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F20101203_AABQAJ powell_k_Page_14thm.jpg 791a4ed0b41261f9699ba6d75419deb5 09b4997e05bb02b1276a4f29def1d116a0628bd0 F20101203_AABPTM powell_k_Page_09.jp2 640b066dd4cff6a223f317610bca855f 84894d495ed3cf7205d2cd4c1a15f79c9a3d4ced PAGE 1 1 CUTTING PERFORMANCE AND STABILITY OF HELICAL ENDMILLS WITH VARIABLE PITCH By KEVIN BRADY POWELL A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2008 PAGE 2 2 Copyright 2008 by Kevin Brady Powell PAGE 3 3 ACKNOWLEDGMENTS I would like to thank my parents and the rest of my family, for their love, support, and encouragement. Their success in life has been a constant inspir ation. I would like to extend a most heartfelt thanks to Am ber Wangle her love has been a great motivator. I would also like to extend a special thanks to Dr. Tony L. Schmitz for giving me the opportunity to work in a great research environment. His endl ess enthusiasm and exceptional knowledge always made it easy to come to wor k. I would also like to thank the rest of my committee, Dr. John K. Schueller and Dr. Gloria J. Wiens. A big thanks goes to Dr. Hitomi Yamaguchi Greenslet and the members of the M achine Tool Research Ce nter whose assistance and friendship proved to be invalu able, especially Raul Zapata who helped in the development of the peaktopeak stab ility lobe diagram. This work would not have been possible w ithout support from Thomas Long and Srikanth Bontha of Kennametal, Inc. PAGE 4 4 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................3 LIST OF TABLES................................................................................................................. ..........6 LIST OF FIGURES.........................................................................................................................7 ABSTRACT...................................................................................................................................10 CHAPTER 1 INTRODUCTION...................................................................................................................12 2 LITERATURE REVIEW.........................................................................................................14 SelfExcited Vibrations in Machining (Chatter).....................................................................14 Prediction and Modeling of Machining Stability...................................................................14 Tool Geometry in Machining Stability...................................................................................15 3 CUTTING FORCE MODEL ...................................................................................................16 Development.................................................................................................................... .......16 Determination.................................................................................................................. .......16 4 TIME DOMAIN SIMULATION ...................................................................................... ......22 Description..............................................................................................................................22 Verification.............................................................................................................................23 5 PEAKTOPEAK STABILITY LOBE IMPLEMENTATION ..............................................25 6 PEAKTOPEAK STABILITY LOBE VERIFICATION ......................................................30 Uniform and Variable Pitch PeaktoPeak Stability Lobe Comparison.................................30 Experimental Setup and Procedure.........................................................................................31 Cutting Tests....................................................................................................................31 Flexure Design.................................................................................................................32 Stability Determination...................................................................................................33 7 STABILITY LOBE VERIFICATION RESULTS ........................................................... .......42 8 CONCLUSION .......................................................................................................................46 APPENDIX A ONCEPERREVOLUTION PLOTS ......................................................................................50 PAGE 5 5 B MATLAB TIME DOMAIN SIMULATION CODE................................................................62 LIST OF REFERENCES ...............................................................................................................69 BIOGRAPHICAL SKETCH .........................................................................................................71 PAGE 6 6 LIST OF TABLES Table page 31: Cutting tests to determ ine cutting force coefficients..............................................................20 32: Cutting force coefficients (Kennametal HEC750S4).............................................................21 PAGE 7 7 LIST OF FIGURES Figure page 31: Cutting force model............................................................................................................ ....17 32: Cutting force co efficient test setup.........................................................................................18 33: Mean cutting force versus feed rate (2000 rpm, ADOC = 2 mm)..........................................18 34: Mean cutting force versus feed rate (6000 rpm, ADOC = 2 mm)..........................................19 35: Mean cutting force versus feed rate (10,000 rpm, ADOC = 2 mm).......................................19 36: Mean cutting force versus feed rate (15,000 rpm, ADOC = 2 mm).......................................20 41 Time domain simulation results for an endmill with variable pitch at 25% radial immersion..........................................................................................................................24 51: Chip thickness variat ion due to cutter vibrations....................................................................25 52: Force versus time for a tool w ith uniform pitch at 7200 rpm and a 4 mm axial depthofcut (unstabl e cutting, chatter)..............................................................................27 53: Force versus time for a tool with variable pitch at 7200 rpm and a 4 mm axial depthofcut (stable cutting)...............................................................................................27 54: Peaktopeak force plot for a tool with uniform pitch...........................................................28 55: Analytical stability lobes [10]............................................................................................... .28 56: Peaktope ak stability lobes (uniform pitch)..........................................................................29 61: Endmill geometry. A) Uniform p itch (Kennametal HEC750S4). B) Variable pitch.............30 62: Peaktopeak force plot (uniform pitch 1:10 mm x .25 mm).................................................34 63: Peaktope ak stability lobes (uniform pitch)..........................................................................34 64: Peaktopeak force plot (variable pitch 1:10 mm x .25 mm).................................................35 65: Peaktope ak stability lobes (variable pitch)..........................................................................35 66: Peaktopeak force plot (uniform pitch, 1:10 mm x .25 mm)................................................36 67: Peaktopeak stability lobes (uniform pitch)..........................................................................36 68: Peaktopeak force plot (variable pitch, 1:10 mm x .25 mm)................................................37 PAGE 8 8 69: Peaktope ak stability lobes (variable pitch)..........................................................................37 610: Flexurebase d cutting test setup...........................................................................................38 611: Cutting stability setup..........................................................................................................38 612: Key notchstyle fl exure dimensions (in mm).......................................................................39 613: Flexure model mesh for modal analysis..............................................................................39 614: Flexure FEM model. a) Sinusoi dal force along the top edge. b) Bottom face constrained in all DOFs. c) FRF determined at top edge.................................................40 615: Flexure and tooltip frequency response functions..............................................................40 616: Stable