Figure 5-4, 𝐹𝐹𝑁𝑁 can be expressed as, 𝐹𝐹𝑁𝑁 =  𝐹𝐹𝑒𝑒 cos πœ‚πœ‚ 𝐹𝐹𝑠𝑠 cos πœ™πœ™π‘–π‘– = sin 𝛽𝛽𝑛𝑛 cos(πœ™πœ™π‘›π‘› + πœƒπœƒπ‘›π‘› ) 𝐹𝐹𝑠𝑠 sin πœ™πœ™π‘–π‘– 𝐹𝐹𝑒𝑒 = sin πœ‚πœ‚  (5.12)  Combining equations (5.10) and (5.12) gives the following relationship: tan πœ™πœ™π‘–π‘– =  tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 cos(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 )  75  (5.13)  Chapter 5. Generalized Mechanics of Metal Cutting And furthermore: cos πœ™πœ™π‘–π‘– = sin πœ™πœ™π‘–π‘– =  cos(πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 )  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛  (5.14)  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  Combining equations (5.7), (5.8), (5.9), and (5.10) with (5.14) results in: 𝐹𝐹𝑋𝑋 𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠 = π‘π‘β„Ž οΏ½  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) πœπœπ‘ π‘  οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  𝐹𝐹𝑍𝑍 𝐼𝐼𝐼𝐼 = 𝐹𝐹𝑠𝑠  = π‘π‘β„Ž οΏ½  cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 )  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 )  (5.15)  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) πœπœπ‘ π‘  οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  Similarly, by substituting equations (5.6) and (5.16) into Equation (5.8): πΉπΉπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 = βˆ’πΉπΉπ‘ π‘   tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 πœπœπ‘ π‘  = π‘π‘β„Ž οΏ½βˆ’ οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  (5.16)  Corresponding specific cutting coefficients in 𝑋𝑋 𝐼𝐼𝐼𝐼 , π‘Œπ‘Œ 𝐼𝐼𝐼𝐼 , and 𝑍𝑍 𝐼𝐼𝐼𝐼 directions can be eva-  luated as:  𝐾𝐾𝑋𝑋 𝐼𝐼𝐼𝐼 = οΏ½  cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) πœπœπ‘ π‘  οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  πΎπΎπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 = οΏ½βˆ’ 𝐾𝐾𝑍𝑍 𝐼𝐼𝐼𝐼 = οΏ½  tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 πœπœπ‘ π‘  οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) πœπœπ‘ π‘  οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  76  (5.17)  Chapter 5. Generalized Mechanics of Metal Cutting The cutting coefficients are transformed from the cutting edge coordinate system (System 2) to chip flow coordinate system (System 4) to calculate the cutting coefficients 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 )𝑇𝑇  Hence:  οΏ½ As a result:  𝐾𝐾𝑋𝑋 𝐼𝐼𝐼𝐼 𝐾𝐾𝑒𝑒 οΏ½ οΏ½ = 𝐢𝐢42 οΏ½πΎπΎπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 οΏ½ 𝐾𝐾𝑣𝑣 𝐾𝐾𝑍𝑍 𝐼𝐼𝐼𝐼  sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ 𝐾𝐾𝑒𝑒 οΏ½=οΏ½ 𝐾𝐾𝑣𝑣 cos 𝛾𝛾𝑛𝑛  βˆ’ sin πœ‚πœ‚ 0  𝐾𝐾𝑋𝑋 𝐼𝐼𝐼𝐼 cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ οΏ½ οΏ½πΎπΎπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 οΏ½ βˆ’ sin 𝛾𝛾𝑛𝑛 𝐾𝐾𝑍𝑍 𝐼𝐼𝐼𝐼  𝐾𝐾𝑒𝑒 = sin 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ 𝐾𝐾𝑋𝑋 𝐼𝐼𝐼𝐼 βˆ’ sin πœ‚πœ‚ πΎπΎπ‘Œπ‘Œ 𝐼𝐼𝐼𝐼 + cos 𝛾𝛾𝑛𝑛 cos πœ‚πœ‚ 𝐾𝐾𝑍𝑍 𝐼𝐼𝐼𝐼 𝐾𝐾𝑣𝑣 = cos 𝛾𝛾𝑛𝑛 𝐾𝐾𝑋𝑋 𝐼𝐼𝐼𝐼 βˆ’ sin 𝛾𝛾𝑛𝑛 𝐾𝐾𝑍𝑍 𝐼𝐼𝐼𝐼  (5.18)  (5.19)  (5.20)  Alternatively, specific cutting coefficients in chip flow coordinate system can be calculated by evaluating the resultant force: 𝐹𝐹 = �𝐹𝐹𝑋𝑋2𝐼𝐼𝐼𝐼 + πΉπΉπ‘Œπ‘Œ2𝐼𝐼𝐼𝐼 + 𝐹𝐹𝑍𝑍2𝐼𝐼𝐼𝐼  (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].  𝐹𝐹 sin π›½π›½π‘Žπ‘Ž π‘π‘β„Ž 𝐹𝐹 cos π›½π›½π‘Žπ‘Ž 𝐾𝐾𝑣𝑣 = π‘π‘β„Ž 𝐾𝐾𝑒𝑒 =  77  (5.22)  Chapter 5. Generalized Mechanics of Metal Cutting Figure 5-6 illustrates and summarizes the procedure outlined above. As result, specific cutting coefficients 𝐾𝐾𝑒𝑒 and 𝐾𝐾𝑣𝑣 on friction and normal directions can be evaluated with the following equations: 𝐾𝐾𝑒𝑒 = 𝐾𝐾𝑣𝑣 =  πœπœπ‘ π‘  οΏ½1 βˆ’ tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 πœπœπ‘ π‘  οΏ½1 βˆ’ tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  sin π›½π›½π‘Žπ‘Ž  (5.23)  𝑐𝑐𝑐𝑐𝑐𝑐 π›½π›½π‘Žπ‘Ž  Or applying the geometrical transformation in Equation (5.20): 𝐾𝐾𝑒𝑒 =  πœπœπ‘ π‘  𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐(sin 𝛾𝛾𝑛𝑛 cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 + cos 𝛾𝛾𝑛𝑛 sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 )) 𝐾𝐾𝑣𝑣 =  cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛 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‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  Shear Parameters From Orthogonal Cutting Data Cutting Conditions  (5.24)  πœ™πœ™π‘π‘ , πœπœπ‘ π‘  , π›½π›½π‘Žπ‘Ž  (5.15) 𝑏𝑏, β„Ž  Tool Geometry And Assumed Chip Flow Direction  (5.6)  𝐹𝐹𝑠𝑠  (5.16)  𝐹𝐹𝑋𝑋𝐼𝐼𝐼𝐼 πΉπΉπ‘Œπ‘ŒπΌπΌπΌπΌ 𝐹𝐹𝑍𝑍𝐼𝐼𝐼𝐼  (5.19)  𝐾𝐾𝑒𝑒 𝐾𝐾𝑣𝑣  πœ†πœ†π‘ π‘  , 𝛾𝛾𝑛𝑛 , πœ‚πœ‚  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 𝐾𝐾𝑒𝑒𝑒𝑒  and 𝐾𝐾𝑣𝑣𝑣𝑣 can be calculated by transforming the edge coefficients from 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate  system to π‘ˆπ‘ˆπ‘ˆπ‘ˆ coordinate system. Alternatively, they can be added when the cutting forces are calculated and transformed back to the 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system or machine  coordinate system. In the simulations and validations shown in the following sections, second method has been used.  78  Chapter 5. Generalized Mechanics of Metal Cutting 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‘Ÿπ‘Ÿ = πΎπΎπ‘Ÿπ‘Ÿ 𝐴𝐴𝑐𝑐 𝐹𝐹𝑑𝑑𝑑𝑑 = 𝐾𝐾𝑑𝑑 𝐴𝐴𝑐𝑐  (5.25)  πΉπΉπ‘Žπ‘Žπ‘Žπ‘Ž = πΎπΎπ‘Žπ‘Ž 𝐴𝐴𝑐𝑐  where 𝐹𝐹𝑖𝑖𝑖𝑖 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 )  79  (5.26)  Chapter 5. Generalized Mechanics of Metal Cutting 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, however, chip