improved spindle dynamics identification technique...
TRANSCRIPT
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IMPROVED SPINDLE DYNAMICS IDENTIFICATION TECHNIQUE FOR RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS
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
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© 2012 Uttara Vijay Kumar
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To my parents, Alka and Vijay and husband, Ashwin
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ACKNOWLEDGMENTS
I extend my sincere gratitude to my advisor, Dr. Tony L. Schmitz, 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 Greenslet 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 Sheffield, 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 unconditional love
and patience.
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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 Response 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
Starragheckert ZT-1000 Super Constellation ................................................... 53
12 mm diameter carbide endmill in a shrink fit holder ................................ 53
16 mm diameter carbide endmill in a shrink fit holder ................................ 54
20 mm diameter carbide endmill in a shrink fit holder ................................ 54
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25 mm diameter carbide endmill in a shrink 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
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LIST OF TABLES
Table page 5-1 Specifications of milling machines tested ........................................................... 62
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 parameters 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 predictions 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 Comparison 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 16 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances .................... 66
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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 diameter 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 long 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) for 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 endmill, 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 artifact 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
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5-32 Comparison metric (m/N) for the FRF 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, overhang 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 spindle 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 predictions 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, overhang 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 spindle 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 Cincinnati 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
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5-48 Average stiffness matrix values for blank-Tribos holders inserted in Mikron UCP-600 Vario ................................................................................................... 73
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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 displacements and rotations at specified coordinate locations. ........................................................................................... 40
3-3 Subassembly I-II composed of tool (I) and holder (II). The generalized force Q1 is applied to U1 to determine G11 and G3a1. ................................................... 41
3-4 Subassembly I-II composed of tool (I) and holder (II). The generalized force Q3a is applied to U3a to determine G3a3a and G13a. .............................................. 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 R3b3b by inverse RCSA. ....................................... 49
5-1 Artifact dimensions for Mikron UCP-600 Vario measurements. .......................... 74
5-2 H22 artifact measurement and E-B fit for Mikron UCP-600 Vario CNC milling machine. ............................................................................................................. 74
5-3 L22/N22 results for the Mikron UCP-600 Vario CNC milling machine. .................. 75
5-4 P22 results for the Mikron 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 H22 short artifact measurement and E-B fit for Starragheckert ZT-1000 Super Constellation milling machine. ............................................................................ 78
5-8 L22/N22 results for the short artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 79
5-9 P22 results for the short artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 80
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5-10 H22 long artifact measurement and E-B fit for Starragheckert ZT-1000 Super Constellation milling machine. ............................................................................ 81
5-11 L22/N22 results for the long artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 82
5-12 P22 results for the long artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. ......................................................................... 83
5-13 H22 short artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling machine. ............................................................................................................. 84
5-14 L22/N22 results for the short artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 85
5-15 P22 results for the short artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 86
5-16 H22 long artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling machine. ............................................................................................................. 87
5-17 L22/N22 results for the long artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 88
5-18 P22 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 H11 tool point measuremen for three flute, 25.4 mm diameter endmill with an overhang length of 99 mm. ......................................... 91
5-21 Comparison between H11 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 Comparison between H11 tool point measurement and prediction for four flute, 19.05 mm diameter endmill, overhang length of 70.4 mm. ........................ 94
5-24 Comparison between H11 tool point measurement and prediction for four flute, 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
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5-26 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (short artifact). ......... 97
5-27 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 55 mm (short artifact). ......... 98
5-28 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (long artifact). .......... 99
5-29 Comparison between H11 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 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (short artifact). ....... 102
5-32 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 65 mm (short artifact). ....... 103
5-33 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (long artifact). ........ 104
5-34 Comparison between H11 tool point measurement nad prediction for four 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 H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (short artifact). ....... 107
5-37 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 75 mm (short artifact). ....... 108
5-38 Comparison between H11 tool point measurement nad prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (long artifact). ........ 109
5-39 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length 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 H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 75 mm (short artifact). ....... 112
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5-42 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 85 mm (short artifact). ....... 113
5-43 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 75 mm (long artifact). ........ 114
5-44 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 85 mm (long artifact). ........ 115
5-45 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (short artifact). ....... 116
5-46 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmil, overhang length of 55 mm (short artifact). ........ 117
5-47 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (long artifact). ........ 118
5-48 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 55 mm (long artifact). ........ 119
5-49 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (short artifact). ....... 120
5-50 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 65 mm (short artifact). ....... 121
5-51 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (long artifact). ........ 122
5-52 Comparison between H11 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 H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (short artifact). ....... 125
5-55 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 75 mm (short artifact). ....... 126
5-56 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (long artifact). ........ 127
5-57 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 75 mm (long artifact). ........ 128
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5-58 Tool point FRF measurement of 25 mm carbide end mill on Cincinnati FTV-5 2500. ................................................................................................................ 129
5-59 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 75 mm (short artifact). ....... 130
5-60 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 85 mm (short artifact). ....... 131
5-61 Comparison between H11 tool point measurement and prediction for four flute, 25 mm diameter endmill, overhang length of 75 mm (long artifact). ........ 132
5-62 Comparison between H11 tool point measurement and prediction for four 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 spindle .......... 134
5-65 Collet holder for Cincinnati FTV-5 2500 spindle ............................................... 135
5-66 Measured and predicted tool point FRF of 12 mm diameter carbide 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 Measured 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 blank 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
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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) .................. 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 point 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 carbide 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 mm (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 flexibly to the holder-spindle-machine (II) to determine the tool point receptance matrix, G11. ............................................................... 159
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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
IMPROVED SPINDLE DYNAMICS IDENTIFICATION TECHNIQUE FOR
RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS
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 modal 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
applied 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
coupled 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 compared 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
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. Measurements 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.
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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 dramatically 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 avoids 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. Even 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
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 ratio 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 methods 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
Analysis (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
frequency responses 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 Timoshenko 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 Chapter 2), relatively less effort has been expended to improve
the identification of the spindle-machine dynamics. Correct identification of spindle-
21
machine dynamics is naturally required for accurate prediction of the tool point
dynamics. This research focuses 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 measurements, a direct FRF refers to a
measurement where the location of the force coincides with the response measurement
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 response 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 synthesis 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 standard 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 Constellation, 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
22
combinations of (thermal) shrink fit tool holders and carbide endmills (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 applying a non-linear least
squares error minimization to the difference between predicted and measured tool point
FRFs. Stiffness values are identified for various diameters carbide blanks (rods)
inserted in shrink fit, collet and Tribos tool holders. The stiffness 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 measurements are presented in
Chapter 5. Finally, Chapter 6 summarizes the dissertation and describes the possible
future work in this area of research.