cutting vi bration (variable pitch, 7300 rpm, 4 mm axial depthofcut)....................41 617: Unstable cutting vi bration (uniform pitch, 7300 rpm, 4 mm axial depthofcut)................41 71: Peaktopeak stability lobes with experimental results (uniform pitch)................................43 72: Peaktopeak stability lobes with experimental results (variable pitch)................................43 73: Uniform pitch, 7300 rpm, 2.5 mm depth of cut (stable)........................................................44 74: Uniform pitch, 7300 rpm, 3 mm depth of cut (unstable).......................................................44 75: Variable pitch, 7300 rpm, 4 mm depth of cut (stable)...........................................................45 76: Variable pi tch, 7300 rpm, 4.5 mm depth of cut (unstable)....................................................45 81: Workpiece chips we lded to cutting teeth...............................................................................48 82: Close up of welded chips...................................................................................................... .49 83: Cutting forces dur ing welded chip cut test.............................................................................49 A1: Uniform pitch, 7225 rpm, 2 mm depth of cut.......................................................................50 A2: Uniform pitch, 7225 rpm, 2.5 mm depth of cut....................................................................50 A3: Uniform pitch, 7225 rpm, 3 mm depth of cut.......................................................................51 A4: Uniform pitch, 7225 rpm, 3.5 mm depth of cut....................................................................51 A5: Uniform pitch, 7300 rpm, 2 mm depth of cut.......................................................................52 A6: Uniform pitch, 7300 rpm, 2.5 mm depth of cut....................................................................52 A7: Uniform pitch, 7300 rpm, 3 mm depth of cut.......................................................................53 PAGE 9 9 A8: Uniform pitch, 7300 rpm, 3.5 mm depth of cut....................................................................53 A9: Uniform pitch, 7300 rpm, 4 mm depth of cut.......................................................................54 A10: Uniform pitch, 11,000 rpm, 5.5 mm depth of cut...............................................................54 A11: Uniform pitch, 11,000 rpm, 7.5 mm depth of cut...............................................................55 A12: Variable pitch, 7225 rpm, 2 mm depth of cut.....................................................................55 A13: Variable pitch, 7225 rpm, 2.5 mm depth of cut..................................................................56 A14: Variable pitch, 7225 rpm, 3 mm depth of cut.....................................................................56 A15: Variable pitch, 7225 rpm, 3.5 mm depth of cut..................................................................57 A16: Variable pitch, 7225 rpm, 4 mm depth of cut.....................................................................57 A17: Variable pitch, 7225 rpm, 4.5 mm depth of cut..................................................................58 A18: Variable pitch, 7225 rpm, 5 mm depth of cut.....................................................................58 A19: Variable pitch, 7300 rpm, 3 mm depth of cut.....................................................................59 A20: Variable pitch, 7300 rpm, 3.5 mm depth of cut..................................................................59 A21: Variable pitch, 7300 rpm, 4 mm depth of cut.....................................................................60 A22: Variable pitch, 7300 rpm, 4.5 mm depth of cut..................................................................60 A23: Variable pitch, 11,000 rpm, 5.5 mm depth of cut...............................................................61 A24: Variable pitch, 11,000 rpm, 7.5 mm depth of cut...............................................................61 PAGE 10 10 Abstract of Thesis Presen ted to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science CUTTING PERFORMANCE AND STABILITY OF HELICAL ENDMILLS WITH VARIABLE PITCH By Kevin Brady Powell May 2008 Chair: Tony L. Schmitz Major: Mechanical Engineering Advancements in machining technology have enabled increasingly aggressive machining operations with the goal of incr easing material removal rate (M RR) to enhance productivity and reduce production cost. In a high speed machining (HSM) operation, spindle speeds are increased to a range which is greater than those traditionally used for a given material in order to achieve an increase in MRR. One mechanism which limits the achievable MRR in machining operations is selfexcited vibrations of the cutt ing tool, known as chatte r. Chatter is caused by variations in the instantaneous chip thickness caused when the vi bration of the tooth currently engaged in the cut is out of phase with the vi bration of the previous tooth. The boundary between stable and unstable combinations of spindle sp eed and axial depth of cu t for a unique machining setup are a function of the workpiece material, tool and workpiece dynamics, and the selected cutting parameters. In some cases, nontraditional t ool geometries (such as serrated tool flutes or variable tooth pitch) can be used to interrupt th e feedback mechanism for the tool vibrations, thus altering the stability of the operation. In this study a simulation was developed with the goal of predicting the milling stability for helical endmills, including cutters with variable tooth pitch. This simulation can be used in PAGE 11 11 the future to develop new cutti ng tools with the goal of maximi zing the material removal rate within a desired spindle speed range. A new way to represent machining stability using the force output of the time domain simulation was also described. By generating a contour plot of the peaktopeak force for a range of axial depths of cut and spindle speeds, a diagram of stable and unstable combinations of axial depth of cut and spindle speed can be developed. This new diagram can be directly compared to traditional stability lobe diagrams. The simulation was validated us ing equal pitch (traditional) and variable pitch endmills. The first task was to determine if cutting force coefficients (for a force model) obtained from the traditional cutting tool could be used to accurately predict the cutting forces of the variable pitch cutting tool. After successful valid ation of this step, stability pr edictions for each of the endmill geometries were completed using the simulation. Through a series of cutting tests, the stability limit for each tool was determined at selected spindle speeds. The predicted stability limit showed good agreement with the experimental limit determined from the cutting tests for both the traditional and the pitch geometries. The simulation can therefore be used for process optimization for a given tool or at the design stag e to predict the performa nce of new geometries. PAGE 12 12 CHAPTER 1 INTRODUCTION The goal of high speed machining is to achieve a significant increase in material removal rate (MRR), which can significan tly reduce production cost and increase production rate. A high MRR is achieved by the combination of increased axial depth of cut and higher spindle speed. Advancements in spindle technology have enab led greater spindle speeds while maintaining the necessary power to perform aggressive cutting operations. In high speed machining operations, the mechanism that limits the achievable MRR is the process instability known as chatter. Chatter is a selfexcited vibrati on caused by variations in inst antaneous chip thickness (the thickness of the material being removed by a toot h at a point in time). When a flexible tool engages a workpiece, the tool begins to vibrate; these vibrations are cut into the new surface, leaving a wavy surface. As the next tooth cuts through the workpiece, the wavy surface creates variations in the instantaneous chip thickness. This, in turn, modulates the force on the cutting tool, creating a feedback mechanism for the tool vi brations. If the current vi bration of the cutting tool is inphase with the wavy surface left by the previous tooth, the instantaneous chip thickness remains nearly constant and vibrations tend to de cay resulting in stable cutting conditions. If the vibration of the cutting tool is outofphase with the previous surface, the variations in the instantaneous chip thic kness can lead to unstable cutting conditions or chatter. The force and vibration levels during chatter are large and can damage the workpiece and/or tool. In some cases, nontraditional tool geometries (such as serrated tool flutes or variable tooth pitch) can be used to