MECHANICS AND DYNAMICS OF MILLING THIN WALLED STRUCTURES

MECHANICS AND DYNAMICS OF MILLING THIN WALLED STRUCTURES

By

Erhan Budak

BSc. Middle East Technical University, Ankara, Turkey 1987;

MSc. Middle East Technical University, Ankara, Turkey 1989

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

in

THE FACULTY OF GRADUATE STUDIES

DEPARTMENT OF MECHANICAL ENGiNEERiNG

1994

© Erhan Budak, 1994

We accept this thesis as conforming

to the required standard

THE UNIVERSITY OF BRITISH COLUMBIA

In presenting this thesis in partial fulfilment of the requirements for an advanced

degree at the University of British Columbia, I agree that the Library shall make it

freely available for reference and study. I further agree that permission for extensive

copying of this thesis for scholarly purposes may be granted by the head of my

department or by his or her representatives, It is understood that copying or

publication of this thesis for financial gain shall not be allowed without my written

permission.

__

Department of fri €cI&Cl4 ca I En InrThe University of British ColumbiaVancouver, Canada

Date l39S

DE-6 (2/88)

Abstract

Peripheral milling of flexible components is a commonly used operation in the aerospace

industry. Aircraft wings, fuselage sections, jet engine compressors, turbine blades and

a variety of mechanical components have flexible webs which must be finish machined

using long slender end mills. Peripheral milling of very flexible plate structures made of

titanium alloys is one of the most complex operations in the aerospace industry and it is

investigated in this thesis.

Flexible plates and cutters deflect statically and dynamically due to periodically vary

ing milling forces and self excited chatter vibrations. Static deflections of the plate and

cutter cause dimensional form errors, whereas forced and chatter vibrations result in poor

surface quality and chipping of the cutting edges. In this thesis, a comprehensive model of

the peripheral milling of very flexible cantilever plates is presented. The plate and cutter

structures are modeled by 8 node finite elements and an elastic beam, respectively. The

cutting forces are shown to be very dependent on the magnitude of the plate and cutter

deformations which are irregular along the helical end mill-plate contact. The interac

tion between the milling process and cutter-plate structures is modeled, and the milling

forces, structural deformations and dimensional form errors left on the finish surface are

accurately predicted by the simulation system developed in this study. A strategy, which

constrains the maximum dimensional form errors caused by static deformations of plate

and cutter by scheduling the feed along the tool path, is developed. The variation of the

plate thickness due to machining and the partial disengagement of the plate and cutter

due to excessive static deflections are considered in the model. The simulation system

is proven in numerous peripheral milling experiments with both rigid blocks and very

flexible cantilevered plates.

11

The self excited vibrations observed during peripheral milling of very flexible struc

tures with multi-degree of freedom dynamics is investigated. A novel analytical model of

milling stability is developed. The stability model requires structural transfer functions

of plate and cutter, milling force coefficients and helical end mill geometry. Chatter

vibration free cutting speeds, axial and radial depths of cut, i.e. stability lobes, are

predicted analytically without resorting to computationally expensive time domain sim

ulations. The analytical chatter stability model is verified in various peripheral milling

experiments, including the machining of plates.

The cutting force and chatter stability models developed in this thesis can be used to

improve the productivity of peripheral milling of thin webs by enabling simulation and

process planning prior to production.

111

Table of Contents

Abstract ii

Table of Contents iv

List of Tables ix

List of Figures x

Acknowledgments xv

Nomenclature xvi

1 Introduction 1

2 Literature Survey 7

2.1 Overview 7

2.2 Geometry of Milling 8

2.3 Milling Force Models 9

2.4 Stability of Dynamic Milling 11

2.4.1 Dynamic Cutting 12

2.4.2 Chatter Stability Models 16

2.5 Peripheral Milling of Flexible Structures 19

3 Structural Modeling of the Workpiece and End Mill 21

3.1 Introduction 21

iv

3.2 Structural Modeling of Workpiece 22

3.2.1 Bending of Thin Plates 23

3.2.2 Series Solutions for Cantilever Plate 28

3.2.3 The Finite Element Modeling of Plate Defiections 34

3.2.4 Dynamics of Cantilever Plate 46

3.2.5 Simulation and Experimental Examples 48

3.3 Structural Modeling of End Mill 51

3.3.1 Cantilever Beam Model for End Mill 52

3.3.2 Simulation and Experimental Examples 54

3.4 Summary 55

4 Modeling of Milling Forces 56

4.1 Introduction 56

4.2 Mechanistic Modeling of Milling Forces 57

4.2.1 Exponential Force Coefficient Model 58

4.2.2 Linear Edge-Force Model 65

4.3 A Mechanics of Milling Approach for Milling Force Prediction . 68

4.3.1 Force Coefficient Expressions 69

4.3.2 Procedure for Milling Force Calculation 81

4.4 Simulation and Experimental Results 83

4.4.1 Cutting Conditions in Milling and Orthogonal Tests 85

4.4.2 Analysis of Orthogonal Data: Identification of Cutting Parameters 86

4.4.3 Prediction of Milling Force Coefficients 93

4.4.4 Accuracy of Milling Force Calculation by Predicted Coefficients 98

4.5 Summary 101

v

• 5 Effects of Milling Conditions on Cutting Forces and Accuracy 104

5.1 Introduction 104

5.2 Surface Generation by Statically Flexible End Mill 104

5.3 Identification of Cutting Conditions for Minimum Dimensional Surface ErrorlO7

5.4 Simulation and Experimental Results 111

5.4.1 Cutting Force and Surface Finish 111

5.4.2 Selection of Optimal Cutting Conditions 113

5.5 Summary 122

6 Static Structure-Milling Process Interaction 123


6.2 Statically Regenerative Milling Force Model (Variation of Chip Thickness

Due to Defiections) 124

6.3 Flexible Milling Force Model (Variation of Radial Depth of Cut Due to

Deflections) 135

6.4 Summary 140

7 Peripheral Milling of Plates 142


7.2 Static Modeling of Plate Milling 144

7.2.1 Structural Model of the Plate . . . 144

7.2.2 Structural Model of the Tool 146

7.2.3 Cutting Force Distribution-Rigid and Flexible Force Models . 147

7.3 Simulation of Peripheral Plate Milling. . . 150

7.3.1 Plate Surface Generation . 151

7.3.2 Control of Accuracy . 152

7.4 Simulation and Experimental Results . . . 153

vi

7.5 Summary.169

8 Analysis of Dynamic Cutting and Chatter Stability in Milling 170


8.2 Formulation of Dynamic Milling Forces 173

8.2.1 Dynamic-Regenerative Chip Thickness 175

8.2.2 Differential Dynamic Milling Forces 176

8.2.3 Total Dynamic Milling Forces 177

8.2.4 Dynamic Displacements of Cutter and Workpiece 184

8.3 Stability Analysis 187

8.3.1 Stability Theory of Periodic Systems 187

8.3.2 Stability Analysis of Milling Using Periodic System Theory . . 189

8.3.3 Milling Stability Analysis Based on the Interpretation of Physics

of Milling Dynamics 193

8.3.4 Truncation of the Characteristic Equation of Dynamic Milling 197

8.3.5 Summary of the Calculation of Milling Stability for the General Case2O2

8.3.6 Accuracy of the Chatter Limit Prediction by the Truncated Char

acteristic Equation 203

8.3.7 Solution of the Characteristic Equation to Determine the Chatter

Stability Limit in Milling 205

8.4 Solutions of Milling Stability Equation for Special Cases 210

8.4.1 Milling of a Single Degree-of-Freedom Workpiece 211

8.4.2 Milling of a Flexible Structure with a Flexible End Mill-Single Ax

ial Element 218

8.4.3 Milling of a Flexible Structure with a Rigid End Mill-Varying Dy

namics in Axial Direction 224

vii

8.5 Dynamic Peripheral Milling of Plates 229

8.6 Summary 239

9 Conclusions 242

Bibliography 247

Appendices 261

A Chip Flow Angle Formulation 262

viii

List of Tables

3.1 Eigenfunction parameters for clamped-free and free-free beams 31

4.1 Helical flute engagement limits to be used in cutting force calculations. 64

4.2 Cutting parameters identified for Ti6A14V from orthogonal cutting tests 92

4.3 Cutting force coefficients for different rake angles as identified and trans

formed from orthogonal data by using the linear edge-force model 96

4.4 Cutting force coefficients for different rake angles as identified from milling

tests by using exponential force model 97

5.1 Cutting conditions for up and down milling tests conducted for surface

error verification and identification of optimal milling conditions 114

7.1 Cutting conditions for experiments 1 and 2. (Material: Titanium Alloy

Ti6A14V) 154

ix

List of Figures

2.1 Dynamic cutting process 12

2.2 Variation of effective clearance angle in dynamic cutting 14

3.1 Internal forces and moments on a differential plate element 24

3.2 Displacement of a point P to F’ due to bending of plate 25

3.3 3D isoparametric solid element 38

3.4 The 3D solid element in the natural coordinates 39

3.5 Flow chart of the developed Finite Element Program 45

3.6 One element example to test the developed FE Program 48

3.7 Structural model for end mill: Cantilever beam with elastically restrained

end 52

4.1 Differential milling forces applied on the cutting tooth 59

4.2 Contact cases of a helical flute with workpiece 63

4.3 Orthogonal and oblique cutting geometries 70

4.4 Orthogonal cutting force diagram 71

4.5 Cutting forces and chip flow geometry on a helical milling cutter 73

4.6 Detailed view of the oblique cutting geometry 74

4.7 The oblique cutting force components in the normal plane 75

4.8 Predicted variations of chip flow angle with rake and friction angles. 80

4.9 Variation of chip flow angle with cutting ratio for different values of friction

angle 80

4.10 The effects of the inclination angle and cutting ratio on the chip flow angle. 81

x

4.11 Generalized milling force prediction algorithm 82

4.12 The orientation of differential milling force components on a ball end mill

flute 84

4.13 Measured cutting and feed forces in orthogonal cutting tests with different

rake angles 87

4.14 Edge forces as identified from the orthogonal cutting tests 88

4.15 Variation of the measured cutting ratio r(= h/he) with chip thickness and

rake angle 89

4.16 The identified values of the shear stress at the shear plane from the or

thogonal cutting tests 91

4.17 The friction angle calculated from the orthogonal cutting forces 92

4.18 Predicted values of the chip flow angle for 300 inclination (helix) angle. . 93

4.19 Variations of the predicted milling force coefficients with the chip thickness. 94

4.20 The statistical error analysis of the milling force predictions 99

4.21 Measured and simulated milling forces (linear-edge force model) 102

4.22 Measured and simulated milling forces (linear-edge force model) 102

4.23 Measured and simulated milling forces (exponential force model) 103

4.24 Measured and simulated milling forces (exponential force model) 103

5.1 Statically flexible end mill model 105

5.2 Variation of the optimum exit angle (for up-milling) with K, 111

5.3 Measured and simulated milling forces for half immersion-up milling. 116

5.4 Measured and simulated milling forces for half immersion-down milling 116

5.5 Simulated and measured surface profiles for half immersion-up milling 117

5.6 Simulated and measured surface profiles for half immersion-down milling 117

xi

5.7 Variation of the predicted m&ximum dimensional surface error due to tool

deflection with the radial depth of cut and the feed per tooth 118

5.8 Variation of the specific-predicted maximum dimensional surface error

(SMSE) with the radial depth of cut and the feed per tooth for up milling.119

5.9 Variation of measured and simulated maximum normal cutting force Fy,nax

with radial depth of cut for different values of feed per tooth 120

5.10 Variation of measured and simulated maximum dimensional surface error

emaa, with radial depth of cut for different values of feed per tooth 121

5.11 Variation of measured and simulated specific maximum surface error SMSE

with radial depth of cut for different values of feed per tooth 121

6.1 Statically regenerative chip thickness geometry in milling 125

6.2 Simulated rigid and statically regenerative milling force in x direction. . 135

6.3 Variation of radial depth of cut and immersion angles due to deflections

in up and down milling 137

7.1 (a) Peripheral down milling of flexible plates, (b) Finite element model of

the plate, (c) Corresponding nodal stations on the tool 145

7.2 Experiment #1 - A sample window of simulated and measured forces. 156

7.3 Experiment #1- (a) Simulated, (b) Measured surface finish dimensions 158

7.4 Experiment #2- (a) Rigid model, (b) Flexible model, (c) Measured surface

finish dimensions 159

7.5 Experiment #2 Predicted and measured surface profiles near the begin

ning, middle and exit feed stations 160

7.6 Experiment #2- (a) Measured and simulated average forces 161

7.7 Experiment #2 - Flexible force model predicted variation of q58t in the

beginning, middle and close to exit feed stations 162

xii

7.8 Scheduled feedrates for Experiment 1 and Experiment 2 for surface error

tolerances of 80 m and 250 IIm, respectively 164

7.9 Experiment # 1 with scheduled feedrate for 80 m tolerance - a) Simulated

surface errors b) Measured surface errors 165

7.10 Experiment # 2 with scheduled feedrate for 250 m tolerance- a) Simu

lated (flexible model) surface errors b) Measured surface errors 166

7.11 The variation of the machining time with tolerance value in Case # 1 and

#2 167

7.12 The simulated variation of the machining time with tool radius and radial

width of cut in Experiment # 2 for tolerance of 250 .tm on the surface. 168

8.1 Dynamic milling process 174

8.2 Node numbering on cutter and workpiece 186

8.3 The effect of number of teeth on peak to peak (AC component) value of

directional coefficient 204

8.4 Variation of chatter frequency with spindle speed as predicted by the or

thogonal cutting chatter theory 207

8.5 Single degree-of-freedom milling system model 212

8.6 The phase angle and transfer function at the chatter stability limit. . . 214

8.7 Variation of the directional milling coefficient a, with the immersion angle

in up and down milling 216

8.8 Comparison of the predicted chatter limit with the published data for the

single degree-of-freedom milling system example 218

8.9 Milling system model with two degree-of-freedom cutter and workpiece. 219

8.10 Experimental and time domain simulated stability limits for end milling

tests. (data by Weck, Altintas and Beer, 1993) 222

xlii

8.11 Analytically predicted stability lobes for the case for which the experimen

tal data and time domain simulations are shown in Figure 8.10 223

8.12 Analytical and time domain stability limit predictions for a case analyzed

by Smith and Tlusty 225

8.13 Stability limit calculation algorithm for flexible workpieces with varying

dynamics in the axial direction 227

8.14 Predicted effect of vibration mode shape on the stability limit 228

8.15 Stability diagram for the cantilever titanium Ti6A14V plate down milled

by 8 flute carbide end mill 230

8.16 Cutting force spectrums in x and y directions for n = 6500 rpm and

a = 0.4 mm 232

8.17 Sound signal and its spectrum at n = 6500 rpm, a = 0.4 mm 233

8.18 Cutting force spectrums in the x and y directions for n = 5000 rpm and

a 0.4 mm 234

8.19 Cutting forces in the y direction for m = 5000 rpm and n 6500 rpm for

a = 0.4 mm 235

8.20 Cutting forces in x and y directions for n = 6500 rpm and a = 2 mm. . 236

8.21 Sound signal and its spectrum for n = 6500 rpm and a = 2 mm 237

8.22 Cutting force spectrums for n = 6500 rpm and a = 2 mm 238

8.23 Sound spectrums for different spindle speeds showing the effect of process

damping on chatter stability, a = 44 mm 240

8.24 Effect of spindle speed and the process damping on the cutting forces.

a=44mm 241

xiv

Acknowledgments

I would like to express my sincere appreciation to my research supervisor Dr. Yusuf

Altinta for his support, guidance and encouragement throughout this work.

I greatly appreciate the assistance of Yetvard Hosepyan and Peter Lee in cutting tests,

and Dr. Ercan Köse with computer software. I thank the Manufacturing Development

Division of Pratt £4 Whitney, Montreal for providing the cutters and workpieces used in

this work. I also appreciate the valuable comments of Dr. Ian Yellowley on my research.

Finally, I am most thankful to my wife and daughter, Asuman and Ece, for their sac

rifice, continuous support and understanding during the course of this study. I dedicate

this work to them.

This research has been supported by the Natural Sciences and Engineering Research

Council of Canada and Pratt £4 Whitney Aircraft of Canada.

xv

Nomenclature

a axial depth of cut

altm limiting axial depth of cut for chatter stability

directional dynamic milling coefficients

[A(t)] directional dynamic milling coefficient matrix

b radial depth of cut

bj(z) effective radial depth of cut at axial position z

[B] strain-displacement matrix for 3D elastic body

d0 cutter diameter

de effective diameter of the cutter

dF, dF, dFa differential tangential, radial and axial cutting forces in

milling

D flexural rigidity

e( k) surface form error at node k

E Young’s modulus of elasticity

[EJ elasticity matrix

F, Ff cutting and feed force in orthogonal cutting

F3,F3,F3 milling forces in feed x, normal y and axial z directions

on flute j

[Ge], [Gm] cutter and workpiece transfer functions

I end mill area moment of inertia of end mill based on the

equivalent diameter

xvi

h uncut chip thickness

ha average chip thickness

h cut chip thickness

[J] Jacobian matrix of coordinate transformation

ICC collet stiffness

[K] stiffness matrix of structure

K, K, K tangential, radial and axial milling force coefficients (ex

ponential force model)

K7,Kac tangential, radial and axial milling-cutting force coeffi

cients (linear-edge force model)

Kte, Kre, Kae tangential, radial and axial milling-edge force coeffi

cients (linear-edge force model)

m rth modal residual

[M] mass matrix of structure

N number of teeth on the cutter

{Q} load vector

r chip thickness ratio or cutting ratio

1? cutter radius

St feed per tooth

t, t, uncut and cut plate thickness

T tooth period

V cutting velocity

x feeding direction coordinate axis

xvii

y normal direction coordinate axis

w deflection of plate

z axial direction coordinate axis

z axial coordinate of the surface generation point

z3,1,z3,2 lower and upper engagement limits for flute ja cutter rake angle in orthogonal cutting

a, a1, normal and velocity rake angles in oblique cutting

/3 friction angle in orthogonal cutting

f3 normal friction angle in oblique cutting

&(k), 6(k) end mill deflections in x and y directions at node k

axial element thickness

, ey, e, linear and shear strains

phase between succesive waves in chatter vibrations

ic chip flow angle

friction coefficient on the rake face

LI Poissons’s ratio

damping ratio

o, a, o stresses in x—

y plane

T shear stress at the shear plane

{} normalized mode shape of the structure

rotation angle of the cutter

4i(z) immersion angle of tooth j at axial position z

cutter pitch angle

xviii

shear angle in orthogonal cutting

normal shear angle in oblique cutting

48t, qes start and exit angles of cut in milling

helix angle

w tooth passing frequency

chatter frequency

spindle speed in (rad/sec)

xix

To Asuman and Ece Polen

xx

Chapter 1

Introduction

The peripheral milling of flexible components is a commonly required operation.

Aircraft wing structures, fuselage sections, jet engine compressors, turbine blades and

precision instrumentation housings all have flexible webs which must be finish machined

by long slender end mills. In general, the majority of the aerospace components listed

here are machined from aluminum or titanium blocks. While the aluminum alloys have a

good machinability rating (due to their low yield stress) the titanium alloys are difficult

to machine because of their poor thermal conductivity. Furthermore, most aerospace and

instrumentation components have tight dimensional tolerances which have to be satisfied

during machining.

The peripheral milling of very flexible, cantilevered plate structures made of titanium

alloys is studied in this thesis. The project was originated and supported by a jet engine

manufacturer, who produces rotors (i.e. blisks), impellers, scrolls and turbine blades

using slender helical carbide end mills. The workpiece material is titanium (Ti6A14V)

for all components. Impellers and blisks are milled on five axis CNC machining centers,

and the cutting time varies from 7 hours to 40 hours depending on the size of each model

component. The wall thickness and the height from the cantilevered bottom (i.e. hub)

of these parts are between 1.0mm to 2.5mm, and 25mm to 75mm, respectively. There

fore, the parts resemble clamped-free-free- free (CFFF) plates and are very flexible. The

carbide end mills used to machine these parts have a diameter of 10mm to 20mm, and

1

Chapter 1. Introduction 2

a gauge length of 35mm to 100mm from the clamping chuck. Because both the plate

type workpiece and the long slender end mill represent very flexible structures, severe

static and dynamic deformations are experienced during peripheral milling of husks and

impellers.

The flexibility of the slender end mill becomes dominant during the roughing of plates

from solid blocks. The end mill removes the material in slotting mode, the resulting forces

have both a strong dc component as well as a dynamic component at the tooth passing

frequency. These forces may then lead to excessive static deformations which may break

the cutter at the shank, as well as severe chatter vibrations which may chip the cutting

edges during roughing.

Flexibilities of both cutter and plate must be considered during semi-finishing and

finishing operations. The end mill is most flexible at its free end which is in contact

with or adjacent to the root, i.e. the most rigid portion of the cantilevered plate. The

plate’s flexibility increases towards the free edge, which is closer to the clamped part

of the end mill. As a result of the milling forces, the cutter statically deflects most at

the plate’s bottom end, and the deflection of the plate increases towards its free edge.

Due to the cutter helix angle, the distribution of loads are very irregular in three carte

sian directions. The amplitudes and direction of the forces change as the cutter rotates.

Furthermore, the cutting forces are dependent on the local chip thickness removed from

the plate, which continuously varies due to static deformations of both plate and tool

structures. The problem is further complicated by the partial disengagement of plate and

tool due to excessive static displacements which are irregular along the axial direction.

In order to understand the physics of the process and constrain the plate deformations

within the required tolerances, a comprehensive model of the milling mechanics, the


structures involved and the interaction between the metal cutting process and structural

deformations have been developed. The plate is represented by its finite element model

with varying thickness, and the cutter is modeled as a continuous elastic beam. The

local cutting forces and displacements of both plate and cutter at each elemental zone

are evaluated by predicting the chip thickness and cutter-plate intersection boundaries.

Furthermore, the model developed in this work is able to predict the chip loads (i.e.

feed rates) along the feed direction so that the static deformations are kept within the

prescribed tolerances as the material is removed from the plate. The developed methods

have been experimentally verified.

The flexibilities of the cutter and the plate produce severe forced and self excited (i.e.

chatter) vibrations during the peripheral milling operations. Forced vibrations can be

avoided by selecting spindle speeds whose corresponding tooth passing frequency harmon

ics do not coincide with the natural modes of the plate. Chatter vibrations are initiated

by transient vibrations, and their stability depends on the axial and radial depths of cut,

cutting speed, workpiece material hardness and structural properties of both tool and

the workpiece. Chatter free axial and radial depths of cut, and cutting speeds have been

determined by time domain simulations of the cutting-structure interaction, explained in

the static case, but including dynamic properties and regeneration of chip thickness at

successive tooth periods. The time domain simulations have been found to be time con

suming, they do not provide a physical insight, and are prone to errors due to sensitivity

of digital differentiation and integration techniques when the chip thicknesses are very

small and the deformations are very large. A novel analytical technique, which predicts

the chatter vibration free stability lobes for multi-degree freedom flexible cutter and flex

ible workpieces, has been developed. The method has been experimentally verified when

milling with flexible end mills and plates.


The peripheral milling of such flexible plates has not been investigated in depth be

fore. The models developed in this thesis for static and dynamic deformations of the

plate during machining are quite comprehensive and are believed to contribute to the

milling literature.

The chapters of the thesis are organized as follows:

In Chapter 2, the relevant literature on milling geometry, milling force modeling,

dynamic milling and chatter stability, and the milling of flexible workpieces is reviewed

in general. Detail reviews are provided when related methods and approaches are used

or introduced in individual chapters.

The static and dynamic modeling of slender end mills and clamped-free-free-free plates

are presented in Chapter 3. The end mill is modeled as a continuous beam and the plate

is represented by a finite element model with 8 node isoparametric elements. Both

structural models are verified using analytical and experimental techniques.

The modeling of milling forces using mechanistic and oblique cutting approaches is

presented in Chapter 4. The influence of edge forces is considered. It is shown that while

mechanistic approaches may provide coefficients to predict the cutting forces and cor

responding structural deformations more accurately, the oblique cutting model provides

more insight to the physics of the process yet it still has sufficient accuracy to predict

the cutting forces in milling. The oblique model reduces the amount of tests required

in mechanistic models as it uses a generalized orthogonal cutting data base to calculate

the milling force coefficients for different milling cutter geometries. Both techniques are

experimentally proven and compared.

The influence of milling conditions, such as feed rate, axial and radial depth of cut

and cutting coefficients, on milling forces and dimensional form errors produced by the


flexible slender end mills are presented in Chapter 5. A method of finding optimal radial

depth of cut to achieve minimum dimensional form errors on finished surfaces is developed

and experimentally proven.

The static regeneration of chip thickness in milling is modeled in Chapter 6. It is

analytically and numerically proven that for a chatter free stable milling operation the

effect of deflections on the chip thickness diminishes in a few tooth periods. Then, the

main mechanism of deflection-milling process interaction is identified as the variation

of the cutter-workpiece immersion boundaries or radial depth of cut when milling very

flexible parts. A flexible milling force model, which uses the effective radial depth of cut

under the deflections, is developed. The model is used in the simulation of the peripheral

milling of plates in Chapter 7.

The simulation model for peripheral milling of flexible plates with flexible end mills

is presented in Chapter 7. The local changes in the radial width of cut, are considered in

• calculating the chip thickness, milling forces and displacements using the model developed

in Chapter 7. The Finite Element modeling of the plate with varying structural properties

due to metal removal, the beam model of the slender end mill, the flexible milling force

model which considers the partial disengagement of the structures along the cutter axis

and milling mechanics are integrated to a comprehensive simulation model. The model

predicts the milling force distribution and surface form errors caused by static flexibilities

of the plate and tool. A feed scheduling technique which constrains the form errors within

the prescribed tolerances has been developed and integrated with the plate simulation

model. The model has been experimentally verified in peripheral milling of very flexible

titanium plates.

A novel general chatter stability model for multi-degree of freedom systems is in

troduced in Chapter 8. The variations in the structural dynamic properties along the

cutter axis, which is the case in plate milling, are considered. The stability model is


analyzed by two different approaches which converge to the same results. The first so

lution is based on the application of the known periodic system theory to the dynamic

milling model introduced, while the second approach is based on the physics of dynamic

milling formulated. The chatter free axial or radial depth of cuts and cutting speeds

are predicted analytically as opposed to being determined using the numerical and time

domain simulation approaches proposed before. The model is verified with experimental

and time domain simulation results for various structures and milling modes including

the peripheral milling of plates. In addition, the forced and chatter vibrations observed

during peripheral milling of plates are presented in Chapter 8.

The thesis concludes with a summary of contributions and suggestions for future work

in Chapter 9.

Chapter 2

Literature Survey

Statics and dynamics of peripheral milling of very flexible webs are highly inter

disciplinary. They include theories and methods of metal cutting, milling mechanics,

structural mechanics and dynamics, and stability of chatter vibrations in milling. Be

cause of this, the relevant literature is cited and explained in each section throughout the

thesis. In this chapter, only a brief review for metal cutting mechanics, milling mechan

ics and dynamics, dynamic cutting and chatter stability literature is given to provide a

theoretical base for the remainder of the work.

2.1 Overview

Although machining processes have been in use in some form or other since the early

ages, it is only during this century in general and since the mid-forties in particular, that

systematic attempts have been made to bring this field into a scientific basis. This can

be attributed to several factors which characterize machining processes: unconstrained

flow of material with large strains, high strain rates, high stresses and temperatures,

and unusual friction conditions. The comprehensive work by F.W. Taylor [1] on the Art

of Cutting Metals published in 1907 was the beginning of serious and systematic stud

ies on the various aspects of metal cutting. However, it was M.E. Merchant’s cutting

process model [2] in 1944 that took the remarkable step from the art of metal cutting

to the science of metal cutting. Since then, progress in machining research has been

considerable. Many models have been developed towards the understanding of the chip

7

Chapter 2. Literature Survey 8

formation, shearing, plastic and elastic contact, friction and wear mechanisms, the pre

diction of forces, stresses, strains and temperatures involved in the machining process.

Also, extensive research efforts have been spent on the understanding and modeling of

dynamic cutting and cutting stability, in the last four decades. These analyses can be

found in several books written on machining and machine tools [3, 4, 5, 6, 7, 8, 9, 10]. In

recent years, more and more emphasis has been put on the modeling of the machining

process because of the increasing demand for untended machining, improved CAD/CAM

systems, advanced process planning, control and monitoring techniques.

2.2 Geometry of Milling

Milling is a multiple point, interrupted cutting operation. Because of the multiple

teeth, each tooth is in contact with the workpiece for a fraction of the total time. The

finished surface, therefore, consists of a series of elemental surfaces generated by the in

dividual cutting edges of the cutter. Due to the nature of relative contact between the

workpiece and the tool, the chip thickness is not constant but starts with a zero thick

ness and increases in up-milling and starts with a finite thickness and decreases to zero

in down milling.

The early research in milling mechanics [11, 12, 13, 14, 15, 16] dealt with the chip

formation mechanism and spindle power estimation. Martelotti [17, 18] showed that the

true path of the milling cutter tooth is trochoidal, but it can be approximated as circular

• if the radius of the cutter is much larger than the feed per tooth. This approximation

simplifies the analysis of the process and, in practice, the necessary condition for the feed

per tooth to radius ratio is usually satisfied. Martelotti also derived an expression for


the amplitude of the tooth marks left on the surface:

where Ii is the height of the tooth mark, St is the feed per tooth and R is the cutter

radius.

2.3 Milling Force Models

Due to the large number of variables involved in the milling geometry, an abundant

amount of data is required for the analysis of milling force and surface finish with em

pirical techniques [19]. Therefore, the analytical or semi-analytical prediction of milling

forces is essential. In early studies, some expressions for the amplitude of the pulsating

cutting force in milling were developed from purely geometrical considerations [14, 13].

Salomon [14] based his equation, for the work done with a straight tooth cutter, on the

assumption that the specific cutting pressure was an exponential function of the chip

thickness. In their analytical milling force expressions, Sabberwal and Koenigsberger

[20, 21] used similar exponential specific milling coefficients (both in tangential and ra

dial directions) which are identified experimentally. This approach for the milling force

coefficients is referred to as the “mechanistic model” which has been adopted by many

researchers in the analysis of the milling process [22, 23, 24, 25]. In another type of mech

anistic modeling, edge milling force coefficients are separated to yield constant milling

force coefficients [26, 27, 28]. In the mechanics of milling models, the milling force coef

ficients are determined by using an oblique cutting model [29, 27, 30, 31].

Milling force and surface generation models can be classified as suggested by Smith

and Tlusty [32]. The simplest milling force model is the average rigid force model which


assumes that the average power cousumed, torque, tangential cuttiug force aud the di

mensional error ou the machined surface are proportional to the material removal rate

[33]. This model though cannot provide accurate results as, in general, there is no simple,

direct relationship between the material removal rate and the cutting forces and cutter

deflections. For accurate predictions, the cutting forces at the tip of the tooth have to

be considered. In the instantaneous rigid force model, the milling force on the helical

cutting edges is computed. The model of Koenigsberger and Sabberwal [21] was the first

complete model in this group. Kline et al. [23, 34, 35] included runout in the milling

force calculations by dividing the end mill into a number of axial elements. Sutherland

et al. [24] and Armarego et al. [36] included the effect of cutter defiections on the chip

thickness calculations by using iterative algorithms.

The accuracy of the milled surfaces was modeled by Kline et al. [35] by calculating

the cutter and workpiece deflections at the surface generations points. Montgomery and

Altintas [37] nsed a dynamic cutting model to simulate the surface produced by a vi

brating end mill. They used true kinematics of milling presented by Martelotti. Ismail

et al. [38] included the effect of tool wear and tool dynamics on the surface generation.

In addition to the force and surface accuracy predictions, milling force models have been

used extensively in adaptive force control [39, 40, 41, 42, 43] and cutter breakage detec

tion [44, 45] of milling operations.

In this thesis, several milling force models are developed to analyze milling forces and

surface accuracy in milling flexible workpieces. The static interaction between the cutter

and workpiece deflections and the milling process is modeled in two ways. First, an

analytical milling force model is developed by formulating the chip thickness under the

effect of static cutter and workpiece deflections. It is shown that the effect of defiections


on the chip thickness diminishes very quickly (in a few tooth periods) for static milling.

Then, a flexible milling force model, which considers the effect of static deflections on

the cutter-workpiece immersion boundaries, is developed and used for peripheral milling

of plates. The model accurately predicts the milling forces and surface errors. It is

shown that if the deflections are not used to update the immersion boundaries (as done

by Kline et al. [35]), the predictions are not accurate, especially in peripheral milling

of very flexible cantilever plates. In order to generalize the milling force prediction for

different cutter geometries, an improved mechanics of milling method is developed. The

accuracy of the model predictions is found to be satisfactory. The model can be used in

milling cutter design, process planning and CAD/CAM systems.

2.4 Stability of Dynamic Milling

Both forced and self-excited vibrations arise in milling operations. Periodic milling

forces excite the cutter and workpiece, and may cause resonance. Self-excited chatter

vibrations occur due to dynamic interactions of the cutting process and structure. Forced

vibrations and chatter stability are particularly important in the peripheral milling of

thin-walled components due to very flexible workpiece and slender end mill. Generally,

there are two approaches used in the analysis of chatter in machining: since chatter is

undesirable, researchers establish stability limits and consider the cutting process as a

black box; the other approach is to understand the mechanics of the cutting process

under dynamic conditions. In the following sections, a brief review of the literature on

dynamic cutting and chatter stability is given.


h

Figure 2.1: Dynamic cutting process.

2.4.1 Dynamic Cutting

Dynamic orthogonal cutting process is shown in Figure 2.1. The tool is removing

chip from an undulated surface which was generated during the previous pass when the

tool’s vibration amplitude was z0 (outer modulation or wave removing). Simultaneously,

the tool is vibrating with amplitude z (inner modulation or wave generation). The pro

cess can be visualized as a superposition of these two distinct mechanisms. There has

been extensive research efforts towards the understanding and modeling of the dynamic

cutting process in 60s and 70s [46, 47, 48, 49, 50, 51, 52, 53]. An excellent review of the

related literature is given by Ilusty [54].

Different mechanisms have been proposed for the dynamic cutting process. Knight

[51] observed that the shear angle oscillates during dynamic cutting. Albrecht [46] con

sidered the oscillations in the shear angle to be a part of the chip segmentation (or cyclic

zoTOOL

x WORKPI ECE


chip formation) mechanism. The shear angle oscillation was considered to be the most

significant source of chatter in [55, 56, 7], as it results in oscillations in the cutting forces

and vice versa. The shear angle oscillation is attributed to the variations of the surface

slope (wave removing) and cutting velocity direction (wave cutting). The variation of

the rake angle due to a vibrating tool and changing cutting velocity direction can also be

considered to have negative damping effects on the dynamic cutting system [54]. Also,

the variation of the friction coefficient between the chip and the rake face of the tool with

the continuously varying cutting velocity in dynamic cutting may have a small effect on

the shear angle variation [49]. The effects of different parameters on the shear angle oscil

lation were investigated by Nigm and Sadek [57]. Their results show that the magnitude

of shear angle oscillation decreases as cutting speed, feed and rake angle increase. The

vibration amplitude does not have a significant effect on the shear angle oscillation which

slightly increases with the frequency of the chip thickness modulation [57]. In both wave

generation and wave removing processes, the clearance angle does not seem to have an

effect on the shear angle oscillation. However, the clearance angle has a strong effect

on the stability of the chatter vibrations. Sisson and Kegg [58] formulated the effective

clearance angle in the dynamic cutting by considering the cutting speed and vibration

velocity normal to the workpiece. The normal and horizontal flank force components

were calculated by assuming simple elastic contact between the flank face and the work

piece. These forces were treated as damping forces (process damping), and the damping

coefficient derived is inversely proportional to the cutting velocity. Sisson and Kegg [58]

could explain the high cutting stability at low speeds, but, by this model, they could not

explain the high speed stability. The mechanism behind the process damping is explained

by Tiusty [54]. In Figure 2.2, for a vibrating tool, the variation of the clearance angle

between the flank face and the cut surface is shown. In the middle of the downward slope

the clearance angle is at minimum, 7mm, and in the middle of the upward slope it is at


Figure 2.2: Variation of effective clearance angle in dynamic cutting.

maximum, 7mar• It has been shown that the decrease of clearance leads to an increase

of the thrust component of the cutting force. Therefore, during the half cycle from (A)

to (C) the normal force is greater than that during the upward motion from (C) to (D).

This variation of the thrust force is 900 out of phase with displacement, thus it represents

a positive damping in the cutting process. The damping coefficient is larger for short

waves as the slope is steeper. The wave length A of the undulations produced on the cut

surface is A = -, v is the cutting speed and f is the vibration frequency. The process

damping coefficients are usually determined empirically, however, there have been a few

attempts to formulate them analytically [58, 59]. In these methods, the definition of

Dynamic Cutting Force Coefficients (DCFC) are used mainly to determine how much

damping arises in the chip formation process. The effects of outer and inner modulations

(wavy surface and vibrating tool) can be superposed by DCFC and the corresponding

transfer function of the dynamic cutting process can be written as follows:

= a(Kdz + Ad0Z0)

= a(Kz+K0z0)

A

z

‘min


where F and F are the normal and tangential cutting forces, a is the width of cut, z

is the amplitude of the vibration normal to the cut surface, z0 is the amplitude of the

modulations on the surface (which is equal to the tool vibration in the previous pass)

and Kd, are direct and cross-inner modulation DCFCs and Kd0,K0 are direct and

cross DCFCs for outer modulation. Unlike the static cutting process, in dynamic cutting

the cutting force coefficients are complex numbers. The real components represent the

cutting stiffness whereas the imaginary components are due to the process damping gen

erated in the cutting process. Tremendous effort has been spent in the determination of

the effects of cutting parameters on DCFCs by using complicated test rigs, some of these

works are cited here [3, 47, 60, 61, 62, 63, 64, 52, 65]. In most of these works, DCFCs

were identified from the controlled dynamic cutting tests as the tool vibration amplitude

and frequency; cutting speed and the tool geometry were varied. The results of these

tests [54] show that the imaginary component of Kd is the largest damping term and

has a special effect on the dynamic cutting process. This also indicates that the process

damping is generated due to the flank contact resulting from vibrations of the tool.

is strongly affected by the cutting speed and wearland on the flank face.

