improved spindle dynamics identification technique...

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IMPROVED SPINDLE DYNAMICS IDENTIFICATION TECHNIQUE FOR RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS

By

UTTARA VIJAY KUMAR

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2012

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© 2012 Uttara Vijay Kumar

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To my parents, Alka and Vijay and husband, Ashwin

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ACKNOWLEDGMENTS

I extend my sincere gratitude to my advisor, Dr. Tony L. Schmitz, for his guidance

and ideas throughout my research. I feel fortunate to have had the opportunity to work

with him.

I would like to thank my committee members Dr. Schueller, Dr. Ifju and Dr. Fuchs

for their support. I am also thankful to Dr. Hitomi Greenslet for her support and

encouragement. I thank my colleagues in the Machine Tool Research Center (MTRC)

for their help, and sense of humor, making the MTRC a fun place to work. I would also

like to acknowledge Dr. Sam Turner at the University of Sheffield, Advanced

Manufacturing Research Center with Boeing, for giving me an opportunity to conduct

experiments on the milling machines for this research.

Last, but not least, I would like to thank my entire family for their unconditional love

and patience.

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TABLE OF CONTENTS page

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

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

LIST OF FIGURES ........................................................................................................ 11

ABSTRACT ................................................................................................................... 17

CHAPTER

1 INTRODUCTION .................................................................................................... 19

Motivation ............................................................................................................... 19

Research Description .............................................................................................. 21

Dissertation Organization ........................................................................................ 22

2 LITERATURE REVIEW .......................................................................................... 25

3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS ................................... 30

Description .............................................................................................................. 30

Frequency Response Function ......................................................................... 30

Three Component Coupling for Tool Point FRF Prediction .............................. 30

Free-Free Beam Receptances ................................................................................ 31

Rigid Coupling of Free-Free Receptances .............................................................. 34

Coupling of Tool-Holder and Spindle-Machine Receptances .................................. 38

4 IDENTIFICATION OF SPINDLE-MACHINE RECEPTANCES ................................ 44

Synthesis Approach ................................................................................................ 45

Finite Difference Approach ..................................................................................... 46

Euler-Bernoulli Method ........................................................................................... 47

5 RESULTS ............................................................................................................... 50

Spindle-Machine Receptances Comparison ........................................................... 50

Tool Point Frequency Response Comparison ........................................................ 51

Mikron UCP-600 Vario...................................................................................... 52

25.4 mm diameter carbide endmill in a shrink fit holder ............................. 52

19.05 mm diameter carbide endmill in a shrink fit holder ........................... 53

Starragheckert ZT-1000 Super Constellation ................................................... 53

12 mm diameter carbide endmill in a shrink fit holder ................................ 53



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Cincinnati FTV-5 2500 ...................................................................................... 55





Introduction of flexible connection between the tool and the holder ........................ 56

Cincinnati FTV-5 2500 ...................................................................................... 58

Mikron UCP-600 Vario...................................................................................... 59

6 CONCLUSION AND FUTURE WORK .................................................................. 154

Conclusion ............................................................................................................ 154

Future Work .......................................................................................................... 156

APPENDIX A:FLEXIBLE COUPLING BETWEEN TOOL AND HOLDER .................... 157

LIST OF REFERENCES ............................................................................................. 160

BIOGRAPHICAL SKETCH .......................................................................................... 164

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LIST OF TABLES

Table page 5-1 Specifications of milling machines tested ........................................................... 62

5-2 E-B fitting parameters for Mikron UCP-600 Vario CNC milling machine spindle. ............................................................................................................... 62

5-3 E-B fitting parameters for the short standard artifact on the Starragheckert ZT-1000 Super Constellation .............................................................................. 63

5-4 E-B fitting parameters for the long standard artifact on the Starragheckert ZT-1000 Super Constellation ................................................................................... 63

5-5 E-B fitting parameters for the short standard artifact on the Cincinnati FTV-5 2500 ................................................................................................................... 64

5-6 E-B fitting parameters for the long standard artifact on the Cincinnati FTV-5 2500 ................................................................................................................... 64

5-7 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 99 mm ......................................................................... 65

5-8 Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 107 mm ....................................................................... 65

5-9 Comparison metric (m/N) for the FRF predictions of 19.05 mm diameter endmill, overhang length 70.4 mm ...................................................................... 65

5-10 Comparison metric (m/N) for the FRF predictions of 19.05 mm diameter endmill, overhang length 76 mm ......................................................................... 65

5-11 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 44.7 mm using short artifact spindle receptances .................... 65


5-13 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 44.7 mm using long artifact spindle receptances ..................... 66

5-14 Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55 mm using long artifact spindle receptances ........................ 66


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5-19 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances. ................... 67


5-21 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances. .................... 67

5-22 Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75 mm using long artifact spindle receptances. ....................... 68










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5-43 Stiffness matrix values of 12 mm diameter blank clamped in a shrink fit holder, Cincinnati FTV-5 2500 ............................................................................ 72

5-44 Average stiffness matrix values for blank-shrink fit holders inserted in Cincinnati FTV-5 2500 ........................................................................................ 72

5-45 Average stiffness matrix values for blank-collet holders inserted in Cincinnati FTV-5 2500......................................................................................................... 72

5-46 Average stiffness matrix values for blank-shrink fit holders inserted in Mikron UCP-600 Vario ................................................................................................... 72

5-47 Average stiffness matrix values for blank-collet holders inserted in Mikron UCP-600 Vario ................................................................................................... 72

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5-48 Average stiffness matrix values for blank-Tribos holders inserted in Mikron UCP-600 Vario ................................................................................................... 73

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LIST OF FIGURES

Figure page 1-1 Example stability lobe diagram ........................................................................... 23

1-2 Standard artifact measurement. A) Direct FRF measurement. B) Cross FRF measurement ...................................................................................................... 24

3-1 Three-component receptance coupling model for the tool (I), holder (II), and spindle-machine (III). .......................................................................................... 39

3-2 Individual components I-II with displacements and rotations at specified coordinate locations. ........................................................................................... 40

3-3 Subassembly I-II composed of tool (I) and holder (II). The generalized force Q1 is applied to U1 to determine G11 and G3a1. ................................................... 41

3-4 Subassembly I-II composed of tool (I) and holder (II). The generalized force Q3a is applied to U3a to determine G3a3a and G13a. .............................................. 42

3-5 The I-II subassembly is rigidly coupled to the spindle-machine (III) to determine the tool point receptance matrix, G11. ............................................... 43

4-1 Artifact model for determining R3b3b by inverse RCSA. ....................................... 49

5-1 Artifact dimensions for Mikron UCP-600 Vario measurements. .......................... 74

5-2 H22 artifact measurement and E-B fit for Mikron UCP-600 Vario CNC milling machine. ............................................................................................................. 74

5-3 L22/N22 results for the Mikron UCP-600 Vario CNC milling machine. .................. 75

5-4 P22 results for the Mikron UCP-600 Vario CNC milling machine. ........................ 76

5-5 Short artifact dimensions for Starragheckert ZT-1000 Super Constellation measurements. ................................................................................................... 77

5-6 Long artifact dimensions for Starragheckert ZT-1000 Super Constellation measurements. ................................................................................................... 77

5-7 H22 short artifact measurement and E-B fit for Starragheckert ZT-1000 Super Constellation milling machine. ............................................................................ 78

5-8 L22/N22 results for the short artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 79

5-9 P22 results for the short artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 80

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5-10 H22 long artifact measurement and E-B fit for Starragheckert ZT-1000 Super Constellation milling machine. ............................................................................ 81

5-11 L22/N22 results for the long artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. .................................................... 82

5-12 P22 results for the long artifact measurement on ZT-1000 Super Constellation Starragheckert milling machine. ......................................................................... 83

5-13 H22 short artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling machine. ............................................................................................................. 84

5-14 L22/N22 results for the short artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 85

5-15 P22 results for the short artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 86

5-16 H22 long artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling machine. ............................................................................................................. 87

5-17 L22/N22 results for the long artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 88

5-18 P22 results for the long artifact measurement on Cincinnati FTV-5 2500 milling machine. .................................................................................................. 89

5-19 Beam model for 25.4 mm diameter, three flute endmill inserted in a tapered shrink fit holder (not to scale).............................................................................. 90

5-20 Comparison between H11 tool point measuremen for three flute, 25.4 mm diameter endmill with an overhang length of 99 mm. ......................................... 91

5-21 Comparison between H11 tool point measurement for three flute, 25.4 mm diameter endmill with an overhang length of 107 mm. ....................................... 92

5-22 Beam model for 19.05 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale).............................................................................. 93

5-23 Comparison between H11 tool point measurement and prediction for four flute, 19.05 mm diameter endmill, overhang length of 70.4 mm. ........................ 94

5-24 Comparison between H11 tool point measurement and prediction for four flute, 19.05 mm diameter endmill, overhang length of 76 mm. ........................... 95

5-25 Beam model for 12 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale).............................................................................. 96

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5-26 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (short artifact). ......... 97

5-27 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 55 mm (short artifact). ......... 98

5-28 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmill, overhang length of 45 mm (long artifact). .......... 99

5-29 Comparison between H11 tool point measuremen and prediction for four flute, 12 mm diameter endmill, overhang length of 55 mm (long artifact). ................. 100

5-30. Beam model for 16 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale)............................................................................ 101

5-31 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (short artifact). ....... 102


5-33 Comparison between H11 tool point measurement and prediction for four flute, 16 mm diameter endmill, overhang length of 55 mm (long artifact). ........ 104

5-34 Comparison between H11 tool point measurement nad prediction for four flute, 16 mm diameter endmill, overhang length of 65 mm (long artifact). ........ 105

5-35 Beam model for 20 mm diameter, two flute endmill inserted in a tapered shrink fit holder (not to scale)............................................................................ 106

5-36 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (short artifact). ....... 107


5-38 Comparison between H11 tool point measurement nad prediction for two flute, 20 mm diameter endmill, overhang length of 65 mm (long artifact). ........ 109

5-39 Comparison between H11 tool point measurement and prediction for two flute, 20 mm diameter endmill, overhang length of 75 mm (long artifact). ........ 110

5-40 Beam model for 25 mm diameter, four flute endmill inserted in a tapered shrink fit holder (not to scale)............................................................................ 111


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5-46 Comparison between H11 tool point measurement and prediction for four flute, 12 mm diameter endmil, overhang length of 55 mm (short artifact). ........ 117







5-53 Tool point FRF measurement of 20 mm carbide end mill on Cincinnati FTV-5 2500. ................................................................................................................ 124





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5-58 Tool point FRF measurement of 25 mm carbide end mill on Cincinnati FTV-5 2500. ................................................................................................................ 129





5-63 Component coordinates for flexible coupling of holder and blank..................... 134

5-64 Various shrink fit holders with blanks for Cincinnati FTV-5 2500 spindle .......... 134

5-65 Collet holder for Cincinnati FTV-5 2500 spindle ............................................... 135

5-66 Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (rigid connection) ................................................ 136



5-69 Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (flexible connection) ............................................ 139



5-72 Collet holder for Mikron UCP-600 Vario ........................................................... 142

5-73 25 mm diameter collet holder and blank for Mikron UCP-600 Vario ................. 143

5-74 Tribos holders for Mikron UCP-600 Vario ......................................................... 143

5-75 Mechanism of tool clamping in a Tribos holder (http://www.us.schunk.com) ... 144

5-76 Beam model for 6.33 mm diameter, 2-flute endmill inserted in a collet holder (not to scale) ..................................................................................................... 144

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5-77 Measured and predicted tool point FRF of 6.33 mm diameter 2-flute carbide endmill in collet holder, overhang length 75 mm (rigid connection) .................. 145

5-78 Measured and predicted tool point FRF of 6.33 mm diameter 2-flute carbide endmill in collet holder, overhang length 75 mm (flexible connection) .............. 146

5-79 Beam model for 19 mm diameter, 4-flute endmill inserted in a collet holder (not to scale) ..................................................................................................... 147

5-80 Measured and predicted tool point FRF of 19 mm diameter carbide 4-flute endmill in collet holder, overhang length 60 mm (rigid connection) .................. 147

5-81 Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in collet holder, overhang length 60 mm (flexible connection) .............. 148

5-82 Beam model for 12.7 mm diameter, 2-flute endmill inserted in a shrink fit holder (not to scale) .......................................................................................... 149

5-83 Measured and predicted tool point FRF of 12.7 mm diameter 2-flute carbide endmill in shrink fit holder, overhang length 66 mm (rigid connection) ............. 149

5-84 Measured and predicted tool point FRF of 12.7 mm diameter 2-flute carbide endmill in shrink fit holder, overhang length 66 mm (flexible connection) ......... 150

5-85 Beam model for 19 mm diameter, 4-flute endmill inserted in a Tribos holder (not to scale) ..................................................................................................... 150

5-86 Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in Tribos holder, overhang length 72 mm (rigid connection) ................. 151

5-87 Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in Tribos, overhang length 72 mm (flexible connection) ....................... 152

5-88 Beam model for 25.4 mm diameter, 4-flute endmill inserted in a shrink fit holder (not to scale) .......................................................................................... 153

5-89 Measured and predicted tool point FRF of 25.4 mm diameter 4-flute carbide endmill in shrink fit holder, overhang length 55 mm (rigid connection) ............. 153

A-1 The tool (I) is coupled flexibly to the holder-spindle-machine (II) to determine the tool point receptance matrix, G11. ............................................................... 159

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Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy

IMPROVED SPINDLE DYNAMICS IDENTIFICATION TECHNIQUE FOR

RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS

By

Uttara V. Kumar

May 2012

Chair: Tony L. Schmitz Major: Mechanical Engineering

Knowledge of the tool point dynamic response is required if milling process models

are to be used to select parameters that avoid chatter, improve surface finish, and

increase part accuracy. The dynamics of the tool-holder-spindle-machine (THSM) can

be obtained by modal testing, but, for the large number of tool-holder combinations in a

production facility, the measurements are time consuming and, at times, inconvenient

(e.g., micro-scale tools).