interrupt the feedback mechanism for the t ool vibrations, thus altering the stability of the operation. It has also been shown that endmills w ith variable tooth pitch can reduce the location error of the finished surface [1]. PAGE 13 13 The boundary between stable and unstable combin ations of spindle sp eed and axial depth of cut for a unique machining setup are a function of the workpiece material, tool and workpiece dynamics, and the selected cutting parameters. The ability to predict the co mbinations of spindle speed and axial depth of cut which can provide the greatest MRR can eliminate the need for expensive and time consuming cutting tests. The goal of this project is to develop and valid ate a numerical algorithm that can be used to predict the stability of variable pitch helical endmills fo r the purpose of tool design. A key component of stability prediction is the relations hip between the cutting force and the uncut chip area, which can be linked by cutting force coeffici ents. The first objective of the project is to verify that previously documented cutting force coefficients of traditional endmills could be used to predict the cutting forces of variable pitch endmills, therefore eliminating the need for cutting force measurements in future predictions. Once the cutting force coefficients of the variable pitch endmill are identified, a timedomain simulation is us ed to develop a stability lobe diagram, or map of stable and unstable spindle speedaxial de pth of cut combinations, which can be used at the cutter design stage to select appropriat e tooth spacing values for improved process performance. PAGE 14 14 CHAPTER 2 LITERATURE REVIEW The literature review focuses on previous res earch in the area of machining stability, outlining work in the implementation of analytical and timedomain simulations for stability predictions along with cutter design with the focus on machining stability in milling. SelfExcited Vibrations in Machining (Chatter) Selfexcited vibration in machining is known as chatter. Chatter can produce large cutting force amplitudes that lead to increased tool wear and degradation of the machined surface. In 1946, Arnold proposed that chatter was the result of selfinduced a nd forced vibrations, which is governed by the internal damping of the tool [2 ]. Later work identified regeneration of waviness as the fundamental cause of self excited vibrations [34]. Regene ration of waviness refers to the variation in chip thickness which results from the interference between the wavy surface left by the vibrating tool and workpiece on the previ ous pass and the vibrating tool and workpiece on the current pass. If the vibrations of the curren t pass are in phase with the vibrations from the previous pass, the chip thickness remains fairly c onstant, as does the cutting force resulting in a stable cut. If the vibrations of the current pa ss are outofphase with the vibrations from the previous pass, the chip thickness can vary grea tly; the variation in chip thickness leads to variation in cutting force which can result in selfexcited vibrations. Prediction and Modeling of Machining Stability With the importance of chatter in machin ing operations, many studies have been performed with the goal of stab ility prediction and modeling [511]. In 1965, Merritt introduced a control system approach to predict stability of a machining operation [5]. Merritt used this approach to develop analytical stability diagrams. In 1983, Tlus ty et al. used a time domain PAGE 15 15 simulation to predict machining stability of helical endmills (including endmills with variable tooth pitch) and develop stability diagrams [6]. Time domain simulations have the ability to handle nonlinear situations in the machini ng operation. In the 1990s, Smith and Tlusty highlighted the use of peaktopeak force diagrams to plot cutting stability [7, 9]. In a peaktopeak force diagram, cutting force is plotted with spindle speed for a given axial depth of cut. In areas where cutting is stable, the force will no t vary with small changes in spindle speed resulting in a horizontal line on the peaktopeak force plot. In areas where cutting is unstable, force will change dramatically with small change s in spindle speed resulting in areas where the plot has a high slope. When multiple series are plotted for a variety of axial depths of cut, favorable combinations of spindle speed and dept h of cut can be identified. Altintas and Budak in 1995 developed an analytical solution to stab ility lobes in milling wh ich accurately predicted the stability of slo tting operations [10]. Tool Geometry in Machining Stability The use of nontraditional tool geometry (such as variable tooth pitch or serrated flutes) can interrupt the feedback mechanism for the tool vibrations, thus altering the stability of the operation [1214]. It has also been shown that en dmills with variable tooth pitch can reduce the location error of the finished su rface, resulting in a more accurate machining operation [1]. It was shown that when using a variable pitch tool geometry, the teeth with the lowest chip load created an accurate cut, reduci ng much of the surface location e rror produced by the teeth with the larger chip load. In 1999, Altintas et al. highlighted an anal ytical solution of stability for endmills with variable pitch [15]. Later, Budak demonstrated an analytic al method that can be used for tool design, resulting in a simple equation to optimize pitch angles [1618]. PAGE 16 16 CHAPTER 3 CUTTING FORCE MODEL Development To predict milling behavior it is necessary to identify relationships between the cutting forces and uncut chip area, A expressed as a product of the axial depth of cut, b, and feed per tooth, ft. A typical force model is provided in Equation 31 [19], where t represents the tangential direction, r represents the ra dial direction, and is the cutter rotation angle (Figure 31). The coefficients ktc, kte, krc, and kre were determined through cutting force measurements using a force dynamometer and preselected cutting conditions (2mm axial depth, 100% radial immersion). By completing tests for a range of feed per toot h values (0.08 to 0.16 mm/tooth in steps of 0.02 mm/tooth), a linear regression can be performed on the resulting mean cutting force values to determine the least squares best fit coefficients. cos sin sin cos sin sinr ty r tx re trcr te ttctFFF FFF bkbfkF bkbfkF (31) Determination The cutting force coefficient tests were performed on a Mikron UPC 600 Vario 5axis CNC mill (Steptec 20,000 rpm spindle). X and Y force data were acquired using a Kistler 3component dynamometer (9257B). The tool used in the test was a 19.05 mm 4flute endmill with uniform pitch (Kennametal HEC750S4). All cutti ng tests were performed on 6061T6 aluminum using a Tribos HSK 63A tool holder (Figure 32). For a selected feed/tooth ( ft ) value, spindle speed ( ) and tooth count ( n) the appropriate feed rate ( fr ) was determined (Eq. 32). Table 31 lists the parameters for each of the cutting tests used to determine the cutting force coefficients. nffrt (32) PAGE 17 17 Table 32 shows the cutting force coefficient results. The values were obtained from the linear regression through the mean cutting forces (see Figure 33 through Figure 36) for each of the four spindle speeds. Each of the points on the mean force versus feed per tooth plot (see Figure 33 through Figure 36) represents the mean cutting for ce for a particular feed rate. The results show a decrease in the tangential cut ting force coefficient as the spindle speed is increased; this trend matches prev ious experimental results. It is theorized that combination of increased strain rate and thermal softening of the workpiece at higher cutting speeds can affect the force required to cut the material, resulting in a change in the cutting force coefficients with spindle speed [20]. Figure 31: Cutting force model. ft fr x y Fr Ft Fx Fy PAGE 18 18 Figure 32: Cutting force coefficient test setup. 