23
Figure 1-1. Example stability lobe diagram
Unstable zone
Stable zone
Chatter
Stable
Spindle Speed
Depth of Cut
24
A
B Figure 1-2. Standard artifact measurement. A) Direct FRF measurement. B) Cross
FRF measurement
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 1906 in the work by
Taylor [11]. The work by Arnold [12] followed by the research by Tlusty, Tobias, and
Merrit [13-15] 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 parameters 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 [16]. Related work includes the mechanistic model
approach for the prediction of the force system [17]. 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 axial depth of cut and spindle
speed [18]. Altintas and Budak used a Fourier series (frequency domain) approach to
approximate the time varying cutting force coefficients for stability lobe diagram
development [19]. A closed form, frequency domain solution for 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 [21]. Modeling approaches based on finite element
analysis [22] and, later, time finite element analysis [23] have also been developed. In
26
all the modeling methods, a description of the system dynamic response, comprised of
the tool-holder-spindle-machine assembly receptance, is required. 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 combination 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 [24] 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 displacement-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 [25]. Many
improvements have been made since then to the RCSA method. Park et al.
incorporated displacement-to-moment, rotation-to-force and rotation-to-moment
receptances in the analysis [26].
Duncan et al. applied RCSA further to investigate the „dynamic absorber effect‟
that results from the interaction of the modes of individual components [27]. 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
27
overhang lengths of the tool. Burns and Schmitz studied the effect of changing tool
overhang length on the connection parameters [28]. The connection 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 [29].
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
using 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 springs 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 [31].
Schmitz et al. extended the RCSA method to three components: 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 stiffness
and damping values at the tool-shrink fit holder connection [32].
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 Ertürk et al. [33]. 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 translational parameters were tuned [34-36].
Further efforts to model the spindle-holder joint interface in THSM assembly
include work by Namazi et al. [37]. They considered translational and rotational springs
uniformly distributed in the holder-spindle interface. In the work by Ahmadi and
Ahmadian [38], 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 incorporating 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 [39]. Park and Chae combined
receptance coupling, finite element analysis, and experimental 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 Özşahin et al [41]. 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 [42].
29
Rezaei et al. used the concept of inverse receptance coupling to extract the
holder-spindle FRFs by removing the portion of the tool outside the holder [43]. The tool
FRFs were determined analytically and subtracted from the measured tool point FRF of
the THSM assembly. This method enables 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 model the cutting tool for RCSA [44].
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
separated into three components: the tool, holder, and spindle-machine. The spindle-
machine receptances, 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 level. 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 system 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
31
Timoshenko beam models (based on the geometry and material properties) with free-
free boundary conditions, while the receptances of the spindle-machine (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
inverse receptance coupling method. Figure 3-1 depicts the three 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:
displacement-to-force, iij
j
xh
f
displacement-to-couple, iij
j
xl
m
rotation-to-force,
iij
j
nf
and
rotation-to-couple,
iij
j
pm
, where i and j are the measurement and force application
coordinate 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
measured 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
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, Rij(ω), are organized in
matrix form in Equation 3-1:
(3-1)
where xi is the substructure displacement at the coordinate location i, θi is the
substructure rotation at the coordinate location i, fj is the force applied to the
substructure at the coordinate location j, and mj is the couple applied to the substructure
at the coordinate location j.
Using this notation, Equations 3-2 to 3-9 describe the direct and cross receptances
for the components I and II at the coordinate locations shown in Figure 3-2. Component
I, the tool, is described using Equations 3-2 through 3-5.
1 1
1 1 11 11
11
11 111 1
1 1
x x
f m h lR
n p
f m
(3-2)
1 1
2 2 12 12
12
12 121 1
2 2
a a a a
a
a a
a a
x x
f m h lR
n p
f m
(3-3)
i i
j j ij ij
ij
ij iji i
j j
x x
f m h lR = =
n pθ θ
f m
33
2 2
2 2 2 2 2 2
2 2
2 2 2 22 2
2 2
a a
a a a a a a
a a
a a a aa a
a a
x x
f m h lR
n p
f m
(3-4)
2 2
1 1 2 1 2 1
2 1
2 1 2 12 2
1 1
a a
a a
a
a aa a
x x
f m h lR
n p
f m
(3-5)
Similarly, component II, the holder, is described by Equations 3-6 through 3-9.
2 2
2 2 2 2 2 2
2 2
2 2 2 22 2
2 2
b b
b b b b b b
b b
b b b bb b
b b
x x
f m h lR
n p
f m
(3-6)
2 2
3 3 2 3 2 3
2 3
2 3 2 32 2
3 3
b b
a a b a b a
b a
b a b ab b
a a
x x
f m h lR
n p
f m
(3-7)
3 3
3 3 3 3 3 3
3 3
3 3 3 33 3
3 3
a a
a a a a a a
a a
a a a aa a
a a
x x
f m h lR
n p
f m
(3-8)
3 3
2 2 3 2 3 2
3 2
3 2 3 23 3
2 2
a a
b b a b a b
a b
a b a ba a
b b
x x
f m h lR
n p
f m
(3-9)
The relationships between displacements/rotations and forces/couples can be
written using the matrix format as shown in Equations 3-10 to 3-17, where ui and qi are
the generalized displacement/rotation and the force/couple vectors, respectively.
34
1 11 11 1
1 11 11 1
x h l f
n p mor 1 11 1u R q (3-10)
1 12 2a au R q (3-11)
2 2 2 2a a a au R q (3-12)
2 2 1 1a au R q (3-13)
2 2 2 2b b b bu R q (3-14)
2 2 3 3b b a au R q (3-15)
3 3 3 3a a a au R q (3-16)
3 3 2 2a a b bu R q (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 receptances are
determined using Equations 3-20 to 3-51. In order to calculate the subassembly
receptances, G11 (direct) and G3a1 (cross) (Equations 3-18 and 3-19, respectively), a
generalized force Q1 (representing both the externally applied force and couple) is
applied at coordinate location 1 (see Figure 3-3).
1 1
1 1 11 11
11
11 111 1
1 1
X X
F M H LG
N P
F M
(3-18)
3 3
1 1 3 1 3 1
3 1
3 1 3 13 3
1 1
a a
a a
a
a aa a
X X
F M H LG
N P
F M
(3-19)
35
The displacement equations for the substructures can be described as follows:
(3-20)
(3-21)
(3-22)
(3-23)
If rigid 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 2a and 2b is given by 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 u2b and u2a in Equation 3-24 gives Equation 3-27.
(3-27)
Equation 3-28 is obtained using Equations 3-25 and 3-26.
(3-28)
Solving for q2b gives Equation 3-29. Given that 2 2a bq q 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
1 11 1 12 2a au R q R q
2 2 2 2 2 1 1a a a a au R q R q
2 2 2 2b b b bu R q
3 3 2 2a a b bu R q
2 2 0b au u
2 2 0b aq q
1 1q Q
2 2 2 2 2 2 2 2 2 1 1 0b a b b b a a a au u R q R q R q
2 2 2 2 2 2 1 1( ) 0b b a a b aR R q R Q
36
expressed as a function of the component receptances. Therefore, given 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 3a and 1 are given by Equations
3-33 and 3-34.
