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Improved Prediction of Spindle-Holder-Tool Frequency Response Functions

Permanent Link: http://ufdc.ufl.edu/UFE0021318/00001

Material Information

Title: Improved Prediction of Spindle-Holder-Tool Frequency Response Functions
Physical Description: 1 online resource (121 p.)
Language: english
Creator: Cheng, Chi Hung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: High speed machining (HSM) offers tremendous capabilities for discrete part manufacturing because it can provide high material removal rates (MRR) in metals, plastics, and composites with good surface finish. To realize these benefits, stability lobe diagrams, which define regions of stable cutting as a function of spindle speed and axial depth of cut, can be used to select appropriate cutting conditions. Computation of these diagrams requires that the dynamics of the cutting system (the machine, spindle, holder, and tool assembly) be known. Typically, impact testing (i.e., exciting the structure with an instrumented hammer and recording the response with a linear transducer) is used to record the required tool point frequency response. However, due to the diversity of tool holders and tools available to end users, it can be prohibitively time-consuming to perform impact testing for each possible combination. Further, it is difficult to measure the responses of 1) small tools using traditional methods; and 2) spindles during high speed rotation. The former is necessary for new micro-milling applications, while the latter is required because the at-speed response for some spindles can differ from the non-rotating response. This study provides a method to address these situations. The tool tip response for a given machine-spindle-holder-tool assembly is predicted by coupling a spindle measurement with finite element models of the holder and tool using the method of receptance coupling substructure analysis (RCSA). RCSA enables a user to analytically couple arbitrary tool-holder combinations to an archived spindle response. Therefore, the user must perform only a single test on the spindle in question. Given this information, the tool point response for any tool-holder can be performed via a ?virtual impact test?. Comparisons of predictions and experimental results are provided for 1) micro-tools; and 2) macro-scale tools coupled to a spindle that exhibits changing dynamics with spindle speed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chi Hung Cheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021318:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021318/00001

Material Information

Title: Improved Prediction of Spindle-Holder-Tool Frequency Response Functions
Physical Description: 1 online resource (121 p.)
Language: english
Creator: Cheng, Chi Hung
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre: Mechanical Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: High speed machining (HSM) offers tremendous capabilities for discrete part manufacturing because it can provide high material removal rates (MRR) in metals, plastics, and composites with good surface finish. To realize these benefits, stability lobe diagrams, which define regions of stable cutting as a function of spindle speed and axial depth of cut, can be used to select appropriate cutting conditions. Computation of these diagrams requires that the dynamics of the cutting system (the machine, spindle, holder, and tool assembly) be known. Typically, impact testing (i.e., exciting the structure with an instrumented hammer and recording the response with a linear transducer) is used to record the required tool point frequency response. However, due to the diversity of tool holders and tools available to end users, it can be prohibitively time-consuming to perform impact testing for each possible combination. Further, it is difficult to measure the responses of 1) small tools using traditional methods; and 2) spindles during high speed rotation. The former is necessary for new micro-milling applications, while the latter is required because the at-speed response for some spindles can differ from the non-rotating response. This study provides a method to address these situations. The tool tip response for a given machine-spindle-holder-tool assembly is predicted by coupling a spindle measurement with finite element models of the holder and tool using the method of receptance coupling substructure analysis (RCSA). RCSA enables a user to analytically couple arbitrary tool-holder combinations to an archived spindle response. Therefore, the user must perform only a single test on the spindle in question. Given this information, the tool point response for any tool-holder can be performed via a ?virtual impact test?. Comparisons of predictions and experimental results are provided for 1) micro-tools; and 2) macro-scale tools coupled to a spindle that exhibits changing dynamics with spindle speed.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chi Hung Cheng.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Schmitz, Tony L.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021318:00001


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IMPROVED PREDICTION OF SPINDLE-HOLDER-TOOL
FREQUENCY RESPONSE FUNCTIONS




















By

CHI HUNG CHENG


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

2007































O 2007 Chi Hung Cheng


































To my family, my grandfather, and Lord Jesus Christ









ACKNOWLEDGMENTS

The author would like to thank his advisor Dr. Tony L. Schmitz, for his understanding,

patience, and unconditional support. Thanks also go to Dr. John Schueller, Dr. Nagaraj Arakere,

Dr. Gloria Wiens, and Dr. Jacob Hammer for serving as committee members.

The author also appreciates Dr. John C. Ziegert for always bringing in new ideas when

there is a bottle neck and Dr. Nam Ho Kim for helping with micro tool finite element simulation.

The thank list extends to the members in Machine Tool Research Center from 2003 to

2007, for all the team works and memorable moments together. With special thanks to Mr. Scott

Payne and Mr. Vadim Tymianski for always being "partners of crimes" when there is the need.

Finally the author would like to thank his family for the support in every aspect, and the

saints in the Church in Gainesville for their loving tender care. Praise the Lord. This work

wouldn't have been done without the sovereign arrangements.











TABLE OF CONTENTS


page

ACKNOWLEDGMENTS .............. ...............4.....


LIST OF TABLES ................ ...............7............ ....

LIST OF FIGURES .............. ...............9.....


AB S TRAC T ........._._ ............ ..............._ 14...


CHAPTER


1 INTRODUCTION ................. ...............16.......... ......


High Speed Machining ................. ...............16................
Chatter And Stability Lobe Diagram ................. ...............16........... ...
Obj ective ................. ...............18.................

2 LITERATURE REVIEW ................. ...............22................


Milling Stability Prediction .............. ...............22....
Experimental Method .............. ........ .. ...............2
Predictive (Non-Experimental) Methods ................. ...............23........... ....
System Dynamics Acquisition .............. ...............23....
Stability Analysis............... ...............25

3 RECEPTANCE COUPLING SUB STRUCTURE ANALYSIS .............. .....................3


Two-Component Receptance Coupling Sub structure Analysis .............. ....................3
Rigid Connection................ .... .. ..........3
Non-Rigid Connection with a Linear Spring............... ...............32.
Non-Rigid Connection with a Linear Spring and a Damper ................. ............... .....33
Non-Rigid Connection with Linear and Rotational Springs and Dampers .....................34
Three-Component Receptance Coupling Substructure Analysis .............. ....................3
Inverse Receptance Coupling Sub structure Analysi s ................ ..............................38
Sub structure Beam Modeling .............. ...............41....
Euler-Bernoulli Beam ................. ...............41........... ....
Timoshenko Beam ................. ...............43........... ....
Fluted Tool Modeling ................. ...............45........... ....

4 ROTATING FREQUENCY RESPONSE FUNCTION PREDICTION ............... .... ........._..54


Runout Signal Filtering............... ...............5
Runout Signal .............. ...............55....
Runout Filtering................. .... .. ................5
The FRF Prediction from Rotating Standard Holder Measurements .................. ...............57












Stability Boundary Validation ................ ...............58........... ....


5 MICRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION..........._...73


Micro Scale Tools on Macro Machine Systems ................. ...............73........... ..
Sensor Options for Micro Tools ................ ...............73........... ...
Modeling Description ................. ...............74.................
Experimental Setup ................. ...............76...
Micro Scale Tools on Micro Spindles .............. ...............78....
The S Value Consideration............... ... ............7
Sensitivity of Standard Artifact Length .................. ...............79........... ...
Micro Tool Frequency Response Function Prediction ................. ............... ...._...80


6 MACRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION ...........95


Variation in Spindle-Base Receptances with Standard Holder Geometry .............................95
Experimental Results ................ ...............95.................
The HSK-63A Interface .............. ...............96....
The CAT-40 Interface .............. ...............97....
The HSK-100A Interface ................. ...............98...._ __ ....


7 CONCLUSIONS AND FUTURE WORK ......____ ..... ........ .....__ ..........15


Conclusions............... ..............11
Future W ork ............ ............ ...............116...


LIST OF REFERENCES ............ ............ ...............118...


BIOGRAPHICAL SKETCH ............ ............ ...............121...










LIST OF TABLES


Table page

3-1 Fixed-free steel rod first mode frequency comparison between different beam
modeling methods. ........._.__ ..... .___ ...............52....

3-2 Average of area section properties for fluted endmills. ................... ...............5

3-3 Average of area section properties for fluted endmills. ................... ...............5

4-1 HSK-63 A short hollow standard holder sub structure dimensions ................. ................7 1

4-2 Solid holder substructure dimensions. ............. ...............71.....

4-3 Material properties used in RCSA modeling. .............. ...............71....

4-4 Substructure dimensions of Regofix collet holder with 12.7 mm diameter, 127 mm
overhang carbide tool blank ................. ...............72................

4-5 Substructure dimensions of Regofix collet holder with 25.4 mm diameter, 127 mm
overhang carbide tool blank ................. ...............72................

4-6 Sub structure dimensions of the endmill-shrink fit holder described in Figure 4-15. ........72

5-1 CAT-40 standard holder artifact substructure section dimensions. ................. ...............93

5-2 CAT-40 ER-25 collet holder and tool substructure section dimensions............................93

5-3 S values for micro standard artifacts ................. ...............94........... ..

5-4 Tapered tool (23.5 mm OH) substructure section dimensions. ........._.._.._ ........_.._.. ...94

6-1 HSK-63A long hollow standard holder artifact sub structure dimensions. ................... ...111

6-2 Substructure dimensions for HSK-63A long shrink fit holder (section I at free end).....111

6-3 Substructure dimensions for HSK-63A long shrink fit holder (section I at free end).....112

6-4 Extended holder lengths of the HSK-63A standard artifacts and shrink fit holders........1 12

6-5 CAT-40 long hollow standard holder artifact substructure dimensions. ................... ......112

6-6 CAT-40 short hollow standard holder artifact sub structure dimensions. ................... .....112

6-7 CAT-40 short solid standard holder artifact substructure dimensions. ................... .........112

6-8 CAT-40 shrink fit holder substructure dimensions ................. ............................112










6-9 CAT-40 long collet holder sub structure dimensions. ................ .......... ...............112

6-10 CAT-40 short collet holder sub structure dimensions. ........._.._.. ................. ...._113

6-11 Extended holder lengths of the CAT-40 standard artifacts and tested holders ...............113

6-12 HSK-100A short hollow standard holder artifact sub structure dimensions. ................... 113

6-13 HSK-100A long soild standard holder artifact substructure dimensions. ................... .....113

6-14 Substructure dimensions of Briney HSK 100ASF-075-43 3 shrink fit holder with
Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head. ...........113

6-15 Sub structure dimensions of Briney HSK 100AE- 125 -472 set screw holder with
Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end............. ............_ ...114

6-16 Extended holder lengths of the HSK-100A standard artifacts and tested holders...........1 14











LIST OF FIGURES


Figure page

1-1 HSM cutting speed ranges for various materials. ............. ...............20.....

1-2 Scheme of (flexible) cutting edge passing through workpiece surface. ............................20

1-3 Stable cut and unstable cut (chatter) on a workpiece surface. ................... ...............2

1-4 Example stability lobe diagram. ............. ...............21.....

1-5 Block diagram for cutting process with regenerated wavy surface. ............. ..................21

2-1 Different approaches to determine stable cutting conditions............... ...............2

2-2 Ball bearing contact angles at high speed rotation ................. ............... ......... ...29

2-3 Impact testing performed on standard artifact with modal hammer and laser
vibrom eter. .............. ...............29....

2-4 Example of time-domain simulation of an unstable cutting process. ............. ..... ........._..30

2-5 Predicted and measured FRF of a 100 mm diameter inserted endmill .............. ...............30

3-1 Components joined with a rigid connection. ................ ...............46..............

3-2 Components connected with a linear spring. ............. ...............47.....

3-3 Assembly with linear spring and damper ................. ...............48........... ..

3-4 Assembly with linear and rotational springs and dampers. ............. .....................4

3-5 Generic case of two sub structures with rigid connection. ................ ............... ...._..50

3-6 Three-component RCSA model for tool-holder-spindle assembly. ............. ..................50

3-7 Spindle and decomposed standard holder artifact. ............. ...............50.....

3-8 Ansys Workbench frequency simulation for 19.1 mm diameter fixed-free steel rod
with 80 mm length. ............. ...............51.....

3-9 Cutting edge of a four fluted flat endmill cutter. ............. ...............51.....

3-10 Solid model of two-fluted endmill cross-section. ....._____ .... ... .__ ..........__......52

4-1 Speed-dependent FRFs of the standard holder at five different spindle speeds: {0,
250 500 a d 1 00 } rpm. ............. ...............60.....










4-2 Example setup for rotating FRF measurement. ............. ...............61.....

4-3 Example of holder and runout signal resultant response. ...........__.....__ ..............61

4-4 Scheme of tachometer-aided runout filtering setup. ........._.._. ....... ........._.._. ...62

4-5 Illustration of once-per-revolution signal identification ........._._.._......_.. ..............62

4-6 Example of shrink fit holder time-domain runout filtering. ............. .....................6

4-7 Frequency response function comparison with/without time-domain filtering. ................64

4-8 Short hollow standard holder geometry and substructure coordinates .............. ...............64

4-9 Experimental setup of standard holder (HSK-63A interface) rotating FRF
measurement. ............. ...............65.....

4-10 Solid holder geometry and substructure coordinates. .................... ..............6

4-11 Solid holder FRF measurement and prediction at {10000, 12000, and 16000} rpm. .......66

4-12 Geometry of Regofix collet holder with tool .......................__ .....__ .........6

4-13 Measured and predicted FRFs for Regofix collet holder with 12.7 mm diameter 127
mm overhang carbide tool blank at 10,000 rpm. ............. ...............67.....

4-14 Measured and predicted FRFs for Regofix collet holder with 25.4 mm diameter 127
mm overhang carbide tool blank at 10,000 rpm. ............. ...............67.....

4-15 Geometry of 19.1 mm diameter, four flutes, carbide endmill with 76.1 mm overhang
length clamped in Command shrink fit holder. ............. ...............68.....

4-16 Stability lobes for FRF measurement at 0 rpm and predictions at {10000 and 16000}
rpm for 19.1 mm diameter, four flute, carbide endmill .............. ...............68....

4-17 Setup for cutting tests. The tool shank deflections were measured using two
orthogonal capacitance probes. .............. ...............69....

4-18 Example results for 8000 rpm slotting cuts. ............. ...............70.....

4-19 Comparison of test cut results to predicted stability boundaries determined from 0
rpmVVV (mesued andVV {10000 and~~ 16000 rpm (prdite) R.............. .................71I

5-1 Example setup of high speed machining with micro tools. ................ ......................82

5-2 Dimension comparison of a 1mm diameter, two-flute micro endmill to a penny. ............82

5-3 CAT-40 standard holder artifact geometry. ............. ...............82.....










5-4 CAT-40 ER-25 collet holder and tool geometry............... ...............83

5-5 Experimental setup for CAT-40 collet holder with 3.18 mm steel tool shank. ........._.......83

5-6 Magnitudes of H41 FRFs measurements (top) and predictions (bottom). ........................84

5-7 Measured and predicted H41 With 46.0 mm tool overhang (marked as 9 in Fig. 5-6)......84

5-8 Measured and predicted H41 With 50.8 mm tool overhang (marked as 12 in Fig. 5-6)....85

5-9 Multiplication factor (M~F) determined from visual fit in relation to the tool overhang
with a linear approximation superimposed. .............. ...............85....

5-10 FRFs ofH44, magnitude measurements (top) and predictions (bottom). ..........................86

5-11 Measured and predicted H44 With 46.0 mm tool overhang (marked as 9 in Fig. 5-10)....86

5-12 Measured and predicted H44 With 50.8 mm tool overhang (marked as 12 in Fig. 5-
10). ............. ...............87.....

5-13 NSK HES 500 electric micro spindle. ............. ...............87.....

5-14 Experimental setup for determination of micro spindle-base receptances. .......................88

5-15 Prediction of tool point FRF for 3.18 mm diameter, 21 mm overhang steel rod by a
17 mm overhang standard artifact with three different S values. ............. ....................89

5-16 FRF prediction of a 3.18 mm diameter, 21 mm overhang steel rod by 20 mm (solid)
and 17 mm (dotted) overhang standard tool artifacts .............. ..... ............... 9

5-17 Example 3.18 mm shank diameter tapered tool (no flutes) with 1.5 mm diameter tool
tip. ............. ...............90.....

5-18 Geometry of NSK HES 500 micro spindle with tapered tool............... ..................9

5-19 Tool tip measurement of tapered tool with an overhang of 23.5 mm compared to
predictions based on different standard artifact overhang lengths............... .................9

5-20 Measurement of tapered tool (with overhang length of 25.5 mm) FRF compared with
selected standard artifact FRF predictions. .............. ...............92....

5-21 FRFs for free-free tapered tool and standard artifact responses coupled to rigid
spindle receptances. ............. ...............93.....

6-1 HSK-63A spindle-base receptances calculated by long and short hollow standard
holders ................. ...............99.................

6-2 Geometry of HSK-63A long hollow standard hollow holder artifact. ................... ..........100










6-3 HSK-63A shrink fit holders: long hollow (left) and short hollow (right) ................... .....100

6-4 HSK-63A FRF predictions for long shrink fit holder by short hollow artifact with
different S values in comparison with measured FRF (heavy solid line).............._._._.....101

6-5 HSK-63A long shrink fit holder FRF predictions using two different standard holders
in comparison with measured FRF (heavy solid line). ............. ......................0

6-6 HSK-63A short shrink fit holder FRF predictions using two different standard
holders in comparison with measured FRF (heavy solid line). ............. ....................10

6-7 Geometry of CAT-40 long hollow standard holder artifact............._ .........___......102

6-8 Geometry of CAT-40 short hollow standard holder artifact.............__ ..........___.....103

6-9 Geometry of CAT-40 short solid standard holder artifact. ......____ ...... .....__.........103

6-10 CAT-40 shrink fit holder FRF predictions in comparison with measured FRF (heavy
solid line). ............. ...............104....

6-11 CAT-40 long collet holder FRF predictions in comparison with measured FRF
(heavy solid line)............... ...............104.

6-12 CAT-40 short collet holder FRF predictions in comparison with measured FRF
(heavy solid line)............... ...............105.

6-13 Geometry of HSK-100A short hollow standard holder artifact. .........._.... ........._......105

6-14 Geometry of HSK-100A long solid standard holder artifact. .........._.... ........_._.. .....106

6-15 Geometry of Briney HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-
19 10 175 carbide adapter and two square insert cutting head. ............. ....................10

6-16 Geometry of Briney HSK 100AE-125-472 set screw holder with Mitsubishi Carbide
MBN 10 1000 TB steel tapered ball end mill with one round carbide insert. .........._......107

6-17 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction........................107

6-18 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction........................108

6-19 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction........................108

6-20 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction........................109









6-21 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction.109

6-22 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction..110

6-23 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction.110

6-24 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction..111









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 PREDICTION OF SPINDLE-HOLDER-TOOL FREQUENCY RESPONSE
FUNCTIONS

By

Chi Hung Cheng

August 2007

Chair: Tony L. Schmitz
Major: Mechanical Engineering

High speed machining (HSM) offers tremendous capabilities for discrete part

manufacturing because it can provide high material removal rates (MRR) in metals, plastics, and

composites with good surface finish. To realize these benefits, stability lobe diagrams, which

define regions of stable cutting as a function of spindle speed and axial depth of cut, can be used

to select appropriate cutting conditions. Computation of these diagrams requires that the

dynamics of the cutting system (the machine, spindle, holder, and tool assembly) be known.

Typically, impact testing (i.e., exciting the structure with an instrumented hammer and recording

the response with a linear transducer) is used to record the required tool point frequency

response. However, due to the diversity of tool holders and tools available to end users, it can be

prohibitively time-consuming to perform impact testing for each possible combination. Further,

it is difficult to measure the responses of 1) small tools using traditional methods; and 2) spindles

during high speed rotation. The former is necessary for new micro-milling applications, while

the latter is required because the at-speed response for some spindles can differ from the non-

rotatmng response.

This study provides a method to address these situations. The tool tip response for a given

machine-spindle-holder-tool assembly is predicted by coupling a spindle measurement with









finite element models of the holder and tool using the method of receptance coupling

substructure analysis (RCSA). RCSA enables a user to analytically couple arbitrary tool-holder

combinations to an archived spindle response. Therefore, the user must perform only a single test

on the spindle in question. Given this information, the tool point response for any tool-holder can

be performed via a 'virtual impact test'. Comparisons of predictions and experimental results are

provided for 1) micro-tools; and 2) macro-scale tools coupled to a spindle that exhibits changing

dynamics with spindle speed.











CHAPTER 1
INTRODUCTION

High Speed Machining

High speed machining (HSM) is an important capability in modern, discrete part

manufacturing. Using the higher cutting speeds made possible by improved cutting tool materials

and coatings, the machine operation time is reduced significantly (Fig. 1-1 [1]). The use ofHSM

makes it possible to efficiently produce complex parts and, therefore, reduce assembly time and

costs relative to the traditional approach where simpler shapes are machined and then

mechanically joined.

High speed machines typically use direct drive spindles, i.e., a spindle shaft with

permanent magnets is driven by a coil located within the surrounding housing. Modern spindle

designs can reach top speeds of 40,000 rpm and higher with powers at the many tens of kW

level. At these higher speeds, micro-milling (or milling with very small diameter cutters) is now

realizable because reasonable cutting speeds, or peripheral velocities of the cutting edge, can still

be maintained even though the cutting edge radius may only be fractions of a millimeter. The

advantage of high speed micro-machining is that it provides a process capable of producing

complex, free form, three-dimensional (3-D) structures from virtually any material. This

provides an alternative to typical MEMS (micro electro mechanical systems) fabrications

techniques, such as silicon etching, that are generally limited to 2-D geometries and specialized

materials. Therefore, it can be expected that the demand of HSM will continue to increase.

Chatter And Stability Lobe Diagram

Surface location error (SLE) and chatter, or unstable machining conditions, impose

limitations on machining efficiency. For any machining operation, the cutting force acting on the









tool causes it to vibrate. This vibration can lead to chatter and corresponding large forces and

potential damage. Even if the operation is stable, however, the vibration state of the tool as it

leaves the newly created surface defines the location of that surface. Variations in the process

parameters can yield either undercut (less material removed than commanded) or overcut (more

material removed) situations; this phenomenon is referred to as surface location error. For either

limitation, it is important to note that the tool-holder-spindle-machine assembly response as

reflected at the tool point (free end of the tool) strongly influences the final behavior.

Chatter occurs due to the inherent feedback mechanism in machining. In turning and

milling operations, the tool cutting edge makes multiple passes through the workpiece surface to

achieve the desired dimension. For each pass, the tool vibrations are imprinted on the surface.

Therefore, the workpiece surface is not uniform and the current chip thickness depends both on

the current tool vibrations and those during the previous pass (Fig. 1-2), which Arnold [2] refers

to as the regeneration of waviness. As the cutting edge removes the wavy surface, the force is

modulated by the varying chip thickness, which leads to further vibration. Depending on the

machining parameters, the feedback system can become unstable and chatter occurs (Fig. 1-3).

The large force and significant tool deflections associated with chatter can be identified audibly.

It not only creates an unacceptable machined surface finish, but can also damage the machine

tool, spindle bearings, tool, and workpiece.

To avoid chatter, stability lobe diagrams can be applied. These diagrams (as shown in Fig.

1-4) enable the machine operator to choose a proper spindle speed-chip width (axial depth for

peripheral end milling) for stable cutting conditions. The concept of the stability lobe diagram

was first developed in 1956 [3]. However, large industrial benefits were not realized until the

high speed machines became commercially available. As can be seen in Fig. 1-4, the width of the









stable regions (beneath the lobes) tends to widen as the spindle speed becomes higher. Therefore,

operating in these higher speed regions increases the material removal rate both by the increased

cutting speed and higher axial depth of cut. Methods used to compute the stability limit are

described in Chapter 2.

The feedback mechanism in machining can also be represented using a block diagram

approach as shown in Fig. 1-5 [4]. The system dynamics are represented by a second order plant

in the forward path. The force is determined by multiplying the difference between the current

and time-delayed chip thickness valued by the gain, represented by the product of the specific

cutting energy and chip width.

Based on this block diagram, the limiting chip width can be expressed for turning

operations as shown in Eq. 1.1 where FRF is the system frequency response function. This

equation emphasizes the importance of the system dynamics in milling performance.

-1
b = (1.1)
2K, Re(F;RF(co))

Objective

Currently, the tool point FRF is measured by impact testing, where an instrumented

hammer (or modal hammer) is used to excite the tool-holder-spindle-machine assembly and the

resulting vibration is measured by an appropriate linear transducer, typically a low mass

accelerometer. Because the assembly dynamics depend on the individual components as well as

their interactions, a new test must be performed for each combination or change in setup (e.g., if

the tool overhang length is changed). In many industrial situations, it is not practical to measure

each combination due to time restrictions. An additional complication is that these tool point

dynamic measurements are necessarily completed with no spindle rotation (zero spindle speed),

but in some situations the system dynamics can vary with spindle speed [5]. For micro-milling,









the situation is even more problematic because, even for zero spindle speed, it can be difficult or

impossible to carry out impact tests on very small diameter endmills (<1 mm).

It is the obj ective of this study to develop a method for predicting tool-holder-spindle

dynamic responses that enables the user to perform a simple test and, subsequently, predict the

dynamic behavior of the assembly. A modified three-component (tool, holder, and spindle)

receptance coupling substructure analysis (RCSA) method is provided, which models the tool

and holder geometries using a Timoshenko beam formulation and couples this result to the

spindle response to predict the tool tip FRF for the situations described in the previous

paragraph.

The scope of this research follows:

1. application of a general RCSA approach, which includes the determination of spindle
dynamics by inverse RCSA and coupling of the spindle response to arbitrary holders and
tools (with emphasis on fluted tools and micro-tools);

2. prediction of the rotating tool point FRF over applicable ranges of spindle speeds;

3. establishing a guideline for standard artifact (used during the inverse RCSA procedure)
geometry based on the selected tool-holder to enable increased accuracy in the tool tip
response prediction; and

4. experimental validation of the general RCSA approach through impact testing and cutting
tests.













liijiiiiil

---,

Iliiiii~iii~jii



Il~ii ~iiii ~,, ,


Nickl-B ase Alloys

Titaumm



Cast Iron



Aluminum
Fib e IRM:cecl


M HS M: Range
E Trazisition Rang
I Normal Rars=e


19~o 190 ~
CuttizgSp~ee: [miin]


199,


Figure 1-1. HSM cutting speed ranges for various materials.


Flexrible
cutting tool


Chip:


Commanded chip:
thickness
Previous pass
Current pass




Thrust f:>rce Instantaneous chip thickness


Figure 1-2. Scheme of (flexible) cutting edge passing through workpiece surface.































Unstable
zone





Stable zone


SSta7ble our


Figure 1-3. Stable cut and unstable cut (chatter) on a workpiece surface.


Cutting
D epth


Spindle Speed


Figure 1-4. Example stability lobe diagram.


disturbarce~


vagying chip thickness


K,: specific cutting energy
b: chip width


Figure 1-5. Block diagram for cutting process with regenerated wavy surface.


21









CHAPTER 2
LITERATURE REVIEW

Milling Stability Prediction

It has been observed in the milling operation that tool vibrations during cutting are

imprinted on the work piece surface. Depending on the phasing between these imprinted waves

and the tool vibrations in the subsequent pass, self-excited vibrations can occur. As seen in Fig.

1-2, the wavy surface created by the previous pass causes the chip thickness during the current

pass to vary, which leads to a varying cutting force acting on the tool. If the waves between the

current and previous pass are aligned, the oscillating component of the force tends to decay.

However, if the waves are out of phase and the chip thickness varies substantially, the vibrations

tend to grow. These vibrations occur at the system natural frequency corresponding to the most

flexible mode, where chatter is mostly observed.

There are two maj or approaches to obtain stable cutting conditions: predictive (non-

experimental) and experimental methods. Predictive methods require knowledge of the cutting

system dynamics (frequency response function, FRF, or modal matrices) and specific cutting

energy values, which relate the cutting force components to the selected chip thickness and

width, for the stability calculation, whereas the experimental method can automatically adjust the

spindle speed to obtain a stable cut based on data obtained during an unstable cut. The methods

to achieve stable cutting conditions are described in Fig. 2-1.

Experimental Method

For on-line spindle speed regulation, Smith [6], and Delio et al. [7] proposed an algorithm

of spindle speed selection for high speed milling machines. In their approach, the sound

spectrum (recorded by a microphone) during an unstable cut is used to identify the chatter

frequency. This chatter frequency depends on the cutting system configuration and is basically










independent of the tooth passing frequency (i.e., the spindle speed multiplied by the number of

cutter teeth) and its harmonics. Once the chatter frequency is identified, the machine operator or

automated control system then adjusts the spindle speed to match the tooth passing frequency to

the chatter frequency. If chatter still occurs, the same procedure is applied until chatter is

eliminated. The result is stable cutting conditions without the knowledge of the system dynamics

and specific cutting energy values. However, this is not a predictive method. Rather, it requires

chatter to occur in a test cut for it to be avoided during actual part production and it does not

always result in the maximum possible material removal rate.

Predictive (Non-Experimental) Methods

The predictive or non-experimental methods require that the system dynamics be identified

by either impact testing or finite element modeling. The system's FRF, or modal parameters, are

then used in a stability analysis algorithm (analytical, semi-analytical, or time-domain

simulation) to determine whether a cutting process is stable. The specific cutting energy, or

cutting coefficient, values must also be determined beforehand. In practice, cutting coefficients

vary with cutting material, tool geometry, etc. The particular values for a given combination may

be determined by experiment (Chapter 4) or values from prior studies may be applied

System Dynamics Acquisition

Acquiring the system dynamics is the first step of predictive methods. Finite element

modeling and impact testing are the most common methods. Modern high speed spindle

configurations include a permanent magnet tightly fit to the spindle shaft with bearings

supporting the front and rear shaft ends. The shaft is powered by the stator mounted inside the

spindle housing. Finite element models therefore require a significant amount of information to

describe this complicated structure. For bearing dynamics, Jones [8] discovered that, due to the

combined effect of the centrifugal force Fc and gyroscopic moment Mg, the contact angle









between the ball and the outer raceway, ao, tends to decrease and the contact angle with inner

raceway, az tends to increase (Fig. 2-2). The variation in contact angles causes the ball bearing

to lose stiffness when rotating at high speed. Researchers such as Shin et al.[9-10] determined

the bearing stiffness and spindle frequency by coupling a simplified Rayleigh beam spindle

model with a bearing model derived from Jones' theory and bearing contact angles measured at

different spindle speeds. The model yielded good correlation with the experimental data.