Periodic cutting forces can cause forced vibrations in milling systems. The Fourier

analysis of the milling forces was done by Gygax [66] and Yellowley [67]. Doolan et al.

[68] designed the optimal pitch angles between the milling cutter teeth to minimize forced

vibrations. Tlusty [69] and Smith [70] used process damping in the modeling of dynamic

milling forces. Montgomery and Altintas [37] developed a comprehensive dynamic milling

model by considering the contact between cutter and workpiece in different zones.


2.4.2 Chatter Stability Models

In his classical paper, On the Art of Cutting Metals (1907) [1], F.W. Taylor states

the following opinion which is based on the experimental observations:

“Chatter is the most obscure and delicate of all problems facing the machinist, and in

the case of castings and forgings of miscellaneous shapes probably no rules or formulae

can be devised which will accurately guide the machinist in taking the maximum cuts and

speeds possible without producing chatter.”

Taylor was partly right in that the first comprehensive treatise on the mechanism of

cutting tool vibration by Arnold [71] appeared four decades later than his paper. Arnold

explained a theory of self- induced vibration based on the decrease in the cutting force

with cutting speed. If the force-speed curve exhibits a negative slope, this implies a nega

tive damping coefficient in the equation of motion and may lead to instability. Hahn [72],

however, pointed out that in general the slope of the force-speed curve is not sufficiently

steep to explain for the self-induced vibration.

In the early stage of the machining chatter research, the existence of negative damping

was considered a necessary condition, and the only source, for chatter to occur. However,

it was later recognized that the most powerful sources of self-excitation, regeneration

and mode coupling, are associated with the structural dynamics of the machine tool

and the feedback between the subsequent cuts. Tlusty and Polacek [73] showed the

importance of the structural dynamics by modeling the machine tool system as a multi

degree-of-freedom structure with positional mode coupling. They analyzed the stability of

mode coupling and regenerative chatter mechanisms and obtained the following classical


equation for the regenerative chatter stability

1aiim

= 2KsR[Gjmin

where aiim is the limit width of cut for the chatter stability, K is the specific cutting force

coefficient and Re[G]mjn is the minimum value of the real part of the structure’s transfer

function, oriented with respect to the cutting force and to the direction of the normal

to the cut surface. Later Tlusty [5] improved this formula to include the lobing effect

by considering the effect of the spindle speed on the chatter frequency. Tobias and Fish-

wick [74] combined the two aspects of the chatter vibrations: the process damping and

dynamics of the machine tool structure. They developed a comprehensive mathematical

theory of chatter of lathe tools by taking into consideration the instantaneous variation

in chip thickness, the penetration effect, and the slope of the cutting force-cutting speed

curve. Merrit used the feedback control theory to develop stability lobe diagrams for

regenerative chatter. Similar to Tiusty, he also neglected the dynamics of the cutting

process. The effect of cutting dynamics on the chatter stability is strong in the slow

cutting speed range where the process damping is high.

Due to the rotating cutter with multiple teeth and the periodically varying direc

tional coefficients, the dynamics and the stability of milling are more complicated than

the orthogonal cutting case. That is why, in the beginning, the stability of chatter in

milling was analyzed using the orthogonal cutting-chatter theory [5, 3]. Later Tlusty et

al. [75, 76, 77] concluded that the time domain simulation of the milling chatter is the

best method to obtain the stability diagrams. Sridhar et al. [78, 79, 80] formulated the

milling dynamics for the straight tooth cutter, and used a numerical algorithm to analyze

the stability of dynamic milling. Minis et al. [81, 82] used Nyquist criterion to analyze

the stability of the milling process and followed a numerical procedure to obtain the


stability limits. There is no analytical method of chatter stability prediction in milling

available in the literature.

Various active and passive chatter suppression methods have been developed. In or

der to prevent the full development of chatter, the phase between the inner and outer

modulation can be disturbed by using variable pitch milling cutters [83, 84] or variable

spindle speeds [85, 86, 87]. Smith [88] used the chatter sound spectrum for the on-line

selection of the spindle speed to utilize the high stability pockets in the stability lobes.

Nachtigal and Srinivasan [89, 90], Shiraishi and Kume [91] and Liu [92] developed feed

back controllers for the control of chatter in turning. Vibration dampers have also been

used in chatter suppression [3, 93].

In this thesis, a comprehensive dynamic milling model is developed by considering

the dynamic displacements of the workpiece and cutter. Unlike the point contact models

considered in the previous milling chatter research [3, 5, 78, 94, 81], the dynamic interac

tion between the cutter and workpiece is modeled along the axial direction by considering

the variations in the dynamics of structures in this direction. A novel stability analysis,

which is based on the physics of dynamic milling, is given. The resulting stability equa

tions are solved analytically by obtaining a relationship between chatter frequency and

spindle speed. The general theory is applied to several common milling cases such as the

peripheral milling of flexible workpieces. Analytical stability conditions are derived for

each case. The analytical solutions are verified by the time domain simulation results.


2.5 Peripheral Milling of Flexible Structures

In end milling operations, a flexible structure may be defined as a workpiece whose

flexibility is significant as compared to the cutter and machine tool. The peripheral

milling of flexible components is a commonly practiced machining operation, especially

in the aerospace industry [95, 96], used in: machining of thin webs, jet engine components

(such as impeller or blisk blades), instrument housings, microwave guides etc.

Kline [35] considered milling of a clamped-clamped-clamped-free (CCCF) plate with

a flexible end mill. He used the Finite Element Method to model the plate. The inter

action between the milling forces and the static and dynamic structural displacements

were neglected in his study. Therefore, the model can only be used for the static milling

of relatively rigid workpieces. Montgomery [97] and Altintas et al. [98] modeled the

dynamic peripheral milling of a plate by employing the true kinematics of the milling

which is trochoidal. The end mill was assumed to be rigid and the plate was modeled by

the Finite Element (FE) method. The dynamic milling forces and the detailed surface

finish were obtained by considering the dynamic regeneration in the chip thickness. Due

to the fact that the plate displacements were obtained by a FE package, off-line with

the developed computer program, and long computer run times, the process is simulated

only at the middle of the plate, along the feed direction. Sagherian et al. [99] improved

Kline’s model by including the dynamic milling forces and the regeneration mechanisms.

However, they did not consider the effect of tool and workpiece deflections on the cutting

geometry, i.e. the radial depth of cut. They also used a numerical force algorithm and the

FE method to simulate cantilever plate displacements. Anjanappa et al. [100] showed

the imprints of high frequency vibrations on thin rib milling.


As a summary, the peripheral milling of plates has been modeled by numerical al

gorithms. The dynamic regeneration mechanism has been included in some of these

models, however, the complete static and dynamic interaction of the milling process and

structural displacements have not been investigated. Also, the stability of milling very

flexible workpieces has not been studied.

In this thesis, the interaction between the flexible plate and cutter is accurately mod

eled. The variation of the immersion boundaries under the deflections is considered.

Finish dimensions of the plate are accurately predicted by an integrated Finite Elements

cutting process model. In order to achieve prescribed tolerances, the peripheral milling

of plates is planned by scheduling feedrates. The simulation system is experimentally

verified. Also, a novel stability model of milling multi degree-of-freedom systems is de

veloped. The model is applied to the peripheral milling of plates and is verified.

Chapter 3

Structural Modeling of the Workpiece and End Mill

3.1 Introduction

Flexible cutter and workpiece structures deflect and vibrate under the milling forces.

If not controlled, static deflections cause dimensional surface errors which may violate the

tolerance requirements on the machined surfaces. The cutter and workpiece vibrations,

on the other hand, result in poor surface quality, chipping of the cutting edges, micro

cracks on the finished surface and may even damage the machine tool if the self excited

chatter vibrations become excessive. The effect of structural deformations on the cutting

process needs to be investigated for the prediction of cutting conditions which result in

the required dimensional accuracy in milling flexible structures. Here, structural mod

els of the flexible workpiece and tool are studied to analyze the milling of very flexible

workpieces.

The flexible workpiece is modeled as a cantilever plate since it represents the most

extreme case for the family of very flexible aerospace components, such as the impeller

and rotor blades. Also, due to its very high flexibility, the cantilever plate is a very good

choice for analyzing the interaction betweell structural deformations and the cutting pro

cess. The thickness and the structural properties of the plate continuously vary due to the

removal of metal during machinillg. Because of the continuous variation of the structural

21

Chapter 3. Structural Modeling of the Workpiece and End Mill 22

properties, and the interaction between the milling process and the structural deforma

tions, the structural and milling force calculations have to be performed together, in the

same computer program. Previously, Kline et al. [35] used a commercial Finite Elements

(FE) software to calculate the workpiece (clamped-clamped- clamped-free, CCCF, plate)

defiections under the milling forces by neglecting the interactions between the deflections

and the milling forces. Altintas et al. [98] used a commercial FE package too in studying

the dynamic milling of a cantilevered plate at the middle section only, so that they did

not have to consider the variation in the plate dynamics. However, they considered the

true kinematics of milling to predict dynamic deformation marks left on the plate surface.

Kline et al. [35] and Sutherland et al. [24] used a beam model for the tool deflections

by defining an equivalent length to account for the clamping stiffness at the collet. In

this study, the plate and tool deformations and the milling forces are calculated in the

integrated algorithms developed for plate milling. A beam model with linear springs at

the fixed end (to account for the collet stiffness) is used for the deflection analysis of the

end mill. This model closely represents the real structure and gives satisfactory results.

In the following, dynamic and static models of the plate and tool are given. These models

are used in Chapters 5,6,7 and 8 to analyze the static and dynamic interactions between

the milling process and the structures, to predict and control dimensional surface error

on the plates, and to study the stability of self excited chatter vibrations.

3.2 Structural Modeling of Workpiece

The workpiece considered in this study is a cantilever (clamped-free-free-free CFFF)

plate. The thickness of the plate is reduced continuously during machining. There

are a number of methods in the literature for the deflection analysis of plates [101,

102]. The accuracies of different methods in the analysis of plate problems depend on


the type of boundary conditions, loading, homogeneity of the thickness and magnitudes

of the deflections considered. There exist exact analytical solutions for some special

boundary conditions and loading types. Unfortunately, there is no closed form solution

for cantilever plate problems. In the following, Plate Theory will be briefly reviewed.

Series solutions for the cantilever plate deflection are formulated. Due to the stepped

thickness and nonuniform loading during milling of the plates, the series methods do

not provide accurate results. Thus, the Finite Elements Method (FEM) is used in the

deflection analysis of the stepped plate. For the constant thickness plate, however, series

solutions may give acceptable results. This applies to the negligible step size which is the

case in some finishing operations. For these cases, the series solution may be preferred, as

it is faster than the FEM. On the other hand, the FE can be used for different boundary

conditions and workpiece geometries and provide accurate results for different types of

loading. In the following sections, the FE formulation and the developed algorithms are

explained. The dynamic analysis of the plate structure is performed by the FEM as well.

The developed models are experimentally verified before being used for milling process

simulation.

3.2.1 Bending of Thin Plates

The basic theory of thin plate bending, so-called Kirchhoff/Love Theory, will be

reviewed. In this theory, it is assumed that the points on the mid surface of the plate

move only in the z direction as the plate deforms in bending, and a line that is straight

and normal to the mid surface before loading is assumed to remain straight and normal

to the mid surface after loading. The theory is applicable to the cases where the plate

deflections (w) are less than half-plate thickness (t/2).

Consider the differential plate element shown in Figure 3.1. In order to be consistent


Mx+MxxdXdy

-

/ M÷Mxy,xdx

Pr

Figure 3.1: Internal forces and moments on a differential plate element.

with the notation used in the plate literature, the z axis will be taken along the thickness

of the plate. The moments, Mr, M and the shear forces, q and qy, their differential

variations and the load distribution in z direction, p(x, y), are shown on the element. If

the plate deformations are small (w < t/2) and if there are no loads at the boundaries

in the plane of the plate (x—

y plane), then the in-plane forces can be neglected. The

transverse shear deformation is assumed to he zero. The first step in the solution is the

moment and force equilibrium equations,

OM 81VI+

pg

PM 8M— (3 1—q

= —p(x,y)

M M

x,u

I Ip(x,y)

s÷

y,vMdy


Undeflected plate

. X

Deflected plate

Figure 3.2: Displacement of a point P to F’ due to bending of plate.

from which the following can be obtained:

82MX 321tJ 82M+ 2

DxDy+

02= —p(x, y) (3.2)

Figure 3.2 shows the displacement of a point F, which is not on the middle plane due

to the bending of the plate. u0(x, y) , v0(x, y) and w(x, y) are the displacements of the

middle surface in x, y and z directions, respectively. Assuming that the straight lines

normal to the middle surface of the plate before bending remain straight and normal after

bending, (u(x, y) and v(x, y)), the displacements of the point F in x and y directions,

can be obtained as follows

U = U0—ZW(3.3)

V = Vo—ZWy

where w = and w = 8?t. From the equations of the linear elasticity,

9v lOu Ov

w(x,y)

uo-z AX

(3.4)


assuming that the mid-plane does not stretch, - and - terms drop out and the

following is obtained:

= U3: = — ZtIisx

Ey = Vy = ZWyy (3.5)

— —r V3:) — —ZW3:

The stress-strain equations reduce to the following form for plane stress (o = 0) and

isotropic, homogeneous and linear elastic material,

= E + vo

= e,,E + vo (3.6)

E—

where v is the Poisson’s ratio and E is Young’s Modulus of Elasticity. If the strains given

by equation (3.5) are substituted in the stress-strain relations given by equation (3.6) the


0•3:

o.y (3.7)

o.z,y

The moment expressions can be obtained by integrations of the stresses along the

thickness of the plate,t/2

M3: Jaxzdz

M= £/2

uzdz (3.8)t/2

M3: I uzdzJ—t/2

from which the following is obtained:

M3: w + vwEt3

‘‘ 12(1—i,2)(3.9)

IvI (1 —


where t is the thickness of the plate. The following governing equation of plate bending

is obtained if the moments given by equation (3.8) are substituted in the equilibrium

equation (3.2),84w 94w ö4w—+2 +-—=- (3.10)8x4 0x28y2 9y4 D

or

= (3.11)

where .cç7” is the biharmonic operator and D is the flexural rigidity and given by

D= 12(1_v2)

(3.12)

The solution of equation (3.10) gives the deflection of the plate, w, under the specified

load, p. The boundary conditions are listed below:

1. Fixed (clamped) edge : Displacement and slope are zero at the edge, i.e. w = 0

and = 0 on boundary C, where n is normal to C.

2. Simple supported edge: Displacement and moment are zero, i.e. w = 0 and

M = —D(w + vw33) = 0, where s is tangent to boundary C.

3. Free edge: Moment and combined-shear are zero, i.e.

M = 0 and q = —D + (2 — v)ww33)= 0.

The exact solution of equation (3.10) exists for a few boundary conditions and

loading cases. As an example, consider a simple supported plate with dimensions (axb)

and sinusoidally varying loading p = Po sin sin A solution of the form w =

w0 sin sin -, which satisfies all the boundary conditions, is assumed and substituted

in equation (3.10) to determine the unknown coefficient

Powo— 1 1Dir4(-+)2


This result can be used to find a solution for different loads on a simple supported

rectangular plate. The method is called the Navier solution in which the load is expressed

by a double Fourier sine series, i.e.

p(x,y) = pmnsinmSi11n (3.13)m=1n=1 a

where4 a b

. irx. ryPmn = —J J p(x,y)sinm—sinn----dxdy (3.14)

ab x=O y=O a b

Then, the solution is given by

00 00Pmn . 7rx . Ky

w(x,y) = 2 2 sin rn—sinn--- (3.15)m=1n=1DW4(m+’)2

a

Another special case is the plate with two opposite sides simple supported and any

combination of boundary conditions on the other sides. For this case, the Levy method

is used for the solution. In this method, the homogeneous and the particular solutions of

equation (3.10) are determined separately. The unknown coefficients in the homogeneous

solution are determined from the boundary conditions. These methods cannot be used

in the analysis of the cantilever plate. The numerical solutions for the cantilever plate

are explained next.

3.2.2 Series Solutions for Cantilever Plate

The numerical methods for the plate deflection analysis discussed in this section are

based on a series type solution. Their accuracy depend on boundary conditions, type

of loading and the approximating functions used in the series. In the following, the

• Galerkin and Ritz formulations are given for the cantilever plate. Although it is shown

that accurate results cannot be obtained for the stepped plate by the series solutions,

the accuracy is acceptable for the constant thickness plate. This may be an acceptable


approximation for some finish milling operations where the radial depth of cut is very

small.

The Galerkin Formulation for the Cantilever Plate

The general form of the approximate solution is in the form of

N

=c1X(x)Y(y) (3.16)

where N is the total number of terms considered in the series, X(x) and Y(y) are admissi

ble functions for plate defiections. Q sign indicates that the solution is an approximation.

The Galerkin formulation requires that at least the kinematic boundary conditions be

satisfied by the approximation functions [103]. The order of the kinematic boundary

conditions, m, is determined by the order of the original differential equation (n), i.e.

m = n/2 —1. In the case of the plate problem, n = 4 thus m = 1. Therefore, the approx

imation functions should satisfy the displacement and the slope boundary conditions. In

the Galerkin weighted residual statement, the unknown coefficients, c, are determined

such that the total domain and boundary weighted error introduced by the approximate

solution is minimum. This is stated by the following weighted residual statement for a

plate with dimensions (axb):

jajbRWdXdY+JRW0 j=1,2,...,N. (3.17)

where RD and RB are the domain and boundary residuals, Wj and W3 are the domain

and boundary weighting functions.The displacement and the slope should be equal to

zero at the clamped edge

th =0(3.18)


The following zero moment and zero equivalent-shear stress boundary conditions ap

ply to the free edges

(w + = 0

(3.19)

D(w + (2— = 0x=O, a

Assume that the loading due to the cutting force in the z axis1 is represented by a

line force applied at the tool position, x =

p(x, y) = F(y)6(x — x1) (3.20)

where 6(x) is the Kronecker delta function which specifies the location of the line force

and the F(y) is the normal cutting force distribution along the y direction of the plate.

The domain residual becomes

RD = D 4 t — F(y)6(x — x1) (3.21)

and the boundary residuals evaluated at the boundaries:

RBr —D + (2— + +

x=O,a (3.22)— D (ti + v) + (z + (2

— v)th)y=b

The residuals can be substituted in equation (3.17) to determine the coefficients, c.

This formulation was followed and the resultant set of equations were solved by computer.

Free-free and cantilever beam eigenfunctions (vibration mode shapes) were used for the

11n order to be consistent with the plate formulation, the normal axis was chosen to be z. In fact,the cutting force in this direction is F as defined in the cutting force model.


Table 3.1: Eigenfunction parameters for clamped-free and free-free beams.

rn am m m1 - 0.734 - 1.8752 - 1.019 - 4.6943 0.982 1.0 4.73 7.8564 1.0 1.0 7.853 11.05 1.0 1.0 11.0 14.1376 1.0 1.0 14.137 (2m — 1)K/2

approximation functions, X and Y, respectively [104]:

Xi =1

X2() = 1—2w

Xm() = cosh EmX + cos mX(3 23)

— am(sinh + sin + sin m) (m = 3,4, ...)Ym(V) = coshm—cosm

— (m=1,2....)

where = and = . The constants in equation (3.23) are given in Table 3.1 for the

first six modes.

In the Galerkin method, the weighting functions are chosen to be the approximation

functions themselves, i.e. Wj = However, the resulting weighted error expressions

should be consistent from the energy point of view. The boundary residuals in equation

(3.21) include both shear force and moment residuals. Therefore, the moment should

be multiplied by the rotational displacements or slopes, w and w. When it is done

so, the formulation becomes exactly the same as the Ritz energy method. The case of

the stepped plate will be discussed before the Ritz formulation is given. One possible

solution for the stepped plate is to consider it as a combination of two plates with different


thicknesses. Thus, two different domains with two different solution functions, say w1

and w2, have to be considered in the formulation. These two solution functions should

satisfy the continuity equations at x = x1. The first two simple continuity requirements

are the continuity of displacement and slope:

= w2(x=x1)(3.24)

wi(x = x1) = = x1)

These are kinematic conditions for the plate deflection formulation, so they have to be

satisfied by the approximating functions before a residual statement can be written.

However, it is not a straightforward task, if it is not impossible, to determine the type

of the functions that would satisfy these conditions. Also, the solution procedure with

additional continuity residuals becomes very lengthy. Thus, the solution of the stepped

plate problem with the Galerkin formulation is not practical and it is dealt with the

Finite Element method.

Ritz Energy Method

The strain energy stored during the bending of a plate is given by [101]:

Dabu = j j [(w + w)2 — 2(1 — v)(ww — w)] dxdy (3.25)

The work done on the plate by the external force, p, is

= jafp(xy)wdxdy (3.26)

The potential energy of a system, H, is defined as the total internal and external

work done in changing the configuration from a reference state to the displaced state.

Then, it is given by

ll,=U+V (3.27)


The approximate solution given by equation (3.16) is to be used in the energy state

ments. Then, the coefficients, c, are determined by using the principle of stationary

potential energy which states that [105]:

“Among all admissible configurations of a conservative system, those that

satisfy the equations of equilibrium make the potential energy stationary with

respect to small admissible variations of displacement.”

Therefore, each approximation function, X and , must be admissible; that is, each

must satisfy compatibility conditions and essential boundary conditions. Then, according

to the principle of stationary potential energy, the equilibrium configuration is defined

by the N algebraic equations

9II 9U 8V=—+=0 (n=1,2 ,N) (3.28)a

Substituting the series solution form given by equation (3.16) and the line force applied

at x = x1 defined by equation (3.20) into equations (3.25-3.26) the following is obtained:

= D ja [ + xiYi”xjYj,,

+ 2vX”YX1” + 2(1 — v)XY’X’] dxdy

(3.29)

=

_j0:j1cixi

= —jF(y)cX(xi)1dy

j=1

where (F) indicates differentiation with respect to x or y. The following matrix equation

is obtained when U and V are substituted in equation (3.28):

[k] {c} = {f} (3.30)


where the elements of the stiffness matrix, [k] and the force vector f are defined as

— Db f j X1”X7YiYj + r”1”XX3

+ 2r(1 — v)X i’’XY’ + vr (x’YxY” + xi”x”) J

f = X(1)jF(y)Yjd

where ra = b/a is the aspect ratio of plate, and = x/a,g = y/a and = xi/a. In

order to make the integrals in equation (3.31) independent of the plate dimensions, the

differentiations indicated by (i) are performed with respect to and . The cantilever and

free-free beam vibration modes given by equation (3.23) are used for the approximating

functions X and Y. In both directions, the first six modes are included in the series. After

the integrals are evaluated numerically, the matrix equation (3.30) is solved to obtain the

coefficient vector {c}. The coefficients and the approximating functions are then used in

equation (3.16) to determine the displacements of the plate at different locations. This

is programmed on computer and the program listing is given in the internal report [106].

Alternative extended Simpson’s rule [107] (page 115) is used to evaluate the numerical

integrals. The force F(y) should be known at the integration points along the y axis to

determine the force vector. However, if the force is a point load or uniform loading, then

F(y) moves outside of the integration.

3.2.3 The Finite Element Modeling of Plate Deflections

The Finite Element method (FEM) is superior to the other presented numerical plate

deflection solutions due to several reasons. First of all, different workpiece geometries

can be modeled by the FEM. In addition, the variable workpiece geometries, such as a

stepped plate, can easily be handled by the FEM. In this section, the FEM formulations

are given for the 3 dimensional (3D)-isoparametric solid element and the structure of the


developed computer program is explained.

Structural Formulations for 3D Elastic Solid

Because of the stepped thickness the workpiece is modeled as a 3D elastic structure.

The stiffness matrix and the force vector for the 3D structure are given by [105]:

[K] =

(3.32)

{Q} = j[N]TFdV+{F}

where the structure is assumed to be free of prescribed surface tractions and

[K] stiffness matrix of the structure

[B] strain-displacement matrix for 3D elastic body

{ u} 3D stress vector

{ Q} consistent load vector

[N] matrix of shape functions

{F} body forces

{ P} concentrated forces on the nodes

[E] elasticity matrix

(1—i’) 1’ 0 0 0

xi (1—v) v 0 0 0

xi ii (1—v) 0 0 0[E] = E’ (3.33)

0 0 0 1—2v 0 0

0 0 0 0 1—2xi 0

0 0 0 0 0 1—2v


where

= (1 + v)(l — 2v)(3.34)

E and v are the modulus of elasticity and the Poisson’s ratio. The strain-displacement

matrix [B] relates the strains, {e}, to the displacements of the structure, {z.}

{} = [B]{} (3.35)

wherea 0 0

n 0U

000[B]= a a (3.36)

7xy

a a7yz 0 — y—

a a7zx ‘T U 7Taz ax

U

{z} = (3.37)

z

where u, v and w are the displacements in x, y and z directions. As [B] includes only

first order derivatives, the shape functions should have C° continuity2.This requirement

is satisfied by the linear shape functions. The convergent rate is determined from the

following expression [105]:

CR 1/2(P+l_m/2) (3.38)

where p is the order of the polynomial used for shape functions, m is the order of the inte

grand in the definition of the stiffness matrix given by equation (3.32) (in = 2 for elastic

solid) and n is the number of finite elements. For linear shape functions the convergence

2Jj general, cr continuity means that the shape functions have continuous r1 order derivatives


rate is 1/n2. In FEM, the shape functions N are used to express the displacements

(u, v, w) in terms of the nodal displacements, {S}. The linear 3D solid element has 8

nodes, thus 8 shape functions. The same shape functions are used in three directions:

ENjuiN1 0 0 N2 ... N8 0 0

{z}= = 0 N1 0 0 N2 ... N8 0 {6} (3.39)

8 ri A A A 7T> U U LV1 U U 1V2 •.. tV8N1w

where {5T}= (u1,v1,w1 , U8,v8, wg). From equations (3.35, 3.36) and (3.39) the

following expression is derived for the matrix [B]:

ON1 ,- ON ON8V V ...

o ON1 0 0 ON2 ON8 0--

ON1 ON ON8ri

— 0 0-—

0 040

— ON1 ON1 ON8 ON8---- V ... ----

ON1 ON1 0 0 ON8 ON8---- ... ----

Element Formulation

The 3D isoparametric element (8 node-brick) is shown in Figure 3.3. Unlike the

3D cubic element, the isoparametric element can have different edge lengths and angles

between the edges. This is, in general, necessary to model three dimensional geome

tries such asa stepped plate, a variable thickness or a variable length plate, without

increasing the number of elements significantly. The same element can also be used

for curved geometries like impeller blades. As it was discussed in the previous section,

an 8 node-3D solid element (also referred to as trilinear isoparametric element or 3D

Chapter 3. Structural Modeling of the Workpiece and End Mill

4

1

Figure 3.3: 3D isoparametric solid element.

x

38

isoparametric-linear tetrahedron) is the simplest 3D element which satisfies the continu

ity requirements. It is simpler to formulate and faster to compute compared to higher

order elements like a quadratic solid element. The quadratic solid element has 20 nodes

(corner nodes and a node on each edge), and a stiffness matrix of (60x60) whereas a

linear-8 node solid element has a (24x24) elemental stiffness matrix. On the other hand,

the convergence rate with the quadratic element is 1/n4, which is obtained at the

cost of longer elemental computations (and formulations). In the developed computer

program, the 8 node-isoparametric solid element was used to model the plate deflections.

In the following, the formulations and the program algorithm will be given for this type

of element. However, the formulation of quadratic or higher elements are quite similar,

and can easily be implemented in the computer program.

y3

7

6

z5

Mathematically, it is very difficult to deal with the irregular element shape shown in


4

1

Figure 3.4: The 3D solid element in the natural coordillates.

Figure 3.3. Thus, another coordinate system is introduced, natural coordinates (, i, ),in which the element looks like a cube, as shown in Figure 3.4. Two coordinate systems

are related to each other by mapping. In isoparametric mapping the global coordinates

(x, y, z) are related to the natural coordinates as follows:

x =xN1(,C) , y = yN(,C) , z = zN(,C) (3.41)

where (xi, y, z) are the nodal coordinates in the global coordinate system. The shape

functions (Ni) are used to define both the coordinates aild the displacements inside the

element, that is why this element is called isoparametric. The serendipity shape functions

[105] are used in the natural coordinates where (—1 > , , < 1):

11

37

C


N1 =

N2 =

N3

N4 (1-e)(1+)(1+C)(342)

N5 =

N6 =

N7 =

N8 =

A shape function N has a unity value oniy at the node i, and zero at the other nodes

of the element. The stiffness matrix defined by equation (3.32) should be calculated in

the natural coordinates as follows:

[Ke] = Je[B1T[E ]dxdydz= I L11 f[B]T[Ej[B}det[J]dddC (3.43)

where [Ke] is the elemental stiffness matrix and [J] is the Jacobian of the transformation

which will be defined in the following formulation. The determinant of [J] can be regarded

as a scale factor that yields volume in the above integral from dxdydz to ddiid [105].

The derivatives of the shape functions with respect to the global coordinates, etc.)

are required to calculate the element stiffness matrix, i.e. in the strain- displacement

matrix [B] (equation 3.40). The derivatives in the natural coordinates can be written as

follows by using the calculus chain rule:

a aa _[i a (344

a a


ôx ô?;

rjl_ öx ôy OzLi

Ox 8?;! Oz

The derivatives of the global coordinates with respect to the natural ones can be

obtained from equation (3.41):

88 8N 88 8N

8 8N 8 0N

8 8N 8 8N

8

8 —1L1

8 8

where [L] = [J]’. Then, the derivatives of the shape functions in the global coordinates

8N--

=[L]

8N--

It should be noted that the derivatives of the shape functions in the natural coordi

nates can be obtained analytically,

ON1= -(1 - )(1 + C); = -(1 - )(1 + ()

where [J] is the Jacobian matrix of the transformation and is given as:

(3.45)

8 IiT.Oz —

Equation (3.44) can be inverted to obtain the following:

(3.46)

(3.47)

are determined as follows:8N--

8N--

(3.48)


and the others are similar. The Jacobian [J], however, has to be inverted numerically.

Finally, the elemental stiffness matrix can be obtained by integrating equation (3.43)

numerically using the Gauss quadrature:

,1 r1 tl

j J J [B]T[E][B]det[J]dd?]d(—1 —1 —1

(3.49)m m m

= YEHiHjHk ([B]T[E][B]det[J])ki=lj=lk=1 ‘

where m is the number of sampling points, H are the weighting for the i point and

(i, ‘ii C) is the coordinate of the sampling point. In the above expression, det[J] and

[B] have to be calculated at the sampling points. The Gauss quadrature is used in the

above integration, as a polynomial of degree (2m — 1) is integrated exactly by rn-point

Gauss quadrature. In FEM, usually the degree of det[J] is considered in the selection

of the number of sampling points for the Gauss quadrature. This is the minimum order

requirement for convergence. The idea is that as the elements become very small, they

only have to represent constant stress, so [B]T[E] [B] goes outside of the integration.

det[J] has quadratic terms in the natural coordinates; therefore, 2 point integration in

each direction is necessary. In three dimensions, 2x2x2 integration is performed with

integration points F(±1/v”, +1/’, +1/i/) and weighting of H = 1.0. The procedure

to obtain the elemental stiffness matrix can be summarized as follows:

• Calculate the Jacobian [J] given by equation (3.45) at the Gauss integration points

Pi.

• Invert [Jj numerically to obtain [L].

• Calculate the strain-displacement matrix [B] (equation 3.40) at P.

• Calculate the elemental stiffness matrix:


[Ke]= ( [BIT[E] [Bidet [J] )

This element has been observed to be too stiff in bending. This is due to the parasitic

shear [105] which is created because of the linear shape functions. This can be improved

by adding the missing modes as internal freedoms [105]:

+ (1 —2)a1+ (1— 2)a2 + (1—

v = Nv+ (1 —2)a4+(1 —2)a5+(1 —2)a6 (3.50)

w = ENw + (1—2)a7+ (1—2)as+ (1—

where the a, are the nodeless degree of freedom. The following elemental equation is

obtained with the inclusion of internal freedoms:

[Kel(24,24) [1(eal(24,9) = {Q}(24)(3.51)

[Kae](9,24) [Ka](gg) I {a}9 J I {0} Jwhere {a} is the nodeless degrees of freedom, and the matrices [Ka], [Keel and [Keel

contain the quadratic terms. The size of the new strain-displacement matrix with the

quadratic terms is (6x33). The nodeless degree of freedom, {a} can be eliminated from

the above equation by static condensation [105]. The resultant matrix equation is as

follows:

[K]{S} = {Q} (3.52)

where the new elemental stiffness matrix is given by

[K] = [[Ke] — [Keal[Ka]’[Kaei] (3.53)


The new element with the quadratic terms gives more accurate results.

Finite Element Program

The formulation of the elemental stiffness matrix has been given. The stiffness ma

trix is the most important part of the FEM, however there are many other functions

in the Finite Element program developed. In this section, description of the different

subroutines in the developed program will be given.

The program flow chart is shown in Figure 3.5. The input data is read from a file

which contains the plate geometry, number of elements in each direction, material con

stants, location of the cutter along the feeding direction and the cutting conditions. Each

subroutine will be briefly described in the following.

MESH: Generates the finite element grid data, i.e. coordinates of the nodes and the

node numbers for elements. It was specifically written for milling of cantilever plates. It

reads the plate dimensions, the number of elements ill each direction, the location of the

cutter and the radial depth of cut from the input file. An uniform mesh is generated in

x and y directions, i.e. the element size is constant which is equal to the length of the

plate in this direction divided by the number of elements. In the z direction, however,

the thickness of the elements are reduced in the cut portion of the plate, which is deter

mined from the location of the cutter. The variable plate thickness can be handled. This

subroutine can be modified to model plates with different boundary conditions.

LAYOUT:Reads all finite element grid data.

STIFF: Generates the stiffness matrix for a 3D isoparametric-linear tetrahedron.


Figure 3.5: Flow chart of the developed Finite Element Program.


Calls a number of subroutines to evaluate the derivatives of the shape functions, the Ja

cobian matrix and its inverse and the stress-displacement matrix. The improved stiffness

matrix given by equation (3.53) is returned to the main program.

SETUP: Places elemental stiffness matrix in the correct position in the total global

stiffness vector. Due to the large size and symmetric-banded nature of the global stiff

ness matrix, only the terms inside the half-band are placed in a vector which makes the

storage possible.

DISCRD: Eliminates the elements from the global stiffness vector for homogeneous

boundary conditions.

DFBAND: Solves the system of linear equations to obtain the displacements. The

force vector contains the lumped forces on the relevant nodes in the cutting zone which

is explained in detail in Chapter 7.

EXPAND: Expands the solution vector back to gross size by placing zeros where

variables were eliminated by DISCRD.

3.2.4 Dynamics of Cantilever Plate

The dynamic response characteristics of the plate are necessary in the analysis of

forced and self excited chatter vibrations. For that, natural modes of the plate should be

determined. Although for relatively rigid workpieces the dynamic modes of the machine

tool have to be studied (usually by experimental modal analysis) for the chatter predic

tion, due to the very high flexibility of the plate, the milling machine table on which the

plate is mounted can be assumed to be rigid. Cantilever plate vibration modes can be


determined by series methods [104]. The vibration modes of stepped plates have been

considered in many studies [108, 109, 110, 111, 112]. However, as the FE formulation is

already used for the static deformations of the plate, dynamic modes can be calculated

by the same method as well. Also, vibration modes of plates with arbitrary dimensions,

e.g. varying thickness and length, milled only certain portion of the length (due to chat

ter limitations) etc., can easily be determined by the 3D element based FE analysis.

Therefore, the mass matrix by FE formulation is given next.

Finite Element Model for Plate Dynamics

The mass matrix for an elastic structure is given by [105]:

[MJ= Jp[N]T[N]dV (3.54)

where [N] is the shape function matrix and p is the mass density. Similar to the stiff

ness matrix formulations, the elemental mass matrix can be integrated in the natural

coordinates by using the Gauss quadrature:

[Me]= ffj p[N]T[N]det[J]dddC = (p[N]T[N]det[J]) (3.55)

where P are the Gauss points. The procedure in the program is the same as for the

stiffness matrix. Again, a vector is used to store the global mass matrix which is banded.

A library subroutine which uses Power’s iteration method is used to determine the eigen

vectors (mode shapes) [q] and eigenvalues {w2} of the structure. The mode shapes can

be normalized to obtain a unity modal mass which is useful in modal analysis:

{qr} = {r}/mr

(3.56)

mr = {r}T[M]{qr}


iN

1 2

Figure 3.6: One element example to test the developed FE Program.

where {q.} is the rth mode shape. The modes of the plate must be updated as the milling

cutter advances in the feeding direction and the thickness of the plate is reduced.