The Receptance Coupling Substructure Analysis (RCSA) approach may be

applied as an alternative to modal testing. In this approach, the THSM assembly is

considered as three separate components: the tool, holder, and spindle-machine. The

modeled tool and holder receptances (or frequency response functions) are analytically

coupled to an archived measurement of the spindle-machine receptance.

In this research, a novel approach to determine the spindle dynamics, referred to

as the Euler-Bernoulli (E-B) method, is proposed. The spindle dynamics obtained by the

new method are compared to two existing methods (referred to here as the synthesis

and finite difference approaches) using a new comparison metric (CM). The subsequent

THSM receptance prediction accuracy for all three spindle dynamics identification

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methods is evaluated using the CM. It is shown that the E-B method is the best

alternative.

Using the E-B method to identify the spindle dynamics, a flexible (rather than rigid)

connection is introduced between the holder and the tool to further improve the

prediction accuracy. Measurements of various tool blank (i.e., a rod with no cutting

flutes), holder, and spindle combinations are performed. A least squares non-linear

error minimization technique is used to determine the stiffness values that represent the

flexible tool-holder connection. The approach is validated using several endmill-holder-

spindle assemblies.

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CHAPTER 1 INTRODUCTION

Motivation

Increased productivity is a common goal in manufacturing environments, where

customer demands drive production requirements. This is true for machining activities,

where reductions in time and cost are desired while maintaining part quality. Advances

in high spindle speeds designs have made higher material removal rates (MRR)

possible. However, the machining process dynamics can dramatically affect productivity

due to unstable cutting conditions (or chatter) and forced vibrations, which can cause

part geometry errors (or surface location errors, SLE) [1-3]. For a particular setup, a

combination of spindle speed and depth of cut that avoids these limitations (chatter and

SLE) while enabling high MRR must be selected. The pre-process milling parameter

selection (depth of cut and spindle speed) is made possible by the use of predictive

process models, including the stability lobe diagram and SLE map.

A stability lobe diagram (SLD) separates the region of stable machining (no

chatter) and unstable machining (self-excited vibrations, or chatter, with poor surface

quality) as shown in Fig. 1-1. The process behavior depends on the tool-workpiece

material combination (this establishes the force model) and the dynamic response of the

tool-holder-spindle-machine (as measured/modeled at the free end of the cutting tool, or

tool point).Given this information, the corresponding SLD can be determined. Even if the

machining is stable, forced vibrations of the flexible tool-holder-spindle-machine

assembly can lead to SLE. Therefore, SLE calculation is also an important

consideration. Again, the tool point dynamic response, or tool point frequency response

function (FRF), and force model are required as input to the process model. The SLE

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calculations are paired with the appropriate SLD to select stable machining parameters

that produce accurate parts.

The tool point FRF can be measured via impact testing [4], where an instrumented

hammer is used to excite the structure at the tool tip and a transducer, often a low-mass

accelerometer, is used to measure the response. The Fourier transforms of the two

time-domain signals are computed. Their frequency-domain ratio gives the desired FRF.

In a production facility where large numbers of tool-holder-spindle combinations are

used, impact testing can be time consuming, costly, and sometimes impossible. It is

therefore a manufacturing research priority to establish methods that limit the number of

required measurements and increase the use of models for pre-process parameter

selection that ensure stable cutting conditions with minimized SLE.

The prediction of the tool point FRF using Receptance Coupling Substructure

Analysis (RCSA) [5-7] is gaining wider acceptance in the field of high speed machining.

In the RCSA approach, the tool-holder-spindle-machine assembly is considered as

three separate components: the tool, holder, and spindle-machine; the individual

frequency responses of these components are then analytically coupled. An archived

measurement of the spindle-machine FRF (or receptance) is analytically coupled to the

free-free boundary condition receptances of the tool and the holder, which are derived

from Timoshenko beam models. RCSA is described in further detail in Chapter 3.

Although significant development work has been completed to improve the tool and

holder modeling techniques and to better understand the connection stiffness and

damping behavior (see Chapter 2), relatively less effort has been expended to improve

the identification of the spindle-machine dynamics. Correct identification of spindle-

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machine dynamics is naturally required for accurate prediction of the tool point

dynamics. This research focuses on identification of an improved technique to obtain

the spindle-machine dynamics for RCSA. The new method of spindle-machine

dynamics is compared to two approaches described in the literature.

Research Description

There are two established methods of spindle dynamics identification. The first is

the synthesis approach [8-9, 46], which requires two FRF measurements (one direct

and one cross) on a standard artifact (a holder with simple geometry which is easy to

model) by impact testing. For these measurements, a direct FRF refers to a

measurement where the location of the force coincides with the response measurement

location as shown in Fig 1-2a. A cross FRF refers to a measurement where the location

of the applied force is not the same as the response measurement location (Fig. 1-2b).

The second method is the finite difference approach [10], where three FRF

measurements (two direct and one cross) are required on the standard artifact. As

opposed to the two standard artifact FRF measurements for the synthesis approach and

the three standard artifact FRF measurements for the finite difference approach, the

novel method proposed in this research, referred to as the Euler-Bernoulli method,

requires only one direct FRF measurement at the free end of the standard artifact. The

calculation of the spindle dynamics using the three approaches is described in detail in

Chapter 4.

In Chapter 5, the spindle dynamics of three different milling machines, a Mikron

UCP-600 Vario, a Starragheckert ZT-1000 Super Constellation, and a Cincinnati FTV-5

2500 are measured and compared for the three approaches. Using these spindle

dynamics, the tool point frequency response functions are then predicted for several

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combinations of (thermal) shrink fit tool holders and carbide endmills (assuming a rigid

connection between the tool and the holder). The predicted tool point FRFs are

compared to measurement results. Furthermore, in order to identify the best method of

spindle dynamics identification, an FRF comparison metric is also defined. The best

approach is then selected and used to introduce a flexible connection between the tool

and the holder in order to improve the accuracy of the tool point frequency response

predictions. The connection stiffness values are obtained by applying a non-linear least

squares error minimization to the difference between predicted and measured tool point

FRFs. Stiffness values are identified for various diameters carbide blanks (rods)

inserted in shrink fit, collet and Tribos tool holders. The stiffness values of different tool-

holder connections are compared. These are then used to predict the tool point FRFs of

actual endmills and the results are compared to measurements.

Dissertation Organization

The dissertation is organized as follows. Chapter 1 provides an introduction to the

research activities. A literature review is completed in Chapter 2. Chapter 3 details the

RCSA approach. Chapter 4 describes the new spindle-machine dynamics identification

technique. Tool points dynamics predictions and measurements are presented in

Chapter 5. Finally, Chapter 6 summarizes the dissertation and describes the possible

future work in this area of research.

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Figure 1-1. Example stability lobe diagram

Unstable zone

Stable zone

Chatter

Stable

Spindle Speed

Depth of Cut

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A

B Figure 1-2. Standard artifact measurement. A) Direct FRF measurement. B) Cross

FRF measurement

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CHAPTER 2 LITERATURE REVIEW

Milling stability has been an active research area for several decades. Recognition

of the process limitations imposed by chatter can be dated back to 1906 in the work by

Taylor [11]. The work by Arnold [12] followed by the research by Tlusty, Tobias, and

Merrit [13-15] led to a fundamental understanding of regenerative chatter. The

regeneration of surface waviness during material removal was identified as the primary

mechanism for self-excited vibration in machining. The source of self-excited vibration is

the variable chip thickness that governs the cutting force and subsequent tool

vibrations. Modeling of the milling process in order to select pre-process parameters for

chatter avoidance and accurate work piece dimensions has been and continues to be a

widely studied topic.

The time marching numerical integration approach to model the milling process is

summarized by Smith and Tlusty [16]. Related work includes the mechanistic model

approach for the prediction of the force system [17]. Frequency domain solutions have

been applied to determine process stability in the form of stability lobe diagrams, which

identify stable and unstable cutting zones as a function of axial depth of cut and spindle

speed [18]. Altintas and Budak used a Fourier series (frequency domain) approach to

approximate the time varying cutting force coefficients for stability lobe diagram

development [19]. A closed form, frequency domain solution for surface location error in

milling was developed by Schmitz and Mann [20]. A numerical method for the stability

analysis of linear time-delayed system based on a semi-discretization technique was

also presented in the literature [21]. Modeling approaches based on finite element

analysis [22] and, later, time finite element analysis [23] have also been developed. In

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all the modeling methods, a description of the system dynamic response, comprised of

the tool-holder-spindle-machine assembly receptance, is required. This response can

be obtained on a case-by-case basis via impact testing, where an instrumented hammer

is used to excite the tool point and the response is measured using (typically) a low-

mass accelerometer. However, because each tool-holder combination must be

measured on each machine, the number of experiments can be excessive. Therefore,

the preferred method is application of an appropriate modeling approach which reduces

the number of required experiments.

The preference of a modeling approach led to the application of receptance

coupling [24] to predict the tool point FRF. In the initial application of receptance

coupling to tool point FRF prediction, an Euler-Bernoulli (E-B) beam model of the

overhung portion of the tool was coupled to the displacement-to-force receptance of the

holder-spindle-machine [5-7]. In this work the fluted portion of the tool was

approximated using the equivalent diameter approach by Kops and Vo [25]. Many

improvements have been made since then to the RCSA method. Park et al.

incorporated displacement-to-moment, rotation-to-force and rotation-to-moment

receptances in the analysis [26].

Duncan et al. applied RCSA further to investigate the „dynamic absorber effect‟

that results from the interaction of the modes of individual components [27]. The

overhang length of the tool can be adjusted to improve the system dynamic stiffness,

resulting in higher removal rates as the critical stability limit is increased. Connection

parameters determined by fitting the predicted FRF to a tool-holder-spindle-machine

(THSM) assembly measurement at a known overhang length were used to predict other

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overhang lengths of the tool. Burns and Schmitz studied the effect of changing tool

overhang length on the connection parameters [28]. The connection parameters were

estimated using a nonlinear least squares algorithm. Schmitz and Duncan also

described the receptance prediction of nested components with a common neutral axis

and studied the sensitivity to noise in the component receptances [29].

Kivanc and Budak modeled endmills as two components, the shank portion and

the fluted portion, taking into account the moment of inertia of the complex cross-section

of the flutes. They incorporated flexible coupling between the tool and holder-spindle

using nonlinear least squares error minimization [30].

Movaheddy and Gerami proposed a receptance coupling method which takes into

account the rotational degrees of freedom responses by a tool and holder-spindle joint

model consisting of two parallel springs without the need to include rotational FRFs in

the receptance coupling equation. The joint parameters were estimated for one

overhang length of the tool using optimization based on genetic algorithm [31].

Schmitz et al. extended the RCSA method to three components: the overhung tool

(i.e. the portion outside the holder), the holder, and the spindle-machine [8-9].The

spindle-machine receptances were archived by measuring direct and cross

displacement-to-force FRF of a simple geometry standard holder and removing the

portion of a holder beyond the flange using inverse receptance coupling approach.

Timoshenko beam models were used to describe the tool and the holder receptances.

