"Applied Science, Faculty of"@en .
"Mechanical Engineering, Department of"@en .
"DSpace"@en .
"UBCV"@en .
"Comak, Alptunc"@en .
"2018-07-16T18:09:43Z"@* .
"2018"@en .
"Doctor of Philosophy - PhD"@en .
"University of British Columbia"@en .
"Recent turn-milling machine tools are capable of carrying out turning, drilling, boring, milling and grinding operations simultaneously, hence they are widely used in industry to produce complex parts in a single set-up. Turn-milling machines have translational axes with a high speed spindle to hold the cutting tool and a low speed spindle to carry the workpiece. The resulting five-axis turn-milling machines can machine parts with complex curved tool paths. This thesis presents the mechanics and dynamics of turn-milling operations to predict cutting forces, torque, power, vibrations, chatter stability and dimensional surface errors in the virtual environment.\r\nFirst, the kinematics of five-axis turn milling operation is modeled using homogenous transformations. The engagement of rotating-moving tool with the rotating workpiece is identified using a commercial graphics system, and used in predicting the chip thickness distribution. The relative vibrations between the tool and workpiece are modeled, and superposed on the chip thickness in the engagement zone. Unlike in regular turning and milling operations with a single spindle which leads to a single and constant delay, turn milling has two time delays contributed by two rotating spindles and three translational feed drives. The regenerative chip thickness with dual delay is used to predict the cutting forces at tool-workpiece engagement zone, which are transformed to three Cartesian directions of the machine. The resulting coupled differential equations with two delays and time periodic coefficients are solved in the semi-discrete time domain to predict chatter stability, cutting forces, vibrations, torque, power and dimensional surface errors simultaneously.\r\nThe thesis presents the first comprehensive digital model of turn milling operations in the literature, and can be used to predict the most productive cutting conditions ahead of costly physical trials currently practiced in the industry."@en .
"https://circle.library.ubc.ca/rest/handle/2429/66507?expand=metadata"@en .
" MECHANICS, DYNAMICS AND STABILITY OF TURN-MILLING OPERATIONS by Alptunc Comak B.S., Istanbul Technical University, 2011 M.Sc., Sabanci University, 2013 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2018 \u00C2\u00A9 Alptunc Comak, 2018 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Mechanics, Dynamics and Stability of Turn-Milling Operations Examining Committee: Yusuf Altintas, Mechanical Engineering Supervisor Hsi-Yung (Steve) Feng, Mechanical Engineering Supervisory Committee Member Farrokh Sassani, Mechanical Engineering University Examiner Steve Cockcroft, Materials Engineering University Examiner submitted by Alptunc Comak in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering iii Abstract Recent turn-milling machine tools are capable of carrying out turning, drilling, boring, milling and grinding operations simultaneously, hence they are widely used in industry to produce complex parts in a single set-up. Turn-milling machines have translational axes with a high speed spindle to hold the cutting tool and a low speed spindle to carry the workpiece. The resulting five-axis turn-milling machines can machine parts with complex curved tool paths. This thesis presents the mechanics and dynamics of turn-milling operations to predict cutting forces, torque, power, vibrations, chatter stability and dimensional surface errors in the virtual environment. First, the kinematics of five-axis turn milling operation is modeled using homogenous transformations. The engagement of rotating-moving tool with the rotating workpiece is identified using a commercial graphics system, and used in predicting the chip thickness distribution. The relative vibrations between the tool and workpiece are modeled, and superposed on the chip thickness in the engagement zone. Unlike in regular turning and milling operations with a single spindle which leads to a single and constant delay, turn milling has two time delays contributed by two rotating spindles and three translational feed drives. The regenerative chip thickness with dual delay is used to predict the cutting forces at tool-workpiece engagement zone, which are transformed to three Cartesian directions of the machine. The resulting coupled differential equations with two delays and time periodic coefficients are solved in the semi-discrete time domain to predict chatter stability, cutting forces, vibrations, torque, power and dimensional surface errors simultaneously. The thesis presents the first comprehensive digital model of turn milling operations in the literature, and can be used to predict the most productive cutting conditions ahead of costly physical trials currently practiced in the industry. iv Lay Summary Advancements in the manufacturing industry parallel with the improvements in material, machine tool and control technologies, demand shorter production times and high quality complex shape parts with competitive costs. Turn milling process is a rapidly emerging technology in aerospace and automotive industries, carrying out simultaneous turning and milling operations to efficiently produce complex parts in a single set up. Currently, the turn milling operations are carried out by relying on costly machining trials and experience due to the lack of process models which are able to simulate and optimize the cutting process in the virtual environment. This thesis develops the novel physics based mathematical models of turn-milling process that predict the most productive cutting conditions with increased part quality. The presented models are expected to broaden the application and planning of turn-milling technology, and reveal further unique assets of the process. v Preface This Ph.D. dissertation proposes a comprehensive mechanics, kinematics, dynamics and stability model of turn milling operations that is applicable to any tool-workpiece geometry and complex toolpath, as the first unified model in the literature. All of the work presented henceforth was conducted by the Ph.D. candidate in the Manufacturing Automation Laboratories at the University of British Columbia, under the supervision of Professor Yusuf Altintas. The research chapters of this dissertation are already published, either currently under peer-review or under preparation. The contributions of the Ph.D. candidate for each chapter are explained in this section. \u00EF\u0082\u00B7 A concise version of Chapter 3 which is about the mechanics model of turn-milling has been published in [1], \u00E2\u0080\u009CComak, A., and Altintas, Y., 2017, Mechanics of turn-milling operations, International Journal of Machine Tools and Manufacturing, 121, pp. 2-9 \u00E2\u0080\u009D. The manuscript was written by me, and edited by my supervisor. I was responsible for all the concept formulation, and simulation of the virtual model of the turn-milling mechanics. Also, the cutting force validation experiments were completely planned, carried out, and analysed by me. \u00EF\u0082\u00B7 Parts of Chapter 4 have been published in [2], \u00E2\u0080\u009CComak, A., Ozsahin, O., and Altintas, Y., 2016, Stability of milling operations with asymmetric cutter dynamics in rotating coordinates, Journal of Manufacturing Science and Engineering, 138(8), p. 081004\u00E2\u0080\u009D. I have developed the conceptual ideas, solved time domain solution of the asymmetric cutter stability, and carried out all the validation experiments. Orkun Ozsahin, former Post-Doctoral Fellow in Manufacturing Automation Laboratories, contributed to the frequency domain solution of the problem. vi \u00EF\u0082\u00B7 Parts of Chapter 4 have been accepted for publication in ASME Journal of Manufacturing Science and Engineering, \u00E2\u0080\u009CComak, A., Altintas, Y., 2018, Dynamics and stability of turn-milling operations with varying time delay in discrete time domain\u00E2\u0080\u009D. I was the primary contributor to all of the research here under the supervision of Prof. Altintas. The conceptual idea of varying time delay in turn-milling process has been developed, and applied to the solution of turn-milling stability solution by me. I was responsible to design all the fixtures used in the experimental validation of the stability model of turn-milling. \u00EF\u0082\u00B7 There will be an another journal paper that proposes the workpiece surface form error model and process optimization for turn-milling operations which are partly explained in Chapters 4 and 5. I contributed to the modelling, and analysis of the surface form error, and process parameter selection methodology in this work. Z. Murat Kilic, former Ph.D. student in Manufacturing Automation Laboratories, provided the vibration solution of the semi discrete time domain method which is used to model the dynamic surface form errors in Chapter 4. vii Table of Contents Abstract ......................................................................................................................................... iii Lay Summary ............................................................................................................................... iv Preface .............................................................................................................................................v Table of Contents ........................................................................................................................ vii List of Tables ................................................................................................................................ xi List of Figures .............................................................................................................................. xii List of Symbols .............................................................................................................................xx List of Abbreviations ............................................................................................................. xxxiii Acknowledgements ................................................................................................................ xxxiv Dedication ............................................................................................................................... xxxvi Chapter 1: Introduction ............................................................................................................... 1 Chapter 2: Literature Review ...................................................................................................... 8 2.1 Overview ........................................................................................................................ 8 2.2 Mechanics of Turning and Milling Operations .............................................................. 8 2.3 Dynamics and Stability of Turning and Milling Operations ........................................ 11 2.3.1 Dynamics and Stability of Low-Immersion Milling Operations ........................ 12 2.3.2 Stability of Milling Operations at Low-Speed Process Damping Region .......... 13 2.3.3 Stability of Milling Operations with Asymmetric Cutter Dynamics .................. 16 2.3.4 Stability of Milling with Multiple and Variable Time Delays ............................ 17 2.4 Kinematics, Mechanics and Dynamics of Turn-Milling Operations ........................... 19 2.5 Summary ...................................................................................................................... 20 Chapter 3: Mechanics of Turn-Milling Processes.................................................................... 22 viii 3.1 Introduction .................................................................................................................. 22 3.2 Kinematics of Turn-Milling ......................................................................................... 23 3.2.1 Screw Theory ...................................................................................................... 24 3.2.2 Denavit-Hartenberg Kinematic Model ............................................................... 30 3.3 Mechanics of Turn-Milling .......................................................................................... 33 3.3.1 Feed Vectors and Static Chip Thickness Distribution ........................................ 33 3.3.2 Static Cutting Force Model in Turn-Milling ....................................................... 42 3.3.3 Cutter-Workpiece Engagement Geometry in Turn-Milling Process .................. 46 3.4 Experimental Verification ............................................................................................ 47 3.5 Summary ...................................................................................................................... 54 Chapter 4: Dynamics and Stability of Turn-Milling Operations ........................................... 55 4.1 Overview ...................................................................................................................... 55 4.2 Stability of Milling Process with Asymmetric Structure Dynamics ............................ 56 4.2.1 Dynamics of Milling in Rotating Coordinates .................................................... 58 4.2.2 Stability of Milling with Asymmetric Cutter Dynamics..................................... 63 4.2.2.1 Frequency Domain Solution of Asymmetric Cutter Dynamics .................... 63 4.2.2.2 Time Domain Solution of Asymmetric Cutter Dynamics ............................ 67 4.2.3 Simulations and Experimental Results ............................................................... 69 4.3 Generalized Dynamics of Turn-Milling ....................................................................... 76 4.3.1 Dynamic Model of Turn-Milling Process ........................................................... 77 4.3.2 Prediction of In-Process Workpiece Dynamics .................................................. 81 4.3.3 Modeling of Dynamic Chip Thickness and Cutting Forces in Turn-Milling ..... 89 4.3.3.1 Dynamic Chip Thickness Model .................................................................. 89 ix 4.3.3.2 Dynamic Cutting Force Model in Turn-Milling Process .............................. 91 4.3.3.3 Dynamic Process Damping Force Model in Turn-Milling Process .............. 94 4.3.4 Modeling of Varying Time Delay in Turn-Milling Process ............................... 98 4.3.4.1 Discrete Tool and Workpiece Motion in Turn-Milling ................................ 98 4.3.4.1.1 Tool Motion............................................................................................. 99 4.3.4.1.2 Workpiece Motion................................................................................. 102 4.3.4.2 Time Delay Model ...................................................................................... 104 4.3.5 Stability of the Turn-Milling Process ............................................................... 111 4.3.6 Time Domain Solution of Force, Vibration and Surface Form Error ............... 126 4.3.6.1 Discrete Time Cutting Force and Vibration Simulation ............................. 127 4.3.6.2 Surface Form Error Model in Turn-Milling................................................ 129 4.3.7 Simulations and Experimental Validations ....................................................... 134 4.3.7.1 Analysis of Position Dependent Machine Dynamics.................................. 135 4.3.7.2 Analysis of the in-process workpiece dynamics ......................................... 144 4.3.7.3 Experimental Validation of Turn-Milling Stability Model ......................... 149 4.4 Summary .................................................................................................................... 172 Chapter 5: Turn-Milling Process Optimization ..................................................................... 173 5.1 Overview .................................................................................................................... 173 5.2 Constraint Based Selection of Turn-Milling Process Parameters .............................. 173 5.2.1 Chip Load Constraint ........................................................................................ 175 5.2.2 Torque and Power Limits of Tool and Workpiece Spindles............................. 177 5.2.3 Stability Constraint ........................................................................................... 178 5.2.4 Surface Form Error Constraint.......................................................................... 180 x 5.2.5 Cutting Speed Constraint .................................................................................. 183 5.3 Case Study for Turn-Milling Process Parameter Selection ........................................ 184 5.4 Summary .................................................................................................................... 190 Chapter 6: Conclusions and Future Research Directions ..................................................... 192 6.1 Summary and Contributions ....................................................................................... 192 6.2 Future Research Directions ........................................................................................ 196 REFERENCES ...........................................................................................................................198 Appendix A Rotation Definitions for Screw Theory ............................................................. 204 A.1 Exponential Coordinates for Pure Rotation ........................................................... 204 A.2 Solution for Paden-Kahan Sub problem 1 ............................................................. 205 A.3 Solution for Paden-Kahan Sub problem 2 ............................................................. 206 xi List of Tables Table 3.1 Cutting conditions for the turn-milling case. ................................................................ 48 Table 3.2 Cutting conditions for the experimental validation case in [80]. .................................. 52 Table 4.1 Modal parameters of asymmetric end mill in principal rotating coordinates (u,v). Tool is 2 fluted cylindrical end mill with 10 mm diameter, 13.5\u00C2\u00B0 rake and 25\u00C2\u00B0 helix angles. ................. 70 Table 4.2 Modal parameters of 4-fluted cylindrical end mill with 12 mm diameter and 30 degree helix angle. .................................................................................................................................... 75 Table 4.3 Simulation parameters for discrete time delay calculation ......................................... 107 Table 4.4 Comparison between the time delays and its discrete approximation in turn-milling and regular milling, turning processes. .............................................................................................. 117 Table 4.5 Stroke limits of translational and rotary drives of Mori Seiki NT 3150 DCG mill-turn center. .......................................................................................................................................... 136 Table 4.6 Material Properties of Al 6061 alloy. ......................................................................... 144 Table 4.7 Modal parameters of the identified chuck FRFs at x and y directions of the machine...................................................................................................................................................... 145 Table 4.8 Material properties of AISI 620 stainless steel. .......................................................... 147 Table 4.9 Modal parameters of tool and workpiece. .................................................................. 153 Table 4.10 Modal parameters of flexible workpiece in case 2. .................................................. 161 Table 4.11 Modal parameters of the 2 fluted cylindrical end mill having 20 mm diameter and 70 mm stick out and AISI 620 stainless steel workpiece. ................................................................ 167 Table 5.1 The modal parameters of the flexible workpiece in XX and YY directions. ............. 185 Table 5.2 MRR for different workpiece spindle speeds. ............................................................ 188 xii List of Figures Figure 1.1 Representation of milling (a), turning (b) and turn-milling (c) operations with their corresponding chip forms: short and spiral chips in milling (d), long and tangled chips in turning (e) and short-comma chip formation in turn-milling (f). ................................................................ 2 Figure 1.2 Configuration of a typical turn-milling machine tool. ................................................... 3 Figure 1.3 Flowchart of the thesis................................................................................................... 5 Figure 3.1 Configuration of turn-milling machine tool and kinematics of the process. ............... 23 Figure 3.2 Kinematic chain of the turn-milling machine tool. ..................................................... 26 Figure 3.3 Sample toolpath for a turn-milling process (a), obtained APT-CL file (b), inverse kinematic solution of the rotary B-axis of the machine (c) and comparison of the results with D-H kinematic model. ........................................................................................................................... 32 Figure 3.4 Multi-axes feed motion in turn-milling process. ......................................................... 34 Figure 3.5 Tool axial discretization and effective workpiece radius calculation (a), Linear workpiece feed representation (b). ................................................................................................ 35 Figure 3.6 Linear Feed vector representation. .............................................................................. 38 Figure 3.7 Angular feed vector due to the spindle head rotational motion. .................................. 40 Figure 3.8 General tool geometry representation. ........................................................................ 41 Figure 3.9 Varying cutter-workpiece engagement geometry in turn-milling (a), 2D representation of CWE in angular and axial coordinates. .................................................................................... 47 Figure 3.10 Experimental set-up to measure cutting forces in turn-milling process. ................... 48 Figure 3.11 Cutter-Workpiece Engagement (a) chip load on each flute of the tool in one spindle period, and T is the tooth passing period. The process parameters are given in Table 3.1. .......... 49 Figure 3.12 Simulated and measured cutting forces in turn-milling for given conditions in Table xiii 3.1.................................................................................................................................................. 51 Figure 3.13 CWE geometry along the tool axis (a) and chip load distribution on each flute (b) for the validation case against the experimental results in [80]. The cutting conditions are given Table 3.2.................................................................................................................................................. 53 Figure 3.14 Simulated and measured [80] resultant cutting forces over a spindle period. The cutting conditions are given in Table 3.2. ................................................................................................. 53 Figure 4.1 Representation of turn-milling machine tool axes....................................................... 56 Figure 4.2 Bearing contact angle at idle and rotating spindle cases (a); spindle mode shifting at high rotational speeds of tool (b). ................................................................................................. 57 Figure 4.3 Tool position at different rotation angles (a); variation of tool tip FRF with respect to tool rotation for symmetric (b) and asymmetric tools (c). ............................................................ 58 Figure 4.4 Dynamics and cutting forces in rotating coordinate frame. ........................................ 59 Figure 4.5 Real and imaginary parts of the measured FRF of the asymmetric end mill in principal rotating coordinates (u,v). ............................................................................................................. 70 Figure 4.6 Chatter stability diagrams in rotating coordinates (a) 50% immersion down milling and (b) 10% immersion down milling. Modal parameters are given in Table 4.1. ............................. 72 Figure 4.7 Experimental verification of the stability for half immersion down milling of asymmetric end mill. Feedrate: 0.2 mm/tooth. Material: Al7050-T7451. Modal parameters of the tool are given in Table 4.1. ........................................................................................................... 73 Figure 4.8 Simulation and experimental results for dynamically symmetric end mill in rotating and fixed coordinate frames. Modal parameters are given in Table 4.2.............................................. 76 Figure 4.9 Structural flexibilities in a typical turn-milling machine tool (a); Dynamic displacements at tool and workpiece (b). ...................................................................................... 77 xiv Figure 4.10 Machine and Tool Coordinate Systems in turn-milling process and total feed vector representation. ............................................................................................................................... 79 Figure 4.11 Turn-milling operations with (b) and without (a) tailstock. ...................................... 82 Figure 4.12 Variation in the workpiece geometry between consecutive passes of milling tool. .. 83 Figure 4.13 Receptance coupling of Structure A and B. .............................................................. 84 Figure 4.14 Decoupling of the dummy workpiece and chuck dynamics identification. .............. 86 Figure 4.15 Receptance coupling of individual beam elements of the shaft mill. ........................ 87 Figure 4.16 Receptance Coupling of the workpiece to the machine\u00E2\u0080\u0099s chuck. .............................. 88 Figure 4.17 The general representation of helical tool geometry on a ball end mill. ................... 90 Figure 4.18 Geometrical representation of discrete chip geometry and axial depth of cut. ......... 92 Figure 4.19 Representation of indented volume under the flank face of the cutting edge. .......... 95 Figure 4.20 Indented volume by the flank face of the tool for short and long wavelengths, corresponding to low speeds (a) and high speeds (b). .................................................................. 96 Figure 4.21 Relative positions of surface points on the rotating workpiece in turn-milling ........ 99 Figure 4.22 Representation of full discrete tool motion in Cartesian coordinates...................... 101 Figure 4.23 Discretization of workpiece motion and resulted phase difference......................... 103 Figure 4.24 Variation of phase difference as a result of workpiece rotation. ............................. 105 Figure 4.25 Length of one full wave imprinted on the workpiece surface. ................................ 106 Figure 4.26 Comparison of discrete time delays in turn-milling and regular milling operations. Simulation inputs are given in Table 4.3. ................................................................................... 108 Figure 4.27 Time delay variation amplitude with different speed and diameter ratios of tool and workpiece. ................................................................................................................................... 109 Figure 4.28 Variation of total time delay amplitude by the speed ratios of tool and workpiece xv spindles. ...................................................................................................................................... 110 Figure 4.29 Calculation of discrete forcing function as a function of axial immersion. ............ 113 Figure 4.30 Approximation of the delayed states by time varying weights [31]. ....................... 116 Figure 4.31 Variation of discrete weights within one time period of the system. ...................... 117 Figure 4.32 Low speed stability envelope defined by limiting depth of cut and asymptotic spindle speed. .......................................................................................................................................... 125 Figure 4.33 Circularity error in turn-milling process.................................................................. 130 Figure 4.34 Surface formation in turn-milling. ........................................................................... 132 Figure 4.35 Overlap effect due to the wiper edge during the surface generation in turn-milling...................................................................................................................................................... 134 Figure 4.36 Representation of functional machine tool volume for NT 3150 DCG Mill-turn center...................................................................................................................................................... 136 Figure 4.37 Location of measured FRFs at (X, Y, Z) = ([0-550], [-125,+125], -470)mm. ........ 137 Figure 4.38 Measured FRFs at different location of ram length and X axis. ............................. 138 Figure 4.39 Peak amplitude variations of the tool FRF along y and z directions of the machine...................................................................................................................................................... 139 Figure 4.40 Variation of modal parameters (natural frequency (a), damping ration (b), dynamic stiffness (c), modal mass (d)) with the increased ram length. .................................................... 140 Figure 4.41 Peak amplitude variation of Y direction FRF of the tool at different z-axis locations...................................................................................................................................................... 141 Figure 4.42 Variations of Y and Z direction FRFs of the dominant tool mode on X-Z plane of the machine. ...................................................................................................................................... 142 Figure 4.43 Stability of the turn-milling cutting process for the most flexible and rigid modes of xvi the tool within the machine functional volume........................................................................... 143 Figure 4.44 Representation of measurement coordinates for the chuck FRF identification. ..... 144 Figure 4.45 Identified Chuck FRFs at XX (a) and YY (b) directions. ....................................... 145 Figure 4.46 Predicted workpiece FRFs of Al alloy at XX and YY directions before and after the machining. ................................................................................................................................... 146 Figure 4.47 Predicted workpiece FRFs of Steel alloy at XX and YY directions before and after the machining. ................................................................................................................................... 148 Figure 4.48 Experimental set-up for turn-milling cutting tests (a); position of fiberoptic displacement sensors on the fixture. ........................................................................................... 149 Figure 4.49 Representation of vibration isolation with (b) and without (a) dampers; Locations of vibrations pads on the fixture (c). ............................................................................................... 150 Figure 4.50 Effect of vibration pads on the amplitude of the cross talk FRFs between the tool tip and fiberoptic sensor casing. ....................................................................................................... 151 Figure 4.51 Sensitivity of the fiberoptic displacement sensor. ................................................... 152 Figure 4.52 Frequency Response Function (FRF) of flexible tool (a) and workpiece (b).......... 153 Figure 4.53 3D and 2D cross sectional views of the stability diagram simulated with the modal parameters listed in Table 4.9. .................................................................................................... 154 Figure 4.54 Stability validation tests for case 1 (a). See Table 4.9 for the dynamic parameters of the turn-milling system. Stability limits and experimental results at c\u00CE\u00A9 = 6 [rev/min] (b); the FFT of sound data (c) and tool motion in feed and normal directions (d) at Point A (stable). Experimental results at c\u00CE\u00A9 = 21 [rev/min] (e) and corresponding sound FFT (f) and tool motion (g) at Point B (chatter). ..................................................................................................................... 155 Figure 4.55 Surface photos of the stable and unstable (chatter) cutting process. ....................... 156 xvii Figure 4.56 Simulated vibrations along the normal (a), feed (b), and axial (c) directions of tool...................................................................................................................................................... 158 Figure 4.57 Simulated axial direction vibrations within the engagement boundaries of the process...................................................................................................................................................... 159 Figure 4.58 Surface location errors contributed by the static and dynamic cutting terms of the system for one spindle period. .................................................................................................... 160 Figure 4.59 Surface location errors left on the workpiece with the corresponding regions of undercut and overcut. .................................................................................................................. 161 Figure 4.60 FRF of the flexible workpiece in case 2. ................................................................. 162 Figure 4.61 Stability validation tests for case 2 when c\u00CE\u00A9 = 6 [rev/min] (a). See Table 4.10 for the modal parameters of the system. Stability limits and experimental results at c\u00CE\u00A9 =12 [rev/min] (b) c\u00CE\u00A9 = 40 [rev/min] (c), c\u00CE\u00A9 =100 [rev/min] (d) and FFT of sound data at Point A (e) and Point B (f)...................................................................................................................................................... 163 Figure 4.62 Stable and unstable regions of cutting with their corresponding stability properties based on the eigenvalues analysis. A: Stable, B: Hopf type chatter, C: Primary Flip type chatter, D: Secondary Flip type chatter. .................................................................................................. 165 Figure 4.63 Stability properties at high speed and low immersion of turn-milling. ................... 166 Figure 4.64 Experimental validation of the process damping models for turn-milling of AISI 620 stainless steel. The solid black line represents the stability limits calculated without considering low-speed process damping forces. The solid purple line shows the stability limits with the process damping model given in Section 4.3.3.3. The red curve defines the low speed process damping stability envelope which is predicted by the asymptotic spindle speed method as expressed in Section 4.3.5................................................................................................................................ 169 xviii Figure 5.1 Effect of workpiece spindle speed and axial depth of cut on MRR in turn-milling process......................................................................................................................................... 174 Figure 5.2 Feasible and non-feasible regions of tool and workpiece spindle speeds for given maximum allowed chip load. Workpiece diameter: 50 mm, Tool diameter: 20 mm, Maximum allowed chip load: 0.4 mm for 2 \u00E2\u0080\u0093fluted cylindrical end mill. ................................................... 176 Figure 5.3 Feasible and non-feasible tool spindle speeds for the cases c\u00CE\u00A9 = 5 rpm (a), and c\u00CE\u00A9 = 30rpm (b). The maximum and minimum allowed chip loads are determined as 0.4 mm and 0.05 mm for the given cutting tool. ............................................................................................................ 177 Figure 5.4 Torque and Power charts for workpiece spindle (a); tool spindle (b) of NT 3150/500C multi axes machine tool. ............................................................................................................. 178 Figure 5.5 The stability boundaries of a turn-milling process, and corresponding tool spindle speeds that correspond to peak of the stability lobes (pockets). ................................................. 179 Figure 5.6 Effect of tool-workpiece spindle speed ratio on the total form error left on the workpiece. ................................................................................................................................... 181 Figure 5.7 The effect of axial depth of cut at different tool spindle speeds for a given workpiece spindle speed. .............................................................................................................................. 182 Figure 5.8 Variation of peak total form error amplitudes at different speed ratios of tool and workpiece spindles. ..................................................................................................................... 183 Figure 5.9 Stability lobes for the given turn-milling case. ......................................................... 185 Figure 5.10 Representation of each process and machine constraint on a turn-milling case. .... 187 Figure 5.11 Feasible and non-feasible cutting regions of turn-milling case (workpiece spindle speed:10 rpm).............................................................................................................................. 188 Figure 5.12 Variation of MRR at different workpiece spindle speeds with their corresponding xix feasible sets of tool spindle speed intervals. ............................................................................... 190 Figure A.1 Pure rotation of a point q around the fixed axis \u00CF\u0089. .................................................. 203 Figure A.2 Rotation about a single axis. .................................................................................... 205 Figure A.3 Rotation about two subsequent axes. ....................................................................... 206 xx List of Symbols ( )tA Directional coefficient matrix at present time A Time invariant average term of directional coefficient matrix at a delay period before 11,xxA , 11,yyA Receptance Matrix of Structure A along xx and yy directions at Point 1 a Axial Depth of cut lima Critical (absolute) axial stability limit arg Argument of the complex eigenvalue ( )tB Directional coefficient matrix at a delay period before B Time invariant average term of directional coefficient matrix at a delay period before ,B C Rotary drives of the turn-milling machine tool 11,xxB , 11,yyB Receptance Matrix of Structure B along xx and yy directions at Point 1 corC Coriolis force matrix pdC Process damping coefficient matrix rC Modal damping matrix of system expressed at rotating coordinates sC Modal Damping matrix of system expressed at stationary coordinates c Commanded feed pdc Process damping coefficient xc , yc Modal damping at x and y directions xxi ,ttc c\u00EF\u0081\u00B1\u00EF\u0081\u00B1 Translational and rotational contact damping D Damping term coefficient for the state space equations tD Tool diameter wD Workpiece diameter d Dynamic displacement vector db Discrete chip width d d,crtaF Differential Dynamic cutting force vector at rta frame d d,pdrtaF Differential Process damping force vector at rta frame d s,crtaF Differential Static cutting force vector at rta frame d s,ertaF Differential edge cutting force vector at rta frame rdF , tdF , adF Differential cutting forces along radial, tangential and axial directions of tool dS Infinitesimal length of a helical cutting edge segment \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSxyzdF Differential cutting forces along x, y, z direction in TCS dz Differential axial disc element TE Total form error ,maxce Maximum scallop height left on the turn milled workpiece i ie\u00EF\u0081\u00B1\u00CF\u0089 Exponential representation of a rigid body rotation resF Total resultant cutting force at x-y plane of the tool ( )s tF Total cutting forces expressed at stationary coordinates xxii ( )tMF Total cutting force vector expressed in Modal space ( )tM,TF Cutting forces acting on tool expressed in Modal Space ( )tM,WF Cutting forces acting on workpiece expressed in Modal Space zF Weight of each discrete axial force element tF Total tangential forces acting on the tool body \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSxyzF Total cutting forces along the x, y, z directions in TCS Af Angular feed of cutting tool L,Tf Total Linear feed vector tf Linear feed of cutting tool Tf Resultant (total) feed vector wf Projected linear feed of rotating workpiece , ,x y zf f f Feed vectors at x, y, z directions of the machine G Rotation-Force Transfer Function g Unit step function btg Homogenous transformation matrix from base frame to tool frame bwg Homogenous transformation matrix from base frame to workpiece frame wtg Homogenous transformation matrix from workpiece to tool frame H Displacement-Force Transfer Function CH Decoupled transfer function matrix of chuck CWPH Coupled transfer function of workpiece-chuck assembly xxiii WPH Transfer function of free-free workpiece jh Chip thickness acting on the tooth, j djh Dynamic chip thickness sjh Static chip thickness ,maxsh Maximum static cusp height totalh Total cusp height including static and dynamic terms oi Oblique angle of the cutter j Tooth Number cK Cutting force coefficient crcK Circularity force matrix cntK Centripetal force matrix eK Edge force coefficient rcK , tcK , acK Cutting force coefficients along radial, tangential and axial directions of tool reK , teK , aeK Edge force coefficients along radial, tangential and axial directions of tool rK Modal stiffness matrix of system expressed at rotating coordinates sK Modal Stiffness matrix of system expressed at stationary coordinates spK Material specific process damping (indentation) coefficient k Period resolution xxiv 1k , 2k , 3k Coefficients of Paden-Kahan sub problem 2 lobek Lobe number ,ttk k\u00EF\u0081\u00B1\u00EF\u0081\u00B1 Translational and rotational contact stiffness xk , yk Modal stiffness at x and y directions L Present time coefficient matrix for the state space equations scL Overlap surface length tpL Length of the surface generated during one tooth passing period wL Flank face wear length wpL Wiper edge length rM Modal mass matrix of system expressed at rotating coordinates sM Modal Mass matrix of system expressed at stationary coordinates m Number of discrete intervals in one spindle period, delay resolution xm , ym Modal mass at x and y directions N Displacement-Moment Transfer Function tN Number of teeth on the tool n Number of joints on the kinematic chain of the machine tool jn The unit outward vector normal to the tool body O Tool orientation vector relative to workpiece coordinate frame BO Base frame for kinematic analysis in Screw Theory P Tool position vector relative to workpiece coordinate frame xxv P Rotation-Moment Transfer Function cP Cutting power transmitted to C axis of the machine tP Cutting power drawn from tool spindle ijtP Point on the workpiece surface left by tooth j at time i *i tjtP \u00EF\u0080\u00AB\u00EF\u0081\u0084 Updated positon of point left by tooth j after one discrete time interval 1i djt TP\u00EF\u0080\u00AB\u00EF\u0080\u00AB Point on the workpiece surface left by tooth j+1 *i djt TP \u00EF\u0080\u00AB Updated position of point left by tooth j+1 after one time period TdP Dynamic displacement vector of tool TfP Full motion vector of tool TrP Rigid body motion vector of tool mp Total number of flexible modes ( )p z axial height of individual discrete element from the tool\u00E2\u0080\u0099s bottom end tp Number of flexible modes of tool TCSp Fictitious rotation axis that passes through the tip of tool wp Number of flexible modes of workpiece ( )sQ Relative displacements between tool and workpiece in Laplace Domain q Number of contact points between the tool and workpiece ,b cq q Point on the rotation axis of the screw for B and C axis of machine ( )tq Vibration vector at present time xxvi ,( )j it \u00EF\u0081\u00B4\u00EF\u0080\u00ADq Vibration vector at one delay period before R Previous time coefficient matrix for the state space equations ( )R z Local tool radius at elevation z 0R Radius at the cylindrical part of the tool w,iR Workpiece radius ew,iR Effective workpiece radius r Number of harmonics in Fourier Transformation otr Tool orientation vector relative to tool coordinate frame ptr Tool position vector relative to tool coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta Coordinate frame defined at radial, tangential and axial directions of tool hr Hone radius of tool S Stiffness term coefficient for the state space equations pS Reference commands of prismatic joints jS Surface generated by tooth j s Laplace Domain Operator 1s z\u00EF\u0080\u00AD\u00EF\u0082\u00AE Transformation operator from Laplace domain to discrete time domain T Transformation matrix between stationary and rotating coordinate frames 1T Transformation matrix that aligns the tool\u00E2\u0080\u0099s orientation with respect to MZ xxvii 2T Transformation matrix between \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t,i t,ix y and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9M MX Y\u00EF\u0080\u00AD cT Cutting torque transmitted to C axis of the machine dT Tooth passing period HT Transformation matrix between \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta and TCS jT Transformation matrix of tooth j between local and principal rotating coordinates of tool sT Spindle period tT Cutting torque