flow angle (πœ‚πœ‚) is nonlinear. To overcome this problem, two new variables, 𝐾𝐾𝑒𝑒1 and 𝐾𝐾𝑒𝑒2 were introduced:  𝐾𝐾𝑒𝑒 1 = 𝐾𝐾𝑒𝑒 cos πœ‚πœ‚ 𝐾𝐾𝑒𝑒 2 = 𝐾𝐾𝑒𝑒 sin πœ‚πœ‚  (5.27)  Substituting Equation (5.27) into Equation (5.26):  where  πΎπΎπ‘Ÿπ‘Ÿ = 𝐢𝐢𝑒𝑒𝑒𝑒 1 𝐾𝐾𝑒𝑒 1 + 𝐢𝐢𝑣𝑣𝑣𝑣 𝐾𝐾𝑣𝑣 𝐾𝐾𝑑𝑑 = 𝐢𝐢𝑒𝑒𝑒𝑒 1 𝐾𝐾𝑒𝑒 1 + 𝐢𝐢𝑒𝑒𝑒𝑒 2 𝐾𝐾𝑒𝑒 2 + 𝐢𝐢𝑣𝑣𝑣𝑣 𝐾𝐾𝑣𝑣 πΎπΎπ‘Žπ‘Ž = 𝐢𝐢𝑒𝑒𝑒𝑒 1 𝐾𝐾𝑒𝑒 1 + 𝐢𝐢𝑒𝑒𝑒𝑒 2 𝐾𝐾𝑒𝑒 2 + 𝐢𝐢𝑣𝑣𝑣𝑣 𝐾𝐾𝑣𝑣  (5.28)  𝐢𝐢𝑒𝑒𝑒𝑒 1 = cos 𝛾𝛾𝑛𝑛 , 𝐢𝐢𝑣𝑣𝑣𝑣 = βˆ’sin 𝛾𝛾𝑛𝑛 , 𝐢𝐢𝑒𝑒𝑒𝑒 1 = cos πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 , 𝐢𝐢𝑒𝑒𝑒𝑒 2 = sin Ξ»s , 𝐢𝐢𝑣𝑣𝑣𝑣 = cos πœ†πœ†π‘ π‘  cos 𝛾𝛾𝑛𝑛 𝐢𝐢𝑒𝑒𝑒𝑒 1 = sin πœ†πœ†π‘ π‘  sin 𝛾𝛾𝑛𝑛 , 𝐢𝐢𝑒𝑒𝑒𝑒 2 = βˆ’cos πœ†πœ†π‘ π‘  , 𝐢𝐢𝑣𝑣𝑣𝑣 = 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: 𝐢𝐢𝑒𝑒𝑒𝑒 1 πΎπΎπ‘Ÿπ‘Ÿ οΏ½ 𝐾𝐾𝑑𝑑 οΏ½ = οΏ½ 𝐢𝐢𝑒𝑒𝑒𝑒 1 πΎπΎπ‘Žπ‘Ž 𝐢𝐢𝑒𝑒𝑒𝑒 1  that:  0  𝐢𝐢𝑒𝑒𝑒𝑒 2 𝐢𝐢𝑒𝑒𝑒𝑒 2  𝐢𝐢𝑣𝑣𝑣𝑣 𝐾𝐾𝑒𝑒 1 𝐢𝐢𝑣𝑣𝑣𝑣 οΏ½ �𝐾𝐾𝑒𝑒 2 οΏ½ 𝐢𝐢𝑣𝑣𝑣𝑣 𝐾𝐾𝑣𝑣  (5.29)  Solving these equations in Eq. (5.29) results in 𝐾𝐾𝑒𝑒1 , 𝐾𝐾𝑒𝑒2 , and 𝐾𝐾𝑣𝑣 . It should be noted 𝐾𝐾𝑒𝑒 = �𝐾𝐾𝑒𝑒 12 + 𝐾𝐾𝑒𝑒 22 πœ‚πœ‚ = tanβˆ’1  𝐾𝐾𝑒𝑒 2 𝐾𝐾𝑒𝑒 1  (5.30) (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 80  Chapter 5. Generalized Mechanics of Metal Cutting 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]: ���𝑐𝑐 = 𝐴𝐴  πœƒπœƒπ‘’π‘’π‘’π‘’  πœƒπœƒπ‘’π‘’π‘’π‘’ 1 οΏ½ 𝐴𝐴 (πœƒπœƒ) 𝑑𝑑𝑑𝑑 βˆ’ πœƒπœƒπ‘ π‘ π‘ π‘  πœƒπœƒπ‘ π‘ π‘‘π‘‘ 𝑐𝑐  (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, tangential, 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 maximum 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‘› ) cos πœ™πœ™π‘–π‘– + sin πœƒπœƒπ‘–π‘– sin πœ™πœ™π‘–π‘– ) 𝐹𝐹𝑠𝑠 = 𝐹𝐹 cos 45Β°  (5.33)  Moreover, another statement of this principle dictates that the projection of the resultant 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‘› ) sin πœ™πœ™π‘–π‘– βˆ’ sin πœƒπœƒπ‘–π‘– cos πœ™πœ™π‘–π‘– ) = 0 Eq. (5.33) and Eq. (5.34) are used to derive the following relationships:  81  (5.34)  Chapter 5. Generalized Mechanics of Metal Cutting πœ™πœ™π‘–π‘– = sinβˆ’1 �√2 sin πœƒπœƒπ‘–π‘– οΏ½  πœ™πœ™π‘›π‘› = cos βˆ’1 οΏ½  tan πœƒπœƒπ‘–π‘– οΏ½ βˆ’ πœƒπœƒπ‘›π‘› tan πœ™πœ™π‘–π‘–  (5.35) (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‘› = cos βˆ’1 οΏ½  1 cos πœ™πœ™π‘–π‘– οΏ½ βˆ’ πœƒπœƒπ‘›π‘› √2 cos 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 οΏ½  tan πœ†πœ†π‘ π‘  cos(πœ™πœ™π‘›π‘› βˆ’ 𝛾𝛾𝑛𝑛 ) βˆ’ cos 𝛾𝛾𝑛𝑛 tan πœ™πœ™π‘–π‘– οΏ½ sin πœ™πœ™π‘›π‘›  (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  𝐹𝐹𝑠𝑠 = πœπœπ‘ π‘  𝐴𝐴𝑠𝑠 = πœπœπ‘ π‘  οΏ½ πœπœπ‘ π‘  = οΏ½  β„Ž 𝑏𝑏 οΏ½ = 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 45Β° οΏ½οΏ½ cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘›  𝐾𝐾𝑒𝑒2 + 𝐾𝐾𝑣𝑣2 cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› 2  (5.40) (5.41)  (5.42)  Results and discussions about this approach are given in the following sections.  