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 180 160 140 120 100 FX (N) ft (m/tooth) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 150 200 250 300 350 FY (N) ft (m/tooth) Figure 33: Mean cutting force versus feed rate (2000 rpm, ADOC = 2 mm) +Z +X +Y Tribos HSK 63A Holder Kistler dynamometer Kennametal HEC750S4 Endmill PAGE 19 19 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 100 90 80 70 FX (N) ft (m/tooth) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 100 150 200 250 300 FY (N) ft (m/tooth) Figure 34: Mean cutting force versus feed rate (6000 rpm, ADOC = 2 mm) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 80 70 60 50 FX (N) ft (m/tooth) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 100 150 200 250 FY (N) ft (m/tooth) Figure 35: Mean cutting force versus feed rate (10,000 rpm, ADOC = 2 mm) PAGE 20 20 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 70 60 50 40 FX (N) ft (m/tooth) 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 x 104 100 150 200 250 FY (N) ft (m/tooth) Figure 36: Mean cutting force versus feed rate (15,000 rpm, ADOC = 2 mm) Table 31: Cutting tests to determ ine cutting force coefficients. Setup Cut Spindle Speed (rpm) Axial Depth (mm) Radial Immersion Feed/Tooth (mm) Feed Rate (mm/min) 1 2000 2 100% 0.08 640 2 2000 2 100% 0.10 800 3 2000 2 100% 0.12 960 4 2000 2 100% 0.14 1120 1 5 2000 2 100% 0.16 1280 1 6000 2 100% 0.08 1920 2 6000 2 100% 0.10 2400 3 6000 2 100% 0.12 2880 4 6000 2 100% 0.14 3360 2 5 6000 2 100% 0.16 3840 1 10000 2 100% 0.08 3200 2 10000 2 100% 0.10 4000 3 10000 2 100% 0.12 4800 4 10000 2 100% 0.14 5600 3 5 10000 2 100% 0.16 6400 1 15000 2 100% 0.08 4800 2 15000 2 100% 0.10 6000 3 15000 2 100% 0.12 7200 4 15000 2 100% 0.14 8400 4 5 15000 2 100% 0.16 9600 PAGE 21 21 Table 32: Cutting force coefficients (Kennametal HEC750S4) rpm 2000 6000 10,000 15,000 Ktc (N/m2) 7.58x108 6.41x108 6.13x108 5.94x108 Kte (N/m) 2.61x104 1.78x104 1.41x104 1.44x104 Krc (N/m2) 3.50x108 1.47x108 1.11x108 8.99x107 Kre (N/m) 2.09x104 1.87x104 1.47x104 1.39x104 PAGE 22 22 CHAPTER 4 TIME DOMAIN SIMULATION Description A time domain simulation was used to determine the cutting forces between the tool and workpiece (the simulation is provided in Appendi x B). The milling simulation implemented in this project is timemarching, using Euler inte gration while moving th rough time in discrete steps. At each step the cutte r is rotated by a small angle,d and it is then determined which teeth are engaged in the cut (the tooth is within the a ngles prescribed by the radial immersion). If the tooth is engaged in the cut, th e instantaneous chip thickness, h, is determined (based on the cut geometry and current system vibrations). If h has a value greater than zero, the cutting force is computed using the force model described in equation (31). If h has a value less than or equal to zero, the tool is said to have moved out of th e cut, and the cutting forces are set to zero. The cutting force simulation requires the input of modal parameters (a description of the system dynamics), the tool geometry, and machining specifications. The modal parameters include the stiffness, damping ratio and natura l frequency for each tool mode in the x and ydirections. The tool geometry includes the numbe r of teeth, helix angle, toothtotooth angle, cutter diameter and the flutetof lute runout. The machining specifi cations needed are the starting and exit angles (a function of cut and radial i mmersion), spindle speed, axial depth of cut and feed per tooth. The feed per tooth of a variable pitch tool, ft,unequal, varies from tooth to tooth as a function of the toothtotooth angle, toothtotooth (deg), the mean feed per tooth, meantf, (m/tooth), and the number of teeth, m. The feed per tooth of a particular tooth is described by Equation 41. 360, ,m f ftoothtotooth meant unequalt (41) PAGE 23 23 Verification The initial goal of the time domain simulation was to determine if cutting force coefficients from a tool with uniform pitch could be used to predict cutting forces of a tool with variable pitch. Figure 41 show s the x and ydirection forces measured during a 25% radial immersion cut in 6160T6 aluminum using a tool with variable pitch, along with the time domain simulation results with and without runout. It can be seen in Fi gure 41 that the model accurately depicts the cutting forces of the cutter with variable pitch using the cutting force coefficients from the uniform pitch cutter (Table 32). The time domain simulation accurately captured the two different dwell times (the time between one tooth leaving the cut and the next tooth entering the cut) corresponding to the two flute separation angles. Flutetoflute runout was added to the time domain simulation by fitting the simulation resu lts to the measured results. Flutetoflute runout accounts for the small variations in toot h radius commonly seen in multiflute endmills. The flutetoflute runout for the tool with variable pitch was between 0 and 15 micrometers. Since cutting forces for a tool with variable pitch can be accurately predicted using the cutting force coefficients from a tool with uniform pitch, the cutting force coefficients shown in Table 32 can be used for future predictions. Ther efore, extensive cutting tests to determine the cutting force coefficients for a tool with variable pitch in various materials would not need to be performed provided that the cutting edge geometry is similar to the geometry used in this study. PAGE 24 24 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 50 0 50 100 150 Fx (N) Model w/ Runout Model w/o Runout Measured 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 100 0 100 200 300 Time (s)Fy (N) Figure 41 Time domain simulation results for an endmill with variable pitch at 25% radial immersion. PAGE 25 25 CHAPTER 5 PEAKTOPEAK STABILITY LOBE IMPLEMENTATION The chip thickness is a function of the cutter vibrations (projected into the normal of the cut surface at that instance in time), fluteto flute runout, and the surface left by the previous tooth (Figure 51). Figure 51: Chip thickness variat ion due to cutter vibrations. Since the tool and workpiece are not rigid, they vibrate as the flutes of the tool move through the workpiece. The vibration results in the tooth leaving a wavy surface behind on the workpiece. The variations in the instantaneous chip thickness result from the phasing between the surface left by the previous tooth and the cu rrent tooth. The magnitude and phase of the vibration are governed by the tool and wo rkpiece dynamics. Depending on the phasing, the forces can grow (unstable cutting, chatter, see Fi gure 52), or remain uniform (stable cutting, see Figure 53). To help visualize the relations hip between cutting stability, axial depth of cut and spindle speed, analytical stability lobes were develope d [6, 10]. Stability lobes are a function of the process parameters, tool geometry, cutting para meters and system dynamics. These analytical stability lobes assume the toothtotooth angle is constant. The time domain simulation was used to determine stability of a tool with variable pitch. PAGE 26 26 The time domain simulation outlined in the previous chapter was used to predict cutting forces for a specified spindle speed and axial de pth of cut. Stability was determined by plotting the peaktopeak (PTP) force values for a ra nge of spindle speed a nd axial depth of cut combinations [7, 9]. Figure 54 is an exampl e of a PTP force plot ranging from 6000 rpm to 18000 rpm and 1 mm to 10 mm axial depth of cut for a tool with uniform pitch. On the PTP force plot, each line represents a different axial depth of cut, b (in increments of 0.5mm). When a cut is in a stable region, the PTP force will no t vary with small changes in spindle speed (see Figure 54, zero slope areas). When a cut is in an unstable region, the PTP force will change dramatically with small changes in spindle speed (see Figure 54, high sl ope areas outlined with the dashed lines). For a tool with uniform pitch, the PTP plot shows regions of instability which agree closely with the regions of instability found in the traditional stability lobe development (see Figure 55). The problem with PTP plots is that they do not give a direct representation of the relationship between cutting stability, axial depthofcut and spindle speed (the parameters of interest). It was possible to make a plot usi ng the PTP force values which provides a direct representation between cutting stability, axial depth of cut and spindle speed by creating a contour plot of the PTP forces w ith respect to these parameters. Figure 56 is a contour plot of the PTP forces shown in figure 54. It can be seen that the contour plot representation of the PTP force values can be used in the same way as traditional analytical st ability lobes. Another advantage of the PTP stability lobes are that th ey allow the PTP cutting force to be displayed along with the traditional st ability lobe information. PAGE 27 27 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 50 100 150 200 250 300 350 400 450 500 Fx (N)Time (s) Figure 52: Force versus time for a tool with uniform pitch at 7200 rpm and a 4 mm axial depthofcut (unstable cutting, chatter). 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 50 100 150 200 250 300 350 400 450 500 Fx (N)Time (s) Figure 53: Force versus time for a tool with variable pitch at 7200 rpm and a 4 mm axial depthofcut (stable cutting). PAGE 28 28 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 0 500 1000 1500 2000 2500 Spindle speed (rpm)PTP Fy (N) Figure 54: Peaktopeak force plot for a tool with uniform pitch. 