drawn from tool spindle BT , CT Transformation matrices for B and C axis rotations of the machine wT Surface generation completion time 0t Offset vector defined in tool frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v Principal rotating frame of tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,l lj ju v Local rotating coordinates of tool \u00EF\u0081\u00BB \u00EF\u0081\u00BD( ), ( )l lj ju t v t Vibration vector expressed in local rotating frame at present time \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 1( ), ( )l lj ju t v t\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD Vibration vector expressed in local rotating frame at one delay period before U Mass normalized mode shape matrix tU Mass normalized mode shape matrix of tool wU Mass normalized mode shape matrix of workpiece xxviii cV Cutting speed vector of workpiece dV Indented volume of workpiece by the flank face of the tool RV Resulting cutting speed tV Cutting speed vector of tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9x y zv , v , v Unit vectors that point the positive direction of the translational axes 0w Offset vector defined in workpiece frame ,a bw Discrete weights for the approximation of the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,X Y Z Translational drives of the turn-milling machine tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A90 0 0, ,X Y Z Initial coordinates of cutter in space \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,M M MX Y Z Cartesian coordinates defined with respect to Machine Coordinate System \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,w w wX Y Z Cartesian coordinates defined with respect to Workpiece Coordinate System , ,x y z Reference position commands for translation drives of machine fx Resulted vibrations at feed direction \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,t t tx y z Cartesian coordinates defined with respect to Tool Coordinate System iY , 1i\u00EF\u0080\u00ABY Augmented state vector at current and next states z Distance vector between the intersection of two rotational axis with their rotational axes z Axial elevation of a discrete axial disc xxix lz , uz Lower and upper levels of discrete axial element 0z Resultant vibrations between flexible tool and workpiece along the tool normal axis \u00EF\u0081\u00A1 Phase angle between the vibration waves by present and previous tooth b\u00EF\u0081\u00A1 Tool bottom face clearance angle n\u00EF\u0081\u00A1 Rake angle at normal plane of oblique cutting tp\u00EF\u0081\u00A1 Real part of residue for the tool mode r\u00EF\u0081\u00A1 Rake angle at orthogonal cutting ( )z\u00EF\u0081\u00A2 Helix angle of tool at axial elevation z n\u00EF\u0081\u00A2 Friction angle at normal plane of oblique cutting tp\u00EF\u0081\u00A2 Imaginary part of residue for the tool mode s\u00EF\u0081\u00A2 Friction angle at orthogonal cutting sp\u00EF\u0081\u00A2 Separation angle ( )s\u00CE\u0093 Modal displacement vector expressed in Modal space ( )t\u00CE\u0093 , ( )t\u00CE\u0093 , (t)\u00CE\u0093 Displacement, velocity and acceleration matrix in modal space fv\u00EF\u0081\u0087 The length of one full vibration \u00EF\u0081\u00A7 Side edge clearance angle of the tool i,t\u00CE\u0094S Displacement vector of tool between two CL points T\u00CE\u0094S Total displacement vector of tool and workpiece during it\u00EF\u0081\u0084 it\u00EF\u0081\u0084 Time elapsed during tool travels between two CL points xxx \u00EF\u0081\u0084 i,w\u00CE\u00B8 Angular displacement of workpiece during it\u00EF\u0081\u0084 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z\u00EF\u0081\u0084 \u00EF\u0081\u0084 \u00EF\u0081\u0084 Discrete feed of cutter in space rb\u00EF\u0081\u00A5 Phase difference due to rigid body rotation of workpiece \u00CE\u00B6 Diagonal damping ratio matrix \u00EF\u0081\u00B8 Rigid body twist of a joint c\u00EF\u0081\u00B8 Twist command for rotary C axis of the machine pd\u00EF\u0081\u00BA Process damping ratio s\u00EF\u0081\u00BA Structural damping term c\u00EF\u0081\u00A8 Chip flow angle ( )t\u00CE\u0098 State matrix at present time ,( ( ))j it t\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00CE\u0098 State matrix at one delay period before fv\u00EF\u0081\u0091 Length of the fractional wave left on the workpiece \u00EF\u0081\u00B1 Rigid body rotation of a joint ,b c\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 Reference command for the rotary axes B and C of the machine ( )z\u00EF\u0081\u00AB Axial immersion angle of tool at elevation z ,r i\u00EF\u0081\u008C \u00EF\u0081\u008C Real and imaginary parts of eigenvalue \u00EF\u0081\u00AD Characteristic multiplier of transition matrix c\u00EF\u0081\u00AD Coulomb friction coefficient \u00EF\u0081\u00B2 The ratio of time spent during cutting and non-cutting \u00EF\u0081\u00B4 Time delay in regenerative vibration mechanism xxxi s\u00EF\u0081\u00B4 Shear stress of workpiece material c,p\u00CF\u0085 Eigenvector \u00CE\u00A6 Total transfer function of system cor\u00CE\u00A6 Transfer function resulted from Coriolis term of the rotation cnt\u00CE\u00A6 Transfer function resulted from centripetal term of the rotation crc\u00CE\u00A6 Transfer function resulted from circularity term of the rotation i\u00CE\u00A6 Transition Matrix between current and next states s\u00CE\u00A6 Structural transfer function T\u00CE\u00A6 Resultant transition matrix over the periodicity of the system \u00EF\u0081\u00A6 Rotation angle of tool c\u00EF\u0081\u00A6 Angle between principal and local rotating coordinate of tool j\u00EF\u0081\u00A6 Instantaneous angular position of tooth, j n\u00EF\u0081\u00A6 Shear angle at normal plane of oblique cutting p\u00EF\u0081\u00A6 Pitch Angle of tool s\u00EF\u0081\u00A6 Shear angle at orthogonal cutting ,z zst ex\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 Start and exit angles of cutter workpiece engagement at axial elevation z w\u00EF\u0081\u00AA Angle between contact point of tooth j with the workpiece and the axis perpendicular to workpiece rotation direction \u00EF\u0081\u00B9 Lag Angle between adjacent teeth of tool a\u00EF\u0081\u0097 Asymptotic spindle speed xxxii c\u00CE\u00A9 Rotational speed of workpiece r\u00EF\u0081\u0097 Ratio of tool and workpiece spindle speeds t\u00EF\u0081\u0097 Rotational speed of tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,b c\u00CF\u0089 \u00CF\u0089 Unit vectors that point the positive direction of the rotary axes c\u00EF\u0081\u00B7 Chatter frequency F\u00EF\u0081\u00B7 Flip bifurcation frequency H\u00EF\u0081\u00B7 Hopf bifurcation frequency dH\u00EF\u0081\u00B7 Dominant hopf bifurcation frequency i\u00CF\u0089 Unit vector of the thi revolute joint n\u00CF\u0089 Diagonal natural frequency matrix n\u00EF\u0081\u00B7 Natural frequency of a flexible mode R\u00EF\u0081\u00B7 Angular velocity of tool axis rotation T\u00EF\u0081\u00B7 Tooth passing frequency \u00EF\u0081\u00B6 Rotation angle between the \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t,i t,ix y and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9M MX Y\u00EF\u0080\u00AD xxxiii List of Abbreviations APT-CL Automatically Programmed Tool-Cutter Location CAM Computer Aided Manufacturing CL Cutter Location CPU Central Processing Unit CVD Chemical Vapor Deposition CWE Cutter-Workpiece Engagement DDE Delay Differential Equation D-H Denavit-Hartenberg FRF Frequency Response Function FT Fourier Transform MCS Machine Coordinate System MQL Minimum Quantity Lubrication PVD Physical Vapor Deposition RC Receptance Coupling TCS Tool Coordinate System WCS Workpiece Coordinate System xxxiv Acknowledgements After an intensive period of five years in Manufacturing Automation Laboratories, I would like to express my sincerest and deepest gratitude to a number of exceptional individuals without whom this dissertation would never have been possible. My heartfelt thanks and appreciation to all of them for being part of this wonderful journey in their own particular way. First and foremost, I am deeply indebted to my supervisor, Prof. Yusuf Altintas, for his guidance, support and continuous encouragement all through my Master\u00E2\u0080\u0099s and PhD years. Words can neither qualify nor quantify to truly express the influence of him on my life. His world-class academic excellence, ideas and vision have influenced and shaped not only my research career but also my personal development. It has been a great pleasure and privilege working, sailing, traveling and having discussions with him. I would like to extend my heartfelt thanks to Altintas family for their great support during my journey in Vancouver. I also would like to thank Prof. Steve Feng, Prof. Farrokh Sassani, Prof. Steve Cockcroft and Prof. Thomas Kurfess for their constructive comments about the dissertation. It has been the privilege to work in a world-class environment in the Manufacturing Automation Laboratories. I would like to express my gratitude to all my colleagues, also the former members of MAL whose cumulative knowledge has been excessively used throughout the dissertation. I have worked with exceptional people from all over the world and it was a precious experience. Among them, I would like to thank Onur, Deniz, Coskun and Oguzhan for their sincere friendship and support. Also, I wish to express thanks to Dr. Z. M. Kilic, Dr. Ozsahin and other post-doctoral fellows in MAL for their collaborations and consultation during various stages of the dissertation. xxxv I would like to thank Dr. Jokin Munoa and IK-4 IDEKO for hosting me and support during my internship in Spain; and the opportunities they made possible. I am deeply grateful to the engineers and technicians at IDEKO for sharing their experience and knowledge. Prof. Erhan Budak \u00E2\u0080\u0093 my former supervisor, was the great mentor and support in not only my Master\u00E2\u0080\u0099s but also PhD years. I owe my special thanks to him. Besides the intellectual help and mentoring, I also would like to express my thanks to the industrial supporters of my laboratory and NSERC-CANRIMT for the support during my PhD and internship. I am thankful to DMG Mori for providing the Mori Seiki NT3150 DCG Mill-Turn Center to MAL; Sandvik Coromant for generously donating the cutting tools used in the experiments. I am deeply thankful to my family for their continuous and unconditional support, encouragement, love and absolute confidence in me throughout my life. My dear mother, Bengi, my father, Zafer and my sister Muazzez, I am grateful to have you all. My love, Burce has been always the emotional support at the difficult times of my PhD, and the driving force for me to complete this dissertation efficiently. This journey would never have been possible without their spiritual support. Lastly, I would like express my gratitude to all people in St. John\u00E2\u0080\u0099s College for making my five years long journey enjoyable. The time I have spent at SJC was one of the most precious experience I have had so far. I shared many fantastic moments, and learned to care and understand different people and their cultures. I have had hundreds of friends, some of them were almost like my brothers and sisters. Among them, I would like specially thanks to Gerson Luis Schwab for his support, encouragement and sharing his life experiences with me. Thank you all. xxxvi Dedication to my family\u00E2\u0080\u00A6 1 Chapter 1: Introduction Machining is a material removal process from the blank workpiece to produce near-net (final) shape parts, thus widely used in the die & mold, automotive, aerospace, and machine manufacturing industries. The cutting tool which penetrates into the workpiece removes chips from the part as a result of the relative motion between the tool and workpiece. Turning, drilling, boring and milling are the most widely used machining operations to produce cylindrical and prismatic parts. A rotating tool (cutter) with multiple cutting edges cuts a prismatic workpiece in milling as seen in Figure 1.1 (a). Each cutting edge of the milling tool periodically enters the workpiece which yields intermittent cuts and short chips. On the other hand, a stationary tool with a single cutting edge cuts a rotating part in turning (see Figure 1.1 (b)), and the cutting edge is always in contact with the workpiece resulting in continuous chips. Turn-milling machine tools which are capable of carrying out turning, drilling, boring, milling and grinding operations in a single set-up are rapidly emerging in the industry because of their multi-functional capabilities in producing complex parts. Turn-milling has the advantage of carrying out turning and milling operations simultaneously on one machine, which reduces the machine set-up time, and improves the productivity. The operation requires two rotating spindles and three Cartesian drives. While a milling tool is mounted on a high-speed spindle, the workpiece is clamped to a rotating chuck which rotates at a slower speed than the spindle (see Figure 1.1 (c)). Since a milling tool performs the chip removal operation, the resulting cutting mechanism is interrupted and has the advantage of chip breaking, reduced tool wear and increased tool life. Also, combining the cutting speeds and feeds of the tool and workpiece spindles, higher material removal rates (MRR) can be achieved with a better surface quality. On the other hand, erroneous selection of cutting parameters results in excessive cutting forces and self-excited, regenerative chatter 2 vibrations which may generate poor surface finish, break the tool, or damage the spindle bearings, thus limiting the full potential for productivity of the turn-milling operations. Therefore, safe and productive cutting parameters must be selected prior to costly production by modeling the mechanics, dynamics and stability of the process which has not been investigated in the literature sufficiently. The turn-milling operations are mainly carried out by relying on costly machining trials and experience. The model needs to consider complex kinematics, dynamics, and tool-workpiece interaction. This thesis proposes a comprehensive mechanics, kinematics, dynamics and stability model of turn-milling operations that is applicable to any tool-workpiece geometry and complex toolpath, as the first unified model in the literature. Figure 1.1 Representation of milling (a), turning (b) and turn-milling (c) operations with their corresponding chip forms: short and spiral chips in milling (d), long and tangled chips in turning (e) and short-comma chip formation in turn-milling (f). In turn-milling, the rotating milling tool is mounted on the tool spindle and positioned in three Cartesian coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9X,Y,Z , and can be tilted around the B-axis of the machine to follow a curved path on the workpiece mounted on the rotating chuck (C-axis) as shown in Figure 1.2. 3 When all five drives operate for turn-milling of curved parts, the chip geometry changes as a function of the machine\u00E2\u0080\u0099s kinematic configuration, tool path dependent rigid body motion of the translational and rotary drives, cutter-workpiece engagement (CWE) geometry, and the angular speeds and geometries of both the tool and workpiece. Also, simultaneous rotations of the tool and workpiece spindles disturb the regenerative vibration and leads to dual delay mechanisms. Figure 1.2 Configuration of a typical turn-milling machine tool. The objective of this thesis is to develop a novel predictive model of the turn-milling process that calculates the chip thickness distribution, cutting forces, power & torque, vibrations between the tool and workpiece, chatter-free cutting conditions, and surface form errors of the finished part. The model is expected to lead to the most productive cutting conditions, and test the operation in 4 the virtual environment without violating the process and machine tool constraints, hence eliminating the need for costly physical trials. The physics based virtual-predictive model of the mechanics and dynamics of turn-milling operations are developed for arbitrary tool and workpiece geometries, and complex toolpaths in this thesis. First, the mechanics of the turn-milling process is modeled to predict the static chip thickness (load), and cutting forces for complex toolpaths and cutter-workpiece engagement geometry. The cutting forces are transformed to tool and machine tool coordinate frames by the developed kinematics model of the turn-milling process. Then, the dynamics of the turn-milling machine tool and its substructures are modeled in modal space to evaluate dynamic chip thickness. The time delay in the regenerative vibration system is modeled by considering the additional time delay contributed by the simultaneous rotations of tool and workpiece spindles. The structural dynamics of the turn-milling machine tool and dynamic cutting forces at tool-workpiece interaction zone are combined, and the system dynamics are modeled as periodic, time-varying delay differential equations. The coupled delay differential equations are solved to predict the relative vibrations, stability limits, and the surface form errors in the turn-milling process. The proposed dynamics and stability model is extended into the nonlinear low-immersion and low-speed regions where the process damping is dominant. The models are experimentally validated for various cases and materials. The desired cutting conditions that maximize the material removal rates are predicted based on the modeled static and dynamic limits of the process and machine tool\u00E2\u0080\u0099s spindles. The flowchart of the overall scheme presented in the thesis is shown in Figure 1.3. 5 Figure 1.3 Flowchart of the thesis. Henceforth, the thesis is organized as follows; The review of the existing turn-milling technology and related literature is presented in Chapter 2. The relevant literature about the mechanics and dynamics of the turning and milling operations is also briefly reviewed. The general mathematical model of the turn-milling mechanics that predicts the static cutting forces and resulting cutting torque and power is presented in Chapter 3. First, the kinematics analysis of the turn-milling machine tool is expressed. Forward and inverse kinematic solutions of the turn-milling process are solved by Screw Theory and Denavit-Hartenberg methods for any five axis toolpath. Then, the cutting forces acting on the tool and workpiece are predicted as a function of the instantaneous chip geometry which may continuously vary as a function of all five axes 6 positions and velocities, as well as the cutter-workpiece engagement (CWE) geometry and angular rotational speeds of tool and workpiece spindles. The discrete cutting forces acting on the differential cutting edges are projected to tool and machine tool coordinates by the kinematic model of the machine tool. The simulated cutting forces are experimentally validated. Chapter 4 is dedicated to the dynamics and the stability of the turn-milling operations. The mechanics model of turn-milling is extended by considering the relative vibrations between the rotating tool and workpiece, and multi-axes feed motion of the machine drives. First, the dynamics and stability of regular milling operations with asymmetric cutting dynamics are modeled to show the basic relationship between the spindle speeds and stable depth of cuts in the stability perspective of the process. Then, the dynamics of the turn-milling machine tool and its substructures are modeled in modal space, and the dynamic chip thickness model is constructed. The presence of tool and workpiece rotations lead to additional time delay in the regenerative vibration mechanism in turn-milling as oppose to turning and milling processes where the time delay is only a function of spindle speed of the tool or workpiece. Hence, the time delay due to the rigid body rotational motion of the workpiece is modeled in discrete time intervals. Then, the governing system equations which are periodic and time-varying delay differential equations are solved in the semi-discrete time domain to predict the stability limits of the turn-milling process. Varying workpiece dynamics due to the mass removal process is studied by a simple Receptance Coupling model, and the flexible in-process workpiece dynamics are predicted analytically. The proposed dynamics and stability model of turn milling is also extended into low-immersion and low-speed process damping region where the non-linear stability properties are dominant. The proposed novel stability model of turn-milling is validated for various cases and materials. 7 Chapter 5 is dedicated to the constraint based selection of turn-milling process parameters. The proposed mechanics and stability models of turn-milling are used to plan the speeds of the tool and workpiece, feeds, and depth of cut without violating the physical limits of machine and cutting tool. The thesis is concluded with a short summary of the presented model, contributions to the literature, and future research directions in Chapter 6. 8 Chapter 2: Literature Review 2.1 Overview The main objective of this thesis is to develop a virtual model of turn-milling operations that predicts the cutting forces and chatter-free cutting conditions without resorting to costly physical trials. This chapter reviews the previous research related to the mechanics, dynamics and stability of regular turning, milling and turn-milling operations. Since the mechanics and dynamics model of turn-milling highly rely on the approaches studied for turning and milling operations, the relevant literature for these two operations is reviewed first. The mechanical models of turning and milling processes are reviewed in Section 2.2. Then, the dynamics and stability of the turning and milling operations are discussed in Section 2.3. Later, the limited research reported for the turn-milling operations is presented in Section 2.4. The chapter is concluded with a summary that points out the gaps in the relevant literature where the fundamental contributions of this thesis are founded. 2.2 Mechanics of Turning and Milling Operations Mechanics of turning and milling operations have been studied extensively by many researchers in the past. Pioneer researcher Merchant [3] presented the shearing effect in orthogonal cutting where the cutting occurs due the shearing action of the chip and expressed the force relationships on the chip-rake face of the cutting edge which is known as Merchant force diagram. Later, Lee and Shaffer [4] applied the plasticity theorem to the chip formation in orthogonal cutting and proposed a slip-line field model to express the plastic deformation of the material and obtained the cutting forces, chip thickness and chip deformation. Shaw et al. [5] studied the shear angle in orthogonal cutting and showed that the shearing mechanism is dependent on the friction between the chip-tool rake face. Palmer and Oxley [6] included the work-hardening effect in the application 9 plasticity theory in the slow orthogonal cutting of the mild steel. Based on their experimental observations, the shear (plastic) zone is the region with considerable width and the deformation between the chip-tool rake face contact is elastic. In addition to the shearing mechanism in the metal cutting process, the indentation of the flank face into the workpiece results in additional forces during the cutting which is known as the ploughing effect. Albrecht [7] studied the effect of ploughing forces in orthogonal cutting process and improved the cutting force diagram developed by Merchant [3] by including the wear land of the tool flank. In the same study, the effect of tool sharpness on the ploughing mechanism is shown as well. The mechanics of the milling process is studied parallel to the advancements in chip formation mechanism in orthogonal cutting. Martelotti [8] developed a geometrical analysis of the milling process and stated that the chip geometry in milling is formed by the rotational and translational motion of the cutter in space. The cutter follows a trochoidal path during cutting and the corresponding thickness of the chip acting on the thj tooth of the cutter \u00EF\u0080\u00A8 \u00EF\u0080\u00A9jh can be expressed by the following geometrical calculation; .sinj jh c \u00EF\u0081\u00A6\u00EF\u0080\u00BD (2.1) where c is the commanded feed value and j\u00EF\u0081\u00A6 is the instantaneous angular position of the tooth j of the cutter. Later, Sabberwal and Koenigsberger [9] related the cutting force and chip area mechanistically and calculated the instantaneous cutting force as; c jF K ah\u00EF\u0080\u00BD (2.2) where cK is the cutting force coefficient, a is the depth of cut and jah represents the chip area. The specific cutting coefficient cK , is obtained experimentally for the workpiece material, and 10 tool geometry pair. Kline et al. [10] applied the mechanistic model of milling and expressed the chip load, and cutting force relationship. Armarego and Epp [11] developed the linear edge cutting force model of milling which separates the cutting and edge forces and expressed the total cutting forces as; c c j eF K ah K a\u00EF\u0080\u00BD \u00EF\u0080\u00AB (2.3) where eK represents the specific edge force coefficient resulting from the static ploughing mechanism of the tool flank face. Experimental tests revealed that the total cutting forces consist of shearing forces and tool-edge forces which can be calculated from the y-intercept of the measured forces at zero feed. They also applied the shear stress and shear angle found by simple orthogonal cutting to the more complex milling process to predict the milling forces analytically using orthogonal to oblique transformation. The mechanistic milling model is improved by including the dynamic deflections of the flexible cutter or workpiece. Tlusty [12] and Sutherland et al. [13] included the cutter or workpiece flexibilities in the chip thickness model and developed a more accurate cutting force estimation for the milling process. These studies also yield the prediction of surface location error and calculation of the true (exact) chip thickness in flexible end milling operations. Sutherland [14] and Montgomery et al. [15] modeled the dynamic chip thickness, thus cutting forces by discretizing the tool and workpiece geometry, kinematics and vibrations. The model presented by Montgomery and Altintas [16] is used to predict the exact chip thickness and time delay in the regenerative chip model in this thesis. The mechanistic model of milling explained in aforementioned studies, is based on the cutting and edge force coefficients which are determined empirically for each cutter geometry. There 11 needs to be numerous experimental tests to apply the specific force constants for different cutter geometry which is not practical and labor expensive. Budak et al. [17] proposed a novel model to predict the force coefficients from the orthogonal cutting data that eliminates the need for experimental milling tests for each cutter geometry. The model uses the shear stress, shear angle and friction coefficient stored in the orthogonal cutting database which are then transformed into oblique cutting geometry. Although it is time consuming to prepare the orthogonal database, it is highly flexible to predict the force coefficients for different cutter geometries and cutting processes. Researchers used the orthogonal database and mechanistic model to predict the cutting forces for ball end mills [18-20], face mills [21], and general milling cutters [22, 23]. In this thesis, both the mechanistic and orthogonal to oblique transformation models are used to predict the force coefficients for different type of cutter geometries. 2.3 Dynamics and Stability of Turning and Milling Operations The stability of one dimensional turning-boring operations have been extensively studied in the past. Pioneer researchers Tobias et al [24] and Tlusty et al. [25] showed the regeneration mechanism due to the relative vibrations between the flexible tool and workpiece, and proposed frequency domain stability models. Later, Merrit [26] modeled the regeneration mechanism by a feedback control block, and used Nyquist law to predict the stability limits. Tlusty [27] and Tobias [28] investigated the non-linear behavior of cutting in time domain when the tool jumps out of the cut due to excessive vibrations. Minis and Yanushevsky [29] modeled the dynamics of milling by two sets of coupled delayed differential equations with periodic coefficients. They used Floquet Theory to consider the effect of spindle speeds, and solved the stability using Nyquist theory. Altintas and Budak [30] averaged the time periodic directional factors, and solved the stability in frequency domain analytically for milling operations. This solution technique, also known as zero-12 order solution, is a fast and accurate method to predict the stability limits in milling operations with high radial immersions. On the other hand, since the cutting forces are time-varying, taking only the average component of the directional factors may not provide high accuracy compared to the time domain solutions. Insperger and Stepan [31, 32] proposed the semi discretization method which efficiently digitizes the system equations in small time intervals within the time period of the system. The main idea in the semi discretization method is that only the delayed states are discretized and approximated by piecewise constant functions while the actual time domain (non-delayed) terms are left in the original form. In a more recent study, Eksioglu et al. [33] approximated the delayed states by high order Lagrange interpolation technique and predicted the stability, cutting forces, vibrations and dimensional surface errors. 2.3.1 Dynamics and Stability of Low-Immersion Milling Operations As the radial immersion of the cutting reduces, the zero-order stability solution which takes only the average of directional factors per tooth period fails to predict the stability limits accurately. Insperger and Stepan [34] modeled the highly interrupted milling dynamics in time domain and also identified the new unstable regions associated with the period-doubling vibrations. Davies et al. [35] coupled the non-cutting (free) vibrations with the forced vibrations of one-dimensional cutting, and mapped them to solve the stability of low immersion milling operations. Budak and Altintas [36] and Merdol and Altintas [37] solved the stability by considering the effect of spindle speed in their multi-frequency domain solution. Later, Insperger et al. [38] investigated the stability and corresponding chatter frequencies in highly interrupted cutting in time domain. As opposed to the turning process where there is single chatter frequency at the unstable region of cutting, they identified four types of frequency sets in milling process: For both the stable and unstable region of cutting, the tooth passing frequency and its higher harmonics, and also the 13 damped natural frequency of the most dominant mode of the system are seen. In the unstable region, two more different frequency sets are observed; Hopf bifurcation frequency and period-doubling (flip bifurcation) frequency. Mann et al. [39] modeled the limit cycle behavior of highly interrupted one dimensional cutting processes and solved the stability by time finite element analysis Later, Gradisek et al. [40] solved the milling stability with the semi discretization method and showed the additional type of instability, period-doubling bifurcation, by simulations and experiments. In-process tool deflection data proved that the chatter can occur periodic and quasi-periodic depending on the type of instability. 2.3.2 Stability of Milling Operations at Low-Speed Process Damping Region At low spindle speeds, shorter undulation waves are generated on the workpiece surface and the contact between the flank face of the tool and vibration waves increase which in turn causes more material to be ploughed by the tool flank. The increased ploughing forces stabilize the process by dissipating vibration energy in the chip thickness direction. Then, the dissipated energy by ploughing forces create an equivalent viscous damping in the system, thus increasing the process stability. The first study reported in the literature by Albrecht [7] states that the additional chip contact mechanism is effective in cutting which is named as ploughing of the material by the tool flank face. In their study, the static edge forces on the wear land of the flank face are considered and the force diagram established by Merchant [3] is reevaluated. Later, Das and Tobias [41] stated that the effect of process damping increases proportionally to the magnitude of total cutting forces in tangential direction, vibration frequency and inversely proportional to cutting speed. As cutting speed is reduced, process damping increases up to a point beyond which chatter can no longer occur at any large width of cut. This phenomenon is known as asymptotic spindle speed and led to more recent studies. Sisson and Kegg [42] explained the physics under the 14 dynamic ploughing mechanism by modeling the additional damping force as a function of tool flank geometry, contact geometry and cutting forces, then derived the expression for the process damping coefficient. Later, Tlusty and Ismail [43] presented the waviness of regeneration mechanism and vibration amplitude and related these quantities to the source of process damping. They compared the clearance angle and slope of the vibration wave left on the surface and stated that up to a certain value of wavelength, the system exhibits regular thrust forces but as the slope increases at low cutting speeds and high frequencies, increased thrust force generates additional damping. Wu [44, 45] developed an elastic-plastic stress field in the tool flank-workpiece contact zone which was then used for an analytical expression that relates the contact forces between the tool flank and workpiece to the total volume of displaced material under the tool. The displaced volume varies with tool vibration and changes the instantaneous cutting direction. Elbestawi et al. [46] improved the model of Wu [44] by considering the phase relationship between the ploughing force and tool vibration and proved that the ploughing force acts in the opposite direction of tool vibration, thus increasing the damping in the dynamic cutting system. In a similar way, Lee et al. [47] and Endres et al. [48] modeled the process damping force as the work material volume pressed by the tool. They discretized the indentation volume at small discrete volumes and integrated over the boundary of tool flank and wavy surface. The tool vibrations which are needed to define the exact contact boundaries are calculated by solving the system equations in an iterative way. Shawky and Elbestawi [49] proposed the process damping effect on the turning process. They decomposed the ploughing forces as static and dynamic components. The static part of the ploughing forces is due to the tool edge radius and feed motion. On the other hand, the dynamic part of the ploughing is due to the surface undulations which are caused by tool deflection or vibration. Chiou and Liang [50] incorporated the dynamic ploughing forces into the chatter 15 stability of turning for the first time in the literature. Process damping coefficient is solved analytically by calculating the indentation volume under the assumption of small vibration amplitudes of the system. Huang and Wang [51] applied the same process damping theory into the dynamic milling process and showed the effect of cutting parameters such as cutting speed, feed, axial-radial depths on the process damping. Altintas et al. [52] identified the process damping coefficients from a controlled orthogonal cutting tests with a fast tool servo which oscillates at desired frequency and amplitude and they solved the stability with process damping by Nyquist law. Parallel to the increased application of process damping phenomena in metal cutting industry especially for difficult-to-cut materials in aerospace and die and mold applications, the necessity for an improved process damping model has arisen. The existing models failed to predict the chatter stability limits especially for milling operations where the vibration amplitude is large and assumptions made by previous researchers to calculate the approximate indentation volume were not valid for such cases. Therefore, the recent researchers aimed to model the indentation volume more accurately by more advanced modeling techniques. Tunc and Budak [53] identified the process damping coefficients directly from experiments and used them to model process damping coefficient through the energy balance in the indentation region. The indented volume is calculated numerically by discretizing the tool flank face and vibration wave and integrate over the contact region for different cutting conditions and tool geometries. The stability at the low speed region is solved in time domain and validated by experimental chatter tests. Ahmadi and Altintas [54] modeled the chatter stability of milling by using the semi discretization method. The process damping coefficients are identified by the frequency domain decomposition method which 16 evaluated the vibration signals measured at two locations of the tool during stable orthogonal cutting. Nonetheless, all the aforementioned process damping models are modeled considering certain assumptions. First, the vibration amplitude is assumed constant at the stability border which is hard to estimate analytically. Another important assumption is the separation angle where the material starts to flow under the tool flank face. Although there are several methods to identify these two quantities, their behavior is highly unpredictable during the dynamic cutting process. Therefore, even under certain assumptions, the predicted stability limits at low speed regions deviate considerably from the experimentally identified stability regions. In a most recent study, Wang et al. [55] proposed a process damping model based on two sets of experiments at chatter conditions in low speed region. The asymptotic spindle speed where the stability limit goes to infinity, and absolute stability limit of the cutting process are analytically calculated by decomposing the eigenvalue equation of the system under certain conditions. Then, the stability envelope at the low speed region is constructed in an efficient way without identifying process damping coefficients, vibration amplitude, and separation angle. In this thesis, the same algorithm is applied to determine the low speed stability envelope in turn-milling process and its effectiveness is compared with the small vibration assumption method [44]. 2.3.3 Stability of Milling Operations with Asymmetric Cutter Dynamics The stability properties of the systems where the asymmetric tool or workpiece flexibilities are dominant, are different than the regular milling operations. Present stability models for regular milling and turning predict the stable and unstable regions assuming that the structural dynamics of the system remain constant at fixed directions and within the operating speed range of the spindle. On the other hand, some spindles cannot maintain the stiffness of bearing at a fixed value 17 due to thermal expansion, centrifugal, and gyroscopic speed effects [56]. Frequently, end mills with two flutes, which have asymmetric dynamic flexibilities in two orthogonal directions [57] are used in high speed milling operations. Li et al. [58] showed that rotating cutter dynamics has significant effect on chatter stability and compared their solution against fixed frame stability solution considering rotating boring bar. Eynian and Altintas [59] solved stability in rotating coordinates using Nyquist stability criterion at low speeds dominated by process damping. Similarly, Comak et al. [2] proposed a time domain stability method by transforming the equation of motion from fixed frame to rotating frame. It is demonstrated that the stability pockets differ significantly when the rotating dynamics of the asymmetric tool are considered. 2.3.4 Stability of Milling with Multiple and Variable Time Delays The aforementioned studies for the solution of milling stability consider the constant spindle speed of tool or workpiece where the corresponding delay in the regeneration mechanism is also constant and time invariant. On the other hand, the cutting operations having variable or multiple time delays are also investigated in the literature. Cutting tools with variable helix and variable pitch angles can be used for improving the stability of the milling process. Variation in the tooth spacing alters the delay in the cutting system disturbing regeneration mechanism. Depending on the type of the variation (alternating or linear) there might be multiple time delays in the regenerative system. The effectiveness of variable pitch cutters in suppressing chatter vibrations in milling was first demonstrated by Slavicek [60]. Assuming an alternating pitch variation, the stability limits are expressed as a function of variation in the pitch angles. Later, Vanherck [61] and Opitz et al. [62] showed that the significant increase in the stability limits can be obtained by variable pitch milling tools. Altintas et al. [63] and Budak [64] proposed optimization methodologies which design the variable pitch tools accurately to 18 increase the chatter stability limits. In the case of variable helix cutters, the time delay is not constant not only between the flutes but also along the tool axis. Sims et al. [65, 66] investigated the chatter stability of variable helix and pitch cutters analytically, and proposed an optimization methodology by comparing different modeling approaches. In a more recent study, Comak and Budak [67] proposed a design guideline for variable pitch and helix cutters to select the best variation, thus time delay combinations to maximize the chatter free material removal rate. Similar approach to variable pitch / helix tools are applied to suppress the chatter vibrations by altering the time delay in dynamic system. Spindle speed modulation is a commonly used technique which continuously changes the spindle speed, thus the time delay, and increases the stability limits. Sexton et al. [68] presented a method of spindle speed variation for single point cutting. Later, Insperger and Stepan [69] applied the semi discretization method to solve the stability of varying spindle speed turning with varying time delays for different modulation frequencies and amplitudes. Different sources of non-constant time delay are investigated as well. Long et al. [70] proposed a dynamic milling model with variable time delay due to the feed motion of the tool and solved the stability of the system in time domain. The variable time delay amplitude was dependent on the ratio of feed to tool radius, and it is found that the model with variable time delay is not much different than the constant time delay model of milling, thus the effect of feed motion on the time delay of the system was neglected. Similarly, Campomones and Altintas [71] presented the chip thickness model considering additional delay caused by the large feed rate. Due to the additional time delay, the regenerative chip thickness is disturbed and as a result, the stability lobes shifted towards higher speeds in their simulations. 19 Finally, existence of multiple delays has been studied in the literature of parallel machining operations. Budak et al. [72], [73] presented the multiple time delays in the stability solution of parallel milling and parallel turning operations. Since the turning and milling tools share the same flexible workpiece, their dynamics, thus the individual time delays are coupled, and affect the regenerative chip thickness mechanism. 2.4 Kinematics, Mechanics and Dynamics of Turn-Milling Operations Despite the well-established research regarding the mechanics and dynamics of milling and turning processes, the turn-milling process is a relatively new technology, thus there are limited works reported in the turn-milling literature. The preliminary studies in turn-milling were only based on experiments. Schulz and Spur [74] proposed two different turn-milling operations, namely coaxial and orthogonal, and carried out experimental trials to identify cutting conditions (i.e. feed, speeds, depth of cut) that result in good surface finish. In another work, Schulz and Kneisel [75] showed the effect of turn-milling process parameters on the tool life and surface quality experimentally. Kopac and Pogacnik [76] studied the influence of cutting conditions on the roughness and compared the results for centric and eccentric turn-milling operations. Similarly, Coudhury and Bajpai [71, 77] and Savas and Ozay [78] presented experimental work to explain the effect of cutting conditions on the surface finish for orthogonal turn-milling and tangential turn-milling operations. All of these studies were based on experimental observations without analytical modeling of the turn-milling cutting process. Zhu et al. [79] proposed a mathematical model to predict the machined surface topography in orthogonal turn-milling, and proposed simple guidelines on the selection of cutting parameters for increased productivity and surface finish. Later, Karaguzel et al. [80] proposed an analytical model to evaluate cutter-workpiece engagement (CWE) for simple tool-workpiece geometries and toolpaths. Also, the effect of ratio of milling and 20 workpiece spindles speeds on the chip thickness, cutting forces and the machined part surface quality has been discussed. They presented analytical expressions to model the surface quality of the machined part by considering the kinematics and static cutting forces for a turn-milling system. The form errors are described as circularity error between the desired and actual workpiece envelope, static cusp height and the circumferential surface roughness of workpiece. Qiu et al. [81] modeled the cutting forces in orthogonal turn-milling for round insert cutters, and expressed the CWE geometry from the discrete tool and workpiece positions. In an another work, Qiu [82] considered the forces generated from the bottom edge of the milling cutter. The cutting force coefficients are identified from the plunge milling experiments and total cutting forces are calculated considering both the side and bottom edges of the tool. Karaguzel et al. [83] showed the effect of tool axis offset on the cutting force, surface quality and tool wear. The surface quality was mathematically calculated and tool wear tests have been conducted to show the tool axis effect on the process outcomes. However, the stability of turn-milling has not been studied in the literature. 