1
3 3 3 2 2 3 2 2 2 2 2 2 1 13 1
1 1 1 1
( )a a a b b a b b b a a aa
U u R q R R R R QG
Q Q Q Q (3-33)
3 1 3 11
3 1 3 2 2 2 2 2 2 1
3 1 3 1
( )a a
a a b b b a a a
a a
H LG R R R R
N P (3-34)
To determine the other two receptances of the sub-assembly I-II, G3a3a and G13a, a
generalized force Q3a is applied to U3a as shown in Figure 3-4.
(3-35) (3-36)
1
2 2 2 2 2 2 1 1( )b b b a a aq R R R Q
11 1 12 21 1
11
1 1 1
a aR q R qU uG
Q Q Q
1
11 1 12 2 2 2 2 2 1 111
1
( )a b b a a aR Q R R R R QG
Q
11 111
11 11 12 2 2 2 2 2 1
11 11
( )a b b a a a
H LG R R R R R
N P
3 3
3 3 3 3 3
3 3
3 3 33 3
3 3
a a
a a a a a
a a
a a aa a
a a
X X
F M H LG
N P
F M
1 1
3 3 13 13
13
13 131 1
3 3
a a a a
a
a a
a a
X X
F M H LG
N P
F M
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 u2a and u2b in Equation 3-41 gives Equation 3-44.
(3-44) Using Equations 3-42 and 3-43 and substituting for q2b and q3a in Equation 3-44 gives
Equation 3-45.
(3-45) Solving for q2a, Equation 3-46 is obtained. Equation 3-47 gives the desired expression
for the subassembly direct receptances.
(3-46) (3-47) By substituting for q2b 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)
1 12 2a au R q
2 2 2 2a a a au R q
2 2 2 2 2 3 3b b b b b a au R q R q
3 3 3 3 3 2 2a a a a a b bu R q R q
2 2 0a bu u
2 2 0b aq q
3 3a aq Q
2 2 2 2 2 2 2 2 2 3 3 0a b a a a b b b b a au u R q R q R q
2 2 2 2 2 2 3 3( ) 0b b a a a b a aR R q R Q
1
2 2 2 2 2 2 3 3( )a b b a a b a aq R R R Q
3 3 3 3 3 3 2 2
3 3
3 3 3
a a a a a a b ba a
a a a
U u R q R qG
Q Q Q
1
3 3 3 3 2 2 2 2 2 2 3 33 3
3
( )a a a a b b b a a b a aa a
a
R Q R R R R QG
Q
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 spindle-machine
(component III) to give the assembly tool point receptances, G11; see Figure 3-5. This
coupling is carried out using Equation 3-52:
(3-52)
where the Rij matrices are the subassembly matrices. Therefore, 11 11R G from the
Equation 3-32 I-II coupling results, 3 1 3 1a aR G from Equation 3-34, 3 3 3 3a a a aR G from
Equation 3-49, and 13 13a aR G from Equation 3-50. The remaining unknown in Equation
3-52 is the spindle-machine receptance matrix, R3b3b. Identification of this receptance
matrix is discussed in Chapter 4.
3 3 3 31
3 3 3 3 3 2 2 2 2 2 2 3
3 3 3 3
( )a a a a
a a a a a b b b a a b a
a a a a
H LG R R R R R
N P
1
12 2 12 2 2 2 2 2 3 31 113
3 3 3 3
( )a a a b b a a b a aa
a a a a
R q R R R R QU uG
Q Q Q Q
13 131
13 12 2 2 2 2 2 3
13 13
( )a a
a a b b a a b a
a a
H LG R R R R
N P
1
11 11 13 3 3 3 3 3 1a a a b b aG R R R R R
39
Figure 3-1. Three-component receptance coupling model for the tool (I), holder (II), and
spindle-machine (III).
40
Figure 3-2. Individual components I-II with displacements and rotations at specified
coordinate locations.
41
Figure 3-3. Subassembly I-II composed of tool (I) and holder (II). The generalized force
Q1 is applied to U1 to determine G11 and G3a1.
42
Figure 3-4. Subassembly I-II composed of tool (I) and holder (II). The generalized force
Q3a is applied to U3a to determine G3a3a and G13a.
43
Figure 3-5. The I-II subassembly is rigidly coupled to the spindle-machine (III) to
determine the tool point receptance matrix, G11.
44
CHAPTER 4 IDENTIFICATION OF SPINDLE-MACHINE RECEPTANCES
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
receptances, 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 G22 is
determined experimentally. Using G22 and a model of the portion of the holder beyond
the flange, the spindle machine receptance R3b3b is calculated. The free end response
for the artifact-spindle-machine assembly is described by Equation 4-1, where the R22,
R23a, R3a3a, and R3a2 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 R3b3b. This step of decomposing
the measured assembly receptances, G22, into the modeled substructure receptances,
R3a2, R22, R23a, and R3a3a, and spindle-machine receptances, R3b3b, is referred to as
“inverse RCSA”. Three approaches for experimentally determining the four spindle-
machine receptances are discussed in the following sections.
(4-2)
1 22 22
22 22 23 3 3 3 3 3 2
22 22
a a a b b a
H LG R R R R R
N P
1
22 22 23 3 3 3 3 3 2
11 1
23 22 22 3 2 3 3 3 3
1
3 2 22 22 23 3 3 3 3
1
3 3 3 2 22 22 23 3 3
a a a b b a
a a a a b b
a a a a b b
b b a a a a
G R R R R R
R R G R R R
R R G R R R
R R R G R R
45
Synthesis Approach
The direct displacement-to-force term, 222
2
XH
F, in the G22 matrix is measured by
impact testing (the TXF software from MLI was used for data acquisition and signal
analysis in this study). In this method, 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 H22 is measured by applying the force and placing the
accelerometer at the same coordinate location (U2 in Figure 4-1). The second
component of the G22 matrix, the rotation-to-force receptance,
222
2
NF
, is calculated by
a first-order backward finite difference approach [8,46] as described in Equation 4-3,
where the cross FRF H2a2 is measured by exciting the assembly at U2 and measuring
the response at coordinate U2a, located a distance S from the artifact‟s free end, as
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 force
at 1), the off-diagonal terms of the G22 matrix may be taken to be equal. See Equation
4-4.
L22 = N22 (4-4)
222 2
22 2 22 2 222
2 2
aa
a
xxx xH Hf fSN
f f S S
46
The measured H22 and derived N22 receptances are used to synthesize P22 as shown in
Equation 4-5.
(4-5)
The four receptances required to populate G22 are now known and Equation 4-2
can be used to obtain R3b3b. Given R3b3b, free-free models for arbitrary tool-holder
combinations can be developed and coupled to the spindle-machine receptances to
predict the tool point FRF, H11, required for milling process simulation. The synthesis
approach thus requires two artifact measurements to determine the G22 matrix. In this
approach, modal fitting of H22 and H2a2 receptances via a peak picking technique [3] can
be applied to reduce the effects of measurement 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 additional direct
displacement-to-force receptance, H2a2a, at the distance, S, from the free end of the
artifact. With the two direct, H22 and H2a2a, and one cross displacement-to-force, H2a2,
receptances, a second-order backward finite difference approach is implemented to
identify the rotation-to-moment receptance [10] (Equation 4-6).
