{"http:\/\/dx.doi.org\/10.14288\/1.0071675":{"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool":[{"value":"Applied Science, Faculty of","type":"literal","lang":"en"},{"value":"Mechanical Engineering, Department of","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider":[{"value":"DSpace","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeCampus":[{"value":"UBCV","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/creator":[{"value":"Eksioglu, Muhittin Caner","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/issued":[{"value":"2011-04-06T15:06:09Z","type":"literal","lang":"en"},{"value":"2011","type":"literal","lang":"en"}],"http:\/\/vivoweb.org\/ontology\/core#relatedDegree":[{"value":"Master of Applied Science - MASc","type":"literal","lang":"en"}],"https:\/\/open.library.ubc.ca\/terms#degreeGrantor":[{"value":"University of British Columbia","type":"literal","lang":"en"}],"http:\/\/purl.org\/dc\/terms\/description":[{"value":"Peripheral milling of thin walled aerospace components takes considerable amount of machining time as blank blocks are cut down to thin webs under excessive structural oscillations during the process. Unstable chatter vibrations and stable forced vibrations cause poor surface finish on the machined part. Predicting the process mechanics in advance eliminates the time consuming trial and error approach in reducing the vibrations which are within the tolerance limits of the part. This thesis presents the mathematical modeling of the peripheral milling of the thin walls with helical end mills. The cutting forces, vibrations and dimensional form errors left on the finish surface are predicted under stable but forced vibration conditions. The chatter stability diagram of the operation is predicted by using both frequency and semi-discrete time domain models. \n\nThe relative vibrations between the flexible part and slender end mill are consi-dered. The tool and the workpiece are discretized along the contact axis to include effect of varying cutting forces and structural dynamics. The differential milling forces are evaluated from the static chip loads contributed by the rigid body motion of the milling operation, and dynamic chip loads caused by the relative vibrations between the flexible tool and flexible thin part. \n\nThe different cylindrical end mill geometries with regular and non-uniform pitch and helix angles, and low speed process damping effects are included in the dynamic force model. The dynamic properties of the flexible structures are represented by expe-rimentally evaluated modal model in order to reduce the number of linear, periodic, delayed differential equations solved in frequency and time domain computations. The periodic, delayed differential equations are solved by the semi discrete time domain method to predict the amplitude of vibrations and forces. The equations of motion are simplified to constant coefficient type ordinary differential equations, and surface location errors are calculated by frequency domain solver. \n\nChatter stability lobes are calculated using semi discrete time domain and fre-quency domain methods. Chatter stability solvers are validated by conducting chatter tests for roughing and finishing stages of thin walled aluminum part at high cutting speeds, and low speed machining of rigid steel block.","type":"literal","lang":"en"}],"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO":[{"value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/33334?expand=metadata","type":"literal","lang":"en"}],"http:\/\/www.w3.org\/2009\/08\/skos-reference\/skos.html#note":[{"value":"MECHANICS AND DYNAMICS OF THIN WALL MACHINING by MUHITTIN CANER EKSIOGLU B.Sc., Istanbul Technical University, Turkey, 2007 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in The Faculty of Graduate Studies (Mechanical Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) April 2011 \u00a9 Muhittin Caner Eksioglu, 2011 \fAbstract Peripheral milling of thin walled aerospace components takes considerable amount of machining time as blank blocks are cut down to thin webs under excessive structural oscillations during the process. Unstable chatter vibrations and stable forced vibrations cause poor surface finish on the machined part. Predicting the process mechanics in advance eliminates the time consuming trial and error approach in reducing the vibrations which are within the tolerance limits of the part. This thesis presents the mathematical modeling of the peripheral milling of the thin walls with helical end mills. The cutting forces, vibrations and dimensional form errors left on the finish surface are predicted under stable but forced vibration conditions. The chatter stability diagram of the operation is predicted by using both frequency and semi-discrete time domain models. The relative vibrations between the flexible part and slender end mill are considered. The tool and the workpiece are discretized along the contact axis to include effect of varying cutting forces and structural dynamics. The differential milling forces are evaluated from the static chip loads contributed by the rigid body motion of the milling operation, and dynamic chip loads caused by the relative vibrations between the flexible tool and flexible thin part. The different cylindrical end mill geometries with regular and non-uniform pitch and helix angles, and low speed process damping effects are included in the dynamic force model. The dynamic properties of the flexible structures are represented by experimentally evaluated modal model in order to reduce the number of linear, periodic, delayed differential equations solved in frequency and time domain computations. The periodic, delayed differential equations are solved by the semi discrete time domain method to predict the amplitude of vibrations and forces. The equations of motion are simplified to constant coefficient type ordinary differential equations, and surface location errors are calculated by frequency domain solver. ii \fChatter stability lobes are calculated using semi discrete time domain and frequency domain methods. Chatter stability solvers are validated by conducting chatter tests for roughing and finishing stages of thin walled aluminum part at high cutting speeds, and low speed machining of rigid steel block. iii \fTable of Contents Abstract ............................................................................................................................ ii Table of Contents ............................................................................................................ iv List of Tables ................................................................................................................... vi List of Figures ................................................................................................................ vii Nomenclature ................................................................................................................... x Acknowledgements ....................................................................................................... xiv 1. Introduction................................................................................................................ 1 2. Literature Review ...................................................................................................... 3 2.1 Overview ............................................................................................................ 3 2.2 Static Forces in Milling ...................................................................................... 3 2.3 Surface Location Error ....................................................................................... 5 2.4 Chatter Stability Prediction Models for Intermittent Milling Operation ........... 9 2.5 Process Damping in Milling ............................................................................ 15 3. Dynamics of Peripheral Milling ............................................................................. 17 3.1 Introduction ...................................................................................................... 17 3.2 Discretized Representation of Process Dynamics ............................................ 18 3.3 3.2.1 Forces in Milling .................................................................................. 18 3.2.2 Equations of Motion in Laplace Domain ......................................... 32 Modal Space Formulation ................................................................................ 35 4. Time Domain Simulation ........................................................................................ 46 4.1 Introduction ...................................................................................................... 46 4.2 Semi Discrete Time Domain Solution ............................................................. 46 4.2.1 4.3 SD Time Domain Solution Example.................................................... 53 Surface Location Error ..................................................................................... 58 iv \f4.4 SLE Example ................................................................................................... 64 5. Chatter Stability in Milling ..................................................................................... 71 5.1 Introduction ...................................................................................................... 71 5.2 Semi Discrete Time Domain Chatter Stability................................................. 71 5.3 Frequency Domain Chatter Stability................................................................ 73 5.4 Chatter Tests..................................................................................................... 77 5.4.1 Thin Wall Roughing Test for Chatter Analysis ................................... 77 5.4.2 Thin Wall Finishing Test for Chatter Analysis .................................... 88 5.4.3 Low Speed Cutting Test for Rigid Workpiece ..................................... 94 5.4.4 Summary of Experiments..................................................................... 96 6. Conclusions............................................................................................................... 97 Bibliography ................................................................................................................... 99 Appendix A: Single Point Formulation of Milling Equation ................................... 105 Appendix B: Fourier Coefficients for Time Dependent Force Terms .................... 110 Appendix C: Variable Helix Cutter Representation ................................................ 113 Appendix D: Accelerometer Mass Elimination ........................................................ 115 v \fList of Tables Table 4-1: Parameters for regular cylindrical end mill .................................................... 54 Table 4-2: Modal parameters of the regular cylindrical end mill .................................... 54 Table 4-3: Least common period and frequency for different cutters ............................. 61 Table 4-4: Modal parameters of Al 7075 plate along its y direction ............................... 65 Table 4-5: Parameters of cylindrical cutter ..................................................................... 66 Table 5-1: Least common period and frequency for different cutters ............................. 72 Table 5-2: Modal Parameters of the 10.5 mm thick plate ............................................... 79 Table 5-3: Modal parameters for tool and workpiece ..................................................... 82 Table 5-4: Tested conditions and results ......................................................................... 83 Table 5-5: Modal Parameters of the 5 mm thick plate .................................................... 90 Table 5-6: Modal parameters of the tool ......................................................................... 91 Table 5-7: Tested conditions for finishing operation ...................................................... 92 Table 5-8: Modal parameters at the tool tip..................................................................... 94 vi \fList of Figures Figure 2-1: Down milling and up milling operations ........................................................ 3 Figure 2-2: Cutter and workpiece deflection ..................................................................... 5 Figure 2-3: Schematic model of dynamic milling ............................................................. 7 Figure 2-4: An example on surface location error, a) chatter stability diagram for the 10% radial immersion down milling, b) the surface location errors in micrometers are calculated along the axial depth of cut (ADOC) for the fixed spindle speed indicated in a) ........................................................................................................................................ 9 Figure 2-5: Low immersion down milling operation with geometric parameters of an end mill ................................................................................................................................... 10 Figure 2-6: Cutting condition parameters that affect the intermittency of the engagement for down milling operation; a,c,e,g are cutting forces in x and y directions, and b,d,f,h are Fast Fourier Transforms of those forces .................................................................... 11 Figure 2-7: Regeneration mechanism .............................................................................. 12 Figure 2-8: Chatter stability lobe and movement of eigenvalues in discrete domain for points A (Hopf bifurcation) and B (Flip bifurcation) ...................................................... 14 Figure 2-9: Tool flank\u2019s indentation into wavy surface in low cutting speeds ............... 15 Figure 3-1: a) Axially discretized general end mill geometry, b) milling forces and angle convention........................................................................................................................ 19 Figure 3-2: Milling Forces Scheme ................................................................................. 21 Figure 3-3 : Demonstration of dynamic and static chip thicknesses ............................... 23 Figure 3-4 : Indentation of flank due to edge radius and undulated surface ................... 25 Figure 3-5: Indentation of flank at several positions of the wavy surface ....................... 28 Figure 3-6 : Discretized representation of the plate and the end mill .............................. 32 Figure 3-7: Relative displacement for axial point r ....................................................... 41 Figure 4-1: Approximation of the delayed term by discretization .................................. 49 Figure 4-2: Discretization of an element in dynamic modal force component matrix with different number of points ............................................................................................... 50 vii \fFigure 4-3: a) Stability chart is taken from [59] (with the permission of Merdol) b)-d) time domain simulation in x and y directions respectively for 7 mm ADOC and 6800 rpm c)-e) simulations in x and y directions respectively for 7 mm ADOC and 7500 rpm ......................................................................................................................................... 56 Figure 4-4: Resultant cutting forces on x-y plane a) unstable cutting b) stable cutting .. 