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MILLING DYNAMICS PREDICTION AND UNCERTAINTY ANALYSIS USING RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS By GREGORY S. DUNCAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006 Copyright 2006 by Gregory Scott Duncan This document is dedicated to my wife Claire. ACKNOWLEDGMENTS I thank my advisor, Dr. Tony Schmitz, and my committee, Dr. John Ziegert, Dr. John Schueller, Dr. Kurtis Gurley, and Dr. Nagaraj Arakere, and the students of the Machine Tool Research Center. I also thank my parents. TABLE OF CONTENTS A C K N O W L E D G M E N T S ................................................................................................. iv LIST OF TABLES .................. .................. ................. ............ .............. .. vii LIST OF FIGURES ......... ........................................... ............ ix ABSTRACT .............. .......................................... xvi CHAPTER 1 IN TR OD U CTION ............................................... .. ......................... .. Justification of W ork ............................................... .. .. .. .... .............. .. 1 L literature R eview .......................................................... 3 Tool Point D ynam ic M odeling.................................... .......................... .......... 3 M achining Stability Investigations................................................ .................. 4 S cop e of W ork .................................................................. ............................. . 5 2 RECEPTANCE COUPLING MODEL DEVELOPMENT.........................................6 B background and N otation .................................................................. .....................6 F lexure M odel ..................................... ............ .. .. ..................... 8 Previous MachineSpindleHolderTool Modeling Technique..............................10 Improved MachineSpindleHolder Tool Modeling Technique..............................12 Spindleholder Base Subassembly Identification..............................................12 Standard test holder receptances ...................................... ............... 13 Extended holder subassembly m odel ................................. ............... 14 Spindleholder base subassembly receptance ........................................... 15 Toolpoint Response Prediction ....................................................................... 16 3 EXPERIMENTAL VALIDATION OF RCSA MODELS.................................21 Experimental Results for Stacked Flexure System................. ............................21 Experimental Results for 30,000 rpm, CAT 40 Spindle...............................22 Standard Test Holder and Finite Difference Method Evaluation........................22 Holder Experimental Verification .................. ............................................... 25 Experimental Results for 24,000 rpm, HSK 63A Spindle..............................25 H older Experim ental V erification .................................................................... 26 T oolpoint R response Prediction ..........................................................................26 Experimental Results for Geared, QuillType CAT 50 Spindle..............................27 Experimental Results for Geared CAT 50 Spindle.......................... .....................27 Experimental Results for Step Tec 20,000 rpm, HSK 63A Spindle.........................28 Tapered Heatshrink Holder and Carbide Tool Blank Results............................29 Collet Holder and Carbide Tool Blank Results .............................................29 F luted T ool R esults............ ............................................................ .. .... .... .. 29 4 STABILITY ANALYSIS UNCERTAINTY .................................. ...............52 Cutting Force Coefficient Determination............................. ............ ..............52 Stability A analysis Techniques ............................................................................. 53 T lu sty M eth o d ................................................... ................ 54 B udak and A ltintas M ethod ....................................................................... ... ...56 U uncertainty of Stability A nalysis.......................................... ........... ............... 58 5 EXPERIMENTAL VALIDATION OF MONTE CARLO SIMULATIONS............63 Experimental Determination of Cutting Force Coefficients......................................64 Aluminum 6061T6 Work Piece and 19.1 mm Diameter, 4 Flute, Carbide T o o l ...................................... ... ............. .......... .. ................................ 6 4 Aluminum 7475T6 Work Piece and 12.7 mm Diameter, 4 Flute, Carbide T ool ....................................... ....................... ..... ..... ........ 66 Stability lobe diagrams for Makino machining center ............................................66 Stability D eterm nation ............................................... ............................ 66 M onte Carlo Simulation Param eters ....................................... ............... 68 Stability L obe R esults............................ .................... .. ........... ...............70 Sensitivity of Budak and Altintas and Tlusty Stability Lobe Prediction T techniques ...................... ......... .......... ......... ... ..... ............ 7 1 Stability Lobe Diagrams for Mikron Machining Center Tests...............................72 Stability D eterm nation ............................................... ............................ 72 M onte Carlo Simulation Param eters ....................................... ............... 75 Stab ility L ob e R esu lts.......... ............................................ .......... ..............77 Spindle Speed Dependent FRF Issues...............................................................79 Sensitivity of the Budak and Altintas Stability Lobe Prediction Techniques when the Threecomponent Spindleholdertool RCSA Model is Used to G generate Toolpoint R response ...................... ................... ..1................81 6 CONCLUSIONS AND FUTURE WORK ........................................................117 C o n clu sio n s................................................... .................. 1 17 F utu re W ork ...................... .. .. ......... .. .. .......... .................................. 119 APPENDIX BEAM RECEPTANCE MODELING.......... .....................120 L IST O F R E F E R E N C E S ...................................................................... ..................... 123 BIOGRAPHICAL SKETCH ............................................................. ............... 129 LIST OF TABLES Table page 21 M ass and inertia adjustm ent ratios. ......................................................................... 20 31. Flexure m odal param eters. ............................................. .............................. 31 32 Standard holder substructure parameters. ..................................... ............... 31 33 Collet holder substructure I, II, and III parameters. ............................................35 34 Standard holder substructure parameters. ..................................... ............... 36 35 Shrink fit holder (25.3 mm bore) substructure parameters. ....................................37 36 Shrink fit holdertool blank substructure parameters........................ ...............39 37 20insert endmill substructure parameters. ................................... ............... 40 38 28insert endmill substructure parameters. ................................... ............... 40 39 16insert facemill substructure parameters. ............. .............................................42 310 Dimensions for tapered heatshrink holder and tool blank assembly ....................43 311 Dimensions for collet holder and tool blank assembly ........................................46 312 Dimensions for collet and tapered heatshrink holders and fluted tools ................49 41 Material statistical properties for holder and tool. ................................................62 51 Cutting coefficients for aluminum 6061T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool............... .... ..................... .........84 52 Statistical properties for cutting coefficients for aluminum 6061T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool............................................85 53 Statistical properties for cutting coefficients for aluminum 7475T6 work piece and 12.7 mm diameter, 4 flute, carbide helix tool............................................85 54 Stability results for slotting cuts on a Mikron Vario machining center .................97 55 Stability results for 50 % radial immersion cuts on a Mikron UCP600 Vario m achining center. ......................... ........... .. ...... .... .............. .. 99 56 Statistical geometric properties of tapered heat shrink holder and 4 flute, 19.1 m m diam eter carbide helix tool.................................... ........................... ......... 100 LIST OF FIGURES Figure page 21 Twocomponent assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s)................... ...............18 22 Twocomponent flexure assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s).............................. 18 23 Previous twocomponent RCSA model.........................................................19 24 Example standard holder for spindleholder base subassembly receptance id entification ................................................. ................ ....................19 25 Standard holder substructures for inverse receptance coupling.............................19 26. Spindleholdertool substructures for tapered thermal heat shrink holder and tool b la n k ...................................... .................................................... 2 0 27 End view of two fluted tool showing rotation angles for area and inertia calculations..................................... ................................ ........... 20 31 Plot of G11 receptances for flexure system..........................................................31 32 Plot of G33 receptances for small holder. ..................................... ............... 32 33 Standard holder geometry comparison............................................. .............32 34 Toolpoint response base on lrstorder finite difference method............................33 35 Toolpoint response base on 2ndorder finite difference method..............................33 36 Measured versus predicted toolpoint response based on 1storder and 2ndorder finite difference method and ten averaged measurement sets ...............................34 37 Collet holder substructure I, II, and III parameters. ............................................34 38 Collet holder H33 predicted and measured results..................................................35 39 Spindle receptances G55(o)) determined from standard holder direct and cross receptance m easurem ents. .............................................. ............................... 36 310 Tapered thermal shrink fit holder (25.3 mm bore) substructure model .................37 311 Measured (two nominally identical holders) and predicted H33 results for tapered thermal shrink fit holder (25.3 mm bore). ...................................... ............... 38 312 Tapered thermal shrink fit holder with 19.1 mm diameter tool blank substructure m o d el. ........................................ ................................... .. 3 8 313 Measured and predicted H11 results for tapered thermal shrink.............................39 314 Measured and predicted H11 results for 20insert endmill. .....................................40 315 Measured and predicted H11 results for 28insert..............................................41 316 Standard holder direct receptances two nominally identical, geared spindles (CAT 50 holderspindle interface). ............................................... ............... 41 317 Measured and predicted H11 results for 16insert facemill. ....................................42 318 Tapered heatshrink holder and tool blank assembly............................................42 319 The FRF for tapered heatshrink holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 1). The overhung tool length was 9 1 .6 m m ......................................................................... 4 3 320 The FRF for tapered heatshrink holder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Number 2). The overhung tool length was 3 8 .7 8 m m ........................................................................ 4 4 321 The FRF for tapered heatshrink holder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 3). The overhung tool length was 8 3 .8 1 m m ........................................................................ 4 4 322 The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 4). The overhung tool length was 132.4 mm. ........45 323 Collet holder and tool blank assem bly ........................................ .....................45 324 The FRF for collet holder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 5). The overhung tool length was 132.4 mm. ........46 325 The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 6). ........................................ ........................ 47 326 The FRF for collet holder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Number 7). The overhung tool length was 73.3 mm. ..........47 327 Tapered heatshrink holder and fluted tool assembly. ...........................................48 328 Collet holder and fluted tool assembly................................ ....................... 48 329 The FRF for tapered heatshrink holder with 19.1 mm diameter, 152.4 mm long carbide, 2 fluted tool assembly. The overhung tool length was 121.5 mm.............49 330 The FRF for collet holder with 25.4 mm diameter, 127.0 mm long carbide, 2 fluted tool assembly. The overhung tool length was 100.8 mm. ..........................50 331 The FRF for tapered heatshrink holder with 19.1 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 79.2 mm............... 50 332 The FRF for tapered heatshrink holder with 25.4 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 80.2 mm...............51 41 Stability lobe diagram ..................... ................ .............................61 42 Geom etry of milling process. ............ ........................................... .....................61 51 Experimental setup for measuring cutting force signals. .......................................82 52 Cutting force signal for 7500 rpm cut with 0.18 mm/tooth chip load ......................83 53 Linear regression for cutting force means at a spindle speed of 7,500 rpm.............83 54 Cutting coefficients versus spindle speed for the Budak and Altintas stability lob e technique e. ..................................................... ................. 84 55 Cutting coefficients versus spindle speed for Tlusty stability lobe technique. ........85 56 Audio signal frequency content for 10,000 rpm spindle speed, 1.02 mm axial depth cut. Cut was determined to be stable. ....................................................... 86 57 Audio signal frequency content results for 19,000 rpm spindle speed, 1.52 mm axial depth cut. Cut was determined to be unstable. .............................................86 58. Stability results for Makino machining center, thermal heat shrink holder, 12.7 mm, 4 flute, carbide helix tool, Aluminum 7475T6 work piece and a 25 percent radial im m version cut .......................................................................... ....... 87 59 Mean and 95 percent (20) confidence intervals for xdirection FRF ......................87 510 Mean and 95 percent confidence intervals forydirection FRF ..............................88 511 Covariance between x andydirection FRFs. .........................................................88 512 Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 percent confidence intervals are shown for the case where the x and ydirection FRFs are not correlated...................................89 513 Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 percent confidence intervals are shown for the case where x and ydirection FRFs are 100 percent correlated.............................89 514 Comparison between Tlusty lobes and experimental results. The mean stability boundary and 95 percent confidence intervals are shown for the case where x and ydirection FRFs are not correlated. ....................... ......... .............. 