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Milling Dynamics Prediction and Uncertainty Analysis Using Receptance Coupling Substructure Analysis


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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 FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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Copyright 2006 by Gregory Scott Duncan

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This document is dedicated to my wife Claire.

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iv ACKNOWLEDGMENTS I thank my advisor, Dr. Tony Schmitz, a nd my committee, Dr. John Ziegert, Dr. John Schueller, Dr. Kurtis Gurley, and Dr. Na garaj Arakere, and the students of the Machine Tool Research Center. I also thank my parents.

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v TABLE OF CONTENTS page ACKNOWLEDGMENTS.................................................................................................iv LIST OF TABLES............................................................................................................vii LIST OF FIGURES...........................................................................................................ix ABSTRACT.....................................................................................................................xv i CHAPTER 1 INTRODUCTION........................................................................................................1 Justification of Work....................................................................................................1 Literature Review.........................................................................................................3 Tool Point Dynamic Modeling..............................................................................3 Machining Stability Investigations........................................................................4 Scope of Work..............................................................................................................5 2 RECEPTANCE COUPLING MODEL DEVELOPMENT..........................................6 Background and Notation.............................................................................................6 Flexure Model...............................................................................................................8 Previous Machine-Spindle-Hold er-Tool Modeling Technique..................................10 Improved Machine-Spindle-Holder Tool Modeling Technique.................................12 Spindle-holder Base Subassembly Identification................................................12 Standard test holder receptances..................................................................13 Extended holder subassembly model...........................................................14 Spindle-holder base subassembly receptance..............................................15 Tool-point Response Prediction..........................................................................16 3 EXPERIMENTAL VALIDAT ION 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 Fin ite Difference Method Evaluation........................22 Holder Experimental Verification.......................................................................25 Experimental Results for 24,000 rpm, HSK 63A Spindle..........................................25 Holder Experimental Verification.......................................................................26

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vi Tool-point Response Prediction..........................................................................26 Experimental Results for Geared, Quill-Type 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 Heat-shrink Holder and Carbide Tool Blank Results............................29 Collet Holder and Carbide Tool Blank Results...................................................29 Fluted Tool Results..............................................................................................29 4 STABILITY ANALYSIS UNCERTAINTY.............................................................52 Cutting Force Coefficient Determination...................................................................52 Stability Analysis Techniques....................................................................................53 Tlusty Method.....................................................................................................54 Budak and Altintas Method.................................................................................56 Uncertainty of Stability Analysis................................................................................58 5 EXPERIMENTAL VALIDATION OF MONTE CARLO SIMULATIONS............63 Experimental Determination of Cutting Force Coefficients.......................................64 Aluminum 6061-T6 Work Piece and 19.1 mm Diameter, 4 Flute, Carbide Tool..................................................................................................................64 Aluminum 7475-T6 Work Piece and 12.7 mm Diameter, 4 Flute, Carbide Tool..................................................................................................................66 Stability lobe diagrams for Makino machining center...............................................66 Stability Determination.......................................................................................66 Monte Carlo Simulation Parameters...................................................................68 Stability Lobe Results..........................................................................................70 Sensitivity of Budak and Altintas and Tlusty Stability Lobe Prediction Techniques.......................................................................................................71 Stability Lobe Diagrams for Mi kron Machining Center Tests...................................72 Stability Determination.......................................................................................72 Monte Carlo Simulation Parameters...................................................................75 Stability Lobe Results..........................................................................................77 Spindle Speed Dependent FRF Issues.................................................................79 Sensitivity of the Budak and Altintas Stability Lobe Prediction Techniques when the Three-component Spindle-hol der-tool RCSA Model is Used to Generate Tool-point Response.........................................................................81 6 CONCLUSIONS AND FUTURE WORK...............................................................117 Conclusions...............................................................................................................117 Future Work..............................................................................................................119 APPENDIX BEAM RECEPTANCE MODELING.....................................................120 LIST OF REFERENCES.................................................................................................123 BIOGRAPHICAL SKETCH...........................................................................................129

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vii LIST OF TABLES Table page 2-1 Mass and inertia adjustment ratios...........................................................................20 3-1. Flexure modal parameters........................................................................................31 3-2 Standard holder substructure parameters.................................................................31 3-3 Collet holder substructure I, II, and III parameters..................................................35 3-4 Standard holder substructure parameters.................................................................36 3-5 Shrink fit holder (25.3 mm bore) substructure parameters......................................37 3-6 Shrink fit holder-tool bla nk substructure parameters...............................................39 3-7 20-insert endmill subs tructure parameters...............................................................40 3-8 28-insert endmill subs tructure parameters...............................................................40 3-9 16-insert facemill substructure parameters..............................................................42 3-10 Dimensions for tapered heat-shri nk holder and tool blank assembly......................43 3-11 Dimensions for collet hold er and tool blank assembly............................................46 3-12 Dimensions for collet and tapered he at-shrink holders and fluted tools..................49 4-1 Material statistical prope rties for holder and tool....................................................62 5-1 Cutting coefficients for aluminum 60 61-T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool.............................................................................................84 5-2 Statistical properties for cutting coefficients for aluminum 6061-T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool....................................................85 5-3 Statistical properties for cutting coefficients for aluminum 7475-T6 work piece and 12.7 mm diameter, 4 flute, carbide helix tool....................................................85 5-4 Stability results for slotting cuts on a Mikron Vario machining center...................97

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viii 5-5 Stability results for 50 % radial im mersion cuts on a Mikron UCP-600 Vario machining center......................................................................................................99 5-6 Statistical geometric properties of tapered heat shrink hol der and 4 flute, 19.1 mm diameter carbide helix tool..............................................................................100

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ix LIST OF FIGURES Figure page 2-1 Two-component assembly. The compone nt responses are coupled through a rigid connection to give the assembly receptance(s)................................................18 2-2 Two-component flexure assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s)................................18 2-3 Previous two-com ponent RCSA model...................................................................19 2-4 Example standard holder for spindl e-holder base subassembly receptance identification.............................................................................................................19 2-5 Standard holder substructures for inverse receptance coupling...............................19 2-6. Spindle-holder-tool substructures for tapered thermal heat shrink holder and tool blank.........................................................................................................................2 0 2-7 End view of two fluted tool show ing rotation angles for area and inertia calculations...............................................................................................................20 3-1 Plot of G11 receptances for flexure system...............................................................31 3-2 Plot of G33 receptances for small holder..................................................................32 3-3 Standard holder geometry comparison.....................................................................32 3-4 Tool-point response base on 1rst-order finite difference method..............................33 3-5 Tool-point response base on 2nd-order finite difference method..............................33 3-6 Measured versus predicted tool-point response based on 1st-order and 2nd-order finite difference method and ten averaged measurement sets..................................34 3-7 Collet holder substructure I, II, and III parameters..................................................34 3-8 Collet holder H33 predicted and measured results....................................................35 3-9 Spindle receptances G55( ) determined from standard holder direct and cross receptance meas urements.........................................................................................36

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x 3-10 Tapered thermal shrink fit holder ( 25.3 mm bore) substructure model...................37 3-11 Measured (two nominally iden tical holders) and predicted H33 results for tapered thermal shrink fit holder (25.3 mm bore).................................................................38 3-12 Tapered thermal shrink f it holder with 19.1 mm diameter tool blank substructure model........................................................................................................................38 3-13 Measured and predicted H11 results for tapered thermal shrink...............................39 3-14 Measured and predicted H11 results for 20-insert endmill.......................................40 3-15 Measured and predicted H11 results for 28-insert.....................................................41 3-16 Standard holder direct receptances two nominally identical, geared spindles (CAT 50 holder-spindle interface)...........................................................................41 3-17 Measured and predicted H11 results for 16-insert facemill. .....................................42 3-18 Tapered heat-shrink holder and tool blank assembly...............................................42 3-19 The FRF for tapered heat-shrink ho lder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Numb er 1). The overhung tool length was 91.6 mm....................................................................................................................43 3-20 The FRF for tapered heat-shrink ho lder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Numb er 2). The overhung tool length was 38.78 mm..................................................................................................................44 3-21 The FRF for tapered heat-shrink ho lder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Numb er 3). The overhung tool length was 83.81 mm..................................................................................................................44 3-22 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 3-23 Collet holder and tool blank assembly.....................................................................45 3-24 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 3-25 The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 6).............................................................................47 3-26 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 3-27 Tapered heat-shrink holder and fluted tool assembly..............................................48

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xi 3-28 Collet holder and fl uted tool assembly.....................................................................48 3-29 The FRF for tapered heat-shrink ho lder with 19.1 mm diameter, 152.4 mm long carbide, 2 fluted tool assembly. The overhung tool length was 121.5 mm.............49 3-30 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 3-31 The FRF for tapered heat-shrink ho lder with 19.1 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 79.2 mm...............50 3-32 The FRF for tapered heat-shrink ho lder with 25.4 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly. The overhung tool length was 80.2 mm...............51 4-1 Stability lobe diagram..............................................................................................61 4-2 Geometry of milling process....................................................................................61 5-1 Experimental setup for meas uring cutting force signals..........................................82 5-2 Cutting force signal for 7500 rpm cu t with 0.18 mm/tooth chip load......................83 5-3 Linear regression for cutting force means at a spindle speed of 7,500 rpm.............83 5-4 Cutting coefficients versus spindle sp eed for the Budak and Altintas stability lobe technique..........................................................................................................84 5-5 Cutting coefficients versus spindle speed for Tlusty stability lobe technique.........85 5-6 Audio signal frequency content fo r 10,000 rpm spindle speed, 1.02 mm axial depth cut. Cut was determined to be stable.............................................................86 5-7 Audio signal frequency content resu lts for 19,000 rpm spindle speed, 1.52 mm axial depth cut. Cut was determined to be unstable................................................86 5-8. Stability results for Makino machin ing center, thermal heat shrink holder, 12.7 mm, 4 flute, carbide helix tool, Alum inum 7475-T6 work piece and a 25 percent radial immersion cut.................................................................................................87 5-9 Mean and 95 percent (2 ) confidence intervals for x -direction FRF.......................87 5-10 Mean and 95 percent confidence intervals for y -direction FRF...............................88 5-11 Covariance between x and y -direction FRFs..........................................................88 5-12 Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 percent c onfidence intervals are shown for the case where the x and y -direction FRFs are not correlated.......................................89

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xii 5-13 Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 percent c onfidence intervals are shown for the case where x and y -direction FRFs are 100 percent correlated...............................89 5-14 Comparison between Tlusty lobes and expe rimental results. The mean stability boundary and 95 percent confidence inte rvals are shown for the case where x and y -direction FRFs are not correlated...................................................................90 5-15 Comparison between Tlusty lobes and expe rimental results. The mean stability boundary and 95 percent confidence inte rvals are shown for the case where x and y -direction FRFs are 100 percent correlated......................................................90 5-16 Budak and Altintas lobes based on cutting coefficient uncertainty.........................91 5-17 Budak and Altintas lobes based on tool-p oint response measurement uncertainty.91 5-18 Tlusty lobes based on cutti ng coefficient uncertainty..............................................92 5-19 Tlusty lobes based on tool-point response measurement uncertainty......................92 5-20 Example of mapping of toolpoint FRF to stability lobe.........................................93 5-21 Experimental setup for measuring chatter based on x and y -direction tool displacement.............................................................................................................93 5-22 8,000 rpm, 2 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................94 5-23 8,000 rpm, 2.5 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................94 5-24 10,000 rpm, 5 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................95 5-25 10,000 rpm, 6 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................95 5-26 15,000 rpm, 2 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................96 5-27 15,000 rpm, 3 mm axial depth slotti ng cut test for Mikron UCP-600 Vario machining center......................................................................................................96 5-28 8,000 rpm, 4 mm axial depth 50 % radial immersion cut test for Mikron UCP600 Vario machining center.....................................................................................97 5-29 8,000 rpm, 5 mm axial depth 50 % radial immersion cut test for Mikron UCP600 Vario machining center.....................................................................................98

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xiii 5-30 10000 rpm, 16 mm axial depth 50 % radi al immersion cut te st for Mikron UCP600 Vario machining center.....................................................................................98 5-31 10000 rpm, 17 mm axial depth 50 % radi al immersion cut te st for Mikron UCP600 Vario machining center.....................................................................................99 5-32 Geometric properties of tapered he at shrink holder and 4 flute, 19.1 mm diameter carbide helix tool.....................................................................................100 5-33 H33 with 95 percent confidence intervals in the x -direction for no holder clamping variation..................................................................................................101 5-34 H33c with 95 percent confidence intervals in the x -direction for no holder clamping variation..................................................................................................101 5-35 H33 with 95 percent confidence intervals in the y -direction for no holder clamping variation..................................................................................................102 5-36 H33c with 95 percent confidence intervals in the y -direction for no holder clamping variation..................................................................................................102 5-37 The x -direction tool-point FRF with 95 pe rcent confidence intervals for no holder clamping force variation.............................................................................103 5-38 The y -direction tool-point FRF with 95 pe rcent confidence intervals for no holder clamping force variation.............................................................................103 5-39 The x -direction tool-point FRF with 95 per cent confidence intervals with holder clamping force variation.........................................................................................104 5-40 The y -direction tool-point FRF with 95 per cent confidence intervals with holder clamping force variation.........................................................................................104 5-41 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 fl ute, carbide helix tool. Holder clamping force variation excluded.........................................................................105 5-42 Tlusty predicted stability lobes w ith 95 percent confidence intervals and measured stability results for slotting cu t with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 fl ute, carbide helix tool. Holder clamping force variation excluded.........................................................................105 5-43 Budak and Altintas predicted stability lo bes with 95 percent confidence intervals and measured stability results for 50 per cent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping fo rce variation excluded....................................................106

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xiv 5-44 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for 50 per cent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping fo rce variation excluded....................................................106 5-45 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 fl ute, carbide helix tool. Holder clamping force variation included..........................................................................107 5-46 Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability results for 50 per cent radial immersion cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide helix tool. Holder clamping fo rce variation included....................................................107 5-47 Experimental setup for measuring rotating FRFs..................................................108 5-48 Magnitude of H33 in x -direction as a functi on of spindle speed.............................108 5-49 Magnitude of predicte d tool-point response, H11, in x -direction as a function of spindle speed..........................................................................................................109 5-50 Predicted x -direction tool-point FRF based on standard holder measurements at 10,000 rpm and chatter frequencies of unstable slotting cuts................................109 5-51 Predicted y -direction tool-point FRF based on standard holder measurements at 10,000 rpm and chatter frequencies of unstable slotting cuts................................110 5-52 The FRF and 95 percent confidence intervals in x -direction for H33 x based on a spindle speed of 10,000 rpm..................................................................................110 5-53 The FRF and 95 percent confidence intervals in x -direction for H33 cx based on a spindle speed of 10,000 rpm..................................................................................111 5-54 The FRF and 95 percent confidence intervals in y -direction for H33y based on a spindle speed of 10,000 rpm..................................................................................111 5-55 The FRF and 95 percent confidence intervals in y -direction for H33 cy based on a spindle speed of 10,000 rpm..................................................................................112 5-56 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

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xv 5-57 Budak and Altintas predicted stability lo bes 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, car bide helix tool. Results based on standard holder measurements performed at 10,000 rpm......................................113 5-58 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 diameter, 4 flute, ca rbide helix tool........................................................................113 5-59 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 5-60 Sensitivity of the Budak and Altintas pred icted stability lobes to input parameter variations including: A) cutt ing coefficients, B) materi als, C) standard holder measurements, and D) geometry varia tions. Results are based on Mikron UCP600 Vario machine, slotting cuts, and sta ndard holder rotational measurements at 10,000 rpm..........................................................................................................115 5-61 Sensitivity of the Budak and Altintas pred icted stability lobes to input parameter variations including: A) cutt ing coefficients, B) materials, C) standard holder measurements, and D) geometry varia tions. Results are based on Mikron UCP600 Vario machine, 50 percent radial immersion cuts, and standard holder rotational measuremen ts at 10,000 rpm..................................................................vii

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xvi Abstract of Dissertation Pres ented 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: Mechanic al and Aerospace Engineering High-speed machining has made significant technological advancements in recent years. Using high-speed machining, incr eased 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 identif y stable and unstable cutting zones. The machining models used to produce these diag rams require knowledge of the tool-point dynamics and application-specific cutting coeffi cients. Tool-point dynamics are typically obtained using impact testing; however, testi ng time is extensive due to the large amount of holder-tool combinations. A technique to predict tool-point dynamics and therefore limit experimental testing time is desirable. This dissertation describes a threecomponent spindle-holder-tool model to predic t tool-point response based on receptance coupling substructure analysis techniques. Experimental validation is provided.

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xvii This dissertation also describes Mont e 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 inve stigated and experimental validation is provided

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1 CHAPTER 1 INTRODUCTION Justification of Work One area of manufacturing research that has made significant technological advancements in recent years is high-speed machining (HSM). Machine improvements include new spindle designs for higher rota tional speed, torque, and power; increased slide speeds and accelerations; direct drive linear motor t echnology; and new machine designs for lower moving mass. The combin ation of new machine technology and tool material/coating developments often makes hi gh-speed machining a viable alternative to other manufacturing processes. A key applic ation example is th e aerospace industry, where dramatic increases in material re moval rates (MRR) made possible using highspeed machining techniques have allowed desi gners to replace assembly-intensive sheet metal build-ups with monolithic aluminum components resulting in substantial cost savings [1, 2]. High-speed machining technol ogy has also been applied to the production of moulds and dies [3-5] and automobile components [6] and has been used to improve the flexibility of manuf acturing systems [7]. Using HSM, increased MRR are achieved through a combination of large axial depths of cut and high spindle speeds (provi ded adequate power is available). One limitation on the allowable axial depth is regenerative chatter and one method of preprocess chatter predic tion 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

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2 spindle speed. However, the machining models used to produce these diagrams require knowledge of the tool-point dynamics and app lication-specific cutting coefficients. The tool-point response is typical ly obtained using impact testing, where an instrumented hammer is used to excite the tool at its fr ee 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 a nd tool length, holder, spindle, and machine) and boundary conditions (e.g., holder force, such as collet torque or shrink fit interface, and draw bar 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 tool-point response ba sed on minimum input da ta is the preferred alternative. An effective tool-point respons e model also creates th e 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 us ers to enter their machining specifications and view the correspo nding stability lobe diagram [8]. The uncertainty associated with sele cting optimum cutting parameters from stability lobe diagrams is al so an important consideration. This uncertainty depends on the technique used to create the stability lo be diagrams and the variation in the model input parameters. A method to evaluate th e sensitivity of various stability lobe algorithms to input parameter variability would enable se lection of the most robust technique. Also, the ability to place uncertainty bounds on stability lobe results would aid the end user in the selecti on of optimum cutting parameters.

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3 Literature Review The literature review proceeds with a su mmary of tool-point dynamic modeling, machining stability investigations, and uncertainty estimation techniques. Tool Point Dynamic Modeling Schmitz et al. [9-12], with the goal of reducing testing requirements, developed a model to predict the tool-point response based on minimum input data using the receptance coupling substructure analysis ( RCSA) method. In these previous studies, a two component model of the machine-spindl e-holder-tool assembly was defined. The machine-spindle-holder displacement-to-for ce receptance was recorded using impact testing, while the tool was modeled analytica lly as an Euler-Ber noulli beam [13]. The tool and machine-spindle-holder substructu re receptances were then coupled through translational and rotational springs and da mpers. The basis of the above-mentioned technique is substructure anal ysis, or component mode synt hesis. These methods have been used for several decades to predict the dynamic response of complicated assemblies using measurements and/or models of the indi vidual components, or substructures. These components can be represented by spatial mass, stiffness, and damping data, modal data, or receptances [e.g., 14-28]. Due to the difficulty in measuring rotational degrees-of-freedom (RDOF) receptances, Schmitz assumed the displacement-to-moment, rotation-to-force, and rotation-to-moment receptances at the end of the holder to be equal to zero. Park et al. [29] describe a technique to determine th e complete receptance matrix, including RDOF, at the end of the holder and th e receptances are incorporated into the tool-point dynamic model.

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4 Machining Stability Investigations Self-excited vibration in metal cutting is known as chatter and has been studied by many researchers [30-62]. Chatter is a condi tion 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 self-excitation in me tal cutting: 1) regeneration of waviness and 2) mode coupling. As the t ool makes a pass, waviness is created on the workpiece surface due to the relative vibrati on between the tool and workpiece. It is possible that the waviness among subsequent cutting passes may be out of phase, thus generating variable chip thic kness and variable cutting forces This condition can cause self-excited vibrations and is known as regeneration of wavi ness. 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 machinin g. For the case of milling, Tlusty made the following assumptions: 1) a linear vibrator y system, 2) a constant direction of the cutting force in relation to the normal cu tting 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 relations hips between the dynamics of the machineholder-tool assembly, the material and geomet ric 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 diag rams. Merrit [52] late r reproduced Tlusty’s results based on control system theory. R ecognizing that nonlinearities may exist in the actual machining process (when the cutter vibr ates out of the workpiece), time domain

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5 simulations of the milling process [46, 62] we re developed to predict stability. Budak and Altintas [55, 56] modeled the cutter a nd workpiece as multiple degree-of-freedom structures, including the axia l direction, and applied periodic system theory to analytically analyze the stab ility of the system. Recent work [57-61] 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 th e tool point FRF using the RCSA method, to provide a more generalized machine-spindleholder-tool model. The generalized model reduces the amount of experimental testing required for various machine-spindle-holdertool combinations. Secondly, the disser tation will investigate and compare the uncertainty of the results and sensitivity to input parameter variation of two popular analytical machining stabili ty prediction techniques.

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6 CHAPTER 2 RECEPTANCE COUPLING MODEL DEVELOPMENT Background and Notation Substructure analysis, or component mode synthesis, methods predict the dynamic response of complicated assemblies using measur ements and/or models of the individual components, or substructures. For an asse mbly consisting of two rigidly connected substructures, as shown in Fig. 2-1, the assembly receptance, Gjk( ), can be expressed as shown in Eq. (2.1), where is the frequency, Xj and j are the assembly displacement and rotation at coordinate j and Fk and Mk are the force and moment applied to the assembly at coordinate k If coordinate j is coincident with coordinate k the receptance is referred to as a direct receptance; otherw ise, it is a cross receptance. Here, the nomenclature Gjk( ) is used to describe the recepta nces that are produced when two substructures (or subassemb lies) are coupled to produce the final assembly. The nomenclature GSjk( ) will replace Gjk( ) in all relevant equations when two substructures (or subassemblie s) are coupled that do not form the final assembly. ()jj jkjk kk jk jjjkjk kkXX HL FM G NP FM (2.1) The substructure receptances, Rjk ( ), are defined in Eq. (2.2), where xj and j are the substructure displacement and rotation at coordinate j and fk and mk are the force and moment applied to the subs tructure at coordinate k [12, 29].