82  Chapter 5. Generalized Mechanics of Metal Cutting Minimum Energy Principle In this approach work done by Shamoto and Altintas [11] is used. In their study, resultant 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‘– cos πœ™πœ™π‘–π‘– + sin πœƒπœƒπ‘–π‘– sin πœ™πœ™π‘–π‘– ] cos πœ†πœ†π‘ π‘  sin π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‘  + sin π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′ =  cos πœƒπœƒπ‘›π‘› tan πœƒπœƒπ‘–π‘– tan πœ†πœ†π‘ π‘  𝑃𝑃𝑑𝑑 = π‘‰π‘‰πœπœπ‘ π‘  π‘π‘β„Ž [cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘› ) cos πœ™πœ™π‘–π‘– + tan πœƒπœƒπ‘–π‘– sin πœ™πœ™π‘–π‘– ] sin π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 =0 πœ•πœ•πœ™πœ™π‘›π‘› πœ•πœ•π‘ƒπ‘ƒπ‘‘π‘‘β€² =0 πœ•πœ•πœ™πœ™π‘–π‘–  (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‘˜ βˆ’ 1) Δ𝑃𝑃′ β„Ξ”πœ™πœ™π‘›π‘› οΏ½ = οΏ½ 𝑛𝑛 οΏ½ βˆ’ 𝜁𝜁 οΏ½ 𝑑𝑑′ οΏ½ πœ™πœ™π‘–π‘– (π‘˜π‘˜) πœ™πœ™π‘–π‘– (π‘˜π‘˜ βˆ’ 1) Δ𝑃𝑃𝑑𝑑 β„Ξ”πœ™πœ™π‘–π‘–  (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‘› ) cos πœ™πœ™π‘–π‘– cos πœƒπœƒπ‘–π‘– + sin πœƒπœƒπ‘–π‘– sin πœ™πœ™π‘–π‘– ]  πœπœπ‘ π‘  = �𝐾𝐾𝑒𝑒2 + 𝐾𝐾𝑣𝑣2 [cos(πœƒπœƒπ‘›π‘› + πœ™πœ™π‘›π‘› ) cos πœ™πœ™π‘–π‘– cos πœƒπœƒπ‘–π‘– + sin πœƒπœƒπ‘–π‘– sin πœ™πœ™π‘–π‘– ] cos πœ†πœ†π‘ π‘  sin πœ™πœ™π‘›π‘› 83  (5.48)  Chapter 5. Generalized Mechanics of Metal Cutting 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 calculation 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‘ˆπ‘ˆ 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.  πΎπΎπ‘Ÿπ‘Ÿ , 𝐾𝐾𝑑𝑑 , πΎπΎπ‘Žπ‘Ž  𝛾𝛾𝑛𝑛 , πœ†πœ†π‘ π‘   Geometrical  𝐾𝐾𝑒𝑒 , 𝐾𝐾𝑣𝑣  Transformation  Max. Shear Stress Min. Energy (5.39), (5.47)  πœ‚πœ‚ 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 transformed 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. 84  Chapter 5. Generalized Mechanics of Metal Cutting 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 =  sin(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) πœπœπ‘ π‘  sin πœ™πœ™π‘›π‘› cos πœ†πœ†π‘ π‘  οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  𝐾𝐾𝑑𝑑 =  πΎπΎπ‘Žπ‘Ž =  cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan πœ†πœ†π‘ π‘  tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 πœπœπ‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  (5.49)  cos(𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) tan πœ†πœ†π‘ π‘  βˆ’ tan πœ‚πœ‚ sin 𝛽𝛽𝑛𝑛 πœπœπ‘ π‘  sin πœ™πœ™π‘›π‘› οΏ½cos 2 (πœ™πœ™π‘›π‘› + 𝛽𝛽𝑛𝑛 βˆ’ 𝛾𝛾𝑛𝑛 ) + tan2 πœ‚πœ‚ sin2 𝛽𝛽𝑛𝑛  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). % 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 =  �𝐾𝐾𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 βˆ’ πΎπΎπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œ οΏ½ Γ— 100 πΎπΎπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œπ‘œ  (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.  