90 515 Comparison between Tlusty lobes and experimental results. The mean stability boundary and 95 percent confidence intervals are shown for the case where x and ydirection FRFs are 100 percent correlated.............. .... .................90 516 Budak and Altintas lobes based on cutting coefficient uncertainty .......................91 517 Budak and Altintas lobes based on toolpoint response measurement uncertainty. 91 518 Tlusty lobes based on cutting coefficient uncertainty........ .............. ............... 92 519 Tlusty lobes based on toolpoint response measurement uncertainty....................92 520 Example of mapping of toolpoint FRF to stability lobe. ............................... ...93 521 Experimental setup for measuring chatter based on x and ydirection tool displaced ent ...................... ......... .................................... ........ 93 522 8,000 rpm, 2 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 94 523 8,000 rpm, 2.5 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 94 524 10,000 rpm, 5 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 95 525 10,000 rpm, 6 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 95 526 15,000 rpm, 2 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 96 527 15,000 rpm, 3 mm axial depth slotting cut test for Mikron UCP600 Vario machining center. ................................... .............. .............. .. 96 528 8,000 rpm, 4 mm axial depth 50 % radial immersion cut test for Mikron UCP 600 V ario m achining center. ............................................................................. 97 529 8,000 rpm, 5 mm axial depth 50 % radial immersion cut test for Mikron UCP 600 V ario m achining center. ............................................................................. 98 530 10000 rpm, 16 mm axial depth 50 % radial immersion cut test for Mikron UCP 600 V ario m achining center. ............................................................................. 98 531 10000 rpm, 17 mm axial depth 50 % radial immersion cut test for Mikron UCP 600 V ario m achining center. ............................................................................. 99 532 Geometric properties of tapered heat shrink holder and 4 flute, 19.1 mm diam eter carbide helix tool. ..... ........................... .....................................100 533 H33 with 95 percent confidence intervals in the xdirection for no holder clam ping variation........... ............................ ........ ........ .. ...... ............ 101 534 H33c with 95 percent confidence intervals in the xdirection for no holder clam ping variation .................. ............................... ........ .. ............ 101 535 H33 with 95 percent confidence intervals in the ydirection for no holder clam ping variation............... ....................... ........ ........ .. ...... ............ 102 536 H33c with 95 percent confidence intervals in the ydirection for no holder clam ping variation .................. ............................... ........ .. ............ 102 537 The xdirection toolpoint FRF with 95 percent confidence intervals for no holder clam ping force variation. ........................................ ........................ 103 538 Theydirection toolpoint FRF with 95 percent confidence intervals for no holder clam ping force variation. ........................................ ........................ 103 539 The xdirection toolpoint FRF with 95 percent confidence intervals with holder clam ping force variation ........... .................................................. ............... 104 540 The ydirection toolpoint FRF with 95 percent confidence intervals with holder clam ping force variation ........... .................................................. ............... 104 541 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for slotting cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clam ping force variation excluded. ............................................. ............... 105 542 Tlusty predicted stability lobes with 95 percent confidence intervals and measured stability results for slotting cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clam ping force variation excluded. ............................................. ............... 105 543 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for 50 percent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping force variation excluded..............................106 544 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for 50 percent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping force variation excluded..............................106 545 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for slotting cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clam ping force variation included ....................................................................... 107 546 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for 50 percent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping force variation included. ................................................107 547 Experimental setup for measuring rotating FRFs. ............................................108 548 Magnitude of H33 in xdirection as a function of spindle speed ............................108 549 Magnitude of predicted toolpoint response, H11, in xdirection as a function of spindle speed. .......................................................................109 550 Predicted xdirection toolpoint FRF based on standard holder measurements at 10,000 rpm and chatter frequencies of unstable slotting cuts. ........................... 109 551 Predicted ydirection toolpoint FRF based on standard holder measurements at 10,000 rpm and chatter frequencies of unstable slotting cuts. ........................... 110 552 The FRF and 95 percent confidence intervals in xdirection for H33x based on a spindle speed of 10,000 rpm ....................................................................... ... 110 553 The FRF and 95 percent confidence intervals in xdirection for H33cx based on a spindle speed of 10,000 rpm .................................................................. ......... 111 554 The FRF and 95 percent confidence intervals inydirection for H33y based on a spindle speed of 10,000 rpm ........................................... ............. ...... 111 555 The FRF and 95 percent confidence intervals inydirection for H33cy based on a spindle speed of 10,000 rpm ............. ....................... .................... 112 556 Budak and Altintas predicted stability lobes with 95 percent confidence intervals for slotting cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Results based on standard holder measurements performed at 10,000 rpm. ............................. .... ............112 557 Budak and Altintas predicted stability lobes with 95 percent confidence intervals for 50 percent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Results based on standard holder measurements performed at 10,000 rpm. .................................... 113 558 Comparison between mean and uncertainty values for Budak and Altintas technique generated stability lobes base on static standard holder measurements and rotating standard holder measurements at 10,000 rpm. Stability lobes are for a slotting cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diam eter, 4 flute, carbide helix tool..................................................................... 113 559 Comparison between mean and uncertainty values for Budak and Altintas technique generated stability lobes base on static standard holder measurements and rotating standard holder measurements at 10,000 rpm. Stability lobes are for a 50 percent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. .................................... 114 560 Sensitivity of the Budak and Altintas predicted stability lobes to input parameter variations including: A) cutting coefficients, B) materials, C) standard holder measurements, and D) geometry variations. Results are based on Mikron UCP 600 Vario machine, slotting cuts, and standard holder rotational measurements at 10,000 rpm ............ ..... ........................................... ........ 115 561 Sensitivity of the Budak and Altintas predicted stability lobes to input parameter variations including: A) cutting coefficients, B) materials, C) standard holder measurements, and D) geometry variations. Results are based on Mikron UCP 600 Vario machine, 50 percent radial immersion cuts, and standard holder rotational measurements at 10,000 rpm ..................................... .............. vii Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MILLING DYNAMICS PREDICTION AND UNCERTAINTY ANALYSIS USING RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS By Gregory S. Duncan August 2006 Chair: Tony Schmitz Major Department: Mechanical and Aerospace Engineering Highspeed machining has made significant technological advancements in recent years. Using highspeed machining, increased material removal rates are achieved through a combination of large axial depths of cut and high spindle speeds. A limitation on the allowable axial depth of cut is regenerative chatter, which is avoided through the use of stability lobe diagrams which identify stable and unstable cutting zones. The machining models used to produce these diagrams require knowledge of the toolpoint dynamics and applicationspecific cutting coefficients. Toolpoint dynamics are typically obtained using impact testing; however, testing time is extensive due to the large amount of holdertool combinations. A technique to predict toolpoint dynamics and therefore limit experimental testing time is desirable. This dissertation describes a three component spindleholdertool model to predict toolpoint response based on receptance coupling substructure analysis techniques. Experimental validation is provided. This dissertation also describes Monte Carlo simulation models that place uncertainty bounds on stability lobe limits produced using two popular analytical techniques developed by Altintas and Tlusty. The sensitivities of the stability limits based on input parameter variation are investigated and experimental validation is provided CHAPTER 1 INTRODUCTION Justification of Work One area of manufacturing research that has made significant technological advancements in recent years is highspeed machining (HSM). Machine improvements include new spindle designs for higher rotational speed, torque, and power; increased slide speeds and accelerations; direct drive linear motor technology; and new machine designs for lower moving mass. The combination of new machine technology and tool material/coating developments often makes highspeed machining a viable alternative to other manufacturing processes. A key application example is the aerospace industry, where dramatic increases in material removal rates (MRR) made possible using high speed machining techniques have allowed designers to replace assemblyintensive sheet metal buildups with monolithic aluminum components resulting in substantial cost savings [1, 2]. Highspeed machining technology has also been applied to the production of moulds and dies [35] and automobile components [6] and has been used to improve the flexibility of manufacturing systems [7]. Using HSM, increased MRR are achieved through a combination of large axial depths of cut and high spindle speeds (provided adequate power is available). One limitation on the allowable axial depth is regenerative chatter and one method of pre process chatter prediction and avoidance is the wellknown stability lobe diagram. Stability lobe diagrams identify stable and unstable cutting zones (separated by stability "lobes") as a function of the chip width (or axial depth in peripheral end milling), a, and spindle speed. However, the machining models used to produce these diagrams require knowledge of the toolpoint dynamics and applicationspecific cutting coefficients. The toolpoint response is typically obtained using impact testing, where an instrumented hammer is used to excite the tool at its free end (i.e., the tool point) and the resulting vibration is measured using an appropriate transducer, typically a low mass accelerometer mounted at the tool point. It should be noted that the measured frequency response function (FRF) is specific to the selected components (e.g., tool and tool length, holder, spindle, and machine) and boundary conditions (e.g., holder force, such as collet torque or shrink fit interface, and drawbar force). If the assembly is altered, a new measurement must generally be performed. Due to the large number of spindle, holder, and tool combinations, the required testing time can be significant. Therefore, a model which is able to predict the toolpoint response based on minimum input data is the preferred alternative. An effective toolpoint response model also creates the potential to expand the use of HSM to a larger audience. For example, a web site application has been created at the University of Florida which allows end users to enter their machining specifications and view the corresponding stability lobe diagram [8]. The uncertainty associated with selecting optimum cutting parameters from stability lobe diagrams is also an important consideration. This uncertainty depends on the technique used to create the stability lobe diagrams and the variation in the model input parameters. A method to evaluate the sensitivity of various stability lobe algorithms to input parameter variability would enable selection of the most robust technique. Also, the ability to place uncertainty bounds on stability lobe results would aid the end user in the selection of optimum cutting parameters. Literature Review The literature review proceeds with a summary of toolpoint dynamic modeling, machining stability investigations, and uncertainty estimation techniques. Tool Point Dynamic Modeling Schmitz et al. [912], with the goal of reducing testing requirements, developed a model to predict the toolpoint response based on minimum input data using the receptance coupling substructure analysis (RCSA) method. In these previous studies, a two component model of the machinespindleholdertool assembly was defined. The machinespindleholder displacementtoforce receptance was recorded using impact testing, while the tool was modeled analytically as an EulerBernoulli beam [13]. The tool and machinespindleholder substructure receptances were then coupled through translational and rotational springs and dampers. The basis of the abovementioned technique is substructure analysis, or component mode synthesis. These methods have been used for several decades to predict the dynamic response of complicated assemblies using measurements and/or models of the individual components, or substructures. These components can be represented by spatial mass, stiffness, and damping data, modal data, or receptances [e.g., 1428]. Due to the difficulty in measuring rotational degreesoffreedom (RDOF) receptances, Schmitz assumed the displacementtomoment, rotationtoforce, and rotationtomoment receptances at the end of the holder to be equal to zero. Park et al. [29] describe a technique to determine the complete receptance matrix, including RDOF, at the end of the holder and the receptances are incorporated into the toolpoint dynamic model. Machining Stability Investigations Selfexcited vibration in metal cutting is known as chatter and has been studied by many researchers [3062]. Chatter is a condition that can limit MRR, degrade the surface quality of the workpiece, and lower the life of the cutting tool. Tobias and Fishweck [53] and Tlusty [62] identified two sources of selfexcitation in metal cutting: 1) regeneration of waviness and 2) mode coupling. As the tool makes a pass, waviness is created on the workpiece surface due to the relative vibration between the tool and workpiece. It is possible that the waviness among subsequent cutting passes may be out of phase, thus generating variable chip thickness and variable cutting forces. This condition can cause selfexcited vibrations and is known as regeneration of waviness. Mode coupling exists when relative vibration between the tool and workpiece exists simultaneously in at least two directions. Tlusty [49] and Tlusty et al. [62] were among the first researchers to analytically