2.5 Summary All the relevant literature in turn-milling process modeling is based on experimental studies or analytical modeling of chip thickness using simple tool geometry and toolpath, thus is restricted only for certain types of turn-milling operations (i.e. orthogonal turn-milling). The cutter workpiece engagement geometry which is used as the boundary condition for the chip thickness model is expressed by geometrical relationships between tool and workpiece. Therefore, the application of the existing chip thickness theories to different toolpaths (B-axis cutting) and turn-milling operations is not possible. On the other hand, the complete kinematic analysis of the turn-milling machine tool is needed to transform displacements, velocities and vibrations from tool -21 workpiece interaction zone to machine coordinates. In the literature, the kinematic analysis of the turn-milling has never been studied. Also, the dynamics and stability of turn-milling operations have never been worked in the literature before, due to its complex regenerative chip mechanism. This thesis presents a generalized chip thickness model that is applicable to any toolpath and tool-workpiece geometries, and kinematic model of turn-milling machine tool. For the first time in the literature, the dynamics and stability model of turn-milling is proposed. The surface location error model is presented based on the mechanics and dynamics of the turn-milling process. Finally, the most productive turn-milling process parameters are modeled based on the constraints of the process and machine tool. 22 Chapter 3: Mechanics of Turn-Milling Processes 3.1 Introduction The prediction of the static cutting forces and resulting cutting torque and power is one of the essential goals of the virtual model of the turn-milling process. The existing work in the literature is restricted to modeling the static chip thickness and cutting forces for certain types of cutter geometries and simple toolpaths. On the other hand, when the milling tool that is attached to the tool spindle follows a curved path on the workpiece mounted on the rotating chuck, the chip mechanism becomes complicated due to the kinematics and tool-workpiece interaction. Hence, the static chip thickness and cutting forces must be modeled accurately instead of resorting to costly machining trials. In this chapter, a general mathematical model of the turn-milling mechanics is presented to predict chip thickness, cutting forces, torque and power which are needed for machine design and process planning aspects of the machining operations. First, the kinematic model of the turn-milling machine tool is presented. Then, the cutting forces acting on the tool and workpiece are predicted as a function of the instantaneous chip geometry which may continuously vary as a function of all five axes positions and velocities, as well as the cutter-workpiece engagement (CWE) and angular rotational speeds of tool and workpiece spindles. The discrete cutting forces acting on the differential cutting edge are projected to tool and machine tool coordinates by the kinematic model of the machine tool. The simulated cutting forces are compared against the experimental data for different workpiece materials. 23 3.2 Kinematics of Turn-Milling A typical turn-milling machine center consists of three linear \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,X Y Z and two rotary drives \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,B C and is capable of performing five-axes turn-milling operations. The linear translational drives are located on the tool spindle which can also be tilted by the rotary B-axis and the workpiece can be rotated at constant speed or indexed at a specified orientation by the C-axis of the machine as shown in Figure 3.1. Thus, a typical turn-milling machine is considered as the hybrid configuration of the five-axis machine tools [84]. When all five drives operate for turn-milling of a curved part, the chip geometry changes as a function of machine\u00E2\u0080\u0099s kinematic configuration, angular speeds and the geometries of the tool and workpiece. Therefore, the kinematics of the turn-milling process must be modeled prior to modeling the chip thickness and cutting forces. Figure 3.1 Configuration of turn-milling machine tool and kinematics of the process. 24 In the turn-milling of a part with a cylindrical sculptured surface, the toolpaths are generated by the CAM systems which indicate the tool tip position \u00EF\u0080\u00A8 \u00EF\u0080\u00A9P and orientation \u00EF\u0080\u00A8 \u00EF\u0080\u00A9O at each cutter location (CL) in the workpiece coordinate system (WCS). The location and orientation of the tool tip at each discrete position are given by GOTO commands of the Automatically Programmed Tools \u00E2\u0080\u0093 Cutter Location file (APT-CL) which is represented for a sample case as follows: GOTO/ P , P , P , O , O , Ox y z i j k The first three columns represent the tool tip position \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y zP P P\u00EF\u0080\u00BDP and the last three columns indicate the tool tip orientation \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,i j kO O O\u00EF\u0080\u00BDO in WCS as shown in Figure 3.2. The corresponding position commands of the linear and rotary drives of the turn-milling machine tool is evaluated by the inverse kinematics model of the machine tool. The inverse kinematics model solves the Cartesian and rotary axis reference commands of the five drives \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , , ,Tb cx y z \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 from the given desired configuration of the tool tip in the workpiece coordinate system. In this thesis, two different inverse kinematics models are applied for the solution of the rotary axes reference commands \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,b c\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 which are the motion B and C rotary axes of the turn-milling machine tool. First, Screw Theory [85] is used to determine the inverse kinematics of the rotary axes and the results are compared against the well-known Denavit-Hartenberg (D-H) model [86]. 3.2.1 Screw Theory Screw Theory provides geometrical description of rigid body motion where all the translational and rotational motions are defined with respect to the base frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9BO as opposed to the D-H model where the joint kinematics are defined for each local coordinate frame on the drives. Since all the translational and rotational motions are described with respect to the base (inertial) frame 25 in Screw Theory, it does not suffer from numerical ill conditions or singularities where it is likely for the D-H model due to the use of local coordinate frames. Also, Screw Theory reduces the number of matrices to be multiplied for the kinematic solution of a system, thus it is efficient to implement for the kinematic modeling of the 5-axis machine tools. According to the Chasles\u00E2\u0080\u0099 Theorem [85], any rigid body motion can be represented as a rotation around an axis combined with a translation along the same axis and such a motion is called a screw motion. In Figure 3.2, \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z are the motions of the prismatic joints which are the X, Y and Z drives of the machine. Similarly, b\u00EF\u0081\u00B1 and c\u00EF\u0081\u00B1 represents the rotary motions of the revolute joints which corresponds to the B and C rotary drives of the machine. \u00EF\u0080\u00A8 \u00EF\u0080\u00A9x y zv , v , v and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,b c\u00CF\u0089 \u00CF\u0089 are the unit vectors that point the positive direction of the translational and rotational axes, respectively. bq and cq are the points on the corresponding rotary axes. The coordinate frames of the base \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,M M MX Y Z , tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,t t tx y z and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,w w wX Y Z are also shown in Figure 3.2. Since the cutting tool and workpiece are assembled at different joints, two open kinematic chains can be constructed from the base frame to the workpiece and tool frames. The workpiece frame starts from the base frame and ends up at the workpiece. Similarly, the tool chain also starts from the base frame and goes to the cutting tool. When these two open kinematic chains are assembled together, they make up the full closed kinematic chain of the machine tool. In Figure 3.2, the workpiece and tool chains are shown for a turn-milling machine tool where the C-axis is on the workpiece and the B-axis is on the tool side. The workpiece chain is from the base frame to the C-axis and the workpiece. The tool chain starts from the base frame and moves along the X, Y and Z drives of the machine, then rotary B-axis and finally to the cutting tool. 26 Figure 3.2 Kinematic chain of the turn-milling machine tool. Considering a rigid body motion consisting of rotation about an axis by \u00EF\u0081\u00B1 radians plus translation along the same axis, the transformation between the thn joint of the machine and the base frame is expressed by the twists \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B8 and rotation \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B1 as [87]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1 2 2 0n nbn n bng e e e g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0080\u00BD (3.1) where the twist can be expressed for the thi revolute joint as [85]; ,ii i i ii\u00EF\u0081\u00B8\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0082\u00B4\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BBvv \u00CF\u0089 q\u00CF\u0089 (3.2) where i\u00CF\u0089 is the unit vector of the thi revolute joint and iq is a point on the rotation axis of the screw. Note that, the unit vector for the revolute joints of the turn-milling machine tools are 27 expressed as \u00EF\u0081\u009B \u00EF\u0081\u009D0 1 0Tb \u00EF\u0080\u00BD\u00CF\u0089 and \u00EF\u0081\u009B \u00EF\u0081\u009D0 0 1Tc \u00EF\u0080\u00BD\u00CF\u0089 for the B and C axes, respectively. On the other hand, the twist for a prismatic joint (pure translation) can expressed as; 0ii\u00EF\u0081\u00B8\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BBv (3.3) Then, the transformation matrix for thi revolute joint can be written as [85]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A93 31 300 1i i i ii iTi i i i i iie ee\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0082\u00B4\u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD \u00EF\u0082\u00B4 \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0082\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00CF\u0089 \u00CF\u0089I \u00CF\u0089 v \u00CF\u0089\u00CF\u0089 v\u00CF\u0089 (3.4) where the exponential of the \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i i\u00EF\u0081\u00B1\u00CF\u0089 is given as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A923 3 sin 1 cosi i i i i ie\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0082\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00CF\u0089I \u00CF\u0089 \u00CF\u0089 (3.5) The detailed derivations of this formula (Rodrigues\u00E2\u0080\u0099 formula) are given in [85] and explained in Appendix A.1. Similarly, the transformation matrix for the prismatic joint can be easily obtained by substituting 0i \u00EF\u0080\u00BD\u00CF\u0089 in Eq. (3.4). The homogenous transformation matrix of the workpiece chain can be written as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 10 andc cbw c bw wb bwg e g g g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD (3.6) where c\u00EF\u0081\u00B8 and c\u00EF\u0081\u00B1 correspond the twist and motion commands of the rotary C-axis on the workpiece chain from base frame to workpiece frame. Similarly, the transformation matrix of the tool chain can be defined as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , , 0y yx x z z B Bbt b btg x y z e e e e g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0080\u00BD (3.7) Note that the bwg and btg are the rigid body transformations of the workpiece and tool coordinate frames relative to the base frame in the reference configuration which is defined by the 28 offset vectors in both coordinate frames \u00EF\u0080\u00A8 \u00EF\u0080\u00A90 0 0 0 0 0 0 0 and T Tx y z x y zw w w\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBw t t t t relative to the base frame as [87]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A93 3 0 3 3 01 3 1 30 ; 00 1 0 1bw btg g\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBI w I t (3.8) Since the workpiece and tool chains are analytically expressed, the full kinematic chain of the turn-milling machine tool can be obtained by combining the Eq. (3.6) and (3.7) as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A91, , , , 0 0y yc c x x b bz zwt c b bw btg x y z g e e e e e g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BD (3.9) Let the tool position and orientation vector relative to tool coordinate frame are given by ptr and otr vectors, the forward kinematics of the turn-milling machine tool can be expressed as [87]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 3 1 3 13 1 3 1 , , , ,1 0 1 0pt otwt c bg x y z\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBr rP O (3.10) where P and O vectors represents the tool\u00E2\u0080\u0099s position and orientation relative to the workpiece frame. Once the forward kinematics are solved, the inverse kinematics model can be presented to find the reference position commands of five drives \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , , ,b cx y z \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 which achieve the given configuration of tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,P O . It is proven in [87] that the prismatic joints do not change the orientation vector regardless of the position and order in Eq. (3.9). Thus, the terms having prismatic joints are removed from the Eq. (3.9) and the reduced matrices yield the forwards kinematics solution for the orientation vector which can be given as; 3 13 10 0c c b bote e\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0082\u00B4\u00EF\u0082\u00B4 \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBrO (3.11) 29 The revolute joint angles b\u00EF\u0081\u00B1 and c\u00EF\u0081\u00B1 are solved for given tool orientation O at each CL point by the Paden-Kahan sub-problem 2 [85] where the detailed derivations are given in Appendix A.2. For the inverse kinematics solution, let the ot\u00EF\u0080\u00BDu r and \u00EF\u0080\u00BDv O , then three coefficients are calculated as [85]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A91 22 22 21 2 1 23 2111 2T T Tc b b cTc bT T Tc b c bTc bTc bc bkkk k k kk\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0082\u00B1\u00EF\u0082\u00B4\u00CF\u0089 \u00CF\u0089 \u00CF\u0089 u \u00CF\u0089 v\u00CF\u0089 \u00CF\u0089\u00CF\u0089 \u00CF\u0089 \u00CF\u0089 v \u00CF\u0089 u\u00CF\u0089 \u00CF\u0089\u00CF\u0089 \u00CF\u0089\u00CF\u0089 \u00CF\u0089 (3.12) Unit vectors of the rotation axes of B and C drives \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,c b\u00CF\u0089 \u00CF\u0089 and their vector products \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c b\u00EF\u0082\u00B4\u00CF\u0089 \u00CF\u0089 are linearly independent, then a new variable z can be defined as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 2 3c b c bk k k\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0082\u00B4z \u00CF\u0089 \u00CF\u0089 \u00CF\u0089 \u00CF\u0089 (3.13) Finally, the rotation angles of B and C drives of the turn-milling machine for given CL data are solved by applying the Paden-Kahan sub-problem 1 [85] as follows; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9' ' ' ' ' '1 1 1' ' ' ' ' '2 2 2atan2 , ,atan2 , ,T T T Tc c c c c cT T T Tb b b b b b\u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0082\u00B4 \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0082\u00B4 \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00CF\u0089 z v z v z z \u00CF\u0089 \u00CF\u0089 z v v \u00CF\u0089 \u00CF\u0089 v\u00CF\u0089 u z u z z z \u00CF\u0089 \u00CF\u0089 z u u \u00CF\u0089 \u00CF\u0089 u (3.14) Note that, the coefficient 3k satisfies two different values (both positive and negative sign). Hence, there exist two b\u00EF\u0081\u00B1 and c\u00EF\u0081\u00B1 for the same CL data. The correct sign is chosen to guarantee the continuity of the toolpath and shorter joint movement. 30 The inverse kinematics solution for the prismatic joints \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z can also be solved. The forward kinematics solution for the prismatic joints can be written by reducing the orientation vector from Eq. (3.10) as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A93 1 3 113 1 3 34 40, , , , 0 01 0 11 1c c b bpt ptpwt c b bw btg x y z g e I e g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0080\u00AD\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0080\u00AD\u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBr rP S (3.15) where the prismatic joints\u00E2\u0080\u0099 reference commands are given as: Tp x y zS S S\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u00AB \u00EF\u0083\u00BBS . Then, Eq. (3.15) can be rewritten and the reference position commands of the prismatic joints are calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 3 13 10 01 1 1c c b bptpbw bte g e g\u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0081\u00B8 \u00EF\u0081\u00B1 \u00EF\u0082\u00B4\u00EF\u0082\u00B4\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBrS P (3.16) 3.2.2 Denavit-Hartenberg Kinematic Model Denavit-Hartenberg (D-H) convention is de-facto in robotics and machine tool kinematic analysis. It describes the kinematic relations between the drive modules by assigning local coordinate frames to each prismatic and revolute joints of the machine tool. The homogenous transformation matrix between thi and \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thi \u00EF\u0080\u00AB drive module is written in a general form as; , , , , , ,, , , , , ,, 1, ,4 4cos sin cos sin sin cossin cos cos cos sin sin0 sin cos0 0 0 1b i b i c i b i c i i b ib i b i c i b i c i i b ii ic i c i iaad\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0080\u00AB\u00EF\u0082\u00B4\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BBT (3.17) where ,b i\u00EF\u0081\u00B1 , ,c i\u00EF\u0081\u00B1 , ia , id are the rotational and translational parameters for the thi drive. For instance, the homogenous transformation matrix for B and C axes where only the pure rotational motion exists around Y and Z axis, respectively, can be expressed as; 31 cos 0 sin 0 1 0 0 00 1 0 0 0 cos sin 0;sin 0 cos 0 0 sin cos 00 0 0 1 0 0 0 1b bc cB Cb b c c\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBT T (3.18) where b\u00EF\u0081\u00B1 and c\u00EF\u0081\u00B1 are angular rotations around B and C drives, respectively. The transformation matrices are assigned to each local prismatic drive module in a similar way. Overall forward and inverse kinematics can be solved by the multiplication of local transformation matrices from workpiece coordinate frame to tool coordinate frame. The inverse kinematics for the rotary drives of the turn-milling machine can be written as; , ,B i C i tooli \u00EF\u0080\u00BDT T O O (3.19) which can be written in the matrix form as; ,,,cos 0 sin 0 1 0 0 0 00 1 0 0 0 cos sin 0 0sin 0 cos 0 0 sin cos 0 10 0 0 1 0 0 0 1 1 1b b x ic c y ib b c c z iOOO\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB (3.20) The reference position commands for the B and C axis of the machine then can be found by solving the Eq. (3.20) for b\u00EF\u0081\u00B1 and c\u00EF\u0081\u00B1 as [89]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2atan2 ,atan2 ,b i j kc j kO O OO O\u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AD (3.21) Note that, the workpiece axis rotates at constant speed, thus the value of the C-axis reference position command \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u00B1 will be 0 or 2\u00EF\u0081\u00B0 for the turn-milling operations. Figure 3.3 (a) shows a sample toolpath of a turn-milling process which is modeled by a commercially available software, NX CAM\u00C2\u00AE [88]. Tool normal axis \u00EF\u0080\u00A8 \u00EF\u0080\u00A9O continuously changes 32 by the B-axis rotational motion of the spindle head at each discrete cutter location. The APT-CL file is then obtained for the given tool path as seen in Figure 3.3 (b). At each line of the CL file, the position \u00EF\u0080\u00A8 \u00EF\u0080\u00A9iP and orientation \u00EF\u0080\u00A8 \u00EF\u0080\u00A9iO of the tool tip can be read. Then, the reference position commands of the revolute and prismatic joints \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , , ,b c x y z\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 , are calculated by the inverse kinematics solution of the Screw Theory by solving the Eq. (3.12) - (3.14). The position command for the rotary B-axis of the machine \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00B1 is plotted for each discrete CL point as shown in Figure 3.3 (c). Finally, the difference between the Screw Theory and D-H inverse kinematics solution are compared in Figure 3.3 (d). The reference position command for the B axis is calculated by 162.2 10\u00EF\u0080\u00AD\u00EF\u0082\u00B4 [rad] error between the two models. Figure 3.3 Sample toolpath for a turn-milling process (a), obtained APT-CL file (b), inverse kinematic solution of the rotary B-axis of the machine (c) and comparison of the results with D-H kinematic model. 33 The inverse kinematics solution will be used to transform the predicted cutting forces and vibrations at the tool-workpiece interaction zone to the tool and machine coordinate frame in the following sections. 3.3 Mechanics of Turn-Milling In this section, the cutting forces acting on the tool and workpiece are predicted as a function of the instantaneous chip geometry. In turn-milling processes, the instantaneous chip area may continuously vary as a function of all five axes positions, velocities, as well as the cutter-workpiece engagement (CWE) geometry due to the kinematics, and the geometry of the process and toolpath. First, the distribution of chip thickness along the tool\u00E2\u0080\u0099s cutting edge is modeled by considering the kinematics of the turn-milling machine, CWE geometry, and speeds of tool and workpiece spindles. Then, the cutting force model is explained in detail and the proposed mechanics model of turn-milling process is validated experimentally. Although any tool geometry can be handled by the generalized model [89], a ball end mill is used to illustrate the prediction of the process mechanics. The tool is divided into differential axial disc elements with a height of dz so that the effect of local ball radius, helix angle and rake angle can be considered in modeling the chip thickness. 3.3.1 Feed Vectors and Static Chip Thickness Distribution The cutting tool is positioned at any orientation by three linear (X, Y, Z) and two rotary (B, C) drives of the machine tool as explained in Section 3.2. The tangential feed, which is the linear travel velocity of the tool, varies along the tool axis as a function of both linear and rotary drive motions similar to the five-axis milling process. The machine is programmed with a feedrate of c [mm/rev/tooth], but the chip thickness distribution changes as the tool rotates and drives move simultaneously. The toolpath is programmed in Workpiece Coordinate System (WCS) while the 34 controller uses the Machine Coordinate System (MCS). In the beginning of the workpiece set-up, WCS is assumed to be aligned with MCS. Tool\u00E2\u0080\u0099s tip travels from discrete cutter location (CL) \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,i i iX Y Z\u00EF\u0080\u00BDiP to \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1 1, ,i i iX Y Z\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0080\u00BDi+1P on the rotating part with a linear feed of tf [mm/min] while the tool axis is oriented by the rotary B-axis from \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , , \u00CB\u0086\u00CB\u0086 \u00CB\u0086, ,x i y i z iO i O j O k\u00EF\u0080\u00BDiO to \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, 1 , 1 , 1 \u00CB\u0086\u00CB\u0086 \u00CB\u0086, ,x i y i z iO i O j O k\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0080\u00BDi+1O leading to an angular feed of Af [rad/min] as shown in Figure 3.4. Figure 3.4 Multi-axes feed motion in turn-milling process. The linear feed vector Lf is composed of the tool\u00E2\u0080\u0099s linear feed tf and the projected workpiece linear feed Wf . The magnitude of the linear feed vector, Lf is evaluated as follows; 2 2( ) ( )L t wf z f f z\u00EF\u0080\u00BD \u00EF\u0080\u00AB (3.22) Tool linear feed is defined in the NC program of the process at the tool tip as; t t tf N c\u00EF\u0080\u00BD \u00EF\u0081\u0097 (3.23) 35 where tN is the number of teeth on the tool and t\u00EF\u0081\u0097 [rev/min] is the rotational speed of the tool. Eq. (3.23) is the total feedrate in the regular milling process. On the other hand, in turn-milling process, the rotational motion of the workpiece (C-axis) also contributes to the feed motion. The tool plunges into the workpiece with a depth of cut \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a as shown in Figure 3.5 (a). While each axial disc element of the tool is ( )p z away from the tool tip, it is also ew,iR away from the workpiece rotation axis at CL point i where the workpiece radius is w,iR . The ew,iR is the effective workpiece radius at CLi and varies along the toolpath. Since the workpiece is mounted on the C-axis which rotates with an angular velocity of c\u00CE\u00A9 [rad/min], it further contributes to the feed as; ( )wf z \u00EF\u0080\u00BD \u00EF\u0082\u00B4ec w,i\u00CE\u00A9 R (3.24) where the effective workpiece radius is calculated as; ( )2ap z\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8ew,i w,iR R (3.25) Note that, the ( )p z is the axial height of individual discrete element from the tool\u00E2\u0080\u0099s bottom end. Figure 3.5 Tool axial discretization and effective workpiece radius calculation (a), Linear workpiece feed representation (b). 36 Since the magnitude of the linear feed vector is calculated, the direction of the linear feed vector should be examined as well. The linear feed direction is found in Tool Coordinate System (TCS) from the two consecutive positions of tool in MCS. While the tool travels by i,t\u00CE\u0094S from location iP to i+1P along the toolpath, the workpiece also rotates by \u00EF\u0081\u0084 i,w\u00CE\u00B8 during the time interval it\u00EF\u0081\u0084 as (see Figure 3.5 (b)); \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 it\u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0082\u00B1 \u00EF\u0081\u0084i,w c\u00CE\u00B8 \u00CE\u00A9 (3.26) where the elapsed time it\u00EF\u0081\u0084 can be calculated as; iL Ltf f\u00EF\u0081\u0084\u00EF\u0080\u00AD\u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00BDi,ti+1 iSP P (3.27) Note that, the sign of the workpiece rotational speed c\u00CE\u00A9 changes with respect to the rotation direction (i.e. clockwise-counter clockwise). If the angular discretization is small, the projected linear distance contributed by the angular rotation of workpiece can be approximated as; \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0082\u00B4ei,w i,w w,iS \u00CE\u00B8 R (3.28) Then, the total distance traveled by tool and workpiece at time interval it\u00EF\u0081\u0084 is updated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00ABT i+1 i i,w\u00CE\u0094S P P \u00CE\u0094S (3.29) The total displacement of the tool and workpiece is calculated in MCS and needs to be transformed to TCS. The transformation between MCS and TCS requires two stages. First, the unit tool orientation vector iO at the cutter location CLi is aligned with z-axis of MCS \u00EF\u0080\u00A8 \u00EF\u0080\u00A9MZ from the kinematic configuration as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , ,, ,M x i y i z iZ O O O\u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0080\u00BDi 1 iO T O i j k (3.30) 37 where the transformation matrix 1T is evaluated from the inverse kinematics of the machine tool configuration as; 3 3cos 0 sin0 1 0sin 0 cosB BB B\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0082\u00B4\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB1T (3.31) where the reference position command of the B-axis of the spindle head B\u00EF\u0081\u00B1 is already solved by Eq. (3.14) or Eq. (3.21). Once the tool normal axis iO at cutter location CLi is aligned with the tool axis \u00EF\u0080\u00A8 \u00EF\u0080\u00A9MZ in MCS, the unit tool axis vectors \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t,i t,ix y can be evaluated in MCS. In this thesis, the unit tool axis vector \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t,ix is positioned on the plane formed by the linear feed direction and unit tool normal axis (Plane-A) like in five-axis milling [90] as follows (see Figure 3.6); ,\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0082\u00B4 \u00EF\u0080\u00BD \u00EF\u0082\u00B4\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8Tt,i t,i i t,i iT\u00CE\u0094Sx y O y O\u00CE\u0094S (3.32) 38 Figure 3.6 Linear Feed vector representation. The rotation matrix between \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t,i t,ix y and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9M MX Y\u00EF\u0080\u00AD can be defined as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A91,211,13 3cos sin 0sin cos 0 ; tan0 0 1\u00EF\u0081\u00B6 \u00EF\u0081\u00B6\u00EF\u0081\u00B6 \u00EF\u0081\u00B6 \u00EF\u0081\u00B6 \u00EF\u0080\u00AD\u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00AB \u00EF\u0083\u00BB-11 t,i2 -11 t,iT xTT x (3.33) where the subscript \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,1 and \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,2 represents the first and second elements of the array, respectively. Once all the Cartesian coordinates of the TCS are aligned with MCS, the total linear feed vector due tool and workpiece linear rigid body motion is evaluated in TCS at discrete cutter location CLi by; ( ) ;i Lz f\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8TL,T, 3 1 2TT\u00CE\u0094Sf T = T TT\u00CE\u0094S (3.34) The angular feed vector due to the rotational motion of the spindle head (B-axis) is modeled from the two consecutive angular orientation of the tool in the space. The discrete angular displacement from location i to \u00EF\u0080\u00A8 \u00EF\u0080\u00A91i \u00EF\u0080\u00AB is calculated as [90]; 39 1, tanA i\u00EF\u0081\u00B1\u00EF\u0080\u00AD\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0082\u00B4\u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0082\u00B4\u00EF\u0083\u00A8 \u00EF\u0083\u00B8i i+1i i+1O OO O (3.35) Then, the angular velocity of the tool axis rotation is evaluated by substituting the time elapsed between CLi and 1CLi\u00EF\u0080\u00AB ((3.27)) as; ,,A iR iit\u00EF\u0081\u00B1\u00EF\u0081\u00B7\u00EF\u0081\u0084\u00EF\u0080\u00BD\u00EF\u0081\u0084 (3.36) The pure rotational motion of the spindle head (B-axis) occurs around a fictitious axis \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSp that is assumed to be passing through the tip of the tool as seen in Figure 3.7 and found from the two subsequent tool normal axis orientations \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i i+1O ,O in TCS as follows; \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0082\u00B4\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0082\u00B4\u00EF\u0083\u00A8 \u00EF\u0083\u00B8i i+1TCS 3i i+1O Op TO O (3.37) Finally, the angular feed vector is evaluated as; ,( ) ( )R iz p z\u00EF\u0081\u00B7\u00EF\u0080\u00BDA TCSf P (3.38) where ( )p z is the axial height at discrete disc element p along the tool normal axis as explained in Figure 3.7. Similar to the linear feed vector, the angular feed vector also depends on the axial elevation and varies along the tool axis. 40 Figure 3.7 Angular feed vector due to the spindle head rotational motion. The total feed in turn-milling process is calculated as the superposition of the linear and angular feed vectors along the tool axis as a function of tool axial height (z) as; ( ) ( ) ( )z z z\u00EF\u0080\u00BD \u00EF\u0080\u00ABT L,T Af f f (3.39) The unit outward vector normal to the tool body at axial height \" \"z and angular position j\u00EF\u0081\u00A6 is expressed for ball and cylindrical end mill as; sin ( , ). cos ( , ). , for cylindrical endmills( )sin ( )sin ( , ). sin ( ) cos ( , ). cos ( ). , for ball end millj jj jt z t zzz t z z t z z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00AB\u00EF\u0080\u00AB\u00EF\u0083\u00AC\u00EF\u0080\u00BD \u00EF\u0083\u00AD\u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0083\u00AEji jni j k (3.40) j\u00EF\u0081\u00A6 is the radial immersion angle of thj tooth of the tool and calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( ) 1 ( )j pz j z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AD (3.41) where p\u00EF\u0081\u00A6 is the pitch angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A92p tN\u00EF\u0081\u00A6 \u00EF\u0081\u00B0\u00EF\u0080\u00BD between the adjacent teeth of the tool and \u00EF\u0081\u00B9 is the lag angle at the axial depth z and can be found as; 41 tan ( )( )( )z zzR z\u00EF\u0081\u00A2\u00EF\u0081\u00B9 \u00EF\u0080\u00BD (3.42) where ( )z\u00EF\u0081\u00A2 and ( )R z represents the local helix angle and local radius at axial depth z . The term ( )z\u00EF\u0081\u00AB in Eq. (3.40) is the axial immersion where it varies along the tool normal axis for ball end mills as depicted in Figure 3.8 (a) and is calculated as; 2 21 00 0( ) ( ) , for ball partsin ( )( ) , for cylindrical partR z R z R zR zR R z R\u00EF\u0081\u00AB \u00EF\u0080\u00AD\u00EF\u0083\u00AC\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00AF \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0083\u00AD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD\u00EF\u0083\u00AF\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00AE (3.43) Figure 3.8 General tool geometry representation. The uncut chip thickness due to the rigid body motion of all five drives of the turn-milling machine is evaluated for thj tooth by projecting the total feed vector ( )zTf onto tool surface normal vector ( )zjn as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ( ) ( )2pj jth z z z\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00B0\u00EF\u0080\u00BD \u00EF\u0082\u00B7\u00EF\u0081\u0097T jf n (3.44) Note that, the uncut chip thickness expressed by Eq.(3.44) is the static chip thickness where all the relative vibrations between the vibrating tool and workpiece are neglected. The static chip thickness is the function of both rotational speeds of tool and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t c\u00EF\u0081\u0097 \u00EF\u0081\u0097 as well as the rigid body motions of all translational and rotary drives of the machine tool. This is the 42 fundamental difference between regular milling and turn-milling operations in evaluating the true (exact) chip thickness. The effect of the relative vibrations on the chip thickness is explained in detail in Chapter 4. 3.3.2 Static Cutting Force Model in Turn-Milling Once the static chip thickness is computed from the tool geometry and kinematics of the turn-milling operation, the cutting forces can be evaluated using an existing knowledge in the literature [91], [92]. First, the cutting tool in turn-milling is axially divided into small discrete elements with height dz as shown in Figure 3.8 (c). The differential chip geometry and corresponding cutting forces generated along the radial, tangential and axial directions of the discrete cutting edge of the thj tooth can be written as the superposition of the shearing and edge cutting components as [92]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,, ,, ,r j rc j j ret j tc j j tea j ac j j aedF z K h z db K dSdF z K h z db K dSdF z K h z db K dS\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB (3.45) where j\u00EF\u0081\u00A6 is the instantaneous angle of tool (Figure 3.8 (b)), dS is the infinitesimal length of a helical cutting edge segment and db is the discrete chip width which can be calculated as follows [22]; 2 21sinulzj jjzdR ddS R dzdz dzdzdb\u00EF\u0081\u00B9\u00EF\u0081\u00AB\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0080\u00BD\u00EF\u0083\u00A5 (3.46) 43 where lz and uz are the lower and upper boundaries of the discrete axial element, respectively. The terms jdRdz and jddz\u00EF\u0081\u00B9 represent the derivatives of the radius and lag angles of the thj tooth with respect to z and the detailed geometrical derivations can be found for general end mills in [22]. The cutting force coefficients \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,rc tc acK K K and edge force coefficients \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,re te aeK K K in radial, tangential and axial directions are obtained either by mechanistic calibration tests or by orthogonal to oblique transformation [92]. In the mechanistic approach, the average forces are measured at different feedrates and expressed by a linear regression of the data. This method is only valid for the given geometry of the cutter and process parameters, thus it cannot be used to predict the force coefficients for different tool geometries and cutting conditions. On the other hand, orthogonal- to-oblique transformation predicts the cutting force coefficients from the database established for the orthogonal cutting process. Then, the orthogonal cutting coefficients are transformed into any oblique cutting process using the cutter geometry. Therefore, it allows to predict the cutting force coefficients without experimental calibration of each cutter geometry and can be applied to more complex tools. The orthogonal database which consists of the shear stress of the workpiece material \u00EF\u0080\u00A8 \u00EF\u0080\u00A9s\u00EF\u0081\u00B4 , friction angle s\u00EF\u0081\u00A2 , rake angle r\u00EF\u0081\u00A1 and shear angle s\u00EF\u0081\u00A6 are transformed to the oblique cutting conditions of the process and the tangential, radial and axial cutting force coefficients on the oblique plane can be calculated as; 44 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2 22 2 22 2 2cos tan tan sinsin cos tan sinsinsin cos cos tan sincos tan tan sinsin cos tan sinn n o c nsrcn n n n c nn nstcn n n n c nn n o c nsacn n n n c niKKiiK\u00EF\u0081\u00A2 \u00EF\u0081\u00A1 \u00EF\u0081\u00A8 \u00EF\u0081\u00A2\u00EF\u0081\u00B4\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A2 \u00EF\u0081\u00A1 \u00EF\u0081\u00A8 \u00EF\u0081\u00A2\u00EF\u0081\u00A2 \u00EF\u0081\u00A1\u00EF\u0081\u00B4\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A2 \u00EF\u0081\u00A1 \u00EF\u0081\u00A8 \u00EF\u0081\u00A2\u00EF\u0081\u00A2 \u00EF\u0081\u00A1 \u00EF\u0081\u00A8 \u00EF\u0081\u00A2\u00EF\u0081\u00B4\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A2 \u00EF\u0081\u00A1 \u00EF\u0081\u00A8 \u00EF\u0081\u00A2\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB (3.47) where n\u00EF\u0081\u00A6 , n\u00EF\u0081\u00A2 , n\u00EF\u0081\u00A1 are the shear, friction and rake angles on the normal plane, respectively and c\u00EF\u0081\u00A8 is the chip flow angle and oi is the oblique angle of the cutter. Considering the Stabler\u00E2\u0080\u0099s rule [93] regarding the chip flow angle, the following assumptions are valid; n sn sn rc i\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A2 \u00EF\u0081\u00A2\u00EF\u0081\u00A1 \u00EF\u0081\u00A1\u00EF\u0081\u00A8\u00EF\u0080\u00BD\u00EF\u0080\u00BD\u00EF\u0080\u00BD\u00EF\u0080\u00BD Therefore, the cutting force coefficients given by Eq. (3.47) become a function of tool geometry \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,r oi\u00EF\u0081\u00A1 and the orthogonal cutting database for the workpiece material \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,s s s\u00EF\u0081\u00B4 \u00EF\u0081\u00A6 \u00EF\u0081\u00A2 . Then, the cutting force coefficients are found for any tool geometry and process parameters. The differential cutting forces acting on the thj tooth in TCS can be evaluated by transforming the cutting forces from \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta frame (Eq. (3.45)) to the \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSxyz tool coordinate frame as below; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSxyzTCS, ,, , ,, ,x j r jj y j t jz j a jdF z dF zdF z dF z dF zdF z dF z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BEHT (3.48) where the transformation matrix between the \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TCSxyz frame is HT expressed as follows; 45 sin ( )sin ( , ) cos ( , ) cos ( )sin ( , )sin ( )cos ( , ) sin ( , ) cos ( )cos ( , )cos ( ) 0 sin ( )j j jj j jz t z t z z t zz t z t z z t zz z\u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6\u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6\u00EF\u0081\u00AB \u00EF\u0081\u00AB\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BBHT (3.49) Considering the axial and angular engagement limits of thj tooth which are denoted by ,0 ,1,j jz z and ,z zst ex\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 in Figure 3.9, the total cutting forces acting on the tool body in TCS can be evaluating by integrating the differential cutting forces contributed by all teeth \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tN over the cutter-workpiece engagement limits as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,1TCS TCS,01, ,jtjzNj jxyz xyzj zF g z dF z dz\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0083\u00A5 \u00EF\u0083\u00B2 (3.50) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,jg z\u00EF\u0081\u00A6 is the unit step function and defined at each discrete force element as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,,0, otherwisez zst j exjg z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0080\u00BD \u00EF\u0083\u00AD\u00EF\u0083\u00AE (3.51) Similar to the cutting forces, the torque and power requirements by tool and workpiece spindles must be calculated for the process planning stage. Cutting torque tT and power tP drawn from the tool spindle are; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D, , Nm, , Wt j t t jt j t t jT z R F zP z T z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0081\u0097 (3.52) where ,t tR F and t\u00EF\u0081\u0097 are the tool radius, tangential force and rotational speed of the tool, respectively. Torque cT and power cP transmitted to the workpiece spindle (C-Axis) are calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D, , Nm, , Wc j w x jc j c c jT z R F zP z T z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0081\u0097 (3.53) 46 Finally, the resulting cutting speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9RV can be calculated as the vectorial superposition of cutting speed vectors of the tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tV and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9cV as; ( , ) sin 2 sin( , ) cos 2 cos 2j j t t jeRj j t t j w cg z RVg z R R\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00B0 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00B0 \u00EF\u0081\u00A6 \u00EF\u0081\u00B0\u00EF\u0080\u00BD \u00EF\u0083\u0097 \u00EF\u0081\u0097\u00EF\u0083\u00AC \u00EF\u0083\u00AC\u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00AD\u00EF\u0080\u00BD \u00EF\u0083\u0097 \u00EF\u0080\u00AB \u00EF\u0081\u0097 \u00EF\u0080\u00AB \u00EF\u0081\u0097\u00EF\u0083\u00AE \u00EF\u0083\u00AER,X tR,X R,YR,Y t cV VV ,VV V V (3.54) 3.3.3 Cutter-Workpiece Engagement Geometry in Turn-Milling Process The cutter-workpiece engagement (CWE) geometry, which determines the chip thickness distribution (see Eq. (3.50)) needs to be identified to calculate the cutting forces, torque, and power that are required for tool and workpiece spindles and the stability of the turn-milling process which is explained in detail in Chapter 4. The simultaneous tool and workpiece rotations, linear (rigid body) motions of the drives and spindle head (B-Axis) tilting may yield a complex and varying intersection geometry between the cutter and workpiece as shown in Figure 3.9 (a). Since the extraction of the CWE is not within the scope of this thesis, it obtained for a given toolpath and blank geometry of the workpiece using MACHPro \u00E2\u0080\u0093 Virtual Machining System developed at our laboratory [94]. First, the toolpath of the turn-milling process is obtained from NX CAM \u00C2\u00AE [88]. The workpiece rotational motion (C-Axis) is assigned to the tool spindle since it does not change the static cutter-workpiece engagement conditions. Thus, the tool follows the curved toolpath and rotates around the workpiece which yields a helical motion. Then, pitch of the corresponding helical motion is calculated and embedded to the CAM software. Once the toolpath of the turn-milling process is obtained, the engagement geometry is obtained for the given toolpath and blank geometry of the workpiece. The tool and workpiece geometries are intersected at each discrete toolpath location CLi and the engagement geometry is mapped to 47 the angular and axial contact zone of the tool with the workpiece Figure 3.9 (b). Later, the CWE is used as a boundary conditions in evaluating the chip thickness (3.44) and hence cutting force distribution (3.50) along the tool normal axis and toolpath. Figure 3.9 Varying cutter-workpiece engagement geometry in turn-milling (a), 2D representation of CWE in angular and axial coordinates. 