The RCSA method was further improved by making use of FEA to estimate the stiffness

and damping values at the tool-shrink fit holder connection [32].

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Timoshenko beam models were also used to model the spindle, holder, and tool

for RCSA and compared to the results obtained by the finite element software in the

work done by Ertürk et al. [33]. The effects of the bearing and interface dynamics,

spindle design, and parameters like tool geometry and holder geometry on the THSM

assembly FRF was also studied. Given knowledge of which mode was affected by

which connection parameters, the translational parameters were tuned [34-36].

Further efforts to model the spindle-holder joint interface in THSM assembly

include work by Namazi et al. [37]. They considered translational and rotational springs

uniformly distributed in the holder-spindle interface. In the work by Ahmadi and

Ahmadian [38], the change in normal contact pressure along the holder and the portion

of the tool inserted in the holder was taken into account by modeling the interface as a

distributed elastic layer. In a recent study incorporating the work by Namazi et al. and

Ahmadi and Ahmadian, a model that couples components through continuous elastic

joints rather than at single points was developed [39]. Park and Chae combined

receptance coupling, finite element analysis, and experimental modal analysis to

determine joint dynamics of modular tools [40].

A closed form approach for the identification of holder-spindle and tool-holder

dynamics was proposed by Özşahin et al [41]. By rearranging the receptance coupling

equation for flexible coupling and obtaining component receptances analytically and

experimentally, the stiffness matrix was obtained. This method was highly sensitive to

measured FRF as well as to the accuracy of the rotational FRFs approximated by

experimental translational FRFs. They further used this procedure to train a neural

network to identify the contact stiffness for different holder and tool combinations [42].

29

Rezaei et al. used the concept of inverse receptance coupling to extract the

holder-spindle FRFs by removing the portion of the tool outside the holder [43]. The tool

FRFs were determined analytically and subtracted from the measured tool point FRF of

the THSM assembly. This method enables the joint parameters to be part of the holder-

spindle FRF and can be used to predict the tool point FRF of any tool with a similar joint

condition (or insertion length). In a recent study, Filiz et al. applied the spectral-

Tchebychev technique to model the cutting tool for RCSA [44].

30

CHAPTER 3 RECEPTANCE COUPLING SUBSTRUCTURE ANALYSIS

Description

This chapter describes the Receptance Coupling Substructure Analysis (RCSA)

approach. In the RCSA approach, the tool-holder-spindle-machine assembly is

separated into three components: the tool, holder, and spindle-machine. The spindle-

machine receptances, or frequency response functions (FRFs), are measured once and

archived. These receptances are then analytically coupled to beam models that

represent the tool-holder to predict the tool point receptances for arbitrary tool-holder

combinations.

Frequency Response Function

A FRF is a (frequency-domain) transfer function, where only the positive

frequencies are considered for the system-specific damping level. An FRF for a system

is expressed as the complex ratio of displacement-to-force (receptance), velocity-to-

force (mobility), or acceleration-to-force (accelerance or inertance) at the specified

coordinate locations. FRFs contain information about the system natural frequencies

and mode shapes and are commonly expressed as the real and imaginary parts or the

magnitude and phase. The receptance of the tool-holder-spindle-machine assembly as

reflected at the tool point is used to produce the desired stability lobe diagram and carry

out the surface location error (SLE) predictions.

Three Component Coupling for Tool Point FRF Prediction

RCSA uses both experimental and modeled FRFs. In the second generation

RCSA method, the assembly was divided into three primary components: the tool, the

holder, and the spindle-machine [8-9].The tool and the holder were described using

31

Timoshenko beam models (based on the geometry and material properties) with free-

free boundary conditions, while the receptances of the spindle-machine (which are

difficult to model based on first principles, primarily due to the difficulty in estimating

damping at interfaces) were calculated by measuring a standard artifact and using the

inverse receptance coupling method. Figure 3-1 depicts the three individual

components of the tool-holder-spindle-machine assembly: the tool (I), the holder (II),

and the spindle-machine (III).

For the tool-holder-spindle-machine RCSA model, four bending receptances are

used to describe each component. They are:

displacement-to-force, iij

j

xh

f

displacement-to-couple, iij

j

xl

m

rotation-to-force,

iij

j

nf

and

rotation-to-couple,

iij

j

pm

, where i and j are the measurement and force application

coordinate locations, respectively.

If i and j are equal, the receptances are referred to as direct receptances; otherwise,

they are cross receptances.

Free-Free Beam Receptances

Because the spindle-machine receptances are difficult to model, they are

measured using a standard holder. The tool and holder receptances, on the other hand,

are convenient to model. In this research, Timoshenko beam elements are used to

32

model the four degrees of freedom (displacement and rotations at both the ends) for

free-free beam receptances of the tool and holder [45]. Equivalent diameter

Timoshenko beam models are used to describe the fluted portion of the tool.

The individual component, or substructure, receptances, Rij(ω), are organized in

matrix form in Equation 3-1:

(3-1)

where xi is the substructure displacement at the coordinate location i, θi is the

substructure rotation at the coordinate location i, fj is the force applied to the

substructure at the coordinate location j, and mj is the couple applied to the substructure

at the coordinate location j.

Using this notation, Equations 3-2 to 3-9 describe the direct and cross receptances

for the components I and II at the coordinate locations shown in Figure 3-2. Component

I, the tool, is described using Equations 3-2 through 3-5.

1 1

1 1 11 11

11

11 111 1

1 1

x x

f m h lR

n p

f m

(3-2)

1 1

2 2 12 12

12

12 121 1

2 2

a a a a

a

a a

a a

x x

f m h lR

n p

f m

(3-3)

i i

j j ij ij

ij

ij iji i

j j

x x

f m h lR = =

n pθ θ

f m

33

2 2

2 2 2 2 2 2

2 2

2 2 2 22 2

2 2

a a

a a a a a a

a a

a a a aa a

a a

x x

f m h lR

n p

f m

(3-4)

2 2

1 1 2 1 2 1

2 1

2 1 2 12 2

1 1

a a

a a

a

a aa a

x x

f m h lR

n p

f m

(3-5)

Similarly, component II, the holder, is described by Equations 3-6 through 3-9.

2 2

2 2 2 2 2 2

2 2

2 2 2 22 2

2 2

b b

b b b b b b

b b

b b b bb b

b b

x x

f m h lR

n p

f m

(3-6)

2 2

3 3 2 3 2 3

2 3

2 3 2 32 2

3 3

b b

a a b a b a

b a

b a b ab b

a a

x x

f m h lR

n p

f m

(3-7)

3 3

3 3 3 3 3 3

3 3

3 3 3 33 3

3 3

a a

a a a a a a

a a

a a a aa a

a a

x x

f m h lR

n p

f m

(3-8)

3 3

2 2 3 2 3 2

3 2

3 2 3 23 3

2 2

a a

b b a b a b

a b

a b a ba a

b b

x x

f m h lR

n p

f m

(3-9)

The relationships between displacements/rotations and forces/couples can be

written using the matrix format as shown in Equations 3-10 to 3-17, where ui and qi are

the generalized displacement/rotation and the force/couple vectors, respectively.

34

1 11 11 1

1 11 11 1

x h l f

n p mor 1 11 1u R q (3-10)

1 12 2a au R q (3-11)

2 2 2 2a a a au R q (3-12)

2 2 1 1a au R q (3-13)

2 2 2 2b b b bu R q (3-14)

2 2 3 3b b a au R q (3-15)

3 3 3 3a a a au R q (3-16)

3 3 2 2a a b bu R q (3-17)

Rigid Coupling of Free-Free Receptances

The free-free tool and holder models are coupled to form the subassembly I-II

identified in Figure 3-3. The component I and II subassembly receptances are

determined using Equations 3-20 to 3-51. In order to calculate the subassembly

receptances, G11 (direct) and G3a1 (cross) (Equations 3-18 and 3-19, respectively), a

generalized force Q1 (representing both the externally applied force and couple) is

applied at coordinate location 1 (see Figure 3-3).

1 1

1 1 11 11

11

11 111 1

1 1

X X

F M H LG

N P

F M

(3-18)

3 3

1 1 3 1 3 1

3 1

3 1 3 13 3

1 1

a a

a a

a

a aa a

X X

F M H LG

N P

F M

(3-19)

35

The displacement equations for the substructures can be described as follows:

(3-20)

(3-21)

(3-22)

(3-23)

If rigid coupling between the two components is assumed, the compatibility condition

that describes the connection between the two components is expressed as shown in

Equation 3-24.

(3-24)

The equilibrium condition at coordinate locations 2a and 2b is given by Equation 3-25.

(3-25)

At coordinate location 1, the external force/couple is applied so the relationship in

Equation 3-26 is obtained.

(3-26)

Substituting for u2b and u2a in Equation 3-24 gives Equation 3-27.

(3-27)

Equation 3-28 is obtained using Equations 3-25 and 3-26.

(3-28)

Solving for q2b gives Equation 3-29. Given that 2 2a bq q from Equation 3-25,

substitution in Equation 3-30 gives Equation 3-31, which can then be written as shown

in Equation 3-32. This equation shows that the sub-assembly receptances can be

1 11 1 12 2a au R q R q

2 2 2 2 2 1 1a a a a au R q R q

2 2 2 2b b b bu R q

3 3 2 2a a b bu R q

2 2 0b au u

2 2 0b aq q

1 1q Q

2 2 2 2 2 2 2 2 2 1 1 0b a b b b a a a au u R q R q R q

2 2 2 2 2 2 1 1( ) 0b b a a b aR R q R Q

36

expressed as a function of the component receptances. Therefore, given the tool and

holder receptances, the tool-holder sub-assembly receptances can be predicted.

(3-29) (3-30) (3-31) (3-32)

Similarly, the cross receptances between coordinates 3a and 1 are given by Equations

3-33 and 3-34.

1

3 3 3 2 2 3 2 2 2 2 2 2 1 13 1

1 1 1 1

( )a a a b b a b b b a a aa

U u R q R R R R QG

Q Q Q Q (3-33)

3 1 3 11

3 1 3 2 2 2 2 2 2 1

3 1 3 1

( )a a

a a b b b a a a

a a

H LG R R R R

N P (3-34)

To determine the other two receptances of the sub-assembly I-II, G3a3a and G13a, a

generalized force Q3a is applied to U3a as shown in Figure 3-4.

(3-35) (3-36)

1

2 2 2 2 2 2 1 1( )b b b a a aq R R R Q

11 1 12 21 1

11

1 1 1

a aR q R qU uG

Q Q Q

1

11 1 12 2 2 2 2 2 1 111

1

( )a b b a a aR Q R R R R QG

Q

11 111

11 11 12 2 2 2 2 2 1

11 11

( )a b b a a a

H LG R R R R R

N P

3 3

3 3 3 3 3

3 3

3 3 33 3

3 3

a a

a a a a a

a a

a a aa a

a a

X X

F M H LG

N P

F M

1 1

3 3 13 13

13

13 131 1

3 3

a a a a

a

a a

a a

X X

F M H LG

N P

F M

37

The component displacement/rotation equations are now described by Equations 3-37

to 3-40.

(3-37) (3-38) (3-39) (3-40) The compatibility equation remains the same as given in Equation 3-24.

(3-41) The equilibrium equations are: and (3-42) (3-43) Substituting for u2a and u2b in Equation 3-41 gives Equation 3-44.

(3-44) Using Equations 3-42 and 3-43 and substituting for q2b and q3a in Equation 3-44 gives

Equation 3-45.

(3-45) Solving for q2a, Equation 3-46 is obtained. Equation 3-47 gives the desired expression

for the subassembly direct receptances.

(3-46) (3-47) By substituting for q2b in Equation 3-47 using Equations 3-42 and 3-46, Equation 3-48 is

obtained. Equation 3-49 gives the final expression after simplification.

(3-48)

1 12 2a au R q

2 2 2 2a a a au R q

2 2 2 2 2 3 3b b b b b a au R q R q

3 3 3 3 3 2 2a a a a a b bu R q R q

2 2 0a bu u

2 2 0b aq q

3 3a aq Q

2 2 2 2 2 2 2 2 2 3 3 0a b a a a b b b b a au u R q R q R q

2 2 2 2 2 2 3 3( ) 0b b a a a b a aR R q R Q

1

2 2 2 2 2 2 3 3( )a b b a a b a aq R R R Q

3 3 3 3 3 3 2 2

3 3

3 3 3

a a a a a a b ba a

a a a

U u R q R qG

Q Q Q

1

3 3 3 3 2 2 2 2 2 2 3 33 3

3

( )a a a a b b b a a b a aa a

a

R Q R R R R QG

Q

38

(3-49) In a similar way, the cross receptances are defined. See Equations 3-50 and 3-51.