85  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-8: Results of Helix Angle Predictions for Ti6Al4V. Table 5-I: Cutting Coefficients and Tool Geometry Used in the Analysis.  Material Ti6Al4V Al7075  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°] 340  1630  608  30  15  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 identification 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.  86  Chapter 5. Generalized Mechanics of Metal Cutting  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.  87  Chapter 5. Generalized Mechanics of Metal Cutting 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 𝑅𝑅𝑅𝑅𝑅𝑅 coordinate system were transformed to π‘ˆπ‘ˆπ‘ˆπ‘ˆ 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 CTGP SCGP MTGN Orthogonal  Rake Angle 𝛾𝛾𝑛𝑛 [Β°] 5 3.5 -5 5  Oblique Angle πœ†πœ†π‘ π‘  [Β°] 0 3.5 -5 0  Friction Angle π›½π›½π‘Žπ‘Ž [Β°] 26.9 32 20.3 28.1  Shear Angle πœ™πœ™π‘›π‘› [Β°] 23 16.5 17.1 31.2  Shear Stress πœπœπ‘ π‘  [MPa] 508 459.2 477.8 549  Chip Flow Angle πœ‚πœ‚ [Β°] 0 3.3 -5.4 0  Comp. Time [ms]  # of Iterations  6.6 ms 6.4 ms 7.4 ms -  2 31 14 -  Comp. Time [ms]  # of Iterations  13.2 14.5 14.4 -  140 115 124 -  Minimum Energy Principle Cutting Tool CTGP SCGP MTGN Orthogonal  Rake Angle 𝛾𝛾𝑛𝑛 [Β°] 5 3.5 -5 5  Oblique Angle πœ†πœ†π‘ π‘  [Β°] 0 3.5 -5 0  Friction Angle π›½π›½π‘Žπ‘Ž [Β°] 26.9 32 20.3 28.1  Shear Angle πœ™πœ™π‘›π‘› [Β°] 33.1 30.1 31.6 31.2  Shear Stress πœπœπ‘ π‘  [MPa] 573.8 596 600 549  Chip Flow Angle πœ‚πœ‚ [Β°] 0 1.72 -3.1 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.  88  Chapter 5. Generalized Mechanics of Metal Cutting Input Angles π›½π›½π‘Žπ‘Ž  Force Relation (5.35) & (5.36)  𝛾𝛾𝑛𝑛 πœ†πœ†π‘ π‘   πœ‚πœ‚ (5.39)  Velocity Relation (5.38)  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‘–  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.  89  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-11: Effect of Chip Thickness on the Cutting Coefficients.  Figure 5-12: Effect of Chip Thickness on the Chip Flow Angle.  90  Chapter 5. Generalized Mechanics of Metal Cutting 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 orthogonal 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 Superslant 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 dynamometer was mounted on the turret as shown in Figure 5-13. Displaying and recording of the measured data were performed with data acquisition software, MalDAQ module of CutPro 8.0. Figure 5-14 shows a sample output screen of MalDAQ.  