3.2.5 Simulation and Experimental Examples

Several simulation and experimental results are presented for static and dynamic

plate displacements. The developed FE program has been tested for different cases. For

that, the results obtained from the program were compared with the ones obtained from

ANSYS. A one-element example is given here. Consider the cubic geometry shown in

Figure 3.6. The material is steel with E = 207 GPa, ii = 0.3. Two (iN) nodal forces are

applied at nodes 3 and 4 as shown in the figure. The displacements of the nodes in the z

direction as obtained from the developed FE program (by using the trilinear stiff element

and the improved 3D element with internal-nodeless degrees of freedom) and ANSYS are

given below in (m):


Node FE (trilinear) FE (improved) ANSYS

1 0.581395D-09 0.72096311-09 0.72096251D-09

2 0.581395D-09 0.72096311-09 0.72096251D-09

3 0.78249311-09 0.949195D-09 0.94919485D-09

4 0.78249311-09 0.949195D-09 0.94919485D-09

The stiffness of a cantilever steel plate has been measured at the middle of the free

end. For this, the plate was mounted on a dynamometer on the milling machine table

which was given small displacements while the tip of a bar clamped in the collet was

touching the plate at the middle of the free end. Therefore, the point force applied

on the plate was measured as a function of the displacement at the same point. The

plate dimensions were (63.5x63.5x3.81 mm). The displacement range of (0-100 tm) was

covered at 5 steps. The stiffness was calculated by linear regression as

kexp = 610N/mm

The stiffness at the same point was calculated by FE and Ritz methods by applying a

• point load at the middle of the free end. (16x16) elements were used in the FE solution

and the first six functions were taken in the series in the Ritz solution. The following

stiffness values were obtained from two methods:

kFE = 714 N/mm ; k2 = 800 N/mm

Therefore, the FE solution is closer to the measurements. The difference of about 15%

(which is more than 30% for the Ritz solution) between the experimental value and

the FE result can be attributed to the measurement error, material uncertainties and

unmodeled surface residual stresses left form the grinding of the plate. The difference

between the FE and Ritz solutions becomes larger for a stepped plate. Consider the case

in which the thickness of the plate is reduced from 3.81 mm to 1.905 mm in the 3/4 of


the plate (between x = 0 and x = 48 mm). In the Ritz solution, an average thickness

can be used. In order to be consistent with the energy approach , the average thickness

can be defined as

ta = /(3 * 1.905 + 3.81)/4 = 2.67mm

as the strain energy is proportional to the thickness of the plate. The stiffness at x =

48, y = 63.5 mm (on the free edge, at the step location). The FE and Ritz solutions give

kFE = 245 N/mm ; k2 = 281 N/mm

If an arithmetical average of the thicknesses is used in the solution, then kRt = 200

N/mm is obtained. This shows that the variation of the plate’s stiffness during the

machining is quite significant and has to be considered in the analysis. Comparing the

results obtained from the FE and Ritz methods it can be concluded that the FE gives

more accurate results than the Ritz solution.

Natural modes of a cantilever plate are considered next. The plate has the dimensions

of 63.5x44x3.81 mm. The plate material is Ti6A14V titanium alloy with volumetric

density of p = 4350 kg/rn3 and E = 110 GPa. The following equation is given in [104]

for cantilever plate natural frequencies:

= a/7 (3.57)

where y = pt is the mass per unit area of the plate, t is the plate thickness, D is the

fiexural rigidity, and a is the length (in y direction) of the plate. ) depends on the aspect

ratio a/b of the plate , where b is the width of the plate. For a/b = 0.7, ). for the first

three modes are : 3.48, 6.63, 15.48. By substituting the values in the frequency equation

(3.57), the first three natural frequencies are obtained. The natural frequencies obtained

from equation (3.57), the FE program and the impact tests are given below in (Hz):

Chapter 3. Structural Modeling of the Workpiece and End Miii 51

Mode Equation 3.57 FE Measurement

1 1658 1657 1525

2 3159 3046 2800

3 7377 7110 6725

The small difference between the predicted and the measured natural frequencies is

partly due to the structural damping. Assume that half of the plate thickness is removed

by milling. The variation of the first 3 natural frequencies of the plate obtained from the

FE program along the feeding direction of the tool is shown below:

Feed position (mm) fi (Hz) f2 f0 1657 3046 7110

16 1571 2790 6000

32 1361 2192 4881

48 1119 1887 4235

63.5 839 1552 3617

Comparing the frequencies of the uncut and the completely machined plate, it is seen

that the natural frequencies linearly vary with the thickness which is also predicted by

equation (3.57).

3.3 Structural Modeling of End Mill

Although the plate is usually very flexible compared to the end mill, long slender end

mill deflections are important especially at the most flexible part (the free end) which

is in contact with the most rigid part of the plate, i.e. the cantilevered end. Therefore,

end mill deflections should be modeled for accurate surface error predictions. However,

dynamic modeling of the cutter may not be necessary in plate milling as the chatter


Figure 3.7: Structural model for end mill: Cantilever beam with elastically restrainedend.

vibrations are expected to occur at the plate modes due to the very high flexibility of

the plates. Also, due to the strong dynamic coupling between the end mill, tool holder

and spindle, it is not possible to model the cutter dynamics by analytical or numerical

means without experimental identification. For that, modal analysis techniques are used

in modeling end mill dynamics [77]. Therefore, only a static model will be given for the

end mill; however, some dynamic testing results are discussed.

3.3.1 Cantilever Beam Model for End Mill

Static deflection tests performed on a vertical milling machine showed that for a

moderate size end mill, the tool deflections are much higher than the deflections of the

spindle and tool holder. The tests were performed by loading the cutter at its free and

fixed ends. The deflections at the fixed and free ends of the cutter and tool holder, and

on the spindle just above the tool holder were measured. As a result, the end mill was

modeled as a cantilever beam with linear springs at the fixed end to account for the

stiffness at the collet (see Figure 3.7). This is because the deflection measurements at

kc/2


the fixed end of the tool were almost the same when it was loaded at the free and fixed

ends separately. This implies that the stiffness at the collet k can be modeled by a

linear spring. Then, the static deflection of the end mill under the cutting forces can be

calculated from the beam theory [113]. The deflection in x or y direction at axial position

zk caused by the force applied at Zm is given by the cantilever beam formulation as

;:i 2

____

F6EI(3im)+7’ ,O<lIk<lJm

S(k, m) = (3.58)

/Fy,ml1713

6E1 — Vm) + V <Vk

where E is the Young Modulus, I is the area moment of inertia of the tool, k is the

experimentally measured tool clamping stiffness in the collet and vk = 1—

ZJ, I being

the gauge length of the cutter. The area moment of the tool is calculated by using an

equivalent tool radius of Re = sR, where s ( 0.8) is the scale factor due to flutes [114].

In the previous studies by Kline et al. [34] and Sutherland et al. [24], the collet stiffness

was taken into consideration by defining an equivalent tool length which is calibrated at

the free end of the tool by using the cantilever beam formula. This does not represent

the real physics and is not practical as each tool, even the same tool with different

clamped-gauge lengths, should be calibrated to determine the equivalent length. The

collet stiffness k, on the other hand, was determined to be the characteristic of the tool

holder and the collet diameter. End mill deflection has been determined using lumped

milling forces by Kline et al. [35], Sutherland et al. [24] and Altintas et al. [98]. Although

this approximation may lead to reasonably accurate results for short end mills, for long

cutters the distribution of the milling forces on the cutter has to be considered for accurate

predictions. In this study, the end mill deflections are calculated by using distributed

milling forces as explained in Chapter 5. The dimensional surface error predictions, by

using the tool deflection model given in this section, are quite accurate as shown in


Chapters 5 and 7.

3.3.2 Simulation and Experimental Examples

Stiffness measurements for two different end mills will be given and the beam model

for the end mill deflections will be verified. The first cutter has 4 flutes which are 300

helical, 25.4 mm diameter, 115 mm gauge length and is made of HSS (High Speed Steel).

The end mill was loaded by using weights (at the free and fixed ends separately) and the

deflections at the loading points were measured by a dial gauge. From linear regression

analysis, the stiffness of the end mill at its free end, kT, and the collet stiffness, k, are

found as

kT = 3580 N/mm ; k = 19600 N/mm

The stiffness of the tool at its free end can be calculated from the beam deflection formula

given by (3.58):

— 3M

where E = 210 GPa for steel, 1 is the gauge length and the area moment of inertia,

I is calculated by using the effective diameter which is identified as de = 0.85d0

for 25.4 mm diameter tools. Then, the stiffness at the free end is found by determining

the equivalent stiffness of the collet and the tool which are connected in series

1.— cb —— 0 mm

CI 6

which is very close to the measured stiffness. The second example is a 4 flute, 30° helical,

19.05 mm diameter carbide end mill with a gauge length of 55.6 mm. k = 19800 N/mm

and kT = 10180 N/mm were measured on the tool. The predicted value is kT = 10450

N/mm (E = 620 GPa for carbide tool).


The effect of support flexibility on the natural modes of beams has been investigated

in many studies [115, 116, 117, 118, 119, 120, 121]. One may think that the mode shapes

and the natural frequencies of end mills can be determined by using the beam model

considered in the static deflection analysis. However, the frequency predictions do not

match with the measured data. For example, for a 4 flute, 00 helix, 19.05 mm diameter

carbide end mill, the first 5 modal frequencies were measured as 113, 625, 2113, 2800

and 3500 Hz in an impact test. The maximum amplitude of the transfer function occurs

at the third mode, 2113 Hz. Another impact test performed on the tool holder revealed

that the first two modes are not related to the tool. The frequencies calculated by using

the equations given in [121] for the natural frequencies of elastically supported beams

(k = 19800 N/mm) are : 875, 1770, 15770 Hz. This suggests that the tool dynamics

cannot be modeled by the simple beam model. A complete analysis of the spindle, tool

holder and end mill assembly is necessary for modeling and prediction purposes. However,

this is not a simple task, and modal testing is used to obtain the modal characteristics

at the free end of end mills [77, 122, 123, 124].

3.4 Summary

The cantilever plate and the end mill structures have been modeled. The static

deformations and the dynamic modes of the plate are calculated by using the Finite

Elements method. The end mill is modeled as a cantilever beam with elastic restraints

at the fixed end. Both methods are verified experimentally.

Chapter 4

Modeling of Milling Forces

4.1 Introduction

The prediction of cutting forces in machining is of fundamental importance as they

are the key factors in determining dimensional surface accuracy, machining power and the

required strength for workpiece holding mechanisms and the cutting tool. Practical and

accurate cutting force prediction methods are also required to optimize process planning

in CAD/CAM environments. In this chapter, different approaches and models are given

for the prediction of milling forces. In peripheral milling operations, the mechanistic

approach is usually used where the cutting forces are related to average chip thickness by

cutting force coefficients calibrated experimentally for a particular workpiece material-

tool pair [35, 32]. Then, the cutting forces produced by the same cutter are predicted

analytically by using the force coefficients as shown by Sabberwal et al. [20, 21], Ar

marego et al. [30], Tlusty et al. [22] and Altintas et al. [25]. The mechanistic model

gives accurate milling force predictions, however, milling forces have to be calibrated for

every tool geometry and cutting conditions and thus it has little use in tool design and

process planning. Armarego et al. [27, 30, 36, 125] used the oblique cutting model for

milling force predictions which is called the mechanics of milling approach. This method

uses an orthogonal cutting database in determining the milling force coefficients without

calibration tests, and therefore it is very practical for optimal tool design and process

planning.

56

Chapter 4. Modeling of Milling Forces 57

4.2 Mechanistic Modeling of Milling Forces

In the mechanistic approach, the cutting force coefficients are directly identified from

the milling tests unlike the mechanics of milling approach which is discussed in the next

section. Two different mechanistic models are considered in this section, the exponential

force coefficient model and the linear edge force model. In the exponential model, the cut

ting forces are proportional to the chip thickness. Although the linear relation between

the chip thickness and the cutting forces in orthogonal cutting was modeled by Merchant

in 1944 [2], Koenigsberger and Sabberwal [21, 20] reported the proportional relationship

between the uncut chip thickness and the tangential milling force component in 1961.

This model has been extended by Tiusty and MacNeil [22] and Kline et al. [23, 34] to

include a radial force component and has been used widely in the milling force analy

sis [24, 126, 25]. The experimental results show that as the chip thickness approaches

zero, cutting forces converge to some values different than zero. Because of this, in the

exponential force model, the cutting force coefficients increase indefinitely as the chip

thickness approaches zero . This is due to the finite sharpness of the tool edge which

results in the ploughing of some material under the tool nose, and the flank contact which

follows it. Flank contact has special importance in the dynamic cutting process as it gen

erates process damping in cutting which is discussed in the chatter stability analysis, in

Chapter 8. Masuko [127], Albrecht [128] and Zorev [4], independently, proposed cutting

models which consider the edge forces. The complicated physics of the ploughing and

the flank contact led the researchers to identify the edge forces by experimental methods.

Zorev [4] proposes different ways of identifying the edge forces, however they are usually

found by extrapolating the cutting forces to zero chip thickness. The linear edge force

model was used by Armarego and Epp [129] in formulating the milling forces for zero

helix cutters and by Yellowley [28] for analytical mean force and torque formulations in


peripheral milling operations.

Exponential and linear edge force models give satisfactory force predictions as long as

the cutting force coefficients are calibrated accurately. The linear edge force model has

the advantage of having linear force coefficients and a better physical interpretation of

the cutting process. However, there is no acceptable model for the prediction of the edge

forces, so they have to be determined experimentally. Also, it may not be completely

correct to assume that the edge forces (ploughing and flank forces) are independent of the

chip thickness as suggested by the linear edge force model, although this is an acceptable

first order approximation [29]. Therefore, in this study, the exponential force coefficient

model is used when the accuracy is of primary importance, such as in surface error

predictions, and the linear edge-force model is used when the interpretation of cutting

data and identification of some cutting characteristics, such as shear stress and friction

• between chip and the tool rake face, are necessary.

4.2.1 Exponential Force Coefficient Model

The elemental tangential ( dF), radial (dFr) and axial (dFa) cutting forces acting

on flute j of a rigid end mill are shown in Figure 4.1, and given by

dFt3(,z) = Kth(q,z)dz ; dF(qS,z) = KdFt3(q,z)(41)

dFaj(5,z) = KadFtj(b,Z)

where is the immersion angle measured from the positive y axis. The uncut chip

thickness, z), can be approximated as

h(ç,z) = St sin çj(z) (4.2)


Figure 4.1: Differential milling forces applied on a milling cutter tooth. Total millingforces are calculated by integrating the differential forces within the engagements limits.Directions of the differential milling forces change as the cutter rotate and also, alongthe axial direction due to helical flutes.

where s is the feed per tooth and (z) is the immersion angle for flute j at axial depth

of cut z. Due to the helix, the immersion for flute j changes along the axial direction z,

= — k1pz (4.3)

where t/’ is the helix angle , = tan /R and q5,, = 2K/N is the cutter pitch angle. R

and N are the radius and number of flutes of the cutter. Cutting parameters I(t, Kr and

Ka are expressed as exponential functions of the average chip thickness per flute period,

ha, as follows

= KTha ; Kr = Kh iç = KAh8 (4.4)

where constants KT, K, KA, p, q and s are determined experimentally for a tool-workpiece

material pair as explained later in this section. The average chip thickness, ha, is defined

y

z


as the ratio of the chip volume removed to the exposed chip area:

I’ exa] stsingdg

Ii =a a(ex— st) (4.5)

— cos st — cos cex— 5t , ,

Yer Yst

where a is the axial depth of cut and q and are the start and exit angles, respec

tively. According to the geometry shown in Figure 4.1, for up milling q 0 whereas

= K for down milling.

In this section only rigid cutting forces, which will shortly be referred to as cutting

forces, are considered, i.e. the effects of the deflections on the cutting geometry are ne

glected. The elemental cutting forces can be resolved in feed, x, and normal, y, directions

as

dF(,z) =

(4.6)

dF,(q5,z) dFt,(b,z) sin qj(z) — dFr,(q5,z)coscj(z)

Substituting equations (4.1) and (4.2) in (4.6), cutting force intensities for flute j are

obtained as

dF(,z)= —Ks [cos(z) + Kr sin(z)] sin(z)

z)= Ks [sin(z) — Kr cos (z)] sin(z) (4.7)

dF2(q,z)= 1(tKastsinq5j(z)

The cutting force intensities for flute j are normalized by the axial depth, but dependent

on feed rate s and cutting force coefficients. The total cutting force contributed by flute

j can be found by integrating the intensities along the in cut portion of the flute,


F,()= { - cos2(z) + Kr(2(z) - sin 2(z))]

3,2

Ks{2(z) — sin 2(z) + ITcos2j(z)]32 (4.8)

4k zj,i(q)

KK z,2()F() = — aS

[cosi(z]Z.()

where z,1() and zj,2(q) are the lower and upper limits of the in cut portion of the flute

j. Total cutting forces on the cutter are found by summing forces contributed by all

flutes which are in cut.

N—i N—i N—i

= F3() ; F(q5) = F(4) ;F(qf) = F,(q) (4.9)j=O j=O j=O

It should be noted that for a zero helix case (b 0), the cutting force intensities given by

equation (4.7) are constant along the axial direction as the local immersion angle j, and

thus the chip thickness does not vary in the axial direction z (see equations 4.2 and 4.3).

Therefore, for zero helix cases the integration of the force intensities is not necessary to

determine the cutting forces as they can directly be obtained by multiplying the force

intensities by the axial depth of cut a.

The cutting force intensities are normalized by the total cutting forces to obtain

purely geometric unit cutting force intensity functions for a cutter whose reference flute

(j = 0) is at immersion q,

df(q,z) — dFr(q,z)/dz df,(,z) — dF(çb,z)/dzdz — F(q5) ‘ dz —

(4.10)

df(q,z) — dF(çf,z)/dzdz —


For example, the unit cutting force intensity in x direction is

j=N-1

4kg, 0.5sin2cbj(z)+Krsin2qj(z)

df(ç,z) —

____________________________________________

dz — jN-1

— cos2q(z) + Kr (2q5(z) — sin2Ai(z))jz,i()

Equation (4.10) shows that the unit cutting force intensity depends on the tool geometry

(, ,), the limits of engagement zj,2 and zj,1, and cutting constant Kr. The cutting force

intensities in x and y directions are particularly important in deflection analysis. For a

multiple flute helical milling case the force intensities vary periodically along the tool axis

due to the engagement of flutes at different axial locations. At some points there may

not be any cutter-workpiece contact resulting in zero intensities. As the cutter rotates,

the force intensities move along the axial direction due to the helix effect.

Limits of Engagement

The limits of engagement are explained in Figure 4.2 which shows the unrolled form

of the cylindrical part surface [25]. The part surface is now bounded by the two vertical

lines qf ç and q = ex, and the two horizontal lines z = 0 and z = a, where a is

the axial depth of cut. Four different possible intersections of the cutter flutes with the

workpiece are shown. As it can be seen from the figure, the engagement limits depend on

the axial depth of cut (a), start and exit angles (qst) and (qex), and the lag angle of the

flute, (ak). The lag angle is the angle between the tip of the flute (z = 0) and the highest

point in the cut (z = a). Equation (4.3) can be inverted as z(qj) = —[ + jq, — 4j]

to obtain the intersections of a flute with the boundaries when necessary. Engagement

limits for the intersections shown in Figure 4.2 are given in Table 4.1.


a

Figure 4.2: Contact cases of a helical flute with workpiece. The contact geometry dependson the axial depth of cut, the immersion angle and the helix angle and it determines thelimits of integration for the milling force calculation which are given in Table 4.1

Average Forces and Identification of Milling Force Coefficients

The cutting force coefficients can be identified from the average cutting forces, i,

and , as they are assumed to be constant over the full rotation of the cutter. The

average cutting forces are independent of the helix angle as the total chip removed in

one rotation of the cutter does not depend on the helix angle. Whitfield [130] integrated

the cutting forces over a rotation of the tool for the different contact cases shown in

Figure 4.2 and obtained the same average force expression for each case. Therefore, the

cutting force intensities given by equation (4.7) are first multiplied by the axial depth of

cut to obtain the cutting forces for the zero helix case, then they are integrated over one

rotation of the cutter and divided by 2r to obtain the average forces as:

1This is equivalent to integrating the forces over one tooth period and dividing by 2ir/N as the flutesare equally spaced.


Table 4.1: Helical flute engagement limits to be used in cutting force calculations. The

numbers shown in the parenthesis correspond to the contact cases in Figure 4.2.

gj(O) zj,1st < j(°) < çtex 0 (1,2) (4j(O) — ak) < qst

Zj,2 = (q5j(0)- ) (1)

(qj(O) — ak) > cst

= z,2 = a (2)qS(O) > ex(0) (q3(O) — ka) < ex (j(O) — ak) < gst

= -(q(0)— ex) (3,4) = q(0)

— g5st) (3)

t < ((O) — akü) <= zj,2 = a (4)

= Kt(PKrQ)

Fy = —K(Q + P1(r) (4.11)

= StKaKtT

where

aNP = —Fcos2l2ir L

aN [ex

Q = — 2ç — sin 2I (4.12)Jst

aN rT = —lcosqSl

27r L


ct and qer are the start and exit angles of the cut, a is the axial depth of cut and N is

the number of teeth. Then, the cutting force coefficients can be obtained as follows:

K — QF+PF—

K — F (4.13)— t(P+QKr)

17 —

_________

1a— sKT

After the cutting coefficients are obtained for different feedrates, they are expressed as

exponential functions of the average chip thickness as shown in equation (4.4). Due to

edge components, the cutting forces converge to a finite value as the feedrate approaches

zero. As a result, the cutting force coefficients are very high at small feeds, and they

decrease exponentially as the feedrate increases.

4.2.2 Linear Edge-Force Model

In the linear edge-force model, the cutting forces are modeled as having two fun

damental components: the edge (due to ploughing and flank contact) and cutting com

ponents [28]. The cutting constants (Kte,Kre,Kae) and (Ktc,Krc,K,) relate the total

cutting forces to ploughing and rubbing, and actual chip cutting mechanisms, respec

tively. The force coefficients are determined experimentally for a certain tool-workpiece

material pair when the mechanistic approach is used.

The elemental tangential, dF, radial, dFr, and axial dFa cutting forces acting on flute


j of an end mill are shown in Figure 4.1, and given by

dFt(,z) = [Kte+Ktchj(q!,z)]dz

dFrj(75, z) = [Kre + Krchj(q’?, z)]dz (4.14)

dFaj(q,z) = [Kae+Kachj(q,z)]dz

where h(q, z) = St sin (z) is the uncut chip thickness and st is the feed rate per tooth.

4 is the immersion angle measured clockwise from the positive y axis to a reference flute

j = 0, which has immersion at its tip z = 0. The elemental forces are resolved into

feed (x) and normal (y) directions and integrated analytically along the in cut portion

of the flute j to obtain the total cutting force produced by the flute.

R[Kte sin (z) — Kre cos (z)

tan

+ [Krc(2(Z) - sin 2(z))-

cos 2(z)]]:= ta [ Kresin(z) — Ktecos(z) (4.15)

+ [Kt(2(z) - sin 2(z)) + Krc cos 2(z))1]

R Z,2()

tan — stKaccos(z)]Z.()

• where zj,i(q) and zj,2(q) are the lower and upper limits of the engagement of the flute

j. The cutting forces contributed by all flutes are calculated and summed to obtain the

total instantaneous forces on the cutter. The determination of the engagement limits is

explained in the previous section.


Average Forces and Identification of Cutting Force Coefficients

The average milling forces per tooth period are F, i, and , and can be found by

integrating equation (4.3) over one full rotation of the cutter,

KteS+KreT(KtcF+KrcQ)

= KteT — KreS + (KCQ + KrcP) (4.16)

= Kae(cexcbst)+stKacT

where F, Q and T are given by equation (4.12) and

aNS=— sinq (4.17)

2ir

c5t and ç are the start and exit angles of the cut, a is the axial depth of cut and N is

the number of teeth. After cutting forces are measured and their averages are found at

different feedrates, they are put into the following form by linear regression

= 1qe + .StFqc (q = x, y, z) (4.18)

where Fqe and Fqc are the edge and cutting components of the forces. Finally, the cutting

force coefficients are identified from equations (4.15) and (4.18) as follows:

K — eS+yeT K —

_______

te —

— S2 + T2 , tc—

p2 + Q2

K — KteS +.

— KP — 4F (4 19)re—

4rc —

2ir F€ .- ae = — 11ac =

Yex —

The cutting force coefficients are directly estimated from the milling data by curve

fitting, hence new milling tests are required for the calibration of each new cutter geom

etry.


4.3 A Mechanics of Milling Approach for Milling Force Prediction

The end mill geometry is complex, having a number of variables such as helix and

rake angles which have to be selected properly in order to improve the machining per

formance. In addition, the cutting edge angles and diameter may vary along the flutes

of some special cutters such as tapered-helical ball end mills. Therefore, a vast amount

of cutting data is necessary to predict the cutting forces for different cutter geometries

which is costly and not practical. Furthermore, the data cannot be generalized as there

is no explicit relationship between the tool geometry, cutting conditions and the cutting

force coefficients in mechanistic millillg force models. Also, an identification procedure

that is based on experiments entirely does not give physical insight, i.e. the relation of

the milling force coefficients to the fundamental machining characteristics of the work-

piece material and tool geometry, such as friction, shear stress, rake and helix angles.

Therefore, although mechanistic cutting force models usually provide accurate predic

tions, they are not practical for tool design, process planning and analysis of cutters with

complex geometries.

The fundamental metal cutting research has mainly concentrated on simple tool ge

ometries, and cannot directly be applied to practical machining operations like peripheral

milling. In this section, a mechanics of milling approach is presented for the milling force

analysis. In this method, an oblique cutting model is used to relate the cutting force

coefficients to the tool geometry, shearing and friction mechanisms. The required cutting

force parameters are identified from orthogonal cutting tests and used in the oblique

cutting model together with the predicted values of the chip flow direction. The same

database can be used in the analysis of cutting forces for different tool geometries which

reduces the number of cutting tests considerably. The use of orthogonal cutting data in


force predictions of drilling and milling operations was first introduced by Armarego and

Whitfield [27]. They used Stabler’s rule for the chip flow angle. In this study, a more

accurate model for the estimation of the chip flow direction is used. The method in the

revised form predicts the cutting force coefficients with more than 85 % accuracy in the

end milling of the titanium alloy Ti6A14V, thus providing an alternative and practical

way of simulating the performance of milling dlltter designs prior to their manufacture

and experimental testing.

4.3.1 Force Coefficient Expressions

The shear stress, shear and friction angles can be identified from orthogonal cutting

• tests for different rake angles and cutting velocities [2]. However, in helical end milling

the edges of the cutter flutes are not orthogonal to the cutting velocity but inclined with

an angle equal to the helix angle as shown in Figure 4.3. Therefore, an oblique cutting

model should be used for the analysis of end milling. The cutting data, however, will be

obtained from the orthogonal cutting tests and used in the oblique cutting model. For

this reason, first the orthogonal cutting model will be briefly discussed and the related

equations for the identification of shear angle, shear stress and friction on the rake face

will be derived.

Consider the force diagram shown for the orthogonal cutting geometry in Figure 4.4.

In thin shear zone models, metal is assumed to deform and form chips in the shear plane.

The angle between the shear plane and the cutting velocity direction is the shear angle,

ç. Shear angle is perhaps the most studied parameter in the metal cutting research, as it

is required in cutting force analysis. Although many models have been proposed for the

shear angle prediction [2, 131, 132, 133], accuracy of the predictions is usually low and

depends on the cutting conditions and the material. This is due to the complex nature of


1’ if’11+1 1

‘7,Orthogonal cutting .

Oblique cutting

Figure 4.3: Orthogonal and oblique cutting geometries. In oblique cutting, the edge ofthe tool is not perpendicular to the cutting velocity which results in a three dimensionalcutting geometry and a chip flow direction which is not parallel to the cutting velocity.

the machining process which involves plastic deformation of metal under extreme friction

effects with high temperature gradients. Despite the extensive research efforts in the last

hundred years, the relations between cutting conditions, tool geometry and the shear and

friction characteristics are still not completely modeled. For this reason, the fundamental

parameters of metal cutting, namely the shear angle, the shear stress at the shear plane

and the friction on the rake face, are identified from the cutting experiments in this

study. The orthogonal cutting model proposed by Merchant [2] is used in the analysis.

In Figure 4.4, the length of the shear plane, AE is

h = h/cosa—htana(4.20)

sm qf cos

orr cos a

tanq8= . (4.21)1 — r sin a

lic ii

where


y

Figure 4.4: Orthogonal cutting force diagram. The force diagram is used to relate thefrictional and normal forces applied on the rake face to the shear and normal forces onthe shear plane. This force diagram is the basis of the Merchant’s orthogonal cuttingforce and shear angle prediction model.

a rake angle

h uncut chip thickness

h chip thickness

shear angle

r = h/he, chip thickness or cutting ratio

Therefore, the shear angle can be identified from the cutting ratio r. The average

shear stress at the shear plane, (r), and the average friction angle on the rake face, (9),

are obtained from the force analysis. The resultant cutting forces on shear plane and

rake face are equal to each other, due to the static equilibrium 2 of the chip. On the

force diagram shown in Figure 4.4, P’ and F,-, are the shear and normal forces on the

2The conservation of momentum is not considered in cutting force analysis due to small velocity andnegligible mass of the chip. However, this effect is important and usually taken into consideration in theanalysis of high speed cutting processes.

x


shear plane whereas N and F are the normal and frictional forces on the rake face. F

and Ff are the cutting and the feed forces which are normal to each other. The friction

coefficient on the rake face can be identified from the ratio of the friction force, F, to

normal force, N which can be expressed as follows

F = Fsino+Ficosa

(4.22)

N = FcosQ—Fsinc

Then, the friction ratio () and the friction angle (3) are given by the ratio

F Fj+Ftanc[Lf=tan8=—= (4.23)N FC—rftano

The shear stress at the shear plane (r) can be identified by dividing the shear force F8

to the area of the shear plane (bh/ sin ),

F cos — Ff sin q= bh

sinqf. (4.24)

where b and h are the width of cut and the uncut chip thickness, respectively.

Both exponential and linear edge cutting force models are used in the analysis of the

orthogonal cutting forces and both methods yield accurate force predictions. For the

analysis and identification of shear stress and friction angle, however, the edge forces

should be separated from the cutting forces as they are not related to the shearing and

rake face friction. Therefore, when the linear edge model is used, the cutting components

of the total cutting forces should be substituted in equations (4.23) and (4.24) . The edge

forces can be identified by extrapolating the cutting forces to zero chip thickness as in

the case of milling force analysis by the linear edge model.


Figure 4.5: Cutting forces and chip flow geometry on a helical milling cutter.

In helical milling, the tooth edge is not orthogonal to the cutting velocity direction

but makes an angle which is equal to the helix angle. This causes the shear plane and

the chip flow direction to be oriented with respect to the edge of the tooth. Therefore,

helical milling force analysis requires the use of an oblique cutting model.

A simple view of a peripheral milling cutter edge geometry is shown in Figure 4.5.

A detailed view of the cutting zone showing the oriented shear plane is given in Figure

4.6. A plane which is normal to the cutting edge is considered for force and velocity

equilibrium equations. The velocity rake angle, c, and the normal rake angle, a, are

shown in different views of the rake face and related as

Y

Vd Fa

d Ft

True view of rake face

d Fr

Normal Plane

V:cutting velocityVn: component ofcutting velocity innormal plane

tan o = tan a cos (4.25)


angleiP,

Figure 4.6: Detailed view of the oblique cutting geometry showing the oriented shearplane and the chip flow on the rake face. The oblique cutting force analysis are performedon the normal plane which is perpendicular to the cutting edge.

_4utti velocity


Figure 4.7: The oblique cutting force components in the normal plane.

Figure 4.7 shows the cutting force components in the normal plane at the tip of the

flute j. dN and dF are the differential normal and friction forces on the rake face of the

tool. The components of the differential tangential, radial, dFrn, and shear, dF8,

forces in the normal plane are related to the milling forces and geometry as follows

dF =

dFrn = dFrj (4.26)

dF— stsinqj(z)

d—

Z

where ç’8 is the shear angle measured in the normal plane and gj(z) is the immersion

angle of the flute j at axial depth z. The normal friction angle, is defined as follows

Vn

tan/3 = tan/3cos77 (4.27)


where /3 is the friction angle at the rake face and is the chip flow angle as shown in

Figure 4.5 (angle between a perpendicular to cutting edge and the direction of the chip

flow over the rake face, as measured in the plane of tool face). The normal shear angle,

q, is obtained from the cutting ratio as in the orthogonal cutting

rt cos a,-,tan = . (4.28)

1— rt sin a,-,

• . . . . . .

. cosiwhere the chip thickness ratio in oblique cutting, rt, is related to r by rt = r cos This

relation is obtained from the mass continuity equation of the chip before and after the cut.

It has been shown [27, 6, 29, 134, 130] that the cutting force coefficients Krc and

Kac can be determined from the oblique cutting analysis with a satisfactory degree of

accuracy. In order to derive the cutting force coefficients, the elemental chip is considered

to be in equilibrium under the action of stresses in the shear plane and at the rake face

of the tool. Based on the oblique cutting model, the cutting coefficients which relate the

cutting forces to tool geometry are expressed by [6]

K — r— Sfl

K — r sin(3 — a) (4 29— sin q’ cos k

— r cos(/3n — a)tanL — tanac

— sin5 k

where

k = \/cos2(q8fl+ /3 — a) + tan2 7?c sin2 [3

Therefore, for a given tool geometry, the cutting force coefficients can be calculated from

equation (4.29) if the shear stress, (r), friction angle, (/3), cutting ratio, (r), the chip flow

angle, (), and the edge force coefficients, Kje,K,-e and Kae, are known. There is no


acceptable theoretical model for the edge component of the cutting forces, they can only

be found from cutting tests. Therefore, prediction of the cutting force coefficients requires

that the shear stress (r), friction angle (/3), cutting ratio (r), chip flow angle () and

the edge force coefficients Kte, Kre and Kae be known. As these parameters depend on

cutting conditions and tool geometry, they should be obtained from the cutting tests in

which chip thickness, rake angle, cutting velocity and the angle of inclination (helix angle)

are varied. Based on the experimental results obtained at the University of Melbourne

[135, 136], Armarego et al. [27, 130] concluded that r, r, /3 and Kte, Kre, Kae are all largely

independent of the tool inclination angle /‘ so that these parameters can be obtained from

simpler orthogonal cutting tests as explained before. This results in an order of magnitude

reduction in the amount of needed machining tests. The prediction of chip flow angle is

crucial for the accurate estimation of cutting forces as explained in the following section.

Armarego et al. [30] and Whitfield [130] used Stabler’s chip flow rule which assumes that

= i/’. This is a crude approximation and may result in significant errors in the cutting

force predictions.

Prediction of Chip Flow Angle

The oblique tool geometry was first rigorously analyzed by Stabler [137]. In the

same study, Stabler also stated the widely accepted Stabler’s rule or chip flow law which

assumes that the chip flow angle is equal to the angle of obliquity. In other words, Sta

bler’s rule assumes that the chip moves parallel to the cutting velocity vector, without

bending after it is cut. The rule does not consider the effect of shear angle, friction and

tool geometry, i.e. rake angle. Shaw et al. [138] experimentally showed that the chip

flow angle varies with the normal rake angle and friction. Therefore, Stabler’s rule may

introduce significant errors in the prediction of the cutting force coefficients depending

on the cutting conditions. Later, Stabler [139] modified the chip flow law to i = k/’


where 0.9 < k < 1.0 varies with the work material and cutting conditions. However,

the modified chip flow law does not change the results considerably. The effects of cut

ting conditions and tool geometry on the chip flow angle were experimentally studied in

several works. Russel and Brown [140] proposed the equation i = tan ‘b cos ce,-, which

indicated dependence of on the rake angle. Zorev [4] considered the effect of cutting

speed on and suggested i = i1’/V°°8 where V is the cutting speed in m/min. Usui et

al. [141] used the minimum energy principle to determine the chip flow angle. Colwell

[142], Zorev [4] and Stabler [139] proposed methods for prediction of which consider

the cutting action of the end cutting edge. The experimental and predicted end cutting

• edge and tool nose radius effects on the chip flow angle are discussed in detail by Oxley

[10]. Whitfield [130] gives a rigorous formulation to obtain the chip flow angle by using

orthogonal cutting data which will be reviewed in the following.

In an early study of metal cutting, Merchant [2] indicates that the direction of shear

is not perpendicular to the cutting edge in the case of oblique cutting, but makes an

angle 6, with the perpendicular. He derived the following equation for 6 from velocity

considerations.tan cos(qs — a) — tan sin

tan 6 = (4.30)cos a,-,

Stabler [137] later formulated the angle, j, that the shear force makes with the cutting

edge normal as follows

sin /3 sintanf = . . (4.31)

cos /3 cos(3—

a) — cos /‘ sin /3 s1n(qsn an)

In general, it is fairly reasonable to assume that the shear force and shear velocity direc

tions are equal. The following expression is obtained when equations (4.30) and (4.31)


are equated to each other (Armarego and Brown, [6])

cos c tan /‘tan(qi3 + i3) = . (4.32)tan — srno tan b

From equations (4.27), (4.28) and (4.32), the following expression for the chip flow angle

, is obtained

A sin — B cos — C sin cos ic + D cos2 1c = E (4.33)

where

A = r cos a + cos tan /3

B = tan/3sinosinb

C = rsintan/3 (4.34)

D = rtan,8tan’b

B = sincoso

The chip flow equation (4.33) can be solved by numerical techniques for cutter geometry

(i/’, a) and (r, /3) which are identified from orthogonal cutting tests. Detailed derivations

of the chip flow equation and a numerical solution procedure are given in Appendix A.