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7 jk jk jk jk k j k j k j k j jkp n l h m f m x f x R (2.2) Based on the coordinates defined in Fi g. 2-1, the equations to determine the assembly direct receptances, Gaa( ) and Gdd( ), and the assembly cross receptances, Gad( )and Gda( ), can be written as a function of the substructure receptances as shown in Eqs. (2.3-2.6), where rigid conne ctions have been applied [27]. 1()()()[()()]()aa aa aaaaabbbccba aa aaXX FM GRRRRR FM (2.3) 1()()()[()()]()dd dd dddddcbbcccd dd ddXX FM GRRRRR FM (2.4) 1()()[()()]()aa dd adabbbcccd aa ddXX FM GRRRR FM (2.5) 1()()[()()]()dd aa dadcbbccba dd aaXX FM GRRRR FM (2.6) In order to populate the substructure receptance matrices, measurement and/or modeling may be applied. Common modeling op tions include closed -form expressions for uniform Euler-Bernoulli beams [13] a nd finite element solutions (which can incorporate the more accurate Timoshenko b eam model [63]). Both approaches are

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8 considered in this study. As a convenience to th e 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, consid er the lumped parameter model for the stacked flexure assembly shown in Fig. 2-2 (an actual, equivalent assembly is also pictured). The base flexure, substr ucture A, is modeled as a single degree-offreedom (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 free-free boundary conditions; it consists of a mass, m1, connected to a massless coordinate, x2, through a spring, k1, and viscous damper, c1. The dynamic response of assembly C to a force, F1, applied at coordinate X1 (which represents the uppermost point on the top flexure) is computed using RCSA. It is assumed th at the substructure rotational receptances, ljk, njk, and pjk, are negligible (by design for flexures) and that the substructures are rigidly connected. The subs tructure receptances are determined from the lumped parameter equations of motion. Fo r substructure A, the motion is described by Eq. (2.7). Assuming a harmonic input force f3( t ) = F3ei t, the corresponding vibration is x3( t ) = X3ei t and the direct receptance h33 can be defined as shown in Eq. (2.8). 3333333()()()() mxtcxtkxtft (2.7) 3 33 2 3 3331 x h f mick (2.8) Complex matrix inversion is applied to determine h12, h21, h11, and h22 for substructure B using the equations of motion provided in Eq. (2.9). Again assuming a solution of the

PAGE 26

9 form xj( t ) = Xjei t for fj( t ) = Fjei t, j = 1, 2, Eq. (2.9) can be written in matrix form as shown in Eq. (2.10). ()()()()()() 11111112121 mxtcxtkxtcxtkxtft (2.9) ()()()()() 111112122 cxtkxtcxtkxtft 2 11 11111 22 1111 x f mickick x f ickick or f x A ) ( (2.10) The receptance matrix for substructure B, GB( ), is obtained by inverting the matrix A ( ) as shown in Eq. (2.11). The direct and cross receptances for substructure B are provided in Eqs. (2.12-2.14). 1 1112 2122()()Bhh GA hh (2.11) 11 2 11 h m (2.12) 12 1221 2 21 11 xx hh ff m (2.13) 2 111 22 2 1111() mick h imcmk (2.14) Substituting Eqs. (2.12-2.14) and Eq. (2.8) into Eq. (2.2) with the appropriate coordinate modifications and RDOF set to zero (flexures mo tion is approximately zero) yields Eqs. (2.15-2.18). Equation (2.19) is obt ained 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. (2.15-2.18) into Eq. (2.19).

PAGE 27

10 11 11 2 11 1 11 11 111 0 0 00 00 xx fm h m R fm (2.15) 1122 2211 2 1221 1 1221 1122 22111 0 00 0000 00 xxxx fmfm hh m RR fmfm (2.16) 22 2 111 22 22 2 22 1111 22 220 0 () 00 00 xx mick fm h R imcmk fm (2.17) 33 33 2 33 333 33 33 331 0 0 00 00 xx fm h mick R fm (2.18) 11 11 1 111112223321 11 11()()()[()()]() XX FM GRRRRR FM (2.19) 1 2 111 22222 11333111111 11111 = (X Fmick mmmickimcmkm (2.20) Previous Machine-Spindle-Holder-Tool Modeling Technique In the previous work of Schmitz et al. [9-12], which describes the tool point frequency response function (or receptance) prediction using the RCSA method, a two-

PAGE 28

11 component model of the machine-spindleholder-tool assembly was defined. The machine-spindle-holder displacement-to-for ce receptance was recorded using impact testing, while the tool was modeled analytica lly as a Euler-Bernoulli beam [13]. The tool and machine-spindle-holder substructure receptances were then coupled through translational and rotational springs and da mpers: see the model in Fig. 2-3, where kx and k are the translational and rotational springs, cx and c are the translational and rotational viscous dampers, component A represents the tool, and component B represents the machine-spindle-holder. The purpose of the springs and dampers between the tool and holder was to capture the effects of a pot entially non-rigid, damped connection. The values of the springs and dampers were determined by measuring the tool point receptance with impact testing and performi ng a nonlinear least squa res fit between the actual measured results and the model. Although it was shown in Schmitz and Burns [12] that this two-component model provide s a valid approximation for a flexible tool clamped in a stiff spindle-holder, it does not o ffer the most generalized solution; i.e., if a new holder is inserted in the spindle, a ne w machine-spindle-holder measurement must be performed. The potential for improvement in the two-component model exists in three areas. First, the model requires an experimental m easurement to determine the receptance at the end of the holder; therefore, multiple spindler-holder combinations each require a separate measurement. A model which identifies, in a single measurement set, the machine-spindle-holder dynamics for all mach ine-spindle-holder combinations would be beneficial. Secondly, the displacement-to-mom ent, rotation-to-force, and rotation-tomoment receptances at the free end of the holder were assumed zero (i.e., perfectly rigid)

PAGE 29

12 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 comp ensate for the fact that these receptances are not truly equal to zero. Fi nally, 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 Machine-Spindle-Holder Tool Modeling Technique In order to enable RCSA predictions for a wider variety of machine-spindle-holdertool combinations, an improved three-component model is presented. In this model, the machine-spindle-holder substructu re is separated into two part s: 1) the machine, spindle, holder taper, and portion of th e holder nearest the spindle wi th standard geometry from one holder to another (hereafter referred to as the spindle-ho lder 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 t echnique for determining the rotation-toforce/moment and displacement-to-moment r eceptances for the free end of the spindleholder base subassembly using only displa cement-to-force measurements is also described. The experimental procedure involv es direct and cross displacement-to-force measurements of a simple geometry ‘standa rd’ holder clamped in the spindle to be modeled. The portion of the standard holder be yond the section with consistent geometry from holder-to-holder is then removed in si mulation using an inverse receptance coupling approach (i.e., decomposition) to identify th e four spindle-holder base subassembly receptances. These receptances are then coupled to models of the actual holder and tool. Spindle-holder Base Subassembly Identification The experimental procedure used to determine the receptances at the free end of the spindle-holder base subassembly, GSjk( ), is described in this s ection. It is composed of

PAGE 30

13 three primary steps. First, the standard holder displacement-to-force direct and cross receptances are determined by impact testi ng. The standard holder geometry, which was selected to approximate a broad range of potential holders, is pr ovided in Figure 2-4. Second, these results are used to determine the three other direct recep tances at the free end of the standard holder. Third, the secti on of the standard holder which is not common to other holders (see Figure 2-5) is re moved using inverse receptance coupling to determine all four spindle-holder base suba ssembly 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 2-4), the four subassembly receptances are determined by measuring the direct, H33, and cross, H33c, and/or H33b, displacement-to-force re ceptances on the standard holder, applying a 1storder or 2nd-order backward finite difference method to find L33 (and, equivalently, N33) [64], and then synthesizing P33. If a 1st-order finite difference method is used, only one cross displacement-to -force receptance, H33c, is required. Both cross displacement-toforce receptances are required if the 2nd-order finite difference method is used. For the cross displacement-to-force 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 signal-to-noise ratio for the H33c measurement (i.e., many of the lower frequency spindle-holder modes resemble a fixed-free f undamental mode shape and have very small amplitudes near the spindle face for the bandwid th 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 H33c measurement provides sufficient signal-tonoise. The receptance L33 is determined from the measured displacement-to-force

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14 receptances using Eq. (2.21) if the 1st-order finite difference me thod is selected or Eq. (2.22) if the 2nd-order 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]. 3333 332cHH L S (2.21) 333333 3334 2bcHHH L S (2.22) 33 2 33 33 33 33 3 3 3 3 3 3 3 3 331 H L N L H F M X X F M P (2.23) 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 finite-differe nce method) and, therefore, P33. To reduce the noise effect, the measured receptance data was smoothed using a Savitzky-Golay 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. 2-5. Equa tions (2.24-2.27) provide the direct and cross extended holder subassembly receptance matrices, where rigid coupling has been applied. These equations are dete rmined from Eqs. (2.3-2.6) by appropriate substitutions.

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15 33 33 1 333333333333 33 33()()()[()()]()aaabbaXX FM GSRRRRR FM (2.24) 44 44 1 444443333334 44 44()()()[()()]()bbbaabXX FM GSRRRRR FM (2.25) 33 44 1 3433333334 33 44()()[()()]()aaabbbXX FM GSRRRR FM (2.26) 44 33 1 4343333333 44 33()()[()()]()baabbaXX FM GSRRRR FM (2.27) Spindle-holder base subassembly receptance The spindle-holder base subassembly receptance matrix, G33( ), can be expressed as shown in Eq. (2.28) by rewriting Eq. (2.3 ). The left-hand side of this equation is known once the steps described in the standa rd test holder receptances section are completed. Also, the extended holder subassembly receptances, GS33, GS44, GS34, and GS43, are determined using the equations pr ovided in the extended holder subassembly model section. Therefore, Eq. (2.28) can be rewritten to solve for the spindle-holder base subassembly receptances, GS55( ). See Eq. (2.29). 3333 1 333334445543 3333()()()[()()]() HL GGSGSGSGSGS NP (2.28)

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16 55 55 1 553433334344 55 55()()[()()]()() xx fm GSGSGSGGSGS fm (2.29) Tool-point Response Prediction To illustrate the technique used to pr edict tool point dynamics, the modeling procedure is applied to a spi ndle using a tapered thermal shri nk holder with a tool blank inserted as shown in Fig. 2-6. The assemb ly is divided into the spindle-holder base subassembly and 13 cylindrical substructures of differing diameters. The spindle-holder base subassembly receptances are determined by the procedure described in the spindleholder base subassembly identification section. The 13 cylindrical substructures are each analytically modeled as a Euler-Bernoulli beam or Timoshenko beam (see Appendix A). To model the receptances of the beams, a composite modulus and mass are substituted for substructures II-VIII to account for poten tial material differences between the holder and the tool blank. Also, the mass expre ssion for these substruc tures (provided in Appendix A) is replaced with the com posite mass shown in Eq. (2.30), where h and t are the density of the holder and tool, respectiv ely. Additionally, the product of the elastic modulus and 2nd area moment of inertia, EI are replaced by the product shown in Eq. (2.31), where Eh is the holder modulus, Et is the tool material modulus, and Ih and It are the 2nd area moments of inertia for th e holder and tool, respectively. 42 2 2L d d d mi t i o h (2.30) 644 4 4i t i o h t t h hd E d d E I E I E EI (2.31)

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17 The next step is to rigidly couple substr uctures I through XIII to produce the direct and cross extended holder-tool subassembly receptances at coordinates 1 and 4. First, Eqs. (2.3-2.6) are used with the appropr iate coordinate modi fications to couple substructure I to substructure II, each s ubstructure having free -free 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 holder-tool subassembly are determined. The final step in the procedure is to pred ict the tool point dynamics by ri gidly coupling the extended holdertool subassembly to the sp indle-holder base subassemb ly. With the appropriate coordinate substitution in Eq.(2.3) the tool point receptance, G11( ), are determined according to Eq. (2.32), where th e receptances associated with coordinates 1 and 4 are the extended holder-tool subassembly direct and cross receptances. 1111 1 111114445541 1111()()()[()()]() HL GGSGSGSGSGS NP (2.32) 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 sec tion are adjusted to account for the actual geometry. Two, three, and four fluted t ools 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. 2-7, and the inertia was calculat ed 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 t ool 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

PAGE 35

18 ratio for the mass of the fluted section. Ta ble 2-1 shows the mass and inertia adjustment ratios. During the substructuring process fo r the extended holder-tool subassembly, the overhung tool section is divided into a shank substructure and a fluted substructure. The fluted substructure is the lengt h of the fluted section. The mass and inertia of the fluted substructure is first calculated as if the sect ion was a solid, cylindrical section and then the mass and inertia results are multiplied by the Table 2-1 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. Assembled system Unassembled system Substructure II Substructure I Fab c Rigid Xa Maaa a b c d d Figure 2-1. Two-component assembly. The co mponent responses are coupled through a rigid connection to give the assembly receptance(s). Figure 2-2. Two-component flexure assembly. The component responses are coupled through a rigid connection to give the assembly receptance(s).

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19 x xc c k kA B Fab c Xa a Figure 2-3. Previous two-component RCSA model. An external force, Fa( t ), isapplied to the free end of the tool (A) to determine the assembly Xa/ Fa receptance. The tool is coupled to the machine-sp indle-holder (B) through springs and dampers. SpindleStandard holder 3 3c3b Acceleromete r SS Cross receptance hammer impact locations Figure 2-4. Example standard holder for spindle-holder base subassembly receptance identification. Hammer imp acts are completed at locat ions 3, 3b, and 3c to identify the required direct and cross receptances. 3a 4 5 3 Spindle Rigid Test HolderSpindle-holder 3 3b I II base subassembly Extended holder subassembly 5 Figure 2-5. Standard holder substruc tures for inverse receptance coupling.

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20 Spindle-holder-tool assembly Coordinate 3 Spindle-holder base 4 5 Substructures subassembly Extended holder-tool subassembly Rigid connections between 1 all substructures XIII . I Figure 2-6. Spindle-holder-t ool substructures for tapered thermal heat shrink holder and tool blank etc... 0 10 20 30 Degree of rotation Figure 2-7. End view of tw o fluted tool showing rotati on angles for area and inertia calculations. Table 2-1. Mass and iner tia adjustment ratios. Tool Flutes Mass Ratio Inertia Ratio 2 0.44 0.38 3 0.33 0.36 4 0.36 0.36

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21 CHAPTER 3 EXPERIMENTAL VALIDATION OF RCSA MODELS Experimental validation of the stacked flexure system and improved threecomponent machine-spindle-holder-tool model fo r five different spin dles is provided. The spindle-holder base subassembly recepta nces were determined for a 36,000 rpm, CAT 40 spindle for three standard holder geom etries to determine the optimum geometry for the standard holder. Once the spindleholder base subassembly receptances were determined, the three-compone nt model was used to comp are predicted and measured results at the end of a collet holder. The spindle-holder base subassembly receptances were also determined for a 24,000 rpm, HSK 63A spindle and Step Tec HVCS 212-X 20,000 rpm, HSK 63A spindle. The three-co mponent model was again used to compare predicted and measured results for spindleholder-tool assembly combinations produced with collet holders, three geometries of tapered heat-shrink holders and various geometries of carbide tool bl anks and fluted tools. The three-component model was also used to compare predicted and measured results for a geared, quilled-type 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. 2-2 was produced by stacking two SDOF flexures. After manuf acture, the modal parameters for the flexures were determined by impact testing. For these measurements, the flexures were adhered to ground (a massive, extremely s tiff machining tombstone was assumed to be ground) with

PAGE 39

22 cyanoacrylate (i.e., quick-bonding cement). An impact hammer was used to excite the flexures at their free end ( over a bandwidth of approximate ly 2 kHz) and the response was measured using a low-mass accelerometer. The modal mass, damping, and stiffness parameters were then extracted from the response by fitting the data using a peak-picking method [25]. The results are shown in Table 3-1. To verify the stacked flexure RCSA mode l, the model parameters from Table 3-1 were substituted into G11( ), provided in Eq. (2-20). Fi gure 3-1 displays measured and predicted G11( ) 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 hol der dimensions on the determination of the spindle-holder 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. 2-5. The dimensions and material properties for each of the holder substructures ar e displayed in Table 3-2, where di and do are the inner and outer diameters, L is the length, is the density, and is the structural damping factor. The structural damping values used in this study were determined experimentally from free-free testing of representative cylindrical rods. During the measurement of the direct and cross receptances for the mounted standard holders, the distance S was select ed 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. 3-2 for the small holder. The spindle-holder base receptance, GS55( ), was also determined for each of the three standard holders by

PAGE 40

23 decoupling the appropriate extended holder subassembly from the spindle-standard holder measurements and the receptances at th e end of each of the th ree standard holders, G33( ), were determined based on the spindleholder base subassembly receptances of the other two holders. Next, the predicted r eceptances at the end of each holder were compared to the measured receptances to select the preferred standard holder geometry. Figure 3-3 displays the magnitudes of the displacement-to-force receptance, H33, for the large holder rebuilt from the small holder spindle-holder 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. 3-3: 1) the sp indle-holder base subassembly receptances, GS55( ), determined from testing the small holde r can be used as a basis to accurately predict the end receptances, G33( ), for the medium and large holders; and 2) the spindle-holder base subassembly receptances determined from testing the large holder do not provide an accurate prediction for the sma ll holder end receptances. It is hypothesized that the small holder produces better result s because the large holder has increased mass which shifts the FRF information to lower fr equencies. Since an accelerometer was used in the FRF measurements of the standard hol der, 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 la rger holder places more information in this frequency range. The machine-spindle-holder base subassembly receptances based on the large holder measurements would therefor e 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 spindle-holder base

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24 subassembly receptances and these receptances were used to predict end receptances for other extended holder-tool subassemblies. Experimental testing was also perfor med to compare tool-point response predictions from the three-component m achine-spindle-holder-tool model when 1st and 2nd-order 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 63 A spindle-holder interface. The machinespindle-holder base receptances were determined and a tool-point response was predicted for a tapered holder-tool blank assembly (see Fig. 2-6) using the methods described in chapter 2. Figure 3-4 shows the pred icted tool-point response when a 1st-order finite difference approach was used based on one, two, four, and ten measur ement set averages. A measurement set consists of ten root-mean-square averaged FRF measurements of the direct and cross standard holder FRFs. Figure 3-5 shows the pr edicted tool-point response when a 2nd-order finite difference approach was used based on one, two, four, and 10 measurement set averages, and Fig. 3-6 shows a comparison between tool-point responses based on the 1st-order and 2nd-order finite difference approaches and the actual measured tool-point response. From th e figures, it can be seen that the 1st-order finite difference method provides the most consistent tool-point response, independent of the number of averaged measurement sets. The 2nd-order finite difference approach produces more variation in the predicted tool-point response based on the number of measurement sets used, and the tool-point response contai ns more noise. Figure 3.6 shows that using the 1st-order finite difference method actually leads to improved accuracy for the model

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25 prediction. Based on this testing, it is recommended to use the 1st-order finite difference method with 5 averaged measurement sets to determine the standa rd holder receptances. Holder Experimental Verification Once the spindle-holder base subassembly receptances, GS55( ), are determined using the 1st order finite difference met hod, it is possible to coupl e this result to arbitrary holder geometries to predict the receptance at any coordinate on the machine-spindleholder assembly. To validate the procedure, a collet holder (19 mm bore) with a CAT 40 spindle interface was divided into three s ubstructures beyond the spindle-holder base subassembly as shown in Fig. 3-7. Each subs tructure 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. 3-7. The first step in predicting the assembly re sponse, as described in chapter 2, was to couple substructures I, II, and III to pr oduce the direct and cross extended holder subassembly receptances at coordinates 3 and 4. The parameters for the free-free substructure receptances are given in Table 33. The next step was to rigidly couple the spindle-holder base subassembly to the ex tended holder subassembly to determine the receptances at the free end of the holder, G33( ). Figure 3-8 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 st eel standard holder again consisted of solid, cylindrical substructures I and II as show n in Fig. 2-6; 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

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26 Table 3-4. The experimental procedure described in chapte r 2 was applied to determine all receptances at coordinate 5, GS55( ), as displayed in Fig. 3-9. Holder Experimental Verification Once the 24,000 rpm/40 kW spindle-holder base subassembly receptances, GS55( ), were determined, it was possible to couple this result to arbitrary holder geometries to predict the receptance at a ny coordinate on the machine-spindle-holder assembly. To validate the procedure, a tapere d thermal shrink fit holder (25.3 mm bore) with an HSK 63A spindle interface was divide d into 12 substructures beyond the spindleholder 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 va lue for that section. Table 3-5 provides the holder geometry and assume d material properties. Following the procedure outlined in chapter 2, the receptances at the free end of the holder, G33( ), were determined. Figure 3-11 shows the predicted H33 result as well as measurements for two nominally identical hol ders. The Euler-Bernoul li beam model was applied to develop the extended hol der receptances in this case. Tool-point Response Prediction To predict the tool-point dynamics, the modeling procedure was again applied to the 24,000 rpm/40 kW spindle using a tapere d thermal shrink holder with a 19.1 mm carbide tool blank inserted as shown in Fig. 3-12. The assembly was divided into the spindle-holder base subassembly and 13 cylindr ical substructures of differing diameters (mean outer diameter values again used for the tapered sections); see Table 3-6. The predicted and measured assembly toolpoint displacement-to-force receptances, H11, are displayed in Fig. 3-13. In this figure, re sults for both Euler-Bernoulli and Timoshenko

PAGE 44

27 (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 Gea red, Quill-Type CAT 50 Spindle In this section, prediction and measurem ent results are provided for two cutters coupled to a geared, quill-type spindle w ith a CAT 50 spindle-holder interface (Big-Plus tool holders were used which include both taper and face contact). The spindle-holder base subassembly receptances were determined using a steel cylindrical standard holder (63.4 mm diameter and 89.0 mm length); th e cross FRF measurements were again recorded at distances of 25.4 mm and 50.8 mm (2nd-order finite difference method was applied) from the free end of the standard holder. The substructure receptances for the solid body tools (i.e., both cut ting tools were composed of solid steel modular bodies with carbide inserts attached) were then co mputed and the tool-point FRF predicted by rigidly coupling the tool models to the spindle measurements. Figure 3-14 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 3-7 (as before substructure, I is nearest the free end of the clamped cutter). Figure 3-15 shows the H11 measurement and prediction for a 28-inse rt facemill (see Table 3-8). In both cases, Euler-Bernoulli beam models were employed to describe the standard holder and cutter bodies. Experimental Results for Geared CAT 50 Spindle In this section, the spindle-holder base subassembly receptances were measured on two nominally identical, geared spindles (CAT 50 holder-sp indle interface). The steel