(4-6)
Modal 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 prediction
results. The fitting strategy is applied in this work.
2
2 2 2 2222 22 22
2 2 2 22 22
1x f NP L N
m f x H H
22 22
22
2 22
2a a a
H H H
SP
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 direct measurement is performed at the free end of the standard artifact in the
new method established in this research. In the new technique, it 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 Equation 4-7 for the displacement-to-force receptance at
the free end of a cylindrical fixed-free beam, where
4 2
1
A
EI i,
2
4
dA ,
4
64
dI , ω is frequency (rad/s), ρ is the density, E is the elastic modulus, η is the solid
damping factor (unitless), d is the “fit” beam diameter, and L is the beam length.
(4-7) The algorithm for fitting each mode is composed of five steps:
1. Determine the natural frequency, fn (Hz), for the mode to be fit from the measured H22 receptance.
2. Select a beam diameter (this is a fitting parameter) and specify the modulus and density (steel values, E = 200 GPa and ρ = 7800 kg/m3, 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 Equation 4-8 [47].
4. Adjust η to obtain the proper slope for the real part of the mode in question.
5. If the subsequent mode magnitude is too large, increase d to dnew and calculate Lnew using Equation 4-9 (to maintain the same natural frequency). If the mode magnitude is too small, decrease d and calculate Lnew.
22 3
sin cosh cos sinh
1 1 cos cosh
L L L LH
EI i L L
48
(4-8)
(4-9)
Once the fit parameters d, L, and η are determined (as well as the selected E and
ρ values), the remaining receptances for the free end of the artifact are calculated as
shown in Equations 4-10 and 4-11. No additional measurements are required.
(4-10) (4-11) The four G22 receptances are then known and Equation 4-2 can be used to obtain R3b3b,
the spindle-machine receptances.
11 2
2 21.87510407
2 16n
d EL
f
newnew
dL L
d
22 22 2
sin sinh
1 1 cos cosh
L LL N
EI i L L
22
cos sinh sin cosh
1 1 cos cosh
L L L LP
EI i L L
49
Figure 4-1. Artifact model for determining R3b3b by inverse RCSA.
50
CHAPTER 5 RESULTS
Spindle-Machine Receptances Comparison
The G22 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 Starragheckert
ZT-1000 Super Constellation, and a Cincinnati 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 G22 identification methods
described in Chapter 4 were completed. The H22 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 L22/N22 receptances from the three methods are displayed in Figure 5-3. Good
agreement in both magnitude and frequency is observed. The P22 receptances obtained
using Equations 4-5 (synthesis), 4-6 (finite difference), and 4-11 (E-B) are displayed in
Figure 5-4. Again, the agreement 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 imaginary part exhibits unexpected positive values near the
anti-resonant frequencies. This is presumably due to the division by the complex-valued
receptance, H22, in Equation 4-5.
Given the G22 receptances (from the three techniques), the corresponding spindle-
machine receptance matrices, R3b3b, were calculated using Eq. 4-2 and a free-free
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
material properties (E = 200 GPa, ρ = 7800 kg/m3, and Poisson‟s ratio = 0.29) to
develop the artifact model.
The H22 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 Figure 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
L22/N22 receptances for both the artifacts are displayed in Figures 5-8 (short) and 5-11
(long) and the P22 receptances in Figures 5-9 (short) and 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 H22 FRF and the E-B fit, as well as a
comparison of the L22/N22 and P22 receptances for the three approaches using both the
short 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 Comparison
The archived spindle-machine receptance matrices, R3b3b, for the three milling
machines were rigidly coupled to Timoshenko beam models of various tool-holder
combinations to predict the corresponding tool point receptances, H11. In these tests,
carbide endmills of different diameters and overhang lengths were clamped in various
52
shrink fit holders. Comparisons between measurements 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, H11, was measured by impact testing and compared to predictions
using the synthesis, finite difference, and E-B R3b3b 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 using an
equivalent diameter, where this diameter was obtained by weighing the carbide tool,
assuming a density (15000 kg/m3), and calculating the solid section equivalent flute
diameter based on the cylindrical dimensions and the tool and flute lengths. The elastic
modulus for the Timoshenko beam model was 550 GPa and Poisson‟s ratio was 0.22.
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 and
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 imag 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)
( ) ( )measured predictedimag H imag HCM
n
53
Tables 5-7 and 5-8 list the CM values for the two overhang 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 endmill 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 compare the predictions
using the synthesis, finite difference, and E-B method R3b3b 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 are 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 methods 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 spindle. 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 Figures 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
54
5-11 to 5-14 list the comparison metric showing that the E-B predictions provide the
closest agreement to measurement, especially for the long artifact predictions. The
imaginary parts of the synthesis approach prediction in the long artifact predictions
(Figures 5-28 and 5-29) show positive values near 3200 Hz. This may be due to the
positive values of the synthesized P22 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 artifacts) 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 5-15 to
5-18.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 measurements were completed on a 20 mm diameter, two flute
endmill clamped 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 were again made using the two standard artifact spindle
receptances. Tool point measurements and predictions for the two overhang lengths are
compared in Figures 5-36 to 5-39 and Tables 5-19 to 5-22 list the CM values and
percent differences. The short artifact predictions for the E-B method slightly outperform
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 5-23 to 5-26 list the CM values, as well as the percent difference with
respect to the smallest CM value. From the figures 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 spindle receptances obtained by
the two standard artifacts. The CM values and percent differences are listed in Tables
5-27 to 5-30. The long artifact predictions using synthesis and finite difference approach
are less accurate than the E-B method predictions.
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 5-31 to 5-34. The accuracy of E-B method is
better than the synthesis and the finite difference 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 point 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 listed in Tables 5-35 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 differences are listed in Tables 5-39
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
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 Schunk 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
complication of the tool‟s fluted portion would not be included in the stiffness
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 kxf , kθf , kxm, and kθm are the displacement-to-force,
rotation-to-force, displacement-to-moment, and rotation-to-moment stiffness values,
respectively and cxf , cθf , cxm, and cθm are the corresponding damping values if viscous
damping is considered at the coupling location (kθf = kxm and cθf = cxm were assumed
due to reciprocity). The derivation of Equation 5-2 is provided in the Appendix A.