develop a method to predict chatter in machining. For the case of milling, Tlusty made the following assumptions: 1) a linear vibratory system, 2) a constant direction of the cutting force in relation to the normal cutting surface, and 3) removal of the time dependency of the chip thickness by analyzing the stability of the system at the mean location between the exit and entry points of the cutter tooth. Based on the results of the analysis, he was able to formulate relationships between the dynamics of the machine holdertool assembly, the material and geometric properties of the work piece and cutter, the geometric properties of the machining operation, chip width, and spindle speed leading to the creation of stability lobe diagrams. Merrit [52] later reproduced Tlusty's results based on control system theory. Recognizing that nonlinearities may exist in the actual machining process (when the cutter vibrates out of the workpiece), time domain simulations of the milling process [46, 62] were developed to predict stability. Budak and Altintas [55, 56] modeled the cutter and workpiece as multiple degreeoffreedom structures, including the axial direction, and applied periodic system theory to analytically analyze the stability of the system. Recent work [5761] has predicted system stability by numerically solving the time delay differential equations produced from the milling model by dividing the time in the cut into a finite number of elements. This technique is known as time finite element analysis (TFEA). Scope of Work The purpose of this dissertation is twofold. The first objective is to build on the previous work of Schmitz, which predicts the tool point FRF using the RCSA method, to provide a more generalized machinespindleholdertool model. The generalized model reduces the amount of experimental testing required for various machinespindleholder tool combinations. Secondly, the dissertation will investigate and compare the uncertainty of the results and sensitivity to input parameter variation of two popular analytical machining stability prediction techniques. CHAPTER 2 RECEPTANCE COUPLING MODEL DEVELOPMENT Background and Notation Substructure analysis, or component mode synthesis, methods predict the dynamic response of complicated assemblies using measurements and/or models of the individual components, or substructures. For an assembly consisting of two rigidly connected substructures, as shown in Fig. 21, the assembly receptance, Gjk(w), can be expressed as shown in Eq. (2.1), where co is the frequency, X, and 9, are the assembly displacement and rotation at coordinate, and Fk and Mk are the force and moment applied to the assembly at coordinate k. If coordinate is coincident with coordinate k, the receptance is referred to as a direct receptance; otherwise, it is a cross receptance. Here, the nomenclature Gjk(co) is used to describe the receptances that are produced when two substructures (or subassemblies) are coupled to produce the final assembly. The nomenclature GSjk(o) will replace Gk(co) in all relevant equations when two substructures (or subassemblies) are coupled that do not form the final assembly. X. X Gk()) Fk Mk H k (2.1) Oj Oj Njk Pjk Fk Mk The substructure receptances, Rjk(o), are defined in Eq. (2.2), where x, and 0, are the substructure displacement and rotation at coordinate, andfk and mk are the force and moment applied to the substructure at coordinate k [12, 29]. Xj X .M ^ = R Jk() fk mk k jk (2.2) fk mk Based on the coordinates defined in Fig. 21, the equations to determine the assembly direct receptances, Gaa(o) and Gdd(wc), and the assembly cross receptances, Gad(w)and Gda(o), can be written as a function of the substructure receptances as shown in Eqs. (2.32.6), where rigid connections have been applied [27]. Xa Xa Gaa ()) a a Raa ()) Rab (m)[Rbb (O)+ Rcc()]Rba () (2.3) Oa Oa Fa Ma Xd Xd Fd Md Gdd() d Od Rdd(J) Rdc(O)[Rbb (o)+ Rcc(o)]l Rcd(o) (2.4) d d Fd Md Xa Xa Fd Md Gad ()) a Rab(O))[Rbb(O))+Rcc())] Rcd (o) (2.5) 00 Qq Fd Md Xd Xd Gda () a Rdc (o)[Rbb (o)+ Rcc(O)lRba () (2.6) Od Od Fa Ma In order to populate the substructure receptance matrices, measurement and/or modeling may be applied. Common modeling options include closedform expressions for uniform EulerBernoulli beams [13] and finite element solutions (which can incorporate the more accurate Timoshenko beam model [63]). Both approaches are considered in this study. As a convenience to the reader, the relevant analytical formulas and finite element Timoshenko stiffness and mass matrices are included in Appendix A. Flexure Model To illustrate the receptance coupling procedure, consider the lumped parameter model for the stacked flexure assembly shown in Fig. 22 (an actual, equivalent assembly is also pictured). The base flexure, substructure A, is modeled as a single degreeof freedom (SDOF) substructure, defined as a mass, m3, connected to ground through a spring, k3, and a viscous damper, c3. The top flexure, substructure B, is modeled with freefree boundary conditions; it consists of a mass, mi, connected to a massless coordinate, x2, through a spring, kl, and viscous damper, c\. The dynamic response of assembly C to a force, Fi, applied at coordinate X1 (which represents the uppermost point on the top flexure) is computed using RCSA. It is assumed that the substructure rotational receptances, Ik, n andpjk, are negligible (by design for flexures) and that the substructures are rigidly connected. The substructure receptances are determined from the lumped parameter equations of motion. For substructure A, the motion is described by Eq. (2.7). Assuming a harmonic input forcef3(t) = F3eCt, the corresponding vibration is x3(t) = X3elwt and the direct receptance h33 can be defined as shown in Eq. (2.8). m333 (t) + c33 (t) + k3x3 (t) = f (t) (2.7) X3 (0() 1 h33 (M)=) ) 23 (2.8) 73(0) o2 m+ioc3+k3 Complex matrix inversion is applied to determine h12, h21, hil, and h22 for substructure B using the equations of motion provided in Eq. (2.9). Again assuming a solution of the form x,(t) = Xjet for f(t) = Fe'w, j = 1, 2, Eq. (2.9) can be written in matrix form as shown in Eq. (2.10). mlx l (t) + cl X (t) + klx (t) c 2 (t) kx2(t) = f () (2.9) c k1(t) klx1(t)+ c2 (t)+ k x2(t) = f2 (t) m)o2 +ico + k, ic1k1 x)1 f+ S ioc k, Jic, k x2 = f J or [A( x))X ={f (2.10) The receptance matrix for substructure B, GB(co), is obtained by inverting the matrix A(c) as shown in Eq. (2.11). The direct and cross receptances for substructure B are provided in Eqs. (2.122.14). h21 h221 GB [A(3) 1 ~[I(2.11) 1 h1i1 = (2.12) m2 m c k 122 M121 (2.14) o02(i om1c + m1k,) Substituting Eqs. (2.122.14) and Eq. (2.8) into Eq. (2.2) with the appropriate coordinate modifications and RDOF set to zero flexuress motion is approximately zero) yields Eqs. (2.152.18). Equation (2.19) is obtained by making the appropriate coordinate modifications to Eq. (2.3), and the linear assembly receptance shown in Eq. (2.20) is determined by substituting Eqs. (2.152.18) into Eq. (2.19). X, X, 01 0 L 0 0 f ml x, x, R12 2 f 2 = 2 f2 m2 X2 R22 f2 ^(= A _A X2 m2 02 m2 X, 01 F, 1 2 0) mI X3 X3 f M3 h33 0 R33 0 ) 3 03 03 f3 m3 X1 M1 01 M1 Sh22 0 0 0 X2 X2 m1 h1 21 0 01 R21 ()= 2 2 = 0 , 0 0 1i ^ C)2 m1 iCOC k, So2(io m1 + 1k,)) 0 1  23 ioc3 + k3 0 (2.16) (2.17) (2.18) = R,, (0o) R12 ()[2 o) + 23, (o)] 1 R (2o) (2.19) ) C 1i 1 1 m I( m0i3+k I I1 k 1 2 mI \m 2 + i32 c + k3 j 2 (i1omc+1 1 m l2 (2.20) Previous MachineSpindleHolderTool Modeling Technique In the previous work of Schmitz et al. [912], which describes the tool point frequency response function (or receptance) prediction using the RCSA method, a two (2.15) component model of the machinespindleholdertool assembly was defined. The machinespindleholder displacementtoforce receptance was recorded using impact testing, while the tool was modeled analytically as a EulerBeroulli beam [13]. The tool and machinespindleholder substructure receptances were then coupled through translational and rotational springs and dampers: see the model in Fig. 23, where kx and ko are the translational and rotational springs, cx and co are the translational and rotational viscous dampers, component A represents the tool, and component B represents the machinespindleholder. The purpose of the springs and dampers between the tool and holder was to capture the effects of a potentially nonrigid, damped connection. The values of the springs and dampers were determined by measuring the tool point receptance with impact testing and performing a nonlinear least squares fit between the actual measured results and the model. Although it was shown in Schmitz and Burns [12] that this twocomponent model provides a valid approximation for a flexible tool clamped in a stiff spindleholder, it does not offer the most generalized solution; i.e., if a new holder is inserted in the spindle, a new machinespindleholder measurement must be performed. The potential for improvement in the twocomponent model exists in three areas. First, the model requires an experimental measurement to determine the receptance at the end of the holder; therefore, multiple spindlerholder combinations each require a separate measurement. A model which identifies, in a single measurement set, the machinespindleholder dynamics for all machinespindleholder combinations would be beneficial. Secondly, the displacementtomoment, rotationtoforce, and rotationto moment receptances at the free end of the holder were assumed zero (i.e., perfectly rigid) due to the difficulty in measuring RDOF and it is likely that the fit values for the springs and dampers between the tool and holder compensate for the fact that these receptances are not truly equal to zero. Finally, the assumption is made that the values of the springs and dampers between the tool and holder are constant as the length of the tool overhang is altered. Improved MachineSpindleHolder Tool Modeling Technique In order to enable RCSA predictions for a wider variety of machinespindleholder tool combinations, an improved threecomponent model is presented. In this model, the machinespindleholder substructure is separated into two parts: 1) the machine, spindle, holder taper, and portion of the holder nearest the spindle with standard geometry from one holder to another (hereafter referred to as the spindleholder base subassembly); and 2) the remaining portion of the holder from the base to the free end (hereafter referred to as the extended holder subassembly). A technique for determining the rotationto force/moment and displacementtomoment receptances for the free end of the spindle holder base subassembly using only displacementtoforce measurements is also described. The experimental procedure involves direct and cross displacementtoforce measurements of a simple geometry 'standard' holder clamped in the spindle to be modeled. The portion of the standard holder beyond the section with consistent geometry from holdertoholder is then removed in simulation using an inverse receptance coupling approach (i.e., decomposition) to identify the four spindleholder base subassembly receptances. These receptances are then coupled to models of the actual holder and tool. Spindleholder Base Subassembly Identification The experimental procedure used to determine the receptances at the free end of the spindleholder base subassembly, GSk(co), is described in this section. It is composed of three primary steps. First, the standard holder displacementtoforce direct and cross receptances are determined by impact testing. The standard holder geometry, which was selected to approximate a broad range of potential holders, is provided in Figure 24. Second, these results are used to determine the three other direct receptances at the free end of the standard holder. Third, the section of the standard holder which is not common to other holders (see Figure 25) is removed using inverse receptance coupling to determine all four spindleholder base subassembly receptances. Each step of the procedure is described in the following sections. Standard test holder receptances Once the standard holder is mounted in a spindle (see Figure 24), the four subassembly receptances are determined by measuring the direct, H33, and cross, H33c, and/or H33b, displacementtoforce receptances on the standard holder, applying a 1st order or 2ndorder backward finite difference method to find L33 (and, equivalently, N33) [64], and then synthesizing P33. If a 1Storder finite difference method is used, only one cross displacementtoforce receptance, H33c, is required. Both cross displacementto force receptances are required if the 2ndorder finite difference method is used. For the cross displacementtoforce measurements, the distance S should be selected to increase the difference in relative amplitudes between H33, H33c, and/or H33b without leading to a poor signaltonoise ratio for the H33, measurement (i.e., many of the lower frequency spindleholder modes resemble a fixedfree fundamental mode shape and have very small amplitudes near the spindle face for the bandwidth of interest). Practically, it has been observed that the finite difference results improve as S is increased; however, care must be taken to ensure that the location of the H33, measurement provides sufficient signalto noise. The receptance L33 is determined from the measured displacementtoforce receptances using Eq. (2.21) if the lstorder finite difference method is selected or Eq. (2.22) if the 2ndorder finite difference method is selected. By reciprocity, N33 can be set equal to L33. The remaining receptance, P33, is synthesized from H33, L33, and N33, as shown in Eq. (2.23) [25]. L33 3 33c (2.21) 2S L33 3H33 4H33b + H33c (2.22) 2S 03 F, X3 03 1 L332 P33 L33N33 33 (2.23) M3 X, M3 F, H33 H33 Due to the subtraction of the similarly scaled H33, H33c, and/or H33b receptances, noise in the measurement data can detrimentally affect the quality of L33 and N33 (produced by the finitedifference method) and, therefore, P33. To reduce the noise effect, the measured receptance data was smoothed using a SavitzkyGolay filter, which performs a local polynomial regression to determine the smoothed value for each data point [65], prior to the application of Eq. (2.21). For this study, filters with polynomial orders of two or three were applied over windows of 31 to 81 data points. Extended holder subassembly model The extended holder subassembly for the steel standard holder consists of solid, cylindrical substructures I and II as shown in Fig. 25. Equations (2.242.27) provide the direct and cross extended holder subassembly receptance matrices, where rigid coupling has been applied. These equations are determined from Eqs. (2.32.6) by appropriate substitutions. X3 X3 F3 M3 GS33(o)= (0 3 =R33(mo)R33a(o)[R3a3a(o)+R3b3b(0o)]1R3a3(0o) (2.24) 03 03 F3 M3 X4 X4 S444 ()4 =R44 (o) R43b(cO)[R3b3b((o)+ R3a3a ()] R3b4(() (2.25) 04 04 F4 M4 X3 X3 F4 AM4 GS 4(o)= 4 = R33i(co)[R^3ai(o)+ R3h3b(o)llR64(co) (2.26) 03 03 X4 X4 GS43 ()= 3 R43b (o)[R3a3a (0) + R3b3b (o)] R3a3 (o) (2.27) 04 04 F3 M3 Spindleholder base subassembly receptance The spindleholder base subassembly receptance matrix, G33(w), can be expressed as shown in Eq. (2.28) by rewriting Eq. (2.3). The lefthand side of this equation is known once the steps described in the standard test holder receptances section are completed. Also, the extended holder subassembly receptances, GS33, GS44, GS34, and GS43, are determined using the equations provided in the extended holder subassembly model section. Therefore, Eq. (2.28) can be rewritten to solve for the spindleholder base subassembly receptances, GS55((). See Eq. (2.29). G33(o)= H33 L33 GS33(0o)GS34(o()[GS44(0)+GS55(o)] GS43(0o) (2.28) _N33 P33 x5 x5 f5 m5 1 GS5() 05 05 GS34(o)[GS33(o)G33(o)1 GS43(~))GS44(o0) (2.29) f5 m5 Toolpoint Response Prediction To illustrate the technique used to predict tool point dynamics, the modeling procedure is applied to a spindle using a tapered thermal shrink holder with a tool blank inserted as shown in Fig. 26. The assembly is divided into the spindleholder base subassembly and 13 cylindrical substructures of differing diameters. The spindleholder base subassembly receptances are determined by the procedure described in the spindle holder base subassembly identification section. The 13 cylindrical substructures are each analytically modeled as a EulerBernoulli beam or Timoshenko beam (see Appendix A). To model the receptances of the beams, a composite modulus and mass are substituted for substructures IIVIII to account for potential material differences between the holder and the tool blank. Also, the mass expression for these substructures (provided in Appendix A) is replaced with the composite mass shown in Eq. (2.30), where ph and pt are the density of the holder and tool, respectively. Additionally, the product of the elastic modulus and 2nd area moment of inertia, El, are replaced by the product shown in Eq. (2.31), where Eh is the holder modulus, E, is the tool material modulus, and Ih and I are the 2nd area moments of inertia for the holder and tool, respectively. Ph(d d2)+p d)L2 mi ((= d ) d) (2.30) 4 Eh ;(d4 d4)+ETd4 EIh= Ehh( +EI, 0 (2.31) 64 The next step is to rigidly couple substructures I through XIII to produce the direct and cross extended holdertool subassembly receptances at coordinates 1 and 4. First, Eqs. (2.32.6) are used with the appropriate coordinate modifications to couple substructure I to substructure II, each substructure having freefree boundary conditions, creating the first subassembly. This subassembly is then coupled to the next substructure to create a subassembly consisting of the first three substructures. This process is continued until all substructures are coupled together and the receptances for the complete extended holdertool subassembly are determined. The final step in the procedure is to predict the tool point dynamics by rigidly coupling the extended holder tool subassembly to the spindleholder base subassembly. With the appropriate coordinate substitution in Eq.