Jorgenson and Shin [10] further extended the prediction of multiple natural frequencies by

modeling the spindle with Timoshenko beam elements.

Cao et al. [l l] used a similar approach as above. With the use of incremental finite element

equations in time domain, Cao's model could predict not only the frequency response function,

but also the time domain spindle response. This model did not incorporate thermal effects, which

are required to completely describe the spindle dynamics.

Thermal issues have been studied mainly to estimate the bearing life. However, in high

speed machining, it is beneficial to be able to predict the spindle response under high speeds

when significant heat can be generated. The heat is mostly generated by the friction of the

bearing contacts and depends on the viscosity of the lubricant and air in the thin gap. The

resulting thermal expansion and the temperature gradient cause the bearing preload to vary,

which results in changes in the bearing stiffness.

Tu et al. [12, 13] developed a thermal model to predict the entire spindle temperature field

and the heat-induced bearing preload. Further, Lin, Tu, and Kamman [14] combined the thermal

model with the consideration of centrifugal force and gyroscopic moment and were able to

predict the overall spindle dynamics. Li and Shin [15] added several spindle bearing

configurations and combined their bearing/spindle model with a thermal model similar to Tu's










approach. Both methods in references [14] and [15] can describe the speed dependent bearing

stiffness, but the frequency response function of the whole spindle was not predicted.

For the previous research studies, the complete finite element (FE) spindle model requires

knowledge of the bearing preload, location, and the assembly tolerance for each component to

define the stiffness. This data is not generally available in production environments.

Furthermore, the model damping cannot be predicted from first principles and requires tuning of

the model by matching the spindle modes to a measured response. In addition, the complex FE

models are often computationally expensive, which is a disadvantage for job shop application.

The second approach to obtain the system dynamics is by impact testing. Fig. 2-3 shows

the FRF measurement of a HSK-63A standard artifact using a modal hammer (instrumented

hammer) and laser vibrometer. Besides a vibrometer (which measures velocity), other commonly

used transducers include accelerometers (acceleration) and capacitance probes (displacement).

Upon impact, the time-domain hammer and transducer signals are recorded and then

converted into the frequency domain using the discrete Fourier transform. The result is the

frequency dependent ratio between response and force ratio, or FRF. The advantage for impact

testing is that it takes relatively short period of time to determine system dynamics and it

captures the system's characteristics (both stiffness and damping) at its current state regardless of

the aging and wear of the components. Once the FRF is known, it can be fit to determine the

system modal matrices.

Stability Analysis

With the system dynamics (FRF or modal matrices) collected in the previous step, there

are three different approaches (time-domain simulation, semi-analytical, and analytical

approach) to determine the stability limit as a function of axial depth of cut (or chip width) and

spindle speed.









Time-domain simulation [16] uses the modal mass, damping, and stiffness matrices to

describe the system dynamics. The milling cutter is dissected in the radial direction to identify

the engaging and disengaging of the tooth-workpiece contact. The instantaneous chip thickness

is calculated and used to estimate the force input. For each tooth rotation, the workpiece surface

profile is recorded and later used to determine the instantaneous chip thickness. The cutter

vibration is computed from the force and system equations of motion using integration. Users are

responsible for providing the depth of cut and spindle speed. The stability of the cutting

condition can be observed after a few iterations in the simulation as shown in Fig. 2-4.

Bayly et al. [17] and Mann et al. [18-19] proposed a semi-analytical method of solving the

milling equations of motion. This technique, Temporal (Time) Finite Element Analysis (TFEA),

divides the cut into a finite number of elements. Since the nature of the milling operation is

discontinuous (free vibration when disengaged and forced vibration when engaged in the cut), a

closed form solution can not be attained. By assuming an approximated solution during forced

vibration and matching it to a time frame where free vibration occurs, a discrete linear matrix of

the dynamic system can be found. The stability is determined from the eigenvalues of this

matrix.

Analytical approaches that provide approximate closed form solutions for the milling

equations are also available. Studies of chatter theory and the development of stability lobe have

been completed by, for example, Tlusty [20], Tobias [21], Polacek [22], Merrit [23],

Koenigsberger [24], and Altintas [25].

Tlusty concluded that a single point cutting process is stable as long as the chip width b is

less than the limit chip width bhm Which is a function of specific cutting energy Ks and spindle

speed ao as shown in Eq. 1.1, where the real part of the frequency response function Re(FRF(co))









is at its minimum. In this frequency-domain solution, the FRF is used directly and there is no

need to extract modal matrices. However, it is still necessary to perform a new impact test to

identify the tool point FRF for each tool and holder combination, which can pose a significant

obstacle on the shop floor.

In order to avoid the tedious testing procedure, Receptance Coupling Substructure

Analysis (RCSA) can be applied to predict the tool point FRF. The concept of receptance

coupling can be traced back to 1947 [26]. RCSA allows multiple components to be analytically

combined after the component models have been defined individually. In some instances,

connections parameters (springs and dampers) between the components, rather than rigid

connections, are required to predict the system overall FRF.

RCSA is convenient to implement for modern spindle-holder-tool combinations because

holder designs with a selected holder-spindle interface are identical from the rear of the holder

(i.e., at the taper that is inserted in the spindle) to the far end of the holder flange. In this study,

the taper types used were CAT and HSK with taper angle approximately 8.2 and 2.9 degrees

respectively. This common geometry enables a consistent gripping area for automatic tool

changes. Therefore, for a given holder-spindle interface, the user can perform an impact testing

on a simple geometry standard artifact to determine the spindle dynamic response, including the

holder-spindle connection. To predict the overall system dynamics, the extended portion of the

holder and the cutting tool (beyond the holder flange) is modeled and then coupled to the spindle

response. Though finite element models of the holder and tool may be applied, they are for rather

simple cylindrical geometries and beam elements are sufficient. The computational time is

therefore low and acceptable for the job shop level. The modular modeling approach aids during






















Predictive (Non-experimental methods)


Stability analysis

Analytical Semi-analytical Time-domain
approach approach simulation


Stable cutting conditions


purchasing and process planning because new holder-tool combinations in a particular spindle

can be quickly and efficiently modeled to test the milling capabilities.

Previous work completed by Schmitz et al. [27-29] shows good correlation between the

analytical prediction and the experiment result. An example result is provided in Fig. 2-5 [30]. In

Chapter 3, the three-component RCSA model [3 1] is described in detail. Modeling of fluted tools

is also discussed.


Figure 2-1. Different approaches to determine stable cutting conditions.


System dynamics acquisition

Impact testing Finite element
model


Experimental
method



n

























Figure 2-2. Ball bearing contact angles at high speed rotation.


Figure 2-3. Impact testing performed on standard artifact with modal hammer and laser
vibrometer.





S I I I l Ilu lll ilff I
01.08 0.1
time(s)


0.1i8


I I I I I I I I


-100 C


0 0.02 0.04 0.06i 0.08
timre8s)


0.1 0.12 0.1f4 0.6 0.18


Figure 2-4. Example of time-domain simulation of an unstable cutting process.



.......... Measured
1 Pedicted





0 500 1000 150 2000 2500

x0-7







0 500 1000 1500 2000 2500
Frequenc~y (Hz)


Figure 2-5. Predicted and measured FRF of a 100 mm diameter inserted endmill.


20:0




ILL
60

0


//I )/I 1/11/]11 Ill~li 111 11111// 11 iii









CHAPTER 3
RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS

In this chapter, the j oining of tool, holder, and spindle dynamics using receptance coupling

substructure analysis (RCSA) is described. Two approaches are detailed, including the basic two-

component coupling and the improved three-component coupling. The latter includes a

decomposition step to identify the spindle dynamics. The decomposition procedure, referred to

as inverse RCSA, is also described.

Two-Component Receptance Coupling Substructure Analysis

RCSA can be applied to combine the dynamics of the cutting system components. A

typical high speed milling machine cutting system includes the machine, spindle (shaft, bearings,

and drawbar for gripping the tool holder), tool holder, and tool. The RCSA concept is to j oin

multiple components (with each of the component' s dynamic response known) by means of rigid

or flexible connections (the latter can be modeled as springs and dampers) and predict the overall

assembly dynamic response.

Rigid Connection

Consider the system shown in Fig. 3-1, where A and B are two components and C is the

assembly of A and B with a rigid connection. The displacements of the components and

assembly, x, can be written as shown in Eqs. 3.1 through 3.3.

x, =Ha (~ ) fa (3.1)

xb =Hb (i b (3.2)

Xe =G.(mi,) Fe (3.3)










Here H, is the receptance, or frequency response function, for component j, G, is the

receptance for assembly j, J is the component force, and F, is the assembly force. All receptances

are functions of frequency, co~. For rigid coupling, the compatibility conditions are;

xa = xb = Xe (3.4)

and the equilibrium conditions are

,fa b =Fe. (3.5)

By substituting Eqs. 3.1 through 3.4 into 3.5, the result in Eq. 3.6 is obtained. The

receptance Go describes the assembly frequency response as a function of the component

receptances.

G = (H,' +H l) 1 (3.6)

Non-Rigid Connection with a Linear Spring

For non-rigid connections between components, consider the case shown in Fig. 3-2. Each

mass is connected to the wall with a linear spring and a viscous damper; the spring-mass-damper

combination is treated as a component. The connection between the components is a linear

spring, ke. To determine the assembly receptances, F, is first applied at assembly coordinate X,

and the equilibrium and compatibility conditions are written as shown in Eqs. 3.7 and 3.8.

f, + f2=F, (3.7)

X, = x,, X2 2 X, and k.(x2 1 2,)=-f (3.8)

Following the same procedure as above, the receptances of the first column of the 2x2 (two

degree of freedom) assembly receptance matrix G are calculated. G, denotes the receptance at

position i under the forcing at position j.









X, 1
G~z(m) H,, -H,, (H22 +H,, +-) H,, (3.9)
R, k

X, 1
G21= _, 2H2(H22 +H,, +-) H,, (3.10)
R, k

In the same manner, if F2 is applied to assembly coordinate X2, the second column of the

receptance matrix can be determined (Eqs 3.11 and 3.12).

X, 1
G12, 1H2(H22,+ H,,+-) H,, (3.11)
F, k

X, 1
G22 2, H2-H22 (H22 +H, +- ) H22 (3.12)
F, k

Here H, and H22 are the receptances of components A and B and are listed below.

1 0
Hz, () = (3.13)


1 0
H22 2 n2 (3.14)
k2 mi, + i2rZ2 n2~i n)22

The complete receptance matrix for two subcomponents is therefore


G(m) = .G (3.15)


Non-Rigid Connection with a Linear Spring and a Damper

Figure 3-3 shows the non-rigid assembly with both a linear spring and a viscous damper.

To find the direct receptance of the assembly at coordinate X2, G22 a force F2 WaS applied at X2.

The equilibrium and compatibility conditions are shown respectively in Eqs. 3.16 and 3.17.

f, + f2 = F2 (3.16)


X,=X,=xY, 2 2, and k.(x, x2 )c. 1, 2 1,)=- (3.17)










Setting F=Fe ""' and x=Xe ""tto denote harmonic motion, Eq. 3.17 can be re-written as

shown in Eq. 3.18.

k e( x, x, )+ iecc (x, x, ) = f, (3 18)

Writing the complex stiffness as K, = k,+imi c., it is obtained


Kc( x, x, )=- f,. (3.19)

For x, = Hz, f, and x, = H,, f,, Eq. 3.19 can be written as

K, (H,, f, H,, f, ) = f, (3.20)

The forces can then be written as


fi = (H,, + H,, +-)-'H,,F, and (3.21)
KC


fi = F, f, = (1 -(H,, + H,, +-)' H, )F (3.22)


The direct receptance at assembly coordinate Xt is then

X xz Hzfi 1
G (m)~ H, H, (H, + H,, + -)'H,,(3.23)


where K = k +imi co

Non-Rigid Connection with Linear and Rotational Springs and Dampers

The milling cutting system can be modeled as shown in Fig. 3-4. As can be seen, the

connections between tool A and holder/spindle B consist of a linear spring kx rotational spring

k,, linear damper cx, and rotational damper c,. The rotational spring and damper are required

to restrict rotation, while the linear spring and damper restrict translation. Rotation must be

considered in this case due to the bending of the tool during application of a force to its free end.










The assembly in Fig. 3-4 has six degrees of freedom. The complete receptance matrix G

(6x6) is shown in Eq. 3.24.

X, G,, G12. G16 F,

XXX F ,F
3 3 (3.24)



1 ,B G, G M,

In the cutting operation, the tool tip is generally considered to be acted upon by a force, but

no moment, so M;1 is set to zero. Notation for the component receptances when both rotations and

translations are to be considered is listed in Eq. 3.25.



R, ~m (c)-, (3.25)




Referring to the components in Fig. 3-4, the following expressions may be written as

x1 = Hz, f, + H12 2 + L12 2 (3.26)

6, = N,, f, + N12f 2 2 2 (3.27)

x2 = H21 1 + H22 2 + L22 2 (3.28)

82 21 1,f 22 2,f P22 2 (3.29)

x3 = H33 3 + L33 3 (3.30)

83 = 33 3 P33 3 (3.31)

To determine the tool tip receptance, a force F, is applied at the tool point (i.e., the free

end of the tool) in the assembly. The corresponding equilibrium conditions are

f, = F,, f2 3 f = 0, and m2 3 m = 0 (3.32)









The compatibility conditions are

x, = X,, O, = O,, i=1, 2, 3 (3.33)

Ikx(x; x ) = f3
~kg (0 6 ) = -nzz334

Here damping in the equations is not included for simplicity of notation. Damping can be

combined with stiffness at the end of derivation as described previously.

By substituting f, = f,, na, = -m,, the following expressions are obtained

Ix, x, = (H, + HZ) fi +(Li + L, )nz -H ,f,
0 3 8, = (N?, + N ,) f + (P,, + P Z)nz NJ f, 35

Substituting Eq. 3.35 into 3.34 and re-grouping gives

k, (H,, + H,,) +1 k, (L,, + L,) f, lf kH,,l~f f .6
kH(N3? + NZ) ks (P7-, + PZ) + (1 nz kh N,, f ,;)


Since f, = F,, the force/moment on the components at coordinate 3 are obtained as



ngj k,N?, 1F, j

Expressing Eqs. 3.26 and 3.27 in matrix form gives


-a1J;: + 2\I~ (3.38)


Letting x, = X,, 8, = O,, f, = F,, f, = fs,n = -m,, and applying Eq. 3.37 gives



= -~ [A] (3.39)


Finally the direct receptances at tool tip are obtained as











-"1 -1_ [AF .jr, rNH2L~rlr"2 (3.40)
G(411 LxN,, Nz Lk,N,,


For combined spring and damper connections, following substitution can be applied.

Ik` JKx I k, + icxc`
->, = w (3.41)

Three-Component Receptance Coupling Substructure Analysis

In previous section, coupling of the milling cutting system was described. Though this

model allows two entities (the spindle-holder and tool) to be joined with linear and rotational

springs and dampers, the model must be recreated for each new holder in a given spindle. An

improved three-component RCSA approach, which is capable of coupling arbitrary holder and

tool combinations to a selected spindle, is now described.

Consider the generic case (Fig. 3-5) of two rigidly coupled substructures that defined an

assembly with end coordinates, a and d, subj ect to forces and moments acting on each.

Follow the two-component RCSA approach, the assembly receptance matrices can be

expressed as shown in Eqs. 3.42 through 3.45, where the 2x2 R,, matrices include the

translational and rotational component receptances.


Goo=L~H No Po Ro-R,,[R, +.R' R (3.42)


G L,H, R,, Ree(R, + R' R (3.43)



Gac/ Had Lad/ ]- R,[R, +R, ] Rec (3.44)










G;, ::;[% dc[b R (3.45)


Using the receptances in Eqs. 3.42 through 3.45, the subscripts can be updated to match the

cutting system described in Fig. 3-6. The three components are the tool (I), the extended portion

of the holder (II), and the spindle base (combination of spindle and the holder from its tapered

interface to the flange, coordinate 3a). To match the geometry shown in Fig. 3-6, components I

and II (with free-free boundary conditions) are first rigidly coupled to form a new component

with end coordinates 1 and 3, similar to the right component in Fig. 3-6. It is by industry

standard that the holders with same type of taper interface have identical flange geometries. The

vendors then design the extended part of the holder depending on the cutting application.

Therefore, it is reasonable to regard the spindle and the standard portion of the holder to be one

entity. By using three-component RCSA, arbitrary holders and tools can be modeled using beam

theory (either Euler-Bernoulli or Timoshenko). The tool point receptance matrix for the three-

component cutting system (Fig. 3-6) is written as


G,, = =,, -R13[R33 3 + R3a3a -1R31 (3.46)


Inverse Receptance Coupling Substructure Analysis

As noted, the tool and holder can be conveniently modeled using beam theory so that the

R11, R13, R33, and R31 receptances in Eq. 3.46 can be determined. However, the spindle base

receptances, R3a3a, are HOt conveniently modeled. In particular, even if the spindle geometry and

tolerances are known, the damping is difficult to estimate from first principles. Therefore,

inverse RCSA is used to analytically decompose a spindle-'standard holder' assembly into its

components (as opposed to the synthesis, or coupling, procedures described in the previous

sections). This method uses the FRF measurement of the assembly and removes the portion of









the simple geometry standard holder beyond the flange to identify the spindle dynamics alone.

This makes it possible to predict any holder-tool combination by coupling the holder and tool

models with the spindle response and eliminates the need to complete an impact test for each

individual assembly.

By taking the advantage of the standard flange geometry for commercial holders with the

same spindle connection type, the dynamic response of the spindle, combined with the holder

taper (inserted inside the spindle) and flange (referred to as the spindle-holder base), by

"detaching" the extended part of the standard holder using inverse RCSA. This makes the

spindle-holder interface (whether the connection is rigid or flexible) a part of the spindle

receptances and further study of the connection parameters is not required. An example standard

holder artifact is shown in Fig. 3-7. Rigid connections between the constant cross-section

sections may be assumed because the standard holder is monolithic by design.

Here notation for different types of receptance matrices is provided

R, single component receptance matrix

RS, coupled substructures (subassembly) receptance matrix

G, final assembly receptance matrix.

The assembly receptance matrix at the holder free end, coordinate 1, is obtained by the

following procedure.

(1) Measure H,, (excite the structure and measure at coordinate 1) and H,,, (excite the

structure at coordinate 1 and measure at la) by impact testing. The distance between the direct

and cross FRFs, respectively, is denoted as S.

(2) Calculate N,, by first order finite difference using H,, and H,,,









H1 H1
11_ 1 0 (3.47)


(3) Assume L,, = N,, by reciprocity.

(4) Synthesize P,, using Eq. 3.48.

L11NI L11
P, (3.48)


(5) Obtain G,, .


G,, =,NLF( (3.49)


The S value must be sufficiently large to yield a sufficient amplitude difference between

the direct receptance H,, and cross receptanceH,, for the finite difference calculation ofN,,.

However, if S is too large, a low H,, amplitude can be obtained, which can lead to decreased

signal to noise ratio. The S distance chosen in this research is generally between one third and

half of the extended holder length.

To Eind the spindle-base direct receptance, R,,ss, the R,, component matrices at each

location (1, 2, 2a, 3) are calculated using free-free beam models. The individual R matrices are

then rigidly coupled to form the RS~ matrices for a rigid coupling of components I and II.

RS,, = R,, R,, (R,, + R2,,0) R,, (3.50)

RS,, = R,, R3,,(Rzz + R2,az)-R2,3 (3.51)

RS,, = R,, (R,, + R2U2U) RU, (3.52)

RS,, = R32,(R,, +R2,,2) 'R,z (3.53)









The receptance matrix of the spindle-base is then determined by inverse RCSA (i.e.,

determine the spindle-base component receptances from an assembly measurement, G11, and

component I-II receptances)

R3a3a = RS31 (RS11 G, z)- RS113 33 RS. (3.54)

Given Raa, to the tool point response for arbitrary holder-tool models (given the geometry

and material properties) can then be predicted using RCSA. Note that the preceding model

assumes rigid connections everywhere. As described previously, the model could be modified to

include flexibility between the spindle and tool-holder, for example.

Substructure Beam Modeling

For each tool-holder section of the RCSA model, the components can be represented by

Euler-Bernoulli or Timoshenko beam models.

Euler-Bernoulli Beam

Johnson and Bishop [32] proposed closed form expressions for the translational and

rotational receptances of uniform cylindrical beam under the application external forces and

moments. For a free-free beam with coordinates j and k on each end, the direct and cross

receptance are given as

-F,
h~ = h = (3.55)
] kkElO + i q) 3 F3


h, = h = (3.56)



1 = -1 =(3.57)


F,
1 = = 10(3.58)
]k kyEl + ir)A2F3










nZ = -n =k (3.59)
D kkEI(+i7)/12F3


nZ~ = -n = 1 (3.60)



p, = pkk _6(3.61)
SEI( + i qAF3


p yk ky 7 (3.62)


where E is the elastic modulus, l is the second area moment of inertia, ris the structural

damping coefficient, and


ii = (3.63)


F, = sin il sinh il (3.64)

F3 = COs ilcosh il- 1 (3.65)

F, = cos il sinh il sin il cosh il (3.66)

F6 = COs il sinh il + sin il cosh il (3.67)

F, = sin il + sinh il (3.68)

F, = sin il sinh il (3.69)

Fo = cos il cosh il (3.70)

The cylindrical beam mass m in Eq. 3.63 is given as

Fr d2~ dr12 L
m = (3.71)









where do and d, are the outer and inner diameters of the beam section, respectively, L is the

section length, p is the density, and co is the frequency (rad/s). The beam's second area moment

of inertia is


I =6 o .(3.72)
64

Timoshenko Beam

The Timoshenko beam model includes rotary inertia and shear effects. It is implemented

using the finite element method [33]. Each element has four degrees of freedom (rotation and

displacement at each end of the beam section). The mass matrix M~ and stiffness matrix K

(including damping) for a free-free beam section [34] are given as


pAl
M=
(1 + ~)2
313 70 2 1 2+1 -9 3 2 "
35 103 210 120 24 70 10 6
1 2 i i2 13 30 2
105 60 120 404 4
--13 7 ,2
35 10 3

Svmmetnic


13 3i+" 2
420 40 24

140 60 120

210 120 24
1 223 i
105 60 120

( 31


30 6 61




15 6 3 )


Svmmetric


(3.73)
where A is the cross-section area, I is the section length, r, is the radius of gyration, and # is the

shear deformation parameter given by


1 6 ~)


2
pAl r,
+
(1 + #)2









12EI(1 + r?)
# = ,(3.74)
k GAl2

where


G, =~ (3.75)
2 (1 + v)

is the shear modulus, v is Poisson's ratio, and k' is the shear coefficient which depends on the

cross-section shape and Poisson's ratio [35].

12 61-a) -12 61~')

K EI(1+il7) 4+-# -612- -
l"(1+#)12 -61
Symmetric (4 + 20 + 0921
S(3.76)
4 21 4 21
k'AG4' I2 21 12
41(1+ Of2 4 -21
Symmetric 12

The element M~ and K matrices are then collected to form the global mass matrix M~ and

global stiffness K matrices. Applying Guyan reduction [33], the equation of motion in the

frequency domain is obtained as shown in Eq. 3.77 for n elements.


0, my
xz fi

1 m.
[Mmj + (3.77)



Xn+1 fn+1
O,1 m

To decide which modeling method more accurately describes typical tool-holder

combinations, a fixed-free solid cylindrical steel rod with diameter do = 19. 1mm and lengths










varying from 60 mm to 110mm (typical tool overhang lengths in milling operations) was

modeled using both Euler-Bernoulli beam and Timoshenko beams (25 elements). The results

were compared to commercial finite element software Ansys Workbench 10.0 resultsb(Table 3-



It is observed that the Timoshenko beam model agrees more closely with the Ansys results,

particularly for shorter beams. This is particularly important since the actual tool/holder

geometry is not typically uniform throughout the length and is sectioned into smaller

components with constant cross-sections. To improve accuracy in this study, only Timoshenko

beam models were applied to describe the tool-holder components in the RCSA predictions

provided in the Chapters 4 through 6.

Fluted Tool Modeling

An important consideration in modeling tool-holder combinations is the fluted portion of

solid body endmills. Although the actual geometry of the helical cutting edge (Fig. 3-9) could be

meshed in a finite element model, this is computationally intensive. As an alternative, the

equivalent diameter of the fluted section can be used to determine the required second area

moment of inertia and beam mass for the Timoshenko beam model.

To identify the equivalent diameter, solid models were constructed using Solidworks 2006

for tools with two (Fig. 3-10), three, and four flutes. The tool profiles were approximated from

SGS high speed steel endmills. It was observed that the two-fluted cutter had a second area

moment of inertia which varied with angle. For the cross-sectional model shown in Fig. 3-10, the

ratio of second area moment of inertia for fluted cross-section relative to the cylindrical shank

was calculated. The ratio was computed over 180 deg in 10 deg increments, where the angle for

plane 2 (Fig. 3-10) was taken to be zero. The results are shown in Table 3-2. The average ratios

for two, three, and four fluted tools are 0.4404, 0.3349, and 0.3594 respectively.








For modeling purposes, only the mean value of the area moment of inertia for two-fluted
endmills was considered. Table 3-3 provides the average value of the fluted tool area section

properties for all three tool geometries.


Assembly


Figure 3-1. Components joined with a rigid connection.


xLZ


Components


x











I I I




1 C, I 1 C2

A 2B


Physical system









Components


Figure 3-2. Components connected with a linear spring.


Assembly















Components


Assembly

Figure 3-3. Assembly with linear spring and damper.








| | I-- --


T~oolX

X ,X2,


Holder


3
,1"


-Y1




~01


Components


I

B :
I


Spindle


Milling Cutting System


X3 X2


CX,


Assembly

Figure 3-4. Assembly with linear and rotational springs and dampers.













~d c Risid b a

Xd X,



Figure 3-5. Generic case of two substructures with rigid connection.



I / I3a
Spindle
|3 2a 2 1




II I

's.. Extended

Spindle-holder base holder To


Figure 3-6. Three-component RCSA model for tool-holder-spindle assembly.



r 3a
Spindle 2 la 1
--- 3 2a

I I
II I
I/ I,_________




Spindle-holder base Extended standard holder


Figure 3-7. Spindle and decomposed standard holder artifact.





























001 DO(m)Z
0 0076 O 022

Figure 3-8. Ansys Workbench frequency simulation for 19.1 mm diameter fixed-free steel rod
with 80 mm length.


Figure 3-9. Cutting edge of a four fluted flat endmill cutter.






























Figure 3-10 Soi oe ftofue ndilcoss ctin.l I

T oabe3-1 Fie-reselrdfrtoefeunycmaionbtendfeetba
moelngmehos
LP~~5~.ength -:'-









Tiohnodeigmtos

Be3592e 6 Hz 272 Hz 206 Hz 163 Hz 1332 Hz 1104 Hz

Euler-
75 -4.67 -3. -2.54 -2.1 0 -z1.6 H -1.3 H
Bernoulli
Timoshenko
0.67 0.266 H 0.5 H 6 z 0.47 0.4 10.4










Table 3-2. Average of area section properties for fluted endmills.
Angle Ratio
0 0.4965
10 0.4328
20 0.3701
30 0.3159
40 0.2767
50 0.2572
60 0.2598
70 0.2843
80 0.3275
90 0.3844
100 0.4480
110 0.5108
120 0.5650
130 0.6042
140 0.6237
150 0.6211
160 0.5966
170 0.5534
180 0.4965

Table 3-3. Average of area section properties for fluted endmills.
Ratio of second Ratio of cross-
area moment of section area
Tool type
inertia (fluted/shank)
(fluted/shank)


6~5~4~44.04% 38.27%

2 flutes


33.49%35.66%

3 flutes



~735.94% 35.88%

4 flutes









CHAPTER 4
ROTATING FREQUENCY RESPONSE FUNCTION PREDICTION

In this chapter, a technique for predicting the rotating, or at-speed, tool point frequency

response using the receptance coupling approach is described. Due to a number of sources (e.g.,

gyroscopic effects, bearings stiffness variation, mass unbalance, thermal changes, and surface

contact effects), the tool-holder-spindle assembly dynamics are known to vary as a function of

spindle speed. Figure 4-1 shows the FRFs for the standard holder-spindle assembly at five

selected spindle speeds for the Step-Tec spindle used in this research. Clearly, the response

varies substantially with rotating speed.

In the following sections a time-domain signal filtering technique is described which was

used to eliminate the runout signals produced by the measured tool surface during rotation.

Runout, in this case, can be visualized as a non-round artifact (elliptical-shaped for example)

rotating about a fixed axis. If a displacement (or velocity) sensor is held fixed relative to the axis

of rotation and targets against the artifact surface, a sinusoidal signal will be obtained with

frequency that varies with the rotating speed. To achieve acceptable signal to noise ratios in FRF

measurements, it is necessary to remove the runout content. After runout filtering was complete,

the inverse RCSA procedure was used to isolate the spindle-base receptances. These receptances

were then coupled (via RCSA) with various holders and tools to predict the at-speed assembly

FRF. These FRFs were finally used to generate analytical stability boundaries and cutting tests

were performed to validate the stability limit. These results are also provided.

Runout Signal Filtering

Runout poses a significant challenge in rotating tool response measurements. This inherent

once-per-revolution signal introduces content at the rotating frequency and its harmonics.

Because this content is related to the imperfections in measurement artifact (the standard holder









in this case), it does not influence the milling process dynamics and must be removed (filtered)

prior to computing the stability limit by frequency domain methods.