(3-50) (3-51)

Coupling of Tool-Holder and Spindle-Machine Receptances

Once the free-free components I and II are (rigidly) coupled to form the

subassembly I-II, this subassembly is then rigidly coupled to the spindle-machine

(component III) to give the assembly tool point receptances, G11; see Figure 3-5. This

coupling is carried out using Equation 3-52:

(3-52)

where the Rij matrices are the subassembly matrices. Therefore, 11 11R G from the

Equation 3-32 I-II coupling results, 3 1 3 1a aR G from Equation 3-34, 3 3 3 3a a a aR G from

Equation 3-49, and 13 13a aR G from Equation 3-50. The remaining unknown in Equation

3-52 is the spindle-machine receptance matrix, R3b3b. Identification of this receptance

matrix is discussed in Chapter 4.

3 3 3 31

3 3 3 3 3 2 2 2 2 2 2 3

3 3 3 3

( )a a a a

a a a a a b b b a a b a

a a a a

H LG R R R R R

N P

1

12 2 12 2 2 2 2 2 3 31 113

3 3 3 3

( )a a a b b a a b a aa

a a a a

R q R R R R QU uG

Q Q Q Q

13 131

13 12 2 2 2 2 2 3

13 13

( )a a

a a b b a a b a

a a

H LG R R R R

N P

1

11 11 13 3 3 3 3 3 1a a a b b aG R R R R R

39

Figure 3-1. Three-component receptance coupling model for the tool (I), holder (II), and

spindle-machine (III).

40

Figure 3-2. Individual components I-II with displacements and rotations at specified

coordinate locations.

41

Figure 3-3. Subassembly I-II composed of tool (I) and holder (II). The generalized force

Q1 is applied to U1 to determine G11 and G3a1.

42

Figure 3-4. Subassembly I-II composed of tool (I) and holder (II). The generalized force

Q3a is applied to U3a to determine G3a3a and G13a.

43

Figure 3-5. The I-II subassembly is rigidly coupled to the spindle-machine (III) to

determine the tool point receptance matrix, G11.

44

CHAPTER 4 IDENTIFICATION OF SPINDLE-MACHINE RECEPTANCES

As discussed previously, the receptances of the spindle-machine (component III of

the tool-holder-spindle-machine assembly) are difficult to model. Therefore, these

receptances are experimentally determined. To identify the spindle-machine

receptances, a standard artifact (i.e., a standard tool holder with a uniform cylindrical

geometry beyond the flange) is inserted in the spindle as shown in Figure 4-1 and G22 is

determined experimentally. Using G22 and a model of the portion of the holder beyond

the flange, the spindle machine receptance R3b3b is calculated. The free end response

for the artifact-spindle-machine assembly is described by Equation 4-1, where the R22,

R23a, R3a3a, and R3a2 matrices are populated using a beam model of the portion of the

artifact beyond the flange. Since the flange geometry is the same for all holders that are

inserted in a particular spindle (to enable automatic tool changes), only the portion of

the holder beyond the flange (towards the tool) is modeled. The flange and the holder

taper (which is inserted in the spindle) are considered part of the spindle-machine.

(4-1) Equation 4-1 is rearranged in Equation 4-2 to isolate R3b3b. This step of decomposing

the measured assembly receptances, G22, into the modeled substructure receptances,

R3a2, R22, R23a, and R3a3a, and spindle-machine receptances, R3b3b, is referred to as

“inverse RCSA”. Three approaches for experimentally determining the four spindle-

machine receptances are discussed in the following sections.

(4-2)

1 22 22

22 22 23 3 3 3 3 3 2

22 22

a a a b b a

H LG R R R R R

N P

1

22 22 23 3 3 3 3 3 2

11 1

23 22 22 3 2 3 3 3 3

1

3 2 22 22 23 3 3 3 3

1

3 3 3 2 22 22 23 3 3

a a a b b a

a a a a b b

a a a a b b

b b a a a a

G R R R R R

R R G R R R

R R G R R R

R R R G R R

45

Synthesis Approach

The direct displacement-to-force term, 222

2

XH

F, in the G22 matrix is measured by

impact testing (the TXF software from MLI was used for data acquisition and signal

analysis in this study). In this method, an instrumented impact hammer is used to apply

the force and an accelerometer (piezoelectric sensor) is used to measure the response;

their ratio is the accelerance (translational acceleration-to-force FRF). The software is

used to twice integrate the accelerance to give the required displacement-to-force

receptance.

The direct FRF H22 is measured by applying the force and placing the

accelerometer at the same coordinate location (U2 in Figure 4-1). The second

component of the G22 matrix, the rotation-to-force receptance,

222

2

NF

, is calculated by

a first-order backward finite difference approach [8,46] as described in Equation 4-3,

where the cross FRF H2a2 is measured by exciting the assembly at U2 and measuring

the response at coordinate U2a, located a distance S from the artifact‟s free end, as

shown in Figure 4-1.

(4-3)

Assuming reciprocity (which states that a cross FRF with a measurement at

coordinate 1 and force at 2 is equal to a cross FRF with a measurement at 2 and a force

at 1), the off-diagonal terms of the G22 matrix may be taken to be equal. See Equation

4-4.

L22 = N22 (4-4)

222 2

22 2 22 2 222

2 2

aa

a

xxx xH Hf fSN

f f S S

46

The measured H22 and derived N22 receptances are used to synthesize P22 as shown in

Equation 4-5.

(4-5)

The four receptances required to populate G22 are now known and Equation 4-2

can be used to obtain R3b3b. Given R3b3b, free-free models for arbitrary tool-holder

combinations can be developed and coupled to the spindle-machine receptances to

predict the tool point FRF, H11, required for milling process simulation. The synthesis

approach thus requires two artifact measurements to determine the G22 matrix. In this

approach, modal fitting of H22 and H2a2 receptances via a peak picking technique [3] can

be applied to reduce the effects of measurement noise on the tool point prediction

results (this strategy is used in this study).

Finite Difference Approach

In this approach, three measurements are required: the direct and cross FRF

measurements as described in the synthesis approach and an additional direct

displacement-to-force receptance, H2a2a, at the distance, S, from the free end of the

artifact. With the two direct, H22 and H2a2a, and one cross displacement-to-force, H2a2,

receptances, a second-order backward finite difference approach is implemented to

identify the rotation-to-moment receptance [10] (Equation 4-6).

(4-6)

Modal fitting of the measurement receptances via a peak picking technique may

again be applied to reduce the effects of measurement noise on the tool point prediction

results. The fitting strategy is applied in this work.

2

2 2 2 2222 22 22

2 2 2 22 22

1x f NP L N

m f x H H

22 22

22

2 22

2a a a

H H H

SP

47

Euler-Bernoulli Method

As an alternative to completing two measurements on the standard artifact for the

synthesis approach and three measurements for the finite difference approach, only a

single direct measurement is performed at the free end of the standard artifact in the

new method established in this research. In the new technique, it is assumed that each

mode within the measurement bandwidth can be approximated as a fixed-free (Euler-

Bernoulli) beam and the individual modes are fit using the closed-form receptance

equation for fixed-free Euler-Bernoulli (E-B) beams presented by Bishop and Johnson

[24]. The fit is completed using Equation 4-7 for the displacement-to-force receptance at

the free end of a cylindrical fixed-free beam, where

4 2

1

A

EI i,

2

4

dA ,

4

64

dI , ω is frequency (rad/s), ρ is the density, E is the elastic modulus, η is the solid

damping factor (unitless), d is the “fit” beam diameter, and L is the beam length.

(4-7) The algorithm for fitting each mode is composed of five steps:

1. Determine the natural frequency, fn (Hz), for the mode to be fit from the measured H22 receptance.

2. Select a beam diameter (this is a fitting parameter) and specify the modulus and density (steel values, E = 200 GPa and ρ = 7800 kg/m3, were used in this research).

3. Calculate the beam length using the closed-form expression for the natural frequency of a fixed-free cylindrical beam; see Equation 4-8 [47].

4. Adjust η to obtain the proper slope for the real part of the mode in question.

5. If the subsequent mode magnitude is too large, increase d to dnew and calculate Lnew using Equation 4-9 (to maintain the same natural frequency). If the mode magnitude is too small, decrease d and calculate Lnew.

22 3

sin cosh cos sinh

1 1 cos cosh

L L L LH

EI i L L

48

(4-8)

(4-9)

Once the fit parameters d, L, and η are determined (as well as the selected E and

ρ values), the remaining receptances for the free end of the artifact are calculated as

shown in Equations 4-10 and 4-11. No additional measurements are required.

(4-10) (4-11) The four G22 receptances are then known and Equation 4-2 can be used to obtain R3b3b,

the spindle-machine receptances.

11 2

2 21.87510407

2 16n

d EL

f

newnew

dL L

d

22 22 2

sin sinh

1 1 cos cosh

L LL N

EI i L L

22

cos sinh sin cosh

1 1 cos cosh

L L L LP

EI i L L

49

Figure 4-1. Artifact model for determining R3b3b by inverse RCSA.

50

CHAPTER 5 RESULTS

Spindle-Machine Receptances Comparison

The G22 receptances determined by the synthesis, finite difference, and Euler-

Bernoulli methods (Chapter 4) are calculated for three spindles and compared. Three

spindle/machine combinations were tested: a Mikron UCP-600 Vario, a Starragheckert

ZT-1000 Super Constellation, and a Cincinnati FTV-5 2500. The specifications of these

machines are listed in Table 5-1.

The Mikron UCP-600 Vario milling machine spindle (HSK-63A interface) was

tested using the steel artifact depicted in Figure 5-1. Direct and cross FRFs were

measured (S = 38.3 mm) using impact testing and the G22 identification methods

described in Chapter 4 were completed. The H22 measurement and 24 mode E-B fit are

presented in Figure 5-2; the E-B fitting parameters are provided in Table 5-2. The

predicted L22/N22 receptances from the three methods are displayed in Figure 5-3. Good

agreement in both magnitude and frequency is observed. The P22 receptances obtained

using Equations 4-5 (synthesis), 4-6 (finite difference), and 4-11 (E-B) are displayed in

Figure 5-4. Again, the agreement is good except at the anti-resonant frequencies

(where the response magnitude is close to zero, near 1580 Hz and 2395 Hz) for the

synthesis approach and at lower frequencies for the finite difference approach. For the

synthesized receptance, the imaginary part exhibits unexpected positive values near the

anti-resonant frequencies. This is presumably due to the division by the complex-valued

receptance, H22, in Equation 4-5.

Given the G22 receptances (from the three techniques), the corresponding spindle-

machine receptance matrices, R3b3b, were calculated using Eq. 4-2 and a free-free

51

boundary condition Timoshenko beam model for the portion of the standard artifact

beyond the flange. The dimensions provided in Figure 5-1 were used together with steel

material properties (E = 200 GPa, ρ = 7800 kg/m3, and Poisson‟s ratio = 0.29) to

develop the artifact model.

The H22 measurement was completed using two artifacts of different lengths for

the Starragheckert ZT-1000 Super Constellation; see Figures 5-5 and 5-6. The direct

and cross FRFs were measured at a distance of S = 32 mm for the short artifact and S

= 50 mm for the long artifact. The 16 mode E-B fit for the short standard artifact and 9

mode E-B fit for the long standard artifact are presented in Figure 5-7 and Figure 5-10;

the E-B fitting parameters for the two artifacts are provided in Tables 5-3 and 5-4. The

L22/N22 receptances for both the artifacts are displayed in Figures 5-8 (short) and 5-11

(long) and the P22 receptances in Figures 5-9 (short) and 5-12 (long). The trends are

similar to those observed for the Mikron UCP-600 Vario data.

Figures 5-13 to 5-18 show the measured H22 FRF and the E-B fit, as well as a

comparison of the L22/N22 and P22 receptances for the three approaches using both the

short and the long standard artifacts (the dimensions are provided in Figures 5-5 and 5-

6) for the Cincinnati FTV-5 2500 milling machine. Tables 5-5 and 5-6 provide the E-B

fitting parameters for the two standard artifacts.

Tool Point Frequency Response Comparison

The archived spindle-machine receptance matrices, R3b3b, for the three milling

machines were rigidly coupled to Timoshenko beam models of various tool-holder

combinations to predict the corresponding tool point receptances, H11. In these tests,

carbide endmills of different diameters and overhang lengths were clamped in various

52

shrink fit holders. Comparisons between measurements and predictions for the three

spindle receptances are provided.