PAGE 45

28 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 3-16 provides standard holder direct FRF measurement re sults for both spindles. Three curves are shown: the solid line represents the averag e of five measurement sets (10 impacts each) completed without removing the holder from th e first spindle (i.e., sp indle 1); the dotted line gives the average of three more spindle 1 measurements after removing and replacing the holder; and the dashed lin e shows the average of five spindle 2 measurements. These curves show that, alt hough the spindles are similar, the difference between the spindle dynamics is larger th an the measurement divergence on a single spindle. Next, a 16-insert solid body facemill was inserted in spindle 1 and the tool-point response recorded. Predictions were finally completed using both the spindle 1 and 2 receptances. This result is provided in Fig. 3-17; the facemill geometry and material properties are given in Table 3-9. It is seen that the prediction completed using the spindle 1 receptances (dotted line) more accu rately 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 consis tency between spindles is manufacturerdependent. Experimental Results for Step Tec 20,000 rpm, HSK 63A Spindle In this section, prediction and measuremen t results are provided for a Mikron Vario CNC machining center with a 20,000 rpm, HSK 63A interfac e Step Tec spindle coupled to a variety of holder tool assemblies. The machine-spindle-holder base receptances were

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29 determined and tool-point responses were predicted using the me thods described in chapter 2. Tapered Heat-shrink Holder and Carbide Tool Blank Results Tool-point responses were measured a nd predicted using the three-component machine-spindle-holder-tool model for the four different holder tool assemblies shown in Fig. 3-18 with the component and assembly dimensions displayed in Table 3-10. The spindle-holder-tool assembly combinations we re produced using three different holders, three different carbide tool bl anks, and four varying tool overhang lengths. The results for the four different test runs displayed in Table 3-10 are shown in Fig. 3-19 through Fig. 3-22. Collet Holder and Carbide Tool Blank Results Tool-point responses were measured a nd predicted using the three-component machine-spindle-holder-tool model for the th ree different holder tool assemblies shown in Fig. 3-23 with the component and assembly dimensions displayed in Table 3-11. The spindle-holder-tool assembly combinations were produced using tw o different carbide blanks and two different tool overhang lengths. In test numbe r 6, the tool blank extended past the collet in the holder and was modeled as a solid su bstructure section. This technique still provided an acceptable tool-point response prediction, as seen in Fig. 3-25. The results for the four differe nt test runs displayed in Ta ble 3-10 are shown in Fig. 3-24 through Fig. 3-26. Fluted Tool Results Tool-point responses were measured a nd predicted using the threecomponent machine-spindle-holder-tool model for the four different holder tool assemblies shown in Fig. 3-27 and 3-28 with the component and as sembly dimensions displayed in Table 3-

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30 12. The spindle-holder-tool assembly combin ations were produced using three different holders, 4 different fluted car bide tools (consisting of both two and four tooth cutters), and two different tool overhang lengths. The ma ss and inertia of the fluted tool sections was modeled using the methods described in Chap ter 2. The results for the four different test runs displayed in Table 3-10 are shown in Fig. 3-29 through Fig. 3-32. The three-component model does a good job of predicting the t ool-point response based on the spindle-holder subassembly recep tances and an analytical model of the extended holder-tool 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, modeli ng the extended holder-tool 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 an alyzing a FEA model. This is especially important with the large va riety of holders and t ools that are currently available and allows external users of the we b-based application [8] at the University of Florida to receive stability l obe diagrams for any holder-t ool combination based simply on their geometries. Predicted results for the tool-point 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 b een modeled as a rigi d connection but in reality has stiffness and damping characteristic s. Including a non-ri gid connection in the overall model could lead to improved results.

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31 Table 3-1. Flexure modal parameters. Modal parameters m k c (kg) (N/m) (kg/s) Large base flexure2.46 8.85x10670.6 Top flexure 0.1455.04x105 1.17 -1 0 1 x 10 -4 Real (m/N) 100 200 300 400 500 -20 -10 0 x 10 -5 Frequency (Hz)Imag (m/N)Predicted from model Actual Predicted from model Actual Figure 3-1. Plot of G11 receptances for flexure system. Table 3-2. Standard holder substructure parameters. Small holder Medium holder Large holder Substructure I II I II I II Coordinate j 3 3b 3 3b 3 3b Coordinate k 3a 4 3a 4 3a 4 di (m) do (m) 0.04440.04440.05100.04440.0510 0.0444 L (m) 0.04000.01730.06620.01730.1157 0.0173 (kg/m3) 7800 E (N/m2) 2x1011 11 0.0015

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32 10-10 10 -8 10 -6 10-4 Log Magnitude 200 600 1000 1400 1800 -200 -100 0 100 200 Phase (degrees)Frequency (Hz) rad P Nm m L Nm m H N 33 33 33 Figure 3-2. Plot of G33 receptances for small holder. 200 600 1000 1400 1800 0 0. 5 1 1. 5 2 2. 5 x10-6 Large Holder Rebuilt Frequency (Hz)Magnitude (m/N) Rebuilt Measured 0 0. 5 1 1. 5 2 2. 5 3 3.5 x10-7 from Small Holder Small Holder Rebuilt from Large Holder Rebuilt Measured 200 60010001400 1800 Frequency (Hz) 0 0. 5 1 1.5 x10-6 Medium Holder Rebuilt from Small Holder Rebuilt Measured 200 60010001400 1800 Frequency (Hz) Figure 3-3. Standard holder geometry compar ison (note: vertical ax is scale is different for the three panes).

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33 -2 -1 0 1 2 x 10 -6 500 1000 1500 2000 2500 -6 -4 -2 0 2 x 10 -6 Frequency (Hz)Real (m/N) Imag (m/N)1 measurement set 2 measurement sets 4 measurement sets 10 measurement sets Figure 3-4. Tool-poi nt response base on 1rst-order finite difference method. -4 -2 0 2 4 x 10 -6 500 1000 1500 2000 2500 -6 -4 -2 0 2 x 10 -6 1 measurement set 2 measurement sets 4 measurement sets 10 measurement sets Frequency (Hz) Figure 3-5. Tool-poi nt response base on 2nd-order finite difference method.

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34 -2 -1 0 1 2 3 x 10 -6 2nd order 1rst order measured 500 1000 1500 2000 2500 -4 -2 0 2 x 10 -6 Figure 3-6. Measured versus pred icted tool-point response based on 1st-order and 2ndorder finite difference method and ten averaged measurement sets. 3 3a 3b 3c 3d 4 I II III 5 Spindle-Holder Assembly 3 Spindle-Holder Base Subassembly Rigid Connection Figure 3-7. Collet holder substruc ture I, II, and III parameters.

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35 Table 3-3. Collet holder substruc ture I, II, and III parameters. Substructure I II III Coordinate j 3 3b 3d Coordinate k 3a 3c 4 di (m) 0.01900.01900.0315 do (m) 0.06300.05000.0496 L (m) 0.02550.02600.0286 (kg/m3) 7800 E (N/m2) 2x1011 11 0.0015 -6 0 6 x 10 -7 200 600 1000 1400 1800 -12 0 x 10 -7 Predicted Measured Imag (m/N) Real(m/N) -6 Frequency (Hz) Figure 3-8. Collet holder H33 predicted and measured results.

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36 Table 3-4. Standard holder substructure parameters. Substructure I II Coordinate j 3 3b Coordinate k 3a 4 do (mm) 63.3 52.7 L (mm) 62.8 16.3 ( k g / m 3 ) 7800 E (N/m2) 2x1011 11 0.0015 0 1000 2000 3000 4000 5000 10-9 10 -8 10 -7 10 -6 10 -5 MagnitudeFrequency (Hz) p55 l55 (rad/Nm) (m/Nm) h55 (m/N ) Figure 3-9. Spindle receptances G55( ) determined from standard holder direct and cross receptance measurements.

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37 Table 3-5. Shrink fit holder (25.3 mm bore) substructure parameters. Substructure di (mm) do (mm) L (mm) (kg/m3) E (N/m2) 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 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 7800 2x1011 0.0015 Spindle-holder assembly Coordinate 3 Spindle-holder base 4 5 Substructures subassembly Extended-holder subassembly Rigid connections between 3 XII . I all substructures Figure 3-10. Tapered thermal shrink fit hol der (25.3 mm bore) substructure model.

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38 500 1000 1500 2000 2500 -1.5 0 1.5 x 10 -7 Real (m/N ) -3 -2 -1 0 x 10 -7 Frequency (Hz)Imag (m/N ) Predicted Holder 1 Holder 2 0 Figure 3-11. Measured (two nominally identical holders) and predicted H33 results for tapered thermal shrink fit holder (25.3 mm bore). Spindle-holder-tool assembly Spindle-holder base 4 5 Substructures subassembly Extended holder-tool subassembly Rigid connections between 1 all substructures XIII . I Figure 3-12. Tapered thermal shrink fit hol der with 19.1 mm diameter tool blank substructure model.

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39 Table 3-6. Shrink fit holder-tool blank substructure parameters. Substructure di (mm) do (mm) L (mm) (kg/m3) E (N/m2) 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 VI 19.1 37.5 5.8 VII 19.1 38.5 5.8 VIII 19.1 39.5 5.8 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 7800 (steel) 14500 (carbide) 2x1011 (steel) 5.85x1011 (carbide) 0.0015 -2 0 2 x 10 -6 Real (m/N ) 100 500 1000 -4 -2 0 Frequency (Hz)Imag (m/N ) Euler Bernoulli Measured Timoshenko x 10-6 Figure 3-13. Measured and predicted H11 results for tapered thermal shrink

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40 -1 0 1.5 Real (m/N ) 100 500 100 0 -2 -1 0 x 10 -7 Frequency (Hz)Imag (m/N ) Measured Predicted x 10 -7 Figure 3-14. Measured and predicted H11 results for 20-insert endmill. Table 3-7. 20-insert endmill substructure parameters. Substructure I II III do (mm) 99.8 80.1 69.9 L (mm) 85.6 94.9 16.8 (kg/m3) 7800 E (N/m2) 2x1011 0.0015 Table 3-8. 28-insert 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 (kg/m3) 7800 E (N/m2) 2x1011 0.0015

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41 -4 0 4 8 x 10 -7 Real (m/N ) 100 500 1000 -10 0 x 10 -7 Frequency (Hz)Imag (m/N ) Measured Predicted -5 Figure 3-15. Measured and predicted H11 results for 28-insert. -6 0 6 x 10 -8 Real (m/N ) 100 1000 2000 -10 -5 0 x 10 -8 Frequency (Hz)Imag (m/N ) 1a 1b 2 Figure 3-16. Standard holder direct receptanc es two nominally identical, geared spindles (CAT 50 holder-spindle interface).

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42 -6 0 6 x 10 -8 Real (m/N ) 100 500 1000 -10 0 x 10 -8 Frequency (Hz)Imag (m/N) Measured Predicted Predicted Figure 3-17. Measured and predicted H11 results for 16-insert facemill. Results are shown for predictions from spindle 1 (dotte d) and spindle 2 (dashed) standard holder measurements. Measurement recorded using spindle 1. Table 3-9. 16-insert facemill substructure parameters. Substructure I II III do (mm) 279.4 63.5 69.9 L (mm) 27.2 88.9 15.9 (kg/m3) 7800 E (N/m2) 2x1011 0.0015 D 1 D 2 D 3 L 1 L 2 L 3 L 4 L 5 Figure 3-18. Tapered heat-shrink holder and tool blank assembly.

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43 Table 3-10. Dimensions for tapered heat -shrink holder and tool blank assembly Test Number 1 2 3 4 Holder Manufacturer Command Command Command Tooling Innovations Holder Part Number H4Y4A0750 H4Y4A0750 H4Y4A1000 HSK63ASF075-315 Tool Distributor McMasterCarr McMasterCarr McMasterCarr McMasterCarr Tool Distributor Number 8788A431 8788A258 8788A263 8788A258 D1 (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 L1 (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 -2 0 2 x 10 Real (m/N) 0 500 1000 1500 2000 2500 -4 -2 0 2 x 10 Frequency (Hz)Imag (m/N) -2 0 2 -6 Real (m/N) Predicted Measured 0 500 1000 1500 2000 2500 -4 -2 0 2-6 Frequency (Hz)Imag (m/N) Figure 3-19. The FRF for tapered heat-shri nk holder with 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (T est Number 1). The overhung tool length was 91.6 mm.

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44 -2 0 2 4 x 10 -7 Real (m/N) 0 500 1000 1500 2000 2500 -5 0 5 x 10 -7 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-20. The FRF for tapered heat-shri nk holder with 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (T est Number 2). The overhung tool length was 38.78 mm. -5 0 5 x 10 -7 Real (m/N) 0 500 1000 1500 2000 2500 -10 -5 0 5 x 10 -7 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-21. The FRF for tapered heat-shri nk holder with 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (T est Number 3). The overhung tool length was 83.81 mm.

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45 -2 0 2 x 10 -6 Real (m/N) 0 500 1000 1500 2000 2500 -5 0 5 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-22. The FRF for collet holder w ith 19.1 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 4) The overhung tool length was 132.4 mm. D 1 D 2 D 3 L 1 L 2 L 3 L 4 L 5 D 4 Figure 3-23. Collet holder and tool blank assembly

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46 Table 3-11. Dimensions for collet holder and tool blank assembly Test Number 5 6 7 Holder Manufacturer Regofix Regofix Regofix Holder Part Number HSKA63/ER40-120 HSKA63/ER40-120 HSKA63/ER40-120 Tool Distributor McMaster-Carr McMaster-Carr McMaster-Carr Tool Distributor Number 8788A431 8788A431 8788A263 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 L1 (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 -1 0 1 x 10 -6 Real (m/N) 0 500 1000 1500 2000 2500 -2 1 0 1 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-24. The FRF for collet holder w ith 25.4 mm diameter, 152.4 mm long carbide tool blank assembly (Test Number 5) The overhung tool length was 132.4 mm.

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47 -1 0 1 x 10 -6 Real (m/N) 0 500 1000 1500 2000 2500 -2 -1 0 1 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-25. The FRF for collet holder w ith 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. -5 0 5 x 10 -7 Real (m/N) 0 500 1000 1500 2000 2500 -5 0 5 x 10 -7 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-26. The FRF for collet holder w ith 19.1 mm diameter, 101.6 mm long carbide tool blank assembly (Test Number 7) The overhung tool length was 73.3 mm.

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48 D 1 D 2 D 3 L 1 L 2 L 6 L 3 L 4 L 5 Figure 3-27. Tapered heat-shrink ho lder and fluted tool assembly. D 1 D 2 D 3 L 1 L 2 L 3 L 4 L 5 D 4 L 4 Figure 3-28. Collet holder and fluted tool assembly.

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49 Table 3-12. Dimensions for collet and tape red heat-shrink holder s and fluted tools. Test Number 8 9 10 11 Holder Manufacturer Command Command Tooling Innovations Regofix Holder Part Number H4Y4A0750 H4Y4A1000 HSK63ASF075-315 HSKA63/ER40-120 Holder Type Tapered Heatshrink Tapered Heatshrink Tapered Heatshrink Collet Holder Tool Manufacturer Dataflute Unknown Unknown Regofix Tool Part Number SH40750 EC100S4 HEC750S2600 ARFST21000 Number of Flutes 4 4 2 2 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 L1 (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 -5 0 5 x 10 -6 Real (m/N) 0 500 1000 1500 2000 2500 -10 -5 0 5 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-29. The FRF for tapered heat-shri nk holder with 19.1 mm diameter, 152.4 mm long carbide, 2 fluted tool assembly The overhung tool length was 121.5 mm.

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50 -1 0 1 x 10 -6 Real (m/N) 0 500 1000 1500 2000 2500 -2 -1 0 1 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-30. The FRF for collet holder w ith 25.4 mm diameter, 127. 0 mm long carbide, 2 fluted tool assembly. The ove rhung tool length was 100.8 mm. -1 0 1 x 10 -6 Real (m/N) 0 500 1000 1500 2000 250 0 -2 -1 0 1 x 10 -6 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-31. The FRF for tapered heat-shri nk holder with 19.1 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly The overhung tool length was 79.2 mm.

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51 -5 0 5 x 10 -7 Real (m/N) 0 500 1000 1500 2000 2500 -10 -5 0 5 x 10 -7 Frequency (Hz)Imag (m/N) Predicted Measured Figure 3-32. The FRF for tapered heat-shri nk holder with 25.4 mm diameter, 101.6 mm long carbide, 4 fluted tool assembly The overhung tool length was 80.2 mm

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52 CHAPTER 4 STABILITY ANALYSIS UNCERTAINTY Uncertainty estimates on milling stability anal ysis results, in the form of confidence intervals on stability lobe di agrams, requires knowledge of the tool-point response and cutting coefficient statistics. For the purpos es of this study, the tool-point response statistics were determined by modal tes ting and prediction from the 3-component machine-spindle-holder-tool RCSA model. The cutting coefficient statistics will be determined through experimental testing. Mont e Carlo methods will be used to generate uncertainty estimates on stability lobe diag rams generated through tw o 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, Fr is the radial cutting force, Ktc is the tangential cutting force coefficient, Krc is the radial cutting force coefficient, Kte is the tangential edge-cutting force coefficient, Kre is the radial edge-cutting force coefficient, a is the axial depth of cut, and h is the instantaneous chip thickness. ttcteFKahKa (4.1) rrcreFKahKa (4.2) The cutting force coefficients in milling are specific to the particular application, and new coefficients are require d 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 mechan istic approach [66]

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53 was used to determine the cutting force coeffi cients. The first step in the procedure was to mount the work piece to a three-axis dynamo meter and select a c onsistent 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 (slot ting), 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 ( x -direction) and orthogonal ( y -direction) 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 coor dinate directions, for the steady state section of the cutting signal and perf orm 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. (4.3-4.6) [66] where N is number of cutter teeth, a is axial depth of cut, Fxe and Fye, are the vertical axis intercept values in the x and y -coordinate directions and Fxc and Fyc are the slope values in the x and y -coordinate directions ye teF K Na (4.3) x e reF K Na (4.4) 4yc tcF K Na (4.5) 4 x c rcF K Na (4.6) Stability Analysis Techniques Stability lobe diagrams (see Fig. 4-1) identify stable and unstable cutting zones (separated by stability ‘lobes’) as a func tion of the chip width (or axial depth in peripheral end milling), alim, and spindle speed. Two analytical methods, based on the

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54 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 th e location of the tooth in th e cut varies with time, the system can be modeled as time invariant by an alyzing 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, (the angle between the cutting force, F and surface normal to the average tooth location in the cut as shown in Fig. 4-2), are required. Based on the mechanistically determined cutting force coeffi cients determined in Eqs. (4.5) and (4.6), values for Ks and can be calculated from Eqs. (4.7) and (4.8) The edge-cutting force coefficients are not included in the analysis. 22 s tcrcKKK (4.7) 1tantc rcK K (4.8) 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, start, and exit, as described in Fig. 4-2. The values of the directiona l factors can be calculated from Eqs. (4.9-4.12). Up milling or slotting:

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55 coscos 22 x avgavgu (4.9) coscosyavgavgu (4.10) Down milling: coscos 22 x avgavgu (4.11) coscosyavgavgu (4.12) Where: 2 s tartexit avg The third step in the procedure is to de termine the frequency response function at the tool tip in the x and y -directions, as defined in Fig. 42. Traditionally, this has been accomplished through impact testing; however, fo r this study, the tool-point FRF will be predicted with both impact testing and the 3-component machine-spindle-holder-tool RCSA technique. The real part of the tool-point FRF in the x -direction and y -direction, Gx and Gy, and the imaginary part of the tool-point FRF in the x -direction and y -direction, Hx and Hy, are required. These terms are substitute d into Eqs. (4.13) and (4.14) with the appropriate directional factors to determine the oriented real, Goriented, and imaginary, Horiented, tool-point FRF. orientedxxyyGuGuG (4.13) orientedxxyyHuHuH (4.14) The final step in the procedure is to calcul ate the axial depth of cut, alim, from Eq. (4.15) and the relative spindle speed, from Eq. (4.16). These values are determined for every chatter frequency, fc (values where Goriented is negative), and plotted relative

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56 to each other. The term, m*, is the average num ber 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. lim *1 2Resorienteda KGm (4.15) *2 2cf mP (4.16) where: 12tanoriented orientedG H and *1 360 degreesexitstartm N Budak and Altintas Method Budak and Altintas [55] provide an alternat ive analytical stability model based on a multi-degree-of-freedom 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 time-varying directional dynamic m illing force coefficient matrix, and ( t ) is the change in time matrix. The tangential cutting coefficient, Kt, is equal to Ktc as defined in Eq. (4.5). 1 {()}[()]{} 2tFtaKAtt (4.17) Due to the periodic nature of [ A ( t )], the matrix terms are expanded into a Fourier series. The average component of the Fourier series expansio n 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 ma trix are based only on the radial cutting coefficient, Kr, and the starting and exit angle of the cut (defined in Fig. 4-2). The radial cutting coefficient, Kr, is defined in Eq. (4.19) in term s of the experimentally determined

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57 cutting coefficients provided in Eqs. (4.5) and (4.6). Again, the edge-cutting coefficients are not included in the analysis, and the ax ial cutting coefficients are not required. [(0)] 2 x xxy yxyyN A (4.18) where: 1 cos22sin2 2exit s tartxxrrKK 1 sin22cos2 2exit s tartxyrK 1 sin22cos2 2exit s tartyxrK 1 cos22sin2 2exit s tartyyrrKK rc r tcK K K (4.19) The next step is to determine the direct tool-point FRF in the x -direction and y direction, xx and yy, respectively, to populate the matrix shown in Eq. (4.20). Traditionally, these values have been determ ined by impact testing but for the purposes of this study, they are both measured a nd predicted with the 3-component machineholder-spindle-tool RCSA method. 0 [] 0xx yy (4.20) 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 0 are determined for each chatter frequency and broken into their real, 1R and 2R, and imaginary components, 1I and 2I.