(5-2) (5-3)
1
11 11 12 2 2 2 2 2 1
1a a a b b aG R R R R R
k
xf xf f f
xm xm m m
k i c k i ck
k i c k i c
58
To identify the stiffness matrix, tool point FRFs were measured for multiple
overhang lengths of the blank on each holder. An optimization procedure based on non-
linear least squares was implemented to find the connection stiffness. The variables
were 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 connection 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. The
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 5-43
lists the stiffness and damping values obtained for three different overhang lengths of
the 12 mm diameter blank inserted into the corresponding shrink fit holder. Figures 5-66
to 5-68 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 lengths 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 5-44 lists the average stiffness
values of the 12 mm, 16 mm, 20 mm, and 25 mm blanks clamped in the shrink fit
min m pH H
59
holder. Similarly, Table 5-45 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 5-46 to 5-48.
Comparison of the stiffness values 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 connection 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 large 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
thermosetting plastic that absorbs shock and reduces vibrations during machining. In
the Timoshenko beam model of the Tribos holder, the section consisting of the
thermosetting plastic chambers and steel was modeled using equivalent values of the
elastic modulus (Eeq), density (ρeq), and Poisson‟s ratio (eq) as shown in Equations 5-5,
5-6, and 5-7, respectively, where Asteel is the cross-sectional area of the steel portion,
Aplastic is the cross-sectional area of the thermosetting plastic chambers, and Atotal is the
total cross-sectional area of the Tribos holder. The values of the elastic modulus,
60
density, and Poisson‟s ratio for steel were the same as described in the previous
sections. The values of the elastic modulus, density and Poisson‟s ratio for the
thermosetting plastic were taken to be 7 GPa, 3700 kg/m3, and 0.5, respectively.
plasticsteel
eq steel plastic
total total
AAE E E
A A (5-5)
plasticsteel
eq steel plastic
total total
AA
A A (5-6)
plasticsteel
eq steel plastic
total total
AA
A A (5-7)
The collet tool-holder connection was the most flexible. This was anticipated due
to the clamping mechanism; a flexible “collet basket” is elastically deformed inside a
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 spindle. 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, and 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 5-46 to 5-48 improves the tool
point FRF prediction as compared to the prediction obtained by the rigid connection
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 that 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.
62
Table 5-1. Specifications of milling machines tested
Manufacturer and model
Geometry Spindle-holder
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 Super 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 fn (Hz) d (m) L (m)
1 550 0.375 0.100 0.6950 2 610 0.520 0.060 0.7771 3 703 0.330 0.060 0.5767 4 795 0.450 0.050 0.6160 5 840 0.565 0.035 0.6903 6 875 0.260 0.050 0.4588 7 975 0.107 0.070 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 0.110 0.2845 15 1872 0.115 0.060 0.2086 16 2040 0.190 0.150 0.2569 17 2620 0.220 0.130 0.2439 18 2985 0.098 0.060 0.1525 19 3060 0.125 0.070 0.1701 20 3205 0.185 0.070 0.2022 21 3800 0.270 0.060 0.2244 22 3975 0.340 0.040 0.2462 23 4150 0.220 0.050 0.1938 24 4310 0.112 0.050 0.1357
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 0.112 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 Super Constellation
Mode fn (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 0.2266 9 3230 0.092 0.045 0.1420
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 14 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
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 - Synthesis 35.21 x 10-9 0.10
Finite Difference 35.83 x 10-9 -1.86
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 5-11. 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 difference 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
66
Table 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
CM (m/N) Percent difference with respect to smallest CM
E-B 146.06 x 10-9 - Synthesis 159.35 x 10-9 -9.10
Finite Difference 163.62 x 10-9 -12.0
Table 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
CM (m/N) Percent difference with respect to smallest CM
E-B 103.15 x 10-9 - Synthesis 152.37 x 10-9 -47.71
Finite Difference 215.11 x 10-9 -108.5
Table 5-14. 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 -66.84
Finite Difference 144.23 x 10-9 -11.92
Table 5-15. 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 -12.69
Finite Difference 47.68 x 10-9 -15.53
Table 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
CM (m/N) Percent difference with respect to 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
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 receptances
CM (m/N) Percent difference with respect to smallest CM
E-B 23.35 x 10-9 - Synthesis 92.36 x 10-9 -295.58
Finite Difference 40.15 x 10-9 -71.95
Table 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
CM (m/N) Percent difference with respect to smallest CM
E-B 31.24 x 10-9 - Synthesis 106.3 x 10-9 -240.31
Finite Difference 62.54 x 10-9 -100.19
Table 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.
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 -48.06
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 5-21. 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 -37.02
Finite Difference 34.42 x 10-9 -20.51
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 -64.15
Table 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.
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 5-24. 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 26.97 x 10-9 -16.79 Synthesis 23.29 x 10-9 -0.84
Finite Difference 23.09 x 10-9 -
Table 5-25. 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 -4.11
Finite Difference 12.17 x 10-9 -
Table 5-26. 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 Synthesis 17.23 x 10-9 -3.81
Finite Difference 16.60 x 10-9 -
69
Table 5-27. 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 -2.76
Finite Difference 120.39 x 10-9 -3.85
Table 5-28. Comparison metric (m/N) for the FRF predictions of 12 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 173.22 x 10-9 - Synthesis 174.46 x 10-9 -0.72
Finite Difference 174.89 x 10-9 -0.96
Table 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.
CM (m/N) Percent difference with respect to smallest CM
E-B 96.35 x 10-9 - Synthesis 231.3 x 10-9 -140.10
Finite Difference 589.4 x 10-9 -511.69
Table 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.
CM (m/N) Percent difference with respect to smallest CM
E-B 142.67 x 10-9 - Synthesis 301.05 x 10-9 -111.01
Finite Difference 208.67 x 10-9 -46.25
Table 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.
CM (m/N) Percent difference with respect to smallest CM
E-B 53.13 x 10-9 - Synthesis 60.23 x 10-9 -13.35
Finite Difference 58.70 x 10-9 -10.48
70
Table 5-32. 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 -405.08
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 using 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 -191.04
Finite Difference 91.43 x 10-9 -26.69
Table 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.
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 -18.61
Table 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.
CM (m/N) Percent difference with respect to smallest CM
E-B 53.69 x 10-9 - Synthesis 60.43 x 10-9 -12.56
Finite Difference 60.33 x 10-9 -12.36
71
Table 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.