(2.3), the tool point receptance, G 1(co), are determined according to Eq. (2.32), where the receptances associated with coordinates 1 and 4 are the extended holdertool subassembly direct and cross receptances. GHl L()1 = GS1l(o)GS14(o)[GS44(o)+GS55(o)] lGS41(o) (2.32) LN,1 P11= For the case where a fluted tool is used instead of a tool blank, the mass and 2nd area moment of inertia of the fluted section are adjusted to account for the actual geometry. Two, three, and four fluted tools were drawn in a computer aided design (CAD) program. The tools were rotated at 10 degree increments in the program through a full rotation, as shown in Fig. 27, and the inertia was calculated by the CAD program about the vertical axis at each rotation. An average of the inertia across all rotations was computed and compared to the inertia of a tool blank section to determine an adjustment ratio for each type of fluted tool. The area at the end of the fluted section was also computed and compared to the area of a tool blank section to determine an adjustment ratio for the mass of the fluted section. Table 21 shows the mass and inertia adjustment ratios. During the substructuring process for the extended holdertool subassembly, the overhung tool section is divided into a shank substructure and a fluted substructure. The fluted substructure is the length of the fluted section. The mass and inertia of the fluted substructure is first calculated as if the section was a solid, cylindrical section and then the mass and inertia results are multiplied by the Table 21 ratios, based on the number of tool flutes, to determine the final fluted substructure mass and inertia prior to the calculation of the substructure receptances. Unassembled system Assembled system Substructure II Substructure I Ma F ( Rigid d c b a a d c o e b a Oa Xa Figure 21. Twocomponent assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s). F, k3 X3 X, t X, k X3 k, X Cj 7, A C1 Substructure A Substructure B Assembly C Figure 22. Twocomponent flexure assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s). Figure 23. Previous twocomponent RCSA model. An external force, Fa(t), isapplied to the free end of the tool (A) to determine the assemblyXa/Fa receptance. The tool is coupled to the machinespindleholder (B) through springs and dampers. Cross receptance hammer impact locations 3c 3b 3 Example standard holder for spindleholder base subassembly receptance identification. Hammer impacts are completed at locations 3, 3b, and 3c to identify the required direct and cross receptances. Spindle 5 3 Test S Holder 3 __ Extended holder subassembly I I 5 3a 3 4 3b Ri I Rigid  Spindleholder base subassembly Figure 25. Standard holder substructures for inverse receptance coupling. Figure 24. Spindleholder base Extended holdertool Spindleholdertool assembly subassembly subassembly Coordinate 51 Rigid connections between 4 all substructures L ~ Substructures XIII ... I Figure 26. Spindleholdertool substructures for tapered thermal heat shrink holder and tool blank Degree of rotation 0 10 20 30 0 etc... Figure 27. End view of two fluted tool showing rotation angles for area and inertia calculations. Table 21. Mass and inertia adjustment ratios. Tool Flutes Mass Ratio Inertia Ratio 2 0.44 0.38 3 0.33 0.36 4 0.36 0.36 CHAPTER 3 EXPERIMENTAL VALIDATION OF RCSA MODELS Experimental validation of the stacked flexure system and improved three component machinespindleholdertool model for five different spindles is provided. The spindleholder base subassembly receptances were determined for a 36,000 rpm, CAT 40 spindle for three standard holder geometries to determine the optimum geometry for the standard holder. Once the spindleholder base subassembly receptances were determined, the threecomponent model was used to compare predicted and measured results at the end of a collet holder. The spindleholder base subassembly receptances were also determined for a 24,000 rpm, HSK 63A spindle and Step Tec HVCS 212X 20,000 rpm, HSK 63A spindle. The threecomponent model was again used to compare predicted and measured results for spindleholdertool assembly combinations produced with collet holders, three geometries of tapered heatshrink holders and various geometries of carbide tool blanks and fluted tools. The threecomponent model was also used to compare predicted and measured results for a geared, quilledtype CAT 50 spindle with a 4 flute, 20 insert endmill and a 28 insert facemill and a geared CAT 50 spindle with a 16 insert, solid body facemill. Experimental Results for Stacked Flexure System The 2DOF flexure assembly shown in Fig. 22 was produced by stacking two SDOF flexures. After manufacture, the modal parameters for the flexures were determined by impact testing. For these measurements, the flexures were adhered to ground (a massive, extremely stiff machining tombstone was assumed to be ground) with cyanoacrylate (i.e., quickbonding cement). An impact hammer was used to excite the flexures at their free end (over a bandwidth of approximately 2 kHz) and the response was measured using a lowmass accelerometer. The modal mass, damping, and stiffness parameters were then extracted from the response by fitting the data using a peakpicking method [25]. The results are shown in Table 31. To verify the stacked flexure RCSA model, the model parameters from Table 31 were substituted into G1i(w), provided in Eq. (220). Figure 31 displays measured and predicted G1(co) results for the stacked flexure system. Experimental Results for 30,000 rpm, CAT 40 Spindle Standard Test Holder and Finite Difference Method Evaluation To evaluate the effect of the standard holder dimensions on the determination of the spindleholder base subassembly receptances, GS55((), three different standard holder geometries were tested in a 36,000 rpm, CAT 40 spindle. The extended holder subassembly for the three steel standard holders consisted of solid, cylindrical substructures I and II as shown in Fig. 25. The dimensions and material properties for each of the holder substructures are displayed in Table 32, where d, and do are the inner and outer diameters, L is the length, p is the density, and r is the structural damping factor. The structural damping values used in this study were determined experimentally from freefree testing of representative cylindrical rods. During the measurement of the direct and cross receptances for the mounted standard holders, the distance S was selected as 25.4 mm. Following the procedure outlined in chapter 2, the measured receptance, H33, and calculated receptances, L33 and P33 were recorded and are shown in Fig. 32 for the small holder. The spindleholder base receptance, GS55((), was also determined for each of the three standard holders by decoupling the appropriate extended holder subassembly from the spindlestandard holder measurements and the receptances at the end of each of the three standard holders, G33(o), were determined based on the spindleholder base subassembly receptances of the other two holders. Next, the predicted receptances at the end of each holder were compared to the measured receptances to select the preferred standard holder geometry. Figure 33 displays the magnitudes of the displacementtoforce receptance, H33, for the large holder rebuilt from the small holder spindleholder base subassembly, the small holder rebuilt from the large holder spindleholder base subassembly, and the medium holder rebuilt from the small holder spindleholder base subassembly. Two conclusions can be drawn from Fig. 33: 1) the spindleholder base subassembly receptances, GS55(o), determined from testing the small holder can be used as a basis to accurately predict the end receptances, G33(c), for the medium and large holders; and 2) the spindleholder base subassembly receptances determined from testing the large holder do not provide an accurate prediction for the small holder end receptances. It is hypothesized that the small holder produces better results because the large holder has increased mass which shifts the FRF information to lower frequencies. Since an accelerometer was used in the FRF measurements of the standard holder, any content below a certain frequency, 140 Hz in this case, was eliminated from the analysis (due to the measurement characteristics of the accelerometer) and the larger holder places more information in this frequency range. The machinespindleholder base subassembly receptances based on the large holder measurements would therefore be based on less information leading to increased error in the prediction. Based on these results, the small standard holder was selected as the experimental apparatus used to identify the spindleholder base subassembly receptances and these receptances were used to predict end receptances for other extended holdertool subassemblies. Experimental testing was also performed to compare toolpoint response predictions from the threecomponent machinespindleholdertool model when 1st and 2ndorder finite difference approaches were used to determine the standard holder receptances. The experimental setup consisted of the standard holder mounted in a 20,000 rpm Step Tec spindle with a HSK 63A spindleholder interface. The machine spindleholder base receptances were determined and a toolpoint response was predicted for a tapered holdertool blank assembly (see Fig. 26) using the methods described in chapter 2. Figure 34 shows the predicted toolpoint response when a 1storder finite difference approach was used based on one, two, four, and ten measurement set averages. A measurement set consists often rootmeansquare averaged FRF measurements of the direct and cross standard holder FRFs. Figure 35 shows the predicted toolpoint response when a 2ndorder finite difference approach was used based on one, two, four, and 10 measurement set averages, and Fig. 36 shows a comparison between toolpoint responses based on the 1storder and 2ndorder finite difference approaches and the actual measured toolpoint response. From the figures, it can be seen that the 1storder finite difference method provides the most consistent toolpoint response, independent of the number of averaged measurement sets. The 2ndorder finite difference approach produces more variation in the predicted toolpoint response based on the number of measurement sets used, and the toolpoint response contains more noise. Figure 3.6 shows that using the 1storder finite difference method actually leads to improved accuracy for the model prediction. Based on this testing, it is recommended to use the 1storder finite difference method with 5 averaged measurement sets to determine the standard holder receptances. Holder Experimental Verification Once the spindleholder base subassembly receptances, GS55(w), are determined using the 1st order finite difference method, it is possible to couple this result to arbitrary holder geometries to predict the receptance at any coordinate on the machinespindle holder assembly. To validate the procedure, a collet holder (19 mm bore) with a CAT 40 spindle interface was divided into three substructures beyond the spindleholder base subassembly as shown in Fig. 37. Each substructure was assumed to be a hollow or solid cylindrical steel beam, as appropriate, even though the geometry of the actual collet holder was slightly more complicated. The assumed holder geometry and model coordinates are also provided in Fig. 37. The first step in predicting the assembly response, as described in chapter 2, was to couple substructures I, II, and III to produce the direct and cross extended holder subassembly receptances at coordinates 3 and 4. The parameters for the freefree substructure receptances are given in Table 33. The next step was to rigidly couple the spindleholder base subassembly to the extended holder subassembly to determine the receptances at the free end of the holder, G33(o). Figure 38 shows the predicted and measured H33 results for the collet holder. Experimental Results for 24,000 rpm, HSK 63A Spindle The extended holder subassembly for the steel standard holder again consisted of solid, cylindrical substructures I and II as shown in Fig. 26; however, the standard holder consisted of a HSK 63A interface mounted in a 24,000 rpm/40 kW spindle. The dimensions and material properties for each of the holder substructures are displayed in Table 34. The experimental procedure described in chapter 2 was applied to determine all receptances at coordinate 5, GS55(c), as displayed in Fig. 39. Holder Experimental Verification Once the 24,000 rpm/40 kW spindleholder base subassembly receptances, GS55((o), were determined, it was possible to couple this result to arbitrary holder geometries to predict the receptance at any coordinate on the machinespindleholder assembly. To validate the procedure, a tapered thermal shrink fit holder (25.3 mm bore) with an HSK 63A spindle interface was divided into 12 substructures beyond the spindle holder base subassembly as shown in Fig. 310. Each substructure was assumed to be a hollow or solid cylindrical steel beam, as appropriate, and the outer diameters for the tapered section substructures were the mean value for that section. Table 35 provides the holder geometry and assumed material properties. Following the procedure outlined in chapter 2, the receptances at the free end of the holder, G33(w), were determined. Figure 311 shows the predicted H33 result as well as measurements for two nominally identical holders. The EulerBernoulli beam model was applied to develop the extended holder receptances in this case. Toolpoint Response Prediction To predict the toolpoint dynamics, the modeling procedure was again applied to the 24,000 rpm/40 kW spindle using a tapered thermal shrink holder with a 19.1 mm carbide tool blank inserted as shown in Fig. 312. The assembly was divided into the spindleholder base subassembly and 13 cylindrical substructures of differing diameters (mean outer diameter values again used for the tapered sections); see Table 36. The predicted and measured assembly toolpoint displacementtoforce receptances, H11, are displayed in Fig. 313. In this figure, results for both EulerBernoulli and Timoshenko (finite element) beam models are provided. It is seen that the finite element model (100 elements were used for each substructure) dominant natural frequency is closer to the measured result, as expected. However, the predicted natural frequency is still approximately 50 Hz higher. Experimental Results for Geared, QuillType CAT 50 Spindle In this section, prediction and measurement results are provided for two cutters coupled to a geared, quilltype spindle with a CAT 50 spindleholder interface (BigPlus tool holders were used which include both taper and face contact). The spindleholder base subassembly receptances were determined using a steel cylindrical standard holder (63.4 mm diameter and 89.0 mm length); the cross FRF measurements were again recorded at distances of 25.4 mm and 50.8 mm (2ndorder finite difference method was applied) from the free end of the standard holder. The substructure receptances for the solid body tools (i.e., both cutting tools were composed of solid steel