57 Figure 4-5: FFT of force data a) unstable cutting b) stable cutting ................................. 58 Figure 4-6: Surface location error a) undercut example b) overcut example .................. 59 Figure 4-7: Flowchart of SLE method ............................................................................. 60 Figure 4-8: Front view of the Al 7075 plate, with four measurement points indicated along its right corner ........................................................................................................ 65 Figure 4-9: Chatter stability for low immersion down milling operation and the region for SLE calculation is indicated with dashed line. .......................................................... 67 Figure 4-10: a) Force in y direction for one spindle rotation, b) Fast Fourier Transform (FFT) of the force signal .................................................................................................. 68 Figure 4-11: Fast SLE calculation algorithm................................................................... 69 Figure 4-12: Three dimensional SLE graph for low immersion down milling of thin plate and SLE at 16580 rpm ..................................................................................................... 70 Figure 5-1: Nyquist plot of the characteristic equation ................................................... 76 Figure 5-2: Side and front views of the plate with dimensions and measurement locations ......................................................................................................................................... 77 Figure 5-3: Frequency response functions of workpiece at different locations and tool tip a) first bending mode of the workpiece, k13,wp \uf03d 2.61\uf0b4106 N\/m, k14,wp \uf03d 2.74 \uf0b4106 N\/m k11,wp \uf03d 1.93 \uf0b4106 N\/m, k12,wp \uf03d 2.18 \uf0b4106 N\/m, b) first torsional mode of the workpiece, k21,wp \uf03d 7.52 \uf0b4106 N\/m, k22,wp \uf03d 2.34 \uf0b4109 N\/m, k23,wp \uf03d 8.58 \uf0b4106 N\/m, k24,wp \uf03d 1.92 \uf0b4109 N\/m second bending mode of the workpiece c) k31,wp \uf03d 1.97 \uf0b4108 N\/m, k31,wp \uf03d 1.97 \uf0b4108 N\/m, k32,wp \uf03d 6.90 \uf0b4107 N\/m, k33,wp \uf03d 2.40 \uf0b4108 N\/m, k34,wp \uf03d 1.67 \uf0b4108 N\/m d) tool\u2019s FRFs at x and y directions.......................................................................................................................... 81 Figure 5-4: Setup for stability test ................................................................................... 82 Figure 5-5: Predicted and measured chatter stability results for the most flexible condition .......................................................................................................................... 84 viii \fFigure 5-6: Y direction measurement and simulation for point A, a) Force measurement b) FFT of force measurement c) simulated vibration displacement d) FFT of displacement .................................................................................................................... 85 Figure 5-7: Y direction measurement and simulation for point B, a) Force measurement b) FFT of force measurement c) simulated vibration displacement d) FFT of displacement .................................................................................................................... 86 Figure 5-8: Y direction measurement and simulation for point C, a) Force measurement b) FFT of force measurement c) simulated vibration displacement d) FFT of displacement .................................................................................................................... 88 Figure 5-9: Side and front views of the plate with designated modal test points ............ 89 Figure 5-10: Predicted stability lobes for the most flexible case and measured points are presented for 5% radial immersion thin wall machining ................................................. 93 Figure 5-11: Two orthogonal direction FRFs measured at the tool tip ........................... 95 Figure 5-12: Chatter stability prediction using different algorithms and test results ...... 96 Figure A-1: Resultant force lumped on point k ............................................................. 105 Figure C-1 : Crest cut end mill with varying helix angle along the flute ...................... 114 Figure D-1: Overlayed measurements indicating the torsional mode of the structure .. 115 Figure D-2: Curve fit for the flexible top right corner, the pink line is the fit curve and the blue line is the measurement. The discrepancy at torsional mode at 1720 Hz is apparent.......................................................................................................................... 116 Figure D-3: The most of the frequencies are 1719 Hz and variation is around 2 Hz. ... 118 Figure D-4: The curve is fitted from the direct FRF and since there is no shift, the errors are less at the top corners of the plate. ........................................................................... 119 ix \fNomenclature a Axial depth of cut \u02c6 A 1, j Differential static cutting force vector of flute j \u02c6 A 2, j Differential dynamic cutting force vector of flute j B\u02c6 1, j Differential static edge force vector of flute j \u02c6 B 2, j Differential process damping force matrix of flute j c Feed per tooth dA, dV Indented area and volume of the flank face dS Length of differential cutting edge dz Differential axial depth of cut D Tool diameter f Force vector in modal force representation Fr , Ft , Fa Radial, tangential and axial milling forces Fx , Fy , Fz Milling forces in Cartesian coordinates G Displacement transfer function matrix h Fourier coefficient index hj Static chip thickness of flute j hd,j Dynamic chip thickness of flute j x \f\u0397 Truncated Toeplitz matrix ij Helix angle of flute j I Identity matrix j Tooth index Krc , K tc , Kac Radial, tangential and axial milling cutting force coefficients Kre , K te , Kae Radial, tangential and axial milling edge force coefficients Ksp Material dependent process damping indentation coefficient KR Residual stiffness L State matrix for current time terms Lw Average flank wear length mt , mwp Total number of modes of tool and workpiece respectively M Transfer function matrix in modal domain MR Residual mass n Spindle speed in rpm N Number of flutes q Differential physical displacement vector in time domain Q(s) Physical displacement vector in Laplace domain containing all axial levels R State matrix for delayed terms xi \fS Input matrix(forces) in state space representation \uf044t Time step T ,Tj Time delay, time delay of flute j T Transformation matrix for Cartesian coordinates u Mass normalized mode shape U Mass normalized mode shape matrix vc Cutting velocity of the edge in cut V Indented volume of the flank face z Axial elevation from the tip of the tool \uf066 Cutter rotation angle \uf066j Instantaneous immersion angle of flute j \uf066p Pitch angle \u03a6 State transition matrix \uf047 Modal coordinates vector \uf047T Delayed modal coordinates vector \u03b3 Mode index \uf06b Axial immersion angle \uf06d Coulomb friction coefficient \u0398 Motion vector containing modal displacements and velocities xii \f\uf074 Least common period \uf077c Chatter frequency \uf077n Natural frequency \uf077p Least common frequency \uf077s Spindle frequency in rad\/s \u03c8j Lag angle of flute j \u03a8 Relative vibrations vector \uf078 Damping ratio [\uf0d7]T Transpose operator det Determinant operator ADOC Axial depth of cut DDE Delayed differential equation DOF Degree of freedom MF Multi-frequency solution SD Semi discretization SLE Surface location error xiii \fAcknowledgements I would like to sincerely express my gratitude to my research supervisor Prof. Yusuf Altintas for the encouragement, support and invaluable guidance he has provided throughout my study at UBC. I would also like to thank my co-supervisor Dr. Doruk Merdol for his continuous support and help. I would like to express my appreciation to our sponsors NSERC-Pratt & Whitney Canada for the Industrial Research Chair Grant, Machine Tool Technologies Research Foundation (MTTRF) for letting us use the Mori Seiki NMV5000 CNC Machining Center, Sandvik Coromant and Data Flute for supplying the tools we used in the machining tests. It was a great privilege to be a member of an active research group in Manufacturing Automation Laboratory (MAL). The discussions I have made with my colleagues and their share of knowledge elaborated this research work. It was also a great fun to be with the brilliant people that came from different parts of the world. Finally, I would like to thank my parents, Nazli and Mahmut Eksioglu. I felt their presence all the time with the support, encouragement, friendship and love they have provided from thousands of miles away. I dedicate this work to them. xiv \fChapter 1. Introduction 1. Introduction Peripheral milling of thin walled components is a common process for aircraft manufacturers. Wing structures, fuselage sections, jet engine compressors and turbine blades are machined from blank blocks to thin webs with slender end mills to meet tight tolerances, achieve complex geometries, and obtain acceptable thermo-mechanical properties. Machining of these parts require large amount of material removal, hence affecting the overall production time and cost. The problems that emerge due to high flexibility of the cutting tool-workpiece system limits the productivity, especially when trial and error based approaches are employed to set cutting conditions. This has motivated researchers to understand the physics of the process, and optimize the parameters to attain higher production rates, and achieve good surface finish quality. The relative vibrations between flexible end mill and workpiece are the source of most of the problems that occur during manufacturing of thin webs. Intermittent engagement of cutter and workpiece excites a wide range of structural natural frequencies that result as unstable chatter vibrations and stable forced vibrations. Effect of cutting conditions, tool geometry, structure dynamics, material characteristics, high speed and low speed machining properties have been studied separately to minimize process vibrations in order to reduce operation cycle times. Most of the available models aim specific application, or general models consume sizable computation time which is not convenient for machining industry. This thesis investigates a general mathematical model for thin wall machining, which is computationally efficient, and has acceptable prediction accuracy within the physical assumptions made. The thesis is organized as follows; Chapter 2 provides the review of existing research on milling mechanics and effect of vibrations on cutting. Brief overview on calculation of static milling forces, the models to predict errors on the finished surface, chatter stability calculation of intermittent milling operation, and process damping effect at low cutting speeds are presented in this chapter. 1 \fChapter 1. Introduction The dynamics of peripheral milling is discussed in Chapter 3 using axially discretized representation. Static milling forces created at each axial level regarding the general end mill geometry are summarized; furthermore the forces emerge from dynamic interactions, such as regeneration and process damping, are included to form general force model. Workpiece and cutter dynamics are presented in axially discretized form which is organized to retrieve the data from experimental modal analysis. Consequently the interactions between milling forces and structure dynamics are expressed via equation of motion in modal domain. The modal equation of motion obtained in Chapter 3 has been solved by semi discrete time domain and frequency domain surface location error algorithms in Chapter 4. The semi discrete time domain method solves linear or linearized periodic coefficient delayed differential equation for given spindle speeds, axial and radial depths. This solution is then compared to time domain solution of CUTPRO to analyze its weaknesses and strengths. For stable cutting conditions, a fast algorithm to obtain surface location errors utilizing the pace of frequency domain solvers is discussed, and a theoretical example is presented. Chapter 5 is dedicated to chatter stability prediction, and experimental validation of theoretical models. Semi discrete chatter stability solution is presented along with frequency domain solver that uses Nyquist stability criterion. High speed milling chatter tests are conducted for machining thin walled part to analyze change of structural dynamics, and to test solvers\u2019 accuracy at different cutting conditions. Finally, a rigid steel block is machined to analyze frequencies dominant at low cutting speeds and verify the process damping model used. The thesis is concluded in Chapter 6, summarizing the contributions made and the future research directions. 2 \fChapter 2. Literature Review 2. Literature Review 2.1 Overview Milling of thin walled components is a common application for aerospace industry, and it is becoming more common for automotive, computer hardware and bioengineering industries. The problems caused by vibrations and static deflections have been studied in details by previous researchers. In this chapter literature survey will be presented on the foundations such as modeling static milling forces, surface location errors, chatter stability prediction models for intermittent engagement, and process damping at low speed cutting. 2.2 Static Forces in Milling In milling, the cutting edge is discontinuously engaged with the workpiece along its periphery. Two different milling techniques are shown in Figure 2-1, which are down milling an up milling. In down milling, chip thickness has the maximum value at the start of the engagement and it reduces to zero as cutter rotates, and in up milling it is the opposite. The amount of material tool removes, hence the forces at each instant vary in time as indicated in Figure 2-1. This brings time dependent characteristic to milling operation unlike turning. Figure 2-1: Down milling and up milling operations 3 \fChapter 2. Literature Review Predicting the forces created in milling is one of the most important tasks for researchers since problems as surface location errors and tool breakage are related on milling forces. Modeling the time dependent chip formation is the first step to calculate cutting forces in milling. The chip load is dependent on kinematics of milling, and fluctuations that are coming from structure dynamics. In this section, milling kinematics is surveyed, which is the foundations of static milling forces. Martelotti considered both rotating and translating motions of the tool and assumed a trochoidal tool path [1]. For planar milling, he expressed that when tool radius is comparatively larger than the feed value, circular tool path approximation has negligible error, thus the formula for static chip thickness simplifies to: h(\uf066) \uf03d c sin(\uf066) , (2.1) where h is the instantaneous chip thickness, c is feed rate per tooth, and \uf066 is instantaneous immersion angle of the cutting tooth. In order to eliminate excessive experimentation, analytical and semi analytical methods have been searched in the past [2]. Experimentally identified specific cutting pressures are presented in the work of Koeingsberger and Sabberwal [3]: F \uf03d Kah , (2.2) where specific cutting pressure is expressed as K \uf03d Ch x . Axial depth of cut (ADOC) is denoted by a . The empirical constants depending on workpiece and tool properties are denoted by C and x . In this thesis, due to convenience in analytical calculations, the linear edge force model [4] is used for milling force predictions: F \uf03d Kc ah \uf02b Kea , (2.3) where K c is specific cutting pressure related with the shearing of material and K e is edge force coefficient which accounts the rubbing between tool flank face and workpiece. These constants can be identified experimentally by fitting a line for federate versus force graphs or analytically calculating by orthogonal cutting theory. 