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PAGE 1 1 IMPROVED SPINDLE DYN AMICS IDENTIFICATION TECHNIQUE FOR RECEPTANCE COUPLING SUBSTRUCTURE ANALYSI S By UTTARA VIJAY KUMAR 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 2012 PAGE 2 2 2012 Uttara Vijay Kumar PAGE 3 3 To my parents, Alka and Vijay and husband, Ashwin PAGE 4 4 ACKNOWLEDGMENTS I extend my sincere gratitude to my advisor, Dr. Tony L. Schm itz for his guidance and ideas throughout my research. I feel fortunate to have had the opportunity to work with him. I would like to thank my committee members Dr. Schueller, Dr. Ifju and Dr. Fuchs for their support. I am also thankful to Dr. Hitomi Gre enslet for her support and encouragement. I thank my colleagues in the Machine Tool Research Center (MTRC) for their help, and sense of humor, making the MTRC a fun place to work I would also like to acknowledge Dr. Sam Turner at the University of Sheffie ld Advanced Manufacturing Research Center with Boeing for giving me an opportunity to conduct experiments on the milling machines for this research. Last but not least I would like to thank my entire family for their u nconditional love and patience PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF TABLES ................................ ................................ ................................ ............ 7 LIST OF FIGURES ................................ ................................ ................................ ........ 11 ABSTRACT ................................ ................................ ................................ ................... 17 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 19 Motivation ................................ ................................ ................................ ............... 19 Research Description ................................ ................................ .............................. 21 Dissertation Organization ................................ ................................ ........................ 22 2 LITERATURE REVIEW ................................ ................................ .......................... 25 3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS ................................ ... 30 Description ................................ ................................ ................................ .............. 30 Frequency Respons e Function ................................ ................................ ......... 30 Three Component Coupling for Tool Point FRF Prediction .............................. 30 Free Free Beam Receptances ................................ ................................ ................ 31 Rigid Coupling of Free Free Receptances ................................ .............................. 34 Coupling of Tool Holder and Spindle Machine Receptances ................................ .. 38 4 IDENTIFICATION OF SPINDLE MACHINE RECEPTANCES ................................ 44 Synthesis Approach ................................ ................................ ................................ 45 Finite Difference Approach ................................ ................................ ..................... 46 Euler Bernoulli Method ................................ ................................ ........................... 47 5 RESULTS ................................ ................................ ................................ ............... 50 Spindle Machine Receptances Comparison ................................ ........................... 50 Tool Point Frequency Response Comparison ................................ ........................ 51 Mikron UCP 600 Vario ................................ ................................ ...................... 52 25.4 mm diameter carbide endmill in a shrink fit holder ............................. 52 19.05 mm diameter carbide endmill in a shrink fit holder ........................... 53 S tarragheckert ZT 1000 Super Constellation ................................ ................... 53 12 mm diameter carbide endmill in a shrink fit holder ................................ 53 16 mm diameter carbide end mill in a shrink fit holder ................................ 54 20 mm diameter carbide endmill in a shrink fit holder ................................ 54 PAGE 6 6 25 mm diameter carbide endmill in a shri nk fit holder ................................ 55 Cincinnati FTV 5 2500 ................................ ................................ ...................... 55 12 mm diameter carbide endmill in a shrink fit holder ................................ 55 16 mm diameter carbide endmill in a shrink fit holder ................................ 56 20 mm diameter carbide endmill in a shrink fit holder ................................ 56 25 mm diameter carbide endmill in a shrink fit holder ................................ 56 Introduction of flexible connection between the tool and the holder ........................ 56 Cincinnati FTV 5 2500 ................................ ................................ ...................... 58 Mikron UCP 600 Vario ................................ ................................ ...................... 59 6 CONCLUSION AND FUTURE WORK ................................ ................................ .. 154 Conclusion ................................ ................................ ................................ ............ 154 Future Work ................................ ................................ ................................ .......... 156 APPENDIX A: FLEXIBLE COUPLING BETWEEN TOOL AND HOLDER .................... 157 LIST OF REFERENCES ................................ ................................ ............................. 160 BIOGRAPHICAL SKETCH ................................ ................................ .......................... 164 PAGE 7 7 LIST OF TABLES Table page 5 1 Specifications of milling machines tested ................................ ........................... 6 2 5 2 E B fitting parameters for Mikron UCP 600 Vario CNC milling machine spindle. ................................ ................................ ................................ ............... 62 5 3 E B fitting parameters for the short standard artifact on the Starragheckert ZT 1000 Super Constellation ................................ ................................ .............. 63 5 4 E B fitting paramet ers for the long standard artifact on the Starragheckert ZT 1000 Super Constellation ................................ ................................ ................... 63 5 5 E B fitting parameters for the short standard artifact on the Cincinnati FTV 5 2500 ................................ ................................ ................................ ................... 64 5 6 E B fitting parameters for the long standard artifact on the Cincinnati FTV 5 2500 ................................ ................................ ................................ ................... 64 5 7 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 99 mm ................................ ................................ ......... 65 5 8 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 107 mm ................................ ................................ ....... 65 5 9 Comparison metric (m/N) for the FRF predictions of 19.05 mm diameter endmill, overhang length 70.4 mm ................................ ................................ ...... 65 5 10 Comparison metric (m/N) for the FRF predicti ons of 19.05 mm diameter endmill, overhang length 76 mm ................................ ................................ ......... 65 5 11 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 44.7 mm using short artifact spindle receptances .................... 65 5 12 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances .................... 66 5 13 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 44.7 mm using long artifact spindle receptances ..................... 66 5 14 Compa rison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55 mm using long artifact spindle receptances ........................ 66 5 15 Comparison metric (m/N) for the FRF predictions of 1 6 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances .................... 66 PAGE 8 8 5 16 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances .................... 66 5 17 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptances ..................... 67 5 18 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances ..................... 67 5 19 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. ................... 67 5 20 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. ................... 67 5 21 Comparison metric (m/N) for the FRF predictions of 20 mm dia meter endmill, overhang length 65.0 mm using long artifact spindle receptances. .................... 67 5 22 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75 mm using l ong artifact spindle receptances. ....................... 68 5 23 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. ................... 68 5 24 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85.0 mm using short artifact spindle receptances. ................... 68 5 25 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75 mm using long artifact spindle receptances. ....................... 68 5 26 Comparison metric (m/N) fo r the FRF predictions of 25 mm diameter endmill, overhang length 85 mm using long artifact spindle receptances. ....................... 68 5 27 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmil l, overhang length 45.0 mm using short artifact spindle receptances. ................... 69 5 28 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using short arti fact spindle receptances. ................... 69 5 29 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 45.0 mm using long artifact spindle receptances. .................... 69 5 30 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptances. .................... 69 5 31 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances. ................... 69 PAGE 9 9 5 32 Comparison metric (m/N) for the F RF predictions of 16 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. ................... 70 5 33 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, o verhang length 55.0 mm using long artifact spindle receptances. .................... 70 5 34 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using long artifact s pindle receptances. .................... 70 5 35 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. ................... 70 5 36 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. ................... 70 5 37 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances. .................... 71 5 38 Comparison metric (m/N) for the FRF pr edictions of 20 mm diameter endmill, overhang length 75.0 mm using long artifact spindle receptances. .................... 71 5 39 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhan g length 75.0 mm using short artifact spindle receptances. ................... 71 5 40 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85.0 mm using short artifact spind le receptances. ................... 71 5 41 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75.0 mm using long artifact spindle receptances. .................... 71 5 42 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85.0 mm using long artifact spindle receptances. .................... 72 5 43 Stiffness matrix values of 12 mm diameter blank clamped in a shrink fit holder, Cincinnati FTV 5 2500 ................................ ................................ ............ 72 5 44 Average stiffness matrix values for blank shrink fit holders inserted in Cincinnat i FTV 5 2500 ................................ ................................ ........................ 72 5 45 Average stiffness matrix values for blank collet holders inserted in Cincinnati FTV 5 2500 ................................ ................................ ................................ ......... 72 5 46 Average stiffness matrix values for blank shrink fit holders inserted in Mikron UCP 600 Vario ................................ ................................ ................................ ... 72 5 47 Average stiffness matrix values for blank collet holders inserted in Mikron UCP 600 Vario ................................ ................................ ................................ ... 72 PAGE 10 10 5 48 Average stiffness matrix values for blank Tribos holders inserted in Mikron UCP 600 Vario ................................ ................................ ................................ ... 73 PAGE 11 11 LIST OF FIGURES Figure page 1 1 Example stability lobe diagram ................................ ................................ ........... 23 1 2 Standard artifact measurement. A) Direct FRF measurement. B) Cross FRF measurement ................................ ................................ ................................ ...... 24 3 1 Three component receptance coupling model for the tool (I), holder (II), and spindle machine (III). ................................ ................................ .......................... 39 3 2 Individual components I II with displaceme nts and rotations at specified coordinate locations. ................................ ................................ ........................... 40 3 3 Subassembly I II composed of tool (I) and holder (II). The generalized force Q 1 is applied to U 1 to determine G 11 and G 3 a 1 ................................ ................... 41 3 4 Subassembly I II composed of tool (I) and holder (II). The generalized force Q 3 a is applied to U 3 a to determine G 3 a 3 a and G 1 3 a ................................ .............. 42 3 5 The I II subassembly is rigidly coupled to the spindle machine (III) to determine the tool point receptance matrix, G11. ................................ ............... 43 4 1 Artifact model for determining R 3 b 3 b by inverse RCSA. ................................ ....... 49 5 1 Artifact dimensions for Mikron UCP 600 Vario measurements. .......................... 74 5 2 H 22 artifact measurement and E B fit for Mikron UCP 600 Var io CNC milling machine. ................................ ................................ ................................ ............. 74 5 3 L 22 / N 22 results for the Mikron UCP 600 Vario CNC milling machine. .................. 75 5 4 P 22 results for the Mikr on UCP 600 Vario CNC milling machine. ........................ 76 5 5 Short artifact dimensions for Starragheckert ZT 1000 Super Constellation measurements. ................................ ................................ ................................ ... 77 5 6 Long artifact dimensions for Starragheckert ZT 1000 Super Constellation measurements. ................................ ................................ ................................ ... 77 5 7 H 22 short artifact measurement and E B fit for Starragheckert ZT 1000 Super Constellatio n milling machine. ................................ ................................ ............ 78 5 8 L 22 / N 22 results for the short artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. ................................ .................... 79 5 9 P 22 results for the short artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. ................................ .................... 80 PAGE 12 12 5 10 H 22 long artifact measurement and E B fit for Starraghe ckert ZT 1000 Super Constellation milling machine. ................................ ................................ ............ 81 5 11 L 22 / N 22 results for the long artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. ................................ .................... 82 5 12 P 22 results for the long artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. ................................ ................................ ......... 83 5 13 H 22 short artifact meas urement and E B fit for Cincinnati FTV 5 2500 milling machine. ................................ ................................ ................................ ............. 84 5 14 L 22 / N 22 results for the short artifact measurement on Cincinnati FTV 5 2500 milling machine. ................................ ................................ ................................ .. 85 5 15 P 22 results for the short artifact measurement on Cincinnati FTV 5 2500 milling machine. ................................ ................................ ................................ .. 86 5 16 H 22 long artifact measurement and E B fit for Cinci nnati FTV 5 2500 milling machine. ................................ ................................ ................................ ............. 87 5 17 L 22 / N 22 results for the long artifact measurement on Cincinnati FTV 5 2500 milling machine. ................................ ................................ ................................ .. 88 5 18 P 22 results for the long artifact measurement on Cincinnati FTV 5 2500 milling machine. ................................ ................................ ................................ .. 89 5 19 Beam model for 25.4 mm diameter, three flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ............. 90 5 20 Comparison between H 11 tool point measuremen for three flute, 25.4 mm diameter endmill with an overhang length of 99 mm. ................................ ......... 91 5 21 Comparison between H 11 tool point measurement for three flute, 25.4 mm diameter endmill with an overhang length of 107 mm. ................................ ....... 92 5 22 Beam model for 19.05 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ............. 93 5 23 C omparison between H 11 tool point measurement and prediction for four flute, 19.05 mm diameter endm ill overhang length of 70.4 mm. ........................ 94 5 24 Comparison between H 11 tool point measurement and prediction for four f lute, 19.05 mm diameter endmill, overhang length of 76 mm. ........................... 95 5 25 Beam model for 12 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ............. 96 PAGE 13 13 5 26 Comparison between H 11 tool point measurement and prediction for four flute, 12 mm diameter endmill overhang length of 45 mm (short artifact ) ......... 97 5 27 Comparison between H 11 tool point measurement and prediction for fo ur flute, 12 mm diameter endmill, overhang length of 55 mm (short artifact) ......... 98 5 28 Comparison between H 11 tool point measurement and prediction for fou r flute, 12 mm diameter endmill, overha ng length of 45 mm (long artifact ) .......... 99 5 29 Comparison between H 11 tool point measuremen and prediction for four flute 12 mm diameter endmill, overhang length of 55 mm (long artifact) ................. 100 5 30. Beam model for 16 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ........... 101 5 31 Compariso n between H 11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (short artifact ) ....... 102 5 32 Comparison between H 11 tool point measurement an d prediction for fou r flute, 16 mm diameter endmill, overhang length of 65 mm (short artifact) ....... 103 5 33 Comparison between H 11 tool point measurement and prediction for four flute, 16 mm diam eter endmill, overhang length of 55 mm (long artifact ) ........ 104 5 34 Comparison between H 11 tool point measurement nad prediction for fou r flute, 16 mm diameter endmill, overhang length of 65 mm ( long artifact) ........ 105 5 35 Beam model for 20 mm diameter, two flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ........... 106 5 36 Comparison between H 11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (short artifact ) ....... 107 5 37 Comparison between H 11 tool p oint measurement and prediction for two flut e, 20 mm diameter endmill, overhang length of 75 mm (short artifact) ....... 108 5 38 Comparison between H 11 tool point measurement nad prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (long artifact ) ........ 109 5 39 Comparison between H 11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang l ength of 75 mm (long artifact ) ........ 110 5 40 Beam model for 25 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale). ................................ ................................ ........... 111 5 41 Comparison between H 11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 75 mm (short artifact ) ....... 112 PAGE 14 14 5 42 Comparison between H 11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 85 mm (short artifact ) ....... 113 5 43 Comparison between H 11 tool point measurement and prediction for fou r flute, 25 mm diameter endmill, overhang length of 75 mm (long artifact ) ........ 114 5 44 Comparison between H 11 tool point measurement and prediction for four flute, 25 mm diamet er endmill overhang length of 85 mm ( long artifact) ........ 115 5 45 Comparison between H 11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (sh ort artifact ) ....... 116 5 46 Comparison between H 11 tool point measurement and prediction for four flute, 12 mm diameter endmil o verhang length of 55 mm (short artifact) ........ 117 5 47 Comparison between H 11 tool point measurement and prediction for fou r flute, 12 mm diameter endmill, overhang length of 45 mm (long artifact ) ........ 118 5 48 Comparison between H 11 tool point measurement and prediction for four flute 12 mm diameter endmill, overhang length of 55 mm (long artifact) ........ 119 5 49 Comparison between H 11 tool poin t measurement and prediction for fou r flute, 16 mm diameter endmill, overhang length of 55 mm (short artifact ) ....... 120 5 50 Comparison between H 11 tool point measurement and prediction for four f lute, 16 mm diameter endmill, overhang length of 65 mm (short artifact ) ....... 121 5 51 Comparison between H 11 tool point measurement and prediction for fou r flute, 16 mm diameter endmill, overhang length of 55 mm (long artifact ) ........ 122 5 52 Comparison between H 11 tool point measurement and prediction for four flute, 16 mm diameter endmill overhang length of 65 mm (long artifact ) ........ 123 5 53 Tool point FRF measurement of 20 mm carbide end mill on Cincinnati FTV 5 2500. ................................ ................................ ................................ ................ 124 5 54 Comparison between H 11 tool point meas urement and prediction for tw o flute, 20 mm diameter endmill, overhang length of 65 mm (short artifact) ....... 125 5 55 Comparison between H 11 tool point measurement and prediction for t wo flute, 20 mm diameter endmill, overhang length of 75 mm (short artifact) ....... 126 5 56 Comparison between H 11 tool point measurement and prediction for tw o flute, 20 mm diameter endmill, overhang length of 65 mm (long artifact) ........ 127 5 57 Comparison between H 11 tool point measurement and prediction for two flute, 20 mm diameter endmill overhang length of 75 mm (long artifact) ........ 128 PAGE 15 15 5 58 Tool point FRF measurement of 25 mm carbide end mill on Cincinnati FTV 5 2500. ................................ ................................ ................................ ................ 12 9 5 59 Comparison between H 11 tool point measurement an d prediction for f ou r flute, 25 mm diameter endmill, o verhang length of 75 mm (short artifact) ....... 130 5 60 Comparison between H 11 tool point measurement and prediction for four f lute, 25 mm diam eter endmill, o verhang length of 85 mm (short artifact ) ....... 131 5 61 Comparison between H 11 tool point measurement and prediction for fou r flute, 25 mm diameter endmill, overhang length of 75 mm (long artifact ) ........ 132 5 62 Comparison between H 11 tool point measurement and prediction for fou r flute, 25 mm diameter endmill, overhang length of 85 mm (long artifact) ........ 133 5 63 Component coordinates for flexible coupling of holder and blank ..................... 134 5 64 Various shrink fit holders with blanks for Cincinnati FTV 5 2500 spin dle .......... 134 5 65 Collet holder for Cincinnati FTV 5 2500 spindle ................................ ............... 135 5 66 Measured and predicted tool point FRF of 12 mm diameter ca rbide blank with overhang length 76 mm (rigid connection) ................................ ................ 136 5 67 Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 71 mm (rigid connection) ................................ ................ 137 5 68 Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (rigid connection) ................................ ................ 138 5 69 Mea sured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (flexible connection) ................................ ............ 139 5 70 Measured and predicted tool point FRF of 12 mm diameter carbide bla nk with overhang length 71 mm (flexible connection) ................................ ............ 140 5 71 Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (flexible connection) ................................ ............ 141 5 72 Collet holder for Mikron UCP 600 Vario ................................ ........................... 142 5 73 25 mm diameter collet holder and blank for Mikron UCP 600 Vario ................. 143 5 74 Tribos holders for Mikron UCP 600 Vario ................................ ......................... 143 5 75 Mechanism of tool clamping in a Tribos holder (http://www.us.schunk.com) ... 144 5 76 Beam model for 6.33 mm diameter, 2 flute endmill inserted in a collet holder (not to scale) ................................ ................................ ................................ ..... 144 PAGE 16 16 5 77 Measured and predicted tool point FRF of 6.33 mm diameter 2 flute c arbide endmill in collet holder, overhang length 75 mm (rigid connection) .................. 145 5 78 Measured and predicted tool point FRF of 6.33 mm diameter 2 flute carbide endmill in collet holder, overhang length 75 mm (flexible connection) .............. 146 5 79 Beam model for 19 mm diameter, 4 flute endmill inserted in a collet holder (not to scale) ................................ ................................ ................................ ..... 147 5 80 Measured and predicted tool point FRF of 19 mm diameter carbide 4 flute endmill in collet holder, overhang length 60 mm (rigid connection) .................. 147 5 81 Measured and predicted tool p oint FRF of 19 mm diameter 4 flute carbide endmill in collet holder, overhang length 60 mm (flexible connection) .............. 148 5 82 Beam model for 12.7 mm diameter, 2 flute endmill inserted in a shrink fit holder (not to scale) ................................ ................................ .......................... 149 5 83 Measured and predicted tool point FRF of 12.7 mm diameter 2 flute carbide endmill in shrink fit holder, overhang length 66 mm (rigid connection) ............. 149 5 84 Measured and predicted tool point FRF of 12.7 mm diameter 2 flute carbide endmill in shrink fit holder, overhang length 66 mm (flexible connection) ......... 150 5 85 Beam model for 19 mm diameter, 4 flute endmill inserted in a Tribos holder (not to scale) ................................ ................................ ................................ ..... 150 5 86 Measured and predicted tool point FRF of 19 mm diameter 4 flute car bide endmill in Tribos holder, overhang length 72 mm (rigid connection) ................. 151 5 87 Measured and predicted tool point FRF of 19 mm diameter 4 flute carbide endmill in Tribos, overhang length 72 m m (flexible connection) ....................... 152 5 88 Beam model for 25.4 mm diameter, 4 flute endmill inserted in a shrink fit holder (not to scale) ................................ ................................ .......................... 153 5 89 Measured and predicted tool point FRF of 25.4 mm diameter 4 flute carbide endmill in shrink fit holder, overhang length 55 mm (rigid connection) ............. 