3.4 Experimental Verification The proposed mechanics model of turn-milling has been verified in turn-milling of cylindrical aluminum alloy Al6061-T6 with a two fluted cylindrical end mill, having 16 mm diameter, 5\u00C2\u00B0 rake and 25\u00C2\u00B0 helix angles. The workpiece diameter and length is selected as 105 mm and 175 mm, respectively, to avoid excessive workpiece vibrations and provide chatter-free cutting for the force measurements. The cutting force coefficients were identified by oblique transformation from the orthogonal database for given tool geometry and workpiece material as \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 930,290,242tc rc acK K K \u00EF\u0080\u00BD MPa and \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 21,25,8te re aeK K K \u00EF\u0080\u00BD N/mm. Cutting tests have 48 been conducted on the Mori Seiki 3150/500C CNC Mill-Turn and the cutting force data along the x, y and z directions are collected by Kistler 9123C1 four component rotary dynamometer as seen in Figure 3.10. The surface of the cylindrical workpiece is also prepared smoothly having 20 m\u00EF\u0081\u00AD radial runout at the tip and 30 m\u00EF\u0081\u00AD axial runout through the rotational axis of the C-Axis. Figure 3.10 Experimental set-up to measure cutting forces in turn-milling process. The cutting conditions used in turn-milling tests are given in Table 3.1. Table 3.1 Cutting conditions for the turn-milling case. Tool Spindle Speed \u00E2\u0080\u0093t\u00EF\u0081\u0097 500 rpm Workpiece Spindle Speed \u00E2\u0080\u0093 c\u00EF\u0081\u0097 2 rpm Tool Axial Feed \u00E2\u0080\u0093 tf 4 mm/min Workpiece Diameter \u00E2\u0080\u0093 wD 105 mm Tool Diameter \u00E2\u0080\u0093 tD 16 mm Depth of cut \u00E2\u0080\u0093 a 1 mm Number of Flutes \u00E2\u0080\u0093 tN 2 Workpiece Material Al6061-T6 The rotating milling cutter is moved \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a into the workpiece radially and fed linearly in a direction parallel to workpiece rotation axis while the workpiece rotates at a constant speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u009749 . Since the tool and workpiece have symmetric geometries, and the tool spindle\u00E2\u0080\u0099s rotation axis (B-axis) does not change along the toolpath, CWE is also same for all cutter locations, CLi . CWE in axial and angular domain of the tool is obtained from MACHPro \u00C2\u00AE [94] as seen in Figure 3.11 (a). The engagement geometry changes along the tool axis as start and exit angles of engagement boundaries vary. The chip thickness (chip load) \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j jh z\u00EF\u0081\u00A6 is calculated for the given APT-CL file of the cutting system by solving the linear feed vector and shown in Figure 3.11 (b). The ratio of time spent during cutting and non-cutting has been evaluated from the engagement geometry given in as 0.1963\u00EF\u0081\u00B2 \u00EF\u0080\u00BD for this particular operation. Figure 3.11 Cutter-Workpiece Engagement (a) chip load on each flute of the tool in one spindle period, and T is the tooth passing period. The process parameters are given in Table 3.1. The predicted and experimentally measured cutting forces are in reasonable agreement as shown in Figure 3.12. The error is less than 5.7% and 9.0% for x and y directions, respectively. On the other hand, the error is around 35% for z direction, and the measured cutting forces vary between consecutive teeth of the milling cutter due to neglected axial run-out of the tool and tool\u00E2\u0080\u0099s bottom contact in the analytical cutting force model. The presence of transient vibrations of both 50 tool and workpiece are due to the highly interrupted cutting nature of the process, which can be minimized by selecting a toolpath and milling cutter with more teeth which lead non-air cutting zones. Especially for high workpiece rotation speeds and large workpiece diameters, the tool axial feed must be kept very small in order to preserve the chip load within feasible values. However, this leads to very narrow engagement limits hence highly interrupted turn-milling processes. The consequences of the highly interrupted cutting process are more severe for the stability of the process which is explained in Chapter 4. 51 Figure 3.12 Simulated and measured cutting forces in turn-milling for given conditions in Table 3.1 Proposed static chip thickness and cutting force model of turn-milling has been also validated against the experimental results reported in [80]. During the experiments, AISI 1040 steel workpiece is turn-milled with a 10 mm diameter cylindrical end mill having 4 flutes. The cutting conditions that are used in the experiments [80] are listed in Table 3.2. The cutting force 52 coefficients for AISI 1040 steel is calculated by orthogonal to oblique transformation and found as; \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 1706,759,577tc rc acK K K \u00EF\u0080\u00BD MPa and \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 0,0,0te re aeK K K \u00EF\u0080\u00BD N/mm. Table 3.2 Cutting conditions for the experimental validation case in [80]. Tool Spindle Speed \u00E2\u0080\u0093 t\u00EF\u0081\u0097 2000 rpm Workpiece Spindle Speed \u00E2\u0080\u0093 c\u00EF\u0081\u0097 20 rpm Tool Axial Feed \u00E2\u0080\u0093 tf 6 mm/min Workpiece Diameter \u00E2\u0080\u0093 wD 90 mm Tool Diameter \u00E2\u0080\u0093 tD 10 mm Depth of cut \u00E2\u0080\u0093 a 0.5 mm Number of Flutes \u00E2\u0080\u0093 N 4 Workpiece Material AISI 1040 Since the CWE geometry was not given explicitly in [80] along the tool axis, it is simulated for the cutting conditions in Table 3.2 using the start and exit angles at the bottom of the tool. Similar to the procedure explained for the previous validation case, the CWE geometry is obtained from the MACHPro \u00C2\u00AE as seen in Figure 3.13 (a). As opposed to the previous validation case, the CWE geometry does not change along the tool axis and the start and exit angles remain constant at different discrete axial height. Since a small diameter milling tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A910 mmtD \u00EF\u0080\u00BD is used to cut a big diameter workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A990 mmwD \u00EF\u0080\u00BD , the intersection geometry between the tool and workpiece does not vary along the tool axis as seen in Figure 3.13 (a). Once the engagement boundaries are obtained, the chip load acting on each flute of the tool is simulated by solving the linear feed vector. The maximum chip load achieves 0.27 mm as shown in Figure 3.13 (b). The ratio of time spent during cutting and non-cutting has also been evaluated for the four fluted milling tool and found as 0.311\u00EF\u0081\u00B2 \u00EF\u0080\u00BD . 53 Figure 3.13 CWE geometry along the tool axis (a) and chip load distribution on each flute (b) for the validation case against the experimental results in [80]. The cutting conditions are given Table 3.2. The simulated resultant cutting forces \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2 2res x yF F F\u00EF\u0080\u00BD \u00EF\u0080\u00AB are compared against the measurements reported in [80] and shown in Figure 3.14. Figure 3.14 Simulated and measured [80] resultant cutting forces over a spindle period. The cutting conditions are given in Table 3.2. 54 The simulated and measured cutting forces are in good agreement. The error is less than 2.5% in the resultant force direction. The deviation between the peak cutting forces at each cutting tooth is due to the tool run out. 3.5 Summary The static chip thickness and cutting force model of turn-milling is presented in this chapter. First, the kinematic analysis of the turn-milling machine tool is presented by Screw Theory and D-H kinematic solution. The forward and inverse kinematic solutions are proposed for any toolpath and geometry of tool and workpiece. Then, the multi-axes feed motion of the milling tool is modeled and resultant feed vector is calculated as the superposition of the linear and angular feed motion of the translational and rotary drives of the machine, respectively. The proposed mechanics model of turn-milling is validated against different cases. Although the proposed chip thickness model is presented for the turn-milling process, it can be extended to any five-axis milling operation as well. 55 Chapter 4: Dynamics and Stability of Turn-Milling Operations 4.1 Overview Turn-milling operations require two rotating spindles, namely the tool and workpiece spindles and three translational drives. The rotating milling tool mounted on the tool spindle can be positioned in three Cartesian coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9X,Y,Z and tilted around the B-axis of the machine tool, and the tool follows a curved path on the workpiece which is mounted on the rotating chuck (C-axis) as seen in Figure 4.1. The resulting kinematics of five axis motion lead to complex chip geometry and cutting forces which have been modeled in the Chapter 3. The dynamics and stability of the turn-milling operations have not been addressed in the literature, thus the cutting parameter selection strategy highly relies on the costly machining trials in industry and hinders the productivity. Erroneous selection of the process parameters such as speeds of tool and workpiece, feed and depth of cut results in the self-excited, regenerative chatter vibrations which lead to excessive cutting forces, poor surface finish, and may damage the machine tool structures. The objective of this chapter is to develop a predictive dynamic model of turn-milling to find the chatter-free cutting conditions without resorting to costly physical trials. First, the dynamics and stability of regular milling processes is solved for asymmetric components of the machine tool in rotating coordinates. The dynamics of the turn-milling process is modeled as a multi-degree of freedom system, then the dynamic chip thickness is modeled as a result of the multi-axes motion of the machine and relative vibrations between the rotating tool and workpiece. The resulting time-varying dynamic system equations are solved to calculate the vibrations and stability of the system in semi-discrete time domain. The surface location error model is developed considering the mechanics and relative vibrations between the flexible tool and workpiece. The stability model of 56 turn-milling is extended to low-immersion and low-speed regions of cutting to predict the stability boundaries. The proposed stability model has been experimentally validated on a turn-milling machine for various cases and materials. 4.2 Stability of Milling Process with Asymmetric Structure Dynamics The turn-milling machine tools have both stationary and rotating structures. While spindle housing, column and ram have stationary dynamics, rotating parts may have both symmetric (i.e. spindle shaft and tool holder) and asymmetric dynamics (i.e. two-fluted end mill) as shown in Figure 4.1. Existing stability models predict the stable and unstable regions of cutting assuming that the structural dynamics of the system remain constant at fixed directions. However, there might be several cases where the tool tip FRF changes such as speed dependent dynamics of the spindle and asymmetric modes of the milling tool. Figure 4.1 Representation of turn-milling machine tool axes. In the case of speed dependent spindle dynamics, centrifugal forces and gyroscopic moments acting on the spindle bearings and rotating shaft change the bearing contact loads and angles which reduces the bearing stiffness at increased rotational speeds as seen in Figure 4.2(a) [95]. As a result, 57 the dominant spindle mode of tool tip FRF shift towards low frequencies. Figure 4.2(b) shows the representative case where the spindle mode of the dummy holder-spindle assembly shifts to lower frequencies due to the increased spindle speed. On the other hand, the nonlinear behaviour of speed-dependent dynamics of the spindle does not lead any dynamic asymmetry between the principal directions of the tool, thus its effect is not within the scope of this thesis. Figure 4.2 Bearing contact angle at idle and rotating spindle cases (a); spindle mode shifting at high rotational speeds of tool (b). Another reason of tool tip FRF alteration is the rotation of non-symmetrical parts of the machine tool (i.e. asymmetric cutting tool or workpiece). Some milling tools have constant dynamics, and the natural frequency and stiffness of the tool tip do not change with respect to angular position of the tool rotation as seen in Figure 4.3 (b). On the other hand, some special milling tools which are widely utilized in turn-milling operations due to their high chip evacuation capacities, may have asymmetric dynamics in two orthogonal directions. As the tool rotates, the tool tip FRF along the orthogonal directions \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,x y changes due to the asymmetric dynamic flexibilities as seen in Figure 4.3(c). As a result, the FRF of the tool-holder-spindle assembly becomes speed and angular orientation dependent. 58 Figure 4.3 Tool position at different rotation angles (a); variation of tool tip FRF with respect to tool rotation for symmetric (b) and asymmetric tools (c). In this section, the stability of the regular milling process is solved in rotating coordinate frame considering only the asymmetric dynamics of the tool. 4.2.1 Dynamics of Milling in Rotating Coordinates Dynamic chip thickness removed by thj tooth of a milling tool can be written in local rotating coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,l lj ju v which is located on the tooth, as (Figure 4.4); 11( ) ( )( ) sin [0 1]( ) ( )l lj jj j l lj ju t u th t cv t v t\u00EF\u0081\u00B4\u00EF\u0081\u00A6\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE (4.1) where c is the feed per revolution per tooth [mm/rev/tooth], j\u00EF\u0081\u00A6 is the immersion angle of thj tooth, \u00EF\u0081\u00BB \u00EF\u0081\u00BD( ), ( )l lj ju t v t and \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 1( ), ( )l lj ju t v t\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD are the vibration vectors at the present time t and previous tooth passing time which has a delay of \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 . In regular milling, the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 is 59 always equal to the tooth passing period dT and remain constant. The static part of the chip thickness \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sin jc \u00EF\u0081\u00A6 does not contribute to the regeneration mechanism, so it is excluded from the stability analysis. Figure 4.4 Dynamics and cutting forces in rotating coordinate frame. The dynamic cutting forces in radial and tangential directions of thj tooth can be written within the cutter-workpiece engagement (CWE) boundary as; ,,( ) ( )r j rcjt j tcF Kg a h tF K\u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00AE \u00EF\u0083\u00BE (4.2) where rcK and tcK are the cutting force coefficients in radial and tangential directions, a is the depth of cut and ( )g \u00EF\u0081\u00A6 is the switching function defined with respect to start \u00EF\u0080\u00A8 \u00EF\u0080\u00A9st\u00EF\u0081\u00A6 and exit \u00EF\u0080\u00A8 \u00EF\u0080\u00A9ex\u00EF\u0081\u00A6 angles of thj tooth as; 1,( ) ;0, otherwisest j exg\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0083\u00AC\u00EF\u0080\u00BD \u00EF\u0083\u00AD\u00EF\u0083\u00AE (4.3) 60 Cutting forces along the radial tangential directions of thj tooth can be written more explicitly by substituting the Eq. (4.1) into Eq. (4.2) as; , 1, 1( ) ( )( ) [0 1]( ) ( )l lr j rc j jl lt j tc j jF K u t u tg aF K v t v t\u00EF\u0081\u00B4\u00EF\u0081\u00A6\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE (4.4) Then, the cutting forces in local rotating frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,l lj ju v are transformed to the principal rotating frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v by considering the tN number of tooth on the cutter as; 110( ) ( )( ) [0 1]( ) ( )tNu rcj j jjv tcF K u t u tg aF K v t v t\u00EF\u0081\u00B4\u00EF\u0081\u00A6\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A5 T T T (4.5) where jT is the orthonormal coordinate transformation matrix for thj tooth which relates the displacements written in local rotating frame to the principal rotating frame as; 2 2cos( ) sin( )( ) ( );sin( ) cos( )( ) ( )lp c p cjj jlp c p cjj ju t u tj jv t v t\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0082\u00B4\u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BET T (4.6) where p\u00EF\u0081\u00A6 is the pitch angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 tN\u00EF\u0081\u00B0 and c\u00EF\u0081\u00A6 is the angle between local \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,l lj ju v and principal \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v rotating frames which are aligned for simplicity in the thesis. The dynamics of classical milling process can be represented in fixed coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,x y by the following periodic, delayed differential equation: \u00EF\u0081\u009B \u00EF\u0081\u009D( ) ( ) ( ) ( ) ( ) ( ) ( )s s s s s st t t t t t t \u00EF\u0081\u00B4\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00ADs s sM q C q K q F A q q (4.7) where the mass sM , damping sC and stiffness sK matrices are constant and the forces ( )s tF and vibrations ( ) [ ]s t x(t); y(t)\u00EF\u0080\u00BDq are defined in fixed coordinates. The directional coefficient matrix \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( )tA is periodic at constant spindle or tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT which corresponds to an constant time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 . 61 The dynamics of a system can be modeled in principal rotating frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v by using the following coordinate transformation. \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9cos sin; ; sin costt tx utt ty v\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0097\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BBT T (4.8) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u00A6 is the tool rotation angle and function of spindle speed of tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 and time \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t . T is the orthonormal transformation matrix that relates the physical quantities written in fixed (inertial) coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,x y to the rotating (non-inertial) coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v . Similar to the displacement transformation in Eq. (4.8), transformation for the velocity and acceleration terms can be written as follows; 22tt tx u uy v vx u u uy v v v\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0081\u0097\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BET TT T T (4.9) where the second and third terms in acceleration expression represent the Coriolis and Centripetal forces, respectively. These terms are proportional to the spindle speed, hence when speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 goes to zero, they vanish. Substituting Eq. (4.8) and Eq. (4.9) into the Eq. (4.7) yields the equation of motion in principal rotating frame as; 2( ) 2 ( ) ( ) ( ) ( ) ( )xt t tyFu u u u u uFv v v v v v\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AB \u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097 \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0081\u0097 \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BEs s sM T T T C T T K T (4.10) The modal parameter matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,s s sM C K are constant and defined in fixed coordinate frame for single degree of freedom system as; 62 0 0 0; ;0 0 0x x xy y ym c km c k\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBs s sM C K (4.11) which can also be transformed to rotating coordinate frame. Considering only the static case as an example; 00kx xky yF k xF k y\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB (4.12) and andkx kuky kvF F x uF F y v\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BET T (4.13) Substituting the Eq. (4.13) into the Eq. (4.12) and multiplying each side by -1T ; 1 100xkuykvkF ukF v\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00A8 \u00EF\u0083\u00B8T T T T (4.14) In this way, the stiffness matrix defined in fixed coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sK is transformed to rotating frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rK as follows; 10 00 0x uy vk kk k\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BBrT T K (4.15) If similar transformation is applied to Eq. (4.10) by multiplying the both sides -1T , the equation of dynamics can be completely represented in the rotating coordinate frame as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 1x uy vF Fu u uF Fv v v\u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BEr r cor r crc cntM C + C K + K + K T (4.16) where the coefficient matrices are given as; 220 2 0 0; ;2 0 0 0t u t u t ut v t v t vm c mm c m\u00EF\u0080\u00AD \u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBcor crc cntC K K (4.17) 63 Note that, the modal parameters given in Eq. (4.17) are represented in the principal direction of the tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v and they remain constant regardless of the angular position of the tool. Equation (4.17) contains two skew-symmetric matrices: one linked with Coriolis force which is presented if a particle is moving in a rotating coordinate frame, and a circulatory matrix [96] which is proportional to tool spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 and adds damping to the system. On the other hand, the matrix that is proportional to 2t\u00EF\u0081\u0097 is the centripetal term and acts as a negative damping to the dynamic system. 4.2.2 Stability of Milling with Asymmetric Cutter Dynamics In this section, the stability of the dynamic cutting process given in Eq. (4.16) for regular milling operations is investigated both in frequency and time domains to obtain the chatter-free cutting conditions for the increased productivity. 4.2.2.1 Frequency Domain Solution of Asymmetric Cutter Dynamics The cutting forces given in Eq. (4.5) can be written in the following; \u00EF\u0081\u009B \u00EF\u0081\u009D( ) ( ) ( ) ( ) ( )t a t t t t \u00EF\u0081\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00ADF A q B q (4.18) where the force ( )tF , displacement ( )tq and periodic directional coefficient matrices ( )tA and ( )tB are given as; 10110( ) ( )( ) ; ( ) ; ( )( ) ( )( ) ( ) [0 1]( ) ( ) [0 1]uvNrcj jj tcNrcj jj tcF u t u tt t tF v t v tKt gKKt gK\u00EF\u0081\u00B4\u00EF\u0081\u00B4\u00EF\u0081\u00B4\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0080\u00AD\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A5\u00EF\u0083\u00A5F q qA T TB T T (4.19) 64 The dynamic cutting forces are transformed from time domain to the frequency domain by taking the Fourier Transform (FT) of (4.18) as; ( ) ( ) ( ) ( , ) ( )j T ta e j\u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BBF A B \u00CE\u00A6 F (4.20) where the vibration vectors at the present time and previous tooth passing period can be transformed from time domain to frequency domain as; ( ) ( , ) ( )( ) ( , ) ( )tj Ttt jt e j\u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B4 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0081\u0097\u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0081\u0097q \u00CE\u00A6 Fq \u00CE\u00A6 F (4.21) ( , )j\u00EF\u0081\u00B7 \u00EF\u0081\u0097\u00CE\u00A6 is the transfer function of the system and depends on the rotational speed as proved in Eq. (4.16). FRFs of the system including the gyroscopic terms can be given in as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A91s cor crc cnt\u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 (4.22) where 22220 20;2 000 0;0 0t u cu c u c us cort v cv c v c vt u t ucrc cntt v t vm ik i c mm ik i c mc mc m\u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD \u00EF\u0081\u0097\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0081\u0097\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0080\u00AD\u00EF\u0081\u0097 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00CE\u00A6 \u00CE\u00A6\u00CE\u00A6 \u00CE\u00A6 (4.23) The directional coefficient matrices ( )tA and ( )tB are periodic at tooth passing frequency, they can be expanded into the Fourier series as; 001( ) ; ( )1( ) ; ( )dT TdT TTr rjr t jr tr rdr rTr rjr t jr tr rdr re t e dtTe t e dtT\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0080\u00BD\u00EF\u0080\u00AB\u00EF\u0082\u00A5 \u00EF\u0080\u00BD\u00EF\u0080\u00AB\u00EF\u0082\u00A5\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0082\u00A5 \u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0082\u00A5\u00EF\u0080\u00BD\u00EF\u0080\u00AB\u00EF\u0082\u00A5 \u00EF\u0080\u00BD\u00EF\u0080\u00AB\u00EF\u0082\u00A5\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0082\u00A5 \u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0082\u00A5\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00A5 \u00EF\u0083\u00A5\u00EF\u0083\u00B2\u00EF\u0083\u00A5 \u00EF\u0083\u00A5\u00EF\u0083\u00B2A A A AB B B B (4.24) where r , T\u00EF\u0081\u00B7 and dT stand for the number of harmonics in Fourier expansion, tooth passing frequency and tooth passing period, respectively. 65 The Eq. (4.20) is an eigenvalue problem and the stability of the system can be solved from the characteristic equation of the system. The method proposed by Altintas and Budak [30] is applied for the eigenvalue problem using the average values of the time varying directional coefficient matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( ) and ( )t tA B and tool tip FRFs ( , )tj\u00EF\u0081\u00B7 \u00EF\u0081\u0097\u00CE\u00A6 defined in principal rotating coordinates. Taking the zeroth term \u00EF\u0080\u00A8 \u00EF\u0080\u00A90r \u00EF\u0080\u00BD of the directional coefficient matrices given in Eq. (4.24), the dynamic milling force equation can be written in frequency domain as follows; ( ) ( , ) ( )j T ta e j\u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BBF A B \u00CE\u00A6 F (4.25) where the time-invariant average terms of directional coefficients are calculated as; 0 01 1( ) ; ( ) , ( 0)d dT Td dt dt t dt rT T\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00B2 \u00EF\u0083\u00B2A A B B (4.26) Assuming the system is vibrating at chatter frequency c\u00EF\u0081\u00B7 at the marginal (critical) point, the characteristic equation of the eigenvalue problem given in Eq. (4.25) can be written as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9det ( , ) 0j T c tI a e j\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BBA B \u00CE\u00A6 (4.27) In regular milling systems written in fixed coordinate system, the eigenvalue of Eq. (4.27) can be calculated for a given chatter frequency, c\u00EF\u0081\u00B7 . However, the FRFs of the dynamic system and characteristic equation written in rotating coordinate frame depend on the spindle speed due to the gyroscopic terms that appear in Eq. (4.16). Thus, the chatter frequency c\u00EF\u0081\u00B7 cannot be included in the unknown eigenvalue of the rotating system, and it is not possible to obtain a closed from solution similar to [30]. Instead, as an alternative to Nyquist solution, the special case that occurs at 2-fluted cutters can be implemented for the analytical solution of Eq. (4.27). For the 2-fluted cutter having uniform pitch angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9p\u00EF\u0081\u00A6 \u00EF\u0081\u00B0\u00EF\u0080\u00BD , special condition occurs for the transformation 66 matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,j j\u00EF\u0080\u00ADT T given in Eq. (4.19) and directional coefficient matrices differ from each other by the phase angle \u00EF\u0081\u00B0 . The characteristic equation can be written as follows; \u00EF\u0080\u00A8 \u00EF\u0080\u00A90det 0I \u00EF\u0080\u00AB\u00EF\u0081\u008C \u00EF\u0080\u00BD\u00CE\u00A6 (4.28) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 01 and ( , )cj Ta e j\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0081\u008C \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0081\u0097\u00CE\u00A6 A\u00CE\u00A6 (4.29) The eigenvalues \u00EF\u0080\u00A8 \u00EF\u0080\u00A9r ij\u00EF\u0081\u008C \u00EF\u0080\u00BD\u00EF\u0081\u008C \u00EF\u0080\u00AB \u00EF\u0081\u008C are calculated from the roots of Eq. (4.28) and the critical axial depth of cut of the cutting process can be found as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 cos sinr ilimc cjaT j T\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u008C \u00EF\u0080\u00AB \u00EF\u0081\u008C\u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AD (4.30) Since the lima is a physical quantity and real number, the imaginary part of the Eq. (4.30) must be zero and leads to, sintan1 cosi cr cTT\u00EF\u0081\u00B7\u00EF\u0081\u00AB\u00EF\u0081\u00B7\u00EF\u0081\u008C\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0081\u0099\u00EF\u0081\u008C \u00EF\u0080\u00AB (4.31) Finally, the critical axial depth of cut lima and spindle speed of cutting tool \u00EF\u0081\u009B \u00EF\u0081\u009Drpmt\u00EF\u0081\u0097 that correspond to stability lobe can be calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9lim1 60; 0,1,2,2 2 2 \u00CF\u0088+kcri t lobeta kN\u00EF\u0081\u00B7\u00EF\u0081\u00AB\u00EF\u0081\u00B0\u00EF\u0081\u008C\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0081\u008C \u00EF\u0081\u0097 \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00AD (4.32) The characteristic equation given in Eq. (4.28) is only valid for the corresponding spindle speed used in FRF definition in Eq. (4.23). Therefore, the spindle speed used for the FRF definition should be picked among the obtained results. 67 4.2.2.2 Time Domain Solution of Asymmetric Cutter Dynamics The dynamic cutting force expression in Eq. (4.5) is substituted into the system equations in Eq. (4.16) which is represented in the principal rotating coordinate as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( ) ( ) ( ) ( ) ( ) ( ) ( )t t t a t t t t \u00EF\u0081\u00B4\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00ADr r cor r crc cntM q C +C q K +K +K q A q B q (4.33) Eq. (4.33) is a delay differential equation (DDE) periodic at tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT , and the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 is constant and equal to the periodicity of the system, dT . The stability of linear DDE can be investigated by checking their infinite number of characteristic roots where the system is assumed asymptotically stable if and only if all the critical characteristic roots have only negative real parts. However, since there is no closed form solution to calculate the characteristic multipliers of the infinite dimensional eigenvalue problem, the linear DDE is approximated by finite dimensional discrete maps (ODEs). In this section, the technique which is called semi-discretization [31] is applied to determine the stability of the given system. First, the Eq. (4.33) is transformed to state space as first-order equations as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00BB \u00EF\u0081\u00BDq q q t \u00EF\u0081\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00ADL R (4.34) where the states and coefficient matrices are given as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A911 2121 22222 22 2 2 2 2 2 2 21 1 12 2( )( ),;( )( )2;20 0 0;( ) ( ) 0u u u u uv v v vr r ru du tu u t uv dtqu dv tv v t vdtvc m k m cmv c c k mIa t a t\u00EF\u0082\u00B4\u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0082\u00B4\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0080\u00AD \u00EF\u0081\u0097 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD\u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBD SL RM A S M D M B (4.35) 68 In semi-discretization method, the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 is divided into m number of discrete intervals, t\u00EF\u0081\u0084 . Since the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B4 and time period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9T of the linear DDE is equal to each other, the delayed states \u00EF\u0080\u00A8 \u00EF\u0080\u00A9q t \u00EF\u0081\u00B4\u00EF\u0080\u00AD are approximated by averaging the past state values at two consecutive time intervals \u00EF\u0081\u009B \u00EF\u0081\u009D1,i it t t \u00EF\u0080\u00AB\u00EF\u0083\u008E as [31]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD12 2i i i m i mq t t q t q qq t\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0081\u00B4 \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0081\u0084 \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00AD \u00EF\u0082\u00BB \u00EF\u0080\u00BD (4.36) Then, for each time interval, the state space equations can be written as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9112i i i i i m i mq q q q\u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00ABL R (4.37) Assuming iL is invertible for all i and the solution of Eq. (4.37) takes the following form (for further details, see Ref. [92]); \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A911 112i it ti i i i i m i mq e q e I q q\u00EF\u0081\u0084 \u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00ABL LL R (4.38) Introducing the augmented state vector as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 2, , ,...,Ti i i i i mq q q q\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BDY (4.39) The linear may between the current \u00EF\u0081\u00BB \u00EF\u0081\u00BDiY and next \u00EF\u0081\u00BB \u00EF\u0081\u00BD1i\u00EF\u0080\u00ABY states can be constructed as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 ,i T i i\u00EF\u0080\u00AB \u00EF\u0080\u00BDY \u00CE\u00A6 Y (4.40) where ,T i\u00CE\u00A6 is the finite-dimensional transition matrix between the thi and \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thi \u00EF\u0080\u00AB states. Repeated multiplications of ,T i\u00CE\u00A6 over the spindle period result the transition matrix connecting the present and past delayed states as; , 1 ,1 ,0...T T m T T\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 (4.41) 69 According to the Floquet Theory, the linear periodic system is unstable if any of the eigenvalues of the transition matrix T\u00CE\u00A6 have modulus greater than one and stable if the modulus is less than unity [31]. Unlike the zero-order frequency domain solution given in Section 4.2.2.1, the stability is searched iteratively by checking trial spindle speed and depth if cuts. Since it considers the time-varying periodic coefficients given at each discrete time interval t\u00EF\u0081\u0084 , the accuracy of semi-discretization is higher than zero-order frequency domain solution. On the other hand, the semi-discretization method is computationally more expensive than the zero-order solution since each spindle speed and depth of cut pair is solved to estimate the stability limit. 4.2.3 Simulations and Experimental Results The proposed stability model for asymmetric cutter dynamics in rotating coordinate frame has been validated in milling Aluminum alloy Al7050-T7451 with a 2-fluted cylindrical end mill, having 10 mm diameter, 45 mm stick out from the shrink-fit tool holder, 13.5\u00EF\u0082\u00B0 rake and 25\u00EF\u0082\u00B0 helix angle. The cutting force coefficients are identified mechanistically [92] by measuring the cutting forces at five different feed speeds in chatter-free cutting conditions. The cutting and edge force coefficients are identified as: 732MPa K 56MPa K 220MPa21.96N/mm K 23.21N/mm K 22N/mmtc rc acte re aeKK\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD The frequency response function of the flexible asymmetric cutting tool is shown in Figure 4.5. Note that the workpiece is assumed as rigid, thus its dynamics are excluded. The modal parameters of the tool are given in Table 4.1 and the most dominant modes are highlighted. 70 Figure 4.5 Real and imaginary parts of the measured FRF of the asymmetric end mill in principal rotating coordinates (u,v). Table 4.1 Modal parameters of asymmetric end mill in principal rotating coordinates (u,v). Tool is 2 fluted cylindrical end mill with 10 mm diameter, 13.5\u00C2\u00B0 rake and 25\u00C2\u00B0 helix angles. v - direction u - direction Modes Frequency [Hz] Damping [%] Mass [kg] Frequency [Hz] Damping [%] Mass [kg] 1 1536 4.296 0.6603 1522 3.824 0.4623 2 2498 3.617 0.1944 2283 4.375 0.0191 3 3207 3.699 0.0104 2612 2.810 0.0469 4 2702 2.500 0.1007 As seen from the Figure 4.5, only the tool modes are dominant and there is a considerable difference between the FRFs of the principal rotating coordinate frame \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v . Although the dynamic flexibilities between the \u00EF\u0080\u00A8 \u00EF\u0080\u00A9u and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9v directions are almost identical, the natural 71 frequencies along these directions are around 1000 Hz away from each other which lead to asymmetric dynamics of the cutting tool. Unbalance mass of the rotating structure causes undesirable vibrations which may damage the machine tool rotary components at high rotational speeds. High-speed machine tool spindles are smoothly balanced by the machine tool manufacturers but the holder and tool assembly may have unbalance mass which should be counter-balanced. Thus, the asymmetric cutting tool and holder assembly is balanced at the nominal speed of 20.000 rpm by adding 1.24 g mass to the holder in Hofmann MC10 PTB unbalance measuring system. The stability of the given system is solved in rotating frame by employing the proposed frequency and time domain methods. The feasible spindle speed range is selected as 14.000-20.000 rpm for half immersion milling operation considering the limitations of machine tool. The most dominant modes \u00EF\u0080\u00A8 \u00EF\u0080\u00A9: 2283Hz; :3207Hzu v of each flexible rotating direction of the tool are considered in the simulated stability diagrams. First, the prediction performances of both methods are compared in rotating coordinates. Figure 4.6(a) shows the stable and unstable regions of cutting for half immersion down-milling operation. The zero-order frequency domain and semi-discrete time domain solutions predict the stability borders similarly but the zero-order solution solves the entire stability region much faster than the semi-discretization method. However, if the radial immersion becomes too low, the periodicity of the milling system becomes too strong to be solved by the zero-order method as presented in [37]. The same cutting conditions are simulated for 10% radial immersion of tool as shown in Figure 4.6(b). As the periodicity of the directional coefficient matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( ), ( )t tA B are strong in the low radial immersion cutting, the zero-order frequency domain method fails to predict 72 the stability border accurately. Highly nonlinear system results bifurcations on the stability border which is unlikely to be seen for zero-order frequency solution. The existence and types of bifurcation in cutting process is thoroughly investigated in Section 4.3. Figure 4.6 Chatter stability diagrams in rotating coordinates (a) 50% immersion down milling and (b) 10% immersion down milling. Modal parameters are given in Table 4.1. The experimental validation of the stability model of asymmetric cutting dynamics is shown in Figure 4.7. The same cutting conditions are used for half immersion down-milling and stability of the system is solved both in principal rotating and fixed coordinate system stated in Eq. (4.7) with semi-discretization method. Cutting tests have been conducted on Quaser UX600 5-axis machining centre, and the vibration and sound data have been measured with an accelerometer and 73 microphone, respectively. The results are classified as stable, marginal, and chatter (unstable) by analyzing the frequency spectrum of the sound data and surface finish of each cutting test. Figure 4.7 Experimental verification of the stability for half immersion down milling of asymmetric end mill. Feedrate: 0.2 mm/tooth. Material: Al7050-T7451. Modal parameters of the tool are given in Table 4.1. 74 As seen from the Figure 4.7 (a), there is a significant difference between the rotating frame solution (black solid line) and the fixed frame solution (red dotted line) for the dynamically asymmetric end mill. The stability lobes shift due to different dynamic properties (natural frequency and modal mass) along the principal rotating coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,u v and the difference is even more evident when the dominant modes are further away from each other. When the system is assumed to be stationary and solved in fixed coordinate frame, the cutting condition at Point A is predicted as stable although the experiments and rotating frame solution indicate otherwise. Taking the FFT of the measured sound data at Point A, the dominant chatter frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A90c\u00EF\u0081\u00B7 is observed at 3052 Hz which is around the dominant flexibility direction of the asymmetric milling cutter (Figure 4.7 (b)). Other dominant peaks different than the tooth passing frequency harmonics are also detected at 2509 Hz and 3592 Hz which are exactly away from the dominant chatter frequency by tooth passing frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 0 0c c T\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0082\u00B1 \u00EF\u0080\u00BD \u00EF\u0082\u00B1 . The situation is opposite at Point B, which is predicted as unstable by fixed frame solution however the rotating frame solution and experiments prove that the cutting is stable at this point (Figure 4.7 (c)). Only the tooth passing frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A90T\u00EF\u0081\u00B7 and its harmonics \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, 1,2,...iT i\u00EF\u0081\u00B7 \u00EF\u0080\u00BD are seen in FFT spectrum of the sound data. Finally, the surface finish is also investigated as a chatter criterion. The stable cutting offers better surface without any chatter marks finish however the poor surface is obtained when the chatter occurs in the system as seen in Figure 4.7 (d)-(e). Proposed solution method is also validated for the symmetric milling cutters where the dynamic properties along the principal rotating coordinates are close to each other. For an example case, a 4-fluted cylindrical end mill having 12 mm diameter is selected and slot milling is employed in the simulations and experiments. Modal parameters of the tool along the flexible directions are 75 listed in Table 4.2 and the most dominant modes are highlighted. As seen from the modal properties of tool, both the natural frequencies and modal masses are close to each other and the system can be called dynamical symmetric system. Table 4.2 Modal parameters of 4-fluted cylindrical end mill with 12 mm diameter and 30 degree helix angle. v \u00E2\u0080\u0093 direction u - direction Modes Frequency [Hz] Damping [%] Mass [kg] Frequency [Hz] Damping [%] Mass [kg] 1 1496 2.902 0.3956 1512 2.822 0.4322 2 2227 4.107 0.0873 2193 8.596 0.0709 3 2879 1.779 0.0602 2891 2.094 0.0641 Stability of the given dynamic system in Table 4.2 is solved in time domain using semi-discretization method both in rotating and fixed coordinate frames. As in the non-symmetric cutter case, only the most dominant tool modes which are located at 2879 and 2891 Hz are considered. As seen from Figure 4.8, both the rotating and fixed frame solutions predict almost the same stability borders for the dynamically symmetric cutting tool. Validation tests are also proved the stability limits. While the cutting at Point A is predicted as chatter by the rotating and fixed frame solutions, it stabilizes for the Point B. FFT analysis of the measured sound spectrum at sample points also confirm the stability of the process. 76 Figure 4.8 Simulation and experimental results for dynamically symmetric end mill in rotating and fixed coordinate frames. Modal parameters are given in Table 4.2. 4.3 Generalized Dynamics of Turn-Milling The dynamics of turn-milling operations are investigated in this section. Mechanics model of turn-milling presented in Chapter 3.3 is extended by considering the relative vibrations between the rotating tool and workpiece when the machine follows a five-axis tool path. First, dynamics of turn-milling machine tool and its substructures are modeled in modal space, and dynamic chip thickness model is presented. Varying in-process workpiece dynamics are predicted with 77 analytical Timoshenko beam theory. Then, the time delay in the regenerative vibration system is modeled by considering the additional delay contributed by the rigid body rotational motion of workpiece. Finally, the resulting periodic, time-varying delay differential equations are solved in semi discrete time domain to predict the vibrations, and stability limits of the turn-milling process. The surface errors of the machined workpiece is modeled considering the mechanics and vibrations of the system The proposed dynamics and stability model of turn-milling is extended to the low-immersion and low-speed region. The model has been experimentally validated for various cases and materials. 4.3.1 Dynamic Model of Turn-Milling Process In a typical turn-milling machine tool, there are several sources of flexibility such as tool, tool holder, spindle, column, ram and workpiece whose vibrations are transmitted to the tool-workpiece contact zone as shown in Figure 4.9 (a). Figure 4.9 Structural flexibilities in a typical turn-milling machine tool (a); Dynamic displacements at tool and workpiece (b). 