Mikron UCP-600 Vario

25.4 mm diameter carbide endmill in a shrink fit holder

A three flute, 25.4 mm diameter carbide endmill was clamped in a shrink fit tool

holder. After inserting this subassembly in the Mikron UCP-600 Vario spindle, the tool

point receptance, H11, was measured by impact testing and compared to predictions

using the synthesis, finite difference, and E-B R3b3b receptance matrices. The

dimensions for the Timoshenko beam tool-holder model are provided in Figure 5-19 for

an overhang length of 99 mm. The fluted portion of the tool was modeled using an

equivalent diameter, where this diameter was obtained by weighing the carbide tool,

assuming a density (15000 kg/m3), and calculating the solid section equivalent flute

diameter based on the cylindrical dimensions and the tool and flute lengths. The elastic

modulus for the Timoshenko beam model was 550 GPa and Poisson‟s ratio was 0.22.

The E-B prediction, synthesis prediction, finite difference prediction, and measurement

are presented in Figure. 5-20. The overhang length was then extended to 107 mm and

the exercise was repeated. The results are shown in Figure. 5-21. A comparison metric

was used to compare the three approaches and quantify which technique provided

better predictions. Equation 5-1 was used to establish the comparison metric, CM,

where imag indicates the imaginary part of the FRF and the absolute value of the

difference was summed over each frequency within the measurements bandwidth and n

is the length of the frequency vector.

(5-1)

( ) ( )measured predictedimag H imag HCM

n

53

Tables 5-7 and 5-8 list the CM values for the two overhang lengths. The percent

difference with respect to the lowest CM value is also specified. The low percent

difference between the three methods suggests that all the three techniques are in good

agreement with each other.

19.05 mm diameter carbide endmill in a shrink fit holder

For these tests, a four flute, 19.05 mm diameter carbide endmill was clamped in a

shrink fit tool holder and this subassembly was inserted in the Mikron UCP-600 Vario

spindle. Tool point measurements were again completed to compare the predictions

using the synthesis, finite difference, and E-B method R3b3b receptance matrices. The

dimensions for the Timoshenko beam tool-holder model are provided in Figure 5-22 for

an overhang length of 70.4 mm. The predictions and measurement are provided in

Figure 5-23. The overhang length was then extended to 76 mm. These results are

shown in Figure 5-24. The comparison metric and percent difference values for the two

overhang lengths are listed in Tables 5-9 and 5-10. Again, all the three methods predict

equally well.

Starragheckert ZT-1000 Super Constellation

12 mm diameter carbide endmill in a shrink fit holder

A four flute, 12 mm diameter carbide endmill was clamped in a shrink fit holder

and the tool-holder was inserted in the machine spindle. Tool point measurements were

completed by impact testing. Figure 5-25 shows the dimensions of the tool-holder

Timoshenko beam model for an overhang length of 44.7 mm. Using the spindle

receptances obtained by measuring two standard artifacts (see Figures 5-5 and 5-6), a

comparison of the measurement and predictions for the three techniques are presented

in Figures 5-26 to 5-29 with two tool overhang lengths of 44.7 mm and 55.0 mm. Tables

54

5-11 to 5-14 list the comparison metric showing that the E-B predictions provide the

closest agreement to measurement, especially for the long artifact predictions. The

imaginary parts of the synthesis approach prediction in the long artifact predictions

(Figures 5-28 and 5-29) show positive values near 3200 Hz. This may be due to the

positive values of the synthesized P22 receptance. These results indicate that the E-B

(single artifact measurement) technique is more robust.


Tool point measurements were performed with a 16 mm diameter, four flute

endmill clamped in a shrink fit holder. The tool-holder dimensions are shown in Figure

5-30. Again, measurements were completed by impact testing with two overhang

lengths (55.0 mm and 65.0 mm) of the tool. Spindle receptances calculated using the

three approaches (for both short and the long artifacts) were coupled to the tool-holder

model to predict the tool point FRF; see Figures 5-31 to 3-34. The comparison metric

and percent difference with respect to the smallest CM value are listed in Tables 5-15 to

5-18.The E-B clearly outperforms the other two approaches for long artifact predictions.


Tool point FRF measurements were completed on a 20 mm diameter, two flute

endmill clamped in a shrink fit holder. Two overhang lengths of 65.0 mm and 75.0 mm

were tested. Figure 5-35 depicts the tool-holder model dimensions for the 65.0 mm

overhang length. Predictions were again made using the two standard artifact spindle

receptances. Tool point measurements and predictions for the two overhang lengths are

compared in Figures 5-36 to 5-39 and Tables 5-19 to 5-22 list the CM values and

percent differences. The short artifact predictions for the E-B method slightly outperform

55

the other two techniques (the percent difference values are large and the long artifact

predictions again predict positive imaginary part values for the synthesis approach).


A 25 mm diameter endmill with four flutes was clamped in a shrink fit holder and

tool point FRFs were measured via impact testing for two overhang lengths of the tool

(75.0 mm and 85.0 mm). The holder-tool model dimensions are shown in Figure 5-40.

Tool point FRF predictions (using both the short and long standard artifact spindle

receptances) for the three approaches and measurements are compared in Figures 5-

41 to 5-44. Tables 5-23 to 5-26 list the CM values, as well as the percent difference with

respect to the smallest CM value. From the figures and tables, all the three approaches

are in good agreement with the measurements.

Cincinnati FTV-5 2500

Using the same 12 mm, 16 mm, 20 mm, and 25 mm carbide endmills clamped in

shrink fit holders with the same overhang lengths, tool point measurements were

completed on the Cincinnati FTV-5 2500 milling machine. The tool-holder model

dimensions were the same as those shown in Figures 5-25, 5-30, 5-35, and 5-40 for the

12 mm, 16 mm, 20 mm, and 25 mm diameter endmills, respectively. The following

sections list the CM values and percent difference for all the four endmills.


Figures 5-45 to 5-48 compare the tool point FRF measurements and predictions

for two overhang lengths 45.0 mm and 55.0 mm with spindle receptances obtained by

the two standard artifacts. The CM values and percent differences are listed in Tables

5-27 to 5-30. The long artifact predictions using synthesis and finite difference approach

are less accurate than the E-B method predictions.

56


Tool point FRF measurements and comparisons are shown in Figures 5-49 to 5-

52 and CM values are listed in Tables 5-31 to 5-34. The accuracy of E-B method is

better than the synthesis and the finite difference approach.


Figure 5-53 shows the experimental setup for the tool point FRF measurement on

the 20 mm diameter carbide endmill clamped in a shrink fit holder. Figures 5-54 to 5-57

display the tool point measurements and predictions for overhang lengths of 65.0 mm

and 75.0 mm using spindle receptances measured by both the short and the long

artifact. CM values for the predictions using the three different approaches for the two

overhang lengths are listed in Tables 5-35 to 5-38. The three techniques can be

considered in good agreement with the measurement for the short artifact predictions,

but positive value of the imaginary in the synthesis approach is again seen for the long

artifact predictions.


The tool point FRF measurements (see Figure 5-58) and predictions are

presented in Figures 5-59 to 5-62 for the two overhang lengths of 75.0 mm and 85.0

mm and the corresponding CM values with percent differences are listed in Tables 5-39

to 5-42. In this case, all the three approaches perform well for both the short and long

artifact predictions.

Introduction of flexible connection between the tool and the holder

It is observed that the predicted natural frequencies for the different tool-holder

combinations are generally higher than the experimental results (i.e., the predicted

modes appear to the right of the measured modes). This is attributed to the assumption

57

of a rigid connection between the tool and the holder. A flexible connection is introduced

between the tool and the holder in this section for the test spindles, Cincinnati FTV-5

2500 and Mikron UCP-600 Vario. The types of tool-holder connections include shrink fit

holders, collet holders, and Schunk Tribos holders. The Euler-Bernoulli method of

spindle identification was used in this study.

Tool point FRFs were completed with carbide blanks (rods) inserted in the shrink

fit holders, collet holders, and Tribos holders. Carbide blanks were used so that the

complication of the tool‟s fluted portion would not be included in the stiffness

identification. The flexible coupling of the components is carried out in two steps: 1) the

spindle-machine is first rigidly coupled to the holder and the portion of the shank inside

the holder; 2) the holder-spindle-machine component is then flexibly coupled to the

portion of the blank that extends outside the holder using translational and rotational

spring constants assembled in the stiffness matrix k (Figure 5-63). The RCSA equation

for the flexible coupling tool point FRF is provided in Equation 5-2. The stiffness matrix

is given by Equation 5-3, where kxf , kθf , kxm, and kθm are the displacement-to-force,

rotation-to-force, displacement-to-moment, and rotation-to-moment stiffness values,

respectively and cxf , cθf , cxm, and cθm are the corresponding damping values if viscous

damping is considered at the coupling location (kθf = kxm and cθf = cxm were assumed

due to reciprocity). The derivation of Equation 5-2 is provided in the Appendix A.

(5-2) (5-3)

1

11 11 12 2 2 2 2 2 1

1a a a b b aG R R R R R

k

xf xf f f

xm xm m m

k i c k i ck

k i c k i c

58

To identify the stiffness matrix, tool point FRFs were measured for multiple

overhang lengths of the blank on each holder. An optimization procedure based on non-

linear least squares was implemented to find the connection stiffness. The variables

were rotation-to-force stiffness (assumed equal to the displacement-to-moment

stiffness) and rotation-to-force damping (assumed equal to the displacement-to-moment

damping) in Equation 5-3 because the holder-tool connection is most effective at limiting

translation (due to the press fit), but can still allow small axial slip and rotational

flexibility due to the finite friction between the tool and internal hole in the holders. The

objective function to be minimized is given by Equation 5-4, where the absolute value of

the difference of the magnitude of the measured (m) and predicted (p) tool point FRFs

was computed.

(5-4) Cincinnati FTV-5 2500

The k matrix was obtained for the various overhang lengths for 12 mm, 16 mm, 20

mm, and 25 mm diameter blanks in shrink fit holders (Figure 5-64) and 12 mm, 16mm,

and 20 mm diameter blanks in a collet holder (Figure 5-65). For example, Table 5-43

lists the stiffness and damping values obtained for three different overhang lengths of

the 12 mm diameter blank inserted into the corresponding shrink fit holder. Figures 5-66

to 5-68 show the measured and predicted carbide blank tool point FRF of the 12 mm

diameter blank clamped in the shrink fit holder for the three overhang lengths assuming

a rigid connection. Figures 5-69 to 5-71 show the measurement and prediction with a

flexible connection for the three overhang lengths. Table 5-44 lists the average stiffness

values of the 12 mm, 16 mm, 20 mm, and 25 mm blanks clamped in the shrink fit

min m pH H

59

holder. Similarly, Table 5-45 lists the average stiffness values of the 12 mm, 16mm, and

20 mm blank clamped in the collet holder.


Tool point FRF measurements for blanks inserted in shrink fit holders, collet

holders (Figure 5-72 and Figure 5-73), and Tribos holders (Figure 5-74) were also

completed on the Mikron UCP-600 Vario. The average stiffness values for different

diameter blanks for the three holders are listed in Tables 5-46 to 5-48.

Comparison of the stiffness values of different diameter blanks in various holders

for both the test spindles (Cincinnati FTV-5 and Mikron UCP-600 Vario) shows that the

connection stiffness values increase with increasing diameter. The increased flexibility

for the smaller diameter tool connection is due to the lower contact surface area with the

holders. The thermal shrink fit holder offers the most rigid connection. For example, the

25 mm diameter shrink fit holder-blank did not require flexible coupling; the holder-tool

interface of large shrink fit diameters can be modeled as a rigid connection. The Tribos

holder offers the next higher connection stiffness; the elastic clamping mechanism is

described in Figure 5-75. The Tribos holder consists of three chambers filled with a

thermosetting plastic that absorbs shock and reduces vibrations during machining. In

the Timoshenko beam model of the Tribos holder, the section consisting of the

thermosetting plastic chambers and steel was modeled using equivalent values of the

elastic modulus (Eeq), density (ρeq), and Poisson‟s ratio (eq) as shown in Equations 5-5,

5-6, and 5-7, respectively, where Asteel is the cross-sectional area of the steel portion,

Aplastic is the cross-sectional area of the thermosetting plastic chambers, and Atotal is the

total cross-sectional area of the Tribos holder. The values of the elastic modulus,

60

density, and Poisson‟s ratio for steel were the same as described in the previous

sections. The values of the elastic modulus, density and Poisson‟s ratio for the

thermosetting plastic were taken to be 7 GPa, 3700 kg/m3, and 0.5, respectively.

plasticsteel

eq steel plastic

total total

AAE E E

A A (5-5)

plasticsteel

eq steel plastic

total total

AA

A A (5-6)

plasticsteel

eq steel plastic

total total

AA

A A (5-7)

The collet tool-holder connection was the most flexible. This was anticipated due

to the clamping mechanism; a flexible “collet basket” is elastically deformed inside a

tapered volume using a clamping nut.