The resulting variations of the chip flow angle, T1D, with the friction angle, cutting ratio,

angle of obliquity and normal rake angle are computed and shown in Figures 4.8, 4.10,

and 4.9.

Figure 4.10 shows the effects of cutting ratio r and angle of obliquity b on. It can

be seen that Stabler’s rule may be a good approximation for only limited ranges of r and

‘. Therefore, Stabler’s rule is a crude approximation and may result in errors when used

in milling force analysis. As experimentally observed by Shaw et al. [138], the solution

indicates that the chip flow angle reduces as the friction increases (see Figure 4.8). In

order to obtain the real variations of chip flow angle, however, one should use cutting

data since friction and shear angle are, in general, functions of rake angle. Hence, one


r

Figure 4.9: Variation of chip flow angle with cutting ratio for different values of frictionangle.

70

60

0,.50

)40

0LI0.20C)

10

0

15 25 35 45

Friction Angie (deg)

Figure 4.8: Predicted variations of chip flow angle with rake and friction angles.

80

5

60

< 400

0

C-)

00.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Chapter 4. Modeling of Milling Forces

80

a)9- 60a)0)C

40U0.

0 20

0

81

Figure 4.10: The effects of inclination angle and cutting ratio on chip flow angle. Thisfigure indicates that Stabler’s rule is an acceptable approximation only in limited rangesof inclination angle and cutting ratio.

cannot follow the constant a, r and 3 lines in Figures 4.8, 4.10 and 4.9 for a particular

cutting operation which can be seen from the experimental results given by Shaw et. al

[138], Brown et. al [29], Pal et. al [143] and Lin et. al [144]. That means orthogonal

cutting data is necessary for the chip flow angle prediction as well.

4.3.2 Procedure for Milling Force Calculation

With the chip flow angle prediction, the procedure is completed for milling force

calculations. The method is summarized in Figure 4.11 where it is referred to as gen

eralized milling force calculation. This is a general method as the same procedure can

be followed to calculate the cutting forces on any milling cutter geometry by using the

same orthogonal data. As an example, consider the ball end mill geometry shown in

Figure 4.12. The cutting velocity and helix angle vary from finite values at the shank

intersection to 00 at the tip of the ball. The three differential force components, dF, dFr

5 15 25 35 45

Inclination Angle, w(deg)


Generalized Milling Force Analysis

Figure 4.11: Generalized milling force prediction algorithm. The procedure is repeatedat a number of points along the cutter flutes if the cutter has variable geometry thateffects the local cutting force coefficients.


and dFa are oriented due to the ball which should be considered when calculating the

cutting forces in x, y and z directions. The local values of the cutting velocity and helix

angle are to be used in the cutting force coefficient calculations. Usually, the normal rake

angle is kept fixed on the flutes, however if the velocity rake is kept fixed, then the local

values of normal rake angle should be considered. Finally, the differential cutting forces

in x, y and z directions can be integrated to obtain the total cutting forces. The ball end

milling forces have been modeled in [145, 146, 147, 148, 149]. Yang et. al [145, 146, 148]

used the orthogonal data to predict the differential resultant force (not the tangential,

axial and radial force components) in ball end milling. They did not consider the he

lix angle and oblique cutting model, which resulted in inaccuracies in their predictions.

Lim et. al [149] used a calibration procedure to identify the cutting force coefficients in

which the cutter was calibrated for different axial depths as the average chip thickness

is a function of the axial immersion in ball end milling. Yucesan and Altintas [147], on

the other hand, used an advanced numerical identification technique for ball end milling

which gives accurate results with increased computation time. Therefore, modeling of

ball end milling forces requires more calibration tests than the end milling. Preliminary

applications of the generalized method to ball end milling gave quite promising results,

however, this is kept outside the scope of this work.

4.4 Simulation and Experimental Results

Several orthogonal cutting and milling tests have been performed on a titanium

alloy (Ti6A14V) for the verification and comparison of the developed method with the

mechanistic model results. (Ti6A14V) is mainly used in aerospace applications, such as jet

engine impellers, due to its high strength to weight ratio even at elevated temperatures.

It accounts for 45 % of the total titanium production. Despite the extensive research


dFa

dFr

Figure 4.12: The orientation of differential milling force components on a ball end millflute. Variable milling coefficients may have to be used as the cutting velocity and helixangle vary along the cutter flutes. All three forces have components in x and y directions.


efforts, titanium is still one of the most difficult materials to machine [150, 151] . It has

a very low thermal conductivity (one-sixth that of steel) which causes high temperature

gradients at the chip tool interface and results in low allowable cutting speeds. Also,

the high strength is maintained at elevated temperatures and this opposes the plastic

deformation needed to form a chip. As it will be seen in the orthogonal cutting data,

titanium’s chip is very thin which results in an unusually small contact area with the

tool causing high contact pressures and temperatures.

4.4.1 Cutting Conditions in Milling and Orthogonal Tests

Full immersion milling tests were conducted with a carbide end mills with single

flute, 30° helix, 19.05 mm diameter. The axial depth of cut and the cutting speed were

5.08 mm aild 30 m/min, respectively. The average chip thickness range of 0.008-0.1

mm was considered. Helical end mills with different normal rake angles were used to

determine the robustness of the method for different geometries. The normal rake angle

was considered to be the angle corresponding to the rake angle in orthogonal cutting [29].

Orthogonal turning tests were performed on titanium tubes to achieve ideal orthogonal

cutting conditions . The outer diameter and the wall thickness of the tubes used were

100 mm and 3.8 mm, respectively. The cutting speed range of 3-47 rn/mm and carbide

tools with different rake angles corresponding to the normal rake angle on the end mills

were used. The feed rate range of 0.005-0.1 mm was covered with 5 steps (0.005, 0.01,

0.03, 0.07, 0.1 mm). Very low cutting speeds were considered in order to see the effect of

cutting speed and also to be able to use the same data base for different milling cutter

geometries such as ball end mills where the cutting speeds are close to zero at the ball

end. For each case, tangential, F, and feed, F, cutting forces were measured by a

Kistler table dynamorneter mounted on the tool holder. In Figures 4.13.a and 4.13.b, the

3The contact between the tooth nose and the finished surface results in a parasitic force in bar turning.


variations of the measured forces with the uncut chip thickness h are shown for cutting

speeds of V 28m/min and V = 47 rn/mm. As it can be seen from these figures, the

forces do not vary with the cutting velocity significantly. This can be attributed to the

counterbalancing of the temperature and strain hardening effects on the shear stress.

4.4.2 Analysis of Orthogonal Data: Identification of Cutting Parameters

The edge forces are shown in Figure 4.14 which are identified by extrapolating the

cutting forces to zero chip thickness. The edge forces are close to each other for differ

ent cutting velocities and rake angles. Therefore, average values of the edge forces are

used in the rest of the analysis. The mean and the standard deviation of the edge forces

were calculated as Kte=24 N/mm with a(Kje)=6.3 and Kre=43 N/mm with (Kre)7.3.

The agreement between the edge force coefficients identified from milling tests and the

orthogonal cutting suggests that the edge forces do not vary with the angle of obliquity.

Also, the edge component of the force coefficients in the axial direction is very small and

therefore it can be neglected in the transformation of the orthogonal cutting data to the

oblique model. These results have also been reported in [135, 136, 27, 130].

The thickness of many chips were measured with a micrometer for each test and the

average value of the measurements is used in calculating the chip ratio r. Figure 4.15

shows the variation of r with the uncut chip thickness and rake angle. No significant

variation was observed with the cutting velocity. The relatively high cutting ratio (thin

chip thickness), and thus higher shear angle, is a characteristic of all titanium alloys and

results in small contact areas between chip and tool. This increases the contact pressures

and temperatures contributing to the wear and chipping of the tool. As it can be seen

from the figure, r exponentially varies with the chip thickness and the following equation


a (deg)800

700.5

600

—. 500•10

400015

- 300

200

100

0• 0.12

a (deg)800

700.5

600AB

500•10

a 4002 °15LL 300

200

100

0

0.12

Figure 4.13: Measured cutting and feed forces in orthogonal cutting tests with differentrake angles (Material: Ti6A14V). a) V=28 rn/mm b) V=47 rn/mm

0 0.02 0.04 0.06 0.08 0.1

h (mm)

V=47 rn/mm

Fc

I

.

.

A

A0

Ff

0 0.02 0.04 0.06 0.08 0.1

t(mm)


EEz

Rake Angle (deg)

Ez

4,6

2.8

(rn/mm)

2.8

V (rn/mm)

88

4.6

15

815

Rake Angle (deg)10

Figure 4.14: Edge forces as identified from the orthogonal cutting tests.


()

15

Rake Angle(deg)

89

Figure 4.15: Variation of the measured cutting ratio r(= h/he) with chip thickness andrake angle.

was identified by curve-fitting as

r = r0ha

r, = 1.755 — 0.028c

a = 0.331 — 0.0082a

(4.35)

Whitfield [130] observed the opposite variations for steel (S1214 and CS1O4O), i.e.

varies exponentially with the velocity and stays constant with the chip thickness. The

drop in cutting ratio at small chip thicknesses has been noted in some previous works,

but it has been neglected in the widely used shear angle relationships, e.g. Merchant

[152], Lee and Shaffer [131] and Shaw et. al [133]. The reasons for this drop can be

attributed to the reduced effective rake angle at small chip thicknesses due to the nose

radius, rubbing and size effect phenomenon of metals. The cutting ratio obtained with

high rake angles was compared with this equation and it was found that this equation

0.0050.01

0.030.07

h (mm) 0.1


is valid for the tools with rake angles 0 — 35°. In fact, equation (4.35) implies that the

cutting ratio r strongly varies with the uncut chip thickness, whereas the variation with

the rake angle is quite small.

Shear stress (r) and friction angle (3) are calculated from equations (4.24) and (4.23),

and shown in Figures 4.16 and 4.17. As it can be seen from Figure 4.16, the shear stress

does not vary significantly with the velocity and the rake angle. This is due to the

opposite effects of the generated heat and the strain rate at the shear zone which are

proportional to the cutting velocity. Therefore, an average value can be used for the shear

stress. The average value and standard deviation of the shear stress were calculated as

r=613 MPa with o(r) = 73. The friction angle, on the other hand, slightly varies with

the rake angle and the cutting velocity. The friction angle identified from the orthogonal

cutting tests is the average value of the friction in the sticking and the sliding regions

between the chip and the rake face of the tool. The friction coefficient is higher in the

sliding region which becomes longer as the rake angle is increased due to the reduced

pressure on the rake face. Hence, the average value of the friction on the rake face

increases with the rake angle. This is recognized as one of the main dilemmas in metal

cutting for the cutting forces reduce whereas the friction increases with the rake angle [9].

The friction angle slightly varies with the cutting velocity as well, but this variation is

mainly in the low cutting velocity range (V < 10 m/min) and it is almost constant in the

practical cutting velocity range. Therefore, the variation of the friction coefficient with

the cutting velocity is more important in the analysis of ball-end milling as the cutting

velocity approaches zero at the center of the ball. Hence, the variation of the friction

angle with the cutting velocity is neglected and the following equation for the friction


0

C,)Cl)

Cl)

ci).

U)

Figure 4.16: The identified values of the shear stress at the shear plane from the orthogonal cutting tests.

angle 3 is obtained by linear regression analysis:

/3 = 19.1 + 0.29a (4.36)

where the rake angle o and the friction angle /3 are in degrees. The average value of the

friction angle is /3 = 21.3° with o() = 3.1.

The mean values of the percentage error between the identified and the calculated

values from the curve- fit equations are determined as

(-0.07 %), /3: (1.6 %), (: -2.3 %), r: (3.8 %), Kie: (-10.9 %), Kre :( 5.1 %)

The chip flow angle (i) for 30° helix angle is computed from the numerical solution of

equation (4.33) by using the orthogonal data, i.e. r and /3 for different rake angles. Figure

4.18 shows that increases with the rake angle and decreases with the chip thickness, i.e.

cutting ratio r, as illustrated in Figures 4.8 and 4.10. The cutting parameters identified

80

28 10 Rake Angle10 15 (deg)

V (rn/mm) 3

c)a)

a)C

C04-C)1.

LI

Figure 4.17: The friction angle calculated from the orthogonal cutting forces.

Table 4.2: Cutting parameters identified for Ti6A14V from orthogonal cutting tests.

from the orthogonal tests are summarized in Table 4.2.

The orthogonal data has also been analyzed by using the exponential force model and

used in [31]. The identified values of shear stress and the friction angle are scattered,

and much higher friction angles are obtained, especially at low chip thickness, due to the

unseparated edge force effects. However, the milling force coefficients calculated by the

exponential force model have acceptable accuracy [31].


2

Rake Angle(deg)

285

V (rn/mm)

r = 613MPa= 19.1 + O.29a

r = r0W’= 1.755 — O.028a

a = 0.331 — 0.0082aKte = 24 N/mmKre = 43 N/mm


F-’

0)ci)

-D

ci)0)C

0U—

Rake Angle(deg)

Average ChipThickness (mm)

Figure 4.18: Predicted values of the chip flow angle for 300 inclination (helix) angle.

4.4.3 Prediction of Milling Force Coefficients

The data obtained from the orthogonal cutting tests and the calculated values of

chip flow angle were used in the transformation equation (4.29) to predict the milling

force coefficients, Krc and Ka. The same edge force coefficients, Kte and Kre, that

were identified from the orthogonal data were used for milling. In a separate analysis, the

edge and the cutting force coefficients were identified from the slot milling tests by using

single flute carbide cutters and a feedrate range of 0.0127-0.1 mm/tooth, and are given

in Table 4.3. Unlike the experimentally identified milling force coefficients, the predicted

milling force coefficients, K and Kac, vary with the feed per tooth as the cutting

ratio r and, thus the chip flow angle , are functions of the uncut chip thickness. The

variations of the predicted coefficients for a = 0 are shown in Figure 4.19.


3500

3000Ktc + Krc A Kac

2500 .cx=Q

o 2000o

ç 1500

°10004 4 4

500

00 0.02 0.04 0.06 0.08 0.1 0.12

Average Chip Thickness (mm)

• Figure 4.19: Variations of the predicted milling force coefficients with the chip thickness.The predicted coefficients vary with the chip thickness due to the strong dependence ofthe cutting ratio on the chip thickness. (Material: Ti6A14V, V=30 rn/mm, a = 00)

As it can be seen from the figure, very low r values result in high shear angles and

force coefficients at low chip thickness. However, for a first order approximation this

variation may be neglected and the average values of the force coefficients can be used as

the edge forces are much higher than the cutting forces in low chip thickness zone. The

critical chip thickness, defined by Yellowley [28], which is the chip thickness at which

the edge force is equal to the cutting force, is useful in determining these zones. From

the orthogonal cutting data, the critical chip thickness can approximately be calculated

as 0.012 mm for the tangential direction. For the radial, or feeding forces, the critical

chip thickness is much higher ( 0.07 mm) due to high edge force and low cutting force

components in this direction. The identified and the predicted milling force coefficients

are shown in Table 4.3. The average values of the predicted milling force coefficients (in

the average chip thickness range of 0.01-0.1 mm) are given. The agreement between the

predicted and the experimentally identified milling force coefficients in tangential and


the axial directions are satisfactory. The average values of edge forces in milling tests are

Kte = 25.2 and Kre = 44, which are very close to the values obtained from the orthogonal

cutting data. The critical chip thickness in the axial direction is approximately 0.005 mm,

and thus neglecting Kae in the predictions does not introduce too much error unless the

chip thickness is very small. The error in the prediction of and K is approximately

within (+ 8 %) with mean values of (0.02 %) and (-2.2 %), respectively. If the Stabler

rule is used in the calculations, i.e. i = 300, the mean values of the errors in and K

increase to (-2.5 % ) and (-13 %), respectively. The increase in the error values associated

with the Stabler rule would be higher for larger helix angles as predicted in Figure 4.10.

The accuracy of prediction is low for high rake angles. As it can be seen from the

table, the predicted values of the radial force coefficient decrease from 646 MPa to 65

MPa as the rake angle increases, from 00 to 20°. The radial or feeding force is equal to

the difference between the components of the rake face friction and the normal forces,

in this direction. This can be expressed by the following equation for the orthogonal

cutting:

Ff = Fcosa — Nsina

or

Ff = Ncosa(tan/3 — tan a) (4.37)

where Ff is the feeding force, F and N are the friction and the normal forces on the rake

face, and /3 and a are the friction and the rake angles. As it can be seen from equation

(4.37), the decrease in feed force with the rake angle depends on the variation of the

friction angle with the rake angle which is identified and given in equation (4.36). The

poor agreement between the identified and the predicted values of Krc for the high rake

angles (> 15°) is due to the inaccuracy of the friction prediction which suggests that more

orthogonal data is required for high rake angles. Fortunately, in the radial direction, the

• Chapter 4. Modeling of Milling Forces 96

Table 4.3: Cutting force coefficients for different rake angles as identified and transformedfrom orthogonal data by using linear edge-force model. Material:Ti6A14V. Note that theaverage values of the predicted force coefficient values are given for the feedrate rangeof s = 0.01 — 0.1 mm/tooth as the transformed force coefficients vary with the chipthickness due to the exponential variation of the cutting ratio r with the chip thickness.The edge-force coefficients are in (N/mm) and cutting force coefficients are in (MPa).

c Test Predicted(deg) Kte Ktc Kre Krc Kae Kac Kc Krc Kac

0 29.7 1825 55.7 770 1.8 735 1963 646 7785 24.7 1698 42.9 438 5.5 591 1805 461 69912 22.7 1731 44.5 317 2.4 623 1619 253 60415 22.3 1630 37.9 340 2.1 608 1544 177 56220 26.8 1439 38.8 376 2.6 604 1434 65 500

critical chip thickness is much larger than it is in the tangential direction, so that the

inaccuracies in Kr have smaller impact on the overall force prediction accuracy. On the

other hand, very high friction values are obtained at large rake angles and it makes the

use of high rake angles impractical in titanium machining.

The milling force coefficients were also identified by the exponential force model (from

equation 4.13) and are given in Table 4.4. For comparison purposes, the predicted values

of exponential force coefficients can be determined from the ones calculated by the linear

edge-force model as follows:

K = Kre/ha + (x = t, r, a) (4.38)

After the cutting force coefficients are obtained for different average chip thickness ha,

then KT, KR, KA,P, q and .s can be obtained by exponential curve-fit.


Table 4.4: Cutting force coefficients for different rake angles as identified from millingtests by using exponential force model. Material: Ti6A14V. (The unit of the coefficientsis MPa.)

a Milling Test(deg) 7? p KR q KA

0 822 0.354 0.268 0.333 0.698 0.2505 799 0.326 0.165 0.402 0.457 0.09612 999 0.268 0.123 0.471 0.482 0.14615 540 0.404 0.134 0.432 0.509 0.15020 631 0.366 0.156 0.400 0.587 0.177Ib


4.4.4 Accuracy of Milling Force Calculation by Predicted Coefficients

The accuracy of the milling force predictions by calculated cutting force coefficients

(linear edge force model) has been tested for over 20 milling experiments. The exper

iments were performed using cutters with different rake angles (0, 5, 12, 15, 20) and

number of teeth (1 and 4), axial depth of cut (5 and 7.5 mm), feed per tooth (0.0127,

0.025, 0.05, 0.1, 0.2 mm/tooth) and radial depth of cut (slotting, up and down milling-

half immersion). The percentage deviations of the average and the maximum cutting

force predictions from the measurements are shown in Figure 4.20. About 80 % of the

force predictions in x and y directions have less than +10 % deviation and the maximum

deviation in all cases is less than 25 %. The predictions for the z direction, however, have

less accuracy as it can be seen from the figure. This can be attributed to the fact that

no edge force has been used for the z direction. Although, the edge force in z direction

is very small compared to x and y directions as shown in Table 4.3, at a small chip

thickness its contribution may become significant. Also, the noise in force measurements

may affect the results as the cutting forces in z direction are relatively small (< 100 N

for most of the cases considered in the statistical analysis).

A sample of half immersion up milling and half immersion down milling tests are

shown to illustrate the accuracy of the instantaneous force predictions. For up milling,

the entry angle is 0° and exit angle is 90°, whereas in the down milling the entry angle

is 90° and the exit angle is 180°. Figure 4.23 shows the measured and the linear edge

model predicted milling forces for a half immersion-up milling cutting test using a 19.05

mm diameter, four flute end mill. The feed per tooth is s = 0.05 mm/tooth and the

normal rake angle is 12°. The axial depth of cut and the cutting speed are 5.08mm and

30m/min, respectively. The force predictions based on the coefficients identified from


Fxmax

Mean-3.9

30

c 250

-4-

0

G) 15U)

-Q0

0

25

99

15

C0

-4-

0

ci)Cd)

.00

10

Io ) 0 u) () 0 U) 0(N - - I -

‘- (NI I

% Deviation

5

0o U) U) 0 U)

I - •-

% Deviation

o U)ccJ

C0-4-

0

ci)U)

.00

50

40

30

20

10

0

Fymax

Mean -0.1

40

2 30-4-

255 2015

50U) 0 it) it)

- - I

% Deviation

0

25

o U) 0 it) it) 0 I1)(N - - I

% Deviation

C0-I

0

ci)Cl)

-o0

15

10

5

0

FzmaxMean -14.7

IIuit) 0 it) 0 it) it) 0 it) 0 it) 00(N (N - - - (N (N C’)

I I

% Deviation

20

C2154-

0

cj 10U)

.005

0

FzaMean -19 — —

--

U)Oit)it)QLOOU)OQ‘- ‘- I - — (N (N C’) ‘

% Deviation

Figure 4.20: The statistical error analysis of the milling force predictions. The percentageerror in the milling force predictions compared to the measured values were determinedfor over 20 different milling tests.


the slot millings tests and those transferred from the orthogonal data by calculating

show good agreement with the measured values. Figure 4.24 shows the predicted and

measured forces for a half immersion-down milling test with a four flute end mill where

= 0.0127 mm/tooth and the rake angle is 0°; the other conditions were the same as

in up milling. The accuracy of the force predictions by using transformed data is almost

the same as the accuracy obtained by the milling test calibrated coefficients.

Figures 4.23 and 4.24 show the cutting forces predicted by the exponential force

model. Although the accuracy of the milling force predictions by the exponential force

model is acceptable, it is low compared to the linear edge model predictions. The expo

nential force coefficient model, in general, gives better results in milling force predictions

by calibration than the ones shown in Figures 4.23 and 4.24. A relatively small axial

depth of cut results in a very high variation of the chip load on the flutes in every flute

period. This results in some inaccuracies as the average cutting force coefficient (which

corresponds to the average chip thickness) is used in the exponential cutting force coef

ficient models. However, statistical analysis and the analysis of the instantaneous forces

show that the accuracy of the cutting force predictions is quite satisfactory with the edge

force models.

The approach allows designing a common orthogonal cutting data base, which can be

used to predict cutting forces in a variety of oblique machining operations. Integration

of such a database to NC tool path generation algorithms in CAD/CAM systems allows

process planners to generate optimal, chatter and tool breakage free tool paths. The data

base is also useful in analyzing the performance of different cutter design geometries prior

to cutting tests.


4.5 Summary

Mechanistic milling force models have been reviewed. An improved mechanics of

milling approach is presented. In this model, the chip flow angle is numerically calculated

which improves the accuracy of predictions. The model predictions are verified by number

of experiments, and compared with the mechanistic model predictions.


Figure 4.21: Measured and simulated milling forces (linear-edge force model). Forceswere calculated by using the transformed milling force coefficients from the orthogonaldata (predicted) and the milling calibration tests (calibrated). (Material: Ti6A14V,half immersion-up milling, t=°O5 mm/tooth, a=5.08 mm, V=30 rn/mm; tool: 4 flutecarbide end mill, a = 12°, d =19.05 mm.)

800

0 45 90 135 180 225

Rotation Angle (deg)

Figure 4.22: Measured and simulated milling forces (linear-edge force model). (Material:Ti6A14V, half immersion-down milling, s=0.0127 mm/tooth, a=5.08 mm, V=30 rn/mm;tool: 4 flute carbide end mill, a = 0°, d =19.05 mm)

800

600

400

200

o 0Ii-

-200

-400

-6000 45 90 135 180 225


270 315 360

zci)0LI

600

400

200

0

-200

-400

270 315 360


z

C)0LI

Figure 4.24: Measured and simulated milling forces (exponential force model). (Material:Ti6A14V, half immersion-down milling, t=00127 mm/tooth, a=5.08 mm, V=30 rn/mm;tool: 4 flute carbide end mill, a = 0, d =19.05 mm)

0 45 90 135 180 225 270 315 360

800

600

400

200ci)C.)0 0LI

-200

-400

-600


Figure 4.23: Measured and simulated milling forces (exponential force model). Forceswere calculated by using the transformed milling force coefficients from the orthogonaldata (predicted) and the milling calibration tests (calibrated). (Material: Ti6A14V,half immersion-up milling, St=O.O5 mm/tooth, a=5.08 mm, V=30 m/min; tool: 4 flutecarbide end mill, a = 12°, d =19.05 mm.)

800

600

400

200

0

-200

-400

0 45 90 135 180 225


270 315 360

Chapter 5

Effects of Milling Conditions on Cutting Forces and Accuracy

5.1 Introduction

Tolerance requirements on machined parts limit material removal rates in finish

end milling operations as end mill and workpiece deflect under milling forces causing

dimensional errors. Surveys have indicated that typical metal removal rates in finish

milling operations are only a fraction of those which should be used for either minimum

cost or maximum production rates as very conservative feedrates are used to ensure that

the part will pass inspection [153, 154]. Therefore, analysis of dimensional accuracy in

milling is necessary to improve the productivity without violating the tolerances. In this

chapter, a surface generation model with flexible end mills is introduced. The effects

of radial depth of cut and the feedrate on the cutting forces and the surface error are

analyzed. A method of constructing optimal cutting conditions which provide minimum

dimensional error is demonstrated. A similar analysis is performed for end milling of

very flexible plates in Chapter 7.

5.2 Surface Generation by Statically Flexible End Mill

The flexible cutter deflects under the periodically varying milling forces which are

modeled in the previous sections. As given in Chapter 3, the end mill is modeled as a

cantilever beam attached to the collet with linear springs in both x and y directions as

shown in Figure 5.1.

104

Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy 105

z

Figure 5.1: Statically flexible end mill model. The tool is modeled as a cantilever beamwith springs attached to its fixed end to account for the clamping stiffness. Due to thehelical flutes on the tool, an effective diameter is used in the deflection analysis.

The tool is divided into a number of elements with equal length. The cutting force

produced by an element of flute j is found from equation (4.8)

= { - cos2(z) + Kr(2j(z) - sin2j(z))]k

(5.1)

= tSt [2(z) — sin 2(z) + Krcos2j(z)]k

where zk represents the z axis boundary of the cutter at node k (see Figure 5.1). The

axial boundaries are modified if they do not match with the nodes on the tool. The

elemental cutting forces are equally split by the nodes k and k — 1 bounding the tool

element i. The deflection in y direction at node k caused by the force applied at node m

2k

-n+1

-2—1

Y


is given by the cantilever beam formulation as [113]

____

L\Fm6EJ(3h1m_)+ k ,O<Vk<Vm

5(k,m) = (5.2)

LFmV2

___

6EJ(3V1m)+ k ,l’m<l’k

where E is the Young Modulus, I is the area moment of inertia of the tool, k is the

experimentally measured tool clamping stiffness in the collet and vk = 1 — z, 1 being

the gauge length of the cutter. The area moment of the tool is calculated by using

an equivalent tool radius of Re = .sR, where s ( 0.8) is the scale factor due to flutes

[114]. In this deflection model, the contact stiffness between the workpiece and the tool is

neglected. Deflections in the x direction can be found similarly. The total static deflection

at nodal station k is calculated by the superposition of the deffections produced by all

(n + 1) nodal forces on the tool

n+1 n+1

= 5(k,m) ; 5,(k) = (k,m) (5.3)m=1 m=1

The finished workpiece surface is generated by points on the helical flutes as they

intersect it. The surface generation occurs as the points on the helical flutes satisfy the

following immersion conditions,

1 0 up — milling(5.4)

( ir down — mzllzng

The axial coordinate (z) of the flute - surface contact point in axial direction z can

be determined as a function of the tool rotation angle, q, as,

z() = up — milling

__________

(5.5)z()

= +JpK down—milling

Note that depending on the cutting geometry, there may be several flutes and contact

points generating the surface and they can be determined from equation (5.5). In the


simulations, the cutter is rotated at 0 angular increments such that the contact point for

a flute jumps from one node of the cutter to the next one, i.e. 0 = and Liz is

the axial length of a tool element. At node k, the deflection of the cutter in y direction,

6(k), is imprinted on the surface as dimensional error e(k).

e(k) = 6(k) (5.6)

The dimensional surface error profile along the axial depth of cut is simulated by deter

mining the error created at each node as the cutter is rotated at 0 angular intervals.

5.3 Identification of Cutting Conditions for Minimum Dimensional Surface

Error

Dimensional surface error magnitudes are the primary concerns in finishing opera

tions. It is common practice to use very low feedrates and radial width of cuts to obtain

the required accuracy. However, this approach presumes that the dimensional surface

error is proportional to the material removal rate (MRR). Researchers such as Wang

[33] attempted to estimate the cutter deflection and surface error in end milling by using

MRR. The agreement with experimental values is poor as the real force components and

their distribution on the cutter flutes are not considered. Therefore, a detailed analysis

of the cutting condition - dimensional surface error relationship is necessary to improve

the process planning in end milling operations. This is partly accomplished by the force

and surface generation models presented in the previous sections and will be extended in

the following.

The material removal rate in an end milling operation depends on the spindle speed. It

is better to use material removed per revolution MFR which is independent of the speed


as a measure of material removal. For end milling operations, MFR can be expressed as

MPR — abstN (5.7)

where a is the axial depth of cut (mm), b is the radial depth of cut (mm), .s is the feed per

tooth (mm), N is the number of teeth, and MFR is the material removed per revolution

(mm3/rev). Equation (5.7) indicates that the amount of material removed is linearly

proportional to a, b and s. The cutting forces and surface error, however, do not vary

linearly with the feed per tooth, .s, and radial width of cut, b, due to two main reasons.

First of all, the cutting coefficients given by equation (4.13) are nonlinear functions of

the average chip thickness which depends on st and b. In addition, the cutting forces

are not proportional to the radial width of cut, b, as it can be seen from equation (4.8).

Therefore, it may be possible to optimize b and .s such that surface errors are less than

the tolerable values without reducing, even increasing, MPR. A suitable index for this

purpose is the ratio of maximum dimensional error, emax, to the MFR which will be

termed as specific maximum surface error (SMSE).

SMSE1b— emax(b,st)

,8t)— MPR(b,s)

In other words, SMSE(b, .St) indicates the maximum error generated on the workpiece

surface in order to remove 1 mm3 material for a specific combination of b and s. There

fore, the optimization procedure reduces to determination of b and s values for which the

SMSE is minimum. The analytical surface generation model described in the previous

section allows the identification of suitable cutting conditions (b, .St).

The optimization procedure applies to both up and down milling operations. However,

in up milling the effect of radial width, b, on the SMSE is enhanced due to a mecha

nism explained as follows. In up milling the components of tangential, F, and radial,

Fr, forces in y axis are in opposite directions as shown in Figure 4.1. Some portions of


the tangential and radial forces cancel each other, resulting in lower normal forces (F).

The same cancellation mechanism results in lower feed forces, F, in down milling, which

does not contribute to surface generation. Up milling is usually not preferred in finishing

operations as it usually produces poor surface finish. In up milling, the chip thickness

is zero when a tooth starts cutting. In the transient part of the process, the tooth rubs

over the surface and then ploughes the metal until the piled up work material is pushed

up along the rake face to form a chip [155]. The surface finish is poor as the surface

is generated during the transient part of the cutting process. However, if the surface is

ground after the finish milling operation, then the dimensional accuracy of the surface

is much more important than the surface quality. The radial width of cut (b) has the

highest effect on the force cancellation. In peripheral milling, the exit angle, can be

used to represent the immersion of cut as,

= cos’(l—

(5.9)

For small radial widths (çe,v < 15°), the normal force is mainly composed of radial force

vectors, therefore it is negative in up-milling. For large radial engagements(qeT 90°),

however, tangential forces contribute to F most and thus it becomes positive. Therefore,

the sign of the average normal force,,

changes from negative to positive as cex varies

between (0° — 90°). The average normal force becomes zero for a particular exit angle

which will be designated by q. At this point the peak normal forces are also minimum.

However, the same comments cannot be made for the dimensional errors on the surface

since they do not only depend on the magnitude of the force, but its distribution along

the helical flutes as well. On the other hand, for relatively small axial depth of cuts it

can be assumed that the maximum surface error generated on the surface is proportional

to the maximum normal cutting force, Fymax An approximate analytical procedure may

be followed to demonstrate the optimal exit angle identification procedure. The average


normal cutting force in peripheral up milling is given by,

=—

sin2q — Kr Sfl2 qex) (5.10)

Since the magnitude of the mean force with respect to the exit angle is desired to be

minimized, the square value should be differentiated with respect to exit angle After

derivations, the following equation for optimal exit angle, , is obtained.

— (0.5 + Kr°x) sin2c5, + 0.25 sin4q — Kr sin2g5

+0.5Kr Sj2 2ç + K sin 2ç& sin2— cos (5.11)

17 2,Ao ()o_T1ki. sin Yex cos —

Equation (5.11) is approximate as the variations of the cutting coefficients, K and Kr, are

neglected in the differentiations. Equation (5.11) can be solved by the Newton-Raphson

iterative technique for different values of Kr as shown in Figure 5.2. As Kr increases,

the optimal exit angle, q, increases. The relation can be expressed by an approximate

equation

= 60.8K,. — 15.5K (5.12)

An average value for Kr should be used in equation (5.12). The analytical formulation

presented above is approximate but it is quite practical for process planning of peripheral

milling operations. If the average value of K,. is known for a material - helical end mill

pair, an approximate optimal exit angle can be directly determined from equation (5.12).

The exact values of an optimal radial depth and feedrate can be determined by using

the cutting force and surface generation models presented in the previous sections. The

variation of maximum dimensional surface error is obtained by milling simulations for a

range of radial depth of cuts and feed-rates. The maximum surface errors, peak normal


60

50

40

10

00 0.2 1.4 1.6

Figure 5.2: Variation of the optimum exit angle (for up-milling) with Kr.

cutting forces and SMSE can be plotted as a function of cutting conditions. The feasible

cutting conditions can be determined from the maximum surface error vs. radial width-

feed graphs according to the tolerance requirements. The procedure is illustrated by

experimental and simulation results in the next section.


Simulation and experimental results are given in two groups. The first group is to

verify the force and deflection models presented . The second group demonstrates the

identification of optimal cutting conditions for minimum dimensional surface error.

5.4.1 Cutting Force and Surface Finish

Several half immersion up and down milling simulations and experiments have been

carried out to verify the surface finish model presented. The cutting forces were mea

sured by a Kistler table dynamometer, and a dial gauge was used for the surface profile

0.4 0.6 0.8 1 1.2


measurements. The work material, 7075 Aluminum alloy, was milled by a four fluted high

speed steel (HSS) end mill with 30 degree helix angle. In both cases, 4 different feedrate

values were used at a spindle speed of 478 rev/mm or a surface speed of 28 m/min. The

cutting constants were identified for up milling and down milling separately. Common

cutting conditions for the tests and simulations presented here are summarized in Table

5.1, under Case # 1 and Case 2.

Sample simulated and measured cutting forces for a feedrate of st = 0.14 mm/tooth

are shown in Figures 5.3 and 5.4 for half immersion up and down milling tests, respec

tively. The agreement between the simulation and experimental force results are quite

satisfactory. It has to be noted that due to a reversed tangential cutting force vector,

the magnitude of the normal forces are about 500 N higher in down milling than in the

case of up milling.

The simulated and measured surface profiles for 4 different feedrates are shown in

Figures 5.5 and 5.6. Here, the surface dimension errors are measured from the intended

reference finish surface, which was marked by using precision machining before the exper

iments. In the up milling operations, because the cutting forces push the cutter towards

the finish workpiece surface, extra material is removed from the desired finish surface

causing overcut conditions. As expected, the maximum surface error occurs at the most

flexible part of the end mill which is its tip. The dimensional errors reduce as the axial

depth location is closer to the cantilevered side of the helical end mill. The dimensional

errors in down milling operations are shown in Figure 5.6. Because the normal cutting

forces deflects the cutter away from the workpiece, extra material is left on the finish

surface causing undercut conditions. The magnitudes of the dimensional errors in down

milling is larger than those in up milling for two reasons. First, the magnitudes of the


normal F cutting forces are larger in down milling. Second, in up milling the tool is

pushed into the material which resists against the deflection. There is no resisting con

tact stiffness in down milling, because the tool deflects away from the workpiece towards

free air. In half immersion down milling, the dimensional errors are almost shifted in

the y axis, whereas in up milling they reduce approximately in a linear fashion. This is

explained as follows: In down milling, cutting force intensities are large at higher axial

depth locations where the rigidity of the cutter is higher as well. This counterbalancing

mechanism results in, somewhat close to, the constant deflection along the axial depth

of cut. The opposite phenomenon occurs in up milling operations, and the large deflec

tions are observed at the tip of the flexible end mill. The maximum difference between

the predicted and measured surface errors is about 15 % in both cases. As it can be

seen from the figures, the surface error predictions are in satisfactory agreement with the

measurements.