Runout Signal

As mentioned, runout is generally present in measurement signals for rotating systems. It

can be caused by an imperfect tool surface, poor tool-holder alignment (dominant in micro

tools), bearing deflection, or a combination of all of these. Although it is not a significant

influence in cutting force perturbation, its amplitude is quite often large enough to saturate the

target vibrations during an impact test, for example. Figure 4-2 shows the setup for a rotating

FRF measurement of a Shrinker V40E-075-138SF CAT-40 shrink fit holder attached to a 36000

rpm, 36 kW, Setco milling spindle. Impact testing was performed at the front (free) end of the

holder with the spindle rotating at 5000 rpm; a capacitance probe (non-contact displacement

sensor) was used to record the resulting vibration. As can be seen in the time trace in Fig. 4-3

(hammer impact at 0.05 s), the holder response to the hammer excitation cannot be easily

separated from the resultant signal (summation of holder response and runout signal). In this

case, the initial response amplitude is similar to the runout amplitude. The details of the time-

domain runout filtering are detailed in the next section.

Runout Filtering

The first step of runout filtering is to separate the once-per-revolution signal from the

combined signal. As shown in Fig. 4-4, this was accomplished using a circular encoder (a black

plate with a small section of white) attached to the holder and a laser tachometer which was

aligned normal to the encoder. During rotation, the tachometer optical pickup sensed the dark-

light transitions and generated a corresponding voltage signal. This signal was used to identify

single revolutions of data in the time-domain.









An illustration of runout signal identification strategy is provided in Fig. 4-5. Here the

heavy solid line sinusoidall wave) represents the runout signal within one revolution. It is

confined between any two tachometer once-per-revolution impulses. It can be seen that there are

several intervals that can be selected; note that only those intervals that do not include the

exponentially decaying holder-spindle response should be considered (i.e., beyond 0.35 s in Fig.

4-5). Therefore, a non-rotating impact test (decaying sinusoid) was typically performed in this

testing to determine this critical point. The once-per-revolution signal identified from a single

interval was reproduced throughout the entire time interval to form the synthesized runout signal.

By subtracting this runout signal from the holder resultant response, the time-domain holder-

spindle response at the selected speed was extracted.

Figure 4-6 shows example results for the data collected using the setup in Fig. 4-2 at a

spindle speed of 5000 rpm. It is seen that the response signal after filtering can be distinguished

from the runout noise. Further, after the response decays to near zero levels, an exponential filter

can be applied to attenuate the remaining system noise. The frequency response function for

these signals is provided in Fig. 4-7. These FRFs were generated from an average of 15

measurements each. It is observed that the runout content generates sharp peaks at regular

intervals, which corrupts the desired spindle-holder response and makes the application of

inverse RCSA to identify the spindle-base receptances impossible. Due to the rotation, the

response recorded was influenced by the rotating angle of the spindle shaft. For example, a tool

natural frequency at 1200 Hz, corresponding to 1500 rpm, will have its magnitude attenuated by

at most 21 percent in one oscillation interval. When operating at slower spindle speed, this error

caused by the changing spindle orientation does not contribute significantly to the data collected

and is therefore neglected.









In addition to the time-domain runout filtering, further smoothing of the measured FRFs

was also implemented in order to reduce the noise. This was important due to the sensitivity of

the finite difference approach used to compute the spindle-holder rotational receptances to noise.

Savitzky-Golay filtering was applied (in the Matlab code with third order Savitzky-Golay

polynomial and a frame size of 31) to carry out the FRF smoothing.

The FRF Prediction from Rotating Standard Holder Measurements

Rotating FRF measurements were performed on a Mikron UCP 600 Vario milling machine

equipped with Step-Tec 20000 rpm, 16 kW cartridge-type spindle (HSK-63A holder interface).

A standard holder (geometry and section dimensions provided in Fig. 4-8 and Table 4-1) was

attached to the spindle and a black/white encoder strip added for once-per-revolution signal

identification. A TTI LT-880 laser tachometer was used to target against the encoder strip. A

Lion Precision C1-C capacitance probe was used to record the vibration response (Fig. 4-9) and a

PCB 086BO3 impact hammer was used to excite the system. A MLI MetalMax 6.0 data

acquisition system was used for data collection. To reduce noise, each FRF was computed from

an average of 50 impacts.

Prior to impact testing, the spindle was warmed to a steady thermal state by running it for

40 minutes. The spindle-base receptance, Raa WaS obtained using inverse RCSA following the

descriptions in Chapter 3. In this case, however, the inverse RCSA was carried out using spindle-

holder data collected at various spindle speeds. The distance, S, between H44 and Ha4 On the

standard artifact was 25.4 mm.

To verify the validity of applying RCSA to rotating tools, a simple structure "step" solid

holder was tested first. This solid holder had no tool attached to it; therefore, the potential

influence of flexible connections between sections could be neglected. The geometry and section









dimensions are provided in Fig. 4-10 and Table 4-2. The receptances for the free-free holder

model (beyond the holder flange) was constructed using 25 Timoshenko beam elements in each

section. The material properties (elastic modulus, E, density, p, structural damping, 5, and

Poisson's ratio, v) used in the modeling for steel and carbide are listed in Table 4-3.

The measured and predicted rotating frequency response functions at three selected spindle

speeds are presented in Fig. 4-11. The 0 rpm FRF is shown in comparison to rotating FRFs. It is

seen that, even with the change in the spindle dynamics, RCSA can still provide reasonable

prediction accuracy.

To further verify this approach, two 152.4 mm carbide tool blanks (12.7 mm and 25.4 mm

diameters) were inserted into a Regofix collet holder. The overhang length was 127 mm in both

cases. The geometry of the tool-holder is shown in Fig. 4-12 and the section dimension of the

tool-holder combinations are listed in Tables 4-4 and 4-5 respectively. The spindle speed was

10000 rpm for both. The results are shown in Fig. 4-13 and 4-14. Again, good agreement

between the predicted and measured FRFs is observed.

Stability Boundary Validation

A four-flute 19.1 mm carbide endmill with 76.1 mm overhang and 48.3 mm cutting length

was inserted into a Command shrink fit holder (Fig. 4-15). Its section dimensions are provided in

Table 4-6. Note that the equivalent diameter of the cutter (section I) is corrected by the second

area moment of inertia developed in Chapter 3 (fluted tool modeling). The connection between

the tool and holder was assumed rigid.

Based on the measured spindle-base receptances over a range of spindle speeds, the

frequency response functions for the tool-holder-spindle were constructed. The corresponding









stability lobes for 100% radial immersion (i.e., slotting) conditions computed from the (0,
10000, and 16000 rpm) FR'Cs are- shown in Fig. 4-16


Figure 4-17 shows the cutting test setup for a four-flute, overhang 76.1 mm, 19.1 mm

carbide endmill clamped in a shrink fit holder. Two capacitance probes were used to record

vibrations of the tool shank in both the x (feed) and y (perpendicular to feed in the plane of the

cut) directions. The tangential and radial cutting force coefficients for the tool/6061-T6

workpiece material combination, kt = 527 N/mm2 and kr = -33 N/mm2, TOSpectively, were

determined from separate dynamometer cutting tests as described in Altintas [36].

Significant disagreement between the stability boundaries is observed. Therefore, cutting tests

were also performed to verify this behavior. A variance ratio, R technique, was developed by

Schmitz [37]. In this method, the x and y displacements are sampled once-per-revolution to

identify synchronous (stable) and asynchronous (unstable) behavior. The statistical variance in

the once-per-revolution data is normalized by the variance in the overall shank motions to obtain

R as shown in Eq. 4. 1, where 02 indicates statistical variance, the x and y subscripts refer to the

x and y directions, respectively, and the rev subscript identifies the once-per-revolution sampled

data. For stable cutting, the tool deflections tend to repeat each revolution so the once-per-

revolution variance values and, subsequently, the R value are small. On the x-y deflection map,

data is observed over a confined region. For unstable cutting, high R values are obtained because

the vibrations occur near the system natural frequency associated with the most flexible mode

and the vibrations are no longer synchronous with rotation. Therefore, the once-per-revolution

data points are spread out on the deflection map.









In Fig. 4-18, example test results are provided for 8000 rpm slotting tests. It was observed

over many tests that cuts were stable when R was < 0.2, marginally stable for 0.2 < R < 0.8, and

unstable when R > 0.8. A comparison between the at-speed and non-rotating stability boundaries

is shown in Fig. 4-19. The lobes generated from the predicted rotating FRFs agree more closely

with experiment. In the spindle speed ranging from 8000 to 10000 rpm, in particular, the stability

boundary, b hm, iS well identified.


x 107







-0 rpm
2500 rpm
-7 5000 rpm
1x 10 -- 7500 rpm
10000 rpm







0 500 1000 1500 2000 2500
Frequency (Hz)

Figure 4-1. Speed-dependent FRFs of the standard holder at five different spindle speeds: {0,
2500, 7500, and 10000} rpm.






















Figure 4-2. Example setup for rotating FRF measurement.


holder response + runout
runout signal


!I



I:i


0
-0.2 ii'
-0.4
-0.6 "


0.05


0.06


0.07
Time (s)


0.08


0.09


Figure 4-3. Example of holder and runout signal resultant response.


II





Modal hammer


Spindle


)7


Laser tachometer


Capacitance
probe


Figure 4-4. Scheme of tachometer-aided runout filtering setup.


5


~OT,


0.1 0.2 0.3 0.4
Time (s)


0.5 0.6 0.7


Figure 4-5. Illustration of once-per-revolution signal identification (solid holder-spindle
response; dotted runout; dashed tachometer signal; and heavy solid line runout in
one revolution).















-response


0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12
Time (s)

Figure 4-6. Example of shrink fit holder time-domain runout filtering.










x 108
I-- urifilItered







-5


x 108




m -



-8
0 500 1000 1500 2000 2500 3000
Frequency (Hz)


Figure 4-7. Frequency response function comparison with/without time-domain filtering.




4 4a 5 Sa 6 6a




I II

Extended holder Spindle-holder base


Figure 4-8. Short hollow standard holder geometry and substructure coordinates.































Figure 4-9. Experimental setup of standard holder (HSK-63A interface) rotating FRF
measurement.





5 5a 66










I I Ill IV V



Figure 4-10. Solid holder geometry and substructure coordinates.














i~;-;F~..L~,-ar~,~,...... ~


x 10-7


x 107


S measured 10000 rpm
-measured 12000 rpm
- measured 16000 rpm
----- predicted 10000 rpm
********** predicted 12000 rpm
Predicted 16000 rpm
-measured 0 rpm


300


600


900 1200
Frequency (Hz)


1500


1800


Figure 4-11. Solid holder FRF measurement and prediction at {10000,~C 12000,C andl 16000}C~ rm.




6a
2a 3 3a 4 4a 5 5a 6--

1 2 --a




I II Ill IV V


Figure 4-12. Geometry of Regofix collet holder with tool.



























-


---measured 10000 rpm -I
Predicted 10000 rpm
-measured 0 rpm


x 105
2


1
Z





-2


x 105
1



-1
E 2


-measured 10000 rpm
Predicted 10000 rpm
-measured 0 rpm


1000 1500
Frequency (Hz)


2000


2500


Figure 4-13. Measured and predicted FRFs for Regofix collet holder with 12.7 mm diameter 127
mm overhang carbide tool blank at 10,000 rpm.


x106


U.3
E
O
m
a,
rr -0.5


x10
5


1000 1500
Frequency (Hz)


2000


2500


Figure 4-14. Measured and predicted FRFs for Regofix collet holder with 25.4 mm diameter 127
mm overhang carbide tool blank at 10,000 rpm.




























7

6

5

E4



2

1

0


0.8 1 1.2 1.4
Spindle speed (rpm)


1.6
x 104


Figure 4-16. Stability lobes for FRF measurement at 0 rpm and predictions at {10000 and
16000} rpm for 19.1 mm diameter, four flute, carbide endmill clamped in Command
shrink fit holder.


I II Ill IV V VI Vll Vlll


Figure 4-15. Geometry of 19. 1 mm diameter, four flutes, carbide endmill with 76.1 mm
overhang length clamped in Command shrink fit holder.




































Figure 4-17. Setup for cutting tests. The tool shank deflections were measured using two
orthogonal capacitance probes. A laser tachometer was used to obtain the once-per-
revolution signal.











x 10-6


X 107
4


B)












1~: 1 100)00


-1 -0.5 0 0.5 1
x (m) x 10 6


3



2

S1.5
1
0.


133 Hz runout
frequency









C500l 1000 I
Frequency (hz)


x 10-6


-1 -0.5 0 0.5 1
x (m) x 10 *


Frequency (hz)


Figure 4-18. Example results for 8000 rpm slotting cuts. A) 2 mm axial depth, x vs. y
displacements with once-per-revolution samples stable with R = 0.16. B) 2 mm,
spectrum of y magnitude only synchronous content is observed (frequencies
identified by diamonds). C) 2.5 mm, x vs. y displacements with once-per-revolution
samples unstable with R = 1.0. D) 2.5 mm, spectrum of y magnitude chatter
frequency is observed at 782 Hz.





6

5

E4



2

1

0


0.8 1


1.2 1.4
Spindle speed (rpm)


1.6
x 104


Figure 4-19. Comparison of test cut results to predicted stability boundaries determined from 0
rpm (measured) and {10000C~C and 16000}~ rpm (npredicted) FR~s.


Table 4-1. HSK-63A
Sections OD (mm)
I 63.3
II 52.7


short hollow
ID (mm)
44.6
0


standard holder substructure dimensions.
Length (mm)
62.8
16.3


Table 4-2. Solid holder substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 22.0 0 11.2
II 33.3 0 13.5
III 41.3 0 13.2
IV 47.4 0 11.6
V 52.6 0 13.1



Table 4-3. Material properties used in RCSA modeling.
Material Elastic modulus, E Density, p Structural damping, 5 Poisson's ratio, v
(N/m 2 ) (kg/m 3)


7800
14400


Steel
Carbide


2x10 "
5.6x10 "


0.0015
0.0015


0.29
0.22










Table 4-4. Substructure dimensions ofRegofix collet holder with 12.7 mm diameter, 127 mm
overhang carbide tool blank.
Sections OD (mm) ID (mm) Length(mm)
I 12.7 0 127.0
II 62.7 0 25.5
III 42.0 12.7 20.6
IV 42.0 35.1 11.5
V 42.0 0 34.0


Table 4-5. Substructure dimensions ofRegofix collet holder with 25.4 mm diameter, 127 mm
overhang carbide tool blank.
Sections OD (mm) ID (mm) Length (mm)
I 25.4 0 127.0
II 62.7 0 25.5
III 42.0 25.4 20.6
IV 42.0 35.1 11.5
V 42.0 0 34.0




Table 4-6. Substructure dimensions of the endmill-shrink fit holder described in Figure 4-15.
Sections OD (mm) ID (mm) Length (mm)
I 14.7 0 48.3
II 19.1 0 27.8
III 34.1 0 13.1
IV 36.2 0 13.1
V 38.3 19.1 13.1
VI 40.4 19.1 13.1
VII 41.4 19.1 10.9
VIII 41.4 0 37.6









CHAPTER 5
MICRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION

The capability of manufacturing complex, free form, three-dimensional structures from

metals places a high demand on micro machining. The ability to quickly produce miniature

metallic components without the need for specialized tooling makes the operation stand out from

other techniques such as silicon etching and electrical discharge machining. Due to the scale of

the features and required tolerances, the allowable margin of error is limited. Therefore, it is

crucial to have a thorough understanding of the micro cutter dynamics prior to the machining

process.

Due to the micro tool size, traditional impact testing cannot be applied to the cutter to

obtain the tool point frequency response function. In this chapter, micro tool/macro holder-

spindle and micro tool/micro spindle frequency response function prediction and experimental

validation is presented. A guideline for the choice of the proper standard tool artifact (for the

measurements required for inverse RCSA) is also provided to improve prediction accuracy.

Micro Scale Tools on Macro Machine Systems

Prior to the availability of micro spindles specifically designed for micro tools, standard

high speed spindles and holders were applied. A common approach was (and remains to be) to

use appropriately sized collets in standard size collet holders (Fig. 5-1 shows a micro endmill

clamped in a Regofix TC-40 ER25 CAT-40 collet holder). The following predictive approach

and results pertain to this scenario. This is followed by predictions for a micro spindle/micro tool

combination.

Sensor Options for Micro Tools

Micro tools are generally defined as having a cutting diameter of 1 mm or less. Figure 5-2

shows a 1mm cutting diameter, two-flute flat micro endmill. Cutting diameters down to 0.005










mm are commercially available, while the tool shank diameters are typically {3, 3.18, or 4} mm.

The shank then tapers down to the cutting edge diameter. With tools in this scale, contact type

sensors accelerometerss) can not be applied because the additional mass would change the

system dynamics drastically. Capacitance (non-contact) sensors can be applied, but corrections

must be made for a non-flat target (sensing) surface. The complicated shape of the small tools

increases the sensor non-linearity and can introduce errors under significant deflections. A laser

vibrometer provides another non-contact sensor option. Because the laser beam can focus on a

small area (1 mm or less depending on the focusing optics), laser vibrometers provide a feasible

option for small tool response measurement. To minimize difficulties associated with targeting

on the fluted edge, the following tests were conducted on tool blank without fluted cutting edges.

Modeling Description

A CAT-40 solid standard holder artifact was used for obtaining the spindle-base

receptances. The geometry and section dimension are described in Fig. 5-3 and Table 5-1. The S

distance between the direct and cross FRF measurements was 50.5 mm.

The geometry and section dimensions of the CAT-40 collet holder with the inserted tool

blank are given in Fig. 5-4 and Table 5-2, where "OH" denotes the overhang length of the tool

shank and "IL" the inserted length into the collet. The material properties used were the same as

listed in Table 4-3 for the steel holder and tool.

Using standard holder measurements and extended holder substructure models, the

spindle-base receptances, R7ava, were computed according to the previously described inverse

RCSA procedure. The tool-holder-spindle assembly receptances (Fig. 5-4) were then determined

using Timoshenko beam models of the tool and collet holder. Experimental results and

predictions for the assembly were compared for two cases: 1) the cross di splacement-to-force

FRF, H41; and 2) the direct displacement-to-force FRF, H44. In both cases, the response was









recorded using a laser vibrometer located at coordinate 4 (on the collet nut) and the system was

excited using an instrumented hammer on coordinates 1 and 4 respectively. [For these tests, the

direct tool point response could not be measured because the vibration range exceeded the

capabilities of the laser vibrometer used in this study.] The corresponding equations are given in

Eqs. 5.1 and 5.2.


G41 41H, 41= S4 G44(S4 +G4a4a -1G~S41 (5.1)



G4 4 4= S4- S4(G4 GSP,4a4a -~1 GS4 (5.2)


The subassembly receptance matrices (GS notation) in Eqs. 5.1 and 5.2 were computed

using Timoshenko beam models of sub structures I-VI (Fig. 5-4) and the spindle-base

receptances. The GS4a4a matrix was obtained by first rigidly coupling sub structures IV-VI in a

sequential manner to obtain RS4a4a, R4a7, R74a, and RS;7, then rigidly coupling this result to the

spindle-holder base receptances, R7ava, to give the direct receptance matrix for the free end

(coordinate 4a) of this subassembly (Eq. 5.3). This was necessary because the receptances must

be known at any coordinate selected for prediction (coordinate 4 in this case).

G;Saa = RSaa RSa7 (RS,, + Ryan,) RS74a (5.3)

For the coupling of free-free substructures I and II-III, however, the procedure was slightly

different. In this case, a rigid connection was not assumed. Rather, the free-free tool (I) was

attached to the free-free holder section (II-III) using a flexible connection in recognition of the

potential for non-rigid coupling at a physical joint (a collet in this case). A scalar stiffness matrix,

Kc, was included in the compatibility equations for the interface at coordinates 2-2a. The

resulting subassembly matrices GS44 and GS41 were then defined as shown in Eqs. 5.4 and 5.5










G~S44 = RS44 RS42a Rn, + RS a~ + K, RSn2a4 (5.4)


GS41 = RS42a R, + RtS 2aa+ K, RK2, (5.5)


where K, = k k and the RS matrices were obtained by rigid coupling of sections II


and III. In the Kc matrix, the four k entries represent stiffness values that restrict displacement (x)

and rotation (0) due to force (f) and moment (m), as identified by the subscript pairs. As noted,

the final step was to rigidly couple the tool-holder and holder-spindle subassemblies to predict

the cross and direct assembly responses using Eqs. 5.1 and 5.2, respectively.

Experimental Setup

Three steel tool blanks (lengths of 66.7 mm, 57.2 mm, and 47.6 mm) were used in the

measurements. Each tool had six insertion lengths from 6.4 mm to 14.3 mm with an increment of

1.6 mm. A PCB 084Al7 impact hammer and a Polytec CLV 1000 laser vibrometer were used to

measure the dynamic response. A total of 18 sets of FRFs with different tool overhangs (ranging

from 33.3 mm to 60.3 mm and numbered 1 to 18) are shown in Fig. 5-6. For these tests, the tool

was excited at its free end (using the instrumented hammer) and the response was measured at

coordinate 4 on the collet holder, attached to a 36000 rpm, 36 kW Setco spindle (not rotating),

using the laser vibrometer. The experimental setup is shown in Fig. 5-5.

It can be observed that in Fig. 5-6 that, at the tool lengths corresponding to measurements

4, 9, 12, and 13, the magnitudes are reduced for both the predictions and measurements. This is

due to the interaction between the tool (cantilever) mode and the spindle-base modes (located at

730 Hz, 950 Hz, and 1410 Hz). Two figures are given below to show the tool-spindle interaction.

As can be seen in Fig. 5-8, the tool mode amplitude is reduced because its frequency falls close









to a spindle natural frequency. Even though the tool is longer in Fig. 5-8 (relative to Fig. 5-7), the

dynamic stiffnesses (or peak to peak amplitude of the real and imaginary parts) are similar.

The contact stiffness matrix K, in Eq. 5.4 was adjusted to visually fit the predict data with

the measurements. For the predicted curves shown in Fig. 5-7 and 5-8 and the bottom panel in

Fig. 5-6, the Ke values were assigned individually to different tool overhangs. The four elements

of K, were assumed to have the relationship shown in Eq. 5.6; k,, and kg- were assumed to be

the same by reciprocity. The order of magnitude of the individual k values were decided by

commercial finite element software Ansys.

k, (N /m) = MF~ 1x 105

kxm~(N/rard)= ktx = MiF -1Ix 104 (5.6)

km(N -m/lrad) = M~F 1x102

The multiplication factor, M~F, was the only value used to adjust the stiffness matrix K, Its

trend is shown in Fig. 5-9. The stiffness matrix can be used to predict the tool point response for

various overhangs using a limited number of assembly measurements (at the longest and shortest

tool overhangs, for example). The overhang range provided in this experiment covers the

practical usage of micro tools.

Measurements and predictions for the 18 H44 direct FRFs are provided in Fig. 5-10. The

linearly decreasing stiffness values were again applied at the tool-holder interface. The

interaction between tool and spindle modes is seen. The real and imaginary parts of the measured

and predicted responses for measurements 9 and 12 are shown in Figs. 5-11 and 5-12,

respectively.









Micro Scale Tools on Micro Spindles

As noted previously, specially designed direct drive micro spindles are commercially

available for micro cutting operation. Figure 5-13 shows a 50000 rpm NSK HES 500 electric

micro spindle (ball bearing spindle support). Air-powered spindles equipped with air bearings

are also available for higher speed operation; the torque is generally quite low, however. The

taper interface design enables the NSK spindle to be coupled to commercial macro spindles

(which are locked from rotation during use of the micro spindle) so the machine axes can be used

to locate the micro spindle relative to the part.

The NSK micro spindle design combines the holder and spindle into one unit (as do other

typical micro spindles). Users only have to insert a proper collet to accommodate tools. The only

varying geometry is the endmills. Therefore, it is reasonable to regard the whole spindle (shown

in Fig. 5-13) as the "spindle-base" as previous described without considering the extended holder

section. In this study, the spindle-base was determined by inserting a blank in the collet,

performing the necessary direct and cross FRF measurements on the "standard artifact" and then

removing the blank in simulation to isolate the spindle-collet response.

The S Value Consideration

Because the dynamics of small scaled tools are very sensitive to the dimensional variation,

particularly length, the proper selection of the standard artifact geometry was explored. First,

comparisons of predictions results from varying S values (distance between the direct and cross

impact testing locations on the standard artifact) for the calculation of the spindle-base

receptances was considered. Second, the influence of the standard artifact length of prediction

accuracy was assessed. This issue was viewed as being potentially very important given the

results from the previous sections that showed a dependence of the connection stiffness on tool

overhang length for the macro scale collet holder-micro tool connection.










The procedure used to calculate the micro spindle receptances was the same as for the

macro systems. Impact testing was performed on the NSK HES 500 electric micro spindle

mounted in a Mikron UCP 600 Vario milling machine Step Tec 20,000 rpm, 16 kW spindle

(HSK-63A interface). A 3.18 mm diameter steel rod was inserted into the micro spindle collet to

act as the standard artifact (Fig. 5-14). To enhance joint stiffness consistency, the insertion length

for all measurements was fixed at 13 mm. A PCB 084Al7 modal hammer was used to excite the

standard artifacts at the free end and a Polytec CLV 1000 laser vibrometer was used to record the

vibration. An MLI MetalMax 6.0 data acquisition system was used for data collection.

Prediction results showed no significant differences with the varying S values. This is

because the flexibility of micro tools is high and, even for large S values (i.e., the cross FRF is

performed closer to the fixed end of standard artifact), the tool FRF can still be distinguished and

the signal to noise ratio is acceptable. Table 5-3 shows the S values tested on three standard
artifacts with overhang (OH) lengths of { 17, 20, and 23} mm, respectively. One example resu~lt


for a 17 mm OH standard artifact used to predict the tool point response of a 21 mm OH tool

blank is presented in Fig. 5-15. The S effect is negligible.

Sensitivity of Standard Artifact Length

Apart from the challenges in the physical FRF measurement, the choice of the proper

standard artifact length poses another challenge in accurate micro tool dynamic prediction. In

Fig. 5-16, two artifacts with OH lengths of 17 mm and 20 mm were used to predict the FRF of a

21 mm OH steel rod (all connections were assumed rigid). It is seen that the 20 mm artifact

yields a closer prediction. In a subsequent test (not shown), a 23 mm artifact was used to predict

the 21 mm OH rod response and the results were again not acceptable. Therefore, it is necessary

to have a guideline for selection of the standard artifact length for each tool of interest. This is

offered as an alternative to including a length-dependent "fitting parameter," such as the MFE









value applied to the connection stiffness in the previous macro scale holder results. Examples are

provided in the following section.

Micro Tool Frequency Response Function Prediction

For the purposes of micro tool tip FRF prediction, steel micro tools were fabricated with a

tapered geometry similar to actual micro endmills, but without the cutting edges (Fig. 5-17). The

"tool tip" diameter was 1.5 mm to provide a reasonable target for the laser vibrometer. The

geometry and section dimensions are given in Fig. 5-18 and Table 5-4.

The experimental steps were the same as previously described. However, five standard
artifacts with OH lengths of {31.5, 25.2, 22, 21, andl8.6} ""~mm w~elrese to calculate the spi~ndle-


base receptances. These receptances were then coupled to the tool model (sections I-IV in Fig. 5-

18) to produce the FRF predictions for 23.5 mm and 25.5 mm OH tapered tools (Figs. 5-19 and

5-20). In general, improved accuracy was obtained when the standard artifact OH length was

closer to the tapered tool OH length, In Fig. 5-19, for the 23.5 mm OH tapered tool, it is seen that

the 31.5 mm OH artifact missed the tool and spindle modes. The 21 mm artifact captured the

natural frequencies, but over predicted the tool mode and interacting spindle mode amplitudes.

The 18.6 mm artifact slightly under predicted the amplitudes of the same modes. This

demonstrates a strong sensitivity of the spindle receptances to moment.

Figure 5-20 shows that the 22 mm OH artifact prediction provided a good match to the

25.5 mm OH tapered tool FRF. Again, it is seen that the artifact overhang length must be close to

the actual tool length to produce a good prediction. It is interesting to see that the best FRF

prediction comes from the artifact with the length which is slightly shorter than the tapered tool

length. To aid in understanding this apparent contradiction, the free-free tapered tool and

standard artifact responses were coupled to an infinitely stiff spindle (i.e., a rigid wall). The

results are provided in Fig. 5-21. By observing the relative locations of the tool natural










frequencies, it is seen why 25.5 mm OH tapered tool FRF can be better predicted by the 22 mm

OH artifact, rather than the 25.2 mm artifact. The improved frequency match between the

tapered tool clamped-free natural frequency and 22 mm OH artifact response yields a more

accurate prediction. Because the tool and artifact models (free-free responses) can be

conveniently computed and coupled to a rigid base, this approach provides an effective means

for standard artifact overhang length selection prior to measurements. The outcome is the ability

to predict the tool point response for arbitrary micro tool geometries (in a selected micro spindle)

that would be otherwise immeasurable by traditional means. This data can then be used for

machining parameter selection to avoid unstable conditions (chatter) and unacceptably large

forced vibrations.





II


I~L;


ii~-


Figure 5-1. Example setup of high speed machining with micro tools. A) High speed machine
with micro endmill clamped into CAT-40 collet holder. B) Schematic detail of a 3.18
mm diameter tool shank inserted into an ER-25 collet.


Figure 5-2. Dimension comparison of a 1mm diameter, two-flute micro endmill to a penny.

7a


"11

Extended holder
Spindle-base


Figure 5-3. CAT-40 standard holder artifact geometry.










7a 4 3a 3 2a
Spindle -1 ,I 7 6a 6 Sa 5 4a





,/ OH




Spindle-base Extended holder


Figure 5-4. CAT-40 ER-25 collet holder and tool geometry.


Figure 5-5. Experimental setup for CAT-40 collet holder with 3.18 mm steel tool shank.


























































"200 400 6500 800 1000 1200 1400 1600 1800
-requency :I-in


2000


Figure 5-7. Measured and predicted H41 With 46.0 mm tool overhang (marked as 9 in Fig. 5-6).


E (I




200 40 00 000 W 1000 1200 1400 1400 1800J 2080





Frequency :nrl


Figure 5-6. Magnitudes of H41 FRFs measurements (top) and predictions (bottom).












~C"~------


x 10


1C


h
Z
-- -1
($
a:


Measuredr
Predicted
16005 1800S 2000


200S 4001 600


800 1000 1200 14800


:[00


1400 1600 1800 12000


400i 500 800 10 1200
Frequency (Hz)


Figure 5-8. Measured and predicted H41 With 50.8 mm tool overhang (marked as 12 in Fig. 5-6).