Comparison between the Cincinnati FTV-5 and Mikron UCP-600 Vario spindles

show that the stiffness values for a particular type of holder with the same diameter

blank give similar results. Therefore, the stiffness values obtained by measurement of

blanks in a selected holder type can be used to predict the tool point FRF of an actual

endmill in that holder when it is inserted in any spindle. The tool point FRF

measurement and prediction for actual endmills in the Mikron UCP-600 Vario spindle

with rigid and flexible couplings are shown in Figures 5-76 to 5-87 for 6.33 mm, 12.7

mm, and 19 mm endmills clamped in the shrink fit, collet, and Tribos holders; the beam

models are also displayed. It can be seen from these figures that the introduction of a

flexible connection between the holder and tool using the average stiffness values

obtained from the blank measurements listed in Tables 5-46 to 5-48 improves the tool

point FRF prediction as compared to the prediction obtained by the rigid connection

61

assumption. The beam model for the 25.4 mm diameter 4-flute endmill clamped in the

shrink fit holder is shown in Figure 5-88. Figure 5-89 shows that the assumption of rigid

connection at the holder-tool interface in case of the 25.4 mm diameter 4-flute endmill

clamped in a shrink fit holder is valid.

62

Table 5-1. Specifications of milling machines tested

Manufacturer and model

Geometry Spindle-holder

connection

Work volume (mm)

Max spindle speed (rpm)

Controller


Vertical (5axis)

HSK63A 600X450X450 20000 Heidenhain iTNC 530

Cincinnati FTV5 2500

Vertical (5 axis)

HSK63A 2540X1003X800 18000 Siemens Fanuc

Starragheckert ZT1000 Super Constellation

Vertical (5 axis)

HSK63A 2000X1600x1600 24000 Siemens

840D

Table 5-2. E-B fitting parameters for Mikron UCP-600 Vario CNC milling machine

spindle.

Mode fn (Hz) d (m) L (m)

1 550 0.375 0.100 0.6950 2 610 0.520 0.060 0.7771 3 703 0.330 0.060 0.5767 4 795 0.450 0.050 0.6160 5 840 0.565 0.035 0.6903 6 875 0.260 0.050 0.4588 7 975 0.107 0.070 0.2788 8 1057 0.208 0.032 0.3734 9 1080 0.255 0.032 0.4090 10 1131 0.107 0.054 0.2589 11 1230 0.173 0.055 0.3157 12 1297 0.206 0.042 0.3354 13 1422 0.196 0.078 0.3125 14 1750 0.200 0.110 0.2845 15 1872 0.115 0.060 0.2086 16 2040 0.190 0.150 0.2569 17 2620 0.220 0.130 0.2439 18 2985 0.098 0.060 0.1525 19 3060 0.125 0.070 0.1701 20 3205 0.185 0.070 0.2022 21 3800 0.270 0.060 0.2244 22 3975 0.340 0.040 0.2462 23 4150 0.220 0.050 0.1938 24 4310 0.112 0.050 0.1357

63

Table 5-3. E-B fitting parameters for the short standard artifact on the Starragheckert ZT-1000 Super Constellation


1 810 0.250 0.080 0.4676 2 900 0.230 0.160 0.4255 3 980 0.430 0.070 0.5575 4 1050 0.197 0.080 0.3646 5 1110 0.256 0.040 0.4042 6 1142 0.205 0.045 0.3566 7 1175 0.187 0.040 0.3358 8 1200 0.200 0.035 0.3436 9 1250 0.810 0.170 0.2143 10 1375 0.117 0.102 0.2455 11 2392 0.112 0.065 0.1821 12 2600 0.260 0.100 0.2662 13 2750 0.350 0.100 0.3003 14 3140 0.430 0.060 0.3115 15 3750 0.550 0.060 0.3223 16 4200 0.087 0.045 0.1211

Table 5-4. E-B fitting parameters for the long standard artifact on the Starragheckert

ZT-1000 Super Constellation


1 772 0.080 0.060 0.2709 2 880 0.077 0.120 0.2490 3 922 0.102 0.070 0.2799 4 1080 0.220 0.110 0.3799 5 1228 0.365 0.050 0.4589 6 1320 0.230 0.130 0.3513 7 2134 0.104 0.060 0.1858 8 3035 0.220 0.060 0.2266 9 3230 0.092 0.045 0.1420

64

Table 5-5. E-B fitting parameters for the short standard artifact on the Cincinnati FTV-5 2500


1 685 0.275 0.090 0.5333 2 1007 0.270 0.075 0.4358 3 1110 0.250 0.060 0.3994 4 1240 0.240 0.080 0.3703 5 1313 0.195 0.045 0.3244 6 1451 0.085 0.040 0.2037 7 1660 0.260 0.060 0.3331 8 1816 0.112 0.070 0.2090 9 2060 0.600 0.110 0.4542 10 2440 0.250 0.080 0.2694 11 2504 0.130 0.055 0.1918 12 3070 0.550 0.040 0.3562 13 3350 1.000 0.040 0.4599 14 3950 0.087 0.066 0.1249

Table 5-6. E-B fitting parameters for the long standard artifact on the Cincinnati FTV-5

2500


1 955 0.080 0.060 0.2435 2 1060 0.089 0.080 0.2439 3 1085 0.250 0.040 0.4040 4 1157 0.104 0.050 0.2523 5 1245 0.340 0.030 0.4398 6 1330 0.270 0.050 0.3792 7 1640 0.230 0.070 0.3152 8 1722 0.173 0.060 0.2668 9 1850 1.400 0.030 0.7322 10 2080 0.500 0.040 0.4127 11 2289 0.125 0.050 0.1967 12 2420 0.530 0.030 0.3939 13 3020 0.128 0.062 0.1733 14 3150 0.128 0.050 0.1697 15 4660 0.149 0.030 0.1505

65

Table 5-7. Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter endmill, overhang length 99 mm

CM (m/N) Percent difference with respect to smallest CM

E-B 115.70 x 10-9 - Synthesis 118.58 x 10-9 -2.49

Finite Difference 118.67 x 10-9 -2.57

Table 5-8. Comparison metric (m/N) for the FRF predictions of 25.4 mm diameter

endmill, overhang length 107 mm


E-B 123.78 x 10-9 - Synthesis 126.10 x 10-9 -1.88



endmill, overhang length 70.4 mm


E-B 35.18 x 10-9 - Synthesis 35.21 x 10-9 0.10



endmill, overhang length 76 mm

CM (m/N) Percent Difference with respect to smallest CM

E-B 179.24 x 10-9 - Synthesis 181.35 x 10-9 -1.17


Table 5-11. Comparison metric (m/N) for the FRF predictions of 12 mm diameter

endmill, overhang length 44.7 mm using short artifact spindle receptances


E-B 102.39 x 10-9 - Synthesis 120.02 x 10-9 -17.22


66

Table 5-12. Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 55.0 mm using short artifact spindle receptances


E-B 146.06 x 10-9 - Synthesis 159.35 x 10-9 -9.10



endmill, overhang length 44.7 mm using long artifact spindle receptances


E-B 103.15 x 10-9 - Synthesis 152.37 x 10-9 -47.71



endmill, overhang length 55 mm using long artifact spindle receptances


E-B 128.86 x 10-9 - Synthesis 214.99 x 10-9 -66.84





E-B 41.26 x 10-9 - Synthesis 46.50 x 10-9 -12.69





E-B 57.34 x 10-9 - Synthesis 65.80 x 10-9 -14.74


67

Table 5-17. Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 55.0 mm using long artifact spindle receptances


E-B 23.35 x 10-9 - Synthesis 92.36 x 10-9 -295.58



endmill, overhang length 65.0 mm using long artifact spindle receptances


E-B 31.24 x 10-9 - Synthesis 106.3 x 10-9 -240.31



endmill, overhang length 65.0 mm using short artifact spindle receptances.


E-B 28.29 x 10-9 - Synthesis 41.51 x 10-9 -46.72





E-B 36.77 x 10-9 - Synthesis 46.02 x 10-9 -25.13



endmill, overhang length 65.0 mm using long artifact spindle receptances.


E-B 28.56 x 10-9 - Synthesis 39.13 x 10-9 -37.02


68

Table 5-22. Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 75 mm using long artifact spindle receptances.


E-B 24.28 x 10-9 - Synthesis 38.65 x 10-9 -59.20





E-B 18.04 x 10-9 -0.56 Synthesis 18.44 x 10-9 -2.77

Finite Difference 17.94 x 10-9 -




E-B 26.97 x 10-9 -16.79 Synthesis 23.29 x 10-9 -0.84



endmill, overhang length 75 mm using long artifact spindle receptances.


E-B 16.40 x 10-9 -34.72 Synthesis 12.67 x 10-9 -4.11



endmill, overhang length 85 mm using long artifact spindle receptances.


E-B 20.06 x 10-9 -20.83 Synthesis 17.23 x 10-9 -3.81


69

Table 5-27. Comparison metric (m/N) for the FRF predictions of 12 mm diameter endmill, overhang length 45.0 mm using short artifact spindle receptances.


E-B 115.92 x 10-9 - Synthesis 119.12 x 10-9 -2.76





E-B 173.22 x 10-9 - Synthesis 174.46 x 10-9 -0.72





E-B 96.35 x 10-9 - Synthesis 231.3 x 10-9 -140.10





E-B 142.67 x 10-9 - Synthesis 301.05 x 10-9 -111.01





E-B 53.13 x 10-9 - Synthesis 60.23 x 10-9 -13.35


70

Table 5-32. Comparison metric (m/N) for the FRF predictions of 16 mm diameter endmill, overhang length 65.0 mm using short artifact spindle receptances.


E-B 87.07 x 10-9 - Synthesis 87.68 x 10-9 -0.70





E-B 30.79 x 10-9 - Synthesis 155.77 x 10-9 -405.08





E-B 72.18 x 10-9 - Synthesis 210.06 x 10-9 -191.04





E-B 30.83 x 10-9 - Synthesis 37.17 x 10-9 -20.55





E-B 53.69 x 10-9 - Synthesis 60.43 x 10-9 -12.56


71

Table 5-37. Comparison metric (m/N) for the FRF predictions of 20 mm diameter endmill, overhang length 65.0 mm using long artifact spindle receptances.


E-B 22.37 x 10-9 - Synthesis 24.16 x 10-9 -7.98





E-B 49.53 x 10-9 Synthesis 70.82 x 10-9 -42.98





E-B 18.90 x 10-9 -13.92 Synthesis 17.38 x 10-9 -4.78





E-B 30.91 x 10-9 - Synthesis 33.32 x 10-9 -7.79





E-B 13.09 x 10-9 -16.85 Synthesis 11.27 x 10-9 -0.58


72

Table 5-42. Comparison metric (m/N) for the FRF predictions of 25 mm diameter endmill, overhang length 85.0 mm using long artifact spindle receptances.


E-B 13.84 x 10-9 -15.06 Synthesis 12.19 x 10-9 -1.36


Table 5-43. Stiffness matrix values of 12 mm diameter blank clamped in a shrink fit

holder, Cincinnati FTV-5 2500

Overhang length (mm) kθf (N/rad) cθf (N-s/rad)

66 3.9 x 106 26 71 4.8 x 106 50 76 4.6 x 106 75

Table 5-44. Average stiffness matrix values for blank-shrink fit holders inserted in


Blank diameter (mm) kθf (N/rad) cθf (N-s/rad)

12 4.4 x 106 50 16 1.7 x 107 153 20 1.9 x 107 665 25 Rigid

Table 5-45. Average stiffness matrix values for blank-collet holders inserted in



12 3.2 x 106 18 16 5.6 x 106 51 20 1.5 x 107 159

Table 5-46. Average stiffness matrix values for blank-shrink fit holders inserted in



12.7 5.4 x 106 30 25 Rigid

Table 5-47. Average stiffness matrix values for blank-collet holders inserted in Mikron

UCP-600 Vario


6.33 2.2 x 105 2 9.5 8.3 x 105 13 12 2.9 x 106 18 19 1.0 x 107 31 25 2.2 x 107 0

73

Table 5-48. Average stiffness matrix values for blank-Tribos holders inserted in Mikron UCP-600 Vario


10 2.1 x 106 0 12 2.9 x 106 2 16 9.0 x 106 0 19 1.9 x 107 0

74

Figure 5-1. Artifact dimensions for Mikron UCP-600 Vario measurements.