78 Relative displacements ( )sQ between the flexible tool and workpiece caused by the cutting force ( )sF acting on the structure can be expressed as follows; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A94 1 4 4 4 1( ) ( ) ( )q q q qs s s\u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0080\u00BDQ \u00CE\u00A6 F (4.42) where subscript q represents the number of contact points between the vibrating tool and workpiece as seen in Figure 4.9 (b). Depending on the type of the turn-milling process (i.e. orthogonal, tangential or co-axial), the contact region between the tool and workpiece may be large, hence multiple number of contact points may be needed to represent the distributed dynamics accurately instead of the lumped dynamics approach [33]. In this thesis, distributed dynamics representation includes dynamic properties of any type of kinematic configuration of the turn-milling process. However, the cases where the contact region is narrow, the lumped dynamics approach can be applied if the number of contact points is selected as one \u00EF\u0080\u00A8 \u00EF\u0080\u00A91q \u00EF\u0080\u00BD . The measured transfer function of a flexible machine tool component can be expressed in Machine Tool Coordinate System (MCS) as; MCS (4 4 ) 21( )2q qss s\u00EF\u0082\u00B4 \u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00ABT2n n\u00CE\u00A6 U UI \u00CE\u00B6\u00CF\u0089 \u00CF\u0089 (4.43) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A94 mq p\u00EF\u0082\u00B4U is the mass normalized mode shape matrix, \u00EF\u0080\u00A8 \u00EF\u0080\u00A9m mp p\u00EF\u0082\u00B4\u00CE\u00B6 and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9m mp p\u00EF\u0082\u00B4n\u00CF\u0089 are the diagonal damping ratio and natural frequency matrices for mp number of flexible modes of the structure, respectively. The modal parameters of machine tool structures can be identified by curve fitting techniques [92]. The displacements and cutting forces are modeled in Tool Coordinate System (TCS) through this chapter, thus the transfer function measured in MCS must be transformed to 79 TCS frame. The transfer function of the system is transformed from MCS \u00EF\u0080\u00A8 \u00EF\u0080\u00A9M M, MX ,Y ,Z to TCS \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t t ix ,y ,O as; TCS MCS 21( ) ( )2s ss s\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8T T T2n n\u00CE\u00A6 T\u00CE\u00A6 T T U U TI \u00CE\u00B6\u00CF\u0089 \u00CF\u0089 (4.44) where T is the transformation matrix obtained from the kinematic model of turn-mill machine tool as given in Chapter 3. The both coordinate systems are shown in Figure 4.10. The tx direction of TCS is aligned on the plane formed by the total feed vector Lf and tool normal axis iO , identical with the approach in the static chip thickness model of turn-milling process. Depending on the toolpath of the five-axis feed motion of turn-milling process, the tool FRF can be transformed to TCS at any cutter location \u00EF\u0080\u00A8 \u00EF\u0080\u00A9iCL . After the proper kinematic transformations of machine tool flexibilities, the equation of motion can be written in TCS as; 2 2( ) ( ) ( ) (4 ) (4 )( ) 2 ( ) ( ) ( )m m m m m m m mp p p p p p q p q ps s s s s s\u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BDT Tn n TCSQ \u00CE\u00BE \u00CF\u0089 Q \u00CF\u0089 Q TU F U T (4.45) Figure 4.10 Machine and Tool Coordinate Systems in turn-milling process and total feed vector representation. 80 The dynamic displacements of the system are solved in the modal space by the following transformation [33]; ( ) ( )s U s\u00EF\u0080\u00BD \u00EF\u0083\u0097Q \u00CE\u0093 (4.46) where ( )s\u00CE\u0093 and U is the modal displacement vector expressed in modal space and mode shape of the structure. Substituting the Eq. (4.46) into Eq. (4.45), the equation of motion can be represented in time domain and modal space as; T( ) 2 ( ) ( ) ( )t t t t\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD2 M Tn n\u00CE\u0093 \u00CE\u00B6\u00CF\u0089 \u00CE\u0093 \u00CF\u0089 \u00CE\u0093 TU F T (4.47) The cutting forces ( )tMF generated at the tool-workpiece contact zone act on both tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( )tM,TF and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( )tM,WF with the same magnitude but in opposite directions as; ( )( )( )ttt\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0080\u00AD\u00EF\u0083\u00AE \u00EF\u0083\u00BEM,TMM,WFFF (4.48) The equation of dynamics of the system at the flexible tool-workpiece contact (cutting) zone can be written as; Tt t t tTw w w w( ) 2 ( ) ( ) ( )( ) 2 ( ) ( ) ( )t t t tt t t t\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AD2 M Tt n,t n,t2 M Tw n,w n,w\u00CE\u0093 \u00CE\u00B6 \u00CF\u0089 \u00CE\u0093 \u00CF\u0089 \u00CE\u0093 TU F T\u00CE\u0093 \u00CE\u00B6 \u00CF\u0089 \u00CE\u0093 \u00CF\u0089 \u00CE\u0093 TU F T (4.49) tU and wU represents the mass normalized mode shape matrices of the tool and workpiece, respectively. Each flexible contact point at tool-workpiece contact zone has 4 flexibility directions both in the translational \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z and torsional \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00B1 modes of the system. Therefore, mode shape matrices are constructed as [33]; 81 1,1,t 1,2,t 1, ,t 1,1,w 1,2,w 1, ,w2,1,t 2,1,wt,1,t ,2,t , ,t ,1,w ,2,w , ,w4 4;t wt wt wp pwq q q p q q q pq p q p\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBu u u u u uu uU Uu u u u u u (4.50) where tp and wp stand for the number of flexible modes for tool and workpiece, respectively and satisfy the condition m t wp p p\u00EF\u0080\u00BD \u00EF\u0080\u00AB . Each element in the tool and workpiece mode shape matrix \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t wU U consists of the individual contributions of translational and torsional mode shapes as; , , , , , ,, , , , , ,, ,t , ,t, , , , , ,, , , , , ,;t wt wt wt wt wx q p t x q p wy q p t y q p wq p q pz q p t z q p wq p t q p wu uu uu uu u\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BEu u (4.51) Note that the individual mass normalized mode shape for translational and torsional directions are obtained from modal analysis of the measured tap tests. The damping \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t w\u00CE\u00B6 ,\u00CE\u00B6 and natural frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A9n,t n,w\u00CF\u0089 ,\u00CF\u0089 matrices in Eq. (4.49) can be constructed similarly. After expressing all the matrices explicitly, Eq. (4.49) is unified in one matrix form in modal space as follows; Tt tt tw ww2 0 0 ( )( ) ( )( )0 2 ( )( ) ( ) 0wtt tttt t\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB2t n,t n,t M T2w n,w n,w\u00CE\u00B6 \u00CF\u0089 \u00CF\u0089 \u00CE\u0093 U\u00CE\u0093 \u00CE\u0093I T F T\u00CE\u00B6 \u00CF\u0089 \u00CE\u0093 U\u00CE\u0093 \u00CE\u0093 \u00CF\u0089 (4.52) In the following section, the forcing term ( )tMF in Eq. (4.52) is modeled in the modal space. 4.3.2 Prediction of In-Process Workpiece Dynamics The workpiece dynamics are usually ignored since their contribution is negligible compared to cutting tool, especially for long slender end mills in regular milling operations. However, in some applications, the workpiece can be as flexible as, or much more flexible than, the cutting tool such as in the case of machining of turbine blades, long shafts and thin-walled parts. In some of 82 the turn-milling machine tools, workpiece is supported by a tailstock from the free end, thus the lateral displacements at the free end are mostly nullified as seen in the Figure 4.11 (b) and the workpiece can be assumed as rigid compared to the cutting tool. Figure 4.11 Turn-milling operations with (b) and without (a) tailstock. In some cases, the workpiece can be mounted only on the rotating chuck like a cantilever beam as seen in Figure 4.11 (a). In this case, the workpiece FRF may be more flexible than the cutting tool, therefore, its dynamics must be included in order to represent the cutting dynamics accurately. Dynamic response of the workpiece may vary continuously during the turn-milling cutting process. As the milling tool removes material from the rotating workpiece, the volume hence the mass of the workpiece changes. Considering the high material removal capability of the turn-milling process compared to turning operations, the flexible workpiece dynamics may change between consecutive passes, thus the stability properties of the cutting system are altered significantly as seen in Figure 4.12. 83 Figure 4.12 Variation in the workpiece geometry between consecutive passes of milling tool. The FRF of the flexible workpiece can be represented as given in Eq. (4.43) as follows; 2,1( )2ss s\u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00ABTW W W2w n,w n w\u00CE\u00A6 U UI \u00CE\u00B6 \u00CF\u0089 \u00CF\u0089 (4.53) As the mass is removed from the workpiece, the natural frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A9n,w\u00CF\u0089 and mass normalized mode shape matrix \u00EF\u0080\u00A8 \u00EF\u0080\u00A9WU changes, hence the stiffness of the workpiece. The modal parameters can be identified by experimental tap tests and initial dynamics of the workpiece can be calculated by Eq. (4.53). However, it is not possible to measure the modal parameters during the cutting process, and in-process workpiece dynamics become unknown. Therefore, the workpiece dynamics must be predicted analytically. Receptance Coupling (RC) theory [97] is used in the literature to predict the tool tip FRF analytically. The spindle, holder and tool is modeled as simple beam elements using the Timoshenko Beam theory, and coupled to each other rigidly or elastically. The same approach is applied here to predict the workpiece tip dynamics which has the most flexible stiffness of the structure. In Figure 4.13, Receptance Coupling of two simple beams is illustrated. The receptances of each beam (A and B) are calculated by Timoshenko Beam theory as follows [97]; 84 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 21 1 1 1 1 1 1 1 1 2 11 1 1 1 1 1 1 11 1 1 1 1 1 1 1, , , , , , , ,, , , , , ,, , , ,, , , ,4 4;x x y x y x x x x x y x y x x xx y y y y y x y x y yx y y y y y x yx x y x y x x xA A A A A A A A A A A A A A A AA A A A A A A A A A A AA A A A A A A AA A A A A A A AH G H G H G H GN P N P N PH G H GN P N P\u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB11 12A A2 1 2 1 21 2 1 2 1 2 1 21 2 1 2 1 2 1 22 1 2 1 2 1 2 12 1 2 1 2 1 2 12 1 2 1 2 1 2 1, ,, , , ,, , , ,4 4, , , ,, , , ,, , , ,y y y x yx y y y y y x yx x y x y x x xx x y x y x x xx y y y y y x yx y y y y y x yA A A AA A A A A A A AA A A A A A A AA A A A A A A AA A A A A A A AA A A A A A A AN PH G H GN P N PH G H GN P N PH G H GN\u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0080\u00BD21A2 2 2 2 2 2 2 22 2 2 2 2 2 2 22 2 2 2 2 2 2 22 1 2 1 2 1 2 1 2 2 2 2 2 2 2 2, , , ,, , , ,, , , ,, , , , , , , ,4 4 4 4;x x y x y x x xx y y y y y x yx y y y y y x yx x y x y x x x x x y x y x x xA A A A A A A AA A A A A A A AA A A A A A A AA A A A A A A A A A A A A A A AH G H GN P N PH G H GP N P N P N P\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB22A (4.54) where H, G, N, P indices represents the displacement-force, rotation-force, displacement-moment and rotation-moment transfer functions of the structure A at point 1 in x and y directions, respectively. They are calculated from the finite element model of Timoshenko Beam theory for given geometry (length, diameter), and material properties (Young modulus, density, passion ratio, loss factor) of the structure. If only the direct transfer functions \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2 matrix\u00EF\u0082\u00B4 of the system are needed ( 11,xxA or 11,yyA ), then the cross-talk terms can be dropped from the Eq. (4.54). Receptance matrix of the beam B can be formulated in a similar way shown in Eq. (4.54). Figure 4.13 Receptance coupling of Structure A and B. 85 The compatibility and continuity conditions at point \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 1A B\u00EF\u0080\u00AD states; 2 ,2 ,12 ,2 ,1A BA BF F FX X X\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00BD (4.55) Then, the displacements at the coupling interface can be calculated as; ,2 ,2,1 ,1A AB BX FX F\u00EF\u0080\u00BD\u00EF\u0080\u00BD2211AB (4.56) Total transfer function at the coupling interface can be expressed as \u00EF\u0080\u00A8 \u00EF\u0080\u00A922 11A +B and substituted into the Eq. (4.56). If the equations are rearranged, the transfer functions of the coupled structure C, are found as follows [97]; \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0081\u009B \u00EF\u0081\u009D111\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB11 11 12 22 11 2112 12 22 11 1221 21 22 11 21-122 22 21 22 11 12C A A A B AC A A B BC B A B AC = B - B A + B B (4.57) Eq. (4.57) represents the direct and cross-talk transfer functions of the coupled structure at point 1 and 2. In turn-milling process, the demonstrated Receptance Coupling theory is applied to the chuck-workpiece structures. First, the dynamics of the chuck (C-axis) is calculated as the initial step. Preferably a short dummy workpiece is mounted on the machine tool chuck and the direct \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,11 ,22,Cdw CdwH H and cross \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,12CdwH FRFs of the structure (Cdw) at point 1 and 2 are measured. Then, the receptance matrix of the dummy workpiece (structure-dw) is calculated using the finite element model of Timoshenko Beam theory for the given geometry and material properties. Finally, the analytically calculated receptances of the structure (dw) is decoupled from the 86 measured FRFs using the Eq.(4.57) and the transfer functions of the structure CH up to the decoupling line are obtained. The decoupling procedure is shown in Figure 4.14. Figure 4.14 Decoupling of the dummy workpiece and chuck dynamics identification. Note that, the dummy workpiece is decoupled until the contact line between the jaws and workpiece. Therefore, the translational and rotational contact stiffness \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,ttk k\u00EF\u0081\u00B1\u00EF\u0081\u00B1 and damping \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,ttc c\u00EF\u0081\u00B1\u00EF\u0081\u00B1 in the contact interface is kept under the chuck dynamics and there is no need for the further joint flexibility analysis as oppose to holder-tool joint interface identification in milling operations. Once the dynamics (FRFs) of the machine\u00E2\u0080\u0099s chuck is obtained, any cylindrical workpiece geometry can be coupled on the calculated FRF of the chuck. Let WPH and CH are the transfer functions of the free-free FRF of the workpiece to be coupled and identified chuck FRF, respectively. The coupled assembly\u00E2\u0080\u0099s transfer function is CWPH . First, the free-free transfer functions of the workpiece is calculated by Timoshenko Beam based on finite element method. Complex shape of the part should be divided into finite number of substructures (beam) as shown in Figure 4.15. 87 Figure 4.15 Receptance coupling of individual beam elements of the shaft mill. Starting from the right end of the subassembly, each substructure is coupled to the adjacent component \u00EF\u0080\u00A8 \u00EF\u0080\u00A9I II\u00EF\u0082\u00AE \u00EF\u0082\u00AE and free-free FRF of the workpiece is obtained. Finally, the free-free workpiece FRF is coupled rigidly on the chuck FRF which is identified previously as follows (see Figure 4.16); \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB-1CWP WP WP WP C WP11 11 12 22 22 21-1CWP WP WP C C12 12 22 22 12-1CWP C WP C WP21 21 22 22 21-1CWP C C WP C C22 22 21 22 22 21H H H H H HH H H H HH H H H HH H H H H H (4.58) Similar calculation can be performed to solve the FRFs in other directions. In Eq. (4.58), the transfer function CWP11H represents the workpiece FRF at the tip. Since the volume of the workpiece changes due to the continuous mass removal in turn-milling process, the new FRFs at the tip can be analytically calculated for given geometry and material properties at any time instant. 88 Figure 4.16 Receptance Coupling of the workpiece to the machine\u00E2\u0080\u0099s chuck. The workpiece dynamics can be analytically predicted through the given toolpath of the turn-milling process. Therefore, the predicted workpiece FRFs can be mapped on the toolpath and the process parameters will be selected more accurately for the increased productivity of the turn-milling process. For instance, the milling cutter may cut deeper at the initial passes of the operation due to the high rigidity of the workpiece compared to the finish part. Then, the depth of cut is gradually decreased as the diameter reduces, hence the dynamic stiffness. Also, as the natural frequencies of the workpiece varies during the cutting process due to the mass removal, the optimum spindle speed that cuts deeper need to be adopted from the prediction of in-process workpiece dynamics. 89 4.3.3 Modeling of Dynamic Chip Thickness and Cutting Forces in Turn-Milling In Section 3.3, the chip thickness and cutting forces are modeled mechanistically considering only the multi-axes rigid body feed motion of the machine drives and tool and workpiece spindles. When the flexible components of the machine tool vibrate, the chip thickness and cutting forces at the tool-workpiece contact region have dynamic parts which are dependent on the present and past time instants. In this section, the exact kinematics of turn-milling systems which considers both rigid body motion of the machine and vibrations are modeled to calculate the dynamic chip thickness and cutting forces. The cutting forces are first modeled in physical coordinates, then transformed to modal space using the same transformation that is applied for the dynamic displacements in Section 4.3.1. 4.3.3.1 Dynamic Chip Thickness Model Considering a milling cutter having tN number of tooth and rotating at constant spindle speed [rev/min]t\u00EF\u0081\u0097 , the instantaneous immersion angle for thj tooth at axial elevation z and time t is written as (see Figure 4.17); \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( , ) ( ) ( 1) ( ) 2 tan (1, 1)j p t tt z t j z z D j N\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A2\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0083\u008E \u00EF\u0080\u00AD (4.59) where ( )t\u00EF\u0081\u00A6 is the angular position of the tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9( ) 2 60tt t\u00EF\u0081\u00A6 \u00EF\u0081\u00B0\u00EF\u0080\u00BD \u00EF\u0081\u0097 , p\u00EF\u0081\u00A6 is the regular pitch angle of the cutter, \u00EF\u0081\u00A2 and tD represents the helix angle and cutter diameter, respectively. The total chip thickness in turn-milling process is expressed by superposing the static \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sjh and dynamic \u00EF\u0080\u00A8 \u00EF\u0080\u00A9djh components as; ( ) ( , ) ( , )s dt j j j jh t h t h t\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AB (4.60) 90 Figure 4.17 The general representation of helical tool geometry on a ball end mill. As mentioned before, the static chip thickness \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sjh is contributed by the multi-axes rigid body feed motion of the tool and workpiece as modeled in Chapter 3. The dynamic chip thickness \u00EF\u0080\u00A8 \u00EF\u0080\u00A9djh is caused by the regenerative vibrations at the present time \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t and one delay period before \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j it \u00EF\u0081\u00B4\u00EF\u0080\u00AD . The dynamic chip thickness can be expressed by projecting the displacement vector on the unit surface vector at axial elevation z as; ( , )dj jh t\u00EF\u0081\u00A6 \u00EF\u0080\u00BD \u00EF\u0082\u00B7 j,zd n (4.61) where d and j,zn are the dynamic displacement and unit surface vectors, respectively. The regenerative displacement vector can be defined in TCS frame as; ,, ,,( ) ( )( ) ( ) ( ) ( )( ) ( )d d d j ij i d d d j id d d j iTCS TCSx x t x tt t y y t y tz z t z t\u00EF\u0081\u00B4\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0081\u00B4\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0081\u0084 \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0084 \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BBd q q (4.62) where ( )tq and ,( )j it \u00EF\u0081\u00B4\u00EF\u0080\u00ADq \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z\u00EF\u0083\u008Eq are the relative vibrations between tool and workpiece at present time \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t and one delay period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 before. Note that the unit tool axis vector \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t,ix at cutter location i is positioned on the plane formed by the resultant feed direction \u00EF\u0080\u00A8 \u00EF\u0080\u00A9Lf and the unit 91 tool normal axis \u00EF\u0080\u00A8 \u00EF\u0080\u00A9iO as seen in Figure 4.10. Hence, the displacement vector represented in TCS frame are already aligned with the feed, cross feed and normal axis of the process coordinate system [98], and there is no further transformation required. The unit surface outward vector j,zn is calculated as a function of tool\u00E2\u0080\u0099s radial \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( , )j t z\u00EF\u0081\u00A6 and axial \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( )z\u00EF\u0081\u00AB immersion angles as (Figure 4.17 (c)); ( , ) sin ( )sin ( , ) . sin ( )cos ( , ) . cos ( ) .z z t z z t z zj j j\u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00ADj,zn i j k (4.63) Note that ( ) 2z\u00EF\u0081\u00AB \u00EF\u0081\u00B0\u00EF\u0080\u00BD for cylindrical end mill. 4.3.3.2 Dynamic Cutting Force Model in Turn-Milling Process The discrete forces acting on the differential chip element can be written in radial \u00EF\u0080\u00A8 \u00EF\u0080\u00A9r , tangential \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t and axial \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a directions of thj tooth by superposing the dynamic and static components as (Figure 4.17 (c)); ( , ) ( , ) ( , ) ( , ) ( , )j j j j jd z d z d z d z d z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00ABt d,c d,pd s,c s,erta rta rta rta rtaF F F F F (4.64) where ( , )jd z\u00EF\u0081\u00A6trtaF is the total differential force vector represented in \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta frame. ( , )jd z\u00EF\u0081\u00A6d,crtaF represents the differential dynamic cutting force vector due to the shearing of the chip from the workpiece and friction between the chip and rake face of the cutting edge. ( , )jd z\u00EF\u0081\u00A6d,pdrtaF is the process damping force vector as a result of the contact between the flank edge of the tool with the newly cut wavy surface which will be explained in detail in Section 4.3.3.3. The static cutting ( , )jd z\u00EF\u0081\u00A6s,crtaF and edge cutting ( , )jd z\u00EF\u0081\u00A6s,ertaF force vectors are neglected since they do not contribute to the stability properties of the system. Differential dynamic cutting forces can be expressed as [92]; 92 ,j ,,( , )( , ) ( , ) ( , ) ( )( , )r j j rcdj t j j tc j ja j j acdF z Kd z dF z K h t db zdF z K\u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u0097 \u00EF\u0083\u0097\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BEd,crta,F (4.65) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,rc tc acK K K are the cutting force coefficients in radial, tangential and axial directions, respectively. ( )db z is the differential contact length and calculated as ( ) sin ( )db z dz z\u00EF\u0081\u00AB\u00EF\u0080\u00BD where dz is the axial length of the differential cutting element and function of the tool spindle\u00E2\u0080\u0099s lead angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9B\u00EF\u0081\u00B1 due to the tilting of spindle B-axis as seen in Figure 4.18. Figure 4.18 Geometrical representation of discrete chip geometry and axial depth of cut. Differential dynamic cutting forces in \u00EF\u0080\u00A8 \u00EF\u0080\u00A9rta frame can be transformed to physical Cartesian coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z of TCS and integrated over the cutter-workpiece engagement (CWE) limits by adding the contributions of each tN number of cutting edges as; 93 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,1,0,( ) ,( ) ( )sin ( )sin ( , ) cos ( , ) cos ( )sin ( , )( , ) ( , ) sin ( )cos ( , ) sin ( , ) cos ( )cos ( , )cos ( ) 0 sin ( )( ) ( , )TCSjTCS TCSjj j jd cxyz j rta j j j j jzxyz z xyz jj zz t z t z z t zdF z dF z z t z t z z t zz zF g dF z\u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB \u00EF\u0081\u00A6\u00EF\u0081\u00AB \u00EF\u0081\u00AB\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0083\u0097 \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AD\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0080\u00BD \u00EF\u0083\u00B21 1T T11,( )0,t z zNst j exzdz gotherwise\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0083\u00AC \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0083\u00AD\u00EF\u0083\u00AE\u00EF\u0083\u00A5 (4.66) Unlike the regular milling process, the CWE varies not only along the tool path but also over the tool normal axis in turn-milling process regardless of the B-axis rotation of tool\u00E2\u0080\u0099s spindle, hence the start zst\u00EF\u0081\u00A6 and zex\u00EF\u0081\u00A6 angles are considered for each individual axial discrete force element. Substituting the Eq. (4.61) and Eq. (4.65) into the Eq. (4.66), the total cutting forces in Cartesian coordinates of TCS are expressed as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,,,( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )x j rc d d j iy j tc d d j iz j ac d d j iTCSF K x t x tF z g K y t y tF K z t z t\u00EF\u0081\u00A6 \u00EF\u0081\u00B4\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00B4\u00EF\u0081\u00A6 \u00EF\u0081\u00B4\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00AB \u00EF\u0083\u00BB1 2T T (4.67) where the rotation matrix is defined as sin cos cotj j\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00AB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB2T . Eq. (4.67) represents the total cutting forces in physical coordinates \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z of TCS but the equation of motion given in Eq. (4.52) is in the modal space. Therefore, the cutting forces are transformed from the physical to modal space using the similar transformation used in the vibration transformation in Section 4.3.1 as follows; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00BB \u00EF\u0081\u00BD ;T Tx j y j z jTCSF F F\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u0097 \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BBM phy phy T WF F U F U U U (4.68) The coefficient matrices Eq. (4.67) are set according to the q number of the contact points at tool-workpiece engagement zone and 4 flexibility directions at each point \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , ,x y z \u00EF\u0081\u00B1 . The resulting generalized dynamics of turn-milling process can be expressed in the modal space as; 94 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tt twwTt t,w w2 0 ( )0( ) ( )0 2 ( )0( ) ( )( ) ( )coswrctc j iBactt ttt tKzg K t tK\u00EF\u0081\u00A6 \u00EF\u0081\u00B4\u00EF\u0081\u00B1\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0081\u0084 \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B82t n,t n,t2w n,w n,wT1 2\u00CE\u00B6 \u00CF\u0089 \u00CE\u0093\u00CF\u0089\u00CE\u0093 \u00CE\u0093I\u00CE\u00B6 \u00CF\u0089 \u00CE\u0093\u00CF\u0089\u00CE\u0093 \u00CE\u0093U UT T T \u00CE\u0093 \u00CE\u0093 TU U (4.69) Eq. (4.69) represents the dynamic systems equations considering the forcing terms resulted from the dynamic cutting forces. On the other hand, the dynamic process damping forces should be taken into account for the cases where the cutting speed is low for the machining of hard to cut materials that leads increased dynamic indentation forces between the flank face of the tool and wavy workpiece surface. Section 4.3.3.3 models the dynamic process damping forces in turn-milling process, and calculates additional forcing term to incorporate into the dynamics model of the turn-milling process. 4.3.3.3 Dynamic Process Damping Force Model in Turn-Milling Process As the workpiece material flows away the workpiece during the chip removal process in turn-milling, the chip separates upwards and downwards directions at the separation point on the honed edge of the tool. While the material above the separation points shears and slides over the rake face of the tool, the material under the tool is indented (ploughed) by the flank face and honed edge of the tool, yielding dynamic indentation forces in radial d,pdrF and tangential d,pdtF directions of the flank face as seen in Figure 4.19. 95 Figure 4.19 Representation of indented volume under the flank face of the cutting edge. At low spindle speeds, shorter undulation vibration waves are generated on the workpiece. As the tool moves on the wavy surface, more material is ploughed by the flank face of the tool, resulting more indentation volume \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dV and increased process damping for the cutting mechanism, hence increases the stable axial depth of cut (see Figure 4.20 (a)). On the other hand, vibration wavelength left on the workpiece are longer at high spindle speeds, thus less material is indented by the flank face and the process damping diminishes in the cutting (see Figure 4.20(b)). Unlike the shearing forces which are proportional with uncut chip thickness as explained in Section 4.3.3.2, the dynamic process damping forces are function of volume extruded by the flank face of the tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dV as follows [45]; sp dcK V\u00EF\u0081\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00BDd,pdrd,pd d,pdt rFF F (4.70) where spK and c\u00EF\u0081\u00AD are the material specific process damping (indentation) coefficient and coulomb friction coefficient, respectively. 96 Figure 4.20 Indented volume by the flank face of the tool for short and long wavelengths, corresponding to low speeds (a) and high speeds (b). Assuming the vibrations during the cutting process have small amplitudes [45], the indented volume can be approximately calculated as; 2 ( )2fwdRd xLV aV dt\u00EF\u0080\u00BD \u00EF\u0080\u00AD (4.71) where a , wL , RV and fx are the depth of cut [mm], flank face wear length [mm], cutting speed [m/min] and vibrations at the feed direction of the cutting, respectively. The indentation coefficient \u00EF\u0080\u00A8 \u00EF\u0080\u00A9spK and coulomb friction coefficient \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u00AD are identified by different methods which are studied by several authors [53], [54]. The flank face wear land length is calculated as a function of hone radius \u00EF\u0080\u00A8 \u00EF\u0080\u00A9hr , separation angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sp\u00EF\u0081\u00A2 and clearance angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00A7 as [54]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sin sin cos cos cotw h sp h hL r r r\u00EF\u0081\u00A2 \u00EF\u0081\u00A7 \u00EF\u0081\u00A7 \u00EF\u0081\u00A2 \u00EF\u0081\u00A7\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AD (4.72) 97 The calculated process damping forces in radial and tangential directions are substituted in Eq. (4.64), and the resulting dynamic equations of motion can be written as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tt twwTt,w2 0 ( )0( ) ( )0 2 ( )0( ) ( )( ) ( )cos0wrctc j iBactt ttt tKzg K t tK\u00EF\u0081\u00A6 \u00EF\u0081\u00B4\u00EF\u0081\u00B1\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A6 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0081\u0084 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00A7 \u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A82t n,t n,t2w n,w n,wd,pdrd,pd1 2 t\u00CE\u00B6 \u00CF\u0089 \u00CE\u0093\u00CF\u0089\u00CE\u0093 \u00CE\u0093I\u00CE\u00B6 \u00CF\u0089 \u00CE\u0093\u00CF\u0089\u00CE\u0093 \u00CE\u0093FUT T T \u00CE\u0093 \u00CE\u0093 FUtw\u00EF\u0083\u00B6\u00EF\u0083\u00B7 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00B7 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00B7\u00EF\u0083\u00B8TUTU (4.73) Note that, the dynamic process damping force terms are function of vibration velocity. When they are expressed at the right hand side of the Eq. (4.73), they provide positive damping to the system, thus increase the damping at low speeds due to the increased indentation volume. The presence of high vibration amplitudes during cutting may cause inaccuracy in calculating the approximate indentation volume given in Eq. (4.71). Although it is not within the scope of this thesis, there are more advanced models in the literature where the indentation volume can be calculated numerically by discretizing the trajectory of the tool and flank face of the tool [53], [54]. Although those models are capable of calculating the indentation volume more accurately, the material specific process damping constant \u00EF\u0080\u00A8 \u00EF\u0080\u00A9spK must be identified by numerous experiments for each workpiece material, flank wear length and flank face geometry even the approximate indentation volume assumption is used. There is a need for a comprehensive database for the process damping coefficients which is costly. In Section 4.3.5, different process damping model is adapted into turn-milling process which relies on less experiments, and analytically calculates the stability envelope at low spindle speeds by solving the asymptotic spindle speed and absolute stability limit. 98 The dynamic flexibilities and process coefficients are modeled in Eq. (4.73). The only unknown is the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 between the present and previous time instants of cutting, which needs to be modeled to solve the DDE for the given set of process parameters. 4.3.4 Modeling of Varying Time Delay in Turn-Milling Process First, the translational and rotational motions of tool and workpiece are discretized in space. Positions of each tooth of the tool and its vibration marks imprinted on the surface of rotating workpiece are evaluated. The additional time delay contributed by the rigid body rotational motion of workpiece is modeled, and the total time delay in the regenerative cutting system is evaluated as follows. 4.3.4.1 Discrete Tool and Workpiece Motion in Turn-Milling Tooth j of the cutter leaves deflection mark at point \u00EF\u0080\u00A8 \u00EF\u0080\u00A9ijtP on the rotating workpiece surface at time it during machining. Then, the same point moves to the surface point \u00EF\u0080\u00A8 \u00EF\u0080\u00A9*i tjtP \u00EF\u0080\u00AB\u00EF\u0081\u0084 after discrete time interval t\u00EF\u0081\u0084 due to the angular motion of workpiece. After one tooth period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT , the tool moves in the commanded feed direction and tooth \u00EF\u0080\u00A8 \u00EF\u0080\u00A91j \u00EF\u0080\u00AB engages with the workpiece at point \u00EF\u0080\u00A8 \u00EF\u0080\u00A91i djt TP\u00EF\u0080\u00AB\u00EF\u0080\u00AB , and the surface point cut by the previous tooth j moves to a new position \u00EF\u0080\u00A8 \u00EF\u0080\u00A9*i djt TP \u00EF\u0080\u00AB as seen in Figure 4.21. Hence, the presence of simultaneous tool and workpiece motions alters the position of vibration waves imprinted on the workpiece surface, resulting in an additional time delay between the present and previous periods. 99 Figure 4.21 Relative positions of surface points on the rotating workpiece in turn-milling The spindle period is divided into m number of discrete points. The linear and angular tool-workpiece motions are digitized, and the time history of point on the machined surface is stored as an array in Cartesian coordinates. The model represents both multi-axes rigid body kinematics and structural dynamic motions of turn-milling system which leads to exact (true) chip thickness history as presented by Montgomery and Altintas for regular milling operations [15]. 4.3.4.1.1 Tool Motion The edge of each tooth moves along a trochoidal trajectory defined by the feed vector contributed by three translational drives of the turn-milling machine tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y zf f f and angular speed of milling cutter \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 . The full motion of the tool \u00EF\u0081\u00BB \u00EF\u0081\u00BDTfP can be represented as the superposition of rigid body motion and relative vibrations between the cutter and workpiece at each discrete time it as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD, , ,TT T Tr d r d r di i i i ix y z\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0081\u009C \u00EF\u0080\u00BDT T T Tf r d r,dP P P P (4.74) where \u00EF\u0081\u00BB \u00EF\u0081\u00BDiTrP and \u00EF\u0081\u00BB \u00EF\u0081\u00BDiTdP are rigid (static) and dynamic displacement vectors of the tool in Cartesian coordinates, respectively. The rigid body motion of thj tooth is evaluated along the toolpath as; 100 \u00EF\u0081\u00BB \u00EF\u0081\u00BD, 0, 0, 0cos( )Y sin( )T Tr j i t jT Tr j i t jiT Tr j iix X X i x Ry Y i y Rz Z Z i z\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0080\u00AB \u00EF\u0081\u0084 \u00EF\u0080\u00AB\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0081\u0084 \u00EF\u0080\u00AB\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AB \u00EF\u0081\u0084\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BETrP (4.75) where tR is the tool radius, \u00EF\u0080\u00A8 \u00EF\u0080\u00A90 0 0, ,X Y Z are the initial coordinates of the cutter in space, and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,x y z\u00EF\u0081\u0084 \u00EF\u0081\u0084 \u00EF\u0081\u0084 are the discrete feed motion of cutter in space evaluated at each discrete time interval, t\u00EF\u0081\u0084 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9. . xi e x f m\u00EF\u0081\u0084 \u00EF\u0080\u00BD . Neglecting the tool\u00E2\u0080\u0099s torsional FRF, dynamic displacements of tool are expressed in Laplace domain as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD( ) ( ) ( ) ( ) ( )( ) y ( ) ( ) ( ) ( ) ( )z ( ) ( ) ( ) ( ) ( )d xx xy xz xd yx yy yz yd zx zy zz zx s s s s F ss s s s s F ss s s s F s\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BBTdP (4.76) where the FRF of the tool ( )ij s\u00CE\u00A6 for tp number of modes is given in Laplace domain as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 21 , ,( ) ; , , ,2tt tt t t tpp pijp p n p n pis i j x y zs\u00EF\u0081\u00A1 \u00EF\u0081\u00A2\u00EF\u0081\u00BA \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00BD\u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00A5\u00CE\u00A6 (4.77) Discrete time response of the cutter \u00EF\u0081\u00BB \u00EF\u0081\u00BDTd d dx (k), y (k),z (k) is evaluated by transforming the FRF(s) and cutting forces from Laplace domain to discrete time using Tustin Method [92] as; 10 1 2 11 0 1 22 2 0 1 2 11 1, , 0 1 2( )2 1( )2 1t tt tt tt t tp pp pijp pp n p n ps zi b z b z b z zs z ss a z a z a z T z\u00EF\u0081\u00A1 \u00EF\u0081\u00A2\u00EF\u0081\u00BA \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0082\u00AE\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0081\u0086 \u00EF\u0082\u00AE \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0080\u00BD\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00A5 \u00EF\u0083\u00A5 (4.78) where the FRF(s) and cutting forces are sampled at T sampling time. The coefficients of Eq. (4.78) are evaluated as; 101 2 202 212 222021224 48 24 4222 2n nnn na T Ta Ta T Tb T Tb Tb T T\u00EF\u0081\u00BA\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0081\u00BA\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00A2 \u00EF\u0081\u00A1\u00EF\u0081\u00A1\u00EF\u0081\u00A2\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AB (4.79) Analytically calculated cutting forces in Section 3.3 are also sampled at the same discrete time. The dynamic displacements of tool calculated in discrete time domain becomes; \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 1 11 1 11 1 1( ) ( ) ( ) ( ) ( )( ) y ( ) ( ) ( ) ( ) ( )z ( ) ( ) ( ) ( ) ( )d xx xy xz xd yx yy yz yd zx zy zz zx i t z z z F i ti t i t z z z F i ti t z z z F i t\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0081\u0084 \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0084\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0084\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0084 \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0086 \u00EF\u0081\u0084\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BBTdP (4.80) Once the discrete tool motion is calculated at each discrete time instant, full motion can be evaluated from Eq. (4.74) (see Figure 4.22). Figure 4.22 Representation of full discrete tool motion in Cartesian coordinates. As the tool moves along the commanded multi-axes feed direction, the position history of each tooth j is stored in an array \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9Tool , , , i tj x y z \u00EF\u0081\u0084 and used for the workpiece motion and time delay analysis. 102 4.3.4.1.2 Workpiece Motion The angular rotation of workpiece is discretized at the same time interval t\u00EF\u0081\u0084 used for the tool discretization but in 2 stages. First, the coordinates of surface points \u00EF\u0081\u00BB \u00EF\u0081\u00BDjiP imprinted by tooth j at time instant i are evaluated in TCS. Then, the surface point \u00EF\u0081\u00BB \u00EF\u0081\u00BDjiP is rotated along the circular surface of the workpiece and moved to the new position \u00EF\u0081\u00BB \u00EF\u0081\u00BD*ji+\u00CE\u0094tP at next discrete time interval \u00EF\u0080\u00A8 \u00EF\u0080\u00A91i \u00EF\u0080\u00AB . This procedure is repeated for each discrete point and tooth until one spindle period is covered. The position history of generated workpiece surface points are stored in an array WP( , )m tj,x, y z \u00EF\u0081\u0084 as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,,,P WPR co. Tool .P WP . Tool .P WP s( (m)).ji xji yji z w wm x m xmj jy m yjm zj j\u00EF\u0081\u00AA\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE\u00EF\u0080\u00BD\u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BEjiP (4.81) where ( )w m\u00EF\u0081\u00AA is the angle between the contact point of tooth j with the workpiece at discrete time interval .m t\u00EF\u0081\u0084 and the axis perpendicular to the workpiece rotation direction \u00EF\u0080\u00A8 \u00EF\u0080\u00A9MZ as (see Figure 4.23); 1,P( ) tan ( ,(R a))w wji ym\u00EF\u0081\u00AA\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AD (4.82) where wR and a are workpiece radius and axial depth of cut, respectively. 103 Figure 4.23 Discretization of workpiece motion and resulted phase difference. The generated surface points are rotated by a discrete rotation angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9w\u00EF\u0081\u00AA\u00EF\u0081\u0084 at discrete time step \u00EF\u0080\u00A8 \u00EF\u0080\u00A91m\u00EF\u0080\u00AB to consider the angular rotation of the workpiece which occurs at \u00EF\u0080\u00A8 \u00EF\u0080\u00A9-M MY Z plane. Therefore, the y and z coordinates of array WP( , )m tj,x, y z \u00EF\u0081\u0084 are updated as shown in Figure 4.23. Note that the x coordinates of the surface points remain same during the update stage, hence the arrays are updated as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9* *1,* *1,* *1,1P WP 1 . WP . P WP 1 .P WP 1 .R sin( (m) )R cos( (m) )ji xji yji zw w ww w wm x m xm ym zj jj mj m\u00EF\u0081\u00AA \u00EF\u0081\u00AA\u00EF\u0081\u00AA \u00EF\u0081\u00AA\u00EF\u0080\u00AB\u00EF\u0080\u00AB\u00EF\u0080\u00AB\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AB\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0081\u0084\u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AF \u00EF\u0083\u00AF \u00EF\u0080\u00AD \u00EF\u0081\u0084\u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00AB \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AE \u00EF\u0083\u00BE*ji+P (4.83) where the discrete rotation angle is calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A960 2w t\u00EF\u0081\u00AA \u00EF\u0081\u00B0\u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0081\u0084C\u00CE\u00A9 (4.84) 104 The algorithm updates all the surface points generated by each tooth of the milling cutter. The sample updating operation that shows the present and past states of the WP( , )m tj,x, y z \u00EF\u0081\u0084 array is shown as follows; 0 00 03 31 20 0 0 01 2 3 30 0 0 01 2 3 31 2 3(3). (3).(1). (2).present states (1). (2). (3). (3).(1). (2). (3). (3).im m m m mWP x x WP x xWP x x WP x xWP y y WP y y WP y y WP y yWP z z WP z z WP z z WP z z\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB1 11 21 11 21 11 2212121111111'(1). '(2).'(1). '(2).'(1). '(2).'(1).past states '(1).'(1).'(1).'(1).'