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58 0[] x xxxxyyy yxxxyyyy (4.21) Now, the limiting axial depths of cut, alim1 and alim2, can be calculated for each eigenvalue based on Eqs. (4.22) and (4.23) where N is the number of cutter teeth. 2 11 1 2 12 1RI lim t Ra NK (4.22) 2 22 2 2 22 1RI lim t Ra NK (4.23) The final step in the procedure is to de termine the relative spindle speed for each axial depth of cut. The spindle speeds, 1 and 2, for each eigenvalue are calculated using Eqs. (4.24) and (4.25) where P is indexed by integer values to produce each lobe and fc is the chatter frequency. 1 160*(2) 2c f NP (4.24) 2 260*(2) 2c f NP (4.25) where: 1 1 1 12tan I R 1 2 2 22tan I R 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 Mont e Carlo method [67, 68]. The Monte Carlo method is a numerical simulation techni que that randomly samples model input parameters, based on their distribution, to dete rmine the uncertainty of the model results. The strategy used to determine the uncertain ty in stability is as follows: 1) use a mechanistic approach to experimentally dete rmine cutting coefficients across a spindle speed range of interest, 2) determine a m ean and standard deviation for the cutting

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59 coefficients based on a sample set consisting of the results at multiple spindle speeds, 3a) obtain multiple measurement sets of the toolpoint response using imp act testing, if the uncertainty of the stability prediction is based on tool -point response measurement variation, or 3b) obtain multiple measurement sets at location H33 and H33c on the standard holder using impact te sting, if the uncertainty of th e stability prediction is based on the three-component machin e-spindle-holder-tool RCSA m odel, 4) use a Monte Carlo simulation that randomly samples, for each simulation run, the cutting coefficient parameters, the tool-point response measurem ent statistics or thre e-component machinespindle-holder-tool inputs, and the stability m odel 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 tool-point response measurements, there are two input parameters to the stability an alysis: the cutting coefficients and the x and y -direction FRF at the tool-point. The statistical properties of both input parameters are based on multiple ex perimental measurements, as outlined in the following chapter. Therefore, the input parameters are assumed to have normal distributions. For the Monte Carlo simulati ons that are based on tool-point responses predicted by the thr ee-component spindle-holder-tool RCS A model, there are four input parameters to the stability analysis: the cutt ing coefficients, the standard holder FRFs, the geometric properties of the holder and tool, a nd the material properties of the holder and tool. The statistical properties of the cu tting coefficients, standard holder FRFs, and geometric properties are again determined e xperimentally and are therefore assumed to

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60 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 4-1 where holder is the density of the steel holder, tool is the density of the carbide tool, Eholder is the modulus of elasticity of the holder, Etool is the modulus of elas ticity of the tool, and is the structural damping factor for both the hol der 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 acro ss 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 li miting axial depth of cut at the selected spindle speed, the limiting axial depth of cu t 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 deviati on of the limiting axial depth of cut at each

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61 spindle speed is calculated allowing uncertainty limits to be placed on the stability lobe diagrams. Spindle Speeda limUnstable Stable Figure 4-1. Stability lobe diagram. Up millingDown milling Cut direction start = 0 exit exit start Rotation direction Rotation direction y x F Surface normal Figure 4-2. Geometry of milling process.

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62 Table 4-1. Material statistical properties for holder and tool. Properties Low High holder (kg/m3) 7.81 x 103 7.83 x 103 tool (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 1010 63.0 x 1010 13.5 x 10-4 16.5 x 10-4

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63 CHAPTER 5 EXPERIMENTAL VALIDATION OF MONTE CARLO SIMULATIONS Experimental validation of the Monte Carlo simulations to place uncertainty bounds on stability lobe diagrams (for bot h 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 7075-T6 work piece, with a 12.7 mm diameter carbide tool mounted in a th ermal heat shrink hold er, and tool-point response variation due to measurement diffe rences at the tool-point. Stability comparisons to the simulation results were generated from 25 percent radial immersion cut tests at a variety of spindle speeds and ax ial depth of cut combinations. Stability was determined for each cut test using two met hods: 1) by qualitative obs ervation, and 2) by measuring the audio signal and determining the chatter frequency content. Simulation results were also generated for a Mikron UC P-600 Vario horizontal machine tool with a Step Tec HVCS 212-X spindle, using an Al uminum 6061-T6 work piece with a 19.1 mm diameter carbide tool mounted in a thermal heat shrink holder, and tool-point response variation based on the three-component machine-spindle-holder-tool 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 y -directions during the cut, and 3) by measuring the chatter frequency of the tool di splacement. Slotting and 50 percent radial immersion down milling machining conditions we re investigated. Simulation results for

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64 the Mikron UPC-600 Vario machining center we re also generated for FRFs measured while the spindle was rotating. Experimental Determination of Cutting Force Coefficients Aluminum 6061-T6 Work Piece and 19.1 mm Di ameter, 4 Flute, Carbide Tool Following the procedure outlined in Chapter 4, cutting tests were performed on an Aluminum 6061-T6 work piece using a 19.1 mm diameter, 4 flute, carbide helix tool (Dataflute part number SH-40750). The cutt ing tool was mounted in a thermal heat shrink holder (Command part number H6Y4A075 0) and tests were performed on a Step Tec 20,000 rpm spindle with an HSK 63A inte rface. 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 ax ial depth of cut (1 mm), and the cutting force signals were recorded with a dynamometer for the dire ction parallel to the cut direction ( x -direction), and orthogonal to th e cut direction ( y -direction). The experimental setup is shown in Fig. 5-1 where the work piece is mounted to a three-axis dynamomete r 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 s hown in Fig. 5-2. The mean cutting force was determined for each signal by averaging the signal across the steady state region (as indicated in Fig. 5-2). 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 y -direction, to determine the slope and vertical axis intercept. For example, Fig. 5-3 shows the linear regression for a spindle speed of 7,500 rpm. Equations (4.4-4.5) were then applied to determine the cutting coeffi cients for the spindle speed of interest.

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65 The procedure was repeated at multiple spindle speeds to determine the cutting coefficients, Ktc and Krc, at nine different spindle speed s across the available spindle speed range and are displayed in Table 5-1. The cutting coefficients used in the Tlusty stability lobe technique, Ks and were determined using Eqs. (4.7-4.8) and the cutting coefficients used in the Altin tas 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 co efficient values, decrease at higher spindle speeds due to thermal softening e ffects. In other word s, it takes less force to cut the material when it is at increased te mperatures. Another inte resting aspect of the cutting coefficient values shown in Table 5-1 is that the radi al 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 techni que. At higher speeds, the slope of the line fit to the x -direction mean force versus feed ra te switches direction and the mean cutting forces in the x-direction increase w ith 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 a nd standard deviation in this case)of the cutting force coefficients were determined based on a sample set comprised of cutting coefficient values from Table 5-1 for spi ndle speeds between 7,500 and 20,000 rpm. As seen in Fig. 5.4 and Fig. 5.5, the cutting co efficients are relatively constant across this range of spindle speeds. Therefore, determ ining the cutting coefficient statistical

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66 properties based on the cutting coefficient valu es across this range is reasonable. Table 5-2 shows the statistical properties of the cutting coeffi cients which were used in the Monte Carlo simulations for the Aluminum 6061-T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool. The variable, , is the standard deviation Aluminum 7475-T6 Work Piece and 12.7 mm Di ameter, 4 Flute, Carbide Tool Again following the procedure outlined in Ch apter 4, cutting tests were performed on an Aluminum 7475-T6 work piece using a 12.7 mm diameter, 4 flute, carbide helix tool (part number CRHEC500S 4R30-KC610M). 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 ax ial 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 6061-T6 work piec e and 19.1 mm diameter, 4 flute, carbide helix tool configuration. To determine th e statistical properties of the cutting force coefficients, the results for the two spindles speeds were averaged and the standard deviations were determined. Table 5-3 show s the statistical properties of the cutting coefficients which were used in the Monte Ca rlo simulations for this tool and work piece combination. Stability lobe diagrams fo r Makino machining center Stability Determination Stability verification experiments were performed for the Makino machine tool, thermal shrink fit holder, using a 12.7 mm diam eter, 4 flute, carbide helix tool, and an

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67 Aluminum 7475-T6 work piece combination. The radial immersion of the cutting tool was 3.175 mm, or 25 percent, and a down milli ng operation was performed. A variety of cut tests were performed at various spindle speeds and axial depth of cut combinations. During the actual testing, stab ility was determined qualitati vely 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 natu ral frequency of the tool-point 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 test ing to avoid damaging th e tool or spindle. In addition to the qualitative assessment of stability, stability was also determined quantitatively for each spindle speed and ax ial 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 determ ined by the experimenter). For the purposes of this experiment, chatter content above 20 signified unstable cutting conditions. Figures 5-6 and 5-7 display examples of a udio signal frequency content for stable and unstable cutting conditions, respec tively. As seen in Fig. 5-6, frequency content does exist at the tooth passing fre quency, as identified, and at additional frequencies due to noise and once-per-revolution runout. This is to be expect ed. 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. 5-8.

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68 Monte Carlo Simulation Parameters The Monte Carlo simulation procedure requir es 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 Krc. The correlation between the two parameters was determined to be 97 percent. High correlation exists between the tangential and ra dial 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 for ce increases, both components also typically increase. During each run of the simulation, values for Ktc and Krc were randomly generated based on a multivariate normal distribution, the cutting coefficient means displayed in Table 5-3, and the covariance matrix shown in Eq. (5.1). The diagonal elements in the covariance matrix represen t the cutting coefficient variances and the offdiagonal elements represent the covariance s between the cutting coefficients. The appropriate cutting coefficients, Ks and for the Tlusty technique were determined in the simulation based on Eqs. (4.7-4.8) and the appropriate cutting coefficients, Kt and Kr, for the Altintas technique were de termined based on Eq. (4.19). 1515 15156.76106.0410 Covariance Matrix= 6.04105.7010xx xx (m2) (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 y -direction and 95 percent confidence intervals are shown in Fig. 59 and Fig. 5-10, respectively.

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69 For the Monte Carlo simulation, 100% corr elation between indi vidual 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 distribu tion) 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 fo r a given iteration of the Monte Carlo simulation, F is its mean value, and F is the standard de viation. The strong correlation between frequencies and real/imag inary parts occurs because the data is collected simultaneously in impact testi ng. It should be noted, however, that the uncertainty was frequency dependent, i.e., F was a function of frequency in Eq. (5.2), and was larger near resonance as shown in Figs. 5-9 and 5-10. F F=F+a (5.2) Another consideration for the FRF data was potential correlation between the x and y -direction measurements. It is possible that if the vari ation between measurements was caused by, for example, a change in th e 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, xy, was evaluated. The result is shown in Fig. 5-11; it is seen that the cova riance is strongly dependent on frequency with the highest values near resonance and near zero levels everywhere else. Because 100% correlation between frequencie s and the real and imaginar y parts for the individual directions was already identif ied, it was not possible to allow a frequency dependent correlation between the two directions. Ther efore, zero and 100 pe rcent correlation was investigated between the x and y directions. The actua l measured correlation was determined to be 93 percent.

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70 Stability Lobe Results One-thousand Monte Carlo simulation r uns were performed for the Makino machining center, thermal heat shrink holde r, 12.7 mm diameter, 4 flute carbide helix tool, and 7475-T6 Aluminum work piece comb ination 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 cu tting coefficients and tool-point response as described in previous s ections. Figures 5-12 and 5-13 show the comparison between predicted stability lobes with 95 percent c onfidence intervals (bas ed on the Budak and Altintas technique) and experimental result s. Figure 5-12 shows the case where the x and y -direction tool-point FRFs are not correlat ed and Fig. 5-13 displays the case where the x and y -direction tool-point FRFs are 100 percent correlated. As seen in the figures, the correlation level between the x and y -direction FRFs does not significantly alter the predicted results. Figures 5-14 and 5-15 show the comparison between predicted stability lobes with 95 percent confidence intervals ba sed on the Tlusty technique as compared to experimental results. Figure 5-14 shows the case where the x and y -direction tool-point FRFs are not correlated, and Fig. 5-15 displays the case where the x and y -direction toolpoint FRFs are 100 percent correlated. As seen in the figures, the correlation level between the x and y -direction FRFs, when using the Tlusty stability technique, does have a significant effect on the uncertainty boundari es of the predicted results. The Tlusty stability analysis technique creates direct ional orientation factors which lump the x and y -direction FRFs into a single, oriented FRF; therefore, the correlation level between the x and y -direction FRFs significantly affects the un certainty levels of the final stability lobes. The Budak and Altintas stability tec hnique does not couple the xand y-direction

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71 FRFs; therefore, the uncertainty levels of the predicted stability lobes are not as sensitive to the x and y -direction FRF correlation. A comp arison between Figs. 5-12 and 5-14 shows that the confidence intervals for the Tl usty lobes are wider for equivalent input mean values, distributions, and correlations as compared to the Budak and Altintas technique generated stability lobes. In addi tion, the Tlusty mean stability boundary tends to over-predict the experimental stability limit for these tests. This may be the result of the asymmetric x and y -direction FRFs. Sensitivity of Budak and Alti ntas 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 th is case) to the uncertainties of each of the input parameters (the cutting coefficients and tool-point response). This is accomplished by selecting random samples from a single i nput variable distribut ion 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 variab les with the greatest contributions can be identified and their variation minimized, if po ssible. 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 he lix tool, a 7475-T6 Aluminum work piece combination, and 25 percent radial immersion down milling, the se nsitivities of the tool-point response and cutting coefficients were determined fo r both the Budak and Altintas and Tlusty analytical stability lobe pr ediction techniques based on 1000 Monte Carlo simulation runs and assuming 100 percent co rrelation be tween the x and y -direction FRFs. Figure 5-16 displays the stability lobe uncertainty base d solely upon cutting coefficient uncertainty

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72 for the Altintas technique, and Fig. 5-17 displa ys stability lobe uncertainty based on toolpoint response uncertainty for the Budak a nd Altintas technique. As shown in the figures, the uncertainty due to both cutting co efficient variation and tool-point response variation is relatively consta nt across the spindle speed range with a slightly larger contribution coming from tool-point response vari ation. Figure 5-18 displays the stability lobe uncertainty based solely upon cutting coefficient uncertainty for the Tlusty technique, and Fig. 5-19 disp lays stability lobe uncertainty based on tool-point response for the Tlusty technique. As shown in the figures, the uncertainty contribution from each input variable is similar; howev er, it is interesting to note in Fig. 5-19 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 toolpoint response (as illustrate d in the tool-point FRF to stability lobe mapping example shown in Fig. 5-20). As seen in Figs. 5-9 and 5-10, 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 UCP-600 Vario machining center, with a thermal heat shrink holder, a 19.1 mm diameter, a 4 flute carbide helix tool, and a 6061-T6 Aluminum work piece combination, two milling conditions were investigated, with slotting and 50 percent radial immersion down milling. A qualita tive assessment of stability was again performed during the experimental process; howev er, a test rig, as di splayed in Fig. 5-21, was attached to the spindle to monitor tool displacement during the milling process. Two

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73 capacitive probes, one in the x -direction and one in the y -direction, measured the displacement of the tool dur ing the experimental cut, and a tachometer used in conjunction with a color mark on the holder identified a cons istent 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., once-per-revolution sampling). For stable cuts, the once-per-rev olution displacement was very consistent; therefore, stability was quantitatively dete rmined by comparing the variance of the onceper-revolution displacement to the variance of the total tool displacement. Since both the x and y -direction displacements were bei ng measured, the joint variance, c2, was determined based on Eq. (5.3) below where x2 and y2 are the variances of the x and y direction displacements, respectively: 222 cxy (5.3) The y -direction (orthogonal to the feed or x -direction) displacement signal was also Fourier transformed to look for chatter c ontent in the frequency domain. The chatter content in the x -and y -direction were identi cal; therefore, the y -direction 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 onceper-revolution displacement and the total cycle displacement of less than 0.2 was a stab le cut, between 0.2 and 0.8 was a marginal cut, and greater than 0.8 was an unstable cu t. The variance ratio ranges to determine stable, marginal, and unstable cuts were sele cted qualitatively based on the surface finish of the milled surface. Further testing is re quired to determine if these variance ratio ranges can be used for other spindle-holder-tool-work piece combinations.

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74 2 once per revolution displacement 2 total displacement c cR (5.4) To illustrate the testing results, the x -direction versus y-dir ection total displacement and once-per-revolution displacement signals and the frequency content of the y -direction displacement are displayed in Figs. 5-22 thr ough 5-27 for slotting experiments. Figure 522 shows the results for an 8,000 rpm, 2 mm axia l depth stable cut. In Fig. 5-22A, the once-per-revolution samples are grouped much more tightly than the total displacement samples shown in the light grey. In Fig. 522B, the frequency conten t of the y-direction displacement is plotted with the synchronous cont ent identified. As seen in Fig. 5-22B, there is no chatter frequency content or, in other words, non-synchronous 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 Fi g. 5-23A, the once-per-revolution samples now have the same spread as the total displacement. Also, Fig. 5-23B shows chatter content at 799 Hz. Figures 5-24 and 5-25 display the re sults for a 10,000 rpm, 5 mm and 6 mm cut, respectively. This is the optimum machini ng speed where the tooth passing frequency is matched to the resonance frequency of the tool-point. In Fig. 5-24A, the once-perrevolution x versus y samples are closely grouped in comparison to the x versus y total displacement, and in Fig. 5-25A, the once-pe r-revolution samples ar e marginally grouped in comparison to the total displacement. Th e advantage of using the once-per-revolution displacement signal and total disp lacement signal plotted in the x -direction versus y direction to determine stability is exemplified in Fig. 5-25B. It is difficult to determine chatter content from the frequency plot because the synchronous frequency and the chatter frequency fall on the same point. Fi gures 5-26 and 5-27 show the results for a

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75 15,000 rpm, 2 and 3 mm axial depth cut, re spectively. The stability results are summarized in Table 5-4 for all spindle sp eed and axial depth of cut combinations. The x -direction versus y -direction total displacement and once-per-revolution displacement signals and the frequency content of the y -direction displacement are again displayed in Figs. 5-28 through 5-31 for 50 pe rcent radial immersi on experiments at a variety of spindle speed and ax ial depth of cut combinations Applying the same criteria as for the slotting cuts, the stability results are summarized in Table 5-5. Monte Carlo Simulation Parameters The tool-point response for the Step Tec sp indle, thermal heat shrink holder, and 4 tooth, carbide helix tool was determined us ing the three-component spindle-holder-tool RCSA model. Therefore, the input parameters to the Monte Carlo simulation were cutting coefficients, standard holder measurem ents, holder and tool material properties, and holder and tool geometries. The first se t of input parameters for the Monte Carlo simulation was the cutting coefficients, Ktc and Krc, for the tool and Aluminum 6061-T6 work piece. The correlation between the two parameters was determined to be 93 percent. During each run of the simulation, values for Ktc and Krc were randomly generated based on a multivariate normal distribution, the cutting coefficient means displayed in Table 5-2, and the covariance ma trix shown in Eq. (5.5). The appropriate cutting coefficients, Ks and for the Tlusty technique were determined in the simulation based on Eqs. (4.7-4.8) and the a ppropriate cutting coefficients, Kt and Kr, for the Budak and Altintas technique were determined base d on Eq. (4.19). As mentioned previously, the diagonal elements in the covariance matr ix represent the cutting coefficient variances, and the off-diagonal elements represent the cova riances between the cutting coefficients.

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76 1314 14157.00101.1010 Covariance Matrix= 1.10109.9110xx xx (m2) (5.5) 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 4-1 with uniform distributions assumed for all parameters. The next set of input parameters were the geometric pr operties of the holder and tool. The relevant dimensions were meas ured multiple times with digital calipers to determine the mean values, as displayed in Fig. 5-32 and Table 5-6. A worst case standard deviation on all measurements usi ng 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 assu med to be equal to the shank diameter. 3m mat R (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 condi tions. 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. 5-33 through 5-36. For the second set of conditions, the standard holder was

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77 removed from the spindle between measuremen ts to account for clamping variation. No appreciable difference was seen in the st andard holder measurements for the two conditions. The measurement technique fo r the case where the standard holder was rotating during measurement will be e xplained in an upcoming section. Another consideration for the FRF data was potential correlation between the xand y-direction measurements. The measur ed correlation was determined to be 93 percent; therefore 100 pe rcent correlation was assumed in the Monte Carlo simulation. As described previously in the Makino sp indle section, 100% correlation between frequencies and the real and imaginary part s for the individual directions was also identified. Stability Lobe Results One-thousand Monte Carlo simulation runs were performed for the Mikron UCP600 machining center, with a th ermal heat shrink holder, a 19.1 mm diameter, a 4 flute carbide helix tool, and a 6061-T6 Aluminum work piece combination. The first result of the simulation was to provide 95 percent uncer tainty bounds on the tool-point response in the x and y -direction as predicted by the three-co mponent spindle-holder-tool RCSA model. Figures 5-37 and 5-38 show the tool-point response in the x and y -direction, respectively for the case where holder clam ping force variation was not included. Figures 5-39 and 5-40 show the tool-point res ponse in the x and y di rection, respectively, for the case where holder clampi ng force variation was included. As seen in the figures, holder clamping force variation does have an effect on the mean tool-point response although the range of the 95 percent conf idence interval bounds is unaffected. Figure 5-41 displays the predicted B udak and Altintas stability lobes and uncertainty as compared to the measured st ability results for slotting for the case where

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78 holder clamping force variation is excluded. As seen in the figure, the predicted stability results are biased to higher spindle speeds th an the measured results. This result is consistent for all cases where the stability results are based on non -rotating measurements of the standard holder FRFs. Future sections w ill show that this bias is due to the spindle dynamics changing with speed. Figure 5-42 disp lays the predicted Tlusty stability lobes and uncertainty as compared to measured st ability 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 de pths 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. 5-42 also show a region between 10,000 and 12,000 rpm where no stability lobes or uncertainty levels exis t. The Tlusty technique does not provide overlapping lobes in this region because the fr equency vectors associated with each lobe produced by the Tlusty stability analysis technique are based only on the negative real part of the tool-point response. In this case, the negative real part of the tool-point response has a very small range, thus creati ng frequency vectors that do not overlap. Figure 5-43 and 5-44 show the predicted stabi lity lobes and uncertainty levels for the Budak and Altintas and Tlusty techniques, re spectively, 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. 5-45 and 5-46 show stability lobe predicted results with uncertainty levels and measured stability results for the Budak and Altintas technique and

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79 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 non-rotating standard holder measurements and the measured stability results. A technique to measure the st andard holder FRFs while the spindle was rotating and process the resulting data to eliminate th e 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. 5-47 where an impact hammer inputs a measured force into the system and a capacitive probe measures the resulting displacement of the standard hol der while the standard holder ro tates at a specified speed. A time signal of the force input and resulting standard holder displa cement 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 si gnal to generate the transient pa rt of the signal. To further eliminate synchronous content, 100 time signals after fitting and s ynchronous component subtraction, were ensemble averaged to pr oduce 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 produ ce the required mean FR F. The standard deviation of the FRF was determined by Four ier-transforming each individual signal (100 signals total) after removing the synchronous content from each signal and determining the standard deviation at each frequenc y. Figure 5-48 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 functi on of spindle speed. Figure 5-49 shows the

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80 predicted tool-point response based on standa rd holder measurements at 0 rpm, 10,000 rpm, and 16,000 rpm. Again, the figure show s that the spindle dynamics are changing as a function of spindle speed. Also of interest is a comparison of the predicted tool-point FRFs based on rotating standard holder measurements versus the locat ion of the measured ch atter frequencies, as displayed in Table 5-4. If th e predicted tool-point FRF is representative of the spindle dynamics, the tool should chatter, during unstabl e cuts, at frequencies near the tool-point FRF modes (natural frequencies). Figure 5-50 and 5-51 display the x and y -direction predicted tool-point FRFs, re spectively, based on standard holder measurements at 10,000 rpm and the chatter frequencies determ ined during unstable cutting conditions for slotting cuts. As expected and seen in the figures, the chatter fre quencies are very close to the tool-point FRF natural frequency for the lower mode. The Monte Carlo simulation was modifi ed 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 sp eeds. Figures 5-52 through 5-55 display the standard holder mean and 95 percent confiden ce 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 pr oduced based on non-rotating measurements. Measuring the standard holder while it is rotating introduces increased uncertainty into the Monte Carlo model.