1000 1500 2000 2500 3000 3500 4000 4500 5000

-4

-2

0

2

4

6

8x 10

-8

Rea

l (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500 5000-10

-8

-6

-4

-2

0

2

x 10-8

Frequency (Hz)

Imag

(m/N

)

Measured

E-B fit

Figure 5-2. H22 artifact measurement and E-B fit for Mikron UCP-600 Vario CNC milling

machine.

75

1000 1500 2000 2500 3000 3500 4000 4500

-2

0

2

4

x 10-7

Re

al (r

ad

/N)

1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

1x 10

-7

Frequency

Ima

g (

rad

/N)

E-B

Synthesis

Finite Difference

Figure 5-3. L22/N22 results for the Mikron UCP-600 Vario CNC milling machine.

76

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4x 10

-6

Re

al (r

ad

/N-m

)

1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

x 10-6

Frequency

Ima

g (

rad

/N-m

)

E-B

Synthesis

Finite Difference

Figure 5-4. P22 results for the Mikron UCP-600 Vario CNC milling machine.

77

Figure 5-5. Short artifact dimensions for Starragheckert ZT-1000 Super Constellation

measurements.

Figure 5-6. Long artifact dimensions for Starragheckert ZT-1000 Super Constellation

measurements.

78

500 1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

x 10-8

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

x 10-8

Frequency (Hz)

Imag

(m/N

)Measured

EB fit

Figure 5-7. H22 short artifact measurement and E-B fit for Starragheckert ZT-1000

Super Constellation milling machine.

79

500 1000 1500 2000 2500 3000 3500 4000 4500-5

0

5x 10

-7

Rea

l (ra

d/N

)

500 1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Imag

(rad

/N)

E-B

Synthesis

Finite Difference

Figure 5-8. L22/N22 results for the short artifact measurement on ZT-1000 Super

Constellation Starragheckert milling machine.

80

500 1000 1500 2000 2500 3000 3500 4000 4500-5

0

5x 10

-6

Rea

l (ra

d/N

-m)

500 1000 1500 2000 2500 3000 3500 4000 4500-6

-4

-2

0

2

x 10-6

Frequency (Hz)

Imag

(rad

/N-m

)

E-B

Synthesis

Finite Difference

Figure 5-9. P22 results for the short artifact measurement on ZT-1000 Super


81

500 1000 1500 2000 2500 3000 3500 4000 4500

-1

0

1

2

x 10-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500

-3

-2.5

-2

-1.5

-1

-0.5

0

x 10-7

Frequency (Hz)

Imag

(m/N

)Measured

E-B

Figure 5-10. H22 long artifact measurement and E-B fit for Starragheckert ZT-1000

Super Constellation milling machine.

82

500 1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

10

x 10-7

Rea

l (ra

d/N

)

500 1000 1500 2000 2500 3000 3500 4000 4500

-15

-10

-5

0

x 10-7

Frequency (Hz)

Imag

(rad

/N)

E-B

Synthesis

Finite Difference

Figure 5-11. L22/N22 results for the long artifact measurement on ZT-1000 Super


83

500 1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6

x 10-6

Rea

l (ra

d/N

-m)

500 1000 1500 2000 2500 3000 3500 4000 4500

-8

-6

-4

-2

0

2

x 10-6

Frequency (Hz)

Imag

(rad

/N-m

)

E-B

Synthesis

Finite Difference

Figure 5-12. P22 results for the long artifact measurement on ZT-1000 Super


84

1000 1500 2000 2500 3000 3500 4000 4500-1

-0.5

0

0.5

1x 10

-7

Rea

l (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500-15

-10

-5

0

x 10-8

Frequency (Hz)

Imag

(m/N

)

Measured

E-B fit

Figure 5-13. H22 short artifact measurement and E-B fit for Cincinnati FTV-5 2500

milling machine.

85

1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6x 10

-7

Rea

l (ra

d/N

)

1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Imag

(rad

/N)

E-B

Synthesis

Finite Difference

Figure 5-14. L22/N22 results for the short artifact measurement on Cincinnati FTV-5

2500 milling machine.

86

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

x 10-6

Rea

l (ra

d/N

-m)

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

2x 10

-6

Frequency (Hz)

Imag

(rad

/N-m

)

E-B

Synthesis

Finite Difference

Figure 5-15. P22 results for the short artifact measurement on Cincinnati FTV-5 2500

milling machine.

87

1000 1500 2000 2500 3000 3500 4000 4500-1.5

-1

-0.5

0

0.5

1

1.5

2x 10

-7

Rea

l (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500-3

-2.5

-2

-1.5

-1

-0.5

0

x 10-7

Frequency (Hz)

Imag

(m/N

)Measured

E-B

Figure 5-16. H22 long artifact measurement and E-B fit for Cincinnati FTV-5 2500 milling

machine.

88

1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

x 10-7

Rea

l (ra

d/N

)

1000 1500 2000 2500 3000 3500 4000 4500-15

-10

-5

0

x 10-7

Frequency (Hz)

Imag

(rad

/N)

E-B

Synthesis

Finite Difference

Figure 5-17. L22/N22 results for the long artifact measurement on Cincinnati FTV-5 2500

milling machine.

89

1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6

x 10-6

Rea

l (ra

d/N

-m)

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

2

x 10-6

Frequency (Hz)

Imag

(rad

/N-m

)E-B

Synthesis

Finite Difference

Figure 5-18. P22 results for the long artifact measurement on Cincinnati FTV-5 2500

milling machine.

90

Figure 5-19. Beam model for 25.4 mm diameter, three flute endmill inserted in a

tapered shrink fit holder (not to scale).

91

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-4

-2

0

2

4

6

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-20. Comparison between H11 tool point measurement, Euler-Bernoulli,

synthesis and finite difference prediction for three flute, 25.4 mm diameter endmill with an overhang length of 99 mm.

92

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-5

0

5

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)


synthesis approach and finite difference prediction for three flute, 25.4 mm diameter endmill with an overhang length of 107 mm.

93

Figure 5-22. Beam model for 19.05 mm diameter, four flute endmill inserted in a

tapered shrink fit holder (not to scale).

94

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-4

-2

0

2

4

6

8x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-12

-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)


synthesis approach and finite difference prediction for four flute, 19.05 mm diameter endmill with an overhang length of 70.4 mm.

95

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-4

-2

0

2

4

6

8

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)


synthesis approach and finite difference prediction for four flute, 19.05 mm diameter endmill with an overhang length of 76 mm.

96

Figure 5-25. Beam model for 12 mm diameter, four flute endmill inserted in a tapered

shrink fit holder (not to scale).

97

500 1000 1500 2000 2500 3000 3500 4000 4500-1

-0.5

0

0.5

1x 10

-6

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

5x 10

-7

Frequency (Hz)

Imag

(m/N

)

Figure 5-26. Comparison between H11 tool point measurement, E-B, Synthesis and

Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 45 mm (short artifact spindle receptances).

98

500 1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

10

x 10-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

x 10-7

Frequency (Hz)

Imag

(m/N

)

Measured

E-B

Synthesis

Finite Difference



99

500 1000 1500 2000 2500 3000 3500 4000 4500

-2

-1

0

1

2x 10

-6

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-4

-3

-2

-1

0

x 10-6

Frequency (Hz)

Imag

(m/N

)


Finite difference approach prediction for four flute, 12 mm diameter endmill with an overhang length of 45 mm (long artifact spindle receptances).

100

500 1000 1500 2000 2500 3000 3500 4000 4500-3

-2

-1

0

1

2

3x 10

-6

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500

-5

-4

-3

-2

-1

0

1x 10

-6

Frequency (Hz)

Imag

(m/N

)

Measured

E-B

Synthesis

Finite Difference



101



102

500 1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

6x 10

-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

x 10-7

Frequency (Hz)

Imag

(m/N

)

Measured

E-B

Synthesis

Finite Difference



103

500 1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6

8

x 10-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500-15

-10

-5

0

x 10-7

Frequency (Hz)

Imag

(m/N

)Measured

EB Predicted

Synth fit

FD fit



104

500 1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6

x 10-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500-10

-5

0

5

x 10-7

Frequency (Hz)

Imag

(m/N

)

Measured

E-B

Synthesis

Finite Difference



105

500 1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

10

x 10-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-15

-10

-5

0

5

x 10-7

Frequency (Hz)

Imag

(m/N

)



106

Figure 5-35. Beam model for 20 mm diameter, two flute endmill inserted in a tapered


107

500 1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Imag

(m/N

)

500 1000 1500 2000 2500 3000 3500 4000 4500

-2

0

2

4x 10

-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference


Finite difference approach prediction for two flute, 20 mm diameter endmill with an overhang length of 65 mm (short artifact spindle receptances).

108

500 1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

2

4

6

x 10-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-12

-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Imag

(m/N

)



109

500 1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

2

4

x 10-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

2

4

6x 10

-7

Frequency (Hz)

Imag

(m/N

)


Finite difference approach prediction for two flute, 20 mm diameter endmill with an overhang length of 65 mm (long artifact spindle receptances).

110

500 1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6

8x 10

-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

2

x 10-7

Frequency (Hz)

Imag

(m/N

)



111



112

500 1000 1500 2000 2500 3000 3500 4000 4500

-2

0

2

4

x 10-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

x 10-7

Frequency (Hz)

Imag

(m/N

)



113

500 1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

6x 10

-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Imag

(m/N

)



114

500 1000 1500 2000 2500 3000 3500 4000 4500-3

-2

-1

0

1

2

3

4x 10

-7

Rea

l (m

/N)

500 1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Imag

(m/N

)

Measured

E-B

Synthesis

Finite Difference



115

500 1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

6x 10

-7

Rea

l (m

/N)

Measured

E-B

Synthesis

Finite Difference

500 1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Imag

(m/N

)



116

1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

10x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



117

1000 1500 2000 2500 3000 3500 4000 4500-1

-0.5

0

0.5

1

1.5x 10

-6

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



118

1000 1500 2000 2500 3000 3500 4000 4500-1

-0.5

0

0.5

1x 10

-6

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



119

1000 1500 2000 2500 3000 3500 4000 4500-2

-1

0

1

2

3x 10

-6

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-4

-3

-2

-1

0

1

2x 10

-6

Frequency (Hz)

Ima

g (

m/N

)



120

1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

2

4

6

8x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-12

-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



121

1000 1500 2000 2500 3000 3500 4000 4500

-1

-0.5

0

0.5

1

x 10-6

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



122

1000 1500 2000 2500 3000 3500 4000 4500

-5

0

5

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-20

-15

-10

-5

0

5x 10

-7

Frequency (Hz)

Ima

g (

m/N

)



123

1000 1500 2000 2500 3000 3500 4000 4500

-1

-0.5

0

0.5

1

x 10-6

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-2.5

-2

-1.5

-1

-0.5

0

x 10-6

Frequency (Hz)

Ima

g (

m/N

)



124

Figure 5-53. Tool point FRF measurement of 20 mm carbide end mill on Cincinnati

FTV-5 2500.

125

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-6

-5

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Ima

g (

m/N

)



126

1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



127

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

2x 10

-7

Frequency (Hz)

Ima

g (

m/N

)



128

1000 1500 2000 2500 3000 3500 4000 4500-10

-8

-6

-4

-2

0

2x 10

-7

Frequency (Hz)

Ima

g (

m/N

)

1000 1500 2000 2500 3000 3500 4000 4500

-4

-2

0

2

4

6x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference



129

Figure 5-58. Tool point FRF measurement of 25 mm carbide end mill on Cincinnati

FTV-5 2500.