modular bodies with carbide inserts attached) were then computed and the toolpoint FRF predicted by rigidly coupling the tool models to the spindle measurements. Figure 314 displays the H11 results for an inserted endmill with 4 'flutes' (20 total inserts, 5 inserts for each tooth). The tool body geometry is defined in Table 37 (as before substructure, I is nearest the free end of the clamped cutter). Figure 315 shows the H11 measurement and prediction for a 28insert facemill (see Table 38). In both cases, EulerBernoulli beam models were employed to describe the standard holder and cutter bodies. Experimental Results for Geared CAT 50 Spindle In this section, the spindleholder base subassembly receptances were measured on two nominally identical, geared spindles (CAT 50 holderspindle interface). The steel cylindrical standard holder was 63.4 mm in diameter and 89.0 mm long. The cross FRF measurement locations were the same as specified previously. Figure 316 provides standard holder direct FRF measurement results for both spindles. Three curves are shown: the solid line represents the average of five measurement sets (10 impacts each) completed without removing the holder from the first spindle (i.e., spindle 1); the dotted line gives the average of three more spindle 1 measurements after removing and replacing the holder; and the dashed line shows the average of five spindle 2 measurements. These curves show that, although the spindles are similar, the difference between the spindle dynamics is larger than the measurement divergence on a single spindle. Next, a 16insert solid body facemill was inserted in spindle 1 and the toolpoint response recorded. Predictions were finally completed using both the spindle 1 and 2 receptances. This result is provided in Fig. 317; the facemill geometry and material properties are given in Table 39. It is seen that the prediction completed using the spindle 1 receptances (dotted line) more accurately identifies the spindle 1 measured frequency content (solid line). Therefore, it would be necessary to measure both spindles to make accurate predictions, rather than relying on manufacturing repeatability. It has been our experience that the dynamic consistency between spindles is manufacturer dependent. Experimental Results for Step Tec 20,000 rpm, HSK 63A Spindle In this section, prediction and measurement results are provided for a Mikron Vario CNC machining center with a 20,000 rpm, HSK 63A interface Step Tec spindle coupled to a variety of holder tool assemblies. The machinespindleholder base receptances were determined and toolpoint responses were predicted using the methods described in chapter 2. Tapered Heatshrink Holder and Carbide Tool Blank Results Toolpoint responses were measured and predicted using the threecomponent machinespindleholdertool model for the four different holder tool assemblies shown in Fig. 318 with the component and assembly dimensions displayed in Table 310. The spindleholdertool assembly combinations were produced using three different holders, three different carbide tool blanks, and four varying tool overhang lengths. The results for the four different test runs displayed in Table 310 are shown in Fig. 319 through Fig. 322. Collet Holder and Carbide Tool Blank Results Toolpoint responses were measured and predicted using the threecomponent machinespindleholdertool model for the three different holder tool assemblies shown in Fig. 323 with the component and assembly dimensions displayed in Table 311. The spindleholdertool assembly combinations were produced using two different carbide blanks and two different tool overhang lengths. In test number 6, the tool blank extended past the collet in the holder and was modeled as a solid substructure section. This technique still provided an acceptable toolpoint response prediction, as seen in Fig. 325. The results for the four different test runs displayed in Table 310 are shown in Fig. 324 through Fig. 326. Fluted Tool Results Toolpoint responses were measured and predicted using the three component machinespindleholdertool model for the four different holder tool assemblies shown in Fig. 327 and 328 with the component and assembly dimensions displayed in Table 3 12. The spindleholdertool assembly combinations were produced using three different holders, 4 different fluted carbide tools (consisting of both two and four tooth cutters), and two different tool overhang lengths. The mass and inertia of the fluted tool sections was modeled using the methods described in Chapter 2. The results for the four different test runs displayed in Table 310 are shown in Fig. 329 through Fig. 332. The threecomponent model does a good job of predicting the toolpoint response based on the spindleholder subassembly receptances and an analytical model of the extended holdertool subassembly. It should be noted that finite element analysis (FEA) models have the potential to provide more accurate model predictions for complicated holder and tool geometries; however, modeling the extended holdertool subassembly as a combination of beam sections has one main advantage: it is less time consuming to simply enter the holder and tool geometries versus creating and analyzing a FEA model. This is especially important with the large variety of holders and tools that are currently available and allows external users of the webbased application [8] at the University of Florida to receive stability lobe diagrams for any holdertool combination based simply on their geometries. Predicted results for the toolpoint tend to be shifted slightly, by up to 50 Hz, to the right on the frequency axis. This is probably due to the connection between the holder and the tool which has been modeled as a rigid connection but in reality has stiffness and damping characteristics. Including a nonrigid connection in the overall model could lead to improved results. Table 31. Flexure modal parameters. Modal parameters m k c (kg) (N/m) (kg/s) Large base flexure 2.46 8.85x106 70.6 Top flexure 0.145 5.04x101 1.17 1 0 ^10 E x 10 20  100 200 300 400 Frequency (Hz) Figure 31. Plot of G1 receptances for flexure system. Table 32. Standard holder substructure parameters. Small holder Medium holder Large holder Substructure I II I II I II Coordinate 3 3b 3 3b 3 3b Coordinate k 3a 4 3a 4 3a 4 d, (m) do (m) 0.0444 0.0444 0.0510 0.0444 0.0510 0.0444 L (m) 0.0400 0.0173 0.0662 0.0173 0.1157 0.0173 p (kg/m3) 7800 E (N/m2) 2x1011 S770.0015 500 200 600 1000 1400 1800 Frequency (Hz) Figure 32. Plot of G33 receptances for small holder. ,25 S2 1 rP 0.51 0 2 6 Large Holder Rebuilt i6 from Small Holder 600 1000 1400 1800 Frequency (Hz) 3 5 2 .5 1 )5 0 2 Small Holder Rebuilt F from Large Holder 00 600 1000 1400 1800 Frequency (Hz) Medium Holder Rebuilt .5x16from Small Holder Rebuilt Measured 1 .5 0 200 600 1000 1400 1800 Frequency (Hz) Figure 33. Standard holder geometry comparison (note: vertical axis scale is different for the three panes). Rebuilt Measured 0 Rebuilt Measured / \ k 1 )0 I ............... 1 measurement set  2 measurement sets ;  4 measurement sets  10 measurement sets 1000 1500 2000 2500 Frequency (Hz) Figure 34. Toolpoint response base on lrstorder finite difference method. o6 x 106 4 2 0 2  4 I x 10 2 S............... measurement set 1 measurement set 2  2 measurement sets ..... 4 measurement sets 4 10 measurement sets A I I 500 1000 1500 Frequency (Hz) 2000 2500 Figure 35. Toolpoint response base on 2ndorder finite difference method. x 106 4 Figure 36. SpindleHol Assembly x 106 500 1000 1500 2000 2500 Measured versus predicted toolpoint response based on l1torder and 2nd order finite difference method and ten averaged measurement sets. Rider Rigid Connection SpindleHolder Base Subassembly Figure 37. Collet holder substructure I, II, and III parameters. x106 I Table 33. Collet holder substructure I, II, and III parameters. Substructure I II III Coordinate 3 3b 3d Coordinate k 3a 3c 4 d,(m) 0.0190 0.0190 0.0315 d (m) 0.0630 0.0500 0.0496 L (m) 0.0255 0.0260 0.0286 p (kg/m3) 7800 E (N/m2) 2x1011 r/ 0.0015 600 1000 1400 1800 Frequency (Hz) Figure 38. Collet holder H33 predicted and measured results. 7 x 10 6 6 7 x 10 0  ^^ 12 L 200 Table 34. Standard holder substructure parameters. Substructure I II Coordinated 3 3b Coordinate k 3a 4 do (mm) 63.3 52.7 L (mm) 62.8 16.3 p (kg/m3) 7800 E (N/m2) 2x101 7 0.0015  P55 (rad/Nm) "...... 155 (m/Nm)  h55 (m/N) 0 1000 2000 3000 4000 5000 Frequency (Hz) Figure 39. Spindle receptances G55(co) determined from standard holder direct and cross receptance measurements. 10 10 . 10 < Table 35. Shrink fit holder (25.3 mm bore) substructure parameters. Substructure d, (mm) do (mm) L (mm) p (kg/m3) E (N/m2) 7 I 25.3 44.2 5.5 II 25.3 45.1 5.5 III 25.3 46.1 5.5 IV 25.3 47.0 5.5 V 25.3 47.9 5.5 VI 25.3 48.9 5.5 11 7800 2x10" 0.0015 VII 25.3 49.8 5.5 VIII 26.0 50.7 5.5 IX 26.0 51.7 5.5 X 26.0 52.6 5.5 XI 26.0 52.6 15.7 XII 52.6 30.3 Spindleholder assembly Spindlehold subassembly r er base Extendedholder subassembly 5 Rigid connections between I 4 all substructures 3 Substructures XII... I Figure 310. Tapered thermal shrink fit holder (25.3 mm bore) substructure model. 7 x 10 II I 1.5 ******* Predicted fec'  Holder 1 S Holder 2 1.5 1I 7 x 10 0 9 2 3 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 311. Measured (two nominally identical holders) and predicted H33 results for tapered thermal shrink fit holder (25.3 mm bore). Spindleholder base Extended holdertool Spindleholdertool assembly subassembly subassembly 5 Rigid connections between 4 all substructures r " Substructures XIII ... I Figure 312. Tapered thermal shrink fit holder with 19.1 mm diameter tool blank substructure model. Table 36. Shrink fit holdertool blank substructure parameters. 500 Frequency (Hz) Figure 313. Measured and predicted H11 results for tapered thermal shrink Substructure d, (mm) do (mm) L (mm) p (kg/m3) E (N/m2) 77 I 19.1 111.9 II 19.1 33.4 5.8 III 19.1 34.4 5.8 IV 19.1 35.4 5.8 V 19.1 36.4 5.8 2x1011 VI 19.1 37.5 5.8 7800 (steel) (steel) VII 19.1 38.5 5.8 14500 5.851011 0.0015 VIII 19.1 39.5 5.8 (carbide) (carbide) IX 19.1 39.5 4.1 X 19.1 40.4 4.1 XI 19.1 41.4 4.1 XII 19.1 41.4 10.6 XIII 41.4 37.4 6 x 10 6 x 10 2 0 2 4 7 x10 500 1000 Frequency (Hz) Figure 314. Measured and predicted H11 results for 20insert endmill. Table 37. 20insert endmill substructure parameters. Substructure I II III do (mm) 99.8 80.1 69.9 L (mm) 85.6 94.9 16.8 p (kg/m3) 17800 E (N/m2) 2x101 7 0.0015 Table 38. 28insert endmill substructure parameters. Substructure I II III IV do (mm) 126.2 130.3 80.0 69.9 L (mm) 55.0 18.3 62.7 18.3 o (kg/m3) 7800 E (N/m2) 2x101 7 0.0015 500 Frequency (Hz) Figure 315. Measured and predicted H11 results for 28insert. 1000 Frequency (Hz) Figure 316. Standard holder direct receptances two nominally identical, geared spindles (CAT 50 holderspindle interface). 7 x 10 8  4 4 7 x 10 0 O 15 10 1000 8 x 10 6 S0 6 x 10 0r & 5 10 L 100 2000 8 x 10 10L 100 500 Frequency (Hz) 1000 Figure 317. Measured and predicted H11 results for 16insert facemill. Results are shown for predictions from spindle 1 (dotted) and spindle 2 (dashed) standard holder measurements. Measurement recorded using spindle 1. Table 39. 16insert facemill substructure parameters. Substructure I II III do (mm) 279.4 63.5 69.9 L (mm) 27.2 88.9 15.9 S(kg/m3) 7800 E (N/m2) 2x1011 77 0.0015 Figure 318. Tapered heatshrink holder and tool blank assembly. Table 310. Dimensions for tapered heatshrink holder and tool blank assembly Test Number 1 2 3 4 Holder Command Command Command Tooling Manufacturer Innovations Holder Part H4Y4A0750 H4Y4A0750 H4Y4A1000 HSK63ASF Number 075315 Tool Distributor McMaster McMaster McMaster McMaster Carr Carr Carr Carr Tool Distributor 8788A431 8788A258 8788A263 8788A258 Number Di (mm) 19.1 19.1 25.4 19.1 D2 (mm) 33.0 33.0 43.9 35.3 D3 (mm) 41.4 41.4 52.6 38.1 Li (mm) 152.4 101.6 152.4 101.6 L2 (mm) 91.63 38.8 83.8 73.3 L3 (mm) 101.9 101.9 101.1 54.0 L4 (mm) 63.5 63.5 71.1 53.8 L5 (mm) 52.6 52.6 46.2 28.2 6 x 10 6 x 10 2 S2 500 1000 1500 2000 Frequency (Hz) 2500 Figure 319. The FRF for tapered heatshrink holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 1). The overhung tool length was 91.6 mm. 1000 1500 Frequency (Hz) 2000 2500 Figure 320. The FRF for tapered heatshrink holder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Number 2). The overhung tool length was 38.78 mm. x 10 1000 1500 Frequency (Hz) 2000 2500 Figure 321. The FRF for tapered heatshrink holder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 3). The overhung tool length was 83.81 mm. Predicted \  Measured I I I I x 10 5 Iw dh%.ei q 500 x 10 5. 0 i 1000 1500 Frequency (Hz) Figure 322. The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 4). The overhung tool length was 132.4 mm. Figure 323. Collet holder and tool blank assembly 2 0 U 2 5 t0o 5 6 <10 500 2000 2500 *2~~~ Table 311. Dimensions for collet holder and tool blank assembly Test Number 5 6 7 Holder Regofix Regofix Regofix Manufacturer Holder Part HSK HSK HSK Number A63/ER40120 A63/ER40120 A63/ER40120 Tool Distributor McMasterCarr McMasterCarr McMasterCarr Tool Distributor 8788A431 8788A431 8788A263 Number D1 (mm) 19.1 19.1 25.4 D2 (mm) 62.7 62.7 62.7 D3 (mm) 35.1 35.1 35.1 D4 (mm) 62.7 62.7 62.7 Li (mm) 152.4 152.4 152.4 L2 (mm) 132.4 99.4 132.4 L3 (mm) 91.5 91.5 91.5 L4 (mm) 46.0 46.0 46.0 L5 (mm) 25.5 25.5 25.5 xlO 10 1 0 Predicted  Measured 'V 6 x 10 0 500 1000 1500 2000 Frequency (Hz) 2500 Figure 324. The FRF for collet holder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 5). The overhung tool length was 132.4 mm. 500 1000 1500 2000 Frequency (Hz) Figure 325. The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 6). The overhung tool length was 99.4 mm producing an insertion length of 53.0 mm extending past the collet to the bottom of the holder. x 10 5 i t0 500 1000 1500 2000 Frequency (Hz) 2500 Figure 326. The FRF for collet holder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Number 7). The overhung tool length was 73.3 mm. 6 x 10 1 1 __ 6 x 10 1 Ctj 2500 Figure 327. Tapered heatshrink holder and fluted tool assembly. Figure 328. Collet holder and fluted tool assembly. Table 312. Dimensions for collet and tapered heatshrink holders and fluted tools. Test Number 8 9 10 11 Holder Command Command Tooling Regofix Manufacturer Innovations Holder Part H4Y4A0750 H4Y4A1000 HSK63ASF HSK Number 075315 A63/ER40120 Holder Type Tapered Heat Tapered Heat Tapered Heat Collet Holder shrink shrink shrink Tool Dataflute Unknown Unknown Regofix Manufacturer Tool Part SH40750 EC100S4 HEC750S2600 ARFST21000 Number Number of 4 4 2 2 Flutes D1 (mm) 19.1 25.4 19.1 25.4 D2 (mm) 33.0 43.9 35.3 62.7 D3 (mm) 41.4 52.6 38.1 35.1 L (mm) 152.4 152.4 101.6 62.7 L2 (mm) 79.2 80.2 121.5 100.8 L3 (mm) 101.9 101.1 54.0 132.4 L4 (mm) 63.5 71.1 53.8 91.5 L5 (mm) 52.6 46.2 28.2 46.0 L6 (mm) 48.3 48.3 94.0 48.3 6 x 10 Predicted S Measured a0  1O_____ II 6 x 10 0 Figure 329. 500 1000 1500 2000 2500 Frequency (Hz) The FRF for tapered heatshrink holder with 19.1 mm diameter, 152.4 mm long carbide, 2 fluted tool assembly. The overhung tool length was 121.5 mm. 6 x 10 1, Figure 330. 