4 \fChapter 2. Literature Review 2.3 Surface Location Error The dimensional accuracy of the finished surface is one of the key objectives of thin wall machining. The static deflections caused by the forces, and relative vibrations between cutting tool and workpiece may result as surface location errors (SLE). Depending on the error values and tolerance limits, another finishing operation can be required for undercuts, and finished part may be scrapped for severe overcuts, thus bringing additional cost to operation. The surface accuracy for milling of thin walled components investigated by Kline et al. [5] considering the relative static deflections between tool and flexible workpiece. Figure 2-2: Cutter and workpiece deflection 5 \fChapter 2. Literature Review The tool is considered as a cantilever beam, and the deflection of the tool in y direction as demonstrated in Figure 2-2, is expressed as: d y,t ( z ) \uf03d Fy 6EI y [(CFY \uf02d z )3 \uf02d ( L \uf02d z )3 \uf02d 3( L \uf02d z ) 2 ( L \uf02d CFY)] , (2.4) where d y,t ( z ) is the deflection of the tool in y direction at point z , Fy is the static y cutting force, CFY is the y force center, I y is the moment of inertia of the tool in y direction, which is calculated from I y \uf03d D4 \/ 64 , E is the modulus of elasticity and D is the diameter of the tool. In their work the deflection of the workpiece is calculated using finite element analysis software and the total deflection is expressed as the sum: d y ( z ) \uf03d d y,t ( z ) \uf02b d y,w ( z ) , (2.5) where d y,w ( z ) and d y ( z ) are workpiece and total displacements at point z . The cutter leaves the error marks when it enters to cut in up milling, and when it exits the cut in down milling at surface generation point. The force and force center are calculated for that instant, the total deflection is obtained by applying Eq. (2.5). Including the effect of helix, the authors plotted the deflections along the z axis. Sutherland and DeVor [6], introduced an improved model where effect of system deflections on the chip load were taken into account. They claimed that previous rigid force models, which calculated the cutting forces neglecting the system deflections, are over-estimating the errors for end milling thin walled cavities as much as 100 to 150 percent. Montgomery and Altintas [7] proposed a dynamic milling model where dynamics of the process is included. The workpiece\u2019s and tool\u2019s dynamics are represented with discrete equivalent masses ( m t , m w ), stiffnesses ( k t , k w ) and damping coefficients ( c t , c w ) as shown in Figure 2-3. The interaction between workpiece and the tool modeled through cutting stiffness k cut , and process damping ccut . 6 \fChapter 2. Literature Review Figure 2-3: Schematic model of dynamic milling The following equations of motions of cutter and workpiece are solved for trochoidal motion of the tool through numerical integration iteratively by updating the chip load with vibrations: \uf026\uf026 t \uf02b c t q\uf026 t \uf02b k t q t \uf03d F mtq , \uf026\uf026 w \uf02b c w q\uf026 w \uf02b k w q w \uf03d F mwq (2.6) where sub-indices t and w indicate tool and workpiece, respectively. q \uf03d (x,y)T is the displacement vector, F \uf03d ( Fx , Fy )T is the force vector, m,c,k are mass, damping and stiffness matrices, in order. After solving the differential equations in time domain for displacement, they obtained the surface errors similar to previous papers. Budak [8] [9] investigated the problem further for thin wall milling for static form errors and used two models which are static deflections and flexible milling model. The static deflections model, similar to Sutherland\u2019s approach, considered change of chip load with static deflections and then plotted the form errors with respect to feed and axial depth. The flexible milling model on the other hand, also introduced change of cutter engagement boundaries; hence change of radial immersion with static deflections. He showed that flexible milling force model matches better with the test measurements, although bringing additional computational effort. Both models excluded effect of structural modes on the form errors. Frequency domain analysis of surface location errors has taken attention in recent studies due to fast calculation of steady state vibrations, and convenience at using measurement frequency response function without necessity to identify modal parame7 \fChapter 2. Literature Review ters. Schmitz and Mann [10] presented a frequency domain SLE approach which only considers forced vibrations at stable cutting conditions without the regeneration effects. The Fourier coefficients of static cutting force are expressed as: Fh \uf03d 1 2\uf070 2\uf070 \uf0f2 F (\uf066)e -ih\uf066 d\uf066 , (2.7) 0 where Fh is the Fourier coefficient of cutting force F , h \uf0ce (\uf02d\uf0a5, \uf0a5) is the coefficient index, and \uf066 is the instantaneous immersion angle of the cutting edge. The displacement in steady state is defined as: N q(t ) \uf03d \uf0e5 \uf0a5 \uf0e5 G(h\u03c9)Fheih(\uf077 t \uf02b2k\uf070\/ N ) , s (2.8) k \uf03d1 h \uf03d\uf02d\uf0a5 where displacement vector is q \uf03d (x,y)T , the Fourier coefficient vector is Fh \uf03d ( Fx,h , Fy,h )T , the number of flutes is N ,and \uf077s is the spindle speed in rad\/s. The transfer function at the tool tip is expressed as a matrix with direct and cross terms: \uf0e9Gxx (h\u03c9) Gxy (h\u03c9) \uf0f9 G (h\u03c9) \uf03d \uf0ea \uf0fa. \uf0eb\uf0eaGyx (h\u03c9) Gyy (h\u03c9) \uf0fb\uf0fa (2.9) At the instant when tool leaves mark on the surface, the location error is stored. The location errors at fixed spindle speed with respect to axial depth of cut (ADOC) are illustrated in Figure 2-6b. The authors [11] found a stable cutting point in a stability pocket as indicated with a dot in Figure 2-6a. Excluding the regeneration forces, cutter runout, effect of vibrations in chip thickness and radial immersion, they calculated the SLE values. Only tool tip transfer functions are taken into account, therefore effect of changing dynamics along the axial depth is neglected. A similar study has taken place helical end mills with semi discretization method [12]. 8 \fChapter 2. Literature Review a) b) Figure 2-4: An example on surface location error, a) chatter stability diagram for the 10% radial immersion down milling, b) the surface location errors in micrometers are calculated along the axial depth of cut (ADOC) for the fixed spindle speed indicated in a) 2.4 Chatter Stability Prediction Models for Intermittent Milling Operation Intermittent cutting forces often occur in thin wall milling applications, especially in the finishing stage, where very low immersion rates cannot be compensated by increasing the flute number and\/or helix angle. The definition of intermittent cutting is having pulse like cutting forces which excites the structure not only in its tooth passing frequency but also at multiple harmonics of the tooth passing frequency. The intermittency of cutting condition is dependent on the immersion rate of the cutting tooth, number of flutes N , and helix angle i . 9 \fChapter 2. Literature Review Figure 2-5: Low immersion down milling operation with geometric parameters of an end mill Immersion rate in milling is defined based on the ratio between radial depth of cut (RDOC) and cutter diameter D . As the ratio becomes smaller (i.e. less than 10%), the cutting forces instantly have changing values between peak to zero, which excites the structure in a broad frequency range as illustrated in Figure 2-6a. When the immersion rate is increased to 30%, the cutting forces become more continuous as demonstrated in Figure 2-6c. Similarly, helix angle and number of flutes in a cutter are inversely proportional to intermittency of the cutting condition. As helix angle increases, the forces become more continuous, the peak values drop and the number of multiple harmonics of tooth passing frequency diminish as in Figure 2-6f. Decreasing the flute number, creates impulsive forces, thus the number of frequencies swept becomes broader. 10 \fChapter 2. Literature Review Parameters Forces 10% FFT of Forces x direction y direction 0\uf06f helix 400 200 0 angle, x 10 1 0 0 N \uf03d4 0.01 5 2 Force [N] immersion, Force [N] 600 0.02 0 1000 2000 3000 4000 5000 Frequency [Hz] Time [s] a) 0\uf06f helix angle, x 10 600 400 200 0 -200 1 0 0 N \uf03d4 0.01 Time [s] 5 2 Force [N] immersion, Force [N] 30% b) 0.02 0 1000 2000 3000 4000 5000 Frequency [Hz] d) c) 10% 45\uf06f helix 400 200 0 angle, 0 N \uf03d4 0.01 1 0 0.02 0 1000 2000 3000 4000 5000 Frequency [Hz] Time [s] e) f) 10% angle, N \uf03d2 400 Force [N] 0\uf06f helix Force [N] x 10 immersion, 200 0 0 5 2 Force [N] immersion, Force [N] x 10 0.01 Time [s] 0.02 5 2 1 0 0 g) 1000 2000 3000 4000 5000 Frequency [Hz] h) Figure 2-6: Cutting condition parameters that affect the intermittency of the engagement for down milling operation; a,c,e,g are cutting forces in x and y directions, and b,d,f,h are Fast Fourier Transforms of those forces The regenerative vibration, or chattering, is reported as one of the most significant problems in milling of thin wall components as high flexibility reduces the stability 11 \fChapter 2. Literature Review limits and intermittent cutting condition brings additional instability. The regenerative chatter vibrations are introduced by Tobias [13] and Tlusty [14] in late 1950s. The idea stated that the chip load fluctuates due to current and past vibrations as illustrated in Figure 2-7. When current and past vibrations are out of phase, then it leads to grow in chip thickness and force in a loop which drives the system to instability. Figure 2-7: Regeneration mechanism Most of the studies until 90s were concentrated on turning stability. Minis and Yanushevsky [15] proposed a comprehensive model where milling operation has been modeled as a two dimensional time dependent delayed differential equation. They solved the problem in frequency domain by truncating the infinite Fourier series, or namely using truncated Toeplitz representation. They obtained the following characteristic equation to identify the stability limit: 1 det[I \uf02d K t a(1 \uf02d e\uf02d\u03c9cT ) \uf0d7 H r \uf02dk (\u03c9c \uf02b ik\u03c9T )] , 2 (2.10) where I is the identity matrix, K t is the tangential cutting force coefficient, \uf077c is the chatter frequency that is searched in the algorithm, T is the time delay, (r , k ) are numbers indicating the harmonic index such as r, k \uf03d 0, \uf0b11, \uf0b12,...., m , m is the number of Fourier terms considered in truncated Toeplitz representation, H r \uf02dk is the oriented transfer function matrix that contains transfer functions, and directional milling coefficients, and 12 \fChapter 2. Literature Review \u03c9T is the tooth passing frequency. The characteristic equation subsequently solved using Nyquist stability criteria to determine instability with respect to axial depth of cut (ADOC) and spindle speed. Followed by Minis\u2019s work, Altintas and Budak [16] introduced fast analytical way to plot stability lobes in frequency domain when only zeroth term of Fourier series considered. Eq. (2.10) has been simplified to: 1 det[I \uf02d K t a(1 \uf02d e\uf02d\u03c9cT ) \uf0d7 H(\u03c9c )] , 2 (2.11) where H is the oriented transfer function of average milling directional coefficients. This approach is named as zero order analytical (ZOA) method. The ZOA method later extended to general helical end mills and non-uniform pitch cutters [17] [18]. Bravo et al. [19], and Thevenot et al. [20] applied ZOA method to extract stability information for thin walled plate machining. Consequent studies showed that in the case of highly intermittent machining, ZOA method cannot account the additional instabilities come from the multiple harmonics of the tooth passing frequency. Multi-frequency (MF) solution which includes multiple terms of Fourier series has been investigated by Budak and Altintas [21], and then later Merdol and Altintas [22] extended it by discussing on the additional instability regions. Temporal finite element method (TFEA) proposed by Bayly et al. [23] and semi discretization introduced by Insperger and Stepan [24] are time domain based chatter stability prediction approaches which are also accurate in intermittent cutting conditions. The cause of additional stability regions explicitly investigated by Bayly et al. [23], and then later by Insperger et al.[25]. The instability in point A is associated with Hopf (Quasi-periodic) bifurcations and B with Flip (period doubling) bifurcations in Figure 2-8. In Hopf bifurcations, two complex conjugate eigenvalues leave the unit circle and the frequency in Hz associated with it is expressed as: fQP \uf03d \uf0b1 \u03c9c kn \uf02b , 2\u03c0 60 13 (2.12) \fChapter 2. Literature Review Figure 2-8: Chatter stability lobe and movement of eigenvalues in discrete domain for points A (Hopf bifurcation) and B (Flip bifurcation) where \u03c9c is chatter frequency, n is the spindle speed in rpm and k is the harmonics, i.e. k \uf03d 0, \uf0b11, \uf0b12,... . The Flip bifurcations are the reasons of additional lobes in intermittent cutting, where only one eigenvalue on the real axis moves out of the unit circle from -1. The frequency of the Flip bifurcations is defined as: f P2 \uf03d n kn \uf02b \u2019 120 60 (2.13) where frequency is related to spindle\u2019s angular speed. Effect of helix angle on the chatter stability in intermittent cutting conditions is modeled in frequency domain using multi frequency solution by Zatarain et al. [26] and chatter islands caused by helix angle are discussed. Similar analysis has been presented by Patel et al. [27] using TFEA method. Non-uniform pitch and non-uniform helix cutters are considered by Sims et al. [28] using semi discretization method. Dombovari et al.[29], also solved stability of serrated cutters by semi discretization. 