130

1000 1500 2000 2500 3000 3500 4000 4500-2

-1

0

1

2

3x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Ima

g (

m/N

)



131

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

x 10-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)



132

1000 1500 2000 2500 3000 3500 4000 4500-2

-1

0

1

2

3x 10

-7

Re

al (m

/N)

Measured

E-B

Synthesis

Finite Difference

1000 1500 2000 2500 3000 3500 4000 4500-5

-4

-3

-2

-1

0

1x 10

-7

Frequency (Hz)

Ima

g (

m/N

)



133

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4

x 10-7

Re

al (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Measured

E-B

Synthesis

Finite Difference



134

Figure 5-63. Component coordinates for flexible coupling of holder and blank

Figure 5-64. Various shrink fit holders with blanks for Cincinnati FTV-5 2500 spindle

135

Figure 5-65. Collet holder for Cincinnati FTV-5 2500 spindle

136

1000 1500 2000 2500 3000 3500 4000 4500-6

-4

-2

0

2

4

6x 10

-6

Rea

l (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500-12

-10

-8

-6

-4

-2

0

x 10-6

Frequency (Hz)

Imag

(m/N

)

Measured

Predicted

Figure 5-66. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (rigid connection)

137

1000 1500 2000 2500 3000 3500 4000 4500-4

-2

0

2

4x 10

-6

Rea

l (m

/N)

1000 1500 2000 2500 3000 3500 4000 4500

-6

-4

-2

0

x 10-6

Frequency (Hz)

Imag

(m/N

)

Measured

Predicted


138

1000 1500 2000 2500 3000 3500 4000 4500

-2

-1

0

1

2

3x 10

-6

Rea

l (m

/N)

Measured

Predicted

1000 1500 2000 2500 3000 3500 4000 4500-5

-4

-3

-2

-1

0

x 10-6

Frequency (Hz)

Imag

(m/N

)


139

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

2

4

x 10-6

Rea

l (m

/N)

Measured

Predicted

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

x 10-6

Frequency (Hz)

Imag

(m/N

)

Figure 5-69. Measured and predicted tool point FRF of 12 mm diameter carbide blank with overhang length 76 mm (flexible connection)

140

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

2

4

x 10-6

Rea

l (m

/N)

Measured

Predicted

1000 1500 2000 2500 3000 3500 4000 4500-8

-6

-4

-2

0

x 10-6

Frequency (Hz)

Imag

(m/N

)


141

1000 1500 2000 2500 3000 3500 4000 4500-3

-2

-1

0

1

2

3x 10

-6

Rea

l (m

/N)

Measured

Predicted

1000 1500 2000 2500 3000 3500 4000 4500-5

-4

-3

-2

-1

0

x 10-6

Frequency (Hz)

Imag

(m/N

)


142

Figure 5-72. Collet holder for Mikron UCP-600 Vario

143

Figure 5-73. 25 mm diameter collet holder and blank for Mikron UCP-600 Vario

Figure 5-74. Tribos holders for Mikron UCP-600 Vario

144

Figure 5-75. Mechanism of tool clamping in a Tribos holder (http://www.us.schunk.com)

Figure 5-76. Beam model for 6.33 mm diameter, 2-flute endmill inserted in a collet holder (not to scale)

http://www.us.schunk.com/

145

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-4

-2

0

2

4

6x 10

-5

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-15

-10

-5

0

x 10-5

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-77. Measured and predicted tool point FRF of 6.33 mm diameter 2-flute carbide endmill in collet holder, overhang length 75 mm (rigid connection)

146

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-4

-2

0

2

4

6x 10

-5

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-12

-10

-8

-6

-4

-2

0

x 10-5

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-78. Measured and predicted tool point FRF of 6.33 mm diameter 2-flute carbide endmill in collet holder, overhang length 75 mm (flexible connection)

147

Figure 5-79. Beam model for 19 mm diameter, 4-flute endmill inserted in a collet holder (not to scale)

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

-0.5

0

0.5

1x 10

-6

Re

al (m

/N)

Measured

Predicted

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-80. Measured and predicted tool point FRF of 19 mm diameter carbide 4-flute endmill in collet holder, overhang length 60 mm (rigid connection)

148

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-1

-0.5

0

0.5

1x 10

-6

Re

al (m

/N)

Measured

Predicted

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000

-15

-10

-5

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-81. Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in collet holder, overhang length 60 mm (flexible connection)

149

Figure 5-82. Beam model for 12.7 mm diameter, 2-flute endmill inserted in a shrink fit holder (not to scale)

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3

-2

-1

0

1

2

3x 10

-6

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5

-4

-3

-2

-1

0

x 10-6

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-83. Measured and predicted tool point FRF of 12.7 mm diameter 2-flute carbide endmill in shrink fit holder, overhang length 66 mm (rigid connection)

150

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-2

-1

0

1

2x 10

-6

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3

-2.5

-2

-1.5

-1

-0.5

0

x 10-6

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-84. Measured and predicted tool point FRF of 12.7 mm diameter 2-flute carbide endmill in shrink fit holder, overhang length 66 mm (flexible connection)

Figure 5-85. Beam model for 19 mm diameter, 4-flute endmill inserted in a Tribos holder (not to scale)

151

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3

-2

-1

0

1

2

3x 10

-7

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5

-4

-3

-2

-1

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-86. Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in Tribos holder, overhang length 72 mm (rigid connection)

152

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-3

-2

-1

0

1

2

3

x 10-7

Re

al (m

/N)

Measured

Predicted

500 1000 1500 2000 2500 3000 3500 4000 4500 5000-5

-4

-3

-2

-1

0

x 10-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-87. Measured and predicted tool point FRF of 19 mm diameter 4-flute carbide endmill in Tribos, overhang length 72 mm (flexible connection)

153

Figure 5-88. Beam model for 25.4 mm diameter, 4-flute endmill inserted in a shrink fit holder (not to scale)

500 1000 1500 2000 2500 3000 3500 4000 4500-1

-0.5

0

0.5

1

1.5

2x 10

-7

Re

al (m

/N)

Measured

E-B

500 1000 1500 2000 2500 3000 3500 4000 4500

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

-7

Frequency (Hz)

Ima

g (

m/N

)

Figure 5-89. Measured and predicted tool point FRF of 25.4 mm diameter 4-flute carbide endmill in shrink fit holder, overhang length 55 mm (rigid connection)

154

CHAPTER 6 CONCLUSION AND FUTURE WORK

Conclusion

In this work a new method of spindle-machine dynamics identification for

Receptance Coupling Substructure Analysis (RCSA), referred to as the Euler-Bernoulli

(E-B) method, is described. In the RCSA approach, the tool-holder-spindle-machine

(THSM) assembly is considered as three separate components: the tool, holder, and

spindle-machine. The individual frequency responses, or receptances, of these

components are then analytically coupled. The spindle-machine receptances are

measured once and archived. Beam models are used to represent the tool-holder

subassembly.

The spindle-machine dynamics were determined using the E-B method, as well as

two other established methods: the synthesis approach and the finite difference

approach. In the synthesis approach, a direct frequency response measurement of a

standard artifact inserted in the test spindle is combined with a cross frequency

response measurement to calculate the required rotational receptances. In the finite

difference approach, two direct and one cross frequency response are measured using

the standard artifact-test spindle combination. Again, these measurement results are

used to determine the rotational frequency response functions (FRFs). In the E-B

method, the direct frequency response measurement is fit using an assumed (fixed-

free) form of each mode within the measurement bandwidth and this fit is used to

determine the rotational receptances (no additional measurements are required).

Standard artifact measurements were performed on three milling machines: a

Mikron UCP-600 Vario, a Starragheckert ZT-1000 Super Constellation, and a Cincinnati

155

FTV-5 2500. The spindle-machine dynamics were determined by the three approaches

and compared. These spindle-machine dynamics were then used to predict the tool

point FRF of various tool-holder combinations (carbide endmills clamped in thermal

shrink fit holders) inserted in the three spindles. The measured tool point FRFs were

compared to the predictions. For these predictions, the connection between the tool and

the holder was assumed to be rigid. The best method to determine the spindle-machine

dynamics was identified by using a new comparison metric.

Based on the comparison metric calculations, it was concluded that the E-B

method provides a robust and accurate identification method for spindle-machine

dynamics. The cross frequency response measurement on the standard artifact in the

synthesis and the finite difference approach may lead to undesired results in the tool

point FRF predictions.

The tool point FRF predictions determined using the rigid connection assumption

between the tool and holder generally predicted higher natural frequencies than the

measurements. Therefore, a flexible connection between the tool and holder was

introduced in order to improve the tool point frequency response prediction accuracy

(the E-B method spindle-machine receptances were used). The stiffness values for the

tool-holder connection were obtained by applying a non-linear least squares error

minimization to the difference between the magnitudes of the predicted and measured

tool point FRFs. Stiffness values were identified for various diameter (for example 10

mm, 16 mm, 20 mm, and 25 mm) carbide blanks (rods) clamped in shrink fit, collet, and

Tribos tool holders. Multiple overhang lengths of the blanks were measured for each

blank-holder set to obtain the stiffness values. The average of these stiffness values

156

was then used to predict the tool point FRFs of actual endmills. The agreement between

the measured and the predicted FRFs improved for the flexible connection (based on

the stiffness values obtained by blank measurements). Therefore, the approach of

identifying the tool-holder connection stiffness values using blanks is valid.

Future Work

The possible future work in this research includes investigation of the Timoshenko

beam model for the endmills. The use of the equivalent diameter to model the

complicated fluted portion of the endmills may not be the best method to identify the

FRFs of the tool. The modeling of the flutes needs to be further studied and analyzed in

order to increase the accuracy of the tool-point FRF predictions. Also, the Tribos holder

Timoshenko beam models requires further study due to the thermosetting plastic

chambers in the holders.

157

APPENDIX A FLEXIBLE COUPLING BETWEEN TOOL AND HOLDER

The free-free tool receptances (R11, R12a, R2a2a and R2a1) and the machine-spindle-

holder receptances (R2b2b) may be coupled using a flexible joint to predict the assembly

tool point receptance. In order to calculate the tool point assembly receptances, G11

(Equation A-1), a generalized force Q1 (representing both the externally applied force,

F, and couple, M) is applied at coordinate location U1 (see Figure A-1), where the

generalized displacement U represents both displacement, X, and rotation, .

(A-1) The displacement equations for the substructures can be described as follows:

(A-2) (A-3) (A-4) For a flexible coupling, the compatibility condition that describes the connection

between the two components is expressed as shown in Equation A-5.

, (A-5)

where the receptance matrix,

xf xf xm xm

f f m m

k i c k i ck

k i c k i c, is composed of four

stiffness values and four damping values that relate the displacement and rotation to the

applied force and couple. The equilibrium condition at coordinate locations 2a and 2b is

given by Equation A-6.

(A-6)

1 1

1 1 11 11

11

11 111 1

1 1

X X

F M H LG

N P

F M

1 11 1 12 2a au R q R q

2 2 2 2 2 1 1a a a a au R q R q

2 2 2 2b b b bu R q

2 2 2( )b a bk u u q

2 2 0b aq q

158

At coordinate location 1, the external force/couple is applied so the relationship in

Equation A-7 is obtained.

(A-7) Substituting for u2b and u2a in Equation A-5 gives Equation A-8.

(A-8) Equation A-9 is obtained using Equations A-7 and A-8.

(A-9)

Solving for q2b gives Equation A-10. Given that 2 2a bq q from Equation A-6,

substitution in Equation A-11 gives Equation A-12, which can then be written as shown

in Equation A-13. This equation gives the assembly receptances expressed as a

function of the component receptances and the stiffness matrix, k. Therefore, given the

tool receptances and the holder-spindle-machine receptances, the tool-holder-spindle-

machine receptances can be predicted using a flexible connection between the tool and

the holder.

(A-10) (A-11) (A-12) (A-13)

1 1q Q

2 2 2 2 2 2 2 2 2 1 1 2( ) ( )b a b b b a a a a bk u u k R q R q R q q

2 2 2 2 2 2 2 1 1 2( )b b b a a a a bk R q R q R Q q

1

2 2 2 2 2 2 1 1

1( )b b b a a aq R R R Q

k

11 1 12 21 111

1 1 1

a aR q R qU uG

Q Q Q

1

11 1 12 2 2 2 2 2 1 1

11

1

1( )a b b a a aR Q R R R R Q

kGQ

11 111

11 11 12 2 2 2 2 2 1

11 11

1( )a b b a a a

H LG R R R R R

N Pk

159

Figure A-1. The tool (I) is coupled flexibly to the holder-spindle-machine (II) to

determine the tool point receptance matrix, G11.

160

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164

BIOGRAPHICAL SKETCH

Uttara Vijay Kumar was born and raised in New Delhi, the capital city of India. She

received her Bachelor of Technology degree in mechanical and automation engineering

from Indira Gandhi Institute of Technology, a constituent college of Guru Gobind Singh

Indraprastha University, Delhi in May 2007. In fall 2007 she began her graduate studies

at the Department of Mechanical and Aerospace Engineering, University of Florida, in

pursuit of her MS degree in mechanical engineering. In spring 2008, she joined the

Machine Tool Research Center under the guidance of Dr. Tony L. Schmitz. She

received her Master of Science degree in December 2009.

improved spindle dynamics identification technique...

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