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 0 1 2 3 4 5 6 7 8 9 10 blim (mm)Spindle Speed (rpm) Figure 55: Analytical stability lobes [10]. b=10 Stable Unstable b=1 Unstable Stable PAGE 29 29 89 146 203 260 317 374 431 488 545 602 659 Spindle Speed (rpm)Axial Depth (m) 0.6 0.8 1 1.2 1.4 1.6 1.8 x 104 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 56: Peaktopeak stab ility lobes (uniform pitch). PAGE 30 30 CHAPTER 6 PEAKTOPEAK STABILITY LOBE VERIFICATION Uniform and Variable Pitch Peakt oPeak Stability Lobe Comparison The PTP force and stability lobe plots were us ed to compare a tool with uniform pitch to a tool with variable pitch to determine if any gains in stability could be achieved. Both tools have the same cutting geometry except for the toothto tooth angle. The toothtotooth angle for the endmill with uniform pitch is 90o between all four teeth (see Fi gure 61 A). The toothtotooth angles for the endmill with variable pitch were 83o and 97o (see Figure 61 B). Both tools were used in the same toolholder w ith the same insertion length so that the dynamic responses were the same. Figure 61: Endmill geometry. A) Uniform pitch (Kennametal HEC750S4). B) Variable pitch. The time domain simulation was performed for each tool (the only change was the toothtotooth angles), using a 6000 rpm to 12000 rpm sp indle speed range and a 1 mm to 10 mm axial depth of cut range, for a 50% radial immers ion upmilling operation (see Figure 62 through Figure 65). The PTP force and stability lobe plots for each tool show that near 7300 rpm, the A B PAGE 31 31 variable pitch tool should perform stable cutting at greater axial depths of cut than the uniform pitch tool. Figure 66 and 67 focus on the lobe near 7300 rp m. It can be seen that the PTP force and stability lobe plots predict that the boundary of instability occurs at a 3.0 mm axial depth of cut for a tool with uniform pitch and a spindle speed of 7300 rpm. The PTP fo rce and stability lobe predictions closely match the stab ility lobe prediction from [10] (as seen in Figure 55). The PTP force and stability lobe predictions were then pe rformed for a tool with variable pitch (see Figure 61 B), all other process and cu tting parameters were unchanged. Figure 68 and 69 focus on the same lobe as Figure 66 and 67 fo r the tool with variable pitch. The PTP force and stability lobe plots for the tool with variable pitch show that instability occurs around 4.5 mm axial depth of cut for a spindle speed of 7300 rpm. The PTP force and stability lobe plots indicate that for a given set of system dynamics, cutting parameters and spindle speed, the tool described in Figure 61 B could achieve a higher material removal rate than the tool described in Figure 61 A, while maintaining stable cutting conditions. The next step was verify the PTP force plots by performing a series of cutting tests for different axial depths of cut and spindle speeds. The stability of each cut was determined and the results were compared to the predictions ma de by the PTP force plots. The bulk of the experimental cuts were performed around 7300 rpm to capture the difference in the stability boundary between the two diffe rent tool geometries. Experimental Setup and Procedure Cutting Tests Cutting tests were performed on a Mikron UPC 600 Vario 5axis CNC mill (Steptec 20,000 rpm spindle). A 6061T6 aluminum wo rkpiece was mounted on a single degreeoffreedom (SDOF) notch style flexure which was, in turn, mounted to the machining table (Figure PAGE 32 32 610). A TTI LT880 laser tachometer was used to obtain a onceperrevolution signal from the spindle. A Polytech CLV 700 laser vibrometer was used to measure the vibration of the workpiece (see Figure 611). For the two selected cutter geometries, the lobe located at 7300 rpm was chosen to be verified. Cuts were taken at 7300 and 7225 rpm beginning around 2 mm and increasing until the cut was determined to be unst able. Cuts were also made at 11,000 rpm to verify the area of increase stability, bu t the stability boundary was not determined. Flexure Design Previous work has shown that the stability beha vior of a particular tool geometry depends on the assembly (tool, tool holder, spindle and workpiece) dynamics. Stability tests were performed with the workpiece mounted to a SDOF flexure, which exhibited higher flexibility than the cutting tool so that th e tool could be considered rigid The benefit of this setup was that multiple tool geometries could be compar ed without the influence of changing dynamics. In previous notchstyle flexure design exerci ses in the Machine Tool Research Center, the theoretical natural frequency was calculated using the analytical solution outlined in [21]. Key parts of the flexure geometry (see Figure 612) were varied until the de sired natural frequency was reached. In this approach, the analytical natural frequency tended to lose accuracy as the natural frequency increased (in general the analytical solution worked well for flexures with a natural frequency below approximately 700 Hz). The flexure used in the cutting tests was th erefore designed using commercial finite element (FE) software (ANSYS Workbench 10.0) to achieve the desired natural frequency. A solid model of the flexure was imported into ANSYS and meshed using 3D quadrilateral elements (see Figure 613). The flexure was constr ained in all degreesoffr eedom at its base (see Figure 614a). For the harmonic analysis, a sinusoi dal force (1000 N) was applied at the top edge PAGE 33 33 (see Figure 614b), and the resulting displacements will be measured at the opposite edge (see Figure 614c). The FEA model was simplified by neglecting the bolted connections which secured the actual flexure to the machining table. It was assumed that the bolted connections resulted in no movement of the base with respect to the machining table. Therefore the base was constrained in all degreesoffreedom as descri bed above. This assumption was deemed to be a reasonable one since the base of the flexure does not contribute an appreciable amount to the flexure dynamics. Another assumption was that the flexure s material properties were isotropic. The flexure used in the cutting tests was desi gned with a natural frequency of 818 Hz, and a stiffness of 1.28 x 106 m/N (approximately 5.5 times more flexible than the most flexible tool mode). Figure 615 shows the imaginary part of the measured flexure and tool frequency response functions (note that the scale on the imaginary axis is different between the two plots). The flexure FRF was measured on the machine tool table along with the rest of the experimental setup to ensure that the dynamics represente d the dynamics when a cutting test was being performed. Stability Determination For each cutting test, the vibrations of the workpiece and onceperrevolution signal of the spindle were recorded. To determine stability, a sample of the workpiece vibration was taken at the same cutter angle for each rotation of th e spindle (from the lase r tachometer onceperrevolution signal). If th e magnitude of the sampled vibration (onceperrevolution) remained close to constant (neglecting transient effects of the cutter entry and cutter exit), the cut was said to be stable (see Figure 616) [ 2224]. If the magnitude of the sampled vibration varied, the cut was said to be unstable (see Figure 617). The on ceperrevolution sample of vibration provides a visual indication of stable or unstable cutting. By increasing ax ial depthofcut for key spindle PAGE 34 34 speeds, the limiting depth of cut can be determin ed and compared to the stability lobes created by the time domain simulation. 6000 7000 8000 9000 10000 11000 12000 0 500 1000 1500 2000 2500 Spindle speed (rpm)PTP Fy (N) Figure 62: Peaktopeak force plot (uniform p itch 1:10 mm x .25 mm). 89 146 203 260 317 374 431 488 545 602 659 Spindle Speed (rpm)Axial Depth (m) 6000 7000 8000 9000 10000 11000 12000 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 63: Peaktopeak stab ility lobes (uniform pitch). PAGE 35 35 6000 7000 8000 9000 10000 11000 12000 0 500 1000 1500 2000 2500 Spindle speed (rpm)PTP Fy (N) Figure 64: Peaktopeak force plot (variable pi tch 1:10 mm x .25 mm). 87 142 197 252 307 362 417 472 527 582 637 Spindle Speed (rpm)Axial Depth (m) 6000 7000 8000 9000 10000 11000 12000 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 65: Peaktopeak stab ility lobes (variable pitch). PAGE 36 36 6000 6500 7000 7500 8000 8500 0 100 200 300 400 500 600 700 800 900 Spindle speed (rpm)PTP Fy (N) 3.5mm 2.5mm 2.0mm 3.0mm 4.0mm Figure 66: Peaktopeak force plot (uniform pitch, 1:10 mm x .25 mm) 89 146 203 260 317 374 431 488 545 602 659 Spindle Speed (rpm)Axial Depth (m) 6000 6500 7000 7500 8000 8500 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 67: Peaktopeak stabil ity lobes (uniform pitch) PAGE 37 37 6000 6500 7000 7500 8000 8500 0 100 200 300 400 500 600 700 800 900 Spindle speed (rpm)PTP Fy (N) 3.5mm 2.5mm 2.0mm 3.0mm 4.0mm 4.5mm 5.0mm Figure 68: Peaktopeak force plot (variable pitch, 1:10 mm x .25 mm) 87 142 197 252 307 362 417 472 527 582 637 Spindle Speed (rpm)Axial Depth (m) 6000 6500 7000 7500 8000 8500 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 69: Peaktopeak stab ility lobes (variable pitch). PAGE 38 38 Figure 610: Flexurebased cutting test setup. Figure 611: Cutting stability setup. Laser vibrometer SDOF flexure Laser tachometer Tool holder Work p iece Single degreeoff reedom notchstyle f lexure Workpiece Machine tool table Selected cutting tool PAGE 39 39 Figure 612: Key notchstyle fl exure dimensions (in mm). Figure 613: Flexure model mesh for modal analysis. PAGE 40 40 Figure 614: Flexure FEM model. a) Sinusoi dal force along the top edge. b) Bottom face constrained in all DOFs. c) FRF determined at top edge. 