500 1000 1500 2000 2500 Frequency (Hz) The FRF for collet holder with 25.4 mm diameter, 127.0 mm long carbide, 2 fluted tool assembly. The overhung tool length was 100.8 mm. 6 x 10 1000 1500 Frequency (Hz) 2000 2500 Figure 331. The FRF for tapered heatshrink holder with 19.1 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 79.2 mm. II I I I 21 'I 2iiii 7 x 10 5 A Predicted Measured 7 x 107 5 10 0 500 1000 1500 2000 2500 Frequency (Hz) Figure 332. The FRF for tapered heatshrink holder with 25.4 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 80.2 mm CHAPTER 4 STABILITY ANALYSIS UNCERTAINTY Uncertainty estimates on milling stability analysis results, in the form of confidence intervals on stability lobe diagrams, requires knowledge of the toolpoint response and cutting coefficient statistics. For the purposes of this study, the toolpoint response statistics were determined by modal testing and prediction from the 3component machinespindleholdertool RCSA model. The cutting coefficient statistics will be determined through experimental testing. Monte Carlo methods will be used to generate uncertainty estimates on stability lobe diagrams generated through two stability analysis techniques, the Tlusty method [62] and the Budak and Altintas method [55]. Cutting Force Coefficient Determination The cutting force model for milling [66] is shown in eqs. (4.1) and (4.2) where, for each tooth in contact with the work piece material, Ft is the tangential cutting force, F, is the radial cutting force, Ktc is the tangential cutting force coefficient, Kr is the radial cutting force coefficient, Kte is the tangential edgecutting force coefficient, Kre is the radial edgecutting force coefficient, a is the axial depth of cut, and h is the instantaneous chip thickness. F, = Kah + K,a (4.1) F = Kah+Ka (4.2) The cutting force coefficients in milling are specific to the particular application, and new coefficients are required for every combination of tool geometry and work piece material. Cutting force coefficient values are also a function of spindle speed, and specific spindle speed ranges also require new coefficients. A mechanistic approach [66] was used to determine the cutting force coefficients. The first step in the procedure was to mount the work piece to a threeaxis dynamometer and select a consistent axial depth of cut, radial depth of cut, and spindle speed for all sample sets. For the case of this study, the radial depth of cut was selected as 100 percent (slotting), and the cutting coefficients for alternative radial depths of cut were assumed to be the same as the slotting case. The second step was to perform a sample milling cut and record the dynamometer force signal parallel (xdirection) and orthogonal (ydirection) to the direction of cut for a minimum of three different feed rates. The third step was to calculate the mean force, in each of the coordinate directions, for the steady state section of the cutting signal and perform a linear regression on average cutting force versus feed rate to determine the slope and vertical axis intercept of the line. Next, the edge coefficients and cutting coefficients were calculated from Eqs. (4.34.6) [66] where Nis number of cutter teeth, a is axial depth of cut, Fxe and Fye, are the vertical axis intercept values in the x and ycoordinate directions and Fx and Fyc are the slope values in the x and ycoordinate directions Kte = e (4.3) Kre xe (4.4) Na Na 4Fc 4Fx Ktc (4.5) Krc (4.6) Na Na Stability Analysis Techniques Stability lobe diagrams (see Fig. 41) identify stable and unstable cutting zones (separated by stability 'lobes') as a function of the chip width (or axial depth in peripheral end milling), a,,,, and spindle speed. Two analytical methods, based on the work of Tlusty [62] and Budak and Altintas [55], are used in this study to generate these diagrams. The methods are described in the following sections. Tlusty Method For the case of milling, Tlusty makes the following five assumptions as the basis of his analysis [62]: 1) the system is linear; 2) the direction of the variable component of cutting force is constant; 3) the variable component of cutting force depends only on vibration in the direction normal to the cut surface; 4) the value of the variable component of cutting force varies proportionally and instantaneously with the variation in chip thickness; and 5) although the location of the tooth in the cut varies with time, the system can be modeled as time invariant by analyzing the system stability at the average location of the tooth in the cut. To develop the stability lobe diagram, the specific cutting force, Ks, and cutting force direction, /f (the angle between the cutting force, F, and surface normal to the average tooth location in the cut as shown in Fig. 42), are required. Based on the mechanistically determined cutting force coefficients determined in Eqs. (4.5) and (4.6), values for Ks and /f can be calculated from Eqs. (4.7) and (4.8). The edgecutting force coefficients are not included in the analysis. K, = Ktc + Krc (4.7) t an Krtc (4.8) Krc The next step in the analysis was to determine the directional factors, ux and uy. These values are dependent on the type of operation, up milling or down milling, and the start and exit location of the cutter teeth, s,tar, and 0ext, as described in Fig. 42. The values of the directional factors can be calculated from Eqs. (4.94.12). Up milling or slotting: ux =cost + Cavg cosI 2 avg (4.9) uy =cos r oag)cos ( avg P) (4.10) Down milling: ux = cos [f + avg os 7 + avg (4.11) u., =cos(r avg)cos( +oavg +3) (4.12) Where: start + exit avg 2 The third step in the procedure is to determine the frequency response function at the tool tip in the x andydirections, as defined in Fig. 42. Traditionally, this has been accomplished through impact testing; however, for this study, the toolpoint FRF will be predicted with both impact testing and the 3component machinespindleholdertool RCSA technique. The real part of the toolpoint FRF in the xdirection and ydirection, Gx and Gy, and the imaginary part of the toolpoint FRF in the xdirection and ydirection, Hx and Hy, are required. These terms are substituted into Eqs. (4.13) and (4.14) with the appropriate directional factors to determine the oriented real, Goriented, and imaginary, Horiented, toolpoint FRF. Goriented = xGx + UyGy (4.13) Horiented = uxHx + uyHy (4.14) The final step in the procedure is to calculate the axial depth of cut, alim, from Eq. (4.15) and the relative spindle speed, Q, from Eq. (4.16). These values are determined for every chatter frequency, fc (values where Goriented is negative), and plotted relative to each other. The term, m*, is the average number of teeth in the cut, and P is the lobe number. The lobe number, P, is indexed by integer values to produce multiple lobes on the stability lobe diagram and N, as noted previously, is the number of cutter teeth. 1 fe alim (4.15) Q0= (4.16) 2Ks Re[Goriented]m 2m* P + 2j where: E = 2 K tan1 Goriented and m* exit + start 1I S'Horiented N 360 degrees Budak and Altintas Method Budak and Altintas [55] provide an alternative analytical stability model based on a multidegreeoffreedom stability model based on applied periodic system theory. The first step of the procedure is to define the cutting force as a function of time, as shown in Eq. (4.17), where a is the axial depth of cut, Kt is the tangential cutting coefficient, A(t) is the timevarying directional dynamic milling force coefficient matrix, and A(t) is the change in time matrix. The tangential cutting coefficient, Kt, is equal to Ktc as defined in Eq. (4.5). {F(t)}= aKt[A(t)]{A(t)} (4.17) 2 Due to the periodic nature of [A(t)], the matrix terms are expanded into a Fourier series. The average component of the Fourier series expansion is used and all other terms in the expansion are assumed to be negligible thus providing for the time invariant matrix provided in Eq. (4.18). The terms in the matrix are based only on the radial cutting coefficient, Kr, and the starting and exit angle of the cut (defined in Fig. 42). The radial cutting coefficient, Kr, is defined in Eq. (4.19) in terms of the experimentally determined cutting coefficients provided in Eqs. (4.5) and (4.6). Again, the edgecutting coefficients are not included in the analysis, and the axial cutting coefficients are not required. N axx Exy [A(0)] = 2 x (4.18) 27 a yx a yy where: a = [cos 2 2K, + K sin 2] xt 2 "start axy = sin 2 2 +Kr cos 2 )]it 2 start y = [ sin 2 + 2 + Kr cos 2. ] eit S2 start ay, = [cos24 2Kr )+Kr, sin24]Pext yy 2 start Kr c (4.19) Ktc The next step is to determine the direct toolpoint FRF in the xdirection and y direction, 4, and yy, respectively, to populate the matrix shown in Eq. (4.20). Traditionally, these values have been determined by impact testing but for the purposes of this study, they are both measured and predicted with the 3component machine holderspindletool RCSA method. #xx o M[]= 1 (4.20) 0 yyI The oriented transfer function is then obtained by multiplying Eq. (4.18) and (4.20) together to obtain Eq. (4.21). The two complex eigenvalues of 4o are determined for each chatter frequency and broken into their real, k1R and k2R, and imaginary components, kil and k21. [0= axxx txyyy (4.21) ayxxx aWyy yy Now, the limiting axial depths of cut, almi and al;m2, can be calculated for each eigenvalue based on Eqs. (4.22) and (4.23) where N is the number of cutter teeth. im 271R + (4.22) aim2 = 22R 1+ (4.23) iml NKt 1R NKt k 2R The final step in the procedure is to determine the relative spindle speed for each axial depth of cut. The spindle speeds, (21 and f2, for each eigenvalue are calculated using Eqs. (4.24) and (4.25) where P is indexed by integer values to produce each lobe andfc is the chatter frequency. 60*(2))f 60_*_(2_)_ f S (4.2460 (27)fc (4.25) SN(N 1 + 2PIT) N(;2 + 2P) where: s; = r2tan1 l 8;2 = T2tan1 K21 SIR ) \ 2R ) Uncertainty of Stability Analysis The goal of the uncertainty analysis is to provide a statement of confidence on the results predicted by the stability analysis. Due to the complexity of the modeling procedure and the multiple input variables to the modeling process, the uncertainty analysis technique selected is the Monte Carlo method [67, 68]. The Monte Carlo method is a numerical simulation technique that randomly samples model input parameters, based on their distribution, to determine the uncertainty of the model results. The strategy used to determine the uncertainty in stability is as follows: 1) use a mechanistic approach to experimentally determine cutting coefficients across a spindle speed range of interest, 2) determine a mean and standard deviation for the cutting coefficients based on a sample set consisting of the results at multiple spindle speeds, 3a) obtain multiple measurement sets of the toolpoint response using impact testing, if the uncertainty of the stability prediction is based on toolpoint response measurement variation, or 3b) obtain multiple measurement sets at location H33 and H33 on the standard holder using impact testing, if the uncertainty of the stability prediction is based on the threecomponent machinespindleholdertool RCSA model, 4) use a Monte Carlo simulation that randomly samples, for each simulation run, the cutting coefficient parameters, the toolpoint response measurement statistics or threecomponent machine spindleholdertool inputs, and the stability model input parameters to generate stability lobes, and 5) determine the statistical properties of the stability lobes based on multiple simulation runs to place uncertainty limits on stability lobe predictions based on both Tlusty and Budak and Altintas analytical techniques. For the Monte Carlo simulations that are based on toolpoint response measurements, there are two input parameters to the stability analysis: the cutting coefficients and the x andydirection FRF at the toolpoint. The statistical properties of both input parameters are based on multiple experimental measurements, as outlined in the following chapter. Therefore, the input parameters are assumed to have normal distributions. For the Monte Carlo simulations that are based on toolpoint responses predicted by the threecomponent spindleholdertool RCSA model, there are four input parameters to the stability analysis: the cutting coefficients, the standard holder FRFs, the geometric properties of the holder and tool, and the material properties of the holder and tool. The statistical properties of the cutting coefficients, standard holder FRFs, and geometric properties are again determined experimentally and are therefore assumed to have normal distributions. The range for the material properties of the holder and tool were determined from the literature [69] and are shown in Table 41 where holder is the density of the steel holder, pool is the density of the carbide tool, Eholder is the modulus of elasticity of the holder, Etool is the modulus of elasticity of the tool, and 7 is the structural damping factor for both the holder and tool. All material property variables were assumed to have a uniform distributions. During each simulation run, the input parameters are sampled and a stability lobe is generated based on either the Budak and Altintas or Tlusty stability prediction technique. It is important to note that both techniques produce multiple, overlapping lobes that may contain many loops; therefore, a technique to trim the lobes such that the limiting axial depths of cut are shown across the spindle speed range of interest is required. To accomplish this, a spindle speed is selected at the minimum of the spindle speed range of interest. The stability prediction techniques produce limiting axial depth of cut vectors for each lobe and relative spindle speed vectors that do not correspond exactly with the selected spindle speed. To determine the limiting axial depth of cut at the selected spindle speed, the limiting axial depth of cut for each lobe is interpolated based on the selected spindle speed and the next lower and higher spindle speeds from the analysis vectors. The minimum interpolated limiting axial depth of cut is selected from the minimum axial depth of cut values for each lobe. The selected spindle speed is now indexed, and the process is repeated multiple times until a limiting axial depth of cut has been selected for the entire spindle speed range of interest. After the simulation runs are completed, the mean and standard deviation of the limiting axial depth of cut at each 61 spindle speed is calculated allowing uncertainty limits to be placed on the stability lobe diagrams. Unstable A A Spindle Speed Figure 41. Stability lobe diagram. Y Up milling Lx F l /= 0 Surface normal .'' Rotation direction Figure 42. Geometry of milling process. Down milling Rotation Cut direction 62 Table 41. Material statistical properties for holder and tool. Properties Low High Holder (kg/m3) 7.81 x 103 7.83 x 103 pool (kg/m3) 1.40 x 104 15.0 x 104 Eholder (N/m2) 20.0 x 1010 20.7 x 1010 Etool (N/m2) 53.8 x 101" 63.0 x 1010 r 13.5 x 104 16.5 x 104 CHAPTER 5 EXPERIMENTAL VALIDATION OF MONTE CARLO SIMULATIONS Experimental validation of the Monte Carlo simulations to place uncertainty bounds on stability lobe diagrams (for both the Tlusty and Budak and Altintas techniques) were performed on two different milling machines. Simulation results for 25 percent radial immersion down milling cuts were generated for a Makino horizontal machine tool at Techsolve, Inc., using an Aluminum 7075T6 work piece, with a 12.7 mm diameter carbide tool mounted in a thermal heat shrink holder, and toolpoint response variation due to measurement differences at the toolpoint. Stability comparisons to the simulation results were generated from 25 percent radial immersion cut tests at a variety of spindle speeds and axial depth of cut combinations. Stability was determined for each cut test using two methods: 1) by qualitative observation, and 2) by measuring the audio signal and determining the chatter frequency content. Simulation results were also generated for a Mikron UCP600 Vario horizontal machine tool with a Step Tec HVCS 212X spindle, using an Aluminum 6061T6 work piece with a 19.1 mm diameter carbide tool mounted in a thermal heat shrink holder, and toolpoint response variation based on the threecomponent machinespindleholdertool model. Stability was determined for each cut test using three methods: 1) by qualitative observation, 2) by measuring the displacement of the tool in the x and ydirections during the cut, and 3) by measuring the chatter frequency of the tool displacement. Slotting and 50 percent radial immersion down milling machining conditions were investigated. Simulation results for the Mikron UPC600 Vario machining center were also generated for FRFs measured while the spindle was rotating. Experimental Determination of Cutting Force Coefficients Aluminum 6061T6 Work Piece and 19.1 mm Diameter, 4 Flute, Carbide Tool Following the procedure outlined in Chapter 4, cutting tests were performed on an Aluminum 6061T6 work piece using a 19.1 mm diameter, 4 flute, carbide helix tool (Dataflute part number SH40750). The cutting tool was mounted in a thermal heat shrink holder (Command part number H6Y4A0750) and tests were performed on a Step Tec 20,000 rpm spindle with an HSK 63A interface. Cutting tests were performed at three different chip loads (0.10 mm/tooth, 0.18 mm/tooth, and 0.25 mm/tooth) for a particular spindle speed and axial depth of cut (1 mm), and the cutting force signals were recorded with a dynamometer for the direction parallel to the cut direction (xdirection), and orthogonal to the cut direction (ydirection). The experimental setup is shown in Fig. 51 where the work piece is mounted to a threeaxis dynamometer for measuring the cutting forces. An example of a recorded cutting signal, based on a 7,500 rpm spindle speed and a 0.18 mm per tooth chip load, is shown in Fig. 52. The mean cutting force was determined for each signal by averaging the signal across the steady state region (as indicated in Fig. 