153 A 1 The tool (I) is coupled flexi bly to the holder spindle machine (II) to determine the tool point receptance matrix, G 11 ................................ ............................... 159 PAGE 17 17 A bstract of Di ssertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IMPROVED SPINDLE DYN AMICS IDENTIFICATION TECHNIQUE FOR RECEPTANCE COUPLING SUBSTRUCTURE ANALYSI S By Uttara V. Kumar May 2012 Chair: Tony L. Schmitz Major: Mechanical Engineering Knowledge of the tool point dynamic response is required if milling process models are to be used to select parameters that avoid chatter, improve surface finish, and increase part accuracy. The dynamics of the tool holder spindle machine (THSM) can be obtained by mod al testing, but, for the large number of tool holder combinations in a production facility, the measurements are time consuming and, at times, inconvenient (e.g., micro scale tools). The Receptance Coupling Substructure Analysis (RCSA) approach may be app lied as an alternative to modal testing. In this approach, the THSM assembly is considered as three separate components: the tool, holder, and spindle machine. The modeled tool and holder receptances (or frequency response functions) are analytically coupl ed to an archived measurement of the spindle machine receptance. In this research, a novel approach to determine the spindle dynamics, referred to as the Euler Bernoulli (E B) method, is proposed. The spindle dynamics obtained by the new method are compa red to two existing methods (referred to here as the synthesis and finite difference approaches) using a new comparison metric (CM). The subsequent THSM receptance prediction accuracy for all three spindle dynamics identification PAGE 18 18 methods is evaluated using the CM. It is shown that the E B method is the best alternative. Using the E B method to identify the spindle dynamics, a flexible (rather than rigid) connection is introduced between the holder and the tool to further improve the prediction accuracy. Me asurements of various tool blank (i.e., a rod with no cutting flutes), holder, and spindle combinations are performed. A least squares non linear error minimization technique is used to determine the stiffness values that represent the flexible tool holder connection. The approach is validated using several endmill holder spindle assemblies. PAGE 19 1 9 CHAPTER 1 INTRODUCTION Motivation Increased productivity is a common goal in manufacturing environments, where customer demands drive production requirements. This is true for machining activities, where reductions in time and cost are desired while maintaining part quality. Advances in high spindle speeds designs have made higher material removal rates (MRR) possible. However, the machining process dynamics can dramati cally affect productivity due to unstable cutting conditions (or chatter) and forced vibrations which can cause part geometry errors (or surface location errors, SLE) [1 3]. For a particular setup, a combination of spindle speed and depth of cut that avoi ds these limitations (chatter and SLE) while enabling high MRR must be selected. The pre process milling parameter selection (depth of cut and spindle speed) is made possible by the use of predictive process models, including the stability lobe diagram and SLE map. A stability lobe diagram (SLD) separates the region of stable machining (no chatter) and unstable machining (self excited vibrations or chatter, with poor surface quality) as shown in Fig. 1 1 The process behavior depends on the tool workpiece material combination (this establishes the force model) and the dynamic response of the tool holder spindle machine (as measured/modeled at the free end of the cutting tool, or tool point).Given this information, the corresponding SLD can be determined. E ven if the machining is stable, forced vibrations of the flexible tool holder spindle machine assembly can lead to SLE. Therefore, SLE calculation is also an important consideration. Again, the tool point dynamic response, or tool point frequency response function (FRF), and force model are required as input to the process model. The SLE PAGE 20 20 calculations are paired with the appropriate SLD to select stable machining parameters that produce accurate parts. The tool point FRF can be measured via impact testing [ 4] where an instrumented hammer is used to excite the structure at the tool tip and a transducer, often a low mass accelerometer, is used to measure the response. The Fourier transforms of the two time domain signals are computed. Their frequency domain r atio gives the desired FRF. In a production facility where large numbers of tool holder spindle combinations are used, impact testing can be time consuming, costly, and sometimes impossible. It is therefore a manufacturing research priority to establish me thods that limit the number of required measurements and increase the use of models for pre process parameter selection that ensure stable cutting conditions with minimized SLE. The prediction of the tool point FRF using Receptance Coupling Substructure A nalysis (RCSA) [ 5 7 ] is gaining wider acceptance in the field of high speed machining. In the RCSA approach, the tool holder spindle machine assembly is considered as three separate components: the tool, holder, and spindle machine ; the individual frequenc y response s of these components are then analytically coupled. An archived measurement of the spindle machine FRF (or receptance) is analytically coupled to the free free boundary condition receptances of the tool and the holder which are derived from Tim oshenko beam models RCSA is described in further detail in Chapter 3. Although significant development work has been completed to improve the tool and holder modeling techniques and to better understand the connection stiffness and damping behavior (see C hapter 2), relatively less effort has been expended to improve the identification of the spindle machine dynamics. Correct identification of spindle PAGE 21 21 machine dynamics is naturally required for accurate prediction of the tool point dynamics. This research fo cuses on identification of an improved technique to obtain the spindle machine dynamics for RCSA. The new method of spindle machine dynamics is compared to two approaches described in the literature. Research Description There are two established methods of spindle dynamics identification The first is the synthesis approach [ 8 9 46 ], which requires two FRF measurements (one direct and one cross) on a standard artifact (a holder with simple geometry which is easy to model) by impact testing. For these mea surements a direct FRF refers to a measurement where the location of the force coincides with the response measuremen t location as shown in Fig 1 2a. A cross FRF refers to a measurement where the location of the applied force is not the same as the respon se measurement location (Fig. 1 2b). The second method is the finite difference approach [ 10 ], where three FRF measurements (two direct and one cross) are required on the standard artifact. As opposed to the two standard artifact FRF measurements for the s ynthesis approach and the three standard artifact FRF measurements for the finite difference approach, the novel method proposed in this research, referred to as the Euler Bernoulli method, requires only one direct FRF measurement at the free end of the st andard artifact. The calculation of the spindle dynamics using the three approaches is described in detail in Chapter 4. In Chapter 5, the spindle dynamics of three different milling machines, a Mikron UCP 600 Vario, a Starragheckert ZT 1000 Super Constell ation, and a Cincinnati FTV 5 2500 are measured and compared for the three approaches. Using these spindle dynamics, the tool point frequency response functions are then predicted for several PAGE 22 22 combinations of (thermal) shrink fit tool holders and carbide en dmills (assuming a rigid connection between the tool and the holder). The predicted tool point FRFs are compared to measurement results. Furthermore, in order to identify the best method of spindle dynamics identification, an FRF comparison metric is also defined. The best approach is then selected and used to introduce a flexible connection between the tool and the holder in order to improve the accuracy of the tool point frequency response predictions. The connection stiffness values are obtained by apply ing a non linear least squares error minimization t o the difference between predicted and measured tool point FRFs. Stiffness values are identified for various diameters carbide blank s (rods) inserted in shrink fit, collet and Tribos tool holders. The stif fness values of different tool holder connections are compared. These are then used to predict the tool point FRFs of actual endmills and the results are compared to measurements. Dissertation Organization The dissertation is organized as follows. Chapter 1 provides an introduction to the research activities. A literature review is completed in Chapter 2 Chapter 3 details the RCSA approach. Chapter 4 describes the new spindle machine dynamics identification technique. Tool points dynamics predictions and m easurements are presented in Chapter 5. Finally, Chapter 6 summarizes the dissertation and describes the possible future work in this area of research. PAGE 23 23 Figure 1 1. Example stability lobe diagram Unstable zone Stable zone Chatter Stable Spindle Speed Depth of Cut PAGE 24 24 A B Figure 1 2. Standard artifact measure ment. A ) Direct FRF measurement. B ) C ross FRF measurement PAGE 25 25 CHAPTER 2 LITERATURE REVIEW Milling stability has been an active research area for several decades. Recognition of the process limitations imposed by chatter can be dated back to 190 6 in the wor k by Taylor [ 1 1 ]. The work by Arnold [12 ] followed by the research by Tlusty, Tobias and Merrit [1 3 1 5 ] led to a fundamental understanding of regenerative chatter. The regeneration of surface waviness during material removal was identified as the primary mechanism for self excited vibration in machining. The source of self excited vibration is the variable chip thickness that governs the cutting force and subsequent tool vibrations. Modeling of the milling process in order to select pre process parameter s for chatter avoidance and accurate work piece dimensions has been and continues to be a widely studied topic. The time marching numerical integration approach to model the milling process is summarized by Smith and Tlusty [1 6 ]. Related work includes the m echanistic model approach for the prediction of the force system [ 1 7 ]. Frequency domain solutions have been applied to determine process stability in the form of stability lobe diagrams, which identify stable and unstable cutting zones as a function of axi al depth of cut and spindle speed [1 8 ]. Altintas and Budak used a Fourier series (frequency domain) approach to approximate the time varying cutting force coefficients for stability lobe diagram development [1 9 ]. A closed form, frequency domain solution fo r surface location error in milling was developed by Schmitz and Mann [ 20 ]. A numerical method for the stability analysis of linear time delayed system based on a semi discretization technique was also presented in the literature [ 2 1 ]. Modeling approaches based on finite element analysis [ 2 2 ] and, later, time finite element analysis [ 2 3 ] have also been developed. In PAGE 26 26 all the modeling methods, a description of the system dynamic response comprised of the tool holder spindle machine assembly receptance is re quired. This response can be obtained on a case by case basis via impact testing, where an instrumented hammer is used to excite the tool point and the response is measured using (typically) a low mass accelerometer. However, because each tool holder combi nation must be measured on each machine, the number of experiments can be excessive. Therefore, the preferred method is application of an appropriate modeling approach which reduces the number of required experiments. The preference of a modeling approach led to the application of receptance coupling [ 2 4 ] to predict the tool point FRF. In the initial application of receptance coupling to tool point FRF prediction, an Euler Bernoulli (E B) beam model of the overhung portion of the tool was coupled to the di splacement to force receptance of the holder spindle machine [ 5 7 ]. In this work the fluted portion of the tool was approximated using the equivalent diameter approach by Kops and Vo [ 2 5 ]. Many improvements have been made since then to the RCSA method. Par k et al. incorporated displacement to moment, rotation to force and rotation to moment receptances in the analysis [ 2 6 ]. Duncan et al. that results from the interaction of the modes of indiv idual components [2 7 ]. The overhang length of the tool can be adjusted to improve the system dynamic stiffness resulting in higher removal rates as the critical stability limit is increased. Connection parameters determined by fitting the predicted FRF to a tool holder spindle machine ( THSM ) assembly measurement at a known overhang length were used to predict other PAGE 27 27 overhang lengths of the tool. Burns and Schmitz studied the effect of changing tool overhang length on the connection parameters [ 2 8 ]. The conn ection parameters were estimated using a nonlinear least squares algorithm Schmitz and Duncan also described the receptance prediction of nested components with a common neutral axis and studied the sensitivity to noise in the component receptances [ 2 9 ]. Kivanc and Budak modeled endmills as two components, the shank portion and the fluted portion taking into account the moment of inertia of the complex cross section of the flutes. They incorporated flexible coupling between the tool and holder spindle u sing nonlinear least squares error minimization [ 