Figure 5-9. Multiplication factor (M~F) determined from visual fit in relation to the tool overhang
with a linear approximation superimposed.


E
8-1(5
E
I `I




























































111111


, 101


200 400 600 800 1000 1200 '1400 1600 1800 2000


Frequency (Hz)


Figure 5-10. FRFs ofH,, magnitude measurements (top) and predictions (bottom).




xlD~"
-- Mc~s4ir3
Ik i?~ b~ I ---; P~dKted



~I


200


Z


4030 600 800 1l000 1200
Freqluency (Hz}


1400 16001 1800 2000)


Figure 5-11. Measured and predicted H, with 46.0 mm tool overhang (marked as 9 in Fig. 5-
10).
























-1 .
E 2


x 10T


'200 400 600 6030 1000 1200
Frequency HU~


1400 1600 1800


2000


Figure 5-12. Measured and predicted H, with
10).


50.8 mm tool overhang (marked as 12 in Fig. 5-


Figure 5-13. NSK HES 500 electric micro spindle.






























Modal
hammer


Figure 5-14. Experimental setup for determination of micro spindle-base receptances.










x 105


-2




Hz

x 105



E measured
-2
I IIV ~-- S = 3mm
E --- S = 6mm
-4
SS = 9 mm
0 2000 4000 6000 8000 10000
Frequency (Hz)

Figure 5-15. Prediction of tool point FRF for 3.18 mm diameter, 21 mm overhang steel rod by a
17 mm overhang standard artifact with three different S values (3 mm dashed, 6 mm
dotted, and 9 mm solid) in comparison to measured data (heavy solid line).












x 10-s


Z

O
co
a,
I~


x105


Z
E
"-2
co
E


2000 4000 6000 8000
Frequency (Hz)


10000


Figure 5-16. FRF prediction of a 3.18 mm diameter, 21 mm overhang steel rod by 20 mm (solid)
and 17 mm (dotted) overhang standard tool artifacts in comparison to measured data
(heavy solid).


Figure 5-17. Example 3.18 mm shank diameter tapered tool (no flutes) with 1.5 mm diameter
tool tip.














II IV


I III V-







Tapered tool
Spindle-base


Figure 5-18. Geometry of NSK HES 500 micro spindle with tapered tool.


x105


x 10-s


0

-5

-10


measurement
-- 31.5mm artifact
S21mm artifact
18.6mm artifact
6000 8000 10000
(Hz)


2000


4000
Frequency


Figure 5-19. Tool tip measurement of tapered tool with an overhang of 23.5 mm compared to
predictions based on different standard artifact overhang lengths.










x 104


0.5 -1



a~-0.5
measurement
S31.5mm artifact
S25.2mm artifact
x 10-4 22mm artifact









0 2000 4000 6000 8000 10000
Frequency (Hz)

Figure 5-20. Measurement of tapered tool (with overhang length of 25.5 mm) FRF compared
with selected standard artifact FRF predictions.





x 10-3


x 10-3


31.5 mm
artifact


25.2 mm
artifact


22 mm
artifact


21 mm 18.6 mm
artifact artifact


0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
Frequency (Hz)


Figure 5-21. FRFs for free-free tapered tool and standard artifact responses coupled to rigid
spindle receptances.


Table 5-1.
Sections


CAT-40 standard holder artifact substructure section dimensions.
OD (mm) ID (mm) Length (mm)


50.5
44.5


66.5
17.4


Table 5-2.
Sections
I
II
III
IV
V
VI


CAT-40 ER-25 collet holder and tool substructure section dimensions.
OD (mm) ID (mm) Length (mm)
3.2 0 OH
42.1 3.2 21.6-IL
42.1 3.2 IL
40.0 12.0 13.4
40.0 18.4 29.6
44.5 5.0 12.0









Table 5-3. S values for micro standard artifacts
OH (mm) S value (mm)
17 3 6 9
20 3 6 9 12
23 3 6 9 12 15


Table 5-4. Tapered tool (23.5 mm OH) substructure section dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 1.5 0 3
II 1.78 0 0.78
III 2.34 0 0.78
IV 2.9 0 0.78
V 3.18 0 18.2/20.2*
for 25.5 mm tapered tool









CHAPTER 6
MACRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION

The application of receptance coupling substructure analysis technique to predict tool tip

frequency response function for rotating tools and micro tools was explored in Chapters 4 and 5.

In this chapter, the focus is macro scale milling tools and the importance of the standard holder

geometry on prediction accuracy. Impact testing was performed with selected standard holders

on various spindles and holder-spindle interfaces to evaluate the influence of standard holder

geometry on dynamics prediction. Based on the results of this testing, a guideline for selecting

the most suitable standard artifact for improved tool response predictions using the three-

component RCSA method.

Variation in Spindle-Base Receptances with Standard Holder Geometry

In theory, the calculated spindle-base receptances for a certain spindle should be

independent of the standard holder used to acquire the system response. [Recall that the inverse

RCSA step removes the extended portion of the standard holder to leave just the spindle-taper

(and flange) base receptances.] However, as shown in Fig. 6-1, the HSK-63A spindle-base

receptances calculated from the longer hollow standard holder exhibit a more flexible spindle

response (left shift in frequencies and larger amplitude) relative to the short hollow standard

holder (holder dimensions are provided in Figs. 6-2 and 4-8). This was also observed for spindle-

base receptances determined from CAT-40 spindle-holder interfaces. As seen in the micro tool

response prediction, macro scale interfaces also exhibit moment-dependence for the interface

stiffness (i.e., the longer standard holder leads to a larger moment for the same force level).

Experimental Results

Impact testing was performed on three spindles with different holder-spindle interfaces

(HSK-63A, CAT-40, and HSK-100A). The test platforms were a 16 kW, 20000 rpm Step-Tec










spindle; a 36 kW, 36000 rpm Secto spindle; and an 18000 rpm, 70 kW Omlat spindle (all were

direct drive, ball bearing type spindles). The procedure used to obtain the spindle-base

receptances and model the substructure beam components was the same as described in Chapter



The HSK-63A Interface

The long hollow standard artifact geometry and section dimensions used for the Step-Tec

spindle testing are presented in Fig. 6-2 and Table 6-1. The short hollow standard is the same as

described in Fig. 4-8 and Table 4-1.

Two shrink fit holders, Command H4Y4A1000 and Shrinker SF-100-374, (long and short

shown in Fig. 6-3) were used for FRF prediction and validation. Their substructure dimensions

are provided in Tables 6-2 and 6-3 respectively. Like the standard holders, shrink fit holders are

constructed from a single piece of steel; therefore, the sections are assumed rigidly connected.

Similar to the micro tool study, the S value (distance between direct and cross FRF

measurements on the standard holders) was varied from 25.2 mm to 50. 8 mm (1 to 2 inches) to

verify that it is not a crucial factor for macro tool FRF predictions. Results for prediction of the

HSK-63A long shrink fit holder FRF by the short hollow standard artifact with three different S

values is presented in Fig. 6-4. It is seen that the three predicted curves overlap each other; no

significant influence from the S value is observed. All subsequent tool point predictions were

also completed with three different S values, but results are presented only for S = 25.4 mm for

brevity. The FRF predictions for long and short shrink fit holders are given in Figs. 6-5 and 6-6,

respectively, using both the long and short hollow standard holder spindle-base receptances.

It is seen that both standard holders (or artifacts) provide accurate predictions for the long

shrink fit holder. However, it is obvious that the short standard holder yields better result in the

short shrink fit holder prediction. Table 6-4 shows the ordered extended holder lengths (i.e., the










length beyond the flange) for all the HSK-63A holders used. The long shrink fit holder has the

length in between the standard artifacts by the similar margins; therefore the predictions are

equally accurate. The length of the short shrink fit holder, however, is less than both artifacts;

therefore, the short standard artifact with the closer holder length gives better results.

The CAT-40 Interface

Three standard holders (Regofix TC40ER40x100AD, Kennametal DV40ER40080, and

Shrinker V40E-075-138SF) were used for CAT-40 interface (Setco spindle) impact testing. Their

geometries and section dimensions are given in Figs. 6-7, 6-8, 6-9 and Tables 6-5, 6-6, 6-7,

respectively, for long hollow, short hollow, and short solid holders. Again, different S values

were applied and it was concluded that it is not a factor for the spindle-base receptances. The

following predictions are based on the S= 25.4 mm results.

For the CAT-40 holder FRF study, the tool point (or free end) FRFs for one shrink fit

holder and two collet holders (with no collets inserted) were predicted and measured. The

section dimensions for these three holders are listed in Tables 6-8, 6-9, and 6-10 with section I

representing the free end. The measured and predicted FRFs are shown in Figs. 6-10, 6-11, and

6-12.

As can be seen in Fig. 6-10, the FRF predictions made by the two shorter standard artifacts

(hollow and solid) matched the extremely short shrink fit holder reasonably well, while the long

hollow standard artifact prediction is less accurate. It is also interesting to note that the solid and

hollow holders provided similar performance. This further suggests that it is a moment

dependence of the spindle-holder interface that influences the prediction results, rather than a

cross-sectional modeling issue in the inverse RCSA step. Table 6-11 gives the extended holder

lengths for the CAT-40 standard artifacts and test holders in length sequence. For the long collet

holder FRF prediction (Fig. 6-11), Table 6-11 shows that the collet holder length falls between









the long and short artifacts; therefore, the predictions are similar between all the artifacts and the

results are all acceptable. Figure 6-12 presents the case where the two short artifacts lengths are

close to the short collet holder length; the predictions are very accurate. The long artifact,

however, did not provide acceptable agreement (recall that any disagreement will reduce the

reliability of the predicted stability boundary). Again, it is observed that it is preferred to match

the standard holder length approximately to the holder in question.

The HSK-100A Interface

For HSK-100A interface impact testing, two standard holders were used. Their geometries

and section dimensions are given in Figs. 6-13 and 6-14 and Tables 6-12 and 6-13, respectively,

for the short hollow and long solid standard artifacts.

Two sets of holder and tool combinations were modeled and tested: an HSK100ASF-075-

433 shrink fit holder with a Sandvik A393.T-19 10 175 carbide adapter and two square insert

cutting head; and an HSK100AE-125-472 set screw holder with a Mitsubishi Carbide MBN 10

1000 TB steel tapered ball end mill with one round carbide insert. Their geometries and

substructure dimensions are given in Figs. 6-15 and 6-16 and Tables 6-14 and 6-15. The impact

testing was performed on both the x and y direction of the Omlat spindle. The predictions are

shown in Figs 6-17 through 6-20 for the shrink fit holder-carbide tool combination and Figs. 6-

21 through 6-24 for the set screw holder-steel tool combination.

For both holder-tool combinations, the long solid artifact length was much closer to

extended lengths (Table 6-16). The predictions in both directions all show better agreement than

those obtained using the short hollow artifact spindle-holder base receptances. [Note that the

FRF asymmetry between the two directions was present because the spindle was mounted to a

high speed machine structure.]










It can be seen that the HSK-100A holder predictions are not as accurate, in general, as

those presented for the HSK-63A and CAT-40 studies, which used only holders and not holder-

tool combinations for prediction validation. For the holder-tool combinations, joint stiffness

between the tools and holders was included in the measurement; the model, however, assumed

rigid connections everywhere. For practical application of the RCSA method in industry, the

prediction agreement shown here (provided the appropriate standard holder length is selected) is

acceptable so the rigid connection assumption is maintained, although investigation of tool-

holder interface stiffness could be pursued as a future research topic.




x 107
-long hollow holder







x 107

S1




0 1000 2000 3000 4000 5000
Frequency (Hz)


Figure 6-1. HSK-63A spindle-base receptances calculated by long and short hollow standard
holders.










Spindle ,







Extended holder
Spindle-base


Figure 6-2. Geometry of HSK-63A long hollow standard hollow holder artifact.

















Figure 6-3. HSK-63A shrink fit holders: long hollow (left) and short hollow (right).



















100











x 107


0.5



-0.5

-1 measurement
Short hollow artifact, S = 1"
-short hollow artifact, S = 1.5"
x 10 short hollow artifact, S = 2"
0.


Z

2


1000 1500
Frequency (Hz)


2000


2500


Figure 6-4. HSK-63A FRF predictions for long shrink fit holder by short hollow artifact with
different S values in comparison with measured FRF (heavy solid line).


x10


U.3

o
m
a,
a -0.5


x10


E -5

- 10

-15
0


-measurement
-long hollow artifact
- short hollow artifact


1000 1500
Frequency (Hz)


2000


2500


Figure 6-5. HSK-63A long shrink fit holder FRF predictions using two different standard holders
in comparison with measured FRF (heavy solid line).


.E -1.













x 107

0.5



rr-0.5




x 108

-5



-10 i" long hollow artifact
--- short hollow artifact
-15
0 500 1000 1500 2000 2500
Frequency (Hz)


Figure 6-6. HSK-63A short shrink fit holder FRF predictions using two different standard
holders in comparison with measured FRF (heavy solid line).













-II I

Extended holder

Spindle-base


Figure 6-7. Geometry of CAT-40 long hollow standard holder artifact.




















V Extended holder
Spindle-base


Figure 6-8. Geometry of CAT-40 short hollow standard holder artifact.


I


Extended holder


Spindle-base


Figure 6-9. Geometry of CAT-40 short solid standard holder artifact.











x 107
1.5








-1.5 measurement
og olo rtfc
sort hollow artifact
-7sothllwatfc
x 10 short solid artifact


1-


1000 1500
Frequency (Hz)


2000


2500


Figure 6-10. CAT-40 shrink fit holder FRF
solid line).


predictions in comparison with measured FRF (heavy


x 10


-4


x 107


-2

.-4

E -6


- measurement


long hollow artifact
short hollow artifact
short solid artifact

2000 2500


1000 1500
Frequency (Hz)


Figure 6-11. CAT-40 long collet holder FRF predictions in comparison with measured FRF
(heavy solid line).













x107
5


500 1000 1500 2000
Frequency (Hz)


2500


Figure 6-12. CAT-40 short collet holder FRF predictions in comparison with measured FRF
(heavy solid line).


Spindle-base


Extended holder


Figure 6-13. Geometry of HSK-100A short hollow standard holder artifact.


T
























Extended holder
Spindle-base

Figure 6-14. Geometry of HSK-100A long solid standard holder artifact.


I YVII VI V IV III II I

Spindle-base Extended holder and carbide tool


Figure 6-15. Geometry of Briney HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-
19 10 175 carbide adapter and two square insert cutting head.


[]: Steel
M : Carb~ide


- n l











[]: Steel


IX ... II I


XII XI X

Spindle-base Extended holder and steeltool


Figure 6-16. Geometry of Briney HSK100AE-125-472 set screw holder with Mitsubishi Carbide
MBN 10 1000 TB steel tapered ball end mill with one round carbide insert.




x 10-6
5 ---------- Predicted
z i Measured




0 1000 2000 3000 4000 5000

x 10-6




m-10


0 1000 2000 3000 4000 5000
Frequency (Hz)

Figure 6-17. HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction (y direction)
using HSK-100A short hollow standard holder artifact.











x 106
5 ~ P
SMeasured



-5t 1
0 1000 2000 3000 4000 5000


-106





~-8
0 1000 2000 3000 4000 50(
Frequency (Hz)


00


Figure 6-18. HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction (x direction)
using HSK-100A short hollow standard holder artifact.


x 10-6


f4

2U


0 1000 2000 3000 4000 5000


x 10-



--8
0 1000 2000 3000
Frequency (Hz)


4000 5000


Figure 6-19. HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction (y direction)
using HSK-100A long solid standard holder artifact.


---Predicted
.:1 -Measured










x 10-6
4 .......... Predicted

2 -i Measured-


0 1000 2000 3000 4000 5000


x 10-6






06 i


1000 2000 3000
Frequency (Hz)


4000 5000


Figure 6-20. HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide
adapter and two square insert cutting head measurement and prediction (x direction)
using HSK-100A long solid standard holder artifact.


x 107


h J
Z
E
" O
co
~ _F;


0 1000 2000 3000 4000 5000


-1


x 107


1000 2000 3000
Frequency (Hz)


4000 5000


Figure 6-21. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction
(y direction) using HSK-100A short hollow standard holder artifact.










x 107

0


0 1000 2000 3000 4000 5000



-107


0 1000 2000 3000
Frequency (Hz)


4000 5000


Figure 6-22. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction
(x direction) using HSK-100A short hollow standard holder artifact.


x 107


z

-


0 1000 2000 3000 4000 5000

x 107


2000 3000
Frequency (Hz)


5000


Figure 6-23. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction
(y direction) using HSK-100A long solid standard holder artifact.










x 107


f5 Measured



0 1000 2000 3000 4000 5000

x 107






0 1000 2000 3000 4000 5000
Frequency (Hz)

Figure 6-24. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB
steel tapered ball end mill with one round carbide insert measurement and prediction
(x direction) using HSK-100A long solid standard holder artifact.



Table 6-1. HSK-63A long hollow standard holder artifact substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 52.7 29.9 100
II 52.7 0 10


Table 6-2. Sub structure dimensions for HSK-63A long shrink fit holder (section I at free end).
Sections OD (mm) ID (mm) Length (mm)
I 44.9 25.4 9.5
II 46.3 25.4 9.5
III 47.8 25.4 9.5
IV 49.3 25.4 9.5
V 50.7 27.4 9.3
VI 52.1 27.4 9.3
VII 52.9 0 44.9










Table 6-3. Sub structure dimensions for HSK-63A long shrink fit holder (section I at free end).
Sections OD (mm) ID (mm) Length (mm)
I 41.9 25.2 10.8
II 42.9 25.2 10.8
III 44 25.2 10.8
IV 44.5 26.2 28.9
V 44.5 6.6 7.8


Table 6-4. Extended holder lengths of the HSK-63A standard artifacts and shrink fit holders.
Short shrink fit Short artifact Long shrink fit Long artifact
69.1 mm 79.1 mm 101.2 mm 120 mm


Table 6-5. CAT-40 long hollow standard holder artifact substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 44.4 25.2 100.2
II 44.4 0 19.7


Table 6-6. CAT-40 short hollow standard holder artifact substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 44.4 25.4 40
II 44.4 0 19.9


Table 6-7. CAT-40 short solid standard holder artifact substructure dimensions.
Section OD (mm) ID (mm) Length (mm)
I 44.6 0 57.3

Table 6-8. CAT-40 shrink fit holder substructure dimensions.
Section OD (mm) ID (mm) Length (mm)
I 44.4 12.7 16.2


Table 6-9. CAT-40 long collet holder substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 48.4 38.4 9.8
II 63 38.4 7.8
III 63 36 7.8
IV 50 33.2 10
V 50 30.1 10
VI 50 28.5 23.2
VII 50 15 6.5




































Table 6-13. HSK-100A long soild standard holder artifact sub structure dimensions.
Section OD (mm) ID (mm) Length (mm)
I 51 0 187


Table 6-14. Substructure dimensions of Briney HSK100ASF-075-433 shrink fit holder with
Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head.
Sections OD (mm) ID (mm) Length (mm)
I 13.4 0 30
II 19.1 0 125.5
III 35.3 19.1 10.6
IV 36.4 19.1 10.6
V 37.6 19.1 10.6
VI 38.1 19.1 17.8
VII 38.1 12 31.6


Table 6-10. CAT-40 short collet holder substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 44.7 34.7 9.7
II 63 39.2 7.9
III 63 36.8 7.9
IV 50 34.5 7.2
V 50 32.4 7.2
VI 44.5 31.3 15.9


Table 6-1 1. Extended holder lengths of the CAT-40 standard artifacts and tested holders.
Shrink fit Short collet Short solid Short hollow Long collet Long hollow
holder holder artifact artifact holder artifact
16.2 mm 55.8 mm 57.3 mm 59.9 mm 75.4 mm 119.9 mm


Table 6-12. HSK-100A short hollow standard holder artifact substructure dimensions.
Sections OD (mm) ID (mm) Length (mm)
I 100 80 20
II 85 0 20










Table 6-15. Substructure dimensions of Briney HSK100AE-125-472 set screw holder with
Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round
carbide insert.
Sections OD (mm) ID (mm) Length (mm)
I 26.1 0 10.9
II 27.5 0 10.9
III 28.9 0 10.9
IV 30.3 0 10.9
V 31.8 0 10.9
VI 33.2 0 10.9
VII 34.6 0 10.9
VIII 36.0 0 10.9
IX 37.4 0 10.9
X 38.1 0 46.5
XI 63.5 0 69.3
XII 63.5 12 21.6


Table 6-16. Extended holder lengths of the HSK-100A standard artifacts and tested holders.
Short hollow Long solid Set screw holder Shrink fit holder
artifact artifact and steel tool and carbide tool
40 mm 187 mm 235.5 mm 236.7 mm









CHAPTER 7
CONCLUSIONS AND FUTURE WORK

The improved receptance coupling substructure analysis (RCSA) method for predictions of

the dynamic response for macro, micro, and rotating tool-holder-spindle combinations was

validated by impact testing and cutting tests. The RCSA technique enables frequency response

functions for arbitrary tools and holders, which are convenient to model, to be coupled with

experimentally-determined spindle responses, which are difficult to model (particularly

regarding damping estimates). This capability was demonstrated for the industrially relevant

cases of 1) spindles that exhibit variations in their dynamic response with the commanded

spindle speed; 2) micro-scale cutting tools and 3) macro-scale cutting tools with different

lengths. The ability to analytically couple arbitrary holder-tool combinations to an archived

spindle response (obtained from the inverse RCSA method) enables job shop management to

determine proper tooling in an a priori manner by using the tool point frequency response

prediction to select preferred operating parameters (to avoid chatter, for example) without the

need for costly trial-and-error cutting tests. For micro tool systems, this technique also provided

a feasible approach to determining the tool point frequency response function using a

combination of traditional impact testing and RCSA modeling. The primary contributions of this

research can be summarized in the following points.

Conclusions

* A limitation of traditional impact testing on macro scale tools is that the measurements
must be carried out when the spindle is not rotating. For spindles that exhibit speed-
dependent dynamics, this can lead to errors in the selected operating conditions. By
measuring the spindle-base receptances at speed using a standard holder without cutting
teeth (so that rotating measurement are possible), predictions of the tool point dynamics
are made possible.

* A primary difficulty to measuring rotating frequency response functions is the inherent
runout in the displacement/velocity signal. A time-domain filtering technique was applied
to remove the runout and yield usable frequency response functions.










* It was shown that Timoshenko beam models are preferred to the analytical Euler-Bernoulli
solutions, despite the increased computational burden.

* It was shown that the diameter of the fluted section of cutting tools can be represented by
an equivalent diameter calculated from the second area moment of inertia determined from
a solid model.

* It was determined that the distance between the direct and cross FRF measurements (S) on
standard holders for computation of the unmeasured spindle receptances is not a crucial
factor. However, the value must be selected to provide a reasonable amplitude difference
between the direct and cross responses based on the measurement signal to noise ratio.

* It was demonstrated that improved FRF prediction for macro tools can be obtained by
strategically choosing the dimension of the artifact to approximately match the desired
tool-holder length. It was hypothesized that this is due to a moment dependence of the
spindle-holder connection stiffness. This was evaluated experimentally by comparing
spindle receptances obtained from long and short standard holders. It was seen that the
longer holder (with higher moment) led to a more flexible spindle response.

* Tool point prediction for micro tapered cutting tools was demonstrated. This presents the
only validated approach to obtaining this information known to the author. It was observed
that the micro-scale spindle-tool combinations also exhibited the moment dependence for
the connection stiffness and improved prediction accuracy was obtained if the artifact
length was selected to have a (fundamental) clamped-free natural frequency near the tool
(fundamental) clamped-free natural frequency.

Future Work


Based on the results obtained in this study, the followings issues warrant further

exploration.

* The moment dependent holder-spindle interface stiffness needs to be studied. Finite
element modeling may provide the necessary capabilities to better understand the
mechani sm for thi s experimentally-ob served condition. An improved understanding could
lead to interface designs that do not exhibit this behavior.

* Similarly, improved accuracy may be obtained if the j oint stiffness between tool and
holder is not assumed to be rigid for all cases. Again, finite element software may be
applied to improve the understanding of this j oint. To be practically beneficial, however, it
will be necessary to develop a method to estimate the j oint stiffness based on the
connection type and geometry without requiring a new model to be developed for every
situation (to limit computation times for the predicted tool point response). Damping must
also be considered.










*In this research the tapered micro tool response measurements was limited to diameters of
1.5 mm and larger due the minimum laser beam diameter available from the laser
vibrometer. Additional tests on smaller diameters, combined with cutting test results,
would be beneficial.









LIST OF REFERENCES


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[7] Delio, T., Tlusty, J., and Smith, S., 1992, "Use of Audio Signals for Chatter Detection and
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[8] Jones, A. B., 1960, "A General Theory for Elastically Constrained Ball and Radial Roller
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[9] Chen, C. H., Wang, K. W., Shin, Y. C., 1994, "An Integrated Approach Toward the
Dynamic Analysis of High Speed Spindles, Part 1: Spindle Model," Transactions of
ASME, Joumnal of Vibration and Acoustics, 116, pp. 506-513.

[10] Jorgenson, B. R., Shin, Y. C., 1998, "Dynamics of Spindle-Bearing Systems at High
Speeds Including Cutting Load Effects," Transactions of ASME, Joumnal of Manufacturing
Science and Engineering, 120, pp. 387-394.

[1l] Cao, Y., Altintas, Y., 2004, "A General Mothod for the Modeling of Spindle-Bearing
Systems," Transactions of ASME, Journal of Mechanical Design, 126, pp. 1089-1 104.

[12] Stein, J. L., Tu, J. F., 1994, "A State-Space Model for Monitoring Thermally Induced
Preload in Anti-Friction Spindle Bearings of High-Speed Machine Tools," Transactions of
the ASME, Journal of Dynamic Systems, Measurement, and Control, 116, pp. 372-386.

[13] Bossmanns, B., Tu, J. F., 2001, "A Power Flow Model for High Speed Motorized
Spindles-Heat Generation Characterization," Transactions of the ASME, Joumnal of
Manufacturing Science and Engineering, 123, pp. 494-505.










[14] Lin, C. W., Tu, J. F., Kamman, J., 2003, "An Integrated Thermo-Mechanical-Dynamic
Model to Characterize Motorized Machine Tool Spindles during Very High Speed
Rotation," International Journal of Machine Tools and Manufacturing, 43, pp. 1035-1050.

[15] Li, H., Shin, Y. C., 2004, "Integrated Dynamic Thermo-Mechanical Modeling of High
Speed Spindles, Part 1: Model Development," Transactions of ASME, Joumnal of
Manufacturing Science and Engineering, 126, pp. 148-158.

[16] Smith, S., Tlusty, J., 1991, "An Overview of Modeling and Simulation of the Milling
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[17] Bayly, P., Halley, J., Mann, B., Davies, M., 2003, "Stability of Interrupted Cutting by
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[25] Budak, E., Altintas, Y., 1998, "Analytical Prediction of Chatter Stability Conditions for
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[28] Schmitz, T., Davies, M., 2001, "Tool Point Frequency Response Prediction for High-
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[29] Schmitz, T., Davies, M., Medicus, K., Snyder, J., 2001, "Improving High-Speed
Machining Material Removal Rates by Rapid Dynamic Analysis," Annals of the CIRP
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[30] Schmitz, T., Duncan, G. S., Zahner C., Dyer, J., Tummond M., 2005, "Improved Milling
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[31] Schmitz, T., Duncan, G. S., 2005, "Three-Component Receptance Coupling Substructure
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[32] Bishop, R., 1955, "The Analysis of Vibrating Systems with Embody Beams in Flexure,"
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[33] Weaver, Jr., Timoshenko, P., Young, D., 1990, Vibration Problems in Engineering, 5th Ed.,
John Wiley and Sons, New York.

[34] Yokoyama, T., 1990, "Vibration of a Hanging Timoshenko Beam under Gravity," Journal
of Sound and Vibration, 141, pp. 245-258.

[35] Hutchinson, J., 2001, "Shear Coefficient for Timoshenko Beam Theory," Journal of
Applied Mechanics, 68, pp. 87-92.

[36] Altintas, Y., 2000, Manufacturing Automation, Cambridge University Press, Cambridge,
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[37] Schmitz, T., 2003, "Chatter Recognition by a Statistical Evaluation of the Synchronously
Sampled Audio Signal," Journal of Sound and Vibration, 262, Issue 3, pp. 721-730.









BIOGRAPHICAL SKETCH

The author was born on March 30, 1971, in Fengyuan, Taiwan. In 1993, he was awarded

Bachelor of Engineering in mechanical engineering from Fengchia University, Taiwan.

Following the college graduation, he served as a patrol sergeant and outboard marine

engine technician in Army Coast Patrol, Taiwan and was discharged in 1995.

His graduate study started in January 1997 in the Mechanical Engineering Department at

the University of Florida. Under the late Dr. Ali A. Seireg' s supervision, he was awarded his

Master of Science in 2000. Soon after graduation, he was employed by China Engine

Corporation in Taiwan, in charge of the component development of GS-1 2.0 liter gas engine.

In 2002, he again j oined the University of Florida for doctoral research. Instructed by Dr.

John C. Ziegert and Dr. Tony L. Schmitz, he was engaged in high speed milling machine

dynamics research and was the teaching assistant for the undergraduate Manufacturing

Engineering class. He was awarded Doctor of Philosophy in 2007.

He is a member of Tau Beta Pi, ASME, and served as the president of Christians on

Campus in the University of Florida in 2006 school year.