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81 The predicted stability lobes based on the Budak and Altintas technique, with 95 percent confidence intervals and measured stab ility results, are shown in Fig. 5-56 for a slotting cut on the Step Tec spi ndle with a tapered heat shrink holder, a 4 tooth, carbide helix tool and an Aluminum 6061-T6 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. Fi gure 5-57 again shows reasonable agreement between predicted and measured stability lo be results for the same system with a 50 percent radial immersion cut. Figures 558 and 5-59 show comparison between predicted stability lobes based on static standard holder measurements versus rotational standard holder measurements at 10,000 rpm for slotti ng and 50 percent radial immersion cuts, respectively. The Tlusty stability lobe tech nique did not produce re alistic 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 Stab ility Lobe Prediction Techniques when the Three-component Spindle-holder-tool RCSA Model is Used to Generate Toolpoint Response. The sensitivity of the stability lobe uncer tainty to the uncertainties of each of the input parameters (the cutting coefficients, ma terial properties, geometries, and standard holder measurements at 10,000 rpm) was determ ined for the Budak and Altintas stability lobes for the Step Tec spindle, thermal heat sh rink holder, 4 tooth, carbide helix tool, and Aluminum 6061 work piece. As shown in Fi g. 5-60 for the slotting case, the geometric properties and material propertie s contribute very little to the stability lobe uncertainty while the cutting coefficient uncertainty is the greatest contributor Figure 5-61 shows the sensitivity results for the 50 percent ra dial immersion cut. Again, the cutting

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82 coefficient uncertainty and standard holder m easurement uncertainty provide the majority of the final stability lobe uncertainty. As explained in the previous section, it was not possible to generate realistic Tlusty stabil ity lobes based on rotational standard holder measurements; therefore, the se nsitivities of the Tlusty te chnique cannot be determined for this particular system setup. Figure 5-1. Experimental setup fo r measuring cutting force signals.

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83 -200 -100 0 100 200 Fx (N) 0 1 2 3 4 5 -200 -100 0 100 200 300 Fy (N)Time (seconds) Steady state region Figure 5-2. Cutting force signal for 7500 rpm cut with 0.18 mm/tooth chip load. 45 40 35 30 Mean Fx (N) 0.100 0.180 0.250 80 100 120 140 160 180 Chip load (mm/tooth)Mean Fy (N) Figure 5-3. Linear regressi on for cutting force means at a spindle speed of 7,500 rpm.

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84 Table 5-1. Cutting coefficients for al uminum 6061-T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool. Spindle Speed (rpm) Ktc (N/m2) Krc (N/m2) Kt (N/m2) Kr (-) Ks (N/m2) (deg.) 1000 8.2 x 108 2.4 x 108 8.2 x 108 2.9 x 10-1 858.0 73.8 2500 7.6 x 108 2.6 x 108 7.6 x 1083.4 x 10-1 805.1 71.4 5000 6.1 x 108 2.8 x 108 6.1 x 1084.7 x 10-2 606.7 87.3 7500 5.2 x 108 -4.0 x 108 5.2 x 108-7.6 x 10-2 525.5 -85.7 10000 5.2 x 108 -4.3 x 108 5.2 x 108-8.3 x 10-2 520.8 -85.3 12500 5.4 x 108 -2.6 x 108 5.4 x 108-4.9 x 10-2 542.6 -87.2 15000 5.2 x 108 -2.8 x 108 5.2 x 108-5.3 x 10-2 523.7 -87.0 17500 5.3 x 108 -4.0 x 108 5.3 x 108-7.5 x 10-2 532.5 -85.7 20000 5.2 x 108 -1.8 x 108 5.2 x 108-3.5 x 10-2 522.3 -88.0 5 6 7 8 9 x 10 -8 K (N/m 2 ) 0 5000 10000 15000 2000 0 -0. 2 0 0. 2 0. 4 0. 6 Spindle Speed (rpm)K t r Figure 5-4. Cutting coefficients versus spi ndle speed for the Budak and Altintas stability lobe technique.

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85 500 600 700 800 900 K s (N/m 2 m ) 0 5000 10000 15000 2000 0 -100 -50 0 50 100 Spindle Speed (rpm) (deg.) Figure 5-5. Cutting coefficients versus spindle speed for Tlusty stability lobe technique. Table 5-2. Statistical properties for cutti ng coefficients for aluminum 6061-T6 work piece and 19.1 mm diameter, 4 flute, carbide helix tool. Ktc (N/m2) Krc mean 5.27 x 108 -3.25 x 107 8.42 x 106 9.73 x 106 Table 5-3. Statistical properties for cutti ng coefficients for aluminum 7475-T6 work piece and 12.7 mm diameter, 4 flute, carbide helix tool. Ktc (N/m2) Krc (N/m2) mean 7.59 x 108 1.97 x 108 8.22 x 107 7.55 x 107

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86 0 1000 2000 3000 4000 5000 0 1 2 3 4 5 6 7 Frequency (Hz)Magnitude Tooth passing frequency Figure 5-6. Audio signal frequency content for 10,000 rpm spindle speed, 1.02 mm axial depth cut. Cut was determined to be stable. 0 1000 2000 3000 4000 5000 0 10 20 30 40 50 60 Frequency (Hz)Magnitude Chatter Frequency Figure 5-7. Audio signal fr equency content results for 19, 000 rpm spindle speed, 1.52 mm axial depth cut. Cut was determined to be unstable.

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87 8000 10000 12000 14000 16000 18000 20000 0 1 2 3 4 5 6 7 8 Spindle Speed (rpm) stable unstable a (mm) lim Figure 5-8. Stability results for Makino machining center, thermal heat shrink holder, 12.7 mm, 4 flute, carbide helix tool Aluminum 7475-T6 work piece and a 25 percent radial immersion cut. -2 0 2 x 10 -6 500 1000 1500 2000 2500 3000 -4 -3 -2 -1 0 1 x 10 -6 Frequency (Hz)Mean 95% Confidence Interval Imag (m/N) Real (m/N)0 Figure 5-9. Mean and 95 percent (2 ) confidence intervals for x -direction FRF.

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88 -2 -1 0 1 2 3 x 10 -6 500 1000 1500 2000 2500 3000 -4 -2 0 2 x 10 -6 0 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-10. Mean and 95 per cent confidence intervals for y -direction FRF. 0 2 4 6 8 x 10-14 (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 0 1 2 3 4 x 10 -6 Frequency (Hz)FRF mag (m/N ) x direction y direction xy 2 2 Figure 5-11. Covariance between x and y -direction FRFs.

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89 8000 10000 12000 14000 16000 18000 20000 0 1 2 3 4 5 6 7 8 Spindle Speed (rpm) predicted stable unstable a (mm) lim Figure 5-12. Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 pe rcent confidence in tervals are shown for the case where the x and y -direction FRFs are not correlated. 0 1 2 3 4 5 6 7 8 a (mm) lim predicted stable unstable 8000100001200014000160001800020000 Spindle Speed (rpm) Figure 5-13. Comparison between Budak and Altintas lobes and experimental results. The mean stability boundary and 95 pe rcent confidence in tervals are shown for the case where x and y -direction FRFs are 100 percent correlated.

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90 -10 -5 0 5 10 15 20 8000100001200014000160001800020000 Spindle Speed (rpm) predicted stable unstable a (mm) lim25 Figure 5-14. Comparison between Tlusty lobe s and experimental results. The mean stability boundary and 95 percent co nfidence intervals are shown for the case where x and y -direction FRFs are not correlated. -5 0 5 10 15 20 25 8000100001200014000160001800020000 Spindle Speed (rpm)a (mm) lim predicted stable unstable Figure 5-15. Comparison between Tlusty lobe s and experimental results. The mean stability boundary and 95 percent co nfidence intervals are shown for the case where x and y -direction FRFs are 100 percent correlated.

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91 0 1 2 3 4 5 6 7 8 8000100001200014000160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-16. Budak and Altintas lobes based on cutting coefficient uncertainty. 0 1 2 3 4 5 6 7 8 8000100001200014000160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-17. Budak and Altintas lobes ba sed on tool-point response measurement uncertainty.

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92 0 2 4 6 8 10 12 14 16 18 20 8000100001200014000160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-18. Tlusty lobes based on cutting coefficient uncertainty. -2 0 2 4 6 8 10 12 14 8000100001200014000160001800020000 Spindle Speed (rpm)a (mm) lim 20 18 16 Figure 5-19. Tlusty lobes based on tool -point response measurement uncertainty.

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93 Spindle Speed (rpm)a (mm) lim Resonance Figure 5-20. Example of mapping of tool-point FRF to stability lobe. Figure 5-21. Experimental setup for measuring chatter based on x and y -direction tool displacement.

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94 1 2 3 4 5 x 10-6 -8 -7 -6 -5 -4 x 10 -6 X (mm)Y (mm) Magnitude (m) Once-per-revolution samplesA B 400 800 1200 1600 2000 0 1 2 3 4 x 10-7 Frequency (hz) Synchronous content Figure 5-22. 8,000 rpm, 2 mm axial depth sl otting cut test fo r Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be stable. -4 -2 0 2 4 6 8 x 10-6 -6 -4 -2 0 2 x 10 -6 X (mm)Y (mm) Magnitude (m) Once-per-revolution samplesA B 0 0.5 1 1.5 2 2.5 3 x 10-7 400800120016002000 0 Frequency (hz) Synchronous content Figure 5-23. 8,000 rpm, 2.5 mm axial depth slotting cut test for Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be unstable.

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95 400 800 1200 1600 2000 0 1 2 3 x 10-8 Frequency (hz)Magnitude (m ) 4 5 6 7 x 10-6 -2 0 2 4 x 10 -6 X (mm)Y (mm)0 Once-per-revolution samples Synchronous contentA B Figure 5-24. 10,000 rpm, 5 mm axial depth sl otting cut test fo r Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be stable. -9 -7 -5 -3 x 10 -6 -4 -2 0 2 x 10 -6 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 1 2 3 4 5 x 10-7 Frequency (hz)Magnitude (m)0samples Figure 5-25. 10,000 rpm, 6 mm axial depth sl otting cut test fo r Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be marginal.

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96 1.2 1.6 2 2.4 2.8 3.2 x 10-6 -8 -4 0 4 8 12 x 10 -7 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 0.4 0.8 1.2 x 10-7 Frequency (hz)Magnitude (m)0 Synchronous content samples Figure 5-26. 15,000 rpm, 2 mm axial depth sl otting cut test fo r Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be marginal. -4 -3 -2 x 10 -6 -2 -1 0 1 x 10 -6 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 2 4 6 8 x 10-8 Frequency (hz)Magnitude (m)0 samples Figure 5-27. 15,000 rpm, 3 mm axial depth sl otting cut test fo r Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be marginal.

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97 Table 5-4. Stability results for slotti ng cuts on a Mikron Vario machining center. Spindle speed (rpm) Axial depth (mm) Variance ratio, R Stability Result Chatter frequency (Hz) 8000 2.0 0.16 Stable 8000 2.5 1.00 Unstable 799 9000 3.0 0.21 Marginal 850 9000 3.5 0.41 Marginal 867 9000 4.0 0.94 Unstable 820 10000 3.0 0.00 Stable 10000 4.0 0.00 Stable 10000 5.0 0.00 Stable 10000 6.0 0.22 Marginal 667 11250 4.0 0.00 Stable 11250 4.5 0.07 Stable 12500 3.0 0.01 Stable 12500 4.0 0.42 Marginal 13750 3.0 0.38 Marginal 13750 3.5 0.99 Unstable 750 15000 2.0 0.67 Marginal 15000 3.0 0.55 Marginal 725 15000 4.0 1.00 Unstable 17500 2.0 0.72 Marginal 17500 2.5 1.00 Unstable 762 20000 2.5 0.88 Unstable 748 400 800 1200 1600 2000 0 1 2 3 x 10-7 Frequency (hz)Magnitude (m) Once-per-revolution Synchronous contentA B 0 2 4 6 x 10-6 -4 -2 0 2 x 10 -6 X (mm)Y (mm)0 samples Figure 5-28. 8,000 rpm, 4 mm axial depth 50 % radial immersion cut test for Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be stable.

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98 0 1 2 x 10 -5 -2 -1 0 1 x 10 -5 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 2 4 6 8 x 10-7 Frequency (hz)Magnitude (m)0 samples Figure 5-29. 8,000 rpm, 5 mm axial depth 50 % radial immersion cut test for Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be unstable. -2 -1 0 x 10 -5 3 4 5 6 7 x 10 -6 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 2 4 6 x 10-7 Frequency (hz)Magnitude (m)0 samples Figure 5-30. 10000 rpm, 16 mm axial depth 50 % radial immersion cut test for Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays freque ncy content of the y displacement. Cutting conditions were determined to be stable.

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99 -2 -1 0 x 10 -5 2 4 6 8 x 10 -6 X (mm)Y (mm) Once-per-revolution Synchronous contentA B 400 800 1200 1600 2000 0 0.4 0.8 1.2 x 10-7 Frequency (hz)Magnitude (m)0 samples Figure 5-31. 10000 rpm, 17 mm axial depth 50 % radial immersion cut test for Mikron UCP-600 Vario machining center. A) displays x displacement versus y displacement, and B) displays frequency content of the y displacement. Cutting conditions were determined to be stable. Table 5-5. Stability results for 50 % radi al immersion cuts on a Mikron UCP-600 Vario machining center. Spindle speed (rpm) Axial depth (mm) Variance ratio, R Stability Result Chatter frequency (Hz) 8000 4.0 0.01 Stable 8000 5.0 1.0 Unstable 760 10000 14.0 0.01 Stable 10000 15.0 0.00 Stable 10000 15.5 0.00 Stable 10000 16.0 0.01 Stable 10000 16.5 0.01 Stable 10000 17.0 0.02 Stable 11250 10.0 0.08 Stable 12500 6.0 0.00 Stable 12500 7.0 0.07 Stable 12500 8.0 0.09 Stable 12500 9.0 0.44 Marginal 766 12500 10.0 0.97 Unstable 723 13750 8.0 0.73 Marginal 13750 9.0 0.96 Unstable 726 15000 8.0 0.03 Stable 15000 9.0 0.81 Unstable 726 17500 5.0 0.13 Stable 17500 6.0 0.11 Stable 17500 8.0 1.0 Unstable 508

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100 D 1 D 2 D 3 L 1 L 2 L 6 L 3 L 4 L 5 Figure 5-32. Geometric properties of tapere d heat shrink holder and 4 flute, 19.1 mm diameter carbide helix tool. Table 5-6. Statistical geometric properties of tapered heat shrink holder and 4 flute, 19.1 mm diameter carbide helix tool. Parameter Mean Standard Deviation D1 (mm) 19.1 D2 (mm) 33.0 D3 (mm) 41.4 L1 (mm) 102.3 L2 (mm) 76.1 L3 (mm) 101.9 L4 (mm) 63.5 L5 (mm) 52.6 L6 (mm) 48.3 0.0254

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101 -1 0 1 x 10 -7 0 1000 2000 3000 40005000 -15 -10 -5 0 5 x 10 -8 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-33. H33 with 95 percent confidence intervals in the x -direction for no holder clamping variation. -1 0 1 x 10 -7 0 1000 2000 3000 4000 5000 -10 -5 0 5 x 10 -8 Mean 95% Confidence Interval Frequency (Hz)Imag (m/N) Real (m/N)-15 Figure 5-34. H33c with 95 percent confidence intervals in the x -direction for no holder clamping variation.

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102 -1 0 1 x 10 -7 0 1000 2000 3000 4000 5000 -15 -10 -5 0 5 x 10 -8 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-35. H33 with 95 percent confidence intervals in the y -direction for no holder clamping variation. -1 0 1 x 10 -7 0 1000 2000 3000 4000 5000 -15 -10 -5 0 5 x 10 -8 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-36. H33c with 95 percent confidence intervals in the y -direction for no holder clamping variation.

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103 -2 0 2 4 6 x 10 -7 500 1000 1500 2000 2500 -6 -4 -2 0 x 10 -7 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N)0 Figure 5-37. The x -direction tool-point FRF with 95 pe rcent confidence intervals for no holder clamping force variation. 0 2 4 x 10-7 500 1000 1500 2000 2500 -6 -4 -2 0 x 10 -7 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N)0 Figure 5-38. The y -direction tool-point FRF with 95 pe rcent confidence intervals for no holder clamping force variation.

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104 -2 0 2 4 6 x 10 -7 500 1000 1500 2000 -6 -4 2 0 2 x 10 -7 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-39. The x -direction tool-point FRF with 95 percent confidence intervals with holder clamping force variation. -1 0 1 x 10 -6 500 1000 1500 2000 -2 -1 0 1 x 10 -6 Frequency (Hz) Mean 95% Confidence Interval Imag (m/N) Real (m/N) Figure 5-40. The y -direction tool-point FRF with 95 percent confidence intervals with holder clamping force variation.

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105 0 1 2 3 4 5 6 predicted stable marginal unstable 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-41. Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability resu lts for slotting cut with Step Tec spindle, tapered heat shrink holder, a nd 19.1 mm diameter, 4 flute, carbide helix tool. Holder clampi ng force variation excluded. 0 5 10 15 20 25 30 predicted stable marginal unstable 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-42. Tlusty predicted stability lobe s with 95 percent conf idence intervals and measured stability results for slotting cu t with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 fl ute, carbide helix tool. Holder clamping force variation excluded.

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106 0 2 4 6 8 10 12 14 16 18 20 80001000012000 14000 160001800020000 Spindle Speed (rpm) predicted stable marginal unstable a (mm) lim Figure 5-43. 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 t ool. Holder clamping fo rce variation excluded. 0 5 10 15 20 25 30 35 40 45 50 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable Figure 5-44. 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 t ool. Holder clamping fo rce variation excluded.

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107 0 1 2 3 4 5 6 7 8 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable Figure 5-45. Budak and Altintas predicted stability lobes with 95 percent confidence intervals and measured stability resu lts for slotting cut with Step Tec spindle, tapered heat shrink holder, a nd 19.1 mm diameter, 4 flute, carbide helix tool. Holder clampi ng force variation included. 0 2 4 6 8 10 12 14 16 18 20 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable U Figure 5-46. 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 t ool. Holder clamping fo rce variation included.

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108 Figure 5-47. Experimental set up for measuring rotating FRFs. 0 0.5 1 1.5x 10 -7 0 rpm 10000 rpm 16000 rpm 500 1000 1500 2000 2500 0 0.5 1 1.5x 10 -7 Freq (Hz)Mag (m/N)0Mag (m/N) Figure 5-48. Magnitude of H33 in x -direction as a functi on of spindle speed.

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109 2 4 6 8 x 10-7 0 500 1000 1500 2000 2500 0 2 4 6 8 x 10 -7 Freq (Hz) 0 0 rp m 10000 rpm 16000 rpm Mag (m/N) Mag (m/N) Figure 5-49. Magnitude of pred icted tool-point response, H11, in x -direction as a function of spindle speed. -1 0 1 2 3 x 10 -7 Real (m/N) chatter frequency 400 800 1200 1600 2000 -3 -2 -1 0 x 10 -7 Freq (Hz)Imag (m/N) Figure 5-50. Predicted x -direction tool-point FRF based on standard holder measurements at 10,000 rpm and chatter frequenc ies of unstable slotting cuts.

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110 -1 0 1 2 3Real (m/N) -3 -2 -1 0Imag (m/N) chatter frequency 400800120016002000 Freq (Hz) x 10 -7x 10 -7 Figure 5-51. Predicted y -direction tool-point FRF based on standard holder measurements at 10,000 rpm and chatter frequenc ies of unstable slotting cuts. -5 0 5 10 x 10 -8 0 500 1000 1500 -10 -5 0 5 x 10 -8 Frequency (Hz)Imag (m/N) Real (m/N)Mean 95% Confidence Interval Figure 5-52. The FRF and 95 per cent confidence intervals in x -direction for H33x based on a spindle speed of 10,000 rpm.

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111 -5 0 5 10 x 10 -8 0 500 1000 1500 -10 -5 0 5 x 10 -8 Frequency (Hz)Imag (m/N) Real (m/N)Mean 95% Confidence Interval Figure 5-53. The FRF and 95 per cent confidence intervals in x -direction for H33cx based on a spindle speed of 10,000 rpm. -5 0 5 10 x 10 -8 0 500 1000 1500 -10 -5 0 5 x 10 -8 Frequency (Hz)Imag (m/N) Real (m/N)Mean 95% Confidence Interval Figure 5-54. The FRF and 95 per cent confidence intervals in y -direction for H33y based on a spindle speed of 10,000 rpm.

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112 -5 0 5 10 x 10 -8 0 500 1000 1500 -6 -4 -2 0 2 x 10 -8 Frequency (Hz)Imag (m/N) Real (m/N)Mean 95% Confidence Interval Figure 5-55. The FRF and 95 per cent confidence intervals in y -direction for H33cy based on a spindle speed of 10,000 rpm. 0 1 2 3 4 5 6 7 8 9 10 8000 1000012000 14000 1600018000 20000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable Figure 5-56. Budak and Altintas predicted stability lobes with 95 percent confidence intervals for slotting cut with Step Te c spindle, tapered heat shrink holder, and 19.1 mm diameter, 4 flute, carbide he lix tool. Results based on standard holder measurements performed at 10,000 rpm.