52). At every chip load, the procedure was repeated three times, for a total of nine cutting tests at each spindle speed of interest. A linear regression was then performed on the nine chip load versus mean cutting force data points, for both the x and ydirection, to determine the slope and vertical axis intercept. For example, Fig. 53 shows the linear regression for a spindle speed of 7,500 rpm. Equations (4.44.5) were then applied to determine the cutting coefficients for the spindle speed of interest. The procedure was repeated at multiple spindle speeds to determine the cutting coefficients, Ktc and Krc, at nine different spindle speeds across the available spindle speed range and are displayed in Table 51. The cutting coefficients used in the Tlusty stability lobe technique, K, and fl, were determined using Eqs. (4.74.8) and the cutting coefficients used in the Altintas stability lobe technique, Kt and Kr, were determined using Eq. (4.19). As can be seen in Fig. 5.4 and Fig. 5.5, the cutting coefficient values decrease with speed up to approximately 8,000 rpm and then become approximately constant across the remaining spindle speed range up to 20,000 rpm. It has been hypothesized that cutting forces, and therefore the cutting coefficient values, decrease at higher spindle speeds due to thermal softening effects. In other words, it takes less force to cut the material when it is at increased temperatures. Another interesting aspect of the cutting coefficient values shown in Table 51 is that the radial cutting coefficient values, Ktc and Kr, become negative at higher spindle speeds. The reason that values are negative is a function of the mechanistic modeling technique. At higher speeds, the slope of the line fit to the xdirection mean force versus feed rate switches direction and the mean cutting forces in the xdirection increase with feed rate thus changing the sign of the radial cutting coefficient to negative. The spindle speed range of interest for the stability study was 7,500 to 20,000 rpm; therefore, the statistical properties (mean and standard deviation in this case)of the cutting force coefficients were determined based on a sample set comprised of cutting coefficient values from Table 51 for spindle speeds between 7,500 and 20,000 rpm. As seen in Fig. 5.4 and Fig. 5.5, the cutting coefficients are relatively constant across this range of spindle speeds. Therefore, determining the cutting coefficient statistical properties based on the cutting coefficient values across this range is reasonable. Table 52 shows the statistical properties of the cutting coefficients which were used in the Monte Carlo simulations for the Aluminum 6061T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool. The variable, ,o, is the standard deviation Aluminum 7475T6 Work Piece and 12.7 mm Diameter, 4 Flute, Carbide Tool Again following the procedure outlined in Chapter 4, cutting tests were performed on an Aluminum 7475T6 work piece using a 12.7 mm diameter, 4 flute, carbide helix tool (part number CRHEC500S4R30KC610M). The cutting tool was mounted in a thermal heat shrink holder and tests were performed on a Makino 24,000 rpm spindle with a CAT 40 interface at Techsolve, Inc. in Cincinnati, Ohio. Cutting tests were performed at four different chip loads (0.03, 0.05, 0.10, and 0.15 mm/tooth) for two spindle speeds (1,000 and 8,900 rpm) and axial depths of cut of 3.0 and 1.0 mm, respectively. The cutting tests were repeated three times at each of the spindle speeds and the cutting coefficients were determined following the same procedure that was applied to the Aluminum 6061T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool configuration. To determine the statistical properties of the cutting force coefficients, the results for the two spindles speeds were averaged and the standard deviations were determined. Table 53 shows the statistical properties of the cutting coefficients which were used in the Monte Carlo simulations for this tool and work piece combination. Stability lobe diagrams for Makino machining center Stability Determination Stability verification experiments were performed for the Makino machine tool, thermal shrink fit holder, using a 12.7 mm diameter, 4 flute, carbide helix tool, and an Aluminum 7475T6 work piece combination. The radial immersion of the cutting tool was 3.175 mm, or 25 percent, and a down milling operation was performed. A variety of cut tests were performed at various spindle speeds and axial depth of cut combinations. During the actual testing, stability was determined qualitatively based on the audio pitch produced during the cut and the surface quality of the machined surface. An unstable cut will produce additional audio content at the natural frequency of the toolpoint in addition to content at synchronous frequencies based on spindle speed selection. Unstable cutting conditions also produce a very rough machined surface. It was necessary to qualitatively evaluate the stability of the cut during testing to avoid damaging the tool or spindle. In addition to the qualitative assessment of stability, stability was also determined quantitatively for each spindle speed and axial depth of cut combination based on the audio signal produced during the cut [44]. As each cutting test was performed, a microphone was used to record the audio signal. The signal was then Fourier transformed to the frequency domain and inspected for content above a specified threshold determined to be chatter (as determined by the experimenter). For the purposes of this experiment, chatter content above 20 signified unstable cutting conditions. Figures 56 and 57 display examples of audio signal frequency content for stable and unstable cutting conditions, respectively. As seen in Fig. 56, frequency content does exist at the tooth passing frequency, as identified, and at additional frequencies due to noise and onceperrevolution runout. This is to be expected. As seen in Fig. 5.7, however, a large amount of content at a particular frequency is associated with chatter. A summary of the stability results is displayed in Fig. 58. Monte Carlo Simulation Parameters The Monte Carlo simulation procedure requires selecting random samples from the input variable distributions and computing the output over multiple iterations. In this study it was necessary to consider the correlation between input variables. The first set of input parameters for the Monte Carlo simulation is the cutting coefficients, Ktc and Kc. The correlation between the two parameters was determined to be 97 percent. High correlation exists between the tangential and radial cutting coefficients because they are based on the tangential and radial cutting forces which are components of the same cutting force vector. As the cutting force increases, both components also typically increase. During each run of the simulation, values for Ktc and K, were randomly generated based on a multivariate normal distribution, the cutting coefficient means displayed in Table 53, and the covariance matrix shown in Eq. (5.1). The diagonal elements in the covariance matrix represent the cutting coefficient variances and the off diagonal elements represent the covariances between the cutting coefficients. The appropriate cutting coefficients, K, and / for the Tlusty technique were determined in the simulation based on Eqs. (4.74.8) and the appropriate cutting coefficients, Kt and K, for the Altintas technique were determined based on Eq. (4.19). C 6.76x1015 6.04x1015' Covariance Matrix= 5(m2) (5.1) 6.04x1015 5.70x1015 (5.1) For the FRF data, five measurement sets were performed in both the x and y directions with the holder removed from the spindle and the tool removed from the holder between each set. The mean FRFs for the x and ydirection and 95 percent confidence intervals are shown in Fig. 59 and Fig. 510, respectively. For the Monte Carlo simulation, 100% correlation between individual frequencies and between the real and imaginary parts of the complex FRF was applied. In other words, a single random value (from a unit variance normal distribution) was used to select the real and imaginary values at each frequency of the FRF. See Eq. (5.2), where a is the random variable, F is the input variable value for a given iteration of the Monte Carlo simulation, F is its mean value, and ca is the standard deviation. The strong correlation between frequencies and real/imaginary parts occurs because the data is collected simultaneously in impact testing. It should be noted, however, that the uncertainty was frequency dependent, i.e., aF was a function of frequency in Eq. (5.2), and was larger near resonance as shown in Figs. 59 and 510. F=F+ax oF (5.2) Another consideration for the FRF data was potential correlation between the x and ydirection measurements. It is possible that if the variation between measurements was caused by, for example, a change in the connection between the holder and spindle, then both directions could be influenced in a similar manner. Therefore, the covariance between the FRFs measured in the two directions, oxy, was evaluated. The result is shown in Fig. 511; it is seen that the covariance is strongly dependent on frequency with the highest values near resonance and near zero levels everywhere else. Because 100% correlation between frequencies and the real and imaginary parts for the individual directions was already identified, it was not possible to allow a frequency dependent correlation between the two directions. Therefore, zero and 100 percent correlation was investigated between the x and y directions. The actual measured correlation was determined to be 93 percent. Stability Lobe Results Onethousand Monte Carlo simulation runs were performed for the Makino machining center, thermal heat shrink holder, 12.7 mm diameter, 4 flute carbide helix tool, and 7475T6 Aluminum work piece combination to place uncertainty boundaries on stability lobe predictions based on both the Budak and Altintas and Tlusty analytical techniques. The predictions were for a 25 percent radial immersion down milling cut and based on randomly generated values for the cutting coefficients and toolpoint response as described in previous sections. Figures 512 and 513 show the comparison between predicted stability lobes with 95 percent confidence intervals (based on the Budak and Altintas technique) and experimental results. Figure 512 shows the case where the x and ydirection toolpoint FRFs are not correlated and Fig. 513 displays the case where the x and ydirection toolpoint FRFs are 100 percent correlated. As seen in the figures, the correlation level between the x and ydirection FRFs does not significantly alter the predicted results. Figures 514 and 515 show the comparison between predicted stability lobes with 95 percent confidence intervals based on the Tlusty technique as compared to experimental results. Figure 514 shows the case where the x and ydirection toolpoint FRFs are not correlated, and Fig. 515 displays the case where the x and ydirection tool point FRFs are 100 percent correlated. As seen in the figures, the correlation level between the x and ydirection FRFs, when using the Tlusty stability technique, does have a significant effect on the uncertainty boundaries of the predicted results. The Tlusty stability analysis technique creates directional orientation factors which lump the x and ydirection FRFs into a single, oriented FRF; therefore, the correlation level between the x and ydirection FRFs significantly affects the uncertainty levels of the final stability lobes. The Budak and Altintas stability technique does not couple the x and ydirection FRFs; therefore, the uncertainty levels of the predicted stability lobes are not as sensitive to the x andydirection FRF correlation. A comparison between Figs. 512 and 514 shows that the confidence intervals for the Tlusty lobes are wider for equivalent input mean values, distributions, and correlations as compared to the Budak and Altintas technique generated stability lobes. In addition, the Tlusty mean stability boundary tends to overpredict the experimental stability limit for these tests. This may be the result of the asymmetric x and ydirection FRFs. Sensitivity of Budak and Altintas and Tlusty Stability Lobe Prediction Techniques The Monte Carlo simulation can be used to determine the sensitivity of the response uncertainty (the stability lobes in this case) to the uncertainties of each of the input parameters (the cutting coefficients and toolpoint response). This is accomplished by selecting random samples from a single input variable distribution while the other input variables are held constant (i.e., their mean values are used in the simulation). In this manner, the contribution of each input parameter's uncertainty to the response uncertainty can be determined and the variables with the greatest contributions can be identified and their variation minimized, if possible. For example, the uncertainty levels of the highest contributing input variables might be reduced by collecting additional data. For the Makino machining center, with a thermal heat shrink holder, a 12.7 mm diameter, a 4 flute carbide helix tool, a 7475T6 Aluminum work piece combination, and 25 percent radial immersion down milling, the sensitivities of the toolpoint response and cutting coefficients were determined for both the Budak and Altintas and Tlusty analytical stability lobe prediction techniques based on 1000 Monte Carlo simulation runs and assuming 100 percent correlation between the x andydirection FRFs. Figure 516 displays the stability lobe uncertainty based solely upon cutting coefficient uncertainty for the Altintas technique, and Fig. 517 displays stability lobe uncertainty based on tool point response uncertainty for the Budak and Altintas technique. As shown in the figures, the uncertainty due to both cutting coefficient variation and toolpoint response variation is relatively constant across the spindle speed range with a slightly larger contribution coming from toolpoint response variation. Figure 518 displays the stability lobe uncertainty based solely upon cutting coefficient uncertainty for the Tlusty technique, and Fig. 519 displays stability lobe uncertainty based on toolpoint response for the Tlusty technique. As shown in the figures, the uncertainty contribution from each input variable is similar; however, it is interesting to note in Fig. 519 that the stability lobe uncertainty is greatest on the left side of the lobes (i.e., to the right of the peaks) and reduces as the spindle speed increases for that particular lobe. This is expected as the lobe peaks at 10,000 and 15,000 rpm correspond to the resonance conditions of the tool point response (as illustrated in the toolpoint FRF to stability lobe mapping example shown in Fig. 520). As seen in Figs. 59 and 510, resonance occurs at approximately 2,000 Hz and the uncertainty is greatest at resonance and decreases with increased frequency. Stability Lobe Diagrams for Mikron Machining Center Tests Stability Determination For the Mikron UCP600 Vario machining center, with a thermal heat shrink holder, a 19.1 mm diameter, a 4 flute carbide helix tool, and a 6061T6 Aluminum work piece combination, two milling conditions were investigated, with slotting and 50 percent radial immersion down milling. A qualitative assessment of stability was again performed during the experimental process; however, a test rig, as displayed in Fig. 521, was attached to the spindle to monitor tool displacement during the milling process. Two capacitive probes, one in the xdirection and one in the ydirection, measured the displacement of the tool during the experimental cut, and a tachometer used in conjunction with a color mark on the holder identified a consistent location during every rotation of the tool. In this manner, it was possible to determine the tool displacement at a specific angle of the holder tool assembly during each rotation (i.e., onceperrevolution sampling). For stable cuts, the onceperrevolution displacement was very consistent; therefore, stability was quantitatively determined by comparing the variance of the once perrevolution displacement to the variance of the total tool displacement. Since both the x and ydirection displacements were being measured, the joint variance, ac2, was determined based on Eq. (5.3) below where ax2 and aU2 are the variances of the x andy direction displacements, respectively: 2 2 2 (5.3) The ydirection orthogonall to the feed or xdirection) displacement signal was also Fourier transformed to look for chatter content in the frequency domain. The chatter content in the xand ydirection were identical; therefore, the ydirection was selected for analysis purposes. Based on the surface quality of the cuts, it was decided that a variance ratio, R as shown in Eq. (5.4), between the onceperrevolution displacement and the total cycle displacement of less than 0.2 was a stable cut, between 0.2 and 0.8 was a marginal cut, and greater than 0.8 was an unstable cut. The variance ratio ranges to determine stable, marginal, and unstable cuts were selected qualitatively based on the surface finish of the milled surface. Further testing is required to determine if these variance ratio ranges can be used for other spindleholdertoolwork piece combinations. 