CM (m/N) Percent difference with respect to smallest CM
E-B 22.37 x 10-9 - Synthesis 24.16 x 10-9 -7.98
Finite Difference 23.98 x 10-9 -7.17
Table 5-38. 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 difference with respect to smallest CM
E-B 49.53 x 10-9 Synthesis 70.82 x 10-9 -42.98
Finite Difference 56.69 x 10-9 -14.46
Table 5-39. 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 -13.92 Synthesis 17.38 x 10-9 -4.78
Finite Difference 16.59 x 10-9 -
Table 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 5-41. Comparison metric (m/N) for 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 -16.85 Synthesis 11.27 x 10-9 -0.58
Finite Difference 11.21 x 10-9 -
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 -15.06 Synthesis 12.19 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) kθf (N/rad) cθf (N-s/rad)
66 3.9 x 106 26 71 4.8 x 106 50 76 4.6 x 106 75
Table 5-44. Average stiffness matrix values for blank-shrink fit holders inserted in
Cincinnati FTV-5 2500
Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)
12 4.4 x 106 50 16 1.7 x 107 153 20 1.9 x 107 665 25 Rigid
Table 5-45. Average stiffness matrix values for blank-collet holders inserted in
Cincinnati FTV-5 2500
Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)
12 3.2 x 106 18 16 5.6 x 106 51 20 1.5 x 107 159
Table 5-46. Average stiffness matrix values for blank-shrink fit holders inserted in
Mikron UCP-600 Vario
Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)
12.7 5.4 x 106 30 25 Rigid
Table 5-47. Average stiffness matrix values for blank-collet holders inserted in Mikron
UCP-600 Vario
Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)
6.33 2.2 x 105 2 9.5 8.3 x 105 13 12 2.9 x 106 18 19 1.0 x 107 31 25 2.2 x 107 0
73
Table 5-48. Average stiffness matrix values for blank-Tribos holders inserted in Mikron UCP-600 Vario
Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)
10 2.1 x 106 0 12 2.9 x 106 2 16 9.0 x 106 0 19 1.9 x 107 0
74
Figure 5-1. Artifact dimensions for Mikron UCP-600 Vario measurements.
1000 1500 2000 2500 3000 3500 4000 4500 5000
-4
-2
0
2
4
6
8x 10
-8
Rea
l (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500 5000-10
-8
-6
-4
-2
0
2
x 10-8
Frequency (Hz)
Imag
(m/N
)
Measured
E-B fit
Figure 5-2. H22 artifact measurement and E-B fit for Mikron UCP-600 Vario CNC milling
machine.
75
1000 1500 2000 2500 3000 3500 4000 4500
-2
0
2
4
x 10-7
Re
al (r
ad
/N)
1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
1x 10
-7
Frequency
Ima
g (
rad
/N)
E-B
Synthesis
Finite Difference
Figure 5-3. L22/N22 results for the Mikron UCP-600 Vario CNC milling machine.
76
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4x 10
-6
Re
al (r
ad
/N-m
)
1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
x 10-6
Frequency
Ima
g (
rad
/N-m
)
E-B
Synthesis
Finite Difference
Figure 5-4. P22 results for the Mikron UCP-600 Vario CNC milling machine.
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.
78
500 1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
x 10-8
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
x 10-8
Frequency (Hz)
Imag
(m/N
)Measured
EB fit
Figure 5-7. H22 short artifact measurement and E-B fit for Starragheckert ZT-1000
Super Constellation milling machine.
79
500 1000 1500 2000 2500 3000 3500 4000 4500-5
0
5x 10
-7
Rea
l (ra
d/N
)
500 1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Imag
(rad
/N)
E-B
Synthesis
Finite Difference
Figure 5-8. L22/N22 results for the short artifact measurement on ZT-1000 Super
Constellation Starragheckert milling machine.
80
500 1000 1500 2000 2500 3000 3500 4000 4500-5
0
5x 10
-6
Rea
l (ra
d/N
-m)
500 1000 1500 2000 2500 3000 3500 4000 4500-6
-4
-2
0
2
x 10-6
Frequency (Hz)
Imag
(rad
/N-m
)
E-B
Synthesis
Finite Difference
Figure 5-9. P22 results for the short artifact measurement on ZT-1000 Super
Constellation Starragheckert milling machine.
81
500 1000 1500 2000 2500 3000 3500 4000 4500
-1
0
1
2
x 10-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500
-3
-2.5
-2
-1.5
-1
-0.5
0
x 10-7
Frequency (Hz)
Imag
(m/N
)Measured
E-B
Figure 5-10. H22 long artifact measurement and E-B fit for Starragheckert ZT-1000
Super Constellation milling machine.
82
500 1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
10
x 10-7
Rea
l (ra
d/N
)
500 1000 1500 2000 2500 3000 3500 4000 4500
-15
-10
-5
0
x 10-7
Frequency (Hz)
Imag
(rad
/N)
E-B
Synthesis
Finite Difference
Figure 5-11. L22/N22 results for the long artifact measurement on ZT-1000 Super
Constellation Starragheckert milling machine.
83
500 1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6
x 10-6
Rea
l (ra
d/N
-m)
500 1000 1500 2000 2500 3000 3500 4000 4500
-8
-6
-4
-2
0
2
x 10-6
Frequency (Hz)
Imag
(rad
/N-m
)
E-B
Synthesis
Finite Difference
Figure 5-12. P22 results for the long artifact measurement on ZT-1000 Super
Constellation Starragheckert milling machine.
84
1000 1500 2000 2500 3000 3500 4000 4500-1
-0.5
0
0.5
1x 10
-7
Rea
l (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500-15
-10
-5
0
x 10-8
Frequency (Hz)
Imag
(m/N
)
Measured
E-B fit
Figure 5-13. H22 short artifact measurement and E-B fit for Cincinnati FTV-5 2500
milling machine.
85
1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6x 10
-7
Rea
l (ra
d/N
)
1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Imag
(rad
/N)
E-B
Synthesis
Finite Difference
Figure 5-14. L22/N22 results for the short artifact measurement on Cincinnati FTV-5
2500 milling machine.
86
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
x 10-6
Rea
l (ra
d/N
-m)
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
2x 10
-6
Frequency (Hz)
Imag
(rad
/N-m
)
E-B
Synthesis
Finite Difference
Figure 5-15. P22 results for the short artifact measurement on Cincinnati FTV-5 2500
milling machine.
87
1000 1500 2000 2500 3000 3500 4000 4500-1.5
-1
-0.5
0
0.5
1
1.5
2x 10
-7
Rea
l (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500-3
-2.5
-2
-1.5
-1
-0.5
0
x 10-7
Frequency (Hz)
Imag
(m/N
)Measured
E-B
Figure 5-16. H22 long artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling
machine.
88
1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
x 10-7
Rea
l (ra
d/N
)
1000 1500 2000 2500 3000 3500 4000 4500-15
-10
-5
0
x 10-7
Frequency (Hz)
Imag
(rad
/N)
E-B
Synthesis
Finite Difference
Figure 5-17. L22/N22 results for the long artifact measurement on Cincinnati FTV-5 2500
milling machine.
89
1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6
x 10-6
Rea
l (ra
d/N
-m)
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
2
x 10-6
Frequency (Hz)
Imag
(rad
/N-m
)E-B
Synthesis
Finite Difference
Figure 5-18. P22 results for the long artifact measurement on Cincinnati FTV-5 2500
milling machine.
90
Figure 5-19. Beam model for 25.4 mm diameter, three flute endmill inserted in a
tapered shrink fit holder (not to scale).
91
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-4
-2
0
2
4
6
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-20. Comparison between H11 tool point measurement, Euler-Bernoulli,
synthesis and finite difference prediction for three flute, 25.4 mm diameter endmill with an overhang length of 99 mm.
92
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-5
0
5
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-21. Comparison between H11 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.