(1).iiimmmWP x x WP x xWP y y WP y yWP z z WP z zWP x xWP y yWP z zWP x xWP y yWP z z\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0083\u00BA\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0080\u00BD\u00EF\u0083\u00AA\u00EF\u0080\u00BD\u00EF\u0080\u00BD\u00EF\u0083\u00AB\u00EF\u0083\u00B9\u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00BB 4.3.4.2 Time Delay Model Discrete motion analysis of tool and workpiece points out that the relative motion between the rotating tool and workpiece alters the coordinates of the surface points on the workpiece and introduces an additional phase difference in the regenerative chip formation mechanism. The additional phase difference due to the angular rotation of workpiece can be defined by the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 between the surface points of consecutive teeth \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD1 *,\u00EF\u0080\u00ABdj ji i-TP P after one tooth passing period. The phase angle between the waves generated by the present and previous tooth at discrete time interval i can be evaluated as; 105 1 *11 *( ) cos ddj ji i Ti j ji i TP PP P\u00EF\u0081\u00A1\u00EF\u0080\u00AB\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00AB\u00EF\u0080\u00AD\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0082\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0082\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.85) At discrete time instant i , \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thj \u00EF\u0080\u00AB tooth of the milling cutter leaves the surface point \u00EF\u0081\u00BB \u00EF\u0081\u00BD1\u00EF\u0080\u00ABjiP on the rotating workpiece and moves to the next discrete point \u00EF\u0081\u00BB \u00EF\u0081\u00BD1\u00EF\u0080\u00ABji+1P after one discrete time interval t\u00EF\u0081\u0084 . Previous tooth \u00EF\u0080\u00A8 \u00EF\u0080\u00A9j of the cutter left the surface point \u00EF\u0081\u00BB \u00EF\u0081\u00BDdji-TP at the same discrete time instant but one tooth passing period before \u00EF\u0080\u00A8 \u00EF\u0080\u00A9di T\u00EF\u0080\u00AD . However, as the workpiece rotates at constant speed, the surface point \u00EF\u0081\u00BB \u00EF\u0081\u00BDdji-TP moves to the new position \u00EF\u0081\u00BB \u00EF\u0081\u00BD* dji-TP after one discrete time interval as seen in Figure 4.24. Therefore, the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 exist between the surface points \u00EF\u0081\u00BB \u00EF\u0081\u00BD1\u00EF\u0080\u00ABjiP and \u00EF\u0081\u00BB \u00EF\u0081\u00BD*dji-TP , and it varies at each time instant due to multi-axes feed motion of the drives, geometries of tool and workpiece. Figure 4.24 Variation of phase difference as a result of workpiece rotation. Since the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 depends on the discrete positions of the tool at the current and past time instants, the corresponding regenerative phase difference \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,rb i\u00EF\u0081\u00A5 due to the workpiece rotation is also time dependent. Therefore, the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 due to phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 can be evaluated 106 by relating the fractional wave length caused by the phase difference \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,rb i\u00EF\u0081\u00A5 to the length of one full vibration wave imprinted on the workpiece surface. The length of one full vibration wave left on the surface \u00EF\u0080\u00A8 \u00EF\u0080\u00A9fv\u00EF\u0081\u0087 can be calculated as (see Figure 4.25); 2fvt c dN f T\u00EF\u0081\u00B0\u00EF\u0081\u0087 \u00EF\u0080\u00BD (4.86) Figure 4.25 Length of one full wave imprinted on the workpiece surface. where tN , cf [Hz] and dT [s] are the number of tooth, chatter frequency and tooth passing period, respectively. The corresponding fractional wave \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,fv i\u00EF\u0081\u0091 and phase difference \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,rb i\u00EF\u0081\u00A5 to the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 can be calculated as; , ,and 2ifv i rb i i i c dNf T\u00EF\u0081\u00A1\u00EF\u0081\u00A5 \u00EF\u0081\u00B0 \u00EF\u0081\u00A1\u00EF\u0081\u0091 \u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0091 \u00EF\u0080\u00BD\u00EF\u0081\u0087 (4.87) Total time delay in the regenerative chip mechanism \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 is evaluated by the superposition of the time delays contributed by the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 and tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT as; ,22 2i t c d c d i t dj i dcN f T f T N TTf\u00EF\u0081\u00A1 \u00EF\u0081\u00B0 \u00EF\u0081\u00A1\u00EF\u0081\u00B4\u00EF\u0081\u00B0 \u00EF\u0081\u00B0\u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB (4.88) 107 Eq. (4.88) shows that the first term is the time varying time delay as a result of workpiece rotation (see Eq. (4.85)) and the second term is the constant time delay corresponding one tooth passing period. Any change in the feed motion and spindle speeds of tool and workpiece alters the phase angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i\u00EF\u0081\u00A1 , thus the time delay in the system. The most notable difference between the turn-milling and the regular milling is the behavior of the time delay in the regenerative chip mechanism. Unlike the regular milling where the time delay is always constant and equal to the tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT\u00EF\u0081\u00B4 \u00EF\u0080\u00BD , the time delay in turn-milling \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 varies at each discrete time instant and does not equal to the tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i dT\u00EF\u0081\u00B4 \u00EF\u0082\u00B9 . Therefore, the stability properties of turn-milling and milling operations differs from itself significantly depending on the amplitude of the time delay variation which is a function of tool and workpiece spindles speeds and geometries. The time varying delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 is substituted into the dynamic of the system in Eq. (4.69) for the stability analysis. Time varying delay has been investigated by discretizing the tool and workpiece motions using the cutting conditions given in Table 4.3. Table 4.3 Simulation parameters for discrete time delay calculation Tool Speed [rpm] Workpiece Speed [rpm] Tool Diameter [mm] Workpiece Diameter [rpm] Number of Tooth Depth of cut [mm] Immersion 6000 600 12 36 4 1 0 \u00EF\u0081\u00B0\u00EF\u0080\u00AD 108 The positions \u00EF\u0081\u00BB \u00EF\u0081\u00BDjiP ,\u00EF\u0081\u00BB \u00EF\u0081\u00BD* dji-TP and \u00EF\u0081\u00BB \u00EF\u0081\u00BD1\u00EF\u0080\u00ABdji-TP are evaluated at each discrete time interval t\u00EF\u0081\u0084 . Then, the phase difference and corresponding time delay due to the workpiece rotation between consecutive teeth are evaluated from Eqs. (4.85)-(4.88) and shown in Figure 4.26. Figure 4.26 Comparison of discrete time delays in turn-milling and regular milling operations. Simulation inputs are given in Table 4.3. As seen from Figure 4.26, the discrete values of total time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,j i\u00EF\u0081\u00B4 periodically vary between the consecutive teeth, leading to time varying but periodic delay differential equations. Considering the same discretization step for each tooth \u00EF\u0080\u00A8 \u00EF\u0080\u00A91, ,...j ji it t \u00EF\u0080\u00AB , the discrete time delays between consecutive teeth \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,( 1)j j\u00EF\u0081\u00B4 \u00EF\u0080\u00AB are almost the same with negligible difference (0.04%) for the given case. Similar behavior of the time delay is also observed for different number of tooth cases. Therefore, the varying time delays are considered to be periodic at tooth passing interval \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT . 109 The variation of delay amplitude is investigated as a function of the tool \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u0097 spindle speeds for the stability analysis. Discrete time delays are normalized by the mean value at tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT and analyzed as a function of the speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9r t c\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 and diameter of tool and workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9r t wD D D\u00EF\u0080\u00BD ratios as shown in Figure 4.27. Figure 4.27 Time delay variation amplitude with different speed and diameter ratios of tool and workpiece. The delay is a function of speeds and diameters of both the tool and the workpiece. When the diameter ratio is 8.33r w tD D D\u00EF\u0080\u00BD \u00EF\u0080\u00BD and the speed ratio is 2r t c\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00BD , the amplitude of the delay does not change more than 0.2% of the mean time delay which is the most extreme case. Thus, the mean value of the discrete time delays can be used for simplicity for the further stability analysis. However, when the speed ratio of 4r t c\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00BD and the diameter ratio of 2r w tD D D\u00EF\u0080\u00BD \u00EF\u0080\u00BD are used, the time delay changes by 1.5% of the tooth period which is more significant than the effect of speed ratio. However, the variation of total time delay amplitude is more severe as tool and workpiece diameters get close to each other which affects the stability of the system. 110 Also, time delay amplitude is as important as the variation amplitude for the stability analysis of the system. Figure 4.28 shows the total discrete time delay amplitudes for different spindle speed ratios. Figure 4.28 Variation of total time delay amplitude by the speed ratios of tool and workpiece spindles. The total time delay in the system is very close to the tooth passing period when the workpiece rotates slowly. On the other hand, as the workpiece speed increases, the total time delay is increased in the system leading to a different dynamic behavior for the regenerative cutting mechanism. For instance, when the speed ratio of the tool and workpiece spindles are 2r t c\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00BD , the total time delay in the turn-milling system varies between the 111 37.51 7.48 10 [ ]s\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0082\u00B4 corresponding to the almost 4000 rpm of tool spindle speed in regular milling process. Therefore, it can be concluded that as the workpiece speed is very slow in such turn-milling operations, the total time delay is very close to tooth passing period and turn-milling time delay converges to the time delay in regular milling operations. As the workpiece speed increases, turn-milling and milling process are dynamically diverged from each other. 4.3.5 Stability of the Turn-Milling Process Stability of the turn-milling process is investigated in time domain and stability diagrams which show the stable and unstable regions of cutting with the corresponding speed and depths, are constructed in this section. It has been previously shown in Section 4.3.4 that the time delay is time-varying as oppose to the regular milling process and the resulting dynamic system equations are special type of time-periodic delay differential equations with varying time delay. Dynamics of turn-milling system with varying time delay can be expanded from Eq. (4.69) into the modal space as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9,( )( ) ( ) ( ) ( )( )cosj iBta(t) t t t tt\u00EF\u0081\u00B4\u00EF\u0081\u00B1\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0081\u0084\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AE \u00EF\u0083\u00BEtzw\u00CE\u0093I \u00CE\u0093 C \u00CE\u0093 D \u00CE\u0093 F \u00CE\u0093 \u00CE\u0093 \u00CE\u0093\u00CE\u0093 (4.89) where the time-varying delay is periodic at tooth passing period , , 1,( ) ( )j i j i j i dt t T\u00EF\u0081\u00B4 \u00EF\u0081\u00B4 \u00EF\u0081\u00B4 \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB and varies as \u00EF\u0081\u009B \u00EF\u0081\u009D, max ,0, , 0 ,j i j i i j\u00EF\u0081\u00B4 \u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0083\u008E \u00EF\u0080\u00BE \u00EF\u0080\u00A2 . The coefficient matrices can be represented as follows; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9Tt tw w2 0 0;0 2 0rcz tcacKg KK\u00EF\u0081\u00A6\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0081\u009C \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB2t n,t n,tpd 2w n,w n,wz T1 2\u00CE\u00B6 \u00CF\u0089 \u00CF\u0089C C D\u00CE\u00B6 \u00CF\u0089 \u00CF\u0089U UF T T K T T KU U (4.90) 112 where pdC term represents the process damping coefficient. As seen from Eq. (4.89), the system equations are solved at each discretized axial depth \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a\u00EF\u0081\u0084 which is increased along the tool normal axis until the system losses it stability at the stability border. Solving the system at each discrete axial depth, the start and exit angles \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,z zst ex\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 of the tool-workpiece engagement must be defined as a boundary condition for the Eq. (4.89). In regular 3-axis milling, the start and exit angles are assumed to be constant and do not vary over the tool axis for certain axial depth of cut. On the other hand, the CWE varies not only along the toolpath but the tool normal axis as shown in Chapter 3. Hence, varying CWE imposes different boundary conditions to DDE at each discretized axial element and forcing term coefficient zF depends on the CWE start and exit angles. Therefore, as the stability limit is sought at increasing axial depths, accurate boundary conditions must be taken into effect. The weight of each discrete axial element zF is calculated by a recursive algorithm and contribution of each element is summed up as the depth of cut increases. At each discrete axial element \u00EF\u0080\u00A8 \u00EF\u0080\u00A9z , the boundary conditions \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,z zst ex\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 are evaluated by the extracted CWE and checked whether the cutter is in cut or not and zF is calculated. Then, at discrete axial element \u00EF\u0080\u00A8 \u00EF\u0080\u00A91z \u00EF\u0080\u00AB , z+1F is calculated as the summation of previous discrete element\u00E2\u0080\u0099s contribution zF , plus the weight of \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thz \u00EF\u0080\u00AB discrete element by checking the boundary conditions \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1,z zst ex\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00AB \u00EF\u0080\u00AB . Mathematical explanation of the recursive algorithm is given as below and shown in Figure 4.29. 113 Figure 4.29 Calculation of discrete forcing function as a function of axial immersion. T 1 11 11 11T 2 22 12 22( ) 1;1; 1. ; ( ) ;( ) 0 ; otherwise ( ) 1;2; 2. ; ( ) ;( ) 0st stst stgz a a gggz a a gg\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00AC \u00EF\u0080\u00BD \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0083\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AE\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0080\u00BD \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8t t1 0 T1 2w wt t2 1 T1 2w wU UF F T T KT T-U -UU UF F T T KT T-U -UT 3 33 13 33T; otherwise ( ) 1;3; 3. ; ( ) ;( ) 0 ; otherwise ( ) 1; . ; ( ) ;st stzz zgz a a gggz z a z a g\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00AC\u00EF\u0083\u00AD\u00EF\u0083\u00AE\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00AC \u00EF\u0080\u00BD \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0083\u00AD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0080\u00BD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AE\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0080\u00BD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0081\u0084 \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8t t3 2 T1 2w wt tz z-1 T1 2w wU UF F T T KT T-U -UU UF F T T KT T-U -U1;( ) 0 ; otherwise z zst stzg\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0083\u00AC \u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0083\u00AD\u00EF\u0080\u00BD\u00EF\u0083\u00AE (4.91) The stability of the dynamic system given in Eq. (4.89) is solved by semi discretization method in time domain [31]. The main idea in semi discretization method is that only the delayed terms are discretized over the periodicity of the system while the non-delayed terms are left in their original form. First, the dynamics equations are represented in state space from as first order equations (ODEs) by the following set of time periodic coefficient matrices [31]; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD,( ) ( ) ( ( ))( ) ( ) ; ( ) ( )j id dt t t tt T t t T t\u00EF\u0081\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00CE\u0098 L \u00CE\u0098 R \u00CE\u0098L L R R (4.92) 114 where the coefficient matrices are; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2( ) 0 0 0( ) ; ;0( )m m m m m m m mm m m mt Itt\u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0082\u00B4 \u00EF\u0082\u00B4\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AF \u00EF\u0083\u00AF\u00EF\u0083\u00AE \u00EF\u0083\u00BE z z\u00CE\u0093\u00CE\u0098 L RF D C F\u00CE\u0093 (4.93) ,( ( ))j it t\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00CE\u0098 represents the states with time-varying delays \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ( )j i t\u00EF\u0081\u00B4 which are periodic at tooth passing intervals dT . ( )t\u00CE\u0098 is the modal states of the tool-workpiece system at time t and consists of the modal displacements ( )t\u00CE\u0093 and modal velocities ( )t\u00CE\u0093 of tool and workpiece. The coefficient matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A9L,R for current and delayed states are periodic at tooth passing intervals dT . It must be noted that the time delay is constant and equal to the tooth period in regular milling, i.e. , ( )j i dt T\u00EF\u0081\u00B4 \u00EF\u0080\u00BD . However, in turn-milling, the time delay \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ( )j i t\u00EF\u0081\u00B4 at each discrete interval \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i t\u00EF\u0081\u0084 is different and time varying but periodic at tooth passing intervals, i.e. , ( )j i dt T\u00EF\u0081\u00B4 \u00EF\u0082\u00B9 . Time period of the system dT is divided into k number of discrete intervals with length t\u00EF\u0081\u0084 as; dTtk\u00EF\u0081\u0084 \u00EF\u0080\u00BD (4.94) where integer number k is also known as the resolution of the corresponding time period of the system. It should be selected as sufficient as to capture at least 15 sampling points in one full vibration wave as a rule of thumb in this thesis for the best solution accuracy. However, at high speeds where the vibration waves are longer, the sampling points might be needed to be increased. Within each discrete time interval t\u00EF\u0081\u0084 or \u00EF\u0081\u009B \u00EF\u0081\u009D1,i it t t \u00EF\u0080\u00AB\u00EF\u0083\u008E , the time periodic coefficient matrices \u00EF\u0080\u00A8 \u00EF\u0080\u00A9L,R and varying time delay ,j i\u00EF\u0081\u00B4 are approximated as; 115 1 1 1,1 1 1( ) ; ( ) ;i i ii i it t tj i jt t tt dt t dt dtt t t\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0082\u00BB \u00EF\u0082\u00BB \u00EF\u0082\u00BB\u00EF\u0081\u0084 \u00EF\u0081\u0084 \u00EF\u0081\u0084\u00EF\u0083\u00B2 \u00EF\u0083\u00B2 \u00EF\u0083\u00B2i iL L R R (4.95) The number of discrete points that cover the time delay ,j i\u00EF\u0081\u00B4 at discrete time interval \u00EF\u0080\u00A8 \u00EF\u0080\u00A9i t\u00EF\u0081\u0084 is rounded to; , 2intj iitmt\u00EF\u0081\u00B4 \u00EF\u0080\u00AB \u00EF\u0081\u0084\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0081\u0084\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.96) where im is used to capture the delayed state in forming the semi-discrete model and called as particular delay resolution. Since the time delay ,j i\u00EF\u0081\u00B4 is different at each discrete time interval and varies, the period resolution k and delay resolution m are not the same integers \u00EF\u0080\u00A8 \u00EF\u0080\u00A9k m\u00EF\u0082\u00B9 as oppose to regular milling where the period and delay resolutions are equal to each other \u00EF\u0080\u00A8 \u00EF\u0080\u00A9k m\u00EF\u0080\u00BD . After introducing the discretization steps for time period and delay, the delayed states ,( ( ))j it t\u00EF\u0081\u00B4\u00EF\u0080\u00AD\u00CE\u0098 are approximated as the weighted sum of the two neighboring delayed states ( )i mt \u00EF\u0080\u00AD\u00CE\u0098 and 1( )i mt \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00CE\u0098 as (see); , , , 1( ( )) (t ) (t )i ij i b i i m a i i mt t w w\u00EF\u0081\u00B4 \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00AD \u00EF\u0082\u00BB \u00EF\u0080\u00AB\u00CE\u0098 \u00CE\u0098 \u00CE\u0098 (4.97) 116 Figure 4.30 Approximation of the delayed states by time varying weights [31]. The weights in Eq. (4.97) also vary within the time period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, [0,1]a bw \u00EF\u0080\u00BD \u00EF\u0083\u008ER because of the varying time delay ,j i\u00EF\u0081\u00B4 and they can be calculated as; , ,, ,0.5 0.5;j i i i j ib i a it m t m t tw wt t\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0080\u00AB \u00EF\u0081\u0084 \u00EF\u0080\u00AD \u00EF\u0081\u0084 \u00EF\u0081\u0084 \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0081\u0084\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0081\u0084 \u00EF\u0081\u0084 (4.98) Note that the discrete weights satisfy the condition, , , 1a i b iw w\u00EF\u0080\u00AB \u00EF\u0080\u00BD . Figure 4.31 shows the variation of discrete weights over the time period of the system. On the other hand, the thick line represents the constant weights in regular milling operation. 117 Figure 4.31 Variation of discrete weights within one time period of the system. As it can be seen in Figure 4.31, the discrete weights are different at each discrete time interval and vary over the time period of the system as oppose to regular milling where the delayed states are approximated by their mean values and the corresponding weights are 0.5a bw w\u00EF\u0080\u00BD \u00EF\u0080\u00BD . Hence, the approximated delayed state ,( )j i\u00EF\u0081\u00B4\u00CE\u0098 has oscillatory behavior between the delayed states at ( )ii mt \u00EF\u0080\u00AD\u00CE\u0098 and 1( )ii mt \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00CE\u0098 . Table 4.4 summarizes the time delay and its discretization differences for turn-milling and regular milling, turning processes. Table 4.4 Comparison between the time delays and its discrete approximation in turn-milling and regular milling, turning processes. Time Delay Time Period Delay & Period Resolutions Discrete weights Turn-milling Varying \u00E2\u0080\u0093 ( ,j i\u00EF\u0081\u00B4 ) Tooth Period \u00E2\u0080\u0093 dT ( )m t k\u00EF\u0082\u00B9 ( ) ( )a bw t w t\u00EF\u0082\u00B9 Milling Constant \u00E2\u0080\u0093 (\u00EF\u0081\u00B4 ) Tooth Period \u00E2\u0080\u0093 dT m k\u00EF\u0080\u00BD 0.5a bw w\u00EF\u0080\u00BD \u00EF\u0080\u00BD Turning Constant \u00E2\u0080\u0093 (\u00EF\u0081\u00B4 ) Spindle Period \u00E2\u0080\u0093 sT m k\u00EF\u0080\u00BD 0.5a bw w\u00EF\u0080\u00BD \u00EF\u0080\u00BD 118 For each discrete time interval \u00EF\u0081\u009B \u00EF\u0081\u009D1,i it t t \u00EF\u0080\u00AB\u00EF\u0083\u008E for 0,1,2, , ( 1)i k\u00EF\u0080\u00BD \u00EF\u0080\u00AD , the state space equations can be expressed to cover the whole time period dT and varying time delay ( i mt \u00EF\u0080\u00AD and 1i mt \u00EF\u0080\u00AD \u00EF\u0080\u00AB ) as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD1( ) ( ) ( ) ( )i ii i i i b i m a i mt t w t w t\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00CE\u0098 L \u00CE\u0098 R \u00CE\u0098 \u00CE\u0098 (4.99) Then, assuming the time periodic coefficient iL is invertible \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9det 0\u00EF\u0082\u00B9 \u00EF\u0081\u009C -1i i iL L L = I , the solution of Eq. (4.99) takes the following form [31]; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1i i i m i m\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB1 2\u00CE\u0098 P \u00CE\u0098 Q \u00CE\u0098 Q \u00CE\u0098 (4.100) where \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A911tti i bti i aee we w\u00EF\u0081\u0084\u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00ADiii,mii,m+1LiL1L2PQ I L RQ I L R (4.101) The states in Eq. (4.100) are expressed r number of times to cover the whole length of the delay time ,j i\u00EF\u0081\u00B4 as an augmented state vector; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u009B \u00EF\u0081\u009D1, , ,Ti i i i r\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BDY \u00CE\u0098 \u00CE\u0098 \u00CE\u0098 (4.102) Since ,j i\u00EF\u0081\u00B4 is time varying, its maximum value which is also known as maximum delay resolution, is considered to capture all states as follows; maxmax max ,int 0.5 where max( )j ir mt\u00EF\u0081\u00B4\u00EF\u0081\u00B4 \u00EF\u0081\u00B4\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0081\u0084\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.103) Then, the linear map which relates the current \u00EF\u0081\u00BB \u00EF\u0081\u00BDiY and next states \u00EF\u0081\u00BB \u00EF\u0081\u00BD1i\u00EF\u0080\u00ABY can be constructed as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD,ii+1 T i\u00EF\u0080\u00BDY \u00CE\u00A6 Y (4.104) 119 where ,T i\u00CE\u00A6 is the transition matrix between thi and \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thi \u00EF\u0080\u00AB states and it can be constructed as; , , 1,1 ( 1)0 0 0 0 00 0 0 0 0 0 010 0 0 0 0 0 00 0 0 0 0 0 0i m i mi iiT im m rII\u00EF\u0080\u00AB\u00EF\u0082\u00AF \u00EF\u0082\u00AF\u00EF\u0082\u00AF\u00EF\u0082\u00AF\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB1 2Q QP\u00CE\u00A6 (4.105) The matrices i,m1Q and i,m+12Q are placed at the \u00EF\u0080\u00A8 \u00EF\u0080\u00A91thim \u00EF\u0080\u00AD and thm columns of the transition matrix ,T i\u00CE\u00A6 , respectively. Size of the transition matrix is defined considering the maximum delay resolution r , thus it does not change over the time period of the system. The stability of the system can be evaluated by forming the transition matrix at k successive intervals covering the principal tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT k t\u00EF\u0080\u00BD \u00EF\u0081\u0084 as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u009B \u00EF\u0081\u009D, 1 ,1 ,0 0 01 stable1 critically stable1 unstableT T k T T k T TI\u00EF\u0081\u00AD\u00EF\u0081\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0082\u00AE \u00EF\u0080\u00BD \u00EF\u0082\u00AE \u00EF\u0080\u00AD \u00EF\u0080\u00BD\u00EF\u0080\u00BC\u00EF\u0083\u00AC\u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AD\u00EF\u0083\u00AF\u00EF\u0080\u00BE\u00EF\u0083\u00AE\u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 \u00CE\u00A6 Y \u00CE\u00A6 Y \u00CE\u00A6 (4.106) Since the system equations are periodic, stability of the system can be evaluated by Floquet Theory [31]. In accordance with the Floquet Theory, the linear periodic system is considered to be unstable if the modulus of any of the characteristic multipliers (eigenvalues) \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00AD of the transition matrix T\u00CE\u00A6 is greater than unity, critically (marginal) stable if the modulus is unity and stable if the modulus is less than unity. Unlike the zero-order frequency domain solution proposed in [30] that gives the critical stability borders analytically, the stability limit must be searched iteratively by checking the trial spindle speeds and depth of cuts. 120 In the case of turn-milling process, the type of instability might be significant for the process identification. Considering the low immersion of cutting because of the process constraints such as diameters and speeds of tool and workpiece and axial feed velocity, the nonlinear behavior of the cutting intensifies as the radial immersion reduces due to the tool jumping out of cut. Depending on the type of instability, the stability lobes might be bifurcated and the measured chatter frequencies will be different. Therefore, the type of the instability is investigated according to the location of critical characteristic multipliers in the complex plane as; i. If the critical characteristic multipliers are complex conjugate \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,2Im 0\u00EF\u0081\u00AD \u00EF\u0082\u00B9 and modulus is greater than one \u00EF\u0080\u00A8 \u00EF\u0080\u00A91,2 1\u00EF\u0081\u00AD \u00EF\u0080\u00BE , the type of the instability is called secondary Hopf bifurcation. The corresponding motion of the chatter vibrations are likely quasi-periodic. The multiple chatter frequencies in Hopf type of bifurcation H\u00EF\u0081\u00B7 can be calculated by \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9maxarg 2 / where qH q q\u00EF\u0081\u00B7 \u00EF\u0081\u00AD \u00EF\u0081\u00B0 \u00EF\u0081\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0083\u008EZ (4.107) Note that, the \u00E2\u0080\u009C arg \u00E2\u0080\u009D operator return the argument (angle) of the complex eigenvalue. The dominant chatter frequency \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9, dH d q\u00EF\u0081\u00B7 \u00EF\u0083\u008E can be obtained by solving the modal velocity terms in Eq. (4.104) and their approximate Fourier coefficients as given in [99]. ii. If the critical multipliers have only negative real part \u00EF\u0080\u00A8 \u00EF\u0080\u00A91Im 0\u00EF\u0081\u00AD \u00EF\u0080\u00BD and less than negative one \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1\u00EF\u0081\u00AD \u00EF\u0082\u00A3 \u00EF\u0080\u00AD , the type of the instability is called Flip bifurcation or period-doubling. If the number of characteristic multiplier having negative real parts is one, then it is called primary flip bifurcation. If there are two characteristic multipliers having negative real part, this case is secondary flip bifurcation. In the case of flip bifurcation, the corresponding 121 motion of chatter vibrations is periodic. The chatter frequency in flip type of bifurcation F\u00EF\u0081\u00B7 can be calculated by; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9/ 30 / 60 and , 1,0,1,iF N i N i\u00EF\u0081\u00B7 \u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0080\u00AB \u00EF\u0081\u0097 \u00EF\u0080\u00BD \u00EF\u0080\u00AD (4.108) The stability properties of the turn-milling process at low cutting speeds is as important as the low immersion cutting cases. At low cutting speed region, process damping forces has huge effect on the stable depth of cuts due to the increased indentation volume and damping between the tool flank face and workpiece contact as explained in Section 4.3.3.3. On the other hand, at low cutting speeds, the number of vibration waves imprinted on the workpiece in one tooth passing period increases. In order to have a reliable solution accuracy, the same number of sampling points in high spindle speeds must be taken in one vibration wave which leads to very small discretization intervals t\u00EF\u0081\u0084 , hence increased computation time. Therefore, the solution speed of the time domain stability algorithm reduces dramatically at low spindle speeds. Computational burden of the low speed stability solution is tackled by the adapting the method developed for the analytical solution of the milling process at process damping region. Following low speed stability solution model is used for turn-milling process to predict the low speed stability envelope. Analytical Prediction of Low Speed Stability Envelope in Turn-Milling The system equation of dynamics represented in Eq. (4.89) are rewritten as follows; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9,( )( ) ( ) ( ) ( )( )cosj iBta(t) t t t tt\u00EF\u0081\u00B4\u00EF\u0081\u00B1\u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0081\u0084\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0081\u009C \u00EF\u0080\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AE \u00EF\u0083\u00BEtzw\u00CE\u0093I \u00CE\u0093 C \u00CE\u0093 D \u00CE\u0093 F \u00CE\u0093 \u00CE\u0093 \u00CE\u0093\u00CE\u0093 (4.109) 122 The total damping coefficient \u00EF\u0080\u00A8 \u00EF\u0080\u00A9C includes the structural damping terms \u00EF\u0080\u00A8 \u00EF\u0080\u00A9s\u00EF\u0081\u00BA of tool and workpiece as well as the process damping terms and ( , )sp cf K V\u00EF\u0080\u00BDpd pdC C . The given set of differential equations are coupled at each direction in modal space, thus the stability limits must be solved iteratively. The basic concept of the simplified method developed by Wang et. al [55] is the system equation are decoupled under certain assumptions: It is assumed that the system has single symmetric dynamics at the most flexible direction. Second assumption is the decoupling is only valid at the radial immersion ratios higher than 0.6. Finally, it is assumed that the start and exit angles are constant along the tool normal axis and the time delay variation is neglected, so only the average time delay \u00EF\u0081\u00B4 term is included in the equation of dynamics. Under the given assumptions, first the system equations are expressed in frequency domain with their zero-order or average terms of the coefficients. Average terms of the periodic coefficient matrices for the forcing term \u00EF\u0080\u00A8 \u00EF\u0080\u00A9zF and process damping \u00EF\u0080\u00A8 \u00EF\u0080\u00A9pdC are given as; ,1 0.25cos 2 0.5 0.25sin 21 0.5 0.25sin 2 0.25cos 221 0.25cos 2 0.5 0.25sin 21 0.5 0.25sin 2 0.25cos 22exstexstrct tcrcrpt sprpckN K akkN K akV\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00B0\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00B0\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A7\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8T Tz,0Tpd 0F TU UTC TU\u00EF\u0083\u00B6\u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00B8TUT (4.110) where rc rc tck K K\u00EF\u0080\u00BD , 1rpk \u00EF\u0081\u00AD\u00EF\u0080\u00BD and \u00EF\u0081\u00AD is the coulomb friction coefficient between the ploughing coefficients at the radial and tangential directions of flank face of the tool. As it can be seen from Eq. (4.110), the matrices which are function of the start and exit angles of immersion are coupled each other. In order to decouple them, the matrices are written with their corresponding eigenvalues \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,c p\u00EF\u0081\u00AC and eigenvectors \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c,p\u00CF\u0085 as [55]; 123 ,1,2,1,201 0.25cos 2 0.5 0.25sin 201 0.5 0.25sin 2 0.25cos 21 00.25cos 2 0.5 0.25sin 21 00.5 0.25sin 2 0.25cos 2exstexstcrcccrcrp pprp pkGkkGk\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00AC\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00AC\u00EF\u0081\u00B1 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00AC\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00AC\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB-1c cp p\u00CF\u0085 \u00CF\u0085\u00CF\u0085 \u00CF\u0085-1 (4.111) Then, the eigenvalues of ,c pG can be written as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A92 2 2,1,22 2 2,1,2; 1 sin2; 1 sin2rc rc rc r rrp rp rp r rkkkk\u00EF\u0081\u00A6 \u00EF\u0081\u00A4\u00EF\u0081\u00AC \u00EF\u0081\u00A4 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A4\u00EF\u0081\u00AC \u00EF\u0081\u00A4 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0082\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00EF\u0082\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD (4.112) where r ex st\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AD . The decoupled reduced system equation with 2-degree of freedom in modal space \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0080\u00A8 \u00EF\u0080\u00A91 2\u00EF\u0080\u00BD \u00EF\u0081\u0087 \u00EF\u0081\u0087\u00CE\u0093 can be written as; 211 1 n222 2 n,1 1 1,2 2 202 0 000 2 00 ( ) ( )0 ( ) ( )2 cospdnpdnctccBcct tNK at t\u00EF\u0081\u00BA\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00BA\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00AC \u00EF\u0081\u00B4\u00EF\u0081\u00AC \u00EF\u0081\u00B4\u00EF\u0081\u00B0 \u00EF\u0081\u00B1\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0081\u0087\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AC \u00EF\u0083\u00BC \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0081\u0087 \u00EF\u0081\u0087\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00AC \u00EF\u0083\u00BC\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD \u00EF\u0083\u00AD \u00EF\u0083\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0081\u0087\u00EF\u0081\u0087 \u00EF\u0081\u0087 \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AE \u00EF\u0083\u00BE \u00EF\u0083\u00AE \u00EF\u0083\u00BE\u00EF\u0083\u00AB \u00EF\u0083\u00BB \u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0081\u0087 \u00EF\u0081\u0087 \u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0081\u0087 \u00EF\u0081\u0087 \u00EF\u0080\u00AD\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00A8 \u00EF\u0083\u00B8T TTU UT (4.113) where the decoupled average process damping coefficient is given as; 2t tppd pcN K acV\u00EF\u0081\u00AC\u00EF\u0081\u00B0\u00EF\u0080\u00BD (4.114) Note that, the presented decoupling is only possible where the eigenvectors in Eq. (4.111) satisfies the condition, 1\u00EF\u0080\u00BD-1c p\u00CF\u0085 \u00CF\u0085 , and this assumption is valid at the radial immersion rates higher than 0.6 according to [55]. In Eq. (4.114), only the real part of the complex eigenvalue p\u00EF\u0081\u00AC yields damping in the system. When the real part of the complex eigenvalue p\u00EF\u0081\u00AC is substituted into the 124 average process damping coefficient term in Eq. (4.114), the specific process damping coefficient is calculated as; '1 and where 8 8t rp r t rp rpd pdc cN K a N KmV mk V mk u\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00BA \u00EF\u0081\u00BA\u00EF\u0081\u00B0 \u00EF\u0081\u00B0\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00BD (4.115) The decoupled system in Eq. (4.113) has two degree of freedom, thus two eigenvalues. However, under the given assumption where the radial immersion rate is higher than 0.6, the complex eigenvalues are same, so only one of the modal coordinate is taken for the stability analysis. Remembering the stability analysis for regular milling process in frequency domain at Section 4.2.2.1, the system equations can be represented as the following form; \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1 ( ) 02 cosst tccBN K ae H s\u00EF\u0081\u00B4 \u00EF\u0081\u00AC\u00EF\u0081\u00B0 \u00EF\u0081\u00B1\u00EF\u0080\u00AD\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00BD (4.116) where ( )H s is the transfer function (FRF) of the system having structural \u00EF\u0080\u00A8 \u00EF\u0080\u00A9s\u00EF\u0081\u00BA and process damping \u00EF\u0080\u00A8 \u00EF\u0080\u00A9'pd\u00EF\u0081\u00BA coefficients. The stability properties of the system given in Eq. (4.116) can be determined by taking the determinant of the characteristic equation similar in Section 4.2.2.1 and the limiting depth of cut can be calculated as; 'lim'(1 )1/ 60s prp r nptc taK fK A D\u00EF\u0081\u00BA \u00EF\u0081\u00BA\u00EF\u0081\u00A6\u00EF\u0081\u00BA\u00EF\u0080\u00AB\u00EF\u0080\u00BD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00AD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0081\u0097\u00EF\u0083\u00AB \u00EF\u0083\u00BB (4.117) where the coefficient A can be written as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A90 10202 cos sin 2 and 1 where tan1c c cccrc rcA ckc\u00EF\u0081\u00AC \u00EF\u0081\u00AC \u00EF\u0081\u00AC\u00EF\u0081\u00AC\u00EF\u0081\u00AC \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A4\u00EF\u0081\u00A6\u00EF\u0081\u00B0 \u00EF\u0081\u00A6\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AB \u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.118) 125 The limiting depth of cut is infinity corresponding to the asymptotic spindle speed [rpm] which can be calculated as; 60rp r natc tK fK A D\u00EF\u0081\u00A6\u00EF\u0081\u0097 \u00EF\u0080\u00BD (4.119) The nf and tD represents the frequency [Hz] and tool diameter, respectively. The limiting depth of cut and asymptotic spindle speed define the lower and left margin of the stability envelope in low spindle speed region of cutting as shown in Figure 4.32. Since the stability lobes are almost impossible to be utilized at low cutting speeds, the upward stability curve is sufficient to define the stable and unstable regions of cutting. Figure 4.32 Low speed stability envelope defined by limiting depth of cut and asymptotic spindle speed. The limiting depth of cut and asymptotic spindle speed can be calculated by Eq. (4.117) and Eq. (4.119), respectively. The inputs are ( )rc tc rck K K , nf , D , r\u00EF\u0081\u00A6 and rpK . All the coefficients except the radial ploughing constant are usually easily identified but identification of the ploughing coefficient is quite challenging. On the other hand, two set of chatter tests yields the prediction of the ploughing coefficient using the Eq. (4.117) and Eq. (4.119). Chatter tests are conducted at two different depth of cut \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 2,a a and spindle speeds \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 2,\u00EF\u0081\u0097 \u00EF\u0081\u0097 within the low speed region and the 126 chatter frequencies are measured \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,1 ,2,c cf f . When the measured chatter frequencies are substituted into the Eq. (4.117) and Eq. (4.119) with their corresponding axial depths and spindle speeds, two set of equations are obtained and ploughing coefficient can be easily calculated. Also, the limiting depth of cut \u00EF\u0080\u00A8 \u00EF\u0080\u00A9lima and asymptotic spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a\u00EF\u0081\u0097 can be calculated as [55]; lim 111;1aaaara ar rrr r\u00EF\u0081\u0097\u00EF\u0081\u0097\u00EF\u0081\u0097\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00AD\u00EF\u0081\u0097 \u00EF\u0080\u00BD \u00EF\u0081\u0097\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00AD\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.120) where 1 2ar a a\u00EF\u0080\u00BD and 1 2r\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 . If the given equations are solved for different spindle speed values, the upward bended stability curve can be constructed. As a result, two sets of chatter tests in the low speed region are sufficient to construct the stability envelope. However, the selection of the low speed region for the chatter tests requires critical engineering justification and the low speed process damping region may vary material to material. 4.3.6 Time Domain Solution of Force, Vibration and Surface Form Error The equation of motion expressed in Eq. (4.89) includes only the dynamic forcing terms which are related with the stability of the dynamic system, and the static force terms are neglected since they do not contribute to the stability as discussed in Section 4.3.3. However, the cutting force and vibration properties of a cutting process depend on the static forcing terms, thus they must be included in the system equations. Recalling the discrete cutting force representation from Section 4.3.3, the total discrete cutting force has been written as; ( , ) ( , ) ( , ) ( , ) ( , )j j j j jd z d z d z d z d z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00ABt d,c d,pd s,c s,erta rta rta rta rtaF F F F F (4.121) 127 where d d,crtaF and dd,pdrtaF are the dynamic cutting and process damping forces. The static cutting d s,crtaF and edge ds,ertaF force terms have been modeled previously in the mechanics model of turn-milling in Chapter 3 as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9( , ) , ( )( , ) ( )sj j jjd z h t db zd z dS z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BDs,crta qcs,erta qeF KF K (4.122) where qcK and qeK are the cutting and edge force coefficients in oblique cutting plane \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,r t a\u00EF\u0080\u00BDq and the static chip thickness sjh results from the multi-axes feed motion of machine drives and calculated from Eq. (3.44). 4.3.6.1 Discrete Time Cutting Force and Vibration Simulation The full equation of motion of the turn-milling system can be written in modal space by including the total static force \u00EF\u0080\u00A8 \u00EF\u0080\u00A9stF as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A9,( ) ( ) ( ) ( ) ( )cosj iBa(t) t t t t t\u00EF\u0081\u00B4\u00EF\u0081\u00B1\u00EF\u0081\u0084\u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00ABz stI \u00CE\u0093 C \u00CE\u0093 D \u00CE\u0093 F \u00CE\u0093 \u00CE\u0093 F (4.123) Force \u00EF\u0080\u00A8 \u00EF\u0080\u00A9zF and vibration \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00CE\u0093 terms in Eq. (4.123) are solved in time domain. First, following the same procedure for stability calculation, Eq. (4.123) is transformed into the state space form as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD,( ) ( ) ( ( )) ( )j it t t t t\u00EF\u0081\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB st\u00CE\u0098 L \u00CE\u0098 R \u00CE\u0098 F (4.124) where the coefficient matrices for current and delayed states \u00EF\u0080\u00A8 \u00EF\u0080\u00A9L,R have the same form as in Eq. (4.93). The solution of cutting forces and vibrations requires the computation of state matrices in one tool spindle rotation since the transition matrices repeat themselves after each tool spindle period. Therefore, the discrete time interval t\u00EF\u0081\u0084 is selected by considering the spindle period of 128 tool sT , instead of tooth passing period dT . Discretizing the state matrices over the spindle period, the solution of Eq. (4.124) can be written as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1i i i m i m\u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB1 2\u00CE\u0098 P \u00CE\u0098 Q \u00CE\u0098 Q \u00CE\u0098 E (4.125) where the coefficient matrices as given as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9111tti i bti i atiee we we\u00EF\u0081\u0084\u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0081\u0084 \u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00ADiii,mii,m+1iLiL1L2L si iPQ I L RQ I L RE I L F (4.126) Then, similar to the stability solution procedure, the state matrices are approximated and the linear map that relates the current \u00EF\u0081\u00BB \u00EF\u0081\u00BDiY and next states \u00EF\u0081\u00BB \u00EF\u0081\u00BDi+1Y are constructed as follows; \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD \u00EF\u0081\u00BB \u00EF\u0081\u00BD,ii+1 T i i\u00EF\u0080\u00BD \u00EF\u0080\u00ABY \u00CE\u00A6 Y E (4.127) where the quasi-static terms \u00EF\u0081\u00BB \u00EF\u0081\u00BDiE is calculated by the \u00E2\u0080\u009CVariation of Constants Theorem\u00E2\u0080\u009D where in-depth mathematical explanation is given in [100]. Note that, the stability analysis for Eq. (4.127) requires the computation of the state transition matrix, ,iT\u00CE\u00A6 that involves only the dynamic (delayed) states as explained in Eq. (4.104). On the other hand, the full time domain solution which solves the displacement and velocity states in modal space, requires the computation of both state transition matrix ,iT\u00CE\u00A6 , and the quasi-static force states \u00EF\u0081\u00BB \u00EF\u0081\u00BDiE within one tool spindle period. The initial conditions for Eq. (4.127) are set to zero before the states are solved recursively. For the current state, \u00EF\u0081\u00BB \u00EF\u0081\u00BDiY takes the following form; 129 \u00EF\u0081\u00BB \u00EF\u0081\u00BD1000iiii r\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0083\u00A9 \u00EF\u0083\u00B9 \u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AA \u00EF\u0083\u00BA \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00CE\u0098\u00CE\u0098Y\u00CE\u0098 (4.128) Considering zero initial states, only the excitation state for the solution of Eq. (4.127) is the forced vibration (static) terms \u00EF\u0081\u00BB \u00EF\u0081\u00BDiE . Then, the consecutive states are solved for each discrete time within spN number of spindle period, and the full history of the modal states i\u00CE\u0098 , are constructed. The details of the recursive state calculations can be found in [101]. Once the modal states are simulated within the desired number of spindle periods, the modal \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,\u00CE\u0093 \u00CE\u0093 and physical \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,Q Q displacement and velocity vectors are found from the modal transformation that is given in Eq. (4.46) as; ( )( )( )( ) . ( )( ) . ( )iiii ii itttt tt t\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0080\u00BD \u00EF\u0083\u00AA \u00EF\u0083\u00BA\u00EF\u0083\u00AB \u00EF\u0083\u00BB\u00EF\u0080\u00BD\u00EF\u0080\u00BD\u00CE\u0093\u00CE\u0098\u00CE\u0093Q U \u00CE\u0093Q U \u00CE\u0093 (4.129) Note that the physical displacement \u00EF\u0080\u00A8 \u00EF\u0080\u00A9Q and velocity \u00EF\u0080\u00A8 \u00EF\u0080\u00A9Q vectors represent the vibrations at q number of contact points along the physical \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , ,x y z \u00EF\u0081\u00B1 directions of tool and workpiece. Once the vibration terms are solved, the cutting forces along the tool-workpiece contact zone are calculated at each discrete time step using Eq. (4.121). 