5.4.2 Selection of Optimal Cutting Conditions

Up milling experiments were performed on free machining steel (AISI 1040) by us

ing a carbide end mill with a 300 helix to demonstrate the optimal selection of cutting

conditions. The spindle speed was 478 rpm. The cutting constants are given in Table

5.1, under Case #3. The absolute values of maximum surface error, emax, generated

and )SMSEI are determined by the simulation for a range of radial width of cuts and

feedrates as shown in Figures 5.7 and 5.8.

Figure 5.7 shows that the simulated maximum surface error remains almost constant

until the radial width is about 3.3 mm. After 3.3 mm the surface error increases con

siderably. Therefore, the feasible region of a radial depth of cut and feed per tooth can

be determined from Figure 5.7 according to the tolerance requirements. The next step


Table 5.1: Cutting conditions for up and down milling tests conducted for surface errorverification and identification of optimal milling conditions. Case # 1 and # 2: MaterialAl7075 with normal chuck. Case # 3: AISI 1040 with stiffer power chuck.

CUTTING PARAMETERS Case #1 Case #2 Case #3Mode of Milling Up Down UpTool diameter - d0 (mm) 19.05 19.05 19.05Tool gauge length - 1 (mm) 54.5 54.5 50.0Clamping stiffness - k (N/mm) 10200 10200 25000Tool material HSS HSS CarbideChuck regular regular powerTangential force coeff. - KT (MPa) 546 437 1140p 0.246 0.343 0.28Radial force coeff. - KR ( - ) 0.270 0.180 0.470q 0.271 0.249 0.079Radial width of cut - b (mm) 9.525 9.525 1, 3.35, 5.5Axial depth of cut - a (mm) 19.05 21.59 19.05Immersion angle of cut -q (deg) 90 90 26.5, 49.5, 65Flute lag angle -ka (deg.) 66.2 75 66.2Number of nodes (n + 1) 50 50 50

is the selection of optimal conditions from the SMSE graph shown iii Figure 5.8. The

optimal conditions correspond to the minimum value of SMSE as explained before.

Figure 5.8 shows that the optimal conditions describe a curve on the radial width-feed

per tooth plane. The graph indicates that SMSE values are very large for small width of

cut and feed per tooth values. This can be attributed to the high percentage of parasitic

edge components in the total cutting forces at a low chip thickness. To verify the results

presented in Figures 5.7 and 5.8, a set of experiments were performed at three different

radial widths (1, 3.35, 5.5 mm) and feedrates at (0.01, 0.06, 0.1 mm/rev-tooth). The

absolute values of maximum normal force, Fymax, and maximum surface error,

obtained from experiments and simulations are shown in Figures 5.9 and 5.10. There

is almost a linear proportionality between Fymax and emax because of a relatively small

Chapter 5. Effects of Milling Conditions on Cutting FOrces and Accuracy 115

axial depth of cut. The agreement between the experimental and simulation results is

quite satisfactory. As it was observed from the previous graphs, Figure 5.10 shows that

at about b = 3.35 mm the surface errors are minimal. Compared to b = 1 mm, the

maximum surface error is almost the same even though the MPR is more than tripled.

Another important point is that at b = 3.35 mm, high feed per tooth values can be

used without increasing the surface error significantly. First of all, use of high feed-rates

further increase the productivity or MFR. Furthermore, the surface finish quality can

be increased by using high feed per tooth values as very rough surfaces are obtained by

small feedrates in up milling due to rubbing. A further increment on the radial width

to 5.5 mm results in approximately tripled dimensional error magnitudes. SMSE for the

same feeds and the radial widths is shown in Figure 5.11. Analysis of the Figures 5.10

and 5.11 confirms that the optimal radial width is 3.35 mm for a feed-rate larger than

0.06 mm.


1500

1000

500

0

-500

-1000

-1500

-2000

-2500 -

0 45 90 135 180 225


Figure 5.3: Measured and simulated milling forces for half immersion-up milling. (Material: 7075 aluminum alloy, St = 0.14 mm/tooth, a = 19 mm, V 28 m/min. Tool: 4flute HSS end miii, 300 helix, diameter d = 19.05 mm)

2500

2000

1500

1000

500

0

0 45 90 135 180 225


270 315 360

Figure 5.4: Measured and simulated milling forces for half immersion-down milling. (Material: 7075 aluminum alloy, s = 0.14 mm/tooth, a = 21.6 mm, V = 28 rn/mm. Tool: 4flute HSS end miii, 30° helix, diameter d = 19.05 mm)

za)20

LI

simulation experiment

270 315 360


Figure 5.5: Simulated and measured surface profiles for half immersion-up milling. (Material: 7075 aluminum alloy, a = 19 mm, V = 28 m/min. Tool: 4 flute HSS end mill, 300

helix, diameter d = 19.05 mm.) (z=0 at the free end of the end mill.)

Surface Error (microns)

Figure 5.6: Simulated and measured surface profiles for half immersion-down milling.(Material: 7075 aluminum alloy, a = 21.6 mm, V = 28 rn/mm. Tool: 4 flute HSS endmill, 30° helix, diameter d = 19.05 mm)

18

15

- 12E-c 900

as

3

00 50 100 150 200 250

Surface Error (microns)

28

24

E20

16

12

4

0-300 -250 -200 -150 -100 -50 0


E ci:,

. Th. ‘ ‘——-

2-0

Figure 5.7: Variation of the predicted maximum dimensional surface error due to tooldeflection with the radial depth of cut and the feed per tooth. For certain values of radialdepth of cut, the error is minimum although the material removal rate is not. This graphis useful in selecting the cutting conditions for improved surface accuracy. (Material:Free machining steel, a = 19 mm, V = 28 rn/mm. Tool: 4 flute carbide end mill, 300

helix, diameter=19.05 mm)

C..

•

•

0

: cc

0 ciS

ITS

E(rn

tc-rev

mn

,J)

02

46

870

/2

0

.CD

—‘ o•

—.

CDCD

—.

CD-q

a-,-

Cl)

C,)

II•

—-

cc—

..

,c—

<C

.n

,

CD

CD

<c-t

—C

0CD

(Cq

CD

0-

CDCD

Cl)

CD

ôC

DcX

cD

’< —

.

CD CDQ

----

,

CD O(I

)• —

—.

$:2

•C

DeC

DC

D

CDCD

0—

.

cCD .

•

CD—

.Ci

)CD

CD-

...

-.

-.

0<

IO(

-

I ItC (

t

(.


2000

1600

. 1200xE>

LA

400

0

Radial Width of Cut (mm)

Figure 5.9: Variation of measured and simulated maximum normal cutting force Fymavwith radial depth of cut for different values of feed per tooth. The tool deflections due tothe normal force are imprinted as dimensional errors on the finished surface. (Material:Free machining steel, a = 19 mm, V = 28 rn/mm. Tool: 4 flute carbide end miii, 300

helix, diameter= 19.05 mm)

0 1 2 3 4 5 6 7 8 9

Chapter 5. Effects of Milling Conditions on Cutting Forces and Accuracy

250

/

/150 /

EE

LUC,)

C’)

121

Simulationst0.1

Experimentst=0.06200

E

100

50

st=0.01 //

//

st=0.01 st=0.06 — — — — st=0.1

00 1 2 3 4 5 6 7 8 9

Radial Depth of Cut (mm)Figure 5.10: Variation of measured and simulated maximum dimensional surface erroremax with radial depth of cut for different values of feed per tooth. The predicted optimumradial depth of cut is verified experimentally. (Material: Free machining steel, up milling,a = 19 mm, V = 28 m/min. Tool: 4 flute carbide end mill, 30° helix, diameter=19.05mm)

12

10

______ ______

8

6

4

2

0• I I I I I I

0 1 2 3 4 5 6 7 8 9Radial Depth of Cut (mm)

Figure 5.11: Variation of measured and simulated specific maximum surface error SMSEwith radial depth of cut for different values of feed per tooth. (Material: Free machiningsteel, up milling, a = 19 mm, V = 28 m/min. Tool: 4 flute carbide end mill, 30° helix,diameter=19.05 mm)


5.5 Summary

Surface generation model with a flexible end mill is presented. The effects of the

feedrate and the radial depth of cut on the surface accuracy and the milling forces are an

alyzed. An optimal milling condition identification method is presented. It is shown that

the surface accuracy can be significantly improved without decreasing, even increasing,

the material removal rate. The simulation results are verified experimentally.

Chapter 6

Static Structure-Milling Process Interaction

6.1 Introduction

In the previous chapter, milling forces and tool deflections are modeled indepen

dently. However, the cutting process and the structural deformations of the tool and

workpiece interact during milling. The dynamic interactions cause chatter and forced

vibrations which are analyzed in Chapter 8. The static interactions result in variations

in the cutting forces, and thus have to be considered for accurate force and deflection

predictions. Sutherland et al. [24] and Armarego et al. [36] analyzed the statically

flexible milling process by numerical models. They determined the chip thickness in a

flexible milling system by considering deflections of the tool in successive tooth periods.

The effects of deflections on the chip thickness and the cutter-workpiece engagement

boundaries are determined together due to the numerical procedure followed. In this

chapter, the effects of static deflections are considered in two groups: local and global

effects. The local effect is the change in the chip thickness due to deflections which is

modeled by the statically regenerative milling force model, whereas the variation in the

cutter-workpiece engagement boundaries is considered as a global effect of the deflections

and it is analyzed in the flexible milling force model. An analytical formulation is given

which shows that the effect of deflections on the chip thickness diminishes very fast. The

regenerative milling mechanism is important for the adaptive control of the milling forces

by feedrate as analyzed in [22, 43]. However, for steady-state milling force and surface

123

Chapter 6. Static Structure-Milling Process Interaction 124

error predictions, only the variation in the engagement boundaries or radial depth of

cut should be considered. Determination of the effective radial depth of cut under the

deflections is explained in this section, however, the application of the method is given

in Chapter 7 for plate milling analysis. Therefore, the analytical formulations for the

structure- milling process interactions given in the following sections eliminate the time

consuming numerical solutions and give more insight to the real physics of the problem.

6.2 Statically Regenerative Milling Force Model (Variation of Chip Thick

ness Due to Deflections)

Figure 6.1 shows the generation of the tooth path by flute j at axial depth of z and

tooth period number i. The tooth period count i is started from the instant when the

reference flute’s tip is at zero immersion (j(O) = 0) in the beginning of the cut. i is

increased by one when the next flute reaches the same location, i.e. one pitch angle (q)

later. As a result of the deflections of the tool in feed (x) and normal (y) directions, the

flutes may lose contact with the workpiece at some points. Therefore, unlike the rigid

milling, the surface being cut by tooth j may not have been created by the previous

tooth only. Here, notation will be used to denote the number of previous tooth

periods which may have an influence on the presently cut surface at tooth period i by

flute j. The value of chailges as the cutter rotates. It is assumed that the value

of remains the same for flute j along the axis z. From hereon, the notation

will be used in place of mj,j(q) for simplicity in the following formulation. In Figure

6.1, O(z), O_,(z), O(z) and O_(z) are the undeflected and deflected tool centers at

axial location z, respectively. Broken and solid lines indicate the tooth paths of the

rigid and flexible cutters. Only tool deflections are considered in this section, however,


Y

Figure 6.1: Statically regenerative chip thickness geometry in milling. The material lefton the cut surface due to the deflections is encountered by the following teeth. Thedeflected positions of the tool in the successive tooth periods have to be considered todetermine the chip thickness. Also, the contact between the cutter flutes and the workmay be lost at some points due to large deflections. This nonlinearity has to be consideredin the analysis as well.

(i — M)tl flexibletooth path

—

(1 i’Y” rigidtooth path

i flexible toothpath

B’

R

i” rigid tooth path

x


the following formulation can easily be extended to include the workpiece deflections by

adding them to the tool deflections. In previous sections, it was shown that the tool

deflections are dependent on the cutting force intensities, and for tooth period i they are

6(,z) = ; (z) (6.1)

where dF(q, z)/dz and dF(q, z)/dz are defined as cutting force intensities at the tooth

period i. Assuming that the axial and radial depths of cut and the cutting parameter

Kr remain the same under flexible and rigid cutting conditions, the unit cutting force

intensity functions are also identical in both regenerative and rigid cutting force mod

els. Considering the changes in the chip load due to the regeneration mechanism, the

regenerative cutting force intensities can be approximated as the product of the total

regenerative cutting force and the unit rigid cutting force intensity

dF(qf,z) dF,(,z)(6.2)

where and F1 are the regenerative cutting forces at the tooth period i, and df(g5, z)/dz

and df(q5, z)/dz are the unit rigid force intensities given by equation (4.10). In the for

mulation, superscript R is used to indicate rigid cutting forces. In this analysis, F is not

considered as it does not take part in the regeneration mechanism. However, F is af

fected by the deflections in x and y directions. Since the cutter deflection at a particular

location is linearly dependent on the force magnitude, equation (6.1) can be scaled with

total force magnitudes,

= F (q) ; 6(gS,z) = (q5)6 (6.3)

where the force magnitude independent unit deflections are defined as

dfR(,z) . dfR(c,z)X — X’’PZ

d ‘ “ —

d


Due to cutter deflections, the chip thickness may change at each flute immersioll

q, and previous deflections or regeneration marks symbolized by counter p. Due to

regeneration, the tooth j experiences an actual chip thickness of A’B’ at tooth period

i, where A’ is on the arc cut at period i—

t, and B’ is on the presently cut arc, see

Figure 6.1. Between two arcs, the cutter center is moved at amount of 1Ust distance in

feed direction. From the exaggerated view of Figure 6.1 it can be argued that when the

point A’ was being cut tooth period before, the immersion angle of the tool, was

not exactly equal to the present immersion çj. However, a careful look at the geometry

would reveal that ç&j as R >> [tSt. Therefore, for the tooth period i the chip

thickness at any z position on the tooth j can be expressed as follows

z) = A’B’ = AB + BB’ — AA’

or

z) = Itst sin b(z) + (g5) cos

cos (z) + z) (6.5)

sin qj(z) — sin qj(z)

The unit deflections defined by equation (6.4) are the same for every tooth period as they

do not consider the regeneration in chip thickness. In equation (6.5), however, possible

tool-workpiece contact losses are considered for tooth periods which are indicated by

unit deflections with subscripts . This is called the basic nonlinearity in the regeneration

mechanism, and it occurs when deflection magnitude BB’ is equal to or greater than

the chip thickness h1,(q, z). Considering the direction sign from the geometry, the tool

material contact loss is modeled iii the formulation as.

sin qj(z) + cosçj(z) = —h,(q5,z) (6.6)


where h,(q) is given by equation 6.5. The above nonlinearity condition is reorganized

as— —h,(,z)

— F) sin (z) + F) () cos

(6.7)— —h,(g,z)

yi — —

F() () sin(z) + F)cos(z)

The chip thickness which includes the effects of tool deflections and the regeneration,

equation (6.5), is substituted in equations (4.1) and (4.6) to obtain the flexible cutting

force intensities as

dF,(,z)= K [cos(z) + Kr sinth(z)] {st sin(z)+

- cos

— sin

(6.8)z)

= K [sin(z) — Kr cos (z)] {‘St sin (z)+

- cos(z)+

[F(1 sinqj(z)}

Equation (6.8) can be integrated along the in cut portion of the tooth to obtain the total


cutting forces realized by the tooth

() = —F (Xj,j () + F(_) ()cX() @)—

(qy () + FX() @)7x(_),, (q5) +

(6.9)

(q) F ()cx,(g5)— (q)+

— +

where the so-called flexibility terms are given by the following expressions

Z32(cb) 1= Ktf() (cos2 (z) + Kr sin 2(z))

fZj,2(,b) /1= KtJ çsin2qj(z) + Kr Sifl2 cbj(b)) 6dz

Zj,i()

(6.10)

= Kj3’2 ( sin2(z)

— Kr cos2 (z))

= KtJ22 (sin2 (z) — Kr sin 2(z))z,j (ç)

The flexibility terms of the (i — )th step are similar except and are to be replaced

by and The above integrals are to be computed numerically by using the tool

deflection values at the nodal points. At this point an approximation will be introduced

for the sake of simplicity by assuming that at the tooth period i and immersion angle q,

is the same for all of the teeth which are in contact with the workpiece. This means

that, at a specific i and çS , if the tooth 0 is cutting the surface which was cut by 2 tooth

periods before, then all of the teeth which are in contact with the workpiece are cutting

the surfaces which were cut 2 tooth periods before. From equation (6.8), the flexibleq)


cutting forces applied on each tooth can be summed up to obtain the following equation

[A()] {F()} - [A()] {F()} + {F()} = {} (6.11)

where the so-called flexibility matrix is given by

[A()j= 1+ () a)

(6.12)7(g) 1—am

whereN-i

=

j=oand the other flexibility terms are similar. The flexible and rigid force terms in equation

(6.11) are defined respectively as,

F) FR(){F(ci)} = ; {FR(q)} = (6.13)

F(q) J F(q) JFor the surface generated by a flute which is free of cutting marks or regeneration,

equation (6.11) reduces to the following form

[A()]1 {F()}1 = {FR()} (6.14)

from which the regenerative cutting forces for the very first tooth period can be obtained

as

{F()}1 = [A()j1{F)} (6.15)

Cutting forces calculated by equation (6.15) include only the current deflections of the

tool. However, the surface left by the first tooth will be regenerated by the successive

teeth of the cutter. If the chip thickness for the first tooth reduces due to the deflections,

the next tooth will have to remove the extra material. After computing the forces for the


first non-regenerative tooth period, (6.11) gives the flexible regenerative cutting forces

for the successive tooth periods

= [A()]’ {{FR()}- {F()} + [A()]1{F()}] (6.16)

t can only be determined in an iterative manner due to two reasons. First, t cannot

be determined independently as the total flexible cutting forces are required in equation

(6.5). The second reason is the nonlinearity defined on the deflections by equations (6.6)

and (6.7). At each iteration step, ji is determined such that it gives the minimum chip

load. The flexible cutting forces calculated in the previous tooth period can be used in

equation (6.5) in the first iteration step. Modifications of deflections are followed by the

redetermination of t in an iterative ioop until the convergence in and the flexible forces

are obtained. Then, the cutter is rotated by a certain incremental amount and the same

iterative loop is repeated.

For an approximate analysis, the nonlinearity expressed by equation (6.7) can be

neglected, i.e. it is assumed that the contact between the workpiece and tool is not lost

during the regeneration process and = 1. Then, the unit deflections are equal for

successive tooth periods

xi =L_1 6Yi—1 (6.17)

It can be seen from equations (6.10) and (6.12), the flexibility matrix remains constant

[A(g5)j = [A(qf)1 = [A(çi)]_1 (6.18)

Substituting into equation (6.11) the following is obtained:

{F()} = [A()]’ {F)} - [[A()]1- [‘1] {F()}1 (6.19)

where [I] is the (2x2) identity matrix. The cutting forces for the first pass are

= {A()]’ {F’)} (6.20)


This can be substituted in equation (6.19) to obtain the forces in the second tooth period

{F()}2 = [A()]’ [2 [I] - [A()]-’] {F)} (6.21)

Similarly, forces in the 3rd and 4th tooth periods are obtained as

{F()}3 = [A()]1 [3 [I]-3 [A()]’ + ([A)r1)2]{FR()}

2 (6.22)[A()]’ [4 [I]-6 [A()]’ +4 ([(A()r’)

- ([A()r1)3]{FR()}

The following general form for the regenerative cutting forces is obtained by intuition:

{F()} = [[I] - [[I]- [A()]1]t] {FR()} (6.23)

Therefore, the regenerative cutting forces can be related to the rigid cutting forces ana

lytically. The basic form of the regenerative cutting forces was proposed by Tlusty [156].

Tomizuka et al. [157] derived the discrete form and Spence and Altintas [43] used it in

the adaptive control of milling forces. Equation (6.23) is more accurate than the previ

ous models as the regeneration of the chip thickness along the helical flutes of the cutter

is modeled, whereas in the previous models, a point contact between the tool and the

workpiece was considered.

A zero helix cutter case will be used to demonstrate the behavior of milling forces

predicted by equation (6.23). For the zero helix case, the coefficients given by equation

(6.10) can easily be determined by multiplying the integrand by the axial depth of cut

a. Let us consider a particular rotation angle of = ir/2. Then,

cr, = 0; = KtKra; a = 0; = Ktar


For relatively short axial depth of cuts, the unit deflections can be approximated as

constant in the cut portion of the flute. Then, unit deflections become equivalent to the

compliance of the cutter

= ; = (6.24)

where k and lc are the stiffness of the cutter in x and y directions, at the tip of the

cutter (or at the midpoint of the axial immersion for a better approximation). Also,

assume that only one flute is in cut when qf = ir/2, e.g. 3 flute cutter in full immersion.

Then,KtKra Kta

7x = ; = (6.25)

and

l+Ya 0[A(ir/2)] = (6.26)

1

Considering practical values of K, Kr, a and k it can be seen that Yx, Yy < 0.01. This

suggests that , [A(ir/2)] [I] and {F(7r/2)} {FR(7r/2)}, and thus the regenerative

cutting forces converge to the rigid cutting forces very fast. The result obtained from

this particular case is generalized by the discussion given below.

For a vibration free cutting process, the cutting forces must stabilize after a transient

period. Here, the stability means that the forces converge to the force values obtained

in the previous tooth period. Therefore, after the steady-state is reached in the flexible

cutting forces, i is always 1 as each tooth has to cut the surface left by the previous

tooth. Mathematically, this can be expressed as

= {F(q5)}_1

and

= [A(q)1_1


If these conditions are imposed on equation (6.11) the following equality is obtained

= {FR()}

Therefore, in a vibration free cutting process the flexible or regenerative cutting forces

converge to the non-regenerative rigid cutting forces after some transients. This is an

expected result as the chip thickness removed with a flexible cutter and a rigid cutter

are the same after some transients. However, as we analyzed in the surface generation

mechanism, some material is left on the finished surface as surface error. This does not

affect the chip thickness as it is not on the machining surface (the surface between st

and qex), but it changes the radial depth of cut as it will be analyzed in the next section.

The formulation given above is programmed into the computer. The nonlinearity

given by equation (6.7) is considered in the numerical solution. As an example, the sim

ulated regenerative and rigid cutting forces in x direction are shown in Figure 6.2. The

axial depth of cut and the immersion angle are 19 mm and 37.8°, respectively. A 4 flute,

300 helix, 19.05 mm diameter end mill with a gauge length of 55 mm was used in the

simulations. The feed per tooth is 0.14 mm/tooth. The workpiece material is aluminum

alloy for which K = 99OMFa and Kr = 0.52 were used. The cutter is divided into

40 axial elements in the simulations. During the first tooth period, the cutter deflects

away from the workpiece as the resultant cutting forces pull the cutter away from the

workpiece. Hence, the tool cuts a very small amount of the total chip load in the first

tooth period resulting in very small cutting forces. In the second tooth period, the cutter

has the additional load due to the material left behind during the first tooth period. The

cutter deflects again, but it removes much more material than the first period, as the

accumulated chip thickness is larger than the tool deflections in this period. As a result,

the regenerative cutting forces approach to the rigid cutting forces and by the fourth


0

-100

-200

z-300

xU-

-400

-500

-600

0 90 180 270 360


Figure 6.2: Simulated rigid and statically regenerative milling force in x direction. Thestatic regeneration mechanism diminishes in a few tooth periods which confirms the analytically obtained results. (K = 990 MPa, Kr = 0.52, a=19 mm, s = 0.14 mm/tooth,up-milling, exit angle: 37.8°. Tool: 4 flute, 300 helix, diameter=19.05 mm)

period they become exactly equal to the rigid forces.

The regenerative forces formulated can be used in modeling the transfer function of

the statically compliant milling process for adaptive control strategies [43, 158]. Also, in

milling force predictions if the feed-rate and workpiece geometry (i.e. axial and radial

depths of cut) change frequently within 3 to four tooth periods, the regenerative force

model has to be used.

6.3 Flexible Milling Force Model (Variation of Radial Depth of Cut Due to

Deflections)

Although the effect of deflections on the chip thickness diminishes very quickly, the

radial depth of cut and thus the immersion boundaries vary due the deflections. This


variation is significant when the deflections are considerable compared to the radial depth

of cut, so both tool and workpiece deflections are considered in this section. The variation

in radial depth of cut and chip thickness due to tool deflections were considered together

in the numerical models in [24, 36], which did not allow to separate the effects of each

mechanism. In this section, the steady-state effect of the deflections is formulated by

considering the variation in the radial depth of cut.

The exit angle, qex, in up milling and the start angle, , in down milling depend on

the radial width of cut, b. In Figure 6.3 the up and down milling geometry under the

effect of deflections is shown. In flexible milling, the radial depth of cut varies along the

axial direction, z, and feed direction x as shown in Figure 6.3 and is given by

bf(z, ) = b + [6(z, ) — w(x, z, q)j (6.27)

where b and bf are the nominal and effective radial depth of cut, respectively. 5,, and w

are the tool and plate deflections in the normal direction (y). The tool deflection in the

(+y) direction (or workpiece deflection in (—y) direction) results in an increased radial

depth of cut in up milling , and a reduced radial depth in down milling. This is the

reason of for having + sign in equation (6.27), which is (+) for up milling and (-) for

down milling. The variable width of cut, bf(z, q), results in different exit or start angles

for each flute which is in cut. In the following, the formulations for the exit and start

angles in up and down milling operations will be given.

Consider the up milling geometry shown in Figure 6.3 where the engagement of tool

and workpiece for deflected and undeflected positions is shown. q denotes the exit angle

for the rigid milling. The exit angle under the deflections can be written as

= cos1(1 — ) (6.28)


Down Milling

Figure 6.3: Variation of radial depth of cut and immersion angles due to deflections inup and down milling.

w

flexible Miffing

Up Milling

y

x

Rigid Mffiing

¶ oy


Note that the start angle, ç5, does not change due to deflections. The cutting forces in

flexible end milling can be calculated from equation (4.8) provided that the lower and

upper limits of the incut portion of a flute, zj,i and zj,2, are determined under the effect

of deflections. The relation between the upper and lower limits and immersion angles

can easily be obtained from equation (4.3), which shows the variation of the immersion

angle, qj(z), in the z axis due to the helix. Then, the upper and lower limits are given

by

Zj,2 =

(6.29)zj = [ + (j - 1)p

- e(Zj,i, )]As it can be seen from equation 6.29 the upper limit does not vary with the deflections

since the start angle remains constant, q 0, as far as the contact between the work-

piece and the tooth is not lost at z,2, i.e. bf(z,2) > 0. If the contact is lost, however, the

upper limit becomes independent of the start angle and it can be calculated from equa

tion (6.27). This is, however, unlikely to occur in up milling operations as the cutting

forces try to deflect the workpiece and tool towards each other if the radial depth of cut

is not too small. In the case where the radial depth of cut is too small the radial forces

are dominant in the normal force which may try to separate the workpiece and the tool.

Therefore, the existence of the contact between the work and the flute should be checked

and if necessary, the new value of z,2 should be determined before calculating the lower

limit. If zj,2 is found to be less than zj,1 there is no need to update the lower limit as

the total contact between flute j and the workpiece is lost. The effect of deflections and,

thus, the variation in the exit angle can be seen in the lower limit. The following implicit

equation is obtained if ex from equation (6.28) is substituted into z,1 in equation (6.29)

F(z,1)= z,1 — + (j — 1) — cos1(1— bf(zj, ))] (6.30)

The above equation can only be solved by iterative techniques as the radial depth bf is


also a function of the lower limit. In the Newton-Raphson method the solution in the

mtII iteration is updated as

z1 = z11 — (6.31)

where— dF — dbf/dz,l

z,1tan —(1— i)2

wheredbf d5 dw

= ——(z,i) — (6.33)

The values and the derivatives of the deflections at zj,1 can be interpolated from the

deflections of nodes k and k + 1 between which z,1 lies:

dS(z,i) —

__________________

dz —

(6.34)

(z,1) = 6 ((k — 1)z) + d(z,1)(z,j — (k — 1)z)

where Lz is the length of an axial element, /z = a/(n — 1). The workpiece deflection

and its derivative are approximated in a similar fashion. The iteration in equation (6.31)

is started by zj,1 of the rigid milling. If z,1 becomes larger than zj,2 the contact between

flute j and the workpiece is lost. This is imposed on the solution by letting z,1 = z,2

when z,i > z,2.

The variation of the start angle for down milling is shown in Figure 6.3. q is the

start angle of the rigid case. The exit angle which is equal to r does not vary under the

deflections if the contact is not lost, as shown in the figure. The comments made for

the upper limit in the up milling case apply to the lower limit in down milling, i.e., the

existence of contact between tool and workpiece should be checked first. Similar to the


exit angle in up milling, Cst in down milling can be expressed as follows

cst = — cos’(l — bf) (6.35)

According to equation (6.29), z,1 remains fixed as the exit angle is always equal to r in

down milling. If ckSt is substituted in equation (6.29) the following implicit equation is

obtained for zj,2

F(z,2)= z,2 — + (j — 1)— + cos’(l

— )] = 0 (6.36)

The solution for z,2 can be obtained iteratively as explained in up milling. There are,

in fact, two iterative loops in the formulation presented above. In the inner ioop the

limits of integration are determined iteratively. The convergence is very fast in this loop.

In the outer loop the cutting forces and defiections of the workpiece and the tool are

calculated by using the limits updated in the inner ioop. The convergence of this loop

depends on the magnitude of the deflections. The iteration process starts by using the

values obtained in the rigid force model. The deflections are used to determine z,1 for

up milling and z,2 for down milling iteratively, as explained above. The new values of

integration limits are used in the cutting force equations, equations (4.8), to update the

forces and deflections to be used in the inner loop again. This procedure is used for

the accurate prediction of dimensional form errors in milling very flexible parts which is

presented in the following chapter.

6.4 Summary

The variations in chip thickness and radial depth of cut due to cutter and workpiece

deflections are analyzed. It is analytically and numerically shown that the chip thickness

approaches the intended value very fast for a stable milling process. A flexible milling


force model which determines the effective radial depth of cut under the deflections is

developed for static milling operations.

Chapter 7

Peripheral Milling of Plates

7.1 Introduction

The peripheral milling of flexible workpieces is complicated, where periodically vary

ing milling forces excite the flexible cutter and workpiece structures both statically and

dynamically. Static deflections produce dimensional form errors, and dynamic displace

ments produce a poor surface finish in milling. The dynamic cutting and stability anal

yses are given in Chapter 8. In this chapter, the static deflections of the plate and end

mill under the milling forces and the resulting dimensional surface errors are consid

ered. The structural models of the plate and the tool described in Chapter 3 and the

milling force models given in Chapters 4 and 6 are used to develop a process simulation

model which helps to produce acceptable tolerances in machining very flexible structures.

A noted previous study on the peripheral milling of flexible structures was carried

out by Kline et al. [35]. Kline considered the milling of a clamped-clamped-clamped-free

(CCCF) plate with a flexible end mill. He used the FE method to model the plate, and

the beam theory for the end mill. However, Kline’s CCCF plate was comparatively rigid,

because it was clamped from the three edges leaving only one edge free for displace

ment. Also, he neglected the effect of the deflected plate and the tool on the immersion

boundaries, which is significant in milling really flexible plates as illustrated in Chapter

6 and in this chapter. Sagherian et al. [99] used a similar model to Kline’s in studying

142

Chapter 7. Peripheral Milling of Plates 143

the milling of a CFFF type plate. Even though the axial depth of cut was considerably

small, Sagherian et al. [99] reported significant defiections of the workpiece and resulting

dimensional errors left on the finish surface. Altintas et al. [98] analyzed cutting forces

and deformations, both dynamically and statically, in the peripheral milling of such a

flexible plate at a particular location by neglecting the time varying structural properties

and the changes in the immersion boundaries. The true kinematics of dynamic milling

[37] were employed in the model in order to track chatter vibration waves left on the

surface. Their study [98] showed that dimensional form errors produced by quasi-static

components of the cutting forces were quite significant. Furthermore, although very low

cutting loads were used in milling the plates, the static deflection of the long slender end

mill was found to be considerable. The previous studies did not investigate any method

which reduces the excessive form errors in milling very flexible structures.

In this chapter, the simulation system developed for the milling of very flexible plates

is explained. The plate structure is modeled by a developed FE code and the cutter is

represented by an elastic beam. The variations in the plate structure and the partial

disengagement of plate from the cutter due to excessive deflections are considered when

predicting the cutting forces and the deformed finish surface dimensions. The Finite

Element code and the milling force calculation routines have to be integrated due to the

interaction between the milling geometry and forces and the plate and tool deflections.

A method of milling very flexible plates within the specified tolerance is developed by

varying the feed along the tool path. In the following sections, modeling of the plate and

tool structures, the cutting force distribution and identification of varying immersion

boundaries, the simulation of plate surface generation and constraint of dimensional

surface errors by feed scheduling are presented. The surface generation and dimensional

accuracy control methods are experimentally proven in machining very flexible plates.


7.2 Static Modeling of Plate Milling

The workpiece considered here is a clamped-free-free-free (CFFF) plate as shown in

Figure 7.1. The plate is down milled by removing a small width of cut with a long slender

helical end mill. As the material is removed and the tool changes its contact position,

the stiffness of the plate changes both in the feed (x) and axial (z) directions. The tool

structure is modeled as a cantilevered beam and it remains fixed during the machining

process. The details of the structural models of the plate and tool are given in Chapter

3. Here, the interaction of tool-plate structures and the machining process is presented.

7.2.1 Structural Model of the Plate

The discontinuity in the plate thickness due to machining requires that the structure

should be considered as a three dimensional object. A finite element (FE) model of

the plate is constructed using 8 node isoparametric elements with thickness control as

presented in Chapter 3. Approximate dimensions of the sample plates used in simulations

and experiments are 63.5mm x35mm (with 2.45mm and 6.3mm thickness). Plates are

divided into an equal number of (n) elements both in feed (x) and tool axis (z) directions.

The number of elements is equal to the plate length divided by the cutter contact length

= R sin q in the immersion zone, which simplifies the force and surface generation

simulation as explained later. Each node is constrained to have three translational degrees

of freedom. The cutting forces acting on the nodes are all zero except at the tool

workpiece contact zone, which is represented by the elements in the immersion zone.

The contact zone elements are bounded by nodal axes boundaries (A — A, B — B) which

are facing the end mill (Fig. 7.1.b). The cutter enters the workpiece in the down milling

mode from the axial nodal line B — B where the plate has an uncut thickness (ta), and


Figure 7.1: (a) Peripheral down milling of flexible plates, (b) Finite element model of theplate, (c) Corresponding nodal stations on the tool.

z

(a)

—I b-’

tu

cutter entry

r%> to

cutter exit

A

(b)(c)


exits from the plate at nodal axis A — A where the plate has an after-cut thickness (ta).

The thickness changes linearly within the isoparametric elements as shown in Figure 7.1.

Since the plate is most flexible in the normal direction, the normal F cutting forces are

applied to the plate at tool-plate contact nodes (A — A, B — B). The cutting forces are

calculated analytically within each element boundary and distributed to four face nodes

equally along the tool’s z axis as explained below. The force distribution is carried out

for all elements representing the tool-plate contact zone in the z direction. The static

deflections of the plate nodes are calculated by solving the following matrix equation,

[K]{w} = {Q} (7.1)

where [Kr] is the square stiffness matrix, and {w} is the displacement vector, and {Q}is the force vector whose elements are zero for all the nodes except the ones at the tool

workpiece contact zone. The elements of {Q} in the contact zone are equated to the

nodal cutting forces, The simulation is carried out at elemental increments along

the feed axis x, and the thickness of the elements are reduced along the tool-plate exit

axis A — A. The material removal is considered by updating the stiffness matrix at each

feed location. The details of the Finite Element model of the plate are given in Chapter 3.

7.2.2 Structural Model of the Tool

As described in Chapter 3, a slender helical end mill with a gauge length of 1 mm from

the clamped chuck end is modeled as a cantilevered beam with an equivalent diameter

of de = . d, where d is the diameter of the cutter and s is the scale factor due to helical

flutes. Experiments showed that s = 0.75 for d = 19.05mm diameter cutter, which is

identified as suggested by Kops et al. [114]. The stiffness of the tool clamping in the

collet is considered by assuming a linear spring between the rigid spindle body and the

• Chapter 7. Peripheral Milling of Plates 147

clamped end of the cutter in the collet. The cutter is divided into equally spaced axial

nodes which correspond to the plate nodes in the axial direction (z). The same nodal

forces {LF} are applied to the tool, but in the opposite direction to the plate nodes.