14 \fChapter 2. Literature Review 2.5 Process Damping in Milling The heat resistant materials such as titanium and steel alloys that are frequently used in thin walled components can only be cut at low cutting speeds. When machining these materials at low speeds, it is known that the chatter stability limit increases due to additional damping caused by the process. Finding a mathematical model to represent this behavior has been searched significantly, and several theories are created to explain it in physical sense. In this thesis, flank indentation approach is adopted. According to this theory, at low speeds vibration marks imprinted on the surface becomes narrower and hence workpiece flank rubs against it that causes the additional damping as illustrated in Figure 2-9. Figure 2-9: Tool flank\u2019s indentation into wavy surface in low cutting speeds Sisson and Kegg [30] explained the low speed cutting behavior by analyzing the tool flank rubbing to wavy inner workpiece surface. They qualitatively searched the effects of parameters such as clearance angle \u03b3 , feed rate and tool wear. Wu [31] claimed that at low cutting speeds plowing forces are dominant due to indentation of the tool edge into workpiece. Since the edge radius is always present, plowing forces are exerted: 15 \fChapter 2. Literature Review Fx \uf03d KspV Fy \uf03d \uf06dFx , (2.14) where Fx is the friction force in feed direction proportional to material dependent on indentation constant Ksp and indented volume V ; Fy is the normal force which is proportional to dry Coulomb friction constant \uf06d . Chiou and Liang [32], utilized Wu\u2019s indentation approach and approximated the indented volume when only effect of tool wear is considered for small vibration amplitudes: L2w a V\uf03d x\uf026 , 2vc (2.15) where Lw is the average flank wear length of the cutter, vc is the cutting velocity of the edge in cut, x\uf026 is the vibration velocity of the surface in radial direction, and a is axial depth of cut. In their work, material dependent indentation coefficient is identified from static indentation tests of the tool edge. Eynian and Altintas [33] applied the same approach to milling problem, and found matching experimental results. In recent studies, indented volume is approximated with more comprehensive models that account clearance angle, edge radius, and amplitudes of the wavy surface. Budak and Tunc [34], calculated the indented volume geometrically over a period, and took the average of energy dissipated to determine average process damping coefficient. Their model included edge radius, separation angle due to indentation, and clearance angle contributions to indented volume. Ahmadi and Ismael [35], similar to Budak\u2019s approach, utilized average energy dissipation method and they also included the effect of vibration wave amplitudes imprinted on the surface. They calculated the stability lobes for maximum amplitude and minimum amplitude of vibrations to determine a region that chatter can happen in low cutting speeds. They analyzed this region by changing feed rate, hence the amplitude of vibrations, to validate their model. 16 \fChapter 3. Dynamics of Peripheral Milling 3. 3.1 Dynamics of Peripheral Milling Introduction Machining thin walled components is a challenge due to undesired static and dy- namic surface location errors and unstable chatter vibrations. In order to mathematically model the problem, the relative motion between flexible workpiece and slender end mill is required. This can be achieved by formulating the periodic milling forces, obtaining the dynamics of the structures, and associating the external loading with the dynamics and the laws of cutting mechanics. In this chapter, the discretized representation of peripheral milling is discussed by considering varying structural properties and cutting mechanics along the cutter axis. The differential milling forces and discrete structural dynamics are presented. The static and dynamic 3-axis cutting forces are formulated using linear edge force model [4][36]. The process damping force is included by considering the tool flank indentation-rubbing approach as presented in the literature [31][32]. Contrary to common methods on modeling the dynamics of the structures via finite element analysis [8][37][19][20], the structural dynamics are obtained through experimental modal analysis. The equations of motion of each discrete point are later collected and subsequently, modal space representation is used to reduce the number of equations in order to shorten the computation time without sacrificing the overall solution accuracy. Finally, modal equations of motion in time domain are expressed for peripheral milling. 17 \fChapter 3. Dynamics of Peripheral Milling 3.2 Discretized Representation of Process Dynamics In the peripheral milling of thin walled structures, the cutter is engaged with the workpiece along its axis. The end mill geometry, structural dynamics, cutting conditions such as radial depth of cut may vary continuously as axial position changes. Thus, discretizing axial depth of cut into sufficient number of points in order to account changing parameters accurately, and constructing the equation of motion considering the effect of each individual point are necessary. In the following subsections, formulation of 3-axis milling forces and equation of motion in frequency domain with the addition of identified structural parameters are provided in discretized form. 3.2.1 Forces in Milling For ideal axially symmetric end mill, the forces are generated when the tool\u2019s flutes remove material. Engagement conditions depend on cutter geometry, tool motion and selected axial and radial depth cuts. After engagement takes place, process mechanics affect the milling forces. In this section, geometrical relations regarding the general end mill, formulated by Engin and Altintas [36], are given followed by the comprehensive cutting mechanics model for three axis milling. General end mills may have varying shape, helix, and pitch angle along the cutter axis. In order to assess each cutting edge\u2019s contribution, the cutter axis is discretized into l number of points as demonstrated in Figure 3-1a. The number of flutes is denoted by N , tooth number is indicated by j , and axial elevation from tip of the tool is represented by z . The relation between z and the discrete points is: z \uf03d kdz , (3.1) where dz is the differential axial depth of cut and k is the axial index of the cutting point. 18 \fChapter 3. Dynamics of Peripheral Milling a) b) Figure 3-1: a) Axially discretized general end mill geometry, b) milling forces and angle convention Instantaneous immersion angle, \uf066 j (t , z ) , is defined to evaluate cutting edge\u2019s position in rotating frame by measuring the angle from fixed y-axis in clockwise direc19 \fChapter 3. Dynamics of Peripheral Milling tion as shown in Figure 3-1b. Instantaneous immersion angle for tooth j at axial elevation z is : \uf0e6 j \uf02d1 \uf0f6 \uf066 j (t , z ) \uf03d \uf066(t ) \uf02b \uf0e7\uf0e7 \uf0e5 \uf066p,e ( z ) \uf0f7\uf0f7 \uf02d \u03c8 j ( z ) , \uf0e8 e \uf03d0 \uf0f8 (3.2) where tooth j \u2019s angular position depends on cutter rotation, \uf066(t ) , lag angle, \u03c8 j ( z ) , and \uf0e6 j \uf02d1 \uf0f6 the summation of the pitch angles, \uf0e7\uf0e7 \uf0e5 \uf066p,e ( z ) \uf0f7\uf0f7 in which the indexing is one less from \uf0e8 e\uf03d0 \uf0f8 the tooth number and zero term is set \uf066p,0 ( z ) \uf03d 0. As spindle rotates with speed n [rev\/min], the angular position of the cutter varies as: \uf066(t ) \uf03d 2\uf070n t. 60 (3.3) Pitch angle, \uf066p,j ( z ) , is the angular spacing between adjacent flutes j and j \uf02b 1 at axial elevation z . For three cases stated below, the pitch angles are given: \uf0ec 2\uf070 uniform helix and pitch \uf0efN \uf0ef \uf066p,j ( z ) \uf03d \uf0ed\uf0660 uniform helix, non-uniform pitch , p, j \uf0ef \uf0ef \uf066 ( z ) \uf03d \uf0660 \uf02b \uf079 ( z ) \uf02d \uf079 ( z ) non-uniform helix and pitch p, j j \uf02b1 j \uf0ee p, j (3.4) where \uf0660p, j is the pitch angle between flutes j and j \uf02b 1 at the tip of the tool. The lag angle \uf079 j ( z ) is the angle between the cutting edge and the end point (at the tool tip) of flute j due to helix of the cutter. It is formulated along the cutting edge for end mills with constant lead or constant helix in [38]. For cylindrical end mills, lag angle is calculated as: \uf079 j ( z) \uf03d 2 z tan(i j ( z )) D , (3.5) where given axial elevation, the cutter diameter is denoted by D( z ) , and the helix angle is represented with i j ( z ) . The diameter of the general end mill is formulated in [38] with 20 \fChapter 3. Dynamics of Peripheral Milling respect to axial position. For uniform helix cutters, the helix angle stays same for each flute and axial element. However, for non-uniform helix cutters, the helix may vary for each flute and\/or axial level and this is illustrated in Appendix C. Total Milling Force, F A. Cutting Forces, Fc A1. Static Cutting Forces, Fcs B. Edge Forces, Fe A2. Dynamic Cutting Forces (Regeneration), Fcd B1.Static Edge Forces, Fes B2. Dynamic Edge Forces (Process Damping), Fed Figure 3-2: Milling Forces Scheme Radial, tangential and axial directions in rotating frame define the milling forces for general end mills as demonstrated in Figure 3-2. The instantaneous differential forces corresponding to axial position are expressed as: N dFrta (t , z ) \uf03d \uf0e5 g (\uf066 j (t , z ))dFrta,j (t , z ) , (3.6) j \uf03d1 where dFrta,j (t , z) \uf03d {dFr,j (t, z) dFt,j (t, z) dFa,j (t, z)}T are differential forces in radial, tangential and axial forces for each flute, respectively. The unit step function g (\uf066 j (t , z )) is used to consider whether the cutting edge is in cut or not. If instantaneous immersion angle \uf066 j (t , z ) is in between predetermined start \uf066st and exit \uf066ex angles, the cutting edge is removing material, hence generating forces: 21 \fChapter 3. Dynamics of Peripheral Milling \uf0ec1 if \uf066st \uf03c \uf066 j (t , z ) \uf03c \uf066ex g (\uf066 j (t , z )) \uf03d \uf0ed \uf0ee0 otherwise . (3.7) Milling forces consist of shearing of material at the rake face, and edge forces caused by the plowing between flank face of the tool and the finished surface as indicated in Figure 3-2. Intended material removal amount and effects coming from structural flexibilities constitute both forces as sum of static and dynamic components. Static part is constructed owing to the general cutting mechanics, treating the workpiece and tool as rigid bodies. In dynamic force formulation, the interaction between structural vibrations and milling forces is considered. The general form in Eq. (3.6) is separated into cutting and edge forces: N c e dFrta (t , z ) \uf03d \uf0e5 g (\uf066 j (t , z ))(dFrta, j (t , z ) \uf02b dFrta,j (t , z )) , (3.8) j \uf03d1 c c c c T where differential cutting forces dFrta, and j (t , z ) \uf03d {dFr,j (t , z ) dFt,j (t , z) dFa,j (t , z)} e e e e T differential edge forces are dFrta, j (t , z ) \uf03d {dFr,j (t , z ) dFt,j (t , z) dFa,j (t , z)} . Cutting forces are proportional to the chip thickness and cutting force coefficients. Static and dynamic forces at axial elevation z and flute number j are expressed for general end mill respectively, using the linear edge force theory [4] [39]: cs c dFrta, j (t , z ) \uf03d K rta h j (t , z )dS ( z ) , (3.9) and cd c dFrta, j (t , z ) \uf03d K rta hd, j (t , z )dS ( z ) , c \uf03d {Krc where cutting force coefficient matrix is K rta Ktc (3.10) Kac }T . Cutting force coeffi- cients can be identified mechanistically from the measured cutting forces by changing feed rate and fitting a linear curve [40], or applying orthogonal to oblique transforma- 22 \fChapter 3. Dynamics of Peripheral Milling tion as described in [4]. Differential contact length is denoted by dS ( z ) and it is ciated with differential axial depth by: dS ( z ) \uf03d dz \/ sin \uf06b( z) , (3.11) where \uf06b( z ) is the axial immersion angle indicating the angle between cutter axis and the normal of the cutting edge at axial elevation z . Axial immersion angle can be calculated for tip, arc and tapered sections of the cutting edge for general end mills as presented in [38]. For cylindrical end mills, it is always ninety degrees which makes differential contact length and differential axial depth equal. Figure 3-3 : Demonstration of dynamic and static chip thicknesses The static chip thickness, h j (t , z ) , and undulations generating the dynamic chip thickness are demonstrated in Figure 3-3. When the feed in x direction is projected on to radial direction of the cutter, static chip thickness is derived [2]: h j (t , z ) \uf03d c sin \uf06b( z )sin \uf066 j (t , z ) . 23 (3.12) \fChapter 3. Dynamics of Peripheral Milling The feed per tooth [mm\/rev\/tooth] is indicated by c( z ) , which is constant for cutters with regular pitch and regular helix. For non-uniform pitch and non-uniform helix cutters, it is calculated via the pitch angle between adjacent flutes [17]: c( z ) \uf03d Feedspeed[mm\/min] \uf066p,j ( z ) . n 2\uf070 (3.13) Substituting Eqs. (3.11) and (3.12) into Eq. (3.9) gives the differential static cutting force for flute j : cs c dFrta, j (t , z ) \uf03d K rta c sin \uf066 j (t , z )dz . (3.14) As milling forces are applied on the flexible end mill and the thin wall, relative displacements occur, predominantly in the principal vibration directions. These vibrations cause undulated surface which changes the overall chip thickness. Regarding the regeneration phenomenon introduced by Tobias [41] and Tlusty [14], the dynamic chip thickness is comprised of instantaneous vibrations and vibration marks imprinted on the surface from the previous tooth. The difference between current and previous vibrations at axial elevation z , assuming that the principal vibration directions are in Cartesian coordinates is: q(t , z) \uf02d q((t \uf02d T j ( z)), z ) \uf03d \uf044q(t , z ) \uf03d \uf044x(t , z)i \uf02b \uf044y(t , z) j \uf02b \uf044z(t , z )k , (3.15) where q(t , z ) \uf03d {x(t , z ) y(t , z ) z(t , z )}T is current vibrations vector in displacement form, q((t \uf02d T j ( z )), z ) is the vibration marks left by the previous tooth. Time delay, indicated by T j ( z ) , provides the instant that previous tooth left vibration marks on the surface. Using pitch angle and time passed in one full revolution, the delay is derived: T j ( z) \uf03d \uf066p,j ( z ) 60 2\uf070 n . (3.16) The dynamic chip thickness is obtained when vibrations are projected on the radial direction of the cutting edge [2]: 24 \fChapter 3. Dynamics of Peripheral Milling hd, j (t , z) \uf03d \uf044x(t , z)sin \uf06b( z)sin \uf066 j (t, z) \uf02b \uf044y(t, z)sin \uf06b( z) cos \uf066 j (t, z) \uf02d \uf044z(t, z) cos \uf06b( z ) . (3.17) Substituting Eqs. (3.11), (3.17) to Eq. (3.10), the differential dynamic cutting force for flute j is expressed: cd c dFrta, j (t , z ) \uf03d K rta {sin \uf066 j (t , z ) cos \uf066 j (t , z ) \uf02d cot \uf06b( z )}\uf044q(t , z) dz . (3.18) Wu [31] states that the contact pressures are exerted on the flank face as a result of tool indentation into workpiece, when finite edge radius exists. Since cutting edge of the tool is not perfectly sharp, the edge or plowing forces are always available. Moreover, vibration marks on the surface affect the amount of material pressed under the tool. Figure 3-4 shows static displacement owing to edge radius and dynamic displacement due to wavy surface. Figure 3-4 : Indentation of flank due to edge radius and undulated surface 25 \fChapter 3. Dynamics of Peripheral Milling The static plowing forces can be calculated through extensive FE models determining the chip separation and elastically deformed volume under the tool flank [42]. Since it is out of this thesis\u2019 scope, the static edge forces are approximated by linear edge force theory for general end mills [4][36]: es e dFrta, j (t , z ) \uf03d K rta \uf0d7 dS ( z ) , e \uf03d {Kre where K rta Kte (3.19) Kae }T are the edge force coefficients [40]. The dynamic edge forces are exerted against the undulated surface, and they are assumed to be a function of the elastically deformed volume. The instantaneous differential dynamic edge force in radial direction is expressed as [31]: dFr,edj (t , z ) \uf03d Ksp dV jd (t , z ) , (3.20) and the differential dynamic edge force in tangential direction is approximated by dry friction model given by Coulomb: dFt,edj (t , z ) \uf03d \uf06ddFr,edj (t , z) , (3.21) where Ksp is the material dependent contact force coefficient. The contact force coefficient can be calculated through tool-material indentation tests as reported in [32][43], or by using fast tool servo mechanism that eliminates regeneration forces but stimulates process damping forces as discussed in [44]. The dynamic Coulomb friction coefficient is \uf06d , and dV jd (t , z ) is instantaneous differential dynamic volume indented into the surface, which is calculated from: dV jd (t , z ) \uf03d dAdj (t , z )dS . (3.22) The instantaneous indented area in radial and tangential plane is denoted by dAdj (t , z) . Slope of the undulated surface, clearance angle \uf067 , geometry of the cutting edge such as edge chamfer, edge radius and tool wear length are the reported factors affecting the area [31][32][34][45][46][47]. In this thesis, the approach of Chiou and Liang [32], who considered the tool wear effect, is used in evaluating the indented area: 26 \fChapter 3. Dynamics of Peripheral Milling dAdj (t , z ) \uf03d p(t , z ) L2w r\uf026j (t , z ) , 2vc (t , z ) (3.23) where Lw is the average flank wear length of the cutter, vc (t , z ) is the cutting velocity of the edge in cut, r\uf026j (t , z ) is the vibration velocity of the surface in radial direction, and p(t , z ) is the unit step function accounting the instants that indentation due to wavy surface took place. Since the dynamic edge force incorporates the vibration velocity and tries to damp out the surface vibrations, it is called process damping force. The indirect proportionality to cutting speed, which comes from the slope of the surface, represents the increasing stability behavior at low cutting speeds. The cutting velocity is expressed as: vc (t , z ) \uf03d \uf066(t ) D( z ) . 