4.3.6.2 Surface Form Error Model in Turn-Milling One of the most important process outcomes of the turn-milling along with the cutting force and chatter stability predictions, is the surface quality of the finished workpiece. As oppose to turning, turn-milling process leads non-cylindrical surfaces due to the simultaneous tool and 130 workpiece rigid body rotations [80]. The form errors in turn-milling occur by several causes such as circularity error, static cusp height and dynamic vibrations between the flexible tool and workpiece. The circularity error is the difference between the desired workpiece shape (perfect circle), and the scallop height due to the tool and workpiece rotations as shown in Figure 4.33. Figure 4.33 Circularity error in turn-milling process. The maximum scallop height can be calculated as [80]; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,max211 and cos 2e cc wt te RN\u00EF\u0081\u00B0\u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0081\u0097\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0081\u0097\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.130) Thus, the minimum scallop height can be achieved when a high ratio of tool to workpiece spindle speed is selected which in turn may reduce the productivity. In addition to circularity error which is a function of spindle speeds of tool and workpiece, the geometry of tool and workpiece also effects the final shape of the machined surface. The contact between the tool\u00E2\u0080\u0099s bottom surface and workpiece within the engagement region leaves cusp heights on the workpiece surface. Also, the steady-state vibrations between the flexible tool and workpiece effect the final form of the workpiece surface. 131 As the tool penetrates into the workpiece at a depth of cut a , it follows a helical path along the workpiece due to angular rotation of workpiece (see Figure 4.34 (a)). Due to the tool\u00E2\u0080\u0099s bottom face clearance angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00A1 the chip is removed only by the side cutting edge of the tool unless there is feed motion perpendicular to the workpiece rotation axis. Therefore, the contact between the bottom face of the tool and workpiece is a point contact. The cutting edge follows a circular arc where the boundaries are determined from the start \u00EF\u0080\u00A8 \u00EF\u0080\u00A9st\u00EF\u0081\u00A6 and exit \u00EF\u0080\u00A8 \u00EF\u0080\u00A9ex\u00EF\u0081\u00A6 angles of tool-workpiece engagement geometry as shown in Figure 4.34 (a). As the side edges remove the chip, the bottom edges which also vibrate along the tool normal axis, are in point contact with the workpiece due to the tool\u00E2\u0080\u0099s bottom face clearance angle. Therefore, the surface is generated at the present pass of the tool, and since the workpiece rotates, it moves by the amount of total feed vector \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tf , hence no surface will be in contact with the tool\u00E2\u0080\u0099s bottom face after a tooth or spindle period of the tool. As the bottom edges generate the surface at the tip of the cutting edge, the cusp exists on the machined workpiece surface between the consecutive tooth passes which have a distance of \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tf as shown in Figure 4.34 (c). The static height of the cusp increases towards the center of the tool due to the clearance angle at the tool\u00E2\u0080\u0099s bottom face \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00A1 , and can be calculated at any discrete location of the bottom edge \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,b ix as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,,max. tan.tans i b i bs t bh xh f\u00EF\u0081\u00A1\u00EF\u0081\u00A1\u00EF\u0080\u00BD\u00EF\u0080\u00BD (4.131) where ,s ih and ,maxsh are the local and maximum cusp height on the surface of the workpiece, respectively. 132 Dynamic displacements or vibrations during turn-milling process also effect the surface errors on the workpiece. Considering a flexible tool-workpiece couple, the resultant vibrations along the tool normal axis \u00EF\u0080\u00A8 \u00EF\u0080\u00A90z changes the cusp height over the cutter-workpiece engagement region. The steady-state vibrations are calculated as discussed in Section 4.3.6.1, and mapped on to the static cusp height between start and exit of the engagement boundary (see Figure 4.34 (d)). Figure 4.34 Surface formation in turn-milling. The total cusp height can be calculated by superposing the static cusp height and dynamic vibrations as; 133 \u00EF\u0080\u00A8 \u00EF\u0080\u00A90, ,total i s i jh h z \u00EF\u0081\u00A6\u00EF\u0080\u00BD \u00EF\u0080\u00AB (4.132) Note that, the width of the static component is defined by the engagement boundary (radial immersion) where the dynamic cutting vibrations \u00EF\u0080\u00A8 \u00EF\u0080\u00A90z are imprinted on the surface. Finally, a three dimensional surface is generated over one tooth passing period. Repeating this procedure for one workpiece spindle period generates the surface for the one helical section on the workpiece which is sufficient unless the steady-state resultant vibrations changes over the toolpath. Presence of bottom face clearance angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00A1 prevents the area contact between the workpiece and tool bottom face which reduces edge forces, but on the other hand leaves cusp heights on the newly generated surface. Wiper edge cutters are also utilized in turn-milling operations where one or multiple bottom edges of the tool has straight edge, and clean the cusp heights from the surface, leading to better surface finish. In the case of wiper edge milling tools or inserts, the static cusp height is eliminated if the length of the wiper edge \u00EF\u0080\u00A8 \u00EF\u0080\u00A9wpL is equal or greater than the total feed per tooth \u00EF\u0080\u00A8 \u00EF\u0080\u00A90s wph L\u00EF\u0080\u00BD \u00EF\u0082\u00AE \u00EF\u0082\u00B3 tf . On the other hand, multiple vibration imprints are possible along the surface generated by the wiper edges. The wiper edge leaves a flat surface \u00EF\u0080\u00A8 \u00EF\u0080\u00A9jS on the workpiece with a length of \u00EF\u0080\u00A8 \u00EF\u0080\u00A9wpL within one tooth passing period. The generated surface translates due to the workpiece rotation by \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tpL which is calculated as; 2 sin60e ctp wL R\u00EF\u0081\u00B0\u00EF\u0081\u0097\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (4.133) 134 After one tooth passing period \u00EF\u0080\u00A8 \u00EF\u0080\u00A9dT , the length of the previously surface \u00EF\u0080\u00A8 \u00EF\u0080\u00A9jS will overlap with the surface generated by the next tooth \u00EF\u0080\u00A8 \u00EF\u0080\u00A91jS \u00EF\u0080\u00AB . The overlapped surface length \u00EF\u0080\u00A8 \u00EF\u0080\u00A9scL is found as; sc w tpL L L\u00EF\u0080\u00BD \u00EF\u0080\u00AD (4.134) The steady-state vibrations along the surface normal causes multiple imprints on the workpiece until surface \u00EF\u0080\u00A8 \u00EF\u0080\u00A9jS is completely out of cut after wT elapsed time as shown in Figure 4.35. Although, more vibrations are imprinted on the surface, the amplitudes remain the same, thus the maximum SLE value calculation is not effected by the multiple vibration imprints. Figure 4.35 Overlap effect due to the wiper edge during the surface generation in turn-milling. 4.3.7 Simulations and Experimental Validations The proposed dynamics and stability model of the turn-milling is validated by experimental cutting tests at different conditions. First, the dynamic model of the machine tool has been experimentally identified, and position-dependent dynamics have been constructed. Then, the 135 prediction of in-process workpiece dynamics has been verified for different workpiece geometries and materials. Turn-milling stability model has been validated for various cases such as for rigid and flexible workpiece dynamics to show effect of workpiece spindle speed on the time delay and stability limits. Finally, the low radial immersion bifurcation analysis and process damping tests have been conducted to validate the proposed stability model of the turn-milling. 4.3.7.1 Analysis of Position Dependent Machine Dynamics Machine tools have various flexible structural components such as column, ram, spindle, holder and cutting tool. Each component introduces additional flexibility to the tool tip which is usually the most flexible component on the machine tools for light cutting operations. On the other hand, the tool tip FRF may vary depending on the position of columns and ram within the machine volume leading to different dynamic responses at different locations of the machine. For the machining of bulky workpieces having a large diameter and long length, the tool tip FRF may even change during the cutting process. Figure 4.36 shows the functional volume of the Mori Seiki NT DCG 3150 type of mill-turn center used for the experimental validation tests through the thesis. It has three translational (X,Y,Z) and two rotary drives (B,C) which have stroke limits as listed in Table 4.5. 136 Figure 4.36 Representation of functional machine tool volume for NT 3150 DCG Mill-turn center. Table 4.5 Stroke limits of translational and rotary drives of Mori Seiki NT 3150 DCG mill-turn center. Stroke Limit Translational Drives Rotary Drives X [mm] Y [mm] Z [mm] B [ ]\u00EF\u0082\u00B0 C [ ]\u00EF\u0082\u00B0 1370 250 570 120\u00EF\u0082\u00B1 \u00EF\u0082\u00B0 \u00EF\u0082\u00B1\u00EF\u0082\u00A5 In order to determine the position dependency of the tool tip FRF, 90 impact tests have been conducted on the machine. The 2 fluted cylindrical end mill having 20 mm diameter and 95 mm stick out length from the tool holder is mounted on the spindle. The tool tip FRFs at YY and ZZ directions of the machine coordinate frame are measured at 45 different locations within the functional machine volume. First, the variation of tool tip FRF is observed with the ram length. As the tool travels along the (+) Y direction of the machine, the ram extends from -62.5 mm to +62.5 mm having a total 137 stroke of 125 mm. The FRFs are measured at the nodes along the ZZ and YY directions as shown in Figure 4.37. Figure 4.37 Location of measured FRFs at (X, Y, Z) = ([0-550], [-125,+125], -470)mm. The measured FRFs are superposed at their corresponding X-axis coordinates. 5 different locations are selected over the stroke limit of the ram (Y-axis); 0 mm, 125\u00EF\u0082\u00B1 mm, 62.5\u00EF\u0082\u00B1 mm and the measured FRFs are shown in Figure 4.38. 138 Figure 4.38 Measured FRFs at different location of ram length and X axis. As the ram extends along the +Y direction of the machine, the flexibility of the dominant tool mode (Z:670 Hz, Y:775 Hz) increases almost twice in Y direction and 45% in the Z direction of the machine. In Figure 4.39, the maximum FRF amplitude of the dominant mode is mapped on the machine volume in the X-Y plane. 139 Figure 4.39 Peak amplitude variations of the tool FRF along y and z directions of the machine. On the other hand, the amplitude of the dominant mode of the tool does not change much as the tool moves along the X direction of the machine. Since the ram can be considered as a cantilever beam structure, its length has great effect on the flexibility of the tool tip and during the 140 X-axis travel, the overall length of the structure remains constant, thus the total compliance at the tip. Also, the modal parameter variations of the dominant tool mode should be defined for further stability analysis. In Figure 4.40, the modal parameter variations are shown with the increased ram length for the Y direction FRF which is more flexible than Z direction FRF of the tool, thus the decisive one for the stability analysis. Figure 4.40 Variation of modal parameters (natural frequency (a), damping ration (b), dynamic stiffness (c), modal mass (d)) with the increased ram length. As it can be seen from Figure 4.40 (a), the natural frequency of the most dominant mode shifts from 705 Hz to 780 Hz as the ram travels along its stroke limit. When the ram length is minimum, the damping ratio measured from the tool tip is around 9% whereas it reduces until 1% as the ram reaches its maximum length due to the increased flexibility of the entire structure. Similarly, the dynamic stiffness reduces around 25% within the stroke limit of the ram. Finally, the modal mass 141 measured from the tip of the tool reduces from 1.2 kg to 0.75 kg. Note that, the dynamic stiffness changes the absolute stability limit of the cutting process whereas the natural frequency defines the locations of the stability pockets where the cutting productivity is substantially increased by cutting deeper and faster. Similar to the previous analysis, the effect of ram length on the tool tip FRF is repeated for the different Z-axis locations of the machine. The peak amplitude of the dominant mode in Y direction increases almost twice when the ram extends to its maximum stroke limit regardless of the Z-axis position of the tool in the machine. Variation of the peak amplitude of Y direction of the tool is shown at different positions \u00EF\u0080\u00A8 \u00EF\u0080\u00A9[ 472, 236,0]mmz \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD of the machine in Figure 4.41. Figure 4.41 Peak amplitude variation of Y direction FRF of the tool at different z-axis locations. In Figure 4.42, the Y and Z direction FRFs of the dominant tool mode are mapped on the X-Z plane of the machine tool. In this analysis, the ram length is kept constant at its maximum stroke 142 limit and X and Z drives of the machine are translated along their stroke limits. As seen from the Figure 4.42, both the Y and Z direction FRF of the tool do not change considerably compared to the ram length extension effect at X-Y plane of the machine. Y and Z direction FRFs of the tool are around 15% and 20% sensitive to the X-Z axes motion of the tool, respectively. Figure 4.42 Variations of Y and Z direction FRFs of the dominant tool mode on X-Z plane of the machine. Finally, the stability of the cutting process with each measured tool tip FRF at different positions on the machine are simulated to show the effect of ram length on the process stability. 143 Slot milling of the aluminum alloy Al7050-T7451 with the same milling tool is selected as the simulation parameters. Note that, the cross-talk FRFs of the tool are omitted in this analysis. The stability limits plotted with rigid lines represent the minimum and maximum stability limits of the most flexible and rigid modes of the tool whereas the remaining modes are plotted as dashed-thin lines. When the ram stroke is minimum, the absolute stability limit of the cutting process is calculated as 4.1 mm as shown in Figure 4.43. As the tool travels along the +Y axis, the extended ram length reduces the process stability down to 2.26 mm which is almost the half of the stability limit of the most rigid position of the tool. The most conservative cutting region for the given process parameters is hatched in the figure. Therefore, the process planner should avoid any process parameter that may yield chatter vibrations considering the variation of tool flexibility along the toolpath. Figure 4.43 Stability of the turn-milling cutting process for the most flexible and rigid modes of the tool within the machine functional volume. 144 4.3.7.2 Analysis of the in-process workpiece dynamics The variation of workpiece FRFs during the machining is significant for the process stability when the workpiece FRFs are flexible. Due to the excessive mass removal during the cutting process, the workpiece FRFs should be predicted at different machining phases. In this section, the proposed in-process FRF prediction algorithm based on the Receptance Coupling Theory is validated for different materials. First, the chuck FRF is identified by decoupling the dummy workpiece geometry from the measured direct and cross FRFs \u00EF\u0080\u00A8 \u00EF\u0080\u00A911 12 22i.e. , ,H H H of the workpiece as presented in Section 4.3.2. The measurements are performed at X and Y directions of the workpiece as shown in Figure 4.44. Figure 4.44 Representation of measurement coordinates for the chuck FRF identification. Aluminum alloy of Al6061 is used as the dummy workpiece material for the chuck identification. The material properties of the dummy workpiece material is given in Table 4.6. Table 4.6 Material Properties of Al 6061 alloy. Density 3kg m\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AB \u00EF\u0083\u00BB Young Modulus \u00EF\u0081\u009B \u00EF\u0081\u009DGPa Poisson Ratio Loss Factor 2700 69.6 0.33 0.0002 145 The geometry of the dummy workpiece is decoupled from the measured FRFs by modeling each cylindrical component as Timoshenko Beam based FE model. The identified chuck FRF is shown in Figure 4.45. The chuck is almost five times more flexible in x-direction of the machine whereas the y-direction has much more damping and high rigidity. Figure 4.45 Identified Chuck FRFs at XX (a) and YY (b) directions. The modal parameters of the identified chuck FRF at x and y directions are listed in Table 4.7. Table 4.7 Modal parameters of the identified chuck FRFs at x and y directions of the machine. Natural Frequency [Hz] Stiffness [N/m] Damping Ratio [%] Modal Mass [kg] XX 359 76.86 10\u00EF\u0082\u00B4 1.5 13.48 YY 345 74.25 10\u00EF\u0082\u00B4 11.13 9.07 After decoupling the dummy workpiece and identify the chuck dynamics in both directions, the workpiece that is used in the experiments needs to be coupled on the identified chuck\u00E2\u0080\u0099s FRFs. Stock workpiece has 120 mm length and 100 mm diameter and the same material with the dummy 146 workpiece (Al6061). The stock geometry is modeled as a single Timoshenko Beam and coupled to the chuck\u00E2\u0080\u0099s FRF. The dynamics at point 1 which is the most flexible location of the stock workpiece is predicted by the Receptance Coupling model in x and y directions of the machine as shown in Figure 4.46. Figure 4.46 Predicted workpiece FRFs of Al alloy at XX and YY directions before and after the machining. 147 As seen from Figure 4.46, the predicted tip FRFs of the stock workpiece has good match with the measured FRFs at point 1. Both the low frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A9313n Hz\u00EF\u0081\u00B7 \u00EF\u0080\u00BD and high frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A91150n Hz\u00EF\u0081\u00B7 \u00EF\u0080\u00BD modes at x and y directions agree well with the measured FRFs at the same point. Then, the material is removed from the stock material and the shaft geometry is changed after several passes of the cutting tool. Four different Timoshenko beams are modeled to represent the final shape of the workpiece and each individual beam is coupled to each other. Then, the assembled beams are coupled on top of the chuck dynamics, and the FRFs at point 1 are calculated and compared with the measured FRFs as shown in Figure 4.46. While the predicted low frequency modes have good accuracy with the measured FRFs, the discrepancy increases at the high frequency modes. The predicted natural frequency of the workpiece is ~25% away from the measured FRF in x direction and the error is ~10% for the y direction. Another set of experiment is conducted on the same machine but with a different workpiece material. Workpiece made of AISI 620 stainless steel is used for the validation tests and the material properties are given in Table 4.8. Table 4.8 Material properties of AISI 620 stainless steel. Density 3kg m\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AB \u00EF\u0083\u00BB Young Modulus \u00EF\u0081\u009B \u00EF\u0081\u009DGPa Poisson Ratio Loss Factor 7860 200 0.29 0.002 The stock workpiece is represented by two beams having different lengths and diameters. The presented coupling model is applied to predict the tip FRF for the stock workpiece and the analytically obtained FRFs are compared with the measurements in Figure 4.47. The amplitude and the natural frequency of the most dominant mode of the workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9251n Hz\u00EF\u0081\u00B7 \u00EF\u0080\u00BD is predicted 148 with a high accuracy. When the material is removed after the machining and the final shape of the workpiece is represented by three Timoshenko beams. The chuck mode, which is around 251 Hz, is still the dominant mode of the system however the high frequency modes starts appearing in the FRF content of the workpiece. The predicted high frequency modes have acceptable agreement with the measurements as shown in Figure 4.47. Figure 4.47 Predicted workpiece FRFs of Steel alloy at XX and YY directions before and after the machining. 149 4.3.7.3 Experimental Validation of Turn-Milling Stability Model The proposed dynamic and stability model of turn-milling given in Section 4.3 has been validated with simulations and experiments conducted on Mori Seiki 3150/500C CNC Mill-Turn center. The experimental turn-milling set-up is shown in Figure 4.48, where sound and tool displacement data during the machining have been collected by a microphone (Shure PG81) and fiber-optic displacement sensors (Philtec RC 20), respectively. Figure 4.48 Experimental set-up for turn-milling cutting tests (a); position of fiberoptic displacement sensors on the fixture. Two pairs of fiberoptic sensors have been placed on the fixture at X and Y directions of the machine to capture the displacements at the shank of the tool. Since the mode shape of the tool can be identified experimentally, the tool tip vibrations (displacements) can also be found by projecting the shank vibrations by the mode shape at the dominant mode of the tool. Fiberoptic displacement sensor is a non-contact and reflectance compensated type of sensor and capable of measuring the displacements of a rotating tool. Precise positioning of the fiberoptic sensor is achieved by a two-axis translational stage which can translate along the X and Y directions by 12.7 150 mm travel stroke with 250 m\u00EF\u0081\u00AD resolution. The fiberoptic sensors haven been placed on top of the translational stage which is also mounted on the fixture as seen in Figure 4.48(c). Since the fiberoptic displacement sensors measure cutting vibrations, it should be as noise-free as possible. The vibrations due to the cutting forces at the tool tip may distort the measured signal by the displacement sensor, thus the vibration isolation of the fixture should be ensured (see Figure 4.49(a)-(b)). In this thesis, vibration isolation pads made of sylomer (polyurethane foam) are used to damp out the transmitted vibrations from tool tip to the sensor casing. Low density sylomer pads are used at the bolted joints of the fixture and high density sylomer pads are placed underneath the translational stages to isolate the vibrations transmitted to the sensor casing as shown in Figure 4.49(c). Figure 4.49 Representation of vibration isolation with (b) and without (a) dampers; Locations of vibrations pads on the fixture (c). Then, the cross talk FRFs between the tool tip and fiberoptic sensor are measured and compared against the case where the vibration pads are not used. As seen in Figure 4.50, the 151 vibration pads reduce the transmitted vibrations from tool tip to the sensor casing by ~ 4 times. The natural frequency of the assembled fixture is around 80 Hz which may correspond to tooth passing frequency when tool rotates at 2500 rpm, thus the damping the noise is crucial for measuring a reliable and accurate displacement value. The transmissibility ratio can be minimized by designing the sylomer pads more accurately. Since modifying the thickness and surface area of the pads changes its stiffness and natural frequency as well, the vibration transmissibility would be improved further. Figure 4.50 Effect of vibration pads on the amplitude of the cross talk FRFs between the tool tip and fiberoptic sensor casing. The sensitivity of the fiberoptic sensor also needs to be calibrated. The tip of the fiberoptic sensor is positioned with respect to the center of the rotating cutting tool. Then, the sensor is moved away from the tool at small increments. At each position of the sensor, the voltage output is noted and plotted as seen in Figure 4.51. The sensitivity of the fiberoptic sensor is calculated as ~ 5.8mV \u00CE\u00BCm in the linear operation range. 152 Figure 4.51 Sensitivity of the fiberoptic displacement sensor. In the first experimental validation case, both the tool and workpiece have comparable flexible modes hence their dynamics are considered. Aluminum alloy Al6061-T6 which has 46mm diameter and 125 mm length is turn milled with a two-fluted cylindrical end mill having 20 mm diameter, 95 mm stick out, and 13.5\u00C2\u00B0 rake and 25\u00C2\u00B0 helix angles. The cutting coefficients are identified by oblique transformation from the orthogonal database for given tool geometry and workpiece material as; \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 930,290,242 MPatc rc acK K K \u00EF\u0080\u00BD . The axial feed of the tool is commanded as 8 /Cmm rev where Crev represents the angular speed of C-axis which carries the workpiece. The modal parameters of each flexible mode of tool and workpiece are measured in MCS by impact modal test where the modal hammer and miniature accelerometer sensitivities are 2.248mV N and 9.9mV g , respectively. The modal parameters of tool and workpiece are listed in Table 4.9. As seen from Figure 4.52, the cross FRFs of the workpiece are comparable with the direct FRF terms, hence they are included in the model. 153 Table 4.9 Modal parameters of tool and workpiece. Tool Workpiece Directions Modes Frequency (Hz) Damping (%) Mass (kg) Frequency (Hz) Damping (%) Mass (kg) XX 1 682 5.32 1.303 332 1.84 4.94 2 800 1.97 1.635 772 1.2 1.63 YY 1 776 3.2 0.873 307 3.42 7.40 2 781 1.1 1.40 XY 1 326 0.72 34.1 2 774 0.20 40.7 YZ 1 787 0.24 19.0 Figure 4.52 Frequency Response Function (FRF) of flexible tool (a) and workpiece (b). The cutter-workpiece engagement (CWE) is evaluated from MACHPRO \u00E2\u0084\u00A2 virtual machining system as presented in [1]. Then, the stability of the system has been simulated within the feasible speed ranges of the tool and workpiece as shown in Figure 4.53. The stability lobes shift towards 154 higher tool speeds as the workpiece spindle speed increases due to the altered time delay in the system. Figure 4.53 3D and 2D cross sectional views of the stability diagram simulated with the modal parameters listed in Table 4.9. The simulated stability diagrams shown in Figure 4.53 are experimentally validated. First, the stability of the given system where the workpiece rotates at 6c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min] is solved, as seen in Figure 4.54.Each simulation is run on a PC having Intel \u00C2\u00AE Core \u00E2\u0084\u00A2 i7 3.40 GHz central processing unit (CPU) and takes ~ 2 hours. The cutting tests have been conducted at different tool spindle speeds. Chatter has been observed at 3350t\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min] at 1.2 mm and 1.4 mm axial depths of cut. Analysing the sound and displacement data at 1.2 mm depth of cut (point A), the system has chattered at 0H 871\u00EF\u0081\u00B7 \u00EF\u0080\u00BD Hz, and its harmonic multipliers are visible at the integer multiples of tooth passing frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A90 0H H , 1,2,i Ti i k\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00BD \u00EF\u0082\u00B1 \u00EF\u0080\u00BD where 0 111.7T\u00EF\u0081\u00B7 \u00EF\u0080\u00BD Hz (see Figure 4.54 (c)). This type of chatter is identified as Hopf bifurcation and the chatter motion is quasi-periodic as confirmed from the tool motion in feed and normal directions (see Figure 4.54 (d)). The tool moves on a torus shape and stroboscopically sampled noise-free deflection data (grey points) form an ellipse which oscillates in time with large vibration amplitudes [40]. 155 Figure 4.54 Stability validation tests for case 1 (a). See Table 4.9 for the dynamic parameters of the turn-milling system. Stability limits and experimental results at c\u00CE\u00A9 = 6 [rev/min] (b); the FFT of sound data (c) and tool motion in feed and normal directions (d) at Point A (stable). Experimental results at c\u00CE\u00A9 = 21 [rev/min] (e) and corresponding sound FFT (f) and tool motion (g) at Point B (chatter). 156 As the workpiece spindle speed is increased to 21c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min], the time delay in the regenerative system \u00EF\u0080\u00A8 \u00EF\u0080\u00A921,j i\u00EF\u0081\u00B4 changed by 4% with respect to the time delay when the workpiece rotates at 6 [rev/min] \u00EF\u0080\u00A8 \u00EF\u0080\u00A96,j i\u00EF\u0081\u00B4 and the stability limits are shifted to right as seen in Figure 4.54 (e). The cutting depths 1.2 mm and 1.4 mm which were unstable at the workpiece spindle speed of 6c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min], became stable when the speed is increased to 21c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min]. The vibrations are dominated only at the tooth passing frequency ( 0 111.7T\u00EF\u0081\u00B7 \u00EF\u0080\u00BD Hz) and its harmonics which indicate stable, forced vibrations as seen in Figure 4.54 (f). Tool oscillates periodically and stroboscopically sampled data form a compact cloud (see Figure 4.54 (g)) which means that the peak vibration amplitudes at feed and normal directions remain approximately constant. Surface photos of the stable and unstable (chatter) cutting points also reveal that the severe chatter marks have been observed at Point A where the process was chatter (see Figure 4.55). On the other hand, when the workpiece spindle speed is increased to 21 rpm, the same point has been stabilized and no chatter marks have been observed on the finished surface of the part. Figure 4.55 Surface photos of the stable and unstable (chatter) cutting process. 157 The vibrations at certain depths and spindle speeds of tool and workpiece are also simulated and the surface location errors are calculated for the given case. The milling tool\u00E2\u0080\u0099s bottom face has ~ 2.73\u00EF\u0082\u00B0 clearance angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00A1 . The simulations are conducted at 0.8 mm depth of cut for 4000 rpm of tool spindle speed and 6 rpm of the workpiece spindle speed. For the given geometry and speeds of tool and workpiece, the total feed magnitude is calculated as 0.4 mm/rev/tooth as discussed in Section 3.3.1. Then, the vibrations between the flexible tool and workpiece along the feed, normal and axial directions are simulated in time domain by solving the state equations given in Eq. (4.127). In order to demonstrate both transient and steady-state behavior of the vibrations, fifty spindle periods are taken for the simulations. Figure 4.56 shows the simulated vibrations along the normal (a), feed (b) and axial (c) directions of the cutting tool. The transient vibrations diminish after 4 spindle periods of the tool. Peak-to-peak values of vibrations are around 10 m\u00EF\u0081\u00AD , 15 m\u00EF\u0081\u00AD , 8 m\u00EF\u0081\u00AD for the normal, feed, and axial directions, respectively. The surface form error of the turn-milled workpiece is also calculated considering static and dynamic components of the cutting process. First, the circularity error between the desired and actual machined surface is calculated as discussed in Section 4.3.6.2. The machined polygon-shaped workpiece deviates from the desired workpiece geometry by 0.3 m\u00EF\u0081\u00AD (see Eq. (4.130)) which is remarkably small. Since the milling tool has the clearance angle, the bottom edge contact between the tool and workpiece is point, thus there is no overlap effect between the generated surfaces at different tooth passing periods. Substituting the bottom edge clearance angle \u00EF\u0080\u00A8 \u00EF\u0080\u00A9b\u00EF\u0081\u00A1 , and total feed magnitude \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tf into the Eq. (4.131), the maximum static cusp height is calculated as ~ 18.8 m\u00EF\u0081\u00AD . 158 Figure 4.56 Simulated vibrations along the normal (a), feed (b), and axial (c) directions of tool. The resultant cusp height is predicted by superposing the cutting vibration terms into the static cusp height. First, the vibrations along the tool normal axis (Z-direction) are calculated within the 159 cutter-workpiece engagement geometry. The start and exit angles are calculated as 0st\u00EF\u0081\u00A6 \u00EF\u0080\u00BD and 1.3694st\u00EF\u0081\u00A6 \u00EF\u0080\u00BD rad for the given process parameters. Then, the axial vibrations are found within the specified engagement boundaries as shown in Figure 4.57. Figure 4.57 Simulated axial direction vibrations within the engagement boundaries of the process. The peak-to-peak amplitude of the axial direction vibrations within the engagement zone is ~ 5 \u00CE\u00BCm , however the indented surface error is calculated by considering the vibrations penetrating into the workpiece which is ~ 3\u00CE\u00BCm in this simulation case. The resultant cusp height or the error left on the workpiece is shown in Figure 4.58 for one spindle period of the cutting process since the steady state vibrations are assumed to stay constant during the toolpath. As the dynamic cutting vibrations overcut the workpiece surface by ~ 3\u00CE\u00BCm , the total cusp height left on the workpiece (undercut) is simulated as ~ 21\u00CE\u00BCm . Note that, the errors printed on workpiece surface at present time rotates on the workpiece due to the workpiece rotation, and a new surface is generated after one tooth passing period as shown in Figure 4.58 (b). 160 Figure 4.58 Surface location errors contributed by the static and dynamic cutting terms of the system for one spindle period. The regions of undercut-overcut surface patterns and their corresponding error amplitudes are plotted in Figure 4.59. Although the desired workpiece surface is circular, it is shown as planar surface at zero amplitude to reference the surface location errors. Since the tip of the tooth\u00E2\u0080\u0099s edge does not yield static cusp height on the surface, it overcuts the workpiece due to the existing vibrations. As the contact region between the tool\u00E2\u0080\u0099s bottom face and workpiece extends behind the edge tip, the static cusp heights and corresponding vibrations undercuts the workpiece. 161 Figure 4.59 Surface location errors left on the workpiece with the corresponding regions of undercut and overcut. A second set of experiments where the workpiece is more flexible than the tool, has also been conducted to validate the proposed model. The cutting tool and the workpiece material was the same as in the previous case. The modal parameters of the flexible workpiece are given in Table 4.10, and the FRFs are plotted in Figure 4.60. Table 4.10 Modal parameters of flexible workpiece in case 2. Direction Mode Frequency [Hz] Damping [%] Mass [kg] XX 1 367 1.49 4.72 2 1135 1.14 0.22 XY 1 1126 0.97 1.15 YY 1 1129 1.12 0.23 YZ 1 1133 0.71 1.28 162 Figure 4.60 FRF of the flexible workpiece in case 2. The cutting tests have been performed at 25% immersion of tool and all the other parameters are kept same as in the previous experimental validation case (Case 1). The simulated stability diagram and experimental results are given in Figure 4.61. The stability lobes shift to higher tool spindle speeds as the workpiece spindle speed increases. The cutting depth 0.6 mm at tool spindle speed 3500t\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min] and with both workpiece speeds 6c\u00EF\u0081\u0097 \u00EF\u0080\u00BD and 12c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min] was predicted to be unstable, and proven by the experiments as seen in Figure 4.61 (b). When the workpiece spindle is increased to 40c\u00EF\u0081\u0097 \u00EF\u0080\u00BD [rev/min], the process has become stable as seen in Figure 4.61 (c). From analysing of the sound and displacement data (Figure 4.61 (e),(f)), only the Hopf bifurcation type of instability has been observed in the system. 163 Figure 4.61 Stability validation tests for case 2 when c\u00CE\u00A9 = 6 [rev/min] (a). See Table 4.10 for the modal parameters of the system. Stability limits and experimental results at c\u00CE\u00A9 =12 [rev/min] (b) c\u00CE\u00A9 = 40 [rev/min] (c), c\u00CE\u00A9 =100 [rev/min] (d) and FFT of sound data at Point A (e) and Point B (f). 164 Previous experimental validations of the proposed stability model of the turn-milling identified the stable and unstable regions of cutting both analytically and experimentally for different workpiece spindle speeds. However, only the Hopf bifurcation type of instability has been observed during the experimental validation tests. In order to determine the other type of instability which is called Flip bifurcation, the given validation cases are simulated at extreme conditions. Since it is known that the nonlinear behavior of the cutting becomes stronger as the radial immersion is reduced, the turn-milling with 5% immersion has been simulated to identify the regions of Hopf and Flip bifurcations. Figure 4.62 shows the stable and unstable regions of cutting with the corresponding type of instability. The eigenvalues are plotted on the complex plane at Point A and it is shown that the magnitudes of the eigenvalues are less than unity in the stable region of the stability diagram. Similarly, the characteristic multipliers in the region B corresponds to Hopf bifurcated chatter where the eigenvalues are complex conjugate and their magnitudes are greater than the unity. On the other hand, real eigenvalues with magnitudes greater than one exist at Points C and D which led to primary and secondary flip type chatters. At point C, there is one real eigenvalue with the magnitude of greater one, thus primary flip type of chatter is observed in the cutting system. At Point D, there are two real eigenvalues and their magnitudes are greater than the unity, thus the system is called secondary flip type of chatter. The corresponding eigenvalues on the complex planes are plotted for each sample point on the stability diagram in Figure 4.62 . It is shown that, flip type of chatter, even though the system is highly interrupted due to very low immersion (5%), does hardly exist in turn-milling for the feasible spindle speed range. In regular milling, however, they are more visible in high speed-low immersion milling operations. 165 Figure 4.62 Stable and unstable regions of cutting with their corresponding stability properties based on the eigenvalues analysis. A: Stable, B: Hopf type chatter, C: Primary Flip type chatter, D: Secondary Flip type chatter. Since the flip bifurcations are barely seen at low spindle speeds of tool in turn-milling, the simulations have been carried out at high tool spindle speeds just to validate their existence. Although they occur at small regions, the bifurcated stability lobes can be seen in Figure 4.63 (a). 166 The type of the instability and vibration is investigated by checking the characteristic multipliers at the critical regions which are located at the left shoulders of the lobes at high speed zones as highlighted on Figure 4.63 (a) [40]. The lens-like flip lobes are much more visible than the previous simulation case where the tool spindle speeds are low. As the tool spindle speed increases, formation of the flip lobes can be seen distinctly. Particular type of flip bifurcation is plotted in Figure 4.63 (b) and (c). However, since the maximum tool spindle is limited to 12000 [rev/min] on the mill-turn machine used in the experiments, the high speed bifurcation validation tests were not possible. Figure 4.63 Stability properties at high speed and low immersion of turn-milling. 167 The proposed stability model of the turn-milling at the low-speed region where the process damping is effective has also validated experimentally. Two different process damping methods explained in Section 4.3.3.3 and Section 4.3.5 are simulated for the given turn-milling case and verified by the experimental chatter tests. A two-fluted coated cylindrical end mill having 20 mm diameter and 70 mm stick out is used to turn-mill the AISI 620 stainless steel workpiece with 120 mm length and 100 mm diameter. The cutting coefficients are identified by oblique transformation from the orthogonal database for given tool geometry and workpiece material as; \u00EF\u0081\u009B \u00EF\u0081\u009D \u00EF\u0081\u009B \u00EF\u0081\u009D, , 2650,1800,125 MPatc rc acK K K \u00EF\u0080\u00BD . The modal parameters of the flexible tool and workpiece are given in Table 4.11. The axial feed of the tool is commanded as c8 mm/rev . Table 4.11 Modal parameters of the 2 fluted cylindrical end mill having 20 mm diameter and 70 mm stick out and AISI 620 stainless steel workpiece. Tool Workpiece Directions Modes Frequency (Hz) Damping (%) Mass (kg) Frequency (Hz) Damping (%) Mass (kg) XX 1 813 3.51 1.99 250 1.90 13.45 2 2322 3.49 0.26 282 3.53 41.83 YY 1 807 3.88 1.31 231 4.28 45.34 2 2319 3.88 0.26 260 2.93 15.58 XY 1 253 1.45 31.66 2 814 2.19 11.51 YZ 1 263 1.78 182 2 815 2.34 36.27 First, the stability of the given system is solved by semi discretization method in time domain without considering the process damping forces. The stability limits without any process damping 168 effect is shown as the solid black line in Figure 4.64. The absolute stability limit is calculated as ~ 0.3 mm and the stability lobes do not bend upward due to the neglected process damping forces at low spindle speeds. Then, the stability of the system is solved for the same cutting conditions but including the process damping forces as expressed in Eq. (4.70). The process damping coefficients for AISI 620 has not been identified but the coefficients which are found for Ti6Al4V titanium alloy in [54] are used since both materials have similar hardness values \u00EF\u0080\u00A8 \u00EF\u0080\u00A9~ 340 HB . The specific indentation force coefficient \u00EF\u0080\u00A8 \u00EF\u0080\u00A9spK and Coulomb friction coefficient \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0081\u00AD are taken as 6 32.69 10 N \u00CE\u00BCm\u00EF\u0082\u00B4 and 0.72 , respectively. The equivalent flank wear land length is calculated as 0.028m for the tool having 60 m\u00EF\u0081\u00AD hone radius and 5\u00EF\u0082\u00B0 clearance angle. The separation angle is also assumed as 50\u00EF\u0082\u00B0 . Note that, the identified specific indentation force coefficient in [54] is adapted for the modal parameters given in Table 4.11 in order to provide the same process damping coefficient for the system. After calculating the process damping forces and solving the Eq. (4.89), the stability limits (solid purple line) are obtained as shown in Figure 4.64. Stability characteristics of two solutions are similar at the high spindle speed region and same stability limits are obtained for high speed stability pockets. As the tool spindle speed lowers, stability limits with process damping effect start to differentiate and divert from the solution without process damping forces. As seen from Figure 4.64, the stability limit goes infinity after the asymptotic spindle speed of 500 rpma\u00EF\u0081\u0097 \u00EF\u0082\u00BB . 169 Figure 4.64 Experimental validation of the process damping models for turn-milling of AISI 620 stainless steel. The solid black line represents the stability limits calculated without considering low-speed process damping forces. The solid purple line shows the stability limits with the process damping model given in Section 4.3.3.3. The red curve defines the low speed process damping stability envelope which is predicted by the asymptotic spindle speed method as expressed in Section 4.3.5. Finally, the same turn-milling system is solved by the Eigen decomposition method that is explained in the Section 4.3.5. Only two different set of chatter experiments are needed to construct 170 the stability envelope of the system. The first experimental test has been performed at ,1 2070 rpmt\u00EF\u0081\u0097 \u00EF\u0080\u00BD - 1 0.4 mma \u00EF\u0080\u00BD and the chatter frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,1cf is measured at 265 Hz. The second experiment has also been conducted at ,2 1625 rpmt\u00EF\u0081\u0097 \u00EF\u0080\u00BD - 2 0.5 mma \u00EF\u0080\u00BD where the chatter frequency \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,2cf was 264.2 Hz. Then, the limiting depth of cut \u00EF\u0080\u00A8 \u00EF\u0080\u00A9lima and asymptotic spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a\u00EF\u0081\u0097 which determine the bottom and left boundaries of the stability envelope are solved by substituting the spindle speeds, depth of cuts and chatter frequencies that are taken from experiments into Eq. (4.120). The limiting depth of cut is calculated as 0.15 mm and the asymptotic spindle speed for the given system is calculated as 1147 rpma\u00EF\u0081\u0097 \u00EF\u0080\u00BD . The specific indentation coefficient is also found as 6 32.51 10 N \u00CE\u00BCm\u00EF\u0082\u00B4 which is very close to the value used for the previous process damping model. The upper boundary for the stability envelope is constructed and shown as the solid red line in Figure 4.64. The validation tests confirm the stability envelope calculated by the eigen decomposition method. The experiments conducted at the tool spindle speeds at 1000 rpm and 1250 rpm were stable as oppose to the stability pockets predicted by the process damping force model. Experiments also validate the asymptotic spindle speed prediction of the eigen decomposition method. Under 1250 rpmt\u00EF\u0081\u0097 \u00EF\u0082\u00BB , the stability of the system goes infinity, leading all the feasible axial depth of cuts to stable cutting points. On the other hand, the asymptotic speed predicted by the process damping force model is ~ 500 rpm which is as not accurate as the predicted value by the eigen decomposition method. In conclusion, the validation tests confirm the accuracy of the shaded stability envelope predicted by the eigen decomposition method. Two sample points are taken from stable and unstable regions and the sound data is analysed. The FFT spectrum shows 171 only the tooth passing frequency and its harmonics at Point A which clearly states the stable cutting conditions. On the other hand, dominant chatter frequency peak and corresponding Hopf bifurcation harmonics can be seen in the FFT spectrum of the Point B where the system was unstable. The validation tests conducted at higher spindle speeds of tool also confirm the proposed stability model of turn-milling. The comparison of two different process damping models leads us to evaluate the capabilities and deficiencies of each model. The process damping force model is widely adopted in the literature of the process damping in machining, and proven by many authors for the cases where the vibration amplitudes during the machining are small. Although it predicts the low-speed stability lobes with an acceptable accuracy, the model highly relies on assumptions such as constant separation angle, flank wear length, specific indentation coefficient and amplitude of the vibration. Even though these constant can be identified empirically, the uncertainty of the estimations is high, and may change during the dynamic cutting process. Also, creating the database for different material and process parameters is labor costly. Therefore, the application of the process damping force model is very restricted, and the results are often unreliable. On the other hand, the eigen decomposition method provides a quick prediction of the stability envelope at the low-speed region. Although it depends on some restrictions as explained in Section 4.3.5, there is no need to identify the separation angle, indentation force coefficient and wear length since they are inherently considered in system dynamics equations. The model only relies on two different sets of experiments and their corresponding chatter frequencies; thus it is robust to apply for different machining cases. In this thesis, the eigen decomposition method is applied in the turn-milling process and validated by experiments. 