30 ] Movaheddy and Gerami proposed a receptance coupling method which takes into account the rotational degrees of freedom responses by a tool and holder spindle joint model consisting of two parallel sprin gs without the need to include rotational FRFs in the receptance coupling equation. The joint parameters were estimated for one overhang length of the tool using optimization based on genetic algorithm [ 3 1 ]. Schmitz et al. extended the RCSA method to thre e comp onents: the overhung tool (i.e. the portion outside the holder), the holder and the spindle machine [ 8 9 ].The spindle machine receptances were archived by measuring direct and cross displacement to force FRF of a simple geometry standard holder and removing the portion of a holder beyond the flange using inverse receptance coupling approach. Timoshenko beam models were used to describe the tool and the holder receptances. The RCSA method was further improved by making use of FEA to estimate the stiff ness and damping values at the tool shrink fit holder connection [ 3 2 ]. PAGE 28 28 Timoshenko beam models were also used to model the spindle, holder and tool for RCSA and compared to the results obtained by the finite element software in the work done by Ert rk et al. [ 3 3 ]. The effects of the bearing and interface dynamics, spindle design and parameters like tool geometry and holder geometry on the THSM assembly FRF was also studied Given knowledge of which mode was affected by which connection parameters, the tra nslational parameters were tuned [ 3 4 3 6 ]. Further efforts to model the spindle holder joint interface in THSM assembly include work by Namazi et al [ 3 7 ]. They considered translational and rotational springs uniformly distributed in the holder spindle int erface. In the work by Ahmadi and Ahmadian [ 3 8 ] the change in normal contact pressure along the holder and the portion of the tool inserted in the holder was taken into account by modeling the interface as a distributed elastic layer. In a recent study in corporating the work by Namazi et al and Ahmadi and Ahmadian, a model that couples components through continuous elastic joints rather than at single points was developed [ 3 9 ]. Park and Chae combined receptance coupling, finite element analysis and exper imental modal analysis to determine joint dynamics of modular tools [ 40 ]. A closed form approach for the identification of holder spindle and tool holder dynamics was proposed by et al [ 4 1 ]. By rearranging the receptance coupling equation for flexible coupling and obtaining component receptances analytically and experimentally, the stiffness matrix was obtained. This method was highly sensitive to measured FRF as well as to the accuracy of the rotational FRFs approximated by experimental translational FRFs. They further used this procedure to train a neural network to identify the contact stiffness for different holder and tool combinations [ 4 2 ]. PAGE 29 29 Rezaei et al used the conce pt of inverse receptance coupling to extract the holder spindle FRFs by removing the portion of the tool outside the holder [4 3 ] The tool FRFs were determined analytically and subtracted from the measured tool point FRF of the THSM assembly. This method e nables the joint parameters to be part of the holder spindle FRF and can be used to predict the tool point FRF of any tool with a similar joint condition (or insertion length). In a recent study, Filiz et al. applied the spectral Tchebychev technique to mo del the cutting tool for RCSA [ 4 4 ]. PAGE 30 30 CHAPTER 3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS Description This chapter describes the Receptance Coupling Substructure Analysis (RCSA) approach. In the RCSA approach, the tool holder spindle machine assembly is se parated into three components: the tool, holder, and spindle machine. The spindle machine receptance s, or frequency response functions (FRFs), are measured once and archived. These receptances are then analytically coupled to beam models that represent the tool holder to predict the tool point receptances for arbitrary tool holder combinations Frequency Response Function A FRF is a (frequency domain) transfer function, where only the positive frequencies are considered for the system specific damping leve l. An FRF for a system is expressed as the complex ratio of displacement to force (receptance), velocity to force (mobility), or acceleration to force (accelerance or inertance) at the specified coordinate locations. FRFs contain information about the syst em natural frequencies and mode shapes and are commonly expressed as the real and imaginary parts or the magnitude and phase. The receptance of the tool holder spindle machine assembly as reflected at the tool point is used to produce the desired stability lobe diagram and carry out the surface location error ( SLE ) predictions. Three Component Coupling for Tool Point FRF Prediction RCSA uses both experimental and modeled FRFs. In the second generation RCSA method, the assembly was divided into three primary components: the tool, the holder, and the spindle machine [ 8 9 ].The tool and the holder were described using PAGE 31 31 Timoshenko beam models (based on the geometry and material properties) with free free boundary conditions, while the receptances of the spindle ma chine (which are difficult to model based on first principles, primarily due to the difficulty in estimating damping at interfaces) were calculated by measuring a standard artifact and using the inv erse receptance coupling method Figure 3 1 depicts the th ree individual components of the tool holder spindle machine assembly: the tool (I), the holder (II), and the spindle machine (III). For the tool holder spindle machine RCSA model, four bending receptances are used to describe each component. They are: dis placement to force, displacement to couple, rotation to force, and rotation to couple, where i and j are the measurement and force application c oordinate locations, respectively. If i and j are equal, the receptances are referred to as direct receptances; otherwise, they are cross receptances. Free Free Beam Receptances Because the spindle machine receptances are difficult to model, they are meas ured using a standard holder. The tool and holder receptances, on the other hand, are convenient to model. In this research, Timoshenko beam elements are used to PAGE 32 32 model the four degrees of freedom (displacement and rotations at both the ends) for free free beam receptances of the tool and holder [ 45 ]. Equivalent diameter Timoshenko beam models are used to describe the fluted portion of the tool. The individual component, or substructure, receptances, R ij are organized in matrix form in Equation 3 1: (3 1) where x i is the substructure displacement at the coordinate location i i is the sub structure rotation at the coordinate location i f j is the force applied to the substructure at the coordinate location j and m j is the couple applied to the substructure at the coordinate location j Using this not at ion, Equation s 3 2 to 3 9 describe the direct and cross receptances for the components I and II at the coordina te locations shown in Figure 3 2 Component I, the tool, is described using Equations 3 2 through 3 5. (3 2) (3 3) PAGE 33 33 (3 4) (3 5) Similarly, component II, the holder, is described by Equations 3 6 through 3 9. (3 6) (3 7) (3 8) (3 9) The relationships between displacements/rotations and forces/couples can be written using the matrix form at as shown in Equations 3 1 0 to 3 17 where u i and q i are the generalized displacement/rotation and the force/couple vectors, respectively. PAGE 34 34 or (3 10) (3 11) (3 12) (3 13) (3 14) (3 15) (3 16) (3 17) Rigid Coupling of Free Free Receptances The free free tool and holder models are coupled to form the subassembly I II identified in Figure 3 3. The component I and II subassembly r eceptances are determined using Equation s 3 20 to 3 51 In order to calculate the subassembly receptances G 11 (direct) and G 3a1 (cross) (Equation s 3 18 and 3 19, respectively), a generalized force Q 1 (representing both the externally applied force and cou ple) is applied at coordinate location 1 (see Figure 3 3) (3 18) (3 19) PAGE 35 35 The displacement equations for the substructures can be described as follows: (3 20) (3 21) (3 22) (3 23) If rigi d coupling between the two components is assumed, the compatibility condition that describes the connection between the two components is expressed as shown in Equation 3 24. (3 24) The equilibrium condition at coordinate locations 2 a and 2 b is given b y Equation 3 25. (3 25) At coordinate location 1 the external force/couple is applied so the relationship in Equation 3 26 is obtained. (3 26) Substituting for u 2 b and u 2 a in Equation 3 24 gives Equation 3 27. (3 27) Equation 3 28 i s obtained using Equations 3 25 and 3 26. (3 28) Solving for q 2b gives Equation 3 29. Given that from Equation 3 25, substitution in Equation 3 30 gives Equation 3 31, which can then be written as shown in Equation 3 32. This equation shows that the sub assembly receptances can be PAGE 36 36 expressed as a function of the component receptances. Therefore, give n the tool and holder receptances, the tool holder sub assembly receptances can be predicted. (3 29) (3 30) (3 31) (3 32) Similarly, the cross receptances between coordinates 3 a a nd 1 are given by Equations 3 33 and 3 34 (3 33) (3 34) To determine the other two receptances of the sub assembly I II, G 3 a 3 a and G 13 a a generalized force Q 3 a is applied to U 3 a as shown in Figure 3 4. (3 35) (3 36) PAGE 37 37 The component displacement/rotation equations are now described by Equations 3 37 to 3 40. (3 37) (3 38) (3 39) (3 40) The compatibility equation remains the same as given in Equation 3 24. (3 41) The equilibrium equations are: and (3 42) (3 43) Substituting for u 2 a and u 2 b in Equation 3 41 gives Equation 3 44. (3 44) Using Equations 3 42 and 3 43 and substituting for q 2 b and q 3 a in Equation 3 44 gives Equation 3 45. (3 45) Solving for q 2 a Equation 3 46 is obtained. Equation 3 47 gives the desired expression for the subassembly direct receptances. (3 4 6) (3 47) By substituting for q 2 b in Equation 3 47 using Equations 3 42 and 3 46, Equation 3 48 is obtained. Equation 3 49 gives the final expression after simplification. (3 48) PAGE 38 38 (3 49) In a similar way, the cross receptances are defined. See Equations 3 50 and 3 51. (3 50) (3 51) Coupling of Tool Holder and Spindle Machine Receptances Once the free free components I and II are (rigidly) coupled to form the subassembly I II, this subassembly is then rigidly coupled to the spindl e machine (component III) to give the assembly tool point receptances, G 11 ; see Figure 3 5. This coupling is carried out using Equation 3 52 : (3 52) where the R ij matrices are the subassembly matrices. Therefore, from the Equatio n 3 32 I II coupling results, from Equation 3 34, from Equation 3 49, and from Equation 3 50. The remaining unknown in Equation 3 52 is the spindle machine receptance matrix, R 3 b 3 b Id entification of this receptance matrix is discussed in Chapter 4. PAGE 39 39 Figure 3 1. Three component receptance coupling model for the tool (I), holder (II), and spindle machine (III). PAGE 40 40 Figure 3 2. Individual components I II with displacements and rotation s at specified coordinate locations. PAGE 41 41 Figure 3 3. Subassembly I II composed of tool (I) and holder (II). The generalized force Q 1 is applied to U 1 to determine G 11 and G 3 a 1 PAGE 42 42 Figure 3 4. Subassembly I II composed of tool (I) and holder (II). The gen eralized force Q 3 a is applied to U 3 a to determine G 3 a 3 a and G 1 3 a PAGE 43 43 Figure 3 5. The I II subassembly is rigidly coupled to the spindle machine (III) to determine the tool point receptance matrix, G11. PAGE 44 44 CHAPTER 4 IDENTIFICATION OF SPINDLE MACHINE RECEPTA NCES As discussed previously, the receptances of the spindle machine (component III of the tool holder spindle machine assembly) are difficult to model. Therefore, these receptances are experimentally determined. To identify the spindle machine receptance s, a standard artifact (i.e., a standard tool holder with a uniform cylindrical geometry beyond the flange) is inserted in the spindle as shown in Figure 4 1 and G 22 is determined experimentally. Using G 22 and a model of the portion of the holder beyond th e flange, the spindle machine receptance R 3 b 3 b is calculated. The free end response for the artifact spindle machine ass embly is described by Equation 4 1, where the R 22 R 23 a R 3 a 3 a and R 3 a 2 matrices are populated using a beam model of the portion of the artifact beyond the flange Since the flange geometry is the same for all holders that are inserted in a particular spindle (to enable automatic tool changes), only the portion of the holder beyond the flange (towards the tool) is modeled. The flange and the holder taper (which is inserted in the spindle) are considered part of the spindle machine. (4 1) Equation 4 1 is rearranged in Equation 4 2 to isolate R 3 b 3 b This step of decomposing the measured assembly receptances, G 22 into the modeled substr ucture receptances, R 3 a 2 R 22 R 23 a and R 3 a 3 a and spindle machine receptances, R 3 b 3 b is referred to as Three approaches for experimentally determining the four spindle machine receptances are discussed in the following sections. (4 2) PAGE 45 45 Synthesis Approach The direct displacement to force term in the G 22 matrix is measured by impact testing (the TXF software from MLI was used for data acquisition and signal analysis in this study ). In this meth od, an instrumented impact hammer is used to apply the force and an accelerometer (piezoelectric sensor) is used to measure the response; their ratio is the accelerance (translational acceleration to force FRF). The software is used to twice integrate the accelerance to give the required displacement to