91  Chapter 5. Generalized Mechanics of Metal Cutting  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.  92  Chapter 5. Generalized Mechanics of Metal Cutting 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‘› = 𝑒𝑒 (0.6123 β„Žβˆ’0.6988)  π›½π›½π‘Žπ‘Ž = 𝑒𝑒 (βˆ’0.1515 β„Žβˆ’0.5453)  (5.51)  πœπœπ‘ π‘  = 𝑒𝑒 0.0417 β„Žβˆ’0.0079 𝑉𝑉+6.2938  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)  𝒃𝒃𝒔𝒔 (Wiper Edge Length) π’“π’“πœΊπœΊ (Corner Radius)  πœΈπœΈπ’‡π’‡ (Radial Rake Angle )  -  πœΈπœΈπ’‘π’‘ (Axial Rake Angle)  -  0Β° 0Β°  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  93  Chapter 5. Generalized Mechanics of Metal Cutting  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.  94  Chapter 5. Generalized Mechanics of Metal Cutting 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  𝒃𝒃𝒔𝒔 (Wiper Edge Length)  -  π’Šπ’Šπ’Šπ’Š (Insert Width)  π’“π’“πœΊπœΊ (Corner Radius)  πœΏπœΏπ’“π’“ (Cutting Edge Angle) 75Β° πœΈπœΈπ’‡π’‡ (Radial Rake Angle )  -  πœΈπœΈπ’‘π’‘ (Axial Rake Angle)  -8Β° -8Β°  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].  95  Chapter 5. Generalized Mechanics of Metal Cutting 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): 𝐢𝐢𝐢𝐢𝐢𝐢π‘₯π‘₯ = 0.613 ,  𝐢𝐢𝐢𝐢𝐢𝐢𝑧𝑧 = 0  (5.54)  Control Points (Points A, B, and D): 𝐴𝐴π‘₯π‘₯ = 0,  𝐡𝐡π‘₯π‘₯ = 0.772,  𝐷𝐷π‘₯π‘₯ = 3.651,  𝐴𝐴𝑧𝑧 = 0  𝐡𝐡𝑧𝑧 = 0.592  (5.55)  𝐷𝐷𝑧𝑧 = 11.336  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  96  (5.56)  Chapter 5. Generalized Mechanics of Metal Cutting  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: 𝑃𝑃𝐴𝐴𝐴𝐴 = 𝑅𝑅 cos 𝑑𝑑 𝑒𝑒 οΏ½βƒ— + 𝑅𝑅 sin 𝑑𝑑 𝑛𝑛�⃗ Γ— 𝑒𝑒 οΏ½βƒ— + 𝐢𝐢, 0 ≀ 𝑑𝑑 ≀ 𝑅𝑅 = 0.8,  πœ‹πœ‹ 2  0.019 βˆ’0.138 βˆ’0.015 𝑒𝑒 οΏ½βƒ— = οΏ½ 0.138 οΏ½, 𝑛𝑛�⃗ = οΏ½βˆ’0.980οΏ½ , 𝐢𝐢 = οΏ½βˆ’0.110οΏ½ βˆ’0.99 βˆ’0.139 0.792  0.015 cos 𝑑𝑑 + 0.792 sin 𝑑𝑑 βˆ’ 0.015 𝑃𝑃𝐴𝐴𝐴𝐴 = οΏ½0.110 cos 𝑑𝑑 βˆ’ 0.111 sin 𝑑𝑑 βˆ’ 0.110οΏ½ 0.792 βˆ’ 0.792 cos 𝑑𝑑  97  (5.57)  Chapter 5. Generalized Mechanics of Metal Cutting  For the linear edge between points B and D, linear parametric equation was used: 𝑃𝑃𝐡𝐡𝐡𝐡 = 𝑃𝑃�⃗𝐡𝐡 + �𝑃𝑃�⃗𝐷𝐷 βˆ’ 𝑃𝑃�⃗𝐡𝐡 οΏ½ 𝑑𝑑 , 0 ≀ 𝑑𝑑 ≀ 1 3.396 0.753 οΏ½βƒ— οΏ½βƒ— 𝑃𝑃𝐡𝐡 = οΏ½βˆ’0.189οΏ½, 𝑃𝑃𝐷𝐷 = οΏ½βˆ’2.070οΏ½ 11.226 0.587 𝑃𝑃𝐡𝐡𝐡𝐡  (5.58)  2.642 𝑑𝑑 + 0.753 = οΏ½βˆ’1.881 𝑑𝑑 βˆ’ 0.189οΏ½ 10.639 𝑑𝑑 + 0.587  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.  98  Chapter 5. Generalized Mechanics of Metal Cutting  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.  