2 (3.24) The surface vibration velocity is obtained when velocities in x,y,z are projected to the radial direction: \uf026 t , z)sin \uf06b( z)sin \uf066 j (t, z) \uf02b y( \uf026 t, z)sin \uf06b( z) cos \uf066 j (t, z) \uf02d z( \uf026 t, z) cos \uf06b( z ) . (3.25) r\uf026j (t , z) \uf03d x( Several methods have been proposed for the unit step function calculation based on the instant of flank contact with workpiece [32][48]. Eynian and Altintas [49] suggested that since static indentation, in volume wise dV js (t , z ) , is happening at all times, the dynamic edge forces are applied against the surface vibrations regardless of uphill or downhill motion as shown in Figure 3-5. In uphill motion, the indentation amount decreases and pushes tool less to bring it close to equilibrium; whereas in downhill motion, the indentation increases and more contact force is applied to push the tool to equilibrium position. The unit step function is set: p(t , z ) \uf03d 1 . 27 (3.26) \fChapter 3. Dynamics of Peripheral Milling Figure 3-5: Indentation of flank at several positions of the wavy surface The process damping force is derived when Eqs. (3.22)- (3.26) are substituted into Eqs. (3.20) and (3.21) consecutively: ed dFrta, j (t , z ) \uf0ec1 \uf0fc Ksp L2w \uf0ef \uf0ef \uf03d \uf0ed\uf06d \uf0fd{sin \uf066 j (t , z ) cos \uf066 j (t , z ) \uf02d cot \uf06b( z )}q\uf026 (t , z )dz \uf066(t ) D( z ) \uf0ef \uf0ef \uf0ee0 \uf0fe (3.27) After defining each component of the Eq. (3.8), the forces in radial, tangential and axial directions are projected to Cartesian coordinates by the transformation matrix T [39]: dFxyz (t , z ) \uf03d T(t , z)dFrta (t , z) , (3.28) where the force vector is dFxyz (t , z) \uf03d {dFx (t , z) dFy (t, z) dFz (t, z)}T . The transformation matrix, transforming the rotating tool coordinates to Cartesian tool coordinates, is expressed as: \uf0e9 \uf02d sin \uf066 j (t , z ) sin \uf06b( z ) \uf02d cos \uf066 j (t , z ) \uf02d sin \uf066 j (t , z ) cos \uf06b( z ) \uf0f9 \uf0ea \uf0fa T (t , z ) \uf03d \uf0ea \uf02d cos \uf066 j (t , z ) sin \uf06b( z ) sin \uf066 j (t , z ) \uf02d cos \uf066 j (t , z ) cos \uf06b( z ) \uf0fa . \uf0ea \uf0fa cos \uf06b( z ) 0 \uf02d sin \uf06b( z ) \uf0eb \uf0fb (3.29) When Eq. (3.14), (3.18), (3.19), (3.27) are substituted into Eq. (3.28), the overall differential milling force in Cartesian tool coordinates for axial elevation z is formulated: 28 \fChapter 3. Dynamics of Peripheral Milling N \u02c6 (t , z ) \uf02b A \u02c6 (t , z )\uf044q(t , z ) \uf02b B\u02c6 (t , z ) \uf02b B\u02c6 (t , z)q\uf026 (t , z)) dz , dFxyz (t , z ) \uf03d \uf0e5 ( A 1, j 2, j 1, j 2, j (3.30) j \uf03d1 Static Dynamic Static Dynamic Cutting Cutting Edge Edge Force Force Force Force (P. Damping) The coefficient matrices for a single axial element are formulated when matrix products are obtained: \u02c6 (t , z ) \uf03d g (\uf066 (t , z )){a (t , z ) a (t , z ) a (t , z )}T A 1, j j 1,x 1,y 1,z a1,x (t , z ) \uf03d c \uf0e9( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z ))(cos 2\uf066 j (t , z ) \uf02d 1) \uf02d K tc sin 2\uf066 j (t , z ) \uf0f9\uf0fb 2\uf0eb (3.31) a1,y (t , z ) \uf03d c \uf0e9 \uf02d( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z )) sin 2\uf066 j (t , z ) \uf02b K tc (1 \uf02d cos 2\uf066 j (t , z )) \uf0f9\uf0fb 2\uf0eb a1,z (t , z ) \uf03d c sin \uf066 j (t , z )( K rc cos \uf06b( z ) \uf02d K ac sin \uf06b( z )) 29 , \fChapter 3. Dynamics of Peripheral Milling \uf0e9 a2,xx (t , z ) a2,xy (t , z ) a2,xz (t , z ) \uf0f9 \uf0ea \uf0fa \u02c6 (t , z ) \uf03d g (\uf066 (t , z )) \uf0ea a (t , z ) a (t , z ) a (t , z ) \uf0fa A 2, j j 2,yx 2,yy 2,yz \uf0ea \uf0fa \uf0eb a2,zx (t , z ) a2,zy (t , z ) a2,zz (t , z ) \uf0fb a2,xx (t , z ) \uf03d 1 \uf0e9( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z ))(cos 2\uf066 j (t , z ) \uf02d 1) \uf02d K tc sin 2\uf066 j (t , z ) \uf0f9\uf0fb 2\uf0eb a2,xy (t , z ) \uf03d 1 \uf0e9 \uf02d( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z )) sin 2\uf066 j (t , z ) \uf02d K tc (1 \uf02b cos 2\uf066 j (t , z )) \uf0f9\uf0fb 2\uf0eb \uf0ec\uf0ef K rc cos \uf06b( z ) sin \uf066 j (t , z ) \uf02b \uf0fc\uf0ef a2,xz (t , z ) \uf03d \uf0ed \uf0fd \uf0ee\uf0ef( K tc cot \uf066 j (t , z ) \uf02b K ac cos \uf06b( z )) cot \uf06b( z ) sin \uf066 j (t , z ) \uf0fe\uf0ef a2,yx (t , z ) \uf03d 1 \uf0e9 \uf02d( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z )) sin 2\uf066 j (t , z ) \uf02d K tc (cos 2\uf066 j (t , z ) \uf02d 1) \uf0f9\uf0fb 2\uf0eb a2,yy (t , z ) \uf03d 1 \uf0e9 \uf02d( K rc sin \uf06b( z ) \uf02b K ac cos \uf06b( z ))(1 \uf02b cos 2\uf066 j (t , z )) \uf02b K tc sin 2\uf066 j (t , z ) \uf0f9\uf0fb 2\uf0eb \uf0ec\uf0ef K rc cos \uf06b( z ) cos \uf066 j (t , z ) \uf02b \uf0fc\uf0ef a2,yz (t , z ) \uf03d \uf0ed \uf0fd \uf0ee\uf0ef(\uf02d K tc tan \uf066 j (t , z ) \uf02b K ac cos \uf06b( z )) cot \uf06b( z ) cos \uf066 j (t , z ) \uf0fe\uf0ef a2,zx (t , z ) \uf03d sin \uf066 j (t , z )( K rc cos \uf06b( z ) \uf02d K ac sin \uf06b( z )) (3.32) a2,zy (t , z ) \uf03d cos \uf066 j (t , z )( K rc cos \uf06b( z ) \uf02d K ac sin \uf06b( z )) a2,zz (t , z ) \uf03d cos \uf06b( z )( K ac \uf02d K rc cot \uf06b( z )) , 30 \fChapter 3. Dynamics of Peripheral Milling \u02c6 (t , z ) \uf03d g (\uf066 (t , z )){b (t , z ) b (t , z ) b (t , z )}T B 1, j j 1,x 1,y 1,z b1,x (t , z ) \uf03d \uf02d( K re \uf02b K ae cot \uf06b( z )) sin \uf066 j (t , z ) \uf02d K te cos \uf066 j (t , z ) \/ sin \uf06b( z ) (3.33) b1,y (t , z ) \uf03d \uf02d( K re \uf02b K ae cot \uf06b( z )) cos \uf066 j (t , z ) \uf02b K te sin \uf066 j (t , z ) \/ sin \uf06b( z ) b1,z (t , z ) \uf03d K re cot \uf06b( z ) \uf02d K ae , \uf0e9b2,xx (t , z ) b2,xy (t , z ) b2,xz (t , z ) \uf0f9 Ksp L2w \uf0ea \uf0fa B\u02c6 2, j (t , z ) \uf03d g (\uf066 j (t , z )) \uf0eab2,yx (t , z ) b2,yy (t , z ) b2,yz (t , z ) \uf0fa \uf066(t ) D( z ) \uf0ea \uf0fa \uf0eb b2,zx (t , z ) b2,zy (t , z ) b2,zz (t , z ) \uf0fb b2,xx (t , z ) \uf03d 1 \uf0e9sin \uf06b( z )(cos 2\uf066 j (t , z ) \uf02d 1) \uf02d \uf06d sin 2\uf066 j (t , z ) \uf0f9\uf0fb 2\uf0eb b2,xy (t , z ) \uf03d 1 \uf0e9 \uf02d sin \uf06b( z ) sin 2\uf066 j (t , z ) \uf02d \uf06d(1 \uf02b cos 2\uf066 j (t , z )) \uf0f9\uf0fb 2\uf0eb b2,xz (t , z ) \uf03d \uf0e9\uf0ebcos \uf06b( z ) sin \uf066 j (t , z ) \uf02b \uf06d cot \uf06b( z ) cos \uf066 j (t , z ) \uf0f9\uf0fb 1 b2,yx (t , z ) \uf03d \uf0e9\uf0eb \uf02d sin \uf06b( z ) sin 2\uf066 j (t , z ) \uf02d \uf06d(cos 2\uf066 j (t , z ) \uf02d 1) \uf0f9\uf0fb 2 b2,yy (t , z ) \uf03d 1 \uf0e9 \uf02d sin \uf06b( z )(1 \uf02b cos 2\uf066 j (t , z )) \uf02b \uf06d sin 2\uf066 j (t , z ) \uf0f9\uf0fb 2\uf0eb b2, yz (t , z ) \uf03d \uf0e9\uf0ebcos \uf06b( z ) cos \uf066 j (t , z ) \uf02d \uf06d cot \uf06b( z ) sin \uf066 j (t , z ) \uf0f9\uf0fb b2,zx (t , z ) \uf03d cos \uf06b( z ) sin \uf066 j (t , z ) b2,zy (t , z ) \uf03d cos \uf06b( z ) cos \uf066 j (t , z ) . 1 b2,zz (t , z ) \uf03d \uf02d cos 2 \uf06b( z ) \/ sin \uf06b( z ) 2 31 (3.34) \fChapter 3. Dynamics of Peripheral Milling 3.2.2 Equations of Motion in Laplace Domain In order to associate forces with structural dynamics, inertial, elastic and damp- ing characteristics of the workpiece and the tool must be identified separately. The structure\u2019s dynamic properties correspond to amplitudes and phases of the calculated frequency response function which is comprised of measured force (input) and motion (output). Accurate representation of the dynamics can be obtained by scanning sufficient range of excitation frequency with modal hammer impact or shaker. Figure 3-6 : Discretized representation of the plate and the end mill In theory, the thin walled element is most flexible in its thickness direction ( y ) and in most of the cases; equations of motion in the other two orthogonal ( x,z ) directions are neglected. The cutter is usually considered rigid compared to thin plate. In this 32 \fChapter 3. Dynamics of Peripheral Milling thesis, for generality, three orthogonal coordinates which are assumed to lie at principal vibration directions are considered, and later simplifications are made for specific applications. In order to model the continuous dynamic nature of the structures, similar to milling force approach, spatial discretization is applied. The thin walled plate is divided into p \uf0b4 w modal nodes and cutter into l \uf0b41 where the differential axial depth, dz kept same for both structures as presented in Figure 3-6. The nodes that are in contact have the same number in workpiece and tool side. Assuming that the transfer functions are identified for each node, the steady state representation of equations of motion for arbitrary structure is expressed in Laplace domain: \uf0e9G xx ( s) G xy ( s) G xz ( s) \uf0f9 \uf0ecF ( s) \uf0fc \uf0ec x( s) \uf0fc \uf0ea \uf0fa\uf0ef x \uf0ef \uf0ef \uf0ef G ( s)Fxyz ( s) \uf03d Q( s) \uf03d \uf0eaG yx ( s) G yy ( s) G yz ( s) \uf0fa \uf0edFy ( s) \uf0fd \uf03d \uf0edy ( s) \uf0fd , \uf0ea \uf0fa\uf0ef \uf0ef \uf0ef z( s) \uf0ef \uf0fe \uf0eb G zx ( s) G zy ( s) G zz ( s) \uf0fb \uf0ee Fz ( s) \uf0fe \uf0ee (3.35) where receptance (displacement) transfer function matrix is denoted by G ( s) , milling force vector is Fxyz ( s) and physical displacement vector is Q( s) . The sub-matrices in transfer function matrix are defined as G ji ( s) , where displacement (response) direction subscript is j \uf03d x,y,z and force direction (excitation) subscript is i \uf03d x,y,z . When j \uf03d i , the term is called direct transfer function for the specific direction; when j \uf0b9 i , the term is called cross transfer function, accounting the geometric coupling effects coming from the other orthogonal directions. The transfer functions in arbitrary directions are identified through measurement for both tool and workpiece, respectively: 33 \fChapter 3. Dynamics of Peripheral Milling \uf0e9 G j1i1,t ( s) G j1i 2,t ( s) \uf0ea \uf0eaG j 2i1,t ( s) G j 2i 2,t ( s) \uf0ea \uf04d \uf04d G ji ,t ( s) \uf03d \uf0ea \uf04c \uf0ea G jqi1,t ( s) \uf0ea \uf04d \uf04d \uf0ea \uf0eaG \uf04c \uf0eb jli1,t ( s ) \uf04c G j1iq,t ( s) \uf04c G j1il ,t ( s) \uf0f9 \uf0fa \uf04c G j 2iq,t ( s) \uf04c G j 2il ,t ( s) \uf0fa \uf0fa \uf04d \uf04d \uf04c \uf04d \uf0fa \uf04c G jqiq ,t ( s) \uf04c G jqil ,t ( s) \uf0fa \uf0fa \uf04d \uf04d \uf04c \uf04d \uf0fa \uf04c G jliq ,t ( s ) \uf04c G jlil ,t ( s) \uf0fa\uf0fb , (3.36) G j1i ( p\uf0b4w),wp ( s) \uf0f9 \uf0fa G j 2i ( p\uf0b4w),wp ( s ) \uf0fa \uf0fa \uf04d \uf0fa G jqi ( p\uf0b4w),wp ( s ) \uf0fa \uf0fa \uf04d \uf0fa G j ( p\uf0b4w)i ( p\uf0b4w),wp ( s ) \uf0fa\uf0fb ( p\uf0b4w)\uf0b4( p\uf0b4w) l\uf0b4l and G ji ,wp ( s) \uf03d G j1i 2,wp ( s) \uf0e9 G j1i1,wp ( s ) \uf0ea G j 2i 2,wp ( s ) \uf0ea G j 2i1,wp ( s ) \uf0ea \uf04d \uf04d \uf0ea \uf04c \uf0ea G jqi1,wp ( s ) \uf0ea \uf04d \uf04d \uf0ea \uf0eaG \uf04c \uf0eb j ( p\uf0b4w)i1,wp ( s) \uf04c G j1iq ,wp ( s) \uf04c \uf04c G j 2iq,wp ( s ) \uf04c \uf04d \uf04d \uf04c \uf04c G jqiq ,wp ( s ) \uf04c \uf04d \uf04d \uf04c \uf04c G j ( p\uf0b4w)iq ,wp ( s) \uf04c (3.37) where looking at the form G jaib ( s) , the subscript a indicates the measurement node, and b represents the node where force is applied. For linear elastic bodies, Maxwell\u2019s reciprocity theory holds [50]: G jaib ( s) \uf03d Gibja ( s) , (3.38) therefore measuring just a column or a row of Eqs. (3.36) and (3.37) is enough to construct whole symmetrical matrix. A sub-vector in force vector, Fi , and sub-vector in displacement vector, i , for i \uf03d x,y,z are expressed, respectively: Fi (s) \uf03d {Fi ,1 ( s) Fi ,2 ( s) \uf04b Fi,q ( s) \uf04b}T , (3.39) i (s) \uf03d {i1 (s) i2 (s) \uf04b iq (s) \uf04b}T , (3.40) where the sizes are l \uf0b41 and ( p \uf0b4 w) \uf0b41 for the tool and workpiece, respectively. 