172 4.4 Summary This chapter presents the dynamics and stability of the turn-milling and regular milling process with asymmetric structural modes. The dynamic chip thickness and cutting forces are modeled as a function of the vibrations between the flexible tool and workpiece, the multi axes rigid body feed motion of the machine drives and cutter-workpiece engagement geometry. The time delay which alters the regeneration mechanism and the stability of the process is modeled by the discrete motions of the tool and workpiece. The resulting time varying delay differential equations are solved in time domain and the stable and unstable regions of cutting are calculated as a function of tool and workpiece spindles speeds and axial depth of cut. It is shown that the workpiece spindle speed alters the regenerative time delay in the chip thickness mechanism and the stability of the turn-milling process depends on the speeds of two spindles, rigid body feed motion of the machine drives. The vibrations during turn-milling process are also solved in time domain, and the surface location errors are modeled considering the tool and workpiece geometries, spindle speeds, and relative vibrations between the tool and workpiece. The proposed model is validated for various cases and materials such as for flexible workpiece dynamics, low-immersion of cutting where the bifurcation analysis has been concluded and the low-speed stability. 173 Chapter 5: Turn-Milling Process Optimization 5.1 Overview The productivity of turn-milling operations can be substantially increased if the process parameters are selected accurately. The material removal rate (MRR) in milling is highest when the machine tool runs at high spindle speeds corresponding to the integer divisions of dominant structure\u00E2\u0080\u0099s natural frequencies. On the other hand, the number of parameters and constraints that effect the productivity of the process are higher in turn-milling compared to turning and milling operations, which makes the process difficult to plan. The process must be planned before the production considering the most productive cutting conditions without violating the machine and process limitations. In this chapter, the constraint-based process parameter selection model is developed by employing the presented mechanics and stability model of the process. First, the machine tool and process based constraints are explained, and the corresponding specifications are given. Then, the parameter selection guideline is explained on a sample case. 5.2 Constraint Based Selection of Turn-Milling Process Parameters The proposed mechanics, dynamics, and stability models of the turn-milling system are employed to plan the speeds of tool and workpiece, feeds, and depth of cut without violating the physical limits of machine and cutting tool. In turn-milling process, the physical constraints include maximum and minimum chip thickness (load), cutting speed range for the machinability of the material, torque and power limits of the machine\u00E2\u0080\u0099s tool and workpiece spindles, the chatter stability limits and maximum surface form error of the finished part. On the other hand, the parameters such as tool life and wear rate which are not in the scope of this thesis should be taken into account to fully represent the process limitations. 174 The ultimate goal of the process planning is to increase the material removal rate (MRR) of the process. In turn-milling, the MRR is defined as the volume of material removed per minute, and can be calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,MRR= c a w wf a D a\u00EF\u0081\u00B0\u00EF\u0081\u0097 \u00EF\u0080\u00AD (5.1) where ,a wf is the axial feed of the tool per workpiece revolution \u00EF\u0080\u00A8 \u00EF\u0080\u00A9cmm rev , and implicitly depends on the workpiece spindle speed. As seen from Eq. (5.1), the MRR in turn-milling does not depend on the tool spindle speed, since the tool feed motion is defined with respect to workpiece rotation speed. In this way, the radial immersion of the tool remains constant regardless of the tool and workpiece spindle speeds. The tool spindle speed, on the other hand, changes the chip load acting on the tool. Figure 5.1 shows the relation between the workpiece spindle speed, axial depth of cut and the MRR of the turn-milling process. Figure 5.1 Effect of workpiece spindle speed and axial depth of cut on MRR in turn-milling process. High MRR values can be achieved for increased depth of cut, tool axial feed and workpiece rotation speed. However, the selection of these parameters strongly depends on the machine and process constraints which are discussed below. 175 5.2.1 Chip Load Constraint The chip load (thickness) \u00EF\u0080\u00A8 \u00EF\u0080\u00A9sjh , distribution along tool normal axis is explained in Section 3.3.1 for turn-milling operations. The maximum chip load is reached at certain radial and axial immersion angles of tool for given axial depth and width of cut. The chip load above a critical value will overload the cutting edges causing chipping and tool breakage. On the other hand, in the case of very low chip load, the chip cannot shear away from the workpiece, and the tool rubs the workpiece material resulting poor tool life and productivity. Also, the cutting force coefficient identified mechanistically, may vary from the linear edge model due to the size effect at the presence of low chip load. Therefore, the chip load must be kept within a certain range as; ,min ,maxs s sj j jh h h\u00EF\u0080\u00BC \u00EF\u0080\u00BC (5.2) where ,minsjh and ,maxsjh represents the lower and upper boundaries of the chip load constraint, respectively. Since the angular rotation of workpiece contributes to feed motion as linear feed at tool-workpiece contact region, increasing the workpiece spindle speed leads to high feed values acting on the tool, hence the tool spindle speed must be increased in order to overcome the high feed velocity. Therefore, the tool and workpiece spindle speeds are proportional with each other \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t c\u00EF\u0081\u0097 \u00EF\u0082\u00B5\u00EF\u0081\u0097 . Considering the tool axial feed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,a wf definition where the radial immersion is kept constant, and the maximum-minimum chip load that is defined by the tool manufacturer, the relation between tool spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t\u00EF\u0081\u0097 and workpiece spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u0097 can be shown as in Figure 5.2. The tool and workpiece speeds are iteratively selected by targeting a maximum and minimum chip thickness. 176 Figure 5.2 Feasible and non-feasible regions of tool and workpiece spindle speeds for given maximum allowed chip load. Workpiece diameter: 50 mm, Tool diameter: 20 mm, Maximum allowed chip load: 0.4 mm for 2 \u00E2\u0080\u0093fluted cylindrical end mill. The maximum and minimum chip loads in Figure 5.2 is determined as 0.4 mm and 0.05 mm for a milling tool having 20 mm diameter and two-flutes, respectively. For instance, the minimum tool spindle speed that does not violate the maximum chip load constraint when the workpiece rotates at 10 rpm, is calculated as 2000 rpm for given diameters of tool and workpiece. As the tool spindle speed increases, the chip load acting on the tool reduces. Figure 5.3 shows the feasible and non-feasible tool spindle speeds for different workpiece spindle speeds. When the workpiece spindle speed rotates at 5 rpm, the tool spindle speeds up to 8000 rpm are feasible. However, as the tool spindle speed increases further, the chip load reduces beyond the lower boundary of the chip load constraint \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,minsjh as shown in Figure 5.3 (a). On the other hand, for the case where the 177 workpiece spindle speed is 30 rpm, the tool spindle speeds below 6000 rpm are not feasible since they violate the maximum limit of the chip load constraint, ,maxsjh (see Figure 5.3 (b)). Figure 5.3 Feasible and non-feasible tool spindle speeds for the cases c\u00CE\u00A9 = 5 rpm (a), and c\u00CE\u00A9 = 30 rpm (b). The maximum and minimum allowed chip loads are determined as 0.4 mm and 0.05 mm for the given cutting tool. 5.2.2 Torque and Power Limits of Tool and Workpiece Spindles The torque and power limits of the tool and workpiece spindles should not be violated during turn milling operation. The peak values of the periodic tangential \u00EF\u0080\u00A8 \u00EF\u0080\u00A9tF and x-direction \u00EF\u0080\u00A8 \u00EF\u0080\u00A9xF turn-milling forces, thus the torque and power of each spindle must be restrained to prevent spindle overload. Cutting torque and power drawn from the tool spindle are calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D, , Nm, , Wt j t t jt j t t jT z R F zP z T z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0081\u0097 (5.3) Similarly, the torque and power transmitted to workpiece spindle (C-axis) are calculated as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D\u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0081\u009B \u00EF\u0081\u009D, , Nm, , Wc j w x jc j c c jT z R F zP z T z\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0081\u00A6 \u00EF\u0081\u00A6\u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0081\u0097 (5.4) 178 The torque and power charts of the workpiece and tool spindles of the NT 3150/500C mill-turn machine are given in Figure 5.4. Figure 5.4 Torque and Power charts for workpiece spindle (a); tool spindle (b) of NT 3150/500C multi axes machine tool. The calculated tool and workpiece spindle\u00E2\u0080\u0099s torque and power should be kept in a certain range as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9,min ,max,min ,max,min ,max,min ,max,,,,t t j tt t j tc c j cc c j cT T z TP P z PT T z TP P z P\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0081\u00A6\u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0080\u00BC \u00EF\u0080\u00BC\u00EF\u0080\u00BC \u00EF\u0080\u00BC (5.5) where the ,mintT , ,maxtT and ,mintP , ,maxtP are the lower and upper boundaries of the torque and power constraint for the tool spindle, respectively and determined from the torque & power chart in Figure 5.4 (b). Similarly, the torque and power limits ,mincT , ,maxcT , ,mincP , ,maxcP for the workpiece spindle are obtained from the values in Figure 5.4 (a). 5.2.3 Stability Constraint Stability of the turn-milling process is one of the most important limiting factors for the process planning and parameter selection stage. The stability of the turn-milling process with varying time 179 delays are explained in detail in Chapter 4, and the mathematical relationships between the depth of cut, tool and workpiece spindle speeds are investigated for given flexible tool and workpiece structures. Inaccurate selection of the process parameters \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,t ca \u00EF\u0081\u0097 \u00EF\u0081\u0097 leads to excessive cutting forces and regenerative chatter vibrations which may damage the spindles, break the milling tool, and result poor surface finish, thus reducing the productivity of the process substantially. On the other hand, the productivity (MRR) can be improved by selecting the most feasible cutting parameters from stability limits of the process which are presented in Figure 5.5. The process is always stable regardless of the spindle speed of the tool for the axial depths under the absolute stability limit (red line). However, the axial depth of cut can be increased further without chatter vibrations for the tool spindle speeds that corresponds to the peak of each stability lobe \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,1,2,...lobe\u00EF\u0081\u0097 Figure 5.5 The stability boundaries of a turn-milling process, and corresponding tool spindle speeds that correspond to peak of the stability lobes (pockets). 180 5.2.4 Surface Form Error Constraint The surface quality of the turn-milled workpiece is very significant for the productivity of the cutting process. The different sources of form errors in turn-milling are studied in Section 4.3.6. The total surface form error left on the workpiece \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TE is calculated by; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9 \u00EF\u0080\u00A8 \u00EF\u0080\u00A911 tancosT w t b o jct tE R a f zN\u00EF\u0081\u00A1 \u00EF\u0081\u00A6\u00EF\u0081\u00B0\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0081\u0097\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0081\u0097\u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 (5.6) where the first two terms represent the circularity error, and static cusp height, respectively, and the last term is the dynamic vibrations along the surface normal of the workpiece. For a given tool and workpiece geometries \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, , ,t w t bN R R \u00EF\u0081\u00A1 , the total form error can be tuned by selecting the proper tool-workpiece spindle speeds \u00EF\u0080\u00A8 \u00EF\u0080\u00A9t c\u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097 and depth of cut \u00EF\u0080\u00A8 \u00EF\u0080\u00A9a . The relationship between the design variables \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,t c a\u00EF\u0081\u0097 \u00EF\u0080\u00AD\u00EF\u0081\u0097 and total form error \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TE should be examined for the parameter selection of the turn-milling process. First, the effect of spindle speeds on the total form error is shown for an example case. Figure 5.6 shows the trend between the tool-workpiece spindle ratios \u00EF\u0080\u00A8 \u00EF\u0080\u00A9r t c\u00EF\u0081\u0097 \u00EF\u0080\u00BD\u00EF\u0081\u0097 \u00EF\u0081\u0097 and total form error \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TE . Increasing the ratio of tool-workpiece speeds reduces the total form error left on the workpiece. 181 Figure 5.6 Effect of tool-workpiece spindle speed ratio on the total form error left on the workpiece. For a given workpiece spindle speed, high tool spindle speeds should be selected to reduce the form error of the machined part. Note that, the chip load at each tool-workpiece spindle speed pair needs to be calculated for the accurate modeling. The effect of axial depth of cut is also investigated for different tool spindle speeds. For a given workpiece spindle speed \u00EF\u0080\u00A8 \u00EF\u0080\u00A960 rpmc\u00EF\u0081\u0097 \u00EF\u0080\u00BD , increasing the axial depth of cut for the certain turn-milling operation reduces the total maximum form error as shown in Figure 5.7. The form errors are the highest at low tool spindle speeds and small axial depth of cuts. 182 Figure 5.7 The effect of axial depth of cut at different tool spindle speeds for a given workpiece spindle speed. The importance of axial depth of cut on the total form errors can be explained more clearly by analysing the variation amplitudes of form errors at different tool-workpiece spindle speed ratios. The standard deviation of the peak amplitudes are calculated and the statistical outputs are plotted as boxplots as shown in Figure 5.8. Boxplots depict the median (red line), minimum, maximum and first-third quartiles of the form error data for each tool-workpiece spindle speed ratios. At low speed ratios, the deviation of the peak form error amplitudes is high from the median meaning that the effect of axial depth of cut on total form error is significant. As the speed ratio increases, the total form error variation reduces, and the effect of axial depth of cut diminishes. It should be noted that, the effect of the tool-workpiece spindle speed ratio is more significant than the axial depth of cut variation on the form errors in turn-milling operations. The speed ratio must be selected high enough to satisfy the maximum allowed form error of the finished workpiece. 183 Figure 5.8 Variation of peak total form error amplitudes at different speed ratios of tool and workpiece spindles. 5.2.5 Cutting Speed Constraint Cutting speed is one of the most significant limiting factors in the machining of high temperature metals such as titanium and nickel alloys which are widely used in aerospace industries. High cutting speeds lead to elevated cutting temperatures at tool-workpiece contact region, which in turn increase the tool wear rate, and reduce the productivity significantly. Turn-184 milling operations offers a two to ten fold increase in tool life compared to conventional turning operations for the machining of hard to cut materials such as Ti6Al4V, Waspaloy, and Inconel 718 [102]. On the other hand, the cutting speed which is correlated with the tool (flank) wear depends on the workpiece material (aluminum, nickel, titanium, steel alloys), coating of the tool (PVD, CVD), and cutting fluid (MQL, flood cooling) used during the operation. Although the turn-milling process offers low tool wear and high cutting speeds compared to conventional turning, the cutting speed for certain tool-workpiece material and coating grade is still restricted. The process parameters must be selected without violating the maximum allowed cutting speed which is given by the tool manufacturers. 5.3 Case Study for Turn-Milling Process Parameter Selection The aforementioned constraints are employed on a turn-milling case to demonstrate the process parameter selection methodology. A 20 mm diameter, two-flute cylindrical end mill having 13.5\u00C2\u00B0 rake and 25\u00C2\u00B0 helix angles is used to turn-mill the 50 mm diameter cylindrical workpiece made of Al6061-T6 where the cutting force coefficients are given in Section 4.3.7. The axial tool feed per workpiece revolution \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,a wf is selected as 8 mm, thus the radial immersion of the tool is kept constant during different tool and workpiece spindle speeds. The modal parameters of the flexible workpiece is listed in Table 5.1 where the milling tool is considered as rigid. The feasible tool and workpiece spindle speeds are selected as 12000 rpm and 100 rpm considering the limits of the machine and geometries of the tool and workpiece. Similarly, the maximum allowed axial depth of cut is limited to 15 mm for the parameter selection algorithm. 185 Table 5.1 The modal parameters of the flexible workpiece in XX and YY directions. Directions Modes Frequency [Hz] Damping [%] Mass [kg] XX 1 250 1.9 13.45 2 282 3.5 41.82 YY 1 231 4.2 45.35 2 260 2.9 15.58 The constraints of the parameter selection algorithm are determined considering the process and machine tool limitations. First, the minimum and maximum chip loads are set to 0.05 mm and 0.4 mm, respectively for the given milling tool. The torque and power limits of tool and workpiece spindles are given in Figure 5.4 for NT 3150/500C turn-milling machine. The stability limits are calculated for the given cutting conditions and dynamics of the workpiece as shown in Figure 5.9 for the constant workpiece spindle rotation \u00EF\u0080\u00A8 \u00EF\u0080\u00A9c\u00EF\u0081\u0097 at 10 rpm. Figure 5.9 Stability lobes for the given turn-milling case. 186 The maximum allowed surface form error \u00EF\u0080\u00A8 \u00EF\u0080\u00A9TE is determined as 40 \u00CE\u00BCm for the semi finish pass of the process. Finally, the maximum cutting speed for the given tool and workpiece material and optimum tool life is selected as 650 m/min. After the constraints are defined for the given turn-milling case, the feasible cutting regions that do not violate the process and machine tool constraints are constructed. First, the combination of tool and workpiece spindle sets which lead to maximum chip thickness \u00EF\u0080\u00A8 \u00EF\u0080\u00A9,max 0.4 mmsjh \u00EF\u0080\u00BD are identified iteratively. Then, the torque and power consumptions of the tool and workpiece spindles are calculated at each axial depth of cut on the stability diagram by solving the corresponding cutting forces at each point. The maximum surface form errors are calculated for the range of feasible tool spindle speed and axial depth of cut by solving the Eq.(5.6). Finally, the maximum allowed cutting speed constraint is employed for the given stability diagram. The feasible cutting regions are presented for different workpiece spindle speeds in Figure 5.10. The feasible region for each workpiece spindle speed is bounded on the right by the maximum cutting speed line, and on the left by the maximum allowed chip load line. On the other hand, the torque and power limits of the tool spindle bound the feasible regions on top. Since the workpiece material is soft, the C-axis\u00E2\u0080\u0099 torque and power limits are greater than the tool spindle limits in this particular example. For instance, the feasible cutting region can be represented for the workpiece spindle speed of 20 rpm as follows; 3932 rpm 10345 rpm0 mm 2.7 mmftfa\u00EF\u0080\u00BC\u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0080\u00BC \u00EF\u0080\u00BC (5.7) where ft\u00EF\u0081\u0097 and fa represent the feasible tool spindle speed and axial depth of cut, respectively. 187 Figure 5.10 Representation of each process and machine constraint on a turn-milling case. The feasible and non-feasible regions of cutting for workpiece spindle speed of 10 rpm is showed in Figure 5.11 as an example. 188 Figure 5.11 Feasible and non-feasible cutting regions of turn-milling case (workpiece spindle speed:10 rpm) The productivity of the given turn-milling case can be improved by tuning the tool-workpiece spindle speeds and axial depth of cut that maximize the MRR within the feasible cutting region. The MRR at different workpiece spindle speeds and feasible axial depth of cuts are calculated from Eq.(5.1) for constant tool axial feed per workpiece revolution, and listed in Table 5.2. Table 5.2 MRR for different workpiece spindle speeds. Workpiece Spindle Speed [rpm] Minimum Feasible Tool Speed ,minft\u00EF\u0081\u0097 [rpm] Maximum Feasible Tool Speed ,maxft\u00EF\u0081\u0097 [rpm] Maximum Feasible Depth of Cut maxfa [mm] MRR 3mm /min\u00EF\u0083\u00A9 \u00EF\u0083\u00B9\u00EF\u0083\u00AB \u00EF\u0083\u00BB 10 1960 10345 5.5 61512 20 3932 10345 2.8 66431 30 5898 10345 1.85 67163 40 7864 10345 1.50 73136 50 9830 10345 1.25 76576 189 Note that, the MRR calculations for workpiece spindle speed of 60 rpm is not included in Table 5.2 since there is no feasible region exist for this particular case. Although the maximum axial depth of cut reduces due to the increased cutting force and torque consumption at high workpiece spindle speeds, the calculated MRR is highest for increased workpiece speeds as seen from Table 5.2. The upper boundary of the feasible regions is constructed either by stability limit or torque-power limits of the tool spindle, thus the corresponding tool spindle speed that leads the highest productivity gain (MRR) is a bounded finite interval. The feasible sets of tool spindle speed intervals at the highest productivity gain can be written as; \u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0081\u00BB \u00EF\u0081\u00BD\u00EF\u0081\u00BB \u00EF\u0081\u00BD1,101,10 2,10 3,10 2,10103,101,20 1,20201,30 1,30301,40 1,40401,50 1,50502370 2570, , | 3350 39005800 8050| 5400 8700| 5898 9500| 7864 10345|10000 10ftf f f ft t t tftf ft tf ft tf ft tf ft tUUUUU\u00EF\u0083\u00AC \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0083\u00AF\u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0081\u0097 \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0083\u00AD\u00EF\u0083\u00AF \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0083\u00AE\u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC\u00EF\u0080\u00BD \u00EF\u0081\u0097 \u00EF\u0080\u00BC \u00EF\u0081\u0097 \u00EF\u0080\u00BC 345 (5.8) where 10,20,...U represents the feasible sets of tool spindle speeds at the highest MRR regions for different workpiece speeds. As the workpiece spindle speed increases, the length of the feasible sets of tool spindle speeds decreases due to the highly restricted right and left boundaries of the feasible cutting regions. Figure 5.12 shows the variation of MRR at different workpiece spindle speeds with their corresponding set of feasible tool spindle speeds. In this particular turn-milling case, the highest feasible workpiece spindle speed provides the most productive cutting conditions. 190 Figure 5.12 Variation of MRR at different workpiece spindle speeds with their corresponding feasible sets of tool spindle speed intervals. 5.4 Summary This chapter presents the process parameter selection strategy for turn-milling operations based on the process and machine tool constraints. The objective of the pre-process machining planning is to obtain the most optimum cutting parameters that maximizes the productivity without violating the constraints. The productivity of a turn-milling process can be increased by reducing the cycle time of the operation where it is only possible by increasing the stable material removal rate (MRR) which is the most evident productivity measure in machining. First, the definition of MRR is given for turn-milling operations. It is shown that, for a constant tool axial feed per workpiece revolution, the MRR is a function of axial depth of cut and workpiece spindle speed. The tool spindle speed, on the other hand, is tuned to change the chip load acting on the milling tool. Then, the process constraints such as maximum chip load, cutting speed, surface form errors, and the stability of the cutting are explained in detail as well as the torque and power limits of the tool and workpiece 191 spindles. The relationships between design variables and each process parameter are justified. Finally, the process parameter selection strategy is demonstrated on an example case, and the most optimum cutting conditions for given tool-workpiece geometries and machine limits are obtained. 192 Chapter 6: Conclusions and Future Research Directions 6.1 Summary and Contributions Turn-milling machines are widely used in aerospace and automotive industries because of their multi-functional capabilities in producing complex parts in one set-up. In turn-milling processes, both milling cutter and workpiece rotate simultaneously while the cutter which can be tilted around the B-axis of the machine, travels in three Cartesian directions, leading to five-axis kinematics with complex chip generation mechanism. When all five drives operate turn-milling of curved parts, the chip geometry changes as a function of machine\u00E2\u0080\u0099s kinematic configuration, tool path dependent rigid body motion of the translational and rotary drives, cutter-workpiece engagement (CWE) geometry, and the angular speeds and geometries of both tool and workpiece. Also, simultaneous rotations of the tool and workpiece spindles disturb the regenerative vibration, and leads to dual delay mechanism which has different stability properties than milling and turning processes. Currently the speeds of two spindles, depth of cut and feeds have been estimated from the costly machining trials in industry. If the selected cutting conditions yield chatter, the process becomes unstable leading to poor surface finish, low productivity, and failure of cutting tool and even machine tool spindle. Therefore, a predictive virtual model of the turn-milling process was presented in this thesis. The physics based mathematical model of the turn-milling simulates the varying chip thickness for multi-axes feed motion of the machine, cutting forces, torque and power for tool and workpiece spindles, relative tool-workpiece vibrations, chatter-free cutting conditions, and surface form errors on the finished workpiece. The virtual model of turn-milling processes has also been utilized to determine the most productive cutting conditions without violating the machine and process limits. 193 The main contributions of this thesis is summarized as: \u00EF\u0082\u00B7 A novel generalized analytical model of the feed rate computation as a function of both tool and workpiece spindle speeds, five-axis kinematics of the machine, and curved tool path has been modeled for turn-milling process. The resultant feed vector is calculated as the superposition of the linear and angular feed motions of the translational and rotary drives of the machine, respectively. Therefore, the chip thickness, cutting forces, torque and power for each spindle have been modeled at any position of the tool and workpiece along the toolpath for a given CAM model of the process. The proposed novel cutting force model of the turn-milling is applicable to any tool-workpiece geometries and complex toolpaths. \u00EF\u0082\u00B7 The kinematics of the turn-milling machine center has been modeled by Screw Theory. The inverse and forward kinematic solutions have been compared with the existing models in the literature. The kinematic solution of the machine was then used to transform the displacements, velocities, and accelerations from tool coordinate system to machine drives. \u00EF\u0082\u00B7 The dynamics of turn-milling processes have been modeled for the first time in the literature. The dynamic chip thickness and cutting forces were modeled as a function of the vibrations between the flexible tool and workpiece, multi axes rigid body feed motion of the machine drives, and cutter-workpiece engagement geometry. \u00EF\u0082\u00B7 A novel time delay model has been presented for turn-milling processes. It is shown that the regenerative vibration mechanism is disturbed by the simultaneous tool and workpiece rotations, and time varying delay exists between the present and previous tooth passing intervals in turn-milling. The time delay is modeled by the discrete motion of the tool and workpiece, and the history of each surface point left by each tooth over the time. It is shown 194 that the time delay varies over the time, and it is not equal to tooth passing period as oppose to regular milling and turning operations where the time delay is constant and equal to the tooth/spindle periods of the operation. The variation amplitude of time delay has been investigated and the relations between the geometries, spindle speeds of tool-workpiece and the resultant time delay in the regenerative vibration system have been studied. \u00EF\u0082\u00B7 Stability of the turn-milling process has been solved for the first time in the literature. The resultant time varying delay differential equations (TV-DDE) are solved by semi discretization method in time domain, and the stable and unstable regions of cutting are calculated as a function of tool and workpiece spindles speeds and axial depth of cut. It is shown that the workpiece spindle speed alters the regenerative time delay in the chip thickness mechanism leading to varying time delays, and the stability of the turn-milling process depends on the speeds of two spindles, and on the rigid body feed motion of the machine drives. As the workpiece spindle speed increases, the total time delay in the regenerative system also increases, and the stability limits shift towards higher tool spindle speeds. \u00EF\u0082\u00B7 Stability of the asymmetric cutter dynamics have been investigated for regular milling operations, and the effect of the dynamically asymmetric milling tools on the milling process stability has been shown by solving the system equations in rotating cutter coordinates. \u00EF\u0082\u00B7 The relative vibrations between flexible tool and workpiece have been calculated for turn-milling processes. The surface location (form) errors resulted from the tool\u00E2\u0080\u0099s bottom face clearance angle and dynamics vibrations have been modeled for the first time in the 195 literature. It is shown that the dynamics of tool is almost negligible since it does not contribute to surface errors imprinted on the workpiece. \u00EF\u0082\u00B7 The low-immersion stability properties of turn-milling have been investigated, and the corresponding chatter frequencies for different bifurcation locations were analyzed. Due to the process constraints, such as maximum chip load, torque, and power of the spindles, the radial immersion of the cutter might be low, which in turn yields a highly interrupted cutting process. The eigenvalues of the dynamic system have been investigated to analyze the type of instability in the cutting, which was validated by experiments. \u00EF\u0082\u00B7 The low-speed process damping model has been developed for turn-milling processes by two different methods. First, the existing analytical process damping model based on small vibration amplitude assumption has been utilized for turn-milling processes, and the additional process damping forces have been modeled which was used for the stability calculation. Then, the experiment based process damping model which was applied to milling processes previously [55], has been adapted to turn-milling processes, and the asymptotic spindle speed and absolute stability limit have been calculated for a given workpiece spindle speed without modeling the contact mechanism between the flank face of the tool and workpiece. It is shown that the experiment based process damping model is more robust than the previous modeling approaches, and does not require any material specific coefficients which are highly sensitive to dynamics of cutting tool/workpiece, tool geometry and other cutting specific constants such as separation angle and vibration amplitude at nonlinear chatter region. \u00EF\u0082\u00B7 In-process flexible workpiece dynamics have been predicted in turn-milling process by Receptance Coupling approach. The complex workpiece geometry has been sliced at 196 different cross sections, and each slice was modeled as simple Timoshenko beam. Then, the receptances of each beam have been coupled, and the dynamics of the workpiece at any cutting state have been predicted. \u00EF\u0082\u00B7 The most productive cutting parameters for a given turn-milling operation have been presented. The static (chip load, torque, power), dynamic (stability, vibration) constraints, and surface tolerance limits have been considered for the analysis, and a simple guideline regarding the selection of turn-milling process parameters have been presented. The presented generalized mechanics, dynamics, and stability models of turn-milling is the state of the art research in the literature. It allows the virtual simulation of turn-milling process physics leading to the selection of the most productive process parameters without costly physical trials. The algorithms of this thesis will be integrated into the CutPro software which was developed at UBC \u00E2\u0080\u0093 MAL. 6.2 Future Research Directions The mathematical models presented in this thesis serve as foundation for the future advanced research in turn-milling technology. Although, the generalized model of turn-milling is capable of carrying out virtual simulations, there are still several aspects of this research that can be further studied as follows: \u00EF\u0082\u00B7 The current cutting force model considers the chip thickness resulted from the side cutting edges of the tool which is the major chip removal mechanism in turn-milling process. On the other hand, the contact forces between the tool\u00E2\u0080\u0099s bottom edge and rotating workpiece should be investigated in operations where the in-workpiece feed motion exists. \u00EF\u0082\u00B7 In-process workpiece dynamics has been modeled by the Receptance Coupling model which might have inaccuracies at the prediction of high frequency modes of the workpiece. 197 Therefore, more sophisticated varying workpiece dynamic models based on finite element analysis can be examined. \u00EF\u0082\u00B7 The effect of dynamically asymmetric milling tools on process stability has been investigated in this thesis. However, in turn-milling processes, the workpiece may also have dynamically asymmetric dominant modes which may disturb the process stability. The asymmetric workpiece dynamics should be incorporated with the presented dynamic model of the turn-milling system. \u00EF\u0082\u00B7 The effect of contact mechanics between the rotating tool and workpiece on the low speed process damping stability should be investigated more deeply in order to model the low speed stability accurately. Since the contact mechanics at tool-workpiece interface is itself a research topic for regular milling operations, the effect of rotating workpiece speed may introduce additional complexity to the cutting mechanism. \u00EF\u0082\u00B7 The tool wear and cutting temperature models must be investigated in order to fully benefit from the turn-milling technology and its advantages over milling and turning processes. 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[102] Karaguzel, U., Olgun, U., Uysal, E., Budak, E., and Bakkal, M., 2015, \"Increasing tool life in machining of difficult-to-cut materials using nonconventional turning processes,\" The International Journal of Advanced Manufacturing Technology, 77(9-12), pp. 1993-2004. 204 Appendix A Rotation Definitions for Screw Theory A.1 Exponential Coordinates for Pure Rotation Let consider a point q attached to a rigid rotating body. If the body is rotated at constant unit velocity around the fixed axis \u00CF\u0089 by \u00EF\u0081\u00B1 radians, the velocity of point q can be written as [85] (see Figure A.1); ( ) ( )q q t q t\u00EF\u0081\u00B7\u00EF\u0080\u00BD \u00EF\u0082\u00B4 \u00EF\u0080\u00BD\u00CF\u0089 (A.1) Figure A.1 Pure rotation of a point q around the fixed axis \u00CF\u0089 . Since the Eq. (A.1) is time invariant, the solution can be written as; ( ) (0)q t e q\u00EF\u0081\u00B1\u00EF\u0080\u00BD \u00CF\u0089 (A.2) where the exponential term e\u00EF\u0081\u00B1\u00CF\u0089 is represented with it Taylor\u00E2\u0080\u0099s series expansion as; 2 32 32! 3!e I\u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB\u00CF\u0089 \u00CF\u0089 \u00CF\u0089 \u00CF\u0089 (A.3) 205 Note that, the matrix \u00CF\u0089 is a skew-symmetric matrix \u00EF\u0080\u00A8 \u00EF\u0080\u00A9T \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00CF\u0089 \u00CF\u0089 . Since the closed form expression of e \u00EF\u0081\u00B1\u00CF\u0089 is needed for computation, Eq. (A.3) can be rewritten by considering formulas for powers; 3 5 2 4 623! 5! 2! 4! 6!e I\u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0083\u00A6 \u00EF\u0083\u00B6 \u00EF\u0083\u00A6 \u00EF\u0083\u00B6\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB \u00EF\u0080\u00AD \u00EF\u0080\u00AB\u00EF\u0083\u00A7 \u00EF\u0083\u00B7 \u00EF\u0083\u00A7 \u00EF\u0083\u00B7\u00EF\u0083\u00A8 \u00EF\u0083\u00B8 \u00EF\u0083\u00A8 \u00EF\u0083\u00B8\u00CF\u0089 \u00CF\u0089 \u00CF\u0089 (A.4) which can be expressed in a simpler form as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A92sin 1 cose I\u00EF\u0081\u00B1 \u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AD\u00CF\u0089 \u00CF\u0089 \u00CF\u0089 (A.5) Eq. (A.5) is also known as Rodrigues\u00E2\u0080\u0099 formula and provides an efficient computing method for the exponential term e\u00EF\u0081\u00B1\u00CF\u0089. A.2 Solution for Paden-Kahan Sub problem 1 Let consider a point p rotates around the given axis \u00EF\u0081\u0096 by \u00EF\u0081\u00B1 radians until it coincides with point q as shown in Figure A.2. The point r is given on the fixed rotation axis \u00EF\u0081\u0096 , then the vectors u and v are constructed as the distance between \u00EF\u0080\u00A8 \u00EF\u0080\u00A9-p r , and \u00EF\u0080\u00A8 \u00EF\u0080\u00A9-q r , respectively. The rotation of from point p to q is defined as; e u v\u00EF\u0081\u00B1 \u00EF\u0080\u00BD\u00CF\u0082 (A.6) For the solution of Eq. (A.6), we define the u\u00EF\u0082\u00A2 and v\u00EF\u0082\u00A2 which are the projections of the vectors u and v on the Plane Q . Let \u00EF\u0081\u00B7 be the unit rotation axis lies on \u00EF\u0081\u0096 , the projection vectors can be calculated as; TTu u uv v v\u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0081\u00B7\u00EF\u0082\u00A2 \u00EF\u0080\u00BD \u00EF\u0080\u00AD\u00EF\u0082\u00A2 \u00EF\u0080\u00BD \u00EF\u0080\u00AD (A.7) It is assumed that the problem has a solution only if u\u00EF\u0082\u00A2 and v\u00EF\u0082\u00A2 have same magnitude and projections u and v on the \u00EF\u0081\u00B7 axis have also same magnitude, such that; 206 T Tu vu v\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00BD\u00EF\u0082\u00A2 \u00EF\u0082\u00A2\u00EF\u0080\u00BD (A.8) If 0u\u00EF\u0082\u00A2 \u00EF\u0082\u00B9 , then the rotation angle \u00EF\u0081\u00B1 can be found as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9sinatan2 ,cosT Tu v u vu v u vu v u v\u00EF\u0081\u00B7 \u00EF\u0081\u00B1\u00EF\u0081\u00B1 \u00EF\u0081\u00B7\u00EF\u0081\u00B1\u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2\u00EF\u0082\u00B4 \u00EF\u0080\u00BD \u00EF\u0083\u00BC\u00EF\u0083\u00AF\u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2\u00EF\u0083\u009E \u00EF\u0080\u00BD \u00EF\u0082\u00B4\u00EF\u0083\u00BD\u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2 \u00EF\u0082\u00A2\u00EF\u0082\u00B7 \u00EF\u0080\u00BD \u00EF\u0083\u00AF\u00EF\u0083\u00BE (A.9) Figure A.2 Rotation about a single axis. A.3 Solution for Paden-Kahan Sub problem 2 Let consider a point p rotates first around the axis 2\u00EF\u0081\u0096 by 2\u00EF\u0081\u00B1 radian, and then rotates around 1\u00EF\u0081\u0096 by 1\u00EF\u0081\u00B1 radian, until it coincides with point q as shown in Figure A.3. If the two axes are not parallel, then the intersection point of two rotation planes, s can be defined as; 2 1e p s e q\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD2 1\u00CF\u0082 \u00CF\u0082 (A.10) Then, similar to the approach for sub-problem 1, the distances between , ,p q s and r are defined \u00EF\u0080\u00A8 \u00EF\u0080\u00A9, ,u p r v q r z s r\u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0080\u00AD \u00EF\u0080\u00BD \u00EF\u0080\u00AD . Substituting these quantities for the rotation equation; 207 2 1e u z e v\u00EF\u0081\u00B1 \u00EF\u0081\u00B1\u00EF\u0080\u00AD\u00EF\u0080\u00BD \u00EF\u0080\u00BD2 1\u00CF\u0089 \u00CF\u0089 (A.11) Figure A.3 Rotation about two subsequent axes. Eq. (A.11) is valid only if, 2 2 1 12 2 2,T T T Tu z v zu z v\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00BD\u00EF\u0080\u00BD \u00EF\u0080\u00BD (A.12) Since the two rotation axes are not parallel, their unit rotation axes 1 2,\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 and their cross product 1 2\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0082\u00B4 are linearly independent. Then, the distance vector z can be expressed with constants as; \u00EF\u0080\u00A8 \u00EF\u0080\u00A91 1 2 2 3 1 22 22 2 21 2 1 2 1 2 3 1 22Tz k k kz k k k k k\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0082\u00B4\u00EF\u0080\u00BD \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0080\u00AB \u00EF\u0082\u00B4 (A.13) Substituting Eq. (A.13) into Eq. (A.12) and solving 23k in Eq. (A.13), three coefficients are calculated as; 208 \u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A9\u00EF\u0080\u00A8 \u00EF\u0080\u00A91 2 2 11 21 21 2 2 12 21 22 2 21 2 1 2 1 23 21 2112T T TTT T TTTu vkv uku k k k kk\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0081\u00B7 \u00EF\u0081\u00B7\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00AD\u00EF\u0080\u00AD \u00EF\u0080\u00AD \u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0082\u00B4 (A.14) For a certain configuration of rotational axes, the coefficients 1 2 3, ,k k k are calculated, and the z , thus s can be solved. Once the vector s is calculated, the problem is reduced to sub-problem 1, and the rotations 1\u00EF\u0081\u00B1 and 2\u00EF\u0081\u00B1 are solved for; 21e p se q s\u00EF\u0081\u00B1\u00EF\u0081\u00B1\u00EF\u0080\u00AD\u00EF\u0080\u00BD\u00EF\u0080\u00BD21\u00CF\u0082\u00CF\u0082 (A.15) Note that, since two points intersect in the case of non-parallel rotation axes, there exist two solutions for this problem. The rotation angle that guarantees continuity along toolpath and shorter joint movement is taken into consideration. "@en .
"Thesis/Dissertation"@en .
"2018-09"@en .
"10.14288/1.0368954"@en .
"eng"@en .
"Mechanical Engineering"@en .
"Vancouver : University of British Columbia Library"@en .
"University of British Columbia"@en .
"Attribution-NonCommercial-NoDerivatives 4.0 International"@* .
"http://creativecommons.org/licenses/by-nc-nd/4.0/"@* .
"Graduate"@en .
"Mechanics, dynamics and stability of turn-milling operations"@en .
"Text"@en .
"http://hdl.handle.net/2429/66507"@en .