force receptance. The direct FRF H 22 is measured by applying the force and placing the accelerometer at the same coordinate location ( U 2 in Figure 4 1). The second component of the G 22 matrix, the rotation to force receptance, is calculated by a first order backward finite difference approach [ 8,46 ] as described in Equation 4 3, where the cross FRF H 2a2 is measured by exciting the assembly at U 2 and measuring the response at coord inate U 2a located a distance S shown in Figure 4 1. (4 3) Assuming reciprocity (which states that a cross FRF with a measurement at coordinate 1 and force at 2 is equal to a cross FRF with a measurement at 2 and a forc e at 1) the off diagonal terms of the G 22 matrix may be taken to be equal. See Equation 4 4. L 22 = N 22 (4 4) PAGE 46 46 The measured H 22 and derived N 22 receptances are used to synthesize P 22 as shown in Equation 4 5. (4 5) The four receptances required t o populate G 22 are now known and Equation 4 2 can be used to obtain R 3 b 3 b Given R 3 b 3 b free free models for arbitrary tool holder combinations can be developed and coupled to the spindle machine receptances to predict the tool point FRF, H 11 required for milling process simulation. The synthesis approach thus requires two artifact measurements to determine the G 22 matrix. In this approach, modal fitting of H 22 and H 2a2 receptances via a peak picking technique [ 3 ] can be applied to reduce the effects of me asurement noise on the tool point prediction results (this strategy is used in this study) Finite Difference Approach In this approach, three measurements are required: the direct and cross FRF measurements as described in the synthesis approach and an a dditional direct displacement to force receptance, H 2a2a at the distance, S from the free end of the artifact. With the two direct, H 22 and H 2a2a and one cross displacement to force H 2a2 receptances a second order backward finite difference approach is implemented to identify the rotation to moment receptance [ 10 ] (Equation 4 6). (4 6) M odal fitting of the measurement receptances via a peak picking technique may again be applied to reduce the effects of measurement noise on the tool point predictio n results The fitting strategy is applied in this work. PAGE 47 47 Euler Bernoulli Method As an alternative to completing two measurements on the standard artifact for the synthesis approach and three measurements for the finite difference approach, only a single d irect measurement is performed at the free end of the standard artifact in the new method established in this research I n the new technique, i t is assumed that each mode within the measurement bandwidth can be approximated as a fixed free (Euler Bernoulli ) beam and the individual modes are fit using the closed form receptance equation for fixed free Euler Bernoulli (E B) beams presented by Bishop and Johnson [ 24 ]. The fit is completed using Eq uation 4 7 for the displacement to force receptance at the free end of a cylindrical fixed free beam, where is frequency (rad/s), is the density, E is the elastic modulus, is the solid damping factor (unitless), d is the beam diameter, and L is the beam length. (4 7) The algorithm for fitting each mode is composed of five steps: 1. Determine the natu ral frequency, f n (Hz), for the mode to be fit from the measured H 22 receptance 2. Select a beam diameter (this is a fitting parameter) and specify the modulus and density (steel values E = 200 GPa and = 7800 kg/m 3 were used in this research ). 3. Calculate the beam length using the closed form expression for the natural frequency of a fixed free cylindrical beam; see Eq uation 4 8 [ 47 ] 4. Adjust to obtain the proper slope for the real part of the mode in question. 5. If the subsequent mode magn itude is too large increase d to d new and calculate L new using Eq uation 4 9 (to maintain the same natural frequency). If the mode magn itude is too small, decrease d and calculate L new PAGE 48 48 (4 8) (4 9) Once the fit parameters d L and are determined (as well as the selected E and values), the remaining receptances for the free end of the artifact are calculated as shown in Eq uations 4 10 and 4 11 No additional measurements are required. (4 10) (4 11) The four G 22 receptance s are then known and Equation 4 2 can be used to obtain R 3 b 3 b the spindle machine receptances. PAGE 49 49 Figure 4 1. Artifact model for determining R 3 b 3 b by inverse RCSA. PAGE 50 50 CHAPTER 5 RESULTS Spindle Machine Receptances Comparison The G 22 receptances determined by the synthesis, finite difference, and Euler Bernoulli methods (Chapter 4) are calculated for three spindles and compared. Three spindle/machine combinations were tested: a Mikron UCP 600 Vario, a Star r agheckert ZT 1000 Super Constellation, and a Cincin nati FTV 5 2500 The specifications of these machines are listed in Table 5 1. The Mikron UCP 600 Vario milling machine spindle (HSK 63A interface) was tested using the steel artifact depicted in Figure 5 1. Direct and cross FRFs were measured ( S = 38.3 mm ) using impact testing and the G 22 identification methods described in Chapter 4 were completed. The H 22 measurement and 24 mode E B fit are presented in Figure 5 2; the E B fitting parameters are provided in Table 5 2 The predicted L 22 / N 22 receptances fr om the three methods are displayed in Figure 5 3. Good agreement in both magnitude and frequency is observed. The P 22 receptances obtained using Equations 4 5 (synthesis), 4 6 (finite difference), and 4 11 (E B) are displayed in Figure 5 4. Again, the agre ement is good except at the anti resonant frequencies ( where the response magnitude is close to zero near 1580 Hz and 2395 Hz) for the synthesis approach and at lower frequencies for the finite difference approach. For the synthesized receptance, the imag inary part exhibits unexpected positive values near the anti resonant frequencies. This is presumably due to the division by the complex valued receptance, H 22 in Equation 4 5. Given the G 22 receptances (from the three techniques), the corresponding spind le machine receptance matrices, R 3 b 3 b were calculated using Eq. 4 2 and a free free PAGE 51 51 boundary condition Timoshenko beam model for the portion of the standard artifact beyond the flange. The dimensions provided in Figure 5 1 were used together with steel ma terial properties ( E = 200 GPa, = 7800 kg/m 3 = 0.29 ) to develop the artifact model. The H 22 measurement was completed using two artifacts of different lengths for the Starragheckert ZT 1000 Super Constellation; see Figures 5 5 and 5 6. The direct and cross FRFs were measured at a distance of S = 32 mm for the short artifact and S = 50 mm for the long artifact. The 16 mode E B fit for the short standard artifact and 9 mode E B fit for the long standard artifact are presented in Figur e 5 7 and Figure 5 10; the E B fitting parameters for the two artifacts are provided in Tables 5 3 and 5 4 The L 22 / N 22 receptances for both the artifacts are displayed in Figures 5 8 (short) and 5 11 (long) and the P 22 receptances in Figures 5 9 (short) a nd 5 12 (long). The trends are similar to those observed for the Mikron UCP 600 Vario data. Figures 5 13 to 5 18 show the measured H 22 FRF and the E B fit, as well as a comparison of the L 22 / N 22 and P 22 receptances for the three approaches using both the s hort and the long standard artifacts ( the dimensions are provided in Figures 5 5 and 5 6) for the Cincinnati FTV 5 2500 milling machine. Tables 5 5 and 5 6 provide the E B fitting parameters for the two standard artifacts. Tool Point Frequency Response Com parison The archived spindle machine receptance matrices, R 3 b 3 b for the three milling machines were rigidly coupled to Timoshenko beam models of various tool holder combinations to predict the corresponding tool point receptances, H 11 In these tests, car bide endmills of different diameters and overhang lengths were clamped in various PAGE 52 52 shrink fit holders Comparisons between measurement s and predictions for the three spindle receptances are provided. Mikron UCP 600 Vario 25.4 mm diameter carbide endmill in a shrink fit holder A three flute, 25.4 mm diameter carbide endmill was clamped in a shrink fit tool holder. After inserting this subassembly in the Mikron UCP 600 Vario spindle, the tool point receptance, H 11 was measured by impact testing and compared to predictions using the synthesis, finite difference and E B R 3 b 3 b receptance matrices. The dimensions for the Timoshenko beam tool holder model are provided in Figure 5 19 for an overhang length of 99 mm. The fluted portion of the tool was modeled usin g an equivalent diameter, where this diameter was obtained by weighing the carbide tool, assuming a density (15000 kg/m 3 ), and calculating the solid section equivalent flute diameter based on the cylindrical dimensions and the tool and flute lengths. The e lastic The E B prediction, synthesis prediction, finite difference prediction, and measurement are presented in Figure. 5 20. The overhang length was then extended to 107 mm an d the exercise was repeated. The results are shown in Figure. 5 21. A comparison metric was used to compare the three approaches and quantify which technique provided better predictions. Equation 5 1 was used to establish the comparison metric, CM where i mag indicates the imaginary part of the FRF and the absolute value of the difference was summed over each frequency within the measurements bandwidth and n is the length of the frequency vector (5 1) PAGE 53 53 Tables 5 7 and 5 8 list the CM values for the two o verhang lengths. The percent difference with respect to the lowest CM value is also specified. The low percent difference between the three methods suggests that all the three techniques are in good agreement with each other 19 05 mm diameter carbide end mill in a shrink fit holder For these tests, a four flute, 19.05 mm diameter carbide endmill was clamped in a shrink fit tool holder and this subassembly was inserted in the Mikron UCP 600 Vario spindle. Tool point measurements were again completed to comp are the predictions using the synthesis, finite difference, and E B method R 3b3b receptance matrices. The dimensions for the Timoshenko beam tool holder model are provided in Figure 5 22 for an overhang length of 70.4 mm. The predictions and measurement ar e provided in Figure 5 23. The overhang length was then extended to 76 mm. These results are shown in Figure 5 24. The comparison metric and percent difference values for the two overhang lengths are listed in Tables 5 9 and 5 10 Again, all the three meth ods predict equally well. Starragheckert ZT 1000 Super Constellation 12 mm diameter carbide endmill in a shrink fit holder A four flute 12 mm diameter carbide endmill was clamped in a shrink fit holder and the tool holder was inserted in the machine spin dle. Tool point measurements were completed by impact testing. Figure 5 25 shows the dimensions of the tool holder Timoshenko beam model for an overhang length of 44.7 mm. Using the spindle receptances obtained by measuring two standard artifacts ( see Figu res 5 5 and 5 6), a comparison of the measurement and predictions for the three techniques are presented in Figures 5 26 to 5 29 with two tool overhang lengths of 44.7 mm and 55.0 mm. Tables PAGE 54 54 5 1 1 to 5 1 4 list the comparison metric showing that the E B pred ictions provide the closest agreement to measurement especially for the long artifact predictions The imaginary part s of the synthesis approach prediction in the long artifact predictions (Figures 5 28 and 5 29) show positive value s near 3200 Hz. This ma y be due to the positive values of the synthesized P 22 receptance These results indicate that the E B (single artifact measurement ) technique is more robust. 16 mm diameter carbide endmill in a shrink fit holder Tool point measurements were performed with a 16 mm diameter four flute endmill clamped in a shrink fit holder. The tool holder dimensions are shown in Figure 5 30. Again measurements were completed by impact testing with two overhang lengths (55.0 mm and 65.0 mm) of the tool. Spindle receptances calculated using the three approaches (for both short and the long artifact s) were coupled to the tool holder model to predict the tool point FRF; see Figures 5 31 to 3 34. The comparison metric and percent difference with respect to the smallest CM value are listed in Tables 5 1 5 to 5 1 8 .The E B clearly outperforms the other two approaches for long artifact predictions. 20 mm diameter carbide endmill in a shrink fit holder Tool point FRF measurement s were completed on a 20 mm diameter two flute endmill c lamped in a shrink fit holder. Two overhang lengths of 65 .0 mm and 75 .0 mm were tested. Figure 5 35 depicts the tool holder model dimensions for the 65 .0 mm overhang length. Predictions we re again made using the two standard artifact spindle receptances. T ool point measurements and predictions for the two overhang lengths are compared in Figures 5 36 to 5 39 and Tables 5 1 9 to 5 2 2 list the CM values and percent difference s The short artifact predictions for the E B method slightly outperform PAGE 55 55 the other two techniques (the percent difference values are large and the long artifact predictions again predict positive imaginary part values for the synthesis approach ) 25 mm diameter carbide endmill in a shrink fit holder A 25 mm diameter endmill with four flutes was clamped in a shrink fit holder and tool point FRFs were measured via impact testing for two overhang lengths of the tool (75.0 mm and 85.0 mm). The holder tool model dimensions are shown in Figure 5 40. Tool point FRF predictions (using both the short and long standard artifact spindle receptances) for the three approaches and measurements are compared in Figures 5 41 to 5 44. Tables 5 2 3 to 5 2 6 list the CM values, as well as the percent difference with respect to the smallest CM value. From the figur es and tables, all the three approaches are in good agreement with the measurements. Cincinnati FTV 5 2500 Using the same 12 mm, 16 mm, 20 mm, and 25 mm carbide endmills clamped in shrink fit holders with the same overhang lengths, tool point measurements were completed on the Cincinnati FTV 5 2500 milling machine. The tool holder model dimensions were the same as those shown in Figures 5 25, 5 30, 5 35, and 5 40 for the 12 mm, 16 mm, 20 mm, and 25 mm diameter endmills, respectively. The following sections list the CM values and percent difference for all the four endmills. 