34 \fChapter 3. Dynamics of Peripheral Milling 3.3 Modal Space Formulation In previous section the equations of motion are presented in Laplace domain. Due to discretized representation, the number of equations to solve can be plenty depending on the axial engagement length. It is known that, analyses in dynamics consume computational effort since the equations need to be evaluated repeatedly in specified time interval [51]. In order to reduce the amount of computation, the reduced degrees of freedom (DOF) are used instead of full DOF. Although, this approach brings errors by disregarding information coming from eliminated DOF, or in other words natural frequencies (modes), in practice most applications have dominant dynamics at certain frequency range. Consequently, selecting sufficient number of DOF for certain frequency bandwidth decreases the computation time dramatically without affecting the solution accuracy. The frequency response function obtained from the measurement with real mode shape assumption has the form [50]: G jaib ( s) \uf03d m u ja,\uf06cuib,\uf06c 1 1 , \uf02b \uf02b \uf0e5 s 2 M Rjaib \uf06c\uf03d1 s 2 \uf02b 2\uf078\uf06c \uf077n,\uf06c s \uf02b \uf077n,\uf06c 2 K Rjaib (3.41) where M Rjaib , K Rjaib are residual mass and stiffness terms respectively, which are included to correct truncation errors coming from the elimination of low and high frequency modes. u ja,\uf06c , uib,\uf06c are the mass normalized mode shapes for nodes in assigned directions ja and ib ; \uf077n,\uf06c , \uf078\uf06c are the natural frequency and the modal damping ratio for specific mode \u03bb , respectively. The displacement is formulated when each term in Eq. (3.41) superposed: m Q( s) \uf03d Qrbody ( s) \uf02b \uf0e5 Q*\uf06c ( s) \uf02b Qst ( s) , \uf06c\uf03d1 35 (3.42) \fChapter 3. Dynamics of Peripheral Milling where Qrbody ( s) implies the rigid body motion behavior at low frequencies [52], Q* ( s) is m number of identified modes\u2019 contribution to displacement, and Qst ( s) represents the displacement correction for high frequency modes, which can be associated with static correction in finite element analysis [51]. It is known that, for the machining processes, the instability condition and the detrimental forced vibrations happen around the dominant natural frequencies of the structure. Selecting sufficient number of dominant modes and neglecting the residual terms, if they have small amplitude in considered frequency range, can be applied without affecting the accuracy. The total displacement in Eq. (3.42) is approximated as the superposition of the modal displacements in physical coordinates as: m Q( s) \uf040 \uf0e5 Q*\uf06c ( s) . (3.43) \uf06c\uf03d1 In practice number of selected modes is smaller than the number of DOF such as mt \uf03c 3l and mwp \uf03c 3( p \uf0b4 w) , where mt is the number of tool\u2019s natural frequencies, and mwp is number of workpiece\u2019s natural frequencies. Benefiting from the linear modal superposition shown in Eq. (3.43), a modal space can be defined to collect all the nodal responses to a modal coordinate, for specific mode \u03bb . This approach reduces the number of equations from 3l and 3( p \uf0b4 w) to mt and mwp for tool and workpiece, respectively. Applying real eigenvalue analysis, Eq. (3.43) is arranged by defining a modal space: Q(s) \uf03d U\uf047(s) , (3.44) where \uf047 t (s) \uf03d {\uf047 t,1 ( s) \uf047 t,2 ( s) \uf04b \uf047 t,mt ( s)}T is modal displacement for the tool, and \uf047 wp (s) \uf03d {\uf047 wp,1 ( s) \uf047 wp,2 ( s) \uf04b \uf047 wp,mwp ( s)}T is modal displacement for the workpiece. The mass normalized modal (mode shape) matrices, which are related with the eigenvectors of the eigenvalue problem, are expressed for tool and workpiece, respectively: 36 \fChapter 3. Dynamics of Peripheral Milling \uf0e9 \uf0e9 ux1,1,t ux1,2,t \uf0ea\uf0ea \uf04d \uf0ea\uf0ea \uf04d \uf0ea \uf0eau uxq ,2,t \uf0ea \uf0eb xq ,1,t \uf0ea \uf04d \uf0ea \uf04d \uf0ea uxl ,1,t uxl ,2,t \uf0ea \uf0ea \uf0e9 u y1,1,t uy1,2,t \uf0ea\uf0ea \uf04d \uf0ea\uf0ea \uf04d \uf0ea \uf0eau U t \uf03d \uf0ea \uf0eb yq ,1,t uyq,2,t \uf0ea \uf04d \uf04d \uf0ea \uf0ea u yl ,1,t u yl ,2,t \uf0ea \uf0ea \uf0e9 uz1,1,t uz1,2,t \uf0ea\uf0ea \uf04d \uf0ea\uf0ea \uf04d \uf0ea \uf0eau uzq ,2,t \uf0ea \uf0eb zq ,1,t \uf0ea \uf04d \uf04d \uf0ea \uf0ea uzl ,1,t uzl ,2,t \uf0eb \uf0f9 \uf04c ux1, mt ,t \uf0f9 \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf0fa \uf04c uxq ,mt ,t \uf0fa\uf0fb q\uf0b4mt \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf04c uxl ,mt ,t \uf0fa \uf04c uy1,mt ,t \uf0f9 \uf0fa \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf0fa \uf0fa \uf04c uyq,mt ,t \uf0fb q\uf0b4mt \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf04c uyl ,mt ,t \uf0fa \uf04c uz1,mt ,t \uf0f9 \uf0fa \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf0fa \uf0fa \uf04c uzq,mt ,t \uf0fb q\uf0b4mt \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf0fa \uf04c uzl ,mt ,t \uf0fb and 37 U\u0302 t , 3l \uf0b4mt (3.45) \fChapter 3. Dynamics of Peripheral Milling U wp \uf0e9 \uf0e9 ux1,1,wp ux1,2,wp \uf0ea\uf0ea \uf04d \uf04d \uf0ea\uf0ea \uf0ea \uf0eau uxq ,2,wp \uf0ea \uf0ea\uf0eb xq ,1,wp \uf0ea \uf04d \uf04d \uf0ea \uf0eau ux( p\uf0b4w),2,wp \uf0ea x( p\uf0b4w),1,wp \uf0ea u u y1,2,wp \uf0ea \uf0e9 y1,1,wp \uf0ea \uf0ea \uf04d \uf04d \uf0ea\uf0ea \uf0ea u yq ,2,wp \uf03d \uf0ea \uf0ea\uf0ebu yq ,1,wp \uf0ea \uf0ea \uf04d \uf04d \uf0ea \uf0eau y( p\uf0b4w),1,wp u y( p\uf0b4w),2,wp \uf0ea uz1,2,wp \uf0ea \uf0e9 uz1,1,wp \uf0ea\uf0ea \uf04d \uf0ea\uf0ea \uf04d \uf0ea \uf0eau uzq ,2,wp \uf0ea \uf0ea\uf0eb zq ,1,wp \uf0ea \uf04d \uf04d \uf0ea \uf0eau uz( p\uf0b4w),2,wp \uf0eb z( p\uf0b4w),1,wp \uf04c \uf04d \uf04c \uf04d \uf04c \uf04c \uf04d \uf04c \uf04d \uf04c \uf04c \uf04d \uf04c \uf04d \uf04c \uf0f9 ux1,mwp ,wp \uf0f9 \uf0fa \uf0fa \uf04d \uf0fa \uf0fa \uf0fa \uf0fa uxq ,mwp ,wp \uf0fa \uf0fb q\uf0b4mwp \uf0fa \uf0fa \uf04d \uf0fa \uf0fa ux( p\uf0b4w),mwp ,wp \uf0fa \uf0fa u y1,mwp ,wp \uf0f9 \uf0fa \uf0fa \uf0fa U\u0302 wp \uf04d \uf0fa \uf0fa uyq ,mwp ,wp \uf0fa\uf0fa \uf0fa \uf0fb q\uf0b4mwp \uf0fa \uf0fa \uf04d \uf0fa uy( p\uf0b4w),mwp ,wp \uf0fa \uf0fa uz1,mwp ,wp \uf0f9 \uf0fa \uf0fa \uf0fa \uf04d \uf0fa \uf0fa \uf0fa \uf0fa uzq,mwp ,wp \uf0fa \uf0fb q\uf0b4mwp \uf0fa \uf0fa , (3.46) \uf04d \uf0fa \uf0fa uz( p\uf0b4w),mwp ,wp \uf0fb 3( p\uf0b4w)\uf0b4mwp where each column of the matrices gives mass normalized displacements in x,y,z directions for specific mode. The q number of nodes is highlighted to indicate the nodes that are in cut. The receptance transfer function matrix is expressed in terms of mass normalized mode shapes and modal parameters, with symmetric mass and stiffness matrices approach [50]: G(s) \uf03d U (Is 2 \uf02b 2\u03b6\u03c9n s \uf02b \u03c92n )\uf02d1U T , (3.47) where I is the identity matrix, \u03b6 is the diagonal damping ratio matrix, and \u03c9 n is the diagonal natural frequency matrix. When Eqs. (3.44) and (3.47) are substituted in Eq. (3.35), the following form appears: U (Is 2 \uf02b 2\u03b6\u03c9n s \uf02b \u03c92n )\uf02d1U T Fxyz (s) \uf03d U\uf047( s) . 38 (3.48) \fChapter 3. Dynamics of Peripheral Milling Eq. (3.48) is pre-multiplied by U \uf02d1 (if mass matrix is introduced, benefiting from the orthogonality properties, U T M can be also used for pre-multiplication [52]): (Is 2 \uf02b 2\u03b6\u03c9n s \uf02b \u03c92n )\uf02d1U T Fxyz ( s) \uf03d \uf047( s) , (3.49) which presents the modal domain model with mt number of equations to solve for tool, and mwp number of equations for workpiece. The time domain model is obtained when Inverse Laplace Transform is applied to Eq. (3.49): 2 T \uf026\uf026 (t ) \uf02b 2\u03b6\u03c9 \uf047 \uf026 \uf047 n (t ) \uf02b \u03c9n \uf047(t ) \uf03d U Fxyz (t ) . (3.50) In general, full size modal matrices are used for modal space representation. However, in this particular problem, only the nodes (1, \uf04b, q) are in cut for both workpiece and the cutter as demonstrated in Figure 3-6. Although, other nodes vibrate due to off diagonal terms in transfer functions, they do not contribute to cutting forces via regeneration and process damping. The effective terms in force vector, in physical domain are identified using Eqs. (3.1), (3.28) and (3.39) for i \uf03d x,y,z : F\u02c6i (t ) \uf03d {dFi (t , dz) \uf04b dFi (t , qdz )}T . (3.51) When the equations are expressed in modal domain, truncating the mode shape vectors after node number q gives the same result with the full modal matrix approach for the force side. The truncated modal matrices, which are shown in Eqs. (3.45) and (3.46), are denoted by U\u02c6 t , U\u02c6 wp for tool and workpiece, where the matrix sizes are 3q \uf0b4 mt and 3q \uf0b4 mwp accordingly. The spatial terms in physical coordinates in the force vector are transformed to \uf026 (t ) \uf03d U\uf047 \u02c6 \uf026 (t ) . However, special care is required modal space by \uf044Q(t ) \uf03d U\u02c6 \uf044\uf047(t ) and Q 39 \fChapter 3. Dynamics of Peripheral Milling for non-uniform helix cutters where delay changes with axial position. The modal coordinate representation of non-uniform helix cutter\u2019s delayed term is: QT (t ) \uf03d U\u02c6 \uf02a \uf047 T (t ) \uf03d \uf0e9[ux1,1 \uf0ea \uf0ea[ux2,2 \uf0ea \uf0ea \uf04d \uf0ea \uf0ea [uxq ,1 \uf0ea \uf0ea [u y1,1 \uf0ea \uf04d \uf0ea \uf0ea [u \uf0ea yq ,1 \uf0ea [u \uf0ea z1,1 \uf0ea \uf04d \uf0ea \uf0ea\uf0eb [u yq ,1 ux1,2 \uf04c ux1,m ] \uf0b4 [\uf047 1 (t \uf02d T j (dz )) \uf047 2 (t \uf02d T j (dz )) ux2,2 \uf04c ux2, m ] \uf0b4 [\uf047 1 (t \uf02d T j (2dz )) \uf047 2 (t \uf02d T j (2dz )) \uf04d \uf04d \uf04d \uf04d \uf04d uxq,2 \uf04c uxq, m ] \uf0b4 [\uf047 1 (t \uf02d T j (qdz )) \uf047 2 (t \uf02d T j (qdz )) u y1,2 \uf04d \uf04c uy1, m ] \uf0b4 \uf04d \uf04d [\uf047 1 (t \uf02d T j (dz )) \uf047 2 (t \uf02d T j (dz )) \uf04d \uf04d uyq ,2 \uf04c u yq ,m ] \uf0b4 [\uf047 1 (t \uf02d T j (qdz )) \uf047 2 (t \uf02d T j (qdz )) uz1,2 \uf04d uz1, m ] \uf0b4 [\uf047 1 (t \uf02d T j (dz )) \uf047 2 (t \uf02d T j (dz )) \uf04d \uf04d \uf04d \uf04d \uf04d uyq ,2 \uf04c u yq ,1 ] \uf0b4 [\uf047 1 (t \uf02d T j (qdz )) \uf047 2 (t \uf02d T j (qdz )) T \uf04b \uf047 m (t \uf02d T j (dz ))] \uf0f9 \uf0fa \uf04b \uf047 m (t \uf02d T j (2dz ))]T \uf0fa \uf0fa \uf04d \uf04d \uf0fa T\uf0fa \uf04c \uf047 m (t \uf02d T j (qdz ))] \uf0fa \uf0fa \uf04c \uf047 m (t \uf02d T j (dz ))]T \uf0fa \uf0fa \uf04d \uf04d \uf0fa \uf04c \uf047 m (t \uf02d T j (qdz ))]T \uf0fa \uf0fa \uf04c \uf047 m (t \uf02d T j (dz ))]T \uf0fa \uf0fa , \uf0fa \uf04d \uf04d \uf0fa \uf04c \uf047 m (t \uf02d T j (qdz ))]T \uf0fa\uf0fb 3q\uf0b41 (3.52) where r \uf03d {1, \uf04b, q} is the axial index of the cutting point. T j ,r is the delay value for the corresponding axial elevation and tooth number j , as calculated in Eq. (3.16). The sign \uf02a indicates row by column multiplication as shown in Eq. (3.52). The detailed discussion on mapping of each term in semi discretization method is discussed in chapter 4. The regenerative displacement and the velocity in forcing function are calculated regarding the relative motion between tool and workpiece as discussed by Bravo et al. [19]. The relative displacement for axial position r is shown in Figure 3-7. In order to define relative regenerative displacement and relative vibration velocity in modal coordinates, the following variables are introduced: \uf0ec U\u02c6 t \uf044\uf047 t (t ) \uf02b U\u02c6 wp \uf044\uf047 wp (t ) uniform helix \uf0ef , \uf044\u03a8(t ) \uf03d \uf0ed \u02c6 \u02c6 \u02c6 \u02c6 U \uf047 ( t ) \uf02d U \uf02a \uf047 ( t \uf02d T ) \uf02b U \uf047 ( t ) \uf02d U \uf02a \uf047 ( t \uf02d T ) non-uniform helix \uf0ef t t j ,r wp wp wp wp j ,r \uf0ee t t (3.53) and \uf026 (t ) \uf03d U\u02c6 \uf047\uf026 (t ) \uf02b U\u02c6 \uf047\uf026 (t ) . \u03a8 t t wp wp 40 (3.54) \fChapter 3. Dynamics of Peripheral Milling Figure 3-7: Relative displacement for axial point r Eq. (3.30) is expanded for all nodes and substituted into Eq. (3.50). Consequently, the time domain modal equations of motion for tool and workpiece are formulated: \uf026\uf026 (t ) \uf02b 2\u03b6 \u03c9 \uf047\uf026 (t ) \uf02b \u03c9 2 \uf047 (t ) \uf03d \uf047 t t n,t t n,t t , (3.55) , (3.56) N \uf026 (t ))dz U\u02c6 tT \uf0e5 ( A1, j (t ) \uf02b A 2, j (t )\uf044\u03a8(t ) \uf02b B1, j (t ) \uf02b B 2, j (t )\u03a8 j \uf03d1 and \uf026\uf026 (t ) \uf02b 2\u03b6 \u03c9 \uf047\uf026 (t ) \uf02b \u03c9 2 \uf047 (t ) \uf03d \uf047 wp wp n,wp wp n,wp wp N T U\u02c6 wp \uf0e5 (A1, j (t ) \uf02b A2, j (t )\uf044\u03a8(t ) \uf02b B1, j (t ) \uf02b B2, j (t )\u03a8\uf026 (t ))dz j \uf03d1 41 \fChapter 3. Dynamics of Peripheral Milling where A1, j (t ), A2, j (t ), B1, j (t ), B2, j (t ) are expanded and arranged forms of corresponding \u02c6 (t , z ), A \u02c6 (t , z), B \u02c6 (t , z), B \u02c6 (t , z) terms in Eq. (3.30). They are expressed as: A 1, j 2, j 1, j 2, j A1, j (t , z ) \uf03d g(\uf066 j (t )) \uf0b4 , {a1,x (t , dz ) \uf04b a1,x (t , qdz ) a1,y (t , dz ) \uf04b a1,y (t , qdz ) a1,z (t , dz ) \uf04b (3.57) a1,z (t , qdz )}3Tq\uf0b41 A 2, j (t ) \uf03d g(\uf066 j (t )) \uf0b4 \uf04c a2,xy (t , dz ) 0 \uf04c a2,xz (t , dz ) 0 \uf04c \uf0e9 a2,xx (t , dz ) 0 \uf0f9 \uf0ea \uf0fa 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf04d 0 a2,xx (t , qdz ) \uf04d 0 a2,xy (t , qdz ) \uf04d 0 a2,xz (t , qdz ) \uf0fa \uf0ea \uf0fa \uf0ea a2,yx (t , dz ) 0 \uf0fa \uf04c a2,yy (t , dz ) 0 \uf04c a2,yz (t , dz ) 0 \uf04c \uf0ea \uf0fa 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf04d 0 a2,yx (t , qdz ) \uf04d 0 a2,yy (t , qdz ) \uf04d 0 a2,yz (t , qdz ) \uf0fa \uf0ea \uf0fa \uf0ea a2, zx (t , dz ) 0 \uf0fa \uf04c a2,zy (t , dz ) 0 \uf04c a2,zz (t , dz ) 0 \uf04c \uf0ea \uf0fa , 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf0fa \uf04d 0 a2,zx (t , qdz ) \uf04d 0 a2,zy (t , qdz ) \uf04d 0 a2,zz (t , qdz ) \uf0fb \uf0eb 3q\uf0b43q (3.58) B1, j (t ) \uf03d g(\uf066 j (t )) \uf0b4 , {b1,x (t , dz ) \uf04b b1,x (t , qdz ) b1,y (t , dz ) \uf04b b1,y (t , qdz ) b1,z (t , dz ) \uf04b b1,z (t , qdz )}T3q\uf0b41 42 (3.59) \fChapter 3. Dynamics of Peripheral Milling B 2, j (t ) \uf03d g(\uf066 j (t )) \uf0b4 \uf04c b2,xy (t , dz ) 0 \uf04c b2,xz (t , dz ) 0 \uf04c \uf0e9b2,xx (t , dz ) 0 \uf0f9 \uf0ea \uf0fa 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf04d 0 b2,xx (t , qdz ) \uf04d 0 b2,xy (t , qdz ) \uf04d 0 b2,xz (t , qdz ) \uf0fa \uf0ea \uf0fa \uf0eab2,yx (t , dz ) 0 \uf0fa \uf04c b2,yy (t , dz ) 0 \uf04c b2,yz (t , dz ) 0 \uf04c \uf0ea \uf0fa 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf04d 0 b2,yx (t , qdz ) \uf04d 0 b2,yy (t , qdz ) \uf04d 0 b2,yz (t , qdz ) \uf0fa \uf0ea \uf0fa \uf0ea b2, zx (t , dz ) 0 \uf0fa \uf04c b2,zy (t , dz ) 0 \uf04c b2,zz (t , dz ) 0 \uf04c \uf0ea \uf0fa 0 \uf04f \uf04d 0 \uf04f \uf04d 0 \uf04f \uf04d \uf0ea \uf0fa \uf0ea \uf0fa \uf04d 0 b ( t , qdz ) \uf04d 0 b ( t , qdz ) \uf04d 0 b ( t , qdz ) 2,zx 2,zy 2,zz \uf0eb \uf0fb . 3q\uf0b43q (3.60) The diagonal unit step function matrix is defined as: g (\uf066 j (t )) \uf03d 0 \uf0e9 g (\uf066 j (t , dz )) 0 \uf0ea 0 \uf04f 0 \uf0ea \uf0ea 0 0 g (\uf066 j (t , qdz )) \uf0ea \uf0ea \uf0ea 0 \uf0ea \uf0ea \uf0ea \uf0ea \uf0ea 0 \uf0ea \uf0ea \uf0eb 0 g (\uf066 j (t , dz )) 0 0 0 \uf04f 0 0 0 g (\uf066 j (t , qdz )) 0 \uf0f9 \uf0fa 0 \uf0fa \uf0fa \uf0fa \uf0fa \uf0fa 0 \uf0fa \uf0fa \uf0fa \uf0fa g (\uf066 j (t , dz )) 0 0 \uf0fa . 