PAGE 1

1 IMPROVED PREDICTION OF SPINDLE-HOLDER-TOOL FREQUENCY RESPONSE FUNCTIONS By CHI HUNG CHENG A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2007

PAGE 2

2 2007 Chi Hung Cheng

PAGE 3

3 To my family, my grandfathe r, and Lord Jesus Christ

PAGE 4

4 ACKNOWLEDGMENTS The author would like to thank his adviso r Dr. Tony L. Schmitz, for his understanding, patience, and unconditional support. Thanks also go to Dr. John Sc hueller, Dr. Nagaraj Arakere, Dr. Gloria Wiens, and Dr. Jacob Hamme r for serving as committee members. The author also appreciates Dr. John C. Zieg ert for always bringing in new ideas when there is a bottle neck and Dr. Nam Ho Kim for help ing with micro tool finite element simulation. The thank list extends to the members in M achine Tool Research Center from 2003 to 2007, for all the team works and memorable moments together. With special thanks to Mr. Scott Payne and Mr. Vadim Tymianski for always being p artners of crimes when there is the need. Finally the author would like to thank his family for the suppo rt in every aspect, and the saints in the Church in Gainesville for their loving tender care. Praise the Lord. This work wouldnt have been done without the sovereign arrangements.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........9 ABSTRACT....................................................................................................................... ............14 CHAPTER 1 INTRODUCTION..................................................................................................................16 High Speed Machining...........................................................................................................16 Chatter And Stability Lobe Diagram......................................................................................16 Objective...................................................................................................................... ...........18 2 LITERATURE REVIEW.......................................................................................................22 Milling Stability Prediction................................................................................................... .22 Experimental Method............................................................................................................ .22 Predictive (Non-Experimental) Methods................................................................................23 System Dynamics Acquisition........................................................................................23 Stability Analysis.............................................................................................................25 3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS............................................31 Two-Component Receptance Coupling Substructure Analysis.............................................31 Rigid Connection.............................................................................................................31 Non-Rigid Connection w ith a Linear Spring...................................................................32 Non-Rigid Connection with a Linear Spring and a Damper...........................................33 Non-Rigid Connection with Linear a nd Rotational Springs and Dampers.....................34 Three-Component Receptance Coupling Substructure Analysis...........................................37 Inverse Receptance Coupling Substructure Analysis.............................................................38 Substructure Beam Modeling.................................................................................................41 Euler-Bernoulli Beam......................................................................................................41 Timoshenko Beam...........................................................................................................43 Fluted Tool Modeling........................................................................................................... ..45 4 ROTATING FREQUENCY RESPO NSE FUNCTION PREDICTION................................54 Runout Signal Filtering........................................................................................................ ...54 Runout Signal..................................................................................................................55 Runout Filtering...............................................................................................................55 The FRF Prediction from Rotating Standard Holder Measurements.....................................57

PAGE 6

6 Stability Boundary Validation................................................................................................58 5 MICRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION..............73 Micro Scale Tools on Macro Machine Systems.....................................................................73 Sensor Options for Micro Tools......................................................................................73 Modeling Description......................................................................................................74 Experimental Setup.........................................................................................................76 Micro Scale Tools on Micro Spindles....................................................................................78 The S Value Consideration..............................................................................................78 Sensitivity of Standard Artifact Length...........................................................................79 Micro Tool Frequency Res ponse Function Prediction....................................................80 6 MACRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION............95 Variation in Spindle-Base Receptances with Standard Holder Geometry.............................95 Experimental Results........................................................................................................... ...95 The HSK-63A Interface..................................................................................................96 The CAT-40 Interface.....................................................................................................97 The HSK-100A Interface................................................................................................98 7 CONCLUSIONS AND FUTURE WORK...........................................................................115 Conclusions.................................................................................................................... .......115 Future Work.................................................................................................................... ......116 LIST OF REFERENCES.............................................................................................................118 BIOGRAPHICAL SKETCH.......................................................................................................121

PAGE 7

7 LIST OF TABLES Table page 3-1 Fixed-free steel rod first mode fre quency comparison between different beam modeling methods..............................................................................................................52 3-2 Average of area section properties for fluted endmills......................................................53 3-3 Average of area section properties for fluted endmills......................................................53 4-1 HSK-63A short hollow standard ho lder substructure dimensions.....................................71 4-2 Solid holder substructure dimensions................................................................................71 4-3 Material properties used in RCSA modeling.....................................................................71 4-4 Substructure dimensions of Regofix collet holder with 12.7 mm diameter, 127 mm overhang carbide tool blank...............................................................................................72 4-5 Substructure dimensions of Regofix collet holder with 25.4 mm diameter, 127 mm overhang carbide tool blank...............................................................................................72 4-6 Substructure dimensions of the endmillshrink fit holder described in Figure 4-15.........72 5-1 CAT-40 standard holder artifact substructure section dimensions....................................93 5-2 CAT-40 ER-25 collet ho lder and tool substructu re section dimensions............................93 5-3 S values for micro standard artifacts..................................................................................94 5-4 Tapered tool (23.5 mm OH) subs tructure section dimensions..........................................94 6-1 HSK-63A long hollow standard holder artifact substructu re dimensions.......................111 6-2 Substructure dimensions for HSK-63A long shrink fit holder (section I at free end).....111 6-3 Substructure dimensions for HSK-63A long shrink fit holder (section I at free end).....112 6-4 Extended holder lengths of the HSK-63A standard artifacts and shrink fit holders........112 6-5 CAT-40 long hollow standard holder ar tifact substructure dimensions..........................112 6-6 CAT-40 short hollow standard holder artifact substructure dimensions.........................112 6-7 CAT-40 short solid standard holder artifact substructu re dimensions.............................112 6-8 CAT-40 shrink fit holder substructure dimensions..........................................................112

PAGE 8

8 6-9 CAT-40 long collet holder substructure dimensions.......................................................112 6-10 CAT-40 short collet holder substructure dimensions......................................................113 6-11 Extended holder lengths of the CAT-40 standard artifacts and tested holders................113 6-12 HSK-100A short hollow standard holder artifact substructure dimensions....................113 6-13 HSK-100A long soild standard holder artifact substructure dimensions.........................113 6-14 Substructure dimensions of Briney HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head............113 6-15 Substructure dimensions of Briney HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end..........................................114 6-16 Extended holder lengths of the HSK-100A standard artifacts a nd tested holders...........114

PAGE 9

9 LIST OF FIGURES Figure page 1-1 HSM cutting speed ranges for various materials...............................................................20 1-2 Scheme of (flexible) cutting e dge passing through workpiece surface.............................20 1-3 Stable cut and unstable cut (chatter) on a workpiece surface............................................21 1-4 Example stability lobe diagram.........................................................................................21 1-5 Block diagram for cutting proce ss with regenerated wavy surface...................................21 2-1 Different approaches to dete rmine stable cutting conditions.............................................28 2-2 Ball bearing contact angles at high speed rotation.............................................................29 2-3 Impact testing performed on standard artifact with modal hammer and laser vibrometer..................................................................................................................... .....29 2-4 Example of time-domain simulati on of an unstable cutting process.................................30 2-5 Predicted and measured FRF of a 100 mm diameter inserted endmill..............................30 3-1 Components joined wi th a rigid connection......................................................................46 3-2 Components connected with a linear spring......................................................................47 3-3 Assembly with linear spring and damper...........................................................................48 3-4 Assembly with linear and rotational springs and dampers................................................49 3-5 Generic case of two substructures with rigid connection..................................................50 3-6 Three-component RCSA model fo r tool-holder-spindle assembly...................................50 3-7 Spindle and decomposed st andard holder artifact.............................................................50 3-8 Ansys Workbench frequency simulation for 19.1 mm diameter fixed-free steel rod with 80 mm length.............................................................................................................51 3-9 Cutting edge of a four fluted flat endmill cutter................................................................51 3-10 Solid model of two-fluted endmill cross-section...............................................................52 4-1 Speed-dependent FRFs of the standard holder at five differe nt spindle speeds: {0, 2500, 7500, and 10000} rpm.............................................................................................60

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10 4-2 Example setup for rotating FRF measurement..................................................................61 4-3 Example of holder and runout signal resultant response...................................................61 4-4 Scheme of tachometer-aided runout filtering setup...........................................................62 4-5 Illustration of once-per-re volution signal identification....................................................62 4-6 Example of shrink fit hold er time-domain runout filtering...............................................63 4-7 Frequency response function comparis on with/without time-domain filtering.................64 4-8 Short hollow standard holder geom etry and substructure coordinates..............................64 4-9 Experimental setup of standard hol der (HSK-63A interf ace) rotating FRF measurement.................................................................................................................... ..65 4-10 Solid holder geometry and substructure coordinates.........................................................65 4-11 Solid holder FRF measurement and prediction at {10000, 12000, and 16000} rpm........66 4-12 Geometry of Regofix collet holder with tool.....................................................................66 4-13 Measured and predicted FRFs for Regof ix collet holder with 12.7 mm diameter 127 mm overhang carbide tool blank at 10,000 rpm................................................................67 4-14 Measured and predicted FRFs for Regof ix collet holder with 25.4 mm diameter 127 mm overhang carbide tool blank at 10,000 rpm................................................................67 4-15 Geometry of 19.1 mm diameter, four flutes, carbide endmill with 76.1 mm overhang length clamped in Command shrink fit holder..................................................................68 4-16 Stability lobes for FRF measurement at 0 rpm and predictions at {10000 and 16000} rpm for 19.1 mm diameter, four flute, carbide endmill.....................................................68 4-17 Setup for cutting tests. The tool shan k deflections were measured using two orthogonal capacitance probes...........................................................................................69 4-18 Example results for 8000 rpm slotting cuts.......................................................................70 4-19 Comparison of test cut results to pred icted stability boundaries determined from 0 rpm (measured) and {10000 and 16000} rpm (predicted) FRFs.......................................71 5-1 Example setup of high speed machining with micro tools................................................82 5-2 Dimension comparison of a 1mm diameter, two-flute micro endmill to a penny.............82 5-3 CAT-40 standard holde r artifact geometry........................................................................82

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11 5-4 CAT-40 ER-25 collet hold er and tool geometry................................................................83 5-5 Experimental setup for CAT-40 collet holder with 3.18 mm steel tool shank..................83 5-6 Magnitudes of 41H FRFs measurements (top) and predictions (bottom).........................84 5-7 Measured and predicted 41H with 46.0 mm tool overhang (marked as 9 in Fig. 5-6)......84 5-8 Measured and predicted 41H with 50.8 mm tool overhang (marked as 12 in Fig. 5-6)....85 5-9 Multiplication factor ( MF ) determined from visual fit in relation to the tool overhang with a linear approximation superimposed........................................................................85 5-10 FRFs of44H, magnitude measurements (top) and predictions (bottom)...........................86 5-11 Measured and predicted 44H with 46.0 mm tool overhang (marked as 9 in Fig. 5-10)....86 5-12 Measured and predicted 44H with 50.8 mm tool overhang (marked as 12 in Fig. 510)............................................................................................................................ ..........87 5-13 NSK HES 500 electr ic micro spindle................................................................................87 5-14 Experimental setup for determinatio n of micro spindle-base receptances........................88 5-15 Prediction of tool point FRF for 3.18 mm diameter, 21 mm overhang steel rod by a 17 mm overhang standard arti fact with three different S values.......................................89 5-16 FRF prediction of a 3.18 mm diameter, 21 mm overhang steel rod by 20 mm (solid) and 17 mm (dotted) overhang standard tool artifacts........................................................90 5-17 Example 3.18 mm shank diameter tapered to ol (no flutes) with 1.5 mm diameter tool tip............................................................................................................................ ...........90 5-18 Geometry of NSK HES 500 micr o spindle with tapered tool............................................91 5-19 Tool tip measurement of tapered tool with an overhang of 23.5 mm compared to predictions based on different stan dard artifact overhang lengths.....................................91 5-20 Measurement of tapered tool (with over hang length of 25.5 mm) FRF compared with selected standard artifact FRF predictions.........................................................................92 5-21 FRFs for free-free tapered tool and st andard artifact responses coupled to rigid spindle receptances............................................................................................................93 6-1 HSK-63A spindle-base receptances calcu lated by long and short hollow standard holders........................................................................................................................ ........99 6-2 Geometry of HSK-63A long hollow standard hollow holder artifact..............................100

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12 6-3 HSK-63A shrink fit holders: long ho llow (left) and shor t hollow (right)........................100 6-4 HSK-63A FRF predictions for long shrink fit holder by short hollow artifact with different S values in comparison with measured FRF (heavy solid line)........................101 6-5 HSK-63A long shrink fit holder FRF predicti ons using two different standard holders in comparison with measured FRF (heavy solid line).....................................................101 6-6 HSK-63A short shrink fit holder FRF pr edictions using two different standard holders in comparison with meas ured FRF (heavy solid line)........................................102 6-7 Geometry of CAT-40 long hollo w standard holder artifact.............................................102 6-8 Geometry of CAT-40 short hollow standard holder artifact............................................103 6-9 Geometry of CAT-40 short so lid standard holder artifact...............................................103 6-10 CAT-40 shrink fit holder FRF predictions in comparison with measured FRF (heavy solid line).................................................................................................................... .....104 6-11 CAT-40 long collet holder FRF predicti ons in comparison with measured FRF (heavy solid line)............................................................................................................. .104 6-12 CAT-40 short collet holder FRF predicti ons in comparison with measured FRF (heavy solid line)............................................................................................................. .105 6-13 Geometry of HSK-100A short hol low standard holder artifact.......................................105 6-14 Geometry of HSK-100A long so lid standard holder artifact...........................................106 6-15 Geometry of Briney HSK100ASF-075433 shrink fit holder with Sandvik A393.T19 10 175 carbide adapter and two s quare insert cutting head........................................106 6-16 Geometry of Briney HSK100AE-125-472 se t screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapere d ball end mill with one round carbide insert..................107 6-17 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter and two square in sert cutting head measurement and prediction........................107 6-18 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter and two square in sert cutting head measurement and prediction........................108 6-19 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter and two square in sert cutting head measurement and prediction........................108 6-20 HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter and two square in sert cutting head measurement and prediction........................109

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13 6-21 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction.109 6-22 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction..110 6-23 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction.110 6-24 HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction..111

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14 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy IMPROVED PREDICTION OF SPINDLE-HOLDER-TOOL FREQUENCY RESPONSE FUNCTIONS By Chi Hung Cheng August 2007 Chair: Tony L. Schmitz Major: Mechanical Engineering High speed machining (HSM) offers trem endous capabilities for discrete part manufacturing because it can provide high material removal rates (MRR) in metals, plastics, and composites with good surface finish. To realize th ese benefits, stability lobe diagrams, which define regions of stable cutting as a function of spindl e speed and axial depth of cut, can be used to select appropriate cutting conditions. Computation of thes e diagrams requires that the dynamics of the cutting system (the machine, spindle, holder, and tool assembly) be known. Typically, impact testing (i.e., ex citing the structure with an in strumented hammer and recording the response with a linear transducer) is used to record the required tool point frequency response. However, due to the diversity of tool ho lders and tools available to end users, it can be prohibitively time-consuming to perform impact te sting for each possible combination. Further, it is difficult to measure the responses of 1) sm all tools using traditional methods; and 2) spindles during high speed rotation. The former is necessa ry for new micro-milling applications, while the latter is required b ecause the at-speed response for some spindles can differ from the nonrotating response. This study provides a method to address these s ituations. The tool tip response for a given machine-spindle-holder-tool assembly is pred icted by coupling a spi ndle measurement with

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15 finite element models of the holder and t ool using the method of receptance coupling substructure analysis (RCSA). RC SA enables a user to analytica lly couple arbitrary tool-holder combinations to an archived spindle response. Th erefore, the user must perform only a single test on the spindle in question. Given this information, the tool point response for any tool-holder can be performed via a virtual impact test. Compar isons of predictions and experimental results are provided for 1) micro-tools; and 2) macro-scale tools coupled to a spindle that exhibits changing dynamics with spindle speed.

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16 CHAPTER 1 INTRODUCTION High Speed Machining High speed machining (HSM) is an important capability in modern, discrete part manufacturing. Using the higher cu tting speeds made possible by improved cutting tool materials and coatings, the machine operation time is reduc ed significantly (Fig. 1-1 [1]). The use of HSM makes it possible to efficiently produce complex parts and, therefore, reduce assembly time and costs relative to the traditional approach wh ere simpler shapes are machined and then mechanically joined. High speed machines typically use direct dr ive spindles, i.e., a spindle shaft with permanent magnets is driven by a coil located within the surrounding housing. Modern spindle designs can reach top speeds of 40,000 rpm and higher with powers at the many tens of kW level. At these higher speeds, micro-milling (or m illing with very small diameter cutters) is now realizable because reasonable cutting speeds, or pe ripheral velocities of th e cutting edge, can still be maintained even though the cutting edge radius may only be fractions of a millimeter. The advantage of high speed micro-machining is that it provides a process capable of producing complex, free form, three-dimensional (3-D) st ructures from virtually any material. This provides an alternative to typi cal MEMS (micro electro mechanical systems) fabrications techniques, such as silicon etch ing, that are generally limited to 2-D geometries and specialized materials. Therefore, it can be expected that the demand of HSM will continue to increase. Chatter And Stability Lobe Diagram Surface location error (SLE) a nd chatter, or unstable machining conditions, impose limitations on machining efficiency. For any ma chining operation, the cutting force acting on the

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17 tool causes it to vibrate. This vibration can l ead to chatter and corresponding large forces and potential damage. Even if the ope ration is stable, however the vibration state of the tool as it leaves the newly created surface de fines the location of that surf ace. Variations in the process parameters can yield either undercut (less materi al removed than commanded) or overcut (more material removed) situations; this phenomenon is referred to as surface location error. For either limitation, it is important to note that the to ol-holder-spindle-machine assembly response as reflected at the tool point (free end of the tool) strongly influences the final behavior. Chatter occurs due to the inherent feedb ack mechanism in machining. In turning and milling operations, the tool cutting edge makes mu ltiple passes through the workpiece surface to achieve the desired dimension. For each pass, the tool vibrations are imprinted on the surface. Therefore, the workpiece surface is not uniform and the current chip thickness depends both on the current tool vibrations and those during the previous pass (F ig. 1-2), which Arnold [2] refers to as the regeneration of waviness. As the cu tting edge removes the wavy surface, the force is modulated by the varying chip thickness, wh ich leads to further vi bration. Depending on the machining parameters, the feedback system can b ecome unstable and chatter occurs (Fig. 1-3). The large force and significant tool deflections asso ciated with chatter can be identified audibly. It not only creates an unacceptable machined su rface finish, but can also damage the machine tool, spindle bearings, tool, and workpiece. To avoid chatter, stability l obe diagrams can be applied. These diagrams (as shown in Fig. 1-4) enable the machine operator to choose a pr oper spindle speed-chip width (axial depth for peripheral end milling) for stable cutting conditio ns. The concept of the stability lobe diagram was first developed in 1956 [3]. However, large industrial benefits were not realized until the high speed machines became commercially available. As can be seen in Fig. 1-4, the width of the

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18 stable regions (beneath the lobes) tends to wide n as the spindle speed becomes higher. Therefore, operating in these higher speed regions increases the material removal rate both by the increased cutting speed and higher axial depth of cut. Me thods used to compute the stability limit are described in Chapter 2. The feedback mechanism in machining can al so be represented using a block diagram approach as shown in Fig. 1-5 [4]. The system dynamics are represented by a second order plant in the forward path. The force is determined by multiplying the difference between the current and time-delayed chip thickness valued by the gain, represente d by the product of the specific cutting energy and chip width. Based on this block diagram, the limiting ch ip width can be expressed for turning operations as shown in Eq. 1.1 where FRF is th e system frequency response function. This equation emphasizes the importance of the sy stem dynamics in milling performance. FRF K bsRe 2 1 (1.1) Objective Currently, the tool point FRF is measured by impact testing, where an instrumented hammer (or modal hammer) is used to excite th e tool-holder-spindle-machine assembly and the resulting vibration is measured by an appropr iate linear transducer, typically a low mass accelerometer. Because the assembly dynamics depend on the individual components as well as their interactions, a new test must be performed for each combination or change in setup (e.g., if the tool overhang length is changed). In many indus trial situations, it is no t practical to measure each combination due to time restrictions. An a dditional complication is that these tool point dynamic measurements are necessarily completed with no spindle rotation (zero sp indle speed), but in some situations the system dynamics can vary with spindle speed [5]. For micro-milling,

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19 the situation is even more problematic because, ev en for zero spindle speed, it can be difficult or impossible to carry out impact tests on very small diameter endmills (<1 mm). It is the objective of this study to develop a method fo r predicting tool-holder-spindle dynamic responses that enables the user to perfor m a simple test and, subsequently, predict the dynamic behavior of the assembly. A modified three-component (tool, holder, and spindle) receptance coupling substructure analysis (RCSA) method is provided, which models the tool and holder geometries using a Timoshenko beam formulation and couples this result to the spindle response to predict the tool tip FRF fo r the situations described in the previous paragraph. The scope of this research follows: 1. application of a general RCSA approach, wh ich includes the determination of spindle dynamics by inverse RCSA and coupling of the spindle response to ar bitrary holders and tools (with emphasis on fluted tools and micro-tools); 2. prediction of the rotating tool point FRF over applicable ra nges of spindle speeds; 3. establishing a guideline for st andard artifact (used during the inverse RCSA procedure) geometry based on the selected tool-holder to enable increased accuracy in the tool tip response prediction; and 4. experimental validation of th e general RCSA approach throu gh impact testing and cutting tests.

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20 Figure 1-1. HSM cutting speed ranges for various materials. Figure 1-2. Scheme of (flexible) cutti ng edge passing through workpiece surface.

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21 Figure 1-3. Stable cut and unstable cut (chatter) on a workpiece surface. Figure 1-4. Example stability lobe diagram. Figure 1-5. Block diagram for cutting pr ocess with regenerated wavy surface. Unstable cut (Chatter mark) Stable cut

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22 CHAPTER 2 LITERATURE REVIEW Milling Stability Prediction It has been observed in the milling operati on that tool vibrations during cutting are imprinted on the work piece surface. Depending on the phasing between these imprinted waves and the tool vibrations in the s ubsequent pass, self-excited vibrati ons can occur. As seen in Fig. 1-2, the wavy surface created by the previous pa ss causes the chip thickness during the current pass to vary, which leads to a varying cutting fo rce acting on the tool. If the waves between the current and previous pass are aligned, the osc illating component of the force tends to decay. However, if the waves are out of phase and the chip thickness varies substantially, the vibrations tend to grow. These vibrations occur at the system natural fre quency corresponding to the most flexible mode, where chatter is mostly observed. There are two major approach es to obtain stable cutti ng conditions: predictive (nonexperimental) and experimental methods. Predicti ve methods require knowledge of the cutting system dynamics (frequency response function, FRF, or modal matrices) and specific cutting energy values, which relate the cutting force components to the selected chip thickness and width, for the stability calcula tion, whereas the experimental me thod can automatically adjust the spindle speed to obtain a stable cut based on da ta obtained during an unstable cut. The methods to achieve stable cutting conditions are described in Fig. 2-1. Experimental Method For on-line spindle speed regulation, Smith [6], and Delio et al. [7] proposed an algorithm of spindle speed selection for high speed m illing machines. In their approach, the sound spectrum (recorded by a microphone) during an unst able cut is used to identify the chatter frequency. This chatter frequency depends on the cutting system configur ation and is basically

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23 independent of the tooth passing frequency (i.e., the spindle speed multiplied by the number of cutter teeth) and its harmonics. On ce the chatter frequency is identified, the machine operator or automated control system then adjusts the spindl e speed to match the tooth passing frequency to the chatter frequency. If chatter still occurs, the same procedure is applied until chatter is eliminated. The result is stable cutting conditi ons without the knowledge of the system dynamics and specific cutting energy values. However, this is not a predictive method. Rather, it requires chatter to occur in a test cut for it to be avoi ded during actual part pr oduction and it does not always result in the maximum po ssible material removal rate. Predictive (Non-Experimental) Methods The predictive or non-experimental methods re quire that the system dynamics be identified by either impact testing or fin ite element modeling. The systems FRF, or modal parameters, are then used in a stability anal ysis algorithm (analytical, semi-analytical, or time-domain simulation) to determine whether a cutting pro cess is stable. The specific cutting energy, or cutting coefficient, values must also be determin ed beforehand. In practice, cutting coefficients vary with cutting material, tool geometry, etc. Th e particular values for a given combination may be determined by experiment (Chapter 4) or values from prior studies may be applied System Dynamics Acquisition Acquiring the system dynamics is the first st ep of predictive methods. Finite element modeling and impact testing are the most common methods. Modern high speed spindle configurations include a permanent magnet tightly fit to the spindle shaft with bearings supporting the front and rear shaf t ends. The shaft is powered by the stator mounted inside the spindle housing. Finite element m odels therefore require a signif icant amount of information to describe this complicated structure. For beari ng dynamics, Jones [8] discovered that, due to the combined effect of the centrifugal force Fc and gyroscopic moment Mg, the contact angle

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24 between the ball and the outer raceway,o tends to decrease and th e contact angle with inner raceway, i tends to increase (Fig. 2-2). The variati on in contact angles causes the ball bearing to lose stiffness when rotating at high speed. Rese archers such as Shin et al.[9-10] determined the bearing stiffness and spindle frequency by coupling a simplified Rayleigh beam spindle model with a bearing model derived from Jones theory and bearing contact angles measured at different spindle speeds. The model yielded goo d correlation with the experimental data. Jorgenson and Shin [10] further extended the prediction of multiple natural frequencies by modeling the spindle with Timoshenko beam elements. Cao et al. [11] used a similar approach as a bove. With the use of incremental finite element equations in time domain, Caos model could pr edict not only the frequency response function, but also the time domain spindle response. This model did not incorporate thermal effects, which are required to completely describe the spindle dynamics. Thermal issues have been studied mainly to estimate the bearing life. However, in high speed machining, it is beneficial to be able to predict the spindle response under high speeds when significant heat can be generated. The h eat is mostly generated by the friction of the bearing contacts and depends on th e viscosity of the lubricant a nd air in the thin gap. The resulting thermal expansion and the temperature gradient cause the bear ing preload to vary, which results in changes in the bearing stiffness. Tu et al. [12, 13] developed a thermal model to predict the entire spindle temperature field and the heat-induced bearing preload. Further, Li n, Tu, and Kamman [14] combined the thermal model with the consideration of centrifugal fo rce and gyroscopic moment and were able to predict the overall spindle dynamics. Li and Shin [15] added several spindle bearing configurations and combined their bearing/spindle model with a thermal model similar to Tus

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25 approach. Both methods in references [14] and [15] can describe the speed dependent bearing stiffness, but the frequency response function of the whole spindle was not predicted. For the previous research studies, the complete finite element (FE) spindle model requires knowledge of the bearing preload, location, and the assembly tolerance for each component to define the stiffness. This data is not ge nerally available in production environments. Furthermore, the model damping cannot be predicte d from first principles and requires tuning of the model by matching the spindle modes to a measured response. In addition, the complex FE models are often computationally expensive, whic h is a disadvantage for job shop application. The second approach to obtain the system dynami cs is by impact test ing. Fig. 2-3 shows the FRF measurement of a HSK-63A standard artifact using a modal hammer (instrumented hammer) and laser vibrometer. Besides a vibrometer (which measures velo city), other commonly used transducers include accelerometers (accelera tion) and capacitance probes (displacement). Upon impact, the time-domain hammer and tr ansducer signals are recorded and then converted into the frequency domain using the di screte Fourier transfor m. The result is the frequency dependent ratio between response and force ratio, or FRF. The advantage for impact testing is that it takes relativ ely short period of time to determine system dynamics and it captures the systems characteristic s (both stiffness and damping) at its current state regardless of the aging and wear of the com ponents. Once the FRF is known, it can be fit to determine the system modal matrices. Stability Analysis With the system dynamics (FRF or modal matr ices) collected in the previous step, there are three different approaches (time-domain simulation, semi-analytical, and analytical approach) to determine the stability limit as a f unction of axial depth of cut (or chip width) and spindle speed.

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26 Time-domain simulation [16] uses the modal mass, damping, and stiffness matrices to describe the system dynamics. The milling cutter is dissected in the radial direction to identify the engaging and disengaging of the tooth-work piece contact. The instantaneous chip thickness is calculated and used to estimate the force i nput. For each tooth rotation, the workpiece surface profile is recorded and later used to determine the instantaneous chip thickness. The cutter vibration is computed from the force and system equations of motion using integration. Users are responsible for providing the depth of cut a nd spindle speed. The stability of the cutting condition can be observed after a few iterations in the simulation as shown in Fig. 2-4. Bayly et al. [17] and Mann et al. [18-19] pr oposed a semi-analytical method of solving the milling equations of motion. This technique, Temporal (Time) Finite Element Analysis (TFEA), divides the cut into a finite number of elements. Since the nature of the milling operation is discontinuous (free vibration when disengaged and forced vibrati on when engaged in the cut), a closed form solution can not be attained. By assuming an approximated solution during forced vibration and matching it to a tim e frame where free vibration occurs, a discrete linear matrix of the dynamic system can be found. The stability is determined from the eigenvalues of this matrix. Analytical approaches that provide approximate closed form solutions for the milling equations are also available. Studi es of chatter theory and the deve lopment of stability lobe have been completed by, for example, Tlusty [ 20], Tobias [21], Pol acek [22], Merrit [23], Koenigsberger [24], and Altintas [25]. Tlusty concluded that a single point cutting pr ocess is stable as long as the chip width b is less than the limit chip widthlimb, which is a function of specific cutting energysK and spindle speed as shown in Eq. 1.1, where the real part of the frequency response function Re(FRF( ))

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27 is at its minimum. In this frequency-domain so lution, the FRF is used directly and there is no need to extract modal matrices. However, it is still necessary to perform a new impact test to identify the tool point FRF for each tool and holder combination, which can pose a significant obstacle on the shop floor. In order to avoid the tedious testing pr ocedure, Receptance Coupling Substructure Analysis (RCSA) can be applied to predict th e tool point FRF. The concept of receptance coupling can be traced back to 1947 [26]. RCSA allows multiple components to be analytically combined after the component models have b een defined individually. In some instances, connections parameters (springs and dampers) between the components, rather than rigid connections, are required to predict the system overall FRF. RCSA is convenient to implement for modern spindle-holder-tool combinations because holder designs with a selected holder-spindle inte rface are identical from the rear of the holder (i.e., at the taper that is inserted in the spindle) to th e far end of the holder flange. In this study, the taper types used were CAT and HSK with taper angle approximately 8.2 and 2.9 degrees respectively. This common geometry enables a consistent gripping area for automatic tool changes. Therefore, for a given holder-spindle inte rface, the user can perform an impact testing on a simple geometry standard artifact to dete rmine the spindle dynamic response, including the holder-spindle connection. To pr edict the overall system dynamics, the extended portion of the holder and the cutting tool (beyond the holder flange) is modeled and then coupled to the spindle response. Though finite element models of the holder and tool may be applied, they are for rather simple cylindrical geometries and beam elemen ts are sufficient. The computational time is therefore low and acceptable for the job shop leve l. The modular modeling approach aids during

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28 purchasing and process planning because new holde r-tool combinations in a particular spindle can be quickly and efficiently modeled to test the milling capabilities. Previous work completed by Schmitz et al. [27-29] shows good corre lation between the analytical prediction and the experiment result. An example result is provided in Fig. 2-5 [30]. In Chapter 3, the three-component RC SA model [31] is described in detail. Modeling of fluted tools is also discussed. Figure 2-1. Different approaches to determine stable cutting conditions. System dynamics acquisition Stability analysis Analytical approach Semi-analytical approach Time-domain simulation Specific cutting energy Experimental method Stable cutting conditions Impact testing Finite element model Predictive (Non-experimental methods)

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29 Figure 2-2. Ball bearing contact angles at high speed rotation. Figure 2-3. Impact testing pe rformed on standard artifact with modal hammer and laser vibrometer. Fc Mg o i Innerrace Modal Hammer Laser Vibrometer Standard Artifact Outerrace

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30 Figure 2-4. Example of time-domain simu lation of an unstable cutting process. Figure 2-5. Predicted and measured FRF of a 100 mm diameter inserted endmill.