The cantilever beam formulation is used to determine the tool deflections at the nodal

stations. The contact stiffness between the workpiece and the tool is neglected. The

deflection at node k caused by the force applied at node m is given by

________

/Fy,m6E1

(3VmZ’k)+ ,O<11k<Vm

6Yk,m = (7.2)

____

ZFm6E1

— , l’ <‘k

where Vk = 1 — zk, E is the Young Modulus, I is the area moment of inertia of the tool,

and k is the tool clamping stiffness in the collet. The total static deflection at nodal

station k is calculated by the superposition of the defiections produced by all (n + 1)

nodal forces,n+1

5(k) = 6Yk,m (7.3)m=1

7.2.3 Cutting Force Distribution-Rigid and Flexible Force Models

Milling forces are calculated as described in Chapter 4. Immersion angles are mea

sured clockwise from the normal (y) axis to a reference flute j = 0, which has immersion

q at its tip z = 0. On flute j, a differential chip element at axial location z has immersion

angle 4(z) = q+jq5—k&z, where = (tan ‘çb)/R and ‘k is the helix angle. The tangen

tial and radial forces acting on the flute element are resolved in x and y directions, and

integrated analytically along the in-cut axial element /c of the flute j which corresponds

to the finite element k on the plate. The axial boundaries of the element k are nodal


stations k — 1 and k,

k) = —sKjk

[sinj(z) cos (z) + Kr sin2 dzZk_ 1

Zk (7.4)k) = stKtJ [sin2 qj(z) — Kr sinj(z)cosj(z)] dz

Zk_1

where zk represents the z axis boundary of the cutter at node k. If a flute element is not in

the contact or in cutting zone, it contributes a zero cutting force. The axial boundaries

are modified if they do not match the nodal stations on the tool. The cutting forces

contributed by all flutes are calculated and summed to obtain the total instantaneous

forces acting on element k. For an end mill with N number of flutes,

N-i N-i

zF(q, k) = k), k) = k) (7.5)j=O j=O

The cutting forces k) are split by the nodal stations k — 1 and k bounding

the tool element k. The same forces are equally split in the opposite direction by the

four nodes of the corresponding plate element k which are facing the tool. The cutting

forces are distributed in a similar fashion to the remaining plate nodes in the contact

zone and at the tool’s nodal stations. The force computation and distribution model is

more accurate than the digitally integrated forces concentrated at the force center of the

structure [35].

For experimental verification, the total cutting forces applied to the whole tool or

plate are calculated by summing the elemental forces.

= k), F()=

k) (7.6)


The effect of tool and workpiece deflections on the chip load and cutting force cal

culations have been neglected so far. The method in this form will be termed as the

rigid model. The analysis is further improved to include the effect of defiections on the

immersion and the cutting force distribution in the flexible model which is formulated

in Chapter 6. Sutherland et al. [24] developed a numerical model which determines the

actual in cut portions of the flutes and the chip thickness under the tool deflections by

employing a regenerative chip thickness algorithm. In the analytical formulation given

in Chapter 6, it is shown that for a static peripheral milling process, which is free of

chatter vibrations, the chip thickness predicted by the regenerative model converges to

the chip thickness in the rigid model after several revolutions of the cutter. The flexible

force model needs to consider only the variations in the immersion boundaries, i.e. start,

cst, and exit, q, angles of the cut along the tool - plate contact zone. çb depends on

the radial width of cut in down milling:

= — cos’(l— -) (7.7)

where b is the actual radial width of cut due to deflections in down milling operations,

• as shown in Figure 6.3. It varies along the z—axis and as the cutter rotates,

b(z, ) = b — 6(z, g) + w(x, z, (7.8)

where b is the desired radial width of cut, 6 and w are the normal cutter and plate

deflections, respectively. Note that due to the different geometrical orientation of down

and up milling operations, the signs in front of the deflection terms in equation (7.8) are

opposite for up milling, as explained in Chapter 6. (7.8) can be used to determine

However, rather than the start angle, the z—axis boundaries are required to calculate

the cutting forces in x and y directions. The varying immersion dependent upper, z,2,


and lower, z,1 axial limits of the immersion for flute j are given by

Zj,2 =

(7.9)Zj,i = + jçp — ex)

zj,1 does not change for down milling as qex remains constant. The equation given for

zj,2 is solved by iterative techniques as q is a function of z. If çb.9 given by equation

(7.7) is substituted in equation (7.9) the following is obtained

1 —1 t1fZJ,2= 7 c-I-Jcp—7r+cos (1— ) (7.10)

Ii

where z,2 is solved with the Newton-Raphson iterative algorithm. The deflection values in

between the nodes can be determined by interpolating the nodal deflections. The iteration

is started with the axial limit z,2 of rigid tool and workpiece. After the convergence in

zj,2 is obtained, its compatibility with the forces and the deflections which were used in

the iteration is checked. If they are not compatible, zj,2 is updated by using the recent

values of deflections. The same procedure is repeated at every angular step.

7.3 Simulation of Peripheral Plate Milling

In milling, the cutting forces are periodic at the flute passing frequency. The distance

between the two nodal stations on the cutter is equal to the plate element height, which

is constant /z = a/n where a is the axial depth of cut. For a helix angle of b, the lag

angle between the two axial nodal stations is 0 = In the rigid model, the cutting

force pulsation for a flute pitch interval (q,) is calculated at helix lag angle increments

of 0 by rotating the tool, and stored in memory for application to the plate-tool contact

nodes later. For this, first the engagement limits are determined as explained by Table

4.1 and Figure 4.2. Then, the elemental and total cutting forces in x and y directions

are calculated from equations (7.4-7.6). The static deflections at the axial nodal stations


of the tool are also calculated once, and stored in memory for the surface generation

simulation as the tool dynamics do not change during milling. In the flexible model,

however, both the force and deflections are modified continuously as the tool and plate

disengage due to deflections. The flexible model therefore significantly differs from the

previous rigid model [35] and the improved rigid model presented here. The cutter is

fed along the x axis, and its centerline is positioned at the axial nodal line A — A. The

position of the cutter is frozen, and the tool is rotated with 0 angular increments. The

cutting forces are distributed to the plate nodes on the nodal lines A — A, B — B at each

angular increment 0. The upper engagement limits for every tooth in cut ,zj,2, is updated

from equation (7.10) by using the calculated plate and tool deflections. This requires an

iterative solution, as equation (7.10) is implicit in z,2. However, the convergence is

quite fast unless the deflections are extremely large. Then, the new values of the upper

engagement limits are used in the cutting force calculation in an iterative manner until

convergence in the milling forces is obtained. The convergence depends on the deflections

and the tolerance value used in the iterations. In order to speed up the convergence by

reducing the stiffness of the iterations, a weighted milling force value obtained in the

previous two iterations are used, i.e. F = (3F_1 + F_2)/4, where i is the iteration

number.

7.3.1 Plate Surface Generation

The finished plate surface is generated by points on helical flutes as they intersect

the contact nodes at the exit nodal axis A — A where the instantaneous immersion is r in

down milling, i.e. q(j, k) = q + jq — kz = ir. Starting with the flute having immersion

= ir, it generates the surface at the bottom contact node (z = 0). Simultaneously,

the following flute touches a nodal point whose height is z = When the cutter

is rotated a b angular increment in the force simulation, the flute jumps to the second


node, and the following flute climbs to the node above its previous position on A — A.

Depending on the width (b) and axial depth of cut (a), there may be more than one

flute generating the surface simultaneously. Using the cutting force distribution at each

cutter rotation increment 0, the deflections of the nodal points touched by the flutes

at the exit axis A — A are calculated from the developed Finite Element routine, and

represented as w(x, k), where x is the cutter center coordinate in feed direction x. In the

flexible force model, tool and workpiece defiections are updated until the convergence

is obtained. Since the forces on the cutter and plate are applied in opposite directions,

they either deflect away or towards each other leaving an overcut or undercut surface.

Therefore, in down milling the final error on the surface node k at cutter feed location x

is

e(x, k) = 6(k) — w(x, k) (7.11)

By repeating the simulation in 0 angular intervals over one flute passing period (q5,), all

the nodal points at the finished plate surface are traversed and the surface errors are

recorded. The cutter is shifted to the next axial nodal line along the feed axis x. The

stiffness matrix of the plate is updated by reducing the thickness at the exit nodal line

A — A, and the simulation is repeated. The solution is continued until the tool leaves the

plate. Peripheral milling of other flexible components with different boundary conditions

can be simulated by the developed algorithm.

7.3.2 Control of Accuracy

It is possible to constrain the magnitude of form errors within the specified tolerances

by predicting the feed along the tool path. The maximum error left on the workpiece

surface is different at each location along the feed direction as a result of the moving

position of the force and the removed material from the plate. For a dimensional tolerance


value (t) on the workpiece surface. it is possible to schedule the feedrate along the feed

axis in order to meet the required accuracy. At each feed location x, the maximum

surface error is determined by using the feedrate obtained in the previous step. The new

value of the feedrate can be approximated as

t.st(x,m) = st(x,m— 1)

max [e(x)J

where m indicates the iteration step and max[e(x)J is the maximum dimensional error

at the feed location x. At the first feed location, x = 0, the iteration is started by a

guessed feedrate value. The iterations are continued until a convergence is obtained in

the feedrate. The cutting coefficients are updated at each step according to the new value

of the average chip thickness. At the following feed locations, the feedrate determined in

the previous feed location is used to start the iteration. After the feedrate is scheduled,

the machining time for the operation can be calculated. The machining time is very

important as flexible plates require very small feedrates resulting in high machining

costs. By using the simulation program, the effects of the tolerance and other cutting

conditions can be analyzed to find feasible conditions as demonstrated in the following

examples.


A number of simulations and experiments have been carried out to demonstrate the

capabilities of the models. Two of the peripheral down milling test results together with

the applied feedrate scheduling are presented below. The plate dimensions, cutting tests

and simulation conditions are summarized in Table 7.1.

The second plate, which is more flexible in the y direction, is similar to the compressor

blades in a jet engine. A spindle speed of 478 rpm was used in both cases. Titanium


Table 7.1: Cutting conditions for experiments 1 and 2. (Material: Titanium AlloyTi6A14V)

CUTTING PARAMETERS Case #1 Case #2KT (MPa) 275 207p 0.6 0.67KR 0.525 1.39q 0.18 0.043Uncut Plate thickness -

t,, (mm) 6.3 2.45Radial width of cut - b (mm) 1.3 0.65Axial depth of cut - a (mm) 35 34Immersion angle of cut -c.s (deg) 30.3 21.3Max. uncut chip (tm) 25.2 2.9Flute lag angle -kua (deg.) 121.6 118.1Simulation angle step (deg) 8.7 6.6Plate mesh size (n x n) 14 x 14 18 x 18

(Ti6A14V) alloyed plates were down milled by a single fluted, 19.05 mm diameter carbide

end mill with a helix angle of b = 300 in dry cutting conditions. The tool gauge length

is 55.6 mm and k0 was measured to be 19800 N/mm. Single flute milling tests eliminate

the effects of runout. In practice, the carbide end mills are ground with the tool holder

in order to eliminate the run-out. Otherwise, tools with the run-out are inefficient in

machining plates at low chip thickness. Flute sections with a shorter radius may not cut

the plate at all, whereas the following flutes experience a larger chip thickness, leading

to larger force [159, 24, 45] and hence unacceptable deformations on the very flexible

plate. The Young’s Modulus of the tool and workpiece materials are 620 GPa and 110

GPa, respectively. Titanium alloys are usually used in aerospace applications because of

their high strength to weight ratio, e.g. (825 MPa/4.4 g/cm3) yield strength to density

ratio for Ti6A14V compared to (530 MPa/7.84 g/cm3) for AISI-1045 cold drawn steel.

However, the modulus of elasticity of Ti6A14V is about the half of that of steel, which


results in low stiffness in addition to the highly flexible plate geometry. During the ex

periments, the plate was rigidly clamped to a Kistler table dynamometer to measure feed

and normal cutting forces. The experiments were carried out on a vertical CNC milling

machine. A spindle mounted dial gauge was used for surface measurements after each

experiment. Chatter vibrations were present during the experiments, but their influence

on the measurements were filtered by passing all the measurements through a 100Hz low

pass filter. Note that even though the chip thickness in plate machining is very small the

static surface form errors are very large due to the high flexibility of the plates.

Case 1:

In this test, it is shown that the improved rigid force model can predict the cutting

forces and the surface errors sufficiently when the plate is relatively rigid. The plate

thickness is to be reduced from 6.3mm to 5mm by using a feedrate of s = 0.05mm feed

per flute which is constant along the feed direction. A sample window of the simulated

instantaneous forces for one cutter revolution shows the agreement between the predic

tions and the measured values (see Figure 7.2). Note that the square wave-like cutting

forces are somewhat distorted due to the filtering.

In Figure 7.3, the simulated and measured dimensional surface errors are shown. Due

to the relatively high rigidity of the plate, both rigid and flexible models give almost

the same force and surface error predictions. The form errors are the smallest at the

cantilevered bottom of the plate, but they are not zero because of tool deflections. Tool

deflections at the plate bottom remain almost the same along the feed axis x, and the

simulation and experimental measurements give approximately 50pm form error here.

The form error magnitudes increase towards the free end of the plate (z — axis), where


600

500

400z

300

LL200

100

0


Figure 7.2: Experiment #1 -

ting conditions: dry cuttingmm/tooth, cutting speed=30mm.

A sample window of simulated and measured forces. Cutin down-milling mode, a =35 mm, b=1.3 mm, st =0.05rn/mm. Tool: carbide end miii, 1 flute, 300 helix, d=19.05

0 45 90 135 180 225 270 315 360


the tool stiffness and plate flexibility increase. At the upper portions of the plate, both

simulation and measurements indicate an increasing trend in the surface form error am

plitudes in the feed direction. This is due to the decreasing stiffness of the plate as a

result of material removal.

The simulation and experimental dimensional errors are in satisfactory agreement

with each other. This shows that if the deflections are relatively small then the rigid

model can predict the surface finish with satisfactory accuracy. When the plate is very

flexible, the influence of deflections on the immersion of the cut along the cutter axis

must be considered as demonstrated in the following test.

Case 2:

Plate thickness is to be reduced from 2.45mm to 1.8mm by using feedrate of s = 8zm

feed per tooth. Experimentally measured and simulated dimensional surface errors are

shown in Figure 7.4. The surface form error at the cantilevered edge of the plate is due

to tool deflection, and it is approximately 3Otm on both the simulated and measured

surfaces. The maximum surface errors predicted by the rigid model are approximately

15Ozm more than the experimental values, which can be clearly observed from the de

tailed views of the deformations shown in the beginning, middle and at the end sections

of the plate (Figure 7.5). The error in the rigid model predictions is due to the immersion

thus force distribution changes as illustrated by the results in Figures 7.6 and 7.7. The

surface error predictions obtained using the flexible model are in very good agreement

with the experimental values as shown in Figure 7.4 and 7.5.

The average cutting forces exponentially increase and decrease in the cutter entry

158

Figure 7.3: Experiment #1- (a) Simulated, (b) Measured surface finish dimensions. Platedimensions: 63.5x35 mm, uncut plate thickness: 6.3 mm, cut plate thickness: 5 mm.

Chapter 7. Peripheral Milling of Plates

(:1

E

L0C,

(30

C

(t

(a)

C),

C

C)

C,

(b)

--I

:-

—‘, ,c:,__

.— .-

159

Figure 7.4: Experiment #2- (a) Rigid model, (b) Flexible model, (c) Measured surfacefinish dimensions. Plate dimensions: 63.5x34 mm, uncut plate thickness: 2.45 mm, cutplate thickness: 1.8 mm.

(b)


L0c

(a),

Db -4-

(C)

c0c

C)

C.

-


35

30

25N

600

35

30E

25N

E30E

25N

20

a)

c1o

0800

Surface Error (mic)

Figure 7.5: Experiment #2 Predicted and measured surface profiles near the beginning,middle and exit feed stations.

0 100 200 300 400 500

100 200 300 400 500

0 200 400 600


100

. 80zC,C)

UciCd1

>

20

0

zC)

0U-

Rotation Angie (deg) (b)

Figure 7.6: Experiment #2- (a) Measured and simulated average forces, (b) A samplewindow of measured and simulated cutting forces. Cutting conditions: dry cutting indown-milling mode, a =34 mm, b=0.65 mm, s =0.008 mm/tooth, cutting speed=30rn/mm. Tool: carbide end miii, 1 flute, 300 helix, d=19.05 mm.

Feed Direction -x (mm) (a)

45 90 135 180 225 270 315 360


172

170

- 168

166

164C,)

162

< 160

158160 180 200 220 240 260 280 300


Figure 7.7: Experiment #2 - Flexible force model predicted variation of ct in the beginning, middle and close to exit feed stations. For the rigid case =l58.7° which variesdue to the deflections of the plate and tool.

and exit transients. The flexible model predicts the average force satisfactorily, whereas

the rigid model results are about 25 % higher than the measured values. After the cutter

reaches the steady state immersion angle ç6. = cos1(1 — b/R) = 21.3°, the average forces

have a slow decreasing trend toward the exit. This is attributed to the changes in the

immersion of cut q due to deflections. Figure 7.7 shows the variation of the start angle

of cut, 4st, predicted by the algorithm described in the flexible force model. In the rigid

case = 180 — = 158.7°. Increase in q5 means a reduction in effective immersion

q which varies as the tool rotates and moves along the feed direction. As the tool ro

tates, the immersed portion of the flute moves up where the plate deflections are larger.

This results in higher qst values. As the cutter approaches the end of the cut, the plate

thickness is reduced, and so is the stiffness. The plate deflects further away from the tool

in the y direction causing a reduction in the effective immersion of the cut. Decreasing

immersion results in reduced cutting forces. q3 remains constant when the intersection

point of the flute with axial line B — B reaches the tip of the plate. The variation of g8


is shown at the beginning, i.e. when x = 0, middle and close to the end of the cut just

before the exit transients. Note that the shape of the variation at each location is very

similar to the surface profile at that point. A sample window of measured and simulated

instantaneous cutting forces (Fig. 7.6.b) shows that the flexible model force predictions

are quite satisfactory compared to the experimental values. Especially for the rotation

angles where the immersed portion of the flute is close to the tip of the plate, the rigid

model-normal force predictions are more than twice the measurements and the flexible

model predictions. It is evident that the previous [35] and the improved rigid model

presented here can not predict the form errors and the cutting forces when the plate is

very flexible.

Control of Accuracy:

The same plates have been machined with a varying feed predicted by the model for

a specified form error limit. The scheduled feed per tooth, s, for Case 1 and Case 2 are

shown in Figure 7.8. The maximum surface errors allowed are 80 im and 250 um for

Case 1 and 2, respectively. The very flexible plate in Case 2 does not allow to obtain

a higher accuracy as it is analyzed later in detail. Due to the relatively high rigidity of

the plate in Case 1, both the flexible and the rigid models give the same results. In Case

2, however, the rigid model estimates an almost three times smaller feed than the flexi

ble model does. Also, as a result of interaction between the deflections and the cutting

forces, the variation of the feedrate along the feed direction is not smooth. The simulated

and the measured surface errors obtained by using the scheduled feedrates are shown in

Figures 7.9 and 7.10 for Case 1 and 2, respectively. Flexible model predictions were

used in Case 2 as they are expected to be more accurate. As the feedrates are deter

mined iteratively in the simulations, the resulting surface error may be slightly different


0.06U • U

0.05

0.04

2 0.03E .E.— .0

0.01.

0 I I I

0 10 20 30 40 50 60

Feed Direction -x (mm)

0.003

0.0025 U U U • • U U

0.002 •• Flexible Model

0.0015 . • Rigid Model

g0•001 ..•..•..

0.0005 • • • U

. U

0 I I I I I

0 10 20 30 40 50 60

Feed Direction -x (mm)

Figure 7.8: Scheduled feedrates for Experiment 1 and Experiment 2 for surface errortolerances of 80 um and 250 m, respectively.


()

E

c

(3

(I)

165

Figure 7.9: Experiment 1 with scheduled feedrate for 80 pm tolerance - a) Simulatedsurface errors b) Measured surface errors

(a)-‘

C)

cc

—--

(b)

‘0


- ct

Figure 7.10: Experiment # 2 with scheduled feedrate for 250 um tolerance- a) Simulated(flexible model) surface errors b) Measured surface errors

(3

E

C—0C.

(313

ri—.C

(a)

—

(b)(3

E

C-0C-.

(5

4—C-


100

902 8070a)E 60

—• 50

2010

0

Figure 7.11: The variation of the machining time with tolerance value in Case # 1 and# 2. The cutting conditions are the same as described before except for the feedratewhich is scheduled to achieve a required tolerance on the surface.

than the tolerance value depending on the error value accepted in the iterations. That

is why the maximum predicted surface error fluctuates around 250 1um in Figure 7.10.

The measured surface errors indicate that the required accuracy is achieved by using

the feedrate scheduling strategy, which would not be possible when the rigid model is

used in milling very flexible plates. Depending on the desired accuracy sometimes the

resulting scheduled feedrates may be very small. This was the situation in Case 2. One

should realize that it may not be possible to achieve the desired accuracy always. If the

workpiece is very flexible, the edge forces may be high enough to create large deflections

even if the feedrate is very small, even zero. The small feedrates result in long machin

ing times increasing the cost. Before deciding on the cutting conditions and tolerance

value one should have an idea about the effect of those parameters on the machining time.

In Figure 7.11 the variation of the machining time obtained from the simulations vs.

the specified dimensional tolerance is shown for Case 1 and 2. They have exponential

0 100 200 300 400 500

Tolerance (mic)


20 2

-

- b=0.65 mm (varying R)16 • 1.6

- -

- R=9.5 mm (varying b) - -

E- E

12 .- 1.2E- ---- -D

U

8 •. -- 0.8

A- -

AI I I 0.4

0 200 400 600 800 1000

Machining Time (mm)

Figure 7.12: The simulated variation of the machining time with tool radius and radialwidth of cut in Experiment 2 for tolerance of 250 ,um on the surface.

shapes similar to general cost vs. accuracy relations. It can be seen from the figure that

for Case 1 the tolerance can be decreased from 300 ,um to 80 tm without increasing

the machining time significantly. Reducing the tolerance constraint from 80 um to 50

tim, however, increases the machining time more than 6 times. Therefore, 80 m was

a very good tolerance selection for Case 1. In Case 2, the accuracy can be improved

by 150 m by reducing the tolerance from 500 1um to 350 m without any significant

increase in the machining time. Increasing the accuracy 100 itm more results in a tripled

machining time. It should be noted that machining times were calculated for a one fluted

cutter and 478 rpm spindle speed. They can be significantly reduced by increasing the

number of flutes and the spindle speed. The effects of tool diameter and radial width

of cut for this case are shown in Figure 7.12, where the radial width of cut is 0.65 mm

and the tool radius is varied. In the cases where the radial width of cut is varied, the

tool diameter is 19.05 mm. In all cases the desired cut thickness of the plate is 1.8

mm and the tolerance is 250 tim. The tool diameter and the radial width of cut define

the immersion angle and average chip thickness. In down milling, peak normal force F


increases as the immersion angle increases if the cutting coefficients are assumed to be

constant. However, the variation of the cutting coefficients is too sharp, especially at a

low average chip thickness. Also, the radial width of cut affects the flexibility of the plate

as the final thickness is fixed. These different factors compete against each other and

the optimal values can be obtained from the analysis of Figure 7.12. As it can be seen

for 250 um accuracy and 0.65 mm radial width of cut, the minimum machining time is

obtained by 15.88 mm (5/8”) diameter tool. The charts which are prepared by the plate

milling simulation system can be used in process planning.

7.5 Summary

A simulation system for the peripheral milling of very flexible, plate type structures

is developed. The plate is cantilevered from the base and is free at the other three edges.

The tool is modeled as a cantilevered beam, and stiffness loss in the collet is considered.

A finite element model with a varying stiffness due to metal removal is used for the plate

structure. The changes in the immersion, both in the feed and cutter axis directions, are

considered in the model. The developed model allows satisfactory prediction of cutting

forces and dimensional surface errors due to deflections of the tool and plate structures

during milling. It is shown that unless the changes in the plate-tool contact boundaries

are considered, the form errors and the cutting forces can not be predicted satisfactorily in

the peripheral milling of very flexible plates. The developed model allows the prediction

and scheduling of feeds along the tool path, and keeps the form errors within the specified

limit.

Chapter 8

Analysis of Dynamic Cutting and Chatter Stability in Milling

8.1 Introduction

In the milling process, cutter, workpiece and machine tool structures are subject

to periodic and transient vibrations due to the intermittent engagement of cutter teeth

and periodically varying milling forces. These effects, however, can be minimized by the

proper selection of cutter geometry and spindle speeds to avoid resonances and large

impact loadings. A more important vibration type in machining is self-excited chatter

vibrations which cause instabilities resulting in poor surface finish and dimensional ac

curacy, chipping of the cutter teeth, and may damage the workpiece and machine tool.

The dynamic milling process-structure interaction is particularly important in milling

thin-walled workpieces due to a highly flexible workpiece and slender end mill. In this

chapter, the chatter stability of milling is analyzed. A general formulation is developed

for the analytical prediction of milling stability and it is applied to several common cases

like the milling of flexible workpieces.

The fundamental chatter theory has been developed by Tobias [3] and Tiusty [5].

Tobias considers the variations in the cutting forces due to the dynamic variations of

chip thickness and cutting velocity. Chip thickness can vary due to either a modulated

surface left from the previous pass (outer modulation or wave removing) or vibrations

of the tool towards the cut surface (inner modulation or wave cutting); whereas cutting

170

Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling 171

velocity can vary due to the vibrations of the tool in the cutting velocity direction. Then,

he considers the total amount of damping in the cutting system, which is a summation

of the structural damping and the damping generated due to variations in the cutting

velocity and the chip thickness, to assess the stability. On the other hand, Tiusty shows

that the cutting system has higher stability against the mode coupling chatter mechanism

(excitation energy generated due to a variation in the cutting velocity) and considers the

regeneration mechanism as the dominant mode of chatter, but includes the process damp

ing generated due to the flank contact. The phase between inner and outer modulations

is the most important factor of the regeneration mechanism. It determines the amount of

periodic variation in the chip thickness and depends on cutting conditions and dynamic

characteristics of the structure. Both approaches by Tobias [3] and Tlusty [5] can be

used to obtain the stability limit (maximum allowable width of cut without chatter) as a

function of cutting velocity which is referred to as the stability diagrams or stability lobes.

A stability analysis of the milling process is much more complicated than the orthogo

nal cutting case. This is mainly due to the rotating milling cutter, multiple cutting teeth

and the dynamically coupled multi degree-of-freedom cutting system. The directional

coefficients (direction cosines to determine the component of the cutting force and the

oriented transfer function in the chip thickness direction) vary as the cutter rotates. In

the early milling stability analysis, Tiusty [5, 77] applied his orthogonal cutting-stability

formula by considering an average direction and number of flutes in cut. Therefore, he

used constant directional coefficients which are calculated in the average direction. There

is no theoretical basis for this approximation and the accuracy is not predictable. Due

to these reasons, later Tlusty et al. [75, 69, 88, 160] and others [37, 98, 161, 162] have

used time domain simulations extensively in milling stability prediction. However, time

domain simulations in chatter are computationally very expensive. In order to obtain


a milling stability diagram, hundreds of simulations have to be carried out at different

spindle speeds by increasing the axial depth of cut and observing the convergence of

the displacements for many revolutions of the cutter. The computational time is even

more critical for the case of milling flexible workpieces where the dynamic characteristics

of the workpiece change due to machining which requires that the chatter simulations

be repeated at a number of stations along the feed direction. The basic nonlinearity in

chatter [75] (loss of contact between the cutting tooth and the workpiece due to large

vibration amplitudes) can be best modeled by time domain simulations, however this is

• not important in predicting the onset of chatter [82]. Opitz et al. [163, 94] replaced the

periodic coefficients with their average values over the time interval during which the

tooth of the cutter is in contact with the workpiece. Then, the oriented transfer function

is calculated by using the average directional coefficients which reduces the coupled dy

namics to a single degree-of-freedom case. However, no theoretical justification is given

for the method.

The first comprehensive theoretical analysis of milling stability has been performed

by Sridhar et al. [78, 79, 80]. They formulated the dynamic milling forces for a straight

tooth cutter. They used a numerical stability algorithm which is based on the numeri

cal evaluation of the system’s state transition matrix. Minis et al. [82, 81] applied the

theory of periodic differential equations on the milling dynamics equations. They used

the Nyquist stability criterion to determine the stability limits. The algorithm depends

on the numerical evaluation of the eigenvalues as the axial depth of cut is increased until

the stability limit is reached. Lee et al. [164, 165] also used the Nyquist criterion to

determine the stability limits numerically.

In this chapter, a comprehensive formulation of dynamic milling forces is given by


neglecting process damping, i.e. the relationship between the milling forces and the chip

thickness is represented by a simple proportionality constant. This is a valid assumption if

the cutting velocity is relatively high (> 100 m/min). The milling cutter and workpiece

are modeled as multi degree-of-freedom structures and the dynamic interaction along

the axial depth of cut is considered in the formulation, unlike the point contact models

used in all of the other chatter formulations. The effects of the variations in cutter and

workpiece dynamics in the axial direction are considered in the model. This is necessary

for the accurate modeling of dynamic milling forces in milling flexible workpieces and

has been neglected in the previous models. The stability analysis is performed by using

two different approaches both of which give the same result. First, the classical periodic

system theory is applied to dynamic milling. Second, a stability analysis which is based on

the physics of the dynamic milling is used. It is realized that the second method is simpler

and gives physical insight to the milling dynamics. An analytical method is developed to

predict the stability limit by deriving a relationship between the chatter frequency and

the spindle speed, for the first time in milling. The application of the general formulation

to some special cases and the accuracy of the predictions are illustrated through examples.

8.2 Formulation of Dynamic Milling Forces

Figure 8.1 shows a crossection of an end mill tooth (j) vibrating and removing a

wavy surface cut by the previous tooth (j — 1). u3 and vj are the rotating tangential and

normal directions at the tip of tooth j, and they can be expressed in terms of the fixed

coordinate system x and y as follows:

u3 = —xcos(z)+ysin4(z)(81)

vj = —xsingj(z)—ycosq!j(z)


Workpiece

vibration marksleft by tooth )

/

/

x

vibration marksleft by tooth (i-i)

Figure 8.1: Dynamic milling process.


or

= —cos(z) sin(z) x(8.2)

( v J —singj(z) —cosq5j(z) L ‘ Jwhere

4(z) =

k — tan?1b- R

2ir—

N is the number of teeth on the cutter, R is the cutter radius and g = Qt is the rotation

angle of the end mill (measured with respect to tooth number j = 0), Q being the

rotational speed of the cutter in (rad/sec).

• 8.2.1 Dynamic-Regenerative Chip Thickness

The chip thickness can be written as a summation of the static and the regenerative

chip thickness as follows:

z) = (v —v)

— (v — v) + s sin q(z) (8.3)

where v, v, and v, v are the dynamic displacements of the cutter and workpiece

in the v direction for current and previous tooth passes (for the rotation angle of the

cutter q, at the axial depth z), respectively. According to the reference system shown

in Figure 8.1, the current cutter displacements in the positive v direction decrease the

chip thickness, whereas the positive cutter displacements in the previous pass increase

the chip thickness. It should be noted that, at this stage the cutter and workpiece

deflections are considered to be in +vj direction, although, in general, they are in the

opposite directions. However, this will be imposed when dynamic displacement-milling

force relations are included to the formulation. Substituting for the rotating coordinates


from equation (8.1), the following is obtained:

z) [(xe — x)— (x — 4)] sin qj(z)

(8 4)+ [(Ycy) — (ywy)]COSj(z)+8tSiflj(Z)

where x, y and x, Yw are the cutter and workpiece displacements in the current pass in

the x and y directions (for the rotation angle of the cutter ç and at the axial depth z),

respectively. Similarly, x, y° and 4, y are the cutter and the workpiece displacements

at the same point in the previous tooth period.

8.2.2 Differential Dynamic Milling Forces

Dynamic differential milling forces can be obtained by using the dynamic chip thick

ness given in equation (8.4). The formulation of these forces is similar to the static milling

force model given in Chapter 6, except the chip thickness dynamically varies in this case.

The effect of vibratory cutting on the milling force coefficients will be neglected. The

milling forces in the x and y directions will be considered as the spindle and the cutter

are quite rigid in the z direction compared to x and y. From Figure 8.1 the tangential

and the radial forces can be resolved in x and y directions as:

dF3(q,z) = [—dFt(,z)cos(z) — dFrj(q,z)sinqj(z)]g(qj(z))(85)

dF(,z) = [dFt(,z)sin(z) — dFrj(,z)cosj(z)]g(j(z))

where

dFt3(ç,z) = Kh(q,z)dz

dFrj (q, z) = KrdF3

and g(gj(z)) determines whether the tooth is in cut. Mathematically, it can be expressed

as follows:

g(j(z)) = 1 st <(z) <e 1 (86)g(j(z))=O or j(Z)>ex J


Then,

dF,(c,z) = —Kjh(,z)[cos(z) + Kr sin q(z)]g((z))dz(87)

dF(,z) = — Kr cos(z)]g((z))dz

Substituting equation (8.4) into (8.7), the following explicit form is obtained for the

differential milling forces:

dF(,z) = —Kt(cosq(z) + Kr sin qS(z)) {[(x — x) — (x — x)]

sin3(z) + [(yc - y) - (y - y)]CO5j(Z) + sjsin(z)}g((z))dz

(8.8)

dF2 (ç, z) = Kt(sin çj(z) — Kr cos (z)) {[(x — x) — (x — x)]

sin çf(z) + [(yc — °) — (Yw — Y,)] cosqj(z) + stsinq!(z)}g(q(z))dz

8.2.3 Total Dynamic Milling Forces

The widely used milling chatter models developed by Tlusty et al. [5, 75, 77] and

Tobias [3], and a relatively recent method by Minis et al. [82, 81], consider a point contact

between the milling cutter and the workpiece. The cutter is modeled as a 2 degree of

freedom structure which is lumped at the tip of the tool. The variation of the cutter

dynamics in the axial direction and the helix angle are neglected in these models which

may cause inaccuracies in the stability limit predictions, especially in end milling where

the axial depth of cut is usually high and the variation of the tool dynamics within the

incut portion of the end mill (or mode shape in a cutter vibration mode) is significant. In

order to consider the real dynamic interaction between the milling cutter and workpiece,

the cutter and workpiece are divided into a number of elements in the axial direction, as

in the case of the static tool and workpiece deflection analysis given in Chapters 5 and

7. This is because the cutter and workpiece dynamics are usually identified by modal

tests at a number of points (or elements). Also, the resulting integrals for the dynamic


milling forces cannot be taken analytically even if the dynamics of the structure can

be determined by analytical means. Then, the differential dynamic milling forces are

integrated within an element i to determine the elemental or nodal milling forces for the

flute j as:zi+1 z1+1

F2(i)= j dF(b,z)dz ; F(i)

= j dF(q,z)dz (8.9)

where z and z1 are the lower and upper z coordinates of the axial element. After the

integration, equation (8.9) can be arranged as follows:

F3(i) = F.(i) — J(t[Ca,() + KrCbj() + Cc,(i) + KrCdj()J

(8.10)

F3(i) = F(i) + Kt[Cb,(i) — 1(rCcz,() + Cd,(i) — IrCcj()]

whereZjf 1

Caj()= j

CZi+1

Cb3(i)= j

(8.11)C,(i) j

Zi+ i

Cd(i) jwhere

/x = (x, — x) (x — x)(8.12)

=

F and F, in equation (8.10) are the milling forces, due to mean chip thickness (St sin qj(z)).

The integrals in equation (8.11), which are to be integrated within each axial element,

contain the dynamic displacements of the cutter and workpiece. These displacements can

be assumed to be constant within each element as the dynamic characteristics and the

response of tool and workpiece structures are determined at discrete nodal points both

by modal tests and the Finite Element solutions. Also, the unit step function g(qj(z))


• will be evaluated at the lower boundary of each axial element, z, and will be assumed to

be invariant within the element when integrating equations (8.11). This does not degrade

the accuracy as it will be shown later, Ca, C6,C, Gd are further integrated for one full

rotation of the cutter in the stability analysis. Then, the integrals can be determined

analytically as follows:

Caj() = + 1) — cos2q(i)]

C6 (i) x(i)g(i)[ZZ + —(sin 2q(i + 1) — sin 2q(i))]

C,(i) zy(i)gj(i)[Zz — —(sin 2(i + 1) — sin 2q(i))}

Cd(i) + 1) — cos2(i)]

where

LSx(i) = [x(i) — x(i)] — [x(i) — x(i)]

qj(i) = j(q,zj)

gj(i) = g(b(z))

=

m is the number of axial elements, x(i),x(i),.. etc. are the nodal displacements of

the tool and the workpiece for the considered rotational angle of the cutter. The above

equations can be simplified by expanding qj(i), qj(i + 1),(cos 2q(i + 1) — cos 2(i)) and

(sin 2(i + 1) — sin 2q3(i)) as follows:

=

=(814)

=

= gf(i) —

Consider Taylor series expansion of cos 2(a + 3) and sin 2(a + 3) around 6:

cos 2(o + 3) = cos 2a — 2/3 sin 2a — 2/32 cos 2a +8 15

sin2(cr+/3) = sin2a+2/3cos2c—2/32cos2o+...