12 mm diameter carbide endmill in a shrink fit holder Figures 5 45 to 5 48 compare the tool point FRF measurements and predictions for two overhang lengths 45.0 mm and 55.0 mm with spind le receptances obtained by the two standard artifacts. The CM values and percent differences are listed in Tables 5 2 7 to 5 30 The long artifact predictions using synthesis and finite difference approach are less accurate than the E B method predictions. PAGE 56 56 16 mm diameter carbide endmill in a shrink fit holder Tool point FRF measurements and comparisons are shown in Figures 5 49 to 5 52 and CM values are listed in Tables 5 3 1 to 5 3 4 The accuracy of E B method is better than the synthesis and the finite diff erence approach. 20 mm diameter carbide endmill in a shrink fit holder Figure 5 53 shows the experimental setup for the tool point FRF measurement on the 20 mm diameter carbide endmill clamped in a shrink fit holder. Figures 5 54 to 5 57 display the tool p oint measurements and predictions for overhang lengths of 65.0 mm and 75.0 mm using spindle receptances measured by both the short and the long artifact. CM values for the predictions using the three different approaches for the two overhang lengths are li sted in Tables 5 3 5 to 5 38 The three techniques can be considered in good agreement with the measurement for the short artifact predictions, but positive value of the imaginary in the synthesis approach is again seen for the long artifact predictions. 25 mm diameter carbide endmill in a shrink fit holder The tool point FRF measurements (see Figure 5 58) and predictions are presented in Figures 5 59 to 5 62 for the two overhang lengths of 75.0 mm and 85.0 mm and the corresponding CM values with percent dif ferences are listed in Tables 5 3 9 to 5 42 In this case, all the three approaches perform well for both the short and long artifact predictions. Introduction of flexible connection between the tool and the holder It is observed that the predicted natural frequencies for the different tool holder combinations are generally higher than the experimental results (i.e., the predicted modes appear to the right of the measured modes). This is attributed to the assumption PAGE 57 57 of a rigid connection between the tool and the holder. A flexible connection is introduced between the tool and the holder in this section for the test spindles, Cincinnati FTV 5 2500 and Mikron UCP 600 Vario. The types of tool holder connections include shrink fit holders, collet holders, and Sch unk Tribos holders. The Euler Bernoulli method of spindle identification was used in this study. Tool point FRFs were completed with carbide blanks (rods) inserted in the shrink fit holders, collet holders, and Tribos holders. Carbide blanks were used so that the identification. The flexible coupling of the components is carried out in two steps: 1) the spindle machine is first rigidly coupled to the holder and the portion of the shank inside the holder; 2) the holder spindle machine component is then flexibly coupled to the portion of the blank that extends outside the holder using translational and rotational spring constants assembled in the stiffness matrix k (Figure 5 63) The RCSA equation for the flexible coupling tool point FRF is provided in Equation 5 2. The stiffness matrix is given by Equation 5 3, where k xf k f k xm and k m are the displacement to force, rotation to force, displacement to moment, and rotation to moment stiffness values, respectively and c xf c f c xm and c m are the corresponding damping values if viscous damping is considered at the coupling location ( k f = k xm and c f = c xm were assumed due to reciprocity) The derivation of Equati on 5 2 is provided in the Appendix A (5 2) (5 3) PAGE 58 58 T o identify the stiffness matrix, tool point FRFs were measured for multiple overhang length s of the blank on each holder. A n optimization procedure based on non linear least squares wa s implemented to fin d the connection stiffness The variables we re rotation to force stiffness (assumed equal to the displacement to moment stiffness) and rotation to force damping (assumed equal to the displacement to moment damping) in Equation 5 3 because the holder tool c onnection is most effective at limiting translation (due to the press fit), but can still allow small axial slip and rotational flexibility due to the finite friction between the tool and internal hole in the holders. T he objective function to be minimized is given by Equation 5 4, where the absolute value of the difference of the magnitude of the measured ( m ) and predicted ( p ) tool point FRFs was computed. (5 4) Cincinnati FTV 5 2500 The k matrix was obtained for the various overhang lengths for 12 mm, 16 mm, 20 mm, and 25 mm diameter blanks in shrink fit holders (Figure 5 64) and 12 mm, 16mm, and 20 mm diameter blanks in a collet holder (Figure 5 65). For example, Table 5 4 3 lists the stiffness and damping values obtained for three different overhang l engths of the 12 mm diameter blank inserted into the corresponding shrink fit holder. Figures 5 6 6 to 5 6 8 show the measured and predicted carbide blank tool point FRF of the 12 mm diameter blank clamped in the shrink fit holder for the three overhang leng ths assuming a rigid connection. Figures 5 69 to 5 71 show the measurement and prediction with a flexible connection for the three overhang lengths. Table 5 4 4 lists the average stiffness values of the 12 mm, 16 mm, 20 mm, and 25 mm blanks clamped in the s hrink fit PAGE 59 59 holder. Similarly, Table 5 4 5 lists the average stiffness values of the 12 mm, 16mm, and 20 mm blank clamped in the collet holder. Mikron UCP 600 Vario Tool point FRF measurements for blanks inserted in shrink fit holders, collet holders (Figure 5 72 and Figure 5 73), and Tribos holders (Figure 5 74) were also completed on the Mikron UCP 600 Vario. The average stiffness values for different diameter blanks for the three holders are listed in Tables 5 4 6 to 5 4 8 Comparison of the stiffness value s of different diameter blanks in various holders for both the test spindles (Cincinnati FTV 5 and Mikron UCP 600 Vario) shows that the connection stiffness values increase with increasing diameter. The increased flexibility for the smaller diameter tool c onnection is due to the lower contact surface area with the holders. The thermal shrink fit holder offers the most rigid connection. For example, the 25 mm diameter shrink fit holder blank did not require flexible coupling; the holder tool interface of lar ge shrink fit diameters can be modeled as a rigid connection. The Tribos holder offers the next higher connection stiffness; the elastic clamping mechanism is described in Figure 5 75. The Tribos holder consists of three chambers filled with a thermo settin g plastic that absorbs shock and reduces vibrations during machining. In the Timoshenko beam model of the Tribos holder, the section consisting of the thermo setting plastic chambers and steel was modeled using equivalent values of the elastic modulus ( E eq ) density ( eq eq ) as shown in Equations 5 5, 5 6, and 5 7, respectively, where A steel is the cross sectional area of the steel portion, A plastic is the cross sectional area of the thermo setting plastic chambers, and A total is the total cross sectional area of the Tribos holder. The values of the elastic modulus, PAGE 60 60 thermo setting plastic were taken to be 7 GPa, 3700 kg/ m 3 and 0.5, respectively. (5 5) (5 6) (5 7) The collet tool holder connection was the most flexible. This was anticipated due tapered volume using a clamping nut. Comparison between the Cincinnati FTV 5 and Mikron UCP 600 Vario spindles show that the stiffness values for a particular type of holder with the same diameter blank give similar results. Therefore, the stiffness values obtained by measurement of blanks in a selected holder type can be used to predict the tool point FRF of an actual endmill in that holder when it is inserted in any s pindle. The tool point FRF measurement and prediction for actual endmills in the Mikron UCP 600 Vario spindle with rigid and flexible couplings are shown in Figures 5 76 to 5 87 for 6.33 mm, 12.7 mm, and 19 mm endmills clamped in the shrink fit, collet, an d Tribos holders; the beam models are also displayed. It can be seen from these figures that the introduction of a flexible connection between the holder and tool using the average stiffness values obtained from the blank measurements listed in Tables 5 4 6 to 5 4 8 improves the tool point FRF prediction as compared to the prediction obtained by the rigid connection PAGE 61 61 assumption. The beam model for the 25.4 mm diameter 4 flute endmill clamped in the shrink fit holder is shown in Figure 5 88. Figure 5 89 shows t hat the assumption of rigid connection at the holder tool interface in case of the 25.4 mm diameter 4 flute endmill clamped in a shrink fit holder is valid. PAGE 62 62 Table 5 1. Specifications of milling machines tested Manufacturer and model Geometry Spindle hold er connection Work volume (mm) Max spindle speed (rpm) Controller Mikron UCP 600 Vario Vertical (5axis) HSK63A 600X450X450 20000 Heidenhain iTNC 530 Cincinnati FTV5 2500 Vertical (5 axis) HSK63A 2540X1003X800 18000 Siemens Fanuc Starragheckert ZT1000 S uper Constellation Vertical (5 axis) HSK63A 2000X1600x1600 24000 Siemens 840D Table 5 2 E B fitting parameters for Mikron UCP 600 Vario CNC milling machine spindle. Mode f n (Hz) d (m) L (m) 1 550 0.375 0.10 0 0.6950 2 610 0.520 0.06 0 0.7771 3 703 0.330 0.06 0 0.5767 4 795 0.450 0.050 0.6160 5 840 0.565 0.035 0.6903 6 875 0.260 0.05 0 0.4588 7 975 0.107 0.07 0 0.2788 8 1057 0.208 0.032 0.3734 9 1080 0.255 0.032 0.4090 10 1131 0.107 0.054 0.2589 11 1230 0.173 0.055 0.3157 12 1297 0.206 0.042 0. 3354 13 1422 0.196 0.078 0.3125 14 1750 0.200 0.11 0 0.2845 15 1872 0.115 0.06 0 0.2086 16 2040 0.190 0.15 0 0.2569 17 2620 0.220 0.13 0 0.2439 18 2985 0.098 0.06 0 0.1525 19 3060 0.125 0.07 0 0.1701 20 3205 0.185 0.07 0 0.2022 21 3800 0.270 0.06 0 0.2244 22 3975 0.340 0.04 0 0.2462 23 4150 0.220 0.05 0 0.1938 24 4310 0.112 0.05 0 0.1357 PAGE 63 63 Table 5 3 E B fitting parameters for the short standard artifact on the Starragheckert ZT 1000 Super Constellation Mode fn (Hz) d (m) L (m) 1 810 0.250 0.080 0.4676 2 900 0.230 0.160 0.4255 3 980 0.430 0.070 0.5575 4 1050 0.197 0.080 0.3646 5 1110 0.256 0.040 0.4042 6 1142 0.205 0.045 0.3566 7 1175 0.187 0.040 0.3358 8 1200 0.200 0.035 0.3436 9 1250 0.810 0.170 0.2143 10 1375 0.117 0.102 0.2455 11 2392 0.11 2 0.065 0.1821 12 2600 0.260 0.100 0.2662 13 2750 0.350 0.100 0.3003 14 3140 0.430 0.060 0.3115 15 3750 0.550 0.060 0.3223 16 4200 0.087 0.045 0.1211 Table 5 4 E B fitting parameters for the long standard artifact on the Starragheckert ZT 1000 Su per Constellation Mode f n (Hz) d (m) L (m) 1 772 0.080 0.060 0.2709 2 880 0.077 0.120 0.2490 3 922 0.102 0.070 0.2799 4 1080 0.220 0.110 0.3799 5 1228 0.365 0.050 0.4589 6 1320 0.230 0.130 0.3513 7 2134 0.104 0.060 0.1858 8 3035 0.220 0.060 0.226 6 9 3230 0.092 0.045 0.1420 PAGE 64 64 Table 5 5 E B fitting parameters for the short standard artifact on the Cincinnati FTV 5 2500 Mode fn (Hz) d (m) L (m) 1 685 0.275 0.090 0.5333 2 1007 0.270 0.075 0.4358 3 1110 0.250 0.060 0.3994 4 1240 0.240 0.080 0.3703 5 1313 0.195 0.045 0.3244 6 1451 0.085 0.040 0.2037 7 1660 0.260 0.060 0.3331 8 1816 0.112 0.070 0.2090 9 2060 0.600 0.110 0.4542 10 2440 0.250 0.080 0.2694 11 2504 0.130 0.055 0.1918 12 3070 0.550 0.040 0.3562 13 3350 1.000 0.040 0.4599 1 4 3950 0.087 0.066 0.1249 Table 5 6 E B fitting parameters for the long standard artifact on the Cincinnati FTV 5 2500 Mode fn (Hz) d (m) L (m) 1 955 0.080 0.060 0.2435 2 1060 0.089 0.080 0.2439 3 1085 0.250 0.040 0.4040 4 1157 0.104 0.050 0.2523 5 1245 0.340 0.030 0.4398 6 1330 0.270 0.050 0.3792 7 1640 0.230 0.070 0.3152 8 1722 0.173 0.060 0.2668 9 1850 1.400 0.030 0.7322 10 2080 0.500 0.040 0.4127 11 2289 0.125 0.050 0.1967 12 2420 0.530 0.030 0.3939 13 3020 0.128 0.062 0.1733 14 3150 0.128 0.050 0.1697 15 4660 0.149 0.030 0.1505 PAGE 65 65 Table 5 7 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 99 mm CM (m/N) Percent difference with respect to smallest CM E B 115.70 x 10 9 Synthesis 118.58 x 10 9 2.49 Finite Difference 118.67 x 10 9 2.57 Table 5 8 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 107 mm CM (m/N) Percent difference with respect to smallest CM E B 123.78 x 10 9 Synthesis 126.10 x 10 9 1.88 Finite Difference 126.27 x 10 9 2.02 Table 5 9 Comparison metric (m/N) for the FRF predictions of 19.05 mm diameter endmill, overhang length 70.4 mm CM (m/N) Percent difference with respect to smallest CM E B 35.18 x 10 9 S ynthesis 35.21 x 10 9 0.10 Finite Difference 35.83 x 10 9 1.8 6 Table 5 10 Comparison metric (m/N) for the FRF predictions of 19.05 mm diameter endmill, overhang length 76 mm CM (m/N) Percent Difference with respect to smallest CM E B 179.24 x 10 9 Synthesis 181.35 x 10 9 1. 17 Finite Difference 181.51 x 10 9 1. 26 Table 5 1 1 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 44.7 mm using short artifact spindle receptances CM (m/N) Percent differenc e with respect to smallest CM E B 102.39 x 10 9 Synthesis 120.02 x 10 9 17. 