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113 0 2 4 6 8 10 12 14 16 18 20 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable Figure 5-57. Budak and Altintas predicted stability lobes with 95 percent confidence intervals for 50 percent radial immersi on cut with Step Tec spindle, tapered heat shrink holder, and 19.1 mm diamet er, 4 flute, carbide helix tool. Results based on standard holder measurements performed at 10,000 rpm. 0 1 2 3 4 5 6 7 Predicted: 0 rpm Predicted: 10000 rpm Stable Marginal Unstable 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim Figure 5-58. 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 w ith Step Tec spindle, tapered heat shrink holder, and 19.1 mm diameter 4 flute, carbide helix tool.

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114 0 2 4 6 8 10 12 14 16 18 80001000012000 14000 160001800020000 Spindle Speed (rpm)a (mm) lim Predicted: 0 rpm Predicted: 10000 rpm Stable Marginal Unstable Figure 5-59. 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 ra dial immersion cut with Step Tec spindle, tapered heat shrink holder, a nd 19.1 mm diameter, 4 flute, carbide helix tool.

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115 0 2 4 6 8 10 800012000 1600020000 Spindle Speed (rpm) 800012000 16000 2000 0 Spindle Speed (rpm) 800012000 16000 20000 Spindle Speed (rpm) 800012000 16000 20000 Spindle Speed (rpm) 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10a (mm) lim a (mm) lim a (mm) lim a (mm) lim predicted stable marginal unstable predicted stable marginal unstable predicted stable marginal unstable predicted stable marginal unstable A B C D Figure 5-60. Sensitivity of the Budak and A ltintas predicted stability lobes to input parameter variations incl uding: A) cutting coefficien ts, 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 measuremen ts at 10,000 rpm.

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116 0 4 8 12 16 20 800012000 1600020000 Spindle Speed (rpm)a (mm) lim predicted stable marginal unstable 0 4 8 12 16 20a (mm) lim predicted stable marginal unstable 800012000 16000 20000 Spindle Speed (rpm) 0 4 8 12 16 20a (mm) lim800012000 16000 20000 Spindle Speed (rpm) 0 4 8 12 16 20a (mm) lim800012000 16000 20000 Spindle Speed (rpm) predicted stable marginal unstable predicted stable marginal unstable A B C D Figure 5-61. Sensitivity of the Budak and A ltintas predicted stability lobes to input parameter variations incl uding: A) cutting coefficien ts, B) materials, C) standard holder measurements, and D) geometry variations. Results are based on Mikron UCP-600 Vario machin e, 50 percent radial immersion cuts, and standard holder rotati onal measurements at 10,000 rpm

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117 CHAPTER 6 CONCLUSIONS AND FUTURE WORK Conclusions The three-component spindle-holde r-tool RCSA method pr ovides accurate predictions for the tool-point response. Experiments, consis ting of direct and cross FRFs, are required on a standard holder. To provi de low noise results, averaging of multiple measurement sets is required. A minimum of 10 measurement sets at each measurement location is preferred. Also, experimental test ing has shown that a smaller standard holder provides better results and a distance of 25 mm between the direct and cross measurements is generally acceptable. Once the experimental measurements have been completed, the entire receptance matrix for the assembly, including rotational degrees of freedom, can be determined at the end of th e standard holder by using a finite difference technique. The extended holder subassembly sections can be modeled analytically as beam sections and removed from the assembly using an inverse RCSA technique. The spindle-holder base receptances are then known and no furthe r experimental testing is required. Any holder tool combination can be analytically modeled as a multitude of beam sections and coupled together usi ng RCSA to produce the receptances for the extended holder subassembly. This subassemb ly is then coupled to the spindle-holder base subassembly to predict the tool-point re sponse. Fluted sections are multiplied by an adjustment ratio to account for lower masses and inertias, and spindles that change dynamically as a function of spindle speed re quire measurement of the standard holder FRFs as the holder rotates at speed. This is an important new technique because the

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118 dynamics of actual cutting tools cannot be meas ured as they rotate due to the motion of the teeth. Experimental va lidation confirmed the accuracy of the model for a large variety of spindle-holde r-tool combinations. Monte Carlo simulations were also co mpleted to place uncertainty limits on analytical stability techniques developed by Budak and Altintas [55] and Tlusty [62]. The first set of simulations pr edicted results based on input va riations due to tool-point response measurements and cutting coeffi cient measurements. The second set of simulations predicted results based on input variations due to geometric properties, material properties, standard holder measurem ents, and cutting coefficients. This set of simulations used the 3-compone nt spindle-holder-tool RCSA m odel to predict tool-point response. Cutting coefficients were determ ined experimentally and the coefficients themselves were shown to decrease with sp eed and reach fairly constant values at approximately 8,000 rpm. For comparison pu rposes, stability was also determined experimentally based on a qualitative assessm ent, the frequency content of the cutting audio signal, a ratio of the once-per-revolution x versus y -direction displacement variance and total x versus y -direction displacement vari ance measured during the cutting process, and the frequency content of the tool displacement during cuts. For the Makino spindle, non-rotating meas urements of the tool-point response combined with the Budak and Altintas technique provided accurate predictions for the system stability lobe diagrams. The Tlusty technique over-predicted the stability limits and generated more uncertainty in the stability predictions For the Mikron sp indle, non-rotating measurements of the standard holder did not pr ovide accurate stability predictions. It was determined that the spindle response was ch anging as a function of spindle speed, and

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119 stability limits predicted by the Altintas tech nique were experimentally validated using rotating standard holder measurem ents. It was also shown th at the stability uncertainty levels were higher for the Tlusty technique than the Budak and Altintas technique, and the output response (stability) of both techniqu es was most sensitive to cutting coefficient variation and either standard hol der or tool-point measurement variation. Variation of the holder and tool material propert ies or geometries had very li ttle effect on the stability lobe uncertainty. All required areas of the scope of wo rk have been completed. The threecomponent spindle-holder-tool RCSA model is currently being impl emented in industry at Techsolve, Inc. in Cincinnati, Ohio and Manufacturing Laboratories, Inc. in Las Vegas, Nevada as a software package. The author also believes that using the model as a basis for predicting tool-point res ponse and machining stability fo r spindles that dynamically change with spindle speed is an important contribution along with providing uncertainty limits on stability lobe diagrams. Future Work There are two areas of future work that would contribute to the field. First, additional investigation into identifying spi ndles that change with spindle speed and optimizing the measurement technique and three-component spindle-holder-tool RCSA model to predict tool-point response and m achining stability for these spindles is desirable. Secondly, a complete finite element model of the spindle system that accurately predicts the speed dependent change in spindle dynamics combined with the three-component spindle-holder-tool RCSA model would reduce the amount of measurement required to predict tool-point re sponse and stability limits at each spindle speed.

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120 APPENDIX BEAM RECEPTANCE MODELING Bishop and Johnson [14] showed that the displacement and rotation-to-force and moment receptances for uniform Euler-Bernou lli beams could be represented by simple closed-form expressions. For a cylindrical free-free beam with coordinates j and k identified at each end, the frequency-dependent direct and cross receptances are given by: 3 3 51 F i EI F h hkk jj 3 3 81 F i EI F h hkj jk (A1) 3 2 11 F i EI F l lkk jj 3 2 101 F i EI F l lkj jk (A2) 3 2 11 F i EI F n nkk jj 3 2 101 F i EI F n nkj jk (A3) 3 61 F i EI F p pkk jj 3 71 F i EI F p pkj jk (A4) where E is the elastic modulus, I is the 2nd area moment of inertia, is the frequencyindependent damping coefficient and: 2 41 m EIiL (A5) L sinh L sin F 1 13 L cosh L cos F L cosh L sin L sinh L cos F 5 L cosh L sin L sinh L cos F 6 (A6) L sinh L sin F 7 L sinh L sin F 8 L cosh L cos F 10

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121 In Eq. (A5), the cylindrical beam mass is given by 42 2 L d d mi o where do is the outer diameter, di is the inner diameter (set equal to zero if the beam is not hollow), L is the length, and is the density; the cylinder’s 2nd area moment of inertia is 644 4i od d I ; and is the frequency (in rad/s). The Timoshenko beam model, which incl udes the effects of rotary inertia and shear, was implemented using finite elements [63]. Each four degree-of-freedom (rotation and displacement at both ends) free-free b eam section was modeled using appropriate mass, M and stiffness, K matrices [71]. The mass matrix was: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 23 6 15 2 Symmetric 2 10 1 5 6 6 6 30 1 2 10 1 3 6 15 2 2 10 1 5 6 2 10 1 5 6 1 120 60 105 1 Symmetric 24 120 11 210 11 3 10 7 35 13 120 60 140 1 24 40 3 420 13 120 60 105 1 24 40 3 420 13 6 10 3 70 9 24 120 11 210 11 3 10 7 35 13 1 l l l l l l l l r Al l l l l l l l Al Mg where A is the cross-sectional area, l is the section length, rg is the radius of gyration, and is a shear deformation parameter given by 2121 EIi kGAl where 1 2 E G is

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122 the shear modulus ( is Poisson’s ratio) and k is the shear coefficient which depends on the cross-section shape and [72 ]. The stiffness matrix (which included damping) was: 2 2 2 2 2 2 2 2 2 2 2 2 3Symmetric 2 4 2 2 4 2 4 1 4 2 4 Symmetric 6 12 2 2 6 2 4 6 12 6 12 1 1 l l l l l l l l AG k l l l l l l l l i EI K (A8) The element M and K matrices were then collected into the global mass, M and stiffness, K matrices using Guyan reduction [67] and th e resulting equation of motion solved in the frequency domain. See Eq. (A9), where n elements have been applied. 1 1 2 2 1 1 1 1 2 2 1 1-n n n nm f m f m f x x x K M2 (A9)

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123 LIST OF REFERENCES [1] Halley, J., Helvey, A., Smith, S., Winf ough, W., 2001, “The Impact of High-speed Machining on the Design and Fabrication of Aircraft Components ,” in: Proceedings of the 17th Biennial Conference on Mechanical Vibration and Noise, ASME Design and Technical Conferences,, 1216 Sept., Las Vegas, Nevada. [2] Zelinski, P., 1999, “Boeing’s One Part Harmony,” Modern Machine Shop Online. Retrieved January, 2004, from http://www.mmsonline.com/articles/089903.html [3] Vivancos, J., Luis, C., Costa, L., Ortiz, J., 2004, “Optimal Machining Parameters Selection in High Speed Milling of Harden ed Steels for Injection Moulds,” Journal of Materials Processing Technology, 155-156, pp. 1505-1512. [4] Urbanski, J., Koshy, P., Dewes, R., Aspinwall, D., 2000, “High Speed Machining of Moulds and Dies for Net Shape Manufacture,” 21, pp. 395-402. [5] Lopez de Lacalle, L., Lamikiz, A., Sa nchez, J., Arana, J. 2002, “Improving the Surface Finish in High Speed Milling of Stamping Dies,” 123, pp. 292-302. [6] Dundas, B., 2003, “Forging a New St rategy,” Modern Machine Shop Online. Retrieved January, 2004, from http://www.mmsonline.com/articles/080304.html [7] Beard, T., 1999, “Fast and Flexible,” Modern Machine Shop Online. Retrieved January, 2004, from http://www.mmsonline.com/articles/080304.html [8] Schmitz, T. L., Tummond, M., Duncan, G., Snyder, J. P., 2003, “ Development of an Internet-based Platform for High-spee d Milling Process Parameter Selection ,” Proceedings of the 18th Annual American Society of Precision Engineering Meeting, Portland, Oregon, October 26th – 31st (On CD). [9] Schmitz, T., Donaldson, R., 2000, “Predi cting High-Speed Machining Dynamics by Substructure Analysis,” Annals of the CIRP, 49 (1), pp. 303-308. [10] Schmitz, T., Davies, M., and Kennedy, M., 2001, “Tool Point Frequency Response Prediction for High-Speed Machining by RCSA,” Journal of Manufacturing Science and Engineering, 123, pp. 700-707. [11] Schmitz, T., Davies, M., Medicus, K., and Snyder, J., 2001, “Improving HighSpeed Machining Material Removal Rates by Rapid Dynamic Analysis,” Annals of the CIRP, 50(1), pp. 263-268. [12] Schmitz, T. and Burns, T., 2003, “Receptance Coupling for High-Speed Machining Dynamics Prediction,” Proc. of the 21st International Modal Analysis Conference, February 3-6, Kissimmee, FL (on CD).

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124 [13] Bishop, R., 1955, “The Analysis of Vibrating Systems which Embody Beams in Flexure,” Proc. Institution of M echanical Engineers (British), 169, pp. 1031-1050. [14] Bishop, R.E.D., Johnson, D.C., 1960, The Mechanics of Vibration, Cambridge University Press, Cambridge. [15] Hurty, W.C., 1965, “Dynamic Analysis of Structural Systems using Component Modes,” AIAA Journal, 3(4), pp. 678-685. [16] Klosterman, A.L., Lemon, J.R., 1969, “Building Block Appro ach to Structural Dynamics,” ASME Publication VIBR-30. [17] Klosterman, A.L., McClelland, Sherlock, W.I., 1977, “Dynamic Simulation of Complex Systems Utilizing Experimental and Analytical Techniques,” ASME Publication 75-WA/Aero-9. [18] Ewins, D.J., 1986, “Analysis of Modi fied or Coupled St ructures using FRF Properties,” Internal Report 86002, Dynamics Section, Department of Mechanical Engineering, Imperial College, London, UK. [19] Craig Jr., R. R., 1987, “A Review of Time-Domain and Frequency Domain Component-Mode Synthesis Methods,” Modal Analysis, 2(2), pp. 59-72. [20] Jetmundsen, B., Bielawa, R.L., Fla nnelly, W.G., 1988, “Generalized Frequency Domain Substructure Synthesis,” Journa l of the American Helicopter Society, 33, pp. 55-64. [21] Otte, D., Leuridan, J., Grangier, H., Aquilina, R., 1991, “Prediction of the Dynamics of Structural Assemblies using Measured FRF Data: Some Improved Data Enhancement Techniques,” Proc. of the 9th International Modal Analysis Conference (IMAC-1991), Flor ence, Italy, pp. 909-918. [22] Farhat, C., Geradin, M., 1992, “A H ybrid Formulation of a Component Mode Synthesis Method,” 33rd SDM Conference, AIAA paper 92-2383-CP, Dallas, TX, pp. 1783-1796. [23] Ren, Y., Beards, C.F., 1993, “A Ge neralized Receptance Coupling Technique,” Proc. of the 11th International Modal Anal ysis Conference (IMAC-1993), Kissimmee, FL, pp. 868-871. [24] Ren, Y., Beards, C.F., 1995, “On Substr ucture Synthesis with FRF Data,” Journal of Sound and Vibration, 185, pp. 845-866. [25] Ewins, D.J., 2000, Modal Testing: Theory, Practice and Application, 2nd Edition, Research Studies Press, Philadelphia, PA.

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125 [26] Lui, W., Ewins, D.J., 2002, “Substructu re Synthesis via Elastic Media,” Journal of Sound and Vibration, 257(2), pp. 361-379. [27] Ferreira, J.V., Ewins, D.J., 1996, “Non linear Receptance Coupling Approach Based on Describing Functions,” Proc. of the 14th International Modal Analysis Conference (IMAC-1996), D earborn, MI, pp. 1034-1040. [28] Yigit, A. S., Ulsoy, A. G., 2002, “Dyna mic stiffness evaluation for reconfigurable machine tools including weak ly non-linear joint ch aracteristics,” Proceedings of the I MECH E Part B Journal of Engineering Manufacture, 216(1), pp. 87-101. [29] Park, S., Altintas, Y., Movahhedy, M ., 2003, “Receptance Coupling for End Mills,” Journal of Machine Tools and Manufacture, 43, pp. 889-896. [30] Fofana, M., Bukkapatnam, S., 2001, “A Nonlinear Model of M achining Dynamics,” in: Proceedings of the 18th Biennial Conference on Mechanical Vibration and Noise, ASME Design Engineering Technical Conferences, 9-13 Sept., Pittsburgh, PA, paper no. DETC2001/VIB-21582. [31] Insperger, T., Stpn, G., Namach chivaya, N., 2001, “C omparisons of the Dynamics of Low Immersion Milling and Cu tting with Varying Spindle Speed,” in: Proceedings of the 18th Biennial Conference on Mechan ical Vibration and Noise, ASME Design Engineering Technical C onferences, 9-13 Sept., Pittsburgh, PA, paper no. DETC2001/VIB-21616. [32] Jayaram, S., Kapoor, S., DeVor, R., 2000, “Analytical Stab ility Analysis of Variable Speed Machining,” Journal of Manufacturing Science and Engineering, Transactions of the ASME, 122, pp. 391-397. [33] Davies, M., Balachandran, B., 2000, “I mpact Dynamics in Milling of Thin-Walled Structures,” Nonlinear Dynamics, 22, pp. 375-392. [34] Corpus, W., Endres, W., 2000, “A Hi gh Order Solution for the Added Stability Lobes in Intermittent Machining,” in: Proceeding of the Symposium on Machining Processes, Orlando, FL, MED-11, pp. 871-878 [35] Smith, S., Jacobs, P., Halley, J., 1999, “The Effect of Drawbar Force on Metal Removal Rate in Milling,” Annals of the CIRP, 48 (1), pp. 293-296. [36] Smith, S., Winfough, W., Halley, J., 1998, “The Effect of Tool Length on Stable Metal Removal Rate in High-Speed Milling,” Annals of the CIRP, 47 (1), pp. 307310. [37] Davies, M., Dutterer, B ., Pratt, J., and Schaut, A., 1998, “On the Dynamics of HighSpeed Milling with Long, Slender Endmills,” Annals of the CIRP, 47 (2), pp. 5560.

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126 [38] Nayfeh, A., Chin, C., Pratt, J., 1997, “A pplications of Perturba tion Methods to Tool Chatter Dynamics,” in: Dynamics and Chaos in Manufacturing Processes, F. C. Moon (ed.), Wiley, New York. [39] Altintas, Y., Lee, P., 1996, “A General Mechanics and Dynamics Model for Helical End Mills,” Annals of the CIRP, 45 (1), pp. 59-64. [40] Tlusty, J., Smith, S., Winfough, W., 1996, “Techniques for the Use of Long Slender End Mills in High-Speed Machining,” Annals of the CIRP, 45 (1), pp. 393-396. [41] Agapiou, J., Rivin, E., and Xie, C., 1995, “Toolholder/Spin dle Interfaces for CNC Machine Tools,” Annals of the CIRP, 44 (1), pp. 383-387. [42] Weck, M., Schubert, I., 1994, “New Interface M achine/Tool: Hollow Shank,” Annals of the CIRP, 43 (1), pp. 345-348. [43] Schultz, H., and Moriwaki, T., 1992, “H igh Speed Machining,” Annals of the CIRP, 41 (2), pp. 637-643. [44] Delio, T., Tlusty, J., Smith, S., 1992, “U se of Audio Signals for Chatter Detection and Control,” ASME Journal of Engineering for Industry, 114, pp.146-157. [45] Shin, Y., 1992, “Bearing Nonlinearity and Stability Analysis in High Speed Machining,” Journal of Engineering for Industry, Transactions of the ASME, 114, pp. 23-30. [46] Smith, S., Tlusty, J., 1991, “An Over view of Modeling and Simulation of the Milling Process,” ASME Journal of Engineering for Industry, 113, pp. 169-175. [47] Minis, I., Yanushevsky, T., Tembo, R ., Hocken, R., 1990, “Analysis of Linear and Nonlinear Chatter in Milling,” Annals of the CIRP, 39, pp. 459-462. [48] King, R., Editor, 1985, Handbook of High-speed Machining Technology, Chapman and Hall, New York. [49] J. Tlusty, J., W. Zaton, W., F. Isma il, F., 1983, “Stability Lobe s in Milling,” Annals of the CIRP, 32 (1), pp. 309-313. [50] Shridar, R., Hohn, R., Long, G., 1 968, “A Stability Algorithm for the General Milling Process,” Journal of Engineering for Industry, Transactions of the ASME, 90, p. 330. [51] Koenisberger, F., Tlusty, J., 1967, M achine Tool Structures-Volume I: Stability Against Chatter, Pergamon Press, Oxford.

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127 [52] Merrit, H., 1965, ‘Theor y of Self-Excited Machine Tool Chatter’, ASME Journal of Engineering for Industry, 87, pp. 447-454. [53] Tobias, S., Fiswick, W., 1958, “Theor y of Regenerative Machine Tool Chatter,” Engineering, London, 258. [54] Tobias, S., Fishwick W., 1958, “The Chatter of Lathe Tools Under Orthogonal Cutting Conditions,” Transactions of the ASME, 80, p. 1079. [55] Budak, E., Altintas, Y ., 1998, “Analytical Prediction of Chatter Stability Conditions for Multi-Degree of Freedom Systems in Milling, Part I: Modeling, Part II: Applications,” ASME Journal of Dynami c Systems, Measurement, and Control, 120, pp. 22-36. [56] Altintas, Y., E. Budak, E., 1995, “A nalytical Prediction of Stability Lobes in Milling,” Annals of the CIRP, 44 (1), pp. 357-362. [57] Bayly, P., Halley, J., Mann, B., Davi es, M., 2001, “Stability of Interrupted Cutting by Temporal Finite Element Analysis,” in: Proceedings of the 18th Biennial Conference on Mechanical Vibration a nd Noise, ASME Design Engineering Technical Conferences, 913 Sept., Pittsburgh, PA, paper no. DETC2001/VIB21581. [58] Mann, B.P., 2003, Dynamics of Milling Pr ocess, Ph.D dissertation, Saint Louis, Mo: Washington University. [59] Mann, B., Bayly, P., Davies, M., Ha lley, J., 2004, “Limit Cycles, Bifurcations, and Accuracy of the Milling Process,” Journal of Sound and Vibration, in press. [60] Mann, B., T. Insperger, T., P.V. Bayly, P., and G. Stepan, G., 2003, “Stability of up-Milling and Down-Milling, Part 2: Experimental Verification,” International Journal of Machine Tools and Manufacture, 43, p. 35. [61] Insperger, T., Mann, B ., Stepan, G., Bayly, P., 2003, “Stability of up-Milling and Down-Milling, Part1: Alternative Analytical Methods,” International Journal of Machine Tools and Manufacture, 43, p. 25. [62] Tlusty, J., 2000, Manuf acturing Processes and Equipm ent, Prentice Hall, Upper Saddle River, NJ. [63] Weaver, Jr., W., Timoshenko, P., and Young, D., 1990, Vibration Problems in Engineering, 5th Ed., John Wiley & Sons, New York. [63] Taylor, B., Kuyatt, C., 1994, “Guide lines for Evaluating and Expressing the Uncertainty of NIST Measurement Re sults,” NIST Technical Note 1297.