99  Chapter 5. Generalized Mechanics of Metal Cutting 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 conducted with 2 mm and 4 mm widths of cut. Using the same procedure described in the previous example, position of each selected 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  𝒃𝒃𝒔𝒔 (Wiper Edge Length)  -  π’Šπ’Šπ’Šπ’Š (Insert Width)  π’“π’“πœΊπœΊ (Corner Radius)  πœΏπœΏπ’“π’“ (Cutting Edge Angle) 75Β° πœΈπœΈπ’‡π’‡ (Radial Rake Angle )  -  πœΈπœΈπ’‘π’‘ (Axial Rake Angle)  -6Β° -6Β°  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  100  Chapter 5. Generalized Mechanics of Metal Cutting  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.  101  Chapter 5. Generalized Mechanics of Metal Cutting  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 acquisition software MalDAQ module of CutPro 9.0. The complete actual testing environment is illustrated in Figure 5-25.  102  Chapter 5. Generalized Mechanics of Metal Cutting Various tests were performed for validation of the proposed force model. Differential 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 mathematical 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  (5.59)  π›½π›½π‘Žπ‘Ž = 25.8772 βˆ’ 1.2837 β„Ž βˆ’ 0.0075 𝑉𝑉 + 0.1818 𝛾𝛾𝑛𝑛  Figure 5-24: Workpiece and 3-Component Dynamometer Fixed to Machine Table for Cutting Tests.  103  Chapter 5. Generalized Mechanics of Metal Cutting  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 selected. The cutter body is Sandvik R790-025HA06S2-16L and the inserts are R790160408PH-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].  104  Chapter 5. Generalized Mechanics of Metal Cutting Control Points in global coordinate system (A, B, D, and E) (Figure 5-27): 𝐴𝐴π‘₯π‘₯ = 11.2, 𝐡𝐡π‘₯π‘₯ = 11.7,  𝐷𝐷π‘₯π‘₯ = 12.5,  𝐸𝐸π‘₯π‘₯ = 12.5,  𝐴𝐴𝑧𝑧 = 0 𝐡𝐡𝑧𝑧 = 0  𝐷𝐷𝑧𝑧 = 0.8  (5.60)  𝐸𝐸𝑧𝑧 = 16  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  (5.61)  𝐸𝐸π‘₯π‘₯ = 12.113, 𝐸𝐸𝑦𝑦 = 2.751, 𝐸𝐸𝑧𝑧 = 15.756  In this example there are two linear edges (|𝐴𝐴𝐴𝐴| and |𝐷𝐷𝐷𝐷|) and one arc (|𝐡𝐡𝐡𝐡|).  Therefore, two linear and one arc parametric equations are used: 𝑃𝑃𝐴𝐴𝐴𝐴 = 𝑃𝑃�⃗𝐴𝐴 + �𝑃𝑃�⃗𝐡𝐡 βˆ’ 𝑃𝑃�⃗𝐴𝐴 οΏ½ 𝑑𝑑 , 0 ≀ 𝑑𝑑 ≀ 1  𝑃𝑃𝐷𝐷𝐷𝐷 = 𝑃𝑃�⃗𝐷𝐷 + �𝑃𝑃�⃗𝐸𝐸 βˆ’ 𝑃𝑃�⃗𝐷𝐷 οΏ½ 𝑑𝑑 , 0 ≀ 𝑑𝑑 ≀ 1 𝑃𝑃𝐴𝐴𝐴𝐴  𝑃𝑃𝐷𝐷𝐷𝐷  0.495 𝑑𝑑 + 11.212 = οΏ½ 0.069 𝑑𝑑 βˆ’ 0.180 οΏ½ 0  βˆ’0.367 𝑑𝑑 + 12.480 = οΏ½ 2.613 𝑑𝑑 + 0.137 οΏ½ 14.969 𝑑𝑑 + 0.787  105  (5.62)  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-27: Illustration of Control Points for the Shoulder Milling Insert.  𝑃𝑃𝐡𝐡𝐡𝐡 = 𝑅𝑅 cos 𝑑𝑑 𝑒𝑒 οΏ½βƒ— + 𝑅𝑅 sin 𝑑𝑑 𝑛𝑛�⃗ Γ— 𝑒𝑒 οΏ½βƒ— + 𝐢𝐢, 0 ≀ 𝑑𝑑 ≀ 𝑅𝑅 = 0.8,  πœ‹πœ‹ 2  0.024 0.137 11.688 𝑒𝑒 οΏ½βƒ— = οΏ½βˆ’0.172οΏ½, 𝑛𝑛�⃗ = οΏ½βˆ’0.975οΏ½ , 𝐢𝐢 = οΏ½ 0.026 οΏ½ βˆ’0.984 0.173 0.787  (5.63)  0.019 cos 𝑑𝑑 + 0.792 sin 𝑑𝑑 + 11.688 𝑃𝑃𝐡𝐡𝐡𝐡 = οΏ½0.111 cos 𝑑𝑑 βˆ’ 0.137 sin 𝑑𝑑 + 0.0262οΏ½ 0.787 βˆ’ 0.787 cos 𝑑𝑑  Using 𝑃𝑃𝐴𝐴𝐴𝐴 , 𝑃𝑃𝐡𝐡𝐡𝐡 , and 𝑃𝑃𝐷𝐷𝐷𝐷 , 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.  