Then, the following is obtained by using the first order Taylor expansions:

cos2q3(i+ 1)— cos2g3(i) = cos2(q(i) — k/z) — cos2c23(i)

= cos2q(i) + 2kz sin 2q(i) — cos2ç(i)

= 2kzsin2(i)

(8.16)

sin2(i + 1) — sin2(i) = sin2(q(i) — k,/z) — sin2(i)

= sin 2(i) — 2kz cos 2q(i) — sin 2gS(i)

= —2kzcos2ç5(i)

The above simplification is necessary in the stability analysis in order to obtain an

explicit expression for the axial depth of cut, or in this case element thickness. If this

is not done at this stage, an implicit stability equation in the axial depth of cut will

be obtained at the end of this analysis. The accuracy of the first order Taylor series

approximation to the above trigonometric expressions depends on the magnitude of the

elemental lag angle (k,Lz). The lag angle becomes smaller as the number of elements

is increased, thus for a sufficiently high number of elements, the accuracy of the first

order approximation is high. Another way of increasing the accuracy is to employ a

high order Taylor series expansion which results in higher powers of Liz. (Note that,

in equation 8.16, second order expansions contain (z2).) Equation (8.13) takes the

following simplified form when (8.16) is substituted:

Caj() = z/x(i)g(i)sin2q(i)

Cb(i) = zLx(i)g(i)(1 — cos2(i))

C2(i) = zzy(i)g(i)(1 + cos2(i))

Cd(i) = zy(i)g(i)sin2q(i)

After the comparison of equations (8.11) and (8.17), it is realized that the first order

Taylor series approximation gives the same result with neglecting the helix angle within


the axial element. This can be obtained by multiplying the integrands in equation (8.11)

by the axial element thickness Liz. It should be noted that the helix effect is considered

between the elements or nodes in equation (8.17). This approach is similar to the one used

in the static milling force model by Kline et al. [35]. However, as discussed above, a high

number of elements can be used to increase the accuracy. The dynamics of the cutter

and workpiece are usually available at a few discrete points along the axial direction.

One may think that dummy elements can be used between the main nodes at which

the dynamic response is known, for the sake of increasing the accuracy in helix angle

modeling. However, in the stability analysis of the dynamic milling forces, it will be

shown that the average values of the directional factors in equation (8.17) are used, and

thus the helix angle disappears. Therefore, increasing the number of elements to improve

helix angle modeling-more than the variation of the cutter and workpiece dynamics in

the axial direction requires-does not increase the accuracy of the overall formulation.

Finally, substituting equation (8.17) into (8.10) the following is obtained for the nodal

dynamic milling forces:

F(i) = F(i)+

a,(i) a(i) Lx(i)(8.18)

( F,,(i) J ( F;.(i) J a(i) a(i) ( y(i) Jwhere

c = (8.19)

In equation (8.18), the matrix elements .., a which relate the dynamic displace

ments to the dynamic milling forces will be called directional dynamic milling coefficients


and they are given as:

a(i) = —g[sin2g5(i)+ Kr(l — cos2(i))]

a,(i) = —g3[(1 + cos2(i)) + Krsin2qj(i)](820)

a(i) = g[(1 — cos 2(i))— Kr sin 2(i)]

a,(i) = g3[sin2q5(i) — Kr(1 + cos2q3(i))j

The total milling forces in each element can be obtained by summing up the cutting

forces on each flute:

N—i N—i

F(i) = F,(i) ; F,(i) = F,,(i) (8.21)j=O j=O

Then, equation (8.18) takes the following form for the total forces

F(i) = F(i) + a(i) a(i) Lx(i)(8.22)

( F,(i) J ( F(i) J a(i) a(i) y(j) Jwhere

N-i

a(i) = (8.23)j=o

and are similar. Equation (8.18) can be written for every axial element, when

these equations are combined the following matrix equation is obtained:

{F} {F}+

[a] [a](8.24)

( {F} J ( {F:} J [a] [a] ( {zy} Jor in more compact form

{F} = {F8} + c[A(t)]{z} (8.25)

where

I {x} 1{zSj = (8.26)


The force and the displacement vectors contain the elemental force and displacements,

e.g. {F}T = {F(1), F(2), ..., F(rn)}. [a], [a,], [a] and [a] are diagonal matrices

with elemental values of a(i), a,(i) etc. in the diagonal, e.g.

a(i) =

(8.27)j#i J

[A(t)] will be referred to as the directional dynamic milling coefficient matrix which is

periodic at the tooth passing frequency w. Unlike the previous models [3, 5, 77, 81],

equation (8.24) gives the dynamic milling forces by including the effects of the helix

angle and the variation in the tool and the workpiece dynamics along the axial direction.

Before the stability analysis of the milling process can be performed, the structural

displacement- milling force relations should be substituted in equation (8.24). The milling

forces, {F}, {F}, will be dropped from equation (8.24) from this point on as they are

not to be considered in the stability analysis of milling since the dynamic milling forces

are generated due to dynamic displacements of the tool and workpiece. Also, according

to linear system theory, external forces are neglected in the stability analysis of linear

systems [166]. However, the static parts of the milling forces should be considered if a

nonlinear analysis is to be done. The basic nonlinearity in machining chatter is the loss of

contact between the cutter and the workpiece due to high amplitude chatter vibrations

[75]. However, this does not affect the stability limit prediction for which the onset

of chatter vibrations is considered. Other nonlinearity which is especially important in

flexible workpiece milling is the variation of cutter-workpiece immersion boundaries due

to static deflections. This is analyzed in Chapter 6 where it is shown that the radial

depth of cut varies along the tool axis. In these cases, the use of average radial depth of

cut in the stability analysis may be an acceptable first order approximation. For exact

analysis of this type of nonlinearity, time domain solutions are required. The next step


in the formulation is to express the dynamic displacements due to the dynamic milling

forces.

8.2.4 Dynamic Displacements of Cutter and Workpiece

The dynamics of the cutter are described by the following linear differential equation:

{r} = [G(D)J{F} (8.28)

where the vectors {r} and {F} contain the cutter displacements and the milling forces

in x and y directions, respectively:

I{x1 I{F}{r} = ; {F} = (8.29)

I {yc} J I {F} J[G(D)] is the dynamic flexibility matrix of the cutter and has the following components:

[G (D)] {G (D)][G(D)] = (8.30)

{G(D)] [G(D)]

where {G (D)j, (D)] represent the direct-dynamic flexibility matrices of the cutter

in x and y directions, [G(D)], [G(D)] are the cross-dynamic flexibility matrices and

D is the differential operator d/dt. In general, the flexibility matrices have the following

form:

[Gj = [[M]D2 + [BCX1D + (8.31)

where [B] and [K] are the mass, damping and stiffness matrices of the cutter

in the x direction, and the other flexibility matrices are similar. It should be noted that

the flexibility matrices are not transfer functions but linear operators. The size of the

flexibility matrices are mxrn, m being the number of nodes on the tool and the workpiece

along the axial direction. The workpiece displacements, {r}, can be written similarly

as:

{r} = —[G(D)]{F} (8.32)


The (-) sign in equation (8.29) is due to the fact that the milling forces applied on the

• cutter and the workpiece are in the opposite directions. The displacement vector and the

flexibility matrix have the same form with those of the cutter:

{r} ;= [G(D)] [G(D)]

(8.33)({y} J [G(D)] [G(D)]

The dynamic displacements of the cutter and the workpiece in the preceding tooth period,

{ r}, {r}, can be expressed as

= {rw(t-T)}=e{r}(8.34)

where T = 2ir/N is the tooth period and e_TD represents the time delay operator.

Then, {x} and {Ly} can be obtained as follows:

{Lx} = ({x} — {x})— ({x} — {%}) = (1 — e_TD)({xc}

— {x})(8 35)

{y} = ({yc} - {y}) - ({Yw} - {y}) = (1-TD)({} - {Yw})

The following is obtained if the displacements of the cutter and workpiece given by

equations (8.28) and (8.33) are substituted in (8.35):

I {x}{z.S} = = (1 —e_TD)([Gc(D)] + {G(D)]){F} (8.36)

I {Ly} JBy substituting equation (8.36) into (8.25), the following eigenvalue equation is obtained

(the static part of the milling forces is ignored as discussed before):

{F} = c(1 —e_TD)[A(t)][G(D)]{F} (8.37)

where [G(D)] = [G(D)] + [G(D)], [A(t)] is defined in equations (8.24) and (8.25), and

c is defined in (8.19). The above expression was more or less derived by Sridhar et al.

[78], but for a structure which had two orthogonal modes. However, the derivation given


node (m+1)

node 1

Figure 8.2: Node numbering on cutter and workpiece.

here is for multi degree-of-freedom workpiece and tool structures. It should be noted

that [G(D)] and [G(D)} contain the dynamic flexibility of the cutter and workpiece

only in the immersed portions of the structures. As shown in Figure 8.2 , the numbering

of the nodes (degree of freedom) start from the free end of the cutter, i.e. node 1

for the cutter is at the bottom of the tool. Workpiece nodes have the same numbers

corresponding to the ones on the cutter. The stability of milling is governed by equation

(8.37) which models multi-degree-of- freedom cutter and workpiece dynamics and helical

cutting flutes. The periodic terms in [A(t)] are due to the time varying directional milling

• coefficients between the local chip thickness directions and the milling forces, and they

have been the main difficulty in milling stability analysis as the standard methods of

stability cannot be used for the time varying systems. For that, Tlusty et al. [5, 167, 77]

approximated the directional coefficients and the stability limit in the direction of the


average resultant force. Similarly, Sankin [168] evaluated the varying coefficients for the

most critical directional orientation. Opitz et al. [163, 94] used the time average values

of the directional coefficients. There is no theoretical justification for these approaches

and the accuracy is not predictable. This was later appreciated by Tlusty et al. [75, 76]

and Smith [169, 70] who used time domain simulations for the milling stability analysis.

A comprehensive model of milling dynamics was developed by Sridhar et al. [78, 79, 80]

for a special and general case of milling, however they used numerical procedures for the

stability analysis. Following the standard procedure for the stability of periodic systems,

Minis et al. [82, 81] used the Fourier analysis and the concept of parametric transfer

functions for the analysis of a two degree-of-freedom milling system

8.3 Stability Analysis

The chatter stability of milling which is governed by equation (8.37) will be allalyzed

by two methods. First, a mathematical stability analysis will be presented by using the

theory of periodic systems. In the following section a brief history and theory of periodic

system stability are given. The second method of the milling stability analysis is based on

an interpretation of the physics of dynamic milling. Both methods yield the same result,

however the second approach is much easier, shorter and gives more physical insight into

the problem.

8.3.1 Stability Theory of Periodic Systems

Equation (8.37) is identified as a linear periodic-differential difference equation.

There exist several methods to study the stability of periodic or delayed (difference)

equations. However, the existence of both periodic and delay terms in equation (8.37)

increases the complexity of the problem. Periodic differential equations arise in many


fields of physics and engineering, including problems in mechanics, astronomy, wave prop

agation, electric circuits, quantum theory of metals and the stability theory of certain

nonlinear differential equations [170, 171]. Homogeneous, linear, second order differential

equations with periodic coefficients are called Hill’s equations, named after Hill due to

his important and lasting contributions to their theory. Hill derived and analyzed the

following general form of Hill’s equation in his study on the motion of the lunar perigee

[172] which was completed in 1877, but published in 1886

y” + [6 — 2Eq(t)]y = 0 (8.38)

where q(t) is a real periodic function. If q(t) = cos 2t the above equation is known as

Mathieu’s differential equation, introduced by Mathieu in 1868, when he determined the

vibration modes of a stretched membrane having an elliptical boundary [173]. The classi

cal stability analysis of Hill’s equation is the procedure of the infinite determinant which

utilizes the Floquet theorem and the Fourier series expansion of the periodic function

q(t) [170, 174, 173, 175]. A similar procedure will be followed for the stability analysis

of the milling, i.e. equation (8.37). There exist other effective methods for the stability

analysis of differential time varying systems including other series type solutions (Bessel

and McLaurin), perturbation and Liapunov methods [170, 173, 176, 177, 178, 179, 174,

171, 180]. The stability of these systems depends on the relative magnitudes of the static

restoring coefficient (designated by 6) and the time varying one, (c), and it can be de

termined from the stability intervals in , 6 plane (usually referred to as Strutt diagram)

[174, 173, 170, 181]. The addition of the periodic restoring force may cause instability

or make the otherwise unstable system stable, depending on its frequecy and amplitude.

This is illustrated by Cooley et al. [175] on some examples of second (stable to unstable)

and third order (unstable to stable) systems. For a physical example, consider oscilla

tions of a pendulum. An inverted pendulum which is unstable at its vertical equilibrium


position can be stabilized, whereas a regular pendulum which is stable around its verti

cal equilibrium position may become unstable by moving the pivot point harmonically

in the horizontal or vertical direction. Almost all the work done in this area deals with

the second order systems due to their importance in dynamic systems analysis. The

literature mentioned here considers mostly single degree of freedom systems, except [179]

which presents a wide spectrum of methods for the analysis of second order-multi degree

of freedom systems.

8.3.2 Stability Analysis of Milling Using Periodic System Theory

Milling stability equation, (8.37), has a delay term, eT), due to the regenera

tion mechanism. Delay or differential difference equations arise in many problems of

physics, engineering, economics and biology [182, 183]. In an early work, Minorsky

[184] analyzed the effect of delay on the self-excited oscillations and stability. Stabil

ity of delayed systems has been studied in many works, some of them are cited here

[185, 186, 183, 187, 188, 189]. Comprehensive analysis of differential difference equations

can be found in [182].

The stability method used in this section is based on the Fourier analysis and the

concept of parametric transfer functions introduced by Zadeh [190] and utilized by Rozen

[191]. The Fourier analysis has traditionally been used in the analysis of periodic

systems [170, 173, 179, 175] and recently by Hall [180]. Cooley et al. [175] used the

Fourier analysis to examine the stability of a class of systems with a sinusoidally varying

parameter.


Equation (8.37) is re-written

{F} = c(i —e_TD)[A(t)][G(D)}{F} (8.39)

where [A(t)] is periodic in time. The contribution of Minis and Yanushevsky [81] is to

apply the stability theory of periodic systems, i.e. the Fourier analysis and parametric

transfer function concept, to the above dynamic milling force expression-for a two degree

of freedom model-presented by Sridhar et al. [80].

Floquet’s theorem (G. Floquet, 1883) [176, 179, 81] states that for a second order

differential equation with periodic and piecewise continuous coefficients like the milling

dynamics equation (8.39), the solutions have the following form:

{F(t)} = eAt{P(t)} (8.40)

where the function {P(t)} is periodic with period T (tooth period). Therefore, the milling

system is stable if the real part of all exponent A’s are negative. In order to obtain explicit

stability conditions, the characteristic equation of the system is derived using the Fourier

analysis. The periodic function {P(t)} is expanded into the following Fourier series:

{P(t)} = {Pk}e (8.41)k=-oo

where the tooth frequency w = 2ir/T = NfZ is the fundamental frequency of [A(t)] and

i = In order to obtain the kth Fourier coefficient {Pk}, equation (8.41) is multiplied

by e_t and integrated. Then, the following is obtained for the Fourier coefficients by

using the properties of orthogonal functions

{Pk} =JTkt

(8.42)


Equations (8.40) and (8.41) are substituted into equation (8.39) to obtain:

{F} = c(1 - eTD) [A(t)] [G(D)] (ext {Pk}e1t)k=-oo

(8.43)

= c(1 — e_TD) [A(t)] [G(D)] {Pk}e(itk=-oo

The result of [G(D)] operating on the exponential function is determined by the

shifting theorem of linear differential operators [192]:

G(D)eat = eatG(a) (8.44)

Then, by using the shifting theorem in equation (8.43) and considering that {Pk} is

a constant vector, the following equation is obtained:

{F} C e(kw)t(l — e_T)[A(t)][G( + ikw)]{Pk} (8.45)k=—oo

where eT(ik) e7’ is substituted as Tw = 2ir. From equations (8.40) and (8.45), it

is concluded that

{P(t)} = c(1 — e_T) eikwt[A(t)][G(A + ikw)]{Pk} (8.46)k=-co

[A(t)] can be expanded into a Fourier series as it is also periodic at the tooth frequency w.

When substituted, equation (8.46) will contain a double Fourier series. This is performed

by multiplying both sides of equation (8.46) by 1/Te_t and integrating from 0 to T:

{Pr} = c(1 — e_T) [Wr_k(A + ikw)]{Pk} (r, k 0, +1, +2, ...) (8.47)k=-oo

where

[Wr_k(A + ikw)] = [Ar_k][G(\ + (8.48)

[Ar_k] is the (r — k)th Fourier coefficient of [A(t)]:

1 T

[Ar_k] = j [A(t)]e_(r_)tdt (8.49)


The linear algebraic system given by (8.47) can be written in the following matrix form:

{P0} [W0(\)] [W_1(.\ + iw)] [W1.— iw)] . . {P0}

{P1} [W1)] [W0+ iw)] [W2X — iw)] . {P1}

{P1} = c(1 - eT) [W1()] [W2(+ iw)] [W0(- iw)] {P1}

(8.50)

Equation (8.50) has nontrivial solutions if the determinant is zero:

det[6Tk[I] — c(1 — e_T)[Wr_k(\ + ikw)]] = 0 (8.51)

where 6rk is the Kronecker delta (i.e. 5rk = 1 if r = k, 6rk = 0 if r k), and [I] is the

(2mx2m) identity matrix.

Equation (8.51) is the characteristic equation of the closed loop milling system.

This equation is an infinite determinant which is a characteristic of periodic systems

[172, 170, 175, 174]. For the system to be stable, all the roots (eigenvalues) of the char

acteristic equation must have negative real parts. Approximate roots of this infinite order

characteristic equation can be obtained by solving its truncated versions. Hill [172], for

example, considered only a 3x3 determinant (first order approximation) in his analysis.

Minis and Yanushevsky [81] derived an expression similar to (8.51) for the two degree

of-freedom milling system model they used. They numerically solved the truncated eigen

value equation by increasing axial depth of cut and determining the sign of the real part

of ). Although this is faster than time domain simulations of milling chatter, still many

iterations have to be done before the stability diagrams can be constructed. Also, the

presented application of the periodic system theory on the milling stability does not


provide physical insight to the dynamic milling process, e.g. relationships between chat

ter frequency, spindle speed and transfer functions. The author performed the stability

analysis by considering the physics of dynamic milling which is presented next.

8.3.3 Milling Stability Analysis Based on the Interpretation of Physics of

Milling Dynamics

The stability analysis can be performed in a much simpler way by considering the

dynamic displacements and milling forces at the stability limit. Start with the dynamic

milling equation without the mean forces {F8}:

{F} = c[A(t)1{} (8.52)

where

{} = ({r} - {r}) - ({r} - {r}) (8.53)

{r} and {r} are the cutter and workpiece displacements. At the chatter stability

limit, cutter and workpiece will vibrate with frequency Of course, this is true for

the zero order approximation, as the solution must include the response to the integer

multiples of w. This is due to the fact that the dynamic milling forces are produced by

the structural vibrations, and vice versa. Then, these forces must also respond to the

periodic variations of the directional milling coefficients matrix [A(t)] which relate the

dynamic displacements to the dynamic milling forces. However, w, stays constant during

a particular milling operation. This is explained by Tlusty [77]:

“The chatter frequency cannot change instantaneously as the vibrations decay

or increase slowly, time is needed for transition.”

If there are close vibration modes, the chatter frequency may shift to another vibration

mode if the process is disturbed, e.g. by changing the spindle speed. Then, for constant


cutting conditions, chatter frequency can be taken as constant.

Zero Order Approximation:

For the zero order approximation, i.e. when the vibrations at the chatter frequency

w are considered oniy, the cutter and workpiece displacements can be determined from

{r} [G(ic)]{F}et , {r} = —[Gw(iwc]{F}e1”t (8.54)

where the harmonics of the tooth passing frequency (ku) have been neglected. The

phase difference between the tooth vibrations in two successive periods is w’T. Then,

substituting equations (8.53) and (8.54) with {r°} = e_T{r} into equation (8.52) the


{F}e = c(1 — e_T)[Ao][G(iwc)]{F}eit (8.55)

where [G] = [Ge] + [Gm]. [A0] is the averaged value of [A(t)j in a tooth period, i.e. the

constant term in the Fourier series expansion of [A(t)]:

[A0] =jT[A(t)]dt (8.56)

The higher Fourier coefficients of [A(t)] are neglected as the response of the milling

forces, cutter and workpiece to the periodic variations of the directional milling coeffi

cients are not considered. Then, equation (8.55) becomes

{F} = c(1 —e_T)[Ao][G(iw)]{F} (8.57)

The characteristic equation is obtained as follows:

det[[I] — c(1 — e_T)[Ao][G(iwc)]] = 0 (8.58)


This is basically the truncated version of the characteristic equation (8.51) for (r, k =

0). However, it should be noted that equation (8.51) contains the eigenvalue A whereas

it has been replaced by (iwo) in equation (8.58). Although the difference seems to be

the simple substitution of A = ±iw0 for the limit of stability (marginal stability), as it

will be shown later, this substitution leads to the analytical determination of the chatter

stability limit in milling. Also, it is realized that, compared to the previous method,

the mathematical analysis presented in this section is shorter and simpler. Sridhar et

al. [78, 79, 80j and Minis et al. [82, 81] numerically determined the chatter limit by

continuously increasing the axial depth of cut in the simulation and determining the

corresponding eigenvalue A for 2 degree-of-freedom milling dynamics models.

Higher Order Approximations:

If the periodic components of [A(t)] are considered in the solution, then the response

of the dynamic forces to these should be included:

iT[A(t)] = [Ar]e [Ar] j [A(t)]e_irwtdt

(8.59)00 00

{ F} = {F}e’ = {Fk}eitk—oo k—00

Hence, the chatter frequency will be superimposed on the integer multiples of the

tooth frequency. It should be noted that the oscillations of the forces with the multiples

of tooth frequency is not because of the sinusoidal variation of the mean chip thickness or

intermittent engagement of the cutter teeth as the mean cutting forces are not considered

here. It is because of the periodically varying directional coefficients contained in [A(t)].

These periodic variations can be regarded as disturbances on the development of chatter.

Whether these periodic fluctuations in the system characteristics can help to increase


the chatter limit compared to a constant dynamics chatter process (like in turning)

by disturbing the phase between the inner and outer modulation (like the methods of

spindle speed variation [85, 86, 87] and irregular tooth pitch [83, 84]) will not be analyzed

here, however, will be discussed in some examples. By using superposition principle the

dynamic displacements can be written as

{r}= k=-00

+ ikw)]{Fk}eit

(8.60)

{r} = — [G,(iw +

Substituting into (8.52) the following is obtained:00 00

= c(1 — eiT) [A(t)] [G(iw + ikw)] {Fk}e’k—oo k=—c,o

(8.61)

= c(1 — e_iT)

k00

[A(t)] [G(i + ik)] {Fk}et

In the above equation, the Fourier coefficients of [A(t)] can be kept under the same

summation with the others by multiplying both sides of the equation by 1/Te_iTwt and

integrating from 0 to T, the following is obtained:

{Fr} = c(1 — e_T) [Wr_k(c + ikw)]{Fk} (r, k = 0, +1, ±2, ...) (8.62)

where

[Wr_k + ikw)] = [Ar_k][G\ + ikw)] (8.63)

[Ar_k] is the (r — k)th Fourier coefficient of [A(t)]:

[Ar_k] = (8.64)

Equation (8.62) has nontrivial solutions if the determinant is zero:

det[Srk[I] — c(1 — e_T)[Wr_k(A + ikw)]] = 0 (8.65)


where 6rk is the Kronecker delta and [I] is the (2mx2m) identity matrix. Equation (8.65)

is the same as (8.51) and, therefore, the simplified analysis of the milling stability without

using the parametric functions gives the same result obtained by classical procedure.

8.3.4 Truncation of the Characteristic Equation of Dynamic Milling

From the simplified stability analysis, or by substituting ) = iw, in equation (8.65),

the characteristic equation at the stability limit takes the following form:

det[6rk[I] — S[Ar_k] [G(iw + ikw)]] = 0 (8.66)

where s = c(1 — eiT), [I] is the identity matrix with size of 2mx2rn. Equation (8.66)

defines an infinite determinant and must be truncated to determine the stability limit.

For example, the zero order approximation is obtained when r = k = 0 in equation

(8.66):

det[[I]— s[Ao][G(iw)]] = 0 (8.67)

where [A0] contains the mean values of the corresponding periodic directional milling

coefficients:

1 T T [a] [a][A0] = j [A(t)]dt

= j dt (8.68)[a] [a]

where [a], are diagonal matrices with the elemental directional coefficients in the

main diagonal:

[apr]kl = apr(k) 1 = k 1(p,r=x,y)

[apr]kl=0 lk Jand

N-i

apr(k) = a(k) (p,r = x,y)j=0


and apr (k) are given in equation (8.20). Before evaluating the integral defined in equation

(8.68), consider the following change of variables:

4(k) g + jq. — kpzk = (t + jT) — kzk =— kzk (8.69)

where r = t + jT, Nw is the angular speed of the spindle in (rad/sec) and w is the

tooth passing frequency. Then, (8.68) becomes

[A0] =[a (t + jT)] [ax (t + jT)]

dtT o o (t + jT)] [ay, (t + jT)]

1 N—i ç(j+i)T (8.70)=

j [AQr)]drj0

1 NT

= j [A(r)]dr

Turning back to angular domain by substituting °k = — kzk the following is obtained:

1 NTI—k,/,zk[A0]

= f [A(Ok)]dOk (8.71)T —kzk

Using T = 2ir/N, equation (8.71) becomes

[A0] =N f21r-kzk = N f2-kzk [a(Ok)] [a(Ok)]

dO (8.72)2ir —k,zk 2ir —kzk [a(Ok)] [a(Ok)]

The elements of the diagonal matrices [a] etc. are written from equation (8.20) as

a(k) = —g(Ok)[sin2Ok + Kr(1 — cos2Ok)]

a(k) = —g(Ok)[(l + cos2Ok) + Kr sin 20k]873

a(k) = g(Ok)[(1 — cos2Ok) — Kr sin 2Ok]

a(k) = g(Ok)[sin20k — Kr(1 + cos2Ok)]


where

g(O) = 1 <Ok <(8.74)

g(Ok) = 0 Ok < st or °k > cex JThus, all the elements of the matrices [a] etc. are non zero oniy in the interval (4st—qex).

Then, the limits of integration become q and q!ex, independent of the element axial

position zk. According to this result, there is no effect of the helix angle on the chatter

stability as predicted by this formulation. Therefore, elements in diagonal of [a] become

equal to each other. Then, equation (8.72) takes the following form:

[A0] = (8.75)2ir c4°)(O)[I] c4?(0)[i]

where [I] is the mxm identity matrix, and

fcrr=j apr(0)dO (p,r=x,y) (8.76)

kst

Finally, the following is obtained by integrating equation (8.73):

if ]ex= L°2° 2KrO + Kr sin 20]

1 r 1ex= {—sin2O—28-f-Krcos2Oj

(8.77)

1 ex= [_sin2O+20+Krcos20]

1 er= {_cos2O_2KrO_Krsin2O]

Then, the zero order approximation to the characteristic determinant, given by equation


(8.67) takes the following form:

N 2 [‘Im [‘]m [G(i)1 [G(iw)]det [112m — = 0 (8.78)

27r c4 [dim0)

[dim [G(iw)] [G(i)]

where [G] = [G] + [G3,]. Equation (8.78) can be further simplified to the following:

det[[I]2m — = 0 (8.79)

where s = c(1 — e_T), [W0] can be regarded as the oriented transfer function in milling

and is given as follows:

a} [G(i.)} + [G(iw)] oj [G(iw)] + cj,) [G(iw)}[W0(zw] = (8.80)

a) [G(iw)] + c4,) [Gyr(ic)] 4} [G(iw)] +

Before the analytical solution of the zero order approximation, higher order Fourier

coefficients of [A(t)] will be given. For the first order approximation r, k = 0, +1, the

following truncated determinant is obtained:

[I] — .s[Ao] [G(iw)] —s[A_i] [G(iw + iL.)] —s[A1][G(iw — i)]

det —s[A1][G(iw)] [I]— .s[Ao][G(iw + ü.)] —s[A2][G(iw — iw)]

—s[A1][G(iw)] —s[A_2][G(ic + iw)] [I]— s[Ao] [G(iw — iw)]

(8.81)

where [Aq] represents the qtI Fourier series coefficient:

1 rT[Aq] =

— j [A(t)]e”tdt (8.82)To

The procedure is very similar to the one followed for [A0]. Similar to the result

obtained in equation (8.72), equation (8.82) takes the following form after the substitution

of 0:

[Aq] = - fex[A(0)]iN8dO(8.83)

27r st


where = Nw has been substituted. As in the case of zero order approximation, [Aq]

becomes independent of the axial position (or element number):

Jm zyN [a [Il [11m 1[Aq] =

— I (8.84)2ir () [Il (q)im [Ij j

Then, [Wr_kj takes the following form:

[Wrkl = [Ar_k][G(Wc + ikw)j

— F (r_k)[(j)] + a(k)[G(iwk)] (r_k)rG (iwk)j + (r_k)[(j)] 1xx

—

[a()) [G (iw)] + a()[G (iwk)] (r_k)rG (iwk)] + (r_k) [C (iwk)] jyx I xy

(8.85)

where wk = w + kw. Expanding e9 = cos qNO — i sin qNO and using trigonometric

identities, the following general form for the coefficients .., o are obtained:

ir 1exP2° I(q) = — J< eiqNO +c1&’° — ce jxx

ir ]cex= { — CKrjC’° +c1e9+ c2exy

(8.86)

ex= [coKreN0 +c1e’0+c2e29]

jf ex(q) Jç_iNO—c1e’0+ C2etP29l

j ‘1st


where

p1=2+Nq , p2=2—Nq

2 Krjc1= (8.87)ivq Pi

— Kr + iC2—

P2

Also, are equal to the complex conjugates of

respectively. Equations (8.84) and (8.86) can be used in truncations of the characteristic

determinant (8.66) to find the chatter limit.

The developed stability formulation predicts no effect of helix angle on the chatter

limit. Of course, this is true when the cutter teeth are equally spaced, the helix angle

is constant along the flutes and the same on each tooth. If the helix angle were lot

approximated within axial elements (equations 8.16- 8.17), matrices [a], .., [afl,] would

have the elemental lag angle k,&/z in the arguments of the trigonometric terms given in

(8.77). This would require an iterative solution for the chatter limit. On the other hand,

theoretically, the element thickness can be taken as infinitesimal in which case the cutter

edge in each axial element approaches to a straight line (zero helix). When the resulting

infinite sized matrices were integrated in equation (8.72), the dependence on the axial

position would be lost again leaving no effect of helix.

8.3.5 Summary of the Calculation of Milling Stability for the General Case

The stability of dynamic milling is predicted by solving the infinite determinant

equation:

d6t[6rk[I1 — SEAr_k] [G(zW + ikw)]] = 0

where


örk: Kronecker delta

[I]: 2mx2m identity matrix

rn: number of axial elements

i=

s = c(1 — eT)

c = LzK

LSz: axial element thickness

K: tangential milling force coefficient

c: chatter frequency

w: tooth (passing) frequency

[G] = [Ge] + [Gm]

[Ge], [Gm]: cutter and workpiece transfer functions

and [Ar_k] which is the (r — k)t Fourier coefficients of the periodically varying direc

tional milling coefficients are given by

[Ar_k] = (8.88)27r dr—k)[I]m

dr—k)[I]m

where a, .., a are given in equation (8.86). For the zero order approximation, the

determinant takes the following form:

det[[I] — s—[W0(iw)]] = 0

where [W0] contains the oriented transfer functions and is defined in equation (8.80).

8.3.6 Accuracy of the Chatter Limit Prediction by the Truncated Charac

teristic Equation

In the rest of the analysis, the zero order approximation of the characteristic equation

will be used to develop an analytical milling stability condition. In general, the accuracy


8

7

6

12

0153045607590


Figure 8.3: The effect of number of teeth on peak to peak (AC component) value ofdirectional coefficient a. (Half immersion-up milling with Kr = 0.5)

of the truncated characteristic determinant depends on whether the Fourier coefficients

of [A(t)] with significant amplitudes are included in the truncated determinant [174].

However, it should be noted that the chatter vibrations occur at a specific chatter fre

quency, and the effect of the directional coefficient oscillations in the harmonics of tooth

passing frequency may not be significant for the stability limit. The Fourier coefficients

of [A(t)] are closely related to the number of teeth in cut. The higher the number of

teeth in cutting is, the smaller the overall variation of the directional coefficients con

tained in [A(t)] becomes. As an example, the variation of the directional coefficient

given by equation (8.20) is shown in Figure 8.3 for 4,8,12 and 24 teeth cutters. A half

immersion-up milling case is considered with Kr = 0.5. As it can be seen from the figure,

the AC component reduces considerably compared to the average value (i.e. zero order

approximation) as the number of teeth is increased. However, as it will be shown by

examples, the zero order approximation predicts the stability limit accurately, including

the cases where AC components are high.


8.3.7 Solution of the Characteristic Equation to Determine the Chatter Sta

bility Limit in Milling

In this section oniy the zero order truncation of the characteristic equation will

be considered as it leads to an analytical method for milling stability. Therefore, the

following eigenvalue equation is to be considered:

det[[I] + A[A0][G(iw]] = 0 (8.89)

where the eigenvalue A is as follows

A = = —Ktz(l — e_T) (8.90)2ir 47r

It should be noted that axial depth of cut is equal to a = mz, where ZXz is the

thickness of axial elements. The unknowns in equations (8.89) and (8.90) are the chatter

frequency w and the axial depth of cut for the defined number of teeth N, radial depth

of cut b, spindle speed (to determine the tooth period T) and milling force constants K

and Kr. Equation (8.90) can be solved for two unknowns as it is a complex equation.

However, aiim cannot be directly obtained from equation (8.90) as it does not appear

explicitly, but /z does. The procedure is explained as follows. In the formulation of the

dynamic milling forces, the variation of the end mill and the workpiece dynamics in the

axial direction were modeled by dividing the total axial depth to number of elements.

The dynamics of the structures were assumed to be constant within each element. The

element thickness /z is selected depending on how strong the variations of the dynamics

of structures in the axial direction are, and the available modal test data or finite element

grid solutions. The difficulty with equation (8.89) is that before the critical axial depth

of cut can be determined, the number of elements in the axial direction, m, must be

known to construct the matrix [W0] which contains the oriented transfer functions of the

cutter and workpiece, respectively. However, nZz is equal to the axial depth of cut aiim


which is being solved for. Therefore, the solution has to be started with an arbitrary m

or a — rn/z. If w and A are known, then equation (8.90) gives the critical depth Zlim

or aiim = m/ziim. [f aiim > a + /-z or aiim a z, the new value of m has to be

determined as

m’ = aijm//z = m/zijrn//z (8.91)

Then, equation (8.89) should be solved again by using m’ to determine the new value

of the critical depth. The presented general milling stability formulation can be used

to analyze the stability of flexible workpiece milling as it considers the variation in the

workpiece dynamics in the axial direction which has been neglected in the other stability

models. This procedure is illustrated on a plate milling example given in section 9.4.3.

Also, the algorithm is given in Figure 8.13. It is realized that a critical radial depth of

cut bum for chatter stability can also be determined from the same equation if the axial

depth of cut is specified. The radial depth of cut is implicitly contained in the elements

of [A0] as it determines the start or exit angles (, q).

Derivation of Chatter Frequency-Spindle Speed Relation

• In order to be able to the calculate eigenvalue of equation (8.89) the chatter frequency

w must be known. The chatter frequency, in general, depends on the spindle speed and

the structural dynamics parameters. The number of waves between subsequent cuts (or

the number of vibration cycles within one tooth period) can be expressed as

k + /2ir = wT (8.92)

where k is the largest possible integer such that < 2ir. In other words, there are k full

vibration cycles and a fraction f/27r of a cycle between the subsequent passes at the same

point. In orthogonal cutting-chatter stability theory, the phase difference at the limit of


1.6

1.5

1.4

1.1

1

Figure 8.4: Variation of chatter frequency with spindle speed as predicted by the orthogonal cutting chatter theory.

stability () is determined as [5]

d2 —1= 2ir — 2tan’(

2d(8.93)

where d = and w and are the undamped natural frequency and damping ratio

of the considered vibration mode. Equations (8.92) and (8.93) define a relationship be

tween the chatter frequency w and the tooth period T. This is plotted in Figure 8.4 in

terms of the frequency ratios, w/w and w/w, for = 0.05. The corresponding spindle

speed n (rpm) can be determined as n = 60/(NT). It should be noted that it is easier

to determine the tooth period T from equation (8.92) when the chatter frequency w is

specified whereas the reverse requires the solution of equation (8.93) which is implicit

in w. A similar equation for milling, which is complex due to the periodically varying

directional coefficients, has not been derived before. In this thesis, chatter stability limits

for milling are derived using the stability formulation presented in the previous sections.

For that, the analysis of chatter frequency-spindle speed relation is necessary and derived

0 0.2 0.4 0.6 0.8 1

w/wn


in the following.

The critical axial depth of cut in terms of the element thickness can be obtained from

equation (8.90) as

Z1jm =— r

A(8.94)

— e_cT)

Substituting A = AR + iA1 and e_T = cos wT — i sin wT in equation (8.94)

4ir AR+iAIZ1im =

— NK [1 — (cos — i sin wT)](8.95)

from which the real and imaginary parts are separated as follows:

47r AR(1 — coswT) + AisinwT .A1(l— coswT) — ARSiIIWCTLZlim

NK (1— coswT)2+ sin2

+(1 — coswT)2+ sin2

(8.96)

/Zlim is a real number, then the imaginary part of equation (8.96) must vanish:

— coswT) — ARsinwT = 0 (8.97)

or,

AR 1—cos(8.98)A1 +/1 — cos2wT

The following quadratic equation is obtained after rearranging the above equation:

(1 +i2)cos2wT—2coswT+(1 — K2)— 0 (8.99)

One of the solutions of equation (8.99) is the trivial solution of = 0 + 2k7r which

corresponds to the case of no regeneration. The nontrivial solution is

1 K2coswT = 2 (8.100)

or,

wT = cos’( ) + 2k (8.101)


Equation (8.101) defines a relationship between the chatter and the tooth frequencies.