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 15 10 5 0 x 107 Flexure Frequency (Hz)Imaginary (m/N) 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 3 2 1 0 1 x 107 Tooltip Frequency (Hz)Imaginary (m/N) Figure 615: Flexure and tooltip frequency response functions. (b) (c) (a) PAGE 41 41 0 0.5 1 1.5 2 2.5 3 60 50 40 30 20 10 0 10 20 30 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 616: Stable cutting vi bration (variable pitch, 7300 rp m, 4 mm axial depthofcut) 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 617: Unstable cutting vi bration (uniform pitch, 7300 rp m, 4 mm axial depthofcut) PAGE 42 42 CHAPTER 7 STABILITY LOBE VERIFICATION RESULTS For the two selected cutter geometries, the lobe located at 7300 rpm was chosen to be verified. Cuts were taken at 7300 and 7225 rp m beginning around 2mm and increasing until the cut was determined to be unstable. Cuts were also made at 11,000 rpm to verify the area of increase stability, but the stab ility boundary was not determined. Figure 71 and 72 show the results of the experimental cutt ing tests overlaid with the peaktopeak cutting force stability lobes. The Os on the plot represent cuts that exhibit stable behavior, while the Xs represent cuts that exhibit unstable behavior. Figure 73 and Figure 74 highlight the transition from stable behavi or (7300 rpm, 2.5 mm depth of cut in Figure 73) to uns table behavior (7300 rpm, 3.0 mm depth of cut in Figure 74) of the uniform pitch tool. Figure 75 and Figure 76 highlight the transition from stable behavior (7300 rpm, 4.0 mm depth of cut in Figure 75) to unstable behavior (7300 rpm, 4.5 mm depth of cut in Figure 76) of th e variable pitch tool. The results of a ll the cutting test results performed in this study can be seen in Appendix A. The PTP stability lobes provide a good indication of th e stable regions for a particular set of system dynamics and cutting conditions for both tools. Improvements to the PTP stability lobe accuracy can be made by increasing the number of revolutions calculated and decreasing the spindle speed and axial depth step size in the time domain simulation at the expense of computation time. PAGE 43 43 89 146 203 260 317 374 431 488 545 602 659 o o x x o o x x x o oSpindle Speed (rpm)Axial Depth (m) 6000 7000 8000 9000 10000 11000 12000 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 71: Peaktopeak stab ility lobes with experimental results (uniform pitch). 87 142 197 252 307 362 417 472 527 582 637 o o o o o x x o o o x o oSpindle Speed (rpm)Axial Depth (m) 6000 7000 8000 9000 10000 11000 12000 1 2 3 4 5 6 7 8 9 10 x 103 PTP Forcey (N) Figure 72: Peaktopeak stab ility lobes with experimental results (variable pitch). PAGE 44 44 0 0.5 1 1.5 2 2.5 3 200 150 100 50 0 50 100 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 73: Uniform pitch, 7300 rpm, 2.5 mm depth of cut (stable). 0 0.5 1 1.5 2 2.5 3 120 100 80 60 40 20 0 20 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 74: Uniform pitch, 7300 rpm, 3 mm depth of cut (unstable). PAGE 45 45 0 0.5 1 1.5 2 2.5 3 60 50 40 30 20 10 0 10 20 30 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 75: Variable pitch, 7300 rp m, 4 mm depth of cut (stable). 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure 76: Variable pi tch, 7300 rpm, 4.5 mm depth of cut (unstable). PAGE 46 46 CHAPTER 8 CONCLUSION The goal of this project was to develop a time domain simulation which can be used to predict the stability behavior of helical endm ills with unequal pitch. This simulation can be implemented for milling process optimization (for a given cutter) or at the design stage when selecting tooth angles for a pa rticular cutting tool (and asso ciated process dynamics). The stability of the machining process was expressed using a new representati on of the peaktopeak force diagram. By generating a cont our plot of the peaktopeak force for a range of axial depths of cut, a diagram of stable and unstable combina tions of axial depth of cut and spindle speed was developed; this diagram provide s the same same information as traditional stability lobe diagrams, but can be applied to cutters with une qual pitch. The benefit of the new diagram over peaktopeak force plots is that the two axes of the contour plot directly correspond to the cutting parameters used to characterize a milling operation, namely spindle speed and axial depth of cut (for a preselected radial immersion). In order to predict cutting forces and, ev entually stability behavior, cutting force coefficients for an appropriate fo rce model are required. The first ta sk was to verify that cutting force coefficients obtained from tests using equa l pitch cutting tools could be used to predict cutting forces for variable pitch cutters. By measuring the forces exer ted on a workpiece during a variety of cutting operations, the cutting force coefficients were determined. The cutting force coefficients were obtained for both equal and variable pitch endmills while keeping all other cutter geometric parameters the same. The cutting force coefficients from the equal pitch endmill were used later in the project to predict the cu tting forces of the variable pitch cutter. The predicted cutting forces matched the measured cu tting forces; therefore, previously documented cutting force coefficients for traditional cutters can be used in future predictions. This is a PAGE 47 47 necessary step for cutting tool design since it was desired to make process predictions for arbitrary designs and avoid producing the cutt er if unfavorable re sults were obtained. Finally, predictions for the variable and equal pitch endmills were made using the simulation. The predictions were then validated by performing a sequence of cutting tests while measuring the workpiece deflections. The vibrations measured during each cutting test was used to determine the stability of the cut. If the vibr ations became larger with the passing of time the cut was said to be unstable. If the vibrations remained constant with time, the cut was said to be stable. The predicted stability in the critical areas tested for both the traditional geometry endmill and the variable pitch endmill matched well with the experimental stability test results. One of the difficulties encountered in this pr oject was a chip evacuation for 100% radial immersion conditions when using the variable pitch endmill. As seen in Figure 81, the problem occurs between the cutting teeth which have the smaller toothtotooth spacing. Figure 82 shows a closeup of the builtup material. It is observed that after the fi rst chip becomes welded to the tooth, subsequent chips are welded to the previous chip leaving layered material affixed to the cutting tooth. The builtup material results in a significant increase in cu tting forces since the there is no longer a sharp tooth to move smoot hly through the workpiece material. The increase in cutting forces can be seen graphically in Figure 83 which is a plot of the force values recorded by the dynamometer during one such inci dent. The first solution that was attempted was to add a jet of compressed air, aimed at the base of the cutter. The hope was that the jet of air would help to evacuate the cut chips before they had the opportunity to become welded to the tooth surface. Unfortunately, adding the compressed air had little effect on preventing the chips from becoming welded. The workaround was to limit th e radial immersion of the cutting tests to 50%. Reducing the radial immersi on provided a more direct path for chip removal and allowed PAGE 48 48 more time for the tooth face to cool while outside the cut. The use of flood coolant may eliminate this phenomenon, but this potential solution was not explored. This is an important observation because it may limit the pitch variation which can be reasonably achieved in pitch tuning exercises for chatter avoidance. In future work, a variety of unique variable pi tch endmills should be tested. In these tests, the effectiveness of various types of coolant a nd lubrication on preven ting the build up of material on cutting teeth should be determined. If the addition of flood coolant or other