22 Finite Difference 122.13 x 10 9 19.28 PAGE 66 66 Table 5 1 2 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using short artifac t spindle receptances CM (m/N) Percent difference with respect to smallest CM E B 1 46.06 x 10 9 Synthesis 159.35 x 10 9 9.10 Finite Difference 163.62 x 10 9 12.0 Table 5 1 3 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endm ill, overhang length 44.7 mm using long artifact spindle receptances CM (m/N) Percent difference with respect to smallest CM E B 103.15 x 10 9 Synthesis 152.37 x 10 9 47.7 1 Finite Difference 215.11 x 10 9 108.5 Table 5 1 4 Comparison metric (m/ N) for the FRF predictions of 12 mm diameter endmill, overhang length 55 mm using long artifact spindle receptances CM (m/N) Percent difference with respect to smallest CM E B 128.86 x 10 9 Synthesis 214.99 x 10 9 66.8 4 Finite Difference 144.23 x 10 9 11.92 Table 5 1 5 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances CM (m/N) Percent difference with respect to smallest CM E B 41.26 x 10 9 Synthesis 46 .50 x 10 9 12.6 9 Finite Difference 47.68 x 10 9 15.5 3 Table 5 1 6 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances CM (m/N) Percent difference with respect t o smallest CM E B 57.34 x 10 9 Synthesis 65.80 x 10 9 14.74 Finite Difference 70.26 x 10 9 22.52 PAGE 67 67 Table 5 17 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptance s CM (m/N) Percent difference with respect to smallest CM E B 23.35 x 10 9 Synthesis 92.36 x 10 9 29 5 58 Finite Difference 40.15 x 10 9 71.9 5 Table 5 1 8 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances CM (m/N) Percent difference with respect to smallest CM E B 31.24 x 10 9 Synthesis 106.3 x 10 9 2 40.31 Finite Difference 62.54 x 10 9 100.19 Table 5 1 9 Comparison metric (m/N) for the FRF predic tions of 20 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 28.29 x 10 9 Synthesis 41.51 x 10 9 46. 72 Finite Difference 41.89 x 10 9 48.0 6 Table 5 20 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 36.77 x 10 9 Synthesis 46.02 x 10 9 25.13 Finite Difference 47.14 x 10 9 28.19 Table 5 2 1 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 28.56 x 10 9 Synthesis 39.13 x 10 9 3 7.02 Finite Difference 34.42 x 10 9 20. 51 PAGE 68 68 Table 5 22 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 24.28 x 10 9 Synthesis 38.65 x 10 9 59. 20 Finite Difference 39.85 x 10 9 64.1 5 Table 5 2 3 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 18.04 x 10 9 0.56 Synthesis 18.44 x 10 9 2.77 Finite Difference 17.94 x 10 9 Table 5 2 4 Comparison metric (m/N) for the FRF predictions of 25 mm diamete r endmill, overhang length 85.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 26.97 x 10 9 16.7 9 Synthesis 23.29 x 10 9 0.8 4 Finite Difference 23.09 x 10 9 Table 5 2 5 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 16.40 x 10 9 34.72 Synthesis 12.67 x 10 9 4.1 1 Finite Difference 12.1 7 x 10 9 Table 5 2 6 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 20.06 x 10 9 20.83 Synthesi s 17.23 x 10 9 3.81 Finite Difference 16.60 x 10 9 PAGE 69 69 Table 5 2 7 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 45.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 115.92 x 10 9 Synthesis 119.12 x 10 9 2.7 6 Finite Difference 120.39 x 10 9 3.85 Table 5 2 8 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptanc es. CM (m/N) Percent difference with respect to smallest CM E B 1 73.22 x 10 9 Synthesis 1 74.46 x 10 9 0.72 Finite Difference 1 74.89 x 10 9 0.96 Table 5 2 9 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang leng th 45.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 96.35 x 10 9 Synthesis 231.3 x 10 9 1 40.10 Finite Difference 589.4 x 10 9 5 11.69 Table 5 30 Comparison metric (m/N) for the FRF pre dictions of 12 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 142.67 x 10 9 Synthesis 301.05 x 10 9 111.0 1 Finite Difference 208.67 x 10 9 46. 25 T able 5 3 1 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 53.13 x 10 9 Synthesis 60.23 x 10 9 1 3.35 Finite Difference 58.70 x 10 9 10.48 PAGE 70 70 Table 5 3 2 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 87.07 x 10 9 Synthesis 87.68 x 10 9 0. 70 Finite Difference 87.63 x 10 9 0. 64 Table 5 33 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 30.79 x 10 9 Synthesis 155.77 x 10 9 405.0 8 Finite Difference 57.53 x 10 9 86.90 Table 5 34 Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm u sing long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 72.18 x 10 9 Synthesis 210.06 x 10 9 19 1.04 Finite Difference 91.43 x 10 9 26. 69 Table 5 3 5 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 30.83 x 10 9 Synthesis 37.17 x 10 9 20.55 Finite Difference 36.57 x 10 9 18.6 1 Table 5 3 6 Com parison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 53.69 x 10 9 Synthesis 60.43 x 10 9 12.5 6 Finite Di fference 60.33 x 10 9 12.36 PAGE 71 71 Table 5 3 7 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 22.37 x 1 0 9 Synthesis 24.16 x 10 9 7. 9 8 Finite Difference 23.98 x 10 9 7. 1 7 Table 5 3 8 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75.0 mm using long artifact spindle receptances. CM (m/N) Percent differen ce with respect to smallest CM E B 49.53 x 10 9 Synthesis 70.82 x 10 9 42.9 8 Finite Difference 56.69 x 10 9 14.4 6 Table 5 3 9 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 75.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 18.90 x 10 9 13.9 2 Synthesis 17.38 x 10 9 4.7 8 Finite Difference 16.59 x 10 9 T able 5 40 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill overhang length 85.0 mm using short artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 30.91 x 10 9 Synthesis 33.32 x 10 9 7.79 Finite Difference 35.06 x 10 9 13.41 Table 5 4 1 Comparison metric (m/N) fo r the FRF predictions of 25 mm diameter endmill, overhang length 75.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 13.09 x 10 9 16.8 5 Synthesis 11.27 x 10 9 0.5 8 Finite Difference 11.21 x 10 9 PAGE 72 72 Table 5 42 Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85.0 mm using long artifact spindle receptances. CM (m/N) Percent difference with respect to smallest CM E B 13.84 x 10 9 15.0 6 Synthesis 12.1 9 x 10 9 1.36 Finite Difference 12.03 x 10 9 Table 5 43 Stiffness matrix values of 12 mm diameter blank clamped in a shrink fit holder, Cincinnati FTV 5 2500 Overhang length (mm) k (N/rad) c f (N s/rad) 66 3.9 x 10 6 26 71 4.8 x 10 6 50 76 4. 6 x 10 6 75 Table 5 4 4 Average stiffness matrix values for blank shrink fit holders inserted in Cincinnati FTV 5 2500 Blank diameter (mm) k (N/rad) c f (N s/rad) 12 4.4 x 10 6 50 16 1.7 x 10 7 153 20 1.9 x 10 7 665 25 Rigid Table 5 4 5 Average stiff ness matrix values for blank collet holders inserted in Cincinnati FTV 5 2500 Blank diameter (mm) k (N/rad) c f (N s/rad) 12 3.2 x 10 6 18 16 5.6 x 10 6 51 20 1.5 x 10 7 159 Table 5 4 6 Average stiffness matrix values for blank shrink fit holders inse rted in Mikron UCP 600 Vario Blank diameter (mm) k (N/rad) c f (N s/rad) 12.7 5.4 x 10 6 30 25 Rigid Table 5 4 7 Average stiffness matrix values for blank collet holders inserted in Mikron UCP 600 Vario Blank diameter (mm) k (N/rad) c f (N s/rad) 6.33 2. 2 x 10 5 2 9.5 8.3 x 10 5 13 12 2.9 x 10 6 18 19 1.0 x 10 7 31 25 2.2 x 10 7 0 PAGE 73 73 Table 5 4 8 Average stiffness matrix values for blank T ribos holders inserted in Mikron UCP 600 Vario Blank diameter (mm) k (N/rad) c (N s/rad) 10 2.1 x 10 6 0 12 2.9 x 10 6 2 16 9.0 x 10 6 0 19 1.9 x 10 7 0 PAGE 74 74 Figure 5 1. Artifact dimensions for Mikron UCP 600 Vario measurements. Figure 5 2. H 22 artifact measurement and E B fit for Mikron UCP 600 Vario CNC milling machine. PAGE 75 75 Figure 5 3. L 22 / N 22 results for the Mikron UCP 600 Vario CNC milling machine. PAGE 76 76 Figure 5 4. P 22 results for the Mikron UCP 600 Vario CNC milling machine. PAGE 77 77 Figure 5 5. Short artifact dimensions for Starragheckert ZT 1000 Super Constellation measurements. Figure 5 6. Long artifact dimensions for Starragheckert ZT 1000 Super Constellation measurements. PAGE 78 78 Figure 5 7. H 22 short artifact measurement and E B fit for Starragheckert ZT 1000 Super Constellation milling machine. PAGE 79 79 Figure 5 8. L 22 / N 22 results f or the short artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. PAGE 80 80 Figure 5 9. P 22 results for the short artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. PAGE 81 81 Figure 5 10. H 22 long artifa ct measurement and E B fit for Starragheckert ZT 1000 Super Constellation milling machine. PAGE 82 82 Figure 5 11 L 22 / N 22 results for the long artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. PAGE 83 83 Figure 5 12. P 22 results for the long artifact measurement on ZT 1000 Super Constellation Starragheckert milling machine. PAGE 84 84 Figure 5 13. H 22 short artifact measurement and E B fit for Cincinnati FTV 5 2500 milling machine. PAGE 85 85 Figure 5 14 L 22 / N 22 results for the short artifact measurem ent on Cincinnati FTV 5 2500 milling machine. PAGE 86 86 Figure 5 15. P 22 results for the short artifact measurement on Cincinnati FTV 5 2500 milling machine. PAGE 87 87 Figure 5 16. H 22 long artifact measurement and E B fit for Cincinnati FTV 5 2500 milling machine. PAGE 88 88 Figure 5 17 L 22 / N 22 results for the long artifact measurement on Cincinnati FTV 5 2500 milling machine. PAGE 89 89 Figure 5 18. P 22 results for the long artifact measurement on Cincinnati FTV 5 2500 milling machine. PAGE 90 90 Figure 5 19. Beam m odel for 25.4 mm diame ter, three flute endmill inserted in a tapered shrink fit holder (not to scale) PAGE 91 91 Figure 5 20. Comparison between H 11 tool point measurement, Euler Bernoulli, synthesis and finite difference prediction for three flute, 25.4 mm diameter endmill with an o v erhang length of 99 mm. PAGE 92 92 Figure 5 21. Comparison between H 11 tool point measurement, Euler Bernoulli, synthesis approach and finite difference prediction for three flute, 25.4 mm diameter endmill with an overhang length of 107 mm. PAGE 93 93 Figure 5 22. Beam model for 19 05 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale) PAGE 94 94 Figure 5 23. Comparison between H 11 tool point measurement, Euler Bernoulli, synthesis approach and finite difference prediction for four flute, 19. 05 mm diameter endmill with an overhang length of 70.4 mm. PAGE 95 95 Figure 5 24. Comparison between H 11 tool point measurement, Euler Bernoulli, synthesis approach and finite difference prediction for four flute, 19.05 mm diameter endmill with an overhang length of 76 mm. PAGE 96 96 Figure 5 25. Beam model for 12 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale). PAGE 97 97 Figure 5 26. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction f or four flute, 12 mm diameter endmill with an overhang length of 45 mm (short artifact spindle receptances) PAGE 98 98 Figure 5 27. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 12 mm dia meter endmill with an overhang length of 55 mm (short artifact spindle receptances) PAGE 99 99 Figure 5 28. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 12 mm diameter endmill with an over hang length of 45 mm (long artifact spindle receptances) PAGE 100 100 Figure 5 29. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 55 mm (long artifact spindle receptances) PAGE 101 101 Figure 5 30. Beam model for 16 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale) PAGE 102 102 Figure 5 31. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 16 mm diameter endmill with an overhang length of 55 mm (short artifact spindle receptances) PAGE 103 103 Figure 5 32. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for f our flute, 16 mm diameter endmill with an overhang length of 65 mm (short artifact spindle receptances) PAGE 104 104 Figure 5 33. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 16 mm diameter endmill with an overhang length of 55 mm (long artifact spindle receptances) PAGE 105 105 Figure 5 34. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute, 16 mm diameter endmill with an overhang l ength of 65 mm (long artifact spindle receptances) PAGE 106 106 Figure 5 35. Beam model for 20 mm diameter, two flute endmill inserted in a tapered shrink fit holder (not to scale) PAGE 107 107 Figure 5 36. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for two flute, 20 mm diameter endmill with an overhang length of 65 mm (short artifact spindle receptances) PAGE 108 108 Figure 5 37. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 75 mm (short artifact spindle receptances) PAGE 109 109 Figure 5 38. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 65 mm (long artifact spindle receptances) PAGE 110 110 Figure 5 39. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances) PAGE 111 111 Figure 5 40. Beam model for 25 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale) PAGE 112 112 Figure 5 41. Comparison between H 11 tool point measur ement, E B Synthesis and Finite difference approach prediction for four flute 25 mm diameter endmill with an overhang length of 75 mm (short artifact spindle receptances) PAGE 113 113 Figure 5 42. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite di fference approach prediction for four flute 25 mm diameter endmill with an overhang length of 85 mm (short artifact spindle receptances) PAGE 114 114 Figure 5 43. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach predicti on for four flute 25 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances) PAGE 115 115 Figure 5 44. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 25 mm di ameter endmill with an overhang length of 85 mm (long artifact spindle receptances) PAGE 116 116 Figure 5 45. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 12 mm diameter endmill with an over hang length of 45 mm (short artifact spindle receptances) PAGE 117 117 Figure 5 46. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 12 mm diameter endmill with an overhang length of 55 mm (shor t artifact spindle receptances) PAGE 118 118 Figure 5 47. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 12 mm diameter endmill with an overhang length of 45 mm (long artifact spindle receptan ces) PAGE 119 119 Figure 5 48. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 12 mm diameter endmill with an overhang length of 55 mm (long artifact spindle receptances) PAGE 120 120 Figure 5 49. Compa rison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 16 mm diameter endmill with an overhang length of 55 mm (short artifact spindle receptances) PAGE 121 121 Figure 5 50. Comparison between H 11 tool poi nt measur ement, E B, Synthesis and Finite difference approach prediction for four flute 16 mm diameter endmill with an overhang length of 65 mm (short artifact spindle receptances) PAGE 122 122 Figure 5 51. Comparison between H 11 tool point measur ement, E B, Synt hesis and Finite difference approach prediction for four flute 16 mm diameter endmill with an overhang length of 55 mm (long artifact spindle receptances) PAGE 123 123 Figure 5 52. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 16 mm diameter endmill with an overhang length of 65 mm (long artifact spindle receptances) PAGE 124 124 Figure 5 53. Tool point FRF measurement of 20 mm carbide end mill on Cincinnati FTV 5 2500. PAGE 125 125 Figure 5 54. Comparison bet ween H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 65 mm (short artifact spindle receptances) PAGE 126 126 Figure 5 55. Comparison between H 11 tool point measur e ment, E B, Synthesis and Finite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 75 mm (short artifact spindle receptances) PAGE 127 127 Figure 5 56. Comparison between H 11 tool point measur ement, E B, Synthesis and F inite difference approach prediction for two flute 20 mm diameter endmill with an overhang length of 65 mm (long artifact spindle receptances) PAGE 128 128 Figure 5 57. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach pr ediction for two flute 20 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances) PAGE 129 129 Figure 5 58. Tool point FRF measurement of 25 mm carbide end mill on Cincinnati FTV 5 2500. PAGE 130 130 Figure 5 59. Comparison between H 11 too l point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 25 mm diameter endmill with an overhang length of 75 mm (short artifact spindle receptances) PAGE 131 13 1 Figure 5 60. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction for four flute 25 mm diameter endmill with an overhang length of 85 mm (short artifact spindle receptances) PAGE 132 132 Figure 5 61. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite diffe rence approach prediction for four flute 25 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances) PAGE 133 133 Figure 5 62. Comparison between H 11 tool point measur ement, E B, Synthesis and Finite difference approach prediction f or four flute 25 mm diameter endmill with an overhang length of 85 mm (long artifact spindle receptances) PAGE 134 134 Figure 5 63. Component c oordinates for flexible coupling of holder and blank Figure 5 64 Various shrink fit holders with blanks for Cinci nnati FTV 5 2500 spindle PAGE 135 135 Figure 5 65 Collet holder for Cincinnati FTV 5 2500 spindle PAGE 136 136 Figure 5 66. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (rigid connection) PAGE 137 137 Figure 5 67. Measured and pred icted tool point FRF of 12 mm diameter carbide blank with overhang length 71 mm (rigid connection) PAGE 138 138 Figure 5 68. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (rigid connection) PAGE 139 139 Figure 5 69. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (flexible connection) PAGE 140 140 Figure 5 70. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 71 mm (flexible connection) PAGE 141 141 Figure 5 71. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (flexible connection) PAGE 142 142 Figure 5 72. Collet holder for Mikron UCP 600 Vario PAGE 143 143 Figure 5 73. 25 mm diameter collet holder and blank for Mikron UCP 600 Vario Figure 5 74. Tribos holders for Mikron UCP 600 Vario PAGE 144 144 Figure 5 75. Mechanism of tool clamping in a Tribos holder ( http://www.us.schunk.com ) Figure 5 76. Beam model for 6.33 mm diameter, 2 flu te endmill inserted in a collet holder (not to scale) PAGE 145 145 Figure 5 77. Measured and predicted tool point FRF of 6.33 mm diameter 2 flute carbide endmill in collet holder, overhang length 75 mm (rigid connection) PAGE 146 146 Figure 5 78. Measured and predicted tool po int FRF of 6.33 mm diameter 2 flute carbide endmill in collet holder, overhang length 75 mm (flexible connection) PAGE 147 147 Figure 5 79. Beam model for 19 mm diameter, 4 flute endmill inserted in a collet holder (not to scale) Figure 5 80. Measured and pred icted tool point FRF of 19 mm diameter carbide 4 flute endmill in collet holder, overhang length 60 mm (rigid connection) PAGE 148 148 Figure 5 81. Measured and predicted tool point FRF of 19 mm diameter 4 flute carbide endmill in collet holder, overhang length 60 mm (flexible connection) PAGE 149 149 Figure 5 82. Beam model for 12.7 mm diameter, 2 flute endmill inserted in a shrink fit holder (not to scale) Figure 5 83. Measured and predicted tool point FRF of 12.7 mm diameter 2 flute carbide endmill in shrink fit holder, overhang length 66 mm (rigid connection) PAGE 150 150 Figure 5 84. Measured and predicted tool point FRF of 12.7 mm diameter 2 flute carbide endmill in shrink fit holder, overhang length 66 mm (flexible connection) Figure 5 85. Beam model for 19 mm diameter, 4 fl ute endmill inserted in a Tribos holder (not to scale) PAGE 151 151 Figure 5 86. Measured and predicted tool point FRF of 19 mm diameter 4 flute carbide endmill in Tribos holder, overhang length 72 mm (rigid connection) PAGE 152 152 Figure 5 87. Measured and predicted tool poi nt FRF of 19 mm diameter 4 flute carbide endmill in Tribos, overhang length 72 mm (flexible connection) PAGE 153 153 Figure 5 88. Beam model for 25.4 mm diameter, 4 flute endmill inserted in a shrink fit holder (not to scale) Figure 5 89. Measured and predicted tool point FRF of 25.4 mm diameter 4 flute carbide endmill in shrink fit holder, overhang length 55 mm (rigid connection) PAGE 154 154 CHAPTER 6 CONCLUSION AND FUTURE WORK Conclusion In this work a new method of spindle machine dynamics identification for Receptance C oupling Substructure Analysis (RCSA), referred to as the Euler Bernoulli (E B) method, is described. In the RCSA approach, the tool holder spindle machine (THSM) assembly is considered as three separate components: the tool, holder, and spindle machine T h e individual frequency response s, or receptances, of these components are then analytically coupled. The spindle machine receptance s are measured once and archived. Beam models are used to represent the tool holder subassembly The spindle machine dynamic s were determined using the E B method, as well as two other established methods: the synthesis approach and the finite difference approach. In the synthesis approach, a direct frequency response measurement of a standard artifact inserted in the test spin dle is combined with a cross frequency response measurement to calculate the required rotational receptances. In the finite difference approach, two direct and one cross frequency response are measured using the standard artifact test spindle combination. Again, these measurement results are used to determine the rotational frequency response functions (FRFs). In the E B method, the direct frequency response measurement is fit using an assumed (fixed free) form of each mode within the measurement bandwidth and this fit is used to determine the rotational receptances (no additional measurements are required) Standard artifact measurements were performed on three milling machines: a Mikron UCP 600 Vario, a Starragheckert ZT 1000 Super Constellation, and a Ci ncinnati PAGE 155 155 FTV 5 2500. The spindle machine dynamics were determined by the three approaches and compared. These spindle machine dynamics were then used to predict the tool point FRF of various tool holder combinations (carbide endmills clamped in thermal shr ink fit holders) inserted in the three spindles. The measured tool point FRFs were compared to the predictions. For these predictions, the connection between the tool and the holder was assumed to be rigid. The best method to determine the spindle machine dynamics was identified by using a new comparison metric. Based on the comparison metric calculations, it was concluded that the E B method provides a robust and accurate identification method for spindle machine dynamics. The cross frequency response mea surement on the standard artifact in the synthesis and the finite difference approach may lead to undesired results in the tool point FRF predictions. The tool point FRF predictions determined using the rigid connection assumption between the tool and hol der generally predicted higher natural frequencies than the measurements. Therefore, a flexible connection between the tool and holder was introduced in order to improve the tool point frequency response prediction accuracy (the E B method spindle machine receptances were used). The stiffness values for the tool holder connection were obtained by applying a non linear least squares error minimization to the difference between the magnitudes of the predicted and measured tool point FRFs. Stiffness values wer e identified for various diameter (for example 10 mm, 16 mm, 20 mm, and 25 mm) carbide blanks (rods) clamped in shrink fit, collet, and Tribos tool holders. Multiple overhang lengths of the blanks were measured for each blank holder set to obtain the stiff ness values. The average of these stiffness values PAGE 156 156 was then used to predict the tool point FRFs of actual endmills. The agreement between the measured and the predicted FRFs improved for the flexible connection (based on the stiffness values obtained by bl ank measurements). Therefore, the approach of identifying the tool holder connection stiffness values using blanks is valid. Future Work The possible future work in this research includes investigation of the Timoshenko beam model for the endmills. The use of the equivalent diameter to model the complicated fluted portion of the endmills may not be the best method to identify the FRFs of the tool. The modeling of the flutes needs to be further studied and analyzed in order to increase the accuracy of the to ol point FRF predictions. Also, the Tribos holder Timoshenko beam models requires further study due to the thermo setting plastic chambers in the holders. PAGE 157 157 APPENDIX A FLEXIBLE COUPLING BETWEEN TOOL AND HOLDER The free free tool receptances ( R 11 R 12a R 2 a2a and R 2a1 ) and the machine spindle holder receptances ( R 2b2b ) may be coupled using a flexible joint to predict the assembly tool point receptance. In order to calculate the tool point assembly receptances G 11 (Equation A 1), a generalized force Q 1 (rep resenting both the externally applied force, F and couple, M ) is applied at coordinate location U 1 (see Figure A 1), where the generalized displacement U represents both displacement, X and rotation, (A 1) The displacement equations for the su bstructures can be described as follows: (A 2) (A 3) (A 4) For a flexible coupling, the compatibility condition that describes the connection between the two components is expressed as shown in Equation A 5. (A 5) where the receptance matri x, is composed of four stiffness values and four damping values that relate the displacement and rotation to the applied force and couple. The equilibrium condition at coordinate locations 2 a and 2 b is given by Equation A 6. (A 6) PAGE 158 158 At coordinate location 1 the external force/couple is applied so the relationship in Equation A 7 is obtained. (A 7) Substituting for u 2 b and u 2 a in Equation A 5 gives Equation A 8. (A 8) Equation A 9 is obtained using Equations A 7 and A 8. ( A 9) Solving for q 2b gives Equation A 10. Given that from Equation A 6, substitution in Equation A 11 gives Equation A 12, which can then be written as shown in Equation A 13. This equation gives the assembly receptances expre ssed as a function of the component receptances and the stiffness matrix, k Therefore, given the tool receptances and the holder spindle machine receptances, the tool holder spindle machine receptances can be predicted using a flexible connection between the tool and the holder. (A 10) (A 11) (A 12) (A 13) PAGE 159 159 Figure A 1. The tool (I) is coupled flexibly to the holder spindle machine (II) to determine the tool point receptance matrix, G 11 PAGE 160 160 LIST OF REFERENCES 1. J. Tlusty, Manufacturing Pr ocesses and Equipment Prentice Hall, Upper Saddle River, NJ, 1999. 2. Y. 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Moradi, Proceedings of the ASME 2010, 10 th Biennial Conference on Engineering Systems Design and Analysis 2010. 44. S. Filiz, C. H. Cheng, K. Powell, T. Schmitz, O.B. Ozdoganlar An improved tool holder model for RCSA tool point frequency response prediction, Precision Engineering 33 (1) (2009) 26 36. 45. W. Weaver Jr., S. Timoshenko, D. Young, Vibration Problems in Engineering 5 th Ed., John Wiley and Sons, New York, NY, Section 5.1 2. 46. S. Sattinger, A method for experimentally determining rotational mobilities of structures, Shock and Vibration Bulletin 50 (1980) 17 27. 47. R. Blevins, Formulas for Natural Frequency and Mode shape Kreiger Publishing Co., Malabar, FL, 2001 PAGE 164 164 BIOGRAPHICAL SKETCH Uttara Vijay Kumar was born and raised in New Delhi, the capital city of India. She received her Bachelor of Technology degree in mechanical and automation engineering from Indira Gandhi Institute of Technology, a constituent college of Guru Gobind Singh Indraprastha University, Delhi in May 2007. In f all 2007 she began her graduate studies at the Department of Mechanical and Aerospace Engineering University of Florida, in pursuit of her M S degree in m echanical e ngineering. In s pring 2008, she joine d the Machine Tool Research Center under the guidance of Dr. Tony L. Schmitz She received her Master of Science d egree in December 2009. 