This equation is the same as the general form of the regeneration expression given in

(8.92), in this case

1—= cos1(

+(8.102)

The following equation for /Zlim is obtained if coswT from equation (8.100) and

sin = — cos2wT are substituted into the real part of equation (8.96) (imaginary

part vanishes):

Z1jm= 2(1 + 1/a) (8.103)

Therefore, given the chatter frequency, the chatter limit in terms of the element thickness

can be determined from equation (8.103). As explained in the beginning of this section,

the corresponding chatter limit can be determined as aiim = m/ziim. If rn’ determined

from equation (8.91) is different than m, then the eigenvalue equation (8.89) is solved

again by using m’ number of axial elements. In general the value of the eigenvalue A can

only be determined by numerically solving the eigenvalue problem defined in equation

(8.89). It should be noted that this is the only part in the developed stability analysis

which requires a numerical solution. However, this is necessary only for the most general

case of the milling stability. As it will be shown in the following sections, the numerical

eigenvalue solution is not necessary for most of the practical cases. These are the cases

where the variation of the cutter and workpiece dynamics in the axial direction can be

neglected, i.e. m = 1. The corresponding tooth period T or spindle speed m to the

computed chatter limit can be determined from equation (8.101). When computing the

value of e, it should be noted that ‘ = 27r — c is also a solution to the (cos1) function

given in equation (8.102). In general, the actual value of the angle can be determined

from the signs of the x (real) and y (imaginary) components of the vector which defines it,


i.e. in which quadrant the vector is. However, this is not possible for (cos) function. One

way of solving this problem is to substitute both solutions into equation (8.96): the one

which gives real /Ztim only, is the solution. Another solution is obtained by considering

the angle defined by the eigenvalue A in the complex plane. The angle subtended by this

vector is

= tan’ = tan’ (8.104)

Then, by substituting = 1/tan into equation (8.100), the following is obtained:

= —cos2 (8.105)

The solutions of the above equation is

wT = (+ir + 2) + 2k7r (8.106)

Considering the solutions obtained from (8.101), the solution is found to be:

= + 2k7r (8.107)

where

= — 2 (8.108)

It should be noted that this solution is more convenient as the actual value of = tan1

can easily be obtained from the sign of AR: ‘ = cp + r, if AR < 0. Therefore, by using

the milling stability formulation developed in this thesis, the relationship between the

chatter frequency and the spindle speed in milling is obtained for the first time. This

also makes the analytical prediction of milling chatter possible.

8.4 Solutions of Milling Stability Equation for Special Cases

The developed general stability formulation considers the dynamic displacements

of the milling cutter and workpiece in x and y directions, and the variation of their


dynamic characteristics in the axial z direction. The variation of the dynamics in the

axial direction is important in milling very flexible workpieces and may affect the stability

limit. This has not been considered in the previous milling chatter models, and it is

modeled in the developed formulation in this thesis. The stability limit and phase (or

spindle speed) formulae require the solution of the eigenvalue A for the stability equation

[[I] + A[A0j[G(iwj]. For the general case, this has to be performed numerically which is

the only numerical part in the formulation. In this section, the general formulation will

be applied to special and practical cases of milling. For these cases, the solution can be

obtained completely analytically. Although a numerical eigenvalue solution is very fast

on computers, the analytical solution provides direct relationships between the milling

conditions, stability limit and chatter frequency.

8.4.1 Milling of a Single Degree-of-Freedom Workpiece

Consider the single degree-of-freedom dynamic milling model shown in Figure 8.5.

This model represents the milling of a flexible part with a relatively rigid cutter. It

should be noted that, although the workpiece is modeled as a single degree-of-freedom

structure, it can have more than one vibration mode in the considered degree-of-freedom

direction. Then, milling of thin walled components such as impellers, thin webs etc. can

also be analyzed by this model if the considered axial depth of cuts are small so that

the dynamics of the workpiece can be assumed constant within the cutting depth. As it

will be shown by some experimental and simulation results in the following sections, the

chatter free axial depth of cuts for flexible workpieces are very small unless low cutting

speeds are used. With moderate and high cutting speeds, the total depth is removed

layer by layer, using very small depths [95]. In practice, usually a constant axial depth of

cut is used throughout, however the dynamics of the workpiece are different for different

layers. Therefore, by the accurate prediction of the stability lobes the number of passes


can be reduced significantly.

For the system shown in Figure 8.5 which has single axial element, i.e. m = 1,

= aiim. Then, the eigenvalue equation (8.89) becomes:

1 — -Ktaijm(1 — e cT)a,yGy(iwc) = 0 (8.109)

where c is the directional coefficient given by equation (8.77). The superscript “(0)”

to denote the zero order approximation will be dropped from the directional milling

coefficient matrix [A0] from this point on for the sake of simplicity. G is the transfer

function of the workpiece in the y direction:

2 /kG (i)

‘ “(8.110)2

—

fl C ‘Y fly C

where and are the stiffness, damping ratio and undamped natural frequency of

Figure 8.5: Single degree-of-freedom milling system model.


the workpiece in the y direction. If the workpiece has more than one vibration mode in

this direction, then the transfer function G can be determined by modal superposition

as:— r

2 w22 ij1 JflyC “Y3 flY”C

where the subscript j denotes the parameter of the th vibration mode. From equation

(8.109), the chatter limit is obtained as

aiim= N

1(8.111)

— ecT)

As aiim is a real number, the imaginary part of the complex term G(i)(1 — e_iT)

must vanish. Hence, the imaginary parts of G(iw)e’Tand G(iw) should be equal to

each other, as shown in Figure 8.6. Then, the real parts will have opposite signs resulting

in

G(i)(1 — e_cT) = 2Re[G(iw)] (8.112)

The real part of the transfer function G(iw) is as follows:

1 1-d2= k,(1 — d)2 +4d

(8.113)

where d = Substituting (8.112) in equation (8.109)

aiim T (8.114)

From Figure 8.6, the phase is obtained as in the orthogonal cutting chatter stability

theory:d2

= 27r — 2tan12cl

(8.115)

Then, the spindle speeds (n) corresponding to the considered chatter frequency in the

different lobes (k) can be found from

k + 6/27r = (8.116)


Figure 8.6: The phase angle and transfer function at the chatter stability limit. Inmilling, chatter frequency may be higher or lower than the modal frequency dependingon the cutting conditions. In both cases G(1 —

6T)= 2Re[G].

Im

N -Re

G -iwTGe


where n = 60/(NT). If the workpiece is flexible oniy in the x direction then the critical

axial depth is determined similarly as:

aiim= N

1(8.117)

These equations derived for milling here are similar to the stability condition devel

oped by Tiusty [5] for single tooth cutting with nonvarying directional coefficients, e.g.

turning and boring. Equations (8.114) and (8.117) suggest that the chatter frequency

can be smaller or higher than the natural frequency depending on the signs of the direc

tional milling coefficients a, and c. If a > 0, then Re[G(i)] must be positive,

i.e. < w,, in order to have aiim > 0, and vice versa. In the classical orthogonal

cutting stability theory of Tlusty, the chatter frequency is always larger than the modal

frequency. This is because of the presumed positive direction for the feed cutting force

which can only be negative for large and impractical rake angles (due to an increased

component of the rake face normal force in the feed direction). However, in milling, the

directional coefficients are not constant as proven here...,o represent averaged

directional milling coefficients, and depending on the radial immersion of the cutter and

Kr, their sign may change. Minis et al. [193] showed that even in turning, the chatter

frequency at the limit of stability can be smaller or larger than the modal frequency

depending on the orientation of the tool holder (right or left-handed), thus the transfer

function. The variations of a with the immersion angle for up and down milling and

different Kr values are shown in Figure (8.7). c is always negative for down milling,

thus the chatter frequency is always higher than the natural frequency of the structure

whereas in up milling, it depends on the exit angle. Similar to the milling force in the

y direction, o, is smaller for up milling resulting in higher stability limits. As it can

be seen from these graphs, the variation of the directional coefficient n and thus the

stability limit aiim with the radial immersion is not linear. These graphs can be used to


1.00

0.50

0.00

-0.50>‘>‘-.

-1.50

-2.00

-2.50

-3.00

-3.50

180

0.00

-0.50

-1.00

-1.50

-2.00

-2.50

-3.00

-3.50

180

stFigure 8.7: Variation of the directional milling coefficient a with the immersion anglein up and down milling. The sign and magnitude of the directional coefficients determinethe chatter frequency and stability limit, respectively.

0 30 60 90 120 150

ex

0 30 60 90 120 150


determine the optimal radial depth of cuts for chatter free milling.

Example: Milling of a SDOF Structure

The stability formula developed in this section is applied to a milling operation in

vestigated by Opitz [163], Sridhar et al. [80] and Minis et al. [81]. The flexibility of the

workpiece in the x direction is represented by a single degree-of-freedom system with the

following parameters:

= 7.152x106 N/rn (39700 ib/in), = 0.0417, = 355 rad/sec

A straight tooth cutter with N = 10 teeth is used and the start and exit angles

are 67° and 139°, respectively. Kr = 0.577 and a is calculated from (8.77) as (-1.6).

Equation (8.117) was used to calculate the stability limit for different values of chatter

frequency w. Then, the corresponding spindle speeds are found from equation (8.92).

Stability lobes are shown in Figure 8.8 in terms of the product of critical depth of cut

aiim and tangential cutting force coefficient K. Also shown in Figure 8.8 is the data

from [81] and [80]. Minis et al. [81] used zero order approximation, but determined the

stability limit for a given spindle speed by increasing the axial depth of cut until the

marginal stability was obtained. Sridhar et al. [80] used the analog simulation data from

[163] to compare with their numerical stability solution results. All three solutions show

excellent agreement except around two peak stability limits. Around the peak stability

points, the chatter frequencies are very close to the natural frequency of the structure

since Re[G(iw] —* 0 as —* . At these frequencies the variation of Re[G(i] with the

frequency or spindle speed is quite steep, and therefore a fine spindle speed or frequency

resolution is necessary for accurate results. It should be noted that the only analytical


6

05000

0100 400

Figure 8.8: Comparison of the predicted chatter limit with the published data for thesingle degree-of-freedom milling system example.

solution is the original contribution proposed here. The resolution can be made as fine

as required without increasing the computation time significantly. Hence, the numerical

solution procedures used in [81] and [163] may have resulted in the differences around

the peaks.

8.4.2 Milling of a Flexible Structure with a Flexible End Mill-Single Axial

Element

Consider the milling system model shown in Figure 8.9. This is the most general case

of a milling system when only one axial element is considered. As in the previous case,

the cutter and workpiece can have more than one vibration mode in the two considered

orthogonal directions, x and y. Then, the general stability equation takes the following

150 200 250 300 350

Spindle Speed (rpm)


Workpiece

kcx

Figure 8.9: Milling system model with two degree-of-freedom cutter and workpiece.


form:

c G(ic4) 0det I [I] + A = 0 (8.118)

c c 0 G(iw) )where .., c are given in equation (8.77). G, = + , G and are the

cutter and workpiece transfer functions which may include contributions of more than

one vibration mode. The cross-transfer functions and have been neglected for

the sake of algebraic simplicity. From equation (8.118) the following quadratic equation

is obtained for A:

a0A2 + a1A + 1 = 0 (8.119)

where

a0 = — abZ)

(8.120)a1 = c,G(iw) +

Then, the eigenvalues A are obtained as:

A = ——(a1+ — 4a0) (8.121)

The critica’ axia’ depth of cut can be obtained by substituting A into equation (8.103):

aiim = —(1 + 1/ic) (8.122)

where ic = . Corresponding spindle speed (n = 60/NT) to a considered chatter

frequency can be found from equations (8.104), (8.107) and (8.108) as:

= +2k7r

where

=— 2ço

and

=tan —=tan —

AR


Example: Milling of a Rigid Workpiece with 2 DOF End Mill

The developed stability method is applied to an end milling operation investigated by

Weck, Altintas and Beer [162] at the Machine Tool Laboratory of Technical University

of Aachen. The workpiece material is aluminum alloy AlZnMgCu 1.5 which is machined

by using 30 mm diameter, three fluted helical end mill with 300 helix angle and 110 mm

gauge length. The measured milling force coefficients are K 600 MPa and Kr = 0.07

(too low radial force coefficient Kr implies that the rake angle on the flutes is very high).

The dynamic parameters of the end mill were determined from modal tests as:

k = 5590 N/mm = 0.039 w = 3788 rad/sec (603 Hz)

k = 5715 N/mm = 0.035 w = 4161 rad/sec (666 Hz)

Hence, the cutter has one dominant vibration mode in each direction. Chatter tests

were conducted at different radial depth of cuts by using a feed per tooth of st = 0.07

mm/tooth. Axial depth of cut and spindle speed increments were 0.5 mm and 200 rpm,

respectively. Discrete time domain chatter simulations were also used to determine the

stability limit. Both experimental and time domain simulation results for quarter im

mersion up milling and slotting tests are shown in Figure 8.10 which is taken from [162].

In chatter simulations, the process was assumed to be unstable when the peak values of

the vibration grow in thirty consecutive oscillation periods. For each spindle speed, this

procedure has to be repeated by increasing the axial depth of cut until the stability limit

at that speed is obtained. Small increments of spindle speed and axial depth should be

used in the simulations to accurately obtain the stability limits, especially the pockets in

the lobes. Then, depending on the spindle speed range of interest and the desired accu

racy, hundreds of simulations may be necessary to obtain a single stability lobe diagram.

Also, the effect of cutting conditions on the stability limit can only be analyzed after the

Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling

E

222

Figure 8.10: Experimental and time domain simulated stability limits for end millingtests. (data by Weck, Altintas and Beer, 1993)

....

....

....

....—-——-

•ts• pq*w•••i••

••• 4.a4.ha.? ad

0000 ooo0t0000

I

0

p0000

b=R12

Legendo :Stablecut• :Lightchatr• :Chatter

Simulated stability

0 1000 2000 3000 4000 5000

Sprndle speed n (rev/mm)

0I I I

100 200 300Cutting speed v (rn/mm)

400

10

00

0a)93

51

01000 2000 3000 4000 5000

Spindle Speed (rpm)

Figure 8.11: Analytically predicted stability lobes for the case for which the experimental• data and time domain simulations are shown in Figure 8.10.

diagrams are obtained. The developed stability method was used to obtain the stability

diagrams shown in Figure 8.11.

The stability limits were calculated at 100 frequencies (or spindle speeds) for each

lobe, thus a total of about 800 frequencies. The total computation time to obtain a

diagram on a IBM 486-66 computer was about 10-15 seconds. As it can be seen from

the figures, the analytical predictions and the time domain simulation results are in good

agreement although the cutter has only three flutes resulting in the directional coeffi

cients with high AC/DC ratio.


b=R12

b=2R

Another example of a 2 DOF milling system is taken from Smith and Tlusty [70].


This is an half immersion-up milling of an aluminum workpiece by a 4 in. diame

ter shell mill with 8 teeth. The modal properties of the cutter were determined as follows:

Mode Frequency (Hz) Stiffness (N/m) Damping Ratio

X 1 260 2.26x108 0.12

2 389 5.54x107 0.04

Y 1 150 2.13x108 0.1

2 348 2.14x107 0.1

The results of the analytical method predictions and the time domain simulations per

formed by Smith and Tiusty [70] are shown in Figure 8.12. As it can also be seen from

this figure, the zero order approximation is in excellent agreement with the time domain

simulations.

8.4.3 Milling of a Flexible Structure with a Rigid End Mill-Varying Dynam

ics in Axial Direction

In this section, the application of the general stability formulation to the milling of

plate-like workpieces is given by considering the variation of the workpiece dynamics in

the axial direction. For this case, the stability equation (8.89) becomes:

det ([I] — Ktz(1 — e_iT)cryy[Gy(iwc)]) = 0 (8.123)

The transfer function of the workpiece in the y direction can be obtained by modal

superposition as:k

[G(iw)] = [u]H (iwo) (8.124)j=1


0.012

0.01

c 0.008

0.006

0.004

0.002

00.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80

Spindle Speed /10000 (rpm)

Figure 8.12: Analytical and time domain stability limit predictions for a case analyzedby Smith and Tiusty.

where

H (i) = . (8.125)— w2 + 2zw,w

and [u3] = {uy,}{uy}T is the j undamped modal matrix of the workpiece, normalized

for unity modal mass . The stability diagrams for flexible workpiece milling can be

generated by using the general stability equation (8.103). However, the formulation is

further simplified if the modes of the workpiece are well separated. In this case, the

stability analysis can be performed around the most flexible mode of the structure by

neglecting the contributions of the other modes. Then, for the rth mode, equation (8.123)

becomes:

det ([I]— A[ttyr]) = 0 (8.126)

1The structure is assumed to be proportionally damped.

k=1 k=0

Analytical• Simulation (Smith & Tiusty)

Chatter

Stable


where

A = Kjz(1 — e_iT)cyyHyr(iw) (8.127)

The imaginary parts in equation (8.127) must vanish to obtain real /z. The eigenvalue

A is a real number as the [uyr] in equation (8.126) is a real matrix. Then, similar to

equation (8.111), the following equation must hold at the limit of stability (see Figure

8.6)

(1 — e_T)Hy = 2Re[Hyr(ic)] (8.128)

The corresponding phase angle and the spindle speeds can be determined same as the sin

gle degree-of- freedom case given by equations 8.115 and 8.116. Substituting in equation

(8.127), the following is obtained for the critical depth of cut in terms of the elemental

thickness /.Ziim:

AZiim = (8.129)

e[ ,r(zc]

As explained before, the critical depth of cut is obtained as aiim = mzjjm, m is the

number of axial elements. For an arbitrary m and axial depth of cut a = m/z, if

aiim a + /z or aiim a — z, the new value of m has to be determined as

m’ = aijm//..z = m/zljm/LSz (8.130)

The solution is started with m = 1. The procedure is outlined in Figure 8.13 and the

method is illustrated with a plate milling example in the following.

Example: Peripheral Milling of a Plate-MDOF Model

Stability limits in milling a cantilever plate (Ti4A16V) with dimensions (63.5x44x3.8

mm) is simulated. The 19.05 mm diameter end mill, which is assumed to be rigid, has

4 flutes, is used in the down milling mode to finish the plate surface. A very small

radial depth of cut with Qst = 175° (which allows relatively high critical depth of cuts so


Consider M differentchatter frequencies aroundthe rth modal frequency.

Start with single axialelement

Solve the elgenvalue problemby using (m) nodes in the modalmatrix Iuyrl (mode shapesat the in-cut nodes only).

Determine the elementalstability limit by using theknown values of N, Kandc which depends on radialdepth of cut and K r

Calculate the limiting axial depthof cut.

Check whether the calculatedlimiting axial depth of cut is insidethe part of the workpiece structureconsidered in the determinationof eigenvalue A.

Determine the correspondingspindle speeds (n) in differentlobes (k).

Figure 8.13: Stability limit calculation algorithm for flexible workpieces with varyingdynamics in the axial direction.

n=60/(NT) (rpm)

k+

Re—arctan

lmIHvr(wc)]


0.014

0M12

o o 008

0.006

0.004

0 0.002

010000 15000 20000 25000 30000 35000 40000 45000 50000

Spindle Speed (rpm)

Figure 8.14: Predicted effect of vibration mode shape on the stability limit. The rigidityof the plate increases in the axial direction towards the fixed end allowing higher stableaxial depth of cuts than the ones predicted by neglecting this variation.

that more than one axial element is in cut) used in the simulations to show the effects

of mode shapes on the stability limits. K 1500 MPa, Kr = 0.7 were used. The

developed Finite Element code was used to determine the normalized modal vectors and

frequencies of the plate. Figure 8.14 shows the simulated stability lobes . Also shown

is the predictions by using the single axial element at the tip of the plate. The rigidity

of the plate increases along the axial direction towards the fixed end, thus as more axial

elements are considered, the predicted stability limit increases. In the stability pocket

shown in the figure, the MDOF model prediction is about 50 % more than the SDOF

model prediction.


8.5 Dynamic Peripheral Milling of Plates

Cutting tests have been performed on cantilever plates in order to analyze the dy

namic milling process and verify the developed milling stability model. Results of several

tests with a titanium (Ti6A14V) cantilever plate of dimensions (63.5x44x3.8 mm) are

presented here. The plate is down-milled by an 8 fluted, 300 helical carbide end mill with

19.05 mm diameter. In all cases, feed per tooth of St = 0.008 mm and radial depth of cut

of 0.325 mm, which corresponds to 15° immersion angle (or = 163°, q = 180°), were

used. The modal parameters of the plate were determined from the developed Fillite

Element program. First three undamped modal frequencies for the unmachined plate

are 1667, 3057 and 7074 Hz. The stability diagram around the first mode is shown in

Figure 8.15. Milling force coefficients of K = 2000 1VIPa and Kr 0.72, and damping

ratio of = 0.05 were used in the simulations. The dominant first mode of the plate is

considered, and chatter stability diagram is computed using the single degree-of-freedom

stability model presented in section 8.4.1. Due to high flexibility, the stable axial depth

of cuts are very small in the peripheral milling of this plate. The highest stability limit is

obtained approximately at 6500 rpm, in the pocket between the lobes k = 1 and k = 2.

The largest pocket (between k = 0 and k = 1) could not be obtained as the maximum

spindle speed of the milling machine was 10000 rpm. The tooth passing frequency with

6500 rpm is 867 Hz which is very close to the half of the first undamped modal frequency,

i.e. 1667/2=833 Hz. The peak stability limits correspond to the tooth passing frequencies

which are integer divisions of the considered modal frequency of the structure. Therefore,

at these spindle speeds while the stability against chatter is maximized, forced vibrations

may become excessive.

2These are average values for Ti6A14V as presented in Chapter 4. Note that edge force componentsshould not be included as they do not depend on the chip thickness, thus produce dc force componentswhich are not considered in chatter analysis.


0.6

E0.5

0.4

0.3aa)90.2C).1

L-010

02000 3000 4000 5000 6000 7000 8000 9000 10000

Spindle Speed (rpm)

Figure 8.15: Stability diagram for the cantilever titanium Ti6A14V plate down milled by8 flute carbide end mill with 19.05 mm diameter. Plate dimensions: 63.5x44x3.8 mm.Radial depth of cut b=zO.325 mm.

In the first test 0.4 mm axial depth of cut was removed at 6500 rpm. As shown in

Figure 8.15, this axial depth is slightly lower than the stability limit, thus a stable cut is

expected. The cutting forces in the x and y direction were measured by a Kistler table dy

namometer whose original bandwidth was 4000 Hz. However, due to the flexibility of the

clamping between the dynamometer and the milling machine table and the mass of the

plate which is mounted on the dynamometer, the real bandwidth is about 1900 Hz in the

y direction and 1750 Hz in the x direction. Also, the sound generated during machining

was recorded by an ordinary microphone which has a wider reliable bandwidth. Figure

8.16 shows the cutting force spectrums in x and y directions. The fundamental frequency

of the cutting force, w = 867 Hz, can be clearly seen in the spectrums. The second peak

at about 1600 Hz is close to the first modal frequency of the plate and second harmonic

of the cutting forces. In order to clarify the source of the vibrations at this frequency,


the following analyses are given. First, consider the sound signal and its spectrum shown

in Figure 8.17. In general, chatter vibrations gradually grow and then diminish when

the vibration amplitudes become so large that the contact between cutting tooth and

workpiece is lost. This pattern is not seen in the sound signal. The plate is milled

at 5000 rpm with the same axial depth of a = 0.4 and radial depth of b = 0.325 mm.

The force spectrums are shown in Figure 8.18. Although at 5000 rpm, a = 0.4 mm is

higher than the predicted stability limit shown in Figure 8.15, the maximum amplitudes

of the force spectrums occur at the tooth passing frequency, = 667 Hz, and its second

harmonic. In Figure 8.19 the cutting forces in y direction are compared for n = 6500

rpm and n = 5000 rpm. Although n = 5000 rpm is in the unstable zone, the peak to

• peak force amplitudes are higher for n = 6500 rpm. This is due to the fact that n = 6500

rpm spindle speed causes resonance in the plate. Therefore, it can be concluded that the

forced vibrations in plate milling are as critical as the chatter vibrations.

The cutting forces for n = 6500 rpm and a = 2 mm are shown in Figure 8.20. In this

case the chatter behavior can be clearly seen as 2 mm and is well above the stability limit.

The cutting force amplitudes grow to very large values and diminish in an exponential-

like manner after the contact between the cutting teeth and the workpiece is lost. The

same behavior can be seen in the sound signal shown in Figure 8.21. The force spec

trums shown in Figure 8.22 indicate that the forced and chatter vibrations exist together.

The chatter free axial depth of cuts are very small in plate milling. Therefore, either

many cuts with small axial depths have to be used to machine the plate without chatter or

very slow cutting speeds should be used to utilize the high process damping generated at

those speeds. In order to show the effect of cutting speed and the process damping on the

chatter behavior, 3 different spindle speeds, n = 6500, 400, 250 rpm, were used to remove


5

.) 4-

o

__________

0 800 1600 2400 3200 400040

30

25

20

15

10

5

0 Li . I I I

0 800 1600 2400 3200 4000

Frequency (Hz)

Figure 8.16: Cutting force spectrums in x and y directions for n = 6500 rpm and a = 0.4mm.


—1

-2

-3

-4

4

3

2

1

0

0

II

25 50 75 100 125

Tooth Period

150

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0

0 800 1600 2400 3200

Frequency (Hz)

4000

Figure 8.17: Sound sigilal and its spectrum at n = 6500 rpm, a = 0.4 mm.


5

=

0 800 1600 2400 3200 4000

10

:

_____ ________________________

0 800 1600 2400 3200 4000

Frequency (Hz)

Figure 8.18: Cutting force spectrums in the x and y directions for n = 5000 rpm anda = 0.4 mm.


60

40

20

0

-20

-40

-60

n=5000 rpm

I Li ii

iiiii

I III

________________________________________________________

II

80 120 160

Tooth Period

Figure 8.19: Cutting forces in the y direction for n = 5000 rpm anda = 0.4 mm.

60

40

20

0

-20

-40

-60

0 40 80 120 160

nUUrpiTI j

200

‘II ‘I I’I I II ‘II liii

0 40 200

n = 6500 rpm for

Chapter 8. Analysis of Dynamic Cutting and Chatter Stability in Milling

400

220

40

-140

-320

-500

500

333

167

-0

-167

-333

-500

0 50 100 150 200 250 300 350

0 50 100 150 200 250 300 350

Tooth Period

236

Figure 8.20: Cutting forces in x and y directions for n = 6500 rpm and a = 2 mm.


•1

Frequency (Hz)

20

15

10

5

0

-5

-10

-15

-20

3.0

2.5

2.0

1.5

1.0

0.5

0.0

0 25 50 75

Tooth Period

100 125 150

0 800 1600 2400 3200 4000

Figure 8.21: Sound signal and its spectrum for n = 6500 rpm and a = 2 mm.


14

12

10

‘—, 4

2

0

0 4000

60

e)50

—

E 0

20

10

0

0 4000

Frequency (Hz)

800 1600 2400 3200

800 1600 2400 3200

Figure 8.22: Cutting force spectrums for n = 6500 rpm and a = 2 mm.


the whole depth of a = 44 mm on the plate with the same radial depth of cut b = 0.325

mm and feedrate s = 0.07 mm. Figure 8.23 shows the sound spectrums for three spindle

speeds. It should be noted that although the predicted chatter limit is higher at n = 6500

rpm compared to n = 250 rpm, no chatter is observed at n = 250 rpm due to process

damping. Figure 8.24 shows the cutting forces for n = 6500 rpm and n = 250 rpm. As it

can also be seen from this figure, much lower peak to peak force amplitudes are obtained

when the resonances are avoided, i.e. harmonics of the tooth frequencies do not match

with the modal frequencies of the plate, and the high process damping zone is utilized.

Then, the peripheral milling of plates at slow spindle speeds with the full axial depth of

cut is more efficient than removing material layer by layer using high spindle speeds with

very small axial depths. In addition to the higher machining times and machining marks

left on the surface in layer cutting, the forced vibrations are excessive in plate milling

when cutting speeds close to high stability pockets are selected.

8.6 Summary

A comprehensive dynamic milling formulation and an analytical chatter stability

model are developed. The cutter and workpiece are modeled as multi degree-of-freedom

structures. A general formula which is able to predict chatter free milling conditions,

i.e. axial and radial depths of cut, and spindle speed, is derived. The formulation is

applied to several special cases including the milling of flexible workpieces. The model

predictions are verified by experimental data and time domain chatter simulations. Also,

dynamic milling tests show that the developed model can be used to determine chatter

free axial depth of cuts in plate milling.


2.5

2.0

1.5

1.0

0.5

0.0

0.20

0.15

— n=6500 rpm

Frequency (Hz)

Figure 8.23: Sound spectrums for different spindle speeds showing the effect of processdamping on chatter stability, a 44 mm

30

25

20

15

10

5

0

n=250 rpm

0.10

0.05

0.00

0 2000 4000 6000 8000 10000


600

400

200

0

-200

-400

-600

-40

-60

-80

-100

-120

-140

-160

-180

Tooth Period

Figure 8.24: Effect of spindle speed and the process damping on the cutting forces.

0 2 4 6 8 10 12

0 2 4 6 8 10 12

a = 44 mm.

Chapter 9

Conclusions

The peripheral milling of very flexible titanium alloy plates under static and dynamic

cutting conditions has been analyzed and modeled in this thesis. The plate is modeled

using 8 node isoparametric finite elements, and the end mill is represented by an elastic

beam. The structural models of the plate and tool are integrated with the models of

peripheral milling mechanics and dynamics developed in the thesis. The models include

accurate and unified formulation of milling mechanics, accurate chip thickness and force

calculations by considering the partial disengagement of flexible cutter and workpiece

structures. The system developed is able to predict the irregular distribution of milling

forces and the dimensional form errors left on the finished surface due to static deforma

tions. A form error constraint algorithm, that schedules the feed along the plate whose

thickness continuously varies due to metal removal, is developed. Another algorithm

has been developed to identify optimal radial depths of cut which allow the metal re

moval rate to be maximized but constrain the normal forces which cause dimensional

form errors. The methods developed are based on the models of peripheral milling and

surface generation proposed in the thesis. The dynamic interaction between the flexible

workpiece and the flexible cutter is formulated by using multi-degree of freedom mod

els. A stability model of self excited chatter vibrations in milling has been formulated,

and solved, using a novel analytical approach which eliminates the use of time domain

numerical simulations of chatter in milling. All models developed and proposed in this

thesis have been verified experimentally.

242

Chapter 9. Conclusions 243

The contributions of this thesis can be summarized in the following main categories

with the related results and conclusions:

• The peripheral milling of very flexible titanium plates is studied in depth for the

first time. There is very little previous work in this area, and in the previous

investigations the plates were either more rigid or as rigid as the cutter itself. The

high flexibility of the plates causes their separation from the cutter, requiring very

small chip loads and the development of different structural and cutting models

than those used for moderately rigid structures.

— A complete simulation system has been developed to predict the milling force

distribution, the static deformation of the very flexible plate and the cut

ter structures as well as their interaction with the milling process. Forces,

surface form errors and milling chatter stability limits are predicted and ex

perimentally proven for the first time for very flexible multi-degree of freedom

milling systems. The physical models of milling mechanics, chip thickness re

generation, surface form error, process-structure interaction, constraint based

optimization of cutting conditions, and stability for chatter vibrations are de

veloped as parts of the plate milling simulation study.

— The milling force coefficients, which relate the milling forces to the chip thick

ness, are expressed using existing mechanistic curve fitting and calibration

techniques. Alternatively, the same coefficients are obtained using an im

proved oblique cutting model with a more accurate chip flow estimation. It

has been demonstrated that the milling force coefficients can be evaluated with

satisfactory accuracy by applying oblique cutting transformation on a set of


standard two dimensional orthogonal cutting test parameters. This method

eliminates the use of calibration tests for each milling cutter geometry.

— It is proven both analytically and by simulation that the effect of static regen

eration of chip thickness on cutting forces diminishes in a few tooth periods,

and the regeneration changes only the dynamic milling forces as it is the pri

mary cause of chatter vibrations. The changes in the static cutting forces

are not due to chip thickness regeneration but rather due to changes in the

immersion boundaries between the flexible tool and workpiece.

— An optimal value of radial depth of cut which minimizes the normal milling

forces and dimensional surface errors is formulated for up milling operations.

It is shown that much higher feedrate values can be used at predicted radial

depths of cut without increasing the forces and deflections significantly, thus

improving the productivity.

— The plate is modeled by 8 node isoparametric finite elements, the cutter is

represented by a continuous elastic beam and the stiffness of tool clamping to

the collet is approximated by a linear spring. The thickness of the elements

is updated as the material is removed in the feed direction. The integrated

modeling of the three interacting structures during peripheral milling is the

most comprehensive found in literature.

— The cutting forces and form errors in peripheral milling of the most flexible

parts ever considered were predicted most accurately. It is shown that in plate

milling, in order to predict the cutting forces and form errors accurately, the

structure-milling process interactions have to be modeled.

— An algorithm has been developed to constrain the maximum form errors

caused by the static deflections of very flexible plates, flexible end mills and


collets. The algorithm is integrated with the plate milling simulation system,

and it automatically identifies the feeds along the tool path.

• A complete analytical stability formula is derived for the first time for multi degree-

of-freedom helical milling operations. Traditionally, milling chatter analyses have

been performed using time domain numerical simulations to allow the considera

tion of periodic excitation and variation of directional coefficients in milling. The

following formulations are introduced to the milling dynamics literature:

— Directional dynamic milling coefficients are introduced in the chatter formu

lation. The coefficients are expressed as functions of cutting conditions (axial

and radial depths of cut, cutting speed), milling force coefficients, and cutter

geometry (helix angle, diameter, number of flutes).

— An analytical relationship between milling conditions, chatter frequency and

spindle speed is derived for the first time. This is obtained by two different

approaches. First, a novel stability analysis of milling dynamics which is

based on the interpretation of the physics of the process is developed. Second,

the stability theory of periodic systems is applied to milling dynamics. Both

approaches yield the same results; however, the former is more practical and

gives more physical insight to the problem.

— For the first time, radial depth of cut explicitly appears in an accurate milling

stability formulation. Hence, both chatter free radial or axial depths of cut

can be directly obtained. It is shown that an optimal radial depth of cut which

maximizes the chatter free material removal rate can be identified using the

introduced directional dynamic milling coefficients.


— The stability method is applied to the most general case of milling flexi

ble workpieces by flexible end mills which have multi degree-of-freedom and

changing dynamics along the cutter axis. The effects of variation of workpiece

dynamics on the stability limit are illustrated.

— The stability formula is applied to various practical cases and verified exper

imentally. It is shown that the accuracy of predicting the chatter stability is

independent of helix angle when zero order approximation is made in deriving

directional factors.

• The system developed is general, it has been developed for the most complex pe

ripheral milling operation (i.e. plate milling), and can be applied to other peripheral

milling operations which generally have more rigid structures and higher chip loads.

Additional research should concentrate on extending the developed static and dy

namic milling process- structure interaction algorithms and the chatter stability method

to complex tool geometries such as variable pitch and taper ball-end mills. The static

model of the plate can be extended to include plates clamped from various points on

the edges. The research algorithms could also be interfaced to an existing CAD/CAM

system for use in the production planning of a variety of components in the aerospace

industry.

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Appendix A

Chip Flow Angle Formulation

The governing equations of the oblique cutting are listed as follows:

tan(3+)=tan)cos

(A.1)tan, — sin c tan L’

dsrj cos c,tan = (A.2)

1— rt sin a,-,

h cosirt= — =r (A.3)

h cosL,

tan3 = tan cos1) (A.4)

where

a,-, normal rake angle

3 friction angle

/3, normal friction angle

chip flow angle

4 normal shear angle

r, rt cutting ratio in orthogonal and oblique cuttingr and 3 can be obtained from the orthogonal cutting data for a specified cutter

geometry, i.e. a and &. Then, substituting equation (A.4) into (A.1), and (A.3) into

(A.2):

262

Appendix A. Chip Flow Angle Formulation 263

— tanqsn+tan/3cos17c— 1—tan8tan/3cos

(A.5)

— tani/’cosa— tan—sinatanb

T COS 71c cos antan q8n = (A.6)

cosij’— rcosilcslnan

Solving equations (A.5) and (A.6) together:

r cos 7)c cos acos — r cos 71c sin a

(A.7)— tancosa —tan/3cosTi(tanTi—sinatan’)

tan 77c — sin a tan b + tan /3 tan L cos a COS 77c

The above equation can be put into the following form:

AsinTi — Bcosq — CsinTicosTi +Dcos2’q= F (A.8)

where

A = rcosa+cosL’tan/9

B = tan/3sinansin/’

C = rsinatanL3 (A.9)

D = rtan/3tan

F sinbcosa

Equation (A.8) is numerically solved for Newton-Raphson algorithm is used for the

solution. The convergence is quite fast for reasonable values of friction angle and the

cutting ratio. The solution does not converge for very high values of the friction angle

(/3> 600) which are not possible physically.

Documents

MECHANICS AND DYNAMICS OF MILLING THIN WALLED STRUCTURES