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128 [64] Sattinger, S., 1980, “A Method for E xperimentally Determining Rotational Mobilities of Structures,” Shock and Vibration Bulletin, 50, pp. 17-27. [65] Mathworks, 2002, Matlab 6.5.0 Release 13: High-Performance Numeric Computation and Visualization So ftware, Natick, Massachusetts. [66] Altintas, Y., 2000, Manufacturing Automa tion: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press, 1rst Edition, Cambridge, UK. [67] Manno, I., 1999, Introduction to th e Monte-Carlo Method, Akademiai Kiado, Budapest, Hungary. [68] Rubinstein, R., 1981 Simulation an d the Monte-Carlo Method, John Wiley & Sons, New York, New York. [69] 1989, Materials Engineeri ng Materials Selector, Penton P ublishing, Inc., Cleveland, OH. [70] Wackerly, D., Mendenhall III, W., and Sc heaffer, R., 2002, Mathematical Statistics with Applications, Duxbury Thomso n Learning, Pacific Grove, CA. [71] Yokoyama, T., 1990, “V ibrations of a Hanging Timoshenko Beam Under Gravity,” Journal of Sound and vibration, 141(2), pp. 245-258. [72] Hutchinson, J., 2001, “Shear Coefficients for Timoshenko Beam Theory,” Journal of Applied Mechanics, 68, pp. 87-92.

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129 BIOGRAPHICAL SKETCH G. Scott Duncan received a Bachelor of Science degree in mech anical engineering from Purdue University in West Lafayette, I ndiana, in 1990. He worked four years in industry as a design engineer for FMC Corporation and eight years as a project engineer for Johnson and Johnson, Inc. He is currently listed as co-inventor on eight patents. He entered the direct Ph.D. program at the Univ ersity of Florida in August of 2002 and is listed as co-author on three journal papers. He has accepted a faculty position starting in the fall of 2006 in the Department of Mechani cal Engineering at Valparaiso University in Valparaiso, Indiana.


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Permanent Link: http://ufdc.ufl.edu/UFE0015544/00001

Material Information

Title: Milling Dynamics Prediction and Uncertainty Analysis Using Receptance Coupling Substructure Analysis
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015544:00001

Permanent Link: http://ufdc.ufl.edu/UFE0015544/00001

Material Information

Title: Milling Dynamics Prediction and Uncertainty Analysis Using Receptance Coupling Substructure Analysis
Physical Description: Mixed Material
Copyright Date: 2008

Record Information

Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
System ID: UFE0015544:00001


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Full Text












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 Machine-Spindle-Holder-Tool Modeling Technique..............................10
Improved Machine-Spindle-Holder Tool Modeling Technique..............................12
Spindle-holder Base Subassembly Identification..............................................12
Standard test holder receptances ...................................... ............... 13
Extended holder subassembly m odel ................................. ............... 14
Spindle-holder base subassembly receptance ........................................... 15
Tool-point 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 ool-point R response Prediction ..........................................................................26
Experimental Results for Geared, Quill-Type 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 Heat-shrink 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 6061-T6 Work Piece and 19.1 mm Diameter, 4 Flute, Carbide
T o o l ...................................... ... ............. .......... .. ................................ 6 4
Aluminum 7475-T6 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 Three-component Spindle-holder-tool RCSA Model is Used to
G generate Tool-point 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

2-1 M ass and inertia adjustm ent ratios. ......................................................................... 20

3-1. Flexure m odal param eters. ............................................. .............................. 31

3-2 Standard holder substructure parameters. ..................................... ............... 31

3-3 Collet holder substructure I, II, and III parameters. ............................................35

3-4 Standard holder substructure parameters. ..................................... ............... 36

3-5 Shrink fit holder (25.3 mm bore) substructure parameters. ....................................37

3-6 Shrink fit holder-tool blank substructure parameters........................ ...............39

3-7 20-insert endmill substructure parameters. ................................... ............... 40

3-8 28-insert endmill substructure parameters. ................................... ............... 40

3-9 16-insert facemill substructure parameters. ............. .............................................42

3-10 Dimensions for tapered heat-shrink holder and tool blank assembly ....................43

3-11 Dimensions for collet holder and tool blank assembly ........................................46

3-12 Dimensions for collet and tapered heat-shrink holders and fluted tools ................49

4-1 Material statistical properties for holder and tool. ................................................62

5-1 Cutting coefficients for aluminum 6061-T6 work piece and 19.1 mm diameter, 4
flute, carbide helix tool............... .... ..................... .........84

5-2 Statistical properties for cutting coefficients for aluminum 6061-T6 work piece
and 19.1 mm diameter, 4 flute, carbide helix tool............................................85

5-3 Statistical properties for cutting coefficients for aluminum 7475-T6 work piece
and 12.7 mm diameter, 4 flute, carbide helix tool............................................85

5-4 Stability results for slotting cuts on a Mikron Vario machining center .................97









5-5 Stability results for 50 % radial immersion cuts on a Mikron UCP-600 Vario
m achining center. ......................... ........... .. ...... .... .............. .. 99

5-6 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

2-1 Two-component assembly. The component responses are coupled through a
rigid connection to give the assembly receptance(s)................... ...............18

2-2 Two-component flexure assembly. The component responses are coupled
through a rigid connection to give the assembly receptance(s).............................. 18

2-3 Previous two-component RCSA model.........................................................19

2-4 Example standard holder for spindle-holder base subassembly receptance
id entification ................................................. ................ ....................19

2-5 Standard holder substructures for inverse receptance coupling.............................19

2-6. Spindle-holder-tool substructures for tapered thermal heat shrink holder and tool
b la n k ...................................... .................................................... 2 0

2-7 End view of two fluted tool showing rotation angles for area and inertia
calculations..................................... ................................ ........... 20

3-1 Plot of G11 receptances for flexure system..........................................................31

3-2 Plot of G33 receptances for small holder. ..................................... ............... 32

3-3 Standard holder geometry comparison............................................. .............32

3-4 Tool-point response base on lrst-order finite difference method............................33

3-5 Tool-point response base on 2nd-order finite difference method..............................33

3-6 Measured versus predicted tool-point response based on 1st-order and 2nd-order
finite difference method and ten averaged measurement sets ...............................34

3-7 Collet holder substructure I, II, and III parameters. ............................................34

3-8 Collet holder H33 predicted and measured results..................................................35

3-9 Spindle receptances G55(o)) determined from standard holder direct and cross
receptance m easurem ents. .............................................. ............................... 36









3-10 Tapered thermal shrink fit holder (25.3 mm bore) substructure model .................37

3-11 Measured (two nominally identical holders) and predicted H33 results for tapered
thermal shrink fit holder (25.3 mm bore). ...................................... ............... 38

3-12 Tapered thermal shrink fit holder with 19.1 mm diameter tool blank substructure
m o d el. ........................................ ................................... .. 3 8

3-13 Measured and predicted H11 results for tapered thermal shrink.............................39

3-14 Measured and predicted H11 results for 20-insert endmill. .....................................40

3-15 Measured and predicted H11 results for 28-insert..............................................41

3-16 Standard holder direct receptances two nominally identical, geared spindles
(CAT 50 holder-spindle interface). ............................................... ............... 41

3-17 Measured and predicted H11 results for 16-insert facemill. ....................................42

3-18 Tapered heat-shrink holder and tool blank assembly............................................42

3-19 The FRF for tapered heat-shrink 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

3-20 The FRF for tapered heat-shrink 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

3-21 The FRF for tapered heat-shrink 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

3-22 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

3-23 Collet holder and tool blank assem bly ........................................ .....................45

3-24 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

3-25 The FRF for collet holder with 19.1 mm diameter, 152.4 mm long carbide tool
blank assembly (Test Number 6). ........................................ ........................ 47

3-26 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

3-27 Tapered heat-shrink holder and fluted tool assembly. ...........................................48









3-28 Collet holder and fluted tool assembly................................ ....................... 48

3-29 The FRF for tapered heat-shrink holder with 19.1 mm diameter, 152.4 mm long
carbide, 2 fluted tool assembly. The overhung tool length was 121.5 mm.............49

3-30 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

3-31 The FRF for tapered heat-shrink holder with 19.1 mm diameter, 101.6 mm long
carbide, 4 fluted tool assembly. The overhung tool length was 79.2 mm............... 50

3-32 The FRF for tapered heat-shrink holder with 25.4 mm diameter, 101.6 mm long
carbide, 4 fluted tool assembly. The overhung tool length was 80.2 mm...............51

4-1 Stability lobe diagram ..................... ................ .............................61

4-2 Geom etry of milling process. ............ ........................................... .....................61

5-1 Experimental setup for measuring cutting force signals. .......................................82

5-2 Cutting force signal for 7500 rpm cut with 0.18 mm/tooth chip load ......................83

5-3 Linear regression for cutting force means at a spindle speed of 7,500 rpm.............83

5-4 Cutting coefficients versus spindle speed for the Budak and Altintas stability
lob e technique e. ..................................................... ................. 84

5-5 Cutting coefficients versus spindle speed for Tlusty stability lobe technique. ........85

5-6 Audio signal frequency content for 10,000 rpm spindle speed, 1.02 mm axial
depth cut. Cut was determined to be stable. ....................................................... 86

5-7 Audio signal frequency content results for 19,000 rpm spindle speed, 1.52 mm
axial depth cut. Cut was determined to be unstable. .............................................86

5-8. Stability results for Makino machining center, thermal heat shrink holder, 12.7
mm, 4 flute, carbide helix tool, Aluminum 7475-T6 work piece and a 25 percent
radial im m version cut .......................................................................... ....... 87

5-9 Mean and 95 percent (20) confidence intervals for x-direction FRF ......................87

5-10 Mean and 95 percent confidence intervals fory-direction FRF ..............................88

5-11 Covariance between x- andy-direction FRFs. .........................................................88

5-12 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 y-direction FRFs are not correlated...................................89









5-13 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 y-direction FRFs are 100 percent correlated.............................89

5-14 Comparison between Tlusty lobes and experimental results. The mean stability
boundary and 95 percent confidence intervals are shown for the case where x-
and y-direction FRFs are not correlated. ....................... ......... .............. 90

5-15 Comparison between Tlusty lobes and experimental results. The mean stability
boundary and 95 percent confidence intervals are shown for the case where x-
and y-direction FRFs are 100 percent correlated.............. .... .................90

5-16 Budak and Altintas lobes based on cutting coefficient uncertainty .......................91

5-17 Budak and Altintas lobes based on tool-point response measurement uncertainty. 91

5-18 Tlusty lobes based on cutting coefficient uncertainty........ .............. ............... 92

5-19 Tlusty lobes based on tool-point response measurement uncertainty....................92

5-20 Example of mapping of tool-point FRF to stability lobe. ............................... ...93

5-21 Experimental setup for measuring chatter based on x- and y-direction tool
displaced ent ...................... ......... .................................... ........ 93

5-22 8,000 rpm, 2 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 94

5-23 8,000 rpm, 2.5 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 94

5-24 10,000 rpm, 5 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 95

5-25 10,000 rpm, 6 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 95

5-26 15,000 rpm, 2 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 96

5-27 15,000 rpm, 3 mm axial depth slotting cut test for Mikron UCP-600 Vario
machining center. ................................... .............. .............. .. 96

5-28 8,000 rpm, 4 mm axial depth 50 % radial immersion cut test for Mikron UCP-
600 V ario m achining center. ............................................................................. 97

5-29 8,000 rpm, 5 mm axial depth 50 % radial immersion cut test for Mikron UCP-
600 V ario m achining center. ............................................................................. 98









5-30 10000 rpm, 16 mm axial depth 50 % radial immersion cut test for Mikron UCP-
600 V ario m achining center. ............................................................................. 98

5-31 10000 rpm, 17 mm axial depth 50 % radial immersion cut test for Mikron UCP-
600 V ario m achining center. ............................................................................. 99

5-32 Geometric properties of tapered heat shrink holder and 4 flute, 19.1 mm
diam eter carbide helix tool. ..... ........................... .....................................100

5-33 H33 with 95 percent confidence intervals in the x-direction for no holder
clam ping variation........... ............................ ........ ........ .. ...... ............ 101

5-34 H33c with 95 percent confidence intervals in the x-direction for no holder
clam ping variation .................. ............................... ........ .. ............ 101

5-35 H33 with 95 percent confidence intervals in the y-direction for no holder
clam ping variation............... ....................... ........ ........ .. ...... ............ 102

5-36 H33c with 95 percent confidence intervals in the y-direction for no holder
clam ping variation .................. ............................... ........ .. ............ 102

5-37 The x-direction tool-point FRF with 95 percent confidence intervals for no
holder clam ping force variation. ........................................ ........................ 103

5-38 They-direction tool-point FRF with 95 percent confidence intervals for no
holder clam ping force variation. ........................................ ........................ 103

5-39 The x-direction tool-point FRF with 95 percent confidence intervals with holder
clam ping force variation ........... .................................................. ............... 104

5-40 The y-direction tool-point FRF with 95 percent confidence intervals with holder
clam ping force variation ........... .................................................. ............... 104

5-41 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

5-42 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

5-43 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









5-44 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

5-45 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

5-46 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

5-47 Experimental setup for measuring rotating FRFs. ............................................108

5-48 Magnitude of H33 in x-direction as a function of spindle speed ............................108

5-49 Magnitude of predicted tool-point response, H11, in x-direction as a function of
spindle speed. .......................................................................109

5-50 Predicted x-direction tool-point FRF based on standard holder measurements at
10,000 rpm and chatter frequencies of unstable slotting cuts. ........................... 109

5-51 Predicted y-direction tool-point FRF based on standard holder measurements at
10,000 rpm and chatter frequencies of unstable slotting cuts. ........................... 110

5-52 The FRF and 95 percent confidence intervals in x-direction for H33x based on a
spindle speed of 10,000 rpm ....................................................................... ... 110

5-53 The FRF and 95 percent confidence intervals in x-direction for H33cx based on a
spindle speed of 10,000 rpm .................................................................. ......... 111

5-54 The FRF and 95 percent confidence intervals iny-direction for H33y based on a
spindle speed of 10,000 rpm ........................................... ............. ...... 111

5-55 The FRF and 95 percent confidence intervals iny-direction for H33cy based on a
spindle speed of 10,000 rpm ............. ....................... .................... 112

5-56 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









5-57 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

5-58 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

5-59 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

5-60 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

5-61 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

High-speed machining has made significant technological advancements in recent

years. Using high-speed 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 tool-point

dynamics and application-specific cutting coefficients. Tool-point dynamics are typically

obtained using impact testing; however, testing time is extensive due to the large amount

of holder-tool combinations. A technique to predict tool-point dynamics and therefore

limit experimental testing time is desirable. This dissertation describes a three-

component spindle-holder-tool model to predict tool-point 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 high-speed 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 high-speed 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 assembly-intensive sheet

metal build-ups with monolithic aluminum components resulting in substantial cost

savings [1, 2]. High-speed machining technology has also been applied to the production

of moulds and dies [3-5] 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 well-known 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 tool-point dynamics and application-specific cutting coefficients. The

tool-point 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 tool-point response based on minimum input data is the preferred

alternative. An effective tool-point 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 tool-point dynamic modeling,

machining stability investigations, and uncertainty estimation techniques.

Tool Point Dynamic Modeling

Schmitz et al. [9-12], with the goal of reducing testing requirements, developed a

model to predict the tool-point response based on minimum input data using the

receptance coupling substructure analysis (RCSA) method. In these previous studies, a

two component model of the machine-spindle-holder-tool assembly was defined. The

machine-spindle-holder displacement-to-force receptance was recorded using impact

testing, while the tool was modeled analytically as an Euler-Bernoulli beam [13]. The

tool and machine-spindle-holder substructure receptances were then coupled through

translational and rotational springs and dampers. The basis of the above-mentioned

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., 14-28].

Due to the difficulty in measuring rotational degrees-of-freedom (RDOF)

receptances, Schmitz assumed the displacement-to-moment, rotation-to-force, and

rotation-to-moment 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 tool-point dynamic

model.









Machining Stability Investigations

Self-excited vibration in metal cutting is known as chatter and has been studied by

many researchers [30-62]. 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 self-excitation 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

self-excited 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-

holder-tool 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 degree-of-freedom

structures, including the axial direction, and applied periodic system theory to

analytically analyze the stability of the system. Recent work [57-61] 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 machine-spindle-holder-tool model. The generalized model

reduces the amount of experimental testing required for various machine-spindle-holder-

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. 2-1, 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. 2-1, 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. (2.3-2.6), 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 closed-form expressions

for uniform Euler-Bernoulli 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. 2-2 (an actual, equivalent assembly

is also pictured). The base flexure, substructure A, is modeled as a single degree-of-

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

free-free 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, -ic1-k1 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. (2.12-2.14).



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. (2.12-2.14) 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. (2.15-2.18). 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. (2.15-2.18) 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( m0-i3+-k I -I1 -k -1
2 mI \-m 2 + i32 c + k3 j 2 (i1omc+1 1 m l2


(2.20)


Previous Machine-Spindle-Holder-Tool Modeling Technique

In the previous work of Schmitz et al. [9-12], which describes the tool point

frequency response function (or receptance) prediction using the RCSA method, a two-


(2.15)









component model of the machine-spindle-holder-tool assembly was defined. The

machine-spindle-holder displacement-to-force receptance was recorded using impact

testing, while the tool was modeled analytically as a Euler-Beroulli beam [13]. The tool

and machine-spindle-holder substructure receptances were then coupled through

translational and rotational springs and dampers: see the model in Fig. 2-3, 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

machine-spindle-holder. The purpose of the springs and dampers between the tool and

holder was to capture the effects of a potentially non-rigid, 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 two-component model provides a valid approximation for a flexible tool

clamped in a stiff spindle-holder, it does not offer the most generalized solution; i.e., if a

new holder is inserted in the spindle, a new machine-spindle-holder measurement must

be performed.

The potential for improvement in the two-component model exists in three areas.

First, the model requires an experimental measurement to determine the receptance at the

end of the holder; therefore, multiple spindler-holder combinations each require a

separate measurement. A model which identifies, in a single measurement set, the

machine-spindle-holder dynamics for all machine-spindle-holder combinations would be

beneficial. Secondly, the displacement-to-moment, rotation-to-force, and rotation-to-

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 Machine-Spindle-Holder Tool Modeling Technique

In order to enable RCSA predictions for a wider variety of machine-spindle-holder-

tool combinations, an improved three-component model is presented. In this model, the

machine-spindle-holder 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 spindle-holder 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 rotation-to-

force/moment and displacement-to-moment receptances for the free end of the spindle-

holder base subassembly using only displacement-to-force measurements is also

described. The experimental procedure involves direct and cross displacement-to-force

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 holder-to-holder is then removed in simulation using an inverse receptance coupling

approach (i.e., decomposition) to identify the four spindle-holder base subassembly

receptances. These receptances are then coupled to models of the actual holder and tool.

Spindle-holder Base Subassembly Identification

The experimental procedure used to determine the receptances at the free end of the

spindle-holder base subassembly, GSk(co), is described in this section. It is composed of









three primary steps. First, the standard holder displacement-to-force 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 2-4.

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 2-5) is removed using inverse receptance coupling to

determine all four spindle-holder 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 2-4), the four

subassembly receptances are determined by measuring the direct, H33, and cross, H33c,

and/or H33b, displacement-to-force receptances on the standard holder, applying a 1st-

order or 2nd-order backward finite difference method to find L33 (and, equivalently, N33)

[64], and then synthesizing P33. If a 1St-order finite difference method is used, only one

cross displacement-to-force receptance, H33c, is required. Both cross displacement-to-

force receptances are required if the 2nd-order finite difference method is used. For the

cross displacement-to-force 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 signal-to-noise ratio for the H33, measurement (i.e., many of the lower frequency

spindle-holder modes resemble a fixed-free 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 signal-to-

noise. The receptance L33 is determined from the measured displacement-to-force









receptances using Eq. (2.21) if the lst-order finite difference method is selected or Eq.

(2.22) if the 2nd-order 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 finite-difference method) and, therefore, P33. To reduce the noise effect,

the measured receptance data was smoothed using a Savitzky-Golay 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. 2-5. Equations (2.24-2.27) provide the

direct and cross extended holder subassembly receptance matrices, where rigid coupling

has been applied. These equations are determined from Eqs. (2.3-2.6) 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)l-lR64(co) (2.26)
03 03


X4 X4

GS43 ()= 3 R43b (o)[R3a3a (0) + R3b3b (o)]- R3a3 (o) (2.27)
04 04
F3 M3

Spindle-holder base subassembly receptance

The spindle-holder base subassembly receptance matrix, G33(w), can be expressed

as shown in Eq. (2.28) by rewriting Eq. (2.3). The left-hand 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 spindle-holder 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


Tool-point 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. 2-6. The assembly is divided into the spindle-holder base

subassembly and 13 cylindrical substructures of differing diameters. The spindle-holder

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 Euler-Bernoulli beam or Timoshenko beam (see Appendix A).

To model the receptances of the beams, a composite modulus and mass are substituted

for substructures II-VIII 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 holder-tool subassembly receptances at coordinates 1 and 4. First,

Eqs. (2.3-2.6) are used with the appropriate coordinate modifications to couple

substructure I to substructure II, each substructure having free-free 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 holder-tool 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 spindle-holder 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 holder-tool 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. 2-7, 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 2-1 shows the mass and inertia adjustment

ratios. During the substructuring process for the extended holder-tool 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 2-1 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 2-1. Two-component 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 2-2. Two-component flexure assembly. The component responses are coupled
through a rigid connection to give the assembly receptance(s).


















Figure 2-3. Previous two-component 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 machine-spindle-holder (B) through springs and
dampers.


Cross receptance
hammer impact
locations
3c 3b 3


Example standard holder for spindle-holder 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 -


Spindle-holder
base subassembly


Figure 2-5. Standard holder substructures for inverse receptance coupling.


Figure 2-4.









Spindle-holder base Extended holder-tool
Spindle-holder-tool assembly subassembly subassembly
Coordinate 51 Rigid connections between
4 all substructures



L ~ Substructures XIII ... I

Figure 2-6. Spindle-holder-tool substructures for tapered thermal heat shrink holder and
tool blank

Degree of rotation
0 10 20 30

0 etc...


Figure 2-7. End view of two fluted tool showing rotation angles for area and inertia
calculations.

Table 2-1. 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 machine-spindle-holder-tool model for five different spindles is provided.