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31 CHAPTER 3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS In this chapter, the joining of tool, holder, and spindle dyna mics using receptance coupling substructure analysis (RCSA) is described. Two approaches are detailed, including the basic twocomponent coupling and the improved three-co mponent coupling. The latter includes a decomposition step to identify the spindle dynami cs. The decomposition procedure, referred to as inverse RCSA, is also described. Two-Component Receptance Coupling Substructure Analysis RCSA can be applied to combine the dynamics of the cutting system components. A typical high speed milling machine cutting system in cludes the machine, spindle (shaft, bearings, and drawbar for gripping the tool holder), tool holder, and tool. The RCSA concept is to join multiple components (with each of the componen ts dynamic response known) by means of rigid or flexible connections (the latter can be modele d as springs and dampers) and predict the overall assembly dynamic response. Rigid Connection Consider the system shown in Fig. 3-1, wh ere A and B are two components and C is the assembly of A and B with a rigid connecti on. The displacements of the components and assembly,jx, can be written as shown in Eqs. 3.1 through 3.3. ax=Ha( ) af (3.1) bx=Hb ( ) bf (3.2) cX=Gc( ) cF (3.3)

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32 Here Hj is the receptance, or frequency response function, for component j, Gjis the receptance for assembly j, fj is the component force, and Fj is the assembly force. All receptances are functions of frequency, For rigid coupling, the compatibility conditions are; ax=bx=cX (3.4) and the equilibrium conditions are af+bf=cF. (3.5) By substituting Eqs. 3.1 through 3.4 into 3.5, the result in Eq. 3.6 is obtained. The receptance Gc describes the assembly frequency re sponse as a function of the component receptances. Gc= (H1 a+H1 b)1 (3.6) Non-Rigid Connection with a Linear Spring For non-rigid connections between components, c onsider the case shown in Fig. 3-2. Each mass is connected to the wall with a linear spri ng and a viscous damper; the spring-mass-damper combination is treated as a component. The c onnection between the comp onents is a linear spring, kc. To determine the assembly receptances, F1 is first applied at assembly coordinate X1 and the equilibrium and compatibility conditions are written as shown in Eqs. 3.7 and 3.8. 2 1f f=F1 (3.7) X1=1x, X2=2x, and kc(1 2x x ) = -2f (3.8) Following the same procedure as above, the rece ptances of the first column of the 2x2 (two degree of freedom) assembly receptance matrix G are calculated. ijGdenotes the receptance at position i under the forcing at position j.

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33 G11( )= 1 1F X= H11-H11(H22+ H11+ ck 1 )1 H11 (3.9) G21( )= 1 2F X= H22( H22+ H11+ ck 1 )1 H11 (3.10) In the same manner, if 2F is applied to assembly coordinate X2, the second column of the receptance matrix can be determined (Eqs 3.11 and 3.12). G12( )= 2 1F X= H22( H22+ H11+ ck 1 )1 H11 (3.11) G22 ( )= 2 2F X= H22H22 ( H22+ H11+ ck 1 )1 H22 (3.12) Here 11Hand 22H are the receptances of components A and B and are listed below. 2 1 1 1 2 2 1 1 112 1 ) (n n ni k H (3.13) 2 2 2 2 2 2 2 2 222 1 ) (n n ni k H (3.14) The complete receptance matrix for two subcomponents is therefore 22 21 12 11) ( G G G G G. (3.15) Non-Rigid Connection with a Linear Spring and a Damper Figure 3-3 shows the non-rigid assembly with both a linear spring a nd a viscous damper. To find the direct receptance of the assembly at coordinate X2,22G, a force 2F was applied at X2. The equilibrium and compatibil ity conditions are shown respec tively in Eqs. 3.16 and 3.17. 2 2 1F f f (3.16) X1=1x, X2=2x, and kc(2 1x x )+ cc(. 2 1x x) = -1f (3.17)

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34 Setting F = F et i and x = X et ito denote harmonic motion, Eq. 3.17 can be re-written as shown in Eq. 3.18. kc(2 1x x )+ icc(2 1x x ) = -1f (3.18) Writing the complex stiffness as Kc= kc+ icc, it is obtained Kc(2 1x x )=-1f. (3.19) For 1 11 1f H x and2 22 2f H x Eq. 3.19 can be written as 1 2 22 1 11) ( f f H f H Kc (3.20) The forces can then be written as 2 22 1 11 22 1) 1 ( F H K H H fc and (3.21) 2 22 1 11 22 1 2 2) ) 1 ( 1 ( F H K H H f F fc (3.22) The direct receptance at assembly coordinate X2 is then 22 1 11 22 22 22 2 2 22 2 2 2 2 22) 1 ( ) (H K H H H H F f H F x F X Gc (3.23) where Kc= kc+ icc. Non-Rigid Connection with Linear a nd Rotational Springs and Dampers The milling cutting system can be modeled as shown in Fig. 3-4. As can be seen, the connections between tool A and holder/ spindle B consist of a linear springxk rotational spring k linear damper xc and rotational damper c The rotational spring and damper are required to restrict rotation, while the linear spring and damper restrict translation. Rotation must be considered in this case due to the bending of the tool during application of a force to its free end.

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35 The assembly in Fig. 3-4 has six degrees of freedom. The complete receptance matrix G (6x6) is shown in Eq. 3.24. 3 2 1 3 2 1 66 61 22 21 16 12 11 3 2 1 3 2 1. . . . . M M M F F F G G G G G G G X X X (3.24) In the cutting operation, the tool tip is genera lly considered to be act ed upon by a force, but no moment, so M1 is set to zero. Notation for the compone nt receptances when both rotations and translations are to be consid ered is listed in Eq. 3.25. ij ij ij ij j i j i j i j i ijP N L H m f m x f x R ) ( (3.25) Referring to the components in Fig. 3-4, th e following expressions may be written as 2 12 2 12 1 11 1m L f H f H x (3.26) 2 12 2 12 1 11 1m P f N f N (3.27) 2 22 2 22 1 21 2m L f H f H x (3.28) 2 22 2 22 1 21 2m P f N f N (3.29) 3 33 3 33 3m L f H x (3.30) 3 33 3 33 3m P f N (3.31) To determine the tool tip receptance, a force 1F is applied at the t ool point (i.e., the free end of the tool) in the assembly. The corresponding equilibrium conditions are 1 1F f 03 2 f f and 03 2 m m (3.32)

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36 The compatibility conditions are i iX x i i i =1, 2, 3 (3.33) 3 2 3 3 2 3) ( ) ( m k f x x kx (3.34) Here damping in the equations is not include d for simplicity of notation. Damping can be combined with stiffness at the end of derivation as described previously. By substituting3 2f f 3 2m m the following expressions are obtained 1 21 3 22 33 3 22 33 2 3 1 21 3 22 33 3 22 33 2 3) ( ) ( ) ( ) ( f N m P P f N N f H m L L f H H x x (3.35) Substituting Eq. 3.35 into 3.34 and re-grouping gives 1 1 21 21 3 3 22 33 22 33 22 33 22 331 ) ( ) ( ) ( 1 ) ( f f N k H k m f P P k N N k L L k H H kx A x x (3.36) Since1 1F f the force/moment on the components at coordinate 3 are obtained as 1 1 21 21 1 3 3F F N k H k A m fx. (3.37) Expressing Eqs. 3.26 and 3.27 in matrix form gives 2 1 12 12 12 12 1 1 11 11 1 1f f P N L H f f N H x. (3.38) Letting1 1X x ,1 1 ,1 1F f ,3 2f f ,3 2m m and applying Eq. 3.37 gives 3 31 1 21 21 1 12 12 12 12 1 1 11 11 1 1 m f xF F N k H k A P N L H F F N H X. (3.39) Finally the direct receptances at tool tip are obtained as

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37 21 21 1 12 12 12 12 11 11 1 1 1 1 41 11N k H k A P N L H N H F F X G Gx. (3.40) For combined spring and damper connections, following substitution can be applied. c i k c i k K K k kx x x x (3.41) Three-Component Receptance Coupling Substructure Analysis In previous section, coupling of the milli ng cutting system was described. Though this model allows two entities (the sp indle-holder and tool) to be join ed with linear and rotational springs and dampers, the model must be recreat ed for each new holder in a given spindle. An improved three-component RCSA approach, which is capable of coupling arbitrary holder and tool combinations to a selected spindle, is now described. Consider the generic case (Fig. 3-5) of two rigidly coupled su bstructures that defined an assembly with end coordinates, a and d, s ubject to forces and moments acting on each. Follow the two-component RCSA approach, the assembly receptance matrices can be expressed as shown in Eqs. 3.42 through 3.45, where the 2x2 Rij matrices include the translational and rotational component receptances. ba cc bb ab aa aa aa aa aa aaR R R R R P N L H G1 (3.42) cd cc bb dc dd dd dd dd dd ddR R R R R P N L H G1 (3.43) cd cc bb ab ad ad ad ad adR R R R P N L H G1 (3.44)

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38 ba cc bb dc da da da da daR R R R P N L H G1 (3.45) Using the receptances in Eqs. 3.42 through 3.45, the subscripts can be updated to match the cutting system described in Fig. 3-6. The thr ee components are the tool (I), the extended portion of the holder (II), and the spindle base (combina tion of spindle and the holder from its tapered interface to the flange, coordinate 3a). To match the geometry shown in Fig. 3-6, components I and II (with free-free boundary c onditions) are first rigidly coupl ed to form a new component with end coordinates 1 and 3, si milar to the right component in Fig. 3-6. It is by industry standard that the holders with same type of ta per interface have identica l flange geometries. The vendors then design the extended part of the holder depending on the cutting application. Therefore, it is reasonable to regard the spindle and the standard portion of the holder to be one entity. By using three-component RCSA, arbitrar y holders and tools can be modeled using beam theory (either Euler-Bernoulli or Timoshenko). Th e tool point receptance matrix for the threecomponent cutting system (F ig. 3-6) is written as 31 1 3 3 33 13 11 11 11 11 11 11R R R R R P N L H Ga a (3.46) Inverse Receptance Coupling Substructure Analysis As noted, the tool and holder can be convenien tly modeled using beam theory so that the R11, R13, R33, and R31 receptances in Eq. 3.46 can be dete rmined. However, the spindle base receptances, R3a3a, are not conveniently modeled. In particular, even if the spindle geometry and tolerances are known, the damping is difficult to estimate from first principles. Therefore, inverse RCSA is used to analytically decompos e a spindle-standard holder assembly into its components (as opposed to the synthesis, or coup ling, procedures described in the previous sections). This method uses the FRF measuremen t of the assembly and removes the portion of

PAGE 39

39 the simple geometry standard holder beyond the flange to identify the spindle dynamics alone. This makes it possible to predict any holder-t ool combination by coupling the holder and tool models with the spindle response and eliminates the need to complete an impact test for each individual assembly. By taking the advantage of the standard flange geometry for commercial holders with the same spindle connection type, the dynamic respons e of the spindle, combined with the holder taper (inserted inside the spi ndle) and flange (referred to as the spindle-holder base), by detaching the extended part of the standard holder using inverse RCSA. This makes the spindle-holder interface (whether the connection is rigid or flexible) a part of the spindle receptances and further study of the connection para meters is not required. An example standard holder artifact is shown in Fig. 3-7. Rigid connections between the constant cross-section sections may be assumed because the sta ndard holder is monolithic by design. Here notation for different types of receptance matrices is provided R, single component receptance matrix RS, coupled substructures (suba ssembly) receptance matrix G, final assembly receptance matrix. The assembly receptance matrix at the holder free end, coordinate 1, is obtained by the following procedure. (1) Measure 11H (excite the structure and m easure at coordinate 1) and 1 1aH(excite the structure at coordinate 1 and measure at 1a) by impact testing. The distance between the direct and cross FRFs, respectively, is denoted as S. (2) Calculate 11Nby first order finite difference using 11H and 1 1aH.

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40 S H H Na 1 1 11 11 (3.47) (3) Assume 11L= 11Nby reciprocity. (4) Synthesize 11P using Eq. 3.48. 11 2 11 11 11 11 11H L H N L P (3.48) (5) Obtain11G. 11 11 11 11 11P N L H G (3.49) The S value must be sufficiently large to yield a sufficient amplitude difference between the direct receptance 11H and cross receptance1 1aH for the finite difference calculation of11N. However, if S is too large, a low1 1aH amplitude can be obtained, which can lead to decreased signal to noise ratio. The S distance chosen in this research is generally between one third and half of the extended holder length. To find the spindle-ba se direct receptance,a aR3 3, the Rij component matrices at each location (1, 2, 2a, 3) are calculated usi ng free-free beam models. The individual R matrices are then rigidly coupled to form the RS matrices for a rigid coupling of components I and II. 21 1 2 2 22 12 11 11) (R R R R R RSa a (3.50) 3 2 1 2 2 22 32 33 33) (a a a aR R R R R RS (3.51) 3 2 1 2 2 22 12 13) (a a aR R R R RS (3.52) 21 1 2 2 22 32 31) ( R R R R RSa a a (3.53)

PAGE 41

41 The receptance matrix of the spindle-base is then determined by inverse RCSA (i.e., determine the spindle-base component rece ptances from an assembly measurement, G11, and component I-II receptances) 33 13 1 11 11 31 3 3) ( RS RS G RS RS Ra a (3.54) Givena aR3 3, to the tool point response for arbitrary holder-tool models (given the geometry and material properties) can then be predic ted using RCSA. Note that the preceding model assumes rigid connections everywhe re. As described previously, th e model could be modified to include flexibility between the spi ndle and tool-holder, for example. Substructure Beam Modeling For each tool-holder section of the RCSA mode l, the components can be represented by Euler-Bernoulli or Timoshenko beam models. Euler-Bernoulli Beam Johnson and Bishop [32] proposed closed form expressions for the translational and rotational receptances of uniform cylindrical beam under the application external forces and moments. For a free-free beam with coordinates j and k on each end, the direct and cross receptance are given as 3 3 51F i EI F h hkk jj (3.55) 3 3 81F i EI F h hkj jk (3.56) 3 2 11F i EI F l lkk jj (3.57) 3 2 101F i EI F l lkj jk (3.58)

PAGE 42

42 3 2 11F i EI F n nkk jj (3.59) 3 2 101F i EI F n nkj jk (3.60) 3 61F i EI F p pkk jj (3.61) 3 71F i EI F p pkj jk (3.62) where E is the elastic modulus, I is the second area moment of inertia, is the structural damping coefficient, and L i EI m 12 4 (3.63) L L F sinh sin1 (3.64) 1 cosh cos3 L L F (3.65) L L L L F cosh sin sinh cos5 (3.66) L L L L F cosh sin sinh cos6 (3.67) L L F sinh sin7 (3.68) L L F sinh sin8 (3.69) L L F cosh cos10 (3.70) The cylindrical beam mass m in Eq. 3.63 is given as 42 2 L d d mi o (3.71)

PAGE 43

43 where od and id are the outer and inner diameters of the beam section, respectively, L is the section length, is the density, and is the frequency (rad/s). The beams second area moment of inertia is 644 4i od d I (3.72) Timoshenko Beam The Timoshenko beam model includes rotary in ertia and shear effects. It is implemented using the finite element method [33]. Each elem ent has four degrees of freedom (rotation and displacement at each end of the beam section). The mass matrix M and stiffness matrix K (including damping) for a free-free beam section [34] are given as (3.73) where A is the cross-section area, l is the section length, gr is the radius of gyration, and is the shear deformation parameter given by 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 23 6 15 2 Symmetric 2 10 1 5 6 6 6 30 1 2 10 1 3 6 15 2 2 10 1 5 6 2 10 1 5 6 1 120 60 105 1 Symmetric 24 120 11 210 11 3 10 7 35 13 120 60 140 1 24 40 3 420 13 120 60 105 1 24 40 3 420 13 6 10 3 70 9 24 120 11 210 11 3 10 7 35 13 1 l l l l l l l l r Al l l l l l l l Al Mg

PAGE 44

44 2 '1 12 GA l k EI (3.74) where 1 2 E G (3.75) is the shear modulus, is Poissons ratio, and 'kis the shear coefficient which depends on the cross-section shape and Poissons ratio [35]. 2 2 2 2 2 2 2 2 2 2 2 2 3Symmetric 2 4 2 2 4 2 4 1 4 2 4 Symmetric 6 12 2 2 6 2 4 6 12 6 12 1 1 l l l l l l l l AG k l l l l l l l l i EI K (3.76) The element M and K matrices are then collected to form the global mass matrix M and global stiffness K matrices. Applying Guyan reduction [33], the equation of motion in the frequency domain is obtained as shown in Eq. 3.77 for n elements. 1 1 2 2 1 1 1 1 2 2 1 1 2n n n nm f m f m f x x x K M (3.77) To decide which modeling method more accu rately describes typical tool-holder combinations, a fixed-free solid cylindrical steel rod with diameter 1 19 od mm and lengths

PAGE 45

45 varying from 60 mm to 110mm (typical tool overhang lengths in milling operations) was modeled using both Euler-Bernoul li beam and Timoshenko beams (25 elements). The results were compared to commercial finite element so ftware Ansys Workbench 10.0 resultsb(Table 31). It is observed that the Timoshenko beam mode l agrees more closely with the Ansys results, particularly for shorter beams. This is partic ularly important since the actual tool/holder geometry is not typically uniform throughout the length and is sect ioned into smaller components with constant cross-sections. To im prove accuracy in this study, only Timoshenko beam models were applied to describe the to ol-holder components in the RCSA predictions provided in the Chapters 4 through 6. Fluted Tool Modeling An important consideration in modeling tool-h older combinations is the fluted portion of solid body endmills. Although the actual geometry of the helical cutting edge (Fig. 3-9) could be meshed in a finite element model, this is com putationally intensive. As an alternative, the equivalent diameter of the fluted section can be used to determin e the required second area moment of inertia and beam mass for the Timoshenko beam model. To identify the equivalent diameter, solid models were constructed using Solidworks 2006 for tools with two (Fig. 3-10), th ree, and four flutes. The tool profiles were approximated from SGS high speed steel endmills. It was observed th at the two-fluted cutter had a second area moment of inertia which varied wi th angle. For the cross-sectiona l model shown in Fig. 3-10, the ratio of second area moment of inertia for fluted cross-section relative to the cylindrical shank was calculated. The ratio was computed over 180 deg in 10 deg increments, where the angle for plane 2 (Fig. 3-10) was taken to be zero. The re sults are shown in Table 3-2. The average ratios for two, three, and four fluted tool s are 0.4404, 0.3349, and 0.3594 respectively.

PAGE 46

46 For modeling purposes, only the mean value of the area moment of inertia for two-fluted endmills was considered. Table 3-3 provides the average value of the fluted tool area section properties for all thr ee tool geometries. Figure 3-1. Components joined with a rigid connection. A B af bf ax bx Components C cF cX Assembly

PAGE 47

47 Figure 3-2. Components connect ed with a linear spring. A B 1f 2f 1x 2x Components Assembly B A X1 X2 kc F1 k1 k2 c1 c2 m1 m2 X1 X2 A B Physical system kc

PAGE 48

48 Figure 3-3. Assembly with linear spring and damper. A B 1f 2f 1x 2x Components Assembly B A X1 X2 kc cc F2

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49 Figure 3-4. Assembly with linear and rotational spri ngs and dampers. Spindle Holder Tool X1 X2 3, X Milling Cutting System B A B A 3f 2f 3x 2x Components Assembly A B X3 X2 kx xc 1f 1x 3 2 1 3m 2m X1 k c 1 2 3 1m

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50 Figure 3-5. Generic case of two s ubstructures with rigid connection. Figure 3-6. Three-component RCSA mode l for tool-holder-spindle assembly. Figure 3-7. Spindle and decompos ed standard holder artifact. Spindle Spindle-holder base Extendedstandardholder S I II 3a 32a 21a1 Spindle Spindle-holder base Extended holder I II 3a 1 2 2a 3 Tool d c a b dX aX a d dM dF aF aM Rigid

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51 Figure 3-8. Ansys Workbench frequency simula tion for 19.1 mm diameter fixed-free steel rod with 80 mm length. Figure 3-9. Cutting edge of a four fluted flat endmill cutter.

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52 Figure 3-10. Solid model of two-fluted endmill cross-section. Table 3-1. Fixed-free steel r od first mode frequency compar ison between different beam modeling methods. Length Beam type 60 mm 70 mm 80 mm 90 mm 100 mm 110 mm EulerBernoulli 3759 Hz 2761 Hz 2114 Hz 1670 Hz 1353 Hz 1118 Hz Timoshenko 3567 Hz 2656 Hz 2051 Hz 1631 Hz 1327 Hz 1100 Hz Ansys 3592 Hz 2672 Hz 2062 Hz 1638 Hz 1332 Hz 1104 Hz Percent difference from Ansys EulerBernoulli -4.6 -3.3 -2.5 -2.0 -1.6 -1.3 Timoshenko 0.7 0.6 0.5 0.4 0.4 0.4

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53 Table 3-2. Average of area section properties for fluted endmills. Angle Ratio 0 0.4965 10 0.4328 20 0.3701 30 0.3159 40 0.2767 50 0.2572 60 0.2598 70 0.2843 80 0.3275 90 0.3844 100 0.4480 110 0.5108 120 0.5650 130 0.6042 140 0.6237 150 0.6211 160 0.5966 170 0.5534 180 0.4965 Table 3-3. Average of area section properties for fluted endmills. Tool type Ratio of second area moment of inertia (fluted/shank) Ratio of crosssection area (fluted/shank) 2 flutes 44.04% 38.27% 3 flutes 33.49% 35.66% 4 flutes 35.94% 35.88%

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54 CHAPTER 4 ROTATING FREQUENCY RESPONS E FUNCTION PREDICTION In this chapter, a technique for predicting th e rotating, or at-speed, tool point frequency response using the receptan ce coupling approach is described. Due to a number of sources (e.g., gyroscopic effects, bearings s tiffness variation, mass unbalance, thermal changes, and surface contact effects), the tool-holde r-spindle assembly dynamics are known to vary as a function of spindle speed. Figure 4-1 shows the FRFs for the standard hold er-spindle assembly at five selected spindle speeds for the Step-Tec spindle used in this research. Clearly, the response varies substantially with rotating speed. In the following sections a time-domain signal filtering technique is described which was used to eliminate the runout signals produced by the measured tool surface during rotation. Runout, in this case, can be visualized as a nonround artifact (elliptica l-shaped for example) rotating about a fixed axis. If a di splacement (or velocity) sensor is held fixed relative to the axis of rotation and targets against the artifact surface, a sinusoida l signal will be obtained with frequency that varies with the rotating speed. To achieve acceptable signal to noise ratios in FRF measurements, it is necessary to remove the runou t content. After runout filtering was complete, the inverse RCSA procedure was used to isolate the spindle-base recepta nces. These receptances were then coupled (via RCSA) w ith various holders and tools to predict the at-speed assembly FRF. These FRFs were finally used to generate analytical stability bound aries and cutting tests were performed to validate the stability limit. These results are also provided. Runout Signal Filtering Runout poses a significant challenge in rotati ng tool response measurements. This inherent once-per-revolution signal introduces content at the rotati ng frequency and its harmonics. Because this content is related to the imperfecti ons in measurement artifact (the standard holder

PAGE 55

55 in this case), it does not influence the milling process dynamics and must be removed (filtered) prior to computing the stability limit by frequency domain methods. Runout Signal As mentioned, runout is genera lly present in measurement signals for rotating systems. It can be caused by an imperfect tool surface, poor tool-holder alignment (dominant in micro tools), bearing deflection, or a combination of all of these. Although it is not a significant influence in cutting force perturbation, its amp litude is quite often larg e enough to saturate the target vibrations during an impact test, for ex ample. Figure 4-2 shows the setup for a rotating FRF measurement of a Shrinker V40E-075-138SF CAT-40 shrink fit holder attached to a 36000 rpm, 36 kW, Setco milling spindle. Impact testing was performed at the front (free) end of the holder with the spindle rotating at 5000 rpm; a capacitance probe (non-contact displacement sensor) was used to record the resulting vibration. As can be seen in the time trace in Fig. 4-3 (hammer impact at 0.05 s), the holder respons e to the hammer excitation cannot be easily separated from the resultant signal (summation of holder response and run out signal). In this case, the initial response amplitude is similar to the runout amplitude. The details of the timedomain runout filtering are detailed in the next section. Runout Filtering The first step of runout filter ing is to separate the once-pe r-revolution signal from the combined signal. As shown in Fig. 4-4, this was accomplished using a circular encoder (a black plate with a small section of white) attached to the holder and a lase r tachometer which was aligned normal to the encoder. During rotation, the tachometer optical pickup sensed the darklight transitions and generated a corresponding voltage signal. This signal was used to identify single revolutions of data in the time-domain.

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56 An illustration of runout signal identification strategy is provided in Fig. 4-5. Here the heavy solid line (sinusoidal wave) represents th e runout signal within one revolution. It is confined between any two tachometer once-per-revol ution impulses. It can be seen that there are several intervals that can be selected; note that only those intervals th at do not include the exponentially decaying holder-spindle response s hould be considered (i.e., beyond 0.35 s in Fig. 4-5). Therefore, a non-rotating impact test (deca ying sinusoid) was typically performed in this testing to determine this critical point. The once-per-revolution signal identified from a single interval was reproduced throughout the entire time in terval to form the synt hesized runout signal. By subtracting this runout signal from the hol der resultant response, the time-domain holderspindle response at the selected speed was extracted. Figure 4-6 shows example results for the data collected using the set up in Fig. 4-2 at a spindle speed of 5000 rpm. It is seen that the response signal after filt ering can be distinguished from the runout noise. Further, after the response decays to near zero leve ls, an exponential filter can be applied to attenuate the remaining sy stem noise. The frequency response function for these signals is provided in Fi g. 4-7. These FRFs were generated from an average of 15 measurements each. It is observed that the r unout content generates sharp peaks at regular intervals, which corrupts the desired spindleholder response and make s the application of inverse RCSA to identify the spindle-base recep tances impossible. Due to the rotation, the response recorded was influenced by the rotating angle of the spindl e shaft. For example, a tool natural frequency at 1200 Hz, corresponding to 1 500 rpm, will have its magnitude attenuated by at most 21 percent in one oscillation interval. Wh en operating at slower sp indle speed, this error caused by the changing spindle orientation does not contribute significantly to the data collected and is therefore neglected.