2 R c, once per revolution displacement (5.4) Oc, total displacement To illustrate the testing results, the xdirection versus ydirection total displacement and onceperrevolution displacement signals and the frequency content of the ydirection displacement are displayed in Figs. 522 through 527 for slotting experiments. Figure 5 22 shows the results for an 8,000 rpm, 2 mm axial depth stable cut. In Fig. 522A, the onceperrevolution samples are grouped much more tightly than the total displacement samples shown in the light grey. In Fig. 522B, the frequency content of the ydirection displacement is plotted with the synchronous content identified. As seen in Fig. 522B, there is no chatter frequency content or, in other words, nonsynchronous content. The synchronous content is always present and is due to tool run out and tooth passing frequencies and their harmonics. Figure 523 shows the results for an 8,000 rpm, 2.5 mm axial depth unstable cut. As seen in Fig. 523A, the onceperrevolution samples now have the same spread as the total displacement. Also, Fig. 523B shows chatter content at 799 Hz. Figures 524 and 525 display the results for a 10,000 rpm, 5 mm and 6 mm cut, respectively. This is the optimum machining speed where the tooth passing frequency is matched to the resonance frequency of the toolpoint. In Fig. 524A, the onceper revolution x versus y samples are closely grouped in comparison to the x versus y total displacement, and in Fig. 525A, the onceperrevolution samples are marginally grouped in comparison to the total displacement. The advantage of using the onceperrevolution displacement signal and total displacement signal plotted in the xdirection versus y direction to determine stability is exemplified in Fig. 525B. It is difficult to determine chatter content from the frequency plot because the synchronous frequency and the chatter frequency fall on the same point. Figures 526 and 527 show the results for a 15,000 rpm, 2 and 3 mm axial depth cut, respectively. The stability results are summarized in Table 54 for all spindle speed and axial depth of cut combinations. The xdirection versus ydirection total displacement and onceperrevolution displacement signals and the frequency content of the ydirection displacement are again displayed in Figs. 528 through 531 for 50 percent radial immersion experiments at a variety of spindle speed and axial depth of cut combinations. Applying the same criteria as for the slotting cuts, the stability results are summarized in Table 55. Monte Carlo Simulation Parameters The toolpoint response for the Step Tec spindle, thermal heat shrink holder, and 4 tooth, carbide helix tool was determined using the threecomponent spindleholdertool RCSA model. Therefore, the input parameters to the Monte Carlo simulation were cutting coefficients, standard holder measurements, holder and tool material properties, and holder and tool geometries. The first set of input parameters for the Monte Carlo simulation was the cutting coefficients, Kt and Kc, for the tool and Aluminum 6061T6 work piece. The correlation between the two parameters was determined to be 93 percent. During each run of the simulation, values for Ktc and K, were randomly generated based on a multivariate normal distribution, the cutting coefficient means displayed in Table 52, and the covariance matrix shown in Eq. (5.5). The appropriate cutting coefficients, K, and /f for the Tlusty technique were determined in the simulation based on Eqs. (4.74.8) and the appropriate cutting coefficients, Kt and Kr, for the Budak and Altintas technique were determined based on Eq. (4.19). As mentioned previously, the diagonal elements in the covariance matrix represent the cutting coefficient variances, and the offdiagonal elements represent the covariances between the cutting coefficients. 7.00xl013 1.10xl014 Covariance Matrix= 1.0 x14 (m2) (5.5) 1.10x1014 9.91x1015 The second set of input parameters to the Monte Carlo simulation were the material properties of the holder and tool. A uniform distribution was assumed since the high and low range values were determined from the literature. Therefore, the standard deviation is shown in Eq. (5.6) [70] where ,mat is the standard deviation of the material property of interest and Rm is the range of the material property of interest. All parameters and their range of values are summarized in Table 41 with uniform distributions assumed for all parameters. The next set of input parameters were the geometric properties of the holder and tool. The relevant dimensions were measured multiple times with digital calipers to determine the mean values, as displayed in Fig. 532 and Table 56. A worst case standard deviation on all measurements using the calipers was determined as 0.0254 mm. Since the measurements were determined experimentally, normal distributions were used for all geometric parameters. Prior to adjustment to the mass and 2nd area moment of inertia, the fluted section diameter was assumed to be equal to the shank diameter. Rmt = (5.6) The final set of input parameters to the Monte Carlo simulation was the x and y direction standard holder measurements, H33 and H33c. For the static case, where the standard holder was not rotating, ten sets of measurements at each standard holder location were performed under two sets of conditions. For the first set of conditions, the standard holder was not removed from the spindle between measurements sets and the mean and 95 percent confidence intervals are shown for the standard holder locations in Figs. 533 through 536. For the second set of conditions, the standard holder was removed from the spindle between measurements to account for clamping variation. No appreciable difference was seen in the standard holder measurements for the two conditions. The measurement technique for the case where the standard holder was rotating during measurement will be explained in an upcoming section. Another consideration for the FRF data was potential correlation between the x and ydirection measurements. The measured correlation was determined to be 93 percent; therefore 100 percent correlation was assumed in the Monte Carlo simulation. As described previously in the Makino spindle section, 100% correlation between frequencies and the real and imaginary parts for the individual directions was also identified. Stability Lobe Results Onethousand Monte Carlo simulation runs were performed for the Mikron UCP 600 machining center, with a thermal heat shrink holder, a 19.1 mm diameter, a 4 flute carbide helix tool, and a 6061T6 Aluminum work piece combination. The first result of the simulation was to provide 95 percent uncertainty bounds on the toolpoint response in the x and ydirection as predicted by the threecomponent spindleholdertool RCSA model. Figures 537 and 538 show the toolpoint response in the x and ydirection, respectively for the case where holder clamping force variation was not included. Figures 539 and 540 show the toolpoint response in the x and y direction, respectively, for the case where holder clamping force variation was included. As seen in the figures, holder clamping force variation does have an effect on the mean toolpoint response although the range of the 95 percent confidence interval bounds is unaffected. Figure 541 displays the predicted Budak and Altintas stability lobes and uncertainty as compared to the measured stability results for slotting for the case where holder clamping force variation is excluded. As seen in the figure, the predicted stability results are biased to higher spindle speeds than the measured results. This result is consistent for all cases where the stability results are based on nonrotating measurements of the standard holder FRFs. Future sections will show that this bias is due to the spindle dynamics changing with speed. Figure 542 displays the predicted Tlusty stability lobes and uncertainty as compared to measured stability results for slotting for the case where holder clamping force variation is excluded. The uncertainty levels are higher for the Tlusty technique versus the Budak Altintas technique and the mean stability lobe predictions are at higher axial depths of cut than measured stability results. These results are consistent with the results for the Makino spindle. The stability lobes based on the Tlusty technique in Fig. 542 also show a region between 10,000 and 12,000 rpm where no stability lobes or uncertainty levels exist. The Tlusty technique does not provide overlapping lobes in this region because the frequency vectors associated with each lobe produced by the Tlusty stability analysis technique are based only on the negative real part of the toolpoint response. In this case, the negative real part of the toolpoint response has a very small range, thus creating frequency vectors that do not overlap. Figure 543 and 544 show the predicted stability lobes and uncertainty levels for the Budak and Altintas and Tlusty techniques, respectively, for 50 percent radial immersion down milling where holder clamping force variation is excluded. The 50 percent radial immersion results follow the same general trends as the slotting stability results. Finally, for comparison purposes, Figs. 545 and 546 show stability lobe predicted results with uncertainty levels and measured stability results for the Budak and Altintas technique and slotting and 50 percent radial immersion cuts, respectively. No significant changes are seen in the uncertainty levels when holder clamping force is included. Spindle Speed Dependent FRF Issues As noted previously, a bias existed between the predicted stability results based on nonrotating standard holder measurements and the measured stability results. A technique to measure the standard holder FRFs while the spindle was rotating and process the resulting data to eliminate the synchronous signal content was created to generate stability lobes based on spindle dynamics at speed. The experimental setup for the measurement technique is shown in Fig. 547 where an impact hammer inputs a measured force into the system and a capacitive probe measures the resulting displacement of the standard holder while the standard holder rotates at a specified speed. A time signal of the force input and resulting standard holder displacement is determined in this manner. To generate a mean FRF, the process was repeated 100 times. A fit of the synchronous part of the time signal (due to standard holder runout) was performed and subtracted from the total signal to generate the transient part of the signal. To further eliminate synchronous content, 100 time signals, after fitting and synchronous component subtraction, were ensemble averaged to produce the final time signals. The final time signals for the force input and standard holder displacement were then Fourier transformed to the frequency domain to produce the required mean FRF. The standard deviation of the FRF was determined by Fouriertransforming each individual signal (100 signals total) after removing the synchronous content from each signal and determining the standard deviation at each frequency. Figure 548 shows the magnitude of H33x as determined without rotation, at 10,000 rpm, and at 16,000 rpm. As seen in the figure, the magnitude changes significantly as a function of spindle speed. Figure 549 shows the predicted toolpoint response based on standard holder measurements at 0 rpm, 10,000 rpm, and 16,000 rpm. Again, the figure shows that the spindle dynamics are changing as a function of spindle speed. Also of interest is a comparison of the predicted toolpoint FRFs based on rotating standard holder measurements versus the location of the measured chatter frequencies, as displayed in Table 54. If the predicted toolpoint FRF is representative of the spindle dynamics, the tool should chatter, during unstable cuts, at frequencies near the toolpoint FRF modes (natural frequencies). Figure 550 and 551 display the x and ydirection predicted toolpoint FRFs, respectively, based on standard holder measurements at 10,000 rpm and the chatter frequencies determined during unstable cutting conditions for slotting cuts. As expected and seen in the figures, the chatter frequencies are very close to the toolpoint FRF natural frequency for the lower mode. The Monte Carlo simulation was modified to accept standard holder FRFs determined at a spindle speed of 10,000 rpm. The standard holder FRFs at 10,000 rpm were used because they were representative of all the FRFs across the spindle speed range of interest and the noise level of the FRFs at this speed were minimal in respect to the FRFs measured at higher spindle speeds. Figures 552 through 555 display the standard holder mean and 95 percent confidence intervals for all measurement locations. As seen in the figures, the variation of the standard holder FRFs increases in speed as compared to standard holder FRFs produced based on nonrotating measurements. Measuring the standard holder while it is rotating introduces increased uncertainty into the Monte Carlo model. The predicted stability lobes based on the Budak and Altintas technique, with 95 percent confidence intervals and measured stability results, are shown in Fig. 556 for a slotting cut on the Step Tec spindle with a tapered heat shrink holder, a 4 tooth, carbide helix tool and an Aluminum 6061T6 work piece. As seen in the figure, the bias has been eliminated and the predicted and measured stability results compare nicely. The uncertainty levels have increased, however, due to the increased uncertainty of measuring the rotating standard holder FRFs. Figure 557 again shows reasonable agreement between predicted and measured stability lobe results for the same system with a 50 percent radial immersion cut. Figures 558 and 559 show comparison between predicted stability lobes based on static standard holder measurements versus rotational standard holder measurements at 10,000 rpm for slotting and 50 percent radial immersion cuts, respectively. The Tlusty stability lobe technique did not produce realistic results as the negative real part of the oriented FRF was very small thus producing only a single stability lobe in the speed range of interest, Sensitivity of the Budak and Altintas Stability Lobe Prediction Techniques when the Threecomponent Spindleholdertool RCSA Model is Used to Generate Tool point Response. The sensitivity of the stability lobe uncertainty to the uncertainties of each of the input parameters (the cutting coefficients, material properties, geometries, and standard holder measurements at 10,000 rpm) was determined for the Budak and Altintas stability lobes for the Step Tec spindle, thermal heat shrink holder, 4 tooth, carbide helix tool, and Aluminum 6061 work piece. As shown in Fig. 560 for the slotting case, the geometric properties and material properties contribute very little to the stability lobe uncertainty while the cutting coefficient uncertainty is the greatest contributor. Figure 561 shows the sensitivity results for the 50 percent radial immersion cut. Again, the cutting coefficient uncertainty and standard holder measurement uncertainty provide the majority of the final stability lobe uncertainty. As explained in the previous section, it was not possible to generate realistic Tlusty stability lobes based on rotational standard holder measurements; therefore, the sensitivities of the Tlusty technique cannot be determined for this particular system setup. Figure 51. Experimental setup for measuring cutting force signals. 200 100 0 100 200 300 200 100 0 100 200 Figure 52. Cutting force signal for 7500 rpm cut with 0.18 mm/tooth chip load. 160 140 S120 100 80 0.100 0.250 0.180 Chip load (mm/tooth) Figure 53. Linear regression for cutting force means at a spindle speed of 7,500 rpm. Steady state region II I ) 1 2 3 4 5 Time (seconds) I Time (seconds) O O I_ 