The spindle-holder 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 spindle-holder base subassembly receptances were

determined, the three-component model was used to compare predicted and measured

results at the end of a collet holder. The spindle-holder base subassembly receptances

were also determined for a 24,000 rpm, HSK 63A spindle and Step Tec HVCS 212-X

20,000 rpm, HSK 63A spindle. The three-component model was again used to compare

predicted and measured results for spindle-holder-tool assembly combinations produced

with collet holders, three geometries of tapered heat-shrink holders and various

geometries of carbide tool blanks and fluted tools. The three-component model was also

used to compare predicted and measured results for a geared, quilled-type 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. 2-2 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., quick-bonding 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 low-mass accelerometer. The modal mass, damping, and stiffness

parameters were then extracted from the response by fitting the data using a peak-picking

method [25]. The results are shown in Table 3-1.

To verify the stacked flexure RCSA model, the model parameters from Table 3-1

were substituted into G1i(w), provided in Eq. (2-20). Figure 3-1 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

spindle-holder 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. 2-5. The dimensions and material properties for

each of the holder substructures are displayed in Table 3-2, 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 free-free 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. 3-2 for the small holder. The spindle-holder

base receptance, GS55((), was also determined for each of the three standard holders by









decoupling the appropriate extended holder subassembly from the spindle-standard

holder measurements and the receptances at the end of each of the three standard holders,

G33(o), were determined based on the spindle-holder 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 3-3 displays the magnitudes of the displacement-to-force receptance, H33, for the

large holder rebuilt from the small holder spindle-holder base subassembly, the small

holder rebuilt from the large holder spindle-holder base subassembly, and the medium

holder rebuilt from the small holder spindle-holder base subassembly. Two conclusions

can be drawn from Fig. 3-3: 1) the spindle-holder 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

spindle-holder 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 machine-spindle-holder 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 spindle-holder base









subassembly receptances and these receptances were used to predict end receptances for

other extended holder-tool subassemblies.

Experimental testing was also performed to compare tool-point response

predictions from the three-component machine-spindle-holder-tool model when 1st and

2nd-order 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 spindle-holder interface. The machine-

spindle-holder base receptances were determined and a tool-point response was predicted

for a tapered holder-tool blank assembly (see Fig. 2-6) using the methods described in

chapter 2. Figure 3-4 shows the predicted tool-point response when a 1st-order finite

difference approach was used based on one, two, four, and ten measurement set averages.

A measurement set consists often root-mean-square averaged FRF measurements of the

direct and cross standard holder FRFs. Figure 3-5 shows the predicted tool-point

response when a 2nd-order finite difference approach was used based on one, two, four,

and 10 measurement set averages, and Fig. 3-6 shows a comparison between tool-point

responses based on the 1st-order and 2nd-order finite difference approaches and the actual

measured tool-point response. From the figures, it can be seen that the 1st-order finite

difference method provides the most consistent tool-point response, independent of the

number of averaged measurement sets. The 2nd-order finite difference approach produces

more variation in the predicted tool-point response based on the number of measurement

sets used, and the tool-point response contains more noise. Figure 3.6 shows that using

the 1st-order finite difference method actually leads to improved accuracy for the model









prediction. Based on this testing, it is recommended to use the 1st-order finite difference

method with 5 averaged measurement sets to determine the standard holder receptances.

Holder Experimental Verification

Once the spindle-holder 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 machine-spindle-

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 spindle-holder base

subassembly as shown in Fig. 3-7. 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. 3-7.

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 free-free

substructure receptances are given in Table 3-3. The next step was to rigidly couple the

spindle-holder base subassembly to the extended holder subassembly to determine the

receptances at the free end of the holder, G33(o). Figure 3-8 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. 2-6; 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 3-4. The experimental procedure described in chapter 2 was applied to determine

all receptances at coordinate 5, GS55(c), as displayed in Fig. 3-9.

Holder Experimental Verification

Once the 24,000 rpm/40 kW spindle-holder 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 machine-spindle-holder

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. 3-10. 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 3-5 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 3-11 shows the predicted H33 result as well as

measurements for two nominally identical holders. The Euler-Bernoulli beam model was

applied to develop the extended holder receptances in this case.

Tool-point Response Prediction

To predict the tool-point 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. 3-12. The assembly was divided into the

spindle-holder base subassembly and 13 cylindrical substructures of differing diameters

(mean outer diameter values again used for the tapered sections); see Table 3-6. The

predicted and measured assembly tool-point displacement-to-force receptances, H11, are

displayed in Fig. 3-13. In this figure, results for both Euler-Bernoulli 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, Quill-Type CAT 50 Spindle

In this section, prediction and measurement results are provided for two cutters

coupled to a geared, quill-type spindle with a CAT 50 spindle-holder interface (Big-Plus

tool holders were used which include both taper and face contact). The spindle-holder

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 (2nd-order 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 tool-point FRF predicted by

rigidly coupling the tool models to the spindle measurements.

Figure 3-14 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 3-7 (as

before substructure, I is nearest the free end of the clamped cutter). Figure 3-15 shows the

H11 measurement and prediction for a 28-insert facemill (see Table 3-8). In both cases,

Euler-Bernoulli beam models were employed to describe the standard holder and cutter

bodies.

Experimental Results for Geared CAT 50 Spindle

In this section, the spindle-holder base subassembly receptances were measured on

two nominally identical, geared spindles (CAT 50 holder-spindle 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 3-16 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 16-insert solid body facemill was inserted in spindle 1 and the tool-point

response recorded. Predictions were finally completed using both the spindle 1 and 2

receptances. This result is provided in Fig. 3-17; the facemill geometry and material

properties are given in Table 3-9. 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 machine-spindle-holder base receptances were









determined and tool-point responses were predicted using the methods described in

chapter 2.

Tapered Heat-shrink Holder and Carbide Tool Blank Results

Tool-point responses were measured and predicted using the three-component

machine-spindle-holder-tool model for the four different holder tool assemblies shown in

Fig. 3-18 with the component and assembly dimensions displayed in Table 3-10. The

spindle-holder-tool 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 3-10 are shown in Fig. 3-19 through

Fig. 3-22.

Collet Holder and Carbide Tool Blank Results

Tool-point responses were measured and predicted using the three-component

machine-spindle-holder-tool model for the three different holder tool assemblies shown

in Fig. 3-23 with the component and assembly dimensions displayed in Table 3-11. The

spindle-holder-tool 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 tool-point response prediction, as seen in Fig. 3-25.

The results for the four different test runs displayed in Table 3-10 are shown in Fig. 3-24

through Fig. 3-26.

Fluted Tool Results

Tool-point responses were measured and predicted using the three- component

machine-spindle-holder-tool model for the four different holder tool assemblies shown in

Fig. 3-27 and 3-28 with the component and assembly dimensions displayed in Table 3-









12. The spindle-holder-tool 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 3-10 are shown in Fig. 3-29 through Fig. 3-32.

The three-component model does a good job of predicting the tool-point response

based on the spindle-holder subassembly receptances and an analytical model of the

extended holder-tool 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 holder-tool 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 web-based application [8] at the University of

Florida to receive stability lobe diagrams for any holder-tool combination based simply

on their geometries. Predicted results for the tool-point 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 non-rigid connection in the

overall model could lead to improved results.









Table 3-1. 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 3-1. Plot of G1 receptances for flexure system.

Table 3-2. 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 3-2. 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 3-3. 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 3-4. Tool-point response base on lrst-order finite difference method.

-o6
x 10-6
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 3-5. Tool-point response base on 2nd-order finite difference method.


x 10-6































-4


Figure 3-6.


Spindle-Hol
Assembly


x 10-6


500 1000 1500 2000


2500


Measured versus predicted tool-point response based on l1t-order and 2nd-
order finite difference method and ten averaged measurement sets.

Rider
Rigid Connection


Spindle-Holder Base
Subassembly


Figure 3-7. Collet holder substructure I, II, and III parameters.


x10-6


I










Table 3-3. 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 3-8. Collet holder H33 predicted and measured results.


-7
x 10
6










-6
-7
x 10
0





-
^^


-12 L
200










Table 3-4. 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 3-9. Spindle receptances G55(co) determined from standard holder direct and
cross receptance measurements.


10



10
. 10
-<-









Table 3-5. 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


Spindle-holder assembly


Spindle-hold
subassembly

r


er base


Extended-holder
subassembly


5 Rigid connections between I
4 all substructures 3



Substructures XII... I


Figure 3-10. 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 3-11. Measured (two nominally identical holders) and predicted H33 results for
tapered thermal shrink fit holder (25.3 mm bore).




Spindle-holder base Extended holder-tool
Spindle-holder-tool assembly subassembly subassembly
5 Rigid connections between
4 all substructures

r "

Substructures XIII ... I


Figure 3-12. Tapered thermal shrink fit holder with 19.1 mm diameter tool blank
substructure model.










Table 3-6. Shrink fit holder-tool blank substructure parameters.


500
Frequency (Hz)


Figure 3-13. 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 3-14. Measured and predicted H11 results for 20-insert endmill.

Table 3-7. 20-insert 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 3-8. 28-insert 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 3-15. Measured and predicted H11 results for 28-insert.


1000
Frequency (Hz)


Figure 3-16. Standard holder direct receptances two nominally identical, geared spindles
(CAT 50 holder-spindle interface).


-7
x 10
8
-


4






-4
-7
x 10

0 O




1-5



-10


1000


-8
x 10
6



S-0



-6-
x 10
0r



& -5


-10 L
100


2000









-8
x 10


-10L
100


500
Frequency (Hz)


1000


Figure 3-17. Measured and predicted H11 results for 16-insert facemill. Results are shown
for predictions from spindle 1 (dotted) and spindle 2 (dashed) standard
holder measurements. Measurement recorded using spindle 1.

Table 3-9. 16-insert 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 3-18. Tapered heat-shrink holder and tool blank assembly.












Table 3-10. Dimensions for tapered heat-shrink 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 075-315
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


S-2


500 1000 1500 2000
Frequency (Hz)


2500


Figure 3-19. The FRF for tapered heat-shrink 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 3-20. The FRF for tapered heat-shrink 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 3-21. The FRF for tapered heat-shrink 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 3-22.


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 3-23. Collet holder and tool blank assembly


2


-0
U


-2

5


t0o


-5


-6
<10


500


2000


2500


*2~--~-~-










Table 3-11. 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/ER40-120 A63/ER40-120 A63/ER40-120
Tool Distributor McMaster-Carr McMaster-Carr McMaster-Carr
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 3-24. 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 3-25.


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 3-26. 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 3-27. Tapered heat-shrink holder and fluted tool assembly.


Figure 3-28. Collet holder and fluted tool assembly.










Table 3-12. Dimensions for collet and tapered heat-shrink holders and fluted tools.
Test Number
8 9 10 11
Holder Command Command Tooling Regofix
Manufacturer Innovations
Holder Part H4Y4A0750 H4Y4A1000 HSK63ASF- HSK-
Number 075-315 A63/ER40-120
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 3-29.


500 1000 1500 2000 2500
Frequency (Hz)
The FRF for tapered heat-shrink 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 3-30.


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 3-31. The FRF for tapered heat-shrink 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 3-32. The FRF for tapered heat-shrink 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 tool-point response and

cutting coefficient statistics. For the purposes of this study, the tool-point response

statistics were determined by modal testing and prediction from the 3-component

machine-spindle-holder-tool 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 edge-cutting force coefficient, Kre is the

radial edge-cutting 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 three-axis 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 (x-direction) and orthogonal (y-direction) 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. (4.3-4.6) [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 y-coordinate directions and Fx and Fyc are the slope values in the x-

and y-coordinate 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. 4-1) 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. 4-2), 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 edge-cutting 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. 4-2. The

values of the directional factors can be calculated from Eqs. (4.9-4.12).

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- andy-directions, as defined in Fig. 4-2. Traditionally, this has been

accomplished through impact testing; however, for this study, the tool-point FRF will be

predicted with both impact testing and the 3-component machine-spindle-holder-tool

RCSA technique. The real part of the tool-point FRF in the x-direction and y-direction,

Gx and Gy, and the imaginary part of the tool-point FRF in the x-direction and y-direction,

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, tool-point 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 tan-1 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

multi-degree-of-freedom 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 time-varying 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. 4-2). 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 edge-cutting 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 tool-point FRF in the x-direction 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 3-component machine-

holder-spindle-tool 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; = r-2tan1 l 8;2 = T-2tan1 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 tool-point response using impact testing, if the

uncertainty of the stability prediction is based on tool-point 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 three-component machine-spindle-holder-tool RCSA model, 4) use a Monte Carlo

simulation that randomly samples, for each simulation run, the cutting coefficient

parameters, the tool-point response measurement statistics or three-component machine-

spindle-holder-tool 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 tool-point response

measurements, there are two input parameters to the stability analysis: the cutting

coefficients and the x- andy-direction FRF at the tool-point. 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 tool-point responses

predicted by the three-component spindle-holder-tool 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 4-1 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 4-1. Stability lobe diagram.

Y Up milling

L-x F l /= 0





Surface normal .'' Rotation direction

Figure 4-2. Geometry of milling process.


Down milling


Rotation




Cut direction






62


Table 4-1. 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 10-4 16.5 x 10-4














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 7075-T6 work piece, with a 12.7

mm diameter carbide tool mounted in a thermal heat shrink holder, and tool-point

response variation due to measurement differences at the tool-point. 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 UCP-600 Vario horizontal machine tool with a

Step Tec HVCS 212-X spindle, using an Aluminum 6061-T6 work piece with a 19.1 mm

diameter carbide tool mounted in a thermal heat shrink holder, and tool-point response

variation based on the three-component machine-spindle-holder-tool 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 y-directions 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 UPC-600 Vario machining center were also generated for FRFs measured

while the spindle was rotating.

Experimental Determination of Cutting Force Coefficients

Aluminum 6061-T6 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 6061-T6 work piece using a 19.1 mm diameter, 4 flute, carbide helix tool

(Dataflute part number SH-40750). 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 (x-direction),

and orthogonal to the cut direction (y-direction). The experimental setup is shown in Fig.

5-1 where the work piece is mounted to a three-axis 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. 5-2. The mean cutting force

was determined for each signal by averaging the signal across the steady state region (as

indicated in Fig. 5-2). 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

y-direction, to determine the slope and vertical axis intercept. For example, Fig. 5-3

shows the linear regression for a spindle speed of 7,500 rpm. Equations (4.4-4.5) 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 5-1. The cutting coefficients used in the Tlusty

stability lobe technique, K, and fl, were determined using Eqs. (4.7-4.8) 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 5-1 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 x-direction mean force versus feed rate switches direction and the mean

cutting forces in the x-direction 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 5-1 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

5-2 shows the statistical properties of the cutting coefficients which were used in the

Monte Carlo simulations for the Aluminum 6061-T6 work piece and 19.1 mm diameter,

4 flute, carbide helix tool. The variable, ,o, is the standard deviation

Aluminum 7475-T6 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 7475-T6 work piece using a 12.7 mm diameter, 4 flute, carbide helix

tool (part number CRHEC500S4R30-KC610M). 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 6061-T6 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 5-3 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 7475-T6 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 tool-point 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 5-6 and 5-7 display examples of audio signal frequency content for stable and

unstable cutting conditions, respectively. As seen in Fig. 5-6, frequency content does

exist at the tooth passing frequency, as identified, and at additional frequencies due to

noise and once-per-revolution run-out. 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. 5-8.









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 5-3, 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. (4.7-4.8) 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 y-direction and 95 percent confidence

intervals are shown in Fig. 5-9 and Fig. 5-10, 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. 5-9 and 5-10.

F=F+ax oF (5.2)

Another consideration for the FRF data was potential correlation between the x-

and y-direction 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. 5-11; 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

One-thousand 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 7475-T6 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 tool-point response

as described in previous sections. Figures 5-12 and 5-13 show the comparison between

predicted stability lobes with 95 percent confidence intervals (based on the Budak and

Altintas technique) and experimental results. Figure 5-12 shows the case where the x-

and y-direction tool-point FRFs are not correlated and Fig. 5-13 displays the case where

the x- and y-direction tool-point FRFs are 100 percent correlated. As seen in the figures,

the correlation level between the x- and y-direction FRFs does not significantly alter the

predicted results. Figures 5-14 and 5-15 show the comparison between predicted stability

lobes with 95 percent confidence intervals based on the Tlusty technique as compared to

experimental results. Figure 5-14 shows the case where the x- and y-direction tool-point

FRFs are not correlated, and Fig. 5-15 displays the case where the x- and y-direction tool-

point FRFs are 100 percent correlated. As seen in the figures, the correlation level

between the x- and y-direction 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

y-direction FRFs into a single, oriented FRF; therefore, the correlation level between the

x- and y-direction FRFs significantly affects the uncertainty levels of the final stability

lobes. The Budak and Altintas stability technique does not couple the x- and y-direction









FRFs; therefore, the uncertainty levels of the predicted stability lobes are not as sensitive

to the x- andy-direction FRF correlation. A comparison between Figs. 5-12 and 5-14

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 over-predict the experimental stability limit for these tests. This may be the result of

the asymmetric x- and y-direction 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 tool-point 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 7475-T6 Aluminum work piece combination, and

25 percent radial immersion down milling, the sensitivities of the tool-point 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- andy-direction FRFs. Figure 5-16

displays the stability lobe uncertainty based solely upon cutting coefficient uncertainty









for the Altintas technique, and Fig. 5-17 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 tool-point response

variation is relatively constant across the spindle speed range with a slightly larger

contribution coming from tool-point response variation. Figure 5-18 displays the stability

lobe uncertainty based solely upon cutting coefficient uncertainty for the Tlusty

technique, and Fig. 5-19 displays stability lobe uncertainty based on tool-point 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. 5-19 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 tool-point FRF to stability lobe mapping example

shown in Fig. 5-20). As seen in Figs. 5-9 and 5-10, 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 UCP-600 Vario machining center, with a thermal heat shrink

holder, a 19.1 mm diameter, a 4 flute carbide helix tool, and a 6061-T6 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. 5-21,

was attached to the spindle to monitor tool displacement during the milling process. Two









capacitive probes, one in the x-direction and one in the y-direction, 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., once-per-revolution

sampling). For stable cuts, the once-per-revolution displacement was very consistent;

therefore, stability was quantitatively determined by comparing the variance of the once-

per-revolution displacement to the variance of the total tool displacement. Since both the

x- and y-direction 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 y-direction orthogonall to the feed or x-direction) displacement signal was

also Fourier transformed to look for chatter content in the frequency domain. The chatter

content in the x-and y-direction were identical; therefore, the y-direction 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 once-per-revolution 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 spindle-holder-tool-work piece combinations.









2
R c, once per revolution displacement (5.4)
Oc, total displacement

To illustrate the testing results, the x-direction versus y-direction total displacement

and once-per-revolution displacement signals and the frequency content of the y-direction

displacement are displayed in Figs. 5-22 through 5-27 for slotting experiments. Figure 5-

22 shows the results for an 8,000 rpm, 2 mm axial depth stable cut. In Fig. 5-22A, the

once-per-revolution samples are grouped much more tightly than the total displacement

samples shown in the light grey. In Fig. 5-22B, the frequency content of the y-direction

displacement is plotted with the synchronous content identified. As seen in Fig. 5-22B,

there is no chatter frequency content or, in other words, non-synchronous content. The

synchronous content is always present and is due to tool run out and tooth passing

frequencies and their harmonics. Figure 5-23 shows the results for an 8,000 rpm, 2.5 mm

axial depth unstable cut. As seen in Fig. 5-23A, the once-per-revolution samples now

have the same spread as the total displacement. Also, Fig. 5-23B shows chatter content at

799 Hz. Figures 5-24 and 5-25 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 tool-point. In Fig. 5-24A, the once-per-

revolution x versus y samples are closely grouped in comparison to the x versus y total

displacement, and in Fig. 5-25A, the once-per-revolution samples are marginally grouped

in comparison to the total displacement. The advantage of using the once-per-revolution

displacement signal and total displacement signal plotted in the x-direction versus y-

direction to determine stability is exemplified in Fig. 5-25B. 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 5-26 and 5-27 show the results for a









15,000 rpm, 2 and 3 mm axial depth cut, respectively. The stability results are

summarized in Table 5-4 for all spindle speed and axial depth of cut combinations.

The x-direction versus y-direction total displacement and once-per-revolution

displacement signals and the frequency content of the y-direction displacement are again

displayed in Figs. 5-28 through 5-31 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 5-5.

Monte Carlo Simulation Parameters

The tool-point response for the Step Tec spindle, thermal heat shrink holder, and 4

tooth, carbide helix tool was determined using the three-component spindle-holder-tool

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 6061-T6

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 5-2, 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. (4.7-4.8) 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 off-diagonal 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 4-1 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. 5-32 and Table 5-6. 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. 5-33 through 5-36. 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 y-direction 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

One-thousand 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 6061-T6 Aluminum work piece combination. The first result of

the simulation was to provide 95 percent uncertainty bounds on the tool-point response in

the x- and y-direction as predicted by the three-component spindle-holder-tool RCSA

model. Figures 5-37 and 5-38 show the tool-point response in the x- and y-direction,

respectively for the case where holder clamping force variation was not included.

Figures 5-39 and 5-40 show the tool-point 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 tool-point response

although the range of the 95 percent confidence interval bounds is unaffected.

Figure 5-41 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 non-rotating measurements

of the standard holder FRFs. Future sections will show that this bias is due to the spindle

dynamics changing with speed. Figure 5-42 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. 5-42 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 tool-point response. In this case, the negative real part of the tool-point

response has a very small range, thus creating frequency vectors that do not overlap.

Figure 5-43 and 5-44 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. 5-45 and 5-46 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

non-rotating 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. 5-47 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 Fourier-transforming each individual signal (100

signals total) after removing the synchronous content from each signal and determining

the standard deviation at each frequency. Figure 5-48 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 5-49 shows the









predicted tool-point 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 tool-point FRFs based on rotating

standard holder measurements versus the location of the measured chatter frequencies, as

displayed in Table 5-4. If the predicted tool-point FRF is representative of the spindle

dynamics, the tool should chatter, during unstable cuts, at frequencies near the tool-point

FRF modes (natural frequencies). Figure 5-50 and 5-51 display the x- and y-direction

predicted tool-point 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 tool-point 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 5-52 through 5-55 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 non-rotating 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. 5-56 for a

slotting cut on the Step Tec spindle with a tapered heat shrink holder, a 4 tooth, carbide

helix tool and an Aluminum 6061-T6 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 5-57 again shows reasonable agreement

between predicted and measured stability lobe results for the same system with a 50

percent radial immersion cut. Figures 5-58 and 5-59 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
Three-component Spindle-holder-tool 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. 5-60 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 5-61 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 5-1. Experimental setup for measuring cutting force signals.









200

100

0

-100

-200
300

200

100

0

-100

-200


Figure 5-2. 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 5-3. 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)


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