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57 In addition to the time-domain runout filteri ng, further smoothing of the measured FRFs was also implemented in order to reduce the noise. This was important due to the sensitivity of the finite difference approach used to compute th e spindle-holder rotational receptances to noise. Savitzky-Golay filtering was a pplied (in the Matlab code with third order Savitzky-Golay polynomial and a frame size of 31) to carry out the FRF smoothing. The FRF Prediction from Rotating Standard Holder Measurements Rotating FRF measurements were performe d on a Mikron UCP 600 Vario milling machine equipped with Step-Tec 20000 rpm, 16 kW cartri dge-type spindle (HSK63A holder interface). A standard holder (geometry and section dimens ions provided in Fig. 4-8 and Table 4-1) was attached to the spindle and a black/white encoder strip adde d for once-per-revolution signal identification. A TTI LT-880 laser tachometer was used to target agains t the encoder strip. A Lion Precision C1-C capacitance probe was used to record the vibration response (Fig. 4-9) and a PCB 086B03 impact hammer was used to exc ite the system. A MLI MetalMax 6.0 data acquisition system was used for data collection. To reduce noise, each FRF was computed from an average of 50 impacts. Prior to impact testing, the spindle was warm ed to a steady thermal state by running it for 40 minutes. The spindle-base receptance,a aR6 6, was obtained using inverse RCSA following the descriptions in Chapter 3. In this case, howev er, the inverse RCSA was carried out using spindleholder data collected at various spindle speeds. The distance, S between 44Hand 4 4aH on the standard artifact was 25.4 mm. To verify the validity of applying RCSA to rota ting tools, a simple structure step solid holder was tested first. This solid holder had no tool attached to it; th erefore, the potential influence of flexible connections between sections could be neglected. The geometry and section

PAGE 58

58 dimensions are provided in Fig. 4-10 and Ta ble 4-2. The receptances for the free-free holder model (beyond the holder flange) was constructe d using 25 Timoshenko beam elements in each section. The material prop erties (elastic modulus, E density, structural damping, and Poissons ratio, ) used in the modeling for steel and carbide are listed in Table 4-3. The measured and predicted rotating frequency response functions at th ree selected spindle speeds are presented in Fig. 4-11. The 0 rpm FRF is shown in comparison to rotating FRFs. It is seen that, even with the change in the spi ndle dynamics, RCSA can still provide reasonable prediction accuracy. To further verify this approach, two 152.4 mm carbide tool blanks (12.7 mm and 25.4 mm diameters) were inserted into a Regofix colle t holder. The overhang length was 127 mm in both cases. The geometry of the tool-holder is shown in Fig. 4-12 and the section dimension of the tool-holder combinations are listed in Tables 4-4 and 4-5 respectively. The spindle speed was 10000 rpm for both. The results are shown in Fig. 4-13 and 4-14. Again, good agreement between the predicted and m easured FRFs is observed. Stability Boundary Validation A four-flute 19.1 mm carbide endmill w ith 76.1 mm overhang and 48.3 mm cutting length was inserted into a Command shrink fit holder (Fig 4-15). Its section dime nsions are provided in Table 4-6. Note that the equiva lent diameter of the cutter (s ection I) is corrected by the second area moment of inertia developed in Chapter 3 (fluted tool modeling). The connection between the tool and holder was assumed rigid. Based on the measured spindle-base receptances over a range of spindle speeds, the frequency response functions for the tool-hol der-spindle were constr ucted. The corresponding

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59 stability lobes for 100% radial immersion (i.e ., slotting) conditions computed from the {0, 10000, and 16000 rpm} FRFs are shown in Fig. 4-16. Figure 4-17 shows the cutting test setup for a four-flu te, overhang 76.1 mm, 19.1 mm carbide endmill clamped in a shrink fit holder. Two capacitance probes were used to record vibrations of the tool shank in both the x (feed) an d y (perpendicular to feed in the plane of the cut) directions. The tangential and radial cutting force coe fficients for the tool/6061-T6 workpiece material combination, kt = 527 N/mm2 and kr = -33 N/mm2, respectively, were determined from separate dynamometer cutt ing tests as described in Altintas [36]. Significant disagreement between the stability boundaries is observed. Therefore, cutting tests were also performed to verify this behavior. A variance ratio, R technique, was developed by Schmitz [37]. In this method, the x and y di splacements are sampled once-per-revolution to identify synchronous (stable) a nd asynchronous (unstable) behavior The statistical variance in the once-per-revolution data is normalized by the variance in the overall shank motions to obtain R as shown in Eq. 4.1, where 2 indicates statistical variance, the x and y subscripts refer to the x and y directions, respectively, and the rev subscript identifies the once-per-revolution sampled data. For stable cutting, the tool deflections tend to repeat each revolution so the once-perrevolution variance values and, subsequently, the R value are small. On the x-y deflection map, data is observed over a confined region. For unstable cutting, high R values are obtained because the vibrations occur near the sy stem natural frequency associated with the most flexible mode and the vibrations are no longe r synchronous with rotation. Ther efore, the once-per-revolution data points are spread out on the deflection map. 2 2 2 2 y x rev y rev xR (4.1)

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60 In Fig. 4-18, example test results are provi ded for 8000 rpm slotting tests. It was observed over many tests that cuts were stable when R was 0.2, marginally stable for 0.2 < R < 0.8, and unstable when R 0.8. A comparison between the at-sp eed and non-rotating stability boundaries is shown in Fig. 4-19. The lobes generated from the predicted rotating FRFs agree more closely with experiment. In the spindle speed ranging fro m 8000 to 10000 rpm, in particular, the stability boundary, blim, is well identified. -1 0 1 x 10-7 Real (m/N) 0 500 1000 1500 2000 2500 -2 -1 0 1 x 10-7 Frequency (Hz)Imag (m/N) 0 rpm 2500 rpm 5000 rpm 7500 rpm 10000 rpm Figure 4-1. Speed-dependent FRFs of the standard holder at five differe nt spindle speeds: {0, 2500, 7500, and 10000} rpm.

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61 Figure 4-2. Example setup for rotating FRF measurement. 0.05 0.06 0.07 0.08 0.09 0.1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Time (s)Response Signal (V) holder response + runout runout signal Figure 4-3. Example of holder and runout signal resultant response.

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62 Figure 4-4. Scheme of tachomet er-aided runout filtering setup. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 -4 -3 -2 -1 0 1 2 3 4 5 Response Signal (V)Time (s) Figure 4-5. Illustration of on ce-per-revolution signal identifi cation (solid holder-spindle response; dotted runout; dash ed tachometer signal; and heavy solid line runout in one revolution). Spindle Modal hammer Capacitance probe Laser tachometer

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63 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 -3 -2 -1 0 1 2 3 4 5 Time (s)Response Signal (V) resultant runout response Figure 4-6. Example of shrink fit ho lder time-domain runout filtering.

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64 -5 0 5 x 10-8 Real m/N 0 500 1000 1500 2000 2500 3000 -8 -6 -4 -2 0 2 x 10-8 Frequency (Hz)Imag m/N unfiltered filtered Figure 4-7. Frequency response function comp arison with/without timedomain filtering. Figure 4-8. Short hollow standard holder geometry and substructure coordinates. 5a 6 I 6a 4 II 4a 5 S Extended holder Spindle-holder base

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65 Figure 4-9. Experimental setup of standa rd holder (HSK-63A in terface) rotating FRF measurement. Figure 4-10. Solid holder geometry and substructure coordinates. 6 6a 5 5a 4 4a 3 3a 1 2 2a I II III IV V

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66 -2 -1 0 1 2 x 10-7 Real (m/N) 300 600 900 1200 1500 1800 -3 -2 -1 0 1 x 10-7 Frequency (Hz)Imag (m/N) measured 10000 rpm measured 12000 rpm measured 16000 rpm predicted 10000 rpm predicted 12000 rpm predicted 16000 rpm measured 0 rpm Figure 4-11. Solid holder FRF measurement and prediction at {10000, 12000, and 16000} rpm. Figure 4-12. Geometry of Regof ix collet holder with tool. I II III IV V 1 2 2a 3 3a 4 4a 5 5a 6 6a

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67 -2 -1 0 1 2 x 10-5 Real (m/N) 500 1000 1500 2000 2500 -3 -2 -1 0 1 x 10-5 Frequency (Hz)Imag (m/N) measured 10000 rpm predicted 10000 rpm measured 0 rpm Figure 4-13. Measured and predicted FRFs for Regofix collet holder with 12.7 mm diameter 127 mm overhang carbide tool blank at 10,000 rpm. -1 -0.5 0 0.5 1 x 10-6 Real (m/N) 500 1000 1500 2000 2500 -15 -10 -5 0 5 x 10-7 Frequency (Hz)Imag (m/N) measured 10000 rpm predicted 10000 rpm measured 0 rpm Figure 4-14. Measured and predicted FRFs for Regofix collet holder with 25.4 mm diameter 127 mm overhang carbide tool blank at 10,000 rpm.

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68 Figure 4-15. Geometry of 19.1 mm diameter, four flutes, carbide endmill with 76.1 mm overhang length clamped in Command shrink fit holder. 0.8 1 1.2 1.4 1.6 x 104 0 1 2 3 4 5 6 7 Spindle speed (rpm)blim (mm) 0 rpm 10000 rpm 16000 rpm Figure 4-16. Stability lobes for FRF measurement at 0 rp m and predictions at {10000 and 16000} rpm for 19.1 mm diameter, four flute, carbide endmill clamped in Command shrink fit holder. I II III IV V VI VII VIII

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69 Figure 4-17. Setup for cutting te sts. The tool shank deflectio ns were measured using two orthogonal capacitance probes. A laser tac hometer was used to obtain the once-perrevolution signal. Laser tachometer Capacitance probe (y dir.) Capacitance probe (x dir.) Toolholder

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70 -1.5 -1 -0.5 0 0.5 1 x 10-5 -20 -15 -10 -5 0 5 x 10-6 x (m)y (m) 500 1000 1500 2000 0 0.5 1 1.5 2 2.5 3 3.5 4 x 10-7 Frequency (hz)y magnitude (m) -1.5 -1 -0.5 0 0.5 1 x 10-5 -20 -15 -10 -5 0 5 x 10-6 x (m)y (m) 500 1000 1500 2000 0 1 2 3 4 5 6 7 x 10-7 Frequency (hz)y magnitude (m) Figure 4-18. Example results for 8000 rpm slotting cuts. A) 2 mm axial depth, x vs. y displacements with once-per-revolution samples stable with R = 0.16. B) 2 mm, spectrum of y magnitude only synchr onous content is observed (frequencies identified by diamonds). C) 2.5 mm, x vs y displacements with once-per-revolution samples unstable with R = 1.0. D) 2.5 mm, spectrum of y magnitude chatter frequency is observed at 782 Hz. A ) B ) C ) D ) 133 Hz runout frequency 782 Hz chatter frequency

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71 0.8 1 1.2 1.4 1.6 x 104 0 1 2 3 4 5 6 7 Spindle speed (rpm)blim (mm) 0 rpm 10000 rpm 16000 rpm stable marginal unstable Figure 4-19. Comparison of test cut results to predicted stability boundaries determined from 0 rpm (measured) and {10000 and 16000} rpm (predicted) FRFs. Table 4-1. HSK-63A short hollow standa rd holder substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 63.3 44.6 62.8 II 52.7 0 16.3 Table 4-2. Solid holder substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 22.0 0 11.2 II 33.3 0 13.5 III 41.3 0 13.2 IV 47.4 0 11.6 V 52.6 0 13.1 Table 4-3. Material properties used in RCSA modeling. Material Elastic modulus, E (N/m2) Density, (kg/m3) Structural damping, Poissons ratio, Steel 2x1011 7800 0.0015 0.29 Carbide 5.6x1011 14400 0.0015 0.22

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72 Table 4-4. Substructure dimensions of Regof ix collet holder with 12.7 mm diameter, 127 mm overhang carbide tool blank. Sections OD (mm) ID (mm) Length(mm) I 12.7 0 127.0 II 62.7 0 25.5 III 42.0 12.7 20.6 IV 42.0 35.1 11.5 V 42.0 0 34.0 Table 4-5. Substructure dimensions of Regof ix collet holder with 25.4 mm diameter, 127 mm overhang carbide tool blank. Sections OD (mm) ID (mm) Length (mm) I 25.4 0 127.0 II 62.7 0 25.5 III 42.0 25.4 20.6 IV 42.0 35.1 11.5 V 42.0 0 34.0 Table 4-6. Substructure dimensions of the endm ill-shrink fit holder described in Figure 4-15. Sections OD (mm) ID (mm) Length (mm) I 14.7 0 48.3 II 19.1 0 27.8 III 34.1 0 13.1 IV 36.2 0 13.1 V 38.3 19.1 13.1 VI 40.4 19.1 13.1 VII 41.4 19.1 10.9 VIII 41.4 0 37.6

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73 CHAPTER 5 MICRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION The capability of manufacturing complex, free form, three-dimensional structures from metals places a high demand on micro machining. The ability to quickly produce miniature metallic components without the ne ed for specialized tooling make s the operation stand out from other techniques such as silicon etching and electrical discharge machining. Due to the scale of the features and required toleran ces, the allowable margin of error is limited. Therefore, it is crucial to have a thorough understanding of th e micro cutter dynamics prior to the machining process. Due to the micro tool size, traditional impact testing cannot be appl ied to the cutter to obtain the tool point frequency response functio n. In this chapter, micro tool/macro holderspindle and micro tool/micro sp indle frequency response functi on prediction and experimental validation is presented. A guidelin e for the choice of the proper standard tool artifact (for the measurements required for inverse RCSA) is also provided to improve prediction accuracy. Micro Scale Tools on Macro Machine Systems Prior to the availability of mi cro spindles specifically design ed for micro tools, standard high speed spindles and holders were applied. A common approach was (and remains to be) to use appropriately sized collets in standard size collet holders (Fig. 5-1 shows a micro endmill clamped in a Regofix TC-40 ER25 CAT-40 colle t holder). The following predictive approach and results pertain to this scenario. This is foll owed by predictions for a micro spindle/micro tool combination. Sensor Options for Micro Tools Micro tools are generally define d as having a cutting diameter of 1 mm or less. Figure 5-2 shows a 1mm cutting diameter, two-flute flat micro endmill. Cutting diameters down to 0.005

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74 mm are commercially available, while the tool sh ank diameters are typically {3, 3.18, or 4} mm. The shank then tapers down to the cutting edge di ameter. With tools in this scale, contact type sensors (accelerometers) can not be applied because the additional mass would change the system dynamics drastically. Capacitance (non-cont act) sensors can be app lied, but corrections must be made for a non-flat target (sensing) su rface. The complicated shape of the small tools increases the sensor non-linearity and can introduce errors under significant deflections. A laser vibrometer provides another non-co ntact sensor option. Because the laser beam can focus on a small area (1 mm or less depending on the focusing optics), laser vibrometer s provide a feasible option for small tool response measurement. To minimize difficulties associated with targeting on the fluted edge, the following tests were conduct ed on tool blank without fluted cutting edges. Modeling Description A CAT-40 solid standard holder artifact wa s used for obtaining the spindle-base receptances. The geometry and section dimension are described in Fig. 5-3 and Table 5-1. The S distance between the direct and cr oss FRF measurements was 50.5 mm. The geometry and section dimensions of the CAT-40 collet holder with the inserted tool blank are given in Fig. 5-4 and Table 5-2, wher e OH denotes the overhang length of the tool shank and IL the inserted length into the collet The material properties used were the same as listed in Table 4-3 for the steel holder and tool. Using standard holder measurements and extended holder substructure models, the spindle-base receptances, R7a7a, were computed according to the previously described inverse RCSA procedure. The tool-holder-spindle assembly receptances (Fig. 5-4) were then determined using Timoshenko beam models of the tool and collet holder. Expe rimental results and predictions for the assembly were compared fo r two cases: 1) the cross displacement-to-force FRF, H41; and 2) the direct displacement-to-force FRF, H44. In both cases, the response was

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75 recorded using a laser vibrometer located at coor dinate 4 (on the collet nut) and the system was excited using an instrumented hammer on coordinate s 1 and 4 respectively. [For these tests, the direct tool point response could not be meas ured because the vibration range exceeded the capabilities of the laser vibrometer used in this study.] The corresponding equations are given in Eqs. 5.1 and 5.2. 41 1 4 4 44 44 41 41 41 41 41 41GS GS GS GS GS P N L H Ga a (5.1) 44 1 4 4 44 44 44 44 44 44 44 44GS GS GS GS GS P N L H Ga a (5.2) The subassembly receptance matrices (GS notation) in Eqs. 5.1 and 5.2 were computed using Timoshenko beam models of substructu res I-VI (Fig. 5-4) and the spindle-base receptances. The GS4a4a matrix was obtained by first rigidl y coupling substruc tures IV-VI in a sequential manner to obtain RS4a4a, RS4a7, RS74a, and RS77, then rigidly coupling this result to the spindle-holder base receptances, R7a7a, to give the direct receptance matrix for the free end (coordinate 4a) of this subassembly (Eq. 5.3). This was necessary because the receptances must be known at any coordinate selected for prediction (coordinate 4 in this case). a a a a a a a aRS R RS RS RS GS74 1 7 7 77 7 4 4 4 4 4 (5.3) For the coupling of free-free substructures I a nd II-III, however, the pr ocedure was slightly different. In this case, a rigid connection was not assumed. Rath er, the free-free tool (I) was attached to the free-free holder se ction (II-III) using a flexible c onnection in recognition of the potential for non-rigid coupling at a physical joint (a collet in this ca se). A scalar stiffness matrix, Kc, was included in the compatibility equations for the interface at coordinates 2-2a. The resulting subassembly matrices GS44 and GS41 were then defined as shown in Eqs. 5.4 and 5.5

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76 4 2 1 1 2 2 22 42 44 44 a c a a aRS K RS R RS RS GS (5.4) 21 1 1 2 2 22 42 41R K RS R RS GSc a a a (5.5) where m f xm xf ck k k k K and the RS matrices were obtained by ri gid coupling of sections II and III. In the Kc matrix, the four k entries represent stiffness values that restrict displacement (x) and rotation () due to force (f) and moment (m), as identified by the subscript pairs. As noted, the final step was to rigidly couple the toolholder and holder-spindle subassemblies to predict the cross and direct assembly responses using Eqs. 5.1 and 5.2, respectively. Experimental Setup Three steel tool blanks (le ngths of 66.7 mm, 57.2 mm, and 47.6 mm) were used in the measurements. Each tool had six insertion lengths from 6.4 mm to 14.3 mm with an increment of 1.6 mm. A PCB 084A17 impact hammer and a Poly tec CLV 1000 laser vibrometer were used to measure the dynamic response. A total of 18 sets of FRFs with different tool overhangs (ranging from 33.3 mm to 60.3 mm and numbered 1 to 18) are shown in Fig. 5-6. For these tests, the tool was excited at its free end (using the instrument ed hammer) and the response was measured at coordinate 4 on the collet holder, attached to a 36000 rpm, 36 kW Setco spindle (not rotating), using the laser vibrometer. The experime ntal setup is shown in Fig. 5-5. It can be observed that in Fig. 5-6 that, at the tool lengths corresponding to measurements 4, 9, 12, and 13, the magnitudes are reduced for both the predictions and measurements. This is due to the interaction between the tool (cantileve r) mode and the spindle-base modes (located at 730 Hz, 950 Hz, and 1410 Hz). Two figures are given be low to show the tool-spindle interaction. As can be seen in Fig. 5-8, the tool mode amp litude is reduced because its frequency falls close

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77 to a spindle natural frequency. Even though the tool is longer in Fig. 5-8 (relative to Fig. 5-7), the dynamic stiffnesses (or peak to peak amplitude of the real and imaginary parts) are similar. The contact stiffness matrix cKin Eq. 5.4 was adjusted to visually fit the predict data with the measurements. For the predicted curves shown in Fig. 5-7 and 5-8 and the bottom panel in Fig. 5-6, the cKvalues were assigned individua lly to different tool overhangs. The four elements of cKwere assumed to have the relationship shown in Eq. 5.6; xmk and fkwere assumed to be the same by reciprocity. The orde r of magnitude of the individual k values were decided by commercial finite element software Ansys. 510 1 ) / ( MF m N kxf 410 1 ) / ( MF k rad N kf xm (5.6) 210 1 ) / ( MF rad m N km The multiplication factor, MF, was the only value used to adjust the stiffness matrixcK. Its trend is shown in Fig. 5-9. The stiffness matrix can be used to predict th e tool point response for various overhangs using a limited number of assemb ly measurements (at the longest and shortest tool overhangs, for example). The overhang rang e provided in this experiment covers the practical usage of micro tools. Measurements and predictions for the 18 H44 direct FRFs are provi ded in Fig. 5-10. The linearly decreasing stiffness values were agai n applied at the tool-holder interface. The interaction between tool and spi ndle modes is seen. The real and imaginary parts of the measured and predicted responses for measurements 9 and 12 are shown in Figs. 5-11 and 5-12, respectively.

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78 Micro Scale Tools on Micro Spindles As noted previously, specially designed di rect drive micro spindles are commercially available for micro cutting operation. Fi gure 5-13 shows a 50000 rpm NSK HES 500 electric micro spindle (ball bearing spindle support). Airpowered spindles equipp ed with air bearings are also available for higher speed operation; the torque is generally quite low, however. The taper interface design enables the NSK spindle to be coupled to commercial macro spindles (which are locked from rotation dur ing use of the micro spindle) so the machine axes can be used to locate the micro spindle relative to the part. The NSK micro spindle design combines the hol der and spindle into one unit (as do other typical micro spindles). Users only have to inse rt a proper collet to accommodate tools. The only varying geometry is the endmills. Therefore, it is reasonable to regard the whole spindle (shown in Fig. 5-13) as the spindle-base as previous described without considering the extended holder section. In this study, the spindle-base was determined by inserting a blank in the collet, performing the necessary direct and cross FRF m easurements on the standard artifact and then removing the blank in simulation to is olate the spindle-collet response. The S Value Consideration Because the dynamics of small scaled tools are very sensitive to the dimensional variation, particularly length, the proper se lection of the standard artifact geometry was explored. First, comparisons of predictions results from varying S values (distance betwee n the direct and cross impact testing locations on th e standard artifact) for the cal culation of the spindle-base receptances was considered. Second, the influenc e of the standard artifact length of prediction accuracy was assessed. This issue was viewed as being potentially very important given the results from the previous secti ons that showed a dependence of the connection stiffness on tool overhang length for the macro scale co llet holder-micro tool connection.

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79 The procedure used to calculate the micro sp indle receptances was the same as for the macro systems. Impact testing was performe d on the NSK HES 500 electric micro spindle mounted in a Mikron UCP 600 Vario milling m achine Step Tec 20,000 rpm, 16 kW spindle (HSK-63A interface). A 3.18 mm di ameter steel rod was inserted in to the micro spindle collet to act as the standard artifact (Fig. 5-14). To enhance joint stiffness consistency, the insertion length for all measurements was fixed at 13 mm. A PC B 084A17 modal hammer was used to excite the standard artifacts at the free e nd and a Polytec CLV 1000 laser vibrom eter was used to record the vibration. An MLI MetalMax 6.0 data acquisi tion system was used for data collection. Prediction results showed no si gnificant differences with the varying S values. This is because the flexibility of micro t ools is high and, even for large S values (i.e., the cross FRF is performed closer to the fixed end of standard arti fact), the tool FRF can still be distinguished and the signal to noise ratio is acceptable. Table 5-3 shows the S values tested on three standard artifacts with overhang (OH) lengths of {17, 20, and 23} mm, respectivel y. One example result for a 17 mm OH standard artifact used to predic t the tool point respons e of a 21 mm OH tool blank is presented in Fig. 5-15. The S effect is negligible. Sensitivity of Standard Artifact Length Apart from the challenges in the physical FRF measurement, the choice of the proper standard artifact length poses a nother challenge in accurate micro tool dynamic prediction. In Fig. 5-16, two artifacts with OH lengths of 17 mm and 20 mm were used to predict the FRF of a 21 mm OH steel rod (all connecti ons were assumed rigid). It is seen that the 20 mm artifact yields a closer prediction. In a subsequent test (not shown), a 23 mm artifact was used to predict the 21 mm OH rod response and the results were agai n not acceptable. Theref ore, it is necessary to have a guideline for selection of the standard artifact length fo r each tool of interest. This is offered as an alternative to including a lengt h-dependent fitting parameter, such as the MF

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80 value applied to the connection stiffness in the previous macro scale holder results. Examples are provided in the following section. Micro Tool Frequency Respon se Function Prediction For the purposes of micro tool tip FRF predicti on, steel micro tools were fabricated with a tapered geometry similar to actual micro endmills, but without the cutting edges (Fig. 5-17). The tool tip diameter was 1.5 mm to provide a reasonable target for the laser vibrometer. The geometry and section dimensions are given in Fig. 5-18 and Table 5-4. The experimental steps were the same as prev iously described. However, five standard artifacts with OH lengths of {31.5, 25.2, 22, 21, and18.6} mm were used to calculate the spindlebase receptances. These receptances were then coupl ed to the tool model (sections I-IV in Fig. 518) to produce the FRF predictions for 23.5 mm and 25.5 mm OH tapered tools (Figs. 5-19 and 5-20). In general, improved accuracy was obtaine d when the standard artifact OH length was closer to the tapered tool OH lengt h, In Fig. 5-19, for the 23.5 mm OH tapered tool, it is seen that the 31.5 mm OH artifact missed the tool and spin dle modes. The 21 mm artifact captured the natural frequencies, but over pred icted the tool mode and intera cting spindle mode amplitudes. The 18.6 mm artifact slightly under predicted the amplitudes of the same modes. This demonstrates a strong sensitivity of the spindle receptances to moment. Figure 5-20 shows that the 22 mm OH artifact prediction provided a good match to the 25.5 mm OH tapered tool FRF. Agai n, it is seen that the artifact overhang length must be close to the actual tool length to produce a good prediction. It is in teresting to see that the best FRF prediction comes from the artifact with the length which is slightly shorter than the tapered tool length. To aid in understanding this apparent contradiction, the freefree tapered tool and standard artifact responses were coupled to an infinitely stiff sp indle (i.e., a rigid wall). The results are provided in Fig. 5-21. By observi ng the relative locations of the tool natural

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81 frequencies, it is seen why 25. 5 mm OH tapered tool FRF can be better predicted by the 22 mm OH artifact, rather than the 25.2 mm artifact The improved frequency match between the tapered tool clamped-free natu ral frequency and 22 mm OH arti fact response yields a more accurate prediction. Because the tool and arti fact models (free-free responses) can be conveniently computed and coupled to a rigid ba se, this approach provides an effective means for standard artifact overhang le ngth selection prior to measurements. The outcome is the ability to predict the tool point response for arbitrary micro tool ge ometries (in a selected micro spindle) that would be otherwise immeasur able by traditional means. This data can then be used for machining parameter selection to avoid unsta ble conditions (chatter) and unacceptably large forced vibrations.

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82 Figure 5-1. Example setup of high speed machin ing with micro tools. A) High speed machine with micro endmill clamped into CAT-40 coll et holder. B) Schematic detail of a 3.18 mm diameter tool shank inserted into an ER-25 collet. Figure 5-2. Dimension comparison of a 1mm di ameter, two-flute micro endmill to a penny. Figure 5-3. CAT-40 standard holder artifact geometry. Spindle S I II Spindle-base 7a Extended holder B) A) Work p iece Micro tool Collet holde r S p indle

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83 Figure 5-4. CAT-40 ER-25 collet holder and tool geometry. Figure 5-5. Experimental setup for CAT-40 co llet holder with 3.18 mm steel tool shank. Spindle Spindle-base 7a Extended holder VIVIVIIIIII OH IL 2 1 3 2a 4 3a 5 4a 6 5a 7 6a Modal hammer Laser vibrometer Tool shank Collet holder Spindle

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84 Figure 5-6. Magnitudes of 41H FRFs measurements (top) and predictions (bottom). Figure 5-7. Measured and predicted 41H with 46.0 mm tool overhang (marked as 9 in Fig. 5-6).

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85 Figure 5-8. Measured and predicted 41H with 50.8 mm tool overhang (marked as 12 in Fig. 5-6). Figure 5-9. Multiplication factor (MF) determined from visual fit in relation to the tool overhang with a linear approximation superimposed.

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86 Figure 5-10. FRFs of44H, magnitude measurements (top) and predictions (bottom). Figure 5-11. Measured and predicted 44H with 46.0 mm tool overhang (marked as 9 in Fig. 510). 1 2 3 12 4 5 6 8 11 9 7 10

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87 Figure 5-12. Measured and predicted 44H with 50.8 mm tool overhang (marked as 12 in Fig. 510). Figure 5-13. NSK HES 500 el ectric micro spindle. Spindle Collet nut Taper interface

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88 Figure 5-14. Experimental setup for determin ation of micro spindle-base receptances. Modal hammer Laser vibrometer NSK micro spindle Standard artifact

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89 -2 0 2 x 10-5 HzReal (m/N) 0 2000 4000 6000 8000 10000 -4 -2 0 x 10-5 Frequency (Hz)Imag (m/N) measured S = 3 mm S = 6 mm S = 9 mm Figure 5-15. Prediction of tool point FRF for 3.18 mm diameter, 21 mm overhang steel rod by a 17 mm overhang standard arti fact with three different S values (3 mm dashed, 6 mm dotted, and 9 mm solid) in comparison to measured data (heavy solid line).

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90 -2 0 2 x 10-5 Real (m/N) 0 2000 4000 6000 8000 10000 -4 -2 0 x 10-5 Frequency (Hz)Imag (m/N) measured 20 mm artifact 17 mm artifact Figure 5-16. FRF prediction of a 3.18 mm diam eter, 21 mm overhang steel rod by 20 mm (solid) and 17 mm (dotted) overhang standard tool ar tifacts in comparison to measured data (heavy solid). Figure 5-17. Example 3.18 mm shan k diameter tapered tool (no flutes) with 1.5 mm diameter tool tip.

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91 Figure 5-18. Geometry of NSK HES 500 micro spindle with tapered tool. -10 -5 0 5 x 10-5 Real (m/N) 0 2000 4000 6000 8000 10000 -10 -5 0 x 10-5 Frequency (Hz)Imag (m/N) measurement 31.5mm artifact 21mm artifact 18.6mm artifact Figure 5-19. Tool tip measurement of tapered to ol with an overhang of 23.5 mm compared to predictions based on different stan dard artifact overhang lengths. Spindle-base Tapered tool I III V II IV

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92 -1 -0.5 0 0.5 1 x 10-4 Real (m/N) 0 2000 4000 6000 8000 10000 -1 -0.5 0 0.5 1 x 10-4 Frequency (Hz)Imag (m/N) measurement 31.5mm artifact 25.2mm artifact 22mm artifact Figure 5-20. Measurement of tapered tool (wit h overhang length of 25.5 mm) FRF compared with selected standard artifact FRF predictions.