types of lubrication prove to be ineffective, the limits of tool geometry and variable pitch cutting parameters should be identified. Also, an error sensitivity analysis should be performed on the various inputs used by the predic tion stability tool to determine the effect of measurement and tool geometry errors on the accuracy of stability predictions using the peaktopeak force method [25]. Figure 81: Workpiece chips welded to cutting teeth PAGE 49 49 Figure 82: Close up of welded chips Figure 83: Cutting forces dur ing welded chip cut test PAGE 50 50 APPENDIX A ONCEPERREVOLUTION PLOTS 0 0.5 1 1.5 2 2.5 3 40 30 20 10 0 10 20 30 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A1: Uniform pitch, 7225 rpm, 2 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 50 40 30 20 10 0 10 20 30 40 50 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A2: Uniform pitch, 7225 rpm, 2.5 mm depth of cut. PAGE 51 51 0 0.5 1 1.5 2 2.5 3 80 60 40 20 0 20 40 60 80 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A3: Uniform pitch, 7225 rpm, 3 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 200 150 100 50 0 50 100 150 200 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A4: Uniform pitch, 7225 rpm, 3.5 mm depth of cut. PAGE 52 52 0 0.5 1 1.5 2 2.5 3 80 60 40 20 0 20 40 60 80 100 120 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A5: Uniform pitch, 7300 rpm, 2 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 200 150 100 50 0 50 100 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A6: Uniform pitch, 7300 rpm, 2.5 mm depth of cut. PAGE 53 53 0 0.5 1 1.5 2 2.5 3 120 100 80 60 40 20 0 20 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A7: Uniform pitch, 7300 rpm, 3 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 200 150 100 50 0 50 100 150 200 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A8: Uniform pitch, 7300 rpm, 3.5 mm depth of cut. PAGE 54 54 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A9: Uniform pitch, 7300 rpm, 4 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A10: Uniform pitch, 11, 000 rpm, 5.5 mm depth of cut. PAGE 55 55 0 0.5 1 1.5 2 2.5 3 250 200 150 100 50 0 50 100 150 200 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A11: Uniform pitch, 11, 000 rpm, 7.5 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 30 20 10 0 10 20 30 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A12: Variable pitch, 7225 rpm, 2 mm depth of cut. PAGE 56 56 0 0.5 1 1.5 2 2.5 3 30 20 10 0 10 20 30 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A13: Variable pitch, 7225 rpm, 2.5 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 40 30 20 10 0 10 20 30 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A14: Variable pitch, 7225 rpm, 3 mm depth of cut. PAGE 57 57 0 0.5 1 1.5 2 2.5 3 100 80 60 40 20 0 20 40 60 80 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A15: Variable pitch, 7225 rpm, 3.5 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 80 60 40 20 0 20 40 60 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A16: Variable pitch, 7225 rpm, 4 mm depth of cut. PAGE 58 58 0 0.5 1 1.5 2 2.5 3 200 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A17: Variable pitch, 7225 rpm, 4.5 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 400 300 200 100 0 100 200 300 400 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A18: Variable pitch, 7225 rpm, 5 mm depth of cut. PAGE 59 59 0 0.5 1 1.5 2 2.5 3 40 30 20 10 0 10 20 30 40 50 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A19: Variable pitch, 7300 rpm, 3 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 120 100 80 60 40 20 0 20 40 60 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A20: Variable pitch, 7300 rpm, 3.5 mm depth of cut. PAGE 60 60 0 0.5 1 1.5 2 2.5 3 60 50 40 30 20 10 0 10 20 30 40 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A21: Variable pitch, 7300 rpm, 4 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A22: Variable pitch, 7300 rpm, 4.5 mm depth of cut. PAGE 61 61 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A23: Variable pitch, 11, 000 rpm, 5.5 mm depth of cut. 0 0.5 1 1.5 2 2.5 3 150 100 50 0 50 100 150 Time (sec)Velocity (mm/s) vibration signal once/rev sample Figure A24: Variable pitch, 11,000 rpm, 7.5 mm depth of cut. PAGE 62 62 APPENDIX B MATLAB TIME DOMAIN SIMULATION CODE % GatorKennaMill_v2.m % Tony Schmitz and Kevin Powell % University of Florida % May 16, 2007 % This is a program to find the forces and deflections in helical peripheral end milling. % It includes tool dynamics, regeneration, runout, variable pitch cutters, and variable % helix angles on different teeth. % X > Y ^ %  % SS CW <Feed  %  clc close all clear all pack % Variables % Specific cutting energy values Ktc = 700e6; % tangential cutting force coefficient, N/m^2 Krc = 210e6; % radial cutting fo rce coefficient, N/m^2 Kte = 4e3; % tangential edge constant, N/m Kre = 3e3; % radial edge constant, N/m % Tool description kx = [8e6]; % N/m zetax = [0.03]; wnx = [600]*2*pi; % rad/s mx = kx./(wnx.^2); % kg cx = 2*zetax.*(mx.*kx).^0.5; % Ns/m x_modes = length(kx); % number of modes in xdirection, integer if length(zetax) ~= x_modes  length(wnx) ~= x_modes disp('Tool data en try error (x modes).') end ky = [7e6]; % N/m zetay = [0.025]; wny = [620]*2*pi; % rad/s PAGE 63 63 my = ky./(wny.^2); % kg cy = 2*zetay.*(my.*ky).^0.5; % Ns/m y_modes = length(ky); % number of modes in ydirection, integer if length(zetay) ~= y_modes  length(wny) ~= y_modes disp('Tool data en try error (y modes).') end m = 2; % number of teeth, integer d = 19.05e3; % diameter, m beta = [30 30]; % helix angle vector (firs t entry is max helix), deg if length(beta) ~= m  max(beta) ~= beta(1) disp('Tool data entry error (beta).') end tooth_angle = [0 180]; % angles of m cutter teeth starting from zero, deg if length(tooth_angle) ~= m  tooth_angle(1) ~= 0 disp('Tool data entry error (angles).') end RO = [0 0]*1e6; % flutetoflute runout relative to largest flute, m if length(RO) ~= m disp('Tool data entry error (RO).') end for cnt = 1:m if RO(cnt) > 0 disp('Tool data entry error (RO sign).') end end % Machining specifications phistart = 0; % starting angle, deg phiexit = 90; % exit end, deg if phistart > phiexit  phi start < 0  phiexit > 180 disp('Machining data entry error (phi).') end % Account for feed/tooth variation due to nonuniform teeth spacing ft_mean = 0.15e3; % mean feed/tooth, m theta = diff([tooth_angle 360]); for cnt = 1:m ft(cnt) = (ft_mean*theta(cnt)*m)/360; end % Grid spacing for multiple simulations low_ss = 5000; % lowest spindle speed, rpm high_ss = 24000; % highest spindle speed, rpm PAGE 64 64 low_ad = 0.5e3; % lowest axial depth, m high_ad = 5.0e3; % highest axial depth, m ss_step = 100; % spindle speed step size, rpm ad_step = 0.5e3; % axial depth step size, m spindle_speed = low_ss:ss_step:high_ss; axial_depth = low_ad:ad_step:high_ad; if (low_ss > high_ss  low_ad > high_ad) disp('Grid spacing data entry error.') end % Simulation specifications rev = 40; % number of revolutions, integer row = length(axial_depth); col = length(spindle_speed); PTP_Fy = zeros(row, col); % Wait bar function to keep track of simulation progress handle = waitbar(0, 'Please wait... simulation in progress.'); for loop1 = 1:row b = axial_depth(loop1); % axial depth of cut, m for loop2 = 1:col waitbar(loop1*loop2/(row*col), handle) omega = spindle_speed (loop2); % spindle speed, rpm % Simulation specifications steps_tooth = ceil(pi*d/(m*tan(beta(1) *pi/180)*ad_step)); % number of steps between teeth as tool rotates, integer steps_rev = m*steps_tooth; % steps per revolution, integer steps = rev*steps_rev; % total number of steps, integer dt = 60/(steps_rev*omega); % integration time step, s dphi = 360/steps_rev; % angular steps size between time steps, deg db = d*(dphi*pi/180)/2/tan(beta(1 )*pi/180); % discretized axial depth, m steps_axial = round(b/db); % number of steps along tool axis % Initialize vectors for cnt = 1:m teeth(cnt) = round(tooth_angle( cnt)/dphi) + 1; end for cnt = 1:steps_rev phi(cnt) = (cnt 1)*dphi; end surf = zeros(steps_axial, steps_rev); % initial surface area for regeneration set equal to zero PAGE 65 65 Forcex = zeros(1, steps); Forcey = zeros(1, steps); xpos = zeros(1, steps); ypos = zeros(1, steps); % Euler integration initial conditions x = 0; y = 0; dp = zeros(1, x_modes); p = zeros(1, x_modes); % xdirection modal displacements, m dq = zeros(1, y_modes); q = zeros(1, y_modes); % ydirection modal displacements, m %************************** MAIN PROGRAM ****************************** for cnt1 = 1:steps % time steps, s for cnt2 = 1:m teeth(cnt2) = teeth(cnt2) + 1; % index teeth pointe r one position (rotate cutter by dphi) if teeth(cnt2) > steps_rev teeth(cnt2) = 1; end end Fx = 0; Fy = 0; for cnt3 = 1:m % sum forces over all teeth, N for cnt4 = 1:steps_axial % sum forces along axial depth of helical endmill, N phi_counter = teeth( cnt3) (cnt41); if phi_counter < 1 % helix has wrapped through phi = 0 deg phi_counter = phi_counter + steps_rev; end phia = phi(phi_counter); % angle for given axial disk using max helix angle, deg phiactual = phi(teeth( cnt3)) (2*(cnt41)*db*tan(bet a(m)*pi/180)/d)*180/pi; % actual angle for selected tooth including local helix lag, deg phi_counter_new = round((phiactualphia)/dphi) + phi_counter ; % counter to select discretized actual phi for selected tooth with local helix, integer if phi_counter_new < 1 % helix has wrapped through phi = 0 deg phi_counter_new = phi_counter_new + steps_rev; end phib = phi(phi_counter_new ); % angle for given ax ial disk using current helix angle, deg if (phib >= phistart) & (phib <= phiexit) % verify that tooth angle is in specified range for current disk, deg PAGE 66 66 w = x*sin(phib*pi/180) y*cos(p hib*pi/180); % vibration normal to surface, out of cut is considered positive, m h = ft(cnt3)*sin(phib*pi/180) + surf(cnt4, phi_counter_new) w + RO(cnt3); % chip thickness includ ing runout effect, m if h < 0 % tooth jumped out of cut ftan = 0; frad = 0; surf(cnt4, phi_counter_new) = su rf(cnt4, phi_counter_new) + ft(cnt3)*sin(phib*pi/180); % update surf vector with current feed, m else % tooth is engaged in cut ftan = Ktc*db*h + Kte*db; frad = Krc*db*h + Kre*db; surf(cnt4, phi_counter_new) = w RO(cnt3); % update surf vector with current vibration and rounout, m end else % to oth angle is outside range bounded by radial immersion ftan = 0; frad = 0; end Fx = Fx fra d*sin(phib*pi/180) ftan*c os(phib*pi/180); % N Fy = Fy frad*cos(phib*pi/180) + ftan*sin(phib*pi/180); end % cnt4 loop end % cnt3 loop Forcex(cnt1) = Fx; Forcey(cnt1) = Fy; % Euler integration for position x = 0; y = 0; % xdirection for cnt5 = 1:x_modes ddp = (Fx cx(cnt5)*dp(cnt5) kx(cnt5)*p(cnt5))/mx(cnt5); dp(cnt5) = dp(cnt5) + ddp*dt; p(cnt5) = p(cnt5) + dp(cnt5)*dt; x = x + p(cnt5); % m end % ydirection for cnt5 = 1:y_modes ddq = (Fy cy(cnt5)*dq(cnt5) ky(cnt5)*q(cnt5))/my(cnt5); dq(cnt5) = dq(cnt5) + ddq*dt; q(cnt5) = q(cnt5) + dq(cnt5)*dt; y = y + q(cnt5); % m PAGE 67 67 end xpos(cnt1) = x; ypos(cnt1) = y; end % cnt1 loop %************************** END OF MAIN PROGRAM ****************************** % Select 2nd half of vectors for peaktopeak calculation s to avoid transients Forcey_trim = Forcey(round(length(Forcey)/2):length(Forcey)); % Calculate peaktopeak values for each set of machining conditions in simulation grid PTP_Fy(loop1, loop2) = max(Forcey_trim) min(Forcey_trim); end % loop 2 end % loop 1 close(handle); % close wait bar if (length(spindle_speed) > 1  length(axial_depth) > 1) figure(1) plot(spindle_speed, PTP_Fy(1,:)) hold on for cnt = 2:length(axial_depth) plot(spindle_speed, PTP_Fy(cnt,:)) end xlabel('Spindle speed (rpm)') ylabel('PTP F_y (N)') con_max = round(min(PTP_Fy(length(axial_depth), :))); con_min = round(min(PTP_Fy(1, :))); con_step = floor((con_max con_min)/10); figure(2) contour(spindle_speed, axial_depth*1e3, PTP_Fy, 300) contourcmap([con_min:con_step: con_max], 'jet', 'colorbar', 'on', 'location', 'vertical') grid on xlabel('Spindle speed (rpm)') ylabel('Axial depth (mm)') hold on h = axes('Position', [0 0 1 1], 'Visible', 'off'); text(0.97, 0.5, 'PTP Force_y (N)', 'rot ation', 90, 'Horizontal Alignment', 'center') end if (length(spindle_speed) == 1 & length(axial_depth) == 1) time = ((1:steps)1)*dt; % simulation time, s figure(3) subplot(211) PAGE 68 68 plot(time, Forcex) title('X force and vibration') ylabel('F_x (N)') subplot(212) plot(time, xpos*1e6) xlabel('Time (s)') ylabel('X Vi bration (\mum)') figure(4) subplot(211) plot(time, Forcey) title('Y force and vibration') ylabel('F_y (N)') subplot(212) plot(time, ypos*1e6) xlabel('Time (s)') ylabel('Y Vi bration (\mum)') end PAGE 69 69 LIST OF REFERENCES [1] Shirase, K., Altintas, Y., (1996), Cu tting Force and Dimensional Surface Error Generation in Peripheral Milling with Variable Pitch Helical End Mills, International Journal of Machine Tools and Manufacturing Vol. 36/5, pp. 567584. [2] Arnold, R. N., (1946), The Mechanism of Tool Vibration in the Cutting of Steel, Proceedings of the Institu tion of Mechanical Engineers Vol. 154/4, pp. 261284. [3] Tobias, S., Fishwick, W., (1958), The Chatter of Lathe Tools under Orthogonal Cutting Conditions, Transactions of the ASME, Vol. 80, pp. 10791088. [4] Tlusty, J., Polocek, M., (1963), The Stability of the MachineTool against SelfExcited Vibration in Machining, Proceedings of the Internati onal Research in Production Engineering Conference, Pittsburgh, PA, ASME: New York; 465. [5] Merritt, H. E., (1965), Theory of Se lfExcited MachineTool Chatter, ASME Journal of Engineering for Industry, Vol. 87, pp. 447454. [6] Tlusty, J., Zaton, W., Ismail, F., (1983), Stability Lobes in Milling, Annals of the CIRP Vol. 32, pp. 309313. [7] Smith, S., Tlusty, J., (1991), An Overview of Modeling and Simulation of the Milling Process, ASME Journal of Engi neering for Industry Vol. 113, pp. 169175. [8] Tlusty, J., Smith, S., Zamudio, C., ( 1991), Evaluation of Cutting Performance of Machining Centers, Annals of the CIRP Vol. 40/1, pp. 405410. [9] Smith, S., Tlusty, J., (1993), Efficient Simulation Programs for Chatter in Milling, Annals of the CIRP Vol. 42, pp. 463466. [10] Altintas, Y., Budak, E., (1995), Analytical Prediction of Stability Lobes in Milling, Annals of the CIRP Vol. 44, pp. 357362. [11] Budak, E., Altintas, Y., (1998), Analytical Pr ediction of Chatter Stability in Milling Part II: Application of the General Formulation to Common Milling Systems, Journal of Dynamic Systems, Measurements, and Control Vol. 120, pp. 3136. [12] Vanherck, P., (1967), Increasing Milling Machine Productivity by Use of Cutters with NonConstant CuttingEdge Pitch, 8th MTDR Conference, Manchester, pp. 947960. [13] Slavicek, J., (1965), The Effect of Irregular Tooth P itch on Stability of Milling, Proceedings of the 6th MTDR Conference Pergamon Press, London. [14] Stone, B. J., (1970), The Effect on the Cha tter Behavior of Machine Tools of Cutters with Different Helix Angles on Adjacent Teeth, Advances in Machine Tool Design and Research Proceedings of the 11th International MTDR Conference University of Birmingham, Vol. A, pp. 169180. PAGE 70 70 [15] Altintas, Y., Engin, S., Budak, E., (1999), Analytical Stability Prediction and Design of Variable Pitch Cutters, Journal of Manufacturing Science and Engineering Vol. 121, pp. 173178. [16] Budak, E., (2003), An Analytical Design Method for Milling Cutters with Nonconstant Pitch to Increase Stability, Part I: Theory, Journal of Manufacturing Science and Engineering Vol. 125, pp. 2934. [17] Budak, E., (2003), An Analytical Design Method for Milling Cutters with Nonconstant Pitch to Increase Stability, Part 2: Application, Journal of Manufacturing Science and Engineering Vol. 125, pp. 3538. [18] Budak, E., (2000), Improving Productivity a nd Part Quality in Milling of Titanium Based Impellers by Chatter Suppression and Force Control, Annals of the CIRP Vol. 49/1, pp. 3136. [19] Tlusty, J., (2000), Manufacturing Processes and Equipment Prentice Hall, Upper Saddle Rive, NJ. [20] Duncan, G. S., (2006), Milling Dynamics Prediction and Uncertainty Analysis Using Receptance Coupling Substructure Analysis, Ph.D. Dissertation, University of Florida, Department of Mechanical and Aerospace Engineering, Gainesville, FL, USA. [21] Smith, S. T., (2000), FlexuresElements of Elastic Mechanisms, Gordon and Breach, Amsterdam. [22] Schmitz, T., Davies, M., Medicus, K ., Snyder, J., (2001), Improving HighSpeed Machining Material Removal Rate s by Rapid Dynamic Analysis, Annals of the CIRP Vol. 50/1, pp. 263268. [23] Schmitz, T., Medicus, K., and Dutterer, B., (2002), Exploring On ceperrevolution Audio Signal Variance as a Chatter Indicator, Machining Scien ce and Technology, Vol. 6/2, pp. 215233. [24] Schmitz, T., (2003), Chatter Recognition by a Statistical Evaluation of the Synchronously Sampled Audio Signal, Journal of Sound and Vibration, Vol. 262/3, pp. 721730. [25] Duncan, G. S., Kurdi, M., Schmitz, T., Snyder, J., (2006), Uncertainty Propagation for Selected Analytical Milling Stability Limit Analyses, Transactions of the NAMRI/SME Vol. 34, pp. 1734. PAGE 71 71 BIOGRAPHICAL SKETCH Kevin Powell was born on June 20th, 1983, in Ga inesville, Florida, to Gregory and Carol Powell. After graduating from Paxon School for Advanced Studies in 2001, he began his collegiate education at the University of Florida, the alma mater of his parents. In 2005, he joined the Machine Tool Research Center (M TRC) under the guidance of Dr. Tony L. Schmitz. After graduating from the University of Florid a with a Bachelor of Science in Mechanical Engineering, the author continued his studies in the MTRC in pursuit of a Master of Science degree. Upon graduation, he will continue his work at Alstom Turbine Technology in Palm Beach Gardens, Florida, where he currently works as a mechanical integrity engineer. 