93
Figure 5-22. Beam model for 19.05 mm diameter, four flute endmill inserted in a
tapered shrink fit holder (not to scale).
94
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-4
-2
0
2
4
6
8x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-12
-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-23. Comparison between H11 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.
95
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-4
-2
0
2
4
6
8
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-24. Comparison between H11 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.
96
Figure 5-25. Beam model for 12 mm diameter, four flute endmill inserted in a tapered
shrink fit holder (not to scale).
97
500 1000 1500 2000 2500 3000 3500 4000 4500-1
-0.5
0
0.5
1x 10
-6
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
5x 10
-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-26. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 45 mm (short artifact spindle receptances).
98
500 1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
10
x 10-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-27. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 55 mm (short artifact spindle receptances).
99
500 1000 1500 2000 2500 3000 3500 4000 4500
-2
-1
0
1
2x 10
-6
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-4
-3
-2
-1
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Figure 5-28. Comparison between H11 tool point measurement, 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 receptances).
100
500 1000 1500 2000 2500 3000 3500 4000 4500-3
-2
-1
0
1
2
3x 10
-6
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500
-5
-4
-3
-2
-1
0
1x 10
-6
Frequency (Hz)
Imag
(m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-29. Comparison between H11 tool point measurement, 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).
101
Figure 5-30. Beam model for 16 mm diameter, four flute endmill inserted in a tapered
shrink fit holder (not to scale).
102
500 1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
6x 10
-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-31. Comparison between H11 tool point measurement, 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).
103
500 1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6
8
x 10-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500-15
-10
-5
0
x 10-7
Frequency (Hz)
Imag
(m/N
)Measured
EB Predicted
Synth fit
FD fit
Figure 5-32. Comparison between H11 tool point measurement, 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).
104
500 1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6
x 10-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500-10
-5
0
5
x 10-7
Frequency (Hz)
Imag
(m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-33. Comparison between H11 tool point measurement, 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).
105
500 1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
10
x 10-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-15
-10
-5
0
5
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-34. Comparison between H11 tool point measurement, 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).
106
Figure 5-35. Beam model for 20 mm diameter, two flute endmill inserted in a tapered
shrink fit holder (not to scale).
107
500 1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Imag
(m/N
)
500 1000 1500 2000 2500 3000 3500 4000 4500
-2
0
2
4x 10
-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
Figure 5-36. Comparison between H11 tool point measurement, 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).
108
500 1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
2
4
6
x 10-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-12
-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-37. Comparison between H11 tool point measurement, 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).
109
500 1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
2
4
x 10-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
2
4
6x 10
-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-38. Comparison between H11 tool point measurement, 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).
110
500 1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6
8x 10
-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
2
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-39. Comparison between H11 tool point measurement, 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).
111
Figure 5-40. Beam model for 25 mm diameter, four flute endmill inserted in a tapered
shrink fit holder (not to scale).
112
500 1000 1500 2000 2500 3000 3500 4000 4500
-2
0
2
4
x 10-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-41. Comparison between H11 tool point measurement, 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).
113
500 1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
6x 10
-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-42. Comparison between H11 tool point measurement, 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).
114
500 1000 1500 2000 2500 3000 3500 4000 4500-3
-2
-1
0
1
2
3
4x 10
-7
Rea
l (m
/N)
500 1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Imag
(m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-43. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 25 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances).
115
500 1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
6x 10
-7
Rea
l (m
/N)
Measured
E-B
Synthesis
Finite Difference
500 1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Imag
(m/N
)
Figure 5-44. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 25 mm diameter endmill with an overhang length of 85 mm (long artifact spindle receptances).
116
1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
10x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-45. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 45 mm (short artifact spindle receptances).
117
1000 1500 2000 2500 3000 3500 4000 4500-1
-0.5
0
0.5
1
1.5x 10
-6
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-46. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 55 mm (short artifact spindle receptances).
118
1000 1500 2000 2500 3000 3500 4000 4500-1
-0.5
0
0.5
1x 10
-6
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-47. Comparison between H11 tool point measurement, 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 receptances).
119
1000 1500 2000 2500 3000 3500 4000 4500-2
-1
0
1
2
3x 10
-6
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-4
-3
-2
-1
0
1
2x 10
-6
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-48. Comparison between H11 tool point measurement, 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).
120
1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
2
4
6
8x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-12
-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-49. Comparison between H11 tool point measurement, 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).
121
1000 1500 2000 2500 3000 3500 4000 4500
-1
-0.5
0
0.5
1
x 10-6
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-50. Comparison between H11 tool point measurement, 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).
122
1000 1500 2000 2500 3000 3500 4000 4500
-5
0
5
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-20
-15
-10
-5
0
5x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-51. Comparison between H11 tool point measurement, 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).
123
1000 1500 2000 2500 3000 3500 4000 4500
-1
-0.5
0
0.5
1
x 10-6
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-2.5
-2
-1.5
-1
-0.5
0
x 10-6
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-52. Comparison between H11 tool point measurement, 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).
124
Figure 5-53. Tool point FRF measurement of 20 mm carbide end mill on Cincinnati
FTV-5 2500.
125
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-6
-5
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-54. Comparison between H11 tool point measurement, 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).
126
1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-55. Comparison between H11 tool point measurement, 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).
127
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
2x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-56. Comparison between H11 tool point measurement, 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).
128
1000 1500 2000 2500 3000 3500 4000 4500-10
-8
-6
-4
-2
0
2x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
1000 1500 2000 2500 3000 3500 4000 4500
-4
-2
0
2
4
6x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
Figure 5-57. Comparison between H11 tool point measurement, 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).
129
Figure 5-58. Tool point FRF measurement of 25 mm carbide end mill on Cincinnati
FTV-5 2500.
130
1000 1500 2000 2500 3000 3500 4000 4500-2
-1
0
1
2
3x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-59. Comparison between H11 tool point measurement, 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).
131
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
x 10-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-60. Comparison between H11 tool point measurement, 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).
132
1000 1500 2000 2500 3000 3500 4000 4500-2
-1
0
1
2
3x 10
-7
Re
al (m
/N)
Measured
E-B
Synthesis
Finite Difference
1000 1500 2000 2500 3000 3500 4000 4500-5
-4
-3
-2
-1
0
1x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-61. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 25 mm diameter endmill with an overhang length of 75 mm (long artifact spindle receptances).
133
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4
x 10-7
Re
al (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Measured
E-B
Synthesis
Finite Difference
Figure 5-62. Comparison between H11 tool point measurement, E-B, Synthesis and
Finite difference approach prediction for four flute, 25 mm diameter endmill with an overhang length of 85 mm (long artifact spindle receptances).