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93 Figure 5-21. FRFs for free-free ta pered tool and standa rd artifact responses coupled to rigid spindle receptances. Table 5-1. CAT-40 standard holder artif act substructure section dimensions. Sections OD (mm) ID (mm) Length (mm) I 50.5 0 66.5 II 44.5 0 17.4 Table 5-2. CAT-40 ER-25 collet holder and tool substructure section dimensions. Sections OD (mm) ID (mm) Length (mm) I 3.2 0 OH II 42.1 3.2 21.6-IL III 42.1 3.2 IL IV 40.0 12.0 13.4 V 40.0 18.4 29.6 VI 44.5 5.0 12.0 -4 -2 0 2 4 x 10-3 Real (m/N) 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 -6 -4 -2 0 x 10-3 Frequency (Hz)Imag (m/N)23.5 mm tool 25.5 mm tool 31.5 mm artifact 25.2 mm artifact 22 mm artifact 21 mm artifact 18.6 mm artifact

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94 Table 5-3. S values for micro standard artifacts OH (mm) S value (mm) 17 3 6 9 20 3 6 9 12 23 3 6 9 12 15 Table 5-4. Tapered tool (23.5 mm OH ) substructure section dimensions. Sections OD (mm) ID (mm) Length (mm) I 1.5 0 3 II 1.78 0 0.78 III 2.34 0 0.78 IV 2.9 0 0.78 V 3.18 0 18.2/20.2* for 25.5 mm tapered tool

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95 CHAPTER 6 MACRO SCALE TOOL FREQUENCY RESPONSE FUNCTION PREDICTION The application of receptance coupling substructu re analysis technique to predict tool tip frequency response function for rotating tools and mi cro tools was explored in Chapters 4 and 5. In this chapter, the focus is macro scale milli ng tools and the importance of the standard holder geometry on prediction accuracy. Impact testing was performed w ith selected standard holders on various spindles and holder-spindle interfaces to evaluate the influence of standard holder geometry on dynamics prediction. Based on the results of this testing, a guideline for selecting the most suitable standard artifact for impr oved tool response predic tions using the threecomponent RCSA method. Variation in Spindle-Base Receptances with Standard Holder Geometry In theory, the calculated spindle-base recep tances for a certain spindle should be independent of the standard holder used to acqui re the system response. [Recall that the inverse RCSA step removes the extended portion of the sta ndard holder to leave just the spindle-taper (and flange) base receptances.] However, as s hown in Fig. 6-1, the HSK-63A spindle-base receptances calculated from the longer hollow sta ndard holder exhibit a more flexible spindle response (left shift in frequencie s and larger amplitude) relative to the short hollow standard holder (holder dimensions are provided in Figs. 6-2 and 4-8). This was also observed for spindlebase receptances determined from CAT-40 spindleholder interfaces. As seen in the micro tool response prediction, macro scale interfaces also exhibit moment-dependence for the interface stiffness (i.e., the longer standard holder leads to a larger mome nt for the same force level). Experimental Results Impact testing was performed on three spindl es with different hol der-spindle interfaces (HSK-63A, CAT-40, and HSK-100A). The test platforms were a 16 kW, 20000 rpm Step-Tec

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96 spindle; a 36 kW, 36000 rpm Secto spindle; and an 18000 rpm, 70 kW Omlat spindle (all were direct drive, ball bearing type spindles). The procedure used to obtain the spindle-base receptances and model the substructure beam components was the same as described in Chapter 3. The HSK-63A Interface The long hollow standard artifact geometry a nd section dimensions used for the Step-Tec spindle testing are presented in Fi g. 6-2 and Table 6-1. The short hollow standard is the same as described in Fig. 48 and Table 4-1. Two shrink fit holders, Comm and H4Y4A1000 and Shrinker SF-100-374, (long and short shown in Fig. 6-3) were used for FRF predicti on and validation. Their su bstructure dimensions are provided in Tables 6-2 and 6-3 respectively. Like the standa rd holders, shrink fit holders are constructed from a single piece of steel; therefore, the sections are assumed rigidly connected. Similar to the micro tool study, the S value (distance between direct and cross FRF measurements on the standard holders) was varied from 25.2 mm to 50. 8 mm (1 to 2 inches) to verify that it is not a crucial factor for macro tool FRF predictions. Results for prediction of the HSK-63A long shrink fit holder FRF by the short ho llow standard artifact with three different S values is presented in Fig. 6-4. It is seen th at the three predicted curves overlap each other; no significant influence from the S value is observed. All subseque nt tool point predictions were also completed with three different S values, but results are presented only for S = 25.4 mm for brevity. The FRF predictions for long and short sh rink fit holders are given in Figs. 6-5 and 6-6, respectively, using both the long and short hollow standard hold er spindle-base receptances. It is seen that both standard holders (or artifacts) provide a ccurate predictions for the long shrink fit holder. However, it is obvious that the short standard hol der yields better result in the short shrink fit holder prediction. Table 6-4 shows the ordered ex tended holder lengths (i.e., the

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97 length beyond the flange) for all the HSK-63A ho lders used. The long shrink fit holder has the length in between the standard artifacts by the similar margins; therefore the predictions are equally accurate. The length of the short shrink fi t holder, however, is le ss than both artifacts; therefore, the short standard artifact with the closer holder length gives better results. The CAT-40 Interface Three standard holders (Regofix TC40ER40x100AD, Kennametal DV40ER40080, and Shrinker V40E-075-138SF) were used for CAT-40 inte rface (Setco spindle) im pact testing. Their geometries and section dimensions are given in Fi gs. 6-7, 6-8, 6-9 and Ta bles 6-5, 6-6, 6-7, respectively, for long hollow, short hollow, and short solid holders. Again, different S values were applied and it was concluded that it is not a factor for the spindle-base receptances. The following predictions are based on the S = 25.4 mm results. For the CAT-40 holder FRF study, the tool po int (or free end) FRFs for one shrink fit holder and two collet holders (with no collets inserted) were predicted and measured. The section dimensions for these three holders are list ed in Tables 6-8, 6-9, and 6-10 with section I representing the free end. The measured and pred icted FRFs are shown in Figs. 6-10, 6-11, and 6-12. As can be seen in Fig. 6-10, the FRF predicti ons made by the two shor ter standard artifacts (hollow and solid) matched the extremely short sh rink fit holder reasonably well, while the long hollow standard artifact prediction is less accurate. It is also interesting to note that the solid and hollow holders provided similar performance. This further suggests that it is a moment dependence of the spindle-holder interface that influences the pr ediction results, rather than a cross-sectional modeling issue in the inverse RCSA step. Table 6-11 gives the extended holder lengths for the CAT-40 standard ar tifacts and test holders in lengt h sequence. For the long collet holder FRF prediction (Fig. 6-11), Table 6-11 show s that the collet holder length falls between

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98 the long and short artifacts; ther efore, the predictions are simila r between all the artifacts and the results are all acceptable. Figure 6-12 presents the case where the two s hort artifacts lengths are close to the short collet holder length; the predictions are ve ry accurate. The long artifact, however, did not provide acceptable agreement (recall that any disagreement will reduce the reliability of the predicted stability boundary). Again, it is observed that it is preferred to match the standard holder length approximate ly to the holder in question. The HSK-100A Interface For HSK-100A interface impact testing, two st andard holders were us ed. Their geometries and section dimensions are give n in Figs. 6-13 and 6-14 and Tabl es 6-12 and 6-13, respectively, for the short hollow and long solid standard artifacts. Two sets of holder and tool combinations were modeled and test ed: an HSK100ASF-075433 shrink fit holder with a Sa ndvik A393.T-19 10 175 carbide adap ter and two square insert cutting head; and an HSK100AE-125-472 set scre w holder with a Mitsub ishi Carbide MBN 10 1000 TB steel tapered ball end mill with one r ound carbide insert. Their geometries and substructure dimensions are given in Figs. 615 and 6-16 and Tables 6-14 and 6-15. The impact testing was performed on both the x and y direct ion of the Omlat spindle. The predictions are shown in Figs 6-17 through 6-20 for the shrink fi t holder-carbide tool combination and Figs. 621 through 6-24 for the set screw holder-steel tool combination. For both holder-tool combinations the long solid artifact le ngth was much closer to extended lengths (Table 6-16). The predictions in both directions all show better agreement than those obtained using the short hol low artifact spindle-holder base receptances. [Note that the FRF asymmetry between the two directions was present because the spindle was mounted to a high speed machine structure.]

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99 It can be seen that the HSK-100A holder pred ictions are not as accura te, in general, as those presented for the HSK-63A and CAT-40 st udies, which used only holders and not holdertool combinations for prediction validation. Fo r the holder-tool combinations, joint stiffness between the tools and holders was included in the measuremen t; the model, however, assumed rigid connections everywhere. For practical application of th e RCSA method in industry, the prediction agreement shown here (p rovided the appropriate standard holder length is selected) is acceptable so the rigid connection assumption is maintained, although investigation of toolholder interface stiffness could be purs ued as a future research topic. -2 -1 0 1 2 x 10-7 Real (m/N) 0 1000 2000 3000 4000 5000 -3 -2 -1 0 1 x 10-7 Frequency (Hz)Imaginary (m/N) long hollow holder short hollow holder Figure 6-1. HSK-63A spindle-ba se receptances calculated by long and short hollow standard holders.

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100 Figure 6-2. Geometry of HSK-63A long ho llow standard hollow holder artifact. Figure 6-3. HSK-63A shrink f it holders: long hollow (left) and short hollow (right). Spindle I II Spindle-base Extended holder

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101 -1 -0.5 0 0.5 1 x 10-7 Real (m/N) 0 500 1000 1500 2000 2500 -2 -1.5 -1 -0.5 0 x 10-7 Frequency (Hz)Imag (m/N) measurement short hollow artifact, S = 1" short hollow artifact, S = 1.5" short hollow artifact, S = 2" Figure 6-4. HSK-63A FRF predic tions for long shrink fit holder by short hollow artifact with different S values in comparison with measured FRF (heavy solid line). -1 -0.5 0 0.5 1 x 10-7 Real (m/N) 0 500 1000 1500 2000 2500 -15 -10 -5 0 x 10-8 Frequency (Hz)Imag (m/N) measurement long hollow artifact short hollow artifact Figure 6-5. HSK-63A long shrink fi t holder FRF predictions using tw o different standard holders in comparison with measured FRF (heavy solid line).

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102 -1 -0.5 0 0.5 1 x 10-7 Real (m/N) 0 500 1000 1500 2000 2500 -15 -10 -5 0 5 x 10-8 Frequency (Hz)Imag (m/N) measurement long hollow artifact short hollow artifact Figure 6-6. HSK-63A short shrink fit holder FR F predictions using two different standard holders in comparison with meas ured FRF (heavy solid line). Figure 6-7. Geometry of CAT-40 l ong hollow standard holder artifact. Spindle I II Spindle-base Extended holder

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103 Figure 6-8. Geometry of CAT-40 s hort hollow standard holder artifact. Figure 6-9. Geometry of CAT-40 shor t solid standard holder artifact. Spindle I Spindle-base Extended holder Spindle I II Spindle-base Extended holder

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104 -1.5 -1 -0.5 0 0.5 1 1.5 x 10-7 Real (m/N) 500 1000 1500 2000 2500 -1 0 1 x 10-7 Frequency (Hz)Imag. (m/N) measurement long hollow artifact short hollow artifact short solid artifact Figure 6-10. CAT-40 shrink fit hol der FRF predictions in compar ison with measured FRF (heavy solid line). -4 -2 0 2 4 6 x 10-7 Real (m/N) 500 1000 1500 2000 2500 -8 -6 -4 -2 0 x 10-7 Frequency (Hz)Imag. (m/N) measurement long hollow artifact short hollow artifact short solid artifact Figure 6-11. CAT-40 long collet holder FRF pred ictions in comparison with measured FRF (heavy solid line).

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105 -10 -5 0 5 x 10-7 Real (m/N) 500 1000 1500 2000 2500 -10 -5 0 5 x 10-7 Frequency (Hz)Imag. (m/N) measurement long hollow artifact short hollow artifact short solid artifact Figure 6-12. CAT-40 short collet holder FRF predictions in comparison with measured FRF (heavy solid line). Figure 6-13. Geometry of HSK-100A s hort hollow standard holder artifact. Spindle Spindle-base Extended holder II I

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106 Figure 6-14. Geometry of HSK-100A l ong solid standard holder artifact. Figure 6-15. Geometry of Briney HSK100ASF075-433 shrink fit holder with Sandvik A393.T19 10 175 carbide adapter and two square insert cutting head. Spindle Spindle-base Extended holder and carbide tool VII VI V IV III II I : Steel : Carbide Spindle Spindle-base Extended holder I

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107 Figure 6-16. Geometry of Briney HSK100AE-125472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tape red ball end mill with one round carbide insert. 0 1000 2000 3000 4000 5000 -5 0 5 x 10-6 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -10 -5 0 x 10-6 Frequency (Hz)Imaginary (m/N) Figure 6-17. HSK100ASF-075-433 shrink fit hold er with Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head measurement and prediction (y direction) using HSK-100A short hollow standard holder artifact. Spindle Spindle-base Extended holder and steeltool XII XI X : Steel IX II I

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108 0 1000 2000 3000 4000 5000 -5 0 5 x 10-6 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -8 -6 -4 -2 0 x 10-6 Frequency (Hz)Imaginary (m/N) Figure 6-18. HSK100ASF-075-433 shrink fit hold er with Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head measurement and prediction (x direction) using HSK-100A short hollow standard holder artifact. 0 1000 2000 3000 4000 5000 -2 0 2 4 x 10-6 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -8 -6 -4 -2 0 x 10-6 Frequency (Hz)Imaginary (m/N) Figure 6-19. HSK100ASF-075-433 shrink fit hold er with Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head measurement and prediction (y direction) using HSK-100A long solid standard holder artifact.

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109 0 1000 2000 3000 4000 5000 -2 0 2 4 x 10-6 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -6 -4 -2 0 x 10-6 Frequency (Hz)Imaginary (m/N) Figure 6-20. HSK100ASF-075-433 shrink fit hold er with Sandvik A393.T-19 10 175 carbide adapter and two square insert cutting head measurement and prediction (x direction) using HSK-100A long solid standard holder artifact. 0 1000 2000 3000 4000 5000 -5 0 5 x 10-7 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -10 -5 0 x 10-7 Frequency (Hz)Imaginary (m/N) Figure 6-21. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction (y direction) using HSK-100A shor t hollow standard holder artifact.

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110 0 1000 2000 3000 4000 5000 -5 0 5 x 10-7 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -10 -5 0 x 10-7 Frequency (Hz)Imaginary (m/N) Figure 6-22. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round carbide insert measurement and prediction (x direction) using HSK-100A shor t hollow standard holder artifact. 0 1000 2000 3000 4000 5000 -5 0 5 x 10-7 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -10 -5 0 x 10-7 Frequency (Hz)Imaginary (m/N) Figure 6-23. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round car bide insert measurement and prediction (y direction) using HSK-100A l ong solid standard holder artifact.

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111 0 1000 2000 3000 4000 5000 -5 0 5 x 10-7 Real (m/N) Predicted Measured 0 1000 2000 3000 4000 5000 -10 -5 0 x 10-7 Frequency (Hz)Imaginary (m/N) Figure 6-24. HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round carbide insert measurement and prediction (x direction) using HSK-100A l ong solid standard holder artifact. Table 6-1. HSK-63A long hollow standard hol der artifact substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 52.7 29.9 100 II 52.7 0 10 Table 6-2. Substructure dimensions for HSK-63 A long shrink fit holder (section I at free end). Sections OD (mm) ID (mm) Length (mm) I 44.9 25.4 9.5 II 46.3 25.4 9.5 III 47.8 25.4 9.5 IV 49.3 25.4 9.5 V 50.7 27.4 9.3 VI 52.1 27.4 9.3 VII 52.9 0 44.9

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112 Table 6-3. Substructure dimensions for HSK-63 A long shrink fit holder (section I at free end). Sections OD (mm) ID (mm) Length (mm) I 41.9 25.2 10.8 II 42.9 25.2 10.8 III 44 25.2 10.8 IV 44.5 26.2 28.9 V 44.5 6.6 7.8 Table 6-4. Extended holder lengths of the HSK63A standard artifacts and shrink fit holders. Short shrink fit Short artif act Long shrink f it Long artifact 69.1 mm 79.1 mm 101.2 mm 120 mm Table 6-5. CAT-40 long hollow standard hol der artifact substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 44.4 25.2 100.2 II 44.4 0 19.7 Table 6-6. CAT-40 short hollow standard hol der artifact substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 44.4 25.4 40 II 44.4 0 19.9 Table 6-7. CAT-40 short solid standard hol der artifact substructure dimensions. Section OD (mm) ID (mm) Length (mm) I 44.6 0 57.3 Table 6-8. CAT-40 shrink fit hol der substructure dimensions. Section OD (mm) ID (mm) Length (mm) I 44.4 12.7 16.2 Table 6-9. CAT-40 long collet holder substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 48.4 38.4 9.8 II 63 38.4 7.8 III 63 36 7.8 IV 50 33.2 10 V 50 30.1 10 VI 50 28.5 23.2 VII 50 15 6.5

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113 Table 6-10. CAT-40 short collet hol der substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 44.7 34.7 9.7 II 63 39.2 7.9 III 63 36.8 7.9 IV 50 34.5 7.2 V 50 32.4 7.2 VI 44.5 31.3 15.9 Table 6-11. Extended holder le ngths of the CAT-40 standard artifacts and tested holders. Shrink fit holder Short collet holder Short solid artifact Short hollow artifact Long collet holder Long hollow artifact 16.2 mm 55.8 mm 57.3 mm 59.9 mm 75.4 mm 119.9 mm Table 6-12. HSK-100A short hollow standard ho lder artifact substructure dimensions. Sections OD (mm) ID (mm) Length (mm) I 100 80 20 II 85 0 20 Table 6-13. HSK-100A long soild standard hol der artifact substructure dimensions. Section OD (mm) ID (mm) Length (mm) I 51 0 187 Table 6-14. Substructure dimensions of Br iney HSK100ASF-075-433 shrink fit holder with Sandvik A393.T-19 10 175 carbide adapter a nd two square insert cutting head. Sections OD (mm) ID (mm) Length (mm) I 13.4 0 30 II 19.1 0 125.5 III 35.3 19.1 10.6 IV 36.4 19.1 10.6 V 37.6 19.1 10.6 VI 38.1 19.1 17.8 VII 38.1 12 31.6

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114 Table 6-15. Substructure dimensions of Br iney HSK100AE-125-472 set screw holder with Mitsubishi Carbide MBN 10 1000 TB steel tapered ball end mill with one round carbide insert. Sections OD (mm) ID (mm) Length (mm) I 26.1 0 10.9 II 27.5 0 10.9 III 28.9 0 10.9 IV 30.3 0 10.9 V 31.8 0 10.9 VI 33.2 0 10.9 VII 34.6 0 10.9 VIII 36.0 0 10.9 IX 37.4 0 10.9 X 38.1 0 46.5 XI 63.5 0 69.3 XII 63.5 12 21.6 Table 6-16. Extended holder lengt hs of the HSK-100A standard artifacts and tested holders. Short hollow artifact Long solid artifact Set screw holder and steel tool Shrink fit holder and carbide tool 40 mm 187 mm 235.5 mm 236.7 mm

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115 CHAPTER 7 CONCLUSIONS AND FUTURE WORK The improved receptance coupling substructure analysis (RCSA) method for predictions of the dynamic response for macro, micro, and ro tating tool-holder-spindle combinations was validated by impact testing and cutting tests. The RCSA technique enab les frequency response functions for arbitrary tools and holders, which are convenient to model, to be coupled with experimentally-determined spindle responses, which are difficult to model (particularly regarding damping estimates). This capability was demonstrated for the industrially relevant cases of 1) spindles that e xhibit variations in their dynami c response with the commanded spindle speed; 2) micro-scale cutting tools a nd 3) macro-scale cutti ng tools with different lengths. The ability to analytically couple arbitr ary holder-tool combinations to an archived spindle response (obtained from the inverse RCSA method) enables job shop management to determine proper tooling in an a priori manner by using the tool point frequency response prediction to select pref erred operating parameters (to avoid chatter, for example) without the need for costly trial-and-error cutting tests. For micro tool systems, this technique also provided a feasible approach to dete rmining the tool point freque ncy response function using a combination of traditional impact testing and RCS A modeling. The primary contributions of this research can be summarized in the following points. Conclusions A limitation of traditional impact testing on m acro scale tools is that the measurements must be carried out when the spindle is not rotating. For spindles that exhibit speeddependent dynamics, this can lead to errors in the selected operating conditions. By measuring the spindle-base receptances at speed using a standard holder without cutting teeth (so that rotating measurement are possibl e), predictions of the tool point dynamics are made possible. A primary difficulty to measuring rotating freq uency response functions is the inherent runout in the displacement/velocity signal. A time-domain filtering technique was applied to remove the runout and yield usable frequency response functions.

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116 It was shown that Timoshenko beam models ar e preferred to the analytical Euler-Bernoulli solutions, despite the increased computational burden. It was shown that the diameter of the fluted section of cutting tools can be represented by an equivalent diameter calculated from the s econd area moment of inertia determined from a solid model. It was determined that the distance between the direct and cross FRF measurements (S) on standard holders for computation of the unmeas ured spindle receptanc es is not a crucial factor. However, the value must be selected to provide a reasonable amplitude difference between the direct and cross responses base d on the measurement signal to noise ratio. It was demonstrated that improved FRF pred iction for macro tool s can be obtained by strategically choosing th e dimension of the artifact to approximately match the desired tool-holder length. It was hypot hesized that this is due to a moment dependence of the spindle-holder connection stiffness. This wa s evaluated experimentally by comparing spindle receptances obtained fr om long and short standard hold ers. It was seen that the longer holder (with higher moment) led to a more flexible spindle response. Tool point prediction for micro tapered cutti ng tools was demonstrated. This presents the only validated approach to obt aining this information known to the author. It was observed that the micro-scale spindle-tool combinati ons also exhibited the moment dependence for the connection stiffness and improved predic tion accuracy was obtai ned if the artifact length was selected to have a (fundamental) cl amped-free natural fre quency near the tool (fundamental) clamped-free natural frequency. Future Work Based on the results obtained in this st udy, the followings issues warrant further exploration. The moment dependent holder-spindle interf ace stiffness needs to be studied. Finite element modeling may provide the necessary capabilities to better understand the mechanism for this experimentally-observe d condition. An improved understanding could lead to interface designs that do not exhibit this behavior. Similarly, improved accuracy may be obtaine d if the joint stiffness between tool and holder is not assumed to be rigid for all case s. Again, finite element software may be applied to improve the understandi ng of this joint. To be practically beneficial, however, it will be necessary to develop a method to estimate the joint stiffness based on the connection type and geometry without requir ing a new model to be developed for every situation (to limit computation times for the predicted tool point response). Damping must also be considered.

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117 In this research the tapered micro tool res ponse measurements was lim ited to diameters of 1.5 mm and larger due the minimum laser beam diameter available from the laser vibrometer. Additional tests on smaller diamet ers, combined with cutting test results, would be beneficial.

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118 LIST OF REFERENCES [1] Schultz, H. and Moriwaki, T., 1992, Hi gh-Speed Machining, Annals of the CIRP, 41, Issue 2, pp. 637-643. [2] Arnold, R. N., V, The Mechanism of Tool Vibration in the Cutting of Steel, Proceedings of the Institution of Mechanical Engineers, 154, Issue 4, pp. 261-284. [3] Tlusty, J., 1956, Theory of Chatter in M achining and Stability Ca lculations in Machine Tools (Russian), Stankii Instru ment, Issue 3 and 4, Moscow. [4] Tlusty, J., 2000, Manufacturing Processes and Equipment, Prentice Hall. [5] Shin, Y.C., 1992, Bearing Nonlinearity and Stability Analysis in High Speed Machining, Journal of Engineering for Industr y, Transactions of the ASME, 114, pp. 23-30. [6] Smith, K. S., 1987, Automatic Selection of the Optim um Spindle Speed in High-Speed Milling, Ph D. Dissertation, University of Florida. [7] Delio, T., Tlusty, J., and Smith, S., 1992, Use of Audio Signals for Chatter Detection and Control, Journal of Engineering for Industry, Transactions of the ASME, 114, pp. 146157. [8] Jones, A. B., 1960, A General Theory for Elastically Constrained Ball and Radial Roller Bearing under Arbitrary Load and Speed Cond itions, Transactions of ASME, Journal of Basic Engineering, pp. 309-320. [9] Chen, C. H., Wang, K. W., Shin, Y. C ., 1994, An Integrated Approach Toward the Dynamic Analysis of High Speed Spindles, Pa rt 1: Spindle Model, Transactions of ASME, Journal of Vibration and Acoustics, 116, pp. 506-513. [10] Jorgenson, B. R., Shin, Y. C., 1998, D ynamics of Spindle-Bearing Systems at High Speeds Including Cutting Load Effects, Transactions of ASME, Journal of Manufacturing Science and Engineering, 120, pp. 387-394. [11] Cao, Y., Altintas, Y., 2004, A Genera l Mothod for the Modeling of Spindle-Bearing Systems, Transactions of ASME, Journal of Mechanical Design, 126, pp. 1089-1104. [12] Stein, J. L., Tu, J. F., 1994, A Stat e-Space Model for Monitoring Thermally Induced Preload in Anti-Friction Spindl e Bearings of High-Speed Mach ine Tools, Transactions of the ASME, Journal of Dynamic Systems, Measurement, and Control, 116, pp. 372-386. [13] Bossmanns, B., Tu, J. F., 2001, A Power Flow Model for High Speed Motorized Spindles-Heat Generation Characterization, Transactions of the ASME, Journal of Manufacturing Science and Engineering, 123, pp. 494-505.

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119 [14] Lin, C. W., Tu, J. F., Kamman, J., 2 003, An Integrated Thermo-Mechanical-Dynamic Model to Characterize Motorized Machine Tool Spindles during Very High Speed Rotation, International Journal of Machine Tools and Manufacturing, 43, pp. 1035-1050. [15] Li, H., Shin, Y. C., 2004, Integrated Dynamic Thermo-Mechanical Modeling of High Speed Spindles, Part 1: Model Developmen t, Transactions of ASME, Journal of Manufacturing Science and Engineering, 126, pp. 148-158. [16] Smith, S., Tlusty, J., 1991, An Overview of Modeling and Simulation of the Milling Process, Transaction of the ASME, J ournal of Engineering for Industry, 113, pp. 169-175. [17] Bayly, P., Halley, J., Mann, B., Davies M., 2003, Stability of Interrupted Cutting by Temporal Finite Element Analysis, Transa ction of the ASME, J ournal of Manufacturing Science and Engineering, 125, Issue 2, pp. 220-225. [18] Mann, B., Bayly, P., Davies, M., Halley, J., 2004, Limit Cycles, Bifurcations, and Accuracy of the Milling Process, Journal of Sound and Vibration, 277, pp. 31-48. [19] Insperger, T., Mann, B., Stpn, G., Bayl y, P., 2003, Stability of Up-Milling and DownMilling, Part 1: Alternative Analytical Met hods, International Jour nal of Machine Tools and Manufacture, 43, pp. 25-34. [20] Tlusty, J., 1954, Self-Excited Vibrations in Cutting Metals (German), Acta Technica Ac. Sc. Hungaricae, Tom VIII, Budapest, pp.319-360. [21] Tobias, S. A., Fishwick, W., 1958, Theor y of Regenerative Machin e Tool Chatter, The Engineer 205. [22] Tlusty, J., Polacek, M. 1963, The Stabil ity of the Machine-Tool against Self-Excited Vibration in Machining, In Proceedings of the International Research in Production Engineering Conference, Pittsburgh, PA, ASME: New York, 465. [23] Merrit, H., 1965, Theory of Self-Excited Machine Tool Ch atter, Transactions of the ASME, Journal of Engineering for Industry, 87, Issue 4, pp. 447-454. [24] Koenisberger, F., Tlusty, J., 1967, Machine Tool Structures-Vol. I: Stability against Chatter, Pergamon Press. [25] Budak, E., Altintas, Y., 1998, Analytical Prediction of Chatter Stability Conditions for Multi-Degree of Systems in Milling. Part I: M odeling, Part II: Appli cations, Transactions of ASME, Journal of Dynamic systems, Measurement and Control, 120, pp. 22-36. [26] Duncan, W.J., 1947, Mechanical Admittances and their Applications to Oscillation Problems, Ministry of Supply, Aeronautical Re search Council Reports and Memoranda No. 2000; His Majestys St ationery Office: London. [27] Schmitz, T., Donaldson, R., 2000, Pre dicting High-Speed Machining Dynamics by Substructure, Analysis. Annals of the CIRP 2000, 49, Issue 1, pp. 303-308.

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120 [28] Schmitz, T., Davies, M., 2001, Tool Point Frequency Response Prediction for HighSpeed Machining by RCSA, Journal of Ma nufacturing Science and Engineering, 123, pp. 700-707. [29] Schmitz, T., Davies, M., Medicus, K., Snyder, J., 2001, Improving High-Speed Machining Material Removal Rates by Rapid Dynamic Analysis, Annals of the CIRP 2001, 50, Issue 1, pp. 263-268. [30] Schmitz, T., Duncan, G. S., Zahner C., Dyer, J., Tummond M., 2005, Improved Milling Capacities through Dynamics Prediction: Th ree Component SpindleHolder-Tool Model, Proceedings of 2005 NSF DM II Grant ees Conference, Scottsdale, AZ. [31] Schmitz, T., Duncan, G. S., 2005, ThreeComponent Receptance Coupling Substructure Analysis for Tool Point Dynamics Predic tion, Journal of Manufacturing Science and Engineering, 127, pp. 781-790. [32] Bishop, R., 1955, The Analysis of Vibrating Systems with Embody Beams in Flexure, Proceeding of the Institution of Mechanical Engineers, 169, pp. 1031-1050. [33] Weaver, Jr., Timoshenko, P., Young, D., 1990, Vibration Problems in Engineering, 5th Ed., John Wiley and Sons, New York. [34] Yokoyama, T., 1990, Vibration of a Hangi ng Timoshenko Beam under Gravity, Journal of Sound and Vibration, 141, pp. 245-258. [35] Hutchinson, J., 2001, Shear Coefficien t for Timoshenko Beam Theory, Journal of Applied Mechanics, 68, pp. 87-92. [36] Altintas, Y., 2000, Manufacturing Automation, Cambridge University Press, Cambridge, UK. [37] Schmitz, T., 2003, Chatter Recognition by a Statistical Evaluation of the Synchronously Sampled Audio Signal, Journa l of Sound and Vibration, 262, Issue 3, pp. 721-730.

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121 BIOGRAPHICAL SKETCH The author was born on March 30, 1971, in Fengyuan, Taiwan. In 1993, he was awarded Bachelor of Engineering in mechanical engi neering from Fengchia University, Taiwan. Following the college graduation, he served as a patrol sergeant and outboard marine engine technician in Army Coast Patrol Taiwan and was discharged in 1995. His graduate study started in January 1997 in the Mechanical Engineering Department at the University of Florida. Under the late Dr. A li A. Seiregs supervision, he was awarded his Master of Science in 2000. Soon after gr aduation, he was employed by China Engine Corporation in Taiwan, in charge of the compone nt development of GS-1 2.0 liter gas engine. In 2002, he again joined the Un iversity of Florida for doctora l research. Instructed by Dr. John C. Ziegert and Dr. Tony L. Schmitz, he was engaged in high speed milling machine dynamics research and was the teaching assi stant for the underg raduate Manufacturing Engineering class. He was awar ded Doctor of Philosophy in 2007. He is a member of Tau Beta Pi, ASME, and served as the president of Christians on Campus in the University of Florida in 2006 school year.


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