0 \uf04f 0 \uf0fa \uf0fa 0 0 g (\uf066 j (t , qdz )) \uf0fb 3q\uf0b43q (3.61) The workpiece and tool equations are coupled to each other due to relative regenerative displacement and velocity terms in the force function. The Eqs. (3.55) and (3.56) are combined by expanding the matrices: \uf026\uf026 (t ) \uf0ef\uf0fc \uf0e9 2\u03b6 t \u03c9 n,t \uf0ef\uf0ec \uf047 t \uf0ed \uf026\uf026 \uf0fd\uf02b \uf0ea \uf047 \uf0ef\uf0ee wp (t ) \uf0ef\uf0fe \uf0eb 0 2 0 \uf0f9 \uf0ef\uf0ec \uf047\uf026 t (t ) \uf0ef\uf0fc \uf0e9\u03c9 n,t \uf0ed \uf0fd\uf02b \uf0ea 2\u03b6 wp\u03c9 n,wp \uf0fb\uf0fa \uf0ef\uf0ee\uf047\uf026 wp (t ) \uf0ef\uf0fe \uf0ea 0 \uf0eb 0 \uf0f9 \uf0ec \uf047 t (t ) \uf0fc \uf0fa\uf0ed \uf0fd \u03c9 2n,wp \uf0fa\uf0fb \uf0ee\uf047 wp (t ) \uf0fe , (3.62) \uf0e9 U\u02c6 tT \uf0f9 N \uf0ec\uf0ef \uf03d \uf0ea T \uf0fa \uf0e5 \uf0ed A1, j (t ) \uf02b A 2, j (t )\uf044\u03a8 e (t ) \uf02b B1, j (t ) \uf02b B 2, j (t ) \uf0e9\uf0ebU\u02c6 t \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb j \uf03d1 \uf0ef\uf0ee 43 \uf0ec\uf0ef \uf047\uf026 t (t ) \uf0ef\uf0fc\uf0fc\uf0ef U\u02c6 wp \uf0f9\uf0fb \uf0ed \uf026 \uf0fd\uf0fd dz \uf0ef\uf0ee\uf047 wp (t ) \uf0ef\uf0fe\uf0ef\uf0fe \fChapter 3. Dynamics of Peripheral Milling where expanded relative regeneration term is defined similar to Eq. (3.53): \uf0ec \uf0ec \uf044\uf047 t (t ) \uf0fc \uf0ef \uf0e9\uf0ebU\u02c6 t U\u02c6 wp \uf0f9\uf0fb \uf0ed \uf0fd \uf044\uf047 wp (t ) \uf0fe \uf0ee \uf0ef \uf044\u03a8 e (t ) \uf03d \uf0ed \uf047 (t ) \uf0fc \uf0ef\uf0e9 \u02c6 \u02c6 \uf0f9\uf0ec t \uf0e9\u02c6 \uf0ef \uf0ebU t U wp \uf0fb \uf0ed\uf047 (t ) \uf0fd \uf02d \uf0ebU t \uf0ee wp \uf0fe \uf0ee uniform helix \uf0ef\uf0ec \uf047 t (t \uf02d T j ,r ) \uf0ef\uf0fc U\u02c6 wp \uf0f9\uf0fb \uf02a \uf0ed \uf0fd non-uniform helix \uf0ef\uf0ee\uf047 wp (t \uf02d T j ,r ) \uf0ef\uf0fe . (3.63) Eq. (3.62) gives the equations of motion of peripheral milling in modal domain. It is simplified as: \uf026\uf026 (t ) \uf02b 2\u03b6\u03c9 \uf047\uf026 (t ) \uf02b \u03c9 2 \uf047(t ) \uf047 n n , (3.64) \uf0e9 U\u02c6 tT \uf0f9 \uf03d \uf0ea T \uf0fa ( f cs (t ) \uf02b f1cd (t )\uf047(t ) \uf02d f 2cd (t )\uf047 T (t ) \uf02b f es (t ) \uf02b f ed (t )\uf047\uf026 (t )) dz \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb where the expanded displacement matrix is \uf047(t ) \uf03d {\uf047 t (t ) \uf047 wp (t )}T . The static cutting force component is: N f cs (t ) \uf03d \uf0e5 A1, j (t ) . (3.65) j \uf03d1 The dynamic cutting force component due to the current vibrations is: \uf0e6 N f1cd (t ) \uf03d \uf0e7 \uf0e5 A 2, j (t ) \uf0e9\uf0ebU\u02c6 t \uf0e7 j \uf03d1 \uf0e8 \uf0f6 U\u02c6 wp \uf0f9\uf0fb \uf0f7 , \uf0f7 \uf0f8 (3.66) The dynamic cutting force component due to the previous vibrations is defined for three cases: \uf0ec \uf0ef cd uniform helix and pitch \uf0ef f\u02c62 (t )\uf047 (t \uf02d T ) \uf0efN \uf0ef f 2cd (t )\uf047 T (t ) \uf03d \uf0ed\uf0e5 f\u02c62,cdj (t ) \uf047(t \uf02d T j ) uniform helix, non-uniform pitch , (3.67) \uf0ef j \uf03d1 \uf0efN q \uf0ef f\u02c62,cdj ,r (t )\uf047 (t \uf02d T j ,r ) non-uniform helix and pitch \uf0e5\uf0e5 \uf0ef j \uf03d1 r \uf03d1 \uf0ee 44 \fChapter 3. Dynamics of Peripheral Milling where f\u02c62,cdj ,r (t ) \uf03d A2, j (t ) \uf0e9\uf0ebU\u02c6 t U\u02c6 wp \uf0f9\uf0fb . (3.68) r The expanded modal matrix for axial element r is: \uf0e9U\u02c6 t \uf0eb 0 \uf04c \uf0e9 0 \uf0eau \uf0ea xr,1,t \uf04c uxr,mt ,t \uf0ea 0 0 0 \uf0ea 0 0 \uf0ea 0 \uf0ea U\u02c6 wp \uf0f9\uf0fb \uf03d \uf0eau yr,1,t \uf04c u yr,mt ,t r \uf0ea 0 0 0 \uf0ea 0 0 \uf0ea 0 \uf0eau \uf0ea zr,1,t \uf04c uzr,mt ,t \uf0ea 0 0 0 \uf0eb \uf0f9 \uf04c uxr ,mwp ,wp \uf0fa\uf0fa \uf0fa 0 0 \uf0fa 0 0 \uf0fa \uf04c u yr ,mwp ,wp \uf0fa\uf0fa . \uf0fa 0 0 \uf0fa 0 0 \uf0fa \uf04c uzr ,mwp ,wp \uf0fa\uf0fa \uf0fa 0 0 \uf0fb 3q\uf0b4( mt \uf02b mwp ) \uf04c \uf04c uxr ,1,wp 0 0 u yr ,1,wp 0 0 uzr ,1,wp 0 0 (3.69) q The other dynamic cutting force components are calculated as f\u02c62,cdj (t ) \uf03d \uf0e5 f\u02c62,cdj ,r (t ) and r \uf03d1 N f\u02c62cd (t ) \uf03d \uf0e5 f\u02c62,cdj (t ) . j \uf03d1 The static edge force component is: N f es (t ) \uf03d \uf0e5 B1, j (t ) . (3.70) j \uf03d1 The dynamic edge force or process damping force component is: f ed \uf0e6 N (t ) \uf03d \uf0e7 \uf0e5 B2, j (t ) \uf0e9\uf0ebU\u02c6 t \uf0e7 j \uf03d1 \uf0e8 45 \uf0f6 U\u02c6 wp \uf0f9\uf0fb \uf0f7 . \uf0f7 \uf0f8 (3.71) \fChapter 4. Time Domain Simulation 4. Time Domain Simulation 4.1 Introduction The methods to solve equations of motion for the peripheral milling process are dis- cussed in this chapter. Semi discretization method proposed by Insperger and Stepan [24] for delayed differential equations with periodic coefficients is extended by including the static and process damping forces for the time domain simulation. Moreover, for stable cutting conditions, surface location error method proposed by Schmitz [10] is used accounting the steady state response of the differential equation. 4.2 Semi Discrete Time Domain Solution The equations of motion derived in previous chapter are delayed differential equ- ations (DDE) with periodic coefficients, where the solution cannot be given in closed form [53]. In order to solve this type of equations, researchers have considered true kinematics model in time domain where they included inner and outer modulations by keeping the past and current surface data and considering the nonlinearities [7] [54]. This model is then solved employing explicit Runge-Kutta methods. However, storing all the geometric data is time consuming. Furthermore, stability of the time stepping scheme and order of accuracy enforce programmers to take small time steps, which adds on to the computation time. Insperger and Stepan [24] proposed semi discretization (SD) method, where they solved the stability of the linear milling equation semi analytically by just taking regenerative force term into account. The objective of SD technique is to approximate delayed and time dependent spatial terms at each discrete time step by piecewise constant values, while keeping actual time domain terms in the original form [53]. In this section, DDE representing the peripheral milling dynamics is solved using time domain SD method by considering all reported force components, while disregarding nonlinearities like runout, tool jump and change of radial immersion due to deflections. 46 \fChapter 4. Time Domain Simulation The modal equation of motion in time domain was derived in section 3.3: \uf026\uf026 (t ) \uf02b 2\u03b6\u03c9 \uf047\uf026 (t ) \uf02b \u03c9 2 \uf047(t ) \uf047 n n (4.1) \uf0e9 U\u02c6 tT \uf0f9 \uf03d \uf0ea T \uf0fa ( f cs (t ) \uf02b f1cd (t )\uf047(t ) \uf02d f 2cd (t )\uf047 T (t ) \uf02b f es (t ) \uf02b f ed (t )\uf047\uf026 (t )) dz \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb The equation is transformed to state space form by dropping the order of DDE one and doubling the number of equations: \uf026 (t ) \uf03d L(t )\u0398(t ) \uf02b R(t )\u0398 (t ) \uf02b S(t ) , \u0398 T (4.2) where the motion vector is: \uf0ec\uf047(t ) \uf0fc , \u0398(t ) \uf03d \uf0ed \uf0fd \uf0ee\uf047\uf026 (t ) \uf0fe2( mt \uf02b mwp )\uf0b41 (4.3) the system matrix is: 0 \uf0e9 \uf0ea T \uf0f6 L(t ) \uf03d \uf0ea\uf0e6 2 \uf0e9 U\u02c6 t \uf0f9 cd \uf0e7 \uf0f7 \uf0ea \uf0fa \uf02d \u03c9 \uf02b f ( t ) dz \uf0ea\uf0e7 n 1 T \u02c6 \uf0f7 U \uf0ea\uf0eb wp \uf0fa\uf0fb \uf0ea\uf0eb\uf0e8 \uf0f8 I \uf0e6 \uf0e9 U\u02c6 T \uf0f9 \uf0e7 \uf02d2\u03b6\u03c9 n \uf02b \uf0ea t \uf0fa T \uf0e7 \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb \uf0e8 \uf0f9 \uf0fa \uf0f6\uf0fa , (4.4) ed f (t )dz \uf0f7 \uf0fa \uf0f7\uf0fa \uf0f8 \uf0fb 2( mt \uf02b mwp )\uf0b42( mt \uf02b mwp ) the static force matrix: 0 \uf0ec \uf0fc \uf0ef\uf0ef \u02c6 T \uf0ef\uf0ef S(t ) \uf03d \uf0ed \uf0e9 U t \uf0f9 cs . \uf0fd es \uf0ef \uf0ea \u02c6 T \uf0fa ( f (t ) \uf02b f (t ))dz \uf0ef \uf0ee\uf0ef \uf0ea\uf0ebU wp \uf0fa\uf0fb \uf0fe\uf0ef2( mt \uf02b mwp )\uf0b41 47 (4.5) \fChapter 4. Time Domain Simulation The product of retarded matrix and delayed motion vector has 3 different forms: \uf0ec \uf0ef \u02c6 (t )\u0398(t \uf02d T ) uniform helix and pitch \uf0efR \uf0efN \uf0ef \u02c6 R (t )\u0398T (t ) \uf03d \uf0ed\uf0e5 R uniform helix, non-uniform pitch j (t )\u0398(t \uf02d T j ) j \uf03d 1 \uf0ef \uf0efN q \uf0ef \u02c6 (t )\u0398(t \uf02d T ) non-uniform helix and pitch R j ,r j ,r \uf0ef \uf0e5\uf0e5 \uf0ee j \uf03d1 r \uf03d1 (4.6) \uf0ec 0 0\uf0f9 \uf0e9 \uf0ef \uf0ea \uf0fa \uf0ec\uf047 (t \uf02d T ) \uf0fc T \uf0ef \uf0ea \uf0e9 U\u02c6 t \uf0f9 \u02c6 cd \uf0fa\uf0ed\uf026 \uf0fd \uf0ef \uf0ea \uf02d \uf0ea \u02c6 T \uf0fa f 2 (t )dz 0 \uf0fa \uf0ee\uf047 (t \uf02d T ) \uf0fe \uf0ef \uf0eb\uf0ea \uf0ea\uf0ebU wp \uf0fa\uf0fb \uf0fb\uf0fa \uf0ef 0 0\uf0f9 \uf0ef N \uf0e9 \uf0ea \uf0fa \uf0ec\uf0ef\uf047 (t \uf02d T j ) \uf0fc\uf0ef \uf0ef T \uf03d \uf0ed \uf0e5 \uf0ea \uf0e9 U\u02c6 t \uf0f9 \u02c6 cd \uf0fa\uf0ed\uf026 \uf0fd , \uf0ea \uf0fa \uf02d f ( t ) dz 0 \uf047 ( t \uf02d T ) 2, j j \uf0ef \uf0ef\uf0fe j \uf03d 1 \uf0ea \u02c6T \uf0fa\uf0ee \uf0ef U \uf0ea \uf0fa wp \uf0ea \uf0fa \uf0eb \uf0fb \uf0ef \uf0eb \uf0fb \uf0ef 0 0\uf0f9 \uf0efN q \uf0e9 \uf0ea \uf0fa \uf0ec\uf0ef\uf047 (t \uf02d T j ,r ) \uf0fc\uf0ef \uf0ef \uf0e9 U\u02c6 tT \uf0f9 \uf0ea \uf0fa\uf0ed\uf026 \uf0fd \uf0e5\uf0e5 cd \u02c6 \uf0ef \uf047 (t \uf02d T j ,r ) \uf0fe\uf0ef \uf0ef j \uf03d1 r \uf03d1 \uf0ea \uf02d \uf0ea \u02c6 T \uf0fa f 2, j (t ) dz 0 \uf0fa \uf0ee \uf0ef \uf0ea\uf0eb \uf0ea\uf0ebU wp \uf0fa\uf0fb \uf0fa\uf0fb \uf0ee where q f\u02c62cd , f\u02c62,cdj , f\u02c62,cdj ,r are discussed in section 3.3. The most general case N \uf0e5\uf0e5 R\u02c6 j,r (t )\u0398(t \uf02d T j,r ) is considered for the following equations. r \uf03d1 j \uf03d1 48 \fChapter 4. Time Domain Simulation Figure 4-1: Approximation of the delayed term by discretization SD method starts the formulation by discretizing the delay term into finite number of points as illustrated in Figure 4-1. The time interval is expressed as: ti \uf03d i\uf044t , i \uf03d 0,1, 2,\uf04b, k . (4.7) The time step \uf044t is a significant factor that affects the accuracy of the numerical solution. Two considerations need to be present when transforming the system to discrete domain. The time step should be small enough to capture all vibrations without having the aliasing problem: \uf044t \uf03c \uf070 \uf077n,max , (4.8) where \uf077n,max is the highest natural frequency of the structure. Moreover, the time varying modal forces or modal force components should be sufficiently represented in discrete domain. Figure 4-2 compares the effect of different number of discretization points on the time varying modal force component. Sudden changes, which mainly occur 49 \fChapter 4. Time Domain Simulation in intermittent cutting conditions, cannot be captured by the method, if there is no point to represent it. This brings sizable numerical error to the solution as discussed in [55]. x 10 6 k=24 k=60 k=119 Directional Modal Force Component] 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 60 65 70 75 80 Cutter Rotation Angle [deg] 85 90 Figure 4-2: Discretization of an element in dynamic modal force component matrix with different number of points The lower limit of the time step is confined by simulation accuracy and the upper limit is confined by computational limitations. The number of arithmetic operations is roughly proportional to 1 ; therefore compromise needs to be made between order of the error \uf044t 4 and the simulation time. The integer number of discretized points is calculated as discussed in [29]: k \uf03d int(Tmax \/ \uf044t \uf02b p \/ 2) , (4.9) where Tmax is the maximum delay in the system, p is the polynomial order of the approximation. It has been reported that the order of approximation of delayed term affects the convergence to exact solution [56][57]. Moreover, when non-uniform helix and pitch cutters are concerned, the delay term can be in between two discretized points. 50 \fChapter 4. Time Domain Simulation In order to increase solution\u2019s accuracy first order interpolation of the delay is reported to be convenient [56], which has the form: \u0398(t \uf02d T j ,r ) \uf03d t \uf02d T j ,r \uf02d (i \uf02d k j ,r )\uf044t \uf044t \u0398(ti \uf02dk j ,r \uf02b1 ) \uf02d t \uf02d T j ,r \uf02d (i \uf02b 1 \uf02d k j ,r )\uf044t \uf044t \u0398(ti \uf02dk j ,r ) , (4.10) where k j ,r \uf03d int(T j ,r \/ \uf044t \uf02b p \/ 2) is mapping the delayed term coming from particular tooth and axial elevation for 2 \uf0a3 k j ,r \uf0a3 k . SD method approximates time dependent matrices in each interval as constant values: 1 L(ti ) \uf03d \uf044t ti \uf02b1 \uf0f2 ti \u02c6 (t ) \uf03d 1 L(t )dt \uf03d Li , R j ,r i \uf044t ti \uf02b1 \uf0f2 ti 1 \u02c6 (t )dt \uf03d R \u02c6 R j ,r j ,r ,i , S(ti ) \uf03d \uf044t ti \uf02b1 \uf0f2 S(t )dt \uf03d Si . ti (4.11) The differential equation in Eq. (4.2) can be treated as constant ordinary differential equation within the interval, ti \uf0a3 t \uf0a3 ti \uf02b1 , when the general solution is attained: \uf0e9\uf0e6 N q \uf0f9 \uf0f6 \u02c6 \u0398(0) \uf02b \uf0f2 eLi .(t \uf02d\uf074) \uf0ea\uf0e7 \uf0e5\uf0e5 R \u0398 ( \uf074 \uf02d T ) \uf02b S \uf0f7 j ,r \uf0f7 i \uf0fa d\uf074 . \uf0e7 j \uf03d1 r \uf03d1 j ,r ,i \uf0ea \uf0fa\uf0fb 0 \uf0f8 \uf0eb\uf0e8 t \u0398(t ) \uf03d e Li .t (4.12) Eq. (4.10) is substituted into Eq. (4.12) and it is assumed that within the interval, \u02c6 ti \uf0a3 t \uf0a3 ti \uf02b1 , L\uf02di 1 , exists and R j ,r ,i stays constant. The following equation is obtained when \u0398(ti \uf02b1 ) is written in terms of \u0398(ti ) , similar to Insperger and Stepan [24]: \uf0e6 N q \uf0f6 \uf0e6 N q \uf0f6 \u0398(ti \uf02b1 ) \uf03d Ci \u0398(ti ) \uf02b \uf0e7 \uf0e5\uf0e5 Di ,0, j ,r \u0398(ti \uf02dk j ,r \uf02b1 ) \uf0f7 \uf02b \uf0e7 \uf0e5\uf0e5 Di ,1, j ,r \u0398(ti \uf02dk j ,r ) \uf0f7 \uf02b Ei (4.13) \uf0e7 j \uf03d1 r \uf03d1 \uf0f7 \uf0e7 j \uf03d1 r \uf03d1 \uf0f7 \uf0e8 \uf0f8 \uf0e8 \uf0f8 where Ci \uf03d eLi .\uf044t , (4.14) 1 \uf0ec \uf0fc\u02c6 Di ,0, j ,r \uf03d \uf0edLi \uf02d1 \uf02b \uf0e9\uf0ebLi \uf02d2 \uf02d (T j ,r \uf02d (k j ,r \uf02d 1)\uf044t )Li \uf02d1 \uf0f9\uf0fb (I \uf02d eLi .\uf044t ) \uf0fd R j ,r ,i , \uf044t \uf0ee \uf0fe (4.15) 51 \fChapter 4. Time Domain Simulation 1 \uf0ec \uf0fc\u02c6 Di ,1, j ,r \uf03d \uf0ed\uf02dLi \uf02d1 \uf02b \uf0e9\uf0eb\uf02dLi \uf02d2 \uf02b (T j ,r \uf02d k j ,r \uf044t )Li \uf02d1 \uf0f9\uf0fb (I \uf02d eLi .\uf044t ) \uf0fd R j ,r ,i , \uf044t \uf0ee \uf0fe (4.16) Ei \uf03d (eLi .\uf044t \uf02d I)Li \uf02d1Si . (4.17) For non-uniform pitch and uniform helix, and uniform pitch and helix cutters, Di ,e, j and Di ,e for e \uf03d 0,1 are introduced, respectively. The terms are defined as q N r\uf03d1 j\uf03d1 Di ,e, j \uf03d \uf0e5 Di ,e, j ,r and Di ,e \uf03d \uf0e5 Di ,e, j . The Eq. (4.14) is mapped using the form \u0398i \uf02b1 \uf03d Zi\u0398i \uf02b Wi : \uf0ec \u0398(ti \uf02b1 ) \uf0fc \uf0e9Ci 0 0 Di ,0, j ,r \uf0ef \u0398(t ) \uf0ef \uf0ea \uf04c i \uf0ef \uf0ef \uf0eaI 0 0 \uf0ef \uf0ef \uf0ea0 I 0 . \uf04c \uf0ef \uf0ef \uf0ea . \uf04d \uf0ed \uf0fd\uf03d\uf0ea \uf04d \uf04c \uf04f \uf0ef \uf0ef \uf0ea\uf04d . \uf04d \uf04c \uf04f \uf0ef \uf0ef \uf0ea . \uf0ef \uf0ef \uf0ea0 0 0 \uf04c \uf0ef\u0398(t \uf0ef \uf0ea ) \uf04c \uf0ee i \uf02d( k j ,r \uf02d1) \uf0fe \uf0eb 0 0 0 Di ,1, j ,r \uf04c \uf04c\uf0f9 \uf0ec \u0398(ti ) \uf0fc \uf0ecEi \uf0fc \uf0ef \uf0ef \uf0ef \uf0fa\uf0ef . \uf04c 0 0 \uf0fa\uf0ef \uf0ef \uf0ef0\uf0ef \uf0ef \uf0ef0\uf0ef . \uf04c 0 0 \uf0fa\uf0ef \uf0fa \uf0ef\uf0ef \uf0ef\uf0ef \uf0ef \uf0ef . \uf04c \uf04c \uf04c\uf0fa \uf0ed \uf0fd\uf02b\uf0ed . \uf0fd, \uf0ef \uf0ef. \uf0ef . \uf04d \uf04d \uf04d \uf0fa \uf0ef\uf0ef \uf0ef \uf0ef \uf0ef \uf0fa I 0 0 \uf0fa \uf0ef\u0398(ti \uf02d( k j ,r \uf02d1) ) \uf0ef \uf0ef . \uf0ef \uf0ef \uf0ef \uf0ef \uf0ef 0 I 0 \uf0fa\uf0fb \uf0ef\uf0ee \u0398(ti \uf02dk j ,r ) \uf0ef\uf0fe \uf0ee 0 \uf0fe (4.18) where given the process mechanics and information on past vibrations, the solution of next time step can be calculated. For uniform pitch and helix cutters, there is only one delay. The matrices associated with the delayed term are located at the first row, k th and (k \uf02b 1) th columns of the transition matrix. When multiple delays exist due to non-uniform helix and pitch angles, the delayed matrices are accumulated at first row and corresponding k j ,r th and (k j ,r \uf02b 1) th columns. SD solution requires 2k (mt \uf02b mwp ) number of initial conditions to start the time domain simulation. Similar to models considering the true kinematics, at the beginning, 52 \fChapter 4. Time Domain Simulation all initial conditions are set to zero. For the first iterations, the only excitation component affecting the solution is forced vibration term Ei . Then algorithm starts to assign previous solutions obtained to corresponding delay elements of the motion vector \u0398i , thus regenerative vibration term becomes effective. After solving for the modal displacements and velocities for determined time interval, the physical displacements for nodes that are in cut are calculated by using transformation from modal space to physical space as: \u02c6 \uf0ec \uf0efQ(t ) \uf0fc \uf0ef \uf0ed \u02c6 \uf0fd \uf03d \uf0e9\uf0ebU\u02c6 t \uf026 \uf0ef \uf0eeQ(t ) \uf0ef \uf0fe U\u02c6 wp \uf0f9\uf0fb \u0398(t ) . (4.19) The size of the coefficient matrix Zi is 2(k \uf02b 1)(mt \uf02b mwp ) \uf0b4 2(k \uf02b 1)(mt \uf02b mwp ) .In order to calculate the next time step, multiplication of first row and the motion vector is sufficient. Moreover, the velocity terms in retarded matrix which have no effect on the solution, can be eliminated and the transition matrix size can be reduced to (k \uf02b 2)(mt \uf02b mwp ) \uf0b4 (k \uf02b 2)(mt \uf02b mwp ) as discussed in [58]. Some other efficient calcula- tion methods for SD algorithm had been proposed by Henninger and Eberhard [55]. After finding the velocities and displacements for each node, differential dynamic force can be calculated by using Eq. (3.18), and differential process damping force can be obtained from Eq. (3.27). 