134
Figure 5-63. Component coordinates for flexible coupling of holder and blank
Figure 5-64. Various shrink fit holders with blanks for Cincinnati FTV-5 2500 spindle
135
Figure 5-65. Collet holder for Cincinnati FTV-5 2500 spindle
136
1000 1500 2000 2500 3000 3500 4000 4500-6
-4
-2
0
2
4
6x 10
-6
Rea
l (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500-12
-10
-8
-6
-4
-2
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Measured
Predicted
Figure 5-66. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (rigid connection)
137
1000 1500 2000 2500 3000 3500 4000 4500-4
-2
0
2
4x 10
-6
Rea
l (m
/N)
1000 1500 2000 2500 3000 3500 4000 4500
-6
-4
-2
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Measured
Predicted
Figure 5-67. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 71 mm (rigid connection)
138
1000 1500 2000 2500 3000 3500 4000 4500
-2
-1
0
1
2
3x 10
-6
Rea
l (m
/N)
Measured
Predicted
1000 1500 2000 2500 3000 3500 4000 4500-5
-4
-3
-2
-1
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Figure 5-68. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (rigid connection)
139
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
2
4
x 10-6
Rea
l (m
/N)
Measured
Predicted
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Figure 5-69. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (flexible connection)
140
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
2
4
x 10-6
Rea
l (m
/N)
Measured
Predicted
1000 1500 2000 2500 3000 3500 4000 4500-8
-6
-4
-2
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Figure 5-70. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 71 mm (flexible connection)
141
1000 1500 2000 2500 3000 3500 4000 4500-3
-2
-1
0
1
2
3x 10
-6
Rea
l (m
/N)
Measured
Predicted
1000 1500 2000 2500 3000 3500 4000 4500-5
-4
-3
-2
-1
0
x 10-6
Frequency (Hz)
Imag
(m/N
)
Figure 5-71. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 66 mm (flexible connection)
142
Figure 5-72. Collet holder for Mikron UCP-600 Vario
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
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-flute endmill inserted in a collet holder (not to scale)
145
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6
-4
-2
0
2
4
6x 10
-5
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15
-10
-5
0
x 10-5
Frequency (Hz)
Ima
g (
m/N
)
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)
146
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6
-4
-2
0
2
4
6x 10
-5
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-12
-10
-8
-6
-4
-2
0
x 10-5
Frequency (Hz)
Ima
g (
m/N
)
Figure 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)
147
Figure 5-79. Beam model for 19 mm diameter, 4-flute endmill inserted in a collet holder (not to scale)
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1
-0.5
0
0.5
1x 10
-6
Re
al (m
/N)
Measured
Predicted
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 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)
148
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1
-0.5
0
0.5
1x 10
-6
Re
al (m
/N)
Measured
Predicted
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
-15
-10
-5
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
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)
149
Figure 5-82. Beam model for 12.7 mm diameter, 2-flute endmill inserted in a shrink fit holder (not to scale)
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3
-2
-1
0
1
2
3x 10
-6
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5
-4
-3
-2
-1
0
x 10-6
Frequency (Hz)
Ima
g (
m/N
)
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)
150
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-2
-1
0
1
2x 10
-6
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3
-2.5
-2
-1.5
-1
-0.5
0
x 10-6
Frequency (Hz)
Ima
g (
m/N
)
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-flute endmill inserted in a Tribos holder (not to scale)
151
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3
-2
-1
0
1
2
3x 10
-7
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5
-4
-3
-2
-1
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
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)
152
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3
-2
-1
0
1
2
3
x 10-7
Re
al (m
/N)
Measured
Predicted
500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5
-4
-3
-2
-1
0
x 10-7
Frequency (Hz)
Ima
g (
m/N
)
Figure 5-87. Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in Tribos, overhang length 72 mm (flexible connection)
153
Figure 5-88. Beam model for 25.4 mm diameter, 4-flute endmill inserted in a shrink fit holder (not to scale)
500 1000 1500 2000 2500 3000 3500 4000 4500-1
-0.5
0
0.5
1
1.5
2x 10
-7
Re
al (m
/N)
Measured
E-B
500 1000 1500 2000 2500 3000 3500 4000 4500
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5x 10
-7
Frequency (Hz)
Ima
g (
m/N
)
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)
154
CHAPTER 6 CONCLUSION AND FUTURE WORK
Conclusion
In this work a new method of spindle-machine dynamics identification for
Receptance Coupling 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. The individual frequency responses, or receptances, of these
components are then analytically coupled. The spindle-machine receptances are
measured once and archived. Beam models are used to represent the tool-holder
subassembly.
The spindle-machine dynamics 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 spindle 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 Cincinnati
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
shrink 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 measurement 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 holder 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 were 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 stiffness values. The average of these stiffness values
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 blank 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 tool-point FRF predictions. Also, the Tribos holder
Timoshenko beam models requires further study due to the thermosetting plastic
chambers in the holders.
157
APPENDIX A FLEXIBLE COUPLING BETWEEN TOOL AND HOLDER
The free-free tool receptances (R11, R12a, R2a2a and R2a1) and the machine-spindle-
holder receptances (R2b2b) may be coupled using a flexible joint to predict the assembly
tool point receptance. In order to calculate the tool point assembly receptances, G11
(Equation A-1), a generalized force Q1 (representing both the externally applied force,
F, and couple, M) is applied at coordinate location U1 (see Figure A-1), where the
generalized displacement U represents both displacement, X, and rotation, .
(A-1) The displacement equations for the substructures 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 matrix,
xf xf xm xm
f f m m
k i c k i ck
k i c k i c, 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 2a and 2b is
given by Equation A-6.
(A-6)
1 1
1 1 11 11
11
11 111 1
1 1
X X
F M H LG
N P
F M
1 11 1 12 2a au R q R q
2 2 2 2 2 1 1a a a a au R q R q
2 2 2 2b b b bu R q
2 2 2( )b a bk u u q
2 2 0b aq q
158
At coordinate location 1, the external force/couple is applied so the relationship in
Equation A-7 is obtained.
(A-7) Substituting for u2b and u2a 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 q2b gives Equation A-10. Given that 2 2a bq q 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 expressed 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)
1 1q Q
2 2 2 2 2 2 2 2 2 1 1 2( ) ( )b a b b b a a a a bk u u k R q R q R q q
2 2 2 2 2 2 2 1 1 2( )b b b a a a a bk R q R q R Q q
1
2 2 2 2 2 2 1 1
1( )b b b a a aq R R R Q
k
11 1 12 21 111
1 1 1
a aR q R qU uG
Q Q Q
1
11 1 12 2 2 2 2 2 1 1
11
1
1( )a b b a a aR Q R R R R Q
kGQ
11 111
11 11 12 2 2 2 2 2 1
11 11
1( )a b b a a a
H LG R R R R R
N Pk
159
Figure A-1. The tool (I) is coupled flexibly to the holder-spindle-machine (II) to
determine the tool point receptance matrix, G11.
160
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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 fall 2007 she began her graduate studies
at the Department of Mechanical and Aerospace Engineering, University of Florida, in
pursuit of her MS degree in mechanical engineering. In spring 2008, she joined the
Machine Tool Research Center under the guidance of Dr. Tony L. Schmitz. She
received her Master of Science degree in December 2009.