106  Chapter 5. Generalized Mechanics of Metal Cutting 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.  107  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-29: Measured and Predicted Forces for Al7050 Slot Milling.  108  Chapter 5. Generalized Mechanics of Metal Cutting Bull-Nose End Mill with Parallelogram Insert In the second case, same workpiece (Al7050 – T451) was used to validate the cutting 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].  109  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-31: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling.  110  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-32: Measured and Predicted Forces for Al7050 Slot Milling.  111  Chapter 5. Generalized Mechanics of Metal Cutting Ball-End Mill with a Circular Insert In the final case of milling validation tests, a ball end mill was selected with a circular (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  Al7050-T541  Al7050-T541  Material:  Spindle Speed: 5000 rpm Feed Rate :  0.1 mm/tooth  2000 rpm 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:  112  Chapter 5. Generalized Mechanics of Metal Cutting  𝑃𝑃𝐴𝐴𝐴𝐴 = 𝑅𝑅 cos 𝑑𝑑 𝑒𝑒 οΏ½βƒ— + 𝑅𝑅 sin 𝑑𝑑 𝑛𝑛�⃗ Γ— 𝑒𝑒 οΏ½βƒ— + 𝐢𝐢, 0 ≀ 𝑑𝑑 ≀ 𝑅𝑅 = 10,  πœ‹πœ‹ 2  0 0 0 𝑒𝑒 οΏ½βƒ— = οΏ½ 0 οΏ½, 𝑛𝑛�⃗ = οΏ½βˆ’1οΏ½ , 𝐢𝐢 = οΏ½ 0 οΏ½ 0 10 βˆ’1  𝑃𝑃𝐴𝐴𝐴𝐴 = οΏ½  (5.65)  10 sin 𝑑𝑑 οΏ½ 0 10 βˆ’ 10 cos 𝑑𝑑  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 experimental values. It can be noted that the measured and predicted cutting forces are found to be in good agreement.  113  Chapter 5. Generalized Mechanics of Metal Cutting  Figure 5-35: Measured and Predicted Forces for Al7050 Half Immersion Up-Milling.  114  Chapter 5. Generalized Mechanics of Metal Cutting  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.  115  Chapter 6. Conclusions 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 geometric 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 geometric-mechanic and kinematic model of the process is presented. The model can be used to predict forces in various cutting operations with general 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 cutting 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 analytically using general geometric model. 116  Chapter 6. Conclusions β€’  The normal and friction forces on the rake face are used as the principal cutting 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 depending on machining operations.  β€’  Generalized coordinate transformation models are developed for both stationary 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 geometries. 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. 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