4.2.1 SD Time Domain Solution Example The SD time domain solution is coded for different types of end mills and validi- ty of the method has been analyzed by comparing it to commercial software (CUTPRO). Standard cutter\u2019s dynamics is tested for stable and unstable cutting cases. The multi point mechanics described in the previous sections is lumped into a single point to exactly mimic the commercial software. The methodology of single point approach is explained in the Appendix A. 53 \fChapter 4. Time Domain Simulation Among the three different cutter types tested, standard uniform pitch and uniform helix cutter example is chosen from Turner et al. [59] due to good experimental match with the simulations. In Table 4-1, the dimensions of the cutter are specified. Table 4-1: Parameters for regular cylindrical end mill Tool Diameter Teeth [mm] ST3 16 Helix 1 Helix 2 Pitch 1 Pitch 2 [deg] [deg] [deg] [deg] 30 30 90 90 4 The modal parameters given in Table 4-2 are experimentally identified for the tool tip in x and y directions, while assuming that the workpiece is considerably rigid compared to cutter. Three modes in each direction are extracted. The axisymmetric geometry of the cutter brings similar frequency modes with mode shape vectors in different directions. It is assumed that the modes in each direction are orthogonal to each other, hence no cross talking exists between the directions. When each x, y and z direction mass normalized mode shape value for the tool tip is placed side by side, lumped mode shape matrix (modal matrix) is constructed: 0 0 0 \uf0f6 \uf0e6 3.543 1.512 1.303 \uf028 \uf0e7 \uf0f7 \uf0e9\uf0ebU t \uf0f9\uf0fb \uf03d 0 0 3.653 1.616 1.483 \uf0f7 . \uf0e7 0 tip \uf0e7 0 0 0 0 0 0 \uf0f7\uf0f83\uf0b46 \uf0e8 (4.20) Table 4-2: Modal parameters of the regular cylindrical end mill Tool and Direction ST3 (x) ST3 (y) Natural Frequency Effective Stiffness [Hz] [N\/m] 2062, 2400, 2956 1.337e7, 9.944e7, 0.0206, 0.0183, 2.031e8 0.0151 1.259e7, 8.624e7, 0.0199, 0.0175, 1.57e8 0.0150 2063, 2388, 2957 54 Damping Ratio \fChapter 4. Time Domain Simulation The material is Al 6061 where mechanistically determined cutting force coeffic \uf03d {160 400 0}T N\/mm2 and edge force coefficient vector is cient vector is K rta e K rta \uf03d {30 26 0}T N\/mm . The process damping is neglected for the selected speeds. The radial immersion is set as 5 mm. The stability chart is for low immersion down milling in Figure 4-3. In order to analyze stable and unstable behavior, 7 mm axial depth of cut (ADOC) with 7500 rpm spindle speed and 7 mm ADOC with 6800 rpm are simulated. The displacement responses in Figure 4-3c,e show that at stable cutting condition CUTPRO and SD results match well. The difference between the results are attributed to circular motion, no runout and change of radial immersion assumptions made by SD method. As vibration magnitude - radial immersion ratio becomes higher; the discrepancy becomes apparent as demonstrated in Figure 4-3b, d. When the system chatters, SD result grows quicker compared to true kinematics model. In CUTPRO simulation, vibrations directly affect the radial immersion, hence the cutting forces, so that the system tries to damp out the grow rate of the oscillations as illustrated in Figure 4-4a. It can be concluded that SD time domain simulation works accurately at less oscillatory stable cutting conditions, and determining the chatter, but it fails at cases where large vibration displacements occur. 55 \fChapter 4. Time Domain Simulation a) X Direction Displacement Displacement [mm] Displacement [mm] X Direction Displacement 0.3 Semi Discretization CUTRPO 0.2 0.1 0 -0.1 -0.2 0 0.005 0.01 0.015 0.02 0.03 0.02 0.01 0 -0.01 0.025 0 c) 0.2 0.1 0 -0.1 -0.2 0.01 0.015 0.02 0.02 0.025 Y Direction Displacement Displacement [mm] Displacement [mm] 0.015 b) Y Direction Displacement 0.005 0.01 Time [s] 0.3 0 0.005 Time [s] 0.025 0.06 0.04 0.02 0 0 0.005 0.01 0.015 Time [s] Time [s] d) e) 0.02 0.025 Figure 4-3: a) Stability chart for standard cutter [59] (with the permission of D. Merdol) b)-d) time domain simulation in x and y directions respectively for 7 mm ADOC and 6800 rpm c)-e) simulations in x and y directions respectively for 7 mm ADOC and 7500 rpm 56 \fChapter 4. Time Domain Simulation Resultant Force on XY Plane Resultant Force on XY Plane 2500 600 Semi Discretization CUTPRO 500 Force [N] Force [N] 2000 1500 1000 500 0 0 400 300 200 100 0.005 0.01 0.015 Time [s] 0.02 0 0 a) 0.02 Time [s] 0.04 b) Figure 4-4: Resultant cutting forces on x-y plane a) unstable cutting b) stable cutting Frequency spectrums of both cases give insight about the modes that cause chatter, and also checks semi discretization method\u2019s validity at determining unstable poles. Figure 4-5a shows the unstable x direction force\u2019s spectrum, and due to difference between force amplitudes, it is presented in logarithmic scale. The main chatter frequency at 2119 Hz and its sub-super harmonics due to tooth passing frequency (453 Hz), which are 1666 Hz, 2569 Hz and 3022 Hz, are identified similarly in both methods. In stable cutting condition as illustrated in Figure 4-5b, the tooth passing frequency at 500 Hz and its harmonics are the main excitation sources. 57 \fChapter 4. Time Domain Simulation FFT of X Direction Force 10 10 6 2119 Hz 2569 Hz 1666 Hz 453 Hz 3022 Hz 5 FFT of Resultant Force Semi Discretization CUTPRO 6 Force [N] Force [N] 10 8 x 10 4 2 4 500 Hz 2 1000 Hz 0 0 1000 2000 3000 4000 Frequency [Hz] a) 0 500 1000 1500 2000 Frequency [Hz] 2500 b) Figure 4-5: FFT of force data a) unstable cutting b) stable cutting In summary, it is noted that the SD accurately solves the response in stable cutting conditions. The advantage of semi discrete time domain solution is that it calculates the whole response by just doing one matrix multiplication as given in Eq. (4.19). If the number of degrees of freedom is high, the simulation time does not increase excessively as it happens in current true kinematics models. On the other hand, if the user needs the accurate behavior where nonlinearities come in, current time domain models should be used. 4.3 Surface Location Error Forced vibrations and static deflections may occur during the stable cutting process. The cutter deviates from the intended surface finish and leaves undercut or overcut regions called the surface location error (SLE) as illustrated in Figure 4-6. This problem necessitates additional finishing passes, or may even cause to scrap the whole part due to tolerance limitations. 58 \fChapter 4. Time Domain Simulation Figure 4-6: Surface location error a) undercut example b) overcut example The surface form errors had been solved in time domain with exact kinematics model as presented in [8][54]. Schmitz [10] extracted the SLE from the steady state response by solving the differential equations in frequency domain. This eliminated lengthy convolution operation used in time domain models. In this section SLE is formulated according to the differential equation derived for the peripheral milling process in chapter 3. The flowchart is demonstrated in Figure 4-7. 59 \fChapter 4. Time Domain Simulation Input static cutting and edge forces (which includes cutting force coefficients and cutting conditions), f cs (t ) , f es (t ) , structural T parameters of tool and workpiece, \u03b6, \u03c9n , U\u02c6 tT ,U\u02c6 wp , set spindle speed (varying) Express forces f cs , f es using Fourier series to transform them to frequency domain. Solve for Fourier coefficients either numerically or analytically. Change Express modal dynamics in frequency domain via transfer spindle function M(i\uf077) . speed, n Calculate the displacement by taking the product of force and transfer function in magnitude-phase form. Write the solution for displacement using Fourier series. Express the displacement in physical domain by taking modal space transformation and find the displacements for each point when \uf066 j (t , z ) equals to 0 (up milling) or \u03c0 (down milling). Store the SLE value for corresponding axial elevation. Figure 4-7: Flowchart of SLE method In stable cutting, the dynamic forces are assumed to have no influence. The peripheral milling equation becomes constant coefficient ordinary differential equation: \uf026\uf026 (t ) \uf02b 2\u03b6\u03c9 \uf047\uf026 (t ) \uf02b \u03c92 \uf047(t ) \uf047 n n \uf0e9 U\u02c6 tT \uf0f9 \uf03d \uf0ea T \uf0fa ( f cs (t ) \uf02b f es (t ))dz . \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb 60 (4.21) \fChapter 4. Time Domain Simulation The time dependent components of the force are expressed by exponential Fourier series in order to have compact form: h \uf03d\uf0a5 \uf0e9 U\u02c6 tT \uf0f9 ih\uf077 t cs es s \uf0ea T \uf0fa ( f (t ) \uf02b f (t ))dz \uf03d f (t ) \uf03d \uf0e5 f hs e p \u02c6 \uf0ea\uf0ebU wp \uf0fa\uf0fb h \uf03d\uf02d\uf0a5 , (4.22) where h is the index of harmonics and i is the imaginary unit. The Fourier coefficients are defined as: \uf074 \uf0e9 \u02c6T \uf0f9 Ut 1 -ih\uf077 t f hs \uf03d dz \uf0f2 \uf0ea T \uf0fa ( f cs (t ) \uf02b f es (t ))e p dt . \u02c6 \uf074 0 \uf0eaU wp \uf0fa \uf0eb \uf0fb (4.23) The least common period, \uf074 , and frequency, \u03c9p , for different cutter geometries are defined in Table 4-3. Table 4-3: Least common period and frequency for different cutters Pitch and Helix Angles Least common period Least common fre- \uf074(sec) quency \uf077 p (rad \/ s) Uniform pitch and helix angle 60 Nn \uf077p \uf03d 2\uf070N n 60 60 ( N \/ \uf065) n \uf077p \uf03d 2\uf070N n 60\uf065 60 n \uf077p \uf03d 2\uf070 \uf074\uf03d Alternating pitch and\/or alternating helix angle i.e. \uf074\uf03d \uf066p,1 \uf03d \uf061, \uf066p,2 \uf03d \uf062, \uf066p,3 \uf03d \uf061, \uf066p,4 \uf03d \uf062 \u03b5 : number of different angles Random pitch and\/or helix angle, \uf074\uf03d axially changing helix angle n 60 The rotation angle is dependent on the spindle speed and time, thus change of variable is applied to the integration. Since \uf066(t ) \uf03d redefined as: 61 2\uf070n 2\uf070n t and d \uf066 \uf03d dt , Eq. (4.23) is 60 60 \fChapter 4. Time Domain Simulation f hs 1 \uf03d dz \uf077s \uf074 \uf077p -ih \uf066 U\u02c6 tT \uf0f9 cs es \uf0ea T \uf0fa ( f (\uf066) \uf02b f (\uf066))e \uf077s d \uf066 , \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb \uf077s \uf074 \uf0e9 \uf0f2 0 (4.24) where \uf066 is the rotation angle, \uf077s \uf03d 2\uf070n \/ 60 is the spindle frequency, and \uf077s \uf074 is the angle corresponding to one period of the forcing function. For complex tools, such as variable pitch, variable helix and non cylindrical cutters, obtaining coefficients via Fast Fourier Transform (FFT) will be faster compared to analytical solution as number of mathematical operations is small. In this section both numerical and analytical approaches are considered. Existence of switching function g(\uf066 j (t )) makes it necessary to redefine the Fourier series for the analytical solution. Eq. (4.23) has the form similar to Schmitz [10]: \uf0e9 U\u02c6 tT \uf0f9 N h\uf03d\uf0a5 f (t ) \uf03d \uf0ea T \uf0fa dz \uf0e5 \uf0e5 ( Fhcs, j +Fhes, j ) \uf06f eih\uf077st , \uf0ea\uf0ebU\u02c6 wp \uf0fa\uf0fb j \uf03d1 h\uf03d\uf02d\uf0a5 s (4.25) where \uf06f is the sign for element wise Hadamard product ( (A \uf06f B)i , j \uf03d Ai, j \uf0d7 Bi , j ),and vector containing the exponential terms is expressed as: \uf0ec ih \uf0e6\uf0e7 \uf077 t \uf02b\uf0e6\uf0e7 j \uf02d1 \uf066 ( dz ) \uf0f6\uf0f7\uf02d \u03c8 ( dz ) \uf0f6\uf0f7 \uf0fc p,e \uf0f7 j \uf0f7 \uf0ef \uf0ef \uf0e7\uf0e8 p \uf0e7\uf0e8 e\uf0e5 \uf03d0 \uf0f8 \uf0f8 \uf0efe \uf0ef \uf0ef \uf0ef \uf04d \uf0ef \uf0ef \uf0ef ih \uf0e6\uf0e7 \uf077 t \uf02b\uf0e6\uf0e7 j \uf02d1 \uf066 ( dz ) \uf0f6\uf0f7\uf02d \u03c8 ( qdz ) \uf0f6\uf0f7 \uf0ef p,e \uf0f7 j \uf0f7\uf0ef \uf0ef \uf0e7\uf0e8 p \uf0e7\uf0e8 e\uf0e5 \uf03d0 \uf0f8 \uf0f8 \uf0efe \uf0ef \uf0ef \uf0e6 \uf0ef j \uf02d 1 \uf0f6 \uf0e6 \uf0f6 \uf0ef ih \uf0e7\uf0e7 \uf077pt \uf02b\uf0e7\uf0e7 \uf0e5 \uf066p,e ( dz ) \uf0f7\uf0f7\uf02d \u03c8 j ( dz ) \uf0f7\uf0f7 \uf0ef \uf0e8 e\uf03d0 \uf0f8 \uf0f8 \uf0ef \uf0efe \uf0e8 \uf0ef \uf0ef e h, j (t ) \uf03d \uf0ed \uf04d \uf0fd \uf0ef \uf0e6 \uf0ef j \uf02d 1 \uf0f6 \uf0e6 \uf0f6 \uf0ef ih \uf0e7\uf0e7 \uf077pt \uf02b\uf0e7\uf0e7 \uf0e5 \uf066p,e ( dz ) \uf0f7\uf0f7\uf02d \u03c8 j ( qdz ) \uf0f7\uf0f7 \uf0ef \uf0e8 e\uf03d0 \uf0f8 \uf0f8\uf0ef \uf0efe \uf0e8 \uf0ef \uf0ef \uf0ef ih \uf0e6\uf0e7 \uf077 t \uf02b\uf0e6\uf0e7 j \uf02d1 \uf066 ( dz ) \uf0f6\uf0f7\uf02d \u03c8 ( dz ) \uf0f6\uf0f7 \uf0ef p,e \uf0f7 j \uf0f7 \uf0ef \uf0ef \uf0e7\uf0e8 p \uf0e7\uf0e8 e\uf0e5 \uf03d0 \uf0f8 \uf0f8 \uf0efe \uf0ef \uf0ef \uf0ef \uf04d \uf0ef \uf0ef j \uf02d1 \uf0f6\uf0ef \uf0f6 \uf0ef ih \uf0e6\uf0e7 \uf077 t \uf02b\uf0e6\uf0e7 \uf0e5 \uf0f7 \uf066 ( dz ) \uf02d \u03c8 ( qdz ) \uf0f7 \uf0f7\uf0ef \uf0ef \uf0e7\uf0e8 p \uf0e7\uf0e8 e \uf03d 0 p,e \uf0f7\uf0f8 j \uf0f8 \uf0eee \uf0fe3q\uf0b41 . 62 (4.26) \fChapter 4. Time Domain Simulation The Fourier coefficient vectors have the form: Fhcs, j \uf03d {Fhcs, j ,x (dz ) \uf04b Fhcs, j ,x (qdz ) Fhcs, j ,y (dz ) \uf04b Fhcs, j ,y (qdz ) Fhcs, j ,z (dz ) \uf04b Fhcs, j ,z (qdz )}3Tq\uf0b41 , (4.27) Fhes, j \uf03d {Fhes, j ,x (dz ) \uf04b Fhes, j ,x (qdz ) Fhes, j ,y (dz ) \uf04b Fhes, j ,y (qdz ) Fhes, j ,z (dz ) \uf04b Fhes, j ,z (qdz )}3Tq\uf0b41. (4.28) Each element in Eqs. (4.27) and (4.28) are calculated for one full spindle revolution and integration limits are simplified since switching function becomes non zero between start and exit angles of the cut: Fhcs, j ,i ( z ) Fhes, j ,i ( z ) 1 \uf03d 2\uf070 1 \uf03d 2\uf070 2\uf070 \uf0f2 a1,i (\uf066, z)e -ih\uf066 0 \uf066 1 ex d\uf066 \uf03d a1,i (\uf066, z )e-ih\uf066d \uf066 , \uf0f2 2\uf070 \uf066 (4.29) st 2\uf070 \uf0f2 b1,i (\uf066, z)e 0 -ih\uf066 \uf066 1 ex d\uf066 \uf03d b1,i (\uf066, z )e-ih\uf066d \uf066 , \uf0f2 2\uf070 \uf066 (4.30) st where i \uf03d x,y,z . The analytical solutions of the integrals are given in Appendix B. Eq. (4.22) can be simply solved for modal displacement in frequency domain by taking the product of transfer function and the modal force: M(i\uf077) f s (i\uf077) \uf03d \u0393(i\uf077) , (4.31) where the modal transfer function is: M(i\uf077) \uf03d (\uf02dI\uf0772 \uf02b 2\u03b6\u03c9n\uf077i \uf02b \u03c92n )\uf02d1 . (4.32) In frequency domain the multiplication is expressed in magnitude-phase form as: \u0393(i\uf077) \uf03d M(i\uf077) f s (i\uf077) \uf03c M(i\uf077) , where the magnitudes are denoted by (4.33) and the phases are represented by \uf03c . The magnitude of the transfer function is: M(i\uf077) \uf03d ((\uf02dI\uf0772 \uf02b \u03c92n )2 \uf02b (2\u03b6\u03c9n\uf077)2 )\uf02d1 , and the phase of the transfer function is: 63 (4.34) \fChapter 4. Time Domain Simulation \uf0e6 2\u03b6\u03c9n \uf077 \uf0f6 . \uf03c M(i\uf077) \uf03d tan \uf02d1 \uf0e7 \uf02d \uf0e7 \uf02dI\uf0772 \uf02b \u03c92 \uf0f7\uf0f7 n \uf0f8 \uf0e8 (4.35) Using Fourier series, solution in Eq. (4.34) is expressed in time domain for numerical and analytical solutions, respectively: \u0393(t ) \uf03d h\uf03d\uf0a5 \uf0e5 h\uf03d\uf02d\